kth sound&vibration book

Sound and Vibration

n

Sund Source

This material is protected by copyright and cannot be used or reproduced in any form without the written permission from: The Marcus Wallenberg Laboratory, KTH, SE-100 44 Stockholm, SWEDEN

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This course material was developed with support from the European Commission Tempus Program JEP-31018-2003 based on the Swedish book “Ljud och Vibrationer” used at KTH. Chapters 1-12 are translated from “Ljud och Vibrationer” by Hans Bodén, Ulf Carlsson, Ragnar Glav, Hans-Peter Wallin, and Mats Åbom by Robert Hildebrand Chapters 13, 15 and 16 are written by Hans Bodén Chapter 14 is written by Ulf Carlsson and Hans Boden Chapter 17 is written by Mats Åbom Edited by Hans Bodén and Tamer Elnady

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CONTENTS

1 INTRODUCTION

1.1 The field of Sound and Vibration 1.2 Job Market for Sound and Vibration Engineers 1.3 Development 1.4 Principles, examples and countermeasures

2 FUNDAMENTAL CONCEPTS

2.1 Fundamental and applied mechanics 2.2 Definitions of sound and vibration fields 2.3 Peak value, mean value RMS-value and power 2.4 Longitudinal waves in gases and liquids

2.4.1 Longitudinal plane waves 2.4.2 Spherical waves

2.5 Diffraction 2.6 Models in room acoustics

2.6.1 Geometrical acoustics 2.7 Waves in solid media 2.8 Frequency analysis of sound

2.8.1 Time and frequency domain 2.9 Levels and DECIBEL 2.10 Filters

2.10.1 Band pass filters 2.10.2 Octave and third octave filters

2.11 Summation of sound fields, interference 2.12 Summation of frequency components 2.13 Important formulas

3 INFLUENCE OF SOUND AND VIBRATION ON MAN AND EQUIPMENT

3.1 The ear and hearing 3.1.1 The ear’s function 3.1.2 Measures of hearing 3.1.3 Measures of noise 3.1.4 Speech and masking 3.1.5 The influence of noise on man 3.1.6 Hearing injuries 3.1.7 Hearing protection 3.1.8 Sound quality

3.2 Effects of shock nd vibration 3.2.1 Machinery and vehicle vibrations

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3.2.2 Effects on man 3.3 Standards

3.3.1 ISO 3740 , ISO 3747, 3.3.2 ISO 2631-1, ISO 2631-2 3.3.3 ISO 5349 3.3.4 ISO 8662 3.3.5 ISO 4866

3.4 Regulations and recommendations 3.4.1 Machines 3.4.2 Vehicles 3.4.3 Work environment 3.4.4 Buildings 3.4.5 External noise

3.5 Important formulas

4 SIGNAL ANALYSIS AND MEASUREMENT TECHNIQUES

4.1 Complex numbers and rotating vectors 4.2 Fourier methods in sound and vibration

4.2.1 Fourier series 4.2.2 Fourier transforms 4.2.3 Parseval’s relationships

4.3 Measurement systems for sound and vibration 4.3.1 The measurement chain 4.3.2 Microphones 4.3.3 Accelerometers 4.3.4 Mounting of accelerometers 4.3.5 Calibration of transducers and measurement systems


5 VIBRATIONS OF SIMPLE MECHANICAL SYSTEMS

5.1 Mechanical power 5.2 Linear systems

5.2.1 One degree of freedom systems 5.2.2 Two degree of freedom systems 5.2.3 Multi degree of freedom systems 5.2.4 Frequency response functions 5.2.5 Damping 5.2.6 Mechanical-electrical circuits

6 THE WAVE EQUATION AND ITS SOLUTIONS IN GASES AND LIQUIDS

6.1 The wave equation in a source-free medium 6.1.1 Equation of continuity 6.1.2 Equation of motion 6.1.3 The thermodynamic equation of state 6.1.4 The homogenous linearised wave equation

6.2 Solutions to the wave equation

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6.2.1 General solution for free one-dimensional wave propagation 6.2.2 Harmonic solution for free one-dimensional wave propagation 6.2.3 Sound intensity for free one-dimensional wave propagation 6.2.4 Energy and energy density for free one-dimensional wave propagation 6.2.5 General solution for free spherical wave propagation 6.2.6 Harmonic solution for free spherical wave propagation 6.2.7 Sound intensity for free spherical wave propagation


7 REFLECTION TRANSMISSION AND STANDING WAVES

7.1 Reflection and transmission of plane waves 7.1.1 Normal incidence against a rigid boundary 7.1.2 Normal incidence at a boundary between two elastic half spaces 7.1.3 Plane wave propagation in three dimensional space 7.1.4 Non-normal incidence at a boundary between two elastic half spaces 7.1.5 Non-normal incidence at a boundary between a fluid and a solid

7.2 Eigen-frequencies and eigen-modes 7.2.1 Eigen-frequencies and eigen-modes in rooms


8 THE WAVE EQUATION AND ITS SOLUTIONS IN SOLIDS

8.1 Introduction 8.2 Wave propagation in infinite and semi-imfinite media 8.3 Quasi-longitudinal waves in beams

8.3.1 The wave equation for quasi-longitudinal waves in beams 8.3.2 Quasi-longitudinal waves in infinite beams 8.3.3 Quasi-longitudinal waves in finite beams 8.3.4 Reflection and transmission of quasi-longitudinal waves at area changes 8.3.5 Standing quasi-longitudinal waves in beams

8.4 Torsional waves in axles 8.4.1 The wave equation for torsional waves in straight cylindrical axles 8.4.2 Torsional waves in straight axles

8.5 Bending waves in beams and plates 8.5.1 The bending wave equation for beams and plates 8.5.2 Bending waves in an infinitely long beams 8.5.3 Bending waves in finite beams 8.5.4 Dispersion 8.5.5 Reflection and transmission at a boundary between two beams 8.5.6 Standing waves in beams 8.5.7 Standing waves in plates

8.6 Mechanical impedance and mobility 8.7 Damping in solid structures

8.7.1 Material damping

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8.7.2 Losses at boundaries 8.7.3 Losses in built up structures 8.7.4 Damping of beams and plates using absorbing material 8.7.5 Mathematical description of damping 8.7.6 Experimental determination of damping


9 ROOM ACOUSTICS

9.1 Energy methods 9.1.1 Energy balance for simple an coupled systems 9.1.2 Relationship between wave theory and energy based methods

9.2 Room acoustics 9.2.1 Sabine’s equation 9.2.2 Sound fields in rooms 9.2.3 Acoustic absorbers 9.2.4 Sound transmission through walls


10 SOUND GENERATION MECHANISMS

10.1 Monopoles 10.2 Dipoles 10.3 Quadropoles

10.3.1 Examples of quadropole sources 10.4 Influence of boundaries

10.4.1 Examples of hard and soft surfaces 10.5 Line sources 10.6 Sound radiation from vibrating structures

10.6.1 Infinite plane surfaces 10.6.2 Finite plates with bending vibrations

10.7 Point excited plates 10.8 Flow generated noise

10.8.1 Scaling laws for flow generated noise 10.8.2 Whistling


11 VIBRATION ISOLATION

11.1 Types of isolation 11.2 General about vibration isolation 11.3 Measures of vibration isolation 11.4 Prediction of vibration isolation 11.5 Models for prediction of vibration isolation

11.5.1 Rigid mass – ideal spring – rigid foundation 11.5.2 Flexible foundation 11.5.3 Wave propagation in the isolator 11.5.4 Non-rigid machine 11.5.5 General expression for vibration isolation

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11.6 Vibration isolation in practice 11.6.1 Design of vibration isolators 11.6.2 Methods for improving vibration isolation 11.6.3 Commercial vibration isolators 11.6.4 Dynamic stiffness


12 SOUND IN DUCTS

12.1 Principles for sound reduction in ducts 12.1.1 Insertion and transmission loss 12.1.2 Requirements on silencers

12.2 Sound propagation in ducts 12.2.1 The modified wave equation

12.3 Reactive silencers 12.3.1 Area changes 12.3.2 Expansion chambers 12.3.3 Side branch resonators

12.4 Electrical – acoustic circuits 12.4.1 Four-pole theory

12.5 Resistive silencers 12.6 Important formulas

13 INDUSTRIAL NOISE AND VIBRATION CONTROL

13.1 Motivation for industrial noise control 13.2 Systematic approach to industrial noise control 13.3 Noise control at the source

13.3.1 Noise generated by fluctuating forces in structures 13.3.2 Noise generated by fluid flow

13.4 Noise control during the propagation path 13.4.1 Control of structure borne sound 13.4.2 Control of airborne borne sound

13.5 Noise control at the receiver 13.6 References

14 MACHINE CONDITION MONITORING

14.1 Introduction 14.2 Basic ideas of machine monitoring 14.3 Typical defects in gears and rolling bearings 14.4 Vibrations of gears and bearings

14.4.1 Vibration characteristics of non-defective gears 14.4.2 Vibration characteristics of non-defective bearings 14.4.3 Vibrations of defective gears 14.4.4 Vibrations of defective bearings

14.5 Monitoring methods 14.5.1 Early time domain methods

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14.5.2 Spectral methods 14.5.3 Cepstral methods 14.5.4 Envelope methods

14.6 Machine condition indicators 14.6.1 RMS-value, peak-value and crest factor 14.6.2 Kurtosis 14.6.3 Defect severity index

14.7 Residual time to failure estimation 14.8 Measurement techniques

14.8.1 Instrumentation 14.8.2 Data acquisition 14.8.3 Signal filtering 14.8.4 Normalized order analysis

14.9 User interface 14.10 Signal processing tools 14.11 References

15 VEHICLE NOISE AND VIBRATION CONTROL

15.1 Motivation for vehicle noise and vibration control 15.2 Character of vehicle noise 15.3 Measurement of exterior vehicle noise 15.4 Vehicle noise sources

15.4.1 Engine noise 15.4.2 Exhaust and intake noise 15.4.3 Cooling system noise 15.4.4 Tyre-road noise 15.4.5 Aerodynamic noise

15.5 Vehicle noise and vibration control 15.5.1 Engine noise control 15.5.2 Exhaust and intake noise control 15.5.3 Interior noise and vibration control

15.6 References

16 NOISE AND VIBRATION IN PIPES AND DUCTS

16.1 Sound generation in pipes and ducts 16.1.1 Turbulent boundary layer generated sound 16.1.2 Sound generation by pipe discontinuities 16.1.3 Control valve sound generation

16.1.3.1 Classification of valves 16.1.3.2 Examples of valve types

16.1.3.3 Valve noise source mechanisms 16.2 Sound transmission in pipes

16.2.1 Fluid-borne sound 16.3.2 Structure-bone sound

16.3 Sound radiation from pipes

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16.3.1 Excitation by structure-borne sound 16.3.2 Excitation by fluid-borne sound 16.3.4 Radiation from pipe openings

16.4 Noise control techniques 16.4.1 Noise control at the source 16.4.2 Noise control during the propagation path 16.4.3 Control of structure borne sound 16.4.4 Reduction of sound radiation

17 SOUND GENERATION FROM FLUID MACHINES

17.1 Classification of fluid machines 17.2 Flow generated sound

17.2.1 The high Mach-number range 17.2.2 The case of liquids discontinuities 17.2.3 The character of the sound

17.3 Noise control

Chapter 1: Introduction

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CHAPTER ONE

INTRODUCTION Acoustics can be regarded as the science of sound and vibration. Sound today refers not only to those mechanical wave motions in air that give rise to sensations of hearing, but even to low-frequency (infrasonic) and high-frequency (ultrasonic) motions that cannot be sensed by hearing, as well as analogous wave motions in, for example, water (underwater acoustics). In solid materials, one speaks instead of vibrations or structure-borne sound. The phenomenon of hearing has fascinated mankind all through the ages. The mathematical theory of sound propagation can be said to have begun with Isaac Newton (1642 - 1727), whose work Principia (1686) contained a mechanical interpretation of sound as pressure pulses propagating in a medium. A more solid mathematical and physical groundwork of theory was provided by Euler (1707 – 1783), Lagrange (1736 – 1813), and d’Alembert (1717 – 1783). That development took place just as continuum mechanics and field theory began to take form, and the wave equation was formulated for functions of space and time. The modern theory of sound and vibrations is the product of the efforts of these mathematical physicists.


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1.1 THE FIELD OF SOUND AND VIBRATION

In today’s well-developed, technological society, the number of systems that emit sound and vibrations is steadily increasing. Examples are machines, vehicles, and processes of all types, in which driving forces and engine power are constantly. Examples are machines, vehicles, and processes of all kinds, in which driving forces and engine power are continually being increased even as simultaneous efforts are made to hold down weight and materials usage. This implies the need for an ever more intensive research and development effort to identify and alleviate noise and vibration disturbances, and satisfy the needs of mankind for an acceptable environment. The fundamentals of Sound and Vibrations are part of the broader field of mechanics, with strong connections to classical mechanics, solid mechanics, and fluid dynamics. The subject of Sound and Vibrations encompasses the generation of sound and vibrations, the distribution and damping of vibrations, how sound propagates in a free field, and how it interacts with a closed space, as well as its effect on man and measurement equipment. Technical applications span an even wider field, from applied mathematics and mechanics, to electrical instrumentation and analog and digital signal processing theory, to machinery and building design. Several of the more important areas are worthy of particular mention:

Figure 1-1 Acoustics spans such diverse areas as biology, art, the natural sciences and technology. The proposed subdivision of the field (adjacent) covers, accordingly, many aspects and fields of science. The unshaded area describes the activity at a typical university sound and vibration laboratory. (Source: From R.B. Lindsay, who published the "acoustic wheel" in the Journal of the Acoustical Society of America 1964, vol 36)

NATURAL SCIENCES

TECHNICAL SCIENCES

LIFE SCIENCES

ART


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Machinery and Vehicle Acoustics deals with constructive measures to bring about machines, vehicles, and processes that are quieter and vibrate less; that requires knowledge of how sound and vibrations are generated, and how that generation relates to such physical parameters as flow velocities, masses, stiffnesses, losses, and geometry, for example. The most common reason that sound arises, in technical applications, is that a time-varying force excites vibrations of a mechanical structure, which then radiates sound. We can convince ourselves of that by knocking on a tabletop. The surrounding air is influenced by the vibrating structure, and responds with contractions and expansions. The mechanical form of energy we call sound has arisen. Analysis of the mechanisms of sound generation helps us to answer the question of why it sounds differently when we hit a tabletop with the soft part of the index finger, than it does when we hit it with the steel tip of a ball-point pen, and why it is louder when we knock the middle of the table with our knuckles than when we do the same thing at the edge of the table. The dull rumble of a Harley – Davidson motorcycle has become a defining characteristic – so important that the factory sought a patent on the sound to prevent competitors from plagiarizing it. That cash register click of a Mercedes door closing shut is something that other auto manufacturers strive for. The concept of Sound Quality is becoming ever more entrenched. Sound should convey a sense of product quality and reliability. The automobile industry has made pioneering efforts in this area and sound quality is given a high priority, along with such other important vehicle characteristics as road performance, safety and design. To describe sound, such subjective designators as “sharpness”, “rawness”, and “boxiness” are used. Several manufacturers have developed software packages that make it possible to modify the sound of products in order to mimic different design variations. The modified sounds are then judged by a “panel of listeners”. In the future, many consumer, industrial, and transportation products will be “sound-designed”.

Force

Vibrations

Acousticradiation

Source Response Acoustic radiation

Technology Model

Mathematical structure

Three stages

Examples:

GeometriesMasses Stiffnesses Losses

Surface area Mode shape (oscillation pattern)ResonancesStructure – air coupling

Roughness forces Gear tooth forces Mechanical imbalances Aerodynamic forces

Force

Roughness

Figure 1-2 Forces that cause vibrations appear in countless situations. It can be a matter of a roller in a bearing exhibiting out-of-roundness, or perhaps small surface irregularities at the contact between a railway wheel and rail. The ability of the forces to induce vibrations depends on mechanical laws in which such parameters as the structure’s geometry, mass distribution, stiffness and losses come into play. The structure’s effectiveness as an acoustic radiator depends on the surface area, its oscillation pattern, and the coupling between the structure and the air at the surface of separation. Sound generation in connection with mechanical structures can be described in three stages: Source – Response - Radiation


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p Z v S Z Y F Sj ij i i ij ki k ik

L

i

N

i

N

= ====∑∑∑

111

Figure 1-3 In a machine as complex as a car, the three stages of sound generation occur in many parallel chains. The velocity v at a specific point, induced by a force F at another specific point, can be described by a so-called frequency response function Y. Similarly, other frequency response functions Z can describe the relationships between the velocities on the surfaces and the sound pressure p at an interior point. The frequency response functions can, moreover, be combined to describe all three stages in the propagation chain. By adding the contributions from all

significant forces, via the dominant radiating surfaces, the total sound pressure at a location of interest in the interior is obtained. The noise in the interior comes mainly from three sources: • The driveline, i.e., the engine, transmission, and drive axles. • The contact zone between the tires and the roadway. • Airflow over the car body. Sound passes from the source into the passenger compartment in two distinct ways: as structure-borne and as air-borne sound. Structure-borne sound has essentially propagated in the form of vibrations from the source to the receiver, whereas air-borne sound had already radiated as sound, e.g., in the engine compartment, before its transmission into the passenger compartment (Sketch: Volvo Technology Report, nr 1 1988)

Forces Car

Vibration i l

Passenger Sound pressure


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Flow Acoustics is the study of the direct generation of sound in an elastic gaseous or liquid medium. The most common sound generation mechanisms are volume, force, and moment fluctuations in the medium, throughout which the elastic energy then spreads. Such phenomena are of great significance for noise from, for example, propellers, jet engines, fans, and vehicles, as well as for the propagation of sound in ducts.

Figure 1-4 Volvo’s wind tunnel is used to find ways to reduce flow-induced sound. Mitigating such sound typically requires an even flow over the body, as well as fine tolerances and good fits, especially at doors and other seals. Airflow over projecting details can cause whistling. The small-scale turbulence around the car body gives a roaring type of noise, and if there are openings an airstream can sometimes create whistling sounds (Photos: Volvo Technology Report, nr 1 1988)

Room Acoustics, or Building Acoustics, deals with how sound fields are built up in various types of rooms and enclosed spaces, as well as how sound is transmitted through different types of structures, such as walls and systems of joists. Figure 1-5 Heavy insulating surfaces effectively mitigate sound transmission from one enclosed space to another, as from, for example, the engine compartment to the cabin of a truck. For vehicles such as ships, cars, trains and aircraft, however, low weight is also a critical demand. Enormous efforts are therefore made in the transportation industry to bring about light, yet effective, designs for sound reduction. In the picture, the roof of a passenger car has been mounted in the opening between a sound reflective “reverberant room” at MWL, KTH (seen), and an adjacent sound absorbing ”anechoic room” (not seen). Using microphones and mathematical models for the sound interaction with the measurement rooms, the sound insulating properties of the car roof can be deduced (Photo: HP Wallin, MWL)


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Signal analysis is an important part of the science of sound and vibrations. The associated experimental methodology is primarily based on transducers, such as microphones and accelerometers, that convert sound and vibrations into equivalent electrical signals for further analysis. Until the 1960’s, most acoustic measurement instrumentation was analogue; then, digital computer technology arrived on a broad front. A numerical algorithm, FFT (Fast Fourier Transform), that transformed signals from the time domain (signal strength as a function of time) to the frequency domain (signal strength as a function of frequency), was a breakthrough in experimental methodology.

The Sound and Vibration subject area offers a wealth of opportunities for both the theoretically-inclined and the more practically / experimentally-oriented who are interested in applying themselves towards the improvement of machinery and vehicle designs, or towards quieter workplaces and societal environments.

Figure 1-6 To solve a noise problem, in which roof vibrations are suspected as the direct source, the car is driven on a chassis dynamometer.. The roof motion is registered as a function of time with the help of a motion sensor - an accelerometer. Transforming the signal to the frequency domain, i.e., expressing its strength as a function of frequency, often greatly facilitates interpretation. Such a transformation is carried out by electrical filters or by a computer. That broad subject is called signal analysis. Its mathematical foundation is derived from mathematical statistics and from Fourier methods. In the case illustrated, a poorly made transmission is to blame. A shaft in the transmission, rotating at 1800 revolutions per minute, is bent or out-of-balance, giving rise to a dominant vibration disturbance at 30 Hz. (Sketch: Brüel & Kjær, Structural Testing, Part 1)

Signal analysis Time

Amplitude

Signal

Frequency


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Figure 1-7 The strength of sound is often given in dB. A change of a single dB is hardly noticeable. An increase

of 8-10 dB is perceived as a doubling of the degree of disturbance. In order to adjust for variations in the sensitivity of the ear at different frequencies, corrected values are often used. The result is called Sound Level, and indicated in units of dB(A). (Diagram: ASF, ”Bullerbekämpning” [in Swedish], 1977, Illustrators: Anette Dünkelberg /Arne Karlsson)

1.2 JOB MARKET FOR SOUND AND VIBRATION ENGINEERS

The task of minimizing sound and vibration in mechanical constructions falls primarily upon those who are responsible for their design and analysis, i.e., primarily engineers with mechanical, vehicle, and machinery backgrounds. Such issues have a great deal of relevance these days, and industry is therefore preoccupied with them. It is usually larger manufacturing companies that have the greatest need for engineers in this field. The vehicle industry employ many sound and vibration engineers but also other industrial sectors that manufacture products in which some type of energy conversion takes place (compressors, separators, turbines, fans, appliances, etc.) have a great need for this kind of knowledge. There are, moreover, many consulting firms in the field. Others that have a need for specialists in this area include national and local authorities that have regulatory responsibilities, as well as national research organizations that have the task of dproducing new knowledge.

Normal conservation Train 100 meters away

(100 km / h) Eccentric press Pain threshold

Manual grinding Air-cooled electric motor 50 kW Spray painting

Weakest perceivable

Sound

Quiet bedroom Impact riveting

Coarse

grinding

Chainsaw

Near a jet plane starting up

Highest sound level that can be

attained

Sound insulated lounge


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1.3 DEVELOPMENT

Figure 1-8 Swedish billboard demonstrating the marketing value of acoustic performance, describing the

“world’s quietest dishwasher” as “Unbelievably unhearable”) (Photo: HP Wallin, MWL) Classical acoustics was summarized by Lord Rayleigh in his fundamental treatise "The theory of sound" 1894-96. That classical theory serves as the basis for the modern science of sound and vibrations. Lord Rayleigh received the Nobel prize in physics in 1904 for his discoveries in the field of optics. The field of noise control engineering started to develop in in the 1930’s. Development on a larger scale came in thethe 1950’s in connection with for instance the effort to build quiet submarines and civil jet engine airplanes. The knowledge acquired spread to other areas of machinery acoustics. In Sweden Atlas Copco was one of the pioneers. They developed quiet pneumatic machinery and compressors, which had a great impact on the noise level along streets and at city squares.


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Today, technical specifications on sound and vibration performance are made for a large share of the products that are delivered by manufacturing industry. The EU has brought about further demands on products to be sold in Europe, with respect to sound and vibration performance. Quality oriented manufacturers consider quiet, vibration-free products as a crucial quality characteristic and a marketable feature, so that all larger manufacturers tend to have departments responsible for sound and vibration performance. In the future, we will see improved and more easily used computer-based computational tools, and a continued development of cheap, personal computer-based systems for measurement and diagnostics. An example of the development of new technology is the so-called active sound and vibration control. The method is based on the use of, for example, microphones registering a disturbing sound field. Making use of automatic control methodologies, an “anti-noise” (phase inverted sound) is emitted that, under some conditions, can eliminate the disturbing field in an effective way. Commercial systems of that type exist already in, for example, ventilation systems and for automobile interiors and airplane cabins. In the future, we can expect corresponding systems on the vibration side as well. Today, vibration isolators, primarily of rubber, are inevitably used to prevent engine vibrations in an automobile from spreading into the car body. By installing parallel electrodynamic vibrators that generate reversed-phase forces, those disturbing forces that manage to pass through the isolators can be counteracted.

Kontrollenhet

VarvtalsgivareSignal från motor Signal från motor

Högtalare

Mikrofon

Figure 1-9 Cross section of the cabin of a turbo-prop SAAB 340 airplane with active noise control. The

microphones register the sound at a number of points and transfer the electrical signals to the control unit. The signals can be treated to drive the speakers in such a way (”antiphase”) as to reduce the noise level in the cabin. The tachometer signals from the engines help to fine tune the automatic control in the control unit. (Source: Ny Teknik)

Signal From motor Signal From motor Tachometer

Controller

Microphone Loudspeaker


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1.4 PRINCIPLES, EXAMPLES AND COUNTERMEASURES

The following pages give some examples of principles and applications in the field of sound and vibrations. On the left side, the underlying principles are presented, and on the facing right side, examples and countermeasures. Source: Asf, Bullerbekämpning (in Swedish), 1977. Rapidness of a process determines the high frequency content

Figure 1-10a (Sketch: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson).

The faster that force, pressure, or velocity changes occur, the higher the frequency content of the resulting sound. A fast ping pong ball gives a high frequency pop when it hits the table, while a slow handball bounces against the floor with a dull, low frequency sound.

Principle Slow impact against the floor – low frequency sound

Rapid impact against the table – high frequency


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Figure 1-10b (Sketch: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson).

Sound level

Angular gear teeth

Rounded gear teeth

High-pitched Low-pitched tones

Example - Countermeasure In the primitive tooth configuration, the teeth slap together so that the forces between them rise and fall rapidly. The high-pitched tones are then strong. The more effort is made to modify the tooth configuration, the more softly they can be made to mesh together. Finally, the forces can be made to rise and fall again slowly. The high-pitched tones are then no longer so dominant. Since the peak force

Force on a tooth

Angular gear

Rounded gear teeth

Force on a tooth

Time

Time


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Make vibrating surfaces as small as possible

Figure 1-11a (Sketches: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson).

An object with small surfaces can vibrate very strongly without radiating a great deal of noise. The lower the frequency of the disturbing tones, the greater the surface area before it becomes a disturbing noise source. Since there is practically always a risk of vibrations when dealing with machinery, the shells and housings used should be as small as possible.

Principle

The electric shaver vibrations are transmitted into the large glass shelf and the resulting noise level is high.

The vibrations are no longer transmitted and the noise diminishes.


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Oil tank

Pump

Motor

Instrument panel

Instrument panel moved to the wall

Example: The hydraulic aggregate was a powerful noise source. Since the wall vibrations of the oil tank were damped by the oil itself, most of the noise was radiated by the instrument panel.

Countermeasure: The panel was separated from the aggregate, reducing the radiating surface area, and thereby even the noise level.


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Densely perforated sheet emits little noise

Figure 1-12a (Sketch: Asf, Bullerbekämpning, 1977 Illustrator: Claes Folkesson).

Large vibrating shells cannot always be avoided. They give off a lot of noise. The reason for their sound radiation is that the vibrating sheet pumps the air to and from, like the piston of a pump. If the sheet is perforated, then it “leaks” and the pumping effect is weakened. The reduction in sound radiation is many times greater than the mere reduction in surface area. Alternatives to perforated sheet are nets and gratings.

Sound level

Principle

Perforated sheet metal Unperforated sheet metal


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Shield over drive belt and flywheel, of unperforated sheet

Perforated sheet

Example: The shield over the flywheel and drive belt constituted a strong noise source. The shield was fabricated of unperforated sheet metal.

Example: The shield over the flywheel and drive belt constituted a strong noise source. The shield was fabricated of unperforated sheet metal.

Countermeasure: A new shield was fabricated from perforated sheet and wire netting. One of the noise sources of the press was thereby eliminated.

Wire mesh netting


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Wind tone is removed by a changed profile or a small disturbing element


When sound flows past an object at certain speeds, a strong pure tone called a Strouhal tone can arise. By extending the dimension of the object in the direction of the airflow, with a “tail” for example, or by disturbing the regularity of the object profile, the tone can be prevented. In a duct, a resonance can amplify a Strouhal tone so much that the duct can be damaged.

Air Flow

Regular pattern of vortices gives a strong tone

Length extension Small disturbing objects

Irregular vortex pattern Irregular vortex pattern

Principle


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Wind

Sheet iron spiral

Chimney

Example: The shield over the flywheel and drive belt constituted a strong noise source. It was fabricated of unperforated sheet metal. Countermeasure: A new shield was fabricated from perforated sheet and wire netting. One of the noise sources of the press was thereby eliminated.


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Pure tones can be cancelled with sound in anti-phase

L2

L1


When the sound only consists of a single tone, or several such within a narrow frequency band, it can be completely or partially cancelled out in an interference muffler. It consists of a branched duct in which sound propagates through two separate paths that subsequently recombine with different time delays. In its simplest variation, shown in the figure, the path difference L1-L2 determines the frequencies at which sound reduction occurs. The time-delayed sound behaves as if in anti-phase.

Principle


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Figure 1-14b (Sketch: Asf, Bullerbekämpning, 1977, Ill: Claes Folkesson).

Interference muffler

Inlet

Outlet

Branches of varying lengths

Example: When the frequency of the disturbing tone, or the temperature of the gas, varies in time, the effective frequency band of the muffler can be widened by a variation of the path length difference through multiple paths. The improvement obtained at the nominal frequency is, however, somewhat less than in the preceding variation. The interference damper is suitable for use


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The ASU Sound & Vibration Lab. At the Faculty of Engineering, Ain Shams University The Sound and Vibration Laboratory at the Faculty of Engineering, Ain Shams University was established in 2004 by a European Union Grant (JEP-31018-2003) within the TEMPUS Program. This is the first laboratory in Egypt specialized in sound and vibration teaching and research. The establishment of the laboratory and the development of the courses were done in cooperation with: - The Marcus Wallenberg Laboratory for Sound and Vibration Research (MWL) at the Royal Institute of Technology (KTH) in Sweden. - Institute of Sound and Vibration Research (ISVR) at the University of Southampton in United Kingdom. Our mission is to produce a new generation of Egyptian engineers and researchers in the field of sound and vibration. ASU-SVL welcomes research institutions, industry and others to cooperate with our staff and use the laboratory facilities. The laboratory hosts the Acoustical Society of Egypt, whose mission is to group all engineers, researchers and students in Egypt interested in the field of sound and vibration. ASU-SVL has joined the X3-Noise Aircraft External Noise Network in Europe acting as the regional focal point for MEDA Countries in this network. X3-Noise is funded within the Sixth Framework Programme. The activities in the lab are divided into three main domains: education, research and consultancy. The lab is equipped with the state-of-the art sound and vibration measurement equipment including different types of analyzers, transducers, calibrators, and accessories. There is an anechoic room (80 m3), a reverberation room (80 m3), and an acoustic flow facility. There is a teaching lab of 13 computers working as data acquisition systems to be used in lab exercises in different courses.

Chapter 2: Fundamental Concepts

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CHAPTER TWO

FUNDAMENTAL CONCEPTS The aim of this chapter is, partly, to give an overview of the subject area of sound and vibrations, before the more detailed descriptions that follow, and partly to present and define a number of important concepts early enough that technically interesting problems may be treated in parallel with the main presentation. In the introduction, we shall relate the subject of sound and vibration (sometimes expressed as "vibroacoustics") to the mechanical sciences as a whole, describe what types of mechanical elastic waves can arise, as well as the conditions for their existence, and define important quantities which specify sound and vibration fields. In the section “Diffraction”, we will examine the conditions under which a sound field can bend around different objects. Using models from the subject of room acoustics, we will make a short preliminary survey of different methods for analyzing how sound propagation and sound fields interact with various types of rooms. Sound and vibration disturbances can only be tracked as functions of time, but are often more effectively analyzed and characterized as functions of frequency. Important concepts, such as the time and frequency domains, are defined, and filtering and frequency analysis are described.


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2.1 BASIC AND APPLIED MECHANICS The science of sound and vibration is a part of applied mechanics. This can be subdivided as in Figure 2-1.

Figure 2-1 Proposed subdivision of mechanics.

In statics, systems are examined in equilibrium states, i.e., the object considered is neither accelerated nor decelerated, and often has, in practice, no velocity. In particle mechanics, one studies the motion of the center of gravity of an object. The object / particle has three degrees of freedom, i.e., translations which can be described in a Cartesian coordinate system. In order to describe rigid body movement, six degrees of freedom are needed: three translations and three rotations. In applied mechanics one studies the behavior of solid, liquid and gaseous media, including deformation, wave propagation, tension, and fracture, when these media are subjected to physical agents of various types, such as forces and temperature shifts. In strength of materials one studies strain and deformation properties with the objective of properly dimensioning a construction for good strength and stability. In fluid mechanics one describes various aspects of the motions of, above all, fluids and gases. A typical problem from this field would be to improve the lift of an airplane wing, for example, or reduce the head loss (pressure drop) across a ventilation duct. Vibroacoustics covers everything from the basics of how sound and vibrations are generated and how they propagate to the question of how vibrations in a solid structure, a combustion engine for example, give rise to acoustic radiation. The division of the fields of mechanics as in Figure 2-1 is a mere simplification which can be adjusted and supplemented in many ways. A model from particle mechanics can, for example, be applied in vibroacoustics to explain the appearance of a certain tone when we blow over the mouth of a bottle, as in Figure 2-2.

Figure 2-2 The tone that arises when we blow over the bottle opening can be determined by methods from particle mech-anics. The phenomenon is call-ed a Helmholz resonator. It was already employed in the amphitheaters of ancient Greece where clay pots were used as resonators, see chapter 10. (Sketch (far left):Asf, Buller-bekämpning, 1977, Ill: Claes Folkesson).

S tatics Particle m echanics

Rigidbody m echanics

Strength of M aterials

Fluid M echanics

Vibroacoustics

Fundam ental M echanics Applied m echanics

Mass

Force

Spring


33

Figure 2-4a A loudspeaker generates longitudinal waves in which particles oscillate parallel to the direction of wave propagation.

Figure 2-4b The hand shaking the rope generates a transverse wave in which particles oscillate perpendicular to the direction of wave propagation.

2.2 DEFINITION OF SOUND AND VIBRATION FIELDS

Sound and vibration waves are mechanical elastic waves, and thus the conditions for their existence are that the medium possess mass and elasticity. If a mass particle is displaced from its equilibrium position, the elastic forces will seek to return it to its original position. The particle influences the surrounding particles and, in this way, a disturbance propagates in the medium. Sound waves transport relatively little mechanical energy; thus, as is familiar to all from everyday life, speech is not a particularly exhausting activity. A speaking person sends out only a few thousandths of a Watt. Nobody is ever going to be able to cook potatoes by yelling at them. This

means, on the other hand, that machines driven at high power levels have almost unlimited capacities to excite acoustic fields; small flaws can convert a share of the available mechanical power into acoustic disturbances. In vibroacoustics there are two classes of waves: longitudinal waves and transverse waves. Longitudinal waves have a particle motion which is parallel to the direction of wave propagation, see Figure 2-4a, while transverse waves have a particle motion which is perpendicular to the direction of propagation as in Figure 2-4b. These shall be described in further detail in later chapters.

Particle Displacement

Wave Propagation

Wave Propagation

Particle displacement

In a gas or a fluid, at a certain point in space r and time t, an acoustic wave can be described by the acoustical field quantities sound pressure ),( trp [Pa] and particle velocity ),( tru [m/s]. Sound pressure variations are normally very small deviations of the ambient pressure, and the particle velocity is small compared to the speed of sound.

Figure 2-3 A disturbance in water on a pond gives rise to a radial wave pattern. These water (or gravitation) waves are nevertheless driven, in contrast to sound and vibration waves, by a balance between the water’s inertia and gravitational effects. (Photo: Klas Persson)

pt Sound Pressure, p

Atmospheric Pressure p0

~ 10 Pa5

t

Figure 2-5 An acoustic field ordinarily implies only small disturbances. Normal speech at a distance of several meters, for example, gives a sound pressure p of a few hundredths Pa superimposed upon atmospheric pressure p0, which is about 105 Pa.


34

2.3 PEAK VALUE, MEAN VALUE, RMS-VALUE, AND POWER Before going any further, we recall some formulas from electrical engineering which are used to characterize signals, for example sound pressure p(t). A harmonic signal is a signal which can be described by a sine or cosine function as

p(t) = p sin(ω t + ϕ), (2-1)

where p(t) is the instantaneous, time-dependent sound pressure, p is the peak value or simply the amplitude, ω = 2πf is the angular frequency, ϕ is the phase angle. The time-averaged mean value of a signal, marked by an overbar, is

dttpT

pT

∫=0

)(1 , (2-2)

where T [s]is the time over which the average value is determined. The root mean square or RMS value is marked by p~ and is defined according to

∫=T

dttpT

p0

2 )(1~ . (2-3)

The rms-value is a very important, and oft-used, form, since it gives information about the time average of the signal power content.

T

t

peak-peak

(t)

p

~

= 2

^ p

p

p

^ p

The relation between the peak value and the rms-value for an arbitrary signal is called the peak factor or crest factor ppTF ~ˆ= . (2-4)

For a harmonic signal, the following relationship applies between the rms-value and the peak value:

2ˆ~ pp = . (2-5)

Figure 2-6 The amplitude characteristics of a harmonic signal, of sound pressure for example, can be described in various ways. The period T is the time required for one complete oscillation.


35

Exercise 2-1 Show that equation 2-5 is valid for a harmonic signal.

n

S

W

Sound Source The instantaneous mechanical power is the product of the instantaneous force and the instantaneous velocity. In acoustics, consequently, the instantaneous acoustic power

),( trW [W], which is transported through a surface S with a normal vector n , as in Figure 2-7, is

∫=S

dSntrutrptrW ),(),(),( . (2-6)

In acoustics, we are typically interested in the time average of power quantities )(rW , see chapter 3. 2.4 LONGITUDINAL WAVES IN GASES AND FLUIDS In gases and fluids, shear stresses are, as a rule, small enough to be neglected. Consequently, only longitudinal mechanical elastic waves can exist in such media. Longitudinal waves are characterized by a particle velocity which is parallel to the direction of wave propagation (see Figure 2-4a). Two cases are considered here for longitudinal waves: plane waves and spherical waves. More complicated wave fields can be constructed from these simple special cases.

2.4.1 Longitudinal plane waves

Longitudinal plane waves are characterized by the condition that points with the same acoustical state, i.e., the same sound pressure and particle velocity, form parallel planes. Figure 2-8 shows the acoustic field of an infinite duct with a harmonically (i.e., sinusoid-ally) oscillating piston at one end.

Direction ofPropagation

λ

Harmonic vibrational velocityv(t) = v sin(2 ft). π

Figure 2-8 A harmonically oscillating piston in an infinitely long duct gives rise to a plane longitudinal acoustic wave that propagates in the duct. The frequency of the wave is the same as the piston oscillating frequency. The quantity 2π f is the so-called angular frequency, designated by ω. The distance λ between two planes in the medium with the same acoustic state is the wavelength in the medium.

Figure 2-7 The acoustic power W over an area S in an acoustic field is the product of the sound pressure, particle velocity, and area.


36

When the piston begins to move forward in the cylinder, it compresses the air before it. That compression propagates at a speed completely independent of the particle velocity, hereafter referred to as the disturbance propagation speed c [m/s]. After a half period, the piston moves in the opposite direction and creates an expansion in the medium, which also moves at the speed c through the medium. After period T [s], the disturbance will have propagated the distance λ [m], so that the following elementary relation applies

fccT ==λ (2-7) or λfc = , (2-8)

where c is the disturbance’s propagation speed or the sound speed, f is (the disturbance’s) frequency, λ is the wavelength.

In air at normal pressure and temperature, c ≈ 340 m/s. For the case of a free plane longitudinal wave, i.e., wave propagation without reflections, there is a very simple relationship between the field quantities sound pressure p and particle velocity ux as well as the time average of the sound power W . Sound pressure and particle velocity are always in phase, i.e., they attain their respective maxima and minima simultaneously. The relation between them can be expressed

),(),( 0 txuctxp xρ= , (2-9)

where ρ0 [kg/m3] is the density in the undisturbed medium (in air at normal pressure and temperature ρ0 ≈ 1.21 kg/m3) and

c is the sound speed.

The time-averaged sound power W becomes, according to (2-2)

dttWT

WT

∫=0

)(1 . (2-10)

Putting (2-6) and (2-9) in (2-10) gives

dtc

StxpT

WT

∫=0 0

2 ),(1ρ

, (2-11)

which, in accordance with (2-3), gives cSpW 0

2~ ρ= . (2-12)

If we study the sound power per unit area, the so-called sound intensity I [W/m2], the time-average of the x-component is obtained as

cpSWI x 02~ ρ== . (2-13)

Thus, for a free plane wave, the time average of the sound intensity is proportional to the square of the rms-value of the sound pressure.


37

In linear vibroacoustics, in more general terms, the time-averaged sound power is proportional to the square of the rms-amplitude of the relevant field quantity, as

22 ~~ ppCW ∝= . (2-14)

This important relation implies that if the constant of proportionality C is known, we can determine the time-averaged sound power from a measurement of the rms sound pressure alone, using a pressure-sensitive microphone. That is usually considerably simpler than determining the particle velocity and making use of (2-6) directly. For a free plane wave, C = S/ρ0c according to (2-12). Note that if the losses in the medium are neglected, and since the wave doesn’t spread through an expanding volume, then time-averaged quantities are independent of distance to the source, i.e., time-averaged sound intensity and sound pressure amplitude are independent of spatial position.

2.4.2 Spherical and cylindrical waves

In the preceding section on plane waves, we were able to show that when the wave does not suffer losses, certain quantities remain independent of the distance to the source. If the source is an arbitrarily placed sphere and all points on its surface oscillate radially at the same amplitude and phase, or if the source radius a is small compared to the sound wavelength, then the source will produce spherical waves, as shown in Figure 2-9.

The mechanical power W emitted into the medium by the pulsating sphere spreads over an ever-expanding spherical area (fig 2-10); thus, the time-averaged sound intensity is

24 rWI r π= . (2-15)

r

2r

3r

λ λaλ λ

Figure 2-9 A spherical source which

oscillates at the same amplitude and phase over its entire surface area gives rise to spherical wave propagation.

Figure 2-10 In spherical wave propagation, sound power is divided over an ever-increasing area. The intensity decreases to one fourth its original value for a doubling of the distance to the source, and to one ninth when the distance is tripled.


38

Examples of real phenomena giving rise to spherical wave propagation are loudspeakers and the outlets of exhaust pipes, provided that the dimensions of the source are small compared to the wavelength, i.e., at sufficiently low frequencies as implied by (2-8). As is evident from Figure 2-9, the curvature of the wave fronts decreases with increasing radius. For engineering purposes, the waves can be considered (locally) plane for radii r > λ / 3, and (2-13) is then applicable even for this spherical wave case. The rms sound pressure can then be expressed in the form

20 4~ rWcp πρ= . (2-16)

If the source is, instead, and infinitely long cylinder, the entire surface of which oscillates with a uniform phase and amplitude, then cylindrical waves arise; see Figure 2-11.

r2r

3r

S2S

3S The mechanical power, often given in units of power per unit length in the case of a line source, is distributed over a cylindrical area, so that the sound intensity can be expressed

rWI r π2'= , (2-17)

where 'W [W/m] is the sound power per unit length. Examples of sound sources that can be regarded as cylindrical are electrical distribution cables, pipes, ducts, transport belts at breweries, and heavily trafficked roads. A pre-requisite is that the distance from the source be small compared to the length of the source.

2.5 DIFFRACTION

Diffraction takes place in all types of wave propagation. Water waves are not noticeably affected by the presence of a thin mooring post in the water. The waves roll on as if the post did not exist. On the other hand, we know that behind a break wall in a protected harbor, or behind a narrow spit of land jutting into the water, a shadow zone free of waves develops. The relationship between the size of the hindrance and the wavelength determines how the wave motion is bent about it. Visible light has an approximate wavelength of 10-7 m, whereas typical human speech has a wavelength of about 1 m. For this reason, we can hear someone speaking from behind a pole without being able to see her at the same time. Thus, wave motions with wavelengths large in relation to the obstacle are little influenced by it and spread behind it. If, instead, the wavelength is small in relation to the obstacle, then a shadow zone develops.

Figure 2-11 An infinitely long cylindrical source oscillating with uniform phase and amplitude over its entire surface gives rise to cylindrical waves.


39

Exercise 2-2 Go to an open area, without any reflecting objects, such as buildings and the like, in the vicinity. Ask a companion to stand about 5 meters away, facing away from you, and take turns voicing an extended (i) ooooooo........ (“oo” as in “food”, not “foot”) (ii) ssssssss........ Note: the oo-sound has a frequency of about 250 Hz and the s-sound is a hiss with a frequency around 6000 Hz. Compare the wavelengths with the diameter of the head. These diffraction phenomena, as well as the persistence of free plane waves as planar and of free spherical waves as spherical, can be explained with the help of Huygen’s principle: "Every point on a wave front can be described as the center of a secondary wave field, a so called elementary wave. The new position [a moment later] of the wave front is the tangent to the set of these elementary waves ", see Figure 2-12.

In Figure 2-13a, it is seen that at such low frequencies that the wavelength is large in relation to the opening in the wall, Huygen’s principle implies a spherical wave propagat-ion from the opening. If the frequency is high, and the wavelength thereby small in relation to the opening, as in Figure 2-13b, then the wave propagates as a beam with shadow zones on each side of it.

a) b)Figure 2-12 Huygen’s principle can be used to show that free

plane waves remain plane, and spherical waves remain spherical as they propagate. In a), the tangent to the so-called elementary waves is seen to build a plane wave front, and in b) the corresponding principle for spherical waves is shown.

λ

λ λ

Shadow zone

a) b)

Spherical propagation

λ

Shadow zone

Figure 2-13 a) shows sound transmission through a hole, small in comparison to the sound wavelength. From the opening, spherical wave propagation occurs. b) shows the case of hole that is large with respect to the wavelength. In the middle, the wave passes relatively unhindered, while shadow zones are built up at the sides.


40

Exercise 2-3 Noise from roads and railways is an enormous problem. A common attempt at a solution is to install noise walls or barriers. How should noise barriers be placed?

a) Is it in the high or the low frequency range that noise barriers can be expected to provide the greatest benefit? b) Is the greatest effect obtained from the barrier when the source is near or far from it? c) Where should the receiver be located with respect to the barrier, to receive the greatest benefit?

Figure 2-14 For highways with average vehicle speeds exceeding 70 km/h, the dominant noise is that generated at the contacts between tires and the road surface.

2.6 ROOM ACOUSTICS MODELS

In many situations, sound interacts with a closed space such as rooms in dwellings, concert halls, or industrial facilities, but even automobile or railway car interiors, or ship or airplane cabins. Knowledge of sound propagation and sound fields in such spaces is therefore an essential part of acoustics. When sound from a source reaches one of the room’s bounding surfaces, a share of the sound power is reflected back into the room and a share absorbed by the wall. In room acoustics, three distinct methods or models are used to describe sound propagation and the sound fields that arise. At low enough frequencies that the wavelength of sound is of the same order of magnitude as the dimensions of the room, wave theoretical room acoustics is a powerful tool. It turns out that when some of the room’s dimensions are whole multiples of half the wavelength, the incident and reflected waves interact such as to bring about standing wave fields, which then dominate the sound field in the room. The particular frequencies at which that occurs are called eigenfrequencies, or resonance frequencies in ordinary speech. The wave pattern, with its characteristic sound maxima and minima, is called an eigenmode, or simply mode; see Figure 2-15.

Figure 2-15 When a dimension of a closed space, an airplane cabin or a vehicle interior for example, is a whole multiple of the half wavelength, then constructive interference arises between incident and reflected waves. A standing wave pattern with nodes and anti-nodes results. The frequencies at which that happens are called eigenfrequencies, and the standing wave patterns are called eigenmodes.

(Sketch: Brüel & Kjær, Technical Review).


41

Direct waveReflected wave

Observation point

Sound source

At higher frequencies, the eigenfrequencies are so tightly spaced that, for practical reasons, we must choose another mathematical description. In chapter 8, we will study statistical energy methods for so called diffuse fields, see Figure 2-16. In practice, the demands for such a field are seldom fulfilled. The actual field is then called a reverberant field. The third method, geometrical room acoustics, is described in the next section.

2.6.1 Reflections and geometrical acoustics

A free wave, incident upon a reflecting surface with irregularities much smaller than a wavelength λ, changes its direction, i.e., is reflected, in a predictable way. In Figure 2-17, the wave fronts of the incident and reflected waves are marked. It is convenient, at times, to indicate a wave by an arrow. The arrow is perpendicular to the wave fronts and points in the direction that the wave propagates. Using Huygen’s principle, we can show that:

(i) Against a plane surface, a plane wave is reflected as a plane wave and a spherical wave as a spherical wave.

(ii) The direction of the incident wave, the normal to the reflecting surface, and the direction of the reflected wave, all lie in the same plane.

(iii) The angle of incidence θi is equal to the angle of reflection θr, see Figure 2-17.

Figure 2-16 Near a source, direct unreflect-ed sound dominates; such sound comprises the so called direct, or free field in which the power from the source is distributed over an ever expanding volume, resulting in a halv-ing of sound pressure for a doubling of dist-ance, according to relation (2-16) between sound power and sound pressure for spheric-al waves. Further away, sound that has been reflected at least once dominates; that sound constitutes the so-called reverberant field. With a stricter demand, that sound pressure be constant and independent of distance to the source, sound must arrive at an observat-ion point from all directions, with random phase; it then fulfills the definition of an (ideal) diffuse field.

Figure 2-17 With the help of Huygen’s principle, we can show that a plane wave, incident on a reflecting surface, is reflected as a plane wave, and that the angle of incidence θi is equal to the angle of reflection θr.

θ i θ r

Incident plane wave

n

Wave fronts

Reflected plane wave

Wave fronts

λ λ


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If the reflecting surface is rigid, the sound pressure amplitudes and phases of the incident and reflected waves are equal the point of reflection. An observer is influenced by both the direct sound from the source and the reflected sound. We can describe the sound field above the reflecting surface by a superimposition of the direct wave field direct from the actual source with the direct field of an identical, imagined mirror source, placed at the same distance from the surface, but on the opposite side; see fig 2-18.

Source

Mirror

source

Wave fronts

Rigid surface

Observer

d

d Reflected spherical wave

Incident spherical wave

In geometrical acoustics, a sound wave is represented by an arrow in the direction of wave propagation, just as a light wave is represented in geometrical optics. We can regard a wave as a beam originating from the source. The method can be used to explain how a parabolic microphone functions; see Figure 2-19. Geometrical acoustics is primarily applied in the design of auditoriums. The purpose is to bring about an even distribution of sound, without focuses and shadow zones; see Figure 2-20. The limitation of the method is that typically only the first, or possibly even the second, reflection can be studied before it becomes impractical to follow the “sound trail”.

Figure 2-19 In a parabolic microphone, the reflected waves are focused into a focal point, at which the microphone is located, to maximize the amplification.

Figure 2-18 A spherical wave incident upon a reflecting surface is reflected as a spherical wave. If the surface is rigid, the reflected wave can be regarded as equivalent to a direct field from an imagined mirror source, identical to the actual source, at the same distance from the surface, but on the opposite side.

Parabola

Microphone

Incident

waves

Stage

Roof

Parquet

Balcony

Figure 2-20 In geometrical acoustics, sound waves are regarded as beams. The method is often used in the design of large musical auditoriums. The limitation is that it is typically only possible to follow the first few reflections of the sound waves. (Source: Brüel & Kjær, Measurements in Building Acoustics.)


43

2.7 WAVE TYPES IN SOLID MEDIA

Solid media can sustain both normal and shear stresses, and thereby resist both volume and shape changes. That implies that in solids, in contrast to gases and liquids, not only longitudinal waves, but even transverse waves, can exist. Transverse and longitudinal waves can also, in combination with each other, build special types of waves, e.g., bending waves. Here, we will acquaint ourselves with some important special cases. Figure 2-21a illustrates a longitudinal wave and Figure 2-21b a transverse wave. When the medium is infinite in the direction of wave propagation, so that no reflections occur, we speak of free wave propagation.

UtbredningsriktningPartikelrörelse Partikelrörelse Utbredningsriktning

a) b) Figure 2-21 Important wave types in solid media. a) Longitudinal wave: particle motion is parallel to the direction of disturbance propagation. b) Transverse wave: particle motion is perpendicular to the direction of disturbance propagation.

In a bounded medium, reflections occur at the boundaries. At certain frequencies, the incident and reflected waves interfere, or interact, such that standing wave patterns arise. These frequencies are called, as in section 2.6, eigenfrequencies or resonance frequencies, and the oscillation pattern is called an eigenmode or mode. Bending oscillations

Longitudinal oscillations

Tuning fork

Bending oscillations

Table-top

Figure 2-22 When a tuning fork is stricken, bending oscillat-ions take place in the legs, essentially in the first eigenmode. That disturb-ance excites a longitudinal oscillation at the same frequency. That latter, in its turn, excites bending waves in the tabletop. As is well known, it isn’t until that large table surface has been put into contact with the tuning fork, and begun vibrating itself, that we begin to hear the tuning fork well.

Particle motion Propagation direction Particle motion Propagation direction


44

Exercise 2-4 Hold a long measuring stick, extended about 1.5 m from your grasp, and shake it transversally at about 1 Hz. It will now (hopefully) be oscillating in its first eigenmode. Increase the frequency until you find the second eigenmode. Observe the antinodes, where there is a large amount of bowing, i.e., deformation, and the nodes, at which there is none.

First eigenmode Second eigenmode

Node

Anti- Node

2.8 CHARACTERIZATION OF SOUND ACCORDING TO FREQUENCY

Sound can be characterized by its frequency content. For mankind, audible sound is that which falls in the 20 – 20 000 Hz range. For frequencies lower that 20 Hz, we speak of infrasound, and over 20 000 Hz of ultrasound. Most acoustic phenomena are frequency-dependent. It is well known that different sound sources have different character, see Figure 2-23. Slowly changing and “soft” events mainly generate low frequency sound, while short and fast events mainly generate high frequency sound. We can be convinced of that by slapping the soft part of the thumb against the tabletop, and then do the same with a ballpoint pen. As is evident from Figure 2-23, the audible region spans over a range of wavelengths from 17 m to 17 mm, which is of fundamental significance for the types of noise control measures that can be undertaken. Analysis of the frequency content of a sound is therefore of great importance in the field of sound and vibration.


45

1 10 20 100 1000 10000 20000

17 17 10 -3.

Frekvens [Hz]

Våglängd [m]

Infraljud Hörbart ljud Ultraljud

Skärande bearbetning

Fartygsmaskiner

Luftblåsning

Sågning

Smältugn

Vindbuller

Ekolod

Bilar, tåg etc

Människa

Hund

Fladdermus

Hörområden

Ljudkällor

340 3,4 0,34

λ

Figure 2-23 Classification of sound according to frequency, and the relation between frequency and wavelength

in air with a sound speed c = 340 m/s. Frequency ranges of various sound generation mechanisms. (Source: Asf, Bullerbekämpning, 1977.)

2.8.1 Time and frequency domains

Figure 2-24a shows a harmonic signal of frequency f0 as a function of time. It can also be described as a function of frequency, as in Figure 2-24b. All power in the signal is concentrated at the frequency f0. We have thus introduced the, for vibroacoustics fundamental, concepts of the time and frequency domains. These are equivalent concepts, and we can choose to describe a signal in either of these domains. We can only record a signal as a function of time. It can thereafter be transformed into the frequency domain by frequency analysis, i.e., by methods from Fourier analysis.

T=1/f 0

Signal

Time

Amplitude

Frequency f 0

a) b) Figure 2-24 Harmonic signal in a) the time domain, and b) the frequency domain.

Infrasound Audible sound Ultrasound

Wavelength [m] 3.4

Frequency [Hz]

340 17 0.34 17*10-3

Audible range Mankind

Dogs

Bats

Wind noise Ship machinery

Cars, Trains, etc.Smelting oven

Cutting toolsSawing

Air blasting

Sound Sources:

Echo


46

Jean Baptiste Fourier (1768-1830) demonstrated more than a century ago that all periodic signals can be described by a summation of harmonic signals. We can illustrate that by means of the three-dimensional Figure 2-25. A periodic signal, e.g., a sound pressure p(t) that repeats with period T, can be written

)()( nTtptp += , n = 1, 2, 3,... (2-18)

Periodic signals are very common in engineering applications. They can be sound pressures or vibrations arising from, for example, rotating machines whose sound and vibration generation is repeated in an identical fashion in cycle after cycle. In Figure 2-25, a signal is shown (heavy line) together with its harmonic components (thin line), at frequencies fn. Every periodic signal can be expressed as the sum of harmonic signals with a so-called Fourier series (see chapter 3) according to

∑=

+=N

nnn tnfptp

10 )2cos(ˆ)( ϕπ , (2-19)

where np is the peak value of the n-th harmonic component, f0 = 1/T and nf0 are the frequencies of the n-th harmonic component, nϕ is the phase angle.

1/T

0

Frequency [Hz]Time [s] T

2T/3

T/3

0

2/T 3/T

4/T

Amplitude

The description of a signal in the frequency domain is usually called a (frequency) spectrum and each line, frequency component, is called a tone or a spectral line. In order to obtain a complete description, the phase angle of each frequency component is also required. Typically, however, we are only interested in the amplitudes of the frequency components. The frequency spectrum is very useful when the signal is to be analyzed, e.g., it provides the opportunity to distinguish every frequency component, even in cases in which a single frequency is dominant in the signal.

Figure 2-25 A periodic signal can be described in both the time and the frequency domains. In the time domain, the signal is represented by its variation in time, and in the frequency domain by the amplitude of the harmonic signals it is built up of. Here, the phase angles ϕn are zero for all frequency components.


47

Example 2-1 With today’s technology, it is possible to discern spectral or frequency components withamplitudes that are a 100 000 - th those of other frequency components present. A rotating machine always gives rise to a strong frequency component corresponding to the rotational speed. Assume that we try to discern weaker signals, such as those of a rollerbearing that is beginning to wear out. We are thus trying to find very small frequencycomponents in the presence of a very strong one. That is considerably easier in the frequency than in the time domain. We can illustrate the foregoing by considering a simple compound signal consisting of two harmonic components, one with frequency f0 = 1/ T and another with a lower amplitude and double the frequency, i.e., 2f0 = 2/ T, as shown in Figure 2-26.

0 0.01 0.02 0.03 0.04 0.05

Time [s]

0 50 100 150 200

Frequency [Hz] a) b)

Amplitude Signal

Figure 2-26 a) A signal in the time domain (heavy line) consisting of two harmonic components (thin lines) with

periods T and T/2. b) Corresponding signal showing two components in the frequency domain. The more complex signal in Figure 2-27 gives rise to more spectral components in the frequency domain.

0 0.05 0.1 0.15 0.2 0.25 0.3

Time [s]

0 20 40 60 80 100

Frequency [Hz] a) b)

Signal Ampl

Figure 2-27 a) A gear that transfers a constant moment and alternates between having two and three teeth

meshing at a time, gives rise to a square wave type of signal in the time domain. b) In the frequency domain, many high frequency components are needed to describe the rapid

variations in the time domain.


48

If the signal is not periodic, but rather transient, as in Figure 2-28, or stochastic as in Figure 2-29, a continuous distribution is instead obtained in the frequency domain. The transformation from the time to the frequency domain, for transient signals, can be described with the help of a Fourier transform; see chapter 3.

Tid [s]

Frekvens [Hz]b)a)

Signal Amp

Figure 2-28 A transient excitation, e.g., a hammer blow against a plate structure, brings about a time-decaying

signal. In the frequency domain, a continuous spectrum is obtained, i.e., the spectral components are infinitely dense. If the vibrations in the structure are dominated by an eigenfrequency, then graphs such as the one given above are obtained.

Frekvens [Hz]Tid [s]a) b)

Signal Amp

Figure 2-29 A purely stochastic process, such as a downpour of rain on a car roof, gives a frequency spectrum of

constant amplitude, a so-called white noise. Another example is the noise caused by the turbulent boundary layer around the fuselage of an airplane.

2.9 LEVELS AND DECIBELS

In example 2-1, we found that it is possible, with modern digital technology, to discern spectral components with amplitudes less than 1/100 000 of those of other spectral components. With that in mind, how can we show the entirety of that spectrum, on a monitor screen for example? If the strong component is 100 mm tall, then the weak one is only 0.001 mm tall in a linear scale, and therefore not even visible. If we want to see the entire spectrum simultaneously, then the amplitude scale must be adapted in some way. The solution is to use a logarithmic scale, which compresses the high amplitudes and expands the low ones.

Time [s]

Frequency [Hz]

Time [s] Frequency [Hz]


49

At the beginning of the 1920’s, it had become practical to carry out routine sound measurements. An acoustic group at Bell Systems in the U.S. introduced a measurement quantity that they called a “sensation unit”, which was based on a logarithmic scale with base 10. The unit Bel (after Alexander Graham Bell), defined as the base-10 logarithm of the quotient between two acoustic power values, eventually proved to be impractically large. Today, the unit commonly used is, instead, a tenth of a Bel. The logarithmic unit deciBel better reflects mankind’s sense of hearing, as we shall see in chapter 3, providing an ideal gradiation of the relevant range of values, neither too coarse nor too fine nor with unwieldy numbers. One deciBel (1 dB) corresponds both to the measurement precision that can typically be obtained in acoustic measurements and to the amount of change that a human can discern in ideal circumstances. The base-10 logar-ithm will henceforth be designated by log. Since the argument of the logarithm, i.e., the ratio of the power value of interest to some chosen reference power value, is dimension-less, then logarithmic quantities are called levels. The Sound Power Level LW indicates the acoustic power with respect to an internationally accepted reference of 10-12 W, as

ref

W WWL log10 ⋅= , (2-20)

where W is the time-averaged sound power, 1210−=refW W is the reference value of sound power.

Figure 2-30 The sound power and Sound Power Level for a number of typical sound sources. Notice that the large linear span is effectively compressed by the logarithmic scale.

(Source: Brüel &Kjær, Acoustic Noise Measurements.)

200

180

160

140

120

100

80

60

40

20

0

100 000 000

1 000 000

10 000

100

1

0.01

0.000 1

0.000 001

0.000 000 01

0.000 000 000 1

0.000 000 000 001

Acoustic power [W]

Sound Power Level L [dB] ref 10 W

W-12 Object

Saturn rocket

Four jetplanes

Large orchestraScream

Typical speech

Whispering

Acoustic power [W]

50 000 000

50 000

10

1

20 10

10

-6

-9

.


50

Example 2-2 The motor of a propeller plane generates a sound power of 0.1 W. Determine the Sound Power Level.

Solution Using the given values in formula (2-20) gives

dB11010log101110log101010log10 11

12

1=⋅⋅=⋅=⋅=

−

−

WL .

The Sound Intensity Level LI is defined as

ref

I IIL log10 ⋅= , (2-21)

where I is the absolute value of the time average of the sound intensity, 1210−=refI W/m2 is the reference value of sound intensity.

Example 2-3 If the Sound Intensity Level in an airplane is 60 dB, what is the sound intensity?

Solution Rewriting formula (2-21) gives

10log I

ref

LI

I= .

Algebraic manipulation, using properties of logarithms, gives 10/10 ILrefII = .

Finally, entering the given values yields 610/6012 101010 −− =⋅=I W/m2.

In linear vibroacoustics, as indicated in formula (2-14), time-averaged power values are proportional to the squared rms-amplitudes of the field variables (e.g., pressure, particle velocity). Thus, to calculate logarithmic levels from the field variables, it is these squared rms-amplitudes that must be used. The Sound Pressure Level Lp (or SPL) is defined as

2

2~log10

refp

p

pL ⋅= , (2-22)

where p~ is the rms-amplitude of the sound pressure,

5102 −⋅=refp Pa is the reference value of sound pressure. The reference value of sound pressure approximately corresponds to the lowest sound pressure that a young person with normal hearing can perceive at 1000 Hz; see chapter 3.


51

Example 2-4 The Sound Pressure Level (SPL) a meter away from a person speaking is about 60 dB. Determine the sound pressure at that SPL. Solution According to (2-22), refrefp ppppL ~log20~log10 22 ⋅=⋅= ,

so that 20/10~ pLrefpp = .

Using the values given above in this relation yields 220/605 10210102~ −− ⋅=⋅⋅=p Pa. The Vibration Velocity Level Lv is defined as

2

2~log10

refv

vvL ⋅= , (2-23)

where v~ is the rms-amplitude vibration velocity, vref = 10-9 m/s is the reference level of vibration velocity.

The Force Level LF is defined as

2

2~log10

refF

FFL ⋅= , (2-24)

where F~ is the rms-amplitude of the force, Fref = 10-6 N is the reference value of force.

2.10 FILTERS

Frequency analysis implies the study of a signal’s distribution along the frequency axis. Such analysis has traditionally been carried out mathematically or by means of analog electrical filters constructed of conventional electrical components. With today’s digital technology, there are two methods that are primarily used: the Fast Fourier Transform (FFT), and digital filtering. Both make use of digitized measurement values. Each type of filter is named after its affect on the signal’s frequency spectrum (see Figure 2-31):

(i) Low pass filter. (ii) High pass filter (iii) Band pass filter (iv) Band stop filter


52

1

frekvens

Amplitud

Förstärkning

f

(i)

(ii)

(iii)

(iv)

f

f

f

A

f

A

f

A

f

A

f

Förstärkning

Förstärkning

Förstärkning

1

1

1

The filter type that is most common is the low pass filter. Such filters are often used at the input to a measurement system to filter away frequency components higher than those to be analyzed. These removed components would otherwise introduce errors during the digitization process by contaminating the low frequency components (“aliasing”). A filter

2.10.1 Band pass filters

An ideal band pass filter, such as the one in Figure 2-32a, suppresses components at all frequencies except those that lie within the bandwidth B (i.e., “passes” those in B). In practice, however, the edges of the band have a certain slope, as shown in Figure 2-32b, which implies that the frequency components immediately outside of the pass band are not completely eliminated. A common way to define the upper fu and lower fl frequency limits of the band is to indicate the frequencies at which the signal is reduced by 3 dB.

Figure 2-32 a) Ideal band pass filter with infinitely steep cutoffs. b) Real filters have imperfect cutoffs. The upper and lower bounding frequencies are then defined by

the frequencies at which the filter reduces the signal by 3 dB.

Amplification [dB] Amplification [dB]

0 0

-3

f f Frequency

B=f -f

a) b) l u

u l

f f l u

Band width,

Frequency

Figure 1-31 Different filters influence on a noise signal’s freq-uency spectrum when the signal passes through them: (i) Low pass filter (ii) High pass filter (iii) Band pass filter (iv) Band stop filter (Source: Brüel & Kjær, course material.)

Amplitude

Frequency

Amplification

Amplification

Amplification

Amplification


53

Band pass filters are named according to how the bandwidth varies along the frequency axis. Filters with bandwidths that do not vary along the frequency axis are called constant absolute bandwidth (CAB) filters; see Figure 2-33a. A filter with a bandwidth proportional to its center frequency, fc, is called a constant relative bandwidth (CRB) filter; see Figure 2-33b.

=100 Hz

0 1k 2k 3k 4k 5k 6k 7k 8k 9k 10kLinjär frekvensskala

Förstärkning [dB]

0

B

Figure 2-33a CAB filter, with a bandwidth that does not vary along the frequency axis; it is typically presented

with a linear frequency axis.

1

Logaritmisk frekvensskala

Förstärkning [dB]

0

B fm= −( )2 266

8 16 31,5 63 125 250 500 1k 2k 4k 8k 16k

mfB )212( 66 −=

Figure 2-33b CRB filter, with a bandwidth that is a certain percentage of the center frequency fc; it is typically

presented with a logarithmic scale. Because of the logarithmic scale, the stacks in the figure do not get wider, moving to the right along the axis. The example in the figure is called a third octave band filter and has a band width that is about 23% of the center frequency.

2.10.2 Third-octave and octave band filters

Third-octave and octave band filters are CRB filters very widely used in the field of sound and vibrations. Center frequencies are standardized, and listed in table 2-2. Both types of filters are named with band numbers, as in table 2-2, or more often by their center frequencies, fc. As is evident from table 2-2, each octave band spans three third-octave bands, which explains the name of this category of filters.

Amplification [dB]

Linear frequency scale

Amplification [dB]

Linear frequency scale 31.5


54

Table 2-1 Definition of third-octave and octave band filters.

Octave band filter Third-octave band filter

Lower frequency limit 2cffl = 6 2cffl =

Upper frequency limit cu ff 2= cu ff 6 2=

Bandwidth ( ) clu fffB 212 −=−= ( ) clu fffB 6 2126 −=−=

Center frequency ulc fff = ulc fff =

Table 2-2 Standardized center frequencies and upper and lower frequency limits of third-octave and octave band

filters. Shading indicates octave bands.

Band no.

Center frequency

fc [Hz]

3rd-octave band filter fl – fu [Hz]

Octave band filter fl – fu [Hz]

Band no.

Center frequency

fc [Hz]

3rd-octave band filter fl – fu [Hz]

Octave band filter

fl – fu [Hz]

1 1.25 1.12 - 1.41 23 200 178 - 224 2 1.6 1.41 - 1.78 24 250 224 - 282 178 - 355 3 2 1.78 - 2.24 1.41 - 2.82 25 315 282 - 355 4 2.5 2.24 - 2.82 26 400 355 - 447 5 3.15 2.82 - 3.55 27 500 447 - 562 355 - 708 6 4 3.55 - 4.47 2.82 - 5.62 28 630 562 - 708 7 5 4.47 - 5.62 29 800 708 - 891 8 6.3 5.62 - 7.08 30 1000 891 - 1120 708 - 1410 9 8 7.08 - 8.91 5.62 - 11.2 31 1250 1120 - 1410

10 10 8.91 - 11.2 32 1600 1410 - 1780 11 12.5 11.2 - 14.1 33 2000 1780 - 2240 1410 - 2820 12 16 14.1 - 17.8 11.2 - 22.4 34 2500 2240 - 2820 13 20 17.8 - 22.4 35 3150 2820 - 3550 14 25 22.4 - 28.2 36 4000 3550 - 4470 2820 - 5620 15 31.5 28.2 - 35.5 22.4 - 44.7 37 5000 4470 - 5620 16 40 35.5 - 44.7 38 6300 5620 - 7080 17 50 44.7 - 56.2 39 8000 7080 - 8910 5620 - 11200 18 63 56.2 - 70.8 44.7 - 89.1 40 10000 8910 - 11200 19 80 70.8 - 89.1 41 12500 11200 - 14100 20 100 89.1 - 112 42 16000 14100 - 17800 11200 - 22400 21 125 112 - 141 89.1 - 178 43 20000 17800 - 22400 22 160 141 - 178


55

2.11 ADDITION OF SOUND FIELDS, INTERFERENCE

The sound pressure level a meter away from a person speaking is about 60 dB. Two persons speaking together, and in fact talking at the same moment, do not bring about a sound pressure level of 120 dB, however. Logarithmic sound pressure levels cannot be added together in that way. In linear vibroacoustics, we can, according to the underlying mathematics, add together the solutions obtained from linear wave equations. The resulting wave motion at a point of interest can therefore be calculated by adding up the contributions from the individual sound waves at that point. For scalar quantities, such as sound pressure, this would be scalar addition, and for vector quantities it would be vector addition. For sound pressure, for example, the fundamental addition rule applies, that the total sound pressure ptot(t) is the sum of the individual waves’ sound pressures pn(t); i.e.,

∑=

=N

nntot tptp

1)()( . (2-25)

In order to determine the SPL of the resulting total sound pressure, we must, in accordance with formula (2-22), use the rms sound pressure. To determine the rms value, we first consider the case of two separate sound waves acting at the same point. The total sound pressure’s squared rms value is, according to equation (2-3),

[ ]

.)()(2~~

)()(1)(1~

021

22

21

0 0

221

22

∫

∫ ∫

++=

=+==

T

T T

tottot

dttptpT

pp

dttptpT

dttpT

p

(2-26)

Considering the third term in (2-26), we can identify three cases:

(i) Uncorrelated sources. In most circumstances in which noise is coming from more than one source, as from several machines in a workshop for example, the sources can be assumed to be statistically uncorrelated , so that the third term in equation (2-26) goes to zero. Hence,

22

21

2 ~~~ ppptot += . (2-27)

Additional uncorrelated sound pressures can then be added to this result. The addition rule for uncorrelated sources is therefore

∑=

=N

nntot pp

1

22 ~~ . (2-28)

For uncorrelated sources, the squared rms sound pressures can evidently be added together. A corresponding addition rule for the SPL is derived from definition (2-22), from which we have

1022 10~ pLrefpp = . (2-29)


56

If two uncorrelated sources give rise to 21

~p and 22

~p , with Sound Pressure Levels Lp1 and Lp2, respectively, then, from (2-22) and (2-28),

⎟⎠⎞⎜

⎝⎛ +=+=

1010222

21

2 21 1010~~~ pp LLreftot pppp , (2-30)

so that

⎟⎠⎞⎜

⎝⎛ +⋅=

1010 21 1010log10 pptot

LLpL . (2-31)

For levels, the addition rule which therefore applies is

∑=

⋅=N

n

Lp

pntot

L1

1010log10 . (2-32)

Example 2-5 A sound source causes a sound pressure level Lp1 at a certain point. What increase in SPL is provided by a second source, equal in strength to, but uncorrelated to, the first?

Solution Formula (2-32) gives

dB 32log10)210log(10)1010log(1011

111 101010+=⋅+=⋅⋅=+⋅= pp

LLLp LLL ppp

tot.

With two equally strong, but uncorrelated, sources, the level is therefore increased 3 dB.

Example 2-6 Elimination of the background level. A typical problem is that the SPL due to a machine must be determined, while other noise sources present cannot be shut down. The solution can therefore be to first measure the background level Lpb without the machine in question operating; then, start the machine and measure the total level Lptot and, from that, back calculate the sound pressure level Lpm due to the machine by itself.

a) Derive a formula for that purpose.

b) Determine dB 83 and dB 90 when ==btotm ppp LLL .

Solution a) Assume uncorrelated sources. Formula (2-28) gives

222222 ~~~,~~~btotmbmtot pppppp −=+=

from which ⎟⎠⎞⎜

⎝⎛ −⋅=

1010 1010log10 bptotpm

LLpL .

b) ( ) 890.891010log10 3.89 ≈=−⋅=mpL dB.


57

(ii) Identical sources. This case is, of course, not so common in noise control situations. Two closely spaced loudspeakers driven with the same signal can be considered identical. In that case, p1(t) = p2 (t) and from (2-26) we receive 2

12 ~4~ pptot = . (2-33)

The total Sound Pressure Level is then

6~4

log1012

21 +=⋅= p

refp L

p

pL

tot dB. (2-34)

So, the Sound Pressure Level increases by 6 dB. (iii) Identical, oppositely phased sources In this case, p1 (t) = - p2 (t), so that equation (2-26) yields 0~ =totp , and Lptot → - ∞. That is called destructive interference, and is a practical means to reduce noise in many cases. One example is so called active noise control, in which a phase-inverted antinoise is emit-ted and interferes with the original noise. In practice, 20-30 dB reductions are obtainable.

2.12 ADDITION OF FREQUENCY COMPONENTS

Section 2.10 discussed different ways to describe a signal’s distribution in the frequency dimension or in frequency bands. Narrow bands give detailed information on the distribution of energy, with relatively low amplitudes in each band. Figure 2-34 shows the same spectrum presented with different bandwidths.

Summation of the sound pressures of individual frequency components is carried out in the same way as for the summation of sound pressures from multiple sources; from (2-28) and (2-32),

∑=

=N

nntot pp

1

22 ~~ (2-35)

Octave band

Third octave band

Narrow band

frequency (log) [Hz]

L [dB] p Figure 2-34 The same sound spectrum presented in narrow band, third octave bands, and octave bands. The bigger the bandwidth, the more frequency components that contribute to any band, giving higher levels. The logarithmic frequency axis causes the CRB filters to have the same apparent width per band over the entire spectrum, while CAB filters, with their fixed band width over the entire spectrum, appear to grow more dense as frequency increases.


58

and ∑=

⋅=N

n

Lp

pntot

L1

1010log10 , (2-36)

where the index n stands for individual frequencies or frequency bands, instead of distinct sources. The proof for each of these formulas is from Parseval’s relations for periodic and non-periodic functions, which in turn comes from Fourier analysis; see chapter 3. Example 2-7 The 1000 Hz octave band includes the 800, 1000, and 1250 Hz third-octave bands. Determine the octave band level, if the third-octave band levels are 79, 86 and 84 dB, respectively.

Solution Formula (2-36) gives Lp = 10⋅log(107.9 +108.6 + 108.4) = 88.6 ≈ 89 dB.

2.14 IMPORTANT RELATIONS

PEAK VALUE, AVERAGE VALUE, RMS AMPLITUDE AND POWER

Time average of a quantity

dttpT

pT

∫=0

)(1 , (2-2)

where T is the averaging time.

RMS amplitude of a quantity

∫=T

dttpT

p0

2 )(1~ . (2-3)

LONGITUDINAL WAVES IN GASES AND LIQUIDS

Longitudinal plane waves

Relation between sound speed, frequency and wavelength. λfc = . (2-8) Time averaged sound intensity

c

pS

WI x0

2~

ρ== . (2-13)

Spherical and cylindrical waves

Sound intensity of spherical waves

24 r

WIrπ

= . (2-15)


59

Sound intensity of cylindrical waves

r

WI r π2'

= . (2-17)

LEVELS AND DECIBELS

Sound Power Level ref

W WWL log10 ⋅= , (2-20)

where 1210−=refW W is the reference value of sound power.

Sound intensity level ref

I IIL log10 ⋅= , (2-21)

where 1210−=refI W/m2 is the reference value of sound intensity.

Sound Pressure Level 2

2~log10

refp

p

pL ⋅= , (2-22)

where 5102 −⋅=refp Pa is the reference value of sound pressure.

Vibration velocity level 2

2~log10

refv

vvL ⋅= , (2-23)

where 910−=refv m/s is the reference level of vibration velocity.

Force level 2

2~log10

refF

FFL ⋅= , (2-24)

where 610−=refF N is the reference level of force. ADDITION OF SOUND FIELDS, INTERFERENCE

Fundamental addition rule for sound pressure

∑=

=N

nntot tptp

1)()( . (2-25)

Addition rule for uncorrelated sources

∑=

=N

nntot pp

1

22 ~~ , (2-28)

∑=

⋅=N

n

Lp

pntot

L1

1010log10 . (2-32)


60

ADDITION OF FREQUENCY COMPONENTS

∑=

=N

nntot pp

1

22 ~~ , (2-35)

∑=

⋅=N

n

Lp

pntot

L1

1010log10 . (2-36)

Chapter 3: Influence of Sound and Vibration on man and equipment

61

CHAPTER THREE INFLUENCE OF SOUND AND VIBRATION ON MAN AND EQUIPMENT This chapter begins with a description of the function of the ear, and how we perceive sound, as a function of its frequency content and strength. That description serves as a foundation, as we then define the indices that are needed to depict how disturbing noise is. One well known such index is Sound Level, given in dB(A), which is intended to serve as a measure of the degree of disturbance. Sound affects not only our hearing, but even the entire body, and psychological disturbances can be more obvious than physical ones at times. Our hearing was developed to best serve our ancestors tens of thousands of years ago; they were in those times hunter-gatherers. Our nerve and gland systems have not changed, and the extra doses of hormones, blood sugar, and blood fats that gave our ancestors courage to fight or strength to flee, merely increase our stress level today. How a product sounds, its sound quality, is one of the characteristics that distinguishes it from competing products. Sound design of products will be an important tool of competition in the future.

Vibrations appear in many situations. We experience them in the home, during transports of different types, and in professional life. Sometimes we even generate vibrations intentionally. In vibratory feeding systems, objects are induced to move forward along a vibrating path. Ultrasonic cleaners are used for sterilization. Vibrating boring machinery is used to bore in rock. In vibration testing, components and entire finished products are exposed to high vibration amplitudes to evaluate their ability to function in their service environment.


62

Nevertheless, vibrations are usually unwanted and harmful. Vibrating production machinery degrades production tolerances and surface finish. An unbalanced turbine can bring about serious fatigue problems leading to breakdowns. Vibrations of hand-held machines can cause blood circulation problems in the hands, the so-called white finger syndrome. Low frequency vibrations in the earth’s crust, earthquakes, can demolish entire cities. Noise and vibrations are annoying, and parliament and authorities pass laws, regulations, norms and guidelines within their respective areas of responsibility. Regulations are more detailed than laws, and must be followed; otherwise the authorities may intervene. Norms and guidelines carry weight when an individual citizen, for example, appeals the decision of a municipal authority to one of the administrative courts that handles the type of issue in question. Verification by measurement is needed to ensure that regulations issued by authorities are fulfilled. To guarantee that such measurements are performed correctly, and that companies declaring the noise and vibration levels generated by their products compete in minimizing such levels rather than in devising misleading measurement techniques, both national and international standards committees have come to agreement on measurement standards in different areas. Standards from the International Organization for Standardization are designated ISO. The European Union, EU, has its own standards designated as EN-standards. These are often in agreement with corresponding ISO-standards. Laws, regulations, norms and guidelines are constantly modified.

Always use the original, in its latest update Short summaries of the type that are presented in this chapter are not only incomplete, but run the risk of becoming out of date as well.

3.1 THE EAR AND HEARING

3.1.1 The ear’s function

Anatomically, the ear is subdivided into three parts: the outer ear, middle ear, and inner ear; see Figure 3-1. The outer ear consists of the pinna, i.e., the visible part of the ear, and the auditory canal. The eardrum separates the outer from the middle ear. The middle ear is an air-filled cavity that obtains its oxygen supply from the eustachian tube, originating in the throat. In the middle ear, there are three auditory bones, or ossicles: the hammer, the anvil, and the stirrup, which connect the eardrum to the oval window of the cochlea, the boundary between the middle and the inner ear. The inner ear consists of the cochlea and the semicircular canals (organ of balance).


63

The main function of the pinna is to, like a funnel, receive sound waves and channel them into the auditory canal, as well as enhance our sound localization abilities by virtue of its shape. The most important clues for sound localization are intensity and time differences in the sound wave that reaches the ear. A sound wave from the right reaches the right ear before it reaches the left, and is also stronger in the right ear than the left. These clues cannot, however, explain our ability to distinguish between sound from before us and sound from behind us, or between sound from above and sound from below. That information comes from the

shape of the pinna, which modifies the incoming sound. The auditory canal leads sound

waves in towards the middle ear. Certain frequency regions are enhanced and others weakened, depending on the direction from which they are coming. Together, the pinna and the auditory canal cause resonances that increase the sound pressure at the eardrum, above all in the 2 - 7 kHz region; see Figure 3-2.

500 1000 2000 5000 10000Frequency [Hz]

15

20

10

-5

0

5

-10

Acoustic amplification [dB]

200

Figure3-1 Schematic of the ear’s anatomy (Source: Brüel & Kjær, course material.)

Figure 3-2 Acoustic amplification caused by reflections around the head and shoulders, in the pinna and the auditory canal. (Source: Pickles, 1988.)

Pinna

Auditory canal

Semicircular canals

Auditory nerve

Cochlea

Eustachian tube

Stirrup

Anvil Hammer

Ear drum

Middle ear

Outer ear Inner ear


64

In the eardrum, acoustic pressure variations are converted into mechanical vibrations, which are then amplified in the middle ear and transferred to the cochlea. Amplification is mainly brought about by a pressure jump due to the much larger area of the eardrum than the contact area of the stirrup against the oval window. The middle ear functions as a so-called impedance transformer, coupling air (low impedance) to the liquid of the cochlea (high impedance). If that transformation mechanism did not take place, then the majority of the incident sound would be reflected back into the auditory canal. There are two muscles in the middle ear. One is fixed to the hammer near the eardrum, and the other is fixed to the stirrup. These muscles firm up the chain of auditory bones. When the muscles are tensed, the stiffness of the ossicle chain is increased, which reduces the transmission of low frequency sound. The stirrup muscle tenses as a reflex against strong sound (about 75 – 95 dB above the weakest sound a person can perceive), which suggests that it serves to protect the inner ear from injury due to noise. The reflex is, however, too slow to protect against impulse sound (strong sound, short in duration, discussed further in section 3.1.6) and the muscles tire quickly due to high frequency noise. The eustachian tube is responsible for supplying air to the middle ear and equalizes static pressure differences between the middle ear and the surrounding air. That occurs when the eustachian tubes open to the throat, as when we swallow or yawn for example. If a static pressure difference nevertheless arises, the eardrum will bow out towards the lowest pressure, i.e., towards either the auditory canal or the middle ear. The increased tension in the eardrum that results causes pain and can, in extreme cases, result in bursting of the eardrum. The cochlea is a rolled-up tube that, is divided along its length into three channels; see Figure 3-3. The upper and lower channels, which are in mutual communication, are filled with a liquid rich in sodium ions, while the middle is filled with a liquid rich in potassium ions. The hearing organ with sensory cells, hair cells, is located on the basilar membrane, which divides the two lower channels.

Stirrup Oval window

Round window Basilar membrane

When the stirrup moves against the oval window of the cochlea, a pressure wave emitted into the upper channel of the cochlea. The wave then moves back along the lower channel towards the round window at the base of the cochlea, the membrane separating the lower channel from the middle ear. At the round window, pressure equalization can occur; thus, the round window moves in antiphase with respect to the oval window. Pressure changes in the channels also affect the basilar membrane. The membrane varies in stiffness and

Figure 3-3 Schematic of the cochlea “unrolled”.


65

width along the length of the cochlea, from stiff and narrow at the base to soft and wide at the apex. Because of these stiffness differences, the mode excited in the cochlea has a maximum at a point that depends on the frequency; see Figure 3-4. High frequency sound has a maximum near the base of the cochlea, while low frequency sound has its maximum near the apex. Thus, a frequency separation of the incoming sound is effected, and the hair cells at different positions along the cochlea react to different frequencies. That implies that a loss of hair cells at the base of the cochlea results in high frequency hearing loss, while a loss of hair cells near the apex results in low frequency hearing loss.

Distance from the oval window [mm]

Amplification

Excitation frequency [kHz]

Cochlea ”unrolled”Sound wave

Figure 3-4 The graph shows the mode shapes (oscillation patterns) of the basilar membrane at various excitation

frequencies. The higher the excitation frequency, the nearer the oval window the maximum amplitude is to be found. (Picture: Brüel & Kjær, course material.)

When the membrane moves, the hair cells are stimulated, and conversion to a nerve signal takes place by means of a sequence of bioelectrical processes based on the differing ion content of the three channels. A difference in potential arises in the hair cells, and that potential difference produces an electrical impulse in the nerve fibers, which are connected to the hair cells. These nerve fibers combine in the auditory nerve, which carries the signal to the hearing centers in the brain, where it is interpreted as sound.

3.1.2 Measure of hearing

The human ear has a wide working range. A young person with normal hearing hears sound in the frequency range 20 - 20 000 Hz, and in the 0 - 130 dB range of sound pressure levels. With advancing age, it is above all the ability to hear weak, high frequency sound that diminishes, as discussed further in section 3.1.6. In many different situations, we must be able to measure hearing or quantify the impression made by a sound. It is also necessary in order to be able to undertake countermeasures in a noisy workplace or to be able to rehabilitate a hearing injury. Our subjective experience of the strength of sound is not in exact agreement with the physically measured sound pressure. The frequency, among other things, affects our perception of sound strength. Figure 3-5 presents a group of curves that connect tones of different frequencies, but with the same Loudness, i.e., tones that are perceived to be equal in strength. The concept of loudness is defined as the sound pressure level a sinusoidal tone at 1000 Hz would have, in order to give the same subjective impression of strength as

0.6 0.3


66

the sound to be assessed. The subjective impression of strength presupposes a person with normal hearing. The unit of loudness is the phone.

20-10

0

6331,5

10

20

30

40

80

60

dBL

[p

50

] 70

100

90

110

120

12500

40

1000

10

20

30

250125 500

Frekvens, [

2000

]

80004000

50

60

70

80

90

100

110 phon C -vägning baseras på 90 phon kurvan

-vägning baseras på 70 phon kurvan

B

-vägning baseras på 40 phon kurvan

A

en normalhörandeHörtröskeln för

person

,

Figure 3-5 Isophone curves. Along a curve, the loudness level is constant, i.e., it is subjectively experienced as

equally strong. The lowermost, dashed curve, is the threshold of hearing for a normally-hearing person. The curves were measured in an acoustically treated room with loudspeakers, and using both ears. Both tones marked in the diagram, at 63 Hz and 1000 Hz, therefore have a loudness of 60 phones. Their respective sound pressure levels, on the other hand, are 75 dB and 60 dB.

The Threshold of hearing is defined as the lowest sound pressure level that induces any sensation of hearing. Even the threshold of hearing varies with frequency as is evident from the lower dashed curve in Figure 3-5. Therefore, frequency-specific sound, usually consisting of sinusoidal tones, is used to determine an individual’s hearing threshold. In cases of impaired hearing, the degree of impairment is indicated relative to a standardized hearing threshold, derived from the averaged hearing thresholds of young people with normal hearing. The most common method to graphically depict these relative hearing impairments is an audiogram, in which the statistical normal hearing level is represented as a zero level, and hearing deficiency is indicated in dB HL (Hearing Level) downwards in the diagram; see Figure 3-6. The region of normal hearing is shaded.

Threshhold of hearing for a person with normal hearing

A-weighting is based on the 40 phone curve

B-weighting is based on the 70 phone curve

C-weighting is based on the 90 phone curve

31.5

Frequency [Hz]

Phones


67

-10 0

10 20 30 40 50 60 70 80 90

100 110 120

-100

102030405060708090

100110120

Frequency [Hz]

Hearing loss [dB HL]

125 250 500 1000 2000 4000 8000

N O R M A L H EARING

Figure 3-6 Audiogram, with the region of normal hearing shaded.

3.1.3 Measures of noise

What we regard as noisy varies from individual to individual. By noise, we usually mean unwanted sound in the audible region.

The strength of sound is measured by a sound level meter, that, in its simplest form, gives the SPL in dB; see section 1.13. The SPL does not, however, take account of the nonlinearity of our perception with respect to frequency, as reflected in the concept of loudness. To better reflect the human perception of sound, sound level meters contain filters, so-called weighting filters, that amplify the microphone signal different amounts at different frequencies; see Figure 3-7.

+ 20 + 10 0 - 10 - 20

- 60 - 50 - 40 - 30

- 70 10 100 1 000 10 000

Frequency [Hz]

Amplification [dB]

Figure 3-7 A, B, C and D-weighting curves. A-weighting is the most common. Under 1000 Hz, the amplication is negative, implying that these frequencies are damped to compensate for the lower sensitivity of mankind to low frequency sound; see Figure 3-5. (Source: Brüel & Kjær, kursmaterial.)


68

A-, B- and C-weighting are taken from the 40, 70, and 90 phone curves in Figure 3-5, respective-ly; see table 3-1. Orig-inally, the thought was that A-weighting would be used at low, B-weighting at inter-mediate, and C-weighting at high SPL, thereby adjusting the measurement results to our perception of sound as it varies in both frequency and strength. Today, however, A-weighting is most often used, although C-weighting is at times applied, particularly in conn-ection with impulsive sound. D-weighting is

primarily used in measuring aircraft noise. A sound pressure level that is measured or determined with weighting filters, is called a Sound Level. Assume that the measured sound level with an A-weighting filter is 75 dB. That is written LA = 75 dB(A). The sound level in dB(A) can be calculated from third-octave and octave band filters as

∑=

Δ+⋅=N

n

ALA

npnL1

10/)(10log10 [dB(A)], (3-1)

where Lpn [dB] is the third-octave or octave band level in band n, ΔAn [dB] is A-weighting in band n.

Example 3-1 Determine, from the f [Hz] 125 250 500 1 k 2 k 4 k 8 k given octave band Lpn [dB] 90 96 92 90 85 86 81

Frequency [Hz]

A-weighting [dB]

B-weighting [dB]

C-weighting [dB]

25 -44.7 -20.4 -4.4 31.5 -39.4 -17.1 -3.0 40 -34.6 -14.2 -2.0 50 -30.2 -11.6 -1.3 63 -26.2 -9.3 -0.8 80 -22.5 -7.4 -0.5

100 -19.1 -5.6 -0.3 125 -16.1 -4.2 -0.2 160 -13.4 -3.0 -0.1 200 -10.9 -2.0 0 250 -8.6 -1.3 0 315 -6.6 -0.8 0 400 -4.8 -0.5 0 500 -3.2 -0.3 0 630 -1.9 -0.1 0 800 -0.8 0 0 1000 0 0 0 1250 +0.6 0 0 1600 +1.0 0 -0.1 2000 +1.2 -0.1 -0.2 2500 +1.3 -0.2 -0.3 3150 +1.2 -0.4 -0.5 4000 +1.0 -0.7 -0.8 5000 +0.5 -1.2 -1.3 6300 -0.1 -1.9 -2.0 8000 -1.1 -2.9 -3.0

10000 -2.5 -4.3 -4.4 12500 -4.3 -6.1 -6.2 16000 -6.6 -8.4 -8.5 20000 -9.3 -11.1 -11.2

Table 3-1 A-, B- and C-weighting for third-octave and octave bands. The octave bands are given in bold.


69

levels, Lpn, the sound level in dB(A) Solution

A-weighting is given in table 3-1, which, from (3-1), gives

ΔAn [dB] Lpn +ΔAn

[dB]

-16.1 73.9

-8.6 87.4

-3.2 88.8

0 90

1.2 86.2

1.0 87.0

-1.1 79.7

95)10101010101010log(10 97.77.862.8988.874.839.7 ≈++++++⋅=AL dB(A).

Equivalent sound pressure level is a form of average sound pressure level during a given period of time. It is defined as the constant sound pressure level that represents the same total sound energy as an actual time varying sound pressure level during a given time period, eight hours for example. This measure is used to characterize a time-varying noise and create a measure of the disturbance or destructive influence of the sound. The length of the measurement period should always be given.

))(1log(100

2

2

, dtp

tpT

LT

refTeq ∫⋅= [dB], (3-2)

where Leq,T is the equivalent sound pressure level during time period T, p(t) is the instantaneous sound pressure, pref = 5102 −⋅ Pa, is the reference sound pressure, T is the length of the measurement period. With the help of the definition of sound pressure level, the expression can be written

)101log(100

10/)(, dt

TL

TtL

Teqp∫⋅= [dB], (3-3)

where Lp(t) is the instantaneous sound pressure level, or, if an A-weighted quantity is intended

)101log(100

10/)(, dt

TL

TtL

TAeqA∫⋅= [dB(A)], (3-4)

where LA(t) is the instantaneous A-weighted sound level.

Equivalent sound level can be registered with an integrating sound level meter or a dosimeter.. It can also be calculated starting from sound level measurements, and with simplifying assumptions on the sound variation during the measurement period. The definition implies that a strong, short-duration sound makes a big contribution to the equivalent sound level. A constant sound level of 100 dB(A) during 15 min corresponds to a constant sound level of 85 dB(A) over 8 hours, i.e., a workday. Table 3-2 shows equivalent values of sound level and exposure time, each corresponding to LA = 85 dB(A) over 8 hours. An increase of the sound level by 3 dB(A) corresponds to a halving of the exposure time for the same equivalent sound level.


70

In many cases, that type of broadband analysis of the sound level does not give a sufficiently detailed picture of the strength of a sound. The sound pressure level is therefore often measured in different frequency bands (octave, third-octave, or narrow band) to identify variations with frequency; see section 1.10.

Table 3-2 Equivalent exposures, i.e., values of sound level, LA, and exposure time, T, giving the same total sound energy.

Sound level LA [dB(A)] Exposure time T

82 16 h 85 8 h 88 4 h 91 2 h 94 1 h 97 30 min

100 15 min 103 7.5 min 106 3.8 min 109 1.9 min 112 1 min 115 0.5 min

3.1.4 Speech and masking

Perhaps the most important function of our hearing is to understand speech. With a normal voice level, the sound level one meter away from a speaker in a quiet environment is 60 - 65 dB(A). With a raised voice it is about 75 dB(A), and with a very loud voice about 85 dB(A). Speech can therefore be said to vary in strength as well as frequency; Figure 3-8 shows the frequency and strength distribution at a distance of 1 meter from a speaker, in an audiogram. The area in the figure constitutes a so-called speech banana. With the help of the speech banana, it is possible to get an idea of what speech sounds a person with a certain hearing impairment can be expected to understand. Vowels are seen to fall in the lower frequency areas, and are relatively strong, while unvoiced consonants, such as “s”, “sh”, and “f”, for example, are high in frequency, and relatively weaker. Vowels carry energy, while consonants carry the most linguistic information. Our ability to comprehend speech depends on many factors. The character of the speech is one such factor. The speech signal normally contains a large excess of information with respect to its linguistic content. That excess is called redundancy. Every language has limits on different linguistic levels. After a certain linguistic sound, only certain other linguistic sounds are allowed to follow. After a certain word, only certain other words can grammatically follow. These limitations give rise to the redundancy. It is that redundancy that permits a person with normal hearing to follow a conversation in a very noisy room, or a speech signal that is severely distorted, as in a bad telephone connection, for example. Listeners take advantage of all of their knowledge of the language, the speaker, and the subject of conversation. A coherent conversation has high redundancy; a solitary word removed from its context has low redundancy.


71

-10 0

10 20 30 40 50 60 70 80 90

100 110 120

-100

102030405060708090

100110120

Frequency [Hz]

Hearing Loss [dB HL]

125 250 500 1000 2000 4000 8000

B

A

C D

The background noise naturally influences our ability to comprehend speech. One of the most obvious effects of noise is that it reduces the possibilities for conversation. This is called masking, and implies that a sound reduces the hearing perception of another sound. We often mean, by masking, that the masking sound makes the masked sound inaudible. Generally, it can be said that a signal is most easily masked by sound with similar or identical frequency content. The phenomenon of masking is asymmetric; narrowband sound masks more upwards than downwards in frequency, as seen in Figure 3-9. That means that strong bass sounds are generally more disruptive of speech comprehension than treble sounds with the same strength.

Noise

Hearing threshhold

without noise

Hearing threshhold with

noise

100

80

60

40

20

0

Lp [dB]

Frequency [Hz]10 100 1000 10000

Figure 3-8 The primary region of the speech spectr-um. Area A is the area for fundamental tones of the human voice, B is the main area for vowels, C for voic-ed consonants, and D for unvoiced consonants. From the figure, we see that the region for vowels, B, goes from 300 Hz to 3000 Hz in frequency, and from 37 dB to 65 dB in the hearing loss dimension. A hearing loss of 65 dB in the 300 - 3000 Hz region therefore implies an inability to comprehend some vowels, and in practice, hardly any other speech sounds either.

Figure 3-9 Masking. The plot shows how a narrowband noise centered at 1000 Hz influences the hearing threshhold of sinusoidal tones. The lower, dashed curve shows the hearing threshhold without masking noise, and the upper curve shows the the hearing threshhold with the masking noise. Note that the masking effect is greater at frequencies higher than the center frequency of the noise than for lower frequencies. (Source: Brüel & Kjær,


72

Vision is an aid for speech comprehension, especially in a loud environment. Despite that a very limited share of speech sounds can be visually deduced, lip reading can nevertheless provide the extra information needed in a difficult listening situation.

3.1.5 The influence of noise on man

Man is influenced by noise in many different ways, and different individuals are influenced to different extents. That applies to both the disturbing effects of noise and to the risk of suffering a hearing injury. A number of physiological effects result from noise exposure, such as contraction of blood vessels, dilation of the pupil, and effects on breathing. Noise reduces attention and can therefore degrade work performance. Even during sleep, our hearing monitors the environment, and sleep disturbances can occur due to noise. Sudden, unexpected, or unknown sounds stimulate the body’s defense mechanisms. This means that, among other things, blood pressure rises, muscle tension and heart frequency both increase, and blood vessels in the skin constrict.

3.1.6 Hearing injuries

Hearing injuries can be of many different types. A way to coarsely, but usefully, categorize hearing injuries is to speak of where the injury is located. One then speaks of hearing injuries of the conduction variety, or of the sensory-neural variety. By a conduction injury, it is meant that something hinders the sound conduction function of the ear somewhere between the pinna and the cochlea. Among the variants that can be named are absence of the auditory canal from birth, larger holes in the eardrum, inflammations in the middle ear, interruption of the chain of auditory bones, or fusing of the stirrup to the oval window. A pure conduction injury can give a maximum hearing loss of 60 dB. Conduction problems cause a weakening, but generally not a distortion, of sound. Hearing losses caused by conduction injuries can, as a rule, be successfully rehabilitated by surgery or with a hearing aid. A sensory-neural hearing injury affects the function of the inner ear or the auditory nerve. Usually, the hair cells in the cochlea are injured or completely absent. Common reasons are advancing age, inherited conditions, noise injury, infections, and tumors. Sensory-neural hearing loss often results in both a weakening and a distortion of sound. Such hearing loss is difficult to rehabilitate. If hair cells are completely absent, a hearing aid is of no help; in principle, such a device simply amplifies the incident sound signal. The indispensable connection to the auditory nerve is lacking in that case. With advancing age, hearing deteriorates. Figure 3-10 shows normal hearing for two age groups. It is primarily the higher frequencies that are impacted by age, and comparing to the speech banana in Figure 3-8 shows that it is the weak, high frequency, unvoiced consonants that are lost first, while the stronger vowels are usually heard well.


73

-100

10 20 30 40 50 60 70 80 90

100 110120

-100

102030405060708090

100110120

Frequency [Hz]


125 250 500 1000 2000 4000 8000

man 50 yrs

man 70 yrs

Noise injuries are a common reason for hearing loss in our society. Men are affected more often than women. It has been debated as to whether that is due to a physiological gender difference, or merely due to greater exposure of men to noise than women, in their work, military service, and recreational activities. Powerful, but short lasting, noise often causes a temporary hearing loss. Hearing returns, as a rule, after a period without noise exposure, but the hair cells can nevertheless have suffered small but irreparable injury. If the noise exposure is repeated enough times, many small such injuries can result in damage that gives a measurable hearing loss. When the ear is exposed to long-lasting and powerful noise, the hair cells in the cochlea are damaged and a permanent hearing loss is the consequence; see Figure 3-11. Research indicates that noise containing impulsive sound is considerably more damaging to hearing than a constant sound with the same equivalent sound level. The explanation for that can be that the stirrup muscle doesn’t have time to tense itself, stiffening the chain of auditory bones, as discussed in section 3.1.1. Impulsive sounds are usually not perceived as especially strong, because of their short durations, but they are all the more damaging.

Figure 3-10 Hearing loss due to age. The median value for men that have not been subjected to dangerous noise levels. Two age groups: 50 and 70 years.(Source: ISO 7029, 1986.)


74

a)

b) An audiogram, depicting hearing loss caused by noise exposure, is shown in Figure 3-12. Noise injuries give rise to hearing loss curves with the characteristic feature that the degradation is most severe in the 4 - 6 kHz frequency range. The dip in the curve there is deepened and widened as the injury progresses. Both age-induced hearing loss and noise injuries are of the sensory-neural type of hearing impairment. It is the hair cells of the cochlea that are damaged or absent, and such conditions cannot be cured.

Figure 3-11 Hair cells photograph-ed with a sweep electron microscope. The photos show three rows of outer hair cells. a) Normal,uninjured hair cells. b) Hair cells injured by noise. (Source: Göran Bredberg, Hörselkliniken, Södersjukhuset.)


75

-10 0

10 20 30 40 50 60 70 80 90

100110 120

-100

102030405060708090

100110120

Frequency [Hz]


125 250 500 1000 2000 4000 8000

3.1.7 Hearing protection

The first step to take at a noisy workplace is to try to reduce noise emission from the noise source itself. Failing sufficient improvement there, measures to improve the acoustics of the facility are the next best alternatives. If the noise level nevertheless remains dangerously high, then use of hearing protection should be adopted. There are two types of hearing protection: earmuffs (see Figure 3-13), and earplugs. Earmuffs enclose the entire outer ear, whereas earplugs are placed in the auditory canal. The latter are of two types: disposable and reusable. There are different models of both earmuffs and earplugs, with varying acoustic properties. Noise measurements at a workplace serve as a basis for the selection of proper hearing protection. Information on the SPL in different frequency bands is important, since different types of hearing protection each have their own frequency dependencies. The protection selected must sufficiently attenuate sound so that the SPL in the auditory canal does not exceed risk levels for hearing damage. It must also fit well; earmuffs should be well-sealed against the head, and earplugs should be properly inserted. Hearing protection must be worn the entire time that one spends in the noisy environment. Even short interruptions drastically diminish the protection provided. It is also important that noise pauses, during which one is away from the noisy environment and can remove the hearing protection, are provided during the workday.

Hearing protection must be maintained to ensure that the sound reduction provided does not diminish with time. The cuff (or ear cushion) of earmuffs should not be damaged, hardened, or dirty. Reusable earplugs must be kept clean; disposable ones must not be reused. In noisy workplaces, regular hearing tests of the employees should be carried out, in order to detect the onset of hearing problems as early as possible.

Figure 3-12 Example of an audiogram of a noise injury. Note the large hearing loss at 4 000 Hz.


76

Figure 3-13 Earmuff-type hearing protection.

(Sketch: Arbetarskyddsnämnden, Buller och vibrationer ombord, Ill: Claes Folkesson.)

3.1.8 Sound quality

In the introduction, an overview of the concept of good sound quality was provided; this is product sound that exudes an impression of quality and reliability. The area, as such, is not new; automobile manufacturers have thought along those same lines for decades. The dilemma, however, has been the need to rely on listening and analyzing prototypes of car models, but without the benefit of reliable analysis tools to evaluate and simulate the effect of design modifications. All modifications had to be physically realized and tested, which is very time demanding. The computer-based analysis and simulation tools available today, however, make it possible to simulate the sound resulting from constructive design changes and play it back to listener panels of potential buyers. This implies that one can reduce the development time of new and improved products. Already today, and especially during the years to come, it will be possible to use the available analysis and simulation tools to listen to how a product will sound, even before a physical prototype exists, in the early planning stages. The purpose of sound quality work is not always to bring about the quietest possible product, but even to derive clear objectives for the product on the basis of the customer’s expectations, transform these into complementary objectives for the various component parts, and then design so as to attain the overall goal for the product sound. The procedure can be divided into four stages: (i) Analyze and understand the characteristic sound content of the design, and

specify the objectives.

(ii) Couple the stated objectives to specific parts and functions of the design.

(iii) Survey and subdivide the composed structure into an source – response – acoustic radiation system, as in figures 1-2 and 1-3 in the introduction.

(iv) Predict the sound content of the product for different design variations, finding that giving the best sound quality.

Obtain hearing protectionthat you really are satisfied with. Make surethat all parts are alwaysworking correctly.

Replaceable sweat pad

Cuff Foam layer Dome liner Dome Headband


77

Predict the sound content of the product for different design variations, and evaluate their relative desirabilities.

Items (i) to (iv) are more thoroughly described below.

(i) Analyze and understand the characteristic sound content of the design, and specify the objectives. Typical consumers don’t describe a product’s sound in terms of decibels, phones, high frequency, and low frequency, but instead more subjective expressions, such as “sharp”, “dull” and “boxy”. The task is therefore to translate these subjective terms into objective and measurable quantities. The objective quantities should, as such, both be measurable and characterize the sound in a way relevant to the product at hand. They can be both conventional vibro-acoustic concepts and other mechanical quantities such as rotational speed. In connection with electric motors that adjust seat position in cars, for instance, it is known that the rotational speed has a large bearing on the subjective experience of sound. That has little to do with man’s physical hearing, but is instead related to the “psychoacoustic” feeling: − Will this mechanism keep working once the warranty has expired? There are a number of methods to transform subjective impressions into objective quantities. The most common are “paired comparison” and “semantic difference analysis”. In the former, two alternative sounds are presented to the experimental subject, who is then asked which one is the most desirable. If there are many subjects, their responses are statistically treated. Finally, relations can be established between the subjective preferences and the measurable characteristics of the sound. The “semantic” difference technique demands that the panel evaluate the sound and rank it with respect to such subjective dimensions as “soft” and “hard”, “sporty” and “luxurious” and so forth. The panel’s evaluation can then be compared to the objective quantities. When the coupling between the subjective concepts and the objective quantities has been established, the objectives can then be formulated. Examples are:

• Modification of the amplitudes of individual tones, in order to bring about a changed hearing threshold; compare to Figure 3-9.

• Altered balance between low and high frequency tire noise, as heard from inside a car.

• Elimination of the rotational speed variations of electric motors and fans.

(ii) Couple the stated objectives to specific parts and functions of the design. In order to transform sound quality objectives into concrete countermeasures, the objectives derived from part (i) above must now be coupled to the designs of individual components and different operational states. A noise problem in which there are strong tonal components can, for example, be attributed to the driveline of a passenger car (engine – transmission – drive shaft – rear axle) for operation in a certain gear at a certain speed.

(iii) Survey and subdivide the complete structure into an source – response –acoustic radiation system, as in figures 1-2 and 1-3 in the introduction.


78

Consider the example of automotive interior noise, again. Beginning with the oscillator element, a first step might be to subdivide the sound into periodic signals from the engine and transmission, and the noise-like sound from the airflow around the car body and from the tires. After that, one might identify how much of the tire noise that comes from each particular wheel. As for the response element, the problem may be dominated by a local resonance in the transmission, for example. Finally, for the radiation element, it might be found that a resonance in the car roof is responsible for the majority of the radiated sound.

(iv) Predict the sound content of the product for different design variations, finding that giving the best sound quality. Beginning with a computational model of the driveline, for example, we can predict the sound that would be generated by the proposed design. By then varying a parameter in the model, such as the engine speed for instance, we can hear when resonances or other undesirable sounds arise.

If our starting point is a measurement rather than a model, then we could use a sound quality program to eliminate resonances or shift them to other engine speeds.

In either case, we or the panel can then listen to the expected sound of the proposed product, before and after the proposed design modifications, and determine the solutions that are preferable.

3.2 EFFECTS OF VIBRATION AND SHOCK

3.2.1 Machinery and vehicle vibrations

Operation of machines and vehicles gives rise to forces. These forces, in turn, generate vibrations. In order to describe the vibrations, we need to know their amplitudes, frequencies, and sometimes even their mode shapes, i.e., the deformation pattern of the structure. In machines, the main sources of vibrations are often forces due to accelerations and retardations of masses. Examples are unbalanced shafts in rotating machinery, reciprocating motions in piston-based machines such as compressors and internal combustion engines, and reciprocating motions in sewing machines. In gears, the contact forces vary as the gear rotates, since the number of teeth in contact is not constant. Shocks and vibrations in gears are also caused by manufacturing variability (tolerances on the tooth geometry), surface roughness, and shaft misalignment. In electrical machines, such as motors and generators, the electromagnetic forces give rise to vibrations. In internal combustion engines, compressors, and other pneumatic and hydraulic machines, pressure variations in the medium are the significant vibration sources. How strong vibrations are acceptable? There are, of course, no definitive answers to that question. Many different factors could come into play, such as surface finish requirements in machining, requirements for assembly precision, or fatigue strength in the most extreme cases. In Figure 3-14, a method is proposed to divide machines into different classes, and to give criteria to judge them on the basis of vibration amplitude.


79

Class I Class II Class III Class IV

to 15 kW Small machines, up

Middle-size machines,15-75 kW or more, up to 300 kW on special foundations

Large machines onstiff and heavy foundations with eigen-frequencies exceedingthe machine’s rotational frequeny

Large machines with a rotational frequency exceeding the eigenfrequency of the

(e.g., turbo-machines)

Good

Good

Good

Good

Allowed

Allowed

Allowed

Allowed

Not allowedNot allowed

Not allowedNot allowed

Barely tolerable

Barely tolerable

Barely tolerable

Barely tolerable

45

28

18

11,2

7,1

4,5

2,8

1,8

1,12

0,71

0,45

0,28

0,18

153

149

145

141

137

133

129

125

121

117

113

109

105

Vibration velocity, rmsVibration velocity level v~ [mm/s] L v [dB] rel. 10 -9 m/s

foundation

Figure 3-14 Proposed division and evaluation of machinery vibrations with respect to machine type and vibration

level. (Source: Brüel & Kjær, information material.)

3.2.2 Effects on man

The human body is influenced by vibrations at any frequency, if their amplitude is high enough. When we study the effects on man, both the purely physical and the psychological effects must be considered. The interesting aspects are:

(i) The characteristics of the human body when subjected to vibrations and shocks.

(ii) Effects of disturbances. (These can be divided into physical, physiological and psychological criteria)

(iii) Acceptable exposure amplitudes for different exposure times and frequencies.

In order to simplify evaluation, international evaluation criteria have been developed. They divide the disturbances into so-called whole body vibrations and hand-arm vibrations.

In the case of whole-body vibrations, the disturbances are assessed according to three criteria:

(i) Health.

(ii) Comfort and feeling.

(iii) Motion sickness.

A lot of work has been carried out to survey and specify these criteria, but because the human body is very complex, and varies from individual to individual, it is a difficult task. Relatively little is known about what amplitudes cause human injuries, since it is difficult


80

to do experiments, for ethical reasons among others. Most studies have therefore been on animals. Animals, however, have different anatomical constructions and sizes than humans, making it uncertain how the results can be extrapolated to man. When the question is comfort and feeling, however, it becomes easier to carry out human experimentation, but even in this case, the results vary, especially due to psychological factors. Most studies have focused on drivers in different transport modes, such as trucks and airplanes. The human body consists of a relatively hard skeleton of bone that is bound together by ligaments, tendons, muscles and other tissues. The soft, vital organs, such as the heart and lungs for example, are protected in the chest cavity or in the abdominal cavity. They are held in place individually by ligaments and membranes, and can therefore undergo relative motions. The combination of soft tissue and bone, and softly fastened inner organs, with masses varying form very small up to a few kilos, constitutes a system that can react very differently at different disturbing frequencies. Low frequencies, under 1 Hz, occur in many transportation situations. These low frequency vibrations affect our sense of balance and bring about effects we usually call motion sickness. From a purely mechanical perspective, the entire human body follows these low frequency oscillations as a single unit; i.e., roughly speaking, a standing human body can be regarded as a rigid body for vertical motions up to about 2 Hz.

In the rest of the low frequency region, 2 - 100 Hz in rough terms, the body can, for some purposes, at low amplitudes, be described as a particle model that describes at the individual organs’ and body parts’ lowest eigenfrequencies; see Figure 3-15. For whole-body vibrations, the vibrations are typically transmitted via the floor for a standing person, and via a seat and seatback for a sitting person. The physical effects of transverse vibration, (perpendicular to the back-bone), differ from the physical effects of axial excitation (parallel to the backbone). One of the most important subsystems that can be put in motion, either for a standing or a sitting person, is the trunk and abdomen, with resonances in the 3-12 Hz region. For resonances in the shoulders,

Figure 3-15 In the low frequency 2 - 100 Hz region, the body can, for some purposes, and for low amplitudes, be regarded as a particle system. The individual organs and body parts have eigen-frequencies, the approximate values of which are indicated in the figure. (Sketch: Brüel & Kjær, informationsmaterial.)

Standing person

Chest cavity (About 60 Hz)

Head (axial motion) (About 25 Hz)

Diaphragm (Axially 10-12

Lung volume

Partial shoulder

(4 – 5 Hz)

Eye, interocular structures

(30 – 80 Hz)

Abdomen (4 – 8 Hz)

Hand (50 – 200 Hz)

Hand

Underarm (16 - 30 Hz)

Leg (Vary from 2 Hz to 20 Hz depending on position)

Sitting person


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abdomen, and hips, see figures 3-16 and 3-17. These low frequency resonances make it very difficult to devise effective vibration isolation elements, for use in a driver’s seat for example, since physical principles dictate that they have even lower eigenfrequencies; they would then risk having a swaying feel. Other important resonances in the body are the head-throat-shoulder system at about 25 Hz, and the eye at about 30-80 Hz.

Frequency [Hz] Figure 3-16 Amplification and damping of vertical vibrations from a vibrating platform under the feet to different

parts of the body, for a standing person. Eigenfrequencies from 3 to 8 Hz give amplified vibrations. Bent knees provide some attenuation. Under about 2 H, the entire body largely behaves as a rigid body. (Plot: Brüel & Kjær, Mechanical Vibration and Shock Measurements, data from Dieckmann/Radke.)

Frequency [Hz]

Figure 3-17 Amplification or attenuation of vertical vibrations from a vibrating platform to different parts of the body, for a sitting person. (Plot: Brüel & Kjær, Mechanical Vibration and Shock Measurements, data from Dieckmann.)

Hip / platform

Shoulders / platform

Head / platform

Platform

Am

plifi

catio

n fa

ctor

Waist / platform Bent knees

Head / platform

Shoulder / platform

Head / shoulder

Am

plifi

catio

n fa

ctor


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At frequencies above the lowest resonances, vibrations are relatively well-damped, assuming that no relatively undamped resonance is excited. Figure 3-18 shows that a tone of 50 Hz is attenuated about 30 dB from the feet to the head, and about 40 dB from the hand to the head. Figure 2-18 Attenuation of vibrations,

at a frequency of 50 Hz, along the human body. The attenuation is given in dB, with the excitation position providing a reference amplitude. The excitation positions are A for the hands, and B for the feet. (Diagram: Brüel & Kjær, Mechanical Vibration and Shock Measurements, data from von Bekesy.)

In the mid-frequency region, above about 80 - 100 Hz, the particle model becomes an all the more inadequate means to describe vibrations in the human body. At these frequencies, it is necessary to regard the body as a continuous medium in which wave propagation of different wave types occurs .. The effects of the vibrations are also dependent on where in the body they occur, and on how they are directed. In the ultra sound region, above several hundred thousand Hertz, the vibratory energy mainly propagates as longitudinal waves. Since the wavelengths at these high frequencies are small in comparison to the dimensions of soft body parts, geometric acoustics methods are applicable (see chapter 2). Hand-arm vibration is the other large problem area. All hand-held machines transfer vibrations to the body via the hand-arm system; see Figure 3-19. Vibrations can arise because of unbalanced rotating elements or reciprocating motions in a machine. Sometimes, however, it is not the machine itself that brings about the strongest effects, but the forces from the work process. That applies to power saws, grinding machines, and such hitting devices as pneumatic boring machines and jackhammers used in road construction.

Vibration attenuation [dB]

Vibration attenuation [dB]

Platform excitation

Head Head


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Figure 3-19 Oscillation pattern in the arm-shoulder (a,c) and hand-arm (b,d) from vibration excited into the right hand (grasping the source). The visual pattern photographed is brought about by double-pulsed holography. In both cases, the excitation is perpendicular to the plane of the photograph, with an acceleration of 30 m/s2. The dark lines, so-called interference fringes, connect points undergoing the same motion. The interference fringes can be compared to isometric (equal altitude) lines on a map; as such, they show a vibration pattern (mode). The displacement difference between two adjacent interference fringes corresponds to about half a light wavelength, i.e., about 3·10-7 m. A large distance implies lower vibration amplitudes, and a short distance implies higher amp-litudes. In a), the arm-shoulder system is excited at 37 Hz. The high-density interference pattern in the soft parts around the shoulder blades shows that the vibrations are transmitted effectively along the entire path from the hand grip via the lower and upper arm. In c), the arm-shoulder system is excited at 152 Hz. The sparse interfer-ence pattern on the upper arm shows that higher frequencies are more severely attenuated. b) and d) show excitation of the hand-arm system at 37 Hz and 152 Hz, respectively. The dense interference pattern in the elbow area at 37 Hz, as compared to 152 Hz, also demonstrates the more effective attenuation of high frequencies. (Hologram and photo: Lennart B M Svensson, Produktionsteknisk Mätteknik, KTH.)

a), b) excitation frequency 37 Hz.

c), d) excitation frequency 152 Hz.


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Vibrations transmitted to a standing or sitting person can cause such problems as motion sickness, reduced comfort, and diminished working performance. Hand-arm vibrations can even cause physical injuries if the vibration amplitudes and exposure times are sufficient. It is often local injuries that occur in the hand-arm system, mainly to blood vessels, nerves, the skeleton, or joints. In the case of blood vessel damage, the walls of the thinnest blood vessels thicken, and the blood flow is reduced. The hands are, to a large extent, the body’s temperature regulator. If the body temperature gets too high, blood flow through the hands increases. If, on the other hand, the body temperature gets too low, the blood flow is reduced by contraction of blood vessels. With injured blood vessels that have thickened walls, however, the blood flow to such peripheral body parts as the fingers can be blocked off completely, and the affected body part whitens. That phenomenon has been extensively studied, as is often called white finger syndrome. The injury is characterized by loss of feeling, numbness, and pricking sensations in the fingers; see Figure 3-20. In the case of nerve injuries, feeling in the hand is reduced.

The vibration exposure sufficient to cause injury is not completely known, neither with respect to the amplitude, the frequency content, or the exposure time. Despite difficulties and lack of quantitative information, the standard presented in section 3.3.3 does provide guidelines to judge the risk that blood flow disturbances of the white finger variety will occur.

Figure 3-20 Vibrations transferred to the hand-arm system can increase the thickness of the walls of narrow blood vessels in the outer body parts, and constrict blood flow. In cases so severe that the blood supply to the fingers is completely cut off, the fingers whiten. That is called white finger syndrome. (Sketch: Brüel & Kjær, Human Vibration.)


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3.3 STANDARDS

In the following sections, a brief description is provided of international standards in the areas of acoustics, and vibrations and shock.

3.3.1 Standard ISO 3740 Acoustics – Determination of sound power levels of sound sources – Guidance for the use of fundamental standards and for working out machine-specific noise measurement methods

Before we delve into the standard itself, let us first say something about its background. In chapter 2, we considered two possible ways to describe noise disturbances – sound pressure and sound power, or alternatively, their logarithmic equivalents, the sound pressure level and the sound power level. The sound pressure level is a useful quantity for describing the sound at a specific place, but not as useful for characterizing the acoustic behavior of a machine. This is because the sound pressure level depends on both the distance and the direction to the machine, as well as the environment in which it is placed. Environment, in an acoustics context, refers to the distance to reflecting surfaces, the room’s volume, and the sound absorption of the reflecting surfaces. The sound power level is, instead, a measure of the total radiated sound power from a machine, and is largely independent of the environment. The sound power and the corresponding sound power level are therefore preferable whenever sound data is to be provided for a certain machine. Based on sound power data, we can:

(i) Calculate the sound pressure level and sound level at a given point in a room, provided we know the distance from the source, the room’s volume, and its sound absorption properties.

(ii) Compare sound data for machines of both the same and different types and sizes.

(iii) Determine whether the machine fulfills agreed-upon specifications.

(iv) Plan for noise control measures that will be needed after the machine is installed.

(v) Sound power data can, moreover, provide a measure of performance of success of efforts to develop quieter products.

A complete description of a machine’s sound emission would consist of the total radiate sound power, and its directivity. The directivity is a measure of a how the machine’s acoustic radiation varies in different directions. Many sources, however, have very little directivity. That is especially true of sources that are small compared to the sound wavelength, which is often the case at low frequencies. ISO 3740 briefly describes the principles of the determination of sound power levels, and gives guidance on choosing the appropriate sound power measurement standard. They are summarized in table 3-3. Table 3-4 gives an overview of the precision of the various methods. If a particular measurement standard exists for the machine type of interest, that would naturally be used.


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Table 3-3 Standard ISO 3741-3747 on different methods to determine the sound power levels of machines and equipment. (Source: ISO 3740-3747 and Brüel & Kjær, Sound Power.)

Interna-tional standard ISO

Class of Method

Test environment

Volume of sound source

Type of noise Measure-able sound power levels

Other information obtainable

3741 Precision Reverberant room with specified data

Continuous, broad band

Third octave, octave

A-weighted sound power level

3742 Should be < 1% of the

Continuous, pure tones, narrow band

3743 Technical Special reverberant room

test room volume

Continuous, broad band, narrow band, pure tones

Octave, A-weighted

Sound power levels with other weighting filters

3744 Indoors in a large room, or outdoors

Largest dimension < 15m

Third octave, octave, A-weighted

Directivity, sound pressure as a function of time, sound power levels with other weighting filters

3745 Precision Anechoic or semi-anechoic room

Should be < 0.5% of the test room volume

All

3746 Survey No special environment

No restrictions. Only limited by the available measurement space

A-weighted

Sound pressure as a function of time, sound power levels with other weighting filters

3747 Survey; only a relative method

No special environment, sound source not moveable

No limitations

Continuous, broad band, narrow band, pure tones

Sound power level in octave bands. Sound pressure level at specified locations


87

Table 3-4 Uncertainty in the determination of sound power levels, expressed as the largest values of the standard deviation in dB. (Source: ISO 3740 and Brüel & Kjær, Acoustics Sound Measurement.)

International standard ISO

Octave band [Hz] 125 250 500 1000-4000

8000 A-weight

Third octave band [Hz]

100-160

200-315

400-630

800-5000

6300-10000

3741 3742

3 2 1,5 3 -

3743 5 3 2 3 2 3744 3 2 1,5 2,5 2

Anechoic room 1 1 0,5 1 - 3745 Semi-anechoic

room 1,5 1,5 1 1,5 -

3746 - - - - - 5 3747 For a source that generates discrete tones 5

For a source that emits an evenly distributed noise over the frequency band of interest

4

From table 3-3, it is evident that three classes of methods exist: precision; technical; and, survey. The precision methods specify rigorous requirements for the measurement environment, demanding access to expensive acoustic measurement laboratories, as shown in figures 3-21 and 3-22; on the other hand, the survey methods do not have such requirements. The standards ISO 3741-3743 are based on measurements in diffuse fields, and ISO 3744-3746 on measurements in free fields. The last, ISO 3747, is intended for machines that cannot be moved, and therefore also makes no special demands on the environment. Figure 3-21 Reverberant room at MWL, KTH (Royal Inst Techn, Sweden). In a reverberant room designed for precision measurements, rigorous requirements are to be met. The floor, walls, and ceiling are to absorb, at most, 6% of the sound power of the incident waves. The volume of the room should fall in the range of about 200 to 300 m3 and its dimensions must have the proportions to ensure that the eigenfrequencies, as discussed in section 5.2, be evenly distributed along the frequency axis. The plexiglass panels that hang from the ceiling are intended to improve the diffuse sound field. Despite all of that, an ideal diffuse sound field is not attainable; thus, several microphone positions are used in practice, or, as in the picture, a rotating microphone boom is used to obtain a spatial average in the room. The measurement object on the floor is a calibrated sound power source of the fan type. (Photo: HP Wallin, MWL.)


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Figure 3-22 Semi-anechoic room at ASU Sound and Vibration Lab., Ain Shams University in Egypt. In the

anechoic measurement room, the same conditions must apply as for propagation in a free field. When the sound waves reach the floor, walls, and ceiling, they must be effectively absorbed. Less that 1% of the incident sound power is permitted to reflect back into the room. In order to be effective at low frequencies, and to be approved for precision measurements, the mineral wool wedges that are commonly used must be at least about a quarter wavelength long; see the detail view. For the room in the picture, that implies a length of about 1.1 m to obtain a lower frequency bound of 80 Hz. The room in the picture has a hard, reflecting floor, and is therefore a so-called semi-anechoic room. In that type of measurement room, the sound field obtained is equivalent to a free field over a hard reflecting plane; that corresponds, for example, to the situation of a vehicle traversing a reflecting roadway. A semi-anechoic measurement room, in combination with a chassis dynamometer, a kind of rolling highway, may therefore be found at any automobile manufacturer. (Photo: HP Wallin, MWL.)

1,1m


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3.3.1.1 Standard ISO 3747 Acoustics – Determination of sound power levels – survey method using a reference sound source

This standard describes a method based on relative measurements using a so-called calibrated sound power source.. The standard permits reflecting surfaces in the vicinity of the measurement object. Before becoming too specific about the contents of the standard, we begin by getting acquainted with some background. For a calibrated sound power source, there should be a calibration protocol that gives the sound power in third octave and octave bands, and with A-weighting included. Two types of calibrated sound power sources exist. Firstly, there are the electroacoustic sources that consist of a generator/amplifier part and a loudspeaker part. Secondly, there are fan-type sound sources that contain a radial fan driven by a powerful electric motor, as in Figure 3-21. The overdimensioned motor supplies a constant rotational velocity which, in turn, generates a stable sound spectrum. An important requirement for such reference sources is that, in addition to a constant sound power, they shoud also have a low directivity, i.e., they should radiate about the same in all directions. The standard does not make any demands on the measurement environment. That means that the sound waves recorded by the measurement microphone can be either direct sound from the source, or reflected sound; see Figure 3-23.

Figure 3-23 The standard for sound power determination ISO 3747 specifies no requirements for the measurement environment, which means that both direct and reflected sound can reach the microphone positions.

According to the preceding section, the sound power level is a measure of the radiated sound power from a machine, and essentially independent of the acoustic environment, whereas the sound pressure level depends on both the distance and the direction to the machine, and the environment in which it is placed. For a certain acoustic environment, i.e., a specific room, and a specific distance to the sound source the machine’s sound power level LW can be related to the resulting sound pressure level Lp at a point, using the so-called room correction K [dB], in the formula

KLL Wp += . (3-5)

The method for the sound power determination with a calibrated sound power source is based on determining the room correction K using the known sound power of that source.

Microphone

Source Direct sound

Reflected sound


90

The sound pressure level from the measurement object is measured first, after which it is replaced by the calibrated reference source, and the sound pressure level measured again at the same microphone positions. The sound power level LW of the measurement object can then be determined from (3-5), as

( )rr pWpW LLLL −+= , (3-6)

where pL is the sound pressure level from the measurement object,

rWL is the sound power level of the reference source,

rpL is the sound pressure level of the reference source.

Usually, the sound pressure level is measured at a number of points on a measurement surface around the measurement object. Then, the spatial average of the sound pressure levels pL is used in (3-6), as

)101log(101

10∑=

⋅=N

n

Lp

pn

NL [dB], (3-7)

where Lpn is the sound pressure level at point n.

Example 3-2 At five measurement points, the sound pressure levels 79, 82, 83, 79 and 81 dB are measured. Determine the spatial average.

Solution (3-7) gives ( ) dB 8109.811010101010

51log10 1.89.73.82.89.7 ≈=⎥⎦

⎤⎢⎣⎡ ++++⋅=pL .

ISO 3747 places no demands on the measurement environment, and is intended for measurement objects that cannot be moved to a measurement environment that gives a higher degree of precision. The overview given below is considerably shortened, and assumes a measurement object placed freely on a reflecting floor. Measurement object: Machines that cannot be moved, and their parts and components.

Running condition is to be specified. Sound Character: Stationary. Broad band, narrow band, discrete tones and

combinations of these. Quantities determined: A-weighted sound power level. Frequency region: Octave bands with center frequencies from 125 Hz up to 8000 Hz. Background levels: The background level for each octave band is to be at least 3 dB

below the corresponding sound pressure level when the


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measurement object or reference source are operated. The correction for the background level is to be made if it is not at least 10 dB lower; see example 1-6.

Placement of reference source: The calibrated reference source can either be placed at the ordinary

position of the measurement object, or at one or more points around the measurement object, depending on its size.

Microphone positions: To find the microphone positions, a hypothetical parallelepiped

reference surface, surrounding the measurement object and reference source positions, is first defined. The microphone positions are then placed on another, typically parallelepiped, surface, with its sides parallel the reference surface, and at a distance of 1 m from it. For measurement objects with horizont-al dimensions of less than a meter, the microphones are placed in the middle of the five measurement surfaces, see Figure 3-24.

1m

1m

1m

Microphone position

Measurement surface

Reference surface Reflecting floor

3.3.2 Standard ISO 2631-1 1997 Vibration and shock – Guidance for evaluating the effects of whole-body vibrations on man. Introduction

Machines and means of transport of all kinds subject man to vibrations that affect comfort, working ability, and, in the worst case, health. Despite the inherent difficulties due to the involved situations, and to some extent the lack of reliable knowledge, an international standard has nevertheless been agreed upon. The standard primarily seeks to simplify the assessment and comparison of measurement results concerned with whole-body vibrations. The standard is concerned with vibrations transferred to the whole body from the surface one stands, sits, or lays upon. It consists, presently, of two parts: Part 1 ISO 2631-1 General requirements

Figure 3-24 The microphone positions on a measurement surface are at distances of 1 m from the reference surface. The reference surface surrounds the measurement object and the reference source.


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Pat 2 ISO 2631-2 Vibration and Shock - Measurement and guideline values for the assessment of comfort in buildings.

The standard includes directions on how to apply the results, and how and what should be measured. In the following two sections, the main features of parts 1 and 2 are briefly described.

2.3.2.1 ISO 2631-1 Part 1: General requirements

Introduction: The primary objective of part 1 is to specify methods to report measurement values used in evaluating whole body vibrations, with respect to:

(i) Health.

(ii) Comfort and sensation.

(iii) Motion sickness.

Part 1, by itself, does not contain any guideline values for evaluating the affects on health, comfort, sensation and motion sickness. It also does not treat the influence of the vibrations on working productivity or the possibility to carry out special work tasks, since those things are highly dependent on the individual in question, the work situation, and the actual work task. Motion sickness is not handled any further in this description, aside from the frequency bands of relevance given below, and the frequency curve in Figure 3-26.

Frequency region: (i) Health, comfort and sensation, 0.5-80 Hz.

(ii) Motion sickness, 0.1-0.5 Hz.

Measurement directions and positions: The vibrations are to be measured in a coordinate system defined by

Figure 3-25, and on the surface that transfers these to the human body:

(i) Standing person – foot surface.

(ii) Sitting person – the seat.

(iii) Lying person – pelvis and back.

The standard states that, for sitting persons and evaluation with respect to comfort, the comfort can, in some cases, be affected by rotational vibrations about the coordinate axes, in the sitting surface, as well as vibrations in the foot support surface. Moreover, for lying persons, without a soft pillow under the head, vibrations under the head are also to be accounted for when evaluating either comfort or health. We will refrain from describing such cases in this more cursory description.

If the strength of the vibrations varies over the support surface, then a spatial average, corresponding to that of section 3.3.1.1 is to be determined.


93

z

z

z

z z

y

y

x

y

x

x

x

y

Figure 3-25 Coordinate systems for measurement of the effects of

vibrations on the human body. Hand transferred vibrations are treated in section 3.3.3. (Sketch: Arbetarskyddsnämden, Buller och vibrationer ombord, Ill: Claes Folkesson.)

Measurement Quantities: The vibration amplitudes are to be weighted by a weighting filter using a method analogous to that of determining the sound level in dB(A); see Figure 3-7.The weighting curves reflect the sensitivity of the body to whole-body vibrations in the vertical and horizontal directions. The measurement quantity is the rms acceleration wa~ [m/s2], in the frequency band of interest. The index w indicates that the quantity is frequency weighted according to

,)~(~1

2∑=

=N

nnnw aWa (3-8)

where Wn is the weighting factor as a function of the frequency, according to Figure 3-26 and table 3-5. The weighting factor Wd,n is used in the x- and y-directions, and Wk,n is used in the z-direction.

na~ [m/s2] is the rms acceleration in the n-th frequency band, e.g., measured in third-octave bands.

N is the number of the frequency band.

Standing

Sitting

Lying down

Hand transferred vibrations


94

Weighting factor Wk,, Wd,, Wf

0.01 0.1 1 10 100 1000 0.00001

0.0001

0.001

0.01

0.1

1

10

Frequency f [Hz]

Wd

Wk

Wf

Figure 3-26 The weighting factors Wk, Wd and Wf reflect the sensitivity of the human body to vibrations in the x-, y- and z-directions. The factor Wd is used in the x-and y-directions, Wk in the z-direction and Wf in the vertical direction, in connection with the assessment of motion sickness. In the standard, there are a total of six weighting curves for different directions and purposes. (Source: ISO 2631-1.)

Table 3-5 The weighting curves Wk and Wd, in third octave bands. (Source: ISO 2631-1.)

Frequency [Hz] Weighting factor Wk Weighting factor Wd

0.5 0.418 0.853 0.63 0.459 0.944 0.8 0.477 0.992 1 0.482 1.011

1.25 0.484 1.008 1.6 0.494 0.968 2 0.531 0.890

2.5 0.631 0.776 3.15 0.804 0.642

4 0.967 0.512 5 1.039 0.409

6.3 1.054 0.323 8 1.036 0.253 10 0.988 0.212

12.5 0.902 0.161 16 0.768 0.125 20 0.636 0.1 25 0.513 0.08

31.5 0.405 0.0632 40 0.314 0.0494 50 0.246 0.0388 63 0.186 0.0295 80 0.132 0.0211


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As discussed in section 2.3, the peak factor is the ratio of the peak value to the rms value of a signal. If the peak factor, calculated with the frequency weighted peak and rms values, does not exceed nine, then the signal’s rms value can be determined in the usual way described in section 2.3. If the vibrations contain shocks that are so strong that the peak factor exceeds 9, then another method that gives greater weight to the peaks should be used instead. In the assessment with respect to health effects, the standard is mainly concerned with sitting, because the effects on lying and standing persons are not sufficiently well known. After calculating the frequency weighted accelerations in the x-, y- and z-directions, from formula (3-8), the values in the x- and y-directions are multiplied by an direction factor K = 1.4. Then, the assessment is made using the highest value obtained. Example 3-3 In the driver’s seat of a work machine, an 8 Hz tone dominates in all three coordinate directions. The following rms-values have been measured:

x-direction; xa~ = 0.84 m/s2, y-direction; ya~ = 0.71 m/s2, z-direction; za~ = 0.48 m/s2.

Which value is to be used to make the assessment with respect to health effects?

Solution Using the weighting filter from table 3-5, or from Figure 3-26, one obtains

Quantity x-direction y-direction z-direction Acceleration a~ [m/s2] 0.84 0.71 0.48

Weighting factor W dW = 0.253 dW = 0.253 kW = 1.036 Direction factor K K = 1.4 K = 1.4 K = 1.0

Weighted acceleration waK ~⋅

[m/s2] 0.30 0.25 0.50

The value in the z-direction is therefore the highest, and should be used in the assessment.

For assessment with respect to comfort, the vector amplitude of the frequency-weighted acceleration, va~ , is determined from

2,

2,

2,

~~~~zwywxwv aaaa ++= , (3-9)

where xwa ,~ , ywa ,

~ and zwa ,~ are the frequency-weighted accelerations in

the respective directions, each obtained from formula (3-8).


96

For assessment with respect to the sensitivity threshold, i.e., the lowest value a normal person can detect, the assessment is to be made with respect to the highest value in any coordinate direction, using formula (3-8).

Vibration type/ spectrum: The vibrations can be periodic, stochastic, transient or combinations of

these, i.e., they can be of both narrow band character, with tonal components, or broad band with noisy character. They can be measured or evaluated with band pass filters of CAB or CRB-type; see section 2.10.1. The latter case includes, for example, a third octave band filter.

Guideline values: As mentioned at the outset, the standard itself does not contain any

guideline values, but some of its appendices give guidance on evaluating the effects of vibration on health, comfort, sensation, and motion sickness. Below, a short summary of the first three is provided.

Health: In the case of daily exposure for 8 hours, no health effects have been reported for 43.0~ ≤⋅ waK m/s2. Moreover, for 73.0~ ≥⋅ waK m/s2, injury is probable. Two methods are also provided to normalize exposure times less than 8 hours per day. The information given applies to completely healthy people, and must be used with extreme caution, as well as the understanding that it usually takes a number of years before any health effects can be observed.

Comfort: Acceptable values for comfort vary according to many factors, such as exposure time, noise, expectations, and the actual activity one is engaged in, such as reading, writing, computer work, or consumption of food and drink. One of the appendices gives the following as probable reactions to the indicated vibration levels experienced by passengers in mass transportation. Table 3-6 Assessment of comfort when traveling by mass transportation.

(Source: ISO 2631-1.)

Acceleration, vector amplitude Degree of comfort

va~ < 0.315 m/s2 Not uncomfortable 0.315 < va~ < 0.63 m/s2 A little uncomfortable

0.5 < va~ < 1.0 m/s2 Rather uncomfortable 0.8 < va~ < 1.6 m/s2 Uncomfortable

1.25 < va~ < 2.5 m/s2 Very uncomfortable

va~ > 2.0 m/s2 Extremely uncomfortable Sensitivity threshold: Half of a healthy, alert population can just barely

sense a vertical acceleration with a peak value wa = 0,015 m/s2, weighted with the weighting filter Wk.


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Example 3-4 On the floor aboard a vehicle, the following vertical and horizontal acceleration values have been measured, in third octave bands.

f [Hz] 2 2.5 3.15 4 5 za~ [m/s2] 0.50 0.70 0.40 0.50 0.40

yx aa ~~ = [m/s2] 0.45 0.60 0.35 0.45 0.30 Other frequencies are negligible in this context, and the peak factor is lower than nine. How will a passenger assess the vibration comfort?

Solution: We apply the frequency-dependent weighting filter from table 3-5, and then calculate the frequency-weighted acceleration in the three coordinate directions, using formula (3-8),

f [Hz] 2 2.5 3.15 4 5 Wk 0.531 0.631 0.804 0.967 1.039 Wd 0.890 0.776 0.642 0.512 0.409

zwa ,~ = 88.0)40.0039.1(...)70.0631.0()50.0531.0( 222 =⋅++⋅+⋅ m/s2,

70.0)30.0409.0(...)60.0776.0()45.0890.0(~~ 222,, =⋅++⋅+⋅== ywxw aa m/s2.

When evaluating with respect to comfort, the vector amplitude va~ of the acceleration, from formula (3-9) is determined.

va~ = 222 88.070.070.0 ++ = 1.32 m/s2.

According to table 3-6, the assessment made is “uncomfortable” to “very uncomfortable”.


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3.3.2.2 ISO 2631-2 Influence of vibration and shock on people in buildings

Frequency range: 1 - 80 Hz. Measurement point & direction: The measurement point is chosen where the vibrations are greatest. Those are normally vertical vibrations in the middle of the floor with the greatest dimensions. Measurement Quantities: The vibration amplitudes are to be weighted by a weighting curve in

the same fashion as for determining the sound level in dB(A); see Figure 3-7. The weighting curve reflects the sensitivity of the body to whole-body vibrations, and is therefore based, in principle, on the curves of that type given in Figure 3-26. The measurement quantity can either be the rms velocity wv~ or the rms acceleration wa~ . The index w indicates that the value is frequency-weighted. The measurement quantity can either be measured with a frequency-weighting filter, or calculated in a corresponding way, based on third-octave bands.

Vibration type/ spectrum: Either narrow or broad band vibrations can be treated. Guidelines: Table 3-7 Guideline values for the assessment of the effect of vibrations and shock on

humans in buildings.

Weighted velocity, wv~

[m/s]

Weighted acceleration, wa~

[m/s2] Moderate disturbance (0.4 - 1.0)·10-3 (14.4 - 36.0)·10-3 Probable disturbance > 1·10-3 > 36·10-3

The guidelines are not intended to be applied to temporary activities such as building and installation work. The standard maintains that few people experience vibration levels below “moderate disturbance” as disturbing. The vibration levels that fall into the “moderate disturbance” category occasionally result in complaints. In the “probable disturbance category”, the vibrations are very noticable and often experienced as disturbing. If the frequency-weighted value is dominated by a particular frequency, which is often the case due to building resonances, then the frequency-weighted value can be replaced by the unweighted rms velocity at the dominant frequency;

the degree of disturbance can be read from Figure 3-27.


99

0.01

Frequency [Hz]

Threshold of sensitivity according to ISO 2631-1 1985

Moderate disturbance

Probable disturbance

0.001

0.0001 1 2 4 8 16 32 64

Speed v~ [m/s]

Figure 3-27 Threshold of sensitivity and regions of moderate disturbance

and probable disturbance as functions of frequency for vibrat-ons in buildings. The curves are based on the rms velocities of vibration without weighting. (Source: SS 460 48 61)

Vibration duration: When the vibrations vary in time, the running conditions of the vibration source are to be recorded. For traffic-induced vibrations, established methods from mathematical statistics can be used.

Example 3-5 In a factory building, a floor resonance is measured to be at 31.5 Hz. The unweighted rms vibration velocity is 1.0 mm/s. Determine whether there is a risk of complaints.

Solution Reading from Figure 3-27, it turns out that the value given falls on the boundary between the regions of moderate disturbance and probable disturbance. Complaints are possible.

3.3.3 Standard ISO 5349 Vibration and shock – Guidelines for the measurement and assessment of hand-transmitted vibration

Frequency region: 5 - 1500 Hz.

Measurement points & directions: The measurement results are to be given in a coordinate system in accordance with Figure 3-25. The measurements are to take place in three directions, in the immediate vicinity of the surfaces that transfer vibrations into the hand, i.e., at the hand grip when applicable. The assessment is based on components in the direction showing the highest measurement values.


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Measurement Quantity: The rms acceleration in one or more directions is measured.

The following options are called out: (i) measurement in third-octave bands, 6.3 - 1250 Hz, (ii) measurement in octave bands, 8 - 1000 Hz, (iii) measurement of frequency-weighted values from 5.6 - 1400 Hz. For the case of frequency weighting, a weighting filter from figure 3-28 is to be used. In the first two cases, the third-octave and octave band values are to be converted to frequency-weighted accelerations. The frequency-weighted acceleration is calculated from

( )∑=

=N

nnhnwh aKa

1

2,,

~~ , (3-10)

where Kn is the weighting factor as a function of frequency, from Figure 3-28, and table 3-8.

nha ,~ [m/s2] is the rms acceleration within the n-th third-octave band or octave band. The index h stands for hand. N is the number of frequency bands.

Vibration duration: The assessment of the effects of hand-transmitted vibrations is based on

a daily exposure time. The total daily exposure time is assumed, when making the assessment, to be 4 hours. To make comparisons to other exposure times possible, an equivalent-energy frequency weighted acceleration for a 4 hour exposure time is used. Then, if the total daily vibration exposure deviates from 4 hours, it can be normalized to the 4-hour equivalent energy value, by the formula

( )∫=τ

0

2,

4)4(,, )(~1~ dtta

Ta wheqwh , (3-11)

where )4(,,~

eqwha is the equivalent-energy, frequency-weighted rms acceleration for a 4-hour exposure.

)(~, ta wh is the rms value as a function of time for the frequency-

weighted acceleration. τ is the total daily work time in hours.

T4 is four hours.

If the exposure time T deviates from four hours, the corresponding equivalent-energy acceleration for four hours is calculated as

)(,,4)4(,,~~

Teqwheqwh aTTa = [m/s2], (3-12)

where )(,,~

Teqwha is the equivalent-energy frequency-weighted acceleration for a time period of T hours, T4 is four hours.


101

0

0.2

0.4

0.6

0.8

1

1.2

6.3 10 16 25 40 63 100 160 250 400 630 1000Frequency [Hz]

Weighting factor, Kn

Figure 3-28 The weighting factor Kn reflects the sensitivity of the

human body to vibrations transferred to the hand-arm system. (Source: ISO 5349)

Example 3-6 For a handheld machine, a frequency-weighted equivalent-energy acceleration value of

)5(,,~

eqwha = 15 m/s2 for a time interval T = 5 h is measured. Determine the corresponding value for a time period of four hours.

Solution (3-12) gives

8.164515~)4(,, ==eqwha m/s2.

Frequency [Hz]

Weighting factor

Kn 6.3 1 8 1 10 1

12.5 1 16 1 20 0.8 25 0.63

31.5 0.5 40 0.4 50 0.3 63 0.25 80 0.2

100 0.16 125 0.125 160 0.1 200 0.08 250 0.063 315 0.05 400 0.04 500 0.03 630 0.025 800 0.02 1000 0.016 1250 0.0125

Table 3-8 Weighting factor Kn in third-octave bands. (Source: ISO 5349.)


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Guidelines: The standard does not provide boundaries for safe exposure, but only guidelines. In an appendix that is not formally a part of the standard, information is given that makes it possible to estimate the probability of developing white finger syndrome, as a function of the frequency-weighted energy equivalent rms value for four hours of daily exposure, )4(,,

~eqwha ,

and the exposure time in years. The curves show the time that passes before signs of white finger syndrome begin to appear for 10, 20, 30, 40 and 50 % of the exposed persons.

Ekvivalent frekvensvägd acceleration,

Expo

nerin

gstid

i an

tal å

r inn

an b

lodf

löde

sstö

rnin

gar

av ty

p "v

ita fi

ngra

r" u

ppträ

der i

pro

cent

uell

ande

l av

expo

nera

de in

divi

der

51

2

2

3

10

5

10

25

20

h,w,eqa

20

(4) , [ / ]

50

%40

%%

%2010

30

%50

Figure 3-29 The relation between equivalent frequency-weighted acceleration and the exposed. In the proposed changes of1999, only the 10% -line is retained.(Source: ISO 5349.)

Exp

osur

e tim

e in

ye

ars

befo

re

bloo

d flo

wdi

stur

banc

es o

f th

e w

hite

fin

ger

vari

ety

appe

arin

a g

iven

per

cent

age

of th

e in

divi

dual

s

Equivalenf frequency weghted acceleration,


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3.3.4 ISO 8662 Hand machines – Handheld motor driven machines – Measurement vibrations in the hand grip

The standard describes the procedure for the measurement of vibrations in the hand grip of handheld, motor-driven machines. The purpose is to describe a laboratory method that, to the extent possible, gives the same results that would be obtained in real working conditions.

ISO 8662 consists of the following parts. • Part 1: General requirements.

This part contains general requirements for the measurement of vibrations in any hand-held machine. The other parts of the standard describe more specific procedures for particular kinds of machines.

• Part 2: Chipping hammers and riveting hammers. • Part 3: Rock drills and rotary hammers. • Part 4: Grinding machines. • Part 5: Pavement breakers and hammers for construction work. • Part 6: Impact drills. • Part 7: Wrenches, screwdrivers, and nut runners. • Part 8: Polishers and orbital sanders. • Part 9: Rammers. • Part 10: Nibblers and shears. • Part 11: Fastener driving tools. • Part 12: Saws and filing machines. • Part 13: Die grinders. • Part 14: Stone-working tools and needle scalers.

3.3.5 Standard ISO 4866 Vibration and shock – Building vibrations - Guidance for measurement of vibrations and assessment of its effects on buildings

Vibrations in a building can originate from many different sources. Examples of sources of building vibrations are earthquakes, rock blasting, wind, and machinery installation. Most of the injuries suffered by buildings are due to human activities resulting in vibrations in the 1 - 150 Hz range. Natural sources, such as earthquakes, primarily cause building damage at very low frequencies, typically 0.1 - 30 Hz. Wind excitation is commonly in the 0.1 - 2 Hz frequency range. The vibration amplitudes that such sources induce vary within a large range. The standard does not provide guideline values, but simply indicates typical amplitudes and frequencies from different types of sources; see table 3-9.


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Table 3-9 Typical frequency ranges and vibration amplitudes of building vibrations caused by various excitation sources. (Source: ISO 4866.)

Source Frequency

range f [Hz]

Displacement ξ~ [μm]

Velocity v~ [mm/s]

Acceleration a~ [m/s2]

Character see symbol

key Traffic road/rail 1 - 80 1 - 200 0.2 - 50 0.02 - 1 C/T

Blasting groundborne 1 - 300 100-2500 0.2 - 500 0.02 - 50 T

Piling groundborne 1 - 100 10 - 50 0.2 - 50 0.02 - 2 T

Machines outdoor, groundborne

1 - 300 10 - 1000 0.2 - 50 0.02 - 1 C/T

Noise traffic, machines outdoors

10 - 250 1 - 1100 0.2 - 30 0.02 - 1 C

Machines Indoor 1 - 1000 1 - 100 0.2 - 30 0.02 - 1 C/T

Earthquake 0.1 - 30 10 - 105 0.2 - 400 0.02 - 20 T Wind 0.1 - 10 10 - 10

5 T Noise Indoor 5 - 500

Symbol key: C = continuous process, T = transient process

3.4 REGULATIONS AND RECCOMENDATIONS

Regulations on noise and vibrations cover many areas of societal concern, but even so, such laws and regulations do not cover all relevant situations. The various authorities involved have limited areas of responsibility, and there are holes in important areas; e.g., for traffic noise, there are requirements for individual vehicles, but not for situations in which vehicles act in concert, i.e., for the areas surrounding highways The inner market of the EU came into being on January 1, 1995. The objective of the inner market is to create conditions of free passage over national borders of goods, services, capital, and people. To bring that about, the EU issues so-called directives, and the national authorities must adapt their respective laws and regulations to ensure that they are not in conflict with those.

3.4.1 Machines

The EU machine directive (89/392 EEC with amendment 91/368 EEC) is one of the more important directives, and deals with a large number of machinery types, from power saws to rock crushers. Certain machines are excepted from the directive, such as cranes, medical machines in direct contact with patients, amusement park machines, boilers, tanks, pressure vessels, fire arms, means of passenger and freight transport, police and military machines, etc. If a product is not specifically excepted as such, the manufacturer is then


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responsible to ensure compliance with the requirements of the directive, in order for the product to have unrestricted access to the EU’s inner market. The machinery directive has two objectives:

(i) To eliminate technical hindrances to trade.

(ii) To incorporate provisions for health and safety in machinery operation.

The directive has many aspects. As far as noise and vibrations are concerned, there are general formulations, such as

"machines are to be designed and produced to minimize noise and vibrations to the lowest possible levels, with regard to technical advances and the availability of noise and vibration countermeasures, above all at the source. "

It also mandates that technical documentation contain the following information:

(i) Noise. • If the equivalent A-weighted sound level LA,eq is lower than 70 dB(A), that is to be indicated. Example: LA,eq < 70 dB(A).

• If LA,eq is greater than 70 dB(A), the actual level is to be specified: Example: LA,eq = 80 dB(A).

• If LA,eq is greater than 85 dB(A), then the A-weighted sound power level is to be given as well. Example: LA,eq = 96 dB(A), LW,A =104 dB(A).

• If the instantaneous C-weighted level, LC, exceeds 130 dB(C), that level is to be indicated. Example: LC = 135 dB(C).

The noise levels given above refer to the operator position. The running condition of the machine, as well as the measurement method, are to be given. The simplest approach is to use an EN-standard for the specific type of machine, or, if that doesn’t exist, a standard from the International Standards Organization, ISO.

(ii) Vibrations. Hand-arm vibrations: • If the frequency-weighted rms acceleration wa~ from section 3.3.3 is less than 2.5 m/s2, that is to be indicated. Example: wa~ < 2.5 m/s2. • If wa~ is greater than 2.5 m/s2, the actual value is to be indicated. Example: wa~ = 2.8 m/s2. Whole body vibrations: • If the frequency-weighted rms acceleration wa~ , based on the relevant standard, is less than 0.5 m/s2, that is to be indicated. Example: wa~ < 0.5 m/s2. • If wa~ is greater than 0.5 m/s2, the actual value is to be indicated. Example: wa~ = 1.2 m/s2.

The manufacturer is to issue an assurance of EU-compliance, if the machine fulfills the requirements, and may then mark the machine with the CE-logo shown in Figure 3-30. For excepted machines, there are other directives that apply.


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3.4.2 Vehicles

3.4.2.1 Motor vehicles

EU directives (70/157 EEC & 96/20 EC);proscribes the highest permissible sound levels for different categories of vehicles. Table 3-10 Highest allowed sound levels of motor vehicles, as proscribed by the EU directives 70/157 EEC, with

amendment 96/20 EC.

Highest allowed sound level, dB(A) Vehicle category EU directives 70/157 EEC &

96/20 EC Passenger car 74 Bus or truck with a total weight up to 3.5 tons with a total weight up to 2 tons

76

with a total weight over 2 tons, but < 3.5 tons 77 Bus with a total weight over 3.5 tons with a motor power 150 kW

78

with a motor power of 150 kW or higher 80 Truck with a total weight over 3.5 tons with a motor power less than 75 kW

77

with a motor power betw 75 kW and 150 kW 78 with a motor power over 150 kW 80 Motorcycle (depending on cylinder volume) Moped

The measurements are, primarily, to be performed in accordance with an international standard, ISO 362. The standard proscribes a specific test path, and that the vehicle is to drive by a microphone at a distance of 7.5 m, as illustrated in Figure 3-31. For motorcycles and mopeds, there are special measurement instructions.

Figure 3-30 CE-logo applied to a machine, which serves as an assurance of compliance with the EU Machinery directive. Guarantees free access to the EU inner market.


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Figure 3-31 Measurements of external vehicle (pass-by) noise are to be carried out in accordance with an

international standard, ISO 362, which states that the vehicle is to drive by a microphone set-up at 7.5 m distance. From the beginning of the test path, A-A, until its end, B-B, the vehicle is to accelerate at full throttle from a certain specified speed. No hindrances to sound are permitted within a radius of 50 m. (Picture: SCANIA, Fordonsakustik och buller.)

3.4.2.2 Airplanes

For aircraft, environmental compliance certificates are issued, and guarantee that the type of airplane in question meets the requirements of the international organization ICAO, International Civil Aviation Organization. The certification is issued in connection with the so-called type-certification. The member countries of the ICAO incorporate its regulations in their respective national requirements for airplane certification. Airplanes with take-off weights exceeding 9000 kg are permitted certain maximum noise values, depending on the weight and number of motors. The three points used in certification pertain to noise data for take-off, side-line, and landing. The take-off value is established at two points. One of those is 6500 m from the “break-release”-point and directly under the flight path. The “break-release”-point is the point at which the air-plane releases the brakes and begins the take-off. The other take-off value is called “side-line”, and is measured where the highest value is obtained, at a distance of 650 m for so-called chapter 2-airplanes and 450 m for chapter 3-airplanes. The landing value is measur-ed at a point 2000 m before the touch-down point, directly beneath the flight path. The so-called chapters correspond to the sharpening of environmental standards by the ICAO. Chapter 3-airplanes, i.e., the airplanes that only meet the requirements at 650 m distance, will be prohibited from flying early in the 2000-decade. No new registrations of that type will be permitted in the EU.

Noise data is measured linearly in third-octave bands, and corrected for, among other things, duration and differences in the levels in each third octave band, before summation. The result is given in units of EPNdB, Effective Perceived Noise.

7.5 m7.5 m

1.2 m


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For airplanes with take-off weights under 9000 kg, only a maximum value during take-off is proscribed. The value is indicated as a sound level in dB(A).

3.4.3 Workplace environment

3.4.3.1 Workplaces, with certain exceptions

EU has issued the directive 86/188 EEC "On the protection of workers from risks due to exposure to noise while working ". It is a so-called minimal directive, which means that the member countries are permitted to issue stricter requirements. Sweden has taken advantage of that option in the proclamation AFS 1992:10 Noise, issued by the labor protection administration. That proclamation dictates, among other things, that if specific limits on noise exposure are exceeded, the reasons are to be investigated, and action is to be taken; see table 3-11.

Table 3-11 Maximum allowed noise levels, as specified by the Swedish Labor Protection Administration proclamation AFS 1992:10 Noise.

Equivalent A-weighted sound level in a typical work day LA,eq = 85 dB(A)

Maximum A-weighted sound level (excepting impulsive sound) LA,max = 115 dB(A)

C-weighted peak value of impulsive sound LC,peak = 140 dB(C)

The workplace safety law, on which the proclamation named above is based, applies to all workplaces with certain exceptions, e.g., aboard ships.

3.4.3.2 Noise aboard ships

Ships are required to comply with the UN’s International Maritime Organization IMO Resolution A: 468 (XXII) Code of Noise Levels Onboard Ships. That recommendation is applied voluntarily by the member states, and contains:

• Maximum allowed sound levels in different spaces, such as the machine room, the navigation spaces, cabins, etc.

• Limiting values for noise exposure. The limit for the equivalent A-weighted level, LA,eq, is 80 dB(A) for a period of 24 hours, which corresponds to 85 dB(A) over 8 hours.

• Requirements on acoustic insulation between cabins.


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3.4.4 Dwellings

There are no EU regulations on the noise in dwellings but there are national regulations. As an example information about Swedish regulations for noise in dwellings will be given. The Swedish health protection law regulates the application of measures to prevent or eliminate sanitary nuisances. A sanitary nuisance is any kind of disturbance that can be injurious to human health, and which is neither minor nor temporary. The assessment of the measures to be undertaken against sanitary nuisances is carried out by the municipal authority responsible for applying the health protection law.

3.4.4.1 New buildings

For newly built dwellings and premises, the Housing Authority (Boverket) building regulation BFS 1998:38 contains directives and general direction on noise from installations inside and outside of buildings, as well as on noise from road traffic. The rules promulgated proscribe, among other things, requirements on insulation against airborne sound, maximum sound levels from footfalls in stairways and rooms above, maximum sound levels from such installed equipment as elevators, etc., and maximum reverberation times. The reverberation time describes how sound reflective a room or space is, as described in detail in chapter 7. Moreover, maximum traffic-induced sound levels, indoors and outdoors, are specified. The rules for dwellings make reference to the Swedish standard SS 02 5267 Building acoustics – sound classification of spaces in buildings -dwellings. The objective of the standard is to simplify the work of the building personnel to improve the acoustic quality of buildings, and to classify buildings for the sake of consumers. The requirements are divided in to four sound classes:

• Sound class A: Very good acoustic conditions.

• Sound class B: Minimum requirements on good living environment; the tenants might still be disturbed, however.

• Sound class C: Meets the requirements of Swedish authorities.

• Sound class D: Used when sound class C cannot be met by renovation work; for example, a 100 year-old stone house.

Table 3-12 gives examples of the highest allowed sound levels of different disturbance sources and spaces.


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Table 3-12 Highest allowed equivalent and maximum sound levels from installations and traffic, indoors and outdoors, at dwellings and other types of premises (Source: BFS 1998:38.)

Room or Space Equivalent sound level (Sound of long duration) LA,eq [dB(A)] LC,eq [dB(C)] For installations, the length of the measurement period is equivalent to the duration of the disturbance; for traffic, it is one day

Highest sound level (Sound of short duration) LA [dB(A)]

Dwellings/ Installations

Kitchen LA,eq = 35 dB(A) LA = 40 dB(A) Bedrooms LA,eq = 30 dB(A)

Additionally, for bedrooms, LC,eq = 50 dB(C)

LA = 35 dB(A)

Dwellings/traffic Kitchen LA,eq = 35 dB(A) Bedrooms LA,eq = 30 dB(A) LA = 45 dB(A)

May be exceeded up to five times per day 2200-0600.

Outside at least half of the bedrooms of an apartment

LA,eq = 54 dB(A)

In at least one outdoor space/balcony connected to the apartment

LA,eq = 54 dB(A)

Other premises/installations

Places of instruction (class-rooms), sleep or rest

LA,eq = 30 dB(A) LA = 35 dB(A)

Other premises/traffic Health care premises, day care and recreation centers, places of instruction (classrooms)

LA,eq = 30 dB(A) LA = 45 dB(A) May be exceeded up to five times per day 2200-0600.

Offices LA,eq = 40 dB(A)


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3.4.4.2 Existing buildings

The Social Administration (Socialstyrelsen) published, in 1996, as part of its collected works, a document titled " Noise indoors and high noise levels, SOSFS 1996:7 (M), General guidance.” That is intended to be used as a tool to help municipalities and others in their work to alleviate different types of societal noise problems occurring indoors in dwellings, and premises for instruction and health care, meeting places, etc. The guidance is intended to suggest when a societal noise problem constitutes a sanitary nuisance in accordance with the health protection law. In some cases, it is not possible to alleviate a sound problem to the extent recommended by that guidance. According to the health protection law, that type of situation is to be handled on a case-by-cases basis, with an investigation into whether it is reasonable to demand full compliance. That investigation is to take into account both economic and technical aspects. This does not apply, however, to noise from air, road, or railway traffic. Table 3-13 gives guideline values for indoor noise. Table 3-14 makes recom-mendations for assessing low frequency noise indoors, and, finally, table 3-15 makes recommendations for assessing sanitary nuisances at high sound levels. Table 3-13 Guideline values for assessing sanitary

nuisances, with respect to indoor noise. (Source: SOSFS 1996:7.)

Comments: The higher guideline value for maximum noise (45 dB(A)) is intended as a protection against sleeping difficulties, waking, etc. To be regarded as a sanitary nuisance, it is sufficient that the guideline value is exceeded a few times, during a night for example. The lower guideline value for maximum noise (35 dB(A)) is used in assessing whether a disturbance, in certain cases, can be regarded as a risk of sanitary nuisance. It should, in that case, be a recurrent phenomenon. The guideline value for equivalent noise (30 dB(A)) refers to the time period at which the disturbing activity takes place. It seeks to protect against sleeping disturbances, speech masking, and perceived disturbance. Table 3-14 Recommendations for assessing the sanitary nuisance, for

equivalent low frequency noise indoors. (Source: SOSFS 1996:7.)

Comments: The recommendations can help to assess whether a perceived disturbance of equivalent low frequency noise is a risk of sanitary nuisance. Table 3-15 Recommendations for assessing sanitary

nuisances at high sound levels. (Source: SOSFS 1996:7.)

Comments: The recommendations on the boundary at which high sound levels constitute a risk of sanitary nuisance are applied to discotheques, concerts, etc. Both indoors and outdoors.

Level Maximum LA,max = 35 – 45 dB(A) Equivalent LA,eq = 30 dB(A)

Third-octave band f [Hz]

Equivalent Third octave bands level

Leq [dB]

31.5 56 40 49 50 43 63 41.5 80 40 100 38 125 36 160 34 200 32

Level Maximum LA,max = 115 dB(A) Equivalent LA,eq = 100 dB(A)


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3.4.5 External noise

3.4.5.1 External industry noise

For external industry noise there are no common EU regulations but as an example information about Swedish national regulations are given in this section. The Swedish Nature Conservation Authority (Naturvårdsverket) has issued, in RR 1978:5 External industry noise – general guidance, guideline values for external industry noise. These are based on equivalent sound levels in dB(A) outdoors and in free field conditions, i.e., no hindrances or buildings are to be located in the immediate vicinity of the measurement point; see table 3-16. Table 3-16 The Nature Conservation Authority guidelines on external industry noise. The figures without

parentheses apply to newly established industrial sites, and with parentheses to already-existing industrial sites. (Source: RR 1978:5.)

Equivalent sound level in dB(A) Highest sound level in dB(A)

Area Daytime hours 07-18

Weekdays 18-22, Sundays and holidays 07-18

Nighttime hours 22-07

Instantaneous nighttime sound, 22-07

Workplaces without noisy activities

60, (65) 55, (60) 50, (55) -

Dwellings and recreational areas adjacent to dwellings, and educational and health buildings

50, (55) 45, (50) 40, (45) 55, (55)

Recreational areas and mobile recreation where exposure to nature is important

40, (45) 35, (40) 35, (40) 50, (50)

If the noise contains audible tones or impact noise, the requirements are made 5 dB(A) stricter. The rules also specify the measurement methodology, and that the noise is to be measured in the free field, at a 1.5 meter elevation, under certain permissible atmospheric conditions and wind speeds, etc.


113


THE EAR AND HEARING

Measures of sound Table 3-17 A-, B- and C-weighting for third-octave and octave bands.

The octave bands are indicated in bold print.

Frequency [Hz]

A-weighting [dB]

B-weighting [dB]

C-weighting

[dB] 25 -44.7 -20.4 -4.4

31.5 -39.4 -17.1 -3.0 40 -34.6 -14.2 -2.0 50 -30.2 -11.6 -1.3 63 -26.2 -9.3 -0.8 80 -22.5 -7.4 -0.5

100 -19.1 -5.6 -0.3 125 -16.1 -4.2 -0.2 160 -13.4 -3.0 -0.1 200 -10.9 -2.0 0 250 -8.6 -1.3 0 315 -6.6 -0.8 0 400 -4.8 -0.5 0 500 -3.2 -0.3 0 630 -1.9 -0.1 0 800 -0.8 0 0 1000 0 0 0 1250 +0.6 0 0 1600 +1.0 0 -0.1 2000 +1.2 -0.1 -0.2 2500 +1.3 -0.2 -0.3 3150 +1.2 -0.4 -0.5 4000 +1.0 -0.7 -0.8 5000 +.5 -1.2 -1.3 6300 -0.1 -1.9 -2.0 8000 -1.1 -2.9 -3.0

10000 -2.5 -4.3 -4.4 12500 -4.3 -6.1 -6.2 16000 -6.6 -8.4 -8.5 20000 -9.3 -11.1 -11.2

Sound level ∑=

Δ+⋅=N

n

ALA

npnL1

10/)(10log10 [dB(A)] (3-1)

Equivalent sound pressure level

))(1log(100

2

2

, dtp

tpT

LT

refTeq ∫⋅= [dB] (3-2)

)101log(100

10/)(, dt

TL

TtL

Teqp∫⋅= [dB] (3-3)


114

115

CHAPTER FOUR

MATHEMATICAL METHODS AND VIBRATIONS OF SIMPLE MECHANICAL SYSTEMS This chapter introduces mathematical methods commonly applied to the analysis of acoustics and vibrations problems. Firstly, it covers rotating vectors, or phasors, and the description of physical quantities using complex numbers; these topics are then applied to the determination of mechanical power and the study of linear oscillations.

4.1 COMPLEX NUMBERS AND ROTATING VECTORS

Using Fourier analysis, complicated, time-dependent events that exactly repeat, or are periodic, can be decomposed into their harmonic components. Doing so amounts to approximating the time-dependent function describing the event by the sum of a set of sine and cosine functions. In more involved cases, the resulting mathematical expressions may be very complicated. As a mathematical tool to simplify the computations in such cases, one may make use of rotating vectors or phasors; see Figure 4-1. If the projection on the x-axis is called the real part and the projection on the y-axis the imaginary part, we can describe the function with the aid of a complex, rotating vector according to )sin()cos( tite ti ωωω += . (4-1)

That relation is usually called Euler’s identity. A summary of formulas pertaining to complex numbers is given in Appendix A.

Chapter 4: Mathematical methods

116

A time-dependent function v(t) = A cosωt + B sinωt can, with the help of (4-1), be written in the form v(t) = ((A - iB) eiωt + (A + iB) e-iωt ) / 2. Thus, a complete description of the event can be obtained by using two rotating vectors with equal, but oppositely directed, rotational velocities: +ωt and -ωt. Because the factor preceding e-iωt is the complex conjugate of the factor preceding eiωt, it suffices to use only one of them in calculations, and results for the other case can always be found by complex conjugation. Thus, in computations, one uses a complex-valued function v(t) = ((A - iB) eiωt = A eiωt, in which complex quantities are indicated by bold print. The actual time function can always be recovered by taking the real part of the final result, e.g., F(t) = Re(F(t)).

Figure 4-1 Rotating vectors are used as aids in the description of harmonic functions. The projection on the x-axis, of a component with a circular frequency of ω, gives the function cos(ωt), and the projection on the y-axis is sin(ωt).

Example 4-1 A harmonically-varying force has a time-dependence of F(t) = F cosωt. The complex force can therefore be written in the form F(t) = F eiωt = F (cosωt + i sinωt), where F(t) = Re( F iωt).

Example 4-2 A harmonically-varying force has a time dependence of F(t) = 1F cosωt - 2F sin ωt. The complex force can therefore be written in the form

=++=+= ))sin())(cos(ˆˆ()ˆˆ()( 2121 titFiFeFiFt ti ωωωF

)),cos(ˆ)sin(ˆ())sin(ˆ)cos(ˆ( 2121 tFtFitFtF ωωωω ++−=

where F(t) = Re( F eiωt).

x

ytωsin

tω

tωcos

tω

tω


117

Example 4-3 As an alternative to the treatment given in example 4-2, the force can be written as

)cos(ˆ)sin(ˆ)cos(ˆ)( 21 ϕωωω +=−= tFtFtFtF ,

where

22

21

ˆˆˆ FFF += and )ˆˆarctan( 12 FF=ϕ .

The complex force can therefore be written as

tiiti eeFt ωϕω FF ˆˆ)( == + ,

where F is a complex amplitude ϕieFˆ =F and ( ))(Re)( ttF F= . Example 4-4 A harmonic vibration with a peak velocity amplitude of v , circular frequency ω and phase angle ϕv at time t = 0 can be written, using the complex vector approach, as

)(ˆ)( vtievt ϕω +=v .

Determine the amplitude and phase of the velocity, acceleration, and displacement, and sketch these in the complex plane.

Solution Velocity: Amplitude: v . Phase t = 0: ϕv. t = t: ωt + ϕv. Acceleration:

=== + )(ˆ)()( vtievdtdt

dtdt ϕωva

=== + vωω ϕω ievi vti )(ˆ

=⎭⎬⎫

⎩⎨⎧ =+== iiei

2sin

2cos2 πππ

)2

()(2 ˆˆπϕωϕω

π

ωω+++ ==

vv

titiieveve .

Transition from velocity to acceleration therefore implies multiplication by iω.

Amplitude: vω . Phase t = t: ωt + ϕv + π/2,

ω t + ϕv

)(ˆ)( vtievt ϕω +=v

Re

Im

v

ωv

−ωv

ω t + ϕv +π/2

ωv

v

v)2(ˆ)( πϕωω ++= vtievta


118

(Phase angle π/2). Displacement:

∫ ∫ === + dtevdttt vti )(ˆ)()( ϕωvx

=−=== ++ )()( ˆ1ˆ1 vv titi evii

evi

ϕωϕωωωω

v

{ }==−=−= − 2/2/ ππ ii eei

)2/()(2/ ˆˆ πϕωϕωπωω

−++− == vv titii eveve .

Transition from velocity to displacement implies, therefore, division by iω.

Amplitude: ωv . Phase t = t: ωt + ϕv - π/2, (Phase shift -π/2).

Example 4-5 The complex conjugate (see appendix A) of the force can be used for calculating both the force amplitude (modulus) and its real part. The complex conjugate of F is indicated by F*. A harmonically-varying force, and its complex conjugate, are shown in the adjacent figure. The modulus of a force is given by

FFeFeF titi ˆˆˆˆ 2* ==== − ωωFFF .

The real part of the force can be calculated from

=−++=+= ))sin(ˆ)cos(ˆ)sin(ˆ)cos(ˆ(21)(

21)Re( * tFitFtFitF ωωωωFFF

).cos(ˆ))cos(ˆ2(21 tFtF ωω ==

F = Fe i tω

ω t

-ω t

F * = −Fe i tω

Re

Im

ω t + ϕv

v( ) ( )t ve i t v= +ω ϕ

Re

Im

v

v

v

v ωω t + ϕv - π/2

−v ωx( ) ( )t v e i t v= + −

ωω ϕ π 2


119

4.2 MECHANICAL POWER

A harmonically-varying force and velocity have respective time dependencies as follow in (4-2) and (4-3):

( ) )ˆRe()cos(ˆ)sin(ˆ)cos(ˆ)( 21 FtiF eFtFtFtFtF ϕωϕωωω +=+=+= ; (4-2)

( ) )ˆRe()cos(ˆ)sin(ˆ)cos(ˆ)( 21vti

v evtvtvtvtv ϕωϕωωω +=+=+= . (4-3)

From fundamental mechanics, the instantaneous mechanical power can be calculated from

=== ))(Re())(Re()()()( tttvtFtW vF

=++= ))()((21))()((

21 ** tttt vvFF

,4/))()()()()()()()(( tttttttt vFvFvFvF **** +++= (4-4)

which can be expressed in the form

=+= 2/)))()(Re())()((Re()( *tttttW vFvF

=+= −++ 2/)ˆˆRe(2/)ˆˆRe( 2 vFvF iiiiti evFevF ϕϕϕϕω

2/))cos()2(cos(ˆˆvFvFtvF ϕϕϕϕω −+++= . (4-5)

Typically, it is the time average of the power that is of interest, and which corresponds to the power that is fed into a mechanical system. The time average of the first term, above, is zero, so that the time-averaged power can be expressed as

)cos(ˆˆ21)(1

0vF

TvFdttW

TW ϕϕ −== ∫ , (4-6)

where the overbar indicates time-averaging. Thus,

)Re(21)Re(

21 ** vFFv ==W . (4-7)

From (4-6), it is evident that maximal power is delivered when force and velocity are in phase, i.e., ϕF = ϕv , whereas no power, at all, is delivered when the phase shift between those quantities is 90º. In that latter case, one can speak of the force being “90º ahead” of the velocity, ϕF - ϕv = 90º, or “90º behind” the velocity, ϕv - ϕF = 90º.


120

4.3 LINEAR SYSTEMS

Often in Vibrations and Acoustics, as well as in other fields for that matter, our interest resides in the calculation of what effect a certain physical quantity, called the input signal, has on another physical quantity, called the output signal; see Figure 4-2. An example is that of calculating what vibration velocity v(t) is obtained in a structure when it is excited by a given force F(t). That problem can often be solved by making use of the theory of linear time- invariant systems.

Linear time-invariant

system Input Signal Output Signal

F’(t),v’(t)F(t),v(t) p’(t),u’(t)p(t),u(t)

From a purely mathematical standpoint, a linear system is defined as one in which the relationship between the input and output signals can be described by a linear differential equation. If the coefficients are, moreover, independent of time, i.e., constant, then the system is also time invariant. A linear system has several important features. The superposition principle implies that if the input signal a(t) gives rise to an output signal b(t), and the input signal c(t) gives rise to an output signal d(t), then the input signal a(t)+c(t) yields the output signal b(t)+d(t). The homogeneity principle states that if the input signal a(t) is multiplied by a constant α, then the output signal is α b(t). A linear system is also frequency-conserving, in the sense that only those frequency components that exist in the input signal can exist in the output signal. Example 4-6 The figure below, from the introduction, shows an example in which the forces that excite an automobile are inputs to a number of linear systems, the outputs from which are vibration velocities at various points in the structure. The vibration velocities are then, in turn, inputs to a number of linear systems, the outputs from which are sound pressures at various points in the passenger compartment. By adding up the contributions from all of the significant excitation forces, the total sound pressures at points of interest in the passenger compartment can be found. In this example, the linear system is described in the frequency domain by so-called frequency response functions, to be described in detail in section 4.3.4. The engine is fixed to the chassis via vibration isolators. If the force F1 that influences the chassis can be cut in half, then, for a linear system, all vibration velocities v1 – vN caused by the force F1 are also halved. In turn, the sound pressures p1 – pN, which are brought about by the velocities v1 – vN, are halved as well. If we make the simplifying assumption that all of the forces acting on the car are uncorrelated, and that those forces are the only sound sources acting, it implies that if all of them can be reduced by 5 dB, then both the linear sound pressure level Lp and the A-weighted sound pressure level LA, are reduced by 5 dB

Figure 4-2 A linear time-invariant system describes the relationship between an input signal and an output signal. For example, the input signal could be a velocity v(t), and the output signal a force F(t), or the input signal an acoustic pressure p(t) and the output signal an acoustic particle velocity u’(t).


121

Yik Zji

(Picture: Volvo Technology Report, nr 1 1988.)

In this chapter, linear oscillations in mechanical systems are considered, i.e., oscillations in systems for which there is a linear relation between an exciting force and the resulting motion, as described by displacements, velocities, and accelerations. Linearity is normally applicable whenever the kinematic quantities can be regarded as small variations about an average value, implying that the relation between the input signal and the output signal can be described by linear differential equations with constant coefficients.

4.3.1 Single Degree of Freedom Systems

In basic mechanics, one studies single degree-of-freedom systems thoroughly. One might wonder why so much attention should be given to such a simple problem. The single degree-of-freedom system is so interesting to study because it gives us information on how a system’s characteristics are influenced by different quantities. Moreover, one can model more complex systems, provided that they have isolated resonances, as sums of simple single degree-of-freedom systems.

Figure 4-3 shows a mechanical single degree-of-freedom (“sdof”) system consisting of a rigid mass m, a spring with spring rate κ, and a viscous damper with a damping coefficient dν. The spring and the viscous damper are located between the mass and the foundation, and are considered to be massless. That implies that the forces on the opposing endpoints of each are equal and oppositely directed, for both elements.

Body Forces Sound Pressure

Vibration velocitiesPassenger compartment


122

κ dv

m

x(t)F(t)

Newton’s Second Law gives the equation of motion of the system, in the form

),)(),(()(2

2t

dttdxtxF

dttxdm x= . (4-8)

Fx contains the spring force, the damper force, and the external exciting force

)()()( tFdt

tdxdtxF vx +−−= κ , (4-9)

where m is mass of the body, κ is the spring constant, dv is the viscous damping coefficient, F(t) is the external excitation, x is the displacement of the mass, dx / dt its velocity, d 2x / dt2 its acceleration. These two equations lead to a second order linear differential equation with constant coefficients,

)()()(2)( 202

2tgtx

dttdx

dttxd

=++ ωδ , (4-10)

in which the following simplifications have been incorporated:

mκω =0 , mdv 2=δ , mtFtg )()( = . (4-11)

where ω0 is the eigenfrequency of the system, and δ is the damping constant. The solution to the differential equation consists of both a homogeneous part xh(t) that corresponds to the homogeneous differential equation, i.e., with the right hand side equal to zero, and a particular solution xp(t) that corresponds to the non-homogeneous differential equation, i.e., with the right hand side non-zero. If there is damping in the system, then the free vibrations can be assumed to have been damped out after a number of periods, and only the forced vibrations remain. Thus, it is usually only the particular solution that is of interest, i.e., the second term in )()()( txtxtx ph += . (4-12)

Because the system is linear, its particular solution, when the exciting force is described by the rotating vector (4-13), represents an oscillation at the excitation frequency, but with a different phase and amplitude. A reasonable assumption for xp is given by (4-14),

Figure 4-3 Single Degree-of-Freedom System.


123

tiegt ωˆ)( =g , tip

tiipp eeext ωωϕ xx ˆˆ)( == . (4-13,14)

That assumed form, substituted into (4-10), provides the following result:

titip

tip

tip egeeie ωωωω ωδωω ˆˆˆ2ˆ 2

02 =++− xxx . (4-15)

The phase and magnitude of the complex amplitude px can now be determined to be

δωωω 2)(

ˆˆ22

0 ig

p+−

=x , (4-16)

ϕipp exx ˆˆ = , (4-17)

2222

0 )2()(

ˆˆδωωω +−

=g

px , (4-18)

πωω

δωϕ n+−

=20

22arctan , n = 0, 1, 2, … . (4-19)

From (4-11) and (4-16), it is apparent that forω « ω0, the stiffness κ determines the displacement. Thus, the low frequency response is stiffness-controlled. On the other hand, for ω » ω0, the mass m determines the displacement response; the high frequency response, therefore, is mass-controlled. Finally, for ω ≈ ω0, the value of the viscous damping coefficient dν is decisive for the displacement; the response at frequencies around the natural frequency is therefore said to be damping-controlled. The magnitude of the amplitude px varies with circular frequency ω. A normalized response, called the amplification factor φ, can be defined as

)0(ˆ

)(ˆ

==

ω

ωφ

p

p

x

x ⇒

20

20

220 )()(4))(1(

1

ωωωδωωφ

+−= . (4-20)

4.3.2 Two degree-of-freedom systems

The simple single degree-of-freedom system can be coupled to another of its kind, producing a mechanical system described by two coupled differential equations; to each mass, there is a corresponding equation of motion (see Figure 4-4). To specify the state of the system at any instant, we need to know time t dependence of both coordinates, x1 and x2, from which follows the designation two degree-of-freedom system.


124

m1 m2

x1(t) x2(t)

F1(t) F2(t)κ1

dv1

κ2

dv2

κ3

dv3

Newton’s second law for each mass gives

⎟⎠

⎞⎜⎝

⎛= t

dttdx

dttdx

txtxFdt

txdm x ,

)(,

)(),(),(

)( 212112

12

1 , (4-21)

⎟⎠

⎞⎜⎝

⎛= t

dttdx

dttdx

txtxFdt

txdm x ,

)(,

)(),(),(

)( 212122

22

2 , (4-22)

)()()()())()(()( 121

21

1212111 tFdt

tdxdt

tdxddt

tdxdtxtxtxF x +⎟⎠

⎞⎜⎝

⎛ −−−−−−= ννκκ , (4-23)

)()()()(

)())()(( 22

321

2232122 tFdt

tdxd

dttdx

dttdx

dtxtxtxF x +−⎟⎠

⎞⎜⎝

⎛ −+−−= ννκκ . (4-24)

(4-21) - (4-24) give

+⎟⎟⎠

⎞⎜⎜⎝

⎛−++

dttdx

dttdx

ddt

tdxd

dt

txdm

)()()()( 212

112

12

1 νν

( ) )()()()( 121211 tFtxtxtx =−++ κκ , (4-25)

−+⎟⎟⎠

⎞⎜⎜⎝

⎛−−

dttdx

ddt

tdxdt

tdxd

dt

txdm

)()()()( 23

2122

22

2 νν

)()())()(( 223212 tFtxtxtx =+−− κκ . (4-26)

Matrix and vector notation can be incorporated into (4-25) and (4-26), which is useful for generalizing to an arbitrary number of degrees-of-freedom. The matrix formulation even makes it possible to solve the system of differential equations using software that performs matrix computations. (4-25) and (4-26) are therefore expressed as

[ ] [ ] [ ] Fxdtxd

dt

xd=⋅+⋅+⋅ KDM

2

2 (4-27)

where

Figure 4-4 Two degree-of-freedom system.


125

[ ] ⎥⎦

⎤⎢⎣

⎡=

2

10

0m

mM , [ ] ⎥

⎦

⎤⎢⎣

⎡+−

−+=

322

221

ννν

νννddd

dddD , (4-28,29)

[ ] ⎥⎦

⎤⎢⎣

⎡+−

−+=

322

221κκκ

κκκK , (4-30)

⎭⎬⎫

⎩⎨⎧

=)()(

)(2

1txtx

tx and ⎭⎬⎫

⎩⎨⎧

=)()(

)(2

1tFtF

tF . (4-31,32)

Once again, let the excitation forces and the particular solutions be expressed by rotating vectors tiet ω

11ˆ)( FF = , tiet ω

22ˆ)( FF = , (4-33,34)

tiet ω1p1p xx ˆ)( = , tiet ω

pp xx 22 ˆ)( = . (4-35,36)

Putting (4-33,34,35,36) into (4-27) gives

[ ] { } [ ] { } [ ] { } { }FxKxDxM ˆˆˆˆ2 =⋅+⋅+⋅− ppp iωω . (4-37)

Solving to the homogeneous equations with the force vector F set equal to zero leads to the system’s eigenfrequencies. Setting, moreover, the damping matrix equal to zero, in order to obtain the undamped eigenfrequencies, the latter are found to be real. Damping, on the other hand, brings about complex-valued eigenfrequencies; the complex values contain information on both the undamped eigenfrequencies and the system damping. The eigenfrequencies ω1 and ω2 are given by the homogeneous equation [ ] { } [ ] { } { }0ˆˆ2 =⋅+⋅− xKxMω . (4-38)

The condition for the existence of solutions to (4-38) is that the system determinant is identically zero, i.e.,

[ ] [ ] 0)det( 2 =+− KMω . (4-39)

For a two degree-of-freedom system, (4-39) has two solutions corresponding to two eigenfrequencies. A system with n degrees-of-freedom has n eigenfrequencies. The eigenfrequencies of the two degree-of-freedom system are

( ) ( )21

32312122

22

232

21

221

2

32

1

212,1 24422 mmmmmm

κκκκκκκκκκκκκκκω

−−−+

++

+±

++

+= .

(4-40)

From linear algebra, it is known that there is an eigenvector corresponding to each eigenvalue (eigenfrequency). These eigenvectors are mutually independent (orthogonal), and contain information on how the system oscillates in the vicinity of their respective


126

eigenfrequencies. Specifically, they describe the system’s mode shapes, as noted in section 1.7. The mode shapes, x1 and x2, are obtained by substituting the eigenfrequencies, i.e., the solutions of (4-39), into (4-38), yielding

[ ] { } [ ] { } { }0ˆˆ 1121 =⋅+⋅− xKxMω , (4-41)

[ ] { } [ ] { } { }0ˆˆ 2222 =⋅+⋅− xKxMω . (4-42)

Example 4-7 A method that provides a respectable amount of isolation of a vibrating machine (see chapter 9) is to use a so-called double-elastic mounting; see Figure 4-5.

m1

m2

x2

x1

κ2

κ1 dv1

dv2

Fstör

Mass

Base

Figure 4-5 A double-layered elastic mounting can provide considerable isolation of a vibrating machine.

Suppose that the vibrating machine is represented by a point mass m2, and that its vibrations are generated by a harmonic excitation force Fexc with a circular frequency ω . To reduce the resulting vibrations in the foundation, the machine is to be isolated by incorporating a spring – mass – spring system between the machine and the foundation, as illustrated in Figure 4-5. The parameters of the model are as follows: m1 = 100 kg, m2 = 500 kg, κ1 = 5⋅106 N/m, κ2 = 1⋅106 N/m, dv1 = 100 kg/s and dv2 = 200 kg/s. a) Determine, for the isolated system, the

(i) undamped eigenfrequencies. Which frequencies, i.e., vibration frequencies generated by the machine, is the mounted machine sensitive to?

(ii) mode shapes. b) In order to quantify the effect of vibration isolation on the foundation, it is common to compare the force on the foundation with and without the isolation system present.


127

Determine (i) the force Fwithout on the foundation without the isolation system in place; (ii) the force Fwith on the foundation with the isolation system in place; (iii) the ratio γ = Fwithout / Fwith. Sketch, in a frequency diagram, that ratio

calculated using three different choices of the spring constant κ2: 1⋅106 N/m; 5⋅106 N/m; and, 1⋅107 N/m. In which frequency band is the vibration isolation effective?

Solution a) In the first task, the undamped ([D] = 0) vibration isolation system’s eigenfrequencies, and corresponding mode shapes, are determined.

(i) The undamped system’s eigenfrequencies can be calculated with the aid of formula (4-40), with the third spring constant κ3 set equal to 0. Thus,

⎩⎨⎧

≈⎩⎨⎧

≈=7.406.245

165760343

2,1ω rad/s

i.e., ⎩⎨⎧

≈48.6

1.392,1f Hz.

(ii) The system’s undamped mode shapes are the solutions to the homogeneous system, with the circular frequency ω set to each of the eigenfrequencies in turn. The undamped ([D] = 0) homogeneous system of equations of motion is, with the assumed harmonic solution forms, exactly as given in (4-38). That, with values entered, becomes

⎪⎩

⎪⎨⎧

=−⋅+⋅−=−⋅−⋅

0ˆ500ˆ101ˆ1010ˆ100ˆ101ˆ106

22

26

16

12

26

16

xxxxxx

n

nωω , n = 1, 2,

in which x1 and x2 (see Figure 4-5) indicate the coordinates of both masses. By multiplying the second of these equations by the factor

6

26

106

100101

⋅

−⋅− nω

,

and substituting in one of the two numerical values for the eigenfrequency, it becomes identical to the first equation. That implies that both equations are linearly dependent, in complete agreement with the theory. A linearly dependent system, with two unknowns, has an infinite number of solutions along a straight line in the x1- x2- plane. In order to solve the system, the equation of that line must be determined. Set the amplitude of x1 to α in the second equation of the set, and solve for the amplitude of x2; that is found to be

αω

⋅−⋅

⋅=

26

6

2500101

101ˆn

x .

If the amplitude of x1 has the value α, then that of x2 must have the value given by the formula above. Then, the eigenvector corresponding to eigenfrequency ωn has the form


128

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

−⋅

⋅⋅=26

6

5001011011

n

nω

αψ ,

where α is an arbitrary constant. Thus, the eigenvector is a vector with a specified direction, but arbitrary length. The physical interpretation of the eigenvector’s direction is the ratio between the amplitudes of motion of the two masses in a resonant oscillation. Putting in the eigenfrequencies as described, with α set to 1 for the sake of convenience, provides an eigenvector or mode form for each,

ωn = ω1 : ⎭⎬⎫

⎩⎨⎧

−=

034.01

1ψ ,

ωn = ω2 : ⎭⎬⎫

⎩⎨⎧

=8.5

12ψ .

The interpretation of the first eigenvector, for example, is that if the system is excited by an excitation frequency near the first eigenfrequency, the system vibrates resonantly with the amplitude of the second mass 0.034 times that of the first. The minus sign, moreover, indicates that the masses move in opposite phase, i.e., in mutually opposing directions.

b) In the second part of the problem, the influence of the vibration isolation system on the foundation is studied; in other words, the force on the foundation, with and without isolation in place, is to be compared.

(i) Without isolators, the entire excitation force is transmitted to the foundation, i.e.,

excwithout FF ˆˆ = .

(ii) With the vibration isolation system, the force on the foundation can be determined from the spring-damper relation for the spring-damper element between mass 1 and the foundation, i.e.,

{ } titiv

titiwith eisubstediee ωωωω ωωκ 1

61111 ˆ)100105(.ˆˆˆ xxxF +⋅==+= ,

where the displacement amplitude of mass 1, in the particular excitation case in which the machine excitation is Fexc, should be used. That displacement amplitude (of mass 1) can, in turn, be calculated by solving the non-homogeneous system of equations (4-37) with the relevant values substituted in, and with the common factor eiω t cancelled out; thus,

⎪⎩

⎪⎨⎧

=⋅++−⋅−−=⋅−−⋅++−

0ˆ101ˆ200ˆ500ˆ101ˆ200

ˆˆ101ˆ200ˆ106ˆ300ˆ100

26

222

16

1

26

216

112

ppppp

excppppp

iiFii

xxxxxxxxxx

ωωωωωω

From the second of these two equations, the displacement amplitude of mass 1 is solved in terms of that for mass 2. Putting that into the first equation, the amplitude of mass 2 can then be expressed in terms of the excitation force amplitude, as


129

excp Fiii

i ˆ)200101()100300106)(500200101(

200101ˆ262626

6

2 ⋅+⋅−−+⋅−+⋅

+⋅=

ωωωωωωx .

The force on the foundation, with the isolation system in place, is therefore

excwith Fiii

ii ˆ)200101()100300106)(500200101(

)200101)(100105(ˆ262626

66

⋅+⋅−−+⋅−+⋅

+⋅+⋅=

ωωωωωωωF ,

which is apparently a function of the frequency of the exciting force.

(iii) The ratio γ between the force on the foundation without and with the isolation, using results from (i) and (ii) above, is

)200101)(100105(

)200101()100300106)(500200101(ˆ

ˆ)( 66

262626

ωωωωωωωω

iiiii

with

without

+⋅+⋅+⋅−−+⋅−+⋅

==F

Fγ .

When the magnitude of the ratio is larger than one, the isolator is effective, i.e., it isolates the foundation from the vibrations of the machine. If the ratio is less that one, the incorporation of the isolator is actually counterproductive, i.e., the force on the foundation is larger than it was without the isolator in place. When a vibration isolator is to be designed, it is a matter of choosing an isolator with the kind of properties that cause the ratio to be greater than 1 for all strong frequencies in the spectrum of the excitation force. In the figure, below, the ratio is plotted for three different values of the spring constant κ2. From the figure, it is evident that the location of the eigenfrequencies is important to the isolation behavior. At the eigen-frequencies, the isolation has a deep minimum, at which it is negative-valued, i.e., the force on the foundation has decreased after incorporation of the vibration isolation system. This example shows that even for a two degree-of-freedom system, solution by hand calculation is a challenge. A numerical method of computation would often be a suitable approach.

0 10 20 30 40 50 60 70 80 90 10010

-3

10-2

10-1

100

101

102

103

104

Frekvens [Hz]

medutan FF ˆˆ

N/m 101 62 ⋅=κ

N/m 105 62 ⋅=κ

N/m 1010 62 ⋅=κ

Område där vibrationsisoleringenökar kraften mot underlaget

Area where the vibration isolation increases the forceon the foundation Formatted: Font


130

4.3.3 System with an arbitrary number of degrees-of-freedom

The results from the two degree-of-freedom system can be generalized to a system with an arbitrary number of masses cascaded, i.e., coupled in series, as in Figure 4-6.

m1

x1(t)

m2

x2(t)

F1(t) F2(t) κ1

dv1

κ2

dv2

κn+1

dvn+1

mn

xn(t)

• • •

Fn(t)

Figure 4-6 System with n cascaded masses. The equations of motion become

+⎟⎟⎠

⎞⎜⎜⎝

⎛−++

dttdx

dttdx

ddt

tdxd

dt

txdm

)()()()( 212

112

12

1 νν

( ) )()()()( 121211 tFtxtxtx =−++ κκ , (4-43)

−⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

dttdx

dttdx

ddt

tdxdt

tdxd

dt

txdm

)()()()()( 323

2122

22

2 νν

,)())()(())()(( 2323212 tFtxtxtxtx =−+−− κκ (4-44)

−⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−− −−−

−−

− dttdx

dttdx

ddt

tdxdt

tdxd

dt

txdm nn

nnn

nn

n)()()()()( 112

121

2

1 νν

,)())()(())()(( 11121 tFtxtxtxtx nnnnnnn −−−−− =−+−− κκ (4-45)

+⎟⎟⎠

⎞⎜⎜⎝

⎛−++ −

+ dttdx

dttdx

ddt

tdxd

dt

txdm nn

nn

nn

n)()()()( 1

12

2

νν

( ) .)()()()( 11 tFtxtxtx nnnnnn =−++ −+ κκ (4-46)


131

The mass matrix, damping matrix, and stiffness matrix, respectively, become

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

nm

mm

0000

0000

2

1

M , (4-47)

[ ]

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+−⋅⋅⋅−+−⋅⋅

⋅•••⋅⋅⋅⋅••−⋅⋅−+−⋅⋅⋅−+

=

+

−−

1

11

3

3322

221

00

00

0

nnn

nnnnddd

dddd

ddddd

ddd

ννν

νννν

ν

νννν

ννν

D , (4-48)

[ ]

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+−⋅⋅⋅−+−⋅⋅

⋅•••⋅⋅⋅⋅••−⋅⋅−+−⋅⋅⋅−+

=

+

−−

1

11

3

3322

221

00

00

0

nnn

nnnnκκκ

κκκκ

κκκκκ

κκκ

K , (4-49)

where non-zero elements not shown in the equations are marked with a •, and zero-valued elements are marked with a ⋅. One can even allow masses to be coupled in parallel, as in Figure 4-7.

m1

x1(t)

F1(t)κ1

dv1

m4

x4(t)

F4(t) κ6

dv6

κ2

dv2 κ3

dv3

κ4

dv4

κ5

dv5

m2

m3

F2(t)

F3(t)

x3(t)

x2(t)

Figure 4-7 System with parallel coupling.


132

The equations of motion become

+⎟⎟⎠

⎞⎜⎜⎝

⎛−++

dttdx

dttdx

ddt

tdxd

dt

txdm

)()()()( 212

112

12

1 νν

( )+−++⎟⎟⎠

⎞⎜⎜⎝

⎛−+ )()()(

)()(21211

313 txtxtx

dttdx

dttdx

d κκν

( ) ,)()()( 1313 tFtxtx =−+κ (4-50)

−⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

dttdx

dttdx

ddt

tdxdt

tdxd

dt

txdm

)()()()()( 424

2122

22

2 νν

,)())()(())()(( 2424212 tFtxtxtxtx =−+−− κκ (4-51)

−⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

dttdx

dttdx

ddt

tdxdt

tdxd

dt

txdm

)()()()()( 435

3132

32

3 νν

,)())()(())()(( 3435313 tFtxtxtxtx =−+−− κκ (4-52)

+⎟⎟⎠

⎞⎜⎜⎝

⎛−++

dttdx

dttdx

ddt

tdxd

dt

txdm

)()()()( 244

462

42

4 νν

( )+−++⎟⎟⎠

⎞⎜⎜⎝

⎛−+ )()()(

)()(24456

345 txtxtx

dttdx

dttdx

d κκν

( ) .)()()( 4345 tFtxtx =−+ κ (4-53)

The mass matrix, damping matrix and stiffness matrix, respectively, become

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

4

3

2

1

000000000000

mm

mm

M , (4-54)

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

++−−−+−−+−

−−++

=

65454

5533

4422

32321

00

00

ννννν

νννν

νννν

ννννν

ddddddddddddd

ddddd

D , (4-55)


133

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

++−−−+−−+−

−−++

=

65454

5533

4422

32321

00

00

κκκκκκκκκκκκκ

κκκκκ

K . (4-56)

The general principle for generating these matrices, for systems in which the directions of forces and velocities are defined as in figures 4-6 and 4-7, can be summarized in the following way:

(i) the mass matrix is diagonal.

(ii) a diagonal element in the stiffness or damping matrix is the sum of the spring rates or damping coefficients, respectively, of all springs / dampers connected to the mass indicated by the row number of the element.

(iii) an off-diagonal element at a specific row and column position in the stiffness or damping matrix has the opposite (negative) of the value of the spring rate or damping coefficient, respectively, for the connection between the mass indicated by the row number and that indicated by the column number.

4.3.4 Frequency response functions

A frequency response function is defined as the relation between an output signal Y(ω) from a linear system, expressed as a function of the circular frequency ω, and the corresponding input signal X(ω), )()()( ωωω XYH = . (4-57)

In other words, it can be interpreted as the proportionality constant in the linear relation between the complex input and output amplitudes. It is one of the most important quantities used in the analysis of sound and vibration problems. If the input signal is a force on a structure, knowledge of the frequency response function permits the computation of the resulting vibration at different points in the structure; if the input signal is a pressure at a point in a ventilation duct, it permits the calculation of the sound pressure at the outlet. Examples of some frequency response functions, commonly used in vibrations and acoustics, are given in table 4-1.


134

Table 4-1 Examples of frequency response functions commonly used in vibrations and acoustics.

Quantity Input quantities Relation

Dynamic flexibility or Receptance H(ω)

Displacement x(ω) Force F(ω)

)()()( ωωω FxH = (4-58)

Mobility or mechanical

admittance Y(ω)

Velocity v(ω) Force F( )ω )()()( ωωω FvY = (4-59)

Accelerance A(ω) Acceleration a(ω) Force F(ω)

)()()( ωωω FaA = (4-60)

Dynamic stiffness κ(ω)

Displacement x(ω) Force F(ω)

)()()( ωωω xF=κ (4-61)

Mechanical impedance Z(ω)

Velocity v(ω) Force F(ω)

)()()( ωωω vFZ = (4-62)

Acoustic impedance Z(ω)

Acoustic volume flow rate Q(ω)Acoustic pressure p(ω)

)()()( ωωω QpZ = (4-63)

Specific impedance Z(ω)

Acoustic particle velocity u(ω)Acoustic pressure p(ω)

)()()( ωωω upZ = (4-64)

In section 4.3.1, the motion of a single degree-of-freedom system was considered. The dynamic flexibility (4-58), i.e., the relation between the displacement x(ω) and the force F(ω), for the single degree-of-freedom system can, as in (4-16), be described by

)(2)(1

1

21

)()(

)(20

20

2 ωωδωω

κ

κδωωωω

ωimim +−

=++−

==Fx

H . (4-65)

The frequency response functions can be presented graphically in a number of different ways. One possibility is to divide them up into real and imaginary parts

))(Im())(Re()( ωωω HHH i+= . (4-66)

For the single degree-of-freedom system considered,

22

022

0

20

)2())(1(

))(1())(Re(

ωωδωω

κωωω

+−

−=H , (4-67)


135

22

022

0

20

)2())(1()(2

))(Im(ωωδωω

κωωδω

+−−=H . (4-68)

Figure 4-8 shows graphs of the real and imaginary parts of H(ω), normalized by 1/κ and with δ as a parameter. For δ = ω0, the system is critically damped, for δ < ω0, it is weakly, or subcritically, damped, and for δ > ω0 it is strongly, or supercritically, damped.

a)

0 1 2 3

- 4

- 3

- 2

- 1

0

1

2

3

4

b) 0 1 2 3

- 5

- 4

- 3

- 2

- 1

0

Figure 4-8 Dynamic flexibility of a single degree-of-freedom system divided into real and imaginary parts.

κRe(H(ω))

δ = 0,7ω0

δ = 0,3ω0

δ = ω0

δ = 0,1ω0

δ = 2ω0

δ = 0

ω /ω0

δ = 2ω0

κIm(H(ω))

δ = 0,7ω0

δ = 0,3ω0

δ = ω0

δ = 0,1ω0

ω /ω0


136

Another possible representation of the frequency response function is in terms of its amplitude and phase angle

22

022

0 )2())(1(

1)(

ωωδωω

κω

+−=H , (4-69)

1)(

2arctan)(

20

20

−=

ωω

ωωδωϕ . (4-70)

A figure in which the amplitude and phase curves are plotted is usually called a Bode diagram, see Figure 4-9.

0 1 2 30

1

2

3

4

5

0 1 2 3

0

− π

Figure 4-9 Bode diagram of the dynamic flexibility of a single degree-of-freedom system, with separate plots of the amplitude and the phase angle.

κ ωH( )

δ = 0

δ = 0,1ω0

δ = 0,3ω0

δ = 0,7ω0

δ = ω0δ = 2ω0

ω /ω0 ϕ [rad]

δ = 0δ = 0,1ω0

δ = 0,7ω0δ = ω0

δ = 2ω0

δ = 0,3ω0

ω /ω0


137

From Figure 4-9, it is apparent that the damping strongly influences the amplitude at a resonance, i.e., ω = ω0. The behavior well away from the resonance frequency is the same, regardless of whether or not there is damping in the system. By considering the magnitude of H(ω) in the vicinity of the resonance frequency, a measure of the losses can be inferred. A polar plot, in which the real part of H(ω) projects onto the x-axis and the imaginary part of H(ω) onto the y-axis, is called a Nyquist diagram, as in Figure 4-10.

-3 -2 -1 0 1 2 3

-5

-4

-3

-2

-1

0

δ = 2ω 0 δ = ω 0 δ = 0,7ω 0 δ = 0,3ω 0 δ = 0,1ω 0

ω

ω → ∞ ω = 0 κ Im[H(ω)]

κ Re[H(ω)]

4.3.5 Damping

For a single degree-of-freedom system, we have thus far only used the viscous damping coefficient dv or the damping constant δ to describe the losses. For structures, it is more common to use the so-called loss factor to describe the influence of different types of damping. In order to demonstrate how the loss factor is defined, it will be useful to first introduce energy quantities for a single degree-of-freedom system. Considering a sinusoidal displacement )sin(ˆ ϕω += txx , the kinetic and potential energies can be expressed, respectively, as

)(cosˆ22

2222

ϕωω +=⎟⎠⎞

⎜⎝⎛= txm

dtdxmEkin , (4-71)

)(sinˆ22

222 ϕωκκ+== txxE pot . (4-72)

Figure 4-10 Nyquist diagram of the dynamic flexibility of a single degree-of-freedom system. The damping can be found using the diameter of the circle. A large circle corresponds to small damping and vice versa.


138

The energy dissipated (i.e., spent or “lost”) in one period, can be expressed as

=+=⎟⎠⎞

⎜⎝⎛=== ∫∫∫∫

T

v

T

vvddis dttxddtdtdxddx

dtdxddxFE

0

222

0

2)(cosˆ ϕωω

,ˆ2ˆ 222 ωπω xdTxd vv == (4-73)

in which we have used the relation ωT = 2π. The lost factor is defined as the dissipated energy per radian, divided by the maximum potential energy,

20

2)max(

2

ω

ωδκ

ωπη === v

pot

dis dE

E . (4-74)1

For the case of the harmonic excitation of a single degree-of-freedom system,

tip et ωxx ˆ)( = , tiet ωFF ˆ)( = , (4-75)

substitution into (4-10) yields

Fx ˆˆ)( 2 =++− κωω vdim . (4-76) Then, from (4-74), Fx ˆˆ))1(( 2 =++− ηκω im . (4-77)

As such, losses can be incorporated by defining a complex spring constant

)1( ηκ i+=κ . (4-78)

4.3.6 Analogue mechanical - electrical circuits

When analyzing the differential equations (4-8), (4-9), and (4-10), which describe the single degree-of-freedom system, similarities to the equations used to describe electrical circuits can be observed. Those similarities are used to define so-called analogue mechanical-electrical circuits. It is even possible to define analogue acoustic-electrical circuits, but we limit the discussion to just mechanical systems at the moment. Assume that electric potential, or voltage, U and current I in an electrical circuit have a harmonic time-dependence.

tieUt ωˆ)( =U , ϕω += tieIt ˆ)(I . (4-79)

In order to describe active circuit elements, i.e., those that provide power to the circuit, two ideal source models are used: an ideal current source, and an ideal voltage source; see 1 For resonant systems with small loss factors, it can be shown that EEpot ≈)max( , where E is the total energy

of the system. For the dissipated energy, TWE disdis = applies. Putting these expressions into (4-74) gives the

following approximate expression for the dissipated power, EWdis ηω= .


139

Figure 4-11. An ideal voltage source delivers a constant voltage, regardless of the circuit to which it is connected, and an ideal current source delivers a constant current, regardless of the circuit to which it is connected.

I

UU I

I

U

The complex ratio of the voltage to the current is called the impedance; see Figure 4-12,

iXR +==IUZ , (4-80)

where R is the resistance and X is the reactance. The most common passive circuit elements are resistors, R, inductors, L, and capacitors, C, shown in Figure 4-13.

R

I

U

I

U C

I

UL

a) Resistance b) Capacitance c) Inductance

Figure 4-13 Three common passive circuit elements are the resistor, the capacitor and the inductor.

Characteristic for passive circuit elements is that they do not supply energy to the system.

Figure 4-11 An ideal voltage source and an ideal current source.

Figure 4-12 General passive circuit element impedance Z.

I

U Z

Impedance

Formatted: Engl


140

For the resistor, IU RtRItU =⇒= )()( . (4-81) For the capacitor,

IUCi

dttIC

tUω1)(1)( =⇒= ∫ . (4-82)

For the inductor,

IU Lidt

tdILtU ω=⇒=)()( . (4-83)

That approach to writing the relation between circuit elements, in complex form, gives simple computational formulas for the determination of the voltage and current in circuits. It is usually called the iω-method. In the complex plane, a representation as in Figure 4-14 is obtained.

Re

Im

90o

CiωIU =

IU R=

IU Liω=I

For the resistor, the voltage and current are in phase. For the capacitor, the voltage lags the current by 90°. For the inductor, the voltage is 90° ahead of the current. In the case of a single degree-of-freedom system with a harmonically-varying velocity

tiev ωˆ=v , (4-84)

the acceleration and displacement are also harmonic

vva ωω ω ievidtd ti === ˆ , (4-85)

vvxωω

ωi

evi

dt ti 1ˆ1=== ∫ . (4-86)

For a mass, Newton’s second law states

dt

tdvmtF )()( = , vF miω= . (4-87)

By comparison to (4-83), it is clear that if the force corresponds to voltage, and the velocity to current, then the mass corresponds to inductance (m ⇔ L). A spring obeys

Figure 4-14 Relation between current and voltage for a resistor, capacitor and inductor, presented in the complex plane.

Formatted: Engl


141

∫= dttvtF )()( κ , vFωκi

= . (4-88)

Comparing that to (4-82) shows that the spring rate is analogous to the reciprocal of an electrical capacitance (κ ⇔ 1 / C). A viscous damper, by definition, follows the law

)()( tvdtF v= , vF vd= . (4-89)

Comparison to (4-81) reveals that the viscous damping coefficient is analogous to an electrical resistance (dv ⇔ R). Table 4-2 Summary of mechanical-electrical circuit equivalents, for passive circuit elements.

Component Equation Equivalence Mass F v= i mω m L↔

Spring F v= κ ω/ i C1↔κ

Viscous damper F v= dv d Rv ↔ Example 4-8 To isolate a machine of mass m = 700 kg, it is mounted upon springs with a combined spring rate κ = 2⋅106 N/m. Due to an imbalance, the machine is subjected to a force F of 50 N, at a frequency of 15 Hz. We would like to calculate the force acting on the foundation, if it can be regarded as completely rigid, i.e., the point impedance ZU as defined in (4-62) is infinitely large.

The problem can be solved by setting up an equivalent mechanical-electrical circuit.

F

miω

∞=UZωκi

v Uv

F

miω

ωκi

v

UF

A rigid mass must have the same velocity at the input as at the output, while the forces at those two points differ. A mass is therefore coupled in-series to the circuit. An ideal mass-less spring, on the other hand, has the same force at its input and output, while the velocities differ. Thus, it is coupled in-parallel to the circuit, i.e., between the relevant voltage (force) and ground. Because the spring is coupled to a rigid foundation with zero-velocity (vU ), the branch that comes after the spring, in-parallel, must have an infinite

κ

m

x(t)F(t)

vu(t)


142

impedance, and can therefore be eliminated. The force Fu acting on the foundation is then obtained by voltage splitting, as

20

22 11

ωωωκ

κωκω

ωκ

−=

−=

+= FFFF

mimii

U ,

where mκω =20 . With the input data entered, one obtains

( )

tititiU eeeF 2.94

62152 7.23

1027001521150ˆ −−=

⋅⋅⋅−==

ππϕF N.


143


MECHANICAL POWER

The time average of the mechanical power

)Re(21)Re(

21 ** vFvF ==W . (4-7)

LINEAR SYSTEMS

Single degree-of-freedom system

Differential equation of motion

)()()(2)( 202

2tgtx

dttdx

dttxd

=++ ωδ (4-10)

where mκω =0

mdv2

=δ m

tFtg )()( = (4-11)

In the frequency plane δωωω 2)(

ˆˆ22

0 ig

p+−

=x (4-16)

where tiegt ωˆ)( =g (4-13)

tipp et ωxx ˆ)( = (4-14)

Two degree-of-freedom system

[ ] [ ] 0)det( 2 =+− KMω (4-39) Frequency response function

Dynamic flexibility

( ) 2

02

02 21

12

1)()()(

ωωδωω

κ

κδωωωωω

imim +−=

++−==

FxH (4-65)


144

Damping

Loss Factor 2

02

)max(2

ω

δωκ

ωπη === v

pot

dis dE

E (4-74)

Complex spring constant κ = κ (1 + iη) . (4-78)

Analogue mechanical-electrical circuits

Mechanical force and velocity correspond to electrical voltage and current, respectively

For a mass, vF miω= (4-87)

so that the mass corresponds to an inductance ( )Lm ↔ .

For a spring, ωκ ivF = (4-88)

so that the spring constant corresponds to the inverse of a capacitance ( )C1↔κ

For a viscous damper vF vd= (4-89)

so that the viscous damping coefficient corresponds to a resistance ( )Rdv ↔ .

Chapter 5: Fourier Methods and Measurement Techniques

145

CHAPTER FIVE

FOURIER METHODS AND MEASUREMENT TECHNIQUES This chapter introduces Fourier methods of analysis permitting the analysis of vibroacoustic problems in the frequency domain. The chapter is concluded with the important subject of measurement, which includes not only transducers, such as microphones and accelerometers, but also a brief description of digital measurement systems. 5.1 FOURIER METHODS IN VIBROACOUSTICS

5.1.1 Fourier series

Most machines emit periodic disturbances to the surroundings, either in the form of fluctuating forces, acting via the machine mounts, or in the form of sound. The reasons for these periodic disturbances can be, to name a few examples, the meshing of gear teeth, imbalances in rotating shafts, or periodic pressure fluctuations that arise in the cylinders of internal combustion engines due to the intake-exhaust cycles; see Figure 5-1a. The disturbances that arise in practice are, as mentioned, periodic, but not usually purely harmonic. They consist of a combination of different frequency components, as exemplified in Figure 5-1b. To analyze the problem in the frequency domain, a method is needed to divide up a measured signal into its harmonic components, so that they can be individually analyzed. For periodic signals, it is possible to use a Fourier series expansion.


146

5.1.1.1 Approximation of signals

As a first step in deriving a method to decompose a periodic signal into its harmonic components, we will study how to best approximate a signal a(t) with a signal b(t). We assume that the fitting of the two signals is to take place during the time interval 0 < t < T.

To carry out the approximation in the simplest possible way, we multiply b(t) by a constant β that is varied to adjust the approximation as well as possible,

)()( tbta β≈ . (5-1) The error in the our approximation then becomes

)()()( tbtate β−= . (5-2)

Next, the averaged squared error is computed over the entire time interval 0 < t < T,

( ) dttbtaT

T

∫ −=0

2)()(1 βε . (5-3)

We minimize ε with respect to β by differentiating, and setting the resulting derivative equal to zero,

∫ =−=T

dttbtatbTd

d

0

2 0))()()((21 ββε . (5-4)

from which

∫∫=TT

dttbdttbta0

2

0

)()()(β . (5-5)

We now introduce the useful concept of orthogonality. If the signals a(t) and b(t) are orthogonal, there is no connection between the two signals, and β equals zero. The orthogonality condition for the two signals a(t) and b(t) on the time interval 0 < t < T is therefore

∫ =T

dttbta0

0)()( . (5-6)

The energy of a signal is proportional to the time integral of the signal squared; compare to formula (2-3). Thus,

∫=T

a dttaE0

2 )(α , (5-7)

in which α is a proportionality constant. Given a signal composed of two parts, v(t)=a(t)+b(t), its energy becomes

( ) ∫∫∫∫ ++=+=TTTT

v dttbtadttbdttadttbtaE00

2

0

2

0

2 )()(2)()()()( αααα . (5-8)

If the signals are orthogonal, we get


147

ba

TT

v EEdttbdttaE +=+= ∫∫0

2

0

2 )()( αα . (5-9)1

Thus, the energy of the combined signal is the sum of the individual energies. If the signals are not orthogonal, equation (5-8) must be used. Possibly, the approximation of the signal a(t) given by (5-1) in combination with (5-5) gives an inadequate fit. To further reduce the error, we incorporate another signal c(t) with a proportionality constant γ , and write the approximation as )()()( tctbta γβ +≈ . (5-10)

The error in that case becomes

)()()()( tctbtate γβ −−= , (5-11)

and the mean squared error is

( ) dttctbtaT

T

∫ −−=0

2)()()(1 γβε . (5-12)

We first minimize ε with respect to β

∫ =−+=∂∂ T

dttbtatctbtbT

0

2 0))()()()()((21 γββε . (5-13)

If we can now choose b(t) and c(t) to be mutually orthogonal, the second term in (5-13) is eliminated, and we obtain

∫∫=TT

dttbdttbta0

2

0

)()()(β , (5-14)

i.e., the same expression as in equation (5-5). If we minimize the error with respect to γ we obtain

∫ =−+=∂∂ T

dttctatctbtcT

0

2 0))()()()()((21 βγγε . (5-15)

If b(t) and c(t) are orthogonal, we can solve for γ in the same way as before,

∫∫=TT

dttcdttcta0

2

0

)()()(γ . (5-16)

By comparison to (5-5), the result obtained is the same as what would have followed from the approximation a(t) = γ c(t). To improve upon that, we therefore only need to add

1 Compare to section 2.1.


148

another orthogonal signal and minimize the mean squared error versus a(t), independently of the other signals included in the approximation. Examples of functions that are orthogonal are sin(nω0t) and cos(nω0t). They are orthogonal for all integer values of n over the time interval T = 2π / ω0. In the next section, we will take advantage of that orthogonality property to decompose a periodic signal into sine and cosine components. That gives the desired decomposition of the signal into its frequency components, given by fn = n / T.

5.1.1.2 Fourier series decomposition

Assume that we have a signal a(t) that is periodic, with period T, and which we wish to approximate with the help of sine and cosine functions,

∑∑∞

=

∞

=++=

110 )sin()cos()(

non

non tntnta ωγωββ . (5-17)

That is called a Fourier series decomposition of the signal a(t). The coefficients βn and γn can be calculated separately, and given by (5-5)

∫∫∫−−−

==2

2

2

2

22

20 )(111)(

T

T

T

T

T

T

dttaT

dtdttaβ , (5-18)

∫∫∫−−−

===2

20

2

20

22

20 ,3,2,1 ,)cos()(2)(cos)cos()(

T

T

T

T

T

Tn ndttnta

Tdttndttnta …ωωωβ ,

(5-19)

…,3,2,1 ,)sin()(2)(sin)sin()(2

20

2

20

22

20 === ∫∫∫

−−−

ndttntaT

dttndttntaT

T

T

T

T

Tn ωωωγ .

(5-20)

The interval of integration in (5-18), (5-19) and (5-20) is -T/2 to T/2, but could just as well have been 0 to T. The coefficient β0 represents the signal’s time average. It can also be shown that corresponding sine and cosine terms can be combined into a single cosine term with a phase angle ϕn.

∑∞

=−+=

100 )cos()(

nnn tnta ϕωδβ , (5-21)

where 22nnn γβδ +=

πβγϕ mnnn += )arctan( , m = 0, 1, 2, …,

where δ1 gives the signal’s amplitude for the first tone, or fundamental tone, δ2 gives the signal’s amplitude for the second tone, or the first overtone, δ3 gives the signal’s amplitude for the third tone, or second overtone, i.e.,


149

a)

0

2

4

6

p [bar]

0 200 Crankshaft angle φ [Degree]

Ignition

Combustion

Exhust

400 600 800 1000 1200 1400

b)

120

140

160

180

200 Lp [dB]

0 200 400 600 800 1000 Frequency [Hz]

Figure 5-1 Cylinder pressure in a single-cylinder, two-cycle engine running at 5500 rpm.

a) The pressure as a function of the crankshaft angle φ. The pressure variation repeats itself periodically after 360° , i.e., after a complete cycle. The different phases are compression, ignition, combustion, exhaust, and intake. For a two-cycle engine, exhaust and intake occur simultaneously, by virtue of the inflowing gases pushing out the exhaust gases. b) The sound pressure level as a function of frequency. The periodic pressure variation, after Fourier series decomposition, results in a frequency spectrum with discrete frequency components For a running speed of 5500 rpm, the fundamental frequency is f0 = 5500/60 = 91.7 Hz.

In Figure 5-1, the Fourier series method is illustrated for the case of pressure in a two-cycle engine. In Figure 5-1a, the pressure is given as a function of the crankshaft angle φ instead of time t. That can be done fairly simply, since there is a relation φ = ω t between these. In Figure 5-1b, the corresponding sound pressure level, Lp, is given as a function of the frequency.

In order to be able to denote each frequency component as a complex, rotating vector, which, as demonstrated in section 4.1, results in simpler computations and a more compact symbolic expression, a complex Fourier series can be defined,

∑∞

−∞==

n

tinneta 0)( ωδ , (5-22)


150

where the complex coefficient δn can be determined from

∫∫∫−

−

−−

− ==2

2

2

2

22

2

00 )(11)(T

T

tinT

T

T

T

tinn dteta

Tdtdteta ωωδ . (5-23)

A table summarizing certain useful signals’ complex Fourier coefficients is given in Appendix D.

In (5-22), there are components with “negative frequencies”. That is because, as shown in section 4.1, a real quantity can be described with the aid of two oppositely rotating complex vectors, each of which is the complex conjugate of the other; see Figure 5-2.

β n

Real

Imag+f

-f

δ n = β n + iγ n

γ n

γ n

β n

δ n* = β n - iγ n

Figure 5-2 Three-dimensional description of the frequency spectrum of a periodic signal. Complex Fourier series

contain components with both negative and positive frequencies. (Source: Brüel &Kjær, Frequency Analysis.)


151

Example 5-1 Consider a rectangular wave, as in the figure below.

From (5-18)

[ ] [ ] 022

11 20

02

2

0

0

20 =+−=+−=+−= −

−∫∫

AAtTAt

TAAdt

TAdt

TT

T

T

T

β

i.e., the time-averaged value is, as expected, equal to 0. Equation (5-19) yields

,0)sin()sin()sin(2)sin(2

)cos(2)cos(2

2

00

00

20

0

2

00

0

20

=+−=⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡−=

=+−=

−

−∫∫

ππ

ππω

ωωω

ωωβ

nnAn

nA

ntn

TA

ntn

TA

dttnAT

dttnAT

T

T

T

Tn

and (5-20) gives

⎪⎩

⎪⎨⎧

=

==−=⎥⎦

⎤⎢⎣

⎡−+⎥

⎦

⎤⎢⎣

⎡=

=+−=

−

−∫∫

,...4,2,0 0

,...5,3,14))cos(1(2)cos(2)cos(2

)sin(2)sin(2

2

00

00

20

0

2

00

0

20

n

nn

An

nA

ntn

TA

ntn

TA

dttnAT

dttnAT

T

T

T

Tn

πππω

ωω

ω

ωωγ

Thus, the Fourier series can be expressed as

⎥⎦⎤

⎢⎣⎡ +++= ...)5sin(

51)3sin(

31)sin(4)( 000 tttAta ωωω

π .

It only consists of the odd sine components. That only sine components appear in the Fourier series is because the rectangular wave is an odd function; odd functions can be decomposed into sine components, since the sine function itself is odd. By definition, a function is odd if it has the property that a(-x) = -a(x), and even if it has the property that a(-x)=a(x). If the rectangular wave is shifted T/4 to the left, making it symmetric about t=0, it becomes an even function instead, and be built up exclusively of cosine functions. How good an approximation of the original rectangular waveform that one obtains, in example 5-1, depends, naturally, on the number of terms included in the Fourier series

-T / 4 T / 4-3T / 4 3T / 4

A

-A

a(t)

t


152

decomposition. In Figure 5-3, the gradual improvement of the curve fit is evident as more and more terms are included in the series.

0 t/ T

1 Componen t 2 Componen ts 8 Componen ts

0.5 1.0 1.5 2.0

0

-1

1

a(t)

0 t/ T

32 Componen ts

0.5 1.0 1.5 2.0

0

-1

1

a(t)

Figure 5-3 Synthesis of a rectangular waveform with different numbers of terms in the Fourier series

decomposition. The graph shows the gradual improvement of the Fourier series approximation when more Fourier components are included in the summation. In the intervals in which the original time function has continuous derivatives, the Fourier series approximation can be arbitrarily close to the original function with enough terms. For discontinuous points, such as those at t/T = 0, 0.5, 1.0, 1.5, etc., there is always a “ripple”, no matter how many terms are included in the approximation. That is called the Gibb’s phenomenon in mathematics. It can be shown that the size of the ripple is, at a maximum, about 18% of the size of the discontinuity.


153

Example 5-2 The nature of the periodic force applications that bring about sound and vibration determines how great are the problems that arise. The figure below shows Fourier series decompositions of a rectangular wave, a triangular wave, and a sine wave.

Square Wave

t

-40

-20

0

LF [dB]

1 5 3 7 Frequency Component, n

-50

-30

-10

9 11 13 15 17 19

Triangular Wave

t

-40

-20

0

LF [dB]


-50

-30

-10

9 11 13 15 17 19

The amplitude of the overtones decays more slowly for the rectangular wave (as 1/n, where n refers to the n-th frequency component) than for the triangle wave (decays as 1/n2), whereas a sine wave only has a single frequency component. Because the overtones often fall in the more disturbing frequency bands, it is a good design principle to always make force applications as soft and “sinusoidal” as possible.

Another phenomenon that occurs in the case of periodic forcing is that the distance Δ f between the frequency components becomes larger, the shorter the period T ; i.e., Δ f = 1 / T. That is illustrated in the figure below, for a periodically repeated rectangular pulse.

-10

-20

-30 0 100 200 300 400 500 600

Frequency [Hz]

0 LF [dB]

T = 0.05 s

TP = 0.8 ms

-10

-20

-30 0 100 200 300 400 500 600

Frequency [Hz]

0 LF [dB]

TP = 0.8 ms

T/5 = 0.01 s

That fact can be used to minimize the number of frequency components excited in sensitive frequency bands, if it is possible to change the period.

Sine

t

-40

-20

0

LF [dB]


-50

-30

-10

9 11 13 15 17 19


154

5.1.2 Fourier transform

Many machines or processes give rise to sound or vibrations which are not periodic, but rather random (stochastic) or transient; see section 2.8. Roughness of the contacting surfaces between a wheel and its path, for instance, or between meshing gear teeth, bring about randomly varying vibrations. Turbulence in flowing media gives rise to randomly varying sound. For non-periodic disturbances, a Fourier series decomposition cannot be made; instead, one must use a so-called Fourier transform. The Fourier transform can be developed from the complex Fourier series (5-22,23) of a periodic train of pulses, as shown in Figure 5-4.

αT

−T T0

F(t)

t

1

Equation (5-23) provides the coefficients of the complex Fourier series,

( )Tin

TniTin

eedteT

TinTinT

T

tinn

0

0

0

2/2/2

2

)2/sin(211 00

0

ωωα

ωδ

αωαωα

α

ω =−

=⋅=−

−

−∫ , (5-24)

but T = 2π /ω0 , which implies that

( )πα

πααδn

nn

sin= . (5-25)

The Fourier series of the pulse train becomes

∑∞

−∞==

n

tinen

nt 0)sin()( ωπα

πααF . (5-26)

The Fourier series coefficients for the cases α = 21 , 41 and 81 are shown in Figure 5-5. From that figure, it is clear that if the pulses are permitted to glide farther and farther apart in the time domain, T → ∞ and the pulse width α T is held constant, from which it follows that α → 0, then the spectral lines approach each other in the frequency domain, i.e., become infinitely dense.

Figure 5-4 Periodic train of pulses with amplitude 1, perod T and pulse width αT.


155

a)

0,2

0

0,4

0,6 0,8

1,0

αδn

8 4-8 -4 0Frequency Component , n

2/1=α

b)

0,2

0

0,4

0,6 0,8

1,0

αδn

16 8-16 -8 0

4/1=α

Frequency Component , n

c)

0,2

0

0,4

0,6 0,8

1,0

αδn

32 16-32 -16 0

α =18

Frequency Component , n

Figure 5-5 Fourier series decomposition of a periodic pulse train with constant pulse width αT. For smaller α

and increasing period T, the frequency components become all the more densely packed a) α = 1 / 2, b) α = 1 / 4 and c) α = 1 / 8.

To derive a relation for a non-periodic event, we therefore consider the limiting case of the period T becoming infinite. If we substitute in the expression for the Fourier coefficients (5-23) into the Fourier series (5-22), we then obtain

∑ ∫∞

−∞= −

−=n

T

T

tintin dtetFeT

tF2

2

00 )(1)( ωω . (5-27)


156

To adapt that to the limiting case when the period T goes to infinity, the interchange ω0 → dω is made because ω0 = 2π / T, and nω0 transforms into a continuous variable ω, i.e., nω0 → ω . The step size in the summation becomes infinitesimally small, and the summation in (5-27) transforms to an integral

ωπ

ωω ddtetFetF tintin∫ ∫∞

∞−

∞

∞−

−= ))((21)( 00 . (5-28)

The expression inside the parentheses is identified as the Fourier transform of the signal,

∫∞

∞−

−= dtetF tiωω )()(F , (5-29)

and the inverse Fourier transform is given by

∫∞

∞−

= ωωπ

ω detF ti)(21)( F . (5-30)

Appendix C contains a table of the Fourier transforms of some useful signals. The Fourier transform is a complex quantity, which, in the case of F(t) representing a force, has the units N/Hz. In order for F(t) to be real, F(-ω ) = F*(ω ) must hold; compare example 4-5. Example 5-3 Calculate the Fourier transform of a single force pulse, with pulse width Tp, as illustrated in the adjacent figure. Applying (5-29) yields

== ∫−

−2

2

ˆ)(p

p

T

T

ti dteF ωωF

=⎟⎠⎞⎜

⎝⎛ −= − 22ˆ

pp TiTi eeiF ωω

ω

( )2

2sinˆp

pp T

TTF

ω

ω= .

The amplitude spectrum becomes

2/

)2/sin(ˆ)(

p

pp T

TTF

ω

ωω =F .

The Fourier transform is real, which implies that the phase spectrum is determined by the sign of sinω Tp. The Fourier transform’s amplitude spectrum and phase spectrum are shown in the figure below, in which a dimensionless frequency ωTp has been incorporated. Note that the transform of the rectangular pulse corresponds to the case in which α → 0 in section 4.4.2. The amplitude spectrum in the figure below is therefore the result obtained in Figure 5-5, in the limit as α → 0. The discrete spectrum for the pulse train has, in the case of a single pulse, transformed into a continuous spectrum.

t

F(t)

-Tp/2

F

Tp/2


157

00,2 0,4

0,6 0,8

1,01,2

1,4

( ) pTFωF

ωΤp −4π −2π 0 2π 4π−6π 6π−8π 8π

0

π/2

π

Fas ϕ [rad/s]

ωΤp −4π −2π 0 2π 4π −6π 6π −8π 8π

Example 5-4 By analysis of the transient forces in the time and frequency domains, respectively, a number of general conclusions, useful in machine and equipment design, can be drawn. The smaller the impulse ( ∫= dttFI )( ), the lower the amplitude in the frequency

domain. The figure below illustrates the effects of two different modifications to the impulse. Note that a dimensionless frequency f Tp is used.

0 1 2

0

-10

10

Tp

I=Ip

f Tp

LF [dB]

0 1 2

-10

0

10

Tp/2

I=Ip/2

f Tp

LF [dB]

0 1 2

0

-10

10

Tp

I=Ip/2

f Tp

LF [dB]

Increased duration or pulse width Tp in the time domain translates into a lowering of the cutoff frequency (the frequency at which the level has fallen 3 dB with respect to the maximum amplitude). By making the pulse longer (increasing duration), a lower frequency


158

excitation is thereby obtained. That can exploited to shift the excitation into a frequency band which is less disturbing or in which the structure is not as effectively excited. The figure below illustrates that effect.

0

0

-10

-20

1 2

t

T 0

f Tp

LF [dB]

0 1 2

0

-10

-20

t

T=Tp

f Tp

LF [dB]

0 1 2

0

-10

-20

t

T = 4 Tp

f Tp

LF [dB]

0 0,5 1 1,5

0

-2

-4

-6

-8

-10

Tp

Tp

f Tp

LF [dB]

If the rise or fall time of the pulse is lengthened, the amplitude decays more rapidly with frequency above the cutoff frequency; see the adjacent figure. That can be exploited to reduce the high frequency content in the excitation. The same also applies to higher time derivatives. The more rounded and “soft” the excitation is in the time domain, the more rapidly the high frequency content decays.


159

5.1.3 Parseval’s relations

Let F1(ω ) and F2(ω ) be Fourier transforms of the time functions F1(t ) and F2(t ). Then, it can be shown that

πωωω

2)()()()( *

21ddttFtF ∫∫

∞

∞−

∞

∞−

= 21 FF , (5-31)

which is called Parseval’s relation. To prove (5-31), put the inverse Fourier transform of F1(t ) into the left hand side of (5-31)

== ∫ ∫∫∞

∞−

∞

∞−

∞

∞−

dtdetFdttFtF tiπωω ω

2)()()()( 1221 F

=⎥⎥⎦

⎤

⎢⎢⎣

⎡= ∫∫

∞

∞−

∞

∞−πωω ω

2)()( 2

ddtetF ti1F

.2

)()( *πωωω d

21 FF∫∞

∞−

=

Parseval’s relation can be used to calculate the mean square value of a quantity from a measured frequency spectrum. The mean square value is given by (2-3), in which we let T → ∞ because we do not have a periodic function,

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= ∫

−∞→

2

2

22 )(1lim

~T

TTdttF

TF . (5-32)

Parseval’s relation yields

πωω

πωωω

2)(

2)()()( 2*2

1dddttF ∫∫∫

∞

∞−

∞

∞−

∞

∞−

== 111 FFF . (5-33)

Putting (5-33) into (5-32), and using the relation F(ω ) = F*(-ω ), yields

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

Ω=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

Ω=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= ∫∫∫

Ω

∞→Ω

Ω

Ω−∞→Ω−∞→

2

0

22

2

22

2

222

)(2lim2

)(1lim)(1

lim~

πωω

πωω dddtt

TF

T

TTFFF .

(5-34)


160

Parseval’s relation for periodic signals can also be derived, as

∑∫∞

−∞=−

=n

nn

T

T

TdttFtF *21

2

221 )()( dd (5-35)

in which d1n and d2n are the Fourier series coefficients of F1(t ) and F2(t ), respectively. Using (5-22),

=⎟⎟⎠

⎞⎜⎜⎝

⎛= ∫ ∑∑∫

−

∞

−∞=

−∞

−∞=−

dteedttFtFT

T m

timm

n

tinn

T

T

2

2

*21

2

221 )()( ωω dd

,2

2

)(*21 dte

T

T

tmni

mm

nn ∫∑∑

−

−∞

−∞=

∞

−∞== ωdd

yet because ⎩⎨⎧

≠=

=∫−

−mnmnT

dteT

T

tmniför 0för 2

2

)( ω ,

then it follows that ∑∫∞

−∞=−

=n

nn

T

T

TdttFtF *21

2

221 )()( dd .

(5-35) can be used to calculate the mean squared value of a periodic signal from its Fourier components, as indicated by formula (2-35)

∑∫∞

−∞=−

==n

n

T

T

dttFT

F 22

2

22 )(1~ d . (5-36)

If the summation is only carried out over the “positive” frequencies, n = 0, 1, 2, … , then 22 2~

nnF d= must be used as the mean square value of the n-th frequency component.

That follows from *nn dd =− , from which nn dd =− . That also provides the basis for

computing the third octave band spectrum, for example, from a narrow band spectrum, or the octave band spectrum from a third-octave band spectrum.


161

Example 5-5 Measurement of the sound pressure level has been carried out in the third octave bands with center frequencies 800 Hz, 1000 Hz and 1250 Hz, from which the results given in the table below were obtained.

f [Hz] 800 1000 1250 Lp [dB] 73.4 69.8 72.1

We now wish to calculate the sound pressure level for the octave band with the center

frequency 1000 Hz.

Solution Calculate, firstly, the mean squared value of the sound pressure in the third octave bands, using formula (2-29). Then, sum up the mean squared values in accordance with Parseval’s relation,

23

22

21

2 ~~~~ ppppoct ++= .

f [Hz] 800 1000 1250 ~p 2 [Pa2] 8.75⋅10-3 3.82⋅10-3 6.48⋅10-3

Calculate the sound pressure level as )~(log10 22refoctp ppL ⋅= .

233323

22

21

2 1091.11048.61082.31075.8~~~~ −−−− ⋅=⋅+⋅+⋅=++= ppppoct ,

8.76)1041091.1(log10)~(log10 10222 =⋅⋅⋅== −−refoctp ppL dB.

5.2 MEASUREMENT SYSTEMS FOR SOUND AND VIBRATIONS

The nature of sound and vibrations to be measured can vary widely. Sound can be “noisy” (roar or hiss-like), like that from a heavily trafficked highway, while vibrations of a machine are often dominated by the rotational frequency and its multiples. A machine under constant loading gives off a stationary noise, while the noise at an airport tends to be intermittent. Moreover, the purpose of measurements varies. If it is merely a question of a noise disturbance survey in an industrial facility, then relatively simple, single-channel instruments are used; see figure 5-6. If the purpose, on the other hand, is to determine the mode shapes of a large structure, such as an airplane fuselage, then a larger measurement system with two or more channels is required.


162

Figure 5-6 The most common portable instrument for measurement and analysis of sound is a sound level meter, which can be found in various makes and models. It measures the total sound pressure level throughout the audible band, but is often also equipped with octave band and third-octave band filters for frequency band analysis. To imitate the sensitivity of human hearing to sound with different frequency contents, so called A, B, and C weighting filters can be used; the measurement quantities obtained are called Sound Levels in dB(A), dB(B) and dB(C), and are discussed in chapter 2. Power (rms) amplitudes, as from (1-3), can normally be displayed for a selected integration time T. Standardized settings are slow, fast, impulse, and peak, with decreasing integration time in the order given, and faster updating of the measurement quantities in that order. Many modern digital sound level meters are of the integrating type, meaning that they can calculate time-averaged values over extended periods, such as hours or days. (Photo: Brüel & Kjær.)

En viktig principPåbörja aldrig en mätning innan syfte och mätsituation är kartlagda - dåligt planlagdamätningar kan sällan eller aldrig utvärderas i efterhand.

Figure 5-7 In vibroacoustics, many quantities are defined in Swedish and international standards and measurement procedures. The intent of standardization is to, among other things, facilitate the comparison of measurement results from different sources and product characteristics from different manufacturers.

(Sketch: Brüel & Kjær, informationsmaterial.)

An important principle: Never begin measurements without first clearly defining the purpose and documenting the circumstances – poorly planned measurements can seldom or never be properly interpreted later.


163

5.2.1 The measurement chain

The measurement systems that are marketed today are primarily digital, i.e., sound pressure and vibrations are converted into digital values for later treatment in more or less advanced signal processors. Ever since the beginning of the 1920’s, when analog measurement began to be practical, a number of methods and measurement quantities have been developed. These are, today, well established and difficult to dispense with. Thus, while digital technology offers ever more sophisticated possibilities, measurement systems are nevertheless often adapted to be able to compare measurement results with those obtained in the past using analog technology. Digital measurement systems have a more complicated structure than analog ones. A flow chart for a digital, single channel system is shown in figure 5-8.

Sound or v ibration signal T ransducer E lectrical signal

Im pedance m atching

A m plification

Signal condition ing

M icrophone

A ccelerom eter

A m plification

F requency

Tim e

S ignal

T im e

D igital

filtering

Fast Fourier

T ransform

A nalog filter A nalog-d igita l conversion Frequency analysis

A m plitude

Frequency

M easurem ent data processing

G raph

Low pass filtering

Exam ple:

A veraging

R M S am plitude determ ination

A -, B -, C -w eighting

Signal

1

Figure 5-8 Flow chart of a digital, single-channel measurement system. For multi-channel systems, the analog

and digital conversions are synchronously controlled for simultaneous sampling of the measurement signals. That improves the precision in computing quantities that depend on several channels, such as frequency response functions; see chapter 3.

5.2.2 Transducers

The types of transducers that are most commonly used in vibroacoustics are microphones to measure sound pressure, accelerometers to measure accelerations of solid structures, and force transducers to measure forces on solid structures. The principles behind force transducers are not described here, but are very similar to those for accelerometers.


164

Transducers convert measured quantities, such as sound pressure, into equivalent electrical signals. A harmonically varying sound pressure induces a harmonic electrical signal at the same frequency. For a certain microphone, the output voltage U(t) [V] due to a certain sound pressure p(t) is

)()( tpCtU = , (5-37)

where the proportionality constant C should be independent of amplitude and frequency, to the extent possible. A number of characteristics are common to all types of transducers:

Sensitivity: Indicates the ratio of electrical output to mechanical input. Example: A microphone’s sensitivity is given in mV/Pa.

Frequency band: Indicates the upper and lower frequency limits, between which the transducer sensitivity varies within a given (small) tolerance range.

Dynamic range: Indicates the upper and lower amplitude limits between which the transducer sensitivity varies within a given (small) tolerance range. The dynamic range is commonly given in dB with respect to a reference value. The lower dynamic boundary is often determined by the transducer’s electrical noise, and the upper boundary by when the transducer is loaded beyond its mechanical linear region.

5.2.3 Microphones

For acoustic measurements with high demands on the precision, condenser microphones are used. Condenser microphones consist of a thin metal membrane called a diaphragm, separated from an opposing “backplate” electrode by an air gap (figure 5-38). The diaphragm and backplate constitute the electrodes of a condenser which is polarized by an electrical charge. When the diaphragm vibrates, due to the sound pressure, the capacitance of the condenser varies and an electrical output signal is generated. That electrical output signal is proportional to the sound pressure. As we established in earlier sections, a sound field is affected by reflection and diffraction when it is disturbed by the presence of objects. When the sound frequency is so high that the wavelength approaches the size of the microphone, then the microphone itself will influence the sound field. A small microphone has a higher upper frequency limit, but a lower sensitivity. Measurement microphones are therefore made in various sizes, and with built-in corrections for different types of sound fields. Microphone sizes are given in inches; typical sizes are 1", 21 ", 41 " and 81 ". A 21 "-microphone is standard in most measurement situations; see table 1-3 for details.

Hål för statisk tryckutjämning

Isolator

Membran

Motelektrod Hylsa

Backplate electrode

Diaphragm

Shield Cavity

Insulator Output terminal

Insulator

Pressure relief vent

Diaphragm

Backplate electrode Casing


165

Figure 5-38 The electrode charges can be brought about in two ways. For externally polarized microphones, an external voltage is applied across the diaphragm and backplate. Pre-polarized microphones are charged by means of a thin electrical material that is placed on the backplate. That solution is usually more expensive, but nevertheless preferred for portable instruments, since it avoids the complications inherent in requiring external electrical voltage. (Source: Brüel & Kjær, Measurement Microphones)

The different types of microphones are:

Free field: Free field microphones are intended for use in direct fields and should therefore be directed towards the sound source. They have built-in corrections that compensate the microphones influence on the sound field.

Pressure: Pressure microphones are mainly intended for calibrations in small cavities and for mounting positions flush with walls and the like. These do not compensate for their own affect on the sound field. They measure the actual sound pressure on the microphone diaphragm.

Diffuse field: Diffuse field microphones are used in diffuse sound fields, i.e., they should have a flat sensitivity curve as a function of frequency for sound that falls in from all directions.

The microphone capsule is directly connected to a pre-amplifier. Its main task is to convert the microphone’s high output impedance to a low one, permitting connection to long cables or to a measurement system with relatively low input impedance.

Table 5-1 Data for some types and sizes of measurement microphones.

Microphone B&K 4145 B&K 4165 B&K 4135 B&K 4138 Diameter 1" (25.4 mm) ½" (12.7 mm) ¼" (6.35 mm) 1/8" (3.2 mm)

Type of sound field Fee field Free field Free and diffuse field

Pressure and diffuse field

Frequency range (±2 dB) 2.6 Hz - 18 kHz 2.6 Hz - 20 kHz 4 Hz - 100 kHz 6.5 Hz - 140 kHz Sensitivity [mV/Pa] 50 50 4 1 Dynamic range [dB] (with recommended Pre-amp)

11 - 146 15 - 146 36 - 164 55 - 168


166

5.2.4 Accelerometers

Piezoelectric accelerometers are the most commonly used vibration transducers. In some measurement situationsError! Bookmark not defined., however, other types may be preferred, such as optical and inductive transducers or strain gauges. The construction of a piezoelectric accelerometer of the compression type can be seen in figure 5-39.

Förspänd fjäder

Rörlig massa

Piezoelektriska element

Bas/monteringsplatta

Vibrerande objektFästskruv

Elektriskutsignal

+_

When the accelerometer, which is firmly mounted to the measurement object, is subjected to vibrations along its axis of symmetry, the mass gives rise to a force that varies as the acceleration varies. That force deforms the piezoelectric discs, which then produce, because of their piezoelectric properties, a charge on the surfaces of the discs proportional to the force, and thereby to the acceleration as well. The piezoelectric effect can be obtained for either compression or shear of the piezoelectric material. Simplified models are shown in figure 5-40.

Figure 5-40 The piezoelectric materials usually are artificially polarized ceramics that give rise to a charge, q, when subjected to a) compression, or b) shear. The accelerometers that are based on the shear principle can be made less sensitive to other types of deformations, as for example, those caused by temperature variations. (Source: Brüel & Kjær, Piezoelectric Accelerometers and Vibration Preamplifiers)

In selecting suitable accelerometers, there are primarily three characteristics of interest: sensitivity, given in charge per unit acceleration [pC/ms-2]; internal resonance frequency; and the accelerometer’s total mass. The sensitivity and resonance frequency are strongly dependent on the mass; see figure 5-41. Accelerometers are therefore available with masses ranging from one gram, for very high vibration levels (shocks), up to 500 grams, intended for very low levels.

+-

F

F

q+-

+-+-+-

+-+-+-+-+-+-

q- +

FF

b)a)

Figure 5-39 Fundamental construction of a piezoelectric accelerometer of the compression variety. The active elements are the piezoelectric discs, on which a mass is resting. The mass is preloaded by a spring and the entire arrangement is enclosed in a metal capsule on a stable mounting plate.

Electrical Output signal

Pre-loaded spring

Seismic mass

Piezoelectric element

Baseplate / mounting plate

Vibrating object

Threaded stud


167

Frequency (log) f f

Useful frequency range

Accelerometer with alarge mass

with a small massAccelerometer

Sensitivity

r r

~f /3 r

Table 5-2 Data for some types and sizes of accelerometers.

Accelerometer B&K 8306 B&K 4370 B&K 4367 B&K 4344 Mass [g] 500 54 13 2 Type Compression Shear Shear Compression Internal resonance frequency [Hz] 4500 18000 32000 70000 Frequency range [Hz] 0.06 - 1250 0.2 - 6000 0.2 – 10600 1 - 21000 Sensitivity [pC/ms-2] 1000 10 2 0.25

Piezoelectric accelerometers are connected to preamplifiers, the primary purpose of which is to match the high output impedance of the accelerometer to the low input impedance of the measurement system. Typically, a charge amplifier is used; despite its name, it does not amplify charge, but rather gives an output voltage proportional to the accelerometer’s charge. The particular advantage of a charge amplifier is that the length of the cable between the accelerometer and the amplifier can be varied without altering the sensitivity. More advanced charge amplifiers even have other functions built in:

(i) Amplification of the signal to a level suited to the measurement system.

(ii) Integration of the acceleration signal to velocity and displacement. Usually, velocity signals are preferred , especially for acoustic measurements.

(iii) Low and high pass filters for damping of frequency components outside of the band to be analyzed, as, for example, around internal resonances and resonances due to the fastening to the measurement object.

It can be an advantage for some applications to use accelerometers with built-in pre-amplifiers. Such accelerometers are normally less sensitive to electrically induced noise in cables and thereby permit the use of long and inexpensive cables.

Figure 5-41 The sensitivity of an accelerometer depends primarily on the seismic mass and the properties of the piezoelectric material. A larger accelerometer normally has a greater sensitivity. The seismic mass and the piezoelectric element constitute a mass-spring system with a resonance frequency fr at which the sensitivity increases dramatically. As a rule of thumb, it can be said that the useful frequency range is below fr /3.


168

5.2.5 Mounting of accelerometers

There are three aspects, above all, that must be considered when mounting accelerometers to measurement objects:

(i) Measurement direction and placement; see figure 5-42. (ii) Mass loading of the measurement object; see figure 5-43. (iii) Attachment to the measurement object; see figure 5-44.

Figure 5-43 Mass loading of the measurement object. The

larger accelerometers have a higher sensitivity and give, therefore, stronger output signals. The accelerometer mass, however, also loads the measurement object itself, modifying its amplitude, especially at high frequencies. A coarse rule of thumb, for measurements on rigid bodies, is that the accelerometer mass should be less than a tenth of the mass of the measurement object. (Sketch: Brüel & Kjær, course material.)

m

< 110m

Figure 5-42 Measurement direction and placement. The accelerometer should be placed so that the meas-urement direction coincides with its axis of maximum sensitivity. Most often, the circumstances dictate what would be a useful placement of the accelerometer on the meas-urement object. If we want to check the condition of a roller bearing, a fundamental principle is to place the transducer as close to the bear-ing as possible. Position A is better than B for axial vibrations, and C is better than D for radial vibrations. (Sketch: Brüel & Kjær, course material)

Axis of principal sensitivity (100%)

Transverse sensitivity (<4%)

A

B

C

D


169

Stålskruv Isolerad skruv Tunnt lager vax

Lim Limbricka med skruvtapp

Magnet

Mätspetsar

1 2 3

4 5 6

Figure 5-44 The method of fixing an accelerometer to a measurement object is very significant. Ideally, it would be rigidly fixed. In practice, however, no fastening method is perfectly rigid. This implies that the accelerometer and the fastening arrangement constitute a spring-mass system with a resonance frequency, in addition to the accelerometer’s internal resonance frequency. That sets an upper limit to the useful frequency range.

(1) Mounting with a threaded steel stud to a smoothly ground surface gives a high resonance frequency. (2) To avoid grounding loops in the measurement system, electrically isolating studs and discs can be used. (3) Wax between the accelerometer and the object is a common method that gives a surprisingly high resonance frequency, but which limits the temperature during to measurements to a maximum of 40-50° C. (4) The accelerometer can be threaded onto special discs that are cemented to the measurement object. (5) Fastening magnets are practical on magnetic surfaces, but give a typical resonance frequency of about 7 kHz, which limits their useful range to about 2 kHz. (6) With the accelerometer placed on a hand-held measurement tip, a low resonance frequency is obtained, so that this method should not be used above 500-1000 Hz. (Sketch: Brüel & Kjær, Mechanical Vibration and Shock Measurements.)

5.2.6 Calibration of transducers and measurement systems

In order to maintain a high standard of measurement quality, transducers and the rest of the measurement system should be calibrated regularly. An important concept in that regard is traceabililty. That is effected by a network of national and international laboratories that guarantee the precision of their own respective instruments and ensure that that precision is traceable to the next level in the network. In practice, this means that Individual laboratories must regularly calibrate their equipment, and have their calibration equipment itself calibrated at the next level in the network. In order to obtain calibrated measurement values on a day-to-day basis, two methods are used:

(i) Calibration based on a transducer’s sensitivity, and knowledge of the amplification of the rest of the measurement system.

(ii) The entire measurement chain is calibrated simultaneously. The transducer is subjected to a known vibration or sound signal, and the system’s output display is adjusted to show the correct value. That is the most common and convenient method to calibrate instrumentation prior to a measurement.

Threaded stud (steel)

Magnet

Thin layer of wax

Cement Cemented disc with threaded stud

Threaded stud (isolating)

Measuring tips


170

Figure 5-45 Piston phones and microphone calibrators are battery driven portable units for the calibration of acoustic measurement systems. Microphones are put into one end of a small cavity. In the other end, there is a piston or a membrane that is excited by an electrical motor or a piezoelectric element. The excitation gives a tone with a known frequency and sound level. (Source: Brüel & Kjær.)

Figure 5-46 To calibrate a vibration measurement system, there are portable, battery-driven electrodynamic vibrators, that generate vibrations of a known frequency and amplitude. (Photo: Brüel & Kjær.)


171


FOURIER METHODS

Fourier series ∑∑∞

=

∞

=++=

110 )sin()cos()(

non

non tntnta ωγωββ (5-17)

where ∫∫

∫

−

−

− ==2

22

2

2

2

20 )(1

1

1)(T

TT

T

T

T dttaT

dt

dtta

β (5-18)

…,3,2,1 ,)cos()(2

)(cos

)cos()(2

202

20

2

2

20

=== ∫∫

∫

−

−

− ndttntaT

dttn

dttntaT

TT

T

T

Tn ω

ω

ω

β (5-19)

…,3,2,1 ,)sin()(2

)(sin

)sin()(2

202

20

2

2

20

=== ∫∫

∫

−

−

− ndttntaT

dttn

dttntaT

TT

T

T

Tn ω

ω

ω

γ (5-20)

Complex Fourier series ∑∞

−∞==

n

tinneta 0)( ωδ (5-22)

where ∫∫

∫

−

−

−

−

−

==2

22

2

2

2

2 0

0

)(1

1

)(T

T

tinT

T

T

T

tin

n dtetaT

dt

dtetaω

ω

δ (5-23)

Fourier transform ∫∞

∞−

−= dtetF tiωω )()(F (5-29)

Inverse Fourier transform


172

∫∞

∞−

= ωωπ

ω detF ti)(21)( F (5-30)

Parseval’s relations

πωωω

2)()()()( *

21ddttFtF ∫∫

∞

∞−

∞

∞−

= 21 FF (5-31)

Calculation of the mean squared value

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

Ω=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

Ω=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= ∫∫∫

Ω

∞→Ω

Ω

Ω−∞→Ω−∞→

2

0

22

2

22

2

222

)(2lim2

)(1lim)(1

lim~

πωω

πωω dddtt

TF

T

TTFFF .

(5-34)

Periodic signals ∑∫∞

−∞=−

=n

nn

T

T

TdttFtF *21

2

221 )()( dd (5-35)

Mean squared value of a periodic signal

∑∫∞

∞−−

== 22

2

22 )(1~n

T

T

dttFT

F d (5-36)

173

CHAPTER SIX

THE WAVE EQUATION AND ITS SOLUTIONS IN GASES

AND LIQUIDS This chapter takes up some of the fundamental elements of acoustics. It begins with a derivation of the homogenous linearized wave equation. Necessary conditions for this equation to be applicable are discussed, and the wave speed is determined for different gases and liquids. General and harmonic solutions are studied for plane, one-dimensional wave propagation and for spherical wave propagation. The important concept of impedance, a relation between sound pressure and particle velocity, is defined. Other important quantities derived include energy, energy density, sound power, and sound intensity. Bearing in mind the fundamental nature of this material, a relatively thorough treatment is called for.

6.1 THE WAVE EQUATION IN A SOURCE-FREE MEDIUM

Acoustic disturbances in fluids, i.e., gases and liquids, which cannot support shear stresses, propagate as longitudinal waves. This type of wave propagation is characterized by motions of fluid particles parallel to the direction in which the disturbance propagates. Alternating regions of compression and rarefaction (expansion) develop in the medium as discussed in section 2.4. Plane wave propagation means that the acoustic variables, such as sound pressure p, have a constant instantaneous magnitude throughout any given plane perpendicular to the direction of wave propagation; see figure 2-4.

Chapter 6: The wave equation and its solutions in gases and liquids

174

Rarefied Compressed

Figure 6-1 Symbolic depiction of plane longitudinal wave propagation for a sinusoidal disturbance. The

fluid particles are sketched in the compressed and in the rarefied regions. A number of simplifying assumptions are needed in the derivation of the source-free linearized wave equation. These assumptions are:

(i) The medium is homogenous and isotropic, i.e., it has the same properties at all points and in all directions.

(ii) The medium is linearly elastic, i.e., Hooke’s law applies.

(iii) Viscous losses are negligible.

(vi) Heat transfer in the medium can be ignored, i.e., changes of state can be assumed to be adiabatic.

(v) Gravitational effects can be ignored, i.e., pressure and density are assumed to be constant in the undisturbed medium.

(vi) The acoustic disturbances are small, which permits linearization of the relations used.

Because compressions and rarefactions of the medium are a consequence of the differences in motions of adjacent regions in the medium, a relation is needed between particle velocity and density. That relation is called the continuity equation. An element cut out of the medium has a mass and is therefore affected by forces due to pressures in the medium. Making use of Newton’s second law, amF = , a relation is obtained between pressure and motion. The thermodynamic equation of state provides a relation between pressure, density, and absolute temperature during a process. The equations will be set up for a volume element cut out of the medium, a so-called fluid particle. The medium can be either a gas or a liquid. More specifically, a fluid particle is a volume element large enough to contain millions of molecules, but small enough that the acoustic disturbances can be regarded as linearly varying within the

Pressure p / p0

Direction of propagation

Wavelength λ

p0

Direction of propagation


175

element. From that line of reasoning, the concept of particle velocity, i.e., the velocity of a considered fluid particle, can be defined. The following quantities are considered: Pressure: ),(),( 0 trpptrpt += , (6-1) where ),( trpt is the total pressure as a function of position ( r ) and time (t), p0 the pressure in the undisturbed medium, and ),( trp the pressure disturbance in the medium, the sound pressure. Particle- zzyyxx eueueutru ++=),( , (6-2)

Velocity: where ),( tru is the particle velocity vector, ux, uy, uz are the corresponding velocity components, zyx eee ,, is the unit vector.

Density: ),(),( 0 trtrt ρρρ += , (6-3) where ),( trtρ is the total density, ρ0 is the density in the undisturbed medium, ),( trρ is the density disturbance in the medium.

Absolute ),( trT is the temperature, in Kelvin. Temperature:

A number of assumptions have been made above, for acoustic wave propagation. The validity of these can only be established by experiment. They have, in fact, been established in that way as valid in normal circumstances. A fundamental assumption is that we consider small disturbances, i.e., small variations in pressure. As a rule of thumb, for air at normal temperature and pressure, the sound pressure level should not exceed 140 dB.

6.1.1 Equations of continuity

As indicated above, the equation of continuity gives a relation between density and particle velocity. The one-dimensional case is studied first, after which we generalize to the three-dimensional case. We consider in and outflow of mass in the x- direction at a given point in time, for a volume element ΔV =ΔxΔyΔz fixed in space, as shown in figure 6-2.

x

yz

+∂ x

ρt ux

)∂(ρt uxρt ux Δ x

Δ y

Δ z

Δ x Figure 6-2 Mass flow in the x-direction through a volume element fixed in space.


176

According to figure 6-2, the mass in the volume element is zyxt ΔΔΔρ , the mass flow into the element is xxt zyu )( ΔΔρ , and the mass flow out is xxxt zyu Δ+ΔΔ )(ρ . The net flow in the element is therefore xxt zyu )( ΔΔρ - (ρtuxΔyΔz)x + Δx , and must equal the mass change

)( zyxt t ΔΔΔ

∂∂ ρ , so that a mass balance is received

xxxtxxtt zyuzyuzyxt Δ+ΔΔ−ΔΔ=ΔΔΔ

∂∂ )()()( ρρρ . (6-4)

The second term on the right-hand side can be expanded into a series, for small variations about the undisturbed equilibrium state, and if higher-order terms can be neglected, then

( ) ⎥⎦⎤

⎢⎣⎡ ΔΔΔ

∂∂

+ΔΔ−ΔΔ=ΔΔΔ∂∂ xzyu

xzyuzyuzyx

t xxtxxtxxtt ρρρρ )()()( . (6-5)

This can be simplified to

( ) 0=∂∂

+∂

∂xt

t uxt

ρρ

. (6-6)

Generalized to the three dimensional case,

( ) ( ) ( ) 0=∂∂

+∂∂

+∂∂

+∂

∂ztytxt

t uz

uy

uxt

ρρρρ

. (6-7)

Defining the del operator, see remark 6-1, as

⎥⎦

⎤⎢⎣

⎡∂∂

+∂∂

+∂∂

=∇z

ey

ex

e zyx , (6-8)

permits a simplified expression of the continuity equation, as

0=)( ut tt ρ

ρ⋅∇+

∂∂

. (6-9)

The second term contains the product of the density and the particle velocity, both acoustic state variables, so that the equation in that form is nonlinear. Putting the total density (6-3) into the equation and taking advantage of the fact that the undisturbed density ρ

0 is

independent of time and position, and ignoring second order terms, that consist of the product of two acoustic disturbances, gives the linearized wave equation

00 =⋅∇+∂∂ u

tρρ

. (6-10)

In one dimension

00 =∂

∂+

∂∂

xu

txρρ

. (6-11)

Remark 6-1 The scalar product of the del operator and a vector field, A⋅∇ , is called the divergence, and can be interpreted as the source strength of the field; see appendix B.


177

6.1.2 Equation of motion

Newton’s second law amF = relates the force on a fluid particle to its motion, its acceleration in particular. By acceleration is meant, a specific fluid particle’s change in velocity when it moves in space, and not the change in velocity of a fluid particle at a specific point in space. Consider a specific fluid particle, with a fixed mass Δm and a fixed volume ΔV = Δ xΔyΔz, as in figure 6-3, that moves with the medium.

( ) ( )

x

yz

+∂0p+p [ ]

∂ x0p+p ( ) 0

p+p

Δy

Δ z

Δx

Δy ΔyΔ z Δ zΔx-

Figure 6-3 Force in the x-direction on a particular fluid particle moving with the medium. The force in the x-direction is

=⎥⎦⎤

⎢⎣⎡ ΔΔΔ+

∂∂

++−ΔΔ+= zyxppx

ppzyppFx ))(()( 000

zyxppx

ΔΔΔ+∂∂

−= )( 0 , (6-12)

where p0 is constant, so that

zyxxp

Fx ΔΔΔ∂∂

−= . (6-13)

In three dimensions, the force vector becomes

zyxzp

eyp

exp

eF zyx ΔΔΔ⎥⎦

⎤⎢⎣

⎡∂∂

+∂∂

+∂∂

−= . (6-14)

Putting in the del operator, see (6-8) and remark 6-2, as

⎥⎦

⎤⎢⎣

⎡∂∂

+∂∂

+∂∂

=∇z

ey

ex

e zyx (6-15)

and using the relation Δ xΔyΔz = ΔV, then (6-14) reduces to

VpF Δ−∇= . (6-16)

Remark 6-2 The product of the del operator and a scalar field, A∇ , is called the grad-ient and indicates the direction in which the field increases most rapidly; see app. B.


178

For a given fluid particle, the velocity u ( ,r t) is a function of position and time. At time t and position (x, y, z), the velocity is u (x, y, z, t). At a later instant t+Δt, the position is (x+Δx, y+Δy, z+Δz) and the velocity is u (x+Δx, y+Δy, z+Δz, t+Δt). The differential change in position (Δx, Δy, Δz) can be written Δ x =uxΔt, Δy = uyΔt and Δz = uzΔt, so that the acceleration can be written

t

tzyxutttuztuytuxua zyx

t Δ

−Δ+Δ+Δ+Δ+=

→Δ

),,,(),,,(lim

0 . (6-17)

The first term is reformulated with the help of a Taylor series, so that equation (6-17) can be rewritten

t

tzyxuttutu

zutu

yutu

xutzyxu

azyx

t Δ

−+Δ∂∂

+Δ∂∂

+Δ∂∂

+Δ∂∂

+=

→Δ

),,,( ),,,(lim

0

… . (6-18)

The acceleration of the fluid particle becomes

tu

zuu

yuu

xuua zyx ∂

∂+

∂∂

+∂∂

+∂∂

= , (6-19)

With simplifying notation this is

uutua )( ∇⋅+

∂∂

= . (6-20)

I one dimension,

x

uu

tu

a xx

xx ∂

∂+

∂∂

= . (6-21)

Remark 6-3 Equation (6-20) gives the total (substantial) acceleration for a special fluid particle, i.e., the acceleration that an observer of a certain fluid particle would measure. It consists of two parts: a partial time derivative that constitutes the local acceleration, velocity change, that the fluid particle passing a certain point experiences; and a part that depends on the particle velocity change in the spatial dimension. Some authors use the notation DtD for this total (substantial) derivative.

For acoustic fields with small disturbances, the second term in (6-20) can be neglected, since it consists of the product of two small quantities, so that

tua

∂∂

= . (6-22)

In one dimension,

t

ua x

x ∂∂

= . (6-23)


179

Making use of (6-16) and (6-22), as well as Δm = (ρ0 + ρ)ΔV , the equation of motion can be formulated as

tuVVp

∂∂

Δ+=Δ∇− )( 0 ρρ . (6-24)

If second order terms can be ignored, then the linear, inviscid equation of motion is

00 =∇+∂∂ p

tuρ . (6-25)

I one dimension,

00 =∂∂

+∂

∂xp

tuxρ . (6-26)

The equation of motion, as we see, gives a relation between pressure and particle velocity in a sound field.

6.1.3 The thermodynamic equation of state

The thermodynamic equation of state relates the inner forces, the pressure, to the deformations, density changes, that arise at different temperatures. For an ideal gas, the ideal gas law applies, i.e., MRTpp )()( 00 ρρ +=+ (6-27)

where (p0 + p) [Pa] is the total pressure, (ρ0 + ρ) [kg/m3] is the total density, R = 8.315 [J/(mol K)] is the ideal gas constant, M [kg] is the mass of a mole of gas, and T [K] is the absolute temperature. Equation (6-27) can be simplified if the thermodynamic process is limited in some way. Two idealizations can be considered: an isothermal process, implying such good heat conduction in the medium that the temperature is constant throughout; or, an adiabatic process, in which no heat conduction occurs. In the latter case, the higher temperatures in a compressed region persist until a rarefaction of the region occurs. In order to determine whether the heat transfer between compressed and rarefied regions is too much to permit the adiabatic approximation, two phenomena must be taken into account. First, the time available for heat flow to occur, and, second, the distance between the (relatively) hot and cold regions, i.e., the half-wavelength. Higher frequencies imply less time available for heat conduction, but, on the other hand, a shorter distance between compressed and rarefied regions. For small disturbances below a certain frequency limit, it can be shown that the process can, to a good approximation, be regarded as adiabatic. For air, that critical frequency lies far above the audible region. And, for an adiabatic change of state,

( ) ( ) γ

ρρρ

⎥⎦

⎤⎢⎣

⎡ +=

+

0

0

0

0p

pp , (6-28)


180

where γ = cp /cv is the ratio of the specific heat of the gas at constant pressure, to that at constant volume. For air, (6-28) is a good approximation, other than at very low temperatures. In order to generalize the derivation somewhat, and in doing so obtain an analogy to longitudinal wave propagation in solid media, the ideal gas assumption is abandoned below. In the general case, the adiabatic change of state is more complex than indicated by (6-28); the total pressure is a function of more variables than just the total density. In order to obtain a relation between pressure and density variations, a Taylor series expansion is made. For the sake of simplicity, pt = (p0 + p) denotes total pressure, and ρt = (ρ0 + ρ) denotes total density, in the series expansion. The total pressure can be expanded as

...21

00

2

22

00 +∂

∂+

∂∂

+=+=== ρρρρ ρ

ρρ

ρ

tt t

t

t

tt

pppppp , (6-29)

where the partial derivatives are constants that remain to be determined for adiabatic disturbances about ρ0, i.e., the density in the undisturbed medium. For small disturbances, second and higher order terms can be neglected, and a linear relation is obtained as

0ρρρ

ρ=∂

∂=

tt

tpp , (6-30)

or 0ρβρ=p , (6-31) where

0

0ρρρ

ρβ=∂

∂=

tt

tp (6-32)

is called the adiabatic bulk modulus.


181

6.1.4 The homogenous linearized wave equation

Working from the linearized expressions for the continuity equation, the equation of motion, and the thermodynamic equation of state, a linearized wave equation is derived. We do so for both the one- and the three-dimensional cases, in parallel.

One dimension Three dimensions

Time derivative of continuity eq. (6-11),

02

02

2=

∂∂∂

+∂

∂tx

u

txρ

ρ . (6-33)

Time derivative of continuity equation (6-

10), using ( ) ⎟⎠⎞

⎜⎝⎛

∂∂

⋅∇=⋅∇∂∂

tuu

t,

002

2=⎟

⎠⎞

⎜⎝⎛

∂∂

⋅∇+∂

∂tu

tρρ . (6-34)

Spatial derivative of eq. of motion (6-26),

02

22

0 =∂

∂+

∂∂∂

xp

txuxρ . (6-35)

Divergence of eq. of motion (6-25),

( ) 00 =∇⋅∇+⎟⎠⎞

⎜⎝⎛

∂∂

⋅∇ ptuρ , (6-36)

abbreviated notation (see remark 6-4),

020 =∇+⎟

⎠⎞

⎜⎝⎛

∂∂

⋅∇ ptuρ . (6-37)

Subtraction of (6-33) from (6-35) gives

02

2

2

2=

∂

∂−

∂

∂

txp ρ . (6-38)

Subtraction of (6-34) from (6-37) gives

02

22 =

∂

∂−∇

tp ρ . (6-39)

Equation of state (6-31) eliminates ρ

02

20

2

2=

∂

∂−

∂

∂

tp

xp

βρ

. (6-40)

Equation of state (6-31) eliminates ρ

02

202 =

∂

∂−∇

tpp

βρ

. (6-41)

The wave equation in one dimension

012

2

22

2=

∂

∂−

∂

∂

tp

cxp

. (6-42)

The wave equation in three dimensions

012

2

22 =

∂

∂−∇

tp

cp . (6-43)

The constant c is defined as 0/ ρβ=c (6-44)

and is the propagation speed of a disturbance in the medium, the speed of sound. According to (6-32),

0

0/

ρρ

ρ∂∂

ρβ

=

==

tt

tpc . (6-45)

For an ideal gas, (6-28) implies


182

)ln(lnlnln 0000

ρργρρ

γ

−=−⇒⎟⎟⎠

⎞⎜⎜⎝

⎛= tt

tt pppp

, (6-46)

so that

t

t

t

t ppρ

γρ

=∂∂

. (6-47)

Put into the expression for the speed of sound (6-45), this yields

0

ρρ

ργ=

=t

ttpc , (6-48)

i.e.,

00 ργ pc = . (6-49)

The temperature dependence of the speed of sound is obtained by putting the ideal gas law (6-27) into (6-48) MRTc γ= . (6-50)

If the speed of sound at 0° C (273 K) is denoted c0, then for other temperatures

2730 Tcc = . (6-51)

The speed of sound increases with temperature, a relationship with great significance for sound propagation outside, where the temperature often varies with distance to the ground. Sound does not propagate in a straight line in that case. It bends, which can result in areas of sound shadow and of sound concentration. A technically important area, in which the increase of the sound speed with temperature must be accounted for, is that of exhaust gas pipes leading from internal combustion engines. Measurements of the speed of sound in air show good agreement with the theoretical values, which supports the adiabatic hypothesis. Example 6-1 Table 6-1 The speed of sound and other data at 0oC (273 K) and 1 atm (1,013·105 Pa) for some types of gases.

Gas γ ρ0 [kg/m3] c0 [m/s] ρ0c0 [kg/m2s] Air (dry) 1.40 1.293 331 428 Carbon dioxide,CO2 1.40 1.977 259 512 Helium, He 1.667 0.178 965 172 Nitrogen, N2 1.40 1.251 334 418 Oxygen, O2 1.40 1.429 316 452

For air at 20° C, using (6-51), c = 333 (293 / 273)1/2 = 343 m/s. The density ρ0 = 1.21 kg/m3.


183

Remark 6-4 In (6-36) we use )( p∇⋅∇ , the divergence of the gradient of the sound pressure p. As before, this is written

,

eeeeee)(

22

2

2

2

2

2

2

2

2

2

2

2pp

zyxz

p

y

p

x

pzp

yp

xp

zyxp zyxzyx

∇=⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂

∂+

∂

∂+

∂

∂=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂

∂+

∂

∂+

∂

∂=

=⎥⎦

⎤⎢⎣

⎡∂∂

+∂∂

+∂∂

⋅⎥⎦

⎤⎢⎣

⎡∂∂

+∂∂

+∂∂

=∇⋅∇

in which we have introduced the symbol ∇2 for the divergence of the gradient of a scalar. That quantity is usually called the Laplacian, or Laplace operator, and can be understood as the scalar product of the del operator with itself, and alternatively written ∇ ⋅ ∇ = ∇2 = Δ. Even the Laplace operator is independent of the coordinate system, and the wave equation (6-43) is therefore valid in other orthogonal coordinate systems as well, such as spherical ones for example; see appendix B for more detail. The definition of c, in (6-44), applies to liquids as well. That means that a liquid’s equation of state is all that is needed to determine the sound speed. Since that equation is typically not available, one must usually refer to experimental data to determine the sound speed in liquids. It is, however, not as easy to determine the adiabatic bulk modulus β, from formula (6-32). It is easier to determine the isothermal bulk modulus βΤ . It can be shown theoretically that the relation between these is β = γ βΤ , after which the sound speed in liquids is found from

0ρβγ Tc = . (6-52)

Example 6-2 Table 6-2 The speed of sound and other data for some liquids.

Liquid Temp [ C ] β [Pa] ρ0 [kg/m3] c [m/s]

Ethyl alcohol 20 - 790 1150

Quicksilver (Mercury)

20 25.3·109 3600 1450

Salt water 13 2.28·109 1026 1500

Fresh water 20 2.18·109 998 1481


184

6.2 SOLUTIONS TO THE WAVE EQUATION

Using the linearized wave equation in the one dimension (6-42), or more generally in three dimensions (6-43), together with the boundary conditions that describe the situation at any boundaries, we can calculate a sound field. Examples of boundaries are rigid walls blocking wave propagation, and transitions to media with a different stiffness and/or density.

6.2.1 General solution for free plane one-dimensional wave propagation

For plane wave propagation in the x-direction, the wave equation (6-42) applies. We assume a general solution form of ( ) ( )cxtgcxtftxp ++−=),( , (6-53)

where f and g are arbitrary functions and (t – x / c) and (t + x / c) are their respective arguments. That assumed solution is known as d’Alembert´s solution. Derivation of (6-53) with respect to x gives

( ) ( )cxtgc

cxtfcx

p+′+−′−=

∂∂ 11 , (6-54)

( ) ( )cxtgc

cxtfcx

p+′′+−′′=

∂

∂222

2 11 , (6-55)

and similarly with respect to t gives

( ) ( )cxtgcxtftp

+′+−′=∂∂ , (6-56)

( ) ( )cxtgcxtft

p+′′+−′′=

∂

∂2

2 . (6-57)

Putting (6-55) and (6-57) into (6-42) shows that the solution fulfills the wave equation. Note that because the wave equation is a linear differential equation, then the sum of any two solutions is also a solution. To interpret (6-53), consider a special point (x1,t1) in the wave, in figure 6-4. It represents a certain sound pressure p1(x1,t1). To have the same sound pressure at another point (x1 + Δ x), at a later instant in time (t1 + Δt), then the arguments have to be the same, i.e.,

( ) ( )cxxttcxt )( 1111 Δ+−Δ+=− , (6-58)

which gives the condition

tcx Δ=Δ . (6-59)

Thus, the solution f (t – x / c) implies wave propagation in the positive direction along the x-axis, with speed c.


185

Similarly, g(t + x / c) implies propagation in the negative x- direction, so that

tcx Δ−=Δ . (6-60)

The propagation speed c of a disturbance is called the speed of sound (or sound speed), wave speed or phase velocity.

6.2.2 Harmonic solution for free, plane, one-dimensional wave propagation

From Fourier analysis, it is known that every periodic process can be built up of the summation of harmonic, sinusoidal processes with different frequencies, the set of which is called a Fourier series. In the preceding section, it was determined that if each individual harmonic process is a solution to the wave equation, then their sum is also a solution. Therefore, only harmonic waves will be treated in the following development. In computations, each individual frequency component is treated independently, and the total sound pressure field can then be obtained by a summation over all frequencies. The harmonic solution we seek for the angular frequency ω = 2π f must, for a certain x-value, say x1 + x, give the same sound pressure at time t as it does one period later at time t + T, where T is called the period. For a harmonic function, that implies that the argument, i.e., the angle, increases by 2π. The solution must also, for a certain time value, say t1 = t, give the same sound pressure at points separated along the x-axis a distance equal to the wavelength λ. Even in that case, the argument must increase by 2π, see figures 6-5a and 6-5b. Thus, we attempt a solution of the form )(cosˆ),( cxtptxp −= + ω , (6-61)

where +p is the amplitude, i.e., the highest value of the sound pressure. The argument ω ( t - x/c) = ω t – kx is called the phase and

k = ω /c (6-62)

is called the wavenumber. The first condition above gives

))((cosˆ)(cosˆ 11 kxTtpkxtp −+=− ++ ωω (6-63) i.e., ω T = 2π , ω = 2π /T . (6-64) The second condition gives

Figure 6-4 Instantaneous picture of the wave propagation in the positive x-direction at time instants t1 and (t1 + Δt). The propagation speed of the disturbance is c.

f(t1 - x1/c) f(t1 + Δt - (x1 + Δx)/c) p(x,t)

x x1 x1 + Δx

p1t = t1

t = t1 + Δt


186

( )( )λωω +−=− ++ xktpkxtp 11 cosˆ)(cosˆ (6-65) i.e., λππλ 2,2 == kk . (6-66)

From (6-62), (6-64) and (6-66), one obtains the relation λfc = , (6-67) which applies to all types of wave propagation, and in which the frequency is

f = 1 / T . (6-68)

Since it is significantly more convenient, from the mathematical perspective, to deal with exponential functions rather than trigonometric ones, the development to follow will make wide use of the complex notation

)()( ˆˆ),( kxtikxti epeptx +−

−+ += ωωp , (6-69)

in which bold print means that the variable concerned is complex. The first term on the right-hand side refers to propagation in the positive x-direction, and the second term to propagation in the negative x-direction. For the sake of physical interpretation, the real part of (6-69) is needed, i.e.,

=−++= +− )ˆˆRe(),( )()( kxtikxti epeptxp ωω

)cos(ˆ)cos(ˆ kxtpkxtp ++−= −+ ωω . (6-70)

The equation of motion (6-26)

00 =∂∂

+∂

∂xp

tuxρ , (6-71)

Figure 6-5a The variation of sound pressure with time at a fixed position x1.

Figure 6-5b The variation of sound pressure with position at a fixed time t1.

p(x1,t)

p+

Period T

t

p(x,t1)

x

Wavelength λ

p+


187

relates the particle velocity to the sound pressure; rearranged, it gives

dtxpu x 1

0∫ ∂

∂−=

ρ . (6-72)

Next, putting in (6-69) gives the particle velocity

( ) ⎥⎦⎤

⎢⎣⎡ +

−−= +

−−

+)()(

0ˆˆ1, kxtikxti

x epiikep

iiktx ωω

ωωρu , (6-73)

Since k/ω = 1/c, the particle can be written

)(

0

)(

0

ˆˆ),( kxtikxti

x ec

pe

cp

tx +−−+ −= ωωρρ

u . (6-74)

The two terms refer to wave propagation in the positive and negative x-directions, respectively. The ratio of pressure to particle velocity is called the specific impedanceError! Bookmark not defined. Z,

xupZ = , (6-75)

and, for the free plane wave case, is therefore

cZ 00 ρ=+ (6-76) and

cZ 00 ρ−=− (6-77)

for propagation in the positive and negative directions, respectively. The quantity ρ0c is called the wave impedance. Example 6-3 For air at 20° C and 1 atm (1.013·105 Pa), ρ0 = 1.21 kg/m3 and c = 343 m/s, so that the wave impedance becomes ms.Pa 41500 == cZ ρ .

6.2.3 Sound intensity for free, plane, one-dimensional wave propagation

The sound intensity is defined as the sound energy per unit time that passes through a unit area perpendicular to the propagation direction. From basic courses in mechanics, it is known that the instantaneous power can be written

)()()W( tutFt ⋅= . (6-78)

A general expression for sound intensity is therefore that

),(),(),( trutrptrI = . (6-79)


188

For propagation in the x-direction, ),(),(),(x txutxptxI x= . (6-80)

The time-averaged sound intensity is

∫=T

xx dttxutxpT

xI0

),(),(1)( . (6-81)

Making use of the expression for pressure, i.e., the real part of (6-69), and the particle velocity, i.e., the real part of (6-74), in (6-81), gives the intensity in the form

cppI x 022 2)ˆˆ( ρ−+ −= . (6-82)

The first term refers to a wave moving in the positive x-direction, and the second term to a wave moving in the negative x-direction. For harmonic waves, the relation between the rms amplitude and the peak value is 2ˆ~ pp = , so that (6-82) can be written

c

pc

pI x

0

2

0

2 ~

~

ρρ−+ −= . (6-83)

Example 6-4 An infinitely large wall oscillates harmonically, with the velocity tvtv ωcosˆ)( = . Determine the radiated sound intensity.

Solution At the separating surface (x = 0), the particle velocities of the plate and the air must be identical. Our boundary condition is therefore v(t) = ux(0,t), from which it follows that

)(ˆ),( kxtix evtx −= ωu .

According to (6-75) and (6-76), the pressure in a plane wave satisfies

)(00 ˆ),(),( kxti

x evctxctx −== ωρρ up .

tvtv ωcosˆ)( = λ Ix

Figure 6-6 Oscillations of the wall radiate compressions and rarefactions into the air, and acoustic power is thereby radiated.


189

Putting that into (6-82) yields 20

20 ~2

ˆvc

vcI x ρ

ρ== . (6-84)

6.2.4 Energy and energy density in free, plane, one-dimensional wave propagation

The energy E associated with a sound wave consists of two parts, the kinetic and the potential energy. The kinetic energy Ek can be related to the velocity of a fluid particle. The potential energy Ep is due to the compression, i.e., the elasticity. Consider a particular mass of gas that has density ρ0 and volume V0 in the undisturbed medium, so that its mass can be written VV tρρ =00 . (6-85)

For wave propagation in the x-direction, its kinetic energy is, from elementary mechanics,

2

),(),(

200 txuV

txE xk

ρ= . (6-86)

The potential energy of the fluid mass comes from the work that expended to compress it. Analogous is the energy stored in a spring as it is loaded. Consider, now, the fluid volume shown in figure 6-7. The force applied to the piston is equal to the product of the piston’s area S and the pressure p against its inside surface, i.e., Fx = p(x,t) S. When the piston moves a distance dx, the differential amount of work dEp = Fx dx = p(x,t) S dx is performed.

F

T vä rs n it tsa re a S

d x

T ryc k px

x Since Sdx = dV , dEp can, with complete generality, be written

dVtxpdE p ),(−= ,

where the minus sign implies that a positive sound pressure p(x,t), which gives a negative volume change (i.e., a volume reduction or compression), corresponds to a positive potential energy. To make further progress, (6-87) must be reformulated as a function of a single var-iable. We choose to express dV as a function of the sound pressure. Differentiating (6-85),

tt

ttt

dVdV

dVVV ρρ

ρρ

ρρ

ρ−≈−=⇒= 2

0000

1 . (6-88)

Since (6-46) states that

Figure 6-7 Work performed to compress a fluid volume.

Cross sectional area S

Pressure p


190

γ

ρρ

⎟⎟⎠

⎞⎜⎜⎝

⎛=

00

ttpp

then

tt

tt

t

t

t

t dpp

dp

dpdγρ

ργρρ

=⇒=1 . (6-89)

(6-88) and (6-89) give tt

dpp

VdVγ

−= . (6-90)

For small pressure and volume disturbances, the undisturbed pressure and the undisturbed volume can be used

dpp

VdV

0

0γ

−= . (6-91)

The work (6-87) can now be written as

dppp

VdE p

0

0γ

= . (6-92)

The potential energy is, finally, obtained by integrating from 0 to the sound pressure p; thus,

2

0

02

ppV

E p γ= . (6-93)

According to (6-49), 002 ργ pc = ; thus, for plane wave propagation in the x-direction, a general expression for the energy is

( ) ),(2

, 22

0

0 txpc

VtxE p

ρ= . (6-94)

The concept of energy density ε [J/m3] refers to the energy per unit volume. Of course this quantity, just as the total energy does, consists of two parts: kinetic and potential energy densities, i.e.,

),(),(),( trtrtr pk εεε += . (6-95)

Using expressions for the kinetic energy (6-86) and the potential energy (6-94), the energy density for propagation in the x-direction is then

20

220

2),(

2),(),(

ctxptxutx x

ρρε += . (6-96)

For plane waves, that expression can be simplified. This is of great importance, because, as will later be seen, spherical waves can, for example, be approximated by plane waves if the distance to the source is sufficiently large; see figure 1-9.


191

According to (6-76) and (6-77), )(),(),( 0ctxptxu x ρ±= applies to plane waves. Using this in (6-96) yields the instantaneous value of the energy density

2

0

2

20

2

20

2 ),(2

),(2

),(),(c

txpc

txpc

txptxρρρ

ε =+= . (6-97)

The time average is obtained by integrating with respect to time

2

0

2

0

)(~),(1)(

cxpdttx

Tx

T

ρεε == ∫ . (6-98)

For a plane wave however, since it doesn’t decay with distance, the rms pressure is independent of position x; that naturally applies to the time-averaged energy density as well, i.e.,

20

2~

c

p

ρε = . (6-99)

A comparison to (6-83) gives the relation between the sound intensity and the energy density of a plane wave, as

cI x ε= . (6-100)

6.2.5 General solution for free spherical wave propagation

The spherical wave is a basic cornerstone in the study of acoustic fields. A natural reason is that the sound fields of many sources can be regarded as spherical. Beyond that, more complex sound fields can be built up of combinations of spherical waves.

Figure 6-8 Symbolic picture of spherical wave propagation from a harmonic disturbance.


192

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rλ

The gradient, divergence and Laplace operators have, in their Cartesian forms, thus far appeared in the derivations of the continuity equation (6-10), the equation of motion (6-25), and the wave equations (6-25) and (6-43). They are, as noted above, general operators, which implies that they have a physical meaning when they operate on a field variable, such as sound pressure or particle velocity for example. They are therefore valid in other coordinate systems besides the Cartesian. A condition for validity is, of course, that the operator be correctly adapted to the coordinate system in which it is to be used. Appendix B provides, without proof, the operators in Cartesian and spherical coordinates. For spherical wave propagation, it is of course more convenient to make use of the latter; see figure 6-9.

x

y

z

φ

θ

r sinθ sinφ

r sinθ cosφ

r cosθ r

(r, θ , φ )

In spherical coordinates, the wave equation (6-43) takes the form

Figure 6-9 Relation between spherical and Cartesian coordinate systems.


193

2

2

22

2

2222

21

sin

1sinsin

11

t

p

cp

rrrr

rr ∂

∂=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂

∂+⎟

⎠⎞

⎜⎝⎛

∂∂

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

φθθθ

θθ . (6-101)

For spherical symmetry, the sound pressure has no angular dependence, and (6-101) reduces to

2

2

22

211

t

p

cp

rr

rr ∂

∂=⎟

⎠⎞

⎜⎝⎛

∂∂

∂∂ (6-102)

or 2

2

22

2 12t

p

crp

rr

p

∂

∂=

∂∂

+∂

∂ . (6-103)

Unlike plane wave propagation, the sound pressure amplitude decays with increasing radius, since the sound power in the wave is divided over an ever-expanding spherical surface of area 4πr2. In a plane wave, the sound intensity is, according to (6-82),

c

pI x0

2

2ˆρ

= , (6-104)

i.e., proportional to the squared sound pressure amplitude. Assuming that applies to spherical waves as well, their amplitude would therefore have to decay at a rate of 1/r according to the energy principle; by analogy to the plane wave case, an assumed solution might therefore take the form

( ) ( )crtgr

crtfr

trp ++−=11),( . (6-105)

Here, the first term represents an outgoing diverging wave and the second term an incident converging wave. The incoming wave seldom exists in connection with acoustic radiation from machines. The solution is verified by inserting it into the wave equation (6-103). On a term-by-term basis, we have

( ) ( ) ( )

( ) ( ) ( )crtgr

crtgcr

crtgrc

crtfr

crtfcr

crtfrcr

p

+++′−+′′+

+−+−′+−′′=∂

∂

322

3222

2

221

221

, (6-106)

( ) ( ) ( ) ( )crtgr

crtgcr

crtfr

crtfcrr

pr

+−+′+−−−′−=∂∂

323222222 , (6-107)

( ) ( )crtgrc

crtfrct

p

c+′′+−′′=

∂

∂222

2

2111 . (6-108)

Putting all of these into (6-103) shows that the assumption does indeed fulfill the wave equation. The equation of motion (6-25) relates the particle velocity to the sound pressure. In spherical coordinates, it takes the form


194

0sin

110 =⎥

⎦

⎤⎢⎣

⎡∂∂

+∂∂

+∂∂

+∂∂ pe

re

re

tu

r φθθρ φθ , (6-109)

With spherical symmetry and rr etruu ),(= , the equations of motion can be reduced to

00 =∂∂

+∂

∂rp

turρ , (6-110)

and the particle velocity expressed as

dtrpur 1

0∫ ∂

∂−=

ρ . (6-111)


195

6.2.6 Harmonic solution for free, spherical wave propagation

In accordance with (6-105), a complex harmonic solution is obtained, as

)()(),( krtikrti er

Ae

rA

tr +−−+ += ωωp . (6-112)

The first term on the right-hand side refers to an outgoing, diverging, wave, and the second to an incident, converging, wave. There is, in many engineering situations, no incident wave. Thus, only the part of the solution representing an outgoing wave need be considered further. The sound pressure amplitude in the outgoing wave is A+/ r, where A+ is a constant. The amplitude is therefore a function of r. From (6-111), the particle velocity becomes

( ) ( ) dterik

rA

dtrr,ttr krti

r 1 1, )(200

−+ ∫∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛ −+

−−=

∂∂

−= ωρρ

pu . (6-113)

Integrating, and applying the relation (6-62), k = ω /c, gives the particle velocity

)(

0

11),( krtir e

rkicrA

tr −+ ⎟⎠⎞

⎜⎝⎛ += ω

ρu . (6-114)

By analogy to the definition (6-75) of the specific impedance of a free plane wave, we define here the complex quantity Z as the ratio of the complex sound pressure to the complex radial particle velocity at a point in a sound field,

ru

pZ = . (6-115)

For an outgoing spherical wave, the specific impedance, using (6-112) and (6-114), becomes

ikr

ikrc

ikr

cr +

=+

==111

100 ρρ

upZ . (6-116)

Multiplying the numerator and denominator by the complex conjugate of the latter, gives

⎟⎟⎠

⎞⎜⎜⎝

⎛

++

+=

+

+=

2222

22

022

22

0111 rk

krirk

rkcrkikrrkc ρρZ , (6-117)

and the specific impedance Z can be divided into a resistive part R and a reactive part X, i.e.,

iXR +=Z . (6-118)


196

kr 201510500

0,8

0,6

0,4

0,2

1,0R/ρ0c

X/ρ0c

Some observations that follow from the preceding are:

(i) Nearfield In the acoustic nearfield (kr = 2πr / λ « 1), i.e., when the radius is small in comparison to the wavelength, both the resistance and the reactance approach zero, but the resistance does so more quickly. The reactance therefore dominates, and the impedance approaches

krcir )( 0ρ≈Z . (6-119)

That means that for a given sound pressure, the particle velocity becomes large and its phase shift approaches 90° with respect to the sound pressure. For kr = 1, both the resistance and the reactance are equally large, ρ0c / 2, and the reactance has its maximum here. (ii) Far field In the acoustic farfield (kr = 2πr / λ » 1), i.e., where the radius is large with respect to the wavelength, the resistance approaches ρ0c and the reactance approaches zero as r goes to infinity. The resistance dominates and the impedance approaches the same expression as for plane waves cr 0)( ρ≈Z . (6-120)

That means that the phase difference between the sound pressure and the particle velocity approaches zero, as is the case for plane waves. The curvature of the spherical waves in the farfield, with increasing distance to the source, becomes all the less significant and the situation asymptotically approaches that of plane waves.

6.2.7 Sound intensity for spherical wave propagation

The time-averaged sound intensity of outgoing spherical waves can be determined by the same methods as for plane waves in section 6.2.3. We can also, in accordance with the expression for power (3-7), write the intensity

2

)*Re(2

)*)(Re()( rr

r rIupup

=⋅

= . (6-121)

Figure 6-10 Normalized resistance R /ρ0c , and normalized reactance X /ρ0c , for outgoing spherical wave propagation.


197

Putting (6-112) and (6-114) into (6-121) gives

2

0

2

20

2

211Re

21)(

crA

ikrcrA

rIrρρ

++ =⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ += . (6-122)

The time-averaged energy flow through a closed, spherical control surface of radius r is

c

ArrrrIW

0

22 24)(

ρππ +== . (6-123)

For a loss-free medium, the sound power is therefore independent of the radius, which is in agreement with the energy principle. The constant A+ can be determined from the phenomenon that generates the spherical wave, as for example the velocity on the surface of a harmonically pulsating sphere. See chapter 8 for further discussion.


198


THE WAVE EQUATION IN A SOURCE-FREE MEDIUM

The linear continuity equation

Three-dimensional 00 =⋅∇+∂∂ u

tρρ (6-10)

One-dimensional 00 =∂

∂+

∂∂

xu

txρρ (6-11)

Linear inviscid equation of motion

Three-dimensional 00 =∇+∂∂ p

tuρ (6-25)

One-dimensional 00 =∂∂

+∂

∂xp

tuxρ (6-26)

Thermodynamic equation of state

Ideal gas law MRTpp )()( 00 ρρ +=+ (6-27)

where R = 8.315 [J/(mol·K)]

The gas law for adiabatic changes of state

( ) ( ) γ

ρρρ

⎥⎦

⎤⎢⎣

⎡ +=

+

0

0

0

0p

pp (6-28)

The wave equation

One-dimensional 012

2

22

2=

∂

∂−

∂

∂

tp

cxp (6-42)

Three-dimensional 012

2

22 =

∂

∂−∇

tp

cp (6-43)

The speed of sound 00 ργ pc = (6-49)

The temperature dependence of the speed of sound

2730 Tcc = (6-51)

where c0 is the speed of sound at 0° C


199

SOLUTIONS TO THE WAVE EQUATION

The general solution for free, plane, one-dimensional wave propagation General solution for the sound pressure

( ) ( )cxtgcxtftxp ++−=),( (6-53)

where f and g are arbitrary functions Harmonic solution for free, plane, one-dimensional wave propagation

Wavenumber k = ω /c (6-62)

Relation between the sound speed, frequency, and wavelength

λfc = (6-67)

Harmonic solution for sound pressure

)()( ˆˆ),( kxtikxti epeptx +−

−+ += ωωp (6-69)

Particle velocity

)(

0

)(

0

ˆˆ),( kxtikxti

x ec

pe

cp

tx +−−+ −= ωωρρ

u (6-74)

Specific impedance xupZ = (6-75)

Specific impedance in the positive x-direction

cZ 00 ρ=+ (6-76)

Specific impedance in the negative x-direction

cZ 00 ρ−=− (6-77)

Sound intensity for free, plane, one-dimensional wave propagation The terms represent waves in the positive and negative x-directions, respectively

c

pc

pI x

0

2

0

2 ~~

ρρ−+ −= (6-83)

The sound intensity for radiation from an oscillating, infinite, planar wall

20

20 ~2

ˆvc

vcI x ρ

ρ== (6-84)


200

Energy and energy density for free, plane, one-dimensional wave propagation

Energy density (time-averaged) 2

0

2~

c

p

ρε = (6-99)

Relation between the sound intensity and the energy density for harmonic waves

cI x ε= (6-100)

General solution for free, spherical wave propagation General solution for the sound pressure

( ) ( )crtgr

crtfr

trp ++−=11),( (6-105)

where f and g are arbitrary functions.

The equation of motion, in the case of spherical symmetry

00 =∂∂

+∂

∂rp

turρ (6-110)

Harmonic solution for free, spherical wave propagation Harmonic solution for sound pressure

)()(),( krtikrti er

Ae

rA

tr +−−+ += ωωp (6-112)

Particle velocity )(

0

11),( krtir e

rkicrA

tr −+ ⎟⎠⎞

⎜⎝⎛ += ω

ρu (6-114)

Complex specific impedance ru

pZ = (6-115)

Complex specific impedance for outgoing waves

ikr

ikrc

ikr

cr +

=⎟⎠⎞

⎜⎝⎛ +

==111

100 ρρ

upZ (6-116)

Sound power and sound intensity for free spherical wave propagation Time averaged sound power in an outgoing wave

c

ArrrrIW

0

22 24)(

ρππ +== (6-123)


201

201

CHAPTER SEVEN

REFLECTION, TRANSMISSION AND STANDING WAVES This chapter covers what happens to sound waves that reach boundaries of some kind. Examples of such boundaries are walls of rooms and vehicles, or the ground surface if the sound propagation occurs outside. The results obtained are of great significance in acoustics. They shed light on how sound absorbent can be effectively incorporated into factories, classrooms, and vehicles, as well as how we should locate sound barriers and build sound walls, and many other things. Reflection of sound waves is also the basis for the development of standing waves, “resonances”, indoors or in vehicle passenger compartments, an effect which is often very significant. If a machine that emits specific sound frequencies is placed in a room with “eigenfrequencies” that match, very high sound levels can easily result. On the other hand, it is important that the eigenfrequencies of a concert hall be evenly distributed with frequency. It is exactly those that “carry” the sound picture when music is played.

Chapter 7: Reflection, transmission and standing waves

202

7.1 REFLECTION AND TRANSMISSION OF PLANE WAVES

Earlier, free wave propagation was described in what were, in principle, infinite media. Of course, such media do not really exist. There are always bounds on the regions in which sound propagation is considered. When a wave comes across a change in the medium, an interface surface, a portion of the wave’s energy is reflected back, and another portion is transmitted through the boundary. A reflected and a transmitted wave are generated by the incident wave. The reflected and transmitted waves do not depend exclusively on the incident wave and its angle of incidence, but also on the properties of the medium behind the interface surface. As a first example, normal incidence against a rigid wall is considered. In that case, total reflection of the incident wave occurs. In other cases, when the wall, the barrier, can itself undergo motion, then transmission will also be possible.

7.1.1 Normal incidence against a rigid barrier

An important special case, normal incidence on a rigid wall or barrier, is treated first. By a rigid wall, a surface that does not undergo any motion whatsoever is implied. The particle velocity normal to the surface is exactly zero, i.e., the specific impedance is infinitely large.

x

Rigid wall

)(ˆ),( kxtiii eutx −= ωu

Figure 7-1 A harmonic plane wave normally incident against a rigid wall.

Since the boundary condition is that the particle velocity normal to the surface must be identically zero, the acoustic field in front of the wall may be expressed as the sum of incident and reflected wave particle velocities

)()( ˆˆ),(),(),( kxtir

kxtiiri eeutxtxtx +− +=+= ωω uuuu , (7-1) 1

where δirr euˆ =u , i.e., a general assumption that permits both the amplitude and the phase

angle to be modified by the reflection at the surface. At the surface x = 0, the particle velocity is equal to zero, i.e., 1 Particle velocity in the x-direction. To not encumber the discussion, the equations in this section are not subscripted by the index x.


203

πirri euu ˆˆˆ −=−= u . (7-2)

Several interesting observations follow from this simple relation. First of all, the amplitude of the reflected wave is equal to that of the incident wave. Thus, the intensity in both waves is equal; no losses occur at the boundary. Second, because δ = π, the reflected wave is phase shifted 180˚ with respect to the incident. The constraints that the boundary condition places on the reflected wave are in accordance with our intuition. If the reflected wave is to eliminate the particle velocity associated with the incident wave, at all time instants, two conditions must hold:

(i) The frequency must be the same for the two waves,

(ii) The amplitude of the reflected wave must be equal to the incident, but in opposite phase, i.e., phase shifted by 180˚.

From (7-1), with ûr = -ûi, one obtains

kxeuieeeutx tii

ikxikxtii sinˆ2)(ˆ),( ωω −=−= −u . (7-3)

In real form, tkxutxu i ωsinsinˆ2),( = . (7-4)

The particle velocity field deviates from that of free wave propagation, as described by (6-74), so that, were we to move as observers at the speed of sound through the field, we would not constantly observe the same phase angle. Instead, for every point x in the field, there is a harmonic time variation of the form sinω t and an x-dependent amplitude function, 2ûisin(kx). That amplitude function is identically zero for certain values of x, implying that the particle velocity at those points is always zero; such points are called nodes. The points at which the amplitude function has its maxima are called antinodes. Nodes and antinodes do not move along the x-axis, so that the entire phenomenon is called a standing wave or standing wave solution, as opposed to free wave propagation. Nevertheless, it bears remembering that the standing wave is built up of free waves, and that both representations are physically equivalent. A condition for the existence of a standing wave is reflection from at least one boundary. In section 7.2, we study a special case of standing waves in which reflection occurs at two or more boundaries. Figure 7-2a shows the amplitude function for particle velocity, normalized by the peak value. The sound pressure in the standing wave can, with reference to the preceding, be determined in two different ways, partly with the aid of the equation of motion (4-26), and partly taking advantage of the specific impedance of free plane waves

xkecutx tii cosˆ2),( 0

ωρ=p . (7-5)

In real form,

tkxcutxp i ωρ coscosˆ2),( 0= . (7-6)

In figure 7-2b, the amplitude function of the sound pressure is also shown; it is also normalized by the peak value. Evidently, the sound pressure nodes and antinodes are shifted a distance of λ /4 with respect to those of the particle velocity.


204

a)

−λ

0

1

-1 x

0,5

- 0,5

u x t u( , ) / 2Velocity maxima

Velocity nodes

2πω −=t

6πω −=t

2πω =t

0=tω

6πω =t

40 λ−=x

−λ 0

1

-1 x x0 = - λ/4

0,5

- 0,5

p x t cu( , ) / ( )2 0ρωt = 0

ωt = π/3

ωt = π/2

ωt = 2π/3

ωt = π

b)

Pressure maxima

Pressure nodes

Figure 7-2 Standing wave pattern for normal incidence against a rigid wall, for a) normalized particle velocity u, and b) normalized sound pressure p.

The first antinode of particle velocity, which is also the first node of sound pressure, occurs at a distance x0 from the wall, obtained from 20 π−=kx , (7-7)

giving x0 = - λ/4. That is of significance, for example, in the selection of the thickness of porous absorbent to be mounted to walls and ceilings. The acoustic absorption is strongly dependent on the particle velocity, or more precisely, on the friction forces that result from the motion of the medium inside the porous absorbent. Finally, it is interesting to consider the energy transport in the standing wave. Making use of the sound intensity expression (3-7), as well as (7-3) and (7-5), it can be shown that the intensity, and thereby the energy transport as well, is zero. That can be understood by considering that the incident and reflected waves are equally strong, i.e., they have the same amplitude, and therefore compensate each other’s energy transport. That is a special case that applies to a rigid wall. If the wall, on the other hand, can


205

undergo motion, a net transport of acoustic energy can take place, and the boundary conditions are modified.

7.1.2 Normal incidence against a boundary between two elastic media

Consider a harmonic plane wave normally incident upon a boundary between two elastic media. These could be gaseous, liquid, or solid media. A solid medium is treated in the same way as a fluid, in which no transverse wave can arise. In accordance with figure 7-3, the boundary between medium 1 and medium 2 is located at x = 0. The two media are characterized by the respective specific impedances Z1 = ρ1c1 and Z2 = ρ2c2.

Medium 1Z1= ρ c1 1

x

Medium 2

)( 1ˆ),( xktiii eptx −= ωp

Z2= ρ2c2

There is an incident wave propagating in the positive x-direction, with sound pressure

)(1ˆ),( xkti

ii eptx −= ωp (7-8) and particle velocity

)(

111

ˆ),( xktii

i ecp

tx −= ωρu , (7-9)

in which the specific impedance ρ1c1, for propagation in the positive x-direction, is used. There is also a reflected wave propagating in the negative direction, with sound pressure

)( 1ˆ),( xktirr eptx += ωp (7-10)

and the particle velocity

)(

111

ˆ),( xktir

r ecp

tx +−= ωρu , (7-11)

in which the negative-valued specific impedance -ρ1c1, applicable to propagation in the neg-ative x-direction, is used. In medium 2, a transmitted wave is assumed with a sound pressure

)( 2ˆ),( xktitt eptx −= ωp (7-12)

and a particle velocity

Figure 7-3 Normal incidence against a boundary between two elastic media.


206

)(

222

ˆ),( xktit

t ec

ptx −= ω

ρu . (7-13)

As in section 7.1.1, we could have assumed complex amplitudes of the sort δiep , i.e., an amplitude p and a phase angle δ. It turns out, however, that if the impedances involved are real, the phase angle is always either 0˚ or 180˚, which is indicated by the sign of the peak value in any case. That being so, it is unnecessary to encumber the derivation with explicitly indicated phase terms for the reflected and the transmitted waves. Since the speeds of sound in the media are generally different, then the wave numbers are also different, i.e., k1 = ω /c1 and k2 = ω /c2 respectively. Two boundary conditions must be fulfilled at all times, and along the entire boundary surface x = 0. Firstly, the total sound pressure in medium 1, built up of the incident and the reflected waves, must be equal to the sound pressure in medium 2 in the form of the transmitted wave, i.e.,

),0(),0(),0( txtxtx tri ===+= ppp . (7-14)

That boundary condition can be shown to hold by setting up the equation of motion of a thin, massless membrane in the boundary plane. We call it, henceforth, the condition of continuity of pressure across the boundary. The other boundary condition is that the particle velocity normal to the surface must be equal on both sides of it, i.e.,

),0(),0(),0( txtxtx tri ===+= uuu . (7-15)

That boundary condition guarantees that both media always remain in contact. It can be demonstrated with the help of the continuity equation (4-11), and is henceforth called the condition of continuity of particle velocity. Putting (7-8), (7-10) and (7-12) into the boundary condition for sound pressure (7-14) gives tri ppp ˆˆˆ =+ . (7-16)

The boundary condition for particle velocity (7-15), by the same approach, yields

221111

ˆˆˆc

pc

pc

p triρρρ

=− . (7-17)

If tp is eliminated from (7-16) and (7-17), one obtains

1122

1122ˆˆ

cccc

pp

i

rρρρρ

+−

= . (7-18)

Defining the reflection coefficient to be the ratio of reflected and incident wave amplitudes,

ˆˆ

i

rpp

R = , (7-19)

then (7-18) can be expressed as


207

1

1

22

11

22

11

1122

1122

cccc

cccc

R

ρρρρ

ρρρρ

+

−=

+−

= . (7-20)

The ratio of the transmitted and the incident wave amplitudes is called the transmission coefficient,

ˆˆ

i

tpp

T = . (7-21)

From (7-16) and (7-17), that becomes

1

22

22

111122

22

cccc

cT

ρρρρ

ρ

+=

+= . (7-22)

From (7-20) and (7-22), we find that, for real specific impedances, the reflection coefficient R is always real, and can be either positive or negative-valued., and the transmission coefficient T is always real and greater than one. That last observation implies that the sound pressure at the boundary in the transmitted wave is always in phase with the incident.

From (7-20), three special cases can be identified

(i) 2211 cc ρρ < . An example of this case would be sound in air, reflecting at the free surface of a body of water. In that case, R > 0 and the sound pressure amplitudes ip and rp have the same sign, i.e., they are in-phase. That implies, for instance, that a positive sound pressure in an incident wave is also reflected as such. In the limit as ρ1c1 / ρ2c2 → 0, it follows that R → 1. The reflected sound pressure amplitude rp is, then, approximately equal to the incident sound pressure amplitude ip . Thus, the sound pressure amplitude in the standing wave field that arises in medium 1 approaches the value 2 rp at the surface. According to the boundary condition, the same sound pressure must exist on both sides of the boundary. That is also clear from (7-22). The resulting particle velocity goes asymptotically to zero on both sides of the boundary. It can also be regarded as rigid, and it was that case that we considered in section 7.1.1.

(ii) 2211 cc ρρ = . In this case, R = 0 and T = 1, i.e., when the specific impedances are equal in both media, the incident wave passes without change of amplitude or phase. No reflected wave arises.

(iii) 2211 cc ρρ > . An example of this case would be sound in water, incident upon air. In this case, R < 0 and the sound pressure amplitudes have differing signs., i.e., they are 180˚ phase-shifted. That implies that, at the boundary, an incident positive sound pressure is reflected as negative.


208

If ρ1c1 / ρ2c2 → ∞, it follows that R → -1. The sound pressure amplitude of the reflected wave has approximately the same magnitude as that of the incident wave, but is negative-valued. The resulting sound pressure in the immediate vicinity of the boundary asymptotically approaches zero as R goes to -1. Thus far, we have assumed real specific impedances for the two media. Typically, however, medium 2 either has losses, or is limited in its extent. If so, then to describe the reflection against medium 2, we can incorporate such characteristics into a complex specific impedance Z at the boundary, by analogy to (4-115)

⊥⋅

=up

upZ =

n , (7-23)

in which p is the sound pressure at the surface, u is the surface velocity vector, and n is a unit vector directed into medium 2, normal to the surface. In order to handle cases of reflection against solid media, (7-23) can be written as

=⊥

=⋅ v

pv

pZn

, (7-24)

where v is the velocity vector at the surface, and n is a unit vector directed into the medium, normal to the surface. The impedances given by (7-23) and (7-24) are generally complex, i.e., they have both real and imaginary parts in accordance with (4-115). Applying (7-23) and the boundary conditions (7-14) and (7-15), the impedance of medium 2 is

),0(),0(),0(),0(

2 txtxtxtx

ri

ri=+==+=

=uupp

Z . (7-25)

Deriving a reflection coefficient as before, we find that it is complex-valued,

ˆ

ˆ

112

112cc

pep

eRi

iri r

r

ρρδ

δ+−

===ZZ

R , (7-26)

i.e., there is a phase difference δr between the amplitudes that can take on values other than 0 and π radians, which were the only possibilities for the case of real wave impedances.

Medium 1 Medium 2

WtWi

Wr

x

Figure 7-4 Symbolic picture of acoustic reflection and transmission at a boundary.


209

When considering incident sound from medium 1, the sound energy that is transmitted into medium 2 can be regarded as absorbed by medium 2. A very central concept in acoustics that describes the absorbing ability of a medium or a boundary is the absorption factor α.. The absorption factor is defined as

i

r

i

ri

i

t

WW

WWW

WW

−=−

== 1α , (7-27)

where Wi is the incident sound pressure, Wr is the reflected sound pressure and Wt is the transmitted sound power. Because the power can be expressed as W = Ix S, where S is the area and Ix is the intensity, which can be expressed as cpI x 0

2 /~ ρ= according to (4-83), the absorption factor can be expressed as

2

2

2

,

, 1ˆˆ

11 R−=−=+=i

r

ix

rx

pp

II

α . (7-28)

7.1.3 Propagation of plane waves in a three-dimensional space

Before analyzing the oblique incidence of a wave against a boundary, we consider how a wave can be described when its direction of propagation doesn’t coincide with a coordinate axis. For sound propagation in the positive x-direction in a Cartesian coordinate system, (4-69) implies that

)(ˆ),( xktieptx ′−=′ ωp , (7-29)

where the ´ (prime) symbol is used to distinguish that coordinate system from coming systems.

Figure 7-5 Plane wave propagation in the

positive x’-direction. The wave fronts are surfaces joining points with identical phase.

y´

x´

λ

W ave fronts

x'e

y'e

To describe multi-dimensi onal propagation, an unprimed coordinate system is introduced. In that system, for simplicity, we begin by studying the propagation in the xy-plane, in order to then generalize to three dimensions. The primed system has been rotated through an angle ϕ1 about the z-axis relative to the unprimed, as shown in figure 7-6.


210

Figure 7-6 Plane wave propagation described in two coordinate systems. One has been rotated through an angle ϕ1 about the z-axis.

In a so-called orthogonal transformation, the description can be transformed from the primed to the unprimed system. The position vector r to a point on the x´-axis is indicated in the respective coordinate systems as yxx eyexexr +=′′= (7-30) i.e., yxxx eeyeexx ⋅′+⋅′=′ , (7-31)

where ),cos( jiji eeee =⋅ in the transformation theory are usually called transformation

coefficients, and are cosines of the angles between the base vectors ie and je . The expression (7-31) can also be stated in the form

1111 sincos)90cos(cos ϕϕϕϕ yxyxx +=−+=′ , (7-32)

and (7-29) transforms in the unprimed system to

)sincos( 11ˆ),,( ϕϕω kykxtieptyx −−=p . (7-33)

To further generalize the discussion, a unit vector n is introduced to designate the direction of propagation; it is expressed the respective coordinate systems as

yyxxx enenen +=′= . (7-34)

From (7-34), applying the orthogonality relations 1=⋅ xx ee and 0=⋅ yx ee , it follows that

1cos),cos( ϕ=′=⋅′= xxxxx eeeen , (7-35)

1sin),cos( ϕ=′=⋅′= yxyxy eeeen . (7-36)

The wave number vector is defined as

nkk ⋅= , (7-37)

ey

y

xλ

Wave fronts

ϕ1

x´

y´

rxeye

xe


211

with a magnitude k = ω /c, and a direction n identical to the direction of propagation; it can be expressed as yxyyxx ekekenenkk 11 sincos)( ϕϕ +=+= . (7-38)

Thus, the components of the wave number vector, i.e., its x and y-axis projections, are

1cosϕkk x = , (7-39) 1sin ϕkk y = , (7-40)

respectively, and we conclude that the most general form of the solution becomes

)(ˆ),( rktieptr ⋅−= ωp , (7-41) or in component form

)(ˆ),( ykxkti yxeptr −−= ωp . (7-42)

In three dimensions, it follows by analogous logic that

)()( ˆ=ˆ),( zkykxktirkti zyxepeptr −−−⋅−= ωωp , (7-43)

where =⋅ rk constant, (7-44)

constitute surfaces of constant phase. Entering (7-43) into the wave equation (4-43)

2

2

22

2

2

2

2

2 1t

p

cz

p

y

p

x

p

∂

∂=

∂

∂+

∂

∂+

∂

∂ (7-45)

provides the condition

222zyx kkk

ckk ++===

ω . (7-46)

That condition is an important relation that will be utilized in the discussion that follows.

7.1.4 Oblique incidence on a boundary between two fluid media

In order to analyze what happens when a plane acoustic wave with a certain angle of incid-ence θi reaches the bounding surface between two fluid media, it is necessary to supplement the types of boundary conditions used up to this point. These boundary conditions, which require continuity of pressure and particle velocity across the boundary surface, are supplemented with the condition that the incident, reflected, and transmitted waves have the same periodicity along the boundary surface, i.e., the plane x = 0 in figure 7-7.

Comment [UC1](4-43)


212

θr

θ iλi

⇒ θ r

Medium 1

pi

pr

x

y

Medium 2

λt

pt

θ tθiλi

λr

λt

x

y

θ t

λt

λi

λr

λr

Z2 =ρ2c2Z1 =ρ1c1

Figure 7-7 Oblique incidence against a boundary surface between two fluid media. From that condition, illustrated in figure 7-7, the projected wavelengths are equal, i.e.,

t

t

r

r

i

iθ

λθ

λθ

λsinsinsin

== . (7-47)

On the side from which the incident wave arrives, the incident and reflected waves traverse the same medium, λi = λr, which yields θi = θr, i.e., the angle of reflection is equal to the angle of incidence. Equation (7-47) can be rewritten with the help of the relation c = f λ. Since the frequency is the same, f = c1/λ1 = c2/λ2 , and (7-47) can therefore be expressed as

sinsin

21

ti

ccθθ

= . (7-48)

That relation is called Snell’s law, and is also known from optics. From Snell’s law as a point of departure, two special cases are considered;

(i) 21 cc > .

Examples of this case are incident waves in water reflecting off an air-water interface. Snell’s law (7-48) dictates that the transmission angle satisfies

it cc

θθ sinsin1

2= . (7-49)

The case c1 > c2 yields θt < θi, i.e., the transmitted wave is redirected in closer to the normal, as shown in figure 7-8a. With the condition c1 > c2, θt has a maximum for grazing incidence, i.e., for θi = 90˚, as


213

1

2max arcsin

cc

t =θ . (7-50)

The relationship is illustrated in figure 7-8b. a) b)

θ r

θ i

c1 c2>

pi

pr

θ t

pt

x

yMedium 1 Medium 2

θ i

c1 c2>

pi x

y

pt

Medium 1 Medium 2Möjligautbrednings-vinklar förtransmitteradvåg

1

2max arcsin

cc

t =θ

Figure 7-8 a) The transmitted wave bends is deflected closer to the normal. b) The illustration of potential angles

of propagation for the transmitted wave. (ii) 21 cc < .

An example of this case would be sound in air, incident on an air-water interface. According to Snell’s law (7-48), θt > θi, i.e., the transmitted wave is deflected away from the normal; see figure 7-9a. For an increasing angle of incidence θi, we reach a boundary case, an angle of incidence θic, where θt = 90˚ and we have a transmitted wave that grazes the boundary surface.

The angle of incidenceθic is given by )arcsin( 21 ccic =θ . (7-51)

For θi > θic, a total reflection of the incident wave occurs. Thus, a transmitted wave only exists for angles of incidence for which θi ≤ θic. The relation is evident in figure 7-9b.

Possible angles of propagation for the transmitted wave


214

a) b)

θ i

c1 c2<

pi

pr

y

θ t

pt

x

Medium 1 Medium 2

Medium 1

θ i c

c 1 c 2 < y

p t

x

Incident waves in this are totally reflected

p i

Medium 1 Medium 2

θt = 90°

Figure 7-9 a) The transmitted wave deflects away from the normal. b) Illustration of possible angles of incidence

such that a transmitted wave exists. We limit the analysis to apply to the case in which a transmitted wave exists. The following assumption can then be made for the sound pressure:

)( 11ˆ),,( ykxktiii

yxeptyx −−= ωp , (7-52)

)()( 1111 ˆˆ),,( ykxktii

ykxktirr

yxyx epReptyx −+−+ == ωωp , (7-53)

)()( 2222 ˆˆ),,( ykxktii

ykxktitt

yxyx epTeptyx −−−− == ωωp . (7-54)

Here, the pairs k1x, k1y and k2x , k2y represent components of the wave number vectors in each medium. R is the reflection coefficient, defined in (7-19),

ir ppR ˆˆ= (7-55)

and T is the transmission coefficient, as defined in (7-21),

it ppT ˆˆ= . (7-56)

It turns out that both R and T are real, since the specific impedances of the media are; thus, to avoid unnecessary complication, they are not expressed as complex. Boundary conditions analogous to those for normal incidence can be set up. Continuity of pressure at x = 0 gives

),,0(),,0(),,0( tri tyxtyxtyx ===+= ppp . (7-57)

Putting the pressure in the incident wave expression (7-52), in the reflected wave expression (7-53), and in the transmitted wave expression (7-54) yields

)()()( 211 ˆˆˆ yktit

yktir

yktii

yyy epepep −−− =+ ωωω , (7-58)

where Snell’s law, (7-48), finally yields


215

tri ppp ˆˆˆ =+ . (7-59)

The boundary condition for continuity of particle velocity normal to the boundary surface is

ttirii tyxtyxtyx θθθ cos),,0(cos),,0(cos),,0( ===+= uuu . (7-60)

For plane waves, according to (4-76) and (4-77),

c0ρpu ±= (7-61)

and, using k1y = k2y, as above, then (7-60) can be expressed as

tt

ir

ii

cp

cp

cp

θρ

θρ

θρ

cosˆ

cosˆ

cosˆ

22111 1=− . (7-62)

From (7-59) and (7-62), the reflection coefficient R takes the form

coscoscoscos

ˆˆ

1122

1122

ti

ti

i

rcccc

pp

Rθρθρθρθρ

+−

== (7-63)

and the transmission coefficient T according to

coscos

cos2ˆˆ

1122

22

ti

i

i

tcc

cpp

Tθρθρ

θρ+

== . (7-64)

In contrast to the derivation in the next section, we have taken account of wave propagation in medium 2 in this case. The surface is said to have a distributed response.

7.1.5 Oblique incidence from a fluid against a solid medium

The case of a plane wave reflecting obliquely against a solid medium is more complex than the corresponding reflection at the boundary surface between two fluids. The reason is that the solid medium, as opposed to the fluid, can support shear stresses. The general case of a stiff elastic medium, such as concrete for example, must account for the propagation of both longitudinal compress ional waves (such as those we treat in this chapter) and shear or transverse waves, in which the particle motion is normal to the direction of propagation. We now consider a special case, a so-called locally-reacting surface. To treat a reflection at a locally-reacting surface, every point on it is regarded as completely isolated from all other such points. For such a surface, the impedance becomes

⊥

= vp

Z , (7-65)

independently of the angle of incidence θi of the sound. The velocity v⊥ of any point on the surface depends exclusively on the sound pressure p acting on that point. If we excite a point on the surface, no other points on the surface move and the acoustic disturbances in the locally-reacting medium only propagate perpendicular to the surface.


216

In summary, a locally-reacting surface can be regarded as such that (7-65) applies to every point with a determined, usually frequency-dependent, value of Z2, independently of the character of the acoustic field; see figure 7-10.

θi

pi

pr

Z2y

pt

x

Medium 1 Medium 2Z1 =ρ1c1

The point impedance of the locally reacting surface, as described by (7-65), must ordinarily be assumed complex. That implies that, in deriving the reflection coefficient, the phase shifts of the reflected and transmitted waves must be accounted for, in contrast to the treatment in section 7.1.2. Continuity of pressure, as stated in (7-14), implies that

tr it

iri epepp δδ ˆˆˆ =+ , (7-66)

and continuity of particle velocity normal to the boundary surface, as expressed in (7-17) and (7-65), that

21111

ˆcos

ˆcos

ˆZ

tr

it

iir

ii ep

ec

pc

p δδ θ

ρθ

ρ=− . (7-67)

Eliminating titep δˆ , yields the reflection coefficient

coscos

ˆˆ

Re112

112cc

pep

i

i

i

iri r

r

ρθρθδ

δ+−

===ZZ

R . (7-68)

The ideal locally-reacting surface, as described above, is a model, i.e., a means to simplify the description of a complicated reality. Below, we discuss the circumstances under which real media can, to a reasonable degree of precision, be described as locally-reacting.

(i) Anisotropic medium.

An anisotropic structure, e.g., a perforated structure as in figure 7-11a, or a honeycomb structure as in figure 7-11b, can effectively block particle motion parallel to the surface, while permitting motion perpendicular to it. That type of structure is used as a sound absorbent. The physical principle is that energy from the incident acoustic field is converted to heat by the viscous forces associated with the particle motion in and out of the cavities.

Figure 7-10 Oblique incidence against alocally-reacting medium.


217

a) b)

Figure 7-11 Sound absorbent in the form of a) perforated panel, and b) honeycomb structure. (ii) Medium with significant losses.

Medium 2 has large losses, i.e., the acoustic disturbances are strongly damped out. Examples are acoustic absorbents, such as various types of mineral wools, as well as earth surfaces with markedly porous character.

(iii) Medium with significant compliance.

We return to section 7.1.4, in which oblique incidence between two fluid media is treated, and especially focus on the case in which c2 « c1. In that case, there is a considerable deflection towards the normal, as in figure 7-8a. The transmission angle θt is relatively independent of θi , and the situation approaches that which applies to a locally-reacting surface. If the specific impedance of medium 2, i.e., ρ2c2 for a surface with a distributed response, and Z2 for a locally-reacting surface, is finite, then R approaches –1, since θi approaches 90° in formulas (7-63) and (7-68). That implies that the amplitude of the reflected wave is unchanged, but phase-shifted by 180°. That situation has important consequences for, among other things, sound propagation along absorbing walls and ceilings; see example 7-1.

Example 7-1

"Bad sound" due to the destructive interference resulting from grazing incidence?

7.2 EIGENFREQUENCIES AND EIGENMODES IN A THREE-DIMENSIONAL SPACE

With the aid of wave-theoretical argumentation, this section will study how characteristic sound fields, standing waves, can be built up in such three-dimensional cavities as dwellings, lecture halls, and vehicle cabins. The method is based upon using solutions of the wave equation to set up mathematical statements of the boundary conditions that apply at the bounding surfaces of the room, such as floors, walls, and ceilings. The difficulty resides in determining these boundary conditions for irregular spaces, such as churches or rooms with bulky contents; as a result, exact analytical solutions only prove possible for


218

especially simple geometries, such as spherical, cylindrical, and parallelepipedic rooms. Despite the practical limitations, the theory is nevertheless essential to the understanding of MANY acoustic phenomena. Some pertinent questions of interest are:

(i) Why does a speaker sound one way in a certain room, and another way elsewhere?

(ii) Why is it that there are very distinct maxima and minima in the sound field of an airplane cabin?

(iii) Where, in a workspace, should a machine be located in order to minimize the resulting acoustic disturbance?

7.2.1 Parallelepipedic room

Let us consider a wave motion in a parallelepipedic room with dimensions as indicated in figure 7-12.

z y

xxlylzl

The medium is loss-free and the walls are infinitely rigid. The boundary condition at all bounding surfaces is that the particle velocity normal to the surface be zero. For plane wave propagation in a three-dimensional space, then in general, according (7-43),

)(ˆ=),( zkykxkti zyxeptr ±±±ωp , (7-69)

where the minus sign stands for propagation in the respective coordinate axis’ positive direction, and the plus sign for propagation in the corresponding negative direction. Because all possible combinations of propagation directions may occur, the sound field in the room will consist of eight waves, i.e.,

++= ++++− )(2

)(1 ˆˆ(),( zkykxkizkykxki zyxzyx epeptrp

++++ +−−−+−+− )(5

)(4

)(3 ˆˆˆ zkykxkizkykxkizkykxki zyxzyxzyx epepep

tizkykxkizkykxkizkykxki eepepep zyxzyxzyx ω)ˆˆˆ )(8

)(7

)(6

−−−−−+− +++ . (7-70)

Putting (7-70) into the wave equation (6-43), expressed in Cartesian coordinates, yields the condition that the components kx, ky and kz of the wave number must satisfy, as in (7-46),

Figure 7-12 Parallelepipedic room with the dimensions lx, ly and lz.


219

zyx kkkc

kk 222 ++===ω . (7-71)

Practice Exercise 7-1 Show, by putting (7-70) into the wave equation (4-43), that the relation given above holds. Because the walls are rigid, the particle velocity perpendicular to the surface must be zero. Thus, the boundary condition is ux= 0 at x = 0 and x = lx,

uy= 0 at y = 0 and y = ly, (7-72)

uz= 0 at z = 0 and z = lz.

According to section 7.1.1, reflection at an infinite rigid surface occurs without change in amplitude or phase, and from that it follows that

87654321 ˆˆˆˆˆˆˆˆ pppppppp ======= . (7-73)

Setting 821 ˆ ,...,ˆ ,ˆ ppp to 8p and developing the exponential terms into the form eiθ = cosθ + isinθ, yields ti

zyx ezkykxkptr ω)cos()cos()cos(ˆ),( =p . (7-74)

The particle velocity can be determined with the help of the equation of motion (4-25)

00 =∇+∂∂ p

tuρ . (7-75)

In component form, ),( trxu is obtained as

xi

dtx

trx ∂∂

−=∂∂

−= ∫ppu

ωρρ 00

1 1),( . (7-76)

Putting the pressure (7-74) into the expression for particle velocity in the x-direction (7-76),

0

ˆ),(ωρik

ptr xx =u sin( xk x ) cos( yky ) cos( zkz ) tie ω . (7-77)

The other velocity components are found by the same approach to be

0

ˆ),(ωρik

ptr yy =u sin( yky ) cos( xk x ) cos( zkz ) tie ω , (7-78)

0

ˆ),(ωρik

ptr zz =u sin( zkz ) cos( xk x ) cos( yky ) tie ω . (7-79)


220

From (7-77), the boundary condition at x = 0 is clearly satisfied. Moreover, to satisfy that at x = lx, the condition sinkxlx = 0 is required; from that, the components of the wave number vector become

xxx lnk π= , (7-80)

satisfied for any integer-valued nx, i.e., nx = 0, 1, 2, ... . In a corresponding wave, the other components are yyy lnk π= , (7-81)

zzz lnk π= , (7-82)

where ny and nz, in the same fashion, can take on the values 0, 1, 2, ... . Putting (7-80), (7-81) and (7-82) into (7-71), and using the relation 2π f = ω = kc, the eigenfrequencies or eigenvalues of the parallelepipedic room are found to be

222

2),,( ⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

z

z

y

y

x

xzyx l

nln

lncnnnf . (7-83)

Thus, the room has an eigenfrequency and a corresponding eigenmode for every combination of the indices nx, ny and nz. The composite sound pressure from all eigenmodes is obtained as

tiziyixi

ii izkykxkptzyx ωe )cos()cos()cos(ˆ),,,( ∑=p , (7-84)

where i stands for the summation over all excited eigenmodes with their respective unique combinations of indices nx, ny and nz, and where ip is the amplitude of the individual mode form. Example 7-2 A so-called reverberant room is a type of acoustic measurement room in which a minimal absorption by walls, floors, and ceilings is sought, as well as a uniform distribution of eigenfrequencies along the frequency axis. Table 7-1 lists all of the eigenfrequencies under 100 Hz calculated for such a room at the Marcus Wallenberg Laboratory for Sound and Vibration Research, MWL, KTH. The room has the dimensions lx = 7.86 m, ly = 6.21 m, lz = 5.05 m. For the sound speed c, a value of 340 m/s has been assumed.


221

Table 7-1 The calculated eigenfrequencies under 100 Hz of a reverberant room with the dimensions 7.86 m × 6.21 m × 5.05 m at MWL, KTH.

nx ny nz f [Hz] nx ny nz f [Hz] nx ny nz f [Hz] nx ny nz f [Hz]

1 0 0 21.6 1 2 0 58.9 1 1 2 75.8 1 2 2 89.4 0 1 0 27.4 2 1 1 61.3 2 2 1 77.5 4 1 0 90.7 0 0 1 33.7 0 2 1 64.3 3 1 1 78.1 3 2 1 91.3 1 1 0 34.9 3 0 0 64.9 2 0 2 80.0 1 3 1 91.4 1 0 1 40.0 0 0 2 67.3 0 3 0 82.1 2 3 0 92.8 2 0 0 43.3 1 2 1 67.8 2 1 2 84.6 4 0 1 92.8 0 1 1 43.4 2 2 0 69.8 3 2 0 84.9 3 0 2 93.5 1 1 1 48.5 3 1 0 70.4 1 3 0 84.9 4 1 1 96.8 2 1 0 51.2 1 0 2 70.7 4 0 0 86.5 2 2 2 97.0 0 2 0 54.7 0 1 2 72.7 0 2 2 86.8 3 1 2 97.4 2 0 1 54.8 3 0 1 73.1 0 3 1 88.7 2 3 1 98.7

Evidently, the lowest eigenfrequency 21.6 Hz is obtained for nx = 1, ny = nz = 0. That implies a standing wave between those walls which stand furthest apart, i.e., lx = 7.86 m. A quick calculation shows that that distance corresponds to a half wavelength at 21.6 Hz. At low frequencies, the eigenfrequencies are relatively sparse, but they become all the more densely spaced as the frequency rises. The eigenfrequencies (0,0,1) at 33.7 Hz and (1,1,0) at 34.9 Hz are only spaced 1.2 Hz apart. Above that, there is a jump of 5.1 Hz to the next eigenfrequency (1,0,1) at 40.0 Hz. Eigenmodes can be divided into three categories:

(i) Axial eigenmodes.

Two of the indices nx, ny and nz are zero. From equation (7-84), it is apparent that the sound pressure only varies in one coordinate direction, and is constant in the other two. The particle velocity, moreover, is parallel to a coordinate axis. Consequently, there are absorption losses only at the walls at which the particle velocity is normal to the surface. Figure 7-13a provides a symbolic depiction of the spatial dependence of the normalized sound pressure, in the form of a graph of the eigenmode (2,0,0), which has the form

xlp

xp

x

π2cosˆ

)(= . (7-85)

In figure 7-13b, the graph of the normalized particle velocity is presented; the expression is

xlu

xu

xx

x π2sinˆ

)(= . (7-86)

Note that where the sound pressure has an antinode (relative maximum), the particle velocity has a node, and vice versa.


222

a)

1

0

-1 x = 0 x = l x

maximum

node

p p ( ) x

b)

0

1

-1x = 0 x = lx

maximum

node

uu( )x

^

Figure 7-13 a) Symbolic picture of the spatical dependences of a) the normalized sound pressure and b) the

normalized particle velocity, for the (2,0,0) mode. Compare these to figure 7-2. (ii) Tangential eigenmodes.

One of the indices nx, ny and nz is zero. The particle velocity is parallel to a pair of the walls and has components in two of the coordinate axes directions. Figure 7-14a illustrates the plane wave fronts and the propagation directions for the mode form (1,1,0). Figure 7-14b shows a symbolic picture of the sound pressure contours for the same eigenmode.

y

x Figure 7-14a Illustration of the plane wave fronts and directions of propagation for the tangential mode form

(1,1,0). (Source: Brüel & Kjær, Architectural Acoustics.)


223

y

x

00 0

0

00,2

0,2

0,2

0,2

0,4

0,4 0,4

0,40,6 0,6

0,60,60,8 0,8

0,8 0,8 1,0

1,01,0

1,0

p/pmax

Figure 7-14b Symbolic picture of the normalized sound pressure contours for the tangential mode form (1,1,0).

(Source: Brüel & Kjær, Architectural Acoustics.) (iii) Oblique eigenmodes.

All of the indices nx, ny and nz are non-zero. The particle velocity has components in all three coordinate axes directions, resulting in reflections from all six walls. That circumstance is decisive for how acoustic absorbents shall be mounted in a room to prevent the development of strong standing wave field, so-called resonances. Figure 7-15 shows a symbolic picture of the sound pressure contours of the eigenmode (1,2,1).

The situation we have studied above, i.e., the determination of mode shapes of sound pressure and particle velocity is the so-called homogeneous problem of free oscillations, in which we have only taken account of initial and boundary conditions. The boundary condition was that the walls be loss-free and rigid, and the initial condition just that the oscillation had already started at t = 0, after which no further excitation or damping took place; i.e., the medium was source-free and lossless.

In reality, it is normally the so-called particular problem with forced oscillations that is of

z

y

x

Figure 7-15 Symbolic picture of the normalized sound pressure contours for the oblique eigenmode (1,2,1). (Picture: Brüel & Kjær, Technical Reveiw)


224

interest, i.e., the room is continually supplied sound power from a source. With boundary conditions as given above, the sound pressures and particle velocities become infinite, because the walls (and the medium) are assumed loss-free. That such does not, in fact, occur, is because even apparently rigid walls absorb some fraction, or some percentage, of the incident sound field. These losses are, however, balanced in stationary sound fields by the energy supplied by the sound source. Bearing that in mind, we can only use the derived relations for sound pressure and particle velocity to draw qualitative conclusions. When a sound source excites a room at certain frequencies, it is only those frequencies that exist in the room. Nevertheless, the sound field is largely built up of the eigenfrequencies and mode shapes that coincide with, or fall in the vicinity of, the frequencies sent out by the sound source. How strongly a particular mode is excited also depends on where in the room the sound source is placed, in relation to the nodes and antinodes of the eigenmode. Provided that the excitation is from a constant velocity source, i.e., a source that has a certain, fixed vibration velocity which is independent of where it is located in the room, the greatest excitation of an eigenmode occurs if the source is located at a particle velocity node. As is evident from figure 7-13, a particle velocity node corresponds to a sound pressure anti-node. Sketching the sound pressure contours of different eigenmodes in accordance with figures 7-13a and 7-14b, it turns out that every eigenmode has a sound pressure antinode in the corner of the room. Moreover, it turns out that only one eighth of all modes have a non-zero sound pressure in the geometric midpoint of the room. A sound source placed in a corner is, therefore, optimally-placed if it is desired to maximally excite the eigenfrequencies of a room. Similarly, if a microphone is placed in a corner, it registers the maximum sound pressure for every excited eigenmode.

_________________________________________________________________________

Example 7-3 In order to investigate how the eigenmodes of a room “color” the sound at a certain listener position, given a loudspeaker at another given location, one can proceed as follows. In a reverberant room with the dimensions 7.86 m × 6.21 m × 5.05 m, a loudspeaker has been placed in one of the corners. A microphone is placed in the diagonally opposite corner. The loudspeaker sends out a noise of relatively constant amplitude in a frequency band of interest, while the sound pressure level is measured by the microphone. The result is shown in figure 7-16.


225

(1,0,0) (0,1,0)(0,0,1)

(1,1,0)

(1,0,1)

(2,0,0) (0,1,1)

(1,1,1)

Lp [dB]

Frekvens [Hz]

100

90

80

70

60

50

40403020 50453525

Figure 7-16 Sound pressure levels measured in a corner of a reverberant room with the dimensions 7.86 m × 6.21 m × 5.05 m. The sound source, a loudspeaker, was placed in a diagonally opposite corner (Compare to the calculated eigenfrequencies in table 7-1, example 7-2).

The indices of the eigenmodes corresponding to each peak in the curve are indicated above them. _________________________________________________________________________ It is often of great interest to know how many eigenfrequencies fall within a certain frequency band, such as a third-octave band for instance. Starting with expression (7-83), we can see that the frequency spacing of purely axial modes in the x, y, or z-directions is

x

zyx lcnnnf

2)0 ,1( ==== , (7-87)

y

zxy lcnnnf

2)0 ,1( ==== , (7-88)

z

yxz lcnnnf

2)0 ,1( ==== . (7-89)

These axial eigenfrequencies may be marked out in a frequency space with the coordinate axes fx, fy and fz as in figure 7-17. From the Pythagorean theorem and relations (7-87), (7-88) and (7-89), it turns out that every eigenfrequency f (nx, ny, nz) is represented in that frequency space and that the distance from the point f (nx, ny, nz), corresponding to the (nx, ny, nz)- mode, to the origin is a measure of the eigenfrequency for that specific mode shape. All eigenmodes at or below a certain frequency f are included in the eighth-sphere in the frequency space lying between the positive fx-, fy-, fz -axes and the spherical surface with radius f.


226

f

fx

fy

fz

c/2ly

c/2lx

c/2lz

The total number of eigenmodes falling below a certain frequency f can be determined by counting the discrete points in the frequency space developed above. That being a time-consuming task, it is more practical to try to develop a formula that provides the same information. To each point representing an eigenmode, according to (7-87), (7-88) and (7-89), there is a corresponding cube with a volume

zyx l

clc

lcV

222=

in the frequency space. The number of eigenmodes N with an eigenfrequency less than f is obtained by dividing the volume of an eighth-sphere (of radius f ) with the volume that a single mode represents, from which

zyx

zyx

lllc

f

lc

lc

lc

f

N3

33

3

4

222

34

81

ππ

=≈ . (7-90)

Formula (7-90) is, however, incomplete. Certain eigenmodes are underrepresented. That applies to the tangential modes with a zero-value index, and which therefore lie on one of the three planes that contain a pair of coordinate axes. It also applies to the axial eigenmodes with two of the indices equal to zero, and which therefore lie along one of the frequency axes. Correcting for those, the total number of eigenmodes with an eigenfrequency lower than f is

Lcf

Sc

fV

c

fN

843

42

2

3

3++≈

ππ , (7-91)

Figure 7-17 Schematic picture of discrete eigen-frequencies f(nx, ny, nz) in a frequency space.


227

where zyx lllV = is the volume of the room,

)(2 zyzxyx llllllS ++= is the bounding surface area of the room, and

)(4 zyx lllL ++= is the perimeter of the room. In rather large rooms at high frequencies, the first term in (7-91) dominates. It can also be shown that as f → ∞, (7-91) also applies to rooms of forms other than parallelepipedic. Example 7-4 Determine, with the aid of (7-91), the number of modes under 100 Hz for the reverberant room with the dimensions 7.86 m × 6.21 m × 5.05 m, which was treated in example 7-2. Compare the calculated number with the actual number that can be counted from table 7-1.

Solution Setting V = 246.49 m3, S = 239.73 m2, L = 76.48 m and c = 340 m/s, then N = 26.27 + 16.29 + 2.81 = 45.37, which is in good agreement with the actual counted result of 44, from table 7-1.

Differentiating equation (7-91) with respect to frequency,

Lc

Sc

fV

c

fdfdN

81

2

423

2++≈

ππ ,

from which it is evident that the number of eigenmodes NΔ in a frequency band B = fu – fl, centered around f, is

BLc

Sc

fV

c

fN

⎟⎟⎠

⎞⎜⎜⎝

⎛++≈Δ

81

2

423

2 ππ . (7-92)

It should be noted that relations (7-91) and (7-92) are only approximate, and that the precision increases with the frequency and the bandwidth. _________________________________________________________________________ Example 7-5 Determine, for the reverberant room of example 7-2, the number of eigenmodes in the 80 Hz third-octave band. It spans the range of frequencies from Hz 3.71280 6 = up to

Hz 8.89280 6 =⋅ . Solution Putting the values from example 7-4, as well as Hz 5.18=B , into equation (7-92), yields

NΔ =14.67. That is in good agreement with the 14 that can be counted from table 7-1. The number of modes in the third-octave band around 1000 Hz is, for the reverberant room of example 7-4, over 18000, which implies more than 78 eigenfrequencies per Hz. Thus, at medium to high frequencies, the eigenfrequencies are so densely spaced that it becomes completely impractical to consider individual eigenfrequencies and modes. Instead, it is fruitful to switch to energy methods, or so-called statistical room acoustics. See chapter 7.


228


Normal incidence against a boundary between two elastic media

Boundary conditions at the surface

Continuity of pressure ),0(),0(),0( txtxtx tri ===+= ppp (7-14)

Continuity of particle velocity

),0(),0(),0( txtxtx tri ===+= uuu (7-15)

Reflection coefficient i

rpp

Rˆˆ

= (7-19)

22

11

22

11

1122

1122

1

1

cccc

cccc

R

ρρρρ

ρρρρ

+

−=

+−

= (7-20)

Transmission coefficient i

tpp

Tˆˆ

= (7-21)

22

111122

22

1

22

cccc

cT

ρρρρ

ρ

+=

+= (7-22)

Complex impedances

In general, ⊥⋅

=up

upZ =

n (7-23)

Solid media ⊥

=⋅ v

pv

pZn

= (7-24)

Reflection coefficient 112

112ˆ

ˆcc

pep

eRi

iri

rr

ρρδ

δ+−

===ZZ

R (7-26)

Absorption factor i

r

i

ri

i

tWW

WWW

WW

−=−

== 1α (7-27)


229

22

,

, 1ˆˆ

11 R−=⎟⎟⎠

⎞⎜⎜⎝

⎛−=+=

i

r

ix

rx

pp

II

α (7-28)

Propagation of plane waves in a three dimensional space

Sound pressure )()( ˆ=ˆ),( zkykxktirkti zyxepeptr −−−⋅−= ωωp (7-43)

Wave number vector zyx kkkc

kk 222 ++===ω (7-46)

Oblique incidence against a boundary surface between two fluid media

Snell’s law ti

ccθθ sinsin21 = (7-48)

Reflection coefficient ti

ti

i

rcccc

pp

Rθρθρθρθρ

coscoscoscos

ˆˆ

1122

1122+−

== (7-63)

Transmission coefficient

ti

i

i

tcc

cpp

Tθρθρ

θρcoscos

cos2ˆˆ

1122

22+

== (7-64)

Oblique incidence from a fluid against a solid medium

Reflection coefficient

112

112coscos

ˆˆ

Recc

pep

i

i

i

iri

rr

ρθρθδ

δ+−

===ZZ

R (7-68)

Eigenfrequencies and eigenmodes in a three-dimensional space

Eigenfrequencies 222

2),,( ⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

z

z

y

y

x

xzyx l

nln

lncnnnf (7-83)

Eigenmodes, sound pressure

∑=i

iptzyx ˆ),,,(p cos( xk xi ) cos( ykyi ) cos( zkzi ) tie ω (7-84)


230

Components of the wave number vector

x

xx l

nk

π= (7-80)

y

yy l

nk

π= (7-81)

z

zz l

nk

π= (7-82)

Number of eigenfrequencies under f [Hz]

Lcf

Sc

fV

c

fN

843

42

2

3

3++≈

ππ (7-91)

Number of eigenfrequencies in frequency band B

BLc

Sc

fVc

fN

⎟⎟⎠

⎞⎜⎜⎝

⎛++≈Δ

81

2

423

2 ππ (7-92)

231

CHAPTER EIGHT

WAVE EQUATIONS AND THEIR SOLUTIONS IN SOLID MEDIA Vibrations in technical systems are, for various reasons, attracting ever more attention. The direct consequences of vibrations, such as durability issues and comfort problems, are well known. There are also indirect consequences of vibrations, such as their impact on function, manufactured product quality, and sound generation. The study of sound is closely connected to the study of vibrations. There are several reasons for that. First of all, vibrations caused by fluctuating forces acting on solid materials are very important sources of acoustic energy. Secondly, vibrations in solid structures are very important carriers of acoustic energy. Thirdly, vibrations of surfaces that bound air volumes generate and radiate sound into the air. Finally, the most effective approaches to mitigating noise are usually those nearest to the source of the noise. A strong foundation of knowledge of vibrations in solid media and structures is, for those reasons, extraordinarily important to both practical and theoretical noise control. This section imparts fundamental knowledge of oscillations and vibrations in solid media. The emphasis of the material is on those parts which are essential to the acoustician. The depiction begins with wave propagation in infinite media. After that, the effects of limits on the spatial extent of the structure are considered. In that context, the important topic of standing waves is also dealt with. Finally, a treatment of losses in solid materials, and their mathematical representation, is provided.

Chapter 8: Wave equations and their solutions in solid media

232

6.1 INTRODUCTION

Vibrations in structures are of great significance in many important technical problems. Comfort and performance of vehicles and vessels are considerably reduced by vibrations. Most noise problems one encounters in technical systems contain a link in which the acoustic disturbances are transmitted in the form of structural vibration; see figure 8-1. Many destructive accidents in technical systems are, directly or indirectly, caused by fatigue failure of parts subjected to vibrations. The functioning of most machines is negatively impacted by vibrations. If, for example, the tool in a machining operation vibrates, the cutting precision is degraded. To o find a rational and effective solution to these problems, it is necessary to be able to analyze the vibrations of a structure. That demands, in turn, fundamental knowledge of wave propagation in solid media.

Ω in

Ω ut

Figure 8-1 Acoustic energy is often transferred in the form of vibrations in solid media. Often, the source

mechanism is that of time dependent contact forces between moving elements of the structure. In the case illustrated, fluctuating forces are generated in the contact zone of the meshing gear teeth. The resulting vibrations spread through the axles and bearings to the rest of the machine. Any attached large, easily excited plates of sheet metal radiate sound strongly.

Because of the great technical significance of vibrations, a number of special disciplines for the analysis of vibrations have arisen in the recent past. In experimental modal analysis, experimentally-determined characteristics of a structure are used to construct a mathematical model of the structure’s vibration behavior. Today, many technical products undergo vibration testing. The product is subjected to a predefined dynamic loading program, using a shaker. Afterwards, possible changes in the properties of the structure are analyzed. By measuring the vibrations at certain, well-selected points, one seeks by machinery monitoring to determine the condition of the measured machine. The idea is that a change in the machine’s condition reveals itself by a change in the machine’s vibrations. If, for example, bearing faults were to develop, certain characteristic frequency


233

components might appear in the vibration spectrum. That information can be used to determine whether or not the machine should be shut down for inspection or service.

8.2 WAVE PROPAGATION IN INFINITE AND SEMI-INFINITE MEDIA

The difference between a fluid and a solid medium is essentially the ability of the latter to support shear stresses. If the medium is regarded as built up of layers, one upon another, there is, in a solid medium, a resistance to sliding of the layers relative to each other. As a result, a solid medium exhibits more types of wave motion than a liquid medium does. In an unbounded solid medium, two types of waves can propagate independently of one another: longitudinal waves and transverse waves. The longitudinal wave is identical to that which exists in gases and liquids; see chapter 4. In longitudinal, or compressive waves, the particle motions are parallel to the direction the wave propagates; see figure 8-2. In transverse, or shear waves, on the other hand, the particle motions are confined to a plane perpendicular to the wave propagation; see figure 8-3. The total displacement of a point in the medium is the sum of both longitudinal and transverse wave contributions.

b)a) Figure 8-2 a) Deformation of a volume element, and b) Particle motions, in a longitudinal wave. The longitudinal

wave is characterized by pure compression of a volume element.

b)a)

Figure 8-3 a) Deformation of a volume element, and b) particle motion, in a transverse wave. The transverse, or

shear, wave is characterized by the shearing of a volume element, without change of volume. If the medium has boundaries, additional wave types can arise. In semi-infinite media, there are, for instance, so-called Rayleigh waves. The Rayleigh wave is an example of a surface wave. Surface waves are characterized by an amplitude that decays with distance from the surface; see figure 8-4.


234

Figure 8-4 Particle motions in a transverse surface wave. The wave propagates to the right in the figure. The seismic waves of greatest import in an earthquake, for example, are of this type. Another example is the velocity field in the air adjacent to a plate vibrating harmonically in bending, at a frequency below the so-called coincidence frequency; see chapter 8.

All waves in solid media consist of a longitudinal and a harmonic part. Even though the particle displacement in a bending wave is largely normal (transverse) to the direction of wave propagation, there is also a longitudinal component. Similarly, a quasi-longitudinal wave in a bar has a transverse component. In practice, all elements of structures are limited in their extents. The waves in them will therefore experience reflections and transmissions at boundaries, and will interfere with other waves. The remainder of this chapter therefore concentrates on structures that can be used to describe and understand wave propagation in different types of vehicles and machines.

8.3 QUASI-LONGITUDINALWAVES IN BARS

Beams are elements of many structures. At frequencies that are not too high, say below 5-10 kHz, vibrations are transmitted in a beam primarily by longitudinal waves, torsional waves, and bending waves. When the beam is subjected to an axial disturbance that is not too high in frequency, a quasi-longitudinal wave is generated, and propagates along the beam axis. The beam is normally called a bar when discussing that wave type1. In a bar subjected to an external harmonic force directed along the bar axis, a wave is generated in synchrony to the force variations, with alternating tensile and compressive regions; the wave propagates along the axis, away from the point of force application. As a tensile region passes through a volume element, the element is extended in the axial direction, and contracted in the transverse direction. If, on the other hand, a compressive region passes, the element is contracted in the axial direction, and extended in the transverse. Such a wave is called a quasi-longitudinal wave; see figure 8-5. A quasi-longitudinal wave is not a pure longitudinal wave. It consists of both longitudinal and transverse components. If the wavelength λL of the oscillations is much greater than the bar thickness d, as in figure 8-5, the longitudinal component of the deformation is considerably larger than the transverse. For a circular cylindrical steel bar with a 5 cm radius, the ratio of the transverse to the longitudinal deformation is 0.01 at an

1 Although they may be geometrically identical, or all 3 wave types may even exist in the same structural element, the convention is to refer to a thin, extended 1D medium as a “beam” in the context of bending waves, a “bar” in the context of quasi-longitudinal waves, and a “shaft” in the context of torsional waves.

Propagation Direction


235

excitation frequency of 1 kHz. In many technical applications, quasi-longitudinal waves are therefore treated as purely longitudinal waves.

d

c L

c L

λ LU n d e f o r m e d b e a m

E x p a n s io n C o m p r e s s io n Figure 8-5 Deformation in a quasi-longitudinal wave. In practice, the deformation in the radial direction is

considerably less than that in the axial direction. Compared to bending waves, the sound radiation from quasi-longitudinal waves is therefore negligible.

Because the displacement perpendicular to the surface of the bar is relatively small, the direct sound radiation from a quasi-longitudinal wave is also relatively small. Despite that, a quasi-longitudinal wave can, indirectly, be very significant for the acoustic properties of a structure. Specifically, when the bar is part of a large composite structure, the quasi-longitudinal wave can transfer acoustic energy to an effective radiator, such as a plate vibrating in bending, for instance.

8.3.1 Wave equation for quasi-longitudinal waves in a bar

We now construct the wave equation for quasi-longitudinal waves in a bar. To begin with, we study the relation between the bar’s deformation, and the internal forces and stresses acting in the bar. Consider a straight prismatic bar. Assume that a longitudinal wave propagates in the positive x-direction. Let ξ ( x,t) indicate the displacement of a particle at position x and time t. Assume that the length of a bar element at rest is Δx. Referring to figure 8-6, its length Δξ when subjected to a normal stress σ ( x,t) is given by

xx

txxx

xx

txtxtxxtx Δ∂∂

≈−⎟⎟⎠

⎞⎜⎜⎝

⎛+Δ

∂

∂+Δ

∂∂

+=−Δ+=Δξ

ξξξ

ξξξξ ),(...)(),(),(),(),( 22

2.

The derivative of the displacement ξ with respect to position x is called the strain ε, i.e.,

x

tx∂∂

=ξ

ε ),( . (8-1)

The deformation, i.e., the strain of the element, is induced by the stress σ acting on the element. The stress can, in turn, be related to the external forces acting on the bar. Thus, we need a relation between the stress and the strain. That so-called constitutive relation is provided by the well-known Hooke’s law: “Stress is directly proportional to strain”,


236

x

EE∂∂

=ξ

ε=σ , (8-2)

where σ and ε are measured on a bar (test piece) subjected to axial load. The so-called elasticity or E-modulus, also called Young’s modulus, is defined by that relation, based on quasi-longitudinal deformation in a bar. The definition also includes a sign convention for σ. The stress is defined positive for positive strains, i.e., when the element extends, and negative when the element contracts2. Thus, stresses are positive when directed out from the element, i.e., when the element is subject to pulling forces; see figure 8-6.

( )ξ x t,

xx

( )ξ x x t+ Δ ,

xtxtxx Δ+−Δ+ ),(),( ξξxx Δ+Δx

),( txx Δ+σ0=σ ),( txσ

Figure 8-6 Displacements and stresses in longitudinal deformation. By setting up the equation of motion for a small element Δx of the bar, we obtain yet another relation between the displacement ξ and the stress σ. Figure 8-7 shows an element cut out from the bar, with a length Δx and a cross sectional area S. The equation of motion for the element, in the x-direction, is given by (8-3):

( )ξ x x t+ Δ 2 ,

( )ξ x x t+ Δ ,

( )Stxx ,Δ+σ( )Stx ,σ

( )ξ x t,

x

Δx/2

x

xStxq Δ),( xStxq Δ),(

xx Δ+

xΔ

Figure 8-7 Bar, and bar element, with forces and stresses indicated.

2 Note that the opposite applies to fluids. In a fluid, the equivalent of stress, i.e., the pressure p, is positive when the volume element is compressed; see chapter 4 for more discussion.


237

xStxStxxStxt

txxxS Δ+Δ++−=

∂

Δ+∂Δ ),(),(),(

),2(2

2qσσ

ξρ . (8-3)

where the displacement of the center of mass is ξ(x + Δx/2,t) and where external forces per unit volume are indicated by q. Expanded in a Taylor series for small Δx,

=⎟⎠

⎞⎜⎝

⎛+

Δ∂∂

+∂

∂Δ

2),(

2

2 xx

txt

xSξ

ξρ

xStxSxx

txStx Δ+⎟⎠⎞

⎜⎝⎛ +Δ

∂∂

++−= ),(),(),( qσσσ . (8-4)

When Δx approaches zero, terms that contain Δx to powers higher than 1 can be neglected. Dividing by SΔx provides the equation of motion

2

2

tx ∂

∂=+

∂∂ ξσ ρq . (8-5)

The desired wave equation for quasi-longitudinal waves is, finally, obtained by eliminating, for example, the stress from equations (8-2) and (8-5),

012

2

22

2=+

∂

∂−

∂

∂Etcx L

qξξ , ρEcL =2 , (8-6)

where cL is the phase velocity of quasi-longitudinal waves in a bar. Note that the wave equation (8-6), with q = 0, has exactly the same form as the wave equation for fluid media, as derived in chapter 4. Example 8-1 Using formula (8-6) and appropriate material data, the wave speed for quasi-longitudinal waves can be determined for some common structural materials. Table 8-1 Material data and quasi-longitudinal wave speed for some common materials.

Material Density ρ [kg/m3]

E-modulus E [N/m2]

Poisson’s ratio3 υ

Quasi-longitudinal wave

speed cL [m/s] Steel 7800 11102 ⋅ 0.3 5100

Aluminum 2800 10103.7 ⋅ 0.3 5100 Concrete 2300 10106.2 ⋅ 0.2 3400 Rubber 1200 66 1040108.0 ⋅−⋅ 0.5 26 - 180

3 Poisson’s number υ and E are the constants that define a homogeneous isotropic linear elastic material.


238

8.3.2 Quasi-longitudinal waves in infinite bars

The solutions to the wave equation (8-6), in the absence of external forces, represent the range of possible quasi-longitudinal motions of an infinite bar. Two solutions of particular interest are d’Alembert’s solution (see chapter 4),

)()(),( tcxgtcxftx LL ++−=ξ , (8-7)

and the plane harmonic wave solution

)()( ˆˆ),( xktixkti LL eetx +−

−+ += ωω ξξξ , (8-8)

where kL is the quasi-longitudinal wave number. The first terms in these solutions describe waves that propagate in the positive x-direction. The plane harmonic wave solution is particularly useful. It can be used, with Fourier methodology as discussed in section 3.4, to build up more complicated solutions. A consequence of the d’Alembert solution is that a wave retains its form as it moves along the bar. Thus, losses of energy from the wave do not occur in systems for which the d’Alembert solution applies; energy losses would imply a change in the amplitude of the wave along its path. Moreover, the retention of form also implies that the d’Alembert solution only holds for systems for which the phase velocity does not depend on frequency. Systems with that property are called non-dispersive; see section 8.5.4. Example 8-2 A quasi-longitudinal wave propagates in a bar. The bar is made of steel, and has a cross-sectional area of 0.5·10-2 m2. The velocity field vx along the bar axis is given by

))(exp(ˆ kxtiv −ω with an amplitude of v = 0.1·10-3 m/s. Calculate the bar’s stress field.

Solution The relation (8-2) between displacement and stress is useful in this context. The displacement is calculated first by performing a time integration of the velocity. The time integration is carried out by dividing by iω. Putting that into (8-2) and differentiating with respect to x then leads to

)()( ˆˆ xkti

L

xktiL

x LL evcEevik

iE

xiE −− −=−=

∂∂ ωω

ωωv

=σ .

After putting in numerical values (cL for steel is taken from example 8-1), the magnitude of the stress amplitude is then calculated to be ≈σ 3.4·103 [N/m2]. That can be compared to the yield stress for steel, 200-300 MN/m2.

8.3.3 Quasi-longitudinal waves in finite bars

Assume that we have a source that generates a plane harmonic wave that propagates in the positive x-direction. If the bar is infinite, the wave propagates ever farther away from the source. If the bar is, on the other hand, finite, the wave will sooner or later reflect against the end. The mechanical properties at the end of the bar determine how the wave is reflected. Mathematically, these properties are formulated as boundary conditions. For


239

example, an unloaded end is described by proscribing that the normal stress be zero at the end. A fixed (rigidly blocked) end is described by proscribing that the displacement be zero there.

The reflection of a wave at a bar endpoint affects the wave in two ways:

(i) The reflected wave amplitude changes with respect to that of the incident wave;

(ii) The reflected wave can undergo a phase shift relative to the incident wave.

That, by analogy to chapter 5, can be expressed mathematically by a reflection coefficient

i

ri reRξ

δˆ

ξ==R . (8-9)

The reflection coefficient is, as such, a practical reformulation of the boundary conditions when the field consists of incident and reflected plane harmonic waves. Example 8-3 Formulate a) the boundary condition, and b) the reflection coefficient for figure 8-8a.

xκ

E,ρ,S

( )xktii Le −ωξ

( )xktii Le +ωξ Rˆa) b)

x

F

( )S x tσ −Δ ,

xΔ

Figure 8-8 a) Bar end coupled to an ideal spring. b) Cross-sectional loads at the bar end. Solution a) The state at the end of the bar is determined by the properties of the spring. Cut out a bar element of width Δ x at the end, as in figure 8-8b. Assume an ideal massless spring with spring rate κ. In that case, the spring load is

),0( tξκ−=F . (8-10)

That force acts on the end of the bar as well. By setting up the equation of motion in the x-direction for the small bar element, the force F can be related to the bar’s normal stress

Stxt

txxS ),(

),2(2

2Δ−−=

∂

Δ−∂Δ σ

ξFρ . (8-11)

If we then allow Δ x to approach zero, we have

),0(),0( tS

St ξσκ

−== F . (8-12)

b) Assume that a plane harmonic wave is incident upon, and reflects from, the end of the bar. The total displacement field is then


240

)()( ˆˆ),( xktir

xktii LL eetx +− += ωωξ ξξ . (8-13)

Formula (8-2) can be used to determine the corresponding total stress field

)ˆ)(ˆ)((),( )()( xktirL

xktiiL LL eikeikE

xEtx +− +−=

∂∂

= ωωξ ξξ

σ . (8-14)

The boundary condition (8-12), above, gives

)ˆˆ()ˆ)(ˆ)(( rirLiL SikikE ξξ +−=+− ξκξ , (8-15)

i.e.,

ESi

ESi

SEik

SEik

L

L

i

r

ρωκ

ρωκ

κ

κ

ξ +

−

=+

−==

1

1

1

1

ˆξ

R . (8-16)

Table 8-2 gives some more examples of different boundary conditions for quasi-longitudinal waves in bars.

Table 8-2 Boundary conditions and reflection coefficients for quasi-longitudinal waves in a bar with some idealized fastening conditions.

Description Boundary Condition Reflection coefficient

Free x ( ) 0,0 =tσ R = 1

Rigidly Fixed

x

( ) 0,0 =tξ R = -1

Spring κ

x

( ) ( )t

St ,0,0 ξσ

κ−= R

ESi

ESi

ρωκ

ρωκ

+

−=

1

1

Damper dvx

( ) ( )

tt

Sd

t v∂

∂−=

,0,0

ξσ R

ESd

ESd

v

v

ρ

ρ

+

−=

1

1

Mass x

m ( ) ( )

2

2 ,0,0t

tSmt

∂

∂−=

ξσ R

ESmi

ESmi

ρω

ρω

+

−=

1

1

If the bar’s length is finite, the wave components are repeatedly reflected against the ends. The motion can then be calculated by specifying a displacement field in the bar in accordance with (8-8), and thereafter determining the amplitudes −ξ and +ξ by putting the solution form into the boundary conditions. Examples that illustrate that method are given in sections 8.3.5.2 and 8.3.5.3.


241

8.3.4 Reflection and transmission of quasi-longitudinal waves at area changes

Reflections of waves at changes along the transmission path can be used to reduce the transfer of mechanical energy in a structure. An example is a quasi-longitudinal wave that is incident upon a joint connecting two bars of differing material and/or cross section; see figure 8-9a.

( )ξ ωi

i t k xeT − 2

E S2 2 2, ,ρ

( )ξ ωi

i t k xe − 1

( )ξ ωi

i t k xeR + 1

x

E S1 1 1, ,ρa)

11 ),( StxΔ−σ

xΔ2

22 ),( StxΔσ

x

b) Figure 8-9 a) Reflection and transmission of a quasi-longitudinal wave at the joint between two semi-infinite

bars. b) Cross sectional areas at the joint. We can analyze the effect of the joint on the wave propagation by considering an incident plane wave, as shown in figure 8-9a. That gives rise to a reflected and a transmitted plane wave. The total displacement field in the coupled beams is therefore given by

⎪⎩

⎪⎨⎧

≥<+= −

+−

0 , ˆ0 ,ˆˆ

),( )(

)()(

2

11

xexeetx xkti

i

xktii

xktii

ω

ωω

ξξξ

TRξ (8-17)

where k1 and k2 are the quasi-longitudinal wave numbers in bars 1 and 2, respectively. It is of great interest to sound propagation to know what fraction of the incident wave is transmitted past the joint. By setting up boundary conditions for the joint, the reflection and transmission coefficients R and T are calculated. The transmission coefficient is defined in the same way as the reflection coefficient,

ti

i

t Te δ

ξ== ˆ

ξT , (8-18)

where δt is the phase shift of the transmitted wave. The first boundary condition requires that the beams remain fastened as the joint, i.e., that the total displacement must be the same on each side of the joint (x = 0), iii ξξξ ˆˆˆ TR =+ . (8-19)

Referring to figure 8-9b, the second condition is a force condition that can be derived by cutting the system on each side of the joint, and specifying dynamic equilibrium of the separated section,

22112

2

2211 ),(),(),0()( StxStxt

txSxS Δ+Δ−−=∂

∂Δ+Δ σσ

ξρρ . (8-20)


242

Allowing the cut to approach the joint, i.e., Δ x → 0, the mass of the element must also approach zero. Thus, the inertial forces on the left hand side of the equation disappear, and the boundary condition reduces to

2211 ),0(),0( StSt σσ = . (8-21)

Using Hooke’s law (8-2), that can be expressed in terms of displacements, as

ii SEkSEk ξξ ˆˆ)1( 222111 TR −=+− . (8-22)

From equations (8-19) and (8-22), the reflection and transmission coefficients are found to be

222

111

222

111

1

1

ˆˆ

ES

ES

ES

ES

Ri

r

ρ

ρ

ρ

ρ

ξ+

+−

==ξ

and

222

111

222

111

1

2ˆ

ˆ

ES

ES

ES

ES

Ti

t

ρ

ρ

ρ

ρ

ξ+

==ξ

, (8-23a,b)

where the wave numbers k1 and k2 are replaced by ω /cL1 and ω /cL2 , respectively. Example 8-4

If two bars of the same material, with cross sectional areas in the ratio 1:2, are coupled together, the reflection and transmission coefficients for the displacement become

33.05.015.01

−=++−

=R and 67.05.015.02

=+⋅

=T .

The amplitude of the part of the wave that is transmitted therefore falls to 67 % of the incident wave amplitude.

8.3.5 Standing quasi-longitudinal waves in bars

If the medium is limited by one boundary in at least one direction, standing waves can arise. Standing waves are a combination, or superposition, of incident and reflected waves; see the comparable discussion in section 5.2.

8.3.5.1 Reflection at a free end

When a harmonic quasi-longitudinal wave is incident upon a free end of a bar, it is reflected back towards the incident wave. The resulting displacement field is then a superposition of the incident and the reflected waves. At certain points, the waves interact constructively, and at others destructively. In stationary conditions, there are certain points, called nodes, at which there are minima of the amplitude. There are also other points, called antinodes, at which there are amplitude maxima. Consider a semi-infinite bar. Assume that a harmonic, quasi-longitudinal wave with a displacement amplitude of ξ is incident upon the free end of the bar at x = 0. At the free end, the total stress must be zero, i.e.,


243

0),0(),0(),0( =+= ttt ri σσσ . (8-24)

Making use of the relation between particle velocity and stress (8-2) in the boundary condition, we find that

ξσσ ˆˆˆ Eik Lir =−= . (8-25)

Therefore, the total stress in the bar is

tiLL

xktiL

xktiL exkEkeEikeEiktx LL ωωω )sin(ˆ2ˆˆ),( )()( ξξξσ −=+−= +− . (8-26)

The total displacement can, using that stress result, be determined from (8-2) to be

tiL exktx ω)cos(ˆ2),( ξξ = . (8-27)

A vibration pattern of that type, in which the spatial and time dependences are separated into two factors in the form f(x)eiωt , can be called a standing wave. The pattern does not move in space. The location of the node points is not a function of time; see figure 8-10.

Figure 8-10 One period of a standing quasi-longitudinal wave in a semi-infinite bar. The dashed lines show the

motions of some particular cross sections of the bar.

ω t = 9π /4

ω t = π /2

ω t = 3π /4

ω t = π

ω t = 5π /4

ω t = 3π /2

ω t = 7π /4

ω t = 2π

Node line 3 Node line 2 Node line 1 Phase

ω t = 5π /2


244

8.3.5.2 Free quasi-longitudinal oscillations of a finite bar

When the length of the bar is finite, a resonant situation can develop. Consider a bar rigidly fixed at one end, as in figure 8-11.

x = 0 x = L F

ξ(0,t) = 0 σ(L,t) = - F/S

If no external forces act on the bar, i.e., F = 0, then equation (8-6) with q = 0 describes the free quasi-longitudinal oscillations of the bar. Because both rightward and leftward moving waves can traverse the bar, a harmonic displacement field is assumed, in the form


−+ += ωω ξξξ . (8-28)

Because no external forces act on the bar, the boundary conditions proscribe that the displacement be zero at the fixed end, and that the stress be zero at the free end,

0 s vd ,0),(och 0),0( =∂∂== =LxxtLt ξσξ . (8-29)

The following system of equations is thereby obtained:

0)ˆˆ( =+ −+tie ωξξ , (8-30a)

0)ˆˆ( =+− −−

+tiLikLik eee LL ωξξ , (8-30b)

where the factor exp(iω t) is always non-zero. The first of these equations gives the result

−+ −= ξξ ˆˆ . If that result is entered into the second equation, then cos(kLL) = 0, i.e.,

,2ππ −= nLk Ln or Lcn Ln )2( ππω −= where n = 1, 2, 3, .... (8-31)

Putting −+ −= ξξ ˆˆ into the assumed displacement field

tin

tiLnn nn exexktx ωω ψ )(ˆ)sin(ˆ),( ξξξ == (8-32)

where ξ is an amplitude. The values ωn are called eigenfrequencies and the correspond-ing functions ψn eigenvectors (mode shapes); see figure 8-12. The displacement field (8-32) is written as a product between a spatially dependent part and eiωt; i.e., it is a standing wave. Every value of n describes a standing wave that can, in principle, exist in the bar without continuously adding energy (external forces are zero). That type of solution is therefore said to be a free oscillation. In practice, energy must, of course, be provided to get such free oscillations started and compensate for the energy lost to heat, by friction for example. Otherwise, the losses in the bar will eventually return it to a state of rest.

Figure 8-11 Bar fixed at one end.


245

ψ1

xL0

ξ

ψ2

xL0

ξ

ψ3

xL0

ξ

ψ4

xL0

ξ

x2x1

ψ4(x2)ψ4(x1)

ψ5

xL0

ξ

ψ6

xL0

ξ

Figure 8-12 Displacements for the first six quasi-longitudinal mode shapes (eigenvectors) ψn of a bar fixed at one

end and free at the other. The free oscillations that constitute the homogeneous solution to the wave equation are also called eigenmodes. The eigenfrequencies represent disturbance frequencies at which the bar is very easily excited. The corresponding mode shapes describe the form of the deformation displayed by the bar when it is excited at that frequency. Of greatest interest is that the free oscillations, i.e., the mode shapes, can be used to build up other types of oscillation patterns using Fourier methods, which were described in section 3.4; for example, oscillating deformation patterns induced by external forces can be decomposed into mode shapes. Table 8-3 below presents the eigenfrequencies and mode shapes for quasi-longitudinal oscillations in finite bars with various end-fastening conditions.


246

Table 8-3 Eigenfrequencies and mode shapes for quasi-longitudinal waves in bars with various end conditions.

Boundary Condition Designation Eigenfrequency fn

[Hz] n = 1, 2, 3,…

Mode shape ψn n = 1, 2, 3, …

xL Free - Free

Lc

n L2

)cos(Lxnπ

xL Fixed - Free

Lc

n L2

)21( − ))

21sin((

Lxn π−

xL

Fixed– Fixed L

cn L

2 )sin(

Lxnπ

xL

κ

Spring - Free

χκχ 1tan ⋅=SE

L )sin(

)cos(cot

Lx

Lx

n

nn

χ

χχ

+

+

xL

κ

Fixed – Spring χ

κχ 1cot ⋅−=SE

L )sin(Lx

nχ

xL

m

Fixed – Mass

χρ

χ ⋅=SLmcot )sin(

Lx

nχ

xL

m

Free – Mass χ

ρχ ⋅=

SLmtan )cos(

Lx

nχ

Here, cL is the phase velocity of quasi-longitudinal waves, L is the bar length, S is the cross-sectional area of the bar, ρ is the density of the bar, E is the Young’s modulus of the bar, κ is the spring constant of a spring, when present, and m is the end mass, when applicable. In the last four cases, the eigenfrequency is determined from the transcendental equation given in the table, and from the formula

nL

n Lc

f χπ2

= .

8.3.5.3 Forced quasi-longitudinal oscillations in finite bars

Assume, now, that a force F excites the bar at its free end, as in figure 8-11. The oscillations that the bar executes are called forced oscillations and constitute the so-called particular solution. The forced oscillations can be calculated by assuming a field of forwards and backwards-moving plane waves, the amplitudes of which are determined from the boundary conditions. The assumed field has the form


−+ += ωω ξξξ . (8-33)

Using Hooke’s law, the total stress can be written as

)ˆˆ(),( )()( xktixktiL

LL eeEiktx +−

−+ +−= ωω ξξσ . (8-34)


247

At the fixed end, x = 0, the total displacement must be zero, i.e.,

0ˆˆ =+ −+ ξξ . (8-35)

x

F

L

Figure 8-13 Cross-sectional quantities near the forced end of the beam. At the free end, the normal stress in the bar must balance the external excitation force F. In order to correctly specify the sign of the stress, we can cut the beam near the end and set up a force balance for the cut-off beam segment, as in figure 8-13.

StxLt

txLxS ),(

),2(2

2Δ−−−=

∂

Δ−∂Δ σ

ξFρ , (8-36)

i.e., if Δ x approaches zero, then

StL ),(σ−=F or StL F−=),(σ . (8-37)

Putting in the total stress yields, after elimination of −ξ ,

SeeEik LikLikL LL F−=+− −

+ξ)( , i.e.,

)cos(2

ˆˆLkESik LL

F=−= −+ ξξ . (8-38)

Substituting that into (8-33) gives the total displacement field

ti

L

L

L

tixikxik

LLe

Lkxk

ESkeee

LkESiktx LL ωω

)cos()sin(

)()cos(2

),( FF−=−= −ξ . (8-39)

Figure 8-14 illustrates that displacement field, both as a function of position for two given excitation frequencies, and as a function of the excitation frequency at two given positions.

σ(L-Δx,t)S

Δx


248

a) b)

c) d) Figure 8-14 Displacement in the bar of figure 8-11 in four different cases. The bar’s length L is 2 m. The material

is steel. a) The excitation frequency is 1200 Hz. b) The excitation frequency is 3000 Hz. c) At the position x = L. d) At the position x = L/2.

8.3.5.4 Forced quasi-longitudinal oscillations in finite bars: modal method

This section will demonstrate how the free oscillations can be used as a tool to determine the forced oscillations. Assume that one end, x = L, of the bar in figure 8-11 above is excited by a harmonic axial force (F/S)δ ( x-L)exp(iω t), where the so-called (Dirac) delta function4 δ(x - x0) is used to mathematically describe a force that acts exactly at the point x = x0. The inhomogeneous equation that describes the forced vibrations of the bar is, according to (8-6) with the external distributed force q replaced by the axial force given above, 4 The delta function has the following integration property: δ ( ) ( ) ( )x x g x dx g x− =∫ 0 0

.

x/L

ξ ξ

0

Frequency [kHz]

1,0

0

x/L

10

0

20

1,0

Frequency [kHz]

ξ

0

ξ

10 20

0

0

10-3

10-2

10-5

10-3

10-2

10-5


249

ti

LeLx

SEF

tcxωδ )(1

2

2

22

2−−=

∂

∂−

∂

∂ ξξ . (8-40)

A standard method to solve this type of problem is to use a summation of the eigenmodes (8-32) as an assumed solution

∑∞

==

1)sin(ˆ),(

n

tiLnn exktx ωξξ , (8-41)

where the so-called modal amplitudes nξ are determined for every eigenmode n. That assumed form is put into the inhomogeneous equation. Then, the time dependence, which is common to all terms, may be cancelled. Both sides of the equation are then multiplied by sin(kLmx) and integrated over x from 0 to L. If we use the fact that the integral5 of the product of two sine functions is L/2 when m = n, and 0 otherwise, we obtain the result

222)sin(2ˆLmL

Lmm

kc

LkSLE

F−

−=ω

ξ , (8-42a)

i.e., ti

n LnL

LnLn ekc

LkxkSLE

Ftx ω

ω⋅

−

⋅⋅−= ∑

∞

=1222

)sin()sin(2),(ξ . (8-42b)

That is a summation of contributions from the eigenmodes of the bar. The amplitudes mξ of the eigenmodes indicate how much each mode contributes to the forced vibrations. Figure 8-15 shows the amplitude mξ as a function of the exciting force frequency. Figure 8-15 shows that an eigenmode makes a large contribution when the excitation frequency is close to its resonance frequency. The amplitude goes to infinity when the excitation frequency is equal to the resonance frequency. In reality, however, the amplitude is of course limited by losses in the bar.

5

⎩⎨⎧

≠=

=∫ nmnmL

dxxkxk m

L

n for 0for 2

)sin()sin(0


250

10-8

10-6

10-5

10-4

10-3

0 50 100 150 200 250 300

η = 0,001

η = 0,01

η = 0,1

Figure 8-15 Idealized appearance of the modal amplitude as a function of the excitation frequency, with the loss factor η as a parameter. The loss factor is a measure of the energy losses in the bar; see section 3.3.5. A high loss factor gives a low amplitude at resonance, and vice versa.

The displacement in (8-42b) is a so-called modal summation, i.e., the displacement is expressed in the form of a sum of contributions from different modes (eigenvectors). With certain reformulations, (8-42b) can be written in the following way

ti

n n

n

nn e

gtxtx ω

ωω∑∑∞

=

∞

= −==

122

1

ˆ),(),( ξξ , (8-43)

where mLkFmfg Lnnn )sin(ˆˆˆ == , m = ρSL,

)sin()sin(

4)12(

)sin(2)sin(8

)12(22

22

2xkLkn

mLkmxk

nL

ES

Ln

Lnstat

Ln

Lnnnn

πκπ

μκω +=

+

== ,

m = ρSL and LESstat =κ .

The mass of the bar is indicated here by m, and its stiffness at 0 Hz, i.e., the static case, by κstat. Compare (8-43) to the expression for the displacement in a loss free (δ = 0) single degree-of-freedom system (3-16). Apparently, every term in the summation (8-43) is equivalent to a single degree-of-freedom system. The parameters κn and μn incorporated above correspond to the mass and the stiffness in a mass-spring system. The dynamic properties of a bar can apparently be represented by an infinite number of single degree-of-freedom systems (mass-spring systems). That is an example of a general principle: Every system with resonances can be regarded as comprised of a set of single degree-of-freedom systems. The so-called modal analysis seeks to mathematically determine models of the system on the basis of this principle.

mξ

Frequency [Hz]

[mm]


251

8.4 TORSIONAL WAVES IN SHAFTS

In many kinds of machines and vehicles, torsional vibrations are a significant problem. They often diminish the machine’s performance, and may a major part of the vibration and (indirectly) sound generation. Torsional waves in shafts of circular cross section are an example of pure transverse or shear waves. One way to interpret torsional waves is to imagine a shaft as comprised of thin discs. When the shaft is subjected to an axial twisting moment, the discs rotate about the axis. If, as a first approximation, the discs are assumed to be rigid, then each disc’s angle of rotation may be used to describe the rotation of the corresponding position in the actual continuously elastic shaft; see figure 8-16.

Undeformed

Deformed

θ

x x

M

Figure 8-16 The twist angle θ ( x) of the cross-section at x along a shaft can be used to describe torsional

oscillations.

8.4.1 Wave equation of a plane torsional wave in a straight circular shaft

Consider a straight shaft with a circular cross section, as in figure 8-16. Assume that an axially-directed twisting moment M acts on the shaft. Cut away a circular disc of length Δ x , as in figure 8-17, and consider it in isolation. Both end sections of the disc are twist-ed with respect to each other, due to the action of the moment. Because of the internal twisting, the shaft element is deformed in pure shear. Because the cross section is assum-ed to rotate without change of shape about the axis of symmetry, the shear angle (= 2ε) increases linearly with the radius r; see figure 8-17. In order to derive an equation for the twist angle θ , we need to express the shear angle in terms of it. From figure 8-17, it is apparent that the length of the circular arc Γ provides a relation between the two angles,

γ Δ x = (θ (x + Δx) − θ (x))r.

Developing that in a Taylor series, truncating non-linear terms, and canceling Δ x ,

x

r∂∂

=θγ . (8-44)

Hooke’s law then relates the shear angle and the shear stress τ on a cross section,

γγυ

ευ

τ GEE=

+=

+=

)1(21

i.e., x

Gr∂∂

=θτ , (8-45)


252

where G is the shear modulus familiar from strength of materials, and υ is Poisson’s ratio.

θ ( x)

θ ( x+Δ x )γ

Δ x

r

aM(x)

M(x+Δ x )Γ

The shear stress at a point on the cross section is, therefore, directly proportional to the distance r from the axis of symmetry. The equation of motion of the shaft for rotation about the shaft axis is

xx

MxMxx

MxMxMxxMt

xJ Δ∂

∂=−Δ

∂∂

+=−Δ+=∂

∂Δ )()()()(

2

2θ ,

i.e., x

Mt

J∂

∂=

∂

∂2

2θ , (8-46)

where J [kgm] is the mass moment of inertia of the shaft per unit length. For a homogeneous circular shaft, it is given by J = ρ a 4π. Finally, we need a relation between the moment M(x) on the section, and the shear stress τ. For any cross section, the net moment must be equal to the resultant moment from all shear stresses over the entire section. Thus, integrating the shear stress over the section, as in figure 8-18, we obtain

x

Gadrrx

GrdrrrdFxMaa

∂∂

=∂∂

=== ∫∫ ∫θπθππτ 4

0

3

02

22)( . (8-47)

r

dr

τ (r)

Figure 8-17 Deformation of a small shaft element subjected to a twisting moment.

Figure 8-18 The moment on a section is calculated by integrating the shear stress distribution over the cross section.


253

The wave equation sought is now obtained by eliminating the local moment in (8-46), by substituting in (8-47). Using, additionally, the expression for the mass moment of inertia,

ρθθ Gctcx

TT

==∂

∂−

∂

∂ 22

2

22

2 , 01 , (8-48)

where cT is the wave speed, or phase velocity, of a pure transverse wave. That suggests, as we have already pointed out, that a torsional wave on a circular shaft is indeed a pure transverse wave.

Example 8-5 Using data on material properties, and formula (8-48), the shear or transverse wave velocity can be determined for some common structural materials.

Table 8-4 Material properties and transverse wave speeds of some common materials.

Material Density ρ [kg/m3]

E-modulus E [N/m2]

Poisson’s ratio υ

Transverse wave speed

cT [m/s] Steel 7800 11102 ⋅ 0.3 3100

Aluminum 2800 10103.7 ⋅ 0.3 3100 Concrete 2300 10106.2 ⋅ 0.2 2200 Rubber 1200 66 1040108.0 ⋅−⋅ 0.5 15 - 110

Note that the phase velocity of transverse waves is lower than the quasi-longitudinal wave speed; see example 8-1. That applies to all materials.

8.4.2 Torsional waves in straight shafts

A solution to the wave equation for torsional waves in a straight bar is a harmonic wave that propagates in the positive x-direction,

)(ˆ),( xkti Tetx −= ωθθ . (8-49)

A geometric interpretation of that solution is obtained if the axis is divided into discs perpendicular to the axis of symmetry. The angle of twist θ of the original shaft can then be interpreted as equivalent to the rotation of the discs about the axis of symmetry. Figure 8-9 shows a torsional wave in a shaft, with the discs marked so as to show their rotations.

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Figure 8-19 Torsional wave in a straight shaft with a circular cross section. P1 and P2 mark two points of constant

phase angle, moving with the wave.

P1

P1

P1 P2

P2

P2

Undeformed

t = 0

t = T / 4

t = T / 2


254

Because, in practice, all shafts are finite, standing waves and resonance phenomena arise from the interference of reflected wave components, just as they do for quasi-longitudinal waves in bars; see sections 8.3.3 – 8.3.5. Example 8-6 The wheel axles of a train are about 1.4 m long. Calculate the lowest torsional resonance frequency of a 1.4 m long free-free steel shaft.

Solution The lowest resonance lies at the frequency at which the torsional half wavelength is equal to the length L of the axle. In this case, the wavelength should be 2.8 m at resonance. The phase velocity of torsional waves in any circular steel shaft is, from example 8-5 above, 3100 m/s. The lowest resonance frequency is therefore

11008.2

31002

≈==L

cf T Hz.

Note that the rotational inertia of the wheels will, in reality, shift the resonance frequency of a real wheel-axle system to lower frequencies than that.

8.5 BENDING WAVES IN BEAMS AND PLATES

Bending waves exist in bodies that exhibit little extension in one or two dimensions. That type of structure is usually idealized by a thin beam or plate. Characteristic for bending waves is that the main deformation is normal to the beam axis or plate surface. In acoustics, bending waves are of particular importance because, for one thing, they are easily excited throughout the audible frequency bands, and for another, they are more strongly coupled to the acoustic medium (e.g., air or water) than other wave types. In this section, bending waves are described using the simplest possible models, the so-called Bernoulli-Euler models.

8.5.1 Bending wave equation for beams and plates

Assume that a slender, straight beam is deformed in a plane, as in figure 8-20. A differential equation for bending waves in beams can be derived by setting up the equations of motion of a small beam element of length Δx. Cut a beam element away from the beam with its end sections perpendicular to the axis of the undeformed beam. Consider that element in isolation, and provide the cross sectional loads needed to maintain it in equilibrium. Let ζ be the beam deformation, or the bending; see figure 8-20.


255

M M

T T

q'

q'

Figure 8-20 Beam element in bending, showing the cross sectional loads; the local moment My and the shear

force T. The beam element is excited by the external force per unit length q’ [N/m]. The cross sectional loads must be related to the deformation of the beam element. In order to understand how that can be done, regard the beam as comprised of layers parallel to the xy-plane, and the undeformed beam axis. When the beam bends, the layers and the axis bend as well, as shown in figure 8-21a. Because the layers exert shear stresses on one another, they normal strain. Layers near the concave side of the axis are shortened, and those near the convex side are extended. The layer with an unchanged length is usually called the neutral axis.

MyMy

Deformed cross section

Layer

Figure 8-21a The beam can be regarded as comprised of layers parallel to the axis of the undeformed beam. When a beam element is subjected to a bending moment, the layers bend. The stresses that thereby arise between the layers are called shear stresses.


256

Figure 8-21b Beam element in bending. In the Bernoulli–Euler theory, it is assumed, for the sake of simplicity,

that a plane cross-section of the beam remains plane and perpendicular to the beam axis when bent. The normal strain is then a linear function of the distance z from the neutral axis. a) View from the side. b) Cross-sectional view.

In general, the originally plane end sections of a beam element are arched, as in figure 8-21a. Establish a coordinate system xyz, in which the neutral axis lies in the xy-plane, as in figure 8-21b. The normal strain ε is, in the general case, a complicated function of y and z. To make any progress, simplifying assumptions must be made. The simplest possible normal strain function would be a constant, but that would imply that shear stresses cannot exist between the layers. The next simplest alternative would be a linear function6 of z of the type

Czz += 0)( εε . (8-50)

What value should be given to the constant C in that case? If we assume that sliding between layers is negligible, we make the classic Bernoulli-Euler assumption. Plane cross sections, initially perpendicular to the neutral axis, remain plane and perpendicular to the neutral axis, even as the beam bends. With that assumption, and using geometrical relations, it can be shown that the normal strain is given by

zx

ρzz2

2

00)(∂

∂−=+=

ζεεε , (8-51)

6 The constant normal strain can be seen as the first term in a Maclaurin series development, i.e., a Taylor series development about the origin, of the actual normal strain function. The linear strain function is a Maclaurin series development with two terms retained.

Neutral axis

Shear Stress

Cross-Section


257

where ρ is the radius of curvature of the beam element. Hooke’s law (8-2) now gives the normal stress over the cross section, as

zx

Ezx

EE2

2

02

2

0∂

∂−=

∂

∂−=

ζσ

ζεσ . (8-52)

The constant part of the normal stress can, essentially, be regarded as a quasi-longitudinal deformation, and ignored hereafter. The linear part of the normal stress that remains can be replaced by a cross-sectional moment My in the y-direction. By integrating the normal stress over the cross section of the beam, the cross sectional moment My can be related to the radius of curvature

2

22

2

2 2

1

2

1

)(x

EIdzzzbx

Ebzdz b

h

h

h

hy

∂

∂−=

∂

∂−== ∫∫

−−

ζζσM , (8-53)

where b is the width of the beam cross-section, and h1 and h2 are the distances of the upper and lower surfaces of the beam to the neutral axis; see figure 8-21b. The integral after the second equality in (8-53) is called the section’s area moment of inertia, and is indicated by Ib. The contribution of the longitudinal stress σ0 to the moment is zero. Formula (8-53) is called the elastic line equation. The quantity EIb is called the beam’s bending stiffness, and designated by D; it has the units [Nm2]. The bending stiffness is a measure of the beam’s ability to withstand a bending moment. A large bending stiffness implies that a large bending moment is required to cause the beam to bow out a certain amount. Table 8-5 gives the cross sectional areas and area moments of inertia for some simple cross sections. Using table 8-5, and the additive properties of some cross sections, the area moment of inertia can be approximately determined for a large number of technically important beam cross sections; see example 8-7.

Table 8-5 Area and area moment of inertia for some simple cross sectional geometries. If bending does not occur about the line of symmetry, the area moment of inertia can be calculated using Steiner’s theorem from mechanics.

Designation Figure Area S [m2]

Area Moment of Inertia Iy [m4]

Circular, radius a

a

πa2 πa4 /4

Elliptical, semi-major axis a and

semi-minor axis b

ab

πab πab3 /4

Rectangular, height h

breadth b b

h

bh bh3/12


258

Example 8-7 Calculate the area moment of inertia for bending about the y-axis, for the beam cross sections illustrated in figure 8-22.

a) A box beam with a rectangular cross section; see figure 8-22a.

b) An I-beam, see figure 8-22b.

b)a)

h

b

t

y

b

tl

h

t2

y

Figure 8-22 Cross sections: a) A box beam with a rectangular cross section; and b) an I-beam.

Solution: The area moments of inertia of both beam cross sections can be calculated using the additive properties of area moments of inertia.

a) This cross section is symmetric about the y-axis. The area moment of inertia can be calculated as the difference between two homogeneous, rectangular cross sections. Based on the area moment of inertia of a rectangular cross section, we obtain

12)2)(2(

12

33 thtbbhI y−−

−= .

The second term is the area moment of inertia of the hollow interior.

b) The area moment of inertia of this cross section, as well, can be handled using that of the rectangular cross section; thus,

31

23

)2(2

)(1212

12th

tbbhI y −−

−= .

The second term, in this case, represents the contributions of the two “empty” rectangular areas on either side of the vertical axis of symmetry. By setting up the equations of motion for the translation ζ in the z-direction, and the rotation θ about the y-axis, the cross sectional loads can be related to the deformations.


259

Equilibrium in the z-direction: xt

Sxtxxtx Δ∂

∂=Δ+Δ++−

2

2'),(),( ζρqTT ,

where T is the shear force on the section, q’ the distributed force per unit length, and S the area of the cross section. Developing that in a Taylor series and letting Δ x go to zero,

2

2'

tS

x ∂

∂=+

∂∂ ζρqT . (8-54)

Rotational equilibrium about the y-axis :

2

22)(

2),(),(),(

txxSΘ

xxxtxtxxtx yy

∂

∂ΔΔ=

ΔΔ′+Δ−Δ++−

θρqTMM ,

where Θ is the rotational moment of inertia about the y-axis, and θ is the rotation of the element about the y-axis. Letting Δ x go to zero, we have

0=+∂

∂− T

M

xy

. (8-55)

That relation is obtained by ignoring the contribution from the rotational inertia, which is small for small Δ x. The shear force T therefore balances the moment difference between the ends of the element. When the moment My and the shear force T are eliminated from the equations above, we obtain the Bernoulli-Euler wave equation for bending waves in slender beams,

qt

Sx

Dx

′=∂

∂+

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂

∂

∂

∂2

2

2

2

2

2 ζζ ρ . (8-56)

This equation can be shown to apply to good precision so long as the bending wavelength is greater than six times the beam height. Note that the bending stiffness D can be a funct-ion of position x. But if it is independent of x, it can be removed from the differential. In deriving the bending equation (8-56), coupling to other wave types was ignor-ed. However, for many beam cross sections, for example, vertical bending waves may be coupled to horizontal bending , torsional, and quasi-longitudinal waves. Only if the beam cross section is symmetric with respect to both the y and the z axes, defined in figure 8-21b, and the axis of symmetry lies in the neutral layer, are the wave types uncoupled and can exist independently of one another. Coupling between the different wave types can also result from asymmetric boundary conditions. If an I-beam, for example, is attached along its entire bottom surface to an elastic foundation, and is excited vertically, then both vertical bending waves and quasi-longitudinal waves will occur. The plate is the two-dimensional analogue of the beam. In the same way as for a beam, a wave equation for bending waves in plates can be derived, and takes the form

),,()(2

222 tyxq

thD p ′′=

∂

∂+∇∇

ζζ ρ , (8-57)


260

where ∇2 is the Laplace operator (see chapter 4), q ′′ is the external force per unit area, and the plate’s bending stiffness is indicated by Dp [Nm] to distinguish it from that of the beam. For a plate of constant thickness h, the stiffness is

Dp = Eh3/(12(1-υ2)), (8-58)

where E is the modulus of elasticity, and υ is Poisson’s ratio.

8.5.2 Bending waves in an infinitely long beam

The form of the bending wave equation (8-56) differs somewhat from the wave equations we have encountered up to this point. The differences are that the second derivative with respect to the position coordinate is replaced by the corresponding fourth derivative, and the sign of that term is the same as that of the time derivative. Physically, those differenc-es have important consequences for the properties and characteristics of bending waves. The solutions to (8-56) represent the way bending oscillations occur in an infin-ite beam. Certain properties of bending waves can be studied using these solutions. If the external load q′ is set to zero, the solutions are the so-called free bending oscillations of an infinite beam. The possible plane wave solutions are obtained by assuming a form

)(ˆ),( xkti Betx −= ωζζ . (8-59)

Putting that into the wave equation (8-56), we see that

DSk B ρω 24 = , (8-60a)

must be fulfilled in order for plane bending waves to be able to propagate along the beam. The so-called bending wave number kB [1/m] has been incorporated here. For a given circular excitation frequency ω , that equation has four different solutions

44,342,1 och DSikDSk BB ρωρω ±=±= . (8-60b,c)

Thus, there are four different solutions of the assumed form to the bending wave equation. The displacement field ζ in a beam vibrating in bending can therefore be expressed as a summation of contributions from the four solutions

tixkxkxikxik eeeeetx BBBB ω)(),( −− +++= DCBAζ , (8-61)

where the constants A, B, C and D are found from the boundary conditions of the beam, i.e., the end fastening. The first two terms correspond to wave solutions propagating in the positive and negative x-direction. The last two terms are so-called near field solutions. These are not waves, as ordinarily meant, but rather oscillations that exponentially decay away from the excitation point. Figure 8-23 illustrates both types of solutions. What is the significance of the near field? It can be shown that it does not transport any energy. Figure 8-23 shows that, just a half wavelength from the excitation point, its amplitude is reduced to a few percent of the original. Apparently, the near field can be very significant near excited areas, i.e., near points of force application and discontinuities, such as area changes and boundaries.


261

Figure 8-24 shows the effect of the near field on a long, point-excited beam. Without the near field, the beam would behave in a fashion similar to the string.

Figure 8-23 Solutions to the bending wave equation near the end of a beam. a) The near field is characterized by

an amplitude that decays exponentially with distance from the excitation point. A reasonable engineering approximation would be to ignore the near field at distances greater than 1/3 of the bending wavelength. The near field is significant near boundaries, force application points, and other discontinuities. b) The bending wave can, if the losses are small, spread over large distances.

F

F

F

F

Figure 8-24 The bending wave near field permits a continuous slope. The near field is a consequence of the

beam’s ability to withstand shear. The string, lacking that ability, instead exhibits a slope discontinui-ty at a point of force application. Thin line – no near field (string). Thick line – incl near field (beam).

Near field

a)

b)

Bending wave

t = 0

t = T /8

t = T /4

t = 3T /8

t = T /2


262

Example 8-8 A railway rail on the Swedish trunk lines has the following properties: the cross sectional area 4107.63 −⋅=S m2 and the area moment of inertia 8102045 −⋅=bI m4. The relevant

material properties of steel are 11102 ⋅=E N/m2, υ = 0.3 and ρ = 7800 kg/m3. Calculate the bending wavelength at the frequencies 100 Hz, 1 kHz and 10 kHz.

Solution The wavelength can be determined from the wave number kB , and the relation λ = 2π /k. For the wave number, we can use formula (8-60a). Substituting that in, we obtain

ffS

EIB

4,42107,63108,710204510222 4

43

8114 ≈

⋅⋅⋅

⋅⋅⋅== −

−πρω

πλ ,

i.e.,

8.5.3 Bending waves in finite beams

In finite beams, the deformation field is complicated by the influence of the beam ends. As for quasi-longitudinal waves, the wave amplitude and phase are affected by a reflection at a boundary. Moreover, a near field develops where a bending wave reflects; see section 8.5.2 above. The near field part of the beam deformation need, normally, only be considered in the immediate vicinity of the beam end. Table 8-6 shows some common models of beam end fastening conditions, and how those are mathematically described as boundary conditions. As for quasi-longitudinal wave propagation, the boundary conditions can be expressed as reflection coefficients for incident plane waves.

Example 8-9

Assume that a bending wave is incident upon a free end, as in figure 8-25.

x)(ˆ xkti

iBe −ωζ

)(ˆ xktiBi

Be +ωζ R

tixkBNi

Be ωζ +Rˆ

a) Formulate the beam’s boundary conditions. b) Express the boundary conditions as a reflection

coefficient.

f 100 Hz 1 kHz 10 kHz λB 4.2 m 1.34 m 0.42 m

Figure 8-25 When a bending wave reaches a beam end, both a bending wave and a near field are reflected. The positive real part of the exponent in the near field component implies that the near field decays exponentially with distance from the boundary.


263

Solution:

a) At a free end of the beam, both the moment and the shear force must equal zero; i.e.,. 0),0( =tyM and 0),0( =tT . (8-62a,b)

According to equations (8-53) and (8-55), that can be written

0),0(2

2=

∂

∂

x

tζ and 0),0(3

3=

∂

∂

x

tζ . (8-62c,d)

b) Specify the response field as

xktiBNi

xktiBi

xktii BBB eeetx ++− ++= ωωω ζζζ RR ˆˆˆ),( )()(ζ .

That response includes the reflection coefficients RB and RBN to provide for a reflected bending wave and a near field. Note that, in the assumed solution form, we have left out the near field term with a negative exponent; on a physical basis, that must clearly be zero-valued. Putting the assumed displacement into the boundary conditions (8-62a,b), we obtain an equation system with the two unknowns RB and RBN. Its solution is

ieii i

i

rBB =⋅=

+−

== 2111

ˆˆ

π

ζ

ζR and 42

12

ˆ

ˆπi

i

rBNBN e

ii

=+

==ζ

ζR . (8-63)

The reflected bending wave has, therefore, the same amplitude as the incident wave, but is phase shifted 90 degrees ahead of it. Table 8-6 provides some additional examples of common idealized boundary conditions for beams in bending vibration.

Table 8-6 Boundary conditions for some common, idealized end fastening conditions.

Designation Boundary Condition

Free

0),0( =tyM i.e., 0),0( 22 =∂∂ xtζ

T( , )0 0t = i.e., 0),0( 33 =∂∂ xtζ

Rigidly Fixed

ζ( , )0 t = 0 0),0( =∂∂ xtζ

Pinned

ζ( , )0 t = 0

M y t( , )0 0= i.e., 0),0( 22 =∂∂ xtζ

Spring Mounted

T( , ) ( , )0 0t t= −κζ M y t( , )0 0=

z

x

z

x

z

x

z

x κ


264

In calculating the bending vibrations of finite beams, we can, therefore, apply the general solution form (8-61) and determine the coefficients A, B, C and D of the four wave (or near field) components using the boundary conditions that apply to the beam in question.

Example 8-10 Set up the system of equations that solves the bending response of the beam in figure 8-26.

Δ xx

( )txL ,Δ−M

L ( )txL ,Δ−T

F F

Figure 8-26 a) A beam with one end rigidly fixed, excited into bending vibrations. b) A cut sections the beam

into two segments, to set up the equations of motion.

Solution: At the rigid end, table 8-6 indicates that the boundary conditions are

0),0( =tζ and 0),0(

=∂

∂x

tζ. (8-64a,b)

At the free, point-excited end, as shown in figure 8-26b, there is no moment

0),(2

2=

∂

∂

x

tLζ (i.e., a moment free end; see equation (8-53)), (8-64c)

and according to equations (8-53) and (8-55),

tib eF

x

tLEItL ωˆ),(),(3

3=

∂

∂−=

ζT . (8-64d)

Putting the assumed solution form (8-61) into (8-64a-d) gives the following system of equations, from which the coefficients of the wave and near field terms may be solved:

0=+++ DCBA ,

0=−+− DCBA ii ,

0=++−− −− LkLkLikLik BBBB eeee DCBA ,

3Bb

LkLkLikLik

kEIFeeeiei BBBB −=−++− −− DCBA .

a) b)


265

If the force in the example were to shift, and excite the beam at an interior point x < L, we would divide the beam into two parts at x. Solutions of the type (8-61) would then be assumed for each of the two parts. The boundary conditions of both beam segments would thereafter give a system of 8 equations, the solutions of which would be the 8 unknown coefficients of the assumed solutions.

8.5.4 Dispersion

The relation (8-60a) between the beam’s wave number and the excitation frequency is the dispersion relation for bending waves. For a given excitation frequency, it specifies what the wave number must be for a bending wave to be able to propagate along the beam. Form the dispersion relation, certain quantities of interest can be determined. Both the phase and the group velocity are, for example, found from the dispersion relation. The phase velocity can be defined using the relation 4 SDkc f ρωω == . (8-65)

The group velocity can be shown to indicate the velocity with which the energy of the wave spreads through the medium. A common definition of the group velocity is

42 SDdkdcg ρωω

== , (8-66)

where the expression after the last equality is specific to a beam. The bending wave equation for a thin plate has the same form as that for a beam. The dispersion relation for a plate can therefore be determined from the dispersion relation (8-60a) by replacing ρS with ρh and D with Dp, as defined in (8-58), from which

2

2224 )1(12

EhDhk pB

ρυωρω

−== . (8-60d)

Note that this dispersion relation only applies to plane bending waves in a thin plate. Similarly, the phase velocity of plane bending waves in a thin plate can be shown to be

42

24

)1(12 ρυωρω

−==

EhhDc pf . (8-65a)

This relation can be used, as will be seen in chapter 8, to study the acoustic radiation from a plate in bending vibration. The equations given above show that bending waves have frequency-dependent phase and group velocities. Waves with that property are said to be dispersive. A physical consequence of dispersion is that vibration fields with multiple frequency components, e.g., pulses, change in appearance with time; see figure 8-27. Another consequence of dispersion is that the bending resonance frequencies of a finite beam are not distributed evenly along the frequency axis.


266

Figure 8-27 When the phase velocity is frequency dependent, i.e., dispersive, the pulse changes shape. The figure

shows the appearance of the pulse in a bar and in a beam at 4 equally-spaced time instants. The original disturbance at t = 0 therefore propagates to both the right and the left. The dispersive bending wave changes shape, while the non-dispersive quasi-longitudinal wave does not. Thick line: Beam (bending wave). Thin line: Bar (quasi-longitudinal wave).

If the excitation frequency is increased to infinity, the expression above shows that the group velocity, as well, goes to infinity. That is not physically acceptable. Energy cannot propagate faster than the speed of sound. That unphysical behavior is caused by the simplifications made in the derivation of the bending wave equation. If the derivation is, on the other hand, carried out in accordance with the so-called Timoshenko theory, which accounts for shear deformation and rotational inertia, a finite group velocity is obtained instead. Figure 8-28 shows the phase velocity for some different kinds of waves.

c

b

a

0 0

2000 4000 6000 8000 10000

200

400

600

800

1000

1200

1400

1600

cf

[m/s]

Frekvens [Hz]

Supersonic area

Figure 8-28 Phase velocity in a steel beam with a 5 cm diameter circular cross section: a) based on the Bernoulli-

Euler theory; b) based on the Timoshenko theory. c) Phase velocity of compressional waves in air. The frequency at which the bending wave phase velocity equals the speed of sound in the surrounding medium is called the coincidence frequency.

t = 0

t = Δt

t = 2Δt

t = 4Δt

Frequency [Hz]


267

Dispersion is of great significance for the radiation behavior of a structure. The phase velocity of a bending wave increases monotonically from zero to infinity, with increasing excitation frequency. Therefore, there must always be a frequency at which the bending wave speed is equal to the frequency-independent speed of sound in air; see figure 8-28. The bending wave is said to be supersonic at that so-called coincidence frequency. Acoustically, that implies that the bending wave, above the coincidence frequency, is a relatively effective radiator, whereas it is an ineffective radiator below coincidence. A more extensive discussion of acoustic radiation, and coincidence phenomena, is made in chapter 8.

8.5.5 Reflection and transmission at the interface between two beams

As in the case of quasi-longitudinal wave propagation, a wave is partially reflected when it encounters a change in the medium. When the beam cross-section or the material changes, the bending stiffness changes accordingly. When a bending wave is incident upon a boundary between two beams with differing bending stiffness, a portion of the energy is therefore reflected back towards the side from which the incident wave came, and a portion is transferred into the receiving side. Reflection not only occurs at material and geometry changes; it also occurs when the ends of two identical beams are joined such that their axes do not coincide. If, for example, a pure bending wave encounters the junction, reflected and transmitted bending waves and near fields arise. Considering figure 8-29, it is also clear that reflected and transmitted quasi-longitudinal waves can be generated at the junction of different beam elements. It is therefore important to remember that, at the junction between different structural elements, wave types can be converted into other wave types. Thus, in general, acoustic energy may be transmitted as bending waves, as quasi-longitudinal waves, and as torsional waves, along different parts of a path. When radiating as sound waves, however, it is bending waves that dominate. Quasi-longitudinal and torsional waves can, nevertheless, be significant to the transport of acoustic energy.

Quasi-longitudinal wave

Bending wave

Deformed Undeformed

a) Figure8-29a A bending wave,

incident upon a corner at which two perpendicular beams join, generates both bending and quasi-longitudinal waves in both beams.


268

)(ˆ xktiiB Be −ωζ )(ˆ xkti

BiB Be +ωζ R

)(ˆ xktiLiB Le +ωζ R)(ˆ xkti

BNiB Be +ωζ R

)(ˆ zktiBiB Be +ωζ T

)(ˆ zktiBNiB Be +ωζ T

)(ˆ zktiLiB Le +ωζ T

x

z Figure 8-29b Schematic illustration of the reflections and transmissions that occur when a bending wave is

incident on a corner. The sudden change in stiffness at the junction causes both quasi-longitudinal and bending waves (and near fields), in the reflected and transmitted fields. The incident bending wave has the amplitude ζiB. The reflected and transmitted waves have amplitudes as designated in the figure, by analogy to the definitions of reflection and transmission coefficients, (8-9) and (8-18).

8.5.6 Standing waves in beams

At boundaries, reflected waves arise and interfere with the incident waves. All semi-infinite and finite beams can therefore exhibit standing waves. The most important type of standing wave is the resonance of a finite system. For broad-band excitation, there is a high probability that the vibrations will be dominated by resonances in the excited frequency band. Therefore, knowledge of mode shapes, and their properties, is often useful in the study of sound and vibrations.

8.5.6.1 Free bending vibrations in a finite beam

By analogy to a finite bar’s quasi-longitudinal vibrations, back and forth moving bending waves can, under certain circumstances, also interact constructively to build up fields with high amplitudes. These fields are the beam’s eigenmodes or free oscillations, i.e., solutions to the homogeneous bending wave equation (8-67) in one dimension, for the case of a constant bending stiffness D,

024

4=− ζ

ζDS

dxd ρ

ω , (8-67)

with boundary conditions. For certain boundary conditions, that equation has an infinite number of non-zero solutions, provided that the frequency ω take on certain specific values


269

ωn. These frequencies are called eigenfrequencies and the corresponding solutions ψn are the so-called eigenvectors or modes.

Figure 8-30 The five first eigenmodes for a beam in bending vibration, rigidly fixed at one end. Table 8-7 provides a summary of the resonances of a finite beam, for various combinations of end conditions.

ψ1 ψ2 ψ3

ψ4 ψ5

270

Table 8-7 Finite beam in bending vibration: eigenvalues knL and eigenmodes ψn, for various common end conditions. (Source: Based on R D Blevins, Formulas for Natural Frequency and Mode Shape. 1979. Van Nostrand Reinhold.)

End Conditions

x k0L k1L k2L k3L k4L k5L knL, n>5 ψn

Fixed – Free - 1.875 4.694 7.855 10.996 14.137 (n-1/2)π cosd(knx) - 1

nσ sind(knx)

Pinned – Pinned - 3.142 6.283 9.425 12.566 15.708 nπ sin(knx)

Pinned –

Free 0 3.927 7.069 10.210 13.352 16.493 (n+1/4)π coss(knx) - 3nσ sins(knx)

Fixed – Pinned - 3.927 7.069 10.210 13.352 16.493 (n+1/4)π cosd(knx) - 4

nσ sind(knx)

Fixed – Fixed - 4.730 7.853 10.996 14.137 17.279 (n+1/2)π cosd(knx) - 5

nσ sind(knx)

Free - Free 0 4.730 7.853 10.996 14.137 17.279 (n+1/2)π coss(knx) - 6nσ sins(knx)

Notation: cosd(x) = cosh(x) - cos(x), coss(x) = cosh(x) + cos(x), sind(x) = sinh(x)-sin(x), sins(x) =sinh(x)+sin(x) and

11σ ≈ 0.734, 1

2σ ≈ 1.018, 13σ ≈ 0.999, 1

4σ ≈ 1.000, ... , 31σ ≈ 1.001, 3

2σ ≈ 1.000, ... , 41σ ≈ 1.001, 4

2σ ≈ 1.000, ... , 51σ ≈ 0.983, 5

2σ ≈ 1.001, 53σ ≈ 1.000, ... ,

61σ ≈ 0.983, 6

2σ ≈ 1.001, 63σ ≈ 1.000, ... .

Otherwise, mnσ ≈ 1.000.


271

Example 8-11 Railway rails are mounted on heavy concrete slabs. The slabs are placed in the track bed with a spacing a of about 0.65 m. Every slab mass acts as a hindrance to the propagation of bending waves, which arise when a train wheel rolls on the rail. At those specific frequencies for which the bending half-wavelength equals a whole number multiple of the slab spacing a, however, the bending waves can propagate, unhindered, great distances from the source. What are the first few (lowest) such frequencies?

Solution Example 8-8 provides some data for Swedish track. The first so-called pass-frequency is determined by setting half the bending wavelength equal to the distance between slabs, and then solving the frequency from the dispersion relation (8-60a; doing so,

10707.450)4(2 22 ≈≈= aSEIaf ρπ Hz.

A system intended to warn about the approach of a train might therefore be based on the detection of vibration signals in the region around 1000 Hz.

8.5.7 Standing waves in plates

Finite plates exhibit resonant properties. Just as for beams in bending vibration, the eigen-frequencies and eigenmodes of a plate depend on the edge fastening conditions. Figure 8-31 shows the six lowest eigenmodes of a plate with simply-supported edges.

(1,1)

(2,1)

(1,2)

(3,1)

(2,2)

(3,2)

Figure 8-31 The six first eigenmodes for a rectangular plate simply supported at all four edges. The eigen-frequencies increase from right to left in the figure. The mode shapes are typically indicated by (m,n) where m gives the number of displacement antinodes as projected along a short edge of the plate, and n gives the number of displacement antinodes as projected along a long edge of the plate.

Compared to the eigenmodes of a beam, those of the plate are complicated by the fact that they also depend on the ratio of the shorter to the longer edge lengths. It is therefore impossible to summarize all of the properties in a single table, like table 8-7 above. In


272

various handbooks and tables, there are collections of tables summarizing the resonance frequencies and mode shapes for various edge mounting conditions, and various aspect ratios. Table 8-8, below, gives some examples, for three different sets of edge conditions. Table 8-8 Eigenfrequencies of rectangular plates with certain boundary conditions. The mode’s indices (m,n)

represent the number of displacement antinodes projected along the short sides (m) and the long sides (n) of the plate, respectively; see figure 8-31. (Source: Based on R D Blevins, Formulas for Natural Frequency and Mode Shape. 1979. Van Nostrand Reinhold.)

Edge Conditions a/b μmn (m,n)

0.4 3.46 (1,3) 5.29 (2,2) 9.62 (1,4) 11.4 (2,3) 18.8 (1,5) 2/3 8.95 (2,2) 9.60 (1,3) 20.7 (2,3) 22.4 (3,1) 25.9 (1,4) 1 13.5 (2,2) 19.8 (1,3) 24.4 (3,1) 35.0 (3,2) 35.0 (2,3)

1.5 20.1 (2,2) 21.6 (3,1) 46.6 (3,2) 50.3 (1,3) 58.2 (4,1) a

bFF

FF

F = Free 2.5 21.6 (3,1) 33.0 (2,2) 60.1 (4,1) 71.5 (3,2) 117.5 (5,1)

0.4 11.4 (1,1) 16.2 (1,2) 24.1 (1,3) 35.1 (1,4) 41.1 (2,1) 2/3 14.3 (1,1) 27.4 (1,2) 43.9 (2,1) 49.4 (1,3) 57.0 (2,2) 1 19.7 (1,1) 49.4 (2,1) 49.4 (1,2) 79.0 (2,2) 98.7 (3,1)

1.5 32.1 (1,1) 61.7 (2,1) 98.7 (1,2) 111.0 (3,1) 128.3 (2,2) a

bSS

SS

S = Simply Supported 2.5 71.6 (1,1) 101.2 (2,1) 150.5 (3,1) 219.6 (4,1) 256.6 (1,2)

0.4 23.6 (1,1) 27.8 (1,2) 35.4 (1,3) 46.7 (1,4) 61.6 (1,5) 2/3 27.0 (1,1) 41.7 (1,2) 66.1 (2,1) 66.6 (1,3) 79.8 (2,2) 1 36.0 (1,1) 73.4 (2,1) 73.4 (1,2) 108.3 (2,2) 131.6 (3,1)

1.5 60.8 (1,1) 93.7 (2,1) 148.8 (1,2) 149.7 (3,1) 179.7 (2,2) a

bCC

CC

C = Clamped

(Fixed) 2.5 147.8 (1,1) 173.9 (2,1) 221.5 (3,1) 291.9 (4,1) 384.7 (5,1)

The eigenfrequencies are calculated from the table parameters, geometric data and material data using the formula

)1(122 2

2

2,υρπ

μ−

=Eh

af mn

nm .

Here, a is the length of a long edge, h the plate thickness, ρ the plate density, E the elastic modulus and υ Poisson’s ratio. Example 8-12 Part of the boundary surface of a machine consists of steel sheet, which can be regarded as a rectangular, simply-supported (at all edges) plate. The thickness of the sheet is 1 mm, and its sides are 0.6 m and 0.4 m long. In order to avoid the excitation of resonances in the plate by the operation of the machine, the eigenfrequencies of the plate need to be known to the designer.

a) Determine the two lowest eigenfrequencies of the plate.

b) How are the eigenfrequencies influenced by doubling the plate thickness?

Solution a) All information needed to solve the problem can be found in tables 8-1 and 8-8. From table 8-1, for steel: E = 2·1011 N/m2, ρ = 7800 kg/m3 and υ = 0.3. According to the problem statement, moreover, a = 0.6 m, a/b = 1.5 and h = 0.001 m. From table 8-8, for the


273

simply-supported boundary conditions along the edges, and for the given parameters, the two lowest eigenfrequencies follow from

μ11 = 32.1 and μ21 = 61.7.

and the deformation patterns for both of these modes are shown in figure 8-31. Putting these values into the formula for the eigenfrequencies beneath table 8-8, the lowest two are found to be

7,21)3,01(780012

001,01026,021,32

2

211

211 ≈−⋅

⋅⋅=

πf Hz

and 8,417,211,327,61

1111

2121 ≈⋅=⋅= ff

μμ

Hz.

b) According to the formula in table 8-8, the eigenfrequency is directly proportional to the plate thickness. When the plate thickness is doubled, the eigenfrequencies are doubled as well,

4.437.21211 =⋅≈f Hz

and 6.838.41221 =⋅≈f Hz.

8.6 MECHANICAL IMPEDANCE AND MOBILITY

The typical machine is a complicated, composite structure. The analysis of composite structures can be considerably simplified by taking advantage of the concepts of impedance or mobility. A mechanical impedance is a frequency response function, as discussed in chapter 3, that describes the relation between an exciting point force and the resulting velocity at a given point. The mechanical impedance and the mobility, are defined as (see table 3-1)

Z(ω, xF, xv ) = F(ω ) /v(ω ) , (8-68a)

and Y(ω, xF, xv ) = v(ω ) /F(ω ), (8-68b)

where xF is the position of the excited force, and xv is the point at which the resulting velocity is referred to. Thus, the mobility is the reciprocal of the impedance. Figure 8-32 exemplifies the concepts of impedance and mobility, for the case of a beam fixed at one end.

xv

tieF ωˆ

tie ωv

vZ ˆ/F= xF

x

Figure 8-32 Quantities that are included in the definitions of mechanical mobility and impedance.


274

Example 8-13 Determine the so-called driving point mobility for quasi-longitudinal waves in a bar fixed at one end, if the excitation and response points are both at the free end of the bar, as in figure 8-11.

Solution We can use formula (8-39) from section 8.3.5.3, which gives the displacement field for that case. The velocity field is obtained by multiplying (8-39) by iω. The mobility is thereafter found by dividing (8-39) by F,

)cos(

)sin()cos()sin(

ˆ)(ˆ

ˆˆ

)(LE

xE

ESi

Lkxk

ESki

Fxi

F L

L

L

x

ρω

ρω

ρωξω

ω −=−===v

Y . (8-69)

Figure 8-33 shows the mobility as a function of the excitation frequency.

Frequency[H z]

0 5000 20000 10000 15000

log Y

Figure 8-33 Mobility for quasi-longitudinal waves in a bar fixed at one end; see figure 8-11. The peaks in the

curve correspond to the eigenfrequencies. Assume that the mobility of a structure is known. It is relatively simple to, using formula (8-68b), calculate the resulting velocity in the structure for different alternative excitation forces. If the mobilities of the parts of a composite structure are known, a system of equations can be specified, the solutions to which are the motions of the composite structure. Chapter 9 shows how the impedance concept can be used to simplify the analysis of vibration isolated machines. Table 8-9, below, provides the impedances of some common mechanical structures.


275

Table 8-9 Point impedances of some common infinite and semi-infinite structural elements.

Mass F v Z = i mω

Spring Fv

Z = κ iω

Longitudinal wave in bar F v Z = S Eρ

Bending wave in an end-excited, slender ½-∞ beam

Fv

)1(5.0 iScB += ρZ

Bending wave in a slender ∞ beam

Fv

Z = +2 1ρSc iB ( )

Edge excited thin ∞ plate

Fv

hp ρDZ 5.3=

Thin ∞ plate excited far from the edges

F

v Z = 8 D p hρ

Example 8-14 Assume that a small motor with a mass of 1 kg is so compliantly mounted that the affects of the surroundings can be neglected at frequencies above 40 Hz. Calculate the velocity if a 100 Hz harmonic force, with an amplitude of 0.5 N, excites the motor.

Solution Because influences from the surroundings can be ignored at 100 Hz, the motor can be treated, as a rigid 1 kg mass, to a first approximation. From table 8-9, the impedance of a rigid mass can be used, so that

41 100,8)11002(5,0 −− ⋅⋅−≈⋅=== iimi πωFFZv m/s.

8.7 LOSSES IN SOLID STRUCTURES

The losses in a mechanical structure have, in a couple of cases, great significance for the vibrations. If, for example, a beam vibrating in bending is much longer than a wavelength, the losses reduce the amplitude of the bending wave considerably as it traverses the beam. In the same way, the losses have very important effects on resonant vibration fields; see figures 8-15 and 8-34. In resonant fields, the propagation path is long, because of the repeated reflections between the ends. Increased losses are therefore an effective way to diminish the reverberant part of a vibration field. It can be shown that the energy in the reverberant field is inversely proportional to the loss factor η

η1∝E . (8-70)


276

Resonance in the blade

Rubber

Steel

Reinforcements

Figure 8-34 The losses in the structure are of great significance to the amplitudes of resonant vibrations. While

grinding the teeth of the circle saw blade, powerful resonance vibrations build up in the blade, causing a strong grating sound. A way to minimize the vibration field, and the resulting noise level, is to apply a rubber-clad disc to the saw blade. That will increase the losses, partly because of the high internal losses in rubber materials, and partly because of friction at the contact between the rubber and the saw blade. (Source: Asf, Bullerbekämpning, 1977, Ill: Claes Folkesson.)

By losses, one typically means irreversible energy conversion from vibrational to other energy forms, e.g., thermal energy. Often, a structure’s losses are divided into two different components: internal losses, i.e., so-called material losses and boundary losses.


277

8.7.1 Material losses

Material losses include those caused by atomic and molecular process within the material itself. Examples of such processes are dislocations, changes in the crystal structure, changes in the molecular structure, and the excitation of oscillations at the molecular level. All of these processes are so-called relaxation phenomena, i.e., they taper off exponentially with time. The characteristic decay time, or relaxation time, can, depending on the phenomenon in question, vary from 10-9 s up to several hours. The relaxation time determines the frequency bands in which the phenomenon has the greatest significance; a short relaxation time implies a strong effect at high frequencies, and vice versa. Thus, it need not be the same dissipation mechanism that dominates in each frequency band. As a result, the loss factor tends to vary with the excitation frequency; see figure 8-35.

♦ ♦ ♦

♦

♦

♦

♦

∗ ∗

∗

∗

∗

0,040

η

0,020

0,030

0,050

0,060

0,070

0,080

0,090

0,100

10 Frequency [Hz]

100 1000 10000

+ +

Figure 8-35 Experimentally determined loss factor for plexiglass. The half-bandwidth method was used to determine the loss factor; see section 8.7.6 for more details.

Material losses are, especially for many metals, relatively small. For a metal, the material loss factor η might typically be of the order 10-4 - 10-3. For a material with relatively high internal losses, e.g., certain plastics, the loss factor might fall in the range 10-2- 10-1. Generally, the internal losses in a material increase with increased shear deformation. Table 8-10 summarizes the typical loss factors of some common materials, and some common structures.


278

Table 8-10 Typical loss factors for some common materials and structures. (Source: E E Ungar, Damping of

Panels, in, L L Beranek. Noise and Vibration Control, McGraw-Hill, 1971)

Material/Structure Loss factor η Steel 0.0002-0.0006

Aluminum 0.0001 Concrete 0.01-0.05

Tile 0.01-0.02 Sand (dry) 0.12-0.6

Gypsum Panel 0.0008-0.003 Wood 0.01

Plywood 0.01-0.013 Particle Board 0.01-0.03

Plastics 0.01-0.1 Carbon-fiber reinforced plastics 0.001

Aluminum structure, riveted 0.01-0.05 Building 0.01

8.7.2 Boundary losses

Boundary losses include those losses that can be attributed to the boundary surfaces of a structure. An important example is surface friction. Surface friction comes about when two surfaces in contact slide relative to one another. In joints and hinges of various kinds, as well as such machine elements as linkages, bearings, and gears, that type of friction is significant. Acoustic radiation from vibrating surfaces to the surrounding media (e.g., air, or water) is another example of a kind of boundary losses. A sometimes important contribution to the total loss factor is that from so-called air pumping. Air pumping occurs in joints in which air is pressed in and out of the small spaces between mating surfaces.

8.7.3 Losses in composite structures

In composite structures, such as machines, vehicles, and buildings, the boundary losses are normally significantly higher than the material losses in the components. The reason for that is, of course, that the losses at the boundary surfaces between the structural elements and the surroundings are significantly higher. It is therefore often the case that a increase in the material losses does not reduce vibrations as much as expected.

8.7.4 Increase of losses in beams and plates by means of absorbers

In two special situations, the losses may be very effectively increased by combining ordinary metallic structural materials with materials that have higher internal losses – so-called absorbers. When the vibrations of parts of a structure, such as attached sheet metal, are predominantly reverberant, that approach may be promising. Another case would be when waves transit a relatively long, reflection-free path, such as a beam. In the first case, the dissipation influences the vibrational energy in the structural member. In the second case, it influences the energy transferred by the beam.


279

To get the full benefit of the added material, it must be suitably placed. The dissipation is maximized when the shear deformation is as large as possible. Thus, the material should be placed in areas with large vibration amplitudes. Plates, shells and beams can be specially adapted to provide high losses. For example, a plate can be clad with a viscoelastic material that has high losses. In order to increase the shear deforma-tion, and by extension the losses, the shear layer can be sandwiched between two metallic sheets (MPM plate); see figure 8-36. Simplified methods to calculate the resulting loss factors of these structural elements can be found in the structural acoustics literature.

M etal

M etal

Viscoelastic layer

Figure 8-36 Plate of metal-plastic-metal type (MPM-plate), which has high losses This type of plate is designed so that the viscoelastic material mainly works in shear. That maximally exploits the material properties, and therefore reduces the layer thickness needed to obtain a desired loss factor.

8.7.5 Mathematical description of dissipation

For strictly periodic processes, it can be shown that relaxation processes cause a phase shift, i.e., a time delay, between stress and strain processes. That implies, by the definition of mechanical work, that some work is performed by each stress-strain cycle. That work is the dissipated energy, i.e., vibrational energy irreversibly converted to other forms. The phase shift also implies that the effect of losses can be described in a mathematically convenient way. Specifically, a phase shift between stress and strain can be introduced by a complex-valued E-modulus (compare to section 3.3.5),

)1( ηiE +=E , (8-71)

where η is the material’s internal loss factor. That relation can be used as a definition of the loss factor. More commonly, however, an equivalent definition based on energy methods is used, as in formula (3-74). Via the E-modulus, even complex phase velocities and wave numbers can be defined. For bending waves in beams, for example, a truncated series development of equation (8-60a), gives an approximation valid for small η, i.e.,

)41()1(

44 ηωη

ρωρ ikiEI

SIS

BB −≈+

==E

k . (8-72)

If that wave number is put into the plane wave solution, then equation (8-61) provides a physical interpretation of the complex wave number,

xikxkxi BBB eeex −−− == 400

ˆˆ)(ˆ ηζζ kζ , (8-73)

i.e., an exponentially decaying wave solution, see figure 8-37.


280

Distance x

0,80

-0,60

0,60

0,40

0,20

0,00

-0,20

-0,40

-0,80

1,00

xikxk BB ee −− 4/0

ˆ ηζ

Figure 8-37 Wave propagation in materials with internal losses. The amplitude of the bending waves decreases with distance from the source. The original, undamped wave amplitude is assumed to be ζ0.

An important consequence of losses is that the vibration amplitude is limited. If one supplies a constant quantity of vibrational energy per unit time to a structure, the structure’s vibrational energy increases successively to a certain level E0. The stationary level occurs when the energy supplied per unit time is balanced by the energy dissipated per unit time. If the energy supply is interrupted, the vibrational energy decays again to zero, as does the amplitude. Chapter 7 derives the result that the energy sustaining vibrations decays according to

teEtE ηω−= 0)( , (8-74)

where E0 is the initial energy value. The loss factor determines how quickly the structure can “suck away” the vibrational energy. A more general discussion of energy storage in systems is given in section 7.1. Example 8-15 The loss factor of a long beam is 0.0005. How much is a bending wave’s amplitude damped per wavelength?

Solution The decay can be studied with the help of the 1st factor in (8-73). After one wavelength, i.e., kB x = 2π, the amplitude is changed by a factor 9992.02/0005.04/2 ≈= −− ππη ee ,


281

i.e., hardly 1 per thousandth. After 100 wavelengths, the amplitude has diminished to about 92 % of its initial value. If an absorber can raise the loss factor to, say, 0.1, then the amplitude would diminish to 85 % of its initial value.

8.7.6 Experimental determination of the loss factor

In many situations, it is important to be able to estimate the loss factor. In practice, theoretical predictions of the loss factor are impossible to make to an acceptable degree of precision. Thus, some kind of experimental method is required. Some such methods are briefly described here.

(i) If the stress and strain are measured simultaneously, the loss factor can be directly determined from the phase shift between them. That method is, due to high demands on the phase precision, mainly useful at low frequencies.

(ii) By recording the decay of vibrational energy as a function of time, the loss factor can be determined from equation (8-74). That is the so-called reverberation time method. The reverberation time T60 is the time it takes for the energy of vibration to decrease by a factor of106, i.e., the velocity level must fall 60 dB; see figure 8-38. The loss factor is then found from the relation

602.2 Tf=η . (8-75)

That method is very useful if η is to be determined in, for example, octave or third-octave bands. Lv [dB]

100

80

60

40

20

Time [s] 0 1 2 3 4 5 6

60 dB

T60

Figure 8-38 The reverberation time T60 is the time it takes for the velocity level to fall by 60 dB. A vibration

source, e.g., an electrodynamic shaker, excites vibrations in the structure. When the vibration velocity reaches stationary conditions, the energy supply i.e., the shaker, is interrupted, and the decay process recorded as a function of time. From that recorded signal, the reverberation time, and thereby the loss factor as well, can be determined; see equation (8-75).

(iii) A narrow band version of method (ii) is performed by exciting the structure with a harmonic force. When the excitation is interrupted, the decay process is


282

recorded. From the relative decrease in energy, i.e., the square of the amplitude, η can be determined.

(iv) Frequency response functions, such as mobility7, i.e., motion normalized by force, see chapter 3 and section 8.6, have peaks at the resonance frequencies. The width of a resonance peak is directly proportional to the loss factor at that frequency. That relation can be used in the so-called half-value bandwidth method in which the loss factor is determined using the formula,

resBres ffff Δ== )(η , (8-76)

where fres is the resonance frequency and ΔB f is the half-value bandwidth between the frequencies, on each side of the peak, at which the mobility has fallen by a factor of the square root of 2 relative to the resonance peak. The name half-value bandwidth comes from the fact that the bandwidth is determined from the points at which the energy is half of that at the resonance peak. By defining the so-called mobility level LY,

2

2log10

refY

YYL ⋅= , (8-77)

where Y is the magnitude of the mobility and Yref = 1·10-3 m/Ns is the reference value for mobility, the loss factor is determined by a measurement of the mobility level. The half-

value bandwidth is then determined by the frequency lines at which the mobility level has decayed by 3 dB with respect to the resonance peak; see figure 8-39.

-80

-81

-82

-83

-84

-85

-86 30 40 50 60 70

3 dB

fu fö

Δb f = fö - fu

LY [dB]

Frequency [Hz]

7 Other such frequency response functions, besides mobility, are the dynamic flexibility (receptance) and the accelerance.


283

Figure 8-39 Calculation of the loss factor from the half-value bandwidth of a resonance peak. The half-value bandwidth is the distance along the frequency axis between the frequencies at which the mobility level has fallen by 3 dB relative to the maximum value at the resonance frequency.

This type of measurement can be quickly and easily performed using the FFT analyzers that have become readily available in this day and age; see figure 8-40. The main disadvantage is that the losses can only be measured at the resonance frequencies of the test object.

a(t)

Frequency

F(t)

Beam

Suspension wires

Excitation hammer

Force transducer Accelerometer

Charge amplifiers

PC

FFT-analyser

Figure 8-40 Instrumentation used in the measurement of the material losses in a test beam vibrating in bending.

The test beam is excited by a light hammer blow. The exciting force is measured with a force transducer, and the resulting vibration velocity in the test beam with an accelerometer. These signals are recorded, and an FFT analyzer calculates the mobility function between the force and velocity. The mobility function may is then plotted, and from the plot, the loss factor can be found by applying the half-value bandwidth method to the resonance peaks seen in the mobility spectrum.

Example 8-16 From a measurement of the mobility of a structure, the half-value bandwidth of a resonance peak is found to be 23 Hz. The resonance frequency is at 1990 Hz. Determine the loss factor of the structure at 1990 Hz.

Solution Putting the given values into formula (8-76) yields 012.0199023 ≈=η .


284


LONGITUDINAL WAVES IN BARS

Strain x

tx∂∂

=ξ

ε ),( . (8-1)

Hooke’s law x

EE∂∂

==ξ

εσ . (8-2)

Equation of motion 2

2

tx ∂

∂=

∂∂ ξ

ρσ . (8-5)

Longitudinal wave equation ρ

ξξ Ectcx

LL

==∂

∂−

∂

∂ 22

2

22

2 , 01 . (8-6)

Plane wave solution )()( ˆˆ),( xktixkti LL eetx +−

−+ += ωω ξξξ . (8-8)

Reflection coefficient ri

i

r eR δ

ξ== ˆ

ξR . (8-9)

TORSIONAL WAVES IN SHAFTS

Hooke’s law x

GrGEE∂∂

==+

=+

=θγγ

υε

υτ

)1(21. (8-45)

Equation of motion x

Mt

J∂

∂=

∂

∂2

2θ . (8-46)

Torsional wave equation ρ

θθ Gctcx

TT

==∂

∂−

∂

∂ 22

2

22

2 , 01 . (8-48)

BENDING WAVES IN BEAMS AND PLATES

Normal strain zx

zz2

2

00)(∂

∂ρ ζεεε −=+= . (8-51)


285

Normal stress zx

Ezx

EE2

2

02

2

0∂

∂−=

∂

∂−=

ζε

ζεσ . (8-52)

Cross-sectional moment 2

2

xEIby

∂

∂−=

ζM . (8-53)

Equations of motion 2

2

tS

x ∂

∂=′+

∂∂ ζρqT and 0=+

∂

∂− T

M

xy . (8-54,55)

Bending wave equation in a beam qt

Sx

Dx

′=∂

∂+

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂

∂

∂

∂2

2

2

2

2

2 ζρ

ζ . (8-56)

Bending wave equation in plates qt

hD p ′′=+∇∇2

222 )(

∂

ζ∂ρζ . (8-57)

Bending stiffness of a plate Dp = Eh3/(12(1-υ2)). (8-58)

Plane wave solution )(ˆ),( xkti Betx −= ωζζ . (8-59)

Dispersion relation for bending waves DSkB ρω 24 = . (8-60)

General solution tixkxkxikxik eeeee BBBB ω)( −− +++= DCBAζ . (8-61)

Phase velocity in a beam 4 SDc f ρω= . (8-65)

Group velocity in a beam 42 SDcg ρω= . (8-66)

Phase velocity in a plate 4 hDc pf ρω= . (8-65a)

MECHANICAL IMPEDANCE AND MOBILITY

Definition of impedance )()(),,( ωωω vFZ =vF xx . (8-68a)

Definition of mobility )()(),,( ωωω FvY =vF xx . (8-68b)


286

LOSSES IN SOLID STRUCTURES

Energy in the reverberant field η1∝E . (8-70)

Complex E-modulus )1( ηiE +=E . (8-71)

Energy decay teEtE ηω−= 0)( . (8-74)

Experimental determination

Reverberation time method 602.2 Tf=η . (8-75)

Half-value bandwidth method resBres ffff Δ== )(η . (8-76)

287

CHAPTER NINE

ENERGY METHODS APPLIED TO ROOM ACOUSTICS This chapter discusses suitable methods for the analysis of sound fields, or of sound transmission, at high enough frequencies that the wavelengths are small compared to typical distances traversed by the sound. Energy-based methods can be used for the analysis of vibrations and acoustics problems in such circumstances. After a short introduction to the subject area, including a bit of historical background, energy balance equations are treated for simple and for coupled acoustic systems. An equation which relates the reverberation time and the loss factor of an acoustic system is then derived. That result is specialized to the case of room acoustics and, with the help of the diffuse field and diffuse intensity concepts, an expression for the reverberation time of a room – Sabine's formula – is derived. We then discuss the measurement of the reverberation time and absorption of a room, and consider the structure of a sound field radiated by an acoustic source. Finally, acoustic absorbers are discussed, and an analysis is made of the sound transmission between two rooms. In that last context, the concepts of the transmission factor and the sound reduction index of partitions (walls) are studied, and some typical examples of insulating partitions are presented.

Chapter 9: Energy Methods Applied to Room Acoustics

288

9.1 OVERVIEW OF ENERGY METHODS

In the study of acoustic systems, we can choose between a number of different methods to describe acoustic fields; see section 1.6. For example, an exact description of the sound field as the sum of eigenfunctions (modes) may be used. For the case of a sound field in a room, as studied in chapter 5, the modal density grows quickly, however, with frequency; see equation (5-92). This implies that exact descriptions of the sound field ordinarily become unreasonably difficult at even low to moderate frequencies. This is a completely general conclusion and applies to all bounded (finite) acoustical systems with low to moderate damping, as, for instance, the case of bending waves on a steel plate. In order to make such an analysis possible at high frequencies, various energy-based methods have been developed instead. These are based on ignoring the wave character of the sound field, and, instead, on treating the field as the superposition of independent sound rays which can be regarded, locally, as propagating plane waves. These waves propagate in accordance with the laws of geometrical acoustics (compare to geometrical optics); i.e., they follow straight-line paths which can be bent if there is a variation in the medium's wave impedance, or which change direction when reflected by a bounding surface (wall of a room). In an energy-based analysis, the field is characterized by its total energy content, and sound rays by their local energy density or (alternatively) by their intensity. If there are several sources, these are treated as incoherent, which implies that the sources' contribution to the sound field at a certain point can be added on a power basis (see equation (1-28)). Because the wave character of the sound is not considered, energy-based methods cannot describe such phenomena as interference and diffraction. In order to classify different acoustic problems and assess whether an energy-based method is suitable, we can resort to a dimensionless number called Helmholtz' Number (He). That number is defined as

klHe = , (9-1)

in which k = 2π/λ is the wave number and l a typical dimension (units of length) for the considered system. The wave number k is dependent on which type of mechanical wave is being considered. For sound in a room, k corresponds, of course, to the wave number for airborne sound, and for bending waves in a plate, for example, k should be interpreted as the bending wave number. Helmotz' number gives a measure of the size of the system as measured in sound wavelengths. In order for a standing wave (a mode) to arise in a system, it is necessary that the system be at least a half wavelength large in some direction. This means that if He is much bigger than π, we can expect a large number of modes in the system, so that energy-based methods are applicable. When He is about the same order of magnitude as π, the system's behavior is dominated by a relatively small number of modes and an exact description of the field is possible. The case in which He is much smaller than π is special, and means that the sound wavelength is much larger than the dimensions of the system, in which case it is no longer reasonable to speak in terms of wave propagation. In that low frequency region, the system's behavior can be modeled as an


289

equivalent discrete mechanical system consisting of masses, springs, and viscous dampers. This type of system is described in chapters 3, 9, and 10. In summary, three different frequency regions can be distinguished, for which we introduce the following designations: no-modes region, few-modes region, and many-modes region. A more precise definition of these terms, as well as a summary of the discussion content, is given in table 9-1. Table 9-1 Classification of finite acoustic systems with low or moderate damping (i.e., resonant systems), using

the Helmholtz number.

Classification Frequency Region Character

No-Modes Region He « π Wave propagation is ignored. The system can be modeled as a discrete mechanical system.

Few-Modes Region He ≈ π A few eigenmodes dominate. A complete mathematical description of the field is possible.

Many-Modes Region He » π

A large number of eigenmodes control the behavior. Analysis is only practicable using energy-based methods.

The classical application of energy methods in acoustics (in the analysis of bounded systems) is the field of room acoustics; it is this application which is discussed in this chapter. The original work in the room acoustic field was done by W C Sabine, W S Franklin, and G Jaeger, at the beginning of the 1900's. As far as systems without boundaries are concerned (infinite systems), and which are really not a part of the subject matter of this chapter, the classical application is geometrical acoustics. The earliest work in that area was also carried out at the beginning of the 1900's and is, when we don't have flow in the medium, analogous to the work done in the field of geometrical optics. The most important application is the study of sound propagation in the atmosphere or in the sea. Geometrical acoustics is also applied in room acoustics to analyze the initial phase of a sound field's development after a sound source is suddenly turned on, an analysis which is of interest for, among other things, the dimensioning of concert halls. The possibility of applying energy-based methods more generally, in solid structures for example, has received increasing attention during the last 20-30 years. Special methods in which a statistical approach is coupled to the energy-based description have been developed (SEA = Statistical Energy Analysis). The statistical approach handles the uncertainty, always present in real problems, as to the exact nature of boundary conditions, as well as to the values of other mechanical parameters (mass, stiffness, etc.).


290

Fundamental contributions to that field were made by, among others, R H Lyon, during the 1960's, so that today SEA is an accepted tool for the vibroacoustic analysis of arbitrary structures. An important application which stimulated the development of the original method was the need to be able to attack complicated vibroacoustic problems in the aerospace industry.

9.1.1 Balance of Energy in Simple and Coupled Acoustical Systems

Consider a bounded linear acoustic system into which we supply power via sources, on the one hand, and remove power by means of losses Wdis (dissipation) in the system, on the other hand. Assuming that the law of conservation of energy is applicable to acoustical energy, then the following balance equation can be written for the system

disin WWdtdE

−= , (9-2)

in which E(t) is the total acoustic energy in the system and W(t) refers to power. In the study of linear acoustic systems, the system's behavior is normally analyzed as a function of frequency. For energy-based analyses and applications in the many-modes region (see table 9-1), a number of adjacent frequency bands (octave and third-octave bands) are normally used. The quantities in equation (9-2), and in the other equations in this chapter, are therefore meant to refer to a specific frequency band. That implies that energy and power are calculated from the primary acoustic quantities (e.g., sound pressure), first after those latter have been band-passed filtered. The assumption of linearity also implies that the system dissipation can be characterized by means of a loss factor in accordance with the definition given in chapter 3; see the footnote to equation (3-74).

EWdis ηω= , (9-3)

in which η is the system’s loss factor. Putting equation (9-3) into (9-2) yields

inWEdtdE

=+ηω . (9-4)

This first order differential equation describes how the system’s energy is built up after an external source has been turned on. A means of determining the loss factor, and hence even the damping, of an acoustic system, is to first excite the system using a source within a certain frequency band (e.g., white noise in an octave band). When the system has thereafter attained a stationary condition, we suddenly turn off the source and measure how the energy decays in time. Assume that we turn off the source at t = 0, and the system’s energy content in the stationary state is E0. For t > 0, equation (9-4) reduces to

0=+ EdtdE ηω . (9-5)1

1 It should be noted that this equation describes a non-stationary (transient) event, whereas the definition of the loss factor (equation (3-74)) is based on a stationary condition. For small loss factors (≈ quasi-stationary events),


291

This equation has the solution

teEtE ηω−= 0)( , t > 0. (9-6)

An event of the type described by equation (9-6) is called, in acoustics, a reverberant event. By studying the reverberant event, it is possible to experimentally determine the loss factor of an acoustic system as a function of frequency. From equation (9-6), we can define a characteristic time T that constitutes a measure of the duration of the reverberant event. In acoustics, T is normally defined as the time it takes for the energy in the system to decay to a value 10-6 (60 dB) times the original value E0; it is ordinarily called the reverberation time, and that particular definition goes back to the early work in the area of room acoustics by W C Sabine. Example 9-1 Determine the relation between a system’s reverberation time T, and its loss factor η.

Solution According to the definitions given above, the relation 10-6 = e-ηωT must apply, from which it follows that 6⋅ln 10 = ηωT , i.e., T = (6⋅ln 10)/ηω ≈ 13.82/ηω. _____________________________________________________________________

To round off this section, we will consider the energy balance for the case of two coupled acoustic systems (1 and 2). That result will later be applied to the study of sound transmission between two rooms. The point of departure for such an analysis is that we apply the energy balance equation (9-2) to the two systems. That gives

disin WWdt

dE,1,1

1 −= , (9-7)

disin WWdt

dE,2,2

2 −= . (9-8)

Referring to figure 9-1, the terms on the right-hand side are now divided up into different contributions. The power input to system 1 can be written in the form

2111,1 WWW in += , (9-9)

in which W11 is the power input into system 1 from a source in system 1, and W21 is the power input into system 1 from system 2. Moreover, the dissipation in system 1 can be expressed as

1211,1 WEW dis += ωη , (9-10)

the distinction is unimportant. For cases in which the loss factors are large, however, equation (9-5) should instead be seen as a definition of the system’s loss factor, distinct from that given in equation (3-74).


292

in which W12 is the power input to system 2 from system 1.

System 2E2

W21

W12

W11

W22

η 1 ω E 1

η 2

ω E 2

System 1E1

Figure 9-1 Two coupled acoustic systems: E indicates total energy and W power transport.

The first term in equation (9-10) represents the portion of the energy loss from system 1 that is completely lost, i.e., converted to thermal energy or radiated to the surroundings. In the same way, and using analogous symbols, we obtain for system 2 the relations

1222,2 WWW in += , (9-11)

2122,2 WEW dis += ωη . (9-12)

Putting equations (9-9) to (9-12) into (9-7) and (9-8) yields the differential equations

211112111 WWWE

dtdE

+=++ ωη , (9-13)

122221222 WWWE

dtdE

+=++ ωη . (9-14)

Equations (9-13) and (9-14) constitute a pair of coupled differential equations that describe how the energy builds up in the two coupled systems after a pair of external sources are turned on.

9.1.2 Relation between Wave Theory and Energy-Based Methods

The equations studied in the preceding section describe how the total acoustic energy of a system changes with time. In formulating these equations, no regard was given to the fact that the energy in the system is transported by waves that propagate, and are reflected, between the boundaries of the system. That aspect will now be given consideration, in order to better understand how a complete field description can, in the right circumstances, be simplified to an energy-based description. In that regard, we shall define the concept of a diffuse field and derive the relation between energy density and incident intensity for


293

such a field. These latter concepts are also necessary precursors to the derivation of the so-called Sabine’s formula in the next section. Our discussion and analysis in this section will be based on the case of a room with hard walls. The results obtained will have general validity and applicability to all types of acoustic systems in the many-modes region. We consider a sound field in the room that falls in the many-modes region, implying that it is built up of a large number of modes (standing waves). Assume that the field, in the immediate vicinity of some arbitrary point in the room, can be regarded as the superposition of propagating plane waves. As is evident from section 5.2, that is the case for a parallelepiped (rectangular prismatic) room. For an arbitrarily-shaped room, it is a reasonable assumption in the many-modes frequency region. Our assumption implies that the sound field, for a specific harmonic component, can be expressed as

) (ˆ rkti

nn ne ⋅−∑= ωpp . (9-15)

An energy-based analysis presupposes that the field can be regarded as built up of independent sound beams that may locally be considered as plane waves. The question is then under what circumstances the plane waves that build up the field in equation (9-15) can be regarded as independent. Since we are concerned with energy quantities, we determine 2p , which takes the form

rkirkin

nm

m

rki

nn

nmn eep ⋅−⋅∗⋅− ∑∑∑ == ppp2

2ˆ . (9-16)

Because the summation contains cross-terms m ≠ n, the waves are not independent, as we would desire. In order to obtain independent (uncorrelated) waves, a spatial averaging of the field is carried out. Assume that we average equation (9-16) over a sufficiently large volume; the cross terms will then disappear because

∫ ≈l

ikx dxel

0

01 ,

for a sufficiently large value of l. Moreover, the small remaining contributions from the cross-terms have differing signs and tend to cancel each other out. The conclusion can therefore be drawn that, after averaging a sufficient volume (with a diameter of the same order of magnitude as a wavelength), one obtains ∑≈

nnpp 22 ˆˆ , (9-17)

in which the particular form of brackets used indicates spatial averaging. Equation (9-17) means that the waves can be added in an energy sense and considered uncorrelated (compare section 1.11). In practice, our result implies that we must spatially average whenever we carry out measurements in connection with energy-based methods. Nevertheless, recall that we restricted our analysis to that of a single harmonic component


294

(pure tone). An alternative to spatial averaging, when broad-band sources are investigated, is averaging over frequency, i.e., taking measurements by frequency band. That is directly evident from equation (9-16), since the phase of the exponential function depends on the product kr.

We turn now to the relation between energy density and the intensity impinging on a wall in a room in which the sound field is completely dominated by modes (standing waves). In acoustics, rooms that have that character, i.e., are bounded by hard walls with little sound absorption, are called reverberant rooms. The opposite is a room with completely absorbing walls – a so-called anechoic room; in the latter, there are no modes, but rather just a free field surrounding any source. Assume that we have a sound field that fulfills equation (9-17), and that, moreover, all plane waves incident on a point in the room have the same strength and are uniformly distributed over all possible angles of incidence. A sound field that fulfills these criteria, and in which all points in the room are equivalent in the sense that they have the same energy density, is called an ideal diffuse field. The rms sound pressure in such an ideal diffuse field is

∑ ==n

nd pp 22 ˆ21~ (same strength of all waves 2

0~p ) 2

0~pN= , (9-18)

where N is the number of plane waves incident on a point in the room. Since equation (9-18) is the sum of plane wave contributions, the energy density ε is obtained by dividing that equation by ρ0c2 – see equation (4-99) – so that 2

02~ cpdd ρε = . The power incident

upon a surface S, belonging to the walls of the room, will now be calculated. We first regard a plane wave that is incident from a certain direction against S (see figure 9-2). The power that meets S is

nn Sc

pW θ

ρcos

~

0

20= , (9-19)

where θ n is the angle of incidence relative to the wall’s normal direction. The total incident power is obtained by adding up all the contributions from the diffuse field

.cos~

0

20 ∑∑

′′

′′ ==

nn

nnd S

cp

WW θρ

(9-20)

The summation in equation (9-20) is only over the modes n´ that are incident on the wall. When the number of modes is large, we can approximate equation (9-20) by an integral over all space angles constituting a hemisphere adjacent to the wall.


295

θn

k1

k 2

k n

Because the density of waves (number/space angle) is N /4π, then against every small space angle increment ΔΩ ,there are a number of waves NΔΩ /4π. For large N, one therefore obtains

∫ Ω≈ πθρ

4/cos~

0

20 NdSc

pWd . (9-21)

That integral can be solved by first expressing the space angle increment, making use of θ, in the form dΩ = 2πsinθ dθ, and then carrying out the integration over the interval 0 to π /2. The result is

c

SpNWd

0

20

4

~

ρ≈ . (9-22)

With the aid of equation (9-18), we can re-express that result in the form

==c

SpW d

d0

2

4

~

ρ(or, alternatively, in terms of energy density)

4cSdε

= (9-23)

in which the equality applies in the limit as the number of modes becomes infinite. We define the diffuse intensity Id as the power, as expressed in (9-23), per unit area, incident on a boundary surface (wall). That provides us the following relation, which is fundamental to classical room acoustics (sometimes called statistical room acoustics):

4

cI d

dε

= . (9-24)

The relation we have derived by considering sound in a room is very general, and applies as an approximation for the many-modes region to all types of wave fields bounded in a three-dimensional (3-D) system. For example, that relation can also be found in the theory of electromagnetic black-body radiation, in which it gives the electromagnetic power incident on the walls of a cavity in thermal equilibrium. The result in equation (9-24)

Figure 9-2 Diffuse sound field incident upon a wall.


296

depends, however, on the dimension number of the enclosed space. Repeating our derivation for the cases of two-dimensional and one-dimensional rooms, respectively, we obtain

π

ε cI d

d = , for 2-D (9-25)

and

2

cI d

dε

= , for 1-D. (9-26)

If equations (9-24) through (9-26) are to be applied to other types waves than airborne sound, it is important to note that the speed of sound is to be interpreted as that speed at which energy propagates, the so-called group velocity (see chapter 6). For sound waves in liquids and gases, that is normally the same as the ordinary sound speed (phase velocity), but for dispersive waves (i.e., those with a frequency-dependent phase velocity), the two wave speeds do differ; that is the case for bending waves, for example.

9.2 ROOM ACOUSTICS

In this section, we will make use of the results already obtained in this chapter to study sound fields in rooms. As mentioned in the introduction to the chapter, that is the classical application of energy-based methods in acoustics. The point of departure for the derivations that are made is a room with hard walls in the many-modes region. Additionally, all sound absorption is assumed to occur at reflections against the various surfaces in the room.

9.2.1 Sabine’s formula

Sabine’s formula provides a relation between the reverberation time T and the acoustic damping (absorption) of a room. In order to derive that formula, we make use of the relation between reverberation time T and the system’s loss factor (see example 9-1)

ηω)/10ln6(= ⋅T . (9-27)

In order to obtain the room’s loss factor, we use equation (9-3)

E

Wdisω

η = . (9-28)

We assume that the sound field in the room can be regarded as an ideal diffuse field with energy density εd. The total acoustic energy in the room can then be written in the form E = Vεd, where V is the volume of the room. When a plane wave with intensity In reaches a surface of the room, a certain sound absorption is obtained. If the surface has the absorption factor α(θ ), then (for the definition of α, see equation (5-27)) nnndisn SIW θα cos, = , (9-29)


297

where S is the surface area and the index n indicates the direction from which the corresponding wave is incident; see figure 9-2. The total absorbed power is obtained by a summation over all waves that are incident in the diffuse field. By analogous reasoning to that which led to equation (9-24), we can derive the following result for the dissipated power Wdis :

θθθαε π

dcS

W ddis ∫=

2

0

2sin)(4

. (9-30)

The average of the absorption factor defined by equation (9-30) is usually called the absorption factor for diffuse incidence,

θθθααπ

dd ∫=2

0

)2sin()( . (9-31)

Note that, because sin(2θ ) has a maximum at θ = 45°, the appearance of α(θ) around that angle is of great significance in the computation of αd. The absorption factors for perpendicular (θ = 0°) and grazing incidence (θ = 90°) have no influence whatsoever on the value of αd. In the case of a room with several absorbing surfaces that differ in their respective absorption characteristics, the total absorbed power becomes

∑=m

mmdd

dis Sc

W ,4α

ε . (9-32)

where the summation includes all surfaces m that contribute to the total absorption of the room as a whole. Sometimes, one even makes use of an average absorption factor for the room, defined as

S

Smm

md

d

∑=

,α

α , (9-33)

where ∑=m

mSS .

Making use of equations (9-28) and (9-32), we can now calculate the loss factor for the entire room as

V

Sc d

ωα

η4

= . (9-34)

That result can, making use of equation (9-27), be expressed as a reverberation time


298

S

VcSc

VTdd αα

161.0m/s)342with()10ln24(===

⋅= . (9-35)

That result is Sabine’s formula for the reverberation time of a room. The first derivation of it resembling ours just given was published in 1903 by W.S.Franklin. Sabine, himself, found the equation empirically after a series of experiments, around the year 1900. When the absorption characteristics of a room are to be described, one ordinarily reports either an average absorption factor, as in (9-33), or, alternatively, the so-called equivalent absorption area A. That area is defined as the total absorbing surface with α = 1, i.e., an acoustic "black hole", which would give the same absorption as the room considered. Mathematically, A can be computed from

mm

md SA ∑= ,α , (9-36)

where the unit for A is usually called [m2 Sabine], or abbreviated to [m2S]. Table 9-2 Absorption data for different materials. (Source: M D Egan, Concepts in Architectural Acoustics,

McGraw-Hill, 1972.)

Material Description Absorption factor αd 125 Hz 250 Hz 500 Hz 1 kHz 2 kHz 4 kHz Tile 0.03 0.03 0.03 0.04 0.05 0.07 Concrete Untreated 0.36 0.44 0.31 0.29 0.39 0.25 Painted 0.10 0.05 0.06 0.07 0.09 0.08 Plywood 1 cm 0.28 0.22 0.17 0.09 0.10 0.11 Window glass

0.35 0.25 0.18 0.12 0.07 0.04

Draperies Pressed thin against wall 0.03 0.04 0.11 0.17 0.24 0.35

Thick, drawn up 0.14 0.35 0.55 0.72 0.70 0.65

Concrete floor

0.01 0.01 0.02 0.02 0.02 0.02

with linoleum layer 0.02 0.03 0.03 0.03 0.03 0.02

with thick mat 0.02 0.06 0.14 0.37 0.66 0.65 Wood floor 0.15 0.11 0.10 0.07 0.06 0.07 Ceiling Gypsum slabs 0.29 0.10 0.05 0.04 0.07 0.09 Plywood 1cm 0.28 0.22 0.17 0.09 0.10 0.11

Besides the surfaces of the room itself, additional absorption of acoustic energy is also obtained from any objects (e.g., furniture) and persons that may be present. These added absorbing entities are normally characterized by a supplemental equivalent absorption energy ΔA. Table 9-3 gives examples of the supplemental absorption energy provided by certain objects.


299

Table 9-3 Added equivalent absorption area from several objects (Source: H Kuttruff, Room Acoustics, Applied

Science, 1973.)

added absorption area [m2S] Object Description 125 Hz 250 Hz 500 Hz 1 kHz 2 kHz 4 kHz With coat 0.17 0.41 0.91 1.30 1.43 1.47 Standing

human Without coat 0.12 0.24 0.59 0.98 1.13 1.12 Student, incl seat Sitting 0.20 0.28 0.31 0.37 0.41 0.42

Chair Cushioned 0.55 0.86 0.83 0.87 0.90 0.87

9.2.1.1 Limitations of the Sabine-Franklins theory and later work

There are a number of limitations of the theory that underlies Sabine’s formula. We will briefly discuss these, as well as touch upon the development of the field of statistical room acoustics after Sabine’s and Franklin’s classical work. One of the most important limitations of the theory is, of course, the assumption of an ideal diffuse field. That idealization has several parts, above all that the field consist of incoherent waves. As indicated in section 9.1.2, that condition is fulfilled in practice by taking spatial and frequency averages of our energy quantities. An ideal diffuse field should be, moreover, homogeneous and isotropic, i.e., all points in the room are to be equivalent, and all directions of propagation equally probable. That requires that the absorption be uniformly distributed throughout the room, so that certain directions of propagation do not become dominant. An example is a rectangular prismatic room in which a wall has a much higher absorption coefficient than all others, leading to a 2-D diffuse field more so than a 3-D one; in consequence, a longer reverberation time is obtained (compare equations (9-24) and (9-25)) than would be expected from a direct application of Sabine’s formula. In order that the assumption of homogeneity remain valid during the reverberation event, it must also not occur too rapidly. Since absorption occurs at surfaces in the room, there is a tendency for the energy density to be smaller in their immediate vicinity than elsewhere in the room. In order for the field to remain homogeneous, the typical time that it takes for sound energy to even itself out throughout the room must be shorter than the reverberation time. It can be shown that that limits the validity of Sabine’s theory to rooms in which <αd> ≤ 0.3. A long series of modified versions of the classical Sabine-Franklin theory of reverberant events have been proposed. For example, the work of R F Norris and C F Eyring was published around 1930. That work takes, as a starting point, a sound beam that propagates and is reflected against the surfaces of the room. The reverberant event is obtained as an average over all possible paths that the beam takes through the room. That approach is also the basis of most later work on energy-based methods for room acoustics. Despite the many advances made after the original development of Sabine’s theory, it nevertheless remains the standard approach normally applied in practice, in routine analysis of the reverberant behavior of a room. Sabine’s formula is also used in standardized measurements of the acoustic absorptive capacity of materials. Such measurements are performed in special reverberant rooms in such a way that Sabine’s theory can be assumed to apply. Measurements of this type must also take account of the


300

phenomenon of absorption (damping) within the medium itself. The effect of such damping is particularly important in rooms that have a large volume, and at high frequencies. Sabine’s formula, incorporating a correction for damping in the medium, is available from most acoustics handbooks, and from the ISO standards for reverberant room measurements; see section 2.3.1.

9.2.1.2 Measurement of reverberation time

Finally, this section will conclude with a discussion of how reverberation time can be determined experimentally. The standard method of measuring reverberation is based on excitation by a broad-band noise source in a certain frequency band (third-octave or octave-band). Because energy density is proportional to the square of the sound pressure, we can record a reverberant event by measuring the time decay of the sound pressure level after a source is abruptly interrupted. As a step in computing the sound pressure level, we must determine the rms sound pressure. For sound fields with an energy content that varies in time, the rms value is calculated by means of a so-called moving average, as

∫−

=t

Ttavav

dpT

tp ττ )(1)(~ 22 ,

where Tav must be much shorter than the reverberation time T. The traditional way to carry out the measurement is to record the sound pressure level on a print-out with a linear SPL scale.

1 0 d B

5 0 0 H z 2 0 0 0 H z 1 0 0 0 H z

1 0 m m /s

T = 4 .1 s T = 4 .6 s T = 5 .4 s

Figure9-3 Examples of reverberation curves measured in third-octave bands in the reverberation room at the

Marcus Wallenberg Laboratory for Sound and Vibration Research, MWL, KTH. If the recording paper unfurls at a constant rate, then an ideal reverberant event, in accordance with Sabine’s theory, would be drawn as a straight line. In practice, deviations from that ideal occur. In part, there are random fluctuations due to the finite averaging interval used to compute the rms value of the pressure; additionally, non-uniform distribution of absorption in the room results in some degree of curvature in the decay curve. An example of reverberation curves measured in a special reverberant room is shown in figure 9-3.


301

9.2.2 Sound fields in rooms

We now analyze a stationary sound field in a room due to a source located within it. A stationary sound field can be interpreted as one with a time-invariant frequency spectrum. Within the framework of an energy-based model, we can divide up the sound field into two parts: the sound beams that have just departed from the source and not yet been reflected by any surfaces; and, the sound beams which have already been reflected one or more times by surfaces. The first part of the field, which is not as yet affected by the room, can be regarded as the same field which would be radiated in a free field environment without bounding surfaces; if the source is, however, in the vicinity of a boundary, e.g., in the middle of the floor or in a corner, then a free field is only obtained in those directions in towards the interior of the room. The part of the total sound field that has, as such, free field character, is usually called the direct field.. The other part of the field, consisting of reflected sound beams, is usually called the reverberant field. The two fields can be considered uncorrelated, so that the total sound field can be obtained by adding the respective energy densities of each. In a stationary situation, the power input to the reverberant field must equal the power lost in reflections at the surfaces of the room. We assume that the reverberant field is an ideal diffuse field, and can then calculate the power dissipated in the room using equations (9-32) and (9-33), which yield

4

ScW dd

disαε

= . (9-37)

Reverberant field

W dir Direct field

Figure 9-4 Sound field in a room: decomposition into a direct field and a reverberant field.

The power Wdis must correspond to the power provided by the direct field minus that which is lost in the first reflection, i.e.,

)1( ddirdis WW α−= , (9-38)

where Wdir is the power sent out by the source. Note that we have chosen to use the absorption factor for diffuse incidence in equation (9-38). That choice was not obvious, and implies, in fact, that we have assumed an isotropic source, and that the direct field spreads out so as to reach the surfaces of the room with a uniform distribution of angles of


302

incidence. For a strongly directional source, or a room with non-uniform distribution of absorbent, that assumption doesn’t hold, and another value of the absorption should be used instead. The value we selected corresponds, nevertheless, to the conventional assumption made in such analyses. The energy density of the direct field, at such distances that sound beams can be regarded as locally plane waves (i.e., in the far field – see (4-100) and (4-123)), is given by

24 cr

WΓ dirdir

πε = , (9-39)

where Γ is the directivity indexError! Bookmark not defined. (direction factor) of the source, which indicates how the sound beam varies in different directions, and r is the distance from a reference point at the source (origin) to an observation point. The directivity index is defined such that the integral of Γ over all space angles that point from the source into the interior of the room, is 4π. For a source that radiates equally in all directions (isotropic source), and is located in the middle of the room, Γ = 1; for the same source on the floor, Γ = 2; for that same source located at an edge between the floor and a wall, Γ = 4; and, for the same source located in a corner of the room, Γ = 8 (see figure 9-5).

Γ = 1 Γ = 2

Γ = 4

Γ = 8

At a distance from all reflecting surfaces Close to one

reflecting surface

At the intersection of two reflecting surfaces

At the intersection of three reflecting surfaces

Figure 9-5 Directivity index Γ (direction factor) for an isotropic source located at various positions in a room with perpendicular surfaces. (Picture: Asf, Bullerbekämpning, 1977, Ill: Claes Folkesson.)

From (9-37), (9-38) and (9-39), the total energy density of the room can be expressed as

ScW

cr

ΓW

d

dirddirddirtot α

α

πεεε

)1(4

4 2

−+=+= . (9-40)


303

Both energy densities in this equation are related to the rms sound pressure in accordance with the plane wave equation (4-99). Equation (9-40) gives, therefore, an rms sound pressure, from a source in the room, of

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −+=+=

Sr

ΓcWpppd

ddirddirtot α

α

πρ

)1(4

4~~~

20222 . (9-41)

Alternatively, expressing the result in terms of sound pressure level in air, taking specific impedance to be ρ0c = 400 Pa s/m, then

⎥⎦

⎤⎢⎣

⎡′

+⋅+=Ar

ΓLL dirW

totp

44

log102π

[dB] (9-42)

where

)1( d

d SA

αα−

=′ ,

is called the room constant. The distance from the source at which the direct field and the reverberant field are equally strong is called the echo radius. By equating the two terms in parentheses in equation (9-42), the echo radius is found to be

21

16⎟⎠⎞

⎜⎝⎛ ′

=πΓAre . (9-43)

Figure 9-6 illustrates the main features of the variation of sound pressure level with distance from a source in a room.


304

Lpd

Lpd − 5

Lpd − 10

Lpd + 5

Lpd + 10

Total sound pressure level

Sound pressure level in reverberant field

Sound pressure level in direct field

3 dB Lp

tot [dB]

1,00 2,0 3,0 Avstånd/Ekoradie r/re

Lptot

Lpd

Lpdir

Figure 9-6 The distance-dependence of sound pressure level, with respect to the level in the reverberant field,

according to equation (9-42). The distance to the source is measured in echo radii; see equation (9-43). Note that when the distance from the source is equal to the echo radius, the direct and reverberant fields contribute equally to the total sound pressure level, which therefore rises by 3 dB.

In practical applications of the results obtained above, A’ is usually interpreted as the equivalent absorption area A (see equation (9-36)), i.e., the effect of the factor (1-<αd>) is ordinarily ignored. In the context of Sabine’s theory, that is in fact reasonable, considering that the influence of that factor on A is proportional to <αd>2, which should be insignificant in the frequency region in which the theory is valid, i.e., when <αd> ≤ 0.3 (see section 9.2.1.1). Finally, we note that the value of the sound power emitted by the source, Wdir, is dependent on the location of the source. As we will see in chapter 8, that is because the radiation impedance experienced by the source depends on its proximity to large (in terms of the Helmholtz number) reflecting surfaces. That also means that the sound power emitted by the source will depend on its location in the room, i.e., whether the source is placed on the floor or in a corner. That is an important consideration to be born in mind in practical applications of the results obtained above.

9.2.3 Acoustic absorbents

The most common measure of the acoustic behavior of a room is the reverberation time. Reverberation time alone, however, is rarely sufficient in and of itself. The distribution of the direct field, as well as reflections that might give rise to echo effects, must also be taken into consideration in, for instance, a lecture hall. Especially demanding venues, such as concert and opera houses, are usually designed with the aid of special computer

Distance / Echo radius


305

programs that, based on geometrical acoustics, predict the distribution of the sound field through the first few reflections.

Volym

StudiosKonferensrum

T[s

]

[ ]

0.5

0

1.0

1.5

2.0

2.5

10 100 1000 10000

KonsertsalarKatolska kyrkor

60,

, Figure 9-7 Optimal reverberation time in various locales, for the frequency range 500-1000 Hz, according to

L.L. Doelle, Environmental Acoustics, McGraw-Hill, New York, 1972. To perform well, locales of different types require different reverberation times, according to their respective functions. To bring about a desired reverberation time in a locale, while minimizing undesirable reflections (echoes), different kinds of acoustic absorbents are used. These absorbents are normally fixed to different reflecting surfaces, such as walls and the ceiling. In this section, we will concentrate on some basic kinds of absorbents, and discuss the mechanisms of sound absorption that they display. The absorption factor concept provides a measure of the performance of an absorbent material; for the special case of normal plane wave incidence, it is defined by equation (5-27). That definition can, moreover, be generalized to arbitrary sound fields without modification. In that case,

i

rWW

−=1α , (9-44)

where the symbols in the numerator and denominator of the second term indicate total reflected and total incident acoustic power, respectively, from/upon a given area S. It is important to note that the absorption factor, in the general case, depends on both the surface characteristics (including geometric form) and on the incident sound field. For the case of plane waves, and a sufficiently large (in terms of Helmholtz number) plane surface (the characteristics of which are identical at all points, and independent of direction, i.e., the surface is acoustically homogeneous and isotropic) the factor is only a function of the angle of incidence, which is specified by the angle θ between the incident wave normal and the surface normal. That assumption is standard in the derivation of formulas in room

Volume [m3]

Catholic churches Concert halls Conference halls Sound studios


306

acoustics, and in the practical application of the latter. For instance, we use that assumption in our derivation of the absorption factor αd for an ideal diffuse field (see equation (9-31)).

9.2.3.1 Porous absorbents

Porous absorbents consist of solid material with a degree of porosity (i.e., materials with cavities interconnected by channels) that permits air to be forced into the interior of the material matrix. Most often, porous absorbents are fibrous materials consisting of thin (2-20 μm) mineral or glass wool fibers, arranged in layers and with random fiber directions in planes parallel to the material surface. Alternatively, fibrous materials based on natural wood fibers are also available for building applications. For high-temperature applications, on the other hand, pressed metal wool might sometimes be preferred. When sound propagates in a porous fiber absorbent, acoustic energy dissipation results from the viscous forces arising as air is forced to flow through the small passages between the fibers. Additionally, heat transfer adds to the effect, to some extent; the temperature fluctuations inherent in a sound wave are evened out by contact with the fibers, which are better heat conductors than the air itself. That process is never completely reversible, implying that losses occur. Another mechanism at work, and which may be significant for sound absorption, is the coupling between sound and vibrations in the porous material’s solid matrix. That phenomenon is normally negligible for fibrous porous materials, except at low frequencies (under 300 Hz). For other types of porous materials (e.g., foam), however, the effect of the ”fluid-structure” coupling may be quite significant. The classical model of a porous absorbent, mainly applied to the fibrous type, is based on considering the absorbent to be an equivalent fluid. In its simplest form, the approach replaces the absorbent by a homogeneous fluid with viscous damping. That viscous damping can be incorporated as an extra term in the equation of motion, equation (4-26),

00 =+∂∂

+ xx xi upu φωρ , (9-45)

in which we have assumed a harmonic signal and written the equation in complex form, and in which φ refers to the flow resistance [Pa s/m2]. Assuming a homogeneous and isotropic absorbent, the equations in the y and z directions are completely analogous. In practice, however, the flow resistance perpendicular to the absorbent surface is usually greater than that parallel to it. As a result, we have to specify a flow resistance for each coordinate direction φx, φy and φz. The flow resistance in our model could be frequency-dependent and complex. Handbook data on fiber materials is usually limited to values of φ for the case of a time-invariant flow (i.e., the case α = 0). Such a measurement is made by measuring the pressure drop of a steady flow over a slab of absorbent material, as in figure 9-8. From equation (9-45), it is evident that the pressure, in that case, varies linearly in x, so that

( )

hpp

u x

211 −=φ , (9-46)

where h is the thickness of the slab.


307

x

p p1 2

ux

Figure 9-9 shows two typical flow resistance (φ) curves for fiber-type absorbent materials. Because the standard unit of φ in the SI-system is very small in magnitude, it is customary to instead use the corresponding unit in the cgs (cm-gram-second) system. Since φ represents a wave resistance per unit length, and the cgs unit of wave resistance is called a Rayl, the cgs unit of φ is therefore a Rayl/cm. To convert to SI units, we note that 1 Rayl/cm = 103 Pa·s/m2. The absorption factor of a porous absorbent is ordinarily determined from measurements specified by ISO standards. Traditionally, the absorption is either measured for the case of perpendicular incidence, using standing waves (in a so-called Kundt’s tube), or it is measured in a reverberant room. By measuring the difference in the reverberation time between the room with, and the room without, the absorbent, and making use of Sabine’s formula, the diffuse field absorption factor can be inferred. Calculating the absorption directly is also possible in some cases. For fibrous absorbents, for example, one might use the equivalent fluid model described in connection with equation (9-45). A necessary condition for such an approach, however, is access to reliable material data on the absorbent, which is often difficult to come by in practice. The most common way to use a porous absorbent in room acoustics is to locate it in front of a hard wall, for the purpose of reducing the wall reflections. We now consider that approach, in order to determine how to choose the flow resistance and absorbent thickness, in order to obtain the best possible sound absorption. The discussion will be based on simulated data obtained from a semi-empirical equivalent fluid model developed by Delany and Bazley (1969).

Figure 9-8 The principle for the measurement of the flow resistance φ of an absorbent material. By measuring the pressure drop p1 – p2 and the flow rate ux over the absorbent test sample, the flow resistance follows from equation (9-46).


308

10 100 1

10

100

Density [kg/m3]

Flow resistance φ, [Rayl/cm]

Figure 9-9 Typical values of the flow resistance φ perpendicular to the surface of glass wool

absorbents (solid line) and rock wool (dashed line). This model is described in acoustics handbooks, and requires, as a prerequisite, the measurement of the flow resistance φ. In figure 9-10, some typical results are provided. As is clear from the figure, “optimal” absorption properties are obtained if we choose a dimensionless flow resistance that satisfies

2/ 0 ≈ch ρφ , (9-47)

where h is the absorbent thickness. The vicinity of the optimum is, nevertheless, quite flat, so that any value in the interval 1 - 3 would be almost as good. Values far outside of that interval, however, would give clearly inferior absorption performance.


309

3 2 1 10

0,1

kh

α ⊥

h 0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0

0,0

0,2

0,4

0,6

0,8

1,0 Absorption factor

Figure 9-10 Calculated absorption factor for a plane wave with normal incidence against a fiber-type absorbent

mounted in front of a hard (completely reflective) wall. The parameter in the diagram is the dimensionless flow resistance φ h/ρ0c.

Equation (9-47) shows that the thinner the absorbent that we select, the greater the flow resistance needed to obtain a good sound absorption. For the optimal case, we obtain α⊥ ≥ 0.9 for a Helmholtz number kh greater than about 1, where k is the wave number in air and α⊥ is the absorption factor for normal incidence. That result is usually expressed as a rule of thumb: that an (optimal) absorbent can be expected to provide good absorption if its thickness is about a quarter wavelength. All curves in figure 9-10 seem to approach an absorption factor α⊥ = 1 at high frequencies, which is equivalent to the reflection coefficient approaching zero. According to equation (5-68), this means that the impedance Zabs of the absorbent must approach the specific impedance of the surrounding air. In other words, at high frequencies, a sound wave passes directly into the absorbent without reflection, and is then gradually converted into thermal energy as it propagates further. Example 9-2 Assume that we wish to dimension an absorbent to be placed in front of a hard wall that is to give α⊥ ≥ 0.9 above 500Hz. Solution The condition that kh ≈ 1 at 500 Hz gives

h = c/ 2πf = {c = 340 m/s} =0.11 m,

and we therefore choose h = 10 cm. Equation (9-47) now provides the optimal value of the flow resistance φ

{ } 230

0 s/mPa100.8s/mPa4002

⋅⋅=⋅==≈ ch

cρ

ρφ .


310

If we choose rock wool as our absorbent, then figure 9-9 shows that the selected flow resistance corresponds to a density of about 45 kg/m3. In figure 9-11, below, the calculated absorption factor in that case (case I) is shown. For comparison, the figure also shows the absorption obtained if we only use an “optimal” absorbent with h = 5 cm (case II). As would be expected, that about doubles the frequency at which α⊥ ≥ 0.9. Note that, for that thickness, double the flow resistance is needed to attain the optimum case. A very similar curve is obtained in case III, in which we move the same absorbent as in case II to a position 5 cm away from the wall. The low frequency damping is hardly affected at all, but we do get some fluctuations at higher frequencies. If the absorbent is moved too far from the wall, say more than 4-5 times its thickness, then while the low frequency performance is certainly good, there can be unacceptably large fluctuations in the high frequency performance. That approach to improving the low frequency performance, for a given absorbent thickness (say 5 cm), is often applied in practice, e.g., by mounting the absorbent to bolts providing something of an air column. An example would be lecture halls and auditoria in which the ceiling absorption is enhanced by mounting absorbent panels a couple of decimeters below the inner ceiling.

Fall I

Fall II

10 cm

5 cm

Fall III 5 + 5 cm

I II III

Frequency, [Hz]

α ⊥

Absorption factor

0 1000 2000 3000 4000 5000 6000 0

0,2

0,4

0,6

0,8

1,0

Figure 9-11 Calculated absorption factor for c = 340 m/s, for the three cases described in the example.

In many applications, the porous absorber must be given some kind of a covering layer. The purpose of that is partly to prevent the loss of fibers to the surroundings, which could constitute a health risk, and partly to protect the absorbent from various external agents, such as humidity and gases with high flow rates. The cover layer normally consists of a thin plastic foil or the like, or perhaps perforated plate. A cover layer normally degrades the acoustic performance of the absorber, especially at high frequencies; that is because the mechanical role of the absorber is that of an equivalent mass, which makes it more difficult for sound to penetrate into the absorbent. Assume that a 20% reduction of the absorption factor, from 1 to 0.8, is permissible at high frequencies. Then, it can be shown that, for a cover layer with surface density m″ and with negligible bending stiffness, the reduction in absorption is inconsequential up to a frequency of


311

mc

f g ′′=

πρ

20 . (9-48)

Example 9-3 Assume that we would like to cover an absorbent by a cover layer, without degrading the absorption characteristics at frequencies under 3000 Hz. What is the maximum permissible surface density?

Solution Using equation (9-48), we find m” ≤ 22 g/m2. For perforated plate cover layers, an inertia effect also occurs; that is attributable to the local acceleration of the fluid at the perforation holes. Moreover, the mechanical characteristics of the plate also have an effect. If the plate is compliant in bending, it provides an added inertia, which should be added to that of the holes. Formulas to calculate the added acoustic mass can be found in most acoustic handbooks; see, also, chapter 10.

Figure 9-12 In hard environments, with long reverberation times, it may be necessary to increase the absorption

by installing added absorbent in the form of, for example, porous absorbers. These are often installed as ceiling baffles, along walls, or in other places where they do not hinder the room’s normal activities. (Picture: Asf, Bullerbekämpning, 1977, Ill: Claes Folkesson.)

9.2.3.2 Resonant Absorbers

This kind of absorbent consists of acoustic systems that provide sound absorption in frequency bands around the vicinity of their resonance frequencies. These are primarily used to obtain sound absorption at low and mid frequencies, up to 500 - 600 Hz. The types

Sound absorbing baffles

Sound absorbing suspended roof

Traverse


312

of absorbers considered up to this point have all been broad-band, with a lower frequency bound approximately corresponding to a quarter wavelength equal to the absorbent thickness. As a result, such absorbers are most suitable for high-frequency applications, above about 500-600 Hz. Porous absorbers that provide good low frequency absorption must be made unreasonably thick (e.g., at 100 Hz, a thickness λ /4 ≈ 75 cm), so that such an approach is only undertaken in very special cases, such as anechoic rooms in laboratories. As an introduction to the discussion of resonance absorbers, a plane surface with an impedance of Z is considered. We assume that that impedance is equivalent to a spring-mass system with damping, giving

dRi

mi ++′′=ωκωZ , (9-49)

where m″ is the mass per unit area, κ is the spring constant per unit area, and Rd is a term that describes the system damping. Note that these parameters can depend on the details of the incident sound field. Otherwise, we have what is called a locally reacting surface; see section 5.1.5. At the resonance frequency ω0, defined as the frequency at which Im(Z) = 0, the impedance is only due to damping: Z = Rd. According to equation (5-68), the reflection coefficient at that frequency, for an incident plane wave, is

id

idcRcR

θρθρ

coscos

0

0+−

=R . (9-50)

As is evident from that equation, we can obtain zero reflection, i.e., R = 0, for a certain angle of incidence, by choosing Rd = ρ0c / cosθ i. ´For the case of normal incidence, θ i = 0, in particular, Rd must be chosen so as to equal the specific impedance of the surrounding fluid. The most common way to bring about a resonant absorber is to create the direct acoustic equivalent of a mass-spring system, a so-called Helmholtz resonator; see chapter 10. Such a resonator consists of a constriction (mouth), which gives rise to an inertial effect, and an enclosed fluid volume, which corresponds to a spring; see figure 9-13. To obtain the desired damping, the resonator can be partially, or completely, filled with porous absorbent material.

The most effective approach is to locate the absorbent in, or near, the constriction. A Helmholtz resonator is a single degree-of-freedom system, and only gives, therefore, a single resonance frequency about which sound absorption is to be expected. Another common way to realize resonant absorption is to place an absorbent behind a compliant plate (panel); figures 9-14 and 9-15 illustrate that type of system, commonly referred to as a panel absorber.

Figur 9-13 Absorbent consisting of a group Helmholtz-resonators.


313

Low frequency Low frequency Low frequency High frequencyHigh frequency High frequency

Influence of the distance between stffeners

Influence of the plate thickness Influence of the distance to the wall

Figure 9-14 By varying such design parameters as the distance between fastening points (i.e., the stiffness), the panel thickness, and distance from the wall, the effective frequency band of a panel absorber can be tuned to some extent. (Picture: Asf, Bullerbekämpning, 1977, Ill: Claes Folkesson).

The lowest resonance frequency of this system occurs when the contained air volume acts as a spring, and the panel as a mass. At higher frequencies, standing waves build up between the wall and the panel, giving rise to a number of resonance frequencies (compare the phenomenon of standing waves in cavities, section 5.2). Normally, however, the systems behavior is merely optimized for the lowest resonance frequency. That resonance can be calculated if we know the spring constant κ of the enclosed volume of air. To calculate κ, consider a harmonic plane wave, with perpendicular incidence; from the equation of continuity (eq (4-11)),

00 =∂

∂+

xi xuρρω , (9-51)

in which we have assumed that the entire space between the panel and the wall is air-filled. If we assume low frequencies (kh « 1), then the particle velocity field in a standing wave, directly in front of a hard wall, varies linearly, while sound pressure, and thereby density as well, are largely independent of x; see figure 5-2. Integration of equation (9-51), at low frequencies, therefore yields

0))0()((0 =−+ xx hhi uuρρω ,

Wi Wr

x wall

absorber

panel 0h

Figure 9-15 Panel absorbent mounted to a hard wall. The panel is assumed to be compliant in bending, and has a surface density m''.


314

where ρ is assumed constant (independent of x). The boundary condition at the plate gives ux(0) = vx, and, at the hard wall, ux(h) = 0. Substituting the relation between sound pressure and density (p = c2ρ, equations (4-31) and (4-44)), finally, gives

)( 2

0chix

ρω

vp = .

From that equation, the spring constant κ (per unit surface area) is directly obtained as

hc 2

0ρκ = . (9-52)

For oblique incidence at an angle θ , analogous logic finds that κ changes by a factor of 1/cos2θ. With absorbent behind the panel, another (lower) speed of sound applies. If the entire space is filled with absorbent, the isothermal sound speed in air is a good approximation for c at low frequencies. That speed can be obtained by dividing the adiabatic sound speed by γ , after which the fundamental resonance frequency of the panel absorber can be calculated according to

{ }

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

′′

′′≈=

′′=

′′=

isothermal,50

adiabatic,60

airfor21

21 2

00

hm

hm

hmc

mf

ρπ

κπ

, (9-5

in which normal temperature and pressure of air have been assumed, when putting in the density and sound speed. For a panel absorber which partially filled with absorbent, the resonance frequency falls somewhere between the two values given by equation (9-53). Figure 9-16 shows the calculated absorption factor, for a normally-incident plane wave, against a panel absorber filled with absorbent. The calculation has been carried out by assuming that the system’s impedance can be written as Z = iωm″ + Zabs , in which the absorbent’s impedance is obtained from the model used earlier in this section (Delany and Bazley(1969)). For comparison, we can calculate the resonance frequency of the system in figure 9-16, from equation (9-53) (isothermal case). That yields

130m 05.0

kg/m 0.350 2

0 ≈⎭⎬⎫

⎩⎨⎧

==′′

=′′

≈h

mhm

f Hz ,

which is somewhat lower than the peak observed in figure 9-16, lying around 135 Hz.


315

α ⊥

Frequency [Hz]

Absorption factor 1,0

0,8

0,6

0,4

0,2

0,0 50 100 150 200 250 300 350 400

Figure 9-16 Calculated absorption factor for a panel absorbent (see figure 9-15). Data: m'' = 3.0 kg/m2, h = 5.0 cm, φ = 3.0 ⋅103 Pa ⋅s/m2 .

Absorbing walls

Engine test benches

Hard wall

Wooden beam Porous board covered with plastic laminate

Figure 9-17 In the low frequency region, porous absorbers are inadequate. Panel absorbers may then serve as

practical alternatives to reduce high levels of noise in, for example, workshop environments. (Picture: Asf, Bullerbekämpning, 1977, Ill: Claes Folkesson.)


316

9.2.4 Sound Transmission through Insulating Partitions

To round off the treatment of room acoustics, consideration is now given to the transmission of sound through insulating partitions used in or between rooms (shields and walls, respectively). The purpose of such a partition is to stop the propagation of sound, mainly by providing an impedance jump that reflects the sound incident upon it. To characterize the ability of an insulating partition to stop sound, the concept of the transmission factor τ and the sound reduction index R are used. The transmission factor of a partition with an area S is defined as

i

tWW

=τ , (9-54)

in which the denominator contains the total incident acoustic power Wi, and the numerator the total transmitted acoustic power Wt. It is important to note that the transmission factor, in general, depends on the mechanical properties of the partition itself, and its geometric form, as well as the specific form of the incident sound field. For the special case of plane waves, and a sufficiently large (measured in Helmholtz numbers) plane partition surface (which is acoustically homogeneous and isotropic), the factor is only a function of the direction of the incident wave relative to the normal direction (the angle of incidence θ). That is the case we consider in this treatment. The transmission factor of most insulating partitions is in the range 10-6 - 10-2. It can be greater at low frequencies, however. The sound reduction index is defined as

τ1log10 ⋅=R . (9-55)

A deeper understanding of the concepts of absorption and transmission factors can be obtained from a consideration of figure 9-18. The figure illustrates the power balance that must hold for a sound field incident upon an insulating partition.

Wi

Wr Wt Wdis

Absorber

Plate

Using the notation from the figure, the power balance can be expressed in the form

distri WWWW ++= ,

Figure 9-18 Power balance for a sound fieldincident upon an insulating partitionconsisting of absorbent attached to aplate.


317

where Wdis is the power lost to damping in the absorber or elsewhere. Using the definitions of α and τ , that can be written as

α = τ + δ , (9-56)

where δ = Wdis / Wi is the dissipation factor. For an absorbent mounted to a plate (figure 9-18), and except in the few-modes region (see section 9.2.4.3, figure 9-23), it is normally the case that τ « δ , which implies that the absorption factor is not significantly influenced by the plate (i.e., the plate can be considered “hard”). The reverse is not true, however. The transmission factor can be significantly altered by covering a wall with an absorbent, if the absorption factor is nearly one. We now consider the sound transmission between two rooms (systems 1 and 2) separated by a wall. Our point of departure is a stationary sound field emitted by a source in one of the rooms, in room 1 for example. Making use of the energy balance equations for two coupled systems, equations (9-13) and (9-14),

21111211 WWWE +=+ωη , (9-57)

122122 WWE =+ωη ; (9-58)

see figure 9-19. The loss factors in the respective systems (rooms) can, if we assume ideal diffuse fields, be calculated from equation (9-34),

1

11 4 V

cAω

η = , (9-59)

2

22 4 V

cAω

η = , (9-60)

where A1 and A2 , are the equivalent absorption areas of the rooms in accordance with equation (9-36).

Figure 9-19 Sound transmission between two rooms: W is sound power, ε energy density, V volume and S the

area of the insulating partition. Compare to figure 9-1.

Source

Room 1 Room 2


318

In order to compute the power transmission between the rooms, we consider the diffuse fields incident upon the wall separating them, and make use of the definition of the transmission factor

SIW dd 1,12 τ= , (9-61)

SIW dd 2,21 τ= . (9-62)

in which Id signifies the diffuse intensity and τd the transmission factor for a diffuse field. That factor can be calculated by an approach analogous to that for αd in equation (9-31). Note that the transmission factor from room 1 to 2 is assumed identical to that from 2 to 1. It can be shown that that reciprocity principle for transmissions factors has general validity. By expressing the field’s energy (E = εV) and diffuse intensity (see equation (9-24)) as energy densities, and putting those into the equations given above, the following relation then it follows from equation 9-58 that

dd

d

SAτε

ε 2

2,

1, 1 += . (9-63)

Because 20

2~ cpdd ρε = , an expression for the difference in the sound pressure level between the rooms is obtained directly from equation (9-63)

)1log(10 22,1,

dpp S

ALL

τ+⋅=− . (9-64)

That result shows that the sound reduction between two rooms depends on both the partition’s sound reduction index and the absorption in the receiving room (room 2). It can be observed that, if the absorption in the receiving room is zero, the same sound level is attained in each room, regardless of how good the insulating wall is. Equation (9-64) is the basis of a technique for the experimental determination of the transmission factor. Assume that we have two reverberant rooms, coupled by a rectangular aperture in their separating wall. A wall element to be tested is then mounted in the aperture. By measuring the difference in the sound pressure level between the rooms, with a sound source in one of them (the sender room, room 1), the transmission factor can then be inferred from equation (9-64). A necessary precondition is, however, that we have first determined the receiver room’s (room 2’s) equivalent absorption area, by a measurement of the reverberation time and an application of Sabine’s formula. In practice, the unit-valued term in the parentheses of equation (9-64) is normally ignored in the application of that equation to the prediction of the sound reduction between two rooms. The equations can then be written in the form

S

ARLL dpp

22,1, log10 ⋅++= . (9-65)

That approximation usually results in an error of less than 1 dB, which is negligible in practice, especially considering that our idealized model (assuming ideal diffuse fields) is already approximate in most rooms.


319

9.2.4.1 Sound reductions across composite partitions

Often, instead of being completely homogeneous, the area of the partition may in fact consist of distinct sub-areas (sub-elements), each of which differs markedly from adjacent areas in its insulating characteristics; for example, a wall with windows is such a composite partition. Additionally, a real partition may have openings or fissures, which, although they may be small in area, have transmission factors that approach 1 at high frequencies; consequently, even relatively small openings can drastically degrade the sound reduction index of the partition. In order to determine the sound reduction index of a composite partition, we assume that the incident sound power is uniformly distributed over the area of all sub-elements of the partition. That is the case for an ideal diffuse field, for instance. This assumption, together with our definition of the transmission factor (equation (9-54)), directly yields

∑∑∑

====n

nni

ninn

i

nnt

i

t SSW

WSS

W

W

WW

)(

)(,

τ

τ

τ , (9-66)

where the index n specifies a sub-element, and ∑=n

nSS . For a sound reduction index R,

one obtains the expression

∑ −⋅

⋅=

n

Rn nS

SR1010

log10 . (9-67)

Example 9-4 A hole is to be made in a 10 m2-wall, with a 40 dB sound reduction index. How large a hole is permissible, if the sound reduction index is not to be degraded by more than 10 dB? Solution Assume that the transmission factor of the hole is 1. That certainly applies at high enough frequencies that the in-plane dimensions of the hole are large, in terms of Helmholtz number. If we let x = Shole /S, then equation (9-67), upon substitution of the pertinent values, yields the expression

)10)1(10log(1030 40 −⋅−+⋅⋅−= xx ,

the solution of which is x = 9⋅10-4 , i.e., Shole = 9⋅10-3 m2. That roughly corresponds to a square hole with 10 cm sides.

9.2.4.2 Flanking transmission

Flanking transmission is a collective designation for the contributions to the sound field in a receiving room made by all transmission paths save the direct one. Figure 9-20 illustrates the situation. Transmission between two rooms can take place by way of a number of different paths: the direct path 1; the indirect path 2; etc. The direct path is that described by the transmission factor given above, in equation (9-54); it corresponds to airborne


320

sound incident on the separating wall, which in turn vibrates, and thereby constitutes a sound radiator in the receiving room. In practice, however, there is always a certain amount of flanking transmission, and the actual sound reduction is therefore somewhat lower than that predicted by considering the direct path alone; that degraded sound reduction index is called the field reduction index.

1

2

3

That field reduction index is usually defined by

1

1

1, )log(10)/log(10 −

≠

− ∑∑ ′+⋅=⋅=′m

mim

mt WWR ττ , (9-68)

in which the index m specifies the transmission path and

i

mtm W

W ,=′τ , m ≠ 1.

9.2.4.3 Simple wall

We now consider the sound reduction index of a homogeneous plate surrounded by a fluid (e.g., air). When a sound wave strikes one side of the plate, vibrations result. If it is a thin plate, its response consists of bending waves. Sound radiation by those bending waves, on the other side, is the mechanism by which the transmitted sound field then arises in the receiving room. In order to analyze the transmission, we assume that we have an infinite plate upon which a plane wave is incident. If we choose a coordinate system so that the incident wave lies in the x-y-plane (see figure 9-21), then the bending wave field in the plate satisfies

))(()( 4

42

trixp iy

Dm pppv −+=∂

∂+′′− ωω , (9-69)

where m” is the mass per unit area of the plate, Dp is the bending stiffness, and we have assumed that all fields have harmonic time-dependencies. Equation (9-69), with the substitution ξ = vx / iω, corresponds to the bending wave equation (6-57) found in section 6.5.1, except that displacement occurs in the x rather than the z-direction.

Figure 9-20 Transmission paths between two rooms. Path 1 is the direct transmission path, while paths 2 and 3 are examples of other, indirect, paths. The transfer of acoustic energy via the indirect paths is usually referred to as flanking transmission.


321

pi

prp t

x

y

Forced bending wave

θ

θ θ

The sound fields in equation (9-69) are symbolized as in section 5.1.3. Accounting for Snell’s law, we obtain the following result for these fields where they contact the plate (i.e., at x = 0),

,ˆ )( ykti ye −= ωαα pp (9-70)

where α = i, r, or t. The particle velocity field vx must have the same y-dependence as the sound field, i.e.,

)(x ˆ ykti

xye −= ωvv . (9-71)

Substituting equations (9-70) and (9-71) into (9-69) gives

trixyp

i

kDmi pppv ˆ)ˆˆ(ˆ)(

4−+=+′′

ωω , (9-72)

where ky = k sinθ and the angle θ is the same for all three sound fields. To solve our problem, we need two additional equations. These are obtained by requiring that the normal velocities of all fields, at the plate, be equal. With the aid of equation (5-62),

c

tx

0

cosˆˆ

ρθp

v = , (9-73)

and tri ppp ˆˆˆ =− . (9-74)

Eliminating xv and rp from equation (9-72), using equations (9-73) and (9-74), leads to the result that

)ˆˆ(2cosˆ

)(0

4

tityp

cikD

mi ppp

−=+′′ρ

θω

ω .

From this equation, we can directly determine the transmission factor as (Note: the power transmission in the x-direction, of both the incident and the direct fields, contains the same factor cosθ / ρ0c, which therefore cancels),

Figure 9-21 Sound transmission through a homogeneous (infinite) thin plate of thickness h. The fluid on each side is assumed to have the same sound speed and specific impedance as that on the other. The incident plane wave excites a forced bending wave with phase velocity c/sinθ.


322

2222

0

2

2

sin12

cos1

1

ˆ

ˆ)(

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛ ′′+

==

c

i

t

cm

p

p

ωθω

ρθω

θτ , (9-75)

in which ωc = 2πfc is the circular frequency corresponding to the so-called coincidence frequency fc ,

p

c Dmcf

′′=

π2

2 . (9-76)

At that frequency, the bending wave speed is equal to c. Evidently, for the case of the lossless simple wall that are analyzing, complete transmission is obtained at a frequency corresponding to fc /sin2θ . At that frequency, the phenomenon of coincidence occurs, i.e., the bending wave speed is identical to the phase velocity (c/sinθ) of the sound wave moving along the plate in the y-direction. It is only for the special case of normal incidence (θ = 0°) that the simple wall does not exhibit a coincidence effect. For normal incidence, no bending waves are excited in the plate; instead, the entire plate moves in unison as a large piston, and only the mass of the plate is “felt” by the sound wave. With respect to frequency, we can distinguish three regimes that characterize the infinite simple wall.

(i) f < fc.

In this regime, the inertia terms dominate in the bending wave equation of the plate. If we ignore the bending stiffness term, then equation (9-75) provides a transmission factor of

{ }2

002

0

2

2

2cos » antag

2cos1

1

ˆ

ˆ)(

−

⎟⎟⎠

⎞⎜⎜⎝

⎛ ′′≈1′′=

⎟⎟⎠

⎞⎜⎜⎝

⎛ ′′+

≈=c

mcm

cmp

p

i

t

ρθωρω

ρθω

θτ .

From that equation, we can obtain a sound reduction index, for normal incidence (θ = 0°), of

{ } cffmfairc

mR <−′′⋅+⋅≈=′′

⋅=⊥ ,42log20log202

log200ρ

ω (9-7

and for diffuse incidence (after integration), of

3−= ⊥RRd , (9-78)

for air at standard temperature and pressure. These equations, in which the only plate parameter accounted for is the mass, are variants of what is commonly referred to as the mass law for simple walls. A salient feature of that law is that the sound reduction index increases by 6 dB for each doubling of the frequency, or of the plate mass. In practice, we


323

always have finite walls that deviate somewhat from the result given above. As long as the plate is large, however, and the frequencies of interest are well below the coincidence frequency, then the mass law is a good approximation of reality; see figure 9-22. “Large”, in this context, refers to both of the wave types involved, i.e., the plate dimensions must be “large” in terms of both bending wavelengths and sound wavelengths in air.

100 1000 0

10

20

30

40

500

Reduction index Rd [dB]

Frequency [Hz]

Figure9-22 Comparison of the calculated (equation (9-78)) – solid line – and measured (in third-octave bands) – stars – sound reduction Rd across a simple wall consisting of a 13 mm thick slab of gypsum (m” = 650 kg/m2, fc =2500 Hz).

(ii) f ≈ fc .

At coincidence, the sound reduction index is controlled solely by the damping in the plate. Without damping, complete transmission, τ = 1, occurs. (iii) f > fc .

In this frequency region, the bending stiffness term in equation (9-75) completely dominates. An expression for the sound reduction across an infinite wall, analogous to that for frequencies below coincidence, can be found. It turns out, however, that finite walls with small to moderate damping do not behave in accordance with the resulting expression. The reason for that is that the boundaries of the finite plate reflect the forced bending waves (direct field), bringing about a reverberant bending vibration field in the wall (compare section 9.2.2 Sound fields in rooms). The bending waves in that reverberant field (or standing wave field) consist of free bending waves that satisfy the homogeneous bending wave equation. As will be shown in section 8.6, sound radiation from free bending waves is negligible below the coincidence frequency fc, whereas very effective radiation


324

occurs above the coincidence frequency. As a result, the sound radiation above the coincidence frequency is dominated by the contribution from the reverberant field of the plate. The direct field and reverberant field radiate equally effectively to the surrounding fluid, but the latter is the dominating component of the plate’s vibration field when there is not a lot of damping. Below coincidence, on the other hand, the reverberant field is normally negligible, because free bending waves are very inefficient radiators at those frequencies. An expression for the sound reduction index of a finite plate, above the coincidence frequency, is provided without proof:

cc

d ffff

cmR >⋅+⋅+

′′⋅= ,log102log10

2log20

0 πη

ρω

(9-79)

where η is the loss factor for the plate. Equation (9-79) accounts only for that part of the sound transmission due to the reverberant field in the plate. The fact that the sound radiation from the reverberant field is normally only significant above the coincidence frequency implies that added damping is only warranted if it is that frequency region in which improved sound insulation is sought. Finally, we briefly touch upon the sound insulation of a plate at low frequencies. With reference to our discussion in the introduction to the chapter, we consider the zero-modes and few-modes regions for a plate (see table 9-1). In the few-modes region, we have only a small number of resonant modes that determine the behavior of the plate. That implies that the sound reduction index varies strongly; at a resonance (Zin ≈ 0), practically no sound insulation at all is obtained, whereas at an anti-resonance (Zin ≈ ∞) very large sound reduction indices are obtained.

log

R[d

B

[ ]ffc,

,]

Figure 9-23 Idealized insulation behavior of a simple wall. In the zero-mode region, the wall is stiffness-

controlled; in the few-mode regions, the behavior is determined by a small number of resonances; and, in the multi-mode region (if, however, f « fc), the wall’s behavior is mass-controlled.

Few-mode region Multi-mode region

Mass law 6 dB / octave

Zero-mode region


325

In the zero-mode region, the plate has no resonances and acts as a pure stiffness κ, the magnitude of which is determined by the plate’s material properties, geometry, and edge mounting (boundary conditions). In that region, the impedance takes the form Zin = κ / iω, which, at low frequencies, results in a very good sound reduction index. Summarizing, to serve as effective sound insulators, plates should be used in either the zero-mode region or the multi-mode region; in the latter, the mass law applies, provided that we are well below the coincidence frequency. In machinery-related applications, for which the insulators are typically metallic materials, the frequencies of interest are usually below coincidence (e.g., fc is 6 kHz for 2 mm steel plate). In building acoustics, on the other hand, the relevant frequencies are normally well above coincidence (e.g., fc is about 200 Hz for 100 mm thick concrete).

9.2.4.4 Double wall

A way to increase the sound insulation of a partition, without too drastic an increase in the mass, is to construct it of two or more layers (walls), with air pockets (possibly absorbent-filled) between the layers. When an acoustic wave passes through such a construction, the total transmission factor, ignoring all reflections (standing waves) between layers, is

Nntot τττττ ......21= , (9-80)

in which we have assumed that there are N layers and the transmission factor of the n-th layer is given by τn. Equation (9-80) holds at sufficiently high frequencies, for large enough distances (in sound wavelengths) between each layer, or given sufficient damping, in the form of absorbent, between layers. At low frequencies, at which that doesn’t apply, standing wave effects (resonances) arise and modify the transmission. For practical reasons, two, or at most three, layers are used in constructions of this type, e.g., windows. The most common case is that of two layers (double walls), which therefore merit further discussion. If we assume that the mass law holds, then equation (9-77) gives the sound reduction index of a finite double wall

cdouble ff

cmm

cm

cmR <

′′′′⋅=

′′⋅+

′′⋅=⊥ ,

4log20

2log20

2log20 22

0

212

0

2

0

1

ρω

ρω

ρω

, (9-8

where m″1 and m″2 are the masses per unit area of each wall. Example 9-5 Express equation (9-81) for the special case of a double wall consisting of two identical elements.

Solution

Set 21 mm ′′=′′ . That yields c

mRdouble

02log40

ρω ′′

⋅=⊥ .

Hence, the sound reduction index increases by 12 dB for each octave band increase in frequency, or for each doubling of the mass. That can be compared to the earlier result of 6 dB for a simple wall.


326

A double wall has a lower frequency bound corresponding to the lowest mechanical resonance of the system. That fundamental resonance corresponds to an oscillation in which the enclosed air acts a stiffness, and the two wall elements as masses. In order for the double wall to work as intended, it must be used at frequencies considerably higher than that resonance frequency. A typical rule of thumb is that the Helmholtz number, based on the thickness of the air pocket, must be greater than one. Finally, we derive an expression for the fundamental resonance frequency of an infinite double wall. By analogy to our analysis of a panel absorber, it follows that for normal plane wave incidence, the enclosed air volume has a stiffness per unit area of

hc 2

0ρκ = ,

in which h is the thickness of the air pocket. The system considered is mechanically equivalent to a system with two masses coupled by springs; see figure 9-24. As we know from basic mechanics, a system of that type has two degrees-of-freedom, but only one non-zero resonance frequency. That resonance frequency is given by

hmm

mmcf

21

212

00

)(21

′′′′′′+′′

=ρ

π . (9-82)

The case of an air pocket filled with absorbent can be treated in the same way as is done in equation (9-53).

k1m'' 2m''

log

R[d

B

[ ]ff0

II III

1m'' 2m''

h

=I

,

,

Figure 9-24 Idealized frequency-dependence of the sound reduction across an infinite double wall obeying the

mass law. See below for a description of regions I –-III. Note that the analogy between a double wall and a mechanical system requires kh « 1, i.e., low frequencies.

Region I corresponds to frequencies less than the fundamental resonance frequency f0. In that region, the enclosed air has a large stiffness, and the wall acts as a simple wall with a mass (m″1 +m″2). Region II corresponds to frequencies around, and including, the fundamental resonance frequency f0. In that region, there are specific frequencies at which


327

the sound reduction index is poor, due to resonances of the enclosed airspace. Region III corresponds to frequencies at which the double wall acts as intended, and we obtain a sound reduction index in accordance with equation (9-81). As for coincidence and finite wall (boundary) effects, the discussion made in connection with simple walls also applies to the double wall, in principle. It is also worth noting that, in practice, double wall constructions are actually made with stiff connectors between the two wall elements; those may be either point or linear features. Such connecting links act as sound bridges between the wall elements, in which structure-borne sound short circuits the insulating air gap. Solid connections are therefore to be avoided, or at least realized in such a way that they have high mobilities compared to the wall elements (compare to vibration isolation, chapter 9).

Figure 9-25 A double wall in the form of two light, single wall elements, separated by an air gap, can insulate

considerably better than a corresponding simple wall. Correctly dimensioned, a double wall can give the same sound reduction as a 5-10 times heavier single wall. (Picture: Asf, Bullerbekämpning, 1977, Ill: Claes Folkesson).

9.2.4.5 Sound reduction across some common insulators Tables 9-4 and 9-5 provide sound insulation data for some commonly occurring insulating partitions.

13 mm gypsum board

Average reduction 37 dB

Average reduction 47 dB

Average reuction 55 db

30 mm mineral wool

30 mm mineral wool


328

Table 9-4 Sound reduction index R across simple wall constructions.

Sound Reduction Index R [dB] by octave band f [Hz] Material

Thickness [mm]

Surface Density [kg/m2] 125 250 500 1000 2000 4000

145 210 34 40 40 46 51 56 Tile, 15 mm plaster 270 350 36 42 48 55 56 59

95 150 31 37 37 34 47 52 105 170 31 39 39 37 49 53 145 260 36 42 42 48 53 58

Brick with 15 mm plaster 270 480 40 46 51 54 59 62

40 95 31 29 27 36 43 48 70 170 30 33 37 44 51 59 120 300 34 38 48 53 61 63 150 350 38 42 47 54 61 64

Concrete

190 430 39 43 50 55 62 66 7 7 17 18 26 28 32 27 Gypsum slab 10 10 19 19 26 31 30 34

10 7 19 19 22 25 25 19 15 11 18 22 24 27 25 32 Plywood 25 15 16 25 26 24 30 36 4 3 14 16 19 21 25 28 Particle

Board 19 15 22 22 27 28 22 24 Al-plate 0.5 1.3 10 12 14 19 25 28

1 8 17 23 30 32 35 38 3.5 28 29 33 36 39 41 31 Steel plate 7 55 33 38 39 40 30 42

50 6 10 16 18 20 24 32 Cover layer: 0.8 mm Al-plate. Core: Polyurethane foam 80 8 12 17 19 16 32 30

Table 9-5 Sound reduction index R of double wall insulators. (Source: W Fasold, W Kraak, W Schirmer, Taschenbuch Akustik, 1982, VEB Verlag.)

Sound Reduction Index R [dB], by octave band f [Hz] Description d1

[mm] dh

[mm] d2

[mm] m ′′

[kg/m2] 125 250 500 1000 2000 4000 40 25 70 275 33 38 43 50 57 55 70 10 70 340 43 44 50 54 55 60 40 50 70 275 35 42 45 53 58 60 70 50 70 340 44 42 48 54 59 58 40 100 70 275 44 42 47 55 58 62

Concrete without damping in the air gap

70 100 70 340 43 41 48 54 59 65 70 110 120 175 42 44 46 48 53 60 70 160 70 135 38 41 42 44 52 60 70 50 70 135 37 43 41 44 55 63

Light concrete with 50 mm mineral wool layer in the air gap

115 80 115 190 45 42 46 59 56 64 60 30 60 100 39 40 40 48 55 64 70 60 70 160 35 40 41 46 56 63 Gypsum with 30 mm mineral wool

layer in the air gap 80 30 80 170 36 41 39 43 52 67

Where d1 thickness of the 1st wall element,

dh thickness of the air gap, d2 thickness of the 2nd wall element, and m ′′ total surface density of the construction.


330

9.3 IMPORTANT RELATIONS ENERGY METHODS IN GENERAL

Helmholtz number klHe = (9-1) Energy balance of single and coupled acoustic systems Energy balance equation of a single system

inWEdtdE

=+ ηω (9-4)

Relation between wave theory and energy-based methods

Diffuse intensity 4

cI d

dε

= (9-24)

ROOM ACOUSTICS Sabine’s formula Relation between reverberation time and loss factor

ηω)/10ln6(= ⋅T (9-27)

Absorption factor for diffuse incidence

θθθααπ

dd ∫=2

0

)2sin()( (9-31)

Absorbed power from a diffuse sound field

∑=m

mmdd

diss Sc

W ,4α

ε (9-32)

Sabine’s formula for reverberation time

S

VcSc

VTdd αα

161.0m/s)342with()10ln24(===

⋅= (9-35)

Equivalent absorption area mm

md SA ∑= ,α (9-36)


331

Sound fields in rooms Sound pressure level emitted by a source in a room

⎥⎦

⎤⎢⎣

⎡′

+⋅+=Ar

ΓLL dirW

totp

44

log102π

(9-42)

Echo-radius 21

16⎟⎠⎞

⎜⎝⎛ ′

=πΓAre (9-43)

Acoustic absorbents Porous absorbents Optimal flow resistance of absorbents mounted to a hard wall

2/ 0 ≈ch ρφ (9-47)

In that case, α┴ ≥ 0.9 when kh is greater than about 1 Influence of a cover layer – lower bound on effective frequency

mc

f g ′′=

πρ

20 (9-48)

Fundamental resonance frequency of a panel absorber

{ }

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

′′

′′≈=

′′=

′′=

isothermal,50

adiabatic,60

airfor21

21 2

00

hm

hm

hmc

mf

ρπ

κπ

(9-53)

Sound transmission through insulators

Transmission factor i

tWW

=τ (9-54)

Sound reduction index τ1log10 ⋅=R (9-55)

Sound transmission between two rooms

)1log(10 22,1,

dpp S

ALLτ

+⋅=− (9-64)


332

If the factor 1 in parentheses in equation (9-64) is neglected, one obtains

SA

RLL dpp2

2,1, log10 ⋅++= (9-65)

Sound Reduction Index across composite walls

∑ −⋅

⋅=

n

Rn nS

SR1010

log10 (9-67)

where ∑=n

nSS

Simple Wall Boundary frequency for coincidence

p

c Dmcf

′′=

π2

2 (9-76)

Sound reduction index

cff < { } 42log20log202

log200

−′′⋅+⋅≈=′′

⋅=⊥ mfairc

mRρ

ω (9-77)

3−= ⊥RR d (9-78)

cff > c

d ff

cmR log102log10

2log20

0⋅+⋅+

′′⋅=

πη

ρω (9-79)

Double wall Sound Reduction Index

cff < , 220

212

0

2

0

1

4log20

2log20

2log20

cmm

cm

cmR double

ρω

ρω

ρω ′′′′

⋅=′′

⋅+′′

⋅=⊥ (9-81)

Fundamental Resonance hmm

mmcf21

212

00

)(21

′′′′′′+′′

=ρ

π (9-82)

333

CHAPTER TEN SOUND RADIATION & GENERATION MECHANISMS This chapter deals with the important question of how sound is really generated and radiated, i.e., what physical mechanisms bring about sound fields. Knowledge of that is crucial, because noise control measures taken at the sound source itself are normally the best and most effective strategies for elimination sound and vibration problems. An ever increasing interest in incorporating sound and vibration requirements into the design of new machines and vehicles also enhances the significance of that topic. In classical acoustics, which was largely developed during the 19-th century, the essential aspects of which were described in chapter 4, sound in a fluid can only be radiated by vibrating solid bodies. Driving the vibration of such bodies are dynamic forces of various kinds, e.g., inertial forces in connection with shocks and electromagnetic forces, as is the case with common loudspeakers. Thus, we will give due attention to sound emission from vibrating structures, particularly large plates. The most important contribution to the sound radiation from a plate is from bending waves. That type of wave primarily involves motions perpendicular to the plane of the plate, and can therefore excite sound waves in a surrounding fluid. It is, however, apparent to all who have listened to water boiling in a kettle, or the sound of airplane starting up, that there must be other mechanisms of sound radiation than vibrating solid bodies.

Chapter 10: Sound Radiation & Generation Mechanisms

334

The last-named examples suggest that flow-induced sound, associated with a time-varying flow field, is yet another important sound generation mechanism. The study of that type of mechanism got underway relatively late, and the first general theories were presented in the 1950’s. The noisy civil jet airplane that began to be used at that time was a driving force in the search for new knowledge in that area. Today, flow acoustics is an established part of the larger field of vibroacoustics, and an introduction to it is included here. With the theory of acoustic waves in fluids (see chapter 4) as a point of departure, this chapter begins by treating the various elementary radiating sources, or point sources. Because the equations of acoustics are linear, the superposition principle applies. It is therefore possible, as in electromagnetics, to first study various point sources and then take advantage of linearity to obtain fields from more general sources by regarding them as distributions of point sources, the fields of which are additive.

10.1 MONOPOLE

The monopole is the simplest type of point source. It can also be called a multipole of zero-th order. The sound field emitted by a monopole is spherically symmetric, as we studied in sections 4.2.5-6. For a harmonic time-variation, (4-112) provides the following general expression for a spherically symmetric field:

)()(),( krtikrti er

er

tr +−−+ += ωω AAp .

With a monopole at the origin, in a free field without boundaries, only an outgoing wave exists. That implies that the second term in the equation given above, representing a wave that moves towards the origin as t increases, can be disregarded. In order to interpret a monopole physically, the amplitude A+ is related to what happens at the source. A clue to how that can be done might be obtained by considering that the physical dimension corresponding to that amplitude is [kg/s2], i.e., it represents the change in mass flow rate [kg/s] per unit time. Thus, in order to determine A+, it is necessary to calculate the acoustic mass flow rate generated by the monopole. For spherical symmetry, at a distance r, the volumetric flow rate is given by

rr r uQ 24π= , (10-1)

where tirr e ωQ=Q ˆ and where ur is the particle velocity in the radial direction. The mass

flow rate is obtained from that expression by multiplying the volume flow rate by the density of the surrounding fluid which, in the context of a linearized theory, can be set to the fluid’s undisturbed density ρ0. According to equation (4-114), the particle velocity in the radial direction is given by

)(

0)11(),( krti

r eikrcr

tr −+ += ωρA

u .


335

Putting that relation into equation (1), and taking the limit of a point source, yields

cki

eikrcr

r ikrr 00

20

04

)11(4limˆρπ

ρπ +−+

→=+=

AAQ .

By taking advantage of the relation ω = kc, the following relation between the mass flow rate and the amplitude of the radiated spherical sound wave is obtained:

π

ωρ4

ˆ00 Q

Ai

=+ . (10-2)

The spherical sound wave radiated from a harmonically oscillating monopole is, as such, given by

ikrm e

ri −=πωρ

400 Q

p , (10-3)

where the index m indicates a monopole and Q0 is called the monopole’s source strength. Assume an arbitrary distribution of monopoles in a free field; the resulting sound field is then obtained through superposition, as

∑ −=n

ikr

n

nm ne

riπωρ

40 Q

p , (10-4)

where nr is the position vector from monopole n to the field point r ; see figure 10-1. If the summation in equation (10-4) is interpreted as an integral, then even cases of a continuous source distribution can be treated.

r

r1

rn

Qn

Q1

The sound power radiated by a monopole is given by equation (4-123), and substituting in A+ from equation (10-2) gives the relation

20

20 ~4

Qck

Wm πρ

= . (10-5)

Figure 10-1 Sound field built up by superposition of monopoles.


336

Finally, we take up the question of which specific physical mechanisms are capable of generating fluctuating mass or volume flow rates in a fluid, and, as such, constitute monopole-type sources. The physical realization of a monopole closest to the ideal is that of a small spherical shell, undergoing pure radial oscillations. The condition that the shell be small can be more specifically stated as a requirement that the wavelength be much larger than the radius of the shell. That is usually expressed by requiring that the Helmholtz number He = ka be much less than 1, where k is the wave number and a the radius of the shell (or, more generally, of the source).

An example of a process that can be modeled as a small oscillating spherical shell is the sound radiation from cavitation bubbles. Such bubbles arise in a flowing liquid when the local pressure is so low that it approaches the liquid’s vapor pressure; see figure 10-2. A cavitation bubble is unstable, and normally implodes shortly after it is generated. That implosion is a strong source of sound, however, because it occurs in a very short span of time, and the sudden local change in volume yields a high value of the volume flow rate Q ~ d(ΔV)/dt , where ΔV is the volume of the bubble before collapse.

valve

collapsing gas bubbles

low pressure Very high pressure

Figure 10-2 For sudden changes of the cross section in a pipe carrying flowing medium, the pressure drop can be so large that cavitation bubbles arise. The sound generated by the imploding cavitation bubbles can be described using the monopole model. Poorly designed valves are examples of that type of sound generation. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

A more widely familiar example of a monopole is that of a loudspeaker mounted in a box, as shown in figure 10-3, where only the front side of the loudspeaker can radiate sound into the surrounding air. At low frequencies, at which the He number, based on the radius of the box, is small, that sources acts as a monopole.


337

Figure 10-3 A loudspeaker element mounted in a box is an example of a sound source approximating a monopole in the low-frequency region. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

A relevant question that might be posed in is how deviations from spherical symmetry of the source region affect the validity of the monopole model. After all, for the case of the speaker in a box, just to take one example, the angular distribution of the flow rate in the source region deviates drastically from the ideal case of a radially oscillating sphere. A closer analysis, however, shows that the only factor of significance at low frequencies is the resultant mass or volume flow delivered by the entirety of the source. The deviations of the flow distribution from spherical symmetry contribute sound fields corresponding to higher order multipoles, such as dipoles and quadrupoles. It can be shown that, outside of the near field of the source, these multipole contributions are in a ratio, to the monopole contribution, of He raised to a power 1, 2, 3, ..., etc, equal to the order of the multipole. At low frequencies (small He numbers) the affect of asymmetry is therefore negligible. Other important examples of monopole sources are fluctuating heat sources, e.g., caused by combustion, as well as inlets and exhaust vents with pulsating flow, e.g., those of internal combustion engines and other reciprocating piston machines; see figure 10-4.


338

Figure10-4 The exhaust gas outlet of a fishing

craft with a low-rpm ignition bulb engine radiates sound that can be described by a monopole model. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

Example 10-1 Investigate whether the exhaust gas outlet from a car can be considered a monopole.

Solution The exhaust gas outlet from the internal combustion engine of a passenger car has a typical radius of about 3 cm. The fundamental frequency f0 of the motor’s pulsations, i.e., the cylinder exhaust frequency, is given by: 60/))2/(( KN × , where N is the crankshaft rotational speed in revolutions per minute (rpm), K is the number of cylinders, and a 4-stroke process is assumed. If, moreover, we assume that there are exactly four cylinders and that the idle speed is 1200 rpm, then f0 = 40 Hz. Because the engine does not generate a perfectly sinusoidal variation of the flow rate, a number of harmonics of the fundamental tone are also obtained: 80 Hz, 120 Hz, 160 Hz, etc. The condition that the exhaust outlet must fulfill to reasonably be considered a monopole is that the He number, based on the diameter of the exhaust pipe, be small. Putting in the values given above, as well as a sound speed of c = 340 m/s, yields He < 0.055 at 100 Hz. The sound radiation from the outlet of the exhaust pipe can therefore be modeled as a monopole, at least up to 400 -500 Hz. (Note: it is the sound speed of the surrounding fluid, into which sound is radiated, rather than that of the hot exhaust gases, that is applicable to this computation. Similarly, the density ρ0 in the equations given above is always to be interpreted as that of the surrounding fluid medium.)


339

10.2 DIPOLE

From the monopole, which is the simplest type of point source (a multipole of zeroth-order), new types of sources can be created by superposition. A systematic way to do so is to first superpose two monopoles with the same source strength, but opposite phases Q and -Q. If the He number based on the distance l between these two monopoles is much less than 1, a dipole is obtained; see figure 10-5. Such a source is also called a multipole of order one. The process can be repeated to obtain multipoles of arbitrary order. For example, a multipole of second order, or a quadrupole as it is usually called, is obtained by the superposition of two dipoles of equal strength, but opposite phase; see section 10.3. To calculate the resultant field from a dipole, and to interpret that source type physically, we now consider two harmonically oscillating monopoles Q and -Q as in figure 10-5.

z

r1

r2

-Q

Q

lrθ

With notation as defined in figure 10-5, equation (10-3) leads to the result

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

−−

21

0 21

4 re

rei ikrikr

tot πωρ Q

p , (10-6)

where zelrr 21

1 −= and zelrr 21

2 += . Incorporating rezyxr ikr π4),,()( −== GG , equation (10-6) can then be written as

)),,(),,(( 21

21

0 lzyxlzyxitot +−−= GGQp ωρ ,

which, for small l, can be approximated by the differential

z

lil

lzyxlzyxlitot ∂

∂−≈

+−−=

GQGG

Qp ωρωρ 021

21

0)),,(),,((

. (10-7)

Equation (10-7) represents the field from a dipole of finite extent. The product lQ is usually called the dipole moment and indicated by zD . To obtain an equality in equation

Figure 10-5 Dipole obtained by superposition of two monopoles (kl « 1).


340

(10-7) (i.e., a point source!), we must let l approach zero. Carrying out that limit while constraining the dipole moment to be constant, one obtains

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂−

==−

→ re

zi ikr

ztot

ld π

ωρ4

lim 00

Dpp , (10-8)

which represents the field from a point source of the dipole type (index d) oriented in the z-direction. To interpret the dipole moment physically we use, as in the monopole case, a dimensional perspective. The dimension of zi D0 ωρ is [N], i.e., that expression can be interpreted as a force. It can be shown that that force acts in the z-direction and corresponds to the force zF that the dipole exerts on the fluid. A dipole is therefore a process that does not provide a net volume flow to the fluid, but only a fluctuating force; that is evident by constructing a control volume around Q and -Q in figure 10-5. In summary, a harmonically oscillating point source ti

zz eF ωˆ=F radiates a sound field of dipole type1 given by

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

⋅−

=−

re

z

ikrz

d π4F

p . (10-9)

We will now study the appearance of the dipole field a little more closely, and therefore begin by calculating the derivative in equation (10-9). That gives

=∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂−

=⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂−

= −−−

)11(44

ikrikrzikr

zd e

zrrze

re

z ππFF

p

ikrz ezr

rik

zr

r−

∂∂

−∂∂

−−

= )1(4 2πF

.

The position vector r has the magnitude 222 zyxr ++= , from which

=++

=∂∂

222 zyx

zzr z

r= {see figure 10-5}= cosθ , (10-10)

and which then leads to the following expression for the sound field:

ikrzd e

ikrrik −+= )11(

4cos

πθF

p . (10-11)

1 A dipole can, of course, be oriented in any direction, and not only in the z-direction relative to some predefined coordinate system.


341

The field consists of two parts, both the near field around the source corresponding to kr « 1 and the far field where kr » 1. In the near field, the sound pressure falls off as a rate 1/r2, and in the far field at a rate 1/r, with increasing distance r from the source.

z

θ zF

In order to find the radiated sound power, we can either use equation (4-121) for the sound intensity and integrate over all directions, or we can consider the far field. The latter is normally the simplest approach, because, for an arbitrary source in a free field, the far field can always be considered locally plane. Thus, the intensity can be calculated with the aid of the plane wave relation, equation (4-83) (with 0~2 =−p ). That relation, and equation (10-11), gives the following expression for the intensity in the far field,

220

222

16cos~

rcFkI z

r πρθ

= . (10-12)

The radiated sound power is obtained by summation of that intensity over a sphere

c

Fkdr

rc

FkdSIW zz

sfärrd

0

22

0

2222

0

22

12

~sincos2

16

~

πρθθθπ

πρ

π=== ∫∫ (10-13)

where the surface element is expressed in spherical coordinates when integrating, i.e., θθπ drdS sin2 2= . The sound power can also be expressed as a dipole moment, using

the relation zz i DF ˆˆ0 ωρ= ,

π

ρ12

~ 240 z

dDck

W = . (10-14)

For a dipole of finite size consisting of two closely spaced monopoles, as in figure 10-5, lz QD ˆˆ = applies; see equation (10-7). For small He numbers, the radiated sound power of

such a “finite” dipole is given very precisely by equation (10-14). We now compare the radiation from such a “finite” dipole with the sound power that a lone monopole of strength Q gives in the free field. Based on equations (10-5) and (10-14),

1 , 3/)( 2 <<= klklWW md . (10-15)

Figure 10-6 The variation of the sound pressure in the far field of a dipole; θcos∝dp .

Note that for θ = 0, 180°, a maximum pressure is obtained, and for θ = 90°,

0=dp holds. The vector Fz represents

the force the dipole exerts on the surrounding medium.


342

That relation implies that, at low frequencies (small He numbers), a dipole emits a considerably lower sound pressure than the corresponding monopole. An example of how that result may be put to use is given by the practice of enclosing the back side of a common woofer (bass loudspeaker) in a sealed box. In that way, a monopole source is obtained, rather than the dipole source that the loudspeaker would otherwise constitute (the front and back sides of the loudspeaker correspond to Q and -Q respectively); without that modification, considerably less sound power is radiated. ________________________________________________________________________ Example 10-2 If a woofer of 30 cm diameter is not enclosed in a sealed box, surrounding the back side of the speaker as in figure 10-3, the radiated sound power is reduced. Estimate that reduction with the aid of equation (10-15), at a frequency of 100 Hz. Figure 10-7 If a loudspeaker is not mounted in a sealed speaker

box, the radiated sound power will be reduced due to the pressure equalization between the front and back sides of the element. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson).

Solution Without an airtight box, the loudspeaker is converted from a monopole to a dipole source. Assume that (as is normally the case), the speaker generates the same volume flow rate even when its back side is enclosed in a box. Equation (10-15) then gives the relation between the sound power emissions for the monopole and dipole cases:

klklWW md ,3/)(/ 2= « 1 .

where l is the distance between the two monopoles of opposite phase. The formula is based on point sources, which means that the source region’s diameter d (for each monopole) should fulfill the condition d « l. In this case, l is the average path length that sound must transit to acoustically bind the loudspeaker’s front (+Q) and back (-Q) sides. The path length l would ordinarily be about equal to the radius of the speaker, i.e., 15 cm. The condition for the source region to be regarded as a concentrated at a point is therefore not fulfilled, so that equation (10-15) must be regarded as a crude approximation. Putting in the values (c = 340 m/s, f = 100 Hz and l = 0.15 m) gives ≈md WW / 0.026 ,

and kl = 0.28. Expressed as a level, the sound power reduction is about 16 dB.

loudspeaker membrane

Pressure equalisation over membrane


343

More sophisticated woofers are usually designed such that the volume of the box, as well as an extra tube-shaped opening in the front side of the box, form a mass-spring system. Above the resonance frequency, the air mass oscillates in phase with the loudspeaker membrane. The volume flow in the extra opening then serves as a radiator, adding a contribution to that from the speaker membrane, amplifying the total volume flow. That solution is usually called a bass reflex box. All physical mechanisms that generate fluctuating forces on a fluid act as acoustic dipoles. By analogy to the case of the monopole, the exact distribution of the force in the source is of little consequence, as long as the He number based on a typical diameter is small. The only important thing is that only a resultant, non-zero, fluctuating force is generated and now mass or volume flows. If there are such flows, a monopole-type source is then also obtained, and that latter would dominate over the dipole source for small He numbers, as indicated by equation (10-15). The most nearly ideal example of a dipole-type source is that of a small (rigid) body undergoing transverse oscillations, e.g., a small sphere oscillating back and forth about an equilibrium position. Such a body generates no net volume flow (compare to the loudspeaker example), but only a fluctuating force corresponding to the inertial force that arises when the oscillating body moves in the surrounding fluid. A transversally vibrating string or beams also serves as an example of a dipoles, if its diameter is small in comparison to the sound wave length. In that case, the field arises by the superposition of a continuous distribution of dipoles along the oscillating string (beam), i.e., a linear dipole. Other important examples of dipole sources are bodies in turbulent flow fields, and propellors. These cases will be discussed in more detail in section 10.8. In the first case, the fluctuating pressure in a turbulent flow field creates a distribution of dipoles at the surface of the body breaking the flow. Moreover, a fluctuating force can also arise due to the turbulent wake trailing off the body in the flow stream. For propellors, the sound generation is primarily due to the blade forces that arise because of the blade rotation. Fluctuations of these forces are caused partly by the rotation of the propellor itself, and partly by turbulent inflow to the propellor.

Dipole Physical situation Sketch

• Transversally oscillating bodies

• Bodies in a flow field

Fluctuating force

• Propellors

Figure 10-8 Example physical situations that give rise to sources of dipole character, at low frequencies (He « 1).


344

10.3 QUADRUPOLE

The quadrupole, or a multipole of second order, is obtained by the superposition of two dipoles with source strengths F and F− . When the He number, based on the distance l between these two dipoles, is much less than 1, a quadrupole is obtained, as illustrated in figure 10-9. As is evident from the figure, there are two basic types of quadrupoles: the longitudinal, for which F and l are parallel; and, the lateral, for which those two vectors are orthogonal. To obtain a quadrupole concentrated at a point, lz must approach zero,

while constraining the product zlF to be a constant value M (called the quadrupole moment) with the dimensions [Nm]. For a lateral quadrupole, that constant represents the (mechanical) moment that the quadrupole exerts on the fluid.

x x

z

y y

z

l z

-l z

F x

F z -

F z l z

-F x -l z

/2

/2 /2

/2

longitudinell M Fzz z zl= longitudinal lateral M Fxz x zl=

Figure 10-9 Quadrupoles obtained by superposition of two dipoles (klz « 1) in each case.

A quadrupole can also be regarded as the superposition of monopoles, as in figure 10-10.

z

0

l z

-l z

-Q

Q

-Q

Q

l z

l z

/2

/2

M Qzz z zi l l = ρ ω0 ′

z

0

-Q Q

l x

Q -Q

l x

l x l z M Qxz x zi l l = ρ ω0

Longitudinal Lateral

Figure10-10 Quadrupoles obtained by the superposition of monopoles ( klx, klz « 1).


345

In calculating the quadrupole moment in figure 10-10, the relation li QF ˆˆ0 ωρ= has been

used. Referring to figure 10-9, the following expressions in spherical coordinates can be derived for the far field of a quadrupole point source (with index q)

longitudinal: +−

= θπ

22

, cos4 r

k zzzzq

Mp near field terms, (10-16)

and

lateral: +−

= φθθπ

cossincos4

2

, rk xz

xzqM

p near field terms. (10-17)

From these results, and using the (locally) plane wave approximation, the intensity in the far field can be found. The radiated sound power can then be calculated by integration over a spherical surface, the result of which is

c

MkW zz

zzq0

24

, 20

~

πρ= , W

k Mcq xz

xz,

~=

4 2

060πρ . (10-18,19)

As long as the He number is small, equations (10-18) and (10-19) give the power radiated by finite quadrupoles to a good degree of precision. Let us therefore compare the radiated sound power from such a “finite” quadrupole, consisting of a force couple as in figure 10-9, with that from a lone dipole of strength F in a free field. Making use of equations (10-13) and (10-18,19), we obtain

1 , 5/)( , 5/)(3 2,

2, <<== klklWWklWW zdlatqzdlongq . (10-20,21)

These relations imply that, at low frequencies (small He numbers), a quadrupole configuration sends out significantly less power than a corresponding dipole.

10.3.1 Examples of sound sources of quadrupole character

The simplest physical realization of a quadrupole is that of a small body turning (swinging) back at forth around an axis of symmetry. Another example is that of two identical loudspeakers (back sides not enclosed) in close proximity, and which are driven in phase and oriented such that their dipole moments are opposed. Yet another example of a system that has quadrupole character at low frequencies is a structure that is rotational symmetric about an axis, and excited asymmetrically, e.g., a church clock.


346

Q

-Q Q

-Q

quadrupole

Physical situation

Body in "rocking" motion . . F -F

Free turbulence

..... “fluctuating forces”

Church bell

Figure 10-11 Examples of physical situations that, at low frequencies (He « 1), constitute sources of quadrupole

character. The most important physical process in this category is, however, free turbulence. Generating the sound in these situations are the fluctuating shear stresses acting on the fluid particles of the turbulent flow fields. These shear stresses comprise oppositely directed force couples, so that each fluid particle in the turbulent field acts as a quadrupole source. Thus, there is a continuous distribution of quadrupoles, and the resultant sound field is obtained by summation (integration) over the entire turbulent field. The sound radiation from jet streams, which will be discussed in greater detail in section 10.8, can be explained by a model of that type; see figure 10-12.

Gasström

Blandningsområde Gas flow

Mixing region

Figure 10-12 Free turbulence arises in so-called mixing zones between gases with differing flow velocities. The

greater the velocity difference, the greater the sound generation. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

10.4 INFLUENCE OF BOUNDARIES

In the earlier sections of this chapter, the influence of boundaries (reflecting surfaces) was completely ignored, and free field conditions were assumed to apply around the sources that were studied. In this section, however, the influence of boundaries will be treated; to begin with, a point source of monopole type will be considered first. Because the classical theory of acoustics is linear, the field generated by a monopole source must be proportional to the source strength ti

rr e ωQ=Q ˆ , i.e.,

QZp ),(),,( yxtyx tr= , (10-22)


347

where Ztr ),( yx is an acoustic transfer (or cross) impedance which describes the relation

between the excitation and the resulting sound field, x is the position vector to the receiver point, and y is the position vector to the source point.

x

y

rQ

The behavior of Ztr depends on the geometry, on the characteristics, or specifically the point impedance, of the boundaries surrounding the source, and on the sound speed and flow of the fluid in which sound propagation occurs. In a wider range of contexts, the function Ztr is usually called the Green’s function. Exact expressions for the Green’s func-tion are only available for a few simple special cases. In general, that function must be calculated numerically, using, for instance, acoustic FEM (Finite Element Methods). An important case in which the Green’s function is simple to determine is that of a reflection against a large (in terms of the He number) plane surface. If such a surface is either ideally hard (infinite impedance) or ideally soft (zero impedance) then the Green’s function can be determined by the mirror source method. The application of that method is illustrated in figure 10-14. The boundary condition for a hard plane (uz (x,y,0) = 0), can be fulfilled by removing the plane z=0 surface, and substituting a mirror source of strength +Q at z = -h. The sound field for z ≥ 0 is obtained by superposing the fields from the original source and the mirror source.

z r1

r2

Q

Q

+/-

r θ h z = 0

Reflecting surface

The case of a soft plane ( p(x,y,0) = 0) can be treated by a mirror source of strength -Q. In summary, the sound field from a lone monopole before a reflecting surface can be expressed, in the region z ≥ 0, as

)44

(21

0

21

re

rei

ikrikr

ππωρ

−−

±= Qp . (10-23)

We now calculate the power radiated, and therefore need an expression for the sound field in the far field. For the vectors 1r and 2r , considering the geometry,

Figure 10-13 Definition of the source point y ,

with respect to the receiver point x .

Figure 10-14 Geometry of the application of the mirror source method to a monopole. The plus sign refers to the case of a hard surface, and the minus sign to that of a soft surface. The height of the source above the plane is h and r is the distance between the observer and the midpoint between the source and the mirror source. Note that the sound speed is assumed constant, and that the fluid doesn’t flow. Moreover, there are no boundaries in the region z > 0.


348

zehrr −=1 and zehrr +=2 ,

where h is the source height above the plane z = 0. By a series development of these expressions for large r and fixed h (i.e., for small h/r),

=+−=−= 2221 cos2 hhrrehrr z θ

...cos)(cos21 2 +−=+−= θθ hrrh

rhr , (10-24)

=++=+= 2222 cos2 hhrrehrr z θ

...cos)(cos21 2 ++=++= θθ hrrh

rhr . (10-25)

Putting equations (10-24) and (10-25) into equation (10-23) gives, if only the far field part (∝1/r) is expressed,

=+±= −− )(4

coscos0 θθπωρ ikhikhikr eeer

i Qp

⎩⎨⎧

+−+

= −)cossin(2

)coscos(240

θθ

πωρ

khikh

er

i ikrQ. (10-26)

With the aid of equation (10-26), the intensity in the far field can be expressed in the radial direction (using the plane wave approximation). The radiated sound power can then be calculated by integrating over a half sphere

)2

2sin1(sin2)cos(sin4)cos(cos4

16

~ 2/

0

22

2

22

220

khkhWdr

khkh

cr

QW m ±⋅=

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

= ∫π

θθπθθ

π

ωρ , (10-27)

where the sign +/- indicates a hard (positive) or soft (negative) plane surface, and mW is the sound power a monopole of strength Q emits into a free field (equation (10-5)). The series development of equation (10-27) for small kh shows that

⎪⎩

⎪⎨⎧

+

+=

planmjukt ,3

)(2planhårt ,2

2khWW m . (10-28)

For the hard plane surface, the sound power is doubled, compared to the corresponding monopole in a free field. Clearly, as kh goes to zero, the sources merge into a monopole of strength 2Q; the source and the mirror source acoustically coincide at the boundary. Thus, four times as much power is radiated as from a monopole of strength Q. But only half of that power is emitted into the half-space z > 0, explaining the result given above.


349

kh

2,0

1,0

0,0 0 1 2 3 4 5 6

Normalised sound power, mWW

For a soft plane surface, a small kh yields a dipole, and the sound power, compared to a monopole in a free field, can be calculated by way of equation (10-15) with l = 2h. If that result is divided by 2, to obtain the power emitted into the region z > 0, the result of equation (10-28) is obtained. From figure 10-15, it is also evident that, for a large value of kh (the He number), the radiated sound power asymptotically approaches the value for a monopole in a free field. For values of kh larger than 1, the variations about the asymptotic value are so small that they normally lack practical significance. The result in equation (10-27) is, moreover, derived for the purely tonal case. For the case of a broad-band source, an averaging of that result over frequency is called for; that averaging causes the variations about the asymptotic value to decay even faster than for the tonal case. The result in equation (10-27) can be interpreted generally, and even applied to structure-borne sound sources, e.g., a force that excites bending waves in a plate. Specifically, the influence of boundaries on the sound radiation from a source can be neglected2 when the distance to all boundaries is large in terms of the He number, based on the wavelength in the medium under consideration. When that condition is fulfilled, the sound power from the source is determined as for a free field, which is the basic assumption behind the energy methods presented in chapter 7.

10.4.1 Examples of surfaces that are acoustically hard or soft

In practice, for airborne sound, it is the hard plane surface (infinite impedance) that is common. Examples would be a source over a concrete floor, an asphalt roadway, or a water surface. In that last example, note that water has a wave impedance about 106 times that of air, making the water surface, as seen from air, “very hard”. The case of a soft plane (zero impedance) practically never occurs for airborne sound. Grassy surfaces and similar “soft” surfaces represent boundary conditions actually falling between the ideally hard and ideally soft, necessitating a more involved model. For sound in water, however, the soft

2 For finite resonant systems, e.g., a plate or a reverberant room, that statement is only valid when considering an average over frequency bands, or, alternatively, over a number of source positions.

Figure 10-15 Sound power emitted by a monopole over a hard (solid curve) and a soft (dashed curve) plane surface. k is the wave number, h is the distance between the monopole and the reflecting plane, and Wm is the sound power from a monopole in a free field.


350

surface model comes into play. At an air-water interface, seen from the water side, there is indeed a boundary to a medium with a very low impedance. Example 10-3 Recalling example 10-1, the exhaust outlet of a vehicle can normally be regarded as a monopole source. Usually, it is located above a hard reflecting surface, the roadway. How much can the sound radiation be reduced, in theory, by placing it at an optimal height? Solution Assuming the outlet acts as an ideal monopole (high impedance), the result in equation (10-27) can be applied. The “hard plane” curve of figure 10-15 shows that there is a minimum of the radiated power for a kh value a little greater than 2. Analyzing equation (10-27), the first minimum is found to be mWW = 0.783 at kh = 2.24, corresponding to h/λ = 0.36. Thus, for a pure 120 Hz tone, the optimal placement would be at h = 1.02 m (c = 340 m/s). Compared to the placement very close to the hard surface (for which

mWW = 2), a sound power reduction of 4 dB is obtained. The optimal height is, however, frequency-dependent, which makes it difficult to apply this strategy in practice, considering that the exhaust gas noise contains a whole range of harmonics. From figure 10-15, it is clear that by choosing a kh value greater than 1, a reduction in the range of 2-4 dB (depending on frequency) results, compared to close placement. Thus, given an exhaust gas spectrum with a 40 Hz fundamental, h should be made greater than 1.35 m. For passenger cars, such a placement is not possible; on the other hand, it certainly is for buses and trucks.

10.5 LINE SOURCES

As described in chapter 1, there are three important basic types of sound fields: plane waves; cylindrical waves; and, spherical waves. Which of these is relevant is determined by the geometry (plane, cylindrical, or spherical) of the source region and boundaries in a given situation. This section takes up the simplest type of cylindrical sound field, namely one that only depends on the radial dimension. Such a field arises if we have a distribution of equal strength monopoles along an infinitely long straight line. Together, they comprise a monopole line source. Based on the line monopole, other line sources of arbitrary order (dipole line source, quadrupole line source, etc.) can be constructed by analogy to the same procedure in the spherical case, using a point monopole. The general expression for the field from a monopole line source is more complicated than that for plane or spherical wave fields. We will therefore be content to study the far field, for a case in which the line monopole consists of uncorrelated parts. To define the source strength, it is assumed that the line source radiates a certain power per unit length W ′ . Every small element Δx of the source is a monopole bringing about the following acoustic intensity in the far field:

24

)(r

xWrI rπ

Δ′=Δ , (10-29)


351

where the index r indicates the radial dimension and the intensity contribution is a time-averaged value; see figure 10-16.

x

h

r

Δx

θ

x1

x2

0

Making use of the plane wave approximation, equation (4-83), the square of the rms sound pressure can be found directly from equation (10-29)

2

02

4~

r

xWcp

π

ρ Δ′=Δ . (10-30)

Recalling equation (1-28), the assumption of uncorrelated sources implies that the contributions 2~pΔ can be added; that summation, as Δx goes to zero, gives

∫∫′

==2

1

2022

4~~

x

xlinje r

dxWcpdp

π

ρ , (10-31)

where we have assumed that the source is infinitely long. From figure 10-16, the relation

222 xhr +=

follows. Putting that into (10-31) gives

( ))(arctan)(arctan44

~12

022

022

1

hxhxhWc

xhdxWc

px

xlinje −

′=

+

′= ∫ π

ρπ

ρ . (10-32)

Expressed in terms of the angle θ = arctan(x/h) (see figure 10-16), we obtain

h

Wcplinje π

θρ4

~ 02 Δ′= , (10-33)

where 12 θθθ −=Δ . That shows that we obtain a sound field that decays at a rate 1/h, and which is proportional to the angle Δθ sweeping a segment of the line source, with its vertex at the observer. For small distances to the source, the angle Δθ approaches π, which

Figure 10-16 Calculation of the sound field from a line source in a free field; h indicates the perpendicular distance from the line source to the receiver.


352

gives a purely cylindrical sound field ( ∝1/h, section 1.4.2). Expressed as a dB level, for the case of air ( 4000 ≈cρ Ns/m3) equation (10-33) becomes

11)log(10)log(10 −Δ⋅+⋅−= ′ θhLL Wp [dB]. (10-34)

LW’ is the sound power level emitted by a 1 m long segment of the source. A practical example of when that model can be applied is the calculation of sound propagation from a highway. Because the sound field in such an application is limited to a half space (≈ totally-reflecting foundation) equation (10-33) should be multiplied by a factor of 2. To investigate, in greater detail, how the sound field varies with distance to the source, the case of an observer facing the middle of it is considered. If the source has a length l, it follows that 2/21 lxx =−= , and putting that into equation (10-32) leads to

⎪⎪⎩

⎪⎪⎨

⎧

′

′

=′

=

.utbredning cylindrisk s vd stort, ,2

.utbredningsfärisk s vd litet, ,4)2arctan(

2~

0

20

02

hlhWc

hlh

lWc

hlhWc

plinje ρπ

ρ

πρ

(10-35)

For large l/h, the source behaves as a line source and radiates a cylindrical sound field, the energy density of which (∝ 2~p ) decays at a rate 1/h, as h increases. For small values of l/h,

a spherical3 sound field, with its energy density decaying at a rate 1/h2 for increasing h, is obtained instead. The conclusion is that sources that sources with an extension in some direction that is large compared to the distance to the listener, can be regarded as line sources. Expressing equation (10-35) as a sound pressure level, for air, yields

( ))2/arctan(log108)log(10 hlhLL Wp ⋅+−⋅−= ′ [dB] . (10-36)

The last term in that equation is plotted in figure 10-17. As is evident from the figure, the sound propagation is cylindrical if l/h > 10, i.e., if the source length is 10 times greater than the perpendicular distance to the listener. Formula (10-35) also shows that the sound pressure level falls off at a rate of 3 dB per doubling of the source-receiver distance.

3 "Spherical", in this context, does not imply "spherically symmetric", but only that the sound field in a certain direction falls off at a rate of 1/(distance)2.


353

h/l

0

5

10

-5

-10

-15

[dB]

10-2 10-1 100 10 102 103

Cylindrical propagation Spherical propagation

(8-35) ‘h/l stort’

(8-35) ‘h/l litet’

l h

)2

log(arctan10hl

⋅

- 3 dB / doublingof distance

Figure 10-17 Depending on the distance between the finite line source and a listener, the sound field has either a

cylindrical or a spherical character. When the length is large in comparison to the distance to the listener, the propagation is cylindrical. If it is short, on the other hand, the propagation is more nearly spherical. In the plots, the source length l is held constant, and the receiver distance h varies.

For distances that fulfill l/h < 1, spherical sound propagation is obtained (6 dB per doubling of distance). Note, however, that the equations derived above are based on the assumption that the listener is in the far field. In practice, that means that kh must be greater than 2-3 for the result to be valid. Our last result can be generalized to sources that are extended in more than one direction. At distances r that fulfill r/d >1, where d is the largest diameter of the source, a sound field that decays at a rate 1/r2 can be expected.

10.6 SOUND RADIATION FROM VIBRATING STRUCTURES

A vibrating structure surrounded by a fluid radiates sound waves into the fluid, be means of the vibration field existing at the boundary surfaces of the structure. It is the normal component of the velocity field at the surface of the structure that is directly responsible for the excitation of sound waves in the fluid; compare equation (4-84). The tangential component radiates a near field dominated by the viscosity of the fluid. The near field is incompressible, which means that it doesn’t radiate sound itself. The viscous near field, however, does provide a dissipation of the vibrational energy of the structure, and thereby contributes to the structure’s loss factor. In practice, it is often the sound power radiated by a vibrating structure that is of greatest interest. That power can be expressed formally using equation (4-121)


354

∫ ∗⊥=

ySydSW )Re(

21 pv , (10-37)

where ⊥v indicates the normal velocity of the bounding surface Sy of the structure; see figure 10-18. By defining a radiation impedance ⊥= vpZ rad , equation (10-37) can also be expressed in the form

∫ ⊥=yS

yrad dSvW 2~)Re(Z . (10-38)

v

⊥

S ydS y y

x

Equation (10-38) is fundamental to the analysis of sound radiation. It can be noted that the value of the radiation impedance normally depends on both the environment in which the structure is located (e.g., proximity to reflecting surfaces) and on the appearance of the structure’s vibration field. An important special case is the situation in which the radiation impedance in equation (10-38) is equal to the specific impedance of the fluid, i.e., Zrad = ρ0c. Then,

ySvcW 200

~⊥= ρ , (10-39)

where ∫ ⊥⊥ =

ySy

ydSv

Sv 22 ~1~ (10-40)

is the spatial average of the squared rms normal velocity. Equation (10-39) corresponds to the sound radiation obtained from one side of a

large (in terms of the He number) plane surface, where all points on the surface oscillate with identical amplitude and phase (a piston). That is because, if free field propagation is assumed, a plane wave field propagating in front of the surface is obtained, implying that Zrad = ρ0c. A means to quantify how effectively a certain structure radiates sound is to normalize equation (10-38) by the sound power radiated by a piston with the same vibration level and area. That measure of effectiveness is ordinarily called the radiation efficiency (s), and given by

Figure 10-18 Sound radiation from a vibrating structure.


355

ySvc

WWWs

200 ~

⊥

==ρ

. (10-41)

Putting in the sound power from equation (10-38) yields

y

yS

rad

Svc

dSv

s y

20

2

~

~)Re(

⊥

⊥∫=

ρ

Z

. (10-42)

The radiation efficiency s depends, in general, on both the environment and the nature of the structure’s vibration field. Thus, differing distributions of velocity response in the vibration field of the structure (e.g., differing mode shapes) give differing radiation efficiencies. That circumstance will be discussed in greater detail in section 10.6.2. In the remainder of this section, the sound radiation from plane surfaces will be studied. The most important application is that of thin plates, for which bending waves are usually the dominating component of vibrations in the plate’s normal direction.

10.6.1 Infinite plane surface

Consider an infinite plane surface that bounds a fluid half-space. A plane harmonic wave (e.g., a bending wave) propagates in the positive x-direction on the surface, as illustrated in figure 10-19. The normal velocity of that wave can be expressed as

)(ˆ),( xktiz setx −= ωvv , (10-43)

where ks is the wave number. If the surface considered is that of a thin plate, the wave number is given by equation (6-60a).

x

z

v z

wave

fluidρ , c

0

For the sound field radiated in the fluid for z > 0, the following assumption can be made:

)(ˆ),,( zxkti zsetzx kpp −−= ω , (10-44)

where Snell’s law has been applied, i.e., the wave number in the x-direction has been set equal to ks. In order for our assumed form to satisfy the wave equation, the wave number must satisfy equation (5-46), giving

22)( sz kc −= ωk . (10-45)

Figure 10-19 Sound radiation from a plane surface with a propagating wave.


356

With the help of the equation of motion, equation (4-25), the sound field’s velocity component in the z-direction can be calculated as

ziz

1

0 ∂∂−

=pu

ωρ .

Putting equation (10-44) into that gives

) (

0

ˆ zxktizz zse kpk

u −−= ωωρ

. (10-46)

The boundary condition at the surface z = 0 requires that the normal velocity be continuous. With the help of equation (10-46), that leads to the following relation:

zk

vp

ˆ ˆ 0ωρ= . (10-47)

The fluid’s acoustic intensity, due to emission by the surface, is then calculated to be

== )Re(21 *

zzI pv {putting in equation (10-47)}= )1Re(~ z2

0 kvωρ , (10-48)

where the wave number in the z-direction kz is given by equation (10-45). Equation (10-48) shows that sound power is only radiated by the surface if kz is real, i.e., if

cks ω≤ , (10-49)

If, instead, ks is larger than that limit, kz takes on an imaginary value, and the sound power radiation becomes zero. Alternatively, equation (10-49) can be restated in terms of the sound speed c,

ss ckc =≤ ω , (10-50)

where cs is the phase velocity of the wave. That implies that a large plane surface can only radiate sound by means of waves that move supersonically (faster than the sound speed in air). For the case in which the equality applies in equation (10-50), i.e., when the wave speed exactly matches the speed of sound, equations (10-45) and (10-48) dictate unlimited radiation. That case is called coincidence. In practice, the radiation is, in fact, limited at coincidence to a finite value by losses in the system. If those are included in the analysis, a certain amount of radiation is even obtained for phase velocities below the speed of sound. For bending waves, in particular, i.e., ks = kB, where kB is given by equation (6-60a), (10-45) specializes to

pz Dmc /)( 2 ′′−= ωωk , (10-51)

where m ′′ is the plate’s surface density (=ρp h). At coincidence, the wave number is zero in the z-direction; for bending waves, that occurs at a particular frequency (coincidence frequency fc). Setting equation (10-51) equal to zero, and solving for the frequency, yields


357

p

cc D

mcf′′

==ππ

ω22

2 . (10-52)

Using equation (10-52), equation (10-51) can be rewritten in the form

ffc cz −= 1ωk . (10-53)

Putting equation (10-53) into (10-48) gives the following expression for the intensity in the field radiated by a bending wave: )11Re(~2

0 ffvcI cz −= ρ . (10-54)

From that equation, and recalling the definition from equation (10-41), the radiation efficiency can be directly expressed as

)11Re( ffs c−= . (10-55)

Equation (10-55) shows that, for an infinite (undamped) thin plate in bending vibration, sound only radiates at frequencies above the plate’s coincidence frequency fc. Moreover, the radiation efficiency asymptotically approaches 1 at high frequencies. At frequencies below fc, the root in equation (10-55) becomes imaginary, and the radiation efficiency identically zero. If losses are included in the analysis, the most important modification is that even some radiation occurs below coincidence; the greater the damping, the greater that radiation, as seen in figure 10-20. Additionally, as mentioned above, the singularity at the coincidence frequency disappears as well, and a finite radiation efficiency is obtained.

10-1

100

10 1 10 -5

10 -4

10 -3

10 -2

10 -1

10 0

10 1

10 2 Radiation efficiency, s

Normalised frequency, f / fc

η = 0,01

η = 0,001

η = 0,0001

Figure 10-20 Radiation efficiency s for an infinite-ly large, thin plate under-going bending oscillations. At frequencies below coincidence, the radiation efficiency is zero if the plate is undamped. For a damped plate, on the other hand, the radiation efficiency is non-zero at all frequencies.


358

Below coincidence, an acoustic near field arises in front of the plate. To study that field more closely, we use equations (10-44) and (10-47) to write the acoustic pressure as

) (0 ˆ ),,( zxkti

zzBetzx k

kv

p −−= ωωρ.

When f « fc, from (10-45), kz ≈ -ikB holds, if the minus sign is chosen when taking the root, so that a field that decays in the positive z-direction is obtained. Putting that relation into the expression for acoustic pressure yields

) (0 ˆ )(),,( xktizk

B BB eeiktzx −−= ωωρ vp , f « fc. (10-56)

That expression shows, firstly, that the acoustic pressure and the velocity field are 90o out of phase at the plate surface (z = 0), and, secondly, that the acoustic field decays exponentially with increasing distance to the plate. In other words, we have a reactive near field, the impedance of which (= p/vz), as seen from the plate into the fluid, is purely imaginary. Example 10-4 Show, using equation (10-52) as a point of departure, that the coincidence frequency of a plate is inversely proportional to the plate’s thickness h. Solution

Equation (10-52) states p

c Dmcf

′′=

π2

2 ,

where, from chapter 6, the surface mass hm pρ=′′ and the bending stiffness per unit

length )1(12 2

3

υ−=

EhD p .

Substitution gives h

KEh

cEh

hcf cppc =

−=

−=

)1( 122

)1( 12

2

22

3

22 υρπ

υρ

π ,

where E

cKp

c⎟⎠⎞

⎜⎝⎛ −

=

22 112

2

υρ

π is a quantity that only depends on the material properties

and the speed of sound in the surrounding fluid. In the table below, the value of Kc is given for some common materials, assuming sound radiation into air ( c = 340 m/s). Table 10-1 Value of Kc for some common construction materials.

Material Glass Concrete Light Concrete Gypsum Steel Aluminum

Kc [m/s] 18 18 38 32 12 12


359

10.6.2 Finite plates in bending

This section treats finite plates in free bending vibrations. The discussion is primarily qualitative, seeking to impart a physical understanding of the factors controlling sound radiation. A point of departure for the logic presented is that a vibrating plate can be modeled as a continuous distribution of monopoles (volume flow sources) over the surface area of the plate; see section 10.1. At frequencies well below coincidence, cB « c, implying that k « kB, where k is the wave number in the fluid in front of the plate. That inequality can also be expressed as

Bkλ « 2π .

In other words, the distance between “crests” and “troughs” in a harmonic bending wave is, acoustically speaking, “small”, and the resulting volume flows tend to cancel each other out, as illustrated in figure 10-21.

v ⊥

plate in bending vibration

fluid

-Q + Q-Q + Q+ Q

large rigid baffle

Figure 10-21 Sound radiation from one side of a plate in bending vibration, below coincidence, modeled as a

distribution of monopoles. Every Q represents the volume flow from a crest (peak) or trough (valley). The arrows indicate the volume flows from peaks to valleys. Note that the “half” volume flows at the edges of the plate are not cancelled out.

For and infinite (undamped) plate, it was shown in section 10.6.1 that such cancellation is complete at frequencies below the coincidence frequency; no sound radiation occurs. For a finite plate, as illustrated in figure 10-21, the cancellation is not as complete as in the middle of the plate. For plates that vibrate in a certain mode below coincidence, two main cases can be distinguished; see figure 10-22. The first case corresponds to a mode with an odd number of antinodes. In principle, there is an “extra” antinode that is not cancelled out, so it is as if a single monopole is radiating. The second case corresponds to a plate oscillating in a mode form with an even number of antinodes. In that case, there is a “+ antinode” to each “– antinode”, implying no net monopole contribution. A certain amount of radiation occurs, nevertheless, because cancellation is not complete at the edges and corners. In any case, “odd” modes radiate considerably more than “even” modes. Therefore, to minimize the sound radiation, plates should be excited so as to give preference to the modes with an even number of antinodes. The difference in radiation between “odd” and “even’ modes is most pronounced at low frequencies, for which the length and width of the plate are much less than the sound wavelength in air. At higher frequencies, the difference in radiation between the different mode types is smaller.


360

+

-

+

Odd mode

+

+

+

-

-

-

Even mode

In the preceding, it was assumed that, in principle, the vibration field had a harmonic spatial variation. In practice, however, the bending wavelength varies over the plate due to the mounting (boundary conditions), and due to stiffeners and inhomogenieties. Thus, the cancellation effect can be expected to diminish, and increased sound radiation results.

Thick plate Thin plate

Pressure equalisation

Exciting force Fexc Exciting force Fexc

Figure 10-23 By replacing a thick plate with a thin plate of the same material, the sound generation was reduced, despite that the plate’s vibration velocity increased. The explanation was that the frequency content of the excitation force was well above the coincidence frequency of the thick plate. By changing to a thinner plate, the coincidence frequency could be raised to well above the frequencies of excitation (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

Another important factor that influences the radiation from a finite plate is the nature of the boundaries that surround the plate. Two basic cases can be distinguished: with and without a baffle; a baffle refers to an acoustically large hard surface in which the plate is mounted, as in figure 10-21. With a baffle, the sound fields on each side of the plate do not interfere, as they do in the absence of a baffle. To illustrate the difference between these two cases, a plate oscillating in its lowest mode is considered; refer to figure 10-24. If the plate is small compared to the sound wavelength, it acts as a monopole-type source with a baffle, or a dipole-type source without. In practice, a plate can be regarded as baffled if it is

Figure 10-22 Illustration of eigenmodes with odd and even numbers of antinodes, respectively. Consider also the (3,1) and (3,2) modes of figure 6-31.


361

mounted such that there is no direct path through the surrounding fluid between the two sides of the plate.

With baffle

Q Monopole Without bafQDipole

-Q

Figure 10-24 Sound radiation from a plate vibrating in its first mode with and without a baffle, respectively.

For frequencies well above the coincidence frequency, the relation analogous to that given above is Bkλ » 2π.

That implies that the “peaks” and “valleys” of a harmonic bending wave are acoustically “far apart” in that case. Therefore, no interference occurs, and the different parts of the plate radiate independently of one another. The size of the plate, and the exact nature of the vibration field, do not influence the radiation (or radiation efficiency). The radiation efficiency derived for the case of an infinite plate, equation (10-55), must also apply to this case. That the radiation factor asymptotically approaches a unit value is also clear because, well above the coincidence frequency, the bending wavelength is much longer than the sound wavelength. Thus, in a local sense, the plate acts as an oscillating piston on the surrounding fluid. Because that “piston” is larger than the sound wavelength, it induces a plane wave in the fluid, from which it follows that the radiation efficiency is 1. Figure 10-25 presents a plot of the gross frequency dependence of the radiation efficiency of a baffled thin plate. In the frequency region f11 < f < fc, the curve is valid so long as the modal density is high, and all modes are equally excited. The first condition means that if the analysis is made in a certain frequency band, e.g., a third octave band, that band should contain at least five modes.

(f / fc)0,5 P λc π2 S

10 log s

0

5 log π P

16 λc ( )

s = 1

β λ2

S λc

4 S

2

monopole corners sides entire plate

fc Frequency, f f11 0,5fc3c/P Figure 10-25 Gross frequency dependence of the radiation efficiency, for the case of a simply supported

(moment free mounting) rectangular plate. The plate is mounted in a baffle, and radiates into a free field. Notation: S is the plate’s area; P is the plate’s perimeter; f11 is the plate’s lowest natural frequency; β is a shape factor that depends on the length and breadth of the plate (see equation (10-57)); λ is the sound wavelength in the fluid medium into which the plate radiates; and λc = c/fc.


362

As is evident from the figure, the plate radiates as a monopole up to the first eigen-frequency f11. The shape factor β, which is equal to 1 for a square plate, is given by

⎟⎠⎞

⎜⎝⎛ +=

ab

ba

21β , (10-57)

where a and b are the edge dimensions of the plate. In the frequency range f11 < f < (3c/P), the radiation is dominated by the corners of the plate. In the region (3c/P) < f < fc /2, it is the edges that give the dominating contribution. In the region fc /2 < f < fc the entire plate begins to contribute to the radiation, and the radiation efficiency increases rapidly. The value of the radiation efficiency can be estimated by interpolation, as suggested in figure 10-25. At coincidence, f = fc, the radiation efficiency has a finite value determined by the plate’s perimeter. Above coincidence, f > fc, the entire plate radiates, and the radiation efficiency is given by the expression derived for the infinite plate, equation (10-55).

10.7 POINT EXCITED PLATES

In this chapter, the sound radiation from a finite, homogeneous, point-excited plate is treated; refer to figure 10-26. Energy-based methods can be used for the analysis, provided that the plate is excited in the many-modes (mode rich) frequency region, referring to table 7-1. Moreover, when averages over frequency bands are considered in the many modes region, the point impedance of a system at the excitation point, as well as the power it radiates, become equal to those of a corresponding infinite system. In the case studied here, the corresponding infinite system is an infinite plate, the thickness and material properties of which are identical to those of the original, finite plate. According to table 6-9, the point impedance of an infinite plate is

mDZ p ′′= 8 , (10-58)

where m” = ρph is the surface density of the plate. Equation (10-58), as well as the results derived below, assume that the acoustic load of the surrounding fluid on the (bending) vibrating plate is negligible. That is normally the case for plates surrounded by a gas (e.g., air), but not for plates radiating into a liquid (e.g., water). By analogy to the treatment of sound fields in rooms (section 7.2.2), the field that is obtained in the plate can be divided up into two parts: the direct field and the reverberant field.

F

Near field

The direct field corresponds to the field that propagates out from the excitation point, and has not been reflected from any boundaries. The reverberant field is the portion of the field that is reflected by boundaries, and which brings about the standing waves (resonant

Figure 10-26 Plate undergoing bending oscillations, excited by a point source. Around the excitation point, there is a bending near field. That has significance for the plate’s sound radiation below the coincidence frequency.


363

modes) that are excited. In an energy-based approach, both of these fields are assumed to be uncorrelated, which permits their respective contributions to the total sound radiation to be added on a power basis. The total radiated power can therefore be expressed as

resdirtot WWW += . (10-59)

The radiation dirW of the direct field is only of significance at excitation frequencies below the plate’s coincidence frequency, at which the sound power is primarily radiated from areas of the plate at which cancellation is ineffective. In the region around the excitation point, a bending near field arises, the amplitude of which falls quickly with increasing radius. These near fields are the main reason for sound radiation by the direct field below coincidence. Above coincidence, the entire plate radiates, and the contribution from the direct field is, for a lightly damped plate, negligible compared to that of the resonant modes, resW . At frequencies well below coincidence, it can be shown that

cm

FWdir 2

20

)(2

~

′′=

π

ρ , f « fc (10-60)

where F is the force a the excitation point. That equation can, in practice, be used up to half of the coincidence frequency without an error exceeding about 1 dB To estimate the radiation of the resonant modes, the mechanical power fed into the plate is first calculated. That is given by equation (3-7),

( ) ===Z

Fv 1Re~*Re21 2FWin {equation (10-58)}

mDF

p ′′=

8

~ 2 . (10-61)

That power builds up the reverberant field in the plate, and in stationary conditions, there must be a balance between inW and the power disW dissipated by the reverberant field. If the total loss factor, including material losses, boundary losses, and radiation losses, is η, then from chapter 3, EWdis ηω= ,

where E is the time averaged value of the total mechanical energy in the plate. For lightly damped resonant systems, E is twice the time-averaged kinetic energy, i.e.,

2~resvSmE ′′= ,

where S is the area of the plate, and vres is the velocity field corresponding to the resonant bending vibrations. In summary, we have

2~resdis vSmW ′′= ηω . (10-62)

From the power balance between the injected and the dissipated power in the system, written with the aid of (10-61) and (10-62),


364

p

resDSm

Fv23

22

)(8

~~

′′=

ηω (10-63)

Using the definition (10-41) of the radiation efficiency, the radiated power becomes

2123

20

)(8

~

pres

Dm

FcsW

′′=

ηω

ρ , (10-64)

where s is the radiation efficiency for the plate’s resonant bending vibrations. If the excitation is considered in frequency bands, that radiation efficiency is available from figure 10-25. For the total radiated power, finally,

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

≥′′

≤′′

+′′

≈+=

cp

cp

resdirtot

ffDm

Fc

ffDm

Fcscm

F

WWW

5.1,)(8

~

5.0,)(8

~

)(2

~

2123

20

2123

20

2

20

ηωρ

ηωρ

πρ

, (10-65,66)

where only the radiation from the reverberant field, with s = 1, is included above the coincidence frequency. A practically significant consequence of the result of equation (10-65) is that increased damping does not always lead to reduced radiation below the coincidence frequency. If the direct field dominates, the sound radiation is not influenced at all by the system’s losses. And when the reverberant field and the direct field are equal contributors to the total sound radiation, an increased damping reduces the sound power emission by, at most, 3 dB. For increased damping to be of any great help, the reverberant field must strongly dominate the sound radiation. An estimation of the limiting loss factor, above which there is no further gain as dissipation increases, can be obtained by setting dirW equal to resW , from equation (10-65), and then solving for η. Assuming the case of edge-dominated radiation (recall figure 10-25), the limiting loss factor is

23 , 2 c

Bgr ffPc

SkP

≤≤≈η , (10-67)

where P is the perimeter of the plate.

10.8 FLOW INDUCED SOUND

That sound generation and flow are connected is apparent from a range of phenomena that we can observe around us. A noteworthy example is that of sound generation by a jet engine. Other examples are sound generation mechanisms of various flow machines, i.e., fans, pumps, compressors, and diesel engines. In all of these cases, the ultimate causes of the sound generation are non-stationary processes in the gases and liquids involved.


365

Figure 10-27 When vehicles move at high speeds, pressure fluctuations are generated, i.e., sound arises, due to

the turbulent boundary layer of the flow field.

p

v

Laminar flow

Turbulent boundary layer

Turbulent eddies

Outside

Inside

Roof

p

Figure 10-28 The pressure fluctuations generated by the turbulent vortices are responsible for noise both inside

and outside of the vehicle. The basis for the analysis of these processes as acoustic sources is available from a theory developed by J.Lighthill in the early 1950’s. Lighthill’s theory of flow acoustic sound radiation is based on the assumption that there are only three basic types of sources that are possible in a fluid, namely monopoles, dipoles, and quadrupoles, and that all flow acoustic sources consist of some combination of these three basic types. Lighthill’s theory also contains a justification of that assumption. The justification is based on a reformulat-ion of the fundamental equations of fluid flow, the equation of motion and the equation of continuity, so that an acoustic wave equation with source terms is obtained. These source terms motivate exactly the three fundamental types mentioned above. A weakness of Lighthill’s theory is that it ignores the interaction between flow and sound. In that theory, the flow field is considered to be a given source that is not influenced by the sound field it generates. That is never really the case, and a certain amount of interaction always occurs. In some cases, that interaction can be strong, and Lighthill’s theory is not applicable. An important example of that is the so-called whistle sound caused by vortex shedding; see section 10.8.2. In many cases, Lighthill’s theory can, nevertheless, be applied successfully, and it is the most commonly-used model for the study of flow-induced sound. The physical mechanisms corresponding to the 3 source types, and examples of when they occur, were discussed earlier in this chapter. Table 10-2 summarizes that.


366

Table 10-2 The three fundamental types of flow acoustic sources.

Source type Physical mechanism Physical situation

Monopole fluctuating volume or mass flow

cavitation, inlets and outlets of piston machines (e.g., valves)

Dipole fluctuating force propellors, fans

Quadrupole fluctuating force couple free turbulence (jet flows)

10.8.1 Scaling laws for flow induced sound

In order to determine how the sound generation from a certain type of source depends on the flow conditions, scaling laws may be used. A scaling law can, among other things, be used to determine how much increase in sound power is obtained due to a change, e.g., when the flow velocity is doubled. Another use is to be able to rank the relative signific-ance of the three source types, i.e., determine which type dominates in a given situation. Equations that compare the sound power generated in a free field, between a dipole and a monopole, or a quadrupole and a dipole, have been derived earlier in this chapter (equations (10-15) and (10-20,21), respectively). These equations show that from small source regions, in terms of the He number,

2)(/ kdWW md ∝ and 2)(/ kdWW dq ∝ , (10-68,69)

where d is a length scale that indicates the size of the source region. To apply that to the case of flow acoustics, we must first be able to determine whether or not the source region is acoustically small. In other words, the wave number k must be known. For flow generated sound, a rule of general validity is that the frequency spectrum of the sound “scales to” (is proportional to) a frequency fst, which is determined by a typical flow velocity U and a typical size d of the source region, as

dUfst /= . (10-70)

That characteristic frequency fst, for flow generated noise, is usually called the Strouhal frequency. The quantities U and d are to be chosen to characterize the source mechanism of interest. Examples of that, for some different cases, are provided in table 10-3.

In the first case (I), there is flow about a cylindrical barrier. Around that barrier, a periodic vortex shedding begins at very small Reynold’s numbers4 (based on the diameter of the cylinder). That shedding gives rise to fluctuating forces, which correspond to dipole sources. The Strouhal frequency is obtained by choosing U as the velocity of the flow field and d as the diameter of the cylinder. The sound generated is relatively narrow banded; except for large Reynold’s numbers, it is a tone-like sound, the Strouhal toneError! Bookmark not defined., with a frequency proportional to fst. 4 Reynold’s number is defined as Re = Ud/v, where v is the kinematic viscosity.


367

Figure 10-29 For a cylindrical pole in an air flow, there is a periodic shedding of vortices that gives rise to

fluctuating forces. Sound generated in that way is called a Strouhal or aeolian tone. The shedding frequency is proportional to the so-called Strouhal frequency, which is determined by the flow velocity and the diameter of the pole. That sound generation can be diminished by reducing, in different ways, the regular generation of vortices. (Picture: Asf , Bullerbekämpning, 1977, Ill: Claes Folkesson.)

In the second case (II), a non-pulsating turbulent jet flow exits a duct. The jet corresponds to a distribution of quadrupole sources. The Strouhal frequency is obtained by choosing U to be the jet’s velocity and d to be the diameter of the duct. The sound generated is broad-banded, with a frequency content that “scales to”, i.e., is proportional to, fst. In the third case (III), a propellor rotates in an otherwise still fluid, at a rotational frequency f0. The blades of the propellor generate time-varying forces on the surrounding fluid, and thereby constitute dipole-type sources. From the perspective of a listener that does not move through the fluid, the time-variation of the forces has two causes: the blade rotation; and, the turbulence in the flow fields around the blades. The rotation brings about a periodic time dependence. If all of the blades are alike, the blade force distribution is repeated every time the propellor rotates through an angle 2 π /K, where K is the number of blades.

Air flow Regular vortex generation giving strong tonal excitation

Extension

Iregular vortex generation Iregular vortex generation

Disturbances


368

Table 10-3 Choice of characteristic velocity, U, and characteristic length, d, for calculating the Strouhal frequency, fst = U /d, in three different cases.

Case I

Periodic vortex shedding

Vortex shedding frequency fvs = 0.2 U/d

Case II

Turbulent jet

Case III Propellor f0 Rotationalfrequency K Number ofblades

If that angle is divided by a rotational frequency 2π f0, the period of rotation, 1/Kf0, is obtained. From that it follows that the fundamental frequency, i.e., the so-called blade pass frequency, becomes Kf0. The sound from a propellor consists of a periodic part (harmonic multiples of the blade pass frequency), and of a broad band part corresponding to the turbulence contribution. For the periodic part, the Strouhal frequency corresponds to the blade passage frequency, and we can, for example, choose U as the peripheral velocity and d as 2πa/K, where a is the radius of the propellor. Equation (10-70) can be used to estimate the size of the source region, measured in the He number, for a flow acoustic source; the result is

{ } MdUfc

dfkdHe st

st ππ

2/2

===== , (10-71)

where M = U/c is the Mach number. From that equation, it is evident that flow acoustic sources are small, acoustically, for small values of the Mach number. For such Mach numbers, equations (10-68,69) can be used to give scaling laws that relate the three fundamental types of sources to each other. Putting the He number from equation (10-71) into these equations gives 2/ MWW md ∝ and 2/ MWW dq ∝ , (10-72,73)

Harmonics of fst


369

Equations (10-72) and (10-73) show that, for small values of the Mach number, the monopole is the most effective flow acoustic source type; after that, there is the dipole, and least effective as a radiator is the quadrupole. Besides ranking the sources, it is also of interest to know how the radiation from each type of source depends on the state of the fluid. Scaling laws that describe that can be obtained by first studying the monopole, and thereafter applying equations (10-72,73). For the monopole, according to equation (10-5),

20

20

~QckWm ρ∝ , (10-74)

where the volume flow Q scales as follows:

2areahastighet~ UdQ =×∝ .

Putting that into equation (10-74) gives 2220

20

20 )(~ dUkdcQckWm ρρ =∝ ; moreover,

the wave number satisfies cd

Ucf

k st ππ 22== , so that

cUdWm42

0ρ∝ . (10-75)

Making use of equations (10-72,73), the corresponding expression for a dipole is

3620 cUdWd ρ∝ , (10-76)

and for a quadrupole, 5820 cUdWq ρ∝ . (10-77)

The equations (10-75) to (10-77) only describe how the motion of the flow field, characterized by the velocity U, can be converted into sound. Physically, that means that these scaling laws describe how the kinetic energy in a flow field is converted to sound energy. For cases in which there are other energy sources in the flow field, e.g., thermal sources caused by combustion, more complicated scaling laws are required. Equation (10-77) is the best known result from Lighthill’s theory, and is usually called Lighthill’s U10-law. For low Mach-numbers, that result describes the sound radiat-ion from a jet in which thermal effects are negligible (a so-called cold jet). Although the equation, as derived, is limited to low Mach numbers, Lighthill assumed that it could also be applied to jet engines of airplanes. A number of experimental investigations have confirmed that assumption; see figure 30. In fact, equation (10-77) works up to a Mach number5 of about 1.5. The insight that jet noise follows the U10-law has been an important factor in the development of quieter jet engines. The strong velocity-dependence implies that noise can be effectively reduced by reducing the engines’ thrust velocities. By simul-taneously increasing the area, that reduction is possible without reducing the total thrust.

5Note that the definition of the Mach number used here corresponds to a “thrust velocity/(sound speed in the surrounding air)”.


370

U 8

U 3

Sound power level (dB) - 20 log(d)

Figure 10-30 Velocity dependence of the sound generation from jet and rocket engines, from Chobotov &

Powell, 1957. Note that the sound power is corrected for the size d of the sources by the scaling law given by equation (10-77).

As is evident from figure 10-30, the exponent of the velocity dependence falls off, and the sound power radiated is proportional to U3 for very high thrust velocities. An explanation for that is that if the kinetic energy is the main energy reserve for sound generation, then the available energy per unit volume in a jet is ρ0U2/ 2. That implies that the available power, corresponding to the outflow of kinetic energy per unit area at the outlet of the jet engine, increases as ρ0U3/ 2. In the limit of high velocities, the maximum sound power that can be generated must asymptotically approach the curve for the available power, i.e., must grow at a rate proportional to the third power of the thrust velocity.


371

Vortices

Jet core

Jet. High flow velocity

Jet. Low flow velocity

Jet

Figure 10-31 By reducing the velocity difference (gradient) in the mixing zone, the sound radiation of the jet can

be reduced. (Picture: Asf , Bullerbekämpning, 1977, Illustrator: Claes Folkesson.) The scaling laws discussed in this section have been derived under the assumption of three-dimensional (3-D) sound fields and free field conditions around the sources. In practice, the source region is often enclosed; consider the example of a throat in a duct with flow. That implies that cases with cylindrical (2-D) sound fields, as well as plane (1-D) sound fields, are also of interest. For the case of a duct, for instance, a plane wave sound field is obtained at low frequencies; see chapter 10. If an analysis corresponding to that carried out here is applied to the cases of 1- and 2-D sound fields, modified versions of the scaling laws (10-75) to (10-77) can be derived. A summary of these scaling laws, for sound fields of arbitrary dimension, is provided in table 10-4.

Table 10-4 Flow induced sound. Scaling laws for sound power in sound fields with different dimensions. U is a characteristic velocity, and d a characteristic length.

Dimension Monopole Dipole Quadrupole 1-D ρ 0

2 2cd U ρ 02 4d U c ρ 0

2 6 3d U c

2-D ρ 02 3d U ρ 0

2 5 2d U c ρ 02 7 4d U c

3-D ρ 02 4d U c ρ 0

2 6 3d U c ρ 02 8 5d U c

10.8.2 Whistle sounds

In some situations, strong interaction can occur between a sound field and a flow field. Examples of such situations are vortex shedding around a body in a flow field, or at a sharp edge. If the flow field is not too turbulent, the shedding is primarily periodic, and corresponds to a certain shedding frequency fvs, which is proportional to the relevant


372

Strouhal frequency, fvs = α fst. For cases in which the vortex shedding frequency coincides with a resonance frequency fres, corresponding to an acoustic mode or a structural mode in a connected system, i.e., resst ff = α , (10-78)

a self-excited acoustic oscillator can result. That condition is necessary, but not sufficient; to actually bring about a self-excited system, a positive feedback must also exist between the flow field and the connected system. When a self-excited system is obtained, the amplitude grows until it is limited by non-linearities and losses. Thus, this type of phenomenon can generate very strong tonal sounds, called whistling, and is normally non-linear to its nature. The high levels are often a problem in a technical context. There are chiefly two ways to eliminate whistling sounds. Either the flow is disturbed, and the degree of turbulence increased, so that the periodic shedding breaks down, or the frequencies fvs and fres are separated by modifying something in the system, e.g., the flow velocity, length, or stiffness. Examples of situations in which whistle tones are generated are shown in figure 10-32.

Figure 10-32 Two examples of how whistle tones can be generated in a flow field.

Case I Periodic vortex shedding from a bar in bending vibration. If the shedding frequency coincides with a bending resonance of the bar, a self-excited acoustic system can arise. A practical example of periodic vortex shedding is that from a high speed electric train’s pantograph (linkages extending from the train to contact and draw power from the trackside electrical net). Case II Periodic vortex shedding at a hole with sharp edges coupled to a resonator. If the shedding frequency coincides with the Helmholtz resonance, a self-excited acoustic oscillator can result.

A practical example of flow-induced whistle noise is that sound which is generated by a garden trimmer. The operation of the trimmer is based upon striking grass stems and thin branches with a thin nylon chord. In the flow field around the nylon chord, there is a periodic vortex shedding that results in a pronounced tone. That tone can be largely eliminated if the cross section of the chord is made elliptical, rather than circular. Wind generated tones arising from high smoke stacks serve as another example of sound caused by periodic vortex shedding. The purpose of the spiral shaped flanges that can be seen wrapped around such smoke stacks is to eliminate that noise source by disturbing the vortex shedding; see figure 10-33.

Beam

Helmholtz resonator

Case 1 Case 2


373

Finally, we note that the howling tone that is sometimes generated by a planer is caused by periodic vortex shedding at an opening coupled to a Helmholtz resonator in the form of a cavity; see figure 10-34. A possible countermeasure is to eliminate the cavity by filling it with rubber filling.

Figure 10-33 The whistling sound generated by a periodic vortex shedding from a smoke stack can be eliminated by wrapping the stack with a spiral band. That effectively disturbs the vortex generation. (Picture: Asf , Bullerbekämpning, 1977, Ill: Claes Folkesson.)

Wind

Spirsl band

Chimney stack


374

o Planer can

Planer

Open cavity

Sharp edge Fixation

Planer

blade

Solution: Modified edge

Rubber filling

Fixation Planer blade

Figure 10-34 A planer can, when rotating, generate a powerful shrieking sound. That sound is generated by a

broad-band vortex shedding along the edge of the planer blade. Certain tones in the generated sound are then strongly amplified in the cavity of the blade. That sound can be eliminated by filling the cavity, and thereby preventing the amplification. After filling the cavity, only a hissing noise remains. (Picture: Asf , Bullerbekämpning, 1977, Ill: Claes Folkesson.)


375


MONOPOLE

Sound pressure ikrm e

ri −=πωρ

400 Q

p (10-3)

Superposition of monopoles ∑ −=n

ikr

n

nm ne

riπωρ

40 Q

p (10-4)

Radiated sound power 20

20 ~4

Qck

Wm πρ

= (10-5)

DIPOLE

Sound pressure ⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

⋅−

=−

re

z

ikrz

d 4πF

p (10-9)

Sound pressure, ikrzd e

ikrrik −⎟

⎠⎞

⎜⎝⎛ +=

114

cosπ

θFp (10-11)

in spherical coordinates

Sound power π

ρ12

~ 240 z

dDck

W = (10-14)

Dipole – monopole radiation ratio

1 , 3/)( 2 <<= klklWW md (10-15)

QUADRUPOLE

Sound pressure in the far field (spherical coordinates)

Longitudinal θπ

22, cos

4 rk zz

zzqM

p −= (10-16)

Lateral φθθπ

cossincos4

2, r

k xzxzq

Mp −= (10-17)


376

Quadrupole – dipole radiation ratio

1 , 5/)( , 5/)(3 2,

2, <<== klklWWklWW zdlatqzdlongq , (10-20,21)

INFLUENCE OF BOUNDARIES

Radiated sound power from a monopole over a plane

⎟⎠⎞

⎜⎝⎛ ±=

khkhWW m 2

2sin1 (10-27)

LINE SOURCE

Sound pressure from an uncorrelated line source

h

Wcplinje π

θρ4

~ 02 Δ′= (10-33)

Sound pressure level 11)log(10)log(10 −Δ⋅+⋅−= ′ θhLL Wp [dB] (10-34)

SOUND RADIATION FROM VIBRATING STRUCTURES

Radiated sound power ∫ ⊥=

ySyrad dSvW 2~)Re(Z (10-38)

Radiation efficiency ySvc

WWWs

200 ~

⊥

==ρ

(10-41)

Coincidence frequency of a thin plate in bending vibration

p

cc D

mcf′′

==ππ

ω22

2 . (10-52)

Radiation efficiency of an infinite thin plate in bending vibration

ff

sc−

=1

1Re . (10-55)


377

POINT EXCITED PLATES Radiated sound power

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

≥′′

≤′′

+′′

≈+=

cp

cp

resdirtot

ffDm

Fc

ffDm

Fcscm

F

WWW

5.1,)(8

~

5.0,)(8

~

)(2

~

2123

20

2123

20

2

20

ηωρ

ηωρ

πρ

(10-65,66)

Boundary loss factor 23 , 2 c

Bgr ffPc

SkP

≤≤≈η , (10-67)

FLOW GENERATED SOUND

Strouhal frequency dUf st /= , (10-70)

Scaling laws for flow generated sound (3-D sound field)

Monopole cUdWm42

0ρ∝ . (10-75)

Dipole 3620 cUdWd ρ∝ , (10-76)

Quadrupole 5820 cUdWq ρ∝ . (10-77)


378

379

CHAPTER ELEVEN

VIBRATION ISOLATION Noise and vibrations have undesirable effects on both human quality of life, and on our material goods. Vibration-generating machinery and processes contribute, to a large extent, to the total noise and vibration exposure. A very useful strategy to reduce noise and vibrations is to interrupt the propagation path between the source and the receiver. Elastic mounting is a simple method to hinder the spread of structural vibrations. In practice, an elastic mounting system is realized by incorporating so-called vibration isolators along the propagation path. Strongly vibrating machines in factories, dwellings, and office buildings can be placed on elastic elements. The passenger compartments in vehicles are isolated from vibrations generated at the wheel-roadway contact by incorporating springs between the wheel axles and the chassis. With properly design and implementation, elastic mounting of machines is both an effective and an inexpensive approach to noise and vibration mitigation. The objective of this chapter is both to provide the essential knowledge required to properly design vibration isolation systems, and to impart a physical understanding of the principles used in vibration isolation.

Chapter 11: Vibration isolation

380

9.1 SOURCE AND SHIELD ISOLATION

Vibration isolation seeks to reduce the vibration level in one or several selected areas. The idea is to hinder the spread of vibrations along the path from the source to the receiver; see figure 11-1. Vibration

source Receiver

Propagation path

Vibration isolation

W iWr

Wt

L p

Figure 11-1 Example of a situation in which the vibrations emanating from a machine are reduced by isolation.

A power Wi impinges on the isolators, a power Wr is reflected back towards the source, and a power Wt is transmitted to the floor.

Of course, there are various options inherent in the actual realization of a vibration isolation system. Firstly, there are options as to where along the path to deploy the isolation; secondly, the isolators themselves can be designed in many different ways. It is essential to locate and design the isolation in the best possible way for the specific situation. Regarding the location of the isolation, one can distinguish two extreme cases: placement near the source; and, placement near the receiver. In the first case, in which the source is isolated from the surroundings, one speaks of source isolation. In the second case, in which the receiver is isolated, one instead speaks of shielding isolation. Both cases are illustrated in figure 11-2. Note that one can, of course, combine shielding and source isolation. If there are very demanding requirements for a low vibration level, it is natural to isolate both the source and the receiver.

a) b)

Figure 11-2 Two different strategies for vibration isolation: a) source isolation of machines; and, b) shielding

isolation of sensitive equipment.


381

11.2 VIBRATION ISOLATION IN GENERAL

A vibration isolation problem is often schematically described by division into substructures: a source structure which is coupled to a receiver structure. The vibration isolation is yet another substructure incorporated between the two structures. The objective of vibration isolation is to reduce the vibrations in some specific portion of the receiver structure. It is apparent that vibration isolation can be realized in many different ways. It therefore falls upon the designer to arrive at an isolation system design well-suited to the specific situation. All practical vibration isolation builds on a single physical principle. When a wave propagating in an elastic medium falls upon an abrupt change (discontinuity) in the properties of the medium, only a portion of the wave passes through that discontinuity. The remaining portion of the wave is reflected back towards the direction from which the incident wave arrives. The magnitude of the reflected portion of the wave depends on the magnitude of the change in properties. Reflection and transmission of simple plane waves in beams is treated in detail in chapter 6. In the case of vibration isolation, one seeks to hinder the propagation of the wave by bringing about such discontinuities in properties along the propagation path. The most common way to accomplish a discontinuity in the properties of the medium is to incorporate an element that is considerably more compliant, i.e., has a lower stiffness than the surrounding medium (see figure 11-3a). That type of element is usually called a vibration isolator. Steel coil springs and rubber isolators in a variety of forms are examples of vibration isolators readily available on the market; see section 11.6 for more details. Note that the stiffness can be changed by incorporating elements that are stiffer than the surrounding parts. W i

W r

W t

κ

a) b) Wi

Wr

Wt

m

Figure 11-3 Two different vibration isolation methods. a) Reflection against a soft element. b) Reflection against

a mass. Wi = incident power, Wr = reflected power and Wt = transmitted power. On further consideration, it is also apparent that one can bring about a reflection by incorporating an element with a differing inertia from that of the medium. Since elements of that type are often idealized as rigid masses, see figure 11-3b, they are referred to as blocking masses. Practical realizations of the concept of blocking masses are, for example, seismic blocks and added masses at compliant points; see figure 11-4.

Considering, however, that the most common construction materials are relatively stiff, such as steel and concrete, it is often simpler to accomplish significant discontinuities in the medium properties by the compliant-element approach. For that reason, it is much more common to use compliant than stiff elements. Nevertheless, for structural reasons, there are some cases in which it is necessary to use stiff elements.


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a)

Seismic block

b)

Added mass

Compliant supporting structure

Figure 11-4 Blocking masses, preferably in combination with elastic elements, give very good vibration isolation

and are frequently used in practice. a) Seismic block. b) Added mass at a compliant point. For a long time, machinery designers have mainly provided for vibration isolation using “trial and error” methods in combination with rough estimates obtained from very simple calculations. That approach tends to yield good results at lower frequencies, up to about 100 Hz, say. On the other hand, in the lion’s share of the audible frequency range, that approach provides little or no control over the actual isolation results obtained. In order to be able to achieve good vibration isolation by design, throughout the entire relevant frequency range, access to more advanced theoretical, as well as experimental, techniques is a necessity.

11.3 MEASURES OF TRANSMISSION ISOLATION

In order to be able to design in optimal vibration isolation, there is a need for, not only the determination of the vibration levels, but also for some measure of the vibration isolation obtained; that latter would permit comparison of alternative isolation strategies that may be applicable in a given situation. A number of different measures are in use for various specific applications. The most universally applied of these is the so-called insertion loss DIL.; it is defined in either of the two following alternative ways1:

[ ]dB afterv

beforev

vIL LLD −= . (11-1a)

[ ]dB afterF

beforeF

FIL LLD −= . (11-1b)

where the velocity and force levels Lv and LF are each defined in chapter 1. The insertion loss is, therefore, defined as the difference in level at a given point before and after the vibration isolation is provided; see figure 11-5. With these definitions as a model, it is of course possible to devise other such measures of the isolation effectiveness based on weighting different frequency components and bands, e.g., using A-weighting for instance.

1 For source isolation, one normally uses (11-1b), i.e., the force level on the foundation is compared before and after. For shielding isolation, one instead uses (11-1a), i.e, the velocity level on the machine is compared before and after isolation is provided. The formulas determined below, for the insertion loss, apply to either definition.


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The choice of the relevant gauge of effectiveness, from (11-1a,b), is ultimately determined by the specific application.

F before

Without isolator With isolator

F after

Figure 11-5 The insertion loss can be defined as the difference in the force level acting on the foundation before

and after the implementation of isolation.

11.4 CALCULATION OF VIBRATION ISOLATION

The design of vibration isolation, for frequencies up to 1000 Hz, calls for relatively complicated calculations. In principle, computational models of machines, isolators, and foundations, which correctly describe the relationship between loads and deformations throughout that entire frequency band, are needed. That is, as quickly becomes apparent to the analyst, a very difficult problem. Even very detailed finite element models are so time-consuming, and give such uncertain results, that they are of doubtful value. In practice, more or less simplified computational models of the different substructures are therefore an inevitable expedient. These should not be relied on for absolute values of the vibration isolation in a narrow frequency band, but they can be used to compare alternative vibration isolation design approaches in octave or third-octave bands. They can also indicated tendencies and suggest how vibration isolation can be improved. By the use of so-called frequency response function methods with input data from measurements, relatively reliable results can be obtained. The measurement of frequency response functions, and especially for isolators, is a whole research area in and of itself.

11.5 SOME VIBRATION ISOLATION COMPUTATIONAL MODELS

In order for computational models to serve as practical tools for the comparison of different vibration isolation alternatives, simplified models must inevitably be used. The problem, in its entirety, includes the transmission of vibrations originating in many degrees of freedom, at each coupling surface between the elements of the structure. To simplify the problem, one can:

(i) Idealize the coupling as occurring via infinitesimally small, i.e., point, contacts.

(ii) Assume that only 1 - 2 degrees-of-freedom contribute to the vibration transmission.


384

(iii) Ignore coupling points that make minor contributions to the vibration transmission.

(iv) Combine parallel transmission paths into a single equivalent path.

That last simplification is only useful if the contact points both between the isolators and the machine, and between the isolators and the foundation, move in phase at the same amplitude.

11.5.1 Rigid body – ideal spring – rigid foundation

At much lower excitation frequencies, considerably simplified models of the components are usable. Assume, for example, that we analyze a machine mounted at four points on a system of concrete joists. Assume, moreover, that the machine has an axle that generates sinusoidal bearing forces at the rotational frequency. At very low disturbance frequencies (i.e., low rotational speeds), the deformations of the machine itself are negligible, i.e., the machine acts as a rigid body. Physically, one can regard the force acting on the machine as so slowly changing in time that all parts of it have time to react to small changes in the force magnitude before the next such change occurs. Mathematically, the machine’s movements can be described by means of equations from rigid body mechanics. The instantaneous state of the machine is then completely described by six degrees-of-freedom, three translational and three rotational. In practice, the number of degrees-of-freedom can normally be further reduced to one or two, eliminating those which are not relevant. As the rotational speed of the axle increases, we eventually arrive at a situation in which the force changes so rapidly that not all parts of the machine have time to react before the force changes again at the point of its application. At that stage, we can begin to speak of wave propagation in the machine. If the rotational speed continues to increase further, we will arrive at a certain excitation frequency at which the amplitude of the machine deformations has a strong peak. At that frequency, the deformation waves and their reflections interact constructively to bring about the maximum in the response. That phenomenon is the so-called resonance phenomenon with which we are already familiar from chapters 5 and 6. At these frequencies, we can no longer regard the machine as a rigid body. A commonly used rule of thumb is that the rigid body assumption is useful up to frequencies of 1/3 of the first resonance frequency, i.e., for low Helmholtz numbers. The rigid body assumption for the machine has an analogue that can be used in the description of the foundation. Consider anew the example of the machine described above. At very low excitation frequencies, the joists respond with a (quasi-)static bending due to the slowly varying force acting at the machine mounting points. If the excitation frequency is so low that the deformation of the joists is so small as to be negligible in comparison to the deformation of the isolators, then the joists can be regarded, from the vibrations perspective, as a rigid foundation. Note that that doesn’t imply that the foundation is not excited into vibration; that would apply no transmission whatsoever. Let the excitation frequency now increase, just as it does when considering the machine. At sufficiently high frequencies, the deformation can no longer be ignored. When the frequency has increased sufficiently, an ever more distinct wave propagation becomes apparent in the foundation. If the geometrical limits of the foundation are far away, then we will eventually reach the first resonance frequency of the foundation. The description


385

of the foundation as rigid can, consequently, only be applied at low frequencies, say up to 1/3 of the first resonance frequency, i.e., once again at low Helmholtz numbers. Assume now that we would like to reduce the vibrations transmitted from the machine into the system of joists by incorporating soft vibration isolators at the mounting positions between the machine and the joists. Under the influence of forces from the machine, the springs are deformed. At low excitation frequencies, all parts of the isolator itself react to the changing of the force. That implies that the cross-sectional load is uniform along the entire isolator. We have, in other words, no considerable wave propagation. Yet another consequence is that the isolator can be considered massless. In contrast to the joists, the isolator is compliant. We can, therefore, not ignore its deformation under load. In these circumstances, the isolator can be regarded as an ideal massless spring. As the frequency increases, the motion in the spring takes on the character of wave propagation more and more. Once again, at a certain point, the situation becomes resonant. In exactly the same way as before, we can adopt the rule of thumb that the spring idealization applies up to about 1/3 of the first resonance frequency. Example 11-1 Consider the machine arrangement illustrated in figure 11-6. An electric motor is elastically mounted, by way of 4 identical isolators, to a 2-mm thick steel plate. When the motor is driven, its rotating parts generate a vertically-oriented, sinusoidal exciting force between the machine and the joists. Calculate the ratio between the total force acting on the foundation with and without the vibration isolators. Carry out the calculations at low frequencies under the assumption that the electric motor, when operating, generates a vertical harmonic exciting force with circular frequency ω and amplitude F . The mass of the motor is 100 kg, and each isolator’s complex stiffness (see chapter 3, section 3.3.5) is (1.0 + 0.01i)·104 N/m.

a) m

4κ

m

κ

x 4F1

F 1

x

F 1

4F1

b) c)

x

Fstör

Fstör

Single isolator

Figure 11-6 a) Electric motor elastically mounted to a large steel plate via four vibration isolators. b) Simplified model of the system in a. c) The system in b represented by its separated subsystems.

Solution Assume that the excitation frequency is so low that: (i) the motor can be considered a rigid body; (ii) the foundation can be regarded as rigid; and, (iii) each isolator can be described as an ideal massless spring. Assume, additionally, that the motors motions are strongly


386

dominated by small-amplitude vertical translations. In these circumstances, the single degree-of-freedom system in figure 11-6b is a useful model to describe the problem.

With isolators. Starting with the system in figure 11-6c, the equation of motion can be constructed for the mass m, as well as Hooke’s law for spring κ Thus,

12

2

4Fx−= excF

dtdm

where 4F1 is the total force acting on the foundation, i.e.,. the force transmitted through all four isolators, and

)0(1 −= xF κ .

Assume a sinusoidal, complex-valued displacement tie ωxx ˆ= and eliminate x using both of the relations given above. Then, the force on the foundation, normalized by the exciting force, is

1

20

2121 1

414

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

ωκωω

mFexc

F,

where ω0 is the machine’s so-called mounting resonance (see chapter 3.3.1), i.e., the resonance frequency of the machine mass on the compliance of the isolators. Note that the first term in the equation only applies to machines with four mounting points. For machines mounted at n points, the term 4κ /m should be replaced by nκ /m.

Without isolators. For the case of no isolators, it becomes evident upon reflection that the force on the foundation is equal to Fexc. The desired ratio between the force with and without isolators is therefore

20

21

ω

ω−=

m

u

FF

.

If an insertion loss is defined on the basis of that ratio, one obtains

m

ImIL m

DY

YY +⋅=−⋅=−⋅= log201log20

41log20

20

22

ωκωω , (11-2)

in which Ym and YI are the machine’s and the isolator’s respective mobilities; see table 6-9 and equation (6-68). Note that that formula even applies to cases of shielding isolation.

The insertion loss, in that case, has several characteristic properties; see figure 11-7. First of all, no isolation is obtained for excitation frequencies far below the mounting resonance f0 corresponding to ω0. Secondly, the insertion loss takes on large, negative values for excitation frequencies near the mounting resonance; there, the force on the foundation is amplified rather than reduced. Finally, large, positive insertion losses are obtained for excitation frequencies well above the mounting resonance. The increase in the


387

insertion loss asymptotically approaches 40 dB/decade, i.e., 40 dB for each increase of the excitation frequency by a factor of 10. The isolation seems, therefore, to be very effective above the mounting frequency. Unfortunately, that effectiveness is largely the result of the grossly simplified model. In reality, the increase in insertion loss is interrupted above, say, 10f0.

1 -40

-20

0

100

80

40

20

60

f0 10 100 1000 Frequency [Hz]

DIL [dB]

Figure 11-7 The insertion loss for a rigid body mounted elastically to a rigid foundation. The mounting resonance

is tuned to f0 = 3.18 Hz. Note the deep trough in the insertion loss at the mounting resonance, and its negative values elsewhere at low frequencies. The vibration isolation system is therefore counterproductive at low frequencies; above all, it is essential that the excitation frequency not fall in the vicinity of the machine’s mounting resonance frequency.

A very important conclusion from example 11-1 is that the vibration isolators must be designed to prevent the coincidence of the machine’s mounting frequency with any important excitation frequency. Moreover, it is clear that a positive effect is obtained from the isolators at frequencies above the mounting frequency. The implication is that as low as possible a mounting resonance frequency must be sought. In practice, machine mounting is often designed so that the mounting resonance frequency falls in the 2-10 Hz band.

11.5.2 Flexible foundation

As the excitation frequency increases, the deformation of the foundation due to the excitation force soon becomes too large to ignore. A model in which the foundation is flexible must then be used. A number of different models with differing characteristics are available for this situation. If, for example, the foundation is a system of joists with considerable dimensions, an infinite plate model might be used to describe the motions of


388

the foundation. If the foundation exhibits a resonance, then a mass-damper system can be used as a first approximation to describe its behavior.

Example 11-2 Consider the machine mounting situation of example 11-1. Assume that an infinite plate would be a valid model of the foundation response. Calculate the ratio between the total force on the foundation with and without isolators.

Solution Assume that the deformation of the foundation is the same at all four machine feet. Additionally, conditions (i) and (iii) from example 11-1 hold.

m

κ4

m

κx

4F1

F1

x

F1 4F1

b)

x

F r

Fs

1

x

1 1

2 x

2 x 2

a) Single isolator

Figure 11-8 Simple model of a machine mounted to a flexible foundation. The equation of motion, Hooke’s law, and the mobility of a plate yield the following system of equations:

121

2

4Fx−= excF

dtdm ,

)( 211 xxF −= κ ,

11

2 4)( FYx platei −= ω . Eliminate x1 and x2 ,

=+−

−=

plateexc

with

imm

F YF

)(441442

21

ωωωκκ

κ

plateIm

m

I

m

imi

YYYY

YY

++=

⎭⎬⎫

⎩⎨⎧

==

=κ4

1:9-6 Table

ωω

.


389

Without isolators, the force on the foundation can be determined by excluding the second of the equations from the system given above, and setting x1 equal to x2. The system then has the solution

platem

m

plateexc

without

mimi

F YYY

YF

+=

+=

ωω

114 1 .

The insertion loss is therefore

platem

plateImILD

YYYYY

+

++⋅= log20 . (11-3)

Formula (11-3) can be shown to even apply to the shielding isolation case. According to table 6-9, the mobility of a 2 cm thick, very large steel plate is

calculated from

5211

3

2

2

1061.2m 0.02=N/m 100.2

0.3=kg/m 7800=4

)1(3 −⋅≈⎭⎬⎫

⎩⎨⎧

⋅==

−=

hEEhplateυρ

ρυ

Y m/Ns.

Putting values into the expression for the insertion loss leads to the graph in figure 11-9. That figure also shows the corresponding results for a rigid foundation. Apparently, the flexible foundation affects the insertion loss in two bands: at the mounting resonance; and, at frequencies over about 50 Hz. At the mounting resonance frequency, the insertion loss increases, due to the ability of the infinite plate to act as an energy sink. Above 50 Hz, the flexible foundation provides a significantly lower insertion loss than the rigid foundation. Here, the isolation obtained is largely determined by the ratio of the isolator mobility to the mobility of the plate. At high frequencies, the insertion loss now asymptotically approaches a 20 dB per decade rate of increase, instead of the 40 dB per decade obtained earlier.


390

1 -40

-20

0

100

80

40

20

60

f0 10 100 1000 Frequency [Hz]

DIL [dB]

Rigid foundation

Compliant foundation

Figure 11-9 Insertion loss for a rigid body elastically mounted to an infinite steel plate. Compared to an ideal,

rigid foundation, the amplification peak at the mounting resonance frequency is reduced and the rate of increase of the insertion loss falls off. That latter effect is due to the diminished mobility or impedance gap between the isolator and the foundation.

11.5.3 Wave propagation in the isolator

When the excitation frequency has increased so much that the deformation field in the isolator is a wave motion, the ideal spring model becomes less and less tenable. Depending on the isolator design, different models for wave propagation in the isolator may be appropriate. In example 11-3, an example is given of a simple wave propagation model of the isolator. Example 11-3 Consider the machine mounting situation illustrated in example 11-2. Assume that the isolator is a circular cylindrical bar undergoing primarily axial deformations. For axial deformation, the motion in the isolator is built up of longitudinal waves. In order to permit a direct comparison between the examples, every isolator is assumed to have a length of 0.05 m and a cross-sectional area of 0.005 m2. The isolator material is assumed to have a density of 2500 kg/m3 and a complex E-modulus 0.1(1 + i 0.01) MPa. For these input values, the isolator stiffness at low frequencies matches that used in examples 11-1 and 11-2 above.

Solution Block one end of the isolator. By calculating the ratio of the force on the blocked end to the excited displacement response at the free end, a frequency-dependent dynamic isolator


391

stiffness κdyn can be calculated. If the stiffness κ in the result from example 11-2 is replaced with this dynamic stiffness, an insertion loss accounting for longitudinal wave propagation is obtained for the isolator. The result can be directly compared to those obtained earlier; see figure 11-10.

Figure 11-10 Insertion loss of isolators. A juxtaposition of all the results from examples 1 - 3. In real vibration

isolation, these results are further modified by the machine’s dynamic behavior. Resonances in the machine, for example, negatively impact the insertion loss. That is because the difference in impedance between the machine and the isolator is reduced at a machine resonance.

For cases in which there is longitudinal wave propagation in the isolators, they become very stiff at certain specific frequencies. That is due to the interaction between waves propagating in opposite directions. At these frequencies, the isolators no longer function as compliant elements. We have frequencies at which little isolation is obtained. In the example given above, these critical frequencies lie at 64 Hz, 130 Hz, etc. Figure 11-10 clearly shows that the first such critical frequency is an upper bound, above which the isolation insertion loss no longer has an unbroken rising trend. Above that frequency, the average isolation remains about constant. In the example given above, the insertion loss above, say, 50 Hz, is in the vicinity of 40 dB, except at the critical frequencies. Example 11-3 clearly shows that wave propagation in the isolator brings the trend of increasing insertion loss to an end after a certain point.

11.5.4 Deformable machine

In examples 11-1 to 11-3, we have assumed that the machine moves along a coordinate direction as a rigid point mass. The insertion loss is then very low at excitation frequencies near the mounting frequency. If several of the machine’s six rigid body degrees-of-

1-40

-20

0

100

80

40

20

60

f0 10 100 1000Frekvens [Hz]

DIL [dB]

Oeftergivligt underlag

Eftergivligt underlag

Vågutbredning iisolatorn

64 Hz

130 Hz

Rigid foundation

Compliant foundation

Wave propagation in the isolator

Frequency [Hz]


392

freedom are taken into account by the model, several mounting resonance frequencies are then exhibited. In the most general case, we therefore have critical frequencies at six different mounting resonances. In fact, every real machine also exhibits internal resonances at certain frequencies. Typically, the first resonance frequency of a compact machine with a 100-kg mass, e.g., a small internal combustion engine, falls in the 100 Hz - 500 Hz range. If the machine is composed of flexibly attached sections, the first resonance can of course lie at even lower frequencies. The possibility of wave propagation in the machine also effects the isolation insertion loss obtained from elastic mounting. That depends, after all, on the relative stiffnesses of the machine and the isolator. If the machine stiffness varies significantly, due to resonances and antiresonances, then even the insertion loss will vary. On average, the isolation performance is degraded above the machine’s first resonance frequency. Figure 11-11 shows the insertion loss of a simple system consisting of a machine with internal resonances. The foundation is rigid and the isolator is the same as in example 11-1.

1 -40

-20

0

100

80

40

20

60

f0 10 100 1000Frequency [Hz]

DIL [dB]

Machine structure with internal resonances

Rigid machine structure

50 kg

Fstör

50 (1 + 0,1 i) MN/m

25 kg

15 kg

10 kg

Maskin med interna resonanser

Figure 11-11 Insertion loss for the machine mounting situation of example 11-1, but with internal resonances in

the machine. The right side of the figure shows a mechanical model of the machine, and the input data used.

In the example, the machine has resonances at 185, 345 and 535 Hz, and antiresonances at 160, 205 and 495 Hz. Figure 11-11 shows that, at the resonance frequencies, at which the machine is compliant, there are insertion loss minima, i.e., frequencies at which the isolation is poor. On the other hand, extra isolation is obtained at the antiresonances, at which the machine is very stiff. For increasing excitation frequency in the region above the first machine internal resonance, the average insertion loss falls off gradually. That effect is due to the ever smaller unsprung mass that takes part in the motions of the point(s) of contact with the isolator(s).

11.5.5 General formula for insertion loss

In the process of solving examples 11-1 to 11-3, we have, without specifically addressing it, developed a general formula for the vibration isolation of an arbitrary structure of the type shown in figure 11-12. To be more specific, we can replace the mobilities of the three structures involved, Ym, YI and Yplate, by the actual, computed or measured, mobilities YM,


393

YI and YF. The result is a formula for the insertion loss of two structures with arbitrary properties, separated by an interposed isolator with arbitrary properties,

FM

FIMILD

YYYYY

+++

⋅= log20 . (11-4)

M achine

Y M

Vibration isolator(s)

Y I Foundation

Y F

Figure 11-12 General vibration isolation problem. Every element of the system is dynamically and acoustically

characterized by its mobility. With the aid of formula (11-4), the isolation insertion loss is estimated from the mobilities of the subsystems. The mobilities can be determined either theoretically, using finite element methods for instance, or experimentally.

Note, however, that the system must still fulfill the condition that the transmission paths between the machine and the foundation can be combined into a single equivalent path. In the more general case, when the transmission paths must be treated as distinct, equation (11-4) is further generalized. It should be noted, finally, that formula (11-4) also applies to the case of shielding isolation.

11.6 VIBRATION ISOLATION IN PRACTICE

By way of a number of examples, we have demonstrated that the high insertion losses predicted for high frequencies by the crudest models are not actually realized. That is because, as we have already pointed out, the assumptions that underlie the simple models are not valid at high frequencies. Machines and foundations are not rigid, and isolators do not remain compliant. In practice, the simpler models often give acceptable results up to about 100 Hz. After that, the trend is that the insertion loss varies around a constant value. In typical machine mounting situations, the average attainable insertion loss at high frequencies is about 20 to 30 dB; see figure 11-13.


394

III

IV

1-20

-10

0

40

20

10

30

f0 10 100 1000Frekvens [Hz]

DIL [dB]

II

I

Figure 11-13 Schematic sequence for a real insertion loss. I) Frequencies below the lowest mounting resonance.

II) Mounting resonances. III) Rigid machine-soft isolator-rigid foundation. IV) Internal resonances in the machine, foundation, and isolators.

In some situations, not even that much isolation is attainable at high frequencies. For a relatively compliant foundation, the high frequency isolation is seldom better than 15 dB. Figure 11-14 shows how the choice of mounting positions affects the insertion loss. The figure shows the insertion loss for a machine mounted elastically to a section of an aluminum ship’s hull. The machine is regarded as a rigid point mass, and the isolator as an ideal spring. Curve a) shows the insertion loss when the mounting positions are taken to be rigid. In both of the other curves, the measured stiffnesses of two alternate mounting positions, one stiff at the intersection of two ribs, and one softer on a single rib, are used. If the softer mounting position is chosen, the isolation obtained at higher frequencies is never more than 7 - 8 dB.

Frequency [Hz]


395

60 80 100 120 140 160 180 200

0

20

40

60

a)

b)

c)

Frekvens [Hz]

DIL [dB]

Figure 11-14 Insertion loss for vibration isolation in a stiffened ship construction. The results clearly show that

an inappropriate choice of the mounting position can degrade the vibration isolation. a) Rigid foundation; b) Flexible foundation; mounting at the intersection of two ribs; c) Flexible foundation, mounting to a single rib.

There are many different ways to design the foundation such that the mounting points have the desired properties, i.e., low mobility. Most of the methods used in practice are based on the use of added masses and stiffening beams applied in an appropriate way. In all such situations, it is important the plan such solutions right from design stage. Their incorporation at a later stage, to give the desired dynamic properties, is in most cases both expensive and time consuming. Figure 11-15 demonstrates a couple of possible ways to design a system of joists in a building with provisions for the incorporation of vibration isolation.

Figure 11-15 Heavy machines are common sources of vibrations and structure-borne sound. If a machine is to be installed in a building, it is important that the machine foundation be designed in an appropriate way. The principle is that the impedance and mobility difference between the isolator system and the foundation be as large as possible. Two alternative approaches can be to provide extra heavy joists in the machine room, or to stiffen them with braces directly supported by the bedrock. (Picture: Asf, Bullerbekämpning, 1977. Ill: Claes Folkesson.)

Frequency [Hz]

Heavy floor Supporting pillars


396

11.6.1 Design of vibration isolators

There are a number of rules of thumb that should be followed when designing vibration isolators. If these are adhered to, the results should be acceptable.

(i) The isolator’s (static) stiffness must be chosen so low that the highest mounting resonance falls far below the lowest interesting excitation frequency.

(ii) The mounting positions on the foundation should be as stiff as possible.

(iii) The points at which the machine is coupled to the isolators should also be as stiff as possible.

Rules (ii) and (iii) are normally not difficult to fulfill at low frequencies; at high frequencies, however, internal resonances make them problematic.

(iv) The isolator should, if possible, be designed so that its first internal anti-resonance falls well above the highest excitation frequency of interest.

That rule is, in practice, very difficult to follow. If it cannot be followed, then one should ascertain by measurements or computations that at least the following alternative rules are fulfilled:

(v) The isolator must be designed so that its internal resonances do not coincide with strong components of the excitation spectrum.

(vi) The isolator must, furthermore, be designed so that its antiresonance frequencies do not coincide with the resonance frequencies of the foundation.

In addition to these rules, there are normally also a number of constraints related to geometric, strength, and stability concerns.

11.6.2 A couple of methods to improve vibration isolation

In some situations, it is possible to significantly improve the isolation performance with relatively modest additional effort. If very good isolation is a requirement, a so-called double layered isolation can be used. That can be regarded as a combination of elastic elements and a blocking mass; see figure 11-16. In practice, a double layered isolation is realized by interposing a large mass between the machine and the foundation. The blocking mass should behave as a rigid body up to frequencies that are as high as possible.

Wi

Wr

Wt

κdyn

m

κdyn

Passenger railway wagons are an example of double elastic mounting. The vibration source, i.e., the wheel-rail contact zone, is isolated first by a primary suspension between the bearings and the frame of the bogies; see figure 11-17. To further improve passenger

Figure 11-16 Schematic illustration of double layered isolation with two compliant elements and one stiff element.


397

comfort and obtain smooth ride characteristics, a secondary suspension, or comfort suspension, is interposed between the bogies and the body of the wagon.

Primary suspension

Secondary suspension

Figure 11-17 An example of double elastic mounting is the attachment of a railway wagon chassis to a bogie.

The so-called primary suspension between the bearings and the and the frame is commonly built up of stiff, so-called chevron elements, of rubber. The secondary suspension, or comfort suspension, which connects the bogie to the chassis of the railcar consists of very compliant air springs or spiral springs in steel.

If the double layer elastic mounting is well-constructed, the insertion loss can be improved. Because a rigid body has been added to the vibration isolation system, it now has six internal rigid body resonances. These cause another set of insertion loss minima at frequencies above the mounting resonance frequency. The isolation should therefore be designed such that those specific frequencies fall below the lowest important excitation frequency. If the added structure is designed to have mass and inertias of the same order of magnitude as those of the machine, then the internal resonances of the isolation system fall in the same range as the mounting resonance. That implies that the lower bound for region III of figure 11-13 occurs at a low frequency; recall that that is the region in which the insertion loss strongly increases. The vibration isolation then becomes fairly effective at low frequencies. If the design of the double elastic system is successful, the isolation increases at a rate of 80 dB per decade in region III, i.e., double the rate of conventional isolation.

In some types of mechanical constructions, machines must be mounted to relatively compliant points. Examples are vehicles of various types. Motors on small boats, such as pleasure craft, are often mounted via vibration isolators directly to a thin hull. In some cases, the vibration isolation becomes completely ineffective as a result. The reason is that the impedance difference between the isolators and the mounting positions is too small. A way to increase that impedance difference is to add so-called added masses at the mounting points; see figure 11-4b. If those are sufficiently large, the insertion loss can be considerably enhanced.


398

11.6.3 Commercially available vibration isolators

The market for vibration isolators is large. Commercially-available vibration isolators can be divided into several important types, including, among others, steel coil springs, rubber isolators, and gas springs; see figure 11-18. The two fundamental properties of an isolator are its dynamic stiffness and loss factor. The stiffness is, as we have seen, the property that largely determines the suitability of an isolator. The loss factor is significant as an amplitude-limiting parameter at resonances. Both of these parameters are dependent on, among other things, the frequency, and are usually experimentally determined.

Figure 11-18 Examples of commercially-available vibration isolators. In practice, vibration isolators are usually either metallic coil springs or rubber blocks of various forms. Coil springs can be made very soft, but provide little damping. Rubber blocks are relatively stiffer, but provide a significant amount of damping. (Picture: Brüel & Kjær.)


399

Steel coil springs can be designed with very small stiffness values. If the lower frequency bound for isolation must be very low, say 2 - 3 Hz, then coil springs may be appropriate. A disadvantage, however, is that coil springs have a very small loss factor. Rubber isolators are the most commonly occurring type of isolator. They can be designed for either shear or compressive loading. In shear, they can be used down to about 3 Hz, and in compression down to about 5 Hz. A typical problem, however, is that the dynamic properties can vary considerably from one sample to the next; a variation of 30 - 40 % in the static stiffness of a certain type of isolator can occur. In critical cases, it can therefore be necessary to measure the actual, individual isolators to be used. Gas springs can be appropriate in situations where especially low resonance frequencies are desirable. Railway wagons and buses sometimes have gas springs that isolate the wagon from the bogie; see figure 11-17.

11.6.4 Isolator dynamic stiffness

an isolator’s stiffness properties can be characterized in several different ways. The performance of the isolator is largely determined by the transfer stiffness. That stiffness is the inverse ratio of a fixed deformation applied to one end, and the resulting force obtained at the other, blocked, end. Reliable dynamic stiffness data is obtained by separate measurements for every individual isolator. The dynamic stiffnesses given in manufacturer’s tables are usually only corrected static stiffnesses. At high frequencies, the deviations between the true dynamic stiffnesses, and the corrected static stiffnesses, are very large. That is illustrated in figure 11-19, in which the measured dynamic stiffness of a common circular cylindrical rubber isolator is shown. The relative deviation between the corrected static stiffness and the true dynamic stiffness can reach several hundred percent.

Static stiffness

Frequency [Hz]

κ

0,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

1,6

1,8

2,0

0 100 200 300 400 500 600 700

Dynamic stffness

[MN/m]

Corrected static stiffness

Figure 11-19 Measured dynamic stiffness of a circular cylindrical rubber isolator, compared to the static

stiffness from the manufacturer’s catalog data, and the corrected static stiffness. Apparently, the corrected static stiffness is only in agreement with the true, measured dynamic stiffness at relatively low frequencies. Stiffness values that are to be used to predict acoustic responses must therefore be determined by more reliable methods, e.g., measured experimentally.


400


MEASURES OF ISOLATION PERFORMANCE

Definition of insertion loss afterv

beforev

vIL LLD −= , (11-1a)

afterF

beforeF

FIL LLD −= , (11-1b)

General formula for insertion loss

FM

FIMILD

YYYYY

+++

⋅= log20 . (11-4)

401

CHAPTER TWELVE SOUND IN DUCTS Many technical systems involve the transport of a liquid or a gas, i.e., pipe, duct, or channel flow. Examples are cooling water in a nuclear power plant, exhaust gases from an internal combustion engine, or fresh air in a ventilation system. Often, the nature of the source also dictates an oscillating “transport”, i.e., sound spreading through the system of channels. That sound is undesirable in practically all cases, except for musical instruments such as trumpets and clarinets. Because the source is primarily designed to perform a more meaningful task than the generation of sound (e.g., to drive an automobile in the case of an automobile engine), and is usually already designed and built when the noise problem is “discovered”, the required solution is a modification to the duct system. Hindrances provided to counter the transmission of sound are commonly called mufflers or silencers, depending on the context. This chapter seeks to provide an introduction to the design of mufflers, and, in the process, illuminate the subject of sound in channels.

Chapter 12: Sound in Ducts

402

12.1 PRINCIPLES OF SILENCERS IN CHANNELS

In many practical applications, the sound source emits, partly, a low frequency sound spectrum comprised of superposed discrete tones, and partly, a higher frequency broad-band spectrum. Consider, for example, a 4-cylinder, 4-stroke automobile engine driven at 3000 revolutions per minute (rpm). The fundamental tone in the exhaust gas noise corresponds to gas expulsions from the cylinders, twice per revolution, giving a frequency of 100 Hz. Additionally, a number of overtones are typically obtained as well, i.e., tones at 200 Hz, 300 Hz, etc.; see figure 3-15. Because of the high velocity of the gas through valve orifices and branch ducts (at temperatures around 900-1200°C), broad-banded sound is also generated, with maximal strength somewhere between a half to a few kHz. The sound generated because of the coupling between the turbulent average flow field and the acoustic field is said to be self-excited; see figure 12-1. Weak vortex excitation at the duct wall

Strong vortex excitation Flow constriction Figure 12-1 The turbulent vortices that develop in the boundary layer between the duct wall and the flowing

medium are said to generate a self-excited noise. That noise is of broad-band character. The self-excitation is enhanced when the flow is disturbed by irregularities in the duct wall. (Picture: Asf, Bullerbekämpning, 1977, Ill Claes Folkesson.)

A similar source spectrum is also obtained from fans, which emit tones corresponding to the blade pass frequency. Corresponding to that bisection of the source spectrum, mufflers and silencers can be said to be based on two different principles: reflection of sound waves and dissipation of acoustic energy. Mufflers based on the first principle are said to be reactive, and are used to mitigate sound consisting of discrete tones, especially in the low frequency region. Mufflers based on the latter principle are called resistive, and are most suited to dealing with high frequency, broad-band noise. In practice, most mufflers are a synthesis of the two types; see figure 12-2.


403

Figure 12-2 This muffler has both resistive and reflective elements. The reflective elements are the three

chambers separated by partitioning walls. When the gas flows between the chambers, through the perforated ducts, part of the acoustic energy is transformed into turbulent vortices that form at the perforations in the duct. (Picture: Asf, Bullerbekämpning, 1977, Ill Claes Folkesson.)

12.1.1 Insertion loss and transmission isolation

In the design of a muffler, the problem is appropriately treated as consisting of three parts:

(i) sound source (e.g., fan, internal combustion engine, compressor);

(ii) duct system (e.g., ventilation channel, exhaust gas system, air pressure duct);

(iii) termination (e.g., register, exhaust gas outlet, nozzle).

Because these three parts, in general, contribute to a coupled problem, they must all be included in any complete analysis. The coupling implies, for example, that a muffler validated for a diesel engine on a test stand doesn’t necessarily work on an actual truck if positioned differently with respect to the engine. That will be covered in section 12.4.1. Until then, however, we can note that in cases in which the source and termination are reflection-free, i.e., a wave incident upon the source/termination is not reflected, the coupling is small and the design can be based exclusively on an analysis of the transmission properties of the muffler. A measure of the latter is the system’s transmission isolation (DTL), defined as the ratio of the incident to the transmitted sound power when the muffler is connected to a reflection-free termination,

( )tiTL WWD log10⋅= . (12-1)

A completely reflection-free termination is, in practice, often difficult to achieve, especially at low frequencies. When measurements are to be carried out without any steady flow, it is, however, often satisfactory to place mineral wool at the termination of the duct to dampen out the reflection there; see figure 12-3.

W i W r WtMuffler

Reflection free termination

Mineral wool

Figure 12-3 Definition of transmission isolation.


404

Because DTL, in the coupled case, can be misleading, some additional measure of the damping properties of the coupled system as a whole is also called for. For that purpose, there are several different measures available; the most common is the insertion loss (DIL), where the sound pressure at some point in the duct system, usually at the outlet, is compared for two different muffler configurations A and B,

ABILD pplog20 ⋅= . (12-2)

Note that, in contrast to DTL, DIL can also be negative; see, for example, figure 12-24.

12.1.2 Muffler performance requirements

The specific design of mufflers is driven by three paramount considerations:

(i) “right” outer geometry;

(ii) low pressure drop;

(iii) sufficient sound attenuation.

The demands on the outer geometry normally, besides limiting the total volume, also imply a specific outer form. Consider, for example, passenger cars, in which the muffler is often to be located behind the rear axle, adjacent to the reserve tire. The demand for a small pressure drop, which is proportional to the square of the average flow velocity, U2, is of course driven by operating costs of the complete system. For the passenger car example, the pressure drop across the entire exhaust system is of the order of magnitude 300 mbar, and since the corresponding volume flow is 10 m3/min, the power loss is about 7.5 hp (i.e., just over 7 mopeds). Both of these demands are, of course, secondary to the last-mentioned, primary demand for adequate sound attenuation. That can be formulated in a more or less detailed way, as a specification for a maximum A-weighted sound level measured at the outlet, or as a narrow band specification curve. From these specifications, the muffler design task can be formulated as follows: arrive at a construction that, within a certain volume of space, gives the desired attenuation without overloading the source.

Yet another implicit demand, in addition to those given above, concerns self-excited noise. In the case of exhaust gas systems, that would primarily be jet noise from the outlet of the duct system, as well as possible acoustic resonances (tones) excited by instabilities in the flow field comparable to those obtained in blowing over the mouth of a beer bottle (see figure 1-2). If the exhaust gas system’s dimensions are chosen to keep down the average flow velocity, say a Mach Number M (= U/c) less than 0.2, and due attention is given to the detailed form of branch points and other sensitive points, the self-excited noise can normally be neglected in comparison to engine noise. In quieter environments, like dwellings and offices, self-excited noise can, nevertheless, be problematic. That applies, for instance, to the sound transmitted through the walls of ventilation ducts, so-called flanking transmission. When the size of the vortices inside the duct is equal to the bending wavelength of the duct wall, very good coupling is obtained, and therefore also high sound transmission; compare to the coincidence phenomenon of chapter 8. It is very difficult to give any exact formulas for self-excited noise, since it is strongly dependent on the actual geometry. Generally, however, it can be said that the flow-induced sound power is


405

proportional to Un, where n varies between 4 and 8, depending on the situation; see the related discussion in chapter 10.

Figure 12-4 Modern luxury class passenger cars have mufflers with very complex structures, in order

to be able to fulfill the stringent comfort demands. (Photo: Kent Lindgren, MWL) Below, we will analyze a number of common muffler configurations, in order to obtain a certain insight into both the acoustic properties and different modelling techniques. We begin, however, with a discussion of sound propagation in a straight duct.

12.2 SOUND PROPAGATION IN A DUCT

The absolutely most common of all geometric configurations that arises in the subject of sound in channels is that of a straight, cylindrical tube. Compared to the 3-dimensional room analyzed in section 5.2, in which the length, breadth, and height are normally of the same order of magnitude, the tube, on the other hand, has a length that is ordinarily much greater than the cross-sectional dimensions. In consequence, below a certain frequency determined by the condition that a typical dimension of the cross-section, e.g., the tube diameter D, be smaller than half the sound wavelength λ, the sound pressure is nearly constant over the cross section. Below that frequency, the only modes (eigenmodes), or propagating waves, that can exist for a considerable distance along the channel, are those that only vary longitudinally (i.e., along the channel). That frequency region is called the plane wave region, and is a very central concept in the analysis of sound propagation in channels. At frequencies above the plane wave region, more complicated wave forms arise; characteristic for these is that the sound pressure varies over the cross section. For

Figure 12-5 Cross section of a straight duct with its coordinate system. The average flow velocity of the fluid is U.

x

yr

U

z

D


406

each of these so-called higher modes, there is a bounding frequency called the cut-on frequency (see table 12-1), below which the wave is strongly damped. In the plane wave region, the higher modes can only exist locally, in the vicinity of geometric discontinuities such as, for example, area changes (see section 12.3.1). Another typical property of sound propagation in a duct is the difference in phase velocity between waves traveling upstream and waves traveling downstream.

12.2.1 The modified wave equation

In order to somewhat clarify the previously discussed characteristics of sound in channels, the wave equation for sound in a channel is now derived and solved. Due to the influence of the steady flow, the wave equation obtained differs from the conventional wave equation, and is therefore often called the modified wave equation. As in the rest of this chapter, small disturbances are assumed (see chapter 4). That assumption is apparently valid in most applications, but even for such a powerful sound source as an internal combustion engine, which may have sound levels up to 160 dB, linear analysis is found to give useful results. Additionally, viscosity and heat transfer within the fluid are ignored, which is also a good assumption except for the case of very small ducts with diameters of the same order of magnitude as the acoustic boundary layers at the walls; in an automobile exhaust system, for example, boundary layers of 0.5 mm thickness are typical. With these assumptions, the only difference between the fluid in a channel and that which was treated in chapter 4 while deriving the conventional wave equation, is that the particle velocity given in section 4.2.1 should now take the form

( ) zzyyxxt eueueuUtru +++= )(, , (12-3)

where we have assumed that the steady flow rate is constant over the cross section. That assumption is motivated by the turbulence of flow fields in most applications. The linearized continuity equation (4-10), in that case, becomes

00 =⋅∇+∂∂

+∂∂

ux

Ut

ρρρ

. (12-4)

In an analogous way, the linearized equation of motion is obtained as

00 =∇+⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂ p

xuU

tuρ . (12-5)

If techniques similar to that used in the table of section 4.1.4 are applied, i.e., letting the material time derivative operate on the continuity equation and the divergence on the equation of motion, subtracting and making use of (4-31) and (4-44), the modified wave equation is then obtained for the flowing fluid within our channel as

01 2

22 =⎟

⎠⎞

⎜⎝⎛

∂∂

+∂∂

−∇ px

Utc

p . (12-6)

That linear differential equation can be solved by separation of variables. Assume, therefore, a solution analogous to (4-69), i.e.,


407

( ) tixi eezyp x ωψ kp ,ˆ= , (12-7)

in which the exponential factors correspond to wave propagation along the channel. In contrast to the plane wave case given in (4-69), the amplitude is allowed to vary over the cross section, according to the factor ψ(y,z). Substituting (12-7) into (12-6) gives

0)2()( 22222

2

2

2=−+++

∂

∂+

∂

∂ ψψ xxx MMkkzy

kkk , (12-8)

which, if ψ is constant and M = 0, gives kx = ± k, i.e., the solution (4-69). Set the expression inside the second parentheses to α2; then,

0)( 22

2

2

2=+

∂

∂+

∂

∂ ψαψzy

. (12-9)

That, together with the boundary condition at the walls of the channel, amounts to an eigenvalue problem, which yields an infinite but countable number of solutions for α and ψ. These solutions are the eigenvectors (modes) of the cross section, and constitute, in many cases, a complete system of orthogonal functions, comparable to the trigonometric Fourier series of chapter 3. Consider, as an example, a rectangular duct with hard walls.

The boundary conditions, that the particle velocities adjacent and perpendicular to the duct walls be zero, can, using (7-56), be written as

0 ,0,0,0

=∂∂

=∂∂

== hzby zypp . (12-10)

Specify

( ) 4)()(, zikzikyikyik zzyy BeeAeezy −− ++=ψ , (12-11)

from which, by analogy to what is obtained for the rigid-walled room in section 5.2.1, the boundary conditions give A, B = 1. Therefore

( ) )cos()cos(, zkykzy zy=ψ , (12-12)

in which ky and kz, as in (5-81) and (5-82), are given by

bnk ny π=, , n = 0, 1, 2, … (12-13)

z

yx

h

b

Figure 12-6 Channel with a rectangular cross section,

hb × .


408

hmk mz π=, , m = 0, 1, 2, … (12-14)

For n, m = 0, these two transverse wave numbers are identically zero, resulting in a constant amplitude over the cross section, which amounts to the plane wave solution. From (12-9), it also follows that 2

,2

,2

mznynm kk +=α . For every combination of n and m, (12-8) provides a relation for the longitudinal wave number,

0)(2)1( 22,

2,

2 =−−−− nmnmxnmx kMkM αkk , (12-15)

which determines the corresponding mode’s propagation velocity and damping. Two types of solutions are obtained, corresponding to propagating waves moving upstream and downstream, respectively,

))1((1

1 2222, nmnmx MkkM

Mα−−−

−=+k , (12-16)

))1((1

1 2222, nmnmx MkkM

Mα−−+

−=−k . (12-17)

From these expressions, it is clear that for every higher mode, there is a limiting frequency, beneath which the longitudinal wave number becomes complex, and the mode is therefore exponentially damped; compare section 8.61. That frequency is the mode’s cut-on frequency. At low enough frequencies, as mentioned above, only the plane wave propagates in the channel; all higher modes are strongly attenuated, i.e., “cut-off”. As the frequency increases and the sound wavelength falls beneath the channel cross-sectional dimensions, more and more higher modes can propagate. That is of great significance to the design of reactive sound mufflers, since a large number of higher propagating modes affords a greater opportunity for sound energy to pass through the muffler. At very high frequencies, with their consequent high modal densities, parallels can be drawn between sound and light; compare the topic of geometric acoustics, section 1.6. Example 12-1 Consider a rectangular ventilation channel, with cross-sectional dimensions b × h = 0.1 × 0.2 m2, and a steady flow at M = 0.1. Determine the plane wave region, i.e., the frequency band in which only plane waves propagate, at c = 340 m/s.

Solution In an infinite channel with a constant cross section, a plane wave can propagate at all frequencies; i.e., the lower limit is 0 Hz. The upper limit is given by the “cut-on” frequency of the first higher mode. That occurs when the wavelength is short enough, or specifically when the half wavelength matches the height h = 0.2 m of the channel, i.e.,

01.000, =⋅== ππ bnk y , (12-18)

71.152.011, ≈⋅== ππ hmkz m-1. (12-19)

“Cut-on” occurs when the root in (12-16,17) goes to 0, i.e.,


409

84112 1,0

2 ≈−= απ

Mcf c Hz. (12-20)

The plane wave region for the given rectangular channel is thus 0 - 841 Hz. Insofar as the cut-on frequency is dependent on the cross sectional shape of the channel, no general formula can be given. To give some rough guidance, however, the lowest modes for rectangular and square cross sections are described in the table that follows.

Table 12-1 Cut-on-frequencies of the lowest modes of a channel with rigid walls, and rectangular or circular cross section. The dashed lines are nodal lines for sound pressure.

Rectangular cross section b

h

Circular cross section D

bcf c 210 = + _

Dcf c π841.101 = +

_

hcf c 201 = +

_

Dcf c π054.302 = +

+__

21

221111

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+=

hbcf c +

+ _ _

Dcf c π832.310 = +

_

bcf c =02 + +_

Dcf c π201.403 = +

+_+

_

_

hcf c =20 ++_

Dcf c π318.504 = +

++

+

__

__

21

222114

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+=

hbcf c + +

+___

Dcf c π331.511 = +_

_+

The phase velocity for the propagation of a mode is given by (12-16,17),

))(Re( , ωω nmxfc k= , (12-21)

i.e., different frequencies propagate at different speeds – the phenomenon of dispersion. From that expression, it is also clear that the higher modes, in the case of no steady flow, always have a higher velocity than the plane wave, which doesn’t exhibit dispersion. In the high frequency limit, however, the phase velocity of all modes approaches that of plane waves, which, for M = 0, is equal to the speed of sound. In the case of flow, the phase velocity depends on the direction of the mode. For example, for the plane wave,

)1( Mcc f ±= , (12-22)

where + corresponds to waves propagating downstream. That implies, among other things, that the eigenfrequencies of a given duct system depend on the Mach number; see example


410

12-2. Thus, a fixed observer doesn’t measure the actual frequency of a source moving with the flow, but rather

)1(0 M±= ωω , (12-23)

where + corresponds to an observer downstream from the source; hence, the phase velocity increases due to the flow.

Example 12-2 Consider a straight ventilation duct between two rooms in an office building. The duct is 3 m long, and has a circular cross section with a 5 cm radius. Calculate the eigenfrequencies of the duct in the plane wave region for STP (standard temperature and pressure).

Solution The plane wave region is bounded by the cut-on of the first eigenmode (see table 12-1): Hz 1082841,1 == Dcf c π . (12-24)

For frequencies below that limit, the sound pressure is given by

( ) tixikxik eepeptx xx ω)ˆˆ(, 00,00,−+ −+ +=p , (12-25)

where, according to (12-16) and (12-17),

Uc

kUc

k xx −=

+−= −+ ωω

00,00, , . (12-26)

If the room is acoustically soft, e.g., contains cushioned furniture and thick carpets and tapestries, then the air, as seen from the ventilation duct, can be regarded as completely compliant, i.e.,

0),(),0( == tLt pp . (12-27)

Thus, one obtains

0ˆˆ =+ −+ pp , (12-28)

0ˆˆ 00,00, =+−+

−+LikLik xx epep , (12-29)

which, after substituting in the expression for the wave number, gives

)1(2 21 MkLie −= . (12-30)

Only frequencies that fulfill that condition give solutions that satisfy the boundary conditions; i.e., LMncf e 2)1( 2−= , n = (0), 1, 2, … (12-31)

which, for the values relevant to the present example, give the eigenfrequencies f e = 57, 113, 170, …, provided that f e < f c .


411

12.3 REACTIVE MUFFLERS

The transmission characteristics of some common reactive mufflers are derived below. Because the Mach number is very low in many practical applications, the influence of the steady flow is, for simplicity, completely ignored. The analysis is also limited to the plane wave region.

12.3.1 Area discontinuity

At any abrupt change in geometry of a channel, an incident wave is partially reflected, and the energy transmitted downstream thereby reduced. It is that effect that we take advantage of in the design of reactive mufflers. The most obvious way to bring about such a discontinuity is a sudden change of cross sectional area; see figure 12-7.

Figure 12-7 An area discontinuity, with the objective of reflecting a part of the incident acoustic energy, can be

realized in many different ways. Any sudden change in cross sectional area reflects a portion of the acoustic energy. (Picture: Asf, Bullerbekämpning, 1977, Ill: Claes Folkesson.)

S1 S2

x

Figure 12-8 A simple area discontinuity (expansion) is the simplest way to bring about a reflection. The reflection properties of the area discontinuity depend only on the relative cross sectional areas of the incoming and outgoing channels.

Duct

Duct

No reflection

Reflection at area change

Reflection at duct branch


412

Ignoring any reflections that might occur at discontinuities downstream of the area change, the expressions for the fields upstream and downstream, respectively, are:

ikxikx epep −−+ += 111 ˆˆp , (12-32)

ikxep −+= 22 ˆp , (12-33)

in which the time factor eiω t is left out. Because we only have plane waves, the corresponding expression for particle velocity follows directly from (4-74),

cepep ikxikxx 0111, )ˆˆ( ρ−−+ −=u , (12-34)

cep ikxx 022, ˆ ρ−+=u , (12-35)

where the fluid’s properties, i.e., c and ρ are assumed to not change across the area discontinuity. Continuity of acoustic pressure and volume flow rate serve as coupling conditions across the discontinuity; see figure 12-8, 21 pp = , (12-36)

22,11, SS xx uu = . (12-37)

A weakness in that formulation of the problem is immediately revealed upon inspection of the latter term. Specifically, in a more complete problem formulation, there would also be a boundary condition describing that portion of the large-section channel’s termination that abruptly ends at a wall. That wall would, in most practical cases, be rigid: ux,2 = 0 over the surface S2 – S1. In a 1-dimensional analysis, that is only possible if the velocity is also zero over S1, i.e., the channel is closed at x = 0. The more complete formulation therefore requires a variation of the velocity (and pressure) over the cross section, which, in turn, requires consideration of higher modes in the analysis. In the low frequency region, however, a 1-dimensional analysis proves adequate; see also section 12.2.3. Putting the expressions for the fields, (12-32)-(12-35), into the equations of continuity (12-36) and (12-37), yields +−+ =+ 211 ˆˆˆ ppp , (12-38)

+−+ =− 22111 ˆ)ˆˆ( pSppS . (12-39)

From these equations, the reflection and transmission coefficients, defined for the area discontinuity in (5-19) and (5-20), are respectively

)()( 2121 SSSSR +−= , (12-40)

)(2 211 SSST += . (12-41)

Notice the similarity between these expressions and those which are obtained in the case of plane wave incidence at the boundary between two different media, (5-20) and (5-22), (exchange S for 1/ρc). From the expression for the intensity of a plane wave (4-82), the transmission isolation for an area discontinuity is obtained as


413

)4)(log(10 212

21 SSSSDTL +⋅= . (12-42)

It is worth noting that that expression is independent of whether the area discontinuity takes the form of an increase or a decrease in the cross sectional area. The transmission isolation is also independent of frequency, since there is no relevant length dimension in the direction of propagation. A closer examination of formula (12-42) shows that the transmission isolation is, in most practical cases, relatively small. In some cases, that situation can be improved by making use of a series of area changes. An example of that is given in figure 12-9, below.

Reflected sound

Silencer

Reflected sound

Fan

Fan

Figure 12-9 Repeated area discontinuities can, in some cases, be used in a constructive way to mitigate sound

propagation in a duct system. In the case illustrated, it was possible to replace the expensive and space-demanding resistive muffler by a series of reflections, provided in a natural way at existing area discontinuities and corners. (Picture: Asf, Bullerbekämpning, 1977, Ill: Claes Folkesson.)

12.3.2 Expansion chamber

In muffler design, there is often an existing system of ducts with nearly constant cross-sectional areas, the dimensions of which may have been selected on the basis of, for


414

example, pressure drop along the flow path. Consequently, an increase in area at some point must necessarily be followed by an equal decrease, yielding an expansion chamber.

S 1

x

S 2

L The derivation of the transmission isolation for that configuration is carried out in the same way as for the area change; for that reason, the details are left as an exercise. Using a 1-dimensional analysis, and assuming continuity of volume flow rate and pressure, the expression for the transmission isolation of an expansion chamber may be found to be

))(sin)22(1log(10 221221 kLSSSSDTL −+⋅= . (12-43)

From that expression, maximal damping is obtained when

,4Lncf = n = 1, 3, 5 …, (12-44)

i.e., when the length of the chamber coincides with an odd multiple of a quarter wavelength. That becomes evident when one considers that the wave reflecting at the area change at the chamber outlet, because of the distance it has propagated across the chamber, contains a phase factor exp(iπ) when it interferes with the field at the inlet – destructive interference. The maximal attenuation increases with increasing magnitude of the area jump, and can, for large jumps S2 / S1 » 1, be estimated by )2log(20 12 SSDTL ⋅≈ . (12-45)

It is important to note that, because formula (12-43) is based on plane wave propagation, the cut-on frequency is an upper limit of its validity. That is shown in the figure below, in which DTL is calculated for an expansion chamber with a circular cross section and eccentrically placed inlet and outlet, with or without the influence of higher modes taken into account. In the expansion chamber above, with an eccentrically placed inlet and outlet, the circumferentially-varying higher modes (see table 12-1) are well-driven, and the lowest cut-on frequency therefore markedly effects the behavior. For a configuration with the inlet and outlet located in the middle of the chamber end walls, a symmetrical problem instead arises, and only radially-varying modes are excited.

Figure 12-10 Expansion chamber. Inlet and outlet ducts have, in this case, the same cross sectional areas. Besides the ratio of the areas of the chamber and connecting ducts, the length L of the chamber and the wavelength of sound, or its frequency, are also design parameters of the expansion chamber.


415

TL

996 Hz 1652 Hz 2073 Hz Cross section

D

[dB]

0 500 1000 1500 2000 0

5

10

15

20

25

30

Frequency [Hz]

TL

2073 Hz Cross section

D

[dB]

0 500 1000 1500 2000 0

5

10

15

20

25

30

Frequency [Hz]

It can be noted that, even if the region of validity of the plane wave model is broader in the symmetrical case, the 1-dimensional analysis is nevertheless not valid up to the first “cut-on”. That is because the attenuation of the higher modes is so low just beneath cut-on that a significant near field develops at the inlet and outlet; compare that to the discussion, above, concerning boundary conditions at an area discontinuity. From both of the above figures, the drastic reduction in the transmission isolation obtained, as more and more higher modes can propagate, is clear. That is, of course, a deficiency of the expansion chamber, but it does have the positive side effect that the simple plane wave formula (12-43) remains valid, in principle, throughout the entire effective range of the muffler. Therefore, in practical design work, the plane wave analysis is normally adequate, when complemented by knowledge of the first cut-on frequency.

Figure 12-11 Transmission isolation DTL for an expansion chamber with an eccentrically located inlet and outlet. 1-D analysis – bold line; 3-D analysis –thin line. The thin vertical dotted lines mark the cut-on- frequencies of the different higher modes.

Figure 12-12 Transmission isolation DTL for an expansion chamber with a centrally located inlet and outlet. 1-D analysis – bold line; 3-D analysis –thin line. The thin vertical dotted line marks the cut-on- frequency of the first radial mode.


416

12.3.3 Side branches

Another very common type of reactive muffler is the side branch, as in figure 12-13. In order to derive a general expression for the transmission isolation of a side branch, in the plane wave region, we assume that the dimensions of the branch inlet, in the wall of the main channel, are considerably less than the wavelength ka<<1. When that applies, the following coupling conditions, between the sound field upstream and that downstream of the branch, can be formulated for pressure and flow rate: +−+ ==+ 211 pppp s , (12-46)

+−+ +=+ 2,1,1, xssxx SS uuuu , (12-47)

in which we have assumed that the cross sectional area of the channel is the same before and after the side branch. The index s is used to refer to quantities pertaining to the side branch. Making use of the equation of motion, (12-47) can be modified into a condition relating pressures, as

+−+ +=− 2011 pZppp sss ScS ρ , (12-48)

in which the properties of the side branch are also contained in the specific impedance

sss upZ = . (12-49)

Substituting (12-46) into (12-48) provides the following expression for the transmission isolation

20 21log10 ssTL ScSD Zρ+⋅= . (12-50)

That formula describes the transmission behavior of a duct with a side branch. The problem that remains is to determine the impedance of different side branch configurations. Before deriving Zs for two of the most common closed side branches (resonators), we observe that maximal damping is obtained when the inlet to the side branch is perfectly compliant, i.e., Zs = 0 ⇔ p = 0, which occurs at a resonance. That is not entirely unreasonable if we remember that such a condition corresponds to the greatest possible deviation between the otherwise hard duct wall, and therefore the greatest possible reflection of the incident sound wave.

a

L S s

S

p 1+p 1

-p 2

+

p s

Figure 12-13 Common practical realization of a side branch muffler: a quarter wave resonator.


417

12.3.3.1 Quarter wave resonator

A quarter wave resonator is the simplest kind of closed side branch; see figure 12-13. It consists of a duct with a constant cross sectional area, terminated by a hard wall; thus, the boundary condition is 0)( =Lsu . (12-51)

The side branch sound pressure and particle velocity in the plane wave region are

ikxs

ikxss ee −−+ += ppp ˆˆ , (12-52)

cee ikxs

ikxss ρ)ˆˆ(1,

−−+ −= ppu . (12-53)

where the boundary condition implies that kLi

ss e 2ˆˆ −+ = pp . (12-54)

From that, the impedance of the quarter wave resonator is found to be

)cot(0 kLcis ρ−=Z . (12-55)

As noted earlier, maximal transmission isolation occurs when Zs = 0, i.e., when

2πnkL = , n = 1, 3, 5, … . (12-56)

i.e., when the length of the resonator coincides with odd multiples of a quarter wavelength (hence the name); see figure 12-14.

20

0

10

400 Frequency [Hz]

200 600 800

DTL [dB]

5

15

25

0 1000

Figure 12-14 Calculated transmission isolation DTL for a quarter wave resonator, L = 0.86 m, at standard

temperature and pressure.


418

12.3.3.2 Helmholtz resonator

Another common variation of a side branch muffler is the Helmholtz resonator (refer, also, to section 7.2.3.2), which is the acoustic equivalent of the mechanical mass-spring system; see figure 1-2. It consists of a closed volume that communicates with the duct system by wave of a small channel – throat, with area SS and length L, as illustrated in figure 12-15. In order to derive the inlet impedance of this type of side branch, we make use of a technique that is effective at low frequencies. Specifically, we approximate the actual acoustic system, in which pressure and particle velocity vary continuously in space, by a system consisting of parts in which either the pressure or the particle velocity is constant; we obtain a particle system, or lumped system. Applied to the case of a Helmholtz resonator, the preceding implies that for wavelengths much greater than the largest diameter of the resonator volume, the pressure is uniform throughout, and therefore only a function of time. If we assume that the acoustic compression of the gas mass V in the resonator volume V0 takes place without heat transfer, then, for an ideal gas, the adiabatic version (4-28) of the ideal gas law applies,

( ) constant0 =+ γVpp V , (12-57)

where pV is the acoustic pressure in the resonator volume. It can be worth noting that, because we have now reduced our time and space-dependent variables to merely time-dependent ones, we can no longer obtain any wave propagation within the resonator volume. That implies, among other things, that a lumped analysis cannot treat the effects of higher modes. The differentiation of (12-57) with respect to time gives

( ) 010 =

∂∂

++∂

∂ −

tVVpp

tp

V VV γγ γ , (12-58)

which, for small disturbances, and making use of (4-49), can be written as

tV

Vc

tpV

∂∂

−=∂

∂

0

02 ρ

. (12-59)

In contrast to the air in the resonator volume, which cannot “glide away”, but rather only change its volume in response to an external load, the air column in the throat can oscillate back and forth like a rigid cylinder – all particles have the same velocity. The air column has the mass Ssρ0L, and the equation of motion therefore gives

t

uLSppS s

sVss ∂∂

=− 0)( ρ . (12-60)

L, S

V0

s

"Spring"

"Mass"

Figure 12-15 Schematic illustration of a Helmholtz resonator.


419

Because the change in resonator volume is given by the motion of the air column, the coupling condition between the throat and the volume is therefore

ssuStV

−=∂∂ , (12-61)

from which (12-59) and (12-60), in the harmonic case, can be written as

02

0 VSci ssV up ρω = , (12-62)

sVs Li upp 0ωρ=− . (12-63)

Substituting (12-62) into (12-63) gives the Helmholtz resonator inlet impedance

02

00 ViScLi ss ωρωρ +=Z . (12-64)

Maximal attenuation is obtained, as noted earlier, when Zs = 0, i.e., when

02 LV

Scf sr π

= . (12-65)

which is the eigenfrequency of the Helmholtz resonator. Depending on how the resonator throat is connected to the channel system and the resonator volume, an incompressible near field is obtained, which, acoustically speaking, increases the effective length L by an amount ΔL. The size of that near field depends on the system geometry. In order to include that in our 1-D analysis, L in the formula given above must be interpreted as the acoustic length L’ , which is the sum of the geometric length and the inner and outer end corrections,

yi LLLL Δ+Δ+=′ . (12-66)

Generally speaking, the end corrections are difficult to calculate, but it can be shown that they are proportional to the cross sectional dimension of the throat. For a circular channel, for example, the following approximate expressions are obtained:

282.0 DL =Δ , (baffled inlet) (12-67)

261.0 DL =Δ , (inlet in a free field) (12-68)

The first of these formulas applies when the throat is placed in a wall that is very large, with respect to the sound wavelength, i.e., a baffle, and the second when the throat is located at great distance from any reflecting surfaces, also as measured in wavelengths. From these expressions, it is apparent that, for a Helmholtz resonator with a throat length of the same order of magnitude as its cross sectional dimensions, the near field term can be decisive; see example 12-3 below.


420

Example 12-3 Consider a Helmholtz resonator to be mounted in the wall of a rectangular ventilation duct, and designed in the form of a rectangular prismatic box, with a circular hole in one of the end walls. The design completely ignores, however, near field effects. Estimate the error in the calculated resonance frequency on account of that oversight. The dimensions of the box are 0.2 × 0.2 × 0.2 m3, with a wall thickness of 1 cm. The radius of the hole is 5 cm, and the connection of the resonator to the ventilation duct can be considered baffled.

Solution The resonance frequency is given by formula (12-65); the only difference between the original calculation and a more correct analysis is the estimation of the acoustic length L’,

Lfr ′∝1 . (12-69)

In the original calculation, the acoustic length is assumed to be equal to the geometric length 01.0==′ LLu . (12-70)

Mounted to a duct system, however, (12-67) suggests a length more accurately given by

05.005.082.001.0 =⋅+=′mL , (12-71)

where the inner end correction has been ignored because the throat’s (hole’s) dimensions are of the same order of magnitude as the resonator volume. As such, one obtains

5.0,, ≈′′= muurmr LLff (12-72)

The computation of the resonance frequency, without taking account of the end corrections, leads to errors in this example of about 100%! Losses have not been included at all in the analysis given above; in practice, however, these may be decisive for the effectiveness of the resonator. Without going into great detail, such losses as friction or radiation damping, for example, can be included in the preceding model by incorporating a force proportional to the velocity of the air column in the resonator throat,

sr R up ⋅−= , (12-73)

which, added to the forces on the gas mass in the throat, gives

ssVs LiR uupp 0ωρ=−− . (12-74)

From that, the inlet impedance of the Helmholtz resonator has the form

02

00 ViScLiR ss ωρωρ ++=Z . (12-75)

Because the impedance can no longer be identically zero, the maximum effectiveness of the resonator is reduced, but with the compensating advantage of a more broad-banded attenuation behavior; see figure 12-16.


421

DTL [dB]

Frequency [Hz]

M = 0,2

M = 0,1 M = 0

18

16

14

12

10

0

4

6

8

2

300 350 400 450 500 550 600

20

Figure 12-16 Transmission isolation DTL for a Helmholtz resonator mounted to a straight duct wall

(see figure 12-15), with turbulence-related losses caused by the steady flow past the throat. The Mach number is M = U/c.

12.4 ANALOGUE ELECTRICAL-ACOUSTICAL CIRCUITS1

In deriving the characteristics of the Helmholtz resonator above, the particle or lumped system idealization was made. In that approach, the continuous fields describing, for instance, pressure or volumetric flow variations in the actual duct system, p(x,t) and Q(x,t), are discretized into elements in which one or the other of the quantities is assumed to be spatially invariant, p(t) or Q(t). The elements, or building blocks, in the lumped systems, also known of as condensed acoustic systems, correspond to the most common types of forces that occur in acoustic systems, i.e., inertial forces, viscous or dissipative forces, and stiffness forces. Consider, for example, the equation of motion (12-63) of the gas mass in the resonator throat above. It can be written in the form

as

Vs MiS

Li ω

ρω ==

− 0Q

pp, (12-76)

in which Ma has been incorporated to designate the acoustic mass. If we momentarily allow the acoustic pressure be replaced by electric potential, and the volume flow rate by electric current, then Ma is analogous to the inductance of a coil. In a similar fashion, (12-62) can be adapted into

ωω

ρiC

Vic a

s

V ==0

20

Qp

, (12-77)

1 See also section 3.3.6.


422

where Ca is the acoustic equivalent of the capacitance of a condenser – an acoustic spring. Naturally, we also have losses given by R, the resistance of a resistor, in (12-73) – an acoustic resistance. The purpose of illuminating this formula-rich analogy between electrical circuit theory and acoustics is, as indicated in section 3.3.6, to simplify the formulation of the applicable differential equations by setting up acoustic circuits in the same way that it is done for electrical circuits. Consider the example of an exhaust gas system connected to a 1-cylinder engine, as in figure 12-17.

1 2 3 4 5 Z L

Figure 12-17 Exhaust gas system connected to a 1-cylinder piston engine. ZL designates the load impedance, i.e., the influence of the surroundings on the sound radiation from the exhaust gas outlet.

The system elements are the acoustic masses in the ducts – elements 1, 3, and 5 ⇔ iωM1, iωM2, iωM3 – as well as the acoustic springs constituted by the volumes – elements 2 and 4 ⇔ C1/iω, C2/iω. The system terminates at a duct outlet open to the surroundings. That might be suitably described by a boundary condition relating pressure and volumetric flow rate, i.e., an acoustic impedance ZL, where the real part determines the sound radiated from the outlet. If we assume that the piston has a constant velocity amplitude, irrespective of the acoustic loading, then the electrical analogue would be constant current source. It then remains to couple together the diverse building blocks into a circuit:

(i) All volume flow (current) produced by the source goes through duct 1; it is therefore in series with the rest of the system.

(ii) The pressure (voltage) at the termination of duct 1 is the same as the input pressure to the first volume (2). The flow rate into the volume is not the same as the flow into the rest of the system (3,4,5 and ZL). Element 2 and the rest of the system are thus coupled in parallel.

(iii) The connecting duct between the two volumes, element 3, is in series with the rest of the system (4, 5 and ZL).

(iv) The pressure (voltage) at the end of the connecting duct (3) is the same as the input pressure to the second volume (4). The flow rate into the volume is not the same as the flow into the last duct and the outlet; thus, element 4 must be connected in parallel with the last duct (5) and the outlet (ZL).

From these observations, an analogue electrical circuit may be constructed; that is presented in figure 12-18, below. This circuit can now be analyzed by the same methods as are used in electrical circuit analysis (current and voltage approaches), in order to obtain a mathematical model of the system behavior.


423

Qs

iω M1 iω M2 iω M3

ω/iC2LZω/iC1

Figure 12-18 Analogue circuit for the exhaust gas system of figure 12-17.

12.4.1 Four-pole theory

In the lumped circuit analysis given above, the acoustic system consists of elements, over which either the pressure, or the volumetric flow rate, is constant. From separate elements put together, combined elements, in which both the pressure and the volumetric flow rate vary, can be devised. Consider, for example, the subsystem of the circuit given above, consisting of elements 1 and 2, as in figure 12-19.

iω M 1

ω/iC1

Qin utQ

inp utp

For that system, the pressure and flow rate at the inlet of the duct (1) are related to the corresponding quantities at the outlet of volume (2) according to

outoutoutin MiCM Qppp 1112 ωω +−= , (12-78)

outoutin Ci QpQ += 1ω . (12-79)

or in matrix form

⎭⎬⎫

⎩⎨⎧

⋅⎥⎦

⎤⎢⎣

⎡ −=

⎭⎬⎫

⎩⎨⎧

out

out

in

in

CiMiCM

Qp

Qp

11

1

1112

ωωω

. (12-80)

Because that equation couples 4 variables/signals to each other, the matrix is often called the system’s 4-pole. Another common name is a 2-port, because it relates the acoustic state at the inlet and outlet of the element under consideration.

p in

Q in Q ut

p ut T

Figure 12-19 Analogue circuit for that part of the exhaust system in figure 12-17 that includes the input duct and the first volume.

Figure 12-20 Schematic representa-tion of a 4-pole. T represents the 4-pole’s matrix.


424

Although the 4-pole described above is derived using the lumped element technique, that is in no way a requirement. The only requirement to derive a 4-pole for a portion of a duct system is that it follow the linear theory, and that the sound propagation be 1-dimensional at the inlet and outlet. How the 4-pole is then arrived at is completely discretionary; see example 12-4, for instance. In some cases, it can even be practical to make use of measured 4-poles. The advantage of using a 4-pole representation is that it offers a very efficient way to organize the analysis of a duct acoustic problem. Consider, for example, a typical automobile exhaust gas system as in figure 12-21.

T 1 T 2 T 3 T 4 T 5

Just as in the lumped element approach, the system is divided up into 5 different parts, each of which is represented by a 4-pole,

( )

( ) [ ]( )

( ) ,⎭⎬⎫

⎩⎨⎧

⋅=⎭⎬⎫

⎩⎨⎧

iout

iout

iiin

iin

Qp

TQp

i = 1, 2, 3, …, (12-81)

such that continuity of pressure and volumetric flow rate is maintained, and the sound field is 1-dimensional at the junctions

( ) ( ) ,1 jin

jout pp =− j = 2, 3, 4, … , (12-82)

( ) ( ) ,1 jin

jout QQ =− j = 2, 3, 4, … . (12-83)

Thus, the system can be represented by a chain of 4-poles, which corresponds, mathematically, to a series of matrix multiplications, as

[ ] [ ] [ ] [ ] [ ] [ ]54321 TTTTTT ⋅⋅⋅⋅= . (12-84)

T therefore completely describes the relation between the acoustic state at the system’s inlet and that at its outlet. Using T, the system’s transmission isolation can also be calculated,

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+++⋅=

2

2221

12114

log10out

inin

outin

outTL Z

ZZ

ZZZ

Dt

ttt , (12-85)

where Zin and Zout represent the acoustic impedance, ρ0c/S, at the inlet and outlet, respectively, and t represents matrix elements.

Figure 12-21 Typical auto-mobile exhaust gas system, and its division into 4-poles.


425

Example 12-4 Derive the 4-pole for a straight duct of length L and cross sectional area S.

Solution The problem’s geometry is defined in figure 12-22.

x=0 x=L

pin

QinQut

put

The sound pressure and volumetric flow rate in the duct are obtained, in the plane wave region, from (12-32) and (12-34), so that, at x = 0, −+ += ppp ˆˆin , (12-86)

cSin 0)ˆˆ( ρ−+ −⋅= ppQ . (12-87) and at x= L, ikLikL

ut ee −−+ += ppp ˆˆ , (12-88)

ceeS ikLikLut 0)ˆˆ( ρ−−+ −⋅= ppQ . (12-89)

The combination of (12-86) and (12-87) gives +p and −p , which, when substituted into (12-88) and (12-89), give the duct’s 4-pole as

[ ] ( ) ( )( ) ( ) ⎥

⎦

⎤⎢⎣

⎡=

kLZkLikLiZkL

cossinsincos

T . (12-90)

where ScZ 0ρ= . In the case of no steady flow, it can be shown that the 4-pole’s determinant must be equal to 1. That result can be useful as a check on the accuracy of calculations made; for example, for the duct above,

[ ] ( ) ( ) ( ) ( ) 1sinsincoscosdet =−= ZkLikLiZkLkLT . (12-91)

Additionally, for a symmetric channel element, i.e., one that looks the same from both directions, the diagonal elements are all equal. That is apparently the case for the duct, but not for the volume element given in (12-80). Because even an automobile engine can be approximately modeled by the electrical-acoustic analogy, the total system engine-exhaust system-outlet can also be represented by a circuit, as in figure 12-23.

Figure 12-22 Straight duct, relating the acoustic quantities of pressure and volumetric flow at the inlet and outlet, respectively.


426

Q K Z K

Z LT

Figure 12-23 An automobile engine and exhaust system, and their equivalent acoustic circuit..

Note that the constant velocity source that represents the engine has been given a so-called source impedance ZS to account for the dependence of the engine, as a sound source, on the muffler to which it is coupled; the volumetric flow rate generated by the constant velocity source is divided between ZS and the exhaust system. The exhaust outlet is represented by an acoustic load impedance ZL. From that circuit representation, the insertion loss, comparing the two exhaust systems A and B, coupled to the same engine and outlet, is formulated as

SLSL

SLSLILD

ZZbZbZbbZZaZaZaa

12112221

12112221log20++++++

⋅= , (12-92)

where aij and bij represent matrix elements for the 4-pole matrices of systems A and B, respectively. From that expression, the coupling mentioned in section 12.1.1, between the source, system, and termination, is apparent. To illustrate the effects of that coupling, figure 12-24 shows the insertion loss for the Helmholtz resonator given in figure 12-15, when mounted at various distances from the outlet in a straight, circular duct which is connected to a reflection-free source, ZS = Zin, and terminates in a free-field, p ≈ 0, in the frequency band of interest. Reference system (B) is a straight duct; i.e., for the frequencies at which the insertion loss is negative, the sound pressure at the outlet is higher with the muffler than without! For the given source and load data, typical of exhaust systems, it is apparently not productive to place a side branch resonator at a distance corresponding to a half-wavelength from the end of the outlet duct. That is because the standing wave generated by the end reflection has a pressure minimum there, p ≈ 0, and the resonator is therefore, in principle, entirely without effect; see section 12.3.3. The greatest possible effect of the resonator is obtained when the length of the outlet duct corresponds to a quarter wavelength.


427

DIL [dB]

Frequency [Hz]

a) b)

c)

25

10

15

20

0

5

-5 300 400 450 500 350

30

Termination

ZL Source

L

Figure 12-24 Insertion loss DIL for a Helmholtz resonator with different lengths L of the outlet duct: a) λr / 2; b)

3λr / 4; c) λr / 4. The wavelength λr is the wavelength at the resonator’s eigenfrequency, fr = 410 Hz. The influence of the steady flow is ignored.

12.5 RESISTIVE MUFFLERS

In contrast to the reactive mufflers treated above, which hinder more so than they damp sound, resistive mufflers convert acoustic energy to heat. In order to accomplish that, the walls of the channel are covered, as was the case in room acoustics, by porous absorbents consisting of, for example, mineral wool or glass fibers (see figure 12-25).

Figure 12-25 This resistive muffler is based on the dissipation of acoustic energy, converting it to heat as it

undergoes repeated reflections against the absorbent-clad walls. (Picture: Asf, Bullerbekämpning, 1977, Ill: Claes Folkesson.)

Outlet

Inlet

Sound absorbing material e.g., mineral wool

Resistive muffler


428

Because damping is mainly obtained from work done by viscous forces (see section 7.2.3), resistive mufflers should be designed to maximize the particle velocities in the porous material. It is, for example, often advantageous to locate absorbent a small distance away from the wall, since the normal velocity at the wall itself is zero; compare cases 2 and 3 in figure 7-11. To damp fan noise in ventilation ducts, a resistive-type muffler, called a baffle muffler, is often used. The reason for that is that ventilation noise is primarily of a broad-band, hissing variety, and the typically large dimensions of the duct permit such a muffler. Figure 12-26 shows some baffle muffler variants, with differing arrangements of the absorbent.

Figure 12-26 Examples of baffle mufflers for ventilation systems. By working with absorbents of different

thicknesses, and distributing these in different ways throughout the cross section, different sound attenuation characteristics can be obtained. (Picture: Asf, Bullerbekämpning, 1977, Ill: Claes Folkesson.)

To optimize the properties of a resistive muffler, in general, a detailed 3-D analysis of the sound field, an often difficult undertaking, is required. On the other hand, in most applications, the plane wave proves to be the most poorly attenuated propagating mode form, so that an estimation of its damping can give valuable information in design work. For rectangular ducts clad with porous material on all walls, the following approximate empirical relations for the transmission isolation, valid in the plane wave region for low steady flow rates, are formulated as

4,105,1 αS

LPDTL = , (12-93)

where P is the free perimeter clad with absorbent, S is the free cross sectional area, L is the length of the absorbent, along the muffler, α is the absorption factor at the frequency under consideration; see (5-27)

Low frequencies

High frequencies

Both low and high frequencies

Thick absorbent

Thin absorbent

Thick absorbent

Large air gap Small air gap Small air gap


429

It should be pointed out that (12-93) completely ignores the reactive attenuation that arises due to the area discontinuities at the inlet and outlet of the dissipative channel; it is, on the contrary, regarded as infinite in length. From the relation given above, it is evident that the ratio of the perimeter to the area is of great significance for the damping properties of the resistive muffler. That is reflected in the appearance of many resistive mufflers, above all in the ventilation industry, in which baffle mufflers are common; see figure 12-27a and 12-27 b).

a) b)

Absorbent

Figure 12-27 Typical cross sectional dimensions of resistive mufflers: a) automobile muffler; b) ventilation

muffler. Formula (12-93) represents the simplest possible model of a resistive muffler; the porous material is replaced by a bounding surface which does not have any influence on the sound field other than the dissipation of acoustic energy; see section 7.2.3. The next stage, in order of increasing modelling refinement, would be to treat the absorbent as an impedance surface, i.e., a boundary with, not only damping, but also mass and stiffness properties; compare the locally-reacting surface concept from section 5.1.5. That type of modelling, which ignores the longitudinal wave propagation in the absorbent, yields good results in many applications. Its usefulness is all the more apparent, considering that many resistive mufflers, to increase the attenuation, are built to hinder the longitudinal wave propagation. Nevertheless, the most refined models also account for longitudinal wave propagation in the absorbent, implying, in principle, a 3-D modelling of two coupled fluids, in which one of them has a complex sound speed and density (to describe the porous material). The resistive muffler has many practical advantages over the reactive type. For example, it is often possible, in a design, to make use of any unallocated spaces for the purpose of sound mitigation. By filling such spaces with absorbent, and directing the flow through them, they become simple resistive mufflers. That approach can be effectively used to attenuate ventilation noise; see figure 12-28.


430

Figure 12-28 In many cases, unused spaces, when clad with absorbent material, can be used as resistive mufflers.

Here, an example shows how that approach can be realized in a building. (Picture: Asf, Bullerbekämpning, 1977, Ill: Claes Folkesson.)

Finally, it is worth noting that, because of the non-reflecting character of resistive mufflers, the coupling effects due to other system elements are not as strong as for reactive mufflers. The performance of a resistive muffler, consequently, is not as sensitive to the muffler’s placement in the system as is the performance of a resonator, for example. That, together with its undramatic frequency dependence, as may be seen in figure 12-29, has made the resistive muffler very widely used. The situation is rarely made worse by the presence of a resistive muffler, and its damping properties also correspond, to some extent, to hearing sensitivity; compare the curve of figure 12-29 to the hearing sensitivity curves in figure 2-5.

Noisy fan

Resistive muffler chamber

Outlet

Hot air outlet

Resistive muffler chamber

Noisy fan Air heater


431

Frequency [Hz] 100 2170

60

40

20

80

0

DTL

[dB]

2400 1940 1710 1480 1250 1020 790 560 330

Figure 12-29 Measured transmission isolation DTL for the resistive rear muffler of a Saab 9000 Turbo.


432


SOUND ATTENUATION IN DUCTS

Insertion Loss and Transmission Isolation

Transmission isolation ( )tiTL WWD log10 ⋅= (12-1)

Insertion loss ABILD pplog20 ⋅= (12-2)

SOUND PROPAGATION IN A DUCT The modified wave equation

01 2

22 =⎟

⎠⎞

⎜⎝⎛

∂∂

+∂∂

−∇ px

Utc

p (12-6)

REACTIVE MUFFLERS

Area discontinuity

Transmission isolation of an area discontinuity

)4)(log(10 212

21 SSSSDTL +⋅= (12-42) Expansion chamber

Transmission isolation of an expansion chamber

))(sin)22(1log(10 221221 kLSSSSDTL −+⋅= (12-43)

Side branches

Transmission isolation of a duct with a side branch

20 21log10 ssTL ScSD Zρ+⋅= (12-50)

Quarter wave resonator

Specific Impedance at the Inlet of a Quarter Wave Resonator

( )kLcis cot0ρ−=Z (12-55) Helmholtz resonator Specific Impedance at the Inlet of a Helmholtz Resonator

02

00 ViScLi ss ωρωρ +=Z (12-64)

Eigenfrequency of a Helmholtz Resonator

02 LV

Scf sr π

= (12-65)

Chapter 13: Industrial Noise and Vibration Control

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CHAPTER 13

INDUSTRIAL NOISE AND VIBRATION CONTROL Noise in industry come frome the machines and appliances used for production. The sources of industrial noise therefore has a wide variety. In this chapter some common noise sources will be discussed along with possible noise control techniques. A systematic approach for analyzing industrial noise problems using the source-path-receiver model is presented. Noie conrol at the source is always rhe preferred option but is usually difficult. Noise control during the propagation path is the second choice and some commonly used techniques are discussed. Noise control at the receiver is the last resort and usually involves hearing protectors in the form of earplugs or earmuffs.

13.1 MOTIVATION FOR INDUSTRIAL NOISE CONTROL

The main purpose of industrial noise control is to protect the hearing of the people working in the production. Most countries have introduced legislation for this purpose. According to present knowledge an equivalent sound level exceeding 85 dB(A) during a 8 hour working day will cause permanent hearing loss. This is based on a dose discussion where it is assumed that sound of different strength and duration contributes according to its contribution to the equivalent sound pressure level defined according to equation (3-3). The equivalent A-weighted sound level is defined according to (3-4).


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⎥⎦

⎤⎢⎣

⎡⋅= ∫ dt

TLogL

T L

Aq

A

0

1010110 dB(A). (13-1)

The European Union has issued directive 2003/10/EC of 6 February 2003 on the minimum health and safety requirements regarding the exposure of workers to the risks arising from noise. Exposure limit values and exposure action values in respect of the daily noise exposure levels and peak sound pressure have been fixed acoording to Table 13-1. Table 13-1 Exposure limit values and exposure action values acording to EU directive 2003/10/EC.

Exposure limit value during a 8 hour working day LA,eq,8h = 87 dB(A)

Exposure limit C-weighted impulse peak value LC,peak = 140 dB(C)

Upper exposure action value during a 8 hour working day LA,eq,8h = 85 dB(A)

Upper exposure action C-weighted impulse peak value LC,peak = 137 dB(C)

Lower exposure action value during a 8 hour working day LA,eq,8h = 80 dB(A)

Lower exposure action C-weighted impulse peak value LC,peak = 135 dB(C)

Under no circumstances shall the exposure exceed the exposure limit values. If, despite the measures taken to implement this Directive, exposures above the exposure limit values are detected, the employer shall:

(a) take immediate action to reduce the exposure to below the exposure limit values;

(b) identify the reasons why overexposure has occurred; and

(c) amend the protection and prevention measures in order to avoid any recurrence.

When applying the exposure limit values, the determination of the worker's effective exposure shall take account of the attenuation provided by the individual hearing protectors worn by the worker. The exposure action values shall not take account of the effect of any such protectors.

In duly justified circumstances, for activities where daily noise exposure varies markedly from one working day to the next, Member States may, for the purposes of applying the exposure limit values and the exposure action values, use the weekly noise exposure level in place of the daily noise exposure level to assess the levels of noise to which workers are exposed, on condition that:

(a) the weekly noise exposure level as shown by adequate monitoring does not exceed the exposure limit value of 87 dB(A); and


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(b) appropriate measures are taken in order to reduce the risk associated with these activities to a minimum.

If the upper exposure action values are exceeded, the employer shall establish and implement a programme of technical and/or organisational measures intended to reduce the exposure to noise, taking into account in particular:

(a) other working methods that require less exposure to noise;

(b) the choice of appropriate work equipment, taking account of the work to be done, emitting the least possible noise, including the possibility of making available to workers work equipment subject to Community provisions with the aim or effect of limiting exposure to noise;

(c) the design and layout of workplaces and work stations;

(d) adequate information and training to instruct workers to use work equipment correctly in order to reduce their exposure to noise to a minimum;

(e) noise reduction by technical means:

(i) reducing airborne noise, e.g. by shields, enclosures, sound-absorbent coverings;

(ii) reducing structure-borne noise, e.g. by damping or isolation;

(f) appropriate maintenance programmes for work equipment, the workplace and workplace systems;

(g) organisation of work to reduce noise:

(i) limitation of the duration and intensity of the exposure;

(ii) appropriate work schedules with adequate rest periods.

Workplaces where workers are likely to be exposed to noise exceeding the upper exposure action values shall be marked with appropriate signs. The areas in question shall also be delimited and access to them restricted where this is technically feasible and the risk of exposure so justifies. Where, owing to the nature of the activity, a worker benefits from the use of rest facilities under the responsibility of the employer, noise in these facilities shall be reduced to a level compatible with their purpose and the conditions of use.

If the risks arising from exposure to noise cannot be prevented by other means, appropriate, properly fitting individual hearing protectors shall be made available to workers and used by under the conditions set out below:

(a) where noise exposure exceeds the lower exposure action values, the employer shall make individual hearing protectors available to workers;


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(b) where noise exposure matches or exceeds the upper exposure action values, individual hearing protectors shall be used;

(c) the individual hearing protectors shall be so selected as to eliminate the risk to hearing or to reduce the risk to a minimum.

The employer shall make every effort to ensure the wearing of hearing protectors and shall be responsible for checking the effectiveness of the measures taken in compliance with this Article.

The employer shall ensure that workers who are exposed to noise at work at or above the lower exposure action values, and/or their representatives, receive information and training relating to risks resulting from exposure to noise concerning, in particular:

(a) the nature of such risks;

(b) the measures taken to implement this Directive in order to eliminate or reduce to a minimum the risks from noise, including the circumstances in which the measures apply;

(c) the exposure limit values and the exposure action values laid down in Article 3 of this Directive;

(d) the results of the assessment and measurement of the noise carried out in accordance with Article 4 of this Directive together with an explanation of their significance and potential risks;

(e) the correct use of hearing protectors;

(f) why and how to detect and report signs of hearing damage;

(g) the circumstances in which workers are entitled to health surveillance and the purpose of health surveillance, in accordance with Article 10 of this Directive;

(h) safe working practices to minimise exposure to noise.

In Egypt there are regulations on Permissible Limits Of Sound Level and Safe Exposure Periods (Egyptian Law 4/1994 - Annex 7). For sound level inside works premises and closed places the maximum permissable limts according to Table 13-2 apply.


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Table 13-2 Maximum permissible limits of sound level inside places of productive activities

No.

TYPE OF PLACE/ACTIVITY

MAXIMUM ALLOWABLE

SOUND LEVEL DECIBEL (A)

1 Work premises with up to 8-hour shifts with the aim of limiting noise hazards on hearing.

90

2 Places of work that requires hearing signals and good audibility of speech.

80

3 Places of work for the follow up, measuring and adjustment of operations, with high performance.

65

4 Places of work with computers or typewriters or similar equipment.

70

5 Places of work for activities that require routine mental concentration.

60

The maximum permissible periods for exposure to noise at work premises (factories and workshops) are as follows:

• The values given hereafter are indicated on the basis of those, which do not affect the sense of hearing.

• Level of noise shall not exceed 90 decibels (A) during a daily work shift (8 hours).

• In case the noise level is higher than 90 decibels (A), the period of exposure shall be reduced according to the following table:

NOISE LEVEL DECIBEL (A) 95 100 105 110 115 PERIOD OF EXPOSURE (HOURS) 4 2 1 1/2 1/4

• The noise level at any one time during working hours shall not exceed 135 decibels.

• In case of exposure to various intensities of noise over 90 decibels the following applies:

For intermittent periods of noise during a shift, the total of

⎟⎠⎞

⎜⎝⎛ ++ ......

22

11

BA

BA

Shall not exceed the number one, where A = is the period of exposure to a specific level of noise per hour B = is the permissible period of exposure at that specific noise level per hour

• In case of exposure to intermittent noise coming from heavy hammers, the exposure period (number of knocks during the daily shift) permitted depends on the noise level, according to the following table:


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Noise level (Decibels)

Number of permissible knocks during daily working hours

135 300 130 1000 125 3000 120 10000 115 30000

Noise coming from heavy hammers shall be considered intermittent if the period between knocks is one second or more. If the period is less than this, the noise shall be considered continuous, in which case the previous four conditions shall apply.

For sound levels in different residential areas the limits according to Table 13-3 apply. Table 13-3 Maximum permissible limits for noise level in different zones

Permissible limits for noise levels (dBA) Day Evening Night

TYPE OF ZONE

from to from to from to Commercial, administrative and downtown area.

55 65 50 60 45 55

Residential areas including some workshops or commercial businesses or on public roads.

50 60 45 55 40 50

Residential areas in the city. 45 55 40 50 35 45 Residential suburbs having low traffic flow.

40 50 35 45 30 40

Rural residential areas (Hospitals and gardens).

35 45 30 40 25 35

Industrial areas (Heavy Industries). 60 70 55 65 50 60

Daytime: from 7am to 6pm

Evening time: from 6 PM to 10 PM

Nighttime: from 10 PM to 7 am

13.2 SYSTEMATIC APPROACH TO INDUSTRIAL NOISE CONTROL

The sources of industrial noise are many and varied which means that almost any imaginable noise control technique may have to be considered. A systematic approach should start with applying the source-path-receiver model. The noise sources can be considered to be of two main types: sources associated with structural vibrations and sources associated with gas fluctuations. An example of the first type is machine surface vibrations causing sound radiation and an example of the second type of noise source is the pulsating exhaust gases from an IC-engine. There are also sources associated with


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interaction of gas flows and structures and sources caused by a free gas flow,i.e., jet noise. In section 13.3 examples of sound sources will be discussed along with possible noise control techniques. When buying new machines or replacing an old machines noise should always be considered.

Figure 13-1 Source-path-receiver model for analyzing noise problems. Noie conrol at the source is always the preferred option but is usually difficult. It is however important to identify the main sources of noise. Noise control during the propagation path is the second choice and some commonly used techniques are discussed. Noise control at the receiver is the last resort and usually involves hearing protectors in the form of earplugs or earmuffs.

Noise control during the propagation path can involve measures such as enclosures, barriers and adding room absorption. Example of possible noise control techniques will be discussed in section 13.4. Noise control at the reciver can involve protecting the worker using hearing protectors in the form of earplugs or earmuffs but can also involve enclosures. Some possible techniques will be discussed in section 13.5.

13.3 NOISE CONTROL AT THE SOURCE

In this section the different types of noise generating mechanisms will be discussed together with possible noise control techniques.

Source Path Receiver


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13.3.1 Noise generated by fluctuating forces in structures

The internal forces in a machine are transferred as structureborne sound to the surface where it is radiated as sound. The forces can be either steady, for instance caused by reciprocating motion in an engine, or transient caused by impacts. The forces can also come from work performed on the work piece by a worker or a machine as exemplified in Figure 13-2. More noise is produced if a task is carried out with great force for a short time than with less force for a longer time.

Figure 13-2 Noisy and quiet bending of a metal strip. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

Another example is a box machine where cardboard is cut with a knife blade, see Figure 13-3. The knife must cut very rapidly and with great force in order for the cut to be perpendicular to the strip and the result is high noise levels.


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Figure 13-3 Noisy method for cutting cardboard. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

Using a blade which travels across the strip, the cardboard can be scored with minimal force for a longer time, see Figure 13-4. Since the cardboard strip continues to move, the knife must travel at an angle in order for the cut to be perpendicular. The cutting is practically noise free.

Figure 13-4 Low noise method for cutting cardboard. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)


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Since the structural vibration will have to radiate as sound from the machine surfaces reduction of the surface area or reduction of the radiation efficiency of he surface can be good noise control techniques. An object with a small surface area may vibrate intensely without a great deal of noise radiation. The higher the frequencies, the smaller the surface must be to prevent disturbance. Since machines always will vibrate to some extent, noise control will be aided if the machines are kept as small as possible.

Figure 13-5 Example showing the importance of the size of the sound radiating surface on the resulting noise generation. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

Another example shows the noise generation from the control panel of a hydraulic system. If the panel is detached from the system itself, the vibrating surface is reduced, and therefore the noise level is decreased.


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Figure 13-6 Example showing the importance of the size of the sound radiating surface on the resulting noise generation. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)


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Large vibrating surfaces cannot always be avoided. The surface vibration pumps air back and forth depending on the vibration patter, and it is this air pumping which causes the sound radiation. If the panel is perforated the air pumping is "short circuited" between the front and back of the plate, and the sound radiation is reduced.

Figure 13-7 Principle for reduction of sound radiation by the use of a perforated plate. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

An example of this type of noise control is shown in Figure 13-8. The protective cover over a flywheel and belt drive of a press is a major noise source. The cover in the example was made of solid sheet metal. A new cover was made of perforated sheet metal and wire mesh. The sound radiation was reduced

Figure 13-8 Example of the reduction of sound radiation by the use of a perforated plate. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)


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Another technique for causing this short circuiting between the front and back of a plate is to change the shape. If the plate has free edges short circuiting takes place at the edges therefore, a long, narrow plate radiates less sound.

Figure 13-9 Principle for reduction of sound radiation by changing the shape of a radiating surface. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

An example of the using of this principle is a belt drive which gives a large amount of low frequency noise because of the vibration of the broad belt. When the broad drive belt was replaced by narrower belts, separated by spacers the noise was reduced.

Figure 13-10 Example of reduction of sound radiation by changing the shape of a radiating surface. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)


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Since a plate with free edges causes less sound radiation compared to a baffled plate because of the pressure equalization or short circuiting between the two sides of the plate, this is a possible noise control method. Clamping the edges prevents pressure equalization and the sound emission is greater, especially at low frequencies. For example, speakers produce more bass if they are enclosed in a cabinet.

Figure 13-11 The sound generation from a loudspeaker is increased by putting it into an enclosure and thus preventing short circuiting of pressure between the front and back of the cone. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

This principle can be used to reduce the noise from a cart which produces noise from the bottom and side plates when the cart is pushed. Sound is also emitted when material is slid down the cart walls. Pressure equalization only takes place at the top edges of the side plates. The walls were replaced by new ones, constructed with a pipe frame. Plates were fastened with a gap between the plates and the frame. Pressure equalization takes place along all the edges, and the low frequency noise is reduced.


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Figure 13-12 Example using the short circuiting of pressure at plate edges to reduce low frequency noise from a cart. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

In some cases noise is generated by impacts between a structure and an object. The sound level generated when a small object hits a plate is determined by the weight of the object and the speed with wich it hits the structure. If the object is dropped onto a plate the noise can be reduced by reducing the drop height.

Figure 13-13 Principle for reduction of sound generation by reducing drop height. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

An example of how noise reduction can be obtained is shown in Figure 13-4 where steel parts are transported from a machine to a storage bin. When the bin is empty, the drop


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height is large and the noise level is high. By introducing a hydraulic system so that the conveyor belt can be raised and lowered the noise can be reduced. Additional noise reduction is obtained if the belt ends in a drum equipped with rubber plates to break the fall of the parts.

Figure 13-14 Example of reduction of sound generation by reducing drop height. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)


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Since sound is generated by structural vibration measures to reduce surface vibration will also give noise reduction. One way is to increase the damping of the structure by adding coatings or intermediate layers with better internal damping.

Figure 13-15 Principle for reduction of sound radiation by introduction of damping layers in a structure. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

An example of noise reduction using this techniqiue is shown in Figure 13-16. The noise from a pump system comes to a large extent from the coupling guard which is made of sheet metal. The noise level was reduced by constructing it of damped metal.


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Figure 13-16 Example of reduction of sound radiation by introduction of damping layers in a pump coupling. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

Another reason for introducing damping is to reduce the effect of structural resonances. They increases noise from a vibrating plate, but can be suppressed by damping the plate. It may often be sufficient to damp only part of the surface, and, in some rare cases, damping of a single point is effective.


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Figure 13-17 Principle for reduction of sound radiation due to structural resonances by introduction of damping. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

An example for an automatic tooth cutter for circular saw blades which generates high sound pressure levels due to structural resonances is shown in Figure 13-17. A urethane rubber coating clamped to the saw blade damps the resonance.

Figure 13-18 Example of reduction of sound radiation caused by structural resonaces in a saw blade by introduction of damping. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)


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It is easier to damp high frequency vibration than low frequency vibration. Large vibrating plates often have low frequency resonances which can be difficult to damp. If the plate is stiffened, the resonance shifts to higher frequency, which can be more easily damped.

Figure 13-19 Principle for reduction of sound radiation by shifting structural resonances to higher frequencies. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

Figure 13-20 shows an example for a machine where low frequency sound comes from the side surfaces of the machine stand. The noise control measure involved stiffening the side plates on the machine with iron straps and installing a damped plate over the braces.

Figure 13-20 Example of sound reduction by shifting structural resonances to higher frequencies. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)


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13.3.2 Noise generated by fluid flow

Flowing gases or liquids can generate high sound pressurelevels when they interact with a solid structure or as a free stream jet. In addition the machines generating the flow, as for instance compressors pumps and IC-engines usually give high pressure pulsations in the connected pipes. Techniques for reducing this type of sound propagating in ducts has been discussed in chapter 12.The pressure pulsations can however also excite the structure and generate structural vibrations which produce sound. Figure 13-21 shows an example of a circulation pump producing pressure pulsations in the water in a heating system. The sound waves are transmitted through the pipes to the radiators, where the large metal surfaces vibrate and radiate sound. This is similar to how the vibrations of the strings in a musical instrument are transmitted through the bridge to the sound box. When the sound box vibrates, sound is transmitted to the air.

Figure 13-21 Principle for sound generation caused by pressure pulsations in a fluid exciting vibrations in a structure and radiating as airborne sound. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

Turbulent fluid flow in pipes also produces sound which can be radiated from the pipes and transmitted to the building structure. This noise can be controlled by reducing the turbulence in the pipe or covering the pipe with sound absorbing material. The vibrations can be isolated from the wall or ceiling with flexible connecting mechanisms.


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Figure 13-22 Example of noise control by reduction of turbulence generated vibrations in pipes. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

Air flowing past objects can generate sound as already discussed in chapter 10. When air passes an object at certain speeds, a strong pure tone, known as a Strohal tone, can be produced. This can be prevented by making the object longer in the direction of flow, such as with a "tail," or by making the object's shape irregular.


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Figure 13-23 Sound generation by air flow past an object in an air stream. For the circular cross section bar a loud Strohal tone is produced. Noise control measures include disturbing the regular production of vortices. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

This type of sound generation can be of importance for instance around chimeneys at certain wind speeds. A possible solution is to mount a strip of sheet metal on the chimeney in a spiral. The pitch of the spiral must not be constant. Regardless of the wind direction, it encounters an irregular object.

Figure 13-24 Noise reduction of a Strohal tone using a shet metal spiral on a chimeney. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)


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When air or another gas blows across the edges of an opening to a hole, loud, pure tones are formed. In this type of sound generation periodic vortice generation interacts with the acoustic resonator formed by the opening and volume. This is how a wind instrument operates and can also be excited in a number of everyday objects, see Figure 13-25. The greater the volume of the hole and the smaller the number of openings, the lower the frequency of the tone will be.

Figure 13-25 Sound generation by blowing air over an opening backed by a cavity. Periodic vortice generation interacts with the acoustic reonator formed for instance by the neck and cavity of the bottle. A more narrow neck and a larger cavity volume give a more low frequency sound. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

An example of this type of sound generation can be found in a cutter wheel revolving under no-load conditions, where sound can arise from the track for holding the plane blade. An air stream is being chopped, creating a siren (pure tone) noise. Minimizing the cavities by filling the empty space in the track with a rubber plate reduces the pumping action and the noise.


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Figure 13-26 Noise control of a cutter wheel by filling the cavity with a rubber material. A strong tonal sound is generated by vortices formed at the edge interacting with the cavity at certain frequencies. After filling the cavity the character of the sound becomes broad band. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

When a gas or liquid flows in ducts or pipes there is always some turbulence exciting the duct walls. The noise from turbulence is increased if the flow must rapidly change direction, if the flow moves at a fast rate, and if objects blocking the flow are close together.


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Figure 13-27 Smooth pipe walls without discontinuities give less turbulence exciting duct wall vibrations and sound. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

Figure 13-28 shows a branch of a steam line having three valves which produce a loud shrieking sound. The branch has two sharp bends which also produce a lot of noise. To control the noise a new branch was created with softer bends. Tubing pieces were placed between the valves, so that turbulence was reduced before the stream reaches the next valve.

Figure 13-28 Noise control of a steam line by introducing softer bends and increasing the distance between valves. Both measures reduce the turbulence incident on the valve. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)


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When a flowing gas mixes with a non-moving gas so called jet noise will be generated. This has already been discussed in chapter 9. A lower outflow speed will produce a lower sound level. For speeds below 300 m/s the sound power is proportional to the flow speed to the power of 8 (U8). A reduction of the speed by half will therefore mean that the sound will be reduced by about 24 dB.

Figure 13-29 Jet noise generating by free stream turbulence. The sound generation is increased by disturbances in the stream. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

An example of this type of sound generation is the exhaust air from a compressed air-driven grinding machine. The air becomes turbulent while leaving the machine through the side handle. To control the noise a new handle was developed, filled with a porous sound-absorbing material between two fine-meshed gauzes. Passage through the porous materials breaks up the turbulence. The air stream leaving the handle is less disturbed, and the exhaust noise is weaker. A straight lined duct-type muffler may also be used.


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Figure 13-30 Noise control for a pneumatic grinder by introduction of a porous muffler inserted in the air stream. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)


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Since the jet noise level is determined by the speed of the jet stream in relation to the speed of the surrounding air, noise production can be greatly reduced by using an air stream with a lower speed outside the jet stream.

Figure 13-31 Principle for jet noise reduction by introducing a secondary air stream around the core jet exhaust to reduce the relatie flow speed difference between the jet stream and the surrounding air. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

This principle can be used to reduce the noise from cleaning of machine parts with compressed air after processing which is often carried out with simple tubular mouthpieces. Very high exit speeds are required, and a strong high frequency noise develops. The simple tubular mouthpiece can be replaced by mouthpieces which produce less noise, such as a dual flow mouthpiece. In this mouthpiece, part of the compressed air moves at a lower speed outside the central stream.


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Figure 13-32 Noise reduction by introducing a secondary air stream around the core jet exhaust in the form of a dual flow mouthpiece. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)


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If the diameter of a gas outlet is large, the noise will peak at the low frequency. If the diameter is small the noise will peak at high frequency. The low frequency noise can be reduced by replacing a large outlet with several small ones. To some extent this will increase the high frequency noise, but this is more easily controlled.

Figure 13-33 Principle for jet noise reduction by dividing the core jet stream into several smaller jet streams. This reduces the turbulent mixing area and the noise generation. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

Steam safety valves may discharge many times each day. Sound production during steam escape can produce high level, low frequency sound.To control the noise a diffuser was formed as a perforated cone. The holes produce many small jet streams and high frequency noise which is absorbed in the down-stream pack.


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Figure 13-34 Jet noise reduction in a steam safety valve by dividing the core jet stream into several smaller jet streams. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.) The inflow to fans is very important for sound generation. If there is an inflow disturbance giving a lot of turbulence the sound will be more intense. The same principle applies, for example, to propellers in water.


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Figure 13-35 Principle fan and propeller sound generation. Inflow disturbances generating inflow turbulence increases the noise generation. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

Fans should therefore not be placed close to any discontinuities in a duct. In Figure 13-31 examples are shown where the fan is placed too close to control vanes, and too close to a sharp bend. The flow is disturbed and the noise at the outlet is increased. To control the noise the control vanes can be moved farther from the fan so that the turbulence has time to die down. In the other case, the bend can be made smoother, and the fan moved away from the bend. Guide vanes can also be used to give a smoother flow through the bend.

Figure 13-36 Fan noise control by increasing the distance between duct discontinuities and the fan. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)


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Turbulence will form if the pressure in a liquid system drops rapidly. Gas is released in the form of bubbles and produces a roaring noise. The pressure drop can be produced by a large, rapid change in volume. Noise is avoided by a slow change in volume.

Figure 13-37 Principle for noise reduction in a liquid filled pipe using smooth duct transitions. Because a rapid pressure rop is avoided less gas bubbles are formed. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.) This principle can be applied to control valves in liquid systems. They often have small valve seats, resulting in large flow speeds with large pressure changes. Twisted flow pathways and sharp edges produce intense turbulence. Sound radiates directly from valves and pipes, and solid sound is conducted to walls. To reduce the noise, control valves with larger cone diameters, straighter flow pathways, and more rounded edges can be used.


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Figure 13-38 Valve noise control by using larger cone diameters, straighter flow pathways, and more rounded edges. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)


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Large pressure drops in liquids can also give cavitation, which is generated when gas bubbles are formed and then collapses. Noise production typically takes place at control valves, at pump pistons, and at propellers when large and rapid pressure drops occur in liquids. This so-called "cavitation" noise is most common in hydraulic systems. Cavitation can be reduced by bringing about the pressure reduction in several smaller steps. In a hydraulic system, the full pump capacity is employed only in exceptional cases. The pressure is generally greatly reduced using a control valve. Cavitation can then arise, producing loud noise from the valve. The noise is conducted as solid-borne sound to connected machines and building structures. To control the noise a pressure reducing insert can be placed in the same pipe as the control valve. The inset has removable plates with different perforations. The plates are selected so that the inset will not produce a greater pressure drop than that required to prevent cavitation.

Figure 13-39 Pressure reduction in several steps to reduce cavitation noise. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)


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High frequency sound is reduced more effectively than low frequency sound by propagation through air. In addition, it is easier to insulate and shield. If the noise source does not cause problems in its immediate vicinity, it may therefore be worthwhile to shift the sound toward higher frequencies. This principle can be applied for external industrial noise. The low frequency noise from roof fans in an industrial building disturbs residents of houses a quarter-mile away. If the rooftop fan is replaced by another one of similar capacity but with a larger number of fan blades, this produces less low frequency noise and more high frequency noise. The low frequency noise no longer causes disturbances, and the high frequency noise is adequately reduced by the distance.

Figure 13-40 Reduction of the community noise from a roof top fan by replacing it with a fan with larger number of blades, thereby shifting the sound to higher frequencies which are more damped when propagating over large distances. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

On the other hand there may be cases where it is beneficial to shift the sound generation to lower frequencies which are less disturbing to the human ear.We are less sensitive to low frequency noise than to high frequency noise. If it is not possible to reduce the noise, it may be possible to change it so that more of it is at lower frequencies.


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An example is the diesel engine in a ship operaing at 125 rpm and directly connected to the propeller. The noise from the propeller is extremely disturbing on board. A differential gear was installed between the motor and the propeller so that the motor speed changed to 75 rpm. The propeller was replaced by a larger one and the noise was shifted to a lower frequency, making it less disturbing.

Figure 13-41 Reduction of the propeller noise disturbance on a ship by reducing the engine spped and thereby shifting the noise generation to lower frequencies. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)


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13.4 NOISE CONTROL DURING THE PROPAGATION PATH

In this section examples of noise control techniques which can be used during the propagation path are given.

13.4.1 Control of structure borne sound

Vibrations in solids can travel a great distance before producing airborne sound. Such vibrations can cause distant structures to resonate. The best solution is to stop the vibration as close to the source as possible.

Figure 13-42 Structural vibrations can be transmitted over large distances and the radiate as airborne sound. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

One example is the vibrations from an elevator whichare transmitted throughout a building. To control this noise the elevator drive can be isolated from the building structure using vibration isolation techniques as described in chapter 11. In chapter 11 the theory and practice of vibration isolation has been discussed. Below some examples and practical advice for application of machine vibration isolation will be given.


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Figure 13-43 Noise control by applying vibration isolation to an elevator drive. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

Knocking on a thin door produces more sound than knocking on a thick wall. For the same reason, noise sources should be mounted on heavy or rigid bases.


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Figure 13-44 Knocking on a thin door produces more sound than knocking on a thick wall. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

A motor-driven oil pump is placed on the side wall of a hydraulic press. The vibrations are transmitted to all plates, which convert the solid-borne sound to loud airborne sound. The oil system is removed from the press and installed in a frame on a heavy base. Sound transmission in the oil line is controlled with an accumulator.

Figure 13-45 Noise control by moving the motor driven oil pump to a separate heavy base floor. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)


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Vibration isolation of machines can reduce the area of excessive noise as shown in Figure 13-46 below. Either the machine or the working area can be isolated.

Figure 13-46 Vibration isolation at the source or at the receiver. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

A machine placed on springs has a so-called "fundamental frequency." Vibrations at or close to the fundamental frequency are greatly intensified. The machine may even break away from its fastenings. Vibrations with lower frequency than the fundamental frequency are not blocked. If the base is very heavy or very rigid, the fundamental frequency is determined entirely by the machine and base weights together with the rigidity of the spring. The lighter the machine and the more rigid the spring, the higher is the fundamental frequency. This reinforcement of vibrations can be avoided by using springs with good internal damping. An application example where this was of importance included two fans used in the same building. Both were vibration isolated with steel springs which have very poor internal damping. The isolation functioned well for both fans during constant operation, but one of the fans was started and stopped frequently. When this happens, the vibration frequency corresponds for a short time with the fundamental frequency, which produces serious disturbance. On the fan with irregular operation, steel dampers were installed with pads which have good internal damping. The isolation was somewhat less, but the disturbance from starting and stopping disappeared.


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Figure 13-47 Vibration isolation with high internal damping to avoid problems when passing through the critical frequency of the vibration isolation. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

A machine and mount with low natural frequency are difficult to vibration isolate unless the floor is very rigid. As shown below, an extra heavy (stiff) or pile-reinforced floor might be necessary.


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Figure 13-48 Measures to improve low frequency vibration isolation by making the foundation more rigid. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)


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A good way to isolate very heavy machines with low natural frequency vibration is to place them on a concrete base plate which rests directly on the ground. Even more effective protection is achieved if the base plate is separated from the remainder of the building by means of a joint. If the ground has a clay layer, it may be necessary to place pilings beneath the plate.

Figure 13-49 Measures to improve low frequency vibration isolation by putting the machine on a separate foundation directly on the ground. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)


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Drive motors with gears and differentials connected to a paper-making machine causes both loud air noise and vibrations in the machines. They require only occasional maintenance which can generally be performed with the machines turned off. Therefore, the machines can be permitted to make large amounts of noise if the noise is prevented from entering the rest of the factory. The engine room has its own thick base plate which is in good contact with the solid ground. The large base plate is also vibration isolated with corrugated rubber mats. Sound is prevented from entering other rooms by means of a brick wall. Holes in the wall for the axles to pass through are sealed with mufflers.

Figure 13-50 Noise and vibration control solution for drive motors for paper-making machines. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

Vibration isolation of a machine may be ineffective if sound is transferred through connections for oil, electricity, water, etc. These connections must be made very flexible. The machine movements will be reduced if a heavy base is selected, and more rigid springs can be used. Figure 13-51 Flexible pipe connections may be necessary for good vibration isolation. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)


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Cooling systems may be serious sources of noise as a result of intense pressure shocks in the liquid from compressors. Compressors may be vibration isolated with steel springs. In addition, flexible connections should be used for all inlet and discharge pipes.

Figure 13-52 Noise and vibration control solution for cooling systems. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

Vibration in machines often results from slippage or loosened bolts. In such cases, the disturbance can be reduced by repair or replacement.


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13.4.2 Control of airborne borne sound

The frequency content of the noise source is of importance for the possibility to reduce airborne sound during the propagation path. High frequency noise is easier to control. When high frequency sound strikes a hard surface, it is reflected much like light from a mirror and it does not travel around corners easily.

Figure 13-53 High frequency sound is reflected by hard surfaces and does not pass corners easily. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

An example of this is high frequency noise from a high-speed riveting machine which travels directly to the worker's ears. To control the noise a sound-insulating hood, open toward the bottom of the machine, was constructed above the hammer. The hood was coated on the inside with sound-absorbing material. The upper portion of the opening was covered with safety glass. As sound travels towards the operator, the glass reflects it against the sound-absorbing walls. The sound level for the machine operator was thus reduced.


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Figure 13-54 Noise control of high frequency sound from a riveting machine by using a hood with sound absorbing material. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)


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Low frequency noise radiates at approximately the same level in all directions and travels around corners and through holes, and then continues to travel in all directions. A shield or barrier has little effect unless it is very large.

Figure 13-55 Low frequency sound radiates in all directions also after passing over a barrier or through a hole in a barrier. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

Compressors and the diesel engines used to drive them produce strong low frequency noise, even if they are provided with effective mufflers at the intake and exhaust. A complete enclosure of damped material lined with sound absorbant will help. The air and exhaust gases must pass through mufflers which are partly made of channels with sound-absorbing walls. Doors for inspection must close tightly.


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Figure 13-56 Noise control of a compressor using a sound absorbing enclosure . (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

If it is not possible to prevent noise, it may be necessary to enclose the machines.

Use a dense material, such as sheet metal or plasterboard, on the outside.

Use a sound absorbant material on the inside. A single hood of this type can reduce the sound level by 15-20 dB(A).

Install mufflers on cooling air openings during enclosure of electric motors, etc.

Install easily opened doors as required for machine adjustment and service.


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Figure 13-57 Enclosure of a hydraulic system requries muffled ventilation openings. Electric motors release both sound and heat, as do the pump and the oil tank. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

The sound reduction number (R ) as defined in chapter 9 describe the ability of a wall to prevent sound incident on one side to be transmitted to the other side. The sound reduction number of a homogeneous single layer wall can be estimated by its mass per surface area.

Figure 13-58 Sound reduction number example. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)


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A sand blast operation creates excess noise. As a noise control measure a separate room is constructed for this operation. The blasting equipment is separated from other work areas with a drapery of lead-rubber fabric, which is heavy but flexible.

Figure 13-59 Noise control of sand blasting operation using lead-rubber draping. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.) In most single layer walls, the coincidence frequency causing a reduction in sound reduction number is close to 100 Hz for a thickness of about 20cm. At higher frequencies, both increased weight and increased rigidity produce greater sound reduction. A cast concrete wall has greater rigidity than a brick wall, and therefore provides greater sound reduction if the two wall weights are equal.


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Figure 13-60 Noise control of sand blasting operation using lead-rubber draping. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

Machines in a large open area in an industrial building create a noise hazard. To control the noise the area containing the machines is surrounded by a brick wall.

Figure 13-61 Noise control using a brick wall enclosure. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)


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Two light walls separated by an air gap provide good sound reduction, increasing with the distance between them up to about 15cm. With sound-absorbing material in between, the sound reduction further increases as the distance between increases. Double walls may provide the same sound reduction as single walls that are five to ten times as heavy.

Figure 13-62 Double wall sound reduction increases with increasing spacing between the walls. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

Figure 13-63 Double wall sound reduction increases with sound-absorbing material in-between the walls. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)


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A double wall provides the best sound reduction if each layer is connected to heavy walls or if there are open joints at both ends. If the layers are fastened to shared studs, the sound reduction is greatly reduced if the studs are close together. The thicker the layers, the farther apart the studs must be in order to avoid substantial reduction of sound reduction.

Figure 13-64 In order not to deteriorate sound reduction there must be a minimum distance between studs connecting double walls. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

It is known from chapter 9 that if a sound source is placed close to reflecting surfaces the emitted sound power will increase. The worst placement is in corners near three surfaces. The best placement is away from the walls.

Figure 13-65 Sound sources should be placed as far away as possible from reflecting surfaces to reduce noise generation. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)


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One technique to reduce the sound level in in rooms, already discussed in chapter 9, is to use sound absorbing material placed on walls or at the ceiling. Porous material through which air can be pressed often makes a good sound absorbant. Examples of such materials include felt, foam rubber, foamed plastic, textile fibers and a number of sintered metals and ceramic materials. If the pores are closed, the absorption is slight. Thin porous absorbants handle high tones. For good effects below 100Hz, the thicknesses required may become impractical. Low frequency absorption is improved with the aid of an air gap behind the absorbant. An example of a workshop with intense lowfrequency noise is shown in Figure 13-66 One part of the shop contains space for hanging absorption baffles, which provide good low frequency absorption and are easily installed. A traverse leaves no room for baffles in the other part of the shop. Instead, horizontal absorbant panels are installed above the traverse, 8 inchesfrom the ceiling, to improve the lowfrequency absorption.

Figure 13-66 Noise control in a factory workshop by mounting sound absorbing baffles in the ceiling and sound absorbing panels on the walls. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

It may be necessay to protect the porous sound absorbing material by using a cover layer. This cover must have sufficient number of openings not to reduce the effectiveness of the absorbant. The thicker the cover layer, the larger the number of perforations that will be required.


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Figure 13-67 Effect of perforated cover layers on the effectiveness of porous sound absorbing material. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

Another type of sound absorber which can be used is the panel absorber discussed in chapter 9. It consists of thin panels, fastened to a system of studs and can absorb low frequencies. The absorption is effective in a narrow frequency range. This range is determined by the stiffness of the panels and the distance between the fastenings. If the panels are fastened to studs on a wall, the distance from the wall also has an effect. A panel with large internal damping gives absorption in a larger frequency range. If a porous absorbing material is used at these low frequencies, it must be very thick.

Figure 13-68 Panel absorbers can be used to absorb sound in a limited frequency range. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)


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Low frequency resonance in an engine room produced a very loud hum near the walls and in the center of the room. When the revolution speed was significantly changed, the hum disappeared completely. The walls were coated with panels on studs to provide the greatest absorption in the range of the loudest tone. In order for the absorbant to continue to function even in the case of slight deviations from the normal rotation speed, a layer with good internal damping was used, which provided a more extensive range with good absorption. As a result, the resonance and the loud hum disappeared.

Figure 13-69 Application example using panel absorbers for low frequency noise control. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)

High frequency noise can be reduced by using barriers or shields in a room. The shield is more effective the higher it is and the closer it is placed to the source. The effect of a shield is considerably reduced if the ceiling is not sound absorbant.

Figure 13-70 It is important to use sound absorbing material in the ceiling when using shields or barriers for noise control in a room. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)


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An example where the use of barriers was effective was in an auto plant with several assembly lines. The work on one line was noisier than the other because grinding work on the bodies produces a shrieking, high frequency sound, disturbing everyone in the plant. To control the noise the other lines were protected from the grinding noise by means of shields on both sides of the line and soundabsorbing baffles suspended above the open area.

Figure 13-71 Noise control of noisy car assembly line using barriers on both sides of the line and soundabsorbing baffles suspended above the open area. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)


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13.5 NOISE CONTROL AT THE RECEIVER

When noise can not be reduced to a sufficient level at the source or during the propagation path noise control at the receiver must be used. This can either consist of building enclosures or control rooms from which the workers control the machines or by using personal hearing protectors in the form of earplugs or earmuffs.

The design considerations for building a control room are similar to for building a machine enclosure discussed in the previous section. Special attention will of course have to be paid to ventilation and visibility issues. Some control measures may include:

• constructing the control rooms with materials having adequate sound reduction number.

• providing good sealing around doors and windows • providing openings for ventilation with passages for cables and piping equipped

with good seals. The control room will need adequate ventilation and possibly air conditioning in hot working areas. Otherwise, there is a risk that the doors will be opened for ventilation, which would spoil the effectiveness of the room in reducing the noise level.


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Figure 13-72 Some aspects which have to be considered when designing a control room to give good noise control. (Picture: Asf, Bullerbekämpning, 1977, Illustrator: Claes Folkesson.)


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If the noise level despite attempting noise control at the source or during the propagation path remains dangerously high, then use of hearing protection should be adopted. There are two types of hearing protection: earmuffs (see figure 13-13), and earplugs. Earmuffs enclose the entire outer ear, whereas earplugs are placed in the auditory canal. The latter are of two types: disposable and reusable. There are different models of both earmuffs and earplugs, with varying acoustic properties. Noise measurements at a workplace serve as a basis for the selection of proper hearing protection. Information on the SPL in different frequency bands is important, since different types of hearing protection each have their own frequency dependencies. The protection selected must sufficiently attenuate sound so that the SPL in the auditory canal does not exceed risk levels for hearing damage. It must also fit well; earmuffs should be well-sealed against the head, and earplugs should be properly inserted. Hearing protection must be worn the entire time that one spends in the noisy environment. Even short interruptions drastically diminish the protection provided. It is also important that noise pauses, during which one is away from the noisy environment and can remove the hearing protection, are provided during the work day.

Hearing protection must be maintained to ensure that the sound reduction provided does not diminish with time. The cuff (or ear cushion) of earmuffs should not be damaged, hardened, or dirty. Reusable earplugs must be kept clean; disposable ones must not be reused. In noisy workplaces, regular hearing tests of the employees should be carried out, in order to detect the onset of hearing problems as early as possible.

Figure 13-73 Earmuff-type hearing protection. (Sketch: Arbetarskyddsnämnden, Buller och vibrationer ombord,

Ill: Claes Folkesson.)

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Chapter 14: Machine Condition Monitoring

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CHAPTER 14

MACHINE CONDITION MONITORING Unexpected production delays cause substantial costs in many branches of industry. This is true in particular for various kinds of process-based industries like the pulp and paper industry, the steel industry and the power generation industry. Quite often the delays are due to failure or breakdown of machines on the production line. Machine vibration monitoring seeks to monitor the health of machines by taking vibration measurements during operation.

A machine consists of many potentially defective elements, such as gears, bearings, shafts etc. During operation, each element contributes to the overall vibration of the machine. Each specific machine element has its own characteristic vibration signature. When a defect e g, a fatigue crack, appears in an element, e g, a gear, its signature is modified. The idea of machine vibration monitoring is to use the modified vibration signature to detect, localise, diagnose and prognosticise the defective element so that relevant maintenance can be planned. Successful machine monitoring requires that the individual signatures from each element can be recovered from the vibration signal measured at the machine surface. Clearly, both engineering knowledge on the principles of machine elements and clever signal processing are needed. This chapter gives a brief overview over the fundamentals of machine vibration monitoring with some emphasis on useful signal analysis tools. Illustrative examples of the vibration monitoring of gears and bearings are given.


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14.1 INTRODUCTION

Machines of all kinds contain elements that suffer from repeated high amplitude dynamic forces. The most typical examples are gears and bearings. Sooner or later the wear of these dynamic forces will cause a failure of an element, the machine and ultimately, in severe cases, the production line. Machine monitoring, or early detection of incipient faults, aims to survey the machine health, or condition, at critical locations, e g, gears and bearings, and possibly predict a future failure. At a certain stage of defect progress or severity, a scheduled stop for maintenance can be made, the damaged element replaced, and production can then continued without unnecessary delays. Basically, machine vibration monitoring uses so called signature analysis, i e the characteristic vibrational signature of the monitored machine element is investigated. In practice, machine monitoring requires some physically measureable signal to monitor. It might be vibration, sound pressure or a temperature signal. Vibrations are the most commonly used monitoring signals. Vibrations are not subject to background disturbances to the same extent as acoustic noise. Vibration sensors, e g, accelerometers, can be placed closer to the source than microphones. It is sometimes claimed that machine monitoring is, in practice, not feasible. There are, however, many examples showing that it can be successfully implemented. The key to success is to know what feature in the signal to monitor, and to implement this in a practical way. This implies that a monitoring system must be individually adapted for its specific application. The development of a monitoring system usually starts with a pilot study including an extensive experimental investigation of the vibration characteristics of the machine. Based on the pilot study suitable vibration signatures are selected, measurement instruments, signal processing strategies, condition indicators and user interfaces are selected and developed.

14.2 BASIC IDEAS OF MACHINE MONITORING

Machine vibration monitoring essentially means the detection of a defect and the assessment of the residual time to machine failure. It includes the following five elements, - to measure and survey characteristic vibration signals, i e signatures, - to process and analyse signatures for early fault detection and localisation, - to survey the progress of the detected faults, - to predict the time to failure or breakdown and - to indicate defects that have grown beyond acceptable limits. The basic premise of machine monitoring is that the appearance of a fault changes the vibration characteristics of the machine. For a successful monitoring it is crucial that characteristic signatures of all sensitive elements are known and measureable. It is also crucial to know how possible defects modify the signatures. Finally, a time to failure prediction requires a model for the defect progress.

The main difficulty in machine monitoring stems from the fact that each machine consists of several vibration sources that contribute to the measured vibrations. Thus, the signature from a specific element, e g, a bearing, is usually hidden in noise and


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contributions from other elements. To overcome this difficulty it is important to place the sensors as close as possible to the monitored element and to use á priori knowledge of the signature to implement a suitable signal analysis procedure that is capable of extracting the signature of the correct element from the measured signal. 14.3 TYPICAL DEFECTS IN GEARS AND ROLLING ELEMENT BEARINGS

Most failures of gears and rolling element bearings start as localised point defects in contact zones between contacting bodies. Gear failures are initiated in the tooth meshing zone and bearing failures in the rolling contact zones. Defects possible to detect and diagnose can be classified in three classes with respect to their cause, see table 14-1.

Table 14-1 Detectable defects divided into three classes.

Class Defect Element Bearing race misalignment Bearing Installation defects

Nonuniform radial or axial loading Bearing, gears Surface wear Bearing, gears

Cracks and spalls Bearing, gears Defects developed in operation Inadequate lubrication Bearing, gears

Shaft wobbling Bearing, gears Jointed coupling defects Bearing, gears

Defects on neighbouring elements which produce dynamic loads on bearing and gears Gear interaction defects Bearing, gears

Cracks and and spalls are the most drastic defects in table 14-1. Spalling is caused by the very high, local, contact stress concentrations localised below the contact surface. Usually a crack is initiated at a material inclusion located within this zone of high stresses. The crack progresses due to the repeated cyclical loading during machine operation. At a certain stage the crack reaches a critical size and the growth becomes unstable. In the unstable phase the crack rapidly grows and eventually reaches the surface, and a piece of material comes loose, creating a spall. The first occurrence of a spall is then rapidly followed by the occurrence of other spalls and cracks, leading to a fast deterioration of the machine health.

All defects in table 14.1 modify the characteristic vibration signature. One of the main tasks of machine monitoring is to identify and localise these defects from measured signatures. Figure 14-1, Figure 14-2 and Figure 14-3 show damaged elements of a planetary gear.


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Figure 14-1 Damaged sun wheel in a planetary gear.

Figure 14-2 Damaged crown wheel.


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Figure 14-3 Damaged pinion.

14.4 VIBRATIONS OF GEARS AND BEARINGS

Most of the effort in machine monitoring has been spent on monitoring bearings and gears. First of all, gears and bearings are parts of almost every rotating machine. Secondly, they are inherently subject to dynamic contact forces with high amplitudes. In the region close to the contact surfaces the material stress is locally very high. Thus, after some time of operation a localised defect growing with time will appear. Any rolling element bearing and gear will, due to imperfections in design and manufacturing, generate time-varying, i e dynamic, forces. In practice all non-defective bearings and gears have their own characteristic signatures determined by their design parameters and their condition. Since the bearing and gear operation process repeats cyclically, each fundamental shaft rotation period, the vibration signals are basically cuclic signals and the corresponding order spectra will have high tonal components. The signatures are characterised by the positions of these tonal frequencies and their amplitudes. These vibration signatures constitute the basis for machine vibration monitoring.


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14.4.1 Vibration characteristics of non-defective gears The purpose of gears is to transmit and transform rotary motion from one shaft to another. For example, the speed and torque needed on the wheels of a truck may differ significantly from the speed and torque on the engine output. Basically the gear design is determined by the specified speed and load capacities. One important parameter is the gear tooth shape, or profile. Ideally the teeth are shaped so that the output shaft speed is a constant multiple of the input shaft speed. In practice, small deviations from the ideal design make the output shaft speed fluctuate around the nominal value. The process in which the gear teeth on the input shaft roll and slide on the output shaft gear teeth is called gear meshing. The time period elapsed between the moment when a tooth goes into meshing and the moment it goes out of meshing is called the tooth meshing period, Tmesh. A gear with K teeth running with shaft rotational period Trot generates a tooth meshing force with period equal to Tmesh. The relation between the shaft rotation period and the tooth meshing period is,

meshrot TKT ⋅= . (14-1)

The characteristic features of gear vibrations are due to the cyclically varying gear meshing forces. These meshing forces generate vibrations that are transmitted to other parts of the machine where they can be measured.

Figure 14-4 Meshing pinion and gear in spur gear. Photo: Ulf Carlsson, MWL. Healthy gears generate meshing forces and vibrations dominated by the gear meshing frequency, fmesh, and its harmonics. A gear with K teeth and a shaft rotating at N revolutions per minute generates a fundamental meshing frequency,


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60

11,

NKKfT

KT

f rotrotmesh

mesh ==== , (14-2)

where frot is the shaft rotational frequency. A completely analogous formula for the outgoing shaft gives the same meshing frequency.

A perfectly sinuousoidal meshing force would imply a perfect single tone vibration spectrum. In real gears, the surfaces are not ideally smooth and the tooth shape are not ideal; hence, the vibration spectrum will always contain harmonics of the meshing frequency, rotmeshnmesh Kfnfnf ⋅=⋅= 1,, , n = 1, 2, 3, … . (14-3)

The nth component of the meshing frequency is frequently called the nth gear order. A gear with teeth perfectly uniform, or identical, in pitch and profile generates a line spectrum described by (14-3). The amplitude of each line depends on the actual shape of the meshing force. In general the amplitudes of the higher harmonics decrease quicker for a smoothly varying force than for an abruptly changing force.

In practice, pitch and profile always vary slightly from tooth to tooth. These variations cause periodic amplitude and phase modulations of the meshing forces. Since the modulation repeats each revolution, the fundamental period of these modulations is given by the rotation period of the shaft Trot. Due to the modulation, each gear order will have repeated uniformly spaced side bands. The frequency increment between the side bands are determined by the repetition frequency of the modulation, thus, it is equal to the shaft rotation frequency, frot. The amplitudes of the side bands are determined by the variation of the pitch and profile from tooth to tooth. A smoothly varying set of teeth will produce low amplitude side bands. In general, the modulation is frequency dependent, i e it varies from order to order.

00

Tooth 1

Time

Vibration

Trot = K TmeshTmesh

Tooth 1Tooth K

Tmesh

• • •

Tooth 2Tooth 2

Figure 14-5 Time history from a gear with K teeth mounted on a shaft rotating with

rotational period Trot.


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Figure 14-5 and Figure 14-6 illustrate the correspondence between the periodicities in the time domain and the harmonic components in the frequency domain. Also note that time domain representation together with tachometer pulses from a known reference position makes it possible to identify the effects of a specific gear tooth.

0

0,5

1

1,5Vibration amplitude

1st meshing: Kfrot

2nd meshing: 2Kfrot

3rd meshing: 3Kfrot

• • •

• • •

• • •• • •

• • •

Side bandsfrot

(K+1)frot

(K+2)frot

(K-1)frot

frot

(2K+1)frot

1 K-2 K K+2 2K2K-2 2K+2 3K-2 3KGear order

Figure 14-6 The vibration spectrum from a gear with K teeth mounted on a shaft rotating with rotational frequency frot. Note that sidebands originating from different meshing orders coincide at the same frequencies.

Exercise 14-1: The meshing force in a particular gear with K teeth is, ideally, perfectly sinuousoidal with frequency Kfrot.

a) Suppose that, due to a systematic error in the manufacturing process, there is a sinuousoidal variation in gear tooth profile along the periferi. Show that this gear defect will generate two side bands to the fundamental gear meshing frequency. Hint: The gear meshing force can be written

Acos(2π frott)sin(2π Kfrott).

What is the fourier transform of this meshing force?

b) Suppose in addition to the tooth profile variation in a) that the pinion of a spur gear is elliptic and that this causes an additional meshing force modulation over the circumference with period 0,5Trot. Calculate the vibration signature of the gear and plot the result in a frequency spectrum. Hint: Assume the meshing force can be written,


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(Acos(2πfrott)+Bcos(2π2frott))sin(2πKfrott).

c) Discuss the vibration signature for a spur gear with a crack on a single tooth. It is assumed that the crack abruptly changes the meshing force. Clearly, even a single gear meshing process with constant load and speed and without other disturbing vibration sources will produce a line spectrum with plenty of lines. The spectrum is further complicated by the fact that the side bands of a specific order may, if they do not decay quickly enough, coincide with the main lines and the side bands of the neighbouring orders. Thus, it might be difficult or even impossible to determine the correct amplitudes from a line spectrum only. Example 14-1 Consider a gear with a driving shaft rotating with period Trot = 0,015 s. Suppose a gear with 15 teeth is coupled to the shaft and that the gear is transmitting a constant torque at constant speed. Figure 14-7 shows the time history of three different gear meshing forces: one perfectly sinuousoidal force, one more realistic with harmonic components and finally one exemplifying a gear with a defective tooth.

0 0,005 0,01 0,0150

2

4

6

8

10

12

Tooth 1Tooth 2

Tooth 15Tooth 14

Tooth 7

Time, [s]

Gear meshing force

a)

b)

c)

Figure 14-7 Time history of three different gear meshing forces. The gear has 15 teeth and rotates with 4000

revolutions per minute, i e with rotational frequency 66,7 Hz. a) Gear generating a purely sinuousoidal meshing force. b) Gear generating a distorted sinuousoidal meshing force. c) Gear meshing force generated by a gear with a localised defect at tooth 7.


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According to the discussion above, the purely sinuousoidal meshing force is a single frequency line located at the 1st meshing frequency,

100060

4000151, =⋅=meshf Hz,

in the frequency spectrum, see Figure 14-8. The distorted, but uniform, force history generates the fundamental meshing component and, in addition, the odd harmonics. Finally the damaged gear generates all odd meshing harmonics and, in addition, side bands mutually spaced 66,7 Hz, see Figure 14-8.

0 1000 2000 3000 4000 5000 6000

0

0,5

1

1,5Force amplitude

Frequency, [Hz]

1st meshing frequency

3rd meshing frequency

5th meshing frequency

Side bands

Figure 14-8 Line spectrum of three different gear meshing forces. a) Purely sinuousoidal gear force, +.

b) Distorted gear force, o. c) Force generated by damaged gear, solid lines. 14.4.2 Vibration characteristics of non-defective bearings Rolling element bearings are another type of machine element used in almost every rotating machine. The purpose of bearings is to support rotating shafts and axles in the machine casing. In rolling element bearings, rolling elements such as balls or rollers are used to reduce the friction forces between the contacting elements.

Ideally, a bearing with a finite number of rolling elements generates dynamic forces with a single characteristic frequency, the rolling element passing frequency. Whenever a rolling element passes close to a machine part vibrations are excited in the part.


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Hence, if a specific rolling element passes a fixed point in space with frequency froll and the number of rolling elements are K, the characteristic passing frequency is K froll.

In practice bearings are produced with finite tolerances implying that there will always be some surface roughness and deviations from the ideal surface profiles. Whenever a contact zone enters regions with imperfections the contact forces will change in a particular way. Since the contact zones rotate cyclically the contact forces will repeat cyclically and certain frequency lines in the contact force spectrum will be strong. A careful analysis of rolling element bearing kinematics shows that there are four different cyclical contact phenomena; - the rolling element spin, - the cage rotation, - the rolling element passing with respect to the outer race and – the rolling element passing with respect to the inner race. The corresponding frequencies are the characteristic frequencies for a rolling element bearing. These frequencies are determined using the shaft speed, N, and the bearing geometry. The rolling element passing frequency or cage rotational frequency is,

)cos1(6021, α

c

rcage d

dNf −⋅

= , (14-4)

the rolling element fault frequency is determined from the element spin frequency as1,

)cos1(602

222

2

1, αc

r

r

cspinelem

d

dddNff −

⋅== , (14-5)

the rolling element passing frequency with respect to the outer race is,

KddNf

c

rout )cos1(

6021, α−⋅

= (14-6)

and the rolling element passing frequency with respect to the inner race is,

KddNf

c

rin )cos1(

6021, α+⋅

= . (14-7)

Here, N is the the shaft speed in revolutions per minute, dr is the diameter of the rolling element, dc is the diameter of the cage, α is the contact angle between the rolling elements and rolling surfaces and K is the number of rolling elements. The geometry of the bearing is explained in Figure 14-9.

1 Each element rotation the defect enters a contact zone twice; once at the inner race and once at the outer race. Hence, the rolling element fault frequency is twice the element spin frequency.


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N

Inner race

Outer raceCage

Shaft

Rolling element

dc

α dr

Figure 14-9 Drawing illustrating bearing design parameters used in formulas (14-4,5,6,7).

Ill: U Carlsson, MWL. These four characteristic bearing frequencies are fundamental frequencies in four harmonics series of frequencies. The amplitudes of the harmonics in each series are a measure of the deviation from sinuousoidal force variation.

In practice the rolling contact forces vary with angular position of the shaft due to external radial and axial loads. This variation causes a modulation of the characteristic bearing rolling contact forces. The modulation frequency is equal to the shaft rotation frequency. In a frequency spectrum the modulation is manifested as a series of side bands, equidistantly spaced with the shaft frequency, appearing on both sides of all harmonic components in the four characteristic bearing frequency series. As in gear vibrations the side bands from different harmonic components usually overlap. They do not, however, coincide because the 4 characteristic tones are not integral multiples of each other.

A localised fault on a single rolling element is subject to another modulation process. Since the fault hits the races in different positions depending on the cage position the rolling element fault signature will be modulated with the cage rotation frequency. Hence, a rolling element fault signature has side bands spaced fcage,1 from the defect frequency.

In summary, a non-defective rolling element bearing is characterised by four different sets of frequencies, or signatures, each consisting of a fundamental component with harmonics, and side bands. The basic assumption in machine monitoring is that a localised defect modifies the signature in a particular way depending on the defect location. Example 14-2 A single line ball bearing with 8 balls is coupled to a shaft rotating with speed 12000 revolutions per minute corresponding to the rotational frequency 200 Hz. The ball diameter is 3 mm, the cage diameter is 12 mm and the contact angle is 0 radian.


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Four vibration signatures measured in a position close to the bearing dominate the vibration spectrum. The fundamental frequencies are,

75)0cos1231(

60212000

1, =−⋅

=cagef Hz,

750)0cos1231(

312

602120002

2

2

1, =−⋅

=elemf Hz,

6008)0cos1231(

60212000

1, =−⋅

=outf Hz

and 10008)0cos1231(

60212000

1, =+⋅

=inf Hz.

Three of the harmonic frequencies of these four sequences has side bands equidistantly spaced with frequency increment 200 Hz. The rolling element fault frequency has side bands spaced with the cage rotation frequency, 75 Hz. Figure 14-10 illustrates the spectrum schematically.

0

0,5

1

1,5Vibration amplitude

fcage

fin

felem

• • •

fin + frot

fout

0 250 500 750 1000 1250Frequency, [Hz]

2fcage

2fout

2fout + frot

fin + 2frot

fin - frot

fin - 2frot

fin - 3frot

fout - frot

fout - 2frot

felem + fcage

felem + 2fcage

felem + 3fcage

felem - fcage

felem - 2fcage

Figure 14-10 A schematic illustration of the four characteristic harmonic bearing sequences. In this

particular case the side bands of the inner and outer race passing frequencies happen to coincide.


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14.4.3 Vibrations of defective gears Extensive experimental investigations have shown that the gear condition influences the vibrations generated by the gear. According to the discussion in paragraph 14.4.1 a gear with perfectly uniform profile and pitch and ideally smooth surfaces has a vibration signature characterised by its gear meshing frequency with harmonics, see formula (14-2). This signature is usually referred to as the harmonic error signature. In practice all gears are manufactured with finite tolerances and the profile and pitch vary slightly from tooth to tooth and the tooth surfaces will have roughnesses. Hence, the harmonic error signal will be modulated by these deviations from the ideal uniform gear. The difference between the actual signature and the harmonic error signature is usually called the residual error signature. The residual error has the same periodicity as the shaft rotation and will cause a series of side bands with frequency step equal to the shaft rotational frequency, see Figure 14-6. An alternative and enlightening interpretation is that the harmonic error signature is a measure of the generated vibrations averaged over all gear teeth. The residual error signature, on the other hand, is a measure of how much the vibrations differ from the average. Using this interpretation, it is clear that the residual error signature should show a strong coupling to the condition of the gear and, hence, that it is the characteristic vibration signature that gear vibration monitoring should focus on. A healthy non-defective gear has a signature consisting of mainly the harmonic error but also a small residual error. When the gear has been in use for a while the contact forces begins to wear the teeth contact surfaces. The wear is manifested in rougher surfaces and, hence, an increase of the residual error signature. When the wear is sufficiently large the increase in the residual error signature, i e the side bands, can be detected by vibration measurements on the gear casing.

The most common types of gear defects are fatigue cracks or spalls in one or more gear teeth. Cracks usually start at the roots of the gear teeth where the material stress due to tooth bending has a maximum. Spalling, on the other hand, usually occurs at the faces of the teeth where the local contact forces have maximum. A crack leads to a reduced tooth stiffness and an increased tooth deflection. Missing material, due to spalling, in the contact surface implies an increased surface roughness and an increased excitation by the contact force. Both these defects will cause irregular behaviour of the contact force as the defective tooth comes into meshing, see Figure 14-11. Since this irregularity will occur once each shaft rotation period, the harmonic error signature will be modulated and side bands spaced at the shaft rotation frequency will appear.


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0,008

0,004

0,0

-0,004

-0,008

Acceleration, [m/s2]

0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 Time, [s]

Figure 14-11 Vibration signal measured on planetary gear with damaged tooth on pinion. Each time the damaged tooth enters the tooth meshing zone the gear will be subject to a sudden excitation force. The graph shows three consequtive occasions when the damaged tooth enters the messhing zone.

Some defect types, such as lubrication problems and shaft misalignment, change the contact conditions between the meshing teeth. Force fluctuations caused by such defects have the same periodicity as the gear meshing. Hence these defects will affect the vibrations at the gear meshing harmonics.

Finally, when both gears have localised defects the vibration signature will, in addition to harmonics of both shaft rotations, contain components with period equal to the tooth hunting period. The tooth hunting period is the time elapsed between two instances when the two defect zones go into simultaneous meshing. Tooth hunting produces a fundamental component with a frequency significantly lower than the shaft rotation frequencies. Example 14-3 Consider the gear in example 14-1. Suppose the contact surfaces are worn and that a defect has developed on the 7th tooth, see the time history of the meshing force in Figure 14-12. The sudden change in contact force occuring when the 7th gear goes into meshing acts as a periodic impulse excitation with a periodicity given by the shaft rotation. In the vibration spectrum side bands of the gear meshing harmonics will appear, see Figure 14-12 and Figure 14-13. The size and position of the side bands is a possible choice of defect indicator.


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0

2

4

6

8

10

12

0 0,005 0,01 0,015 Time, [s]

Meshing force

1st tooth

7th tooth

Figure 14-12 Time history of gear meshing force. The random nature small-scale variation between the vibration

caused by individual teeth is due to variation in pitch and profile. Gear is running with constant load and constant speed.

0

0,5

1

1,5Amplitude of meshing force

0 500 1000 1500 2000 2500 3000 3500 4000Frequency, [Hz]

1st meshing frequency

3rd meshing frequency

5th meshing frequency

Side bands

Figure 14-13 Line spectrum of gear meshing force. Gear is running with constant load and constant speed. Gear

pitch and profile varies randomly. At higher frequencies, e g higher than 3500 Hz, the contribution from the variation in pitch and profile is of the same order as the gear meshing components, cf Figure 14-14.


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2000 2200 2400 2600 2800 3000 3200 3400 3600 3800 40000

0,01

0,02

0,03

0,04

0,05

0,06Gear meshing force

Frequency, [Hz] Figure 14-14 Line spectrum of gear meshing force. Gear is running with constant load and constant speed. Gear

pitch and profile varies randomly. A comparison of gear meshing forces between an ideal gear with no variation in pitch and profile and a gear with random variation in pitch and profile. Thin line represents gear without variation in pitch and profile. Clearly some signal averaging is needed to suppress the contribution from the unavoidable spread in pitch and profile.

One important topic for gear vibration monitoring is to determine how the residual error signature changes as different types of defects are initiated and progress and how vibration signals measured on the gear casing can be processed in order to detect, diagnose and track a defect in an unambigous way. Several methods, some based on time-domain information, some on frequency-domain information, have been developed during the years. 14.4.4 Vibrations of defective bearings Most rolling element bearing defects appear on the rolling contact surfaces; i e the rolling elements, the inner race and the outer race. A localised surface defect, such as a crack or a spall will, each time a rolling contact surface passes generate a wide band impulse. At constant speed the bearing and its seating are periodically excited with contact impulses. A very important and useful consequence of this is that the defect can be localised using information on the periodicity of the defect signatures, see section 14.4.2. A defect on a rolling element has a certain periodicity as has a defect on the inner race etc. Thus, characteristic frequency lines appearing in the vibration signature carry information on the defect location.


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A detailed investigation shows that each contact force impulse has a very sharp leading edge that ensures excitation energy at very high frequencies. In practice all bearings and their supporting structures have several eigen-frequencies in the excited frequency range. Each impulse excitation is therefor succeeded by a reverberant or ringing vibration. The decay time or reverberation time is a measure of the structural losses; rapid decay implies high losses whereas slow decay implies low losses, see Figure 14-14. Thus, the high frequency envelope, or time domain signature, of a damaged bearing consists of a periodic sequence of exponentially decaying vibration ringings or impulse responses. The task of machine monitoring is to identify and diagnose the bearing defect from this signature.

0 0,01 0,02 0,03 0,04 0,05 0,06 -1

-0,8

-0,6

-0,4

-0,2

0

0,2

0,4

0,6

0,8

1 Acceleration, [mm/s2]

Time, [s]

Figure 14-15 A spall located in the roller bearing rolling surface generates impulse excitation forces. The reverberant decay of the vibrations caused by this excitation is a measure of the structural losses.

Defects developed during operation are, at early stages, manifested as modified or modulated rolling contact friction forces. Surface wear, smoothly distributed over the rolling contact surfaces, produces a smoothly amplitude modulated rolling contact force whereas localised defects, such as cracks and spalls, produce an abruptly changing force modulation. At early defect stages, the vibration signature changes are most significant at high frequencies. Also, the contributions from other, distant, sources are usually small at high frequencies. Therefore most modern monitoring methods use high frequency vibration for defect detection and diagnostics.


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14.5 MONITORING METHODS

The defect signature energy is usually distributed over a wide frequency interval and is therefore easily masked by energy from other sources. Several time domain as well as frequency domain methods have been developed over the years to cope with this problem and provide an accurate defect detection method. Time domain methods dealing with localised defects involve indicators sensitive to impulsive oscillations. Well-known examples are the peak value, root mean square (rms-) value, crest factor and Kurtosis. Intelligent signal filtering is crucial for the success of all these methods.

Early methods, developed before FFT analysers appeared, used peak and rms detecting instruments in combination with various filters. After birth of the FFT analyser, methods focused on frequency domain properties like harmonic sequences of the characteristic defect frequencies. Finally, during the last two decades of the 20th century, methods such as high frequency envelope detection, relying on advanced signal processing and stochastic signal analysis have emerged. 14.5.1 Early time domain methods The classical machine monitoring method, used by machine operators since the 1920’s, is to press a screw driver tip to the machine casing and the handle to the skull bone and listen to the machine vibrations, taking advantage of bone conduction. This is basically an instrument which uses the screw driver as vibration sensor and the operators auditory system as the signal analyser. In the 1950´s, efforts were made to summarise these experiences in automatic measurement systems. At that time the available instruments were simple accelerometers, analog filters, root mean square indicating voltmeters and oscilloscopes indicating peak values. So the early machine monitoring methods used various combinations of vibration rms and peak values. Several methods used either the peak or the rms-value; others used their ratio, the crest factor. Unfortunately indicators based on the rms-value are insensitive to incipient defects. Crest factor indicators, on the other hand, are sensitive for early defects when the peak value is high and the rms-value is low. At later stages of defect life the overall rms-value increases significantly and the crest factor is reduced. This fact has led to the common misconception that defect severity decreases after the early stage when it in reality is progressing towards the final failure.

The early methods suffer from some severe disadvantages: - early defect detection is difficult in a noisy environment, - defect localisation and - defect identification are in practice impossible. A solution to these problems requires intelligent filtering and signal processing. 14.5.2 Spectral methods The character of the vibration signature is most prominent in the frequency domain. For this reason many monitoring techniques are based on frequency domain characteristics, the spectrum, of the vibration signature. Spectral methods are most succesfully applied to the low and intermediate part of the frequency range.


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The strategy is to extract the signature from a single gear or bearing from the total vibration measured on the machine casing and to monitor the characteristic defect frequency amplitudes. The basic assumption is, of course, that a growing defect implies increasing amplitudes at the defect frequencies. Some defect indicators, like the defect severity index, focus on the difference in amplitude at the defect frequency and the amplitude of the background. Other alternative methods focus on the side bands and seek to construct indicators measuring the relative amplitudes of the side bands.

The main problem of spectral methods is, as always, to suppress the contributions from other vibration sources to such an extent that they do not interfere with the investigated signature. In some cases the signature dominates the measured vibrations whereas in other cases the signature may be hidden in other contributions. In such cases some signal processing is needed to extract the signature and suppress contributions from other sources. One useful signal processing tool is time domain averaging or synchronised time domain averaging1. Synchonised averaging means that the signal is ensemble averaged in the time domain with each time record synchronised with a tachometer signal tracking the shaft rotation period. This averaging will, if properly performed, cancel out all frequency components except the ones synchronous with the shaft, i e the ones periodic in the time window. It is common practice to normalise the frequency axis with a characteristic frequency, e g the shaft rotation frequency, to get an order spectrum. The use of order spectra has several advantages as compared to standard frequency spectra. 14.5.3 Cepstral methods The signal cepstrum is useful for detecting repetitive patterns in a signal. In principle, the signal cepstrum is defined as the Fourier transform of the magnitude of the signal spectrum. Hence, vibration signatures including harmonic side bands are relatively easy to detect in a signal cepstrum. On the other hand several experimental investigations have shown that monitoring the cepstrum may give ambiguous results, especially regarding the defect progress. 14.5.4 Envelope methods The main difficulty with spectral and cepstral methods is that they focus on low and mid-frequencies where the influence from other sources is usually large. Thus, it may be very difficult to extract the correct signature from the large number of frequency lines found at low frequencies. Envelope methods aims to avoid this problem by focusing on high frequencies, e g for bearings from 20 kHz to 30 kHz. The fundamental idea of high frequency envelope methods is to use the high frequency part of the defect modulated vibration signal and demodulate it to obtain the vibration signature. To ensure that the vibration signal is dominated by contributions from the correct element, the accelerometer is placed as close as possible to the monitored element. Contributions from other distant sources will then be attenuated before they reach the accelerometer.

Obviously, high frequency envelope methods require the excitation to have a substantial amount of energy at high frequencies. This is true for localised defects, such as


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cracks and spalls, but also for surface wear in fluid film bearings. When a ball in a ball bearing rolls over a spall on the bearing inner race an abruptly changing contact force is experienced. This contact force has impact character and excites vibrations from low frequencies up to ultrasound frequencies. One of the crucial elements of envelope methods is to select a suitable high frequency region and band-pass filter the accelerometer signal. Some methods suggest that the filter should be centered around a structural resonance frequency. Others recommend regions free from strong tones due to resonances or harmonic components. A second important element is to demodulate or calculate the vibration signature from the band-pass filtered vibration signal.

High frequency resonance techniques use the large amplitude vibration in the neighbourhood of a structural high frequency resonance. After band-pass filtering the vibration signal its time history consists of a series of exponentially decaying impulse responses. Each impulse response consists of a narrowband resonant vibration signal acting as carrier modulated with a series of impacts produced by the defect. This implies that information on the defect, its periodicity etc, is available in the signal modulation. The defect signature is obtained by removing the carrier wave, that is, by extracting the envelope, from the band-passed signal. Figure 14-16 below describes the defect signature extraction procedure schematically.

Vibrationsensor

Band-passfilter

Envelopdetector

Condition indicatorbased on time or

frequency domaininformation

Figure 14-16 Schematic description of defect signature extraction using the high frequency resonance technique. When the residual error signature has been extracted by demodulating the band-pass filtered vibration signal it can be processed in various ways to detect, identify and diagnose the defect. Several indicators exist. In the time-domain, abrupt changes in the envelope can be detected and quantified using rms-value, peak-value, crest factor or statistical measures like Kurtosis. In the frequency domain peaks in the envelope spectrum coinciding with characteristic defect frequencies can identify and localise a defect. A damaged gear tooth can be localised using phase information. Experience shows that phase is more sensitive than amplitude in detecting and localising gear tooth cracks. A frequently used indicator is for instance the severity index defined in section 14.6.3.


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Example 14-4 The details of a defect detection and diagnosis algorithm depend largely on the defect type and the monitored element characteristics. The following example provides some insights into a typical detection and diagnosis algorithm developed for monitoring fatigue cracks in gear teeth. Suppose vibrations are measured at a gear casing and that a tachometer is used to track the pinion speed or rotation frequency. The tachometer signal provides a means to estimate the instantaneous pinion speed and synchronise the sampling with the shaft rotation using a resampling procedure. At least 150 – 200 vibration time histories are acquired covering a single revolution each. This ensemble of single revolution time histories is ensemble averaged to enhance the signal components periodic in the time domain, i e with frequencies, n/Trot, n = 1, 2, 3, … . This set of lines includes the characteristic gear meshing harmonics with side bands. All other signal components are effectively suppressed. The next step is to extract a suitable part of the gear signature. The time domain averaged signal is band-pass filtered around a dominant gear tooth meshing harmonic in the high frequency region. The filtered signal can be approximated with a high frequency amplitude and phase modulated signal. The gear signature is contained in the modulation signal. The presence of the high frequency carrier wave complicates both visual and numerical signature inspections. Therefor the signature is demodulated to remove the carrier wave and keep the modulation signal. Demodulation can be accomplished using any envelope detector. The demodulation procedure provides two modulation functions: one amplitude modulation function, and one phase modulation function, as functions of time or equivalently of pinion rotation angle. The final step is to analyse the modulation signal and search for information relevant to defect detection and diagnosis. This analysis can take place in either the frequency domain or the time domain. A frequency domain analysis is preceeded by a Fourier transformation of the enveloped time domain signature. In the time domain a localised defect such as a crack will impose a sudden change is the signature. It has been demonstrated that this change is most clearly visible, as a phase jump, in the phase modulation function. An automatic detection procedure is implemented using a Kurtosis analysis, which is sensitive to the existence of sudden jumps in the signal. The detection sensitivity is set as a suitable Kurtosis alarm level. Finally, the crack location is determined from the pinion rotation angle at the phase jump. Figure 14-17 gives a schematic description of the detection algorithm.


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Accelerometer signal.Acquire 150-200 complete

pinion revolutions.

Tachometer signal.Synchronised with

pinion speed

Numerical resamplingat rate synchronised

with tachometer.

Synchronisedaveraging

Band-passfilter

Envelopdetector

Kurtosis(Phase) > Alarm level :Look for phase jump

Localisecracked zone

Raw data

Data sampledin synchr withpinion speed

Ampl and phasemodulation functions

Defect indication

Defect localisation

Signal average

Defect signature

Figure 14-17 A gear crack detection algorithm based on high frequency envelope techniques.

14.6 MACHINE CONDITION INDICATORS

Machine condition monitoring requires an easily interpreted and unambiguous condition indicator, or defect severity index, that can be calculated from the extracted vibration signature. A useful indicator should allow defect growth tracking and provide a means to define indicator levels indicating acceptable and unacceptable machine conditions. The following section summarises the machine condition indicators introduced in section 14.5 above. 14.6.1 Rms-value, peak-value and crest factor The first developed condition indicators simply used the rms-value, the peak-value or combinations thereof to detect and measure the severity of a defect. The crest factor, defined as the ratio of the peak-value to the rms-value of the measured vibration, is one popular combination indicator. All of these three indicators are simple, but blunt, and sometimes fail to detect a defect at a sufficiently early stage of development. Tuning of these indicators is accomplished by measuring a large number of healthy elements and using the indicator levels as reference values. By correlating indicator changes to defect growth levels indicating unacceptable conditions, alarm levels or critical levels, can be set.


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15.6.2 Kurtosis Most defects in machinery develop where two surfaces meet in rolling and sliding contact. A normal healthy surface has a small-scaled roughness distribution whose deviation from the nominal ideal surface exhibits a Gaussian normal distribution over the surface. When two healthy surfaces repeatedly interact by a combined sliding and rolling contact the roughness distribution is successively changed. Defects like wear and fatigue cracks have been shown to produce a more narrow and peaked roughness distribution. Assuming that the roughness is directly correlated to the generated vibration, the vibration time history can be statistically analysed to give information on the surface condition.

In statistical analysis, the normalised fourth statistical moment, denoted Kurtosis, is known to be a good measure of the peakedness of the probability density function. Thus, Kurtosis has become a popular peakedness indicator for rougness distributions and is frequently used as a machine condition indicator. The normalised fourth moment, or Kurtosis of a signal v, is defined as,

2

1

2

1

4

))((1

))((1

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−

=

∑

∑

=

=

N

nn

N

nn

vtvN

vtvN

K , (14-8)

where v, in this application is the vibration displacement, velocity or acceleration and overbar – denotes the average value. For healthy rolling element bearings, experience has shown that the Kurtosis is close to 3,0. A defect at early stages has a Kurtosis value in the range 3,5 to 4,0, whereas severly damaged bearings show Kurtosis values in the range 4,0 to 9,0, see [3]. It is also possible to identify the defect type by measuring the Kurtosis and the rms-value of the acceleration. It has been demonstrated [3] that each combination of Kurtosis and rms acceleration value can uniquely identify the defect. The main difficulty in applying the Kurtosis indicator is, as usual, to prepare a vibration history suitable for analysis. The analysed vibration signature must be characteristic for the bearing or gear that is being monitored without significant contributions from other sources. As usual this requires smart instrumentation and signal filtering. 14.6.3 Defect severity index As mentioned in paragraph 14.5.2 the defect severity index measures the severity of the defect by comparing the overall background vibration level with the level at the characteristic defect frequency. One possible definition of a severity index, m, is [4],

filter

analyserLf

fm

Δ

Δ−= Δ )110( 10/ , 0 < m < 1, (14-9)


521

where ΔL is the level difference, Δfanalyser is the frequency linewidth and Δffilter is the bandwidth of the filter used to extract the vibration signature. For bearings experience has shown that a severity index equal to say 15 % is suitable as a treshold value for early defect detection when the high frequent part of the vibration history is used. 14.7 RESIDUAL TIME TO FAILURE ESTIMATION

Most maintenance strategies of today demand replacement of faulty parts as soon as vibration levels increase above the normal. Experience has shown, however, that the time between defect detection and failure in many cases is fairly large. Hence, substantial savings in maintenance costs can be achieved if the time to replacement can be prolonged. This life time extension can be accomplished by using estimates of the residual time to failure. Residual time to failure estimates can be based on analytical models or empirical models using statistical data from large data bases collected for various machine elements and failure modes. Most residual life time estimation methods use mathematical models for fatigue life prediction in combination with stochastic models for cumulative damage or trending analysis. The experience on the use of these methods is, however, still limited. One of the main problems is that the methods available for fatigue life prediction are developed for simple elements, not for elements in complex built-up structures. An example of this type of model is Paris’ law, developed in fracture mechanics, which can be applied to model the growth of a fatigue crack in, for instance, a gear tooth. A fair conclusion is that the present methodology has not reached the point for practical use. Much research effort is still needed before any residual time to failure estimation can be performed with confidence. 14.8 MEASUREMENT TECHNIQUES

Machine monitoring can be performed either off-line or on-line. On-line monitoring means that the monitoring is performed in real-time, i e on fresh, just-acquired data, whereas off-line monitoring is performed on data stored on a secondary medium such as a tape. Note also that on-line systems can be designed to be either remote or on-site controlled. Regardless of what philosophy is used, the vibrations must be sensed and acquired in a proper way. 15.8.1 Instrumentation In principle, only a vibration signal is required for machine monitoring. It has been found, however, that tachometer signals, synchronised with various axle speeds of interest, give valuable information for the vibration signal processing.

Since machine monitoring requires high quality vibration signals, it is recommended to spend some time on selecting suitable sensors and sensor locations. In general, the sensors should be placed as close to the monitored element as possible. Also, the sensor should be oriented in a direction where the vibration from the monitored element


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is most significant. A short distance and a suitable orientation is crucial to ensure that the relevant vibration signature can be identified from the measured vibration. The speed of all shafts engaged in the monitored element should be measured using tachometers. A tachometer giving several synchronised pulses each shaft rotation is preferred since it provides a more accurate shaft speed determination. A typical remote controlled on-line monitoring system is designed in the following way, see Figure 14-18. The sensor signals are acquired and processed by a digital signal processor (DSP) placed close to the machine. Normally the data acquisition takes place according to a predefined scheme, for instance 15 minutes record time once every day. Within the DSP the signals are sampled and processed to produce the necessary monitoring information. The purpose of the processing is to reduce the amount of data transferred to the host computer. Finally the reduced vibration data is transferred to the host computer for presentation to the operator by GSM.

Accelerometer

Signal conditioner

Tachometer

Speed

Samplingfrequency forsynchronised

sampling

A/D converter

Resampler

Synchronisedaveraging

Band pass filter Operator

Envelop detector FFT Cepstrum • • •

Indicator(Alarm levels)GSM

Remote monitor

DSP

Figure 14-18 A schematic picture showing a remote controlled on-line monitoring system. The data storage and

computation intensive parts are preferrably performed within a Digital Signal Processor (DSP).


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14.8.2 Data acquisition The most important signal acquisition system parameters are the frequency resolution, the bandwidth and the dynamic range. A successful monitoring often requires high frequency bands measured with very high resolution in order to resolve and identify the vibration signature components. Also the high frequency components are often weak as compared to the low frequency components. In some cases the energy in the band of interest may be 1000 times lower than the low frequent energy. Accurate results in these cases require a data acquisition system with sufficiently high dynamic range. Both these problems are particularly severe for low speed bearings, [1]. 14.8.3 Signal filtering Very often the signal quality can be significantly improved by signal filtering. A filter tuned to remove as much as possible of the unwanted signal components is in fact usually the key to a successful detection of an incipient fault. The filter tuning makes it possible to adapt a defect detection system to a specific machine by taking advantage of, for example, the existence of a structural eigen-frequency. In practice it has proven useful to band-pass filter the signal through a filter with certain frequency limits. 14.8.4 Normalised order analysis As explained in section 14.4 the characteristic gear and bearing frequencies are directly proportional to some shaft rotation frequency frot. The proportionality constant depends on the element design parameters, such as the number of rolling elements or gear teeth. Since the shaft rotation frequency normally varies significantly during the data acquisition, it is common practice to work with vibration history data with time normalised with the shaft rotation period or with vibration spectra with the frequency axis normalised with shaft rotation frequency. Spectra normalised in this manner are denoted normalised order spectra, see Figure 14-6. In contrast to ordinary frequency spectra the vibration signature peak positions in normalised order spectra are speed invariant. The peaks in an ordinary spectrum will shift in frequency in correspondance to the change in shaft speed. This invariance property simplifies interpretation and facilitates unambiguous ensemble averaging. 14.9 USER INTERFACE

An easily interpreted presentation of the results is another important issue for machine monitoring. A good alternative is to indicate a healthy machine with, say, a green light signal and a detected defect with a red light signal. When the detected defect signal is positive, the operator can investigate the details of the condition indicators and diagnose the defect. Suppose a gear is being monitored. When a local defect is localised it is informative to correlate the residual error signature to the geometry of the gear. Since the signature is periodic with a rotational period T it can be plotted in a polar diagram with one period, or revolution, corresponding to 360 degrees in the polar plot. The defect then


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appears as a distinct peak at a specific angle in the polar diagram. In a sequence of polar signature plots synchronised with the defect rotation, the defect will be localised at the same relative phase angle. The angular location of the peak corresponds to the position of the defect zone relative to the trigger position of the tachometer, see Figure 14-19. Another useful facility in modern computer systems is the possibilty to produce sound signals from given signatures. The human ear is in fact very sensitive to sudden irregularities in the signature. Thus, listening to the residual error signature very often proves to be the most sensitive defect indicator. The sudden changes in signature amplitude and phase then appear as a beating sound.

a)

2,4975

4,9951

7,4926

9,9901

30

210

60

240

90

270

120

300

150

330

180 0

b)

2.4975

4.9951

7.4926

9.9901

30

210

60

240

90

270

120

300

150

330

180 0

Figure 14-19 Polar diagrams of time domain averaged signatures are useful for visual inspection and interpretation. This example shows a time domain signature covering one revolution, 360 degrees rotation, of a pinion shaft. Each peak in the polar plot corresponds to a specific tooth going into meshing. Hence, in this example, the pinion has 20 teeth. If the tachometer signal is triggered at 0 degrees rotation, close to tooth number 1, the diagram in b indicates a localised defect, such as a crack, at tooth number 2. In order to avoid false alarms due to environmental disturbances the time domain averaged signatures should be ensemble averaged. A real defect is then enhanced since it occurs at the same angular position in each signature. In this case Kurtosis, indicating the presence of sudden peaks, could be used as a defect indicator.

14.10 SIGNAL PROCESSING TOOLS

In section 14.4 it was explained that useful information on the gear or bearing condition is available in the residual error signature. In section 14.8, techniques for non-intrusive measurement of these vibrations were discussed. When the vibration and tachometer signals have been acquired the next step is to process the signals to extract the desired signatures for each machine element.

The choice of analysis tool is largely dependent on the properties of the monitored element. The standard technique for bearing monitoring is the high frequency envelope


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technique. Gear monitoring, on the other hand, often involves order tracking or synchronised time domain averaging. Exercise 14-2 Explain why time domain averaging is not as useful for extracting bearing signatures as for gear signatures. REFERENCES

1 A Barkov, N Barkova and A Azovtsev Peculiarities of Slow Rotating Element Bearings Condition Diagnostics. http://www.vibrotek.com/articles

2 P D McFadden 1987 Mechanical Systems and Signal Processing 1 (2), p 173-183,

Examination of a Technique for the Early Detection of Failure in Gears by Signal Processing of the Time Domain Average of the Meshing Vibration.

3 H R Martin and F Honarvar 1995 Applied Acoustics, 44, p 67-77, Application of

Statistical Moments to Bearing Failure Detection. 4 A Barkov and N Barkova Condition Assessmentand Life Prediction of Rolling

Element Bearings – Part 1. http://www.vibrotek.com/articles.


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Chapter 15: Vehicle Noise and Vibration

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CHAPTER 15

VEHICLE NOISE AND VIBRATION Road vehicles are the major source if environmental noise worl wide. Manufacturers of cars and trucks have to meet ever stricter legislative noise limits in order to be able to sell their products. Interior noise and vibration issues are also important sales arguments. This chapter gives the fundamentals of road vehicle noise and vibration. It begins with a section on legal demands and acoustic standards and measurement techniques. The major sources of noise and vibration are discussed as well as noise control techniques. Finally some advanced simulation tools are briefly discussed.

15.1 MOTIVATION FOR VEHICLE NOISE AND VIBRATION CONTROL

Vehicle manufacturers work with noise and vibration control to fulfill legistlation demands and to meet customer requirements. International and national legislation puts increasingly strict demands on the allowable noise level for road vehicles. The exterior noise level is measured using the standardized ISO 362 drive-by test, which will be descibed in section 15.3. The exterior noise control work is mainly motivated by legislation demands while interior noise and vibration control work is motivated by driver and passenger noise and vibration comfort requirements. The motivation for reducing traffic noise is that it is the most important environmental noise source in Europe and in the rest of the world. According to [1] 25 % of the population in Europe is exposed to transportation noise with an equivalent sound level over 65 dB(A). At this sound level


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sleep is seriously disturbed and most people become annoyed [2]. The same study [1] showed that in four European countries (France, Germany, Great Britain and the Netherlands) 20-25 % of the population are disturbed by road traffic noise and 2-4 % are disturbed by railway noise. In an OECD report [3] the social cost of transport noise is estimated to be 0.4 % of the GNP of the OECD countries.

Figure 15-1 25 % of the European population is exposed to equivalent sound levels above 65 dB(A)

and get there sleep disturbed. (SCANIA, Fordonsakustik och buller) The relative importance of interior and exterior noise for road vehicles depends on the type of vehicles and where in the market segment they are located. All vehicles have to meet the legislative demands according to trhe standardized pass-by-noise test. The importance of interior noise depends on which type of customer the company in question is selling to and the prize level. If the customer is a private person buying a car there is a much higher tolence for noise when looking at a low prize car compared to an expensive car. Trucks and buses are usually sold to companies who may have their own demands on driver or passenger comfort. These demands may also be different for different countries or different parts of the world.

15.2 CHARACTER OF VEHICLE NOISE

Exterior road traffic noise results from the combined contributions from a large number of different vehicles. The noise generation from these vehicles vary depending on their type and how they are operated. Trucks are typically noisiest followed by buses and motorcycles while cars are the quietest. The contribution of cars to the overall traffic noise level is however great because of their large numbers. Cars account for about 80% of the road traffic [4]. The individual vehicle noise also depends on how it is operated especially the speed and load. For lower speeds, below 40-50 km/h, engine noise including exhaust and intake noise dominates for cars. For higher speeds, above 70 km/h, tyre-road noise dominates the car extrerior noise generation. For heavier vehicles the


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engine noise is dominat under most conditions. A typical engine noise spectrum is shown in figure 15-2 showing high levels from around 50 Hz to 3 kHz. Typically there are peaks caused by engine harmonics and resonances in the exhaust or intake system or body around 50-300 Hz. The level is then fairly constant up to around 3 kHz where it starts to fall off. Heavier vehicles with Diesel engines typically have a more low frequency character compared to personal cars with petrol engines. The individual noise sources are discussed in more detail in section 15.4.

Figure 15-2 Typical road vehicle exterior noise spectra measured using ISO 362.

Internal vehicle noise is caused by airborne and structure borne noise from the engine and tyre-road interaction noise. If exterior noise is reduced interior noise will typically also be reduced. This especially true for airborne engine noise and tyre-road noise.


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15.3 MEASUREMENT OF EXTERIOR VEHICLE NOISE

The highest allowable levels according to the EU directive 96/20 EC is shown in Table 15.1. Tabel 15-1 Highest allowable sound level for road vehicles according to the EU directive 96/20 EC.

Type of vehicle Sound level dB(A)

Personal car 74 Bus and truck with total weight below 3,5 ton with total weight below 2 ton

76

with total weight above 2 ton but below 3,5 ton 77 Bus with total weight above 3,5 ton with engine power below 150 kW

78

with engine power 150 kW or above 80 Truck with total weight above 3,5 ton with engine power below 75 kW

77

with engine power below between 75 kW and 150 kW

78

with engine power above 150 kW 80 Motor cycle (depending on cylinder volume) Moped

The measurements are to be made according to the international standard ISO 362. This the so called pass-by-noise test where the vehicle is driven past a microphone on 7.5 m distance from the road, see figure 15-3. The vehicle vehicle must be accelerated using full throttle from the start of the test section, A-A, until the end, B-B, starting at a specified speed. There must not be any disturbing objects within a 50 m radius.

Figure 15-3 Measurement of exterior vehicle noise according to ISO 362, where the vehicle is driven past a

microphone mounted 1.2 m above ground and 7.5 m from the center of the test road. ( SCANIA, Fordonsakustik och buller.)


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15.4 VEHICLE NOISE SOURCES

Figure 15-4 shows the main noise sources on a road vehicle. The engine and exhaust and intake noise dominates as far as the result of the pass-by noise test is concerned. Tyre-road noise is of importance for environmental disturbances at higher speeds. In the following the different noise sources are discussed in greater detail.

Figure 15-4 Main vehicle noise sources. [4]

15.4.1 Engine noise

Figure 15-5 shows the main sources of engine noise as established by e.g. Priede [5]. The two main types of noise sources are combustion-induced noise and mechanical noise.

Figure 15-5 Main engine noise sources. [4]

Force generaton Vibration transmisson

Radiators of noise

Air/fuel -mixing Ignition delay Heat Release Cylinder pressure

Gas forces

Inertia forces

Fuel injection

Valve-gear Torques

Cylinder pressure pulses

Piston slap impacts

Timing gear impacts

Cylinderhead

Piston, Connection rod and crankshaft

Cylinder wallsrWater jacket

Front of Crankcase

Rocker cover Intake Manifold

Crankcase panels

Crankshaft Pulley

Water Jacket

Side covers Sump

Timing cover


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Combustion noise is caused by the periodic variation of cylinder pressure acting on the engine structure. Mechanical noise is caused by piston slap impacts and and other mechanical impacts in gears and bearings. Combustion noise is more pronounced for diesel engines where the cylinder pressures are high. The cycle in a Diesel engine can be divided into for steps: air intake, compression, fuel injection and combustion, exhaust. Figure 15-6 below show the different parts of a Direct Injection Diesel engine operation cycle. A typical figure showing cylinder pressure versus crank angle for the compression and combustion parts of the engine cycle is shown in figure 15-7.

Step one: Air intake, the piston moving down while taking in air.

Step two: Compression, The piston is now moving up and compresses the air

Step three: Fuel is injected and ignites by the pressure and heat.

Step four: Exhaust, the piston moves up and presses out the emissions.

Figure 15-6 The four steps of a typical four-stroke Diesel engine. [8]


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Figure 15-7 Typical Diesel Engine Pressure Trace showing x: start of injection, y: start of combustion, 1:

ignition delay, 2: uncontrolled combustion, 3: controlled combustion, 4: combustion during expansion stroke [4] Noise generation strongly depends on two parameters: how abrupt the pressure rise is, and pressure peak rate and width. The noise generated is also strongly dependent on the structure attenuation which is the difference between cylinder pressure and sound pressure at an outside measurement point. The characteristics of the engine structure is represented by the attenuation curve. The typical shape of the cylinder pressure frequency spectrum for four different engine speeds is shown in figure 15-8. The level decreases with frequency but the decrease rates vary with engine type in the range 30-60 dB/decade. In the figure to the left sound pressure level is plotted agains frequency while the figure on the right shows sound pressure level plotted against engine orders, i.e., frequency normalized by the engine speed expressed in Hz. It can be seen that the main effect of changing the engine speed is a frequency shift,


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Figure 15-7 Typical shape of cylinder pressure frequency spectrum. [4] Since mechanical noise is caused my mechanical impact in the piston and in gears and bearings it is much more difficult to locate and analyse. The most common sources of mechanical noise are:

• Piston slap which is generated when the piston moves from side to side during the engine cycle.

• Timing gear rattle which is generated when the teeth of the gears impact. It is coupled to torsional vibrations of the crankshaft.

• Bearing noise which is generated by impact and vibration caused by fluctuating forces during the engine cycle.

The main engine surfaces and components from which engine noise is radiated are: the crankcase, the oil sump, the front timing cover, the cylinder head and the gearbox. The crankcase and cylinder block have vibration modes which in combination with the radiation efficiency of the surfaces determines the noise generation. The fist bending mode of the engine block can for large engines be around 200 Hz. The oil sump is attached to the bottom of the crankcase where vibration levels are high, often have a large radiating area and can have structural resonances around 1 kHz. The front timing cover and rocker cover are also attached to parts of the cylinder block with high vibration levels and can contribute significantly to the noise radiation above a frequency of 1 kHz. The gearbox can generate noise caused gear meshing impacts but also radiate engine noise.

15.4.2 Exhaust and intake noise

Exhaust and intake noise is caused by the periodic operation of the engine. Each time the exhaust port opens to the exhaust manifold a high pressure pulse is emitted. This means that strong engine harmonics are generated. The fundamental frequency will for a n cylinder four stroke engine running with speed N rpm be

6020 ⋅⋅

=nNf , (15-1)

where N/60 is the engine speed expressed in Hz and the division with two has to do with the fact that the crankshaft makes two rotations per engine cycle. Exhaust and intake


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noise spectra are often expressed in engine orders where the frequency has been normalized by N/60, the engine speed expressed in Hz. For a four-cylinder engine the 2nd order would therefore be the fundamental and for a six-cylinder engine it would be the 3rd order. A typical exhaust noise spectrum is shown in figure 15-9. This example is a 6-cylinder diesel engine running at 1600 rpm and it can be seen that the fundamental frequency according to 15-1 is the dominating component. The dominating components are at 8026016003 =⋅⋅ Hz and the 3rd order. The sound pressure level of the engine harmonics decreases for higher frequencies. Above 500-1000 Hz broadband flow generated noise typically dominates the spectrum, especially on the exhaust side. This effect can not be seen in figure 15-9 since signal proessing has been used to extract the periodic part of the spectrum in this case.

Figure 15-9 Typical shape of exhaust noise spectrum.

15.4.3 Cooling system noise

Cooling fan noise can be an important source of vehicle noise even though it is normally only operated during part of the vehicle journey. Especially for heavy vehicles where large air flows are needed cooling fan noise may be significant. Axial fans are the dominating type of fan used even though centrifugal and mixed flow fans may be considered in special cases. Fan noise typically increases with fan-tip speed and fan power. They should therefore be run at the lowest speed possible considering the required air flow. The fan noise typically increases linearly with volume flow rate and approximately proportionally to the square of the rotational speed.

15.4.4 Tyre-road noise

Tyre-road noise is the dominating external sound source at higher speeds, above 70-100 km/h. The main parameters which affects tyre noise are: vehicle speed, vehicle weight, tyre properties and road surface properties. Sound level measured during pass-by noise testing increases proportional to the engine speed raised to a coefficient determined by the tyre and road surface characteristics. An increase in vehicle weight causes an increase in


536

tyre-road noie. The tyre structure, tread pattern, tread material and tyre wear all influence the noise level. The road surface texture is also very important which means that this is not a problem which can be solved by vehicle manufacturers on their own. Recently interesting tests have been made with absorbing road surfaces which gave a substantial reduction of tyre-road noise. When introducing new road surface materials factors like wear, clogging of pores and water drainage will off course have to be considered. The main sources of tyre-road noise are: air pumping and tyre vibration. When the tyre is in contact with the road surface quantities of air may be trapped in the tread groove and then released. The pressure fluctuation caused in this way is called air pumping. Radiation of noise from tyre wall vibration is a major source of tyre noise. Since these vibrations are caused at the tyre road contac patch the details of tyre and road surface characteristics are important for the noise generation.

Figure 15-10 Tyre road interaction noise at 800 Hz for a car driving at 90 km/h.

15.4.5 Aerodynamic noise

At higher vehicle speeds turbulent boundary layer noise is important for interior noise but can also contribute to the exterior noise level. Flow separation can also occur at objects sticking out from the body, like rear view mirrors, giving rise to additional noise generation. Figure 15-11 Simulation of the flow field around a car causing aerodynamic noise generation at high vehicle speeds.


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15.5 VEHICLE NOISE AND VIBRATION CONTROL

15.5.1 Engine noise control

Engine noise is, as briefly described in section 15.4.1, caused by various types of force generation within the engine and is transmitted to the radiating outer surfaces. The transmission path properties is determined by the vibration modes of the structure. The properties of the outer surface will also influence the sound radiation. There are therefore a number of ways in which the final sound radiation may be influenced:

• Reduction at the source of combustion forces and mechanical forces. • Reduction of the vibration transmission between the sources and the outer

surface. • Reduction of the sound radiation of the outer surface.

Reduction of combustion pressures is intimately coupled to changes in the combustion process or combustion chamber shape. Since any changes to the design of the combustion chamber or to the combustion process will also have an effect on engine performance and exhaust gas emissions this is a difficult path for a noise control engineer. Unfortunately most design changes which would reduce noise would also increase exhaust emissions. Piston slap can be reduced by redesign of the piston and cylinder or by oil film injection. Also in this case a number of other engine performance considerations will have to be made. Gear and bearing noise can be reduced by improved design of these components for instance attention to gear tooth profiles and bearing clearances. Manufacturing cost and componenr performance will certainly be an issue in these cases. To reduce the transmission of vibrations to the engine outer surface the crank case and cylinder block can be redesigned. One example is strengthening and stiffening of the structure. Other more advanced redesigns can be made involving extensive simulation the dynamics using finite element moelling. The oil sump and other covers used on the engine usually make a substantial ontribution to the radiated noise. The vibration of these covers can be reduced by using damped materials, such as mpm plates. The vibration transmission can be reduced by isolating the cover rom the main engine structure by vibration isolation at the coupling points, see figure 15-9 for some example designs.


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Figure 15-12 Examples of vibration isolation techniques for engine covers. [4]

The above mentioned noise reduction techniques can usually only be realized if the noise control engineer is involved early in the engine design process. If the noise problem must be fixed at a later stage shields, enclosures and engine compartmemt absorption may be used. Encapsulation of the engine is an effective noise control technique but complete encapsulation is not possible because of engine cooling and availability for maintenance. Highly radiating surfaces can be covered with close-fitting radiation shields with a low radiation efficiency compared to the engine surface. They should be resiliently mounted and must be well sealed. A technique to reduce the noise transmitted to the outside is to put absorbing material on the walls of the engine compartment. Porous materials has been used for this purpose but there is a problem with contamination of the material by dirt and oil. An interesting alternative which has been introduced recently is to use micro-perforated metal sheets as sound absorbers. Figure 15-13 shows an example of a micro-perforate used for engine compartment absorption.

15.5.2 Exhaust and intake noise control

Exhaust and intake system noise originates, as explained in section 164.2, from the pressure pulsations caused by the operation of the engine and additional flow generated noise. To control the noise generation at the source involves making changes to the combustion process, combustion chamber or exhaust or intake valves which also influences engine performance and exhaust gas emissions. Instead one often use mufflers or silencers which are devices used in a flow duct to prevent sound from reaching the openings of the duct and radiating as far-field sound. Reactive silencers do this by reflecting sound back towards the source while absorptive silencers attenuate sound using absorbing material. They are necessary components in the design of any exhaust or intake system for internal combustion (IC) engines. No car or truck can pass the standard noise tests


539

required by legislation or compete on the market without them. There are three basic requirements for a modern exhaust systems; compact outer geometry, sufficient attenuation and low pressure drop. The aim of this section is to discuss acoustical design and analysis of IC-engine exhaust and intake systems. Most emphasis is put on exhaust system design. Modern intake systems made from plastic material with non-rigid walls are are not discussed because not much has been published on this subject. The theory and techniques presented can be used also for other applications such as compressors and pumps and to some extent also for air-conditioning and ventilation systems. Two different physical principles are used for sound reduction in mufflers. Sound can be attenuated by the use of sound absorbing materials in which sound energy is converted into heat mainly by viscous processes. Typical sound absorbing materials used are rock wool, glass wool and plastic foams. To force the exhaust flow through the absorbing material would create a large pressure drop so the material is usually placed concentrically around the main exhaust pipe, see figure 15-10. To protect the absorbing material and prevent it from being swept away by the flow a perforated pipe is usually inserted between the main pipe and the absorbing material. Sometimes a thin layer of steel wool is included for additional protection. In some cases the outer chamber containing the absorbing material is flattened because of space limitations in fitting the muffler under a car, see figure 15-11.

perforated sheet steel mineral woll

steel woll

oo

oo

oo o

oo

oo

oo

oo

Figure 15-10 Typical exhaust system absorption muffler [8].

Figure 15-11 Absorption muffler of flat oval type [8].

z

y

x

xrϕS

1

2

S

z = 0 z =

Γ1

2Γ

inlet outletcentral passage

lining

n

n

mineral wool

steel wool


540

The other physical principle used is reflection of sound which is caused by area changes or use of different kinds of acoustic resonators, see figure 15-12. These types of mufflers are called reactive. If the acoustic energy is reflected back towards the source then the question is what happens with it once it reaches the source. It could of course be that the source is more or less reflection free but this is not usually the case. If multiple reflections in-between the source and the reactive muffler occurs the sound pressure level should build up in this region and cause an increase further downstream too. The answer to this apparent paradox is that a reactive muffler when properly used causes a mismatch in the acoustic properties of the exhaust system and the source to actually reduce the acoustic energy generated by the source.

VSn

a)

b)

ng

Figure 15-12 Different types of resonators in a typical reactive automobile silencer; a) λ/4 resonator and b) Helmholtz resonator [8].

There are also cases where resistive and reactive properties are combined in the same muffler element. All reactive muffler elements do in fact cause some loss of acoustic energy in addition to reflecting a significant part of the acoustic energy back towards the source. The losses can be increased for instance by reducing the hole size of perforates especially if the flow is forced through the perforates. Figure 15-13 shows some typical perforate muffler elements, where especially the cross flow and reverse flow type have a significant resistive as well as reactive character.


541

c)

b)

a)

Figure 15-13 Different types of perforated muffler elements having both reactive and resistive character; a) through flow, b) cross flow and c) reverse flow [8].

In order to assess the success of a new muffler design there is a need for measures to quantify the sound reduction obtained. There are at least three such measures in common use: transmission loss, insertion loss and noise reduction. The transmission loss is defined as the ratio between the sound power incident to the muffler (Wi) and the transmitted sound power (Wt) for the case that there is a reflection free termination on the downstream side

( )ti WWTL log10 ⋅= . (15-2) This makes it difficult to measure transmission loss since an ideal reflection free termination is difficult to build especially if measurements are to be made with flow. There are measurement techniques2,3 which can be used to determine transmission loss by using multiple pressure transducers upstream and downstream of the test object. It is also necessary to make two sets of measurements either by using two acoustic sources, one downstream and one upstream of the test object, or by using two different downstream acoustic loads. The advantage of using transmission loss is on the other hand that it only depends on the properties of the muffler itself. It does not depend on the acoustic properties of the upstream source or the downstream load. Transmission loss can therefore also be calculated if the acoustic properties of the muffler is known without having to consider the source or load characteristics. Since the transmitted sound power


542

can never be larger than the incident the transmission loss must always be positive. A high transmission loss value tells us that the muffler has the capacity to give a large sound reduction at this frequency. It will not tell us how big the reduction will be since this depends on the source and load properties. Insertion loss is defined as the difference in sound pressure level at some measurement point in the pipe or outside the opening when comparing the muffler element under test to a reference system

( )rm ppIL ~~log20 ⋅= , (15-3) where mp~ is the rms-value of the sound pressure for the muffler under test and rp~ is the rms-value of the sound pressure for the reference system. It is common that the reference system is a straight pipe with the same length as the muffler element under test but it could also be a base-line muffler design against which new designs are tested. Insertion loss is obviously easy to measure as it only requires a sound pressure level measurement at the chosen position for the two muffler systems. It does however depend on both upstream acoustic source characteristics and downstream acoustic load characteristics. Insertion loss is therefore difficult to calculate since especially the source characteristics are difficult to obtain. Methods for determining source data will be further discussed in section 82.6. Insertion loss has the advantage that it is easy to interpret. A positive value means that the muffler element under test is better than the reference system while a negative value means that it is worse. Sound reduction is defined as the difference in sound pressure level between one point upstream of the muffler and one point downstream

( )du ppSR ~~log20 ⋅= , (15-4) where up~ is the rms-value sound pressure upstream of the muffler and dp~ is the rms-value of the sound pressure downstream of the muffler. Just as insertion loss sound reduction is easy to measure but difficult to calculate since it depends on source and load properties. The interpretation is less clear compared to transmission loss and insertion loss. It does tell us the difference in sound pressure level over the muffler for the test case but the result may depend heavily on where the measurement positions are placed. Development of computer programs for acoustical design and analysis of flow ducts can be said to have started in the 1970's, although some codes were certainly in existence earlier. The low frequency region is usually most important for IC-engine applications. This means that a one-dimensional or plane wave approach is sufficient for the main exhaust and intake pipes. Most of the codes developed have used the so called transfer matrix method which is described below. This is a linear frequency domain method which means that any non-linearity in the sound propagation caused by high sound pressure levels is neglected. Local non-linearity at for instance perforates can however be handled at least approximately. The assumption of linear sound propagation has experimentally


543

been shown to be reasonably good. Numerical methods such as FEM can also be used to solve the linear equations. They are especially useful for mufflers with complicated geometry and large cross-sectional area where the plane wave propagation is no longer sufficient to describe the sound propagation in the frequency range of interest, but do increase the complexity of the calculations significantly. FEM should therefore only be considered as complement to the analytical methods for cases where they fail. Non-linear time domain techniques such as the method of characteristics or CFD techniques have also been suggested. They are usually linked to a non-linear model of the gas exchange process of the engine and are not really adapted for modelling muffler components even though some interesting attempts have been made. There are commercial codes available for simulating the engine gas exchange process which are in use by the automotive industry. They are an interesting alternative for obtaining information about the engine as an acoustic source. The transfer matrix method is an effective way for the analysis of sound propagation inside a duct network, especially if most of the acoustic elements are connected in cascade. The exhaust system of an internal combustion engine does in many cases have this kind of transmission line character. The method which is often referred to as the 4-pole method, was originally developed for the theory of electric circuits. In order to get a complete model accurate for analysis and design of exhaust systems, we must also take the influence of the sound source and the termination of the system into account. That is the sound generation and acoustic reflection characteristics of the engine as an acoustic source and the sound reflection and radiation characteristics of the termination. Using the assumptions of linearity and plane waves, the actual physical system with engine, exhaust system and outlet can be described by a sound source, transmission line and acoustic load, see figure 15-14.

Figure 15-14 IC-engine equipped with muffler, and its acoustic representation.


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The source is fully determined by the source strength Ps, and the source impedance Zs which reveals how the source reacts to an arbitrary outer load such as an exhaust system. The load is described by a termination impedance. The source data and load data will be discussed further below. Three basic assumptions concerning the sound field inside the transmission line are made in the transfer matrix method. First, the field is assumed to be linear, i.e. the acoustic pressure is typically less than one percent of the static pressure. This allows the analysis to be carried out in the frequency domain and transfer function formulations can be used to describe the physical relationships. The assumption of linearity does not however mean that no non-linear acoustic phenomena inside the system can be modelled. Some local non-linear problems can for example be solved in the frequency domain by iteration techniques. The second assumption requires that the system within the black box is passive, i.e. no internal sources of sound are allowed. Finally, only the fundamental acoustic mode, the plane wave, is allowed to propagate at the inlet and outlet sections of the system. Provided the above mentioned assumptions are valid there exists a complex 2 2× matrix T, one for each frequency, which completely describes the sound transmission within the system.

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

2

2

2221

1211

1

1ˆˆ

ˆˆ

VP

tttt

VP (15-5)

where 1P and 2P are the temporal Fourier transforms of the acoustic pressures and 1V

and 2V are the temporal Fourier transforms of the volume velocity at the inlet and the the outlet, respectively. The major advantage with the transfer matrix method is the simplicity with which the transfer matrix for the total system is generated from a combination of subsystems, each described by its own transfer matrix, see figure 15-15.

A 1 A 2 A3 A 4 A 5

Figure 15-15 Exhaust system modelled with the transfer matrix method [8].

The transfer matrix for a number of elements A A A1 2, , ,… N connected in cascade is obtained from repeated matrix multiplication.

T A==∏ jj

N

1

(15-6)


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This is a procedure which for long transmission lines is much more effective than solving a large system of equations as will be the alternative if the mobility matrix formulation is used. The division of the total system into more easily analysed subsystems can be done in many different ways as long as the coupling sections between the elements fulfil two conditions. First, there must be continuity in acoustic pressure and volume velocity. This is achieved by choosing a suitable formulation of the transfer matrix, where the effects of discontinuities are included within the described element. Second, the coupling sections must not allow any higher order modes to propagate. This condition implies that the allowed frequency range for the classical transfer matrix method has an upper limit which coincides with the cut-on frequency for the first higher order mode in the coupling section. With modal decomposition the number of modes can easily be extended by increasing the dimension of the transfer matrix and accordingly the frequency range14. Once the division of the system into acoustical elements has been done the final task is to generate the total transfer matrix. This is done in analogy with the theory of electric circuits, i.e. by regarding the system as a network of cascade or parallel coupled elements. In exhaust systems, most of the elements are usually connected in cascade and the transfer matrix formulation is therefore especially powerful for this application. To illustrate the applicability of the linear frequency domain analysis of a silencer for a small 2-stroke IC-engine with varying displacement is presented below. This analysis constitutes one step in the design of an improved silencer for a portable rock drill machine. Due to the complexity of the engine it was not possible to obtain any reliable acoustic source data and the analysis is restricted to transmission loss. This may seem as a major drawback but the guidance given by this analysis is still useful in many practical design applications. As all measurements were made at room temperature and pressure (using a loudspeaker source), so are the simulations. The Mach number is at typical running conditions less than 0.05 and therefore all mean flow effects are neglected. The silencer is divided into acoustic elements as shown in figure 15-16 and specified in Table 15-2.

1

234

56 789

Figure 15-16 Geometry and acoustic elements of the reactive rock drill silencer. [8]


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Element no. Type Data [cm, cm2 ] Comments

1 straight pipe = + =3 5 1 1 5 6. . , .* S *end correction from (71)

2 λ/4 resonator = = ℜ =3 6 18 2 1. , . ,Sr no yielding walls

3 straight pipe = =1 7 18 2. , .S

4 λ/4 resonator = = ℜ =1 7 15 7 1. , . ,Sr

5 straight pipe = + ⋅ =4 6 2 0 4 2 5. . , .* S *end corrections from (71)

6 λ/4 resonator = = ℜ =3 4 15 7 1. , . ,Sr

7 straight pipe = =0 6 18 2. , .S

8 λ/4 resonator = = ℜ =2 5 15 7 1. , . ,Sr

9 straight pipe = + =10 6 0 4 2 5. . , .* S *end correction from (71)

Table 15-2 Type and data for the elements which has been used to model the 2-stroke IC-engine silencer. The length given for each element is the actual geometrical length, i.e. no end corrections apart from those specified

are added. [8] From these elements connected in cascade from inlet to outlet (element no 1 to 9), the transfer matrix and the corresponding transmission loss for the silencer has been generated. The transmission loss has also been measured using the 2-microphone method and comparison between theory and experiments is thus possible.

45

40

35

30

25

20

15

10

5

0 500 1000 1500 2000

TL (dB)

Frequency (Hz)

Figure 15-17 Calculated and measured (∗) transmission loss for the rock drill silencer. [8]


547

As seen from figure 15-17 the agreement between calculated and measured transmission loss is reasonably good. The discrepancies which occurs for higher frequencies are most likely due to near field effects which as seen from the table above either have been neglected or estimated from a similar geometry. Especially the effects in the region between elements 5-9 where the "flow" is reversed, is difficult to estimate. As the inlet and outlet to this region, i.e. element no 5 and 9, are located eccentrically and further also on "opposite sides", the geometry is well suited to excite the first higher order mode of the main cross-section and increased accuracy thus requires 3-D analysis. Nevertheless, the simulation is still useful revealing the poor attenuation below 500 Hz which, regarding the character of the exhaust noise emitted from most IC-engines, shows that this prototype needs to be improved.

15.5.3 Interior noise and vibration control

Interior noise control work is motivated by driver and passenger sound and vibration comfort. Sound level in dB(A) is therefore not a good measure for interior noise. Instead sound quality measures such as loudness or harshness are used, see section X.X. Interior noise in road vehicles is often dominated by engine noise. At higher speeds a large contribution also comes from aerodynamic, turbulent boundary layer, noise. Other sources of interior noise are tyre-road noise, exhaust noise, intake noise, cooling system noise ande ventilation system noise. Engine noise either enters the cabin as airborne sound or structure borne sound. The wall between the engine compartment and the cabin is the key to reducing airborne sound transmission. It is typically made from several layers with absorbing material towards the engine compartment. It is also very important that the sealing beween engine compartment and cabin is good to avoid sound leakage. Structure borne sound transmission can be reduced by vibration isolation and damping of vibrating structures. The resulting sound field in the cabin is determined by the acoustic modes of the passenger cavity. When one of the first engine harmonics coincides with an acoustic cavity resonance an intense low frequency sound is sometimes generated called a cabin boom. To reduce the cabin resonances absorbing trim material can be used on the cabin walls and ceiling. The amount of absorbing material will however have to be balanced to not have a negative effect on cabin sound quality.


548

Experimental methods

• Interior noise measurement and analysis. • Methods for analysis of airborne and structure-borne sound transmission. • Use of on-road and static tests; use of engine noise simulator.


549

15.7 References 1. J. Lambert and M. Vallet. 1994 INRETS, LEN report no. 9420..Study related to the

preparation of a communication on a future EC noise policy. 2. B. Berglund and T. Lindvall ed. 1995 Community noise. Document prepared for the

World Health Organization. Archives of the Center for Sensory Research, vol. 2, issue 1.

3. OECD. The social cost of land transport. Environment monograph N32. Paris. 1990. 4. P. Nelson ed. Transportation Noise Reference Book. Butterworth & Co. Ltd., 1987. 5. T. Priede. In Search of Origins of Engine Noise – an Historical Review. SAE

Congress, Detroit, Paper 800534 (1980). 6. www.howstuffswork.com 7. P.M Sagdeo “Frequency spectra of Diesel engine heat release rate, pressure

development and engine noise: An experimental investigation of cause and effect relationship” Journal/Conf: Papers presented at the SAE international off-highway and power plant congress and exposition. 14p, 1987.

8. R. Glav 1994 Doctoral thesis KTH. On Acoustical modelling of silencers.


550

Chapter 16: Noise and Vibration in Pipes and Ducts

551

CHAPTER 16

NOISE AND VIBRATION IN PIPES AND DUCTS This chapter discusses noise and vibration control for pipes conveying fluids, i.e. gases, liquids and vapors, used for instance in the process and power generation industry. Noise and vibration problems associated with the machines generating the flow: pumps, compressors, fans etc, are treated in Chapter 17. The chapter starts with describing sound generation mechanisms, then discussess sound transmission, sound radiation and finally gives an overview over noise control methods. Noise from pipes and ducts is important in many industrial sectors but also in residential buildings. Sound generated by fluid flow in pipes can propagate as fluid-borne sound or structure-borne. I may also radiate directly from openings or pipe walls or by connected structural components.

15.1 SOUND GENERATION IN PIPES WITH FLUID FLOW

For a source operating under free field conditions the source strength can be described uniquely by the emitted sound power. For in duct sources the presence of boundaries will affect the source output. It is only for high frequencies well above the plane wave range in a duct that a source can be uniquely characterized by acoustic power. The frequency limit for the plane wave range in a circular duct with diameter D and speed of sound c can be calculated from

Dcfc π

84.1= , (16-1)


552

which is the cut-on frequency for the first acoustic cross-mode. In practice power can be used to characterize a broad-band source, such as valve noise, when f > (2-3) fc1. Normally for industrial valves the dominating part of the spectrum will be well above the plane wave range which means that this criteria is satisfied. It can be mentioned that for valves and flow regulators working at low Mach-numbers (<0.1), e.g., components in ventilation systems, the plane wave range is often important. Then more general source characterization methods must be used such as active acoustic 1- and 2-ports [1]. In the following sub-sections a short review of the different noise generating mechanisms will be given. 16.1.1 Turbulent boundary layer generated sound An important sound source is the turbulent boundary layer flow. The interaction with the pipe wall gives sound generation of dipole character. The internal sound power is then determined by [2]

SMUKW DD33ρ= , (16-2)

where 510122 −×−≈DK , U is the flow speed, ρ is the density, M is the Mach-number and S is the pipe cross section area. The internal sound power level can be calculated from [2]

dBNTNT

PP

SS

UUKLWi

,log60log25log10

log10log60

010

010

010

010

010

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+≈

κκ

(16-3)

where ( ) UK ×−≈ 816.0 , U0 = 1 m/s, S0 = 1 m2, P0 = 101325 Pa, N0 = 287 J/kgK, T0 = 273 K, κ0 = 1.4. To calculate the sound power level in octave bands the following empirical relationship can be used

fWiWOctW LLL ,,, Δ+= dB. (16-4) where the correction term scales with octave center requency (fm) divided by flow speed according to

5, −=Δ fWL dB for 5.12<Ufm

(16-5)

⎟⎠⎞⎜

⎝⎛−=Δ U

fL mfW 10, log5.1512 dB for 5.12≥U

fm ,


553

which is illustrated in figure 16-1 Using these expressions the A-weighted sound power level can also be calculated by applying the appropriate weighting factor to each octave band.

Figure 16-1 Frequency spectrum correction for turbulent boundary layer noise.

16.1.2 Sound generation by pipe discontinuities Any flow disturbance caused by the pipe design will result in flow separation and increased sound generation. Examples of pipe discontinuities causing increased flow separation are sudden area changes and flanges extending inside the pipe.

Figure 16-2 Pipe discontinuities cause flow separation and increased sound generation.


554

Objects such as gratings, struts or metering devices projecting into the flow can lead to periodic pressure fluctuations which can cause tonal sound generation. So called Strohal tones are caused by period flow separation around obstacles as discussed in Chapter 10. They will have the frequency

dUStf = , (16-6)

where U is the flow speed, d is a typical cross dimension of the object and St is the Strouhal number which will take on different values depending on the geometry.

Figure 16-3 Periodic flow separation around objects protruding into the flow cause Strohal tone generation. Tonal sound from pipes can also have its origin in resonances of connected structures excited by the turbulent pipe flow. 16.1.3 Control valve sound generation A main noise source in the process and power generation industry is control valves or flow regulators. These devices serve several purposes from the basic to start and stop a flow to regulation of flow rates and pressures. In particular at high flow rates and high pressure drop ratios across a flow regulator, large in duct sound powers can be created exciting vibrations in the surrounding pipe structure and associated high levels of radiated sound. In this chapter an overview of this potential noise problem will be presented starting with a classification of valves and the basic noise generating mechanisms, then discussing available methods to predict the radiated noise and finally giving an overview over noise control methods. 16.1.3.1 Classification of valves Valves can be classified after the mode of operation and after the design. Concerning the operation the two main categories are: • Stop valves are used to shut off or partially shut off the flow of fluid. • Check valves are non-return valves which means that there is flow only in one direction. Besides stop and check valves a number of other operating modes exist mainly related to the control of pressure or flow rate (throttling). Concerning the valve design all valves can be split into the following main parts: the body, the plug, the seat and the stem. The body


555

being the outer shell of the valve, the plug the moving part that changes the valve geometry, the seat the part of the body on which the plug rests when the valve is closed and the stem is the part creating the motion of the plug. The name of the moving part can vary depending on the design e.g. gate, vane, disk and ball. 16.1.3.2 Examples of valve types • Globe valves are one of the most common valve types with a linear valve motion and a rounded body. This valve type has the advantage to offer precise throttling andcontrol and operation at high pressures. Examples of globe valve bodies are shown in Figure 1. Globe valves are often used as pressure control valves with the valve opening controlled by a pressurized diaphragm. • Gate valves (also known as knife or slide valves) are linear motion valves in which a flat or wedge shaped closure element slides into the flow to provide shut-off. This valve type does not change the direction of the flow and is primarily designed to either be fully opened or closed. Throttling or operation with a partially open valve is not recommended for high flow rates due the risk for strong flow induced vibrations. • Butterfly valves consist of a circular disc or vane with its pivot axis at right angles to the flow in the pipe which when rotated seals against seats in the valve body. This type is normally used as a throttling device to control flow rate. An alternative design is the use of two semi-circular discs hinged on a common spindle. • Ball valves use a ball to stop or start the flow of fluid. The ball performs the same function as the disk in the globe valve. When the valve handle is operated to open the valve, the ball rotates to a point where the hole through the ball is in line with the valve body inlet and outlet. When the valve is shut, which requires only a 90-degree rotation of the handwheel for most valves, the ball is rotated so the hole is perpendicular to the flow openings of the valve body, and flow is stopped.


556

Figure 16-4 Examples of globe valve bodies. The valve plug (not Shown) will move linearly to gradually open the flow passage.

Figure 16-5 Example of a gate valve with a rising stem. Note, the wedge shaped gate which can flex to avoid camping in the valve seat at high temperatures.

Figure 16-6 Example of a butterfly valve.

Figure 16-7 Example of a ball valve.


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16.1.3.3 Valve noise source mechanisms Aerodynamic noise: Based on Lighthills theory for aerodynamic sound [3] the sound generation due to the unsteady flow in a valve can be split into three source mechanisms: 1) fluctuating volume flow (monopole source), 2) fluctuating surface pressure (dipole source) and 3) free turbulence (quadrupole source). For a given unsteady flow the sound power (W) generated under free field conditions by these three mechanisms scale with the Mach-number M as

53321 :::: MMMWWW ∝ (16-7)

For cases with a compressible medium (gases) the Mach-number will often be close to 1 at the cross-section where the flow stream has the minimum area (vena contracta). Equation (16-7) then implies that all three source mechanisms could play a role for the aerodynamic sound production. But detailed investigation carried out by, e.g., Reethof [4] show that for the subsonic range (M < 1) fluctuating sources of dipole type dominate the sound. After the flow at the vena contracta has become sonic shock waves develop down stream of the valve. In this range the noise production is dominated by the shock-cell turbulence interaction as first described by Powell [5]. Hydrodynamic noise: For cases with a “incompressible” medium (liquids) the Mach-number is normally very small and it can be expected that the monopole type of mechanism, see equation (16-7), will dominate [4]. In liquids there is also the possibility for cavitation, i.e., the creation of vapor filled bubbles which implode and represent a monopole source. The rapid collapse of the bubbles can create very high local pressure peaks with levels up to 1010 Pa that can result in mechanical damage [5]. When the flow is accelerated towards the vena contracta of a valve the speed increases and the static pressure drops in accordance with Bernoullis equation, see Figure 5. Cavitation starts when the local static pressure reaches a certain critical limit Pcr, the value of which depends on the temperature and the amount of solved gas in the liquid. The minimum value for the critical pressure is the vapor pressure in the liquid Pv. To describe the cavitation process a cavitation number can be introduced

212

11

UPP v

ρσ

−= , (16-8)

where P1 is the upstream static pressure, Pv the vapor pressure, ρ the density and U1 the upstream flow speed. For a fixed upstream pressure the downstream pressure can be lowered until the critical pressure is reached somewhere in the valve and cavitation starts. For a given valve this corresponds to a certain upstream flow speed and a critical cavitation number σcr. A further decrease in the downstream pressure will create higher flow speeds and a further reduction in the cavitation number. The more the solved gas content increase the larger the critical cavitation number becomes. The main alternatives to reduce the value of σcr are to reduce the temperature to lower the vapor pressure, degas the liquid and increase the static pressure in the system. Also use of a too narrow vena contracta section should be avoided by for instance using a multiple stage valve arrangement to create the necessary total pressure drop.


558

Figure 16-8 Principle behavior of cavitation in a valve, P1 and P2 denote upstream and downstream static pressure, respectively. When the minimum static pressure is less than a certain critical value (Pcr) cavitation starts. When the downstream pressure P2 is less than the vapor pressure a phenomenon called “flashing” can occur. In this case a liquid-gas mixture approaches the vena contracta and partial vaporization of the liquid occurs during acceleration of the flow.

Figure 16-9 The use of a multi-stage arrangement to create a certain pressure drop without reaching the critical cavitation pressure in the system.


559

The principle behavior of cavitation noise [7] can be illustrated by figure 16-10. For σ < σcr the emitted sound increases rapidly and reaches a maximum at σmax after which it drops. This corresponds to the region where single bubble cavitation dominates the noise production.Concerning the spectrum created by cavitation noise it is broadband with a peak corresponding to the dominating size of the imploding bubbles [7].

Figure 16-10 Typical variation of external sound pressure level. 1) Cavitation starts; 2) Maximum sound production; 3) Region where flashing can occur.

A further decrease in the cavitation number will lead to sheet cavitation, i.e., the bubbles form continuous gas filled layers [8]. Fur sufficiently small cavitation numbers flashing can occur, see Figure 16-10. In this case there is no implosion of vapor bubbles and the strong dominating monopole mechanism is no longer found [6]. Mechanical noise: This normally originates from the valve plug and is mainly a problem in liquid filled systems. One mechanism is so called water hammers which is created by the rapid closing of a stop valve. This creates a travelling pressure wave (“shock wave”) with an amplitude of the order of

cUp ρ∝maxˆ , (16-9) where ρ is the density, c the speed of sound and U the mean flow speed. For the case of water this can easily produce pressure peaks of the order of 106-7 Pa which can lead to noise problems and mechanical damage. To reduce the effect of water hammers special designs of the valve seat or body can be used and also the upstream pipe can be fitted with a flexible section or expansion tank which can absorb the pressure pulse. In high pressure steam power plants so called condensed steam water hammers where high pressure steam undergoes a rapid condensation can be created. This is potentially very dangerous since very high pressure loads can result that can blow up a pipe wall and cause serious accidents [9]. Another mechanism is periodic flow separation creating fluctuating fluid forces which excite structural vibrations in the valve plug + stem, e.g.,


560

bending modes. A particular dangerous situation is when a periodic flow phenomenon around the valve plug, characterized by a Strouhal-frequency fSt, is close to a structural eigenfrequency fMek. This can create a selfsustained oscillator which means that the two phenomena form a positive feed-back loop,where energy from the mean flow is fed into the structural eigenfrequency. A growing oscillation at a dominating frequency will then by created limited in amplitude only by losses or non-linear effects [10,11]. This type of phenomenon is normally referred to as valve screech and can create very high vibration amplitudes with risk for mechanical failure as well as high emitted noise levels. Screech can also be created by interaction between an acoustical mode in the pipe system and a structural valve mode. Also for this case the energy feeding the structural and acoustical modes is taken from the mean flow via the fluid forces acting on the valve plug. To eliminate valve screech there exist two main alternatives: i) to disturb or reduce the amplitude of the periodic flow phenomenon at the valve plug; ii) to damp or move (change mass/stiffness/length) the mechanical eigenfrequency. Concerning the first alternative typical methods are based on increasing the upstream turbulence level or using geometrical modifications to reduce flow instabilities. 16.1.3.4 Sound generation by solids in the fluid Solids may be transported by the fluid in a number of applications including: pneumatic conveyance of grains and plastic granulates and coal conveyed in water. Additional noise is created by contact of the particles with the pipe wall and with each other. The noise spectrum is of high frequency character with a maximum typically between 2 kHz and 16 kHz. The sound pressure level depends on the flow rate, the pipe material and the type of solid transported. For conveyance of plastic granulates experience has shown that the A-weighted sound level can be 85 dB(A) and 100 dB(A) at a distance of 1m from a steaight pipe section. Figure 16-11 shows the sound power level in octave bands for a 90o pipe bend and a straight pipe section. Figure 16-12 shows the sound pressure level at a distance of 1m from the straight pipe section. The air volume flow velocity was 0.28 m3/s (1000 m3/h) and 6000 kg of granulate, with dimensions 3x4 mm, was transported per hour. The A-weighted sound power level was 102 dB(A) for the 90o bend and 92 dB(A) for the straight pipe section. The sound level at 1 m distance from the straight pipe section was 85 dB(A). When coal is conveyed in water the resulting sound generation will depend on parameters such as: particle size, concentration of solids, pipe internal diameter as well as on flow spped. Common flow sppeds are 1-6 m/s and solid concentration 0.3-0.6 kg/dm3. As an example for a pipe with 0.3 m diameter and flow speed 4 m/s the measured sound level at a distance of 0.2 m was 69-72 dB(A).


561

Figure 16-11 Sound power level in octave bands for 3 m long steel pipes with width DN 100 and wall thickness 3 mm conveying plastic granulates at an air volume flow speed of 0.28 m/s and transporting 6000 kg granulate per hour: full line - 90o pipe bend, dashed line straight pipe section [2].

Figure 16-12 Sound pressure level in octave bands at a distance of 1 m from a steel pipe with width DN 100 and wall thickness 3 mm conveying plastic granulates at an air volume flow speed of 0.28 m/s and transporting 6000 kg granulate per hour [2].


562

16.2 SOUND TRANSMISSION IN PIPES Sound can be transmitted either through the fluid, liquid or gas, or through the pipe wall. Fluid-borne sound and structure-borne sound in pipes are treated separately in this section even though there may be a significant coupling between the fluid-borne and structure-borne sound in the case of liquid filled pipes. In the most simple case this coupling may be modeled as a change in the speed of sound in the fluid caused by the fact that the pipe walls can not be regarded as perfectly rigid. Sound in ducts has already been discussed in Chapter 11 so this section will not cover the theory for sound propagation in ducts and pipes but rather give practical advice regarding applications of industrial relevance. 16.2.1 Fluid-borne sound Sound propagates as longituidinal waves in fluids. Depending on the relationship between the acoustic wavelength and the pipe dimensions different modes may propagate in the pipe as shown in Table 16-1. The wavelength is calculated as the ratio between the speed of sound and the frequency so the speed of sound is a key quantity in working with sound propagation in pipes. The general expression for the speed of sound is given by

0ρβ

=c , (16-10)

where β is the adiabatic modulus of compression and ρ0 the density. For gases this reduces to

Μ==

RTpc γ

ργ

0

0 , (16-11)

where γ is the ratio of specific heats ( )vp cc , p0 is the static pressure, R is the general gas constant, T is the temperature and M is the molar mass. Table 16-1 and 16.2-gives the properties of some gases and liquids at a pressure of 100 kPa. Table 16-1 Properties of gases at: p0 = 105 Pa, T = 273 K. Gas Speed of

sound c [m/s] Density ρ [kg/m3]

Ratio of specific heats γ

Molar mass M g/mol

Air 332 1.293 1.40 28.96 Carbon dioxide 260 1.977 1.31 44.01 Carbon monoxide

336 1.250 1.39 28.01

Helium 964 0.179 1.64 4.00 Hydrogen 1258 0.090 1.41 2.02 Natural gas 399 0.83 1.32 - Oxygen 312 1.429 1.37 32.00 Water vapour (at 100 oC)

478 0.598 1.33 18.02


563

Table 16-2 Properties of liquids at: p0 = 105 Pa, T = 293 K. Liquid Speed of

sound c [m/s] Density ρ [kg/m3]

Adiabitic modulus of compression

Acetone 1190 792 Ethanol 1180 789 Gasoline 1166 750 Hydraulic oil 1050 900 Mineral oil 130 – 1520 1040 – 700 Mercury 1451 13551 Methanol 1123 792 Sea water (3.2 % salt)

1481 1020

Water 1440 1000 Transformer oil 1425 895 In liquid filled pipes the speed of spound can be substantially reduced compared to the values given in Table 16-2. For plane wave propagation the following equation may be used to estimate the effective speed of sound (ce) in the liquid filled pipe

2

2

211

21121

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

⎟⎟⎠

⎞⎜⎜⎝

⎛−+

+

=

o

o

e

dt

dt

E

cc

β

(16-12)

where E is the modulus of elasticity for the pipe material, t is the wall thickness and do is the pipe outer diameter. Equation 16-12 apply only if there are no undisolved gases in the liquid. If this is the case a substantial reduction of the speed of sound occur. The speed of sound in gas liquid mixture can be calculated from

2

1

1

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎟⎠

⎞⎜⎜⎝

⎛

++

⎟⎟⎠

⎞⎜⎜⎝

⎛

++

=

l

g

lg

g

l

gg

lg

gg

l

l

mmm

mmm

c

ρρ

ρρ

β

ρ

βρ

(16-13)


564

where index g stand for gas, index l stands for liquid, mg is the mass of gas and ml is the mass of liquid.

Sound propagating in an infinite medium is attenuated but except for sound propagation over long distances outdoors this attenuation can normally be neglected. The attenuation increases with frequency and is higher inside pipes and ducts compared to in free space. Results from natural gas pipelines of 2-inch diameter showed for instance an attenuation of 0.2 dB/octave in the frequency range 300-1200 Hz. The following empirical relations can be used to estimated the damping in dB/m in acoustically narrow pipes

ρμπ ′

=Δ f

cdlL

i

p 37.17, (16-14)

where di is the pipe inner diameter and μ′ is given by

2

11⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−+=′

μτ

γγμμ

pc, (16-15)

where γ is the ratio of specific heats (cp/cv) and t is the thermal conductivity. If the viscosity variation with temperature is considered the following alternative empirical relationships can be used

( )MTP

fdl

L

i

p 111293

10109.44

54

+×

=Δ −

, for 14.0≤M (16-16)

ργ ′′

=Δ f

UdlL

i

p 104.111 , for 14.0>M (16-17)

where

76.0

273⎟⎠⎞

⎜⎝⎛=′′ Tγγ , for gases (16-18)

and 09.1

273⎟⎠⎞

⎜⎝⎛=′′ Tγγ , for water vapour. (16-19)


565

In liquids the attenuation is smaller. The attenuation in a 50 mm diameter water pipe is about 0.045 dB/m. The effect of duct or pipe area changes cause acoustic reflections discussed in section 11.3.1. If the pipe brances it can to a first approximation be assumed that the sound power is divided in the different branches proportionally to the ratio between the cross-sectional area of the branches. Pipe bends may cause additional reflections giving reductions in transmitted sound power. For 90o bends in circular cross section pipes an estimate may be obtained from Table 16-3. Table 16-3 Sound power level reduction for 90o bends in circular cross section pipes. Wavelength/Pipe inner diameter (λ/di) Sound power level reduction ΔLW [dB]

< 0.7 3 0.7 – 1.4 2 1.4 – 2.8 1

>2.8 0 Rectangular cross section ducts give larger sound power reduction compared to circular cross section ducts. Far below the first cut-off frequency for higher modes there will only be a level reduction if the cross sectional are is different before (S1) and after (S2) the bend:

⎟⎟⎠

⎞⎜⎜⎝

⎛ +=Δ

21

22

21

10 2log10

SSSS

LW . (16-20)

There will therefore be no sound power reduction if the areas are equal. For higher frequencies there will be substantially higher reductions. A rule of thumb is that the average reduction in total level will be

⎟⎟⎠

⎞⎜⎜⎝

⎛+=Δ 1

2log10

2

110 S

SLW . (16-21)

In the plane wave range, below the first cut-off frequency, reductions from several bends or other discontinuities can normally not be added since there is a strong interaction between them. For frequencies well above the first cut-off frequency this is permissible. 16.3.2 Structure-bone sound Propagation of structure borne sound in pipe walls is more complicated than propagation of fluid borne sound. A number of different wave types can propagate even though bending waves in the pipe walls normally dominates. Table 16-4 shows speed of sound, loss factors, surface mass and modulus of elasticity of some common pipe materials.


566

Table 16-4 Properties of some pipe materials at T = 293 K. Material Longitudinal

speed of sound cL [m/s]

Loss factor Density [kg/m3]

Modulus of elasticity E [N/m2]

Aluminium 5100 0.0001 2700 70x109 Cast iron 3400 0.0015 7600 100x109 Copper 3600 0.002 8900 115x109 Glass (industrial)

5000 0.004 2500 80x109

Lead 1250 0.02 11300 5x109 Low-pressure polyethylene

1100 0.10 950 0.8x109

Polyester resin 2300 0.14 2200 4.5x109 PVC (rigid) 1600 0.04 1300 2.7x109 Steel 5100 0.0001 7800 200x109 For gas filled pipes natural frequencies can be calculated by neglecting interaction between pipe wall vibration and sound propagation in the fluid. This is not true for liquid filled pipes where interaction has to e taken into account making the problem more complicated. Natural frequencies determined by the design and location of pipe attachments may be of importance. As a first approximation the pipe may be treated as a beam attached at discrete points. 16.3 SOUND RADIATION FROM PIPES Sound radiation from pipes can either occur from the pipe walls or openings. In this section the pipe wall radiation will further be divided into the cases when the excitation comes from structure borne sound excitation or fluid borne sound excitation. Most results are for gas filled pipes. 16.3.1 Excitation by structure-borne sound The radiation efficiency (s) , defined in section 9.6, is a measure of how efficient the structure is as sound radiator compared to a plane pipston of the same surface area. If the vibration velocity distribution in the pipe is known the radiated sound power level can be calculated from

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟

⎟⎠

⎞⎜⎜⎝

⎛= ⊥

01010

2

10 log10log10~

log10SS

svv

L P

refW , (16-22)

where ⊥v~ is the r.m.s.-value of the pipe surface normal velocity, 910−=refv m/s, SP is the pipe surface area and S0 = 1 m2. For straight pipe an upper limit for the radiation efficiency can be estimated rom


567

( ) 3

41

1

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=

fdc

fs

O

(16-23)

where c is the speed of sound in the surrounding medium and dO is the pipe outer diameter. Figure 16-13 shows ( ))(log10 10 fs as a function of normalized frequency,

( )cdf O4 .

Figure 16-13 ( ))(log10 10 fs as a function of normalized frequency, ( )cdf O4 [2].

16.3.2 Excitation by fluid-borne sound Sound radiated from pipes can be related to air-borne sound inside pipes through the pipe sound reduction index. The sound reduction indes for walls was defined in section 8.2.4. The sound transmission of a cylindrical pipe wall has maxima at the frequencies corresponding to the cut-on frequencies for acoustic modes inside the pipe [12,13], see Figure 16-14. For broadband sources the exact modeling of these maxima is not so important instead a frequency averaged model is used. Based on this model the sound reduction index of the pipe wall (Rp) is given by


568

( )⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛×−= −

1415

106.7log1022

2

2710

y

x

ppP

Gc

Gft

cRρ

, (16-24)

where standard atmospheric conditions (P0=101,325 kPa) are assumed in the air surrounding the pipe and tp is the pipe wall thickness and the G functions are defined in Table 16-5.

Figure 16-14 Typical behavior of the sound transmission through a pipe wall. Local minima occur in the sound reduction at the cut-on frequencies (fcn) for acoustic modes in the downstream duct and at the pipe ring frequency (fr). Table 16-5 Frequency factors Gx and Gy. Here fc1 is the cut-on for the first acoustic mode in the downstream pipe (see equation (1)),= is the ring frequency where cp is longitudinal wave speed in the pipe wall and the Di the internal diameter,0 3 = is the coincidence frequency for bending waves and c0 is the speed of sound in the outside air.

Radiated noise: The sound level just outside the pipe wall is given by


569

( )2100 1log165 MRLL ppA −−−+= + [dB(A)], (16-25)

where +

pL and Rp is obtained from equations (7) and (16-24), respectively and M2 is the downstream Mach-number. Assuming cylindrical wave spreading and a free field the sound level at a radial distance of x meters will be

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+

++−=

pi

piA

xA tD

xtDLL

222

log10 100 [dB(A)], (16-26)

16.3.4 Radiation from pipe openings In the plane wave range the sound radiation from pipe openings can to a first approximation be treated as an acoustic monopole excited by the fluctuating volume flow at the opening as discussed in section 9.1. Information about sound radiation from pipe openings for higher frequencies or more complex geometries can be found in [x]. 16.4 NOISE CONTROL TECHNIQUES Noise control can as in other cases be applied either at the source, during the propagation path or at the receiver. A lot can be gained if noise control can be considered from the beginning when designing a new plant or when improving an existing plant. 16.4.1 Noise control at the source Fans, pumps and compressors are major noise sources in pipes. They have however already been discussed in Chapter 13. In this section we will therefore concentrate on control of noise associated with the flow in pipes. Setting the flow rate as low as possible for the application in question is therefore an important noise control technique. Noise created by a high speed ( > 40 m/s for gases) turbulent flow in a pipe must be considered. The internal sound power created by the turbulent boundary layer in a straight pipe is according to VDI 3733 given by

( ) ( ) ⎟⎠⎞

⎜⎝⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛−××+=+ U

fTN

NTUPAfLW 101

110

0010

610 log5.15

4.1log15log25log1020 γ , (16-27)

where A is the cross-sectional area of the pipe in m2, P the static pressure in Pa, U the flow speed in m/s, N the gas constant (N0=287 J/kg K), T the absolute temperature (T0=273 K), γ the specific heat ratio and f is the octave band mid-frequency in Hz. The formula is valid in the range 12.5 ≤ f /U ≤ 800. In addition bends and regions with flow separation, e.g., area expansions, can represent important sources of flow induced noise. To avoid excessive flow separation and noise generation expansions should be in the form


570

of conical sections with an angle not exceeding 15 degrees. Rapid area changes, flanges and other disturbances protruding into the flow should be avoided.

Figure 16-15 Sudden area changes increase the level of turbulence and should be avoided. Bends should also be separated from the outlet jet (> 5 Dj) of a valve to avoid a strong excitation of wall vibrations and sound radiation. It can also be noted that bends will have a higher sound transmission than a straight pipe section of the same diameter and wall thickness due to mode conversion.Narrow bends should therefore be avoided and disturbing objects such as valves should not be placed close together, see Figure 16-16. In cases where high flow velocities can not be avoided flow guides can be used to keep the flow from hitting the pipe wall..

Figure 16-16 Noise control by using soft bends and keeping a sufficient distance between control valves. Steady state noise associated with control valves will be discussed in some detail below. From a thermodynamic point of view a control valve converts pressure energy into heat in order to control the mass flow. The heat conversion normally takes place via turbulent flow losses with associated noise generation. Of course it is possible to design control valves where the heat production is created via laminar flow losses, which would result in


571

a virtually noise free operation. Standard valve designs are however based on turbulent dissipation and for such valves the generated noise is proportional to the pressure drop ΔP . Based on a study of orifice plates Jenvey [14] concluded that the acoustic power scales as

4.2iAPW βΔ∝ , (16-28)

where Aj is the outlet jet area. The value of the exponent β is 4 for the subsonic case and 3 for a choked flow. Equation (16-28) suggest two methods for reducing the source strength. The first alternative is so called single flow path multistage valve trims where a desired pressure drop is split into a number of steps. Assuming an unchanged jet area and N equal steps that act as independent sources, the total power compared to the single step arrangement is

β−= 1NWWN . (16-29)

For instance with three steps the reduction in sound power will be between 10-14 dB depending on the value for the exponent. Noise reduction on a given valve can be achieved by a series of carefully designed downstream throttling plates as described for instance by Hynes [15].


572

Figure 16-17 Noise control by using pressure reducing inserts placed in the same pipe as the control valve. The plates are selected so that the inserts will not produce a greater pressure drop than that required to prevent cavitation.

The second alternative is so called single stage multiple flow path valve trims where the outlet jet is split into a number of smaller interacting jets. This procedure will lead to a decrease in the power which, assuming N independent paths (sources), will be proportional N1.4 . Another effect of decreasing the jet size is that the peak frequency of the spectrum will be shifted upwards. This has a positive effect due to the reduced sensitivity of the human ear at higher frequencies. Of course it is also possible to combine both alternatives and design multipath, mulitistage valve trims. 16.4.2 Noise control during the propagation path In gas filled systems silencers can be used to reduce noise. There are two basic types of silencers reflective and dissipative. Reflective silencers create a reflection of waves by an impedance mismatch, e.g., by an area change or a side-branch resonator. The reflective silencer type is primarily intended for the plane wave range and is efficient for stopping single tones. Dissipative silencers are based on dissipation of acoustic energy into heat via porous materials such as fiberglass or steel wool. These silencers are best suited for broad-band sources and for the mid- or high frequency range. Different types of silencers have been discussed in Chapter 11.


573

Transmission of structure borne sound to pipes from surrounding structures or vice versa should be avoided by flexible couplings at the pipe clamping points. These can be rubber inserts or different types of springs. Pipes should furthermore be secured to structures at points with high input impedance. If the main propagation takes place in the pipe wall blocking masses can be used. In the case of liquid filled pipes this may not be a good solution because the pulsation in the liquid may excite the pipoe wall immediately after such a structure-borne sound barrier. Flexible pipe connections, see Figure 16-x, can also be used to reduce both structure-borne and fluid-borne sound transmission.

Figure 16-18 Noise control by using flexible pipe connections.

16.4.4 Reduction of sound radiation For a given internal sound power in a pipe the radiated sound depends on the sound reduction index of the pipe wall. A representative value [16,17] for the sound reduction index in the mid frequency range between fc1 and fr, where maximum transmission occurs, is given by the first two terms in equation (30)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+=

i

pppW Dc

tcfR

2210log1010

ρρ

. (16-30)

From this equation it follows that an increase of the wall thickness with a factor 2 will reduce the radiated sound level with 3 dB. It follows that changing the wall thickness is not a very efficient way of reducing the radiated sound. Instead so called “acoustic lagging” is to prefer when large reductions (> 10 dB) are needed [18,19]. This basic idea is similar to a vibration isolation and aims at shielding the original pipe with a structure that has a reduced vibration level. This is done by wrapping the pipe with a porous material and covering this with a limp and impervious top sheet. The porous material


574

provides together with the enclosed air stiffness and damping. The damping is important to reduce the effect of acoustic modes between the pipe wall and the top sheet. Assuming that the top sheet is masscontrolled the increase in sound reduction index for frequencies well above the fundamental mass-spring resonance will be

2

0010 2

log10 ⎟⎟⎠

⎞⎜⎜⎝

⎛ ′′=Δ

ρωcm

R p , (16-32)

where ω is the angular frequency, m′′ is the mass per m2 of the top sheet, ρ0 is the density of air and c0 the speed of sound in air.

Figure 16-19 Extra sound reduction from acoustical lagging assuming air at 20 C. The mass per unit area is: - - - - - 5 kg/m2; …….. 10 kg/m2; 20 kg/m2.


575

References 1. H. Bodén and M. Åbom 1995 Acta Acustica 3, 549-560. Modelling of fluid machines as sources of sound in duct and pipe systems. 2. VDI-Richtlinien 1996, VDI-3733. Noise at pipes. 3. M.J. Lighthill 1952 Proc. Royal Society A211, 564-587. On sound generated aerodynamically. I General theory. 4. G.C. Chow and G. Reethof 1980 ASME paper 80-WA/NC-15. A study of valve noise generation: Processes for compressible fluids. 5. A. Powell 1953 Proc. Phys. Soc. London, Sect. B, Vol. 66, 1039-1056. On the mechanism of choked jet noise. 6. H.D. Baumann and G.W. Page 1995 Noise Control Eng. J. 43(5), 145-158. A method to predict sound levels from hydrodynamic sources associated with flow through throttling devices. 7. D.W. Jorgensen 1961 J. Acoust. Soc. Amer. 33, 1334-1338. Noise from cavitating submerged water jets. 8. R.T. Knapp 1955 ASME 77, 1045-1054. Recent investigations of the mechanics of cavitation and cavitation damage. 9. E.B. Woodruff, H.B Lammers, T.F. Lammers. Steam-Plant Operation, 7:th ed.. Mc Graw-Hill, ISBN 0-07-036150-9. 10. W.K. Blake. Mechanics of flow-induced sound and vibration. XXX 11. U. Ingard 1977. Valve noise and vibration, Report no. 40 prepared for Värmeforsk, Sweden (www.varmeforsk.se). 12. G. Reethof and W.C. Ward 1986 ASME July -86, 00329. A theoretically based valve noise prediction method for compressible fluids. 13. G. Reethof 1977 Noise Control Eng. J. 9(2), 74-85. Control valve and regulator noise generation, propagation and reduction. 14. P.L. Jenvey 1975 J. Sound and Vib. 41(4), 506-509. Gas pressure reducing valve noise. 15. K.M. Hynes 1971 ISA Trans. 10(4), 416-421. The development of a low-noise constant area throttling device. 16. L. Cremer 1955 Acustica 5, 245-256. Theorie der schalldämung zylindrischer schalen. 17. M. Heckl 1958 Acustica 8, 259-265. Experimentelle untersuchungen zur schalldämung von zylindern. 18. M.E. Hale and B.A. Kugler 1975 ASME paper 75 WA/Pet-2. The acoustic performance of pipe wrapping systems. 19. W.H. Bruggeman and L.L. Faulkner 1975 ASME paper 75 WA/Pwr-7. Acoustic transmission of pipe wrapping systems.


576

Chapter 17: Sound Generation From Fluid Machines

577

CHAPTER 17

SOUND GENERATION FROM FLUID MACHINES

In general a machine can be defined as a device that converts one form of energy to another,

where normally one of the energy forms is mechanical energy or work. Fluid machines are a

special class of machines that involve a fluid (liquid or a gas) in their working cycle.

Important examples of fluid machines are fans, pumps, compressors, turbines and internal

combustion engines. Fluid machines are one of the most common machine types and used in

transportation systems (cars, trucks, ships and trains), in the process industry to transport

fluids in pipes and ducts, in home appliances and in buildings. In this chapter we will describe

the mechanisms that generate sound in fluid machines. The focus will be on fluid generated

sound and structure born sound sources and vibrations are not addressed. The chapter also

describes noise control and here the focus is noise control at the source rather than along the

transmission path. The structure of the chapter is that is starts with a classification of fluid

machines based on their mode of operation. Then follows a description of the mechanisms


578

that can produce sound with some examples related to common fluid machines. The chapter is

ended with a discussion of noise control.

17.1 Classification of fluid machines Fluid machines involve the exchange of mechanical energy or work with a fluid medium.

There are three basic ways in which this exchange can take place: via mechanical forces

(pressure), via volume displacements or via heating. Of course in many machines more than

one of these ways for energy exchange is involved. The mechanical force is normally created

by rotating surfaces, e.g., propellers, blades or screws. Examples of machines using this

principle are fans, turbines and compressors. Depending on the main flow direction in the

machine relative to the driving axis rotating fluid machines can be split into: axial, radial and

mixed flow. Figure 17-1 shows an example of these three types for the case of fans.

Figure 17-1: The three basic types of fans: axial, radial and mixed flow. The classification relates to the main

flow direction relative the driving axis. In the mixed flow fan the flow moves both in the axial and radial

directions.

Regarding volume displacement machines this can be created by pistons, screws and gears.

Examples of machines using this principle are compressors and pumps. Finally fluid

machines involving heating are in most cases driven by combustion. Often these machines are

a combination also involving energy exchange via forces or volume displacements. For

instance a gas turbine is driven by combustion which then drives the compressor and the

turbine. Another example is the internal combustion engine which in its standard version is a

piston machine driven by combustion.

Axial fan Radial fan Mixed flow fan


579

17.2 Flow generated sound From chapter 10 we know that based on Lighthills theory (1952) for aerodynamic sound there

are 3 basic types of fundamental acoustic source mechanisms in unsteady flows. These are:

1) Fluctuating (unsteady) volume flows;

2) Fluctuating (unsteady) fluid forces;

3) Free turbulence or fluctuating (unsteady) shear stress on fluid particles;

The three source mechanisms are also referred to as monopole, dipole and quadrupole type of

aero-acoustic sources or in general (both gases and liquids) fluid dynamic acoustic sources,

see chapter 10 and Table 10-2. From chapter 10 we can also obtain the relative strength in

terms of the acoustic power (W ) generated by each of these sources, see table 10-4

2 4: : 1: :m d qW W W M M∝ , (17-1)

where M =U/c is a characteristic Mach-number of the flow process producing the sound.

Since many applications, outside the aeronautical field and high speed turbines and

compressors, involve Mach-numbers much less than 1 equation (17-1) imply a strong

ordering of the source mechanisms. This means if we study a machine in the low Mach-

number range we first look for monopole type of mechanisms. If there are no fluctuating

volume processes involved then the second or dipole type of mechanism will dominate. This

mechanism involves unsteady fluid forces produced by moving (rotating surfaces), but

unsteady fluid forces also occur around objects with flow separation. Therefore even if there

are no moving propellers or blades involved, we can still get contribution from the dipole part

from flow separation. Finally the weakest source for the low Mach-number case is the

quadrupole corresponding to free turbulence. Since we normally always have flow separation

involved this source is often impossible to observe for the low Mach-number case.

Example 17-1 The internal combustion (IC-) engine

This is a good example of a low Mach-number application. The engine is a heat driven piston

machine producing a volume flow in the intake and exhaust systems. This volume flow has a

steady part plus an unsteady part related to the number of cylinders and the details of the


580

engine design. The Mach-numbers involved (in the intake or exhaust pipes) are typically less

than 0.3. From equation (17-1) this implies that there is 10 dB difference between the

monopole and dipole type of sources. This is not very large so one should not entirely neglect

sound from flow separation in the intake or exhaust system. However, the effect of the

jetnoise at the outlet of the exhaust pipe should be of the order of 20 dB less than the

monopole, so this is safe to neglect. One can also note that if we consider the entire car or

truck the only monopole type of source comes from the IC-engine. Since for low Mach-

number applications this source is expected to dominate the first thing one must put on an

automobile is a muffler! It can be noted here that the pulsations on the exhaust side are

stronger and also the intake side normally does not radiate freely, which makes it less

important.

Figure 17-2 Illustration of an IC-engine exhaust system producing a pulsating volume flow and representing a

monopole type of source.

17.2.1 The high Mach-number range

Turning now to the high Mach-number range or Mach-numbers of the order of 1, we find that

all three source mechanisms are equally important. For Mach-numbers larger than 1 one can

obtain so called shock-wave phenomena, i.e., discontinuities in the flow field related to the

fact that information in the flow field cannot travel faster than the local sound speed. These

shock-waves are often non-stationary that is they oscillate and change their shape or position

in space. During these motions the shock-wave surface will create both unsteady volume

flows and pressures, i.e., it represents a combination of both monopole and dipole sources.

Noise produced by such moving shock-waves is common on today’s high performance

compressors for instance on aero-engines and is referred to as buzz-saw noise.

Example 17-2 The jet engine

This is a good example of a high Mach-number application. The most common type of

modern jet engines are the high by-pass-ratio engines where a high thrust is achieved using an

exhaust jet with a large area but with a reduced speed. Compared to the old type of jet engines

IC-engine

Muffler Muffler


581

in the 1960´s this have reduced the noise by up to 20 dB for the same thrust. The engine

consists of an inlet stage with a turbo-fan creating a by-pass flow outside the gas turbine part.

This is followed by the inlet compressor stages, the combustion chamber and then the outlet

turbine stages. In this case all three source mechanisms can play a role. The combustion or

unsteady heat release will generate changes in pressure and density and is a monopole type of

source. The turbo-fan and the compressor/turbine blades represent dipole types of source.

Then there are also guide vane arrangements in the compressor/turbine sections which will

generate unsteady fluid forces and contribute to the dipole sound. Finally there is the exhaust

jet which is a quadrupole type of source. Regarding the combustion it is normally not a

dominating contribution since the degree of unsteadiness is relatively small. However, under

certain conditions the heat release from the combustion can couple to acoustic modes in the

combustion chamber and form positive feed-back loops. This can lead to significant

oscillation levels which could damage the engine. To avoid this it is important to control the

level of damping in the system.

Figure 17-3 An example of a modern high-by-pass ratio jet engine.

16.2.2 The case of liquids

The sound generating mechanisms are of course the same. But since the speed of sound in

most liquids are higher than in gases typically with a factor 4-5, see Table 6-1 and 6-2, and

the flow speeds involved in liquid systems are smaller, Mach-numbers are often much smaller

than 0.1. This implies that sound produced by free turbulence or quadrupole sound is not of

interest. However another phenomenon becomes important in the liquid case and that is the

possibility of a phase transition or cavitation. Cavitation involves the creation of vapor

bubbles in the liquid, when the local (static) pressure is reduced to values close to or below

the vapor pressure. Based on Bernoulli’s equation this will occur in regions where the local


582

flow speed is large, e.g., at constrictions or at moving surfaces (propellers). The formation of

a cavitation bubble is of course a monopole type of source mechanism. But the process is

slow compared to the collapse or implosion of the unstable bubbles, which can be a strong

and high frequency monopole noise source. The pressure peaks generated by the rapid

implosion can also lead to mechanical wear (erosion).

17.2.3 The character of the sound

The periodic operation of a fluid machine implies that the acoustic source processes also will

be periodic. Assuming that the machine has a working cycle that repeats itself with the time

period T0, then it will produce sound (and vibration) spectra which contain harmonics of a

fundamental frequency f0=1/T0. Besides the periodic content there will always be non-periodic

or random contributions to the sound from the turbulent part of the flow in the machine. This

will produce a broad-band contribution to the spectrum.

Figure 17-4 Typical spectrum from a fluid machine.

Example 17-3

Assume we have a fan with B equal blades which are uniformly spaced. If the fan rotates

with N rpm the rotation frequency is f=N/60. Since the process repeats itself each time a blade

passes a fixed position in space we must multiply this with B the number of blades to obtain

the fundamental frequency: 0 60f B N= × .

Example 17-4

For a four stroke internal combustion engine with B cylinders rotating with N rpm we get:

f [Hz]

Level (dB)

f0 2f0 3f0

Broad-band part


583

10 2 60 120f B N B N= × = × . Here the factor ½ comes from the fact that a four stroke engine

only opens its valves once during two revolutions. Furthermore the result assumes that the

cylinders are phase shifted uniformly around one revolution and produce equal acoustic

pulses. In practice this is of course not perfectly satisfied so besides these so called main

engine orders (harmonics) also other harmonics of the rpm will exist. But the main engine

orders can be expected to be the strongest.

17.3 Noise control From a scientific point of view the best or most efficient is always to modify the sources.

However, in practice due to design limitations, economy or other reasons one is often forced

to do noise control along the transmission paths or at the receivers.

Figure 17-5 The chain source-transmission-receiver. Ideally noise control should be done as early as possible in

the chain.

Recalling the scaling laws for flow generated sound from chapter 10 we see that (Eqs. (10-75)

– (10-77))

2W d U α∝ , (17-2)

where α = 4, 6 or 8 for the monopole, dipole and quadrupole case, respectively. This equation

shows that the most important for noise control of the source is to reduce the speed. This is

most pronounced for the jet noise case, where a 50% reduction in flow speed gives 24 dB

reduction of the sound power.

Besides reducing the flow speed one can try to reduce the periodicity and the amplitude of the

periodic fluctuations by changing the geometry. One example of this is to use fans with equal

but unevenly spaced blades. This will reduce the fundamental to the fan rpm and will also

give smaller amplitudes of the harmonics. Of course there will be a new problem to properly

balance the fan and there can also be negative effects on the aerodynamic performance.

Another example of a geometry change is on IC-engines, where the intake or exhaust

Sources Transmission paths

Receivers


584

manifolds connecting the cylinders can be used to shift the phase of the engine pulses to

cancel certain engine harmonics. For the intake side this kind of approach is not mainly used

for noise control but to tune the engine sound, e.g., to make it more sporty.

Appendix A COMPLEX NUMBERS

The imaginary unity

1−=i (A-1)

A complex number can be written

ϕϕϕ ieiiyx zzz =+=+= )sin(cos (A-2)

where x is the real part of z, Re(z), and y is the imaginary part of z, Im(z). The absolute value and phase of z can be calculated by

22 yx +=z (A-3)

xy

=ϕtan (A-4)

The complex conjugate of z is given by

ϕϕϕ ieiiyx −=−=−= zzz )sin(cos* (A-5)

Some rules for complex numbers:

)()( 212121 yyixx +++=+ zz (A-6)

212121

ϕϕ iie +⋅= zzzz (A-7)

21

2

1

2

1 ϕϕ iie −=zz

zz

(A-8)

The Moivres Formula

( ) ϕϕϕ innnn enin zzz =+= sincos (A-9)

Euler’s Formula

ϕϕϕ sincos ie i += (A-10)

2

cosϕϕ

ϕii ee −+

= (A-11)

iee ii

2sin

ϕϕϕ

−−= (A-12)

Appendix B OPERATORS IN VECTOR ANALYSIS

Cartesian Coordinates

zp

yp

xpp zyx ∂

∂+

∂∂

+∂∂

=∇ eee

zu

yu

xu

u zyx∂

∂+

∂

∂+

∂∂

=⋅∇

2

2

2

2

2

22

zp

yp

xpp

∂

∂+

∂

∂+

∂

∂=∇

Spherical Coordinates

φθθ φθ ∂∂

+∂∂

+∂∂

=∇p

rep

re

rpep r sin

11

( ) ( )φθ

θθθ

φθ ∂

∂+

∂∂

+∂∂

=⋅∇u

ru

rur

rru r sin

1sinsin11 2

2

2

2

2222

22

sin1sin

sin11

φθθθ

θθ ∂

∂+⎟

⎠

⎞⎜⎝

⎛∂∂

∂∂

+⎟⎠

⎞⎜⎝

⎛∂∂

∂∂

=∇p

r

p

rrp

rrr

p

x

y

z

φ

θ

r sinθ sinφ

r sinθ cosφ

r cosθ r

(r, θ , φ )

x

y

z

(x, y, z)

Appendix C FOURIER TRANSFORM

Definition:

The Fourier Transform F of a function F, and their inverse transform are defined by,

∫∞

∞−

−= dtetF tiωω )()(F (C1)

and ∫∞

∞−

= ωωπ

ω detF ti)(21)( F . (C2)

Properties:

Function F – Transform F Graph – Comment

F )()( 21 tbFtaF + 1

F )()( 21 ωω FF ba + Superposition

F )(atF , 0≠a

2 F )(1 a

aωF

F )( TtF − 3

F Tie ωω −)(F Time Delay

F tietF Ω)( 4

F )( Ω−ωF

F ∫∞

∞−

−= τττ dGtFGF )()( 5

F )()( ωω GF

Convolution

F )(tδ Dirac Pulse

6 F 1

1F(ω)

ω

F 1 1

F(t)

t 7

F )(2 ωπδ Dirac Pulse

F )cos( tΩ 1

F(t)

t 8

F ))()(( Ω−+Ω+ ωδωδπ π

F(ω)

ωΩ−Ω

F )sin( tΩ 1

F(t)

t 9

F ))()(( Ω−−Ω+ ωδωδπi π

F(ω)/i

ωΩ−Ω

−π

F tie Ω

10 F )(2 Ω−ωπδ

2πF(ω)

ωΩ

F ⎩⎨⎧ <<

sother timefor ,00 ,1 Tt

T

1F(t)

t 11

F 2

2)2sin( Tie

TT

T ω

ωω −

TF(ω)

ω2π/T 4π/T

F

⎪⎪⎩

⎪⎪⎨

⎧

+<<−+<<<<

sother timefor ,02,)(

10,

2121121

21

11

TTtTTtTTTTtTTtTt

, TTT =+ 212

T

1F(t)

tT2T1 12

F 2/)(21

12

21

2sin

2sin4 TTieTT

T+− ωωω

ω

F(ω)

ω2π/T 4π/T

13 F

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

<<−−

<<

<<

sother timefor ,0

,

,1

0 ,

22

21

11

TtTTTtT

TtT

TtTt

T

1F(t)

tT2T1

F +− − 2

1

1

1

2

2sin

TieT

Ti ω

ω

ω

ω2)(

2

2

2

2)(

2)(sin

TTieTT

TTi +−

−

−ω

ω

ω

ω

TF(ω)

ω2π/T 4π/T

Ex: 31 TT = ,

652 TT =

F

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

<<−−

+

<<

<<−

sother timefor ,0

,21))cos(1(

,1

0 ,21))cos(1(

22

21

11

TtTTTtT

TtT

TtTt

π

π

T

1F(t)

tT2T1

14

F

−

−

+−

21

22

1

2

1

2cos)1(

Ti

eT

T

ii

ωω

ωπω

ω

⋅

−−

+− )

)(

1(2

22

2ωπ

ωω

TT

ii

2)(

22

2)(

cosTT

ie

TT +−−

ωω

TF(ω)

ω2π/T 4π/T

Ex: 31 TT = ,

652 TT =

F

⎪⎪

⎩

⎪⎪

⎨

⎧

<<−−

<<

<<

TtTTTTt

TtT

TtTt

22

2

21

11

,)(2)(cos

,1

0 ,2

sin

π

π

T

1F(t)

tT2T1

15

F

+

−

−+

−−

−−−

22

1

21

4

21

12

ωπ

ωπ

ω

ωωω

T

eiT

iee

TiTiTi

22

2

22

)(4

)(22

ωπ

ωπ ωω

−−

−−

+

−−

TT

eieTTi

TiTi

TF(ω)

ω2π/T 4π/T

16 F )sin( 2tΩ , Ω > 0 1

F(t)

t-1

F )44

cos(2 πωπ +

ΩΩ , Ω > 0

π/Ω F(ω)

ω Zeros: Zn ∈

)14(,0 +⋅Ω= nn πω

Maximum: Zn ∈ )14(max, −⋅Ω= nn πω

Appendix D COMPLEX FOURIER SERIES

Definitions:

A periodic signal F with periodic time T can be written in a complex Fourier series with Fourier coefficients Fn according to,

∑∞

−∞=

=n

tinnetF 0)( ωF , ∫

−

−=2

2

0)(1 T

T

tinn dtetF

TωF ,

Tπω 2

0 = (D1)

Function F – Fourier Coefficient Fn Graph - Comments

F )()( 21 tbFtaF + Linearity 1

Fn )()( 21 ωω FF ba +

F )(atF , 0≠a

2 Fn )(1

aaωF

F )( TtF − 3

Fn Tie ωω −)(F

F tietF Ω)( 4

Fn )( Ω−ωF

F ∑∞

−∞=−

nnTt )(δ Dirac pulse train

5

Fn ∑∞

−∞=−

n Tn

T)2(1 πωδ

F 1 Constant 6

Fn )(2 ωπδ

F )cos( tΩ 1

F(t)

t

7

Fn ))()(( Ω−+Ω+ ωδωδπ

F )sin( tΩ 1

F(t)

t

8

Fn ))()(( Ω−−Ω+ ωδωδπi

F tie Ω 9

Fn )(2 Ω−ωπδ

F ⎩⎨⎧

<<−<<∨−<<−−

4414242,1

TtTTtTTtT

t

1 F(t)

-1

-T/2 · T/2

10

Fn ⎩⎨⎧

⋅−

even 0,odd ,12 1

nn

nin

π

2/πFn

n1−1 2 3 4••

F ⎪⎩

⎪⎨

⎧

<<+⋅−

<<−+⋅

20,14

02,14

TttT

tTtT

t

1F(t)

-1

-T/2 ·T/2

11

Fn ⎩⎨⎧

⋅even 0,odd ,14

22 nn

n π

4/π2 Fn

n1−1 2 3 4••

F 22,2 TtTTt <<− t

1F(t)

-1-T/2

·

T/2 12

Fn πin

n 1)1( +− 1/π

Fn

n1−1 2 3 4

F

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

<<

<<−

+−<<−

<<−

−+<<

TtT

TtTTT

TTtTtT

TtTTT

tTTTt

4

4334

34

32

2112

12

1

,1

,)(2

,1

,2

0 ,1

T3

1F(t)

tT2

T1

-1TT4

·

13

Fn

−−

−

+−

TTTin

eT

TTnTTn

iT 21)(sin)()(2

12

122

πππ

TTTin

eT

TTnTTn

iT 43)(sin

)()(234

342

+−−

−−

πππ

F

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

<<−

−

−<<−−

++<<−

−<<−−<<−

2112

1

1212

21

2

2

11

),cos(

),22

sin(

2,12,1

,1

TtTTT

Tt

TtTTT

TTtTtT

TtTTtT

π

π t

1F(t)

-1-T/2

·

T/2-T1 T1

-T2 T2

14

Fn )cos()sin()

4)(

82( 2121

222

12

22 T

TTnT

TTnn

TTT

nn

−+

−−

+ ππππ

ππ

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