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1994

Practical application of random vibration signalanalysis on structural dynamicsYu-Chin L. Allen

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Recommended CitationAllen, Yu-Chin L., "Practical application of random vibration signal analysis on structural dynamics" (1994). Thesis. RochesterInstitute of Technology. Accessed from

PRACTICAL APPLICATION OF RANDOM

VIBRATION SIGNAL ANALYSIS IN

STRUCTURAL DYNAMICS

Yu-Chin L. Allen

A Thesis Submitted in Partial Fulfillment

of the Requirements for 'the Degree of

Master of Science

in Mechanical Engineering

Approved by: Professor J. S. Torok (Thesis Advisor)

Professor N. Berzak

Advisor D. S. Aldridge (AC Rochester)

Professor C. W. Haines (Dept. Head)

Department of Mechanical Engineering

College of Engineering

Rochester Institute of Technology

Rochester, New York

May 1994

PERMISSION GRANTED:

I, Yu-Chin Allen hereby grant permission to the Wallace Memorial Library of the

Rochester Institute of Technology to reproduce my thesis entitled Practical

Application of Random Vibration Signal Analysis in Structural Dynamics in

whole or in part. Any reproduction will not be for commercial use or profit.

May 17, 1994

Yu-Chin L. Allen

Acknowledgements

I would like to thank my family, my husband Stu for all his patience and

understanding over the past six years as we endeavored to juggle our work,

school, and family. Our children, Russell and Susan, for behaving so I could

work and encouraging me to finish so we can play.

I would also like to thank Dr. Torok for all his support, countless

weekends, and all the course work over the years.

in

Abstract

The use of random signal analysis in vibration and structural analysis is

discussed and documented. The need to minimize product performance and

durability risks due to competitive pressure in the marketplace has forced

industry to streamline methods of product design, processing, and evaluation.

Engineers need to be experienced in much more advanced techniques than

they did even a decade ago. Random vibration signals are used to simulate real

world environments and are another tool in structural analysis. This

investigation outlines the basics of random vibrations, frequency analyzers,

data analysis, along with associated noise and errors problems. A case study

using an aluminum beam is used to demonstrate the concepts presented in this

work.

IV

Table of Contents

Paae

List of Symbols vii

List of Tables ix

List of Figures X

1. Introduction 1

2. Background 4

2.1 Fourier Analysis 5

2.2 Modern Frequency Analyzers 9

2.3 Noise and Error 9

3.0 Random Vibration 10

3.1 Basic Probability and Statistics in Random

Vibration 10

3.2 Correlation 16

3.3 Random Environment Definitions 17

4.0 Data Analysis 22

4.1 Frequency Analyzers 22

4.2 Measurement Functions 23

4.2.1 Time Domain Analysis 23

4.2.2 Amplitude Domain Analysis 23

4.2.3 Frequency Domain Analysis 24

5.0 Noise and Error 27

5.1 Accuracy of Measurements 27

5.2 Fourier Transform Algorithm Error 28

5.3 Error Classification 32

5.4 Noise 33

5.5 Frequency Response Errors 40

5.5.1 Random Errors in Frequency Response 40

5.5.2 Bias Errors in Frequency Response 41

6.0 Theoretical Analysis 43

6.1 Exact Solution of Free Vibration 43

6.2 Approximation Method 56

7.0 Experimental Analysis 66

7.1 Equipment 66

7.2 Damping Analysis 67

7.3 Spectral Analysis and Correlation 70

7.4 Mode Shapes 77

8.0 Conclusion 80

8.1 Theoretical Analysis Application 80

8.2 Conclusions and Recommendations 88

References 91

VI

List of Symbols

F(t) Fourier Transform

Frequency

CO

T Time Period

p(x) Probability Density Function

Standard Deviation

E [ ] Expected Value

RxxCO : Autocorrelation Function

Rxy(T): Cross Correlation Function

Gxx(T): Autospectral or Power Spectral Density (one sided)

Gxy(T): Cross spectral Density (one sided)

Sxx(T): Autospectral or Power Spectral Density (two sided)

Sxx(T): Cross spectral Density (two sided)

H(f) Frequency Response Function

2

Yxv(f)J

Coherence Function

Be Filter Bandwidth

u(t) True Input Signal

v(t) True Output Signal

m(t) Input Noise

vii

n(t) : Output Noise

M Moment

V Shear Force

PMass Density

A Area

HMass per Unit Length

KE Kinetic Energy

U Potential Energy

D Dissipation Function

Qi Generalized Force

[M] : Mass Matrix

[C] : Damping Matrix

[K] : Stiffness Matrix

1>iNormal Mode

qi Generalized Coordinate

nd Number of Averages

? iTrial Function

VIII

List of Tables

Table Page

Table 1 : Typical Values of S(f) and G(f) 21

Table 2: Determination of Log Decrement 67

Table 3: Determination of Damping, C 68

Table 4: Comparison of Natural Frequencies 80

Table 5: Frequency Response Estimates for Mode 2 84

Table 6: Frequency Response Estimates for Mode 3 84

Table 7: Frequency Response Estimates for Mode 4 84

Table 8: Frequency Response Estimates for Mode 5 85

Table 9: Frequency Response Estimates for Mode 6 85

Table 10: Frequency Response Gains from Experimentation 85

Table 1 1 . Frequency Response of First Mode Approximation with Six

Functions at Ten times the Damping Constant 88

Table 12: Frequency Response of First Mode Approximation with Six

Functions at One Hundred times the Damping Constant 88

IX

List of Figures

Figure Page

1 Periodic Signal for Fourier Transform Example 6

2 Fourier Transform Example 8

3 Random Signal 1 1

4 Probability Density Curve 1 2

5 Random Signals Time Aligned 16

6 Typical Random Vibration Spectrum 20

7 Analyzer Block Diagram 22

8 Waterfall Analysis Example 26

9 Sine Wave for Fourier Transform Example 28

10 Sine Wave Used in Discrete Fourier Transform 30

1 1 Ideal Single Input/Single Output System 33

12 System with Noise at Input and Output 35

13 System Model with Noise at Output Only 38

14 Beam Model for Continuous System 42

1 5 Mode Shape 1 with Exact Solution 52

1 6 Mode Shape 2 with Exact Solution 52

1 7 Mode Shape 3 with Exact Solution 53

1 8 Mode Shape 4 with Exact Solution 53

1 9 Mode Shape 5 with Exact Solution 54

20 Mode Shape 6 with Exact Solution 54

x

1.0 Introduction

Fierce competition in the worldwide marketplace has forced industries to

revolutionize their processes. From concept selection to design, prototype

fabrication to mass production, every phase of a product has been scrutinized

and waste of any form (material, process, overhead) eliminated. No longer

does industry have the luxury of long lead times, oligopolies, and product

recalls. Consumers are better educated, have more choices, and will not

accept a second rate product. Industry relies on engineers to be able to

produce functional, durable, esthetically pleasing products which perform

better than the competition in order to stay in business.

To stay ahead of competition, the engineer must make use of many tools

available to streamline the job. The invention of the integrated circuit (IC) has

given today's engineer important tools. Made available to the masses are

reliable and affordable analyzers, data loggers, sensors, and computers as a

result of the IC revolution. No longer does computing power reside in the halls

of universities and government research labs. Powerful computing is available

even to the smallest contract engineering firms and engineering students. Now

we are able to redesign and improve product without hundreds of costly

hardware iterations.

One role engineers must play in today's marketplace is that of the

structural dynamics analyst. As a structural dynamics analyst the engineer must

study the motion of the product to ensure that the product will behave

1

appropriately in the field to guarantee success. Products must withstand their

everyday use without cracking, buckling, or failing their performance. There are

many tools available to the analyst today. Theoretical analysis and modeling

are enhanced by computers armed with sophisticated software packages which

are able to solve systems of complex equations in seconds. Traditional

transducers for force, acceleration, pressure, temperature, etc. are

complimented with computers and various analyzers, which aid in defining the

product and its field of operation. The structural dynamics analyst is

responsible for characterizing a product's dynamic behavior in every

appropriate environment. This makes today's product better suited for fierce

competition.

Random vibration analysis is used by the structural analyst in many

ways. Many environments products are subjected to are considered random,

such as the beating of waves on oil rigs, to the noise emanating from

automobiles. A random environment simulation in product evaluation is

instrumental in predicting and assuring durability. Random vibration is also

used in excitation of products to determine dynamic characteristics, such as the

natural frequency, mode shapes and damping factor. A thorough

understanding of random vibration is invaluable to the analyst.

The study of random vibration is not trivial. This type of environment is

not simply described with just one or two words, but through the definition of

varying frequency spectra. To describe the environment acceleration,

frequency, and statistics must all be utilized. This is usually done with dynamic

signal analyzers which involve complex calculations. Measurement of signals,

characterization of the environment, and simulation are all possible with

sophisticated computer systems which involve analog- to- digital conversions

(A/D), signal processing, and Fast Fourier Transforms (FFT). The terms

correlation and spectral analysis are used used to describe the randomness of

the data and the environment. All of this is necessary to understand the

random vibration environment.

As with any measurements taken with sophisticated equipment, the

question arises as to the validity of the data. The popular adage "garbage in-

garbageout"

has haunted every data taking engineer. Especially with random

vibration where noise can look just as viable as the environment itself, care

must be taken in the winnowing process to separate the data from the noise.

Much work has been done in the field of random vibration to mentor the

inexperienced.

The goal of this thesis is threefold. First, random vibration basics are

detailed, and their use in structural dynamics analysis demonstrated. Second,

noise and error when related to random vibration and signal processing is

detailed. Third, the concepts are all tied together and demonstrated in an

example involving a beam. An aluminum beam is presented, which is evaluated

using several different modeling techniques. The beam is then subjected to a

rigorous dynamic evaluation, and the results compared to the various models.

Recommendations to aid in further work with both random vibration and signal

processing are presented at the end.

2.0 Background

The powerful Integrated Circuit (IC) revolution has changed the job of

engineers. No longer bogged down to solve complicated third and forth order

equations and systems, computer programs are available to determine the

behavior of components through modeling only theorized twenty years ago.

Finite element analysis is widely used by industry to model complex systems

and improve designs without wasting valuable time physically evaluating

hardware iterations. This has spawned the branch of engineering called

computer aided engineering (CAE). The area of test and measurement has

also prospered with the IC revolution with the introduction of affordable signal

processing and analyzers. 1 970 marked the year of the first commercially

available FFT analyzer by Time Data Corporation. General Radio Corporation

was instrumental in developing the algorithms used today to create random

vibration in the laboratory. Virtually any type of vibration environment can be

recreated in the lab from mixed mode vibration to triaxial shaker excitation.

Today the market is supplied by many manufacturers of analysis and test

equipment who compete for the business of structural dynamics analysts. The

IC has replaced the once expensive analog computer and filter equipment

available to the engineer with affordable high speed digital ones.

Random vibration is extensively used in engineering to describe

environments. With random vibration, the exact values cannot be precisely

determined for a given instant of time. Instead, a statistically valid estimate of

the environment is made to describe amplitude occurrences. The study of

random vibration involves probability as well as its application to

measurements. Many events are described by random signals, such as jet

noise and earthquake intensity, since there is no deterministic model that can

describe the unpredictable nature of the events.

Random vibration can also be used when studying the dynamics of a

system. Using random vibration, all frequencies in a predetermined frequency

bandwidth can be simultaneously excited. This can provide a quick method of

identifying natural frequencies of the system by examining the system input and

output. All natural frequencies are excited at once, which allows the

interactions between them to be examined. To the structural dynamics analyst,

this can be of vital importance when resonances have destructive interference

in a random vibration environment.

2.1 Fourier Analysis

A random signal appears as a complex waveform in the time domain. It

is not a trivial task to characterize. Fourier Analysis is used to interpret random

data since the frequency domain is better suited to describe random

environments. The assumption is made that all motion can be described in

terms of sines and cosines of varying frequencies which are harmonically

related. This was discovered first by J. Fourier (1768-1830). Functions are

described by the period and associated amplitude, which are combined to

create the Fourier series. For an example, consider the periodic waveform in

Figure 1. The Fourier series used to describe the signal consists of the

equation:

Periodic Signal

0n

c

*-0.

1-0.

-0.

n-0

.2

1

|.4-

0

z-\

4

6-

8-

1

-1.2 1 1 1 1 1 1 1 1

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

Time (sec)

Figure 1. Periodic Signal Example for Fourier Transform Example

F( l) := a0+ X! (a COS

("'W

0'

*)+ b

nsin

(n"<

0'

l))

where:

2- jr.

coq:= = fundamental frequency

T

a0-

1

rT

F(t) dt

n'

yF( t)-cosfn-co Q-tj

dt

T

F( t)-sinfn-co0-t) dt

For the example presented in Figure 1 :

ao = an = 0

n"

T

~2

F q-

sinin (n-

coq-

t) dt

bn:-

-F0-(i -cos(n-co0--

T-n-co0

bn'= F q- (l cos

(n-k)J

n-n

n'r

0n^jt

^0F(t)

:=4-

31 g(l-sin(nco0T))

F(t)=4-

ji

sin ( coq-

tj + sin(3-

coq-

t) + -

sin(5-

coq-

t] +

The Fourier spectrum would be appear as in Figure 2.

1 -I

0.9-

F 0.8-

u 0.7-

n 0.6-

c 0.5-

* 0.4-

10.3-

0.1 -

0-

Discrete Spectrum Example

* t

%t

m

A T0 0.5 1 1.5 2 2.5

Frequency (Hz)

1 1

3 \ 4.5

Figure 2: Fourier Transform Example

The operation itself is quite laborious and prohibitive when performed by hand

on a complex signal, but has been assimilated to the modern computer age

through the use of Discrete Fourier Transforms (DFT) where large sets of data

can be manipulated using this algorithm on computers.

The Discrete Fourier Transform is an approximation of the continuous

Fourier transform:

X(f) :=

00

-00

x(t)-e"j23tftdt

Continuous data is sampled at discrete time increments, dt. Then, Fourier

coefficients (Xm) are approximated on the sampled data(amplitude of

measurement at time interval dt) (xn) at frequency fm by:

8

x

1 -j2scf( ndt)

In 1965 J. W. Cooley and J. W. Tukey introduced a new method for determining

coefficients, involved with Fourier transform algorithm, which greatly reduced

the number of data points needed. This is the Fast Fourier Transform (FFT)

which forms the basics for much of the computations in equipment today . For

more information on DFT and FFT see references 2 and 7.

2.2 Modern Frequency Analyzers

Today, the dynamic signal analyzer is a basic tool used by anyone

studying structural dynamics. Analyzers can be multichannel, allowing for the

investigation of multiple inputs simultaneously. Not only can the motion of a

single point be determined, but the interactions between points can be found.

Time and frequency domain measurements are made in a matter of seconds.

Analyzers are flexible; parameters are manipulated to suit the data under

investigation. If peaks value are the main interest of the user, then averaging

techniques and windowing are easily manipulated to reflect peak hold data and

flat top windowing. Data is stored in basic formats for transfer into spreadsheets

for more complex mathematical manipulation through standardization of file

formats such as the Universal File Format (UFF) or Standard File Format (SDF).

Manipulation of data is not a roadblock for today's engineer. Armed with the

appropriate transducer, signal conditioning, and analyzer, engineers can make

any type of measurement.

2.3 Noise and Error

Noise and error plague engineering data. Sophisticated equipment is

haunted by ubiquitous power line noise (usually 60-Hz). The most meticulous

of measurements can be destroyed by a noisy environment or errors. The

recognition of errors and noise and methods to affect them are necessary to

assure viable data. Simple techniques exist, such as increasing statistical

accuracy by increasing the number of measurements used in averages, which

must be weighted against the cost of doing so in the competitive marketplace. It

is difficult to obtain several samples of a one time event, such as the launching

of a satellite. Engineering judgment must be relied on to make a "goodenough"

decision for the current atmosphere.

10

3.0 Random Vibration

Random vibration is a vital part of the analysis of dynamic systems. A few

applications of the concept of random vibration are: resonance identification,

modeling of complex systems, description of realistic environments. Definition

of basic terms used to describe random events is needed before proceeding

further into correlation and spectral analysis.

Random signal analysis is used to identify situations in which precise values

cannot be determined to identify the system, so best estimation is used.

Probability and statistics are needed to describe the properties of the

environment . The study of random vibrations deals with determining the

statistical characteristics of a system and how the output depends on the

dynamic properties of the system under investigation. Several terms are

needed to describe a random signal, such as average, mean square, variance,

and correlation. To understand them, a brief introduction to a few concepts in

statistics will be presented.

3.1 Basic Probability and Statistics in Random Vibration

A fundamental term in statistics relating to random signal analysis is probability

density function (PDF), which is given the symbol p(x). PDF is defined as the

length of time an event occurs between two given limits divided by the total

length of the event under observation. PDF defines the probability an event wil

occur over a given time period. In mathematical terms, it would be:

10

p(xo) = prob ( x0 < x < x0 + dx) = dt/T

where:

dt= the time xo<x<xrj -i-dx

T = the total period of investigation.

r x

Prob (xi<x<x2)

:=p( x) dx

To help understand the concepts of random vibration, a random sample can be

used to identify basic terms. Figure 3 displays a typical random event recorded

over a period of time. The probability that the x value will be in the range of xi

and X2 during the time period T identified is indicated by the shaded area.

1.6

s

i 1.2

g

n

I

0.8

0.4

Random Signal

i i i i i i i i i i i i i i i i i i

0 0.10.20.30.40.50.60.70.80.9 1 1.11.21.31.41.51.61.71.8

Time

Figure 3: Random Signal Example

11

For the example if the period is between 1 and 1 .6 for the time period,

and the probability that the signal is greater than 1 for the identified time, then:

T = 1.6- 1 =0.6

dt = 0.2 (shaded area)

then : p(x) = .2 / .6 =.33.

A Probability Density Curve is a plot of PDF vs. the event . See Figure 4

for an example. The area under the probability density curve gives the

probability. Referring to the example set in Figure 3, the probability the x value

being between x-| and X2 during the time period T is shown by the shaded

region in Figure 4. The probability density curve portrays the probability of x

without the time dependence.

Probability Density Curve

0.4

0.99 0.80 0.95 0.72 0.42 0.45 0.35 0.96 0.310.10 1 2

Signal

Figure 4: Probability Density Curve

12

Measurements of probability density and the probability density curve

can be made with a probability analyzer. The event under investigation is

sampled quickly over a period of time, then displayed. The values for the limits

are then entered and the probabilities determined. Many naturally occurring

events, when sampled over a long period of time, will display a bell shaped

curve which has been been termed the Gaussian Distribution. The formula for

the Gaussian Distribution is:

-,N2

(x

m)

p(x):=-Lr.e

where:

m = mean value of x

a= standard deviation

The Gaussian distribution is also referred to as the normal distribution. In

random vibration theory, the characteristics of many systems are approximated

by this distribution. Random Environments are described by a few key formulas,

such as mean value and standard deviation which will be defined next.

The mean value is a fundamental characteristic of the environment. When p(x)

can be determined for a random process, then a variety of characteristics can

be defined for the process defined as x(t). Allowing the expression E [ ] to

signify the expected value of the term in the brackets, the following definitions

are defined:

Mean value of x ,or average value of x, m ,

is given by:

13

E[x] = m=i:- x(t) dt

Relating the p(x) to the mean, since

dtZt:=p(x)- dx

Then :

E[x] =

00

-00

x-p( x) dx

The mean value gives the central tendency of a function or signal.

Similarly, the mean square value of x = E [ x2

] : average value of x 2

E [ xA2 ] = x(t)2dl

00

x -p(x) dx

Standard Deviation of x, a,and variance, a can then be defined:

a =E[(x-E[x])*2]

14

The variance is the square of the deviation of the x from the mean level

E [ x ] . It can also be shown that the variance is equal to the mean square

minus the mean squares or:

2

a=E[x2]-(E[x])2

Variance provides an indication of the dispersion of the variable about the

mean value.

Now that the basic terms used to describe random processes have been

defined, they can be used to describe environments. A sample function is a

single time history used to describe a random process. A collection of sample

functions is commonly referred to as an ensemble. The mean value of a

sample function can be found through summing the instantaneous values of

each sample function at the same time point, then dividing by the number of

functions(See Figure 5). The term stationary random process refers to a

process in which the characteristics (mean value and variance) of the ensemble

do not depend on absolute time. Stationary also means ensemble average is

equal to each sample average.

15

Acceleration

(g)

Time(sec)

Figure 4: Random Signals Time Aligned

3.2 Correlation

Correlation is used to determine how a function relates to another

function at a later point in time. Two types of correlation are instrumental to the

study of random vibration, autocorrelation and cross correlation.

Autocorrelation relates a function to itself at a later fixed instant of time. The

formula for Autocorrelation is:

Rxx(T) = E[x(t)x(t + T)]

or

RxxW =lim 1rT

T?oo Tx( t)-x(t + x) dt

Autocorrelation has the following properties:

16

1 . if the process x(t) is stationary, then the mean and standard

deviation are independent of t

2. it is an even function

For Cross correlation relates two different functions at a fixed later time:

Rxy(T) = E [ x(t) y(t+T)] and Ryx(T) = E [ y(t) x(t+T)]

or

rT

RXy(x) = lim j_.

T-^-oo Tx(t)-y(t + x) dt

In general, cross correlation if not even. It the time separation T is very large,

no correlation is expected. Correlation is commonly used to detect periodicity in

functions, predict signals in noise, and aid in the measurement of time delays.

3.3 Random Environment Definitions

With the definition of correlation, other terms used to describe random

vibration can be introduced. A nonstationary random process is one in which

the mean and autocorrelation vary with time. The terms weakly stationary or

stationary in the wide sense describe a special case in random vibration where

the mean is constant, but the autocorrelation depends on the time separation

(T) only and not time (t). Strongly stationary or stationary in the strict sense

implies that the correlation is time invariant. A nonstationary random process

has characteristics which depend on time; to characterize it, an instantaneous

average over the ensemble must be used. An ergotic random process is one

17

where the mean and autocorrelation computed for a sample are the same for

the entire function of a stationary random process.

Autospectral or Power Spectral Density (PSD) (Gxx (0) for a stationary

record represents the rate of change of the mean square value with respect to

the frequency. This term is used extensively in describing random vibration

processes, as it relates the vibration amplitude with probability, hence it is

instrumental to random processes. The values are estimated by computing the

mean square value in a narrow frequency bandwidth at central frequencies,

then dividing the value by the frequency bandwidth. The total of all mean

square values divided by frequency gives the total area of the function and is

the total mean square value of the function.

Spectral density functions are commonly used for: determination of

system properties when random vibration is involved, specification of random

vibration test profiles, and identification of energy and noise sources.

Fourier analysis ties together the time and frequency domains of a

signal. The coefficients of the Fourier series can be plotted against frequency

producing the Fourier spectrum. Computers are instrumental in performing

Fourier transforms utilizing the fast Fourier Transform (FFT) algorithm which

minimizes the time to compute values.

The Fourier integral is defined as:

18

X(f) :=

00

x(t)-e~i2ltftdt

with the inverse transform:

x(t):=

00

-00

X(f)-ei2jlftdt

where the integral can be considered as a summation of the continuous

spectrum of sinusoids.

Spectral Density is defined as the Fourier transform of the correlation

function. There is autospectral density defined as:

S(f) :=

XXRxx(x)e l2*fTdx

with the inverse:

RxxG):=

and cross spectral density defined as:

Sxx(

f)-e5tT

dx

Sxy(f) :=

00

Rxy (x) e

1 % x

dx

with the inverse:

19

Rxy(,):=

00

Sxy(f)-ei2:ifTdx

The units for spectral density are the (mean square value)/ (unit of frequency).

Used commonly in vibration is acceleration spectral density or power spectral

density described as gA2/Hz. A graph of a typical random environment of this

type is shown in Figure 6. The mean square value of a stationary random

process is the area under this graph which is also Rxx(O).

Power

Spectral

Density

(gA2/Hz)

0

Frequency (Hz)

Figure 6: Typical Random Vibration Spectrum

Two common terms used to describe processes are narrow band and

broad band. As the names imply, a narrow band process has spectral density

which occupies only a narrow band of frequencies. A broad band process

consists of a wide frequency band of excitation. These types of processes can

be used in dynamics analysis to excite a structure. If a particular frequency

20

range is known then narrow band excitation can be used to excite a small

range. If there is no indication of frequency range to use, then a broad band is

typically selected to provide an indication of further excitation needs.

If the frequency range of interest is limited to positive frequencies only ( from 0

to positive infinity) then one sided spectral densities are of interest. These are

defined as Gxx( f ) and Gyy( f ):

Gxx(f) = 2Sxx(f) andGyy(f) = 2Syy(f).

in which f > 0. This is feasible, since S(f) is even. Typical values of S(f) and G(f)

are presented in Table 1.

Table 1 : Typical values of S(f) and G(f)

Frequency(Hz) S(f) (gA2/H;z)G(f)(gA

0 0.002 0.004

5 0.0052 0.0104

10 0.036 0.072

15 0.009 0.018

20 0.6 1.2

25 0.003 0.006

30 0.0025 0.005

35 0.058 0.116

40 0.006 0.012

45 0.0063 0.0126

50 0.351 0.702

21

4.0 Data Analysis

Data analysis for the structural dynamics analyst typically involves the

frequency response of a structure instrumented with transducers. Basic terms

and applications are presented in this section. Examples are displayed in

Chapter 7 with the aluminum beam example.

4.1 Frequency Analyzer

Frequency analyzers are used throughout industry as a tool in

understanding system behavior. Whether the system is a vibrating mass or a

production line a dynamic signal analyzer will make a physical measurement

then present it in a format the user requires.

A frequency analyzer consists of a bank of variable frequency narrow

band filters with an rms meter to display the filter output. The first analyzers

were analog, which could be very cumbersome and large depending on the

functions it had to perform. Many analyzers today accomplish the filtering

through digital signal processing (dsp) chips, so analyzers can be portable

hand held units. See Figure 7 for a block diagram of a typical frequency

analyzer setup.

Input x(t) Narrow Bandpass filter w/

bandwidth = df and center

frequency= to.

Square

and

Average

Divide by dfGxx(fQ)

Obtain x(fo, df, t)

Figure 7. Analyzer Block Diagram

22

4.2 Measurements Functions

Analyzers are flexible instrumentation. With the appropriate hardware

configuration, structures as large as buildings and as small as electronic

boards have their dynamic characteristics identified. Three common

measurement domains are the time, amplitude, and frequency. Other

measurements are available as special features of equipment manufacturers,

such as Bruel and Kjaer's Cepstrum analysis.

4.2.1 Time Domain Analysis

Time domain analysis usually consists of a time record and the

autocorrelation and cross correlation functions associated with the signals. This

is the type of trace which can be found from an ordinary oscilloscope. Time

domain analysis can be used as a quick method of monitoring signal integrity.

When starting any analysis, a quick view of the transducer prior to measurement

can indicate problems with instrumentation setup. Time domain analysis

(sometime a referred to as running mode) and operating mode deflection are

two analysis types which takes a structure operating in its normal condition and

displays the dynamics as a function of time. It is distinguished from modal

analysis, in that it is time based instead of frequency based. It has the ability to

show product interactions during operation. Analyzers start with data from the

time domain to form frequency and amplitude domain measurements.

4.2.2 Amplitude Domain Analysis

Amplitude domain analysis consist of histogram, probability density

functions, and cumulative probability density functions (See chapter 3 for

23

details). It can provide an indication of data nature ( is it Gaussian or not ?) or

give minimum and maximum data values over the data record. This involves

general statistical analysis.

4.2.3 Frequency Domain Analysis

Frequency domain analyses are important to the structural dynamics

analyst in understanding the dynamic characteristics of the structure. Power

spectrum (both cross and auto), linear spectrum, frequency response,

coherence, and spectral maps are all typical measurements. Power spectra (for

details see chapter 3) are one-sided, versus linear spectra which are two sided.

Linear spectra are the result of direct FFT of a time signal. Spectral

measurements are useful in identifying frequencies of interest ( resonances,

operating frequencies, cutoff frequencies, etc. ).

Frequency response is a two or more channel measurement. One

channel is typically designated the input, and all others the outputs. The

frequency response function of a system, H(f), is defined as:

H(f) :=

00

h (x) eJ T

dx

where:

h ( x ) = the unit impulse function

describing the system

Using complex polar notation to describe frequency response:

H(f) :=|H(f)|-e~j*(f)

where:

24

I H( f ) | = gain factor of a system

<K f )= phase factor of a system

H( f ) is the Fourier tranform of h(x ).

Many engineers use the terms "frequencyresponse"

and "transferfunction"

interchangeably. This should not be practiced, since a transfer function more

accurately describes the Laplace transform of the unit impulse response

function.

The coherence function is also used quite commonly in the study of

system dynamics. Coherence is the ratio of the absolute value of the cross-

spectral density function squared, to the product of the two comprising

autospectral density functions:

^ GXX(f)-Gyy(f)

Coherence is a measure of the accuracy of the linear input/output model

assumed for the system under investigation. Coherence is used successfully in

identifying errors, as will be discussed in chapter 5.

Spectral maps, or waterfall analyses are a three dimensional displays of

data. Typically another parameter, such as revolutions per minute (rpm) for a

rotating piece of machinery, is used to align the typical power spectrum while

conditions are varied. This type of analysis is useful in distinguishing between

system natural frequency and forced vibration. A natural frequency will be

constant on the Waterfall, following the skew of the alignment, while a forced

25

vibration will vary following the third parameter (rpm). See Figure 8 for a

spectral map example.

6320 RPM Spectrum from spectral map

I 13Spectral Map Chan 9 6.32kRPM

Mag

Peak!

gA2fflz

JL A A JUma JL <& iffli

6.48k

0 Functn Lin Hz g^larkerg RCLD

Spectral Map Chan 9

RPM

1.064k

Vibration

Level

g'2/hz

Frequency [hz]

ilA.

3.2k

Figure 8: Waterfall Analysis Example

26

5.0 Noise and Error

The accuracy of measurements made concerning random vibration

signals must be examined to understand the nature of the environment. When

dealing with random data sampling, this type of investigation is extremely

important, due to the unpredictable nature of the subject. The common scenario

of "garbage in / garbageout"

can be the pitfall of an inexperienced engineer.

Error is inherent to random measurements, since measurements are only a

sample of the environment. The sampling technique also produces error in the

data.

5.1 Accuracy of Measurements

Errors are a fundamental problem in any type of experimentation.

Repeatability and reproducibility is a problem with random data. How much

variation can be allowed to characterize the process must be determined.

When analyzing data, only a sample of the infinite ensemble is used. With the

constraints of the analysis hardware ( memory, speed, timing, etc), only a limited

length of a given sample can be analyzed.

When measuring spectral density, a basic dilemma must be resolved

with each measurement. In order to achieve high resolution, the filter

bandwidth of the analyzer, Be, must be small. Filter bandwidth refers to the

frequency span a filter in the frequency analyzer is made to measure. If a filter

is set to measure between 20 to 25 Hz, the Be would be 25-20 = 5 Hz. But, to

achieve good statistical reliability, Be, must be large when compared to the

27

frequency. To compensate for this dilemma, averaging time must be set to a

large value, so the sample function must last for a long time. With many

functions this is possible, but for some events, such as rocket launches, long

samples and repetition can be impossible.

Frequency analyzers are also susceptible to a steady state or bias error

when the filter bandwidth covers a range of frequencies where the spectral

density is changing rapidly over the frequency band. Again the solution to this

is to use an analyzer with narrow enough frequency band and take data over a

long period of time. The speed of digital techniques have made them very

popular in the field of frequency analysis. Digital speed compared to analog

filtering becomes a major factor since it allows for longer time records to be

made reducing error. Taking longer samples with more averages can answer

some error problems, but there is much more to understanding frequency

analysis techniques.

5.2 Fourier Transform Algorithm Error

To understand how a frequency analyzer calculates the Fourier

Transform of a signal, consider the sine wave in Figure 9.

28

A

m

P 1 1

Sine Wave

x,o 11 XJ2.If...

1

i

t

XfJtJt

(- 1 ' ^

u

d

e

i 0^<o3

XjKit-*

U "1 w i

n 0 15 30 45 60 75 90 105120135150165180195

GTime (msec)

Xr = X(t = r*dt)

Discrete Time Series { Xr }, r= .... -1.0, 1.2, ...

Figure 9. Sine Wave for Fourier Transform Example.

The analyzer will determine the statistical characteristics of the function

by manipulating discrete values of Xf. Given the equation for discrete Fourier

transforms:

x(t):=

ag + 2-

Vj [ajc-cos(2-jfk-

j + b j,-

sin f 2- % k-

]]k=i

where:

ak- x(t)-

cos[2- ji- k-

] dt

0

bk:=T x(t)-sin(2-3i-k--Jdt

If the continuous time series is unknown, and only equally spaced samples are

known; we have a sequence of samples (xr), where r = 0,1,2,... (n-1) where and

t =r*

dt. dt=T/N.

29

Replace the above integral with a summation:

1 ^ -i(2,.|).(r.dt)Xk'=

T2-.Xr'C dt

The Xk represents the whole area under the curve (See Figure 10). If T=N*dt is

substituted then:

A

m

P i -,

i

t

u

d

e

n

G

Sine Wave

x( t)-cosxio *" x:2

0 15 30 45 60 75 90 105120135150165180195

Time (msec)

Figure 10: Sine Wave used in DFT

H-l "i-2-jtk-r

Xk:=^SXre N

r=o

The discrete Fourier transform (DFT) of the series { xr}, r= 0, 1, 2, ..., (N-1) is:

Hi)M-/

Xk:=^X>fe

k-o

withk=0, 1,2, 3,..., (N-1).

The inverse discrete Fourier transform (IDFT) is then:

30

xr :=X!XkeHi)

with r= 0, 1, 2, 3,... (N-1). With using the DFT method, the smaller the sampling

interval, the closer the approximation.

Problems arise when trying to calculate values for x|< greater than (N-1).

If k=N+L, then:

.

i-

Xn + L:=N"EXre

2jtl(N+ L)N

Simplifying,

Xn + L:=irZXr.P \ N /. -i-2-KT

but,

i 1 Jt-r ._

no matter the value of r, so x n+[_= x |_. Thus the coefficients of x k just repeat

themselves for k>(N-1) causing distortion, which is called aliasing. The Nyquist

frequency is defined as:

At =sampling time

Sampling Frequency^=Sampling Rate

At

Nyquist Frequency= --SamP|in9 Frequency

2- At

31

This is the maximum frequency that can be detected from data sampled at time

spacing dt. The physical result of sampling at a frequency too low for a system

is that higher frequencies are missed in measurements.

The DFT method of Fourier transform was replaced when J. W. Cooley

and J. W. Tukey introduced the Fast Fourier Transform (FFT) in 1965. This

algorithm streamlined the computation of the Fourier Transform by changing the

method used to interpolate constants in the trigonometric polynomial. This in

combination with the integrated circuit revolution have made frequency

analyzers available to industry. The FFT equation computes the complex

coefficients Ck instead of ak and bk with the formula:

2m- 1

F(x):=l-Vck-eikx

k=o

where:

jiijk

c^:=yi-em

k

k

for each k = 0, 1, 2 2m -1.

Time and frequency domains are usually related through FFT and IFFT in

the frequency analyzers today instead of DFT and IDFT.

An outline of the process frequency analyzers use for spectral estimates

is:

1 . A discrete time series is created by the results of sampling the

continuous event under observation. Data generated are {Xr} and {Yr>

32

for r= 0, 1, 2, 3,..., (N-1) at a time interval dt=T/N.

2. The Fourier Transforms for {Xr} and {Yr} are calculated.

3. Discrete series for spectral densities are determined from {Xr} and

{Yr>. Appropriate spectra are then derived.

4. IFFT is then used to determine the corresponding correlation

functions.

5.3 Error Classification

There are two classes of error when dealing with random vibration, bias

error and random error. Random error is the result of the nature of the data, the

unpredictability from one sampling to the next. The incident of random error is

high when sampling is limited. Bias errors are consistent errors in data which

are due to the processing of the data to the appropriate format for the user. This

can be due to calculation inefficiencies. Combined these errors pollute

measurements being made. This is one reason why they must be identified and

an understanding of how to work with them must be attained by the user of

spectral analysis and correlation

5.4 Noise

To understand noise in a system, use a ideal single input/ single output

system as modeled in Figure 11.

33

x(t)h(x),H(f)

y(t)

Figure 11: Ideal Single Input/ Single Output System

Under ideal conditions:

y(t):=

00

h (x) x (t - x) dx

where h(x) is the transfer function for the linear system

Evaluation of a system frequency response function can be performed from

measurement of x(t) and y(t) using the equation:

H(f)= Gxy(f)/Gxx(f)

H(f) is a complex number which can be represented by:

H(f) = HR (f)- i H i (f)

Frequency response polar form is:

34

H(f) := |H(f)|-ei*(f)

|H(f)| :=(HR(f)2

+H!(f)2)

(|)(f):= atan

:the gain factor

Hi(0\

HR(f)j': the phase factor

If the assumption is that the input x(t) is from a stationary random process, and

data is accumulated over sufficient time so that the transients die out, the input/

output auto spectrum relation is:

Syy(f) = I H(f) 1 2 Sxx (f) : two sided

Gyy(f) = I H(f) 1 2 Gxx (f) :one sided

And the input/output cross spectrum relation is:

SXy(f) = H(f) Sxx (f) : two sided

Gyy(f) = H(f) Gxx (f) :one sided

These are true when there is no extraneous noise, the system is not time

varying, and there are no nonlinear characteristics.

Noise is introduced to the system as in Figure 12:

35

n(t)

u(t)h(T),H(f)

V(t)

^ y(t)

m(Wlux(t)

where: u(t) and v(t) are the true signals

m(t) and n(t) are noise terms.

Figure 12: System with Noise at Input and Output.

Now the terms x(t) and y(t) consist of:

x(t) = u(t) + m(t) and y(t) = v(t) +n(t)

With noise,

Gum(f) = GVn(f)=Gmn(f)=0.

The system spectral relation is:

Gw(f)= IH(f)|2Guu(f)

And the cross spectrum is:

36

GUv<f) = H(f) Guu(f).

The original system spectral density functions are now:

Gxx(f) = Guu(f)+Gmm(f)>Guu(f)

Gyy(f) = Gvv(f) + Gnn(f)>Gw(f)

Gxy(f)=Guv(f)

The coherence is defined by the following:

2._(|Guv(f)[):

Yuv(f)!=

Gmi(f)-Gvv(f)uuvw ~vw -

=s true coherence

vm*._ QGxy(nD

XV -("^|f|is measured coherence.

In an ideal case, coherence is unity. In actual experimentation, coherence

varies from zero to one due to:

1 . Noise in the measurement.

2. Bias errors in the spectral estimate.

3. Nonlinearity in the relationship between x(t) and y(t).

4. The output y(t) affected by inputs other than x(t)

simultaneously.

37

Relating the two:

IGxy(f)l2= IGuv(f)l 2 = IH(f)|2 Guu2(f)= Gw(f)Guu(f)

IH(f)|2= Gw(f)/Guu(f)

2.

Ouu(f)-Gyv(f)Yxy(0

"=

(Guu(f)+Gmm(f))-(Gvv(f)+Gnn(f))

which is less than or equal to 1 .

If input noise exists in a system, then there are two cases:

1 . The noise will pass through the system and:

-Hxy(f) will be a biased estimate of H(f) in gain and phase.

-If Gxm(f) is zero, Hxy(f) is unbiased.

2. The input noise does not pass through the system then:

- gain will be a biased estimate of H(f), but phase will not be

biased.

- phase will however depend on the ratio Gmm(f)/Guu(f) known as

a signal to noise ratio.

- GXy(f) will not be influenced, and hence will give a true estimate

of Guv(f)-

38

If the system has noise in the output only (See Figure 13):

n(t)

x(t)h(T),H(f)

v(t)

"*- y(t)

Figure 13: System Model with Output Noise only.

For the model:

input is virtually noise free

the output is composed of v(t) which is due to x(t) and other

deviations present.

From the coherence of the noise in the system, eliminating input noise, the

relationships:

Gyy(f) := Gvv(f) +Gnn(f)

Gvv(f) :=Yxy(f) -Gyy(f)

Gnn(f) := (i-Yxy(f)2)-Gyy(f)

Thus coherence when there is noise at the output is between zero and one, it

cannot be one.

39

5.5 Frequency Response Errors

Estimation for frequency response in single input/ single output systems

involves both random and bias errors. It is extremely important that these errors

are understood, and if possible minimized, for data analysis to be credible.

Coherence functions will aid in the detection and classification of errors, so it is

good practice to analyze coherence whenever making frequency response

measurements.

5.5.1 Random Errors in Frequency Response

Random error in frequency response is due to these common sources:

measurement noise in transducers, computational noise in data processing,

other unwanted inputs, and nonlinearity between the input and the output.

Random error is related to the coherence and number of averages used in

spectral density calculation. To minimize random error these steps should be

utilized:

1. Minimize noise in instrumentation by using appropriate equipment

(high quality transducers when available, watch grounding loops,

etc.)

2. If the coherence falls broadly over a frequency range and the

frequency response is relatively low, extraneous noise at the output

should be suspected. This can be due to uncorrelated inputs as well.

3. If the coherence falls broadly over a frequency range, the

frequency response is not relatively low,and the input power spectral

density is small, then there could be input noise being measured.

40

Bias errors could also be caused by this scenario.

4. If the gain is sharply peaked, as in the case of a system with slight

damping, coherence will peak at corresponding frequencies. This is

due to high signal to noise ratios at these frequencies. If the

coherence does not peak or even worse notches, then nonlinearities

can be suspected. These results may also be indicative of bias errors

in the data.

5.5.2 Bias Errors in Frequency Response Functions

There are several causes of bias error in frequency response function

estimates. Some of the main ones are: noise at the input which does not pass

through the system, spectral density estimation errors due to resolution

problems, unmeasured inputs, and system nonlinearities. To detect and

minimize these bias errors the following guidelines can be followed:

1 . Minimize the noise at the system input when possible.

2. Choose a sufficiently narrow resolution bandwidth, Be, for the

measurement.

3. If coherence falls broadly over a frequency range where the frequency

response is not at a small value and the input spectral density is low,

then input noise is indicated.

4. If coherence notched sharply at the frequency response sharp notch

or peak, then error can be due to either inadequate spectral resolution

or system nonlinearity. This can be easily determined by varying the

spectral resolution and repeating testing. Improved resolution with an

41

increase in coherence will confirm resolution problems. Nonlinearity

is the case otherwise.

42

6.0 Theoretical Analysis

A small aluminum beam is analyzed using techniques presented in the

previous sections. This section details an exact analysis of a cantilever beam

and an assumed modes approximation. The beam is considered a system with

continuously distributed mass and elasticity. It is assumed to be homogeneous

and isotropic with displacements in the elastic region described by Hooke's

Law. The natural frequency and mode shapes of the beam are determined for

the continuous beam system using analytical techniques. Random vibration is

used to excite the beam at the base and the frequency response of the beam

determined using assumed mode methods.

6.1 Exact Solution with Free Vibration

Examining the lateral deflections of a beam without rotary inertia, the

following model is developed. Consider the beam in Figure 14.

/

/

/

/gw V-

dx dx

M +

J

/6M\ck

jdx

Figure 14: Beam Model for Continuous System

Summing all forces in the Y direction gives,

43

p-A-dX-

2

6 Y

6t

,6V,

V + | | dX

SX,

+ V

Where:

p = Mass/ Volume

A = Cross Sectional Area

Simplifying,

p-A-dX-

2

8 Y

St,8X;

dX

p-A-

6 Y 5V

6t6X

Summing the moments about point o yields the following:

SM dX / SV \ dX-dX + M-M- V- - V + dX

SX2 \ SX /

2

If higher order terms are ignored, the equation is simplified to:

44

SMV :=

SX

From strength of materials, M is known to be:

2

6 Ym .=

m-

2

SX

Making the substitution,

2 2/2

8 Y 6 / S-Y'

pA- := - H-r

2 2 1 2

St St \ SX

which is the governing differential equation of a beam when rotary inertia is

insignificant.

If the beam is considered continuous with uniform material, El is constant, the

following equation is derived:

2 4

6 -Y H S Y+ := o

pa

Assume that the solution is of the form:

45

Y(X,t) := Y(X)-SIN((ot)

Substituting into the equation:

2_ e s4y-co Y + := o

PA

This equation eliminates the dependence on time, and has one independent

variable. The partial derivative becomes an ordinary derivative:

. 2

d4-Y pAcoHY := 0

dX4 e

Make the substitution:

P:=

2

pAco

e

And the equation is sirrlplified to

d4-Y

dX4

4

The solution of this equation is assumed to be of the form,

46

_ a.X

Y :=C.-e1

1

Then,

CaD4- e_1"

_ ci- e_l

, a.X 4 a.XCYaV-e1

:= o

Solving this equation,

D2*1/?

or

^

The four roots to the equation are:

aj:= 6 a2

:= ~B

a3

:= iB a4:= -iB

So the solution becomes:

Y(X) := C +C2-e"pX

+C3-eipx

+C4-e"ipX

47

Examining the exponentials, the following relationship can be used to evaluate

the equation:

e"px

:= cosh(BX + sinh(Bx))

e"ipx

:=cos(bx) + sin(px)

And thus,

Y(X) :=

Acosh(Bx) + Bsinh(Bx) + Ocos(bx) + Dsin(Bx)

where A, B, C, and D are all constants of integration.

For the cantilever beam used in the investigation the boundary conditions:

Y(o,t) :=o=>Y(0)

= 0

SY

-(o,t):= o

SX=> Y'(0)=0

2

8 Y

M(L,t) := H- - 0

^Y"(L)=0

3

8 Y

V(L,t) :=e-^-0

8X=> Y'"(L)=0

Making this set of substitutions yields:

Y(0)= 0 = A +C= A = -C

48

Y(0) :=

p-(A-cosh(px) +Bsinh(px)) -Ccos(px) -Dcos(px)=Q

0 := p-(B + D)= B = _D

Y(L) := p2- (Acosh(pL) + B-sinh(pL))-

C-cos(pl)-

Dsin(pL) _

= 0

Simplifying into previous equations and solving for A/B and equating results

yields:

cosh (pi.) + cos (pL) _

sinh (pL) + sin (pL)

(sinh(pL) -sin(pL)) (cosh(pL) +cos(pI_))

This results in the following equation:

cosh (pL) cos (pL) + l := o

The first six roots are:

p i-L:= 1.875

49

2"L:= 4.695

P o-L'= 7.855

P 4-L := 10.996

P trh := 14.137

P fiL "= 17.279

Solving back to frequency yields:

a, =PL2' S

4

pAL

The corresponding eigenvalues are:

coj:= 3.52 '

H

4

pAL

50

eto o

:= 22.0-

I I 4

pAL

to3

:= 61.70-e

pAL

ea,4:= 120.91- I

J pAL

to5

:= 199.86-

e

pAL

to6

:= 298.56-

e

pAL

The eigenfunctions are determined displaying the deflection shape at the

natural frequency. For a specific frequency make the appropriate substitutions:

51

Y j( X ) := A-cosh (& -

x)+ B-

sinh (p {- X\ + C-cos (p -

x)+

D-sin (p {- X)

since C = -A and D =-B, then

Yi(X) := (cosh (p fX)-

cos (p j-

x)) ^ + sinh (p f x)-

sin (p fX)B

Y j( X ) := A-cosh fp - X) + B-

sinh (p {X\ - A-

cos (p - X\ - B-sin (p - X\

The mode shape is given by this function, not exact values, thus B can be

eliminated and the shapes defined by:

Yi(X) :=(cosh(prL-) -cosfpi-L-^)+ sinh (p f L-^ -sin(pfL-B L

For the identified frequencies the following mode shapes are identified:

52

Mode Shape 1

z=x/L

z :=o,.i.. lPL := 1.875

Y j(z) :=cosh(PLz)-cos(pLz)+ ( -.734095 )( sinh( pL-z)-

sin( 0L-z) )

z * lU)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.034

0.128

0.273

0.46

0.679

0.922

1.182

1.451

1.725

2

Figure 15: Mode Shape 1 from Exact Solution

Mode Shape 2

z=x/L

z :=0,.l.. 1PL := 4.694

Y 2( z)=

cosh( PL-z)-

cos( pLz) +" 1.018467-

( sinh( PLz)-

sin( pLz) )

z Y2(z)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.185

0.602

1.052

1.367

1.427

1.179

0.634

-0.14

-1.047

-2

Figure 16: Mode Shape 2 from Exact Solution

53

Mode Shape 3

z=x/L

z :=o,.l.. 1PL := 7.855

Y 3( z):=

cosh( PL-z)-

cos(PL-

z) +- 1.00-

( sinh( PL z)-

sin(PL-

z) )

Y3(z)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.456

1.208

1.509

1.043

0.019

-0.992

-1.41

-0.997

0.002

1.001

Figure 17: Mode Shape 3 from Exact Solution

Mode Shape 4

z=x/L

z :=o,.i.. lPL := 10.996

Y4(z) :=cosh(PL-z)-

cos(PLz) + "1.00-( sinh( fSL-z)-

sin(PLz))

z * 4U)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.77

1.508

0.868

-0.63

-1.41

-0.64

0.832

1.397

0.437

-1

Figure 18: Mode Shape 4 from Exact Solution

54

Mode Shape 5

z=x/L

z:=0,.l.. 1pL := 14.137

Y 5( z):=

cosh( PLz)-

cos( PL-z) +- 1.00-

( sinh( pL-z)-

sin( PLz) )

Y5(z)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

2

-2

1.074

1.3191 1 1 1

1 1 ! 1

-0.423

-

1.393

7.335- 10~4

1.397

0.437

-1.26 0 0.2 0.4

z

0.6 0.8 1

-0.831

1

Figure 19: Mode Shape 5 from Exact Solution

Mode Shape 6

z=x/L

z :=o,.i.. lPL := 17.279

Y 6( z):=

cosh( PLz)-

cos(PL-

z) +- 1.00-

( sinh( pL z)-

sin( PL z) )

z *6u)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

1.322

0.674

-

1.339

-0.22

1.414

-0.221

-1.345

0.642

1.144

-1

Figure 20: Mode Shape 6 from Exact Solution

55

6.2 Approximate Methods for Base Excitation

Random vibration excitation is used to determine the dynamic

characteristics of the beam. Consider the beam model in Figure 21 with the

base excited in the direction perpendicular to the beam.

* x

Yb

y(x, t)

/////////

Figure 21 : Cantilever Beam with Base Motion Excitation Perpendicular to Beam

The.assumed mode method assumes that the solution to the boundary-

value problem is of the form:

y(x,t):=yb + X(t)i(x)'qi(t)

i

where:

9 j Cx ) = shape functions

q i ( t ) = generalized coordinates

Lagrange's equations are used to approximate the equations of motion. Using

the following definitions:

56

p,(x) = mass/length

E = Young's Modulus

I = Area Moment of Inertia

f(x, t)= force/length

The equations for energy for a cantilever beam with base excitation can be

defined. For a continuous system kinetic energy is:

rL

KE := -

2/5^q i<t> A -nO) dx

Potential energy is:

rL

U := -

2

e f?" 9 A -p(x) dx

To accommodate damping, use the Dissipation Function:

L

2 (?q

i"

<t> A -c(x) dx

For generalized forces use:

rL

Qi:= f(x,t)-(|)-(x)dx

Lagrange's Equations is then defined as:

57

d /6KE\_

6KE

dtUqJ Sqj

6U+ :=Qi

qi

where i = 1, 2, ...

Since the KE does not depend on coordinate q j (t) and U does not depend on

*

velocity q j (t) Lagrange's equation becomes with damping and generalized

forces:

d /ske\ sd su

dt^Sq^ fiqj 6qi

In matrix form:

[M]{af(t)} + [C]{dr(t)} + [ K]{cT(t)}=[Q]

Set:

y(x,t)=

yb + ^]q1(t)-i|)1(x)

or

y=

yb +X^i'^i

where:

i|) ( x )= Normal Mode or a shape function

This creates a system of N generalized coordinates or degrees of freedom:

58

q 1 (t). q 2(t). q 3(t) qn(t)

Which provide N equations of motion:

* .

[ M ] nxn 'q + [ C ] nxnq + [ K] nxnq = Q

with:

M |x(x)-i|ji(x)-i() j(x) dx

L

c(x)-t|j j(x)-i|jj(x) dx

K

L

e-ip-(x)-ii>j(x) dx

To effectively perform calculations, let z= x/L ,a dimensionless variable and

replace integration limits from 0 to 1,instead of x= 0 to x= L.Thus:

l

M^L-

H"9 i(z)"9 j(z) dz

Clj:=L-c-ijj i< z)-i|> (z) dz

Kr=

Ict

d d . , x d d, , v jH-

i|>i(z)-

i|j :(z) dz

dz dz dz dzJ

59

r 1

Q i:= L-

f(z,t)-i|)(z) dz

To analyze the vibration of the beam, let:

"! i( z):=

<j>i(z)

Using the results of the exact analysis for normal modes, and substituting into

the defined equations:

K .=

1 0 0 0 0 0

0 1 0 0 0 0

M;-

0

0

0

0

1

0

0

1

0

0

0

0

0 0 0 0 1 0

0 0 0 0 0 1

'

1 0 0 0 0 0"

0 1 0 0 0 0

C :=

0

0

0

0

1

0

0

1

0

0

0

0

0 0 0 0 1 0

0 0 0 0 0 1

12.360 0 0 0 0 0

0 485.481 0 0 0 0

0 0 3806.629 0 0 0

0 0 014617.05-

0 0

0 0 0 0 39941.929 0

0 0 0 0 0 87094.658

60

Q:=

.783

.434

.254

.182

.141

.116

Since the normal modes are orthogonal, the equations are decoupled to form

the following set of equations to describe the beam with base excitation:

jiLMj-qi + cLCj-qj + qj:=-pLQj-y b

L

for i = 1 to 6.

This system can be simplified by dividing through by L and substituting values

for Mj and C\:

H-K

p-qj + c-qj + qi:=-pQryb

The frequency response function of this system can be determined by takingthe Fourier transforms ( FT( ) )of both sides:

FTK

a>% + ica> +-i)-FT(q>) ^-pQj-FT^y^)

-M-Qi

2 EIKi-

co p + icco +

Make the following substitution,

61

FT

EIKjk, := -

1L4

then,

(qj) ._

"HQi

FT(yb)2

-

co p + icco + kj

Furthermore,

Fr(q*i)

FT(yb)

2-

pQ- co

2-

co p + icco + kj

If the Frequency response function H is defined as:

FT [ output ] = H ( co)*FT[input]

where: FT [ ] Fourier transform of |

then,

dHwir, (*V-Qi>2

-

-2

( kj p co J (ceo)

can be used to solve for the response of the system when excited with random

vibration at the base. See Figure 22 for example.

62

A^Ks "^-1 H I A2 *Syb

hkSyb

Frequency

So

tFrequency

Figure 22: Frequency Response Function Example with Random Vibration Input

for Single Degree of Freedom System

SqW:= (|H(co)|)2Sy(co)

Vibration measurements are typically measured using acceleration units

instead of displacements. The equations derived can be transformed to

acceleration equivalents:

y:=

y b+ ^ i( x)q i( x>

Displacement

y:=yb + ^i(x)-qi(x)

Acceleration

The Fourier Transform FT [ ] can be represented by:

FT[y]= FT[yb]+ ^/FTtqj]

For 1 function, the acceleration shape function is:

63

FT[y]= FT[ybr l + (-coj-ij) x(x-pQ

)

-

co p + icoc + (kj)

thus,

Hv(co):= 1+ (-co2) -9 1(x)-T

-pQ

-

co p + ICOC + (K ])

For two functions:

Y(x,t)=yb(t) +^ j(x)-q j(t) +H)2(x)-q2(t)

and

y:=yb + ^ i(x)-q 1

+ iMx)'q2

FT[y] = FT[yb]+ ^l(x)FT[q*1] + ^2(x)FT[q2]

which can be shown to be:

FT[y] =FT[yb]*

1 +

co -pQ j-tIj x(x) co -pQ2"9 2(x)

2 2

co p + icoc + (k -A-

co p + icoc + A:9)

and

Hv(co):= 1 +

co -pQ j-ip x(x) co-pQ2-^2(x)

-

co p + icoc + (k j)-

co p + icoc + A:2)

64

This technique can be expanded for six functions:

FT[y] = FT[yb]*i+l>n<x>-pco Qn

-

co p + icco + ^kn"\

and

Hy(to) := i + ^]^n(X).pco Qn

co p + icco + (k ^

65

7.0 Experimental Analysis

An aluminum beam is fixed to a small modal shaker through an

aluminum support (See Figure 23). The beam is instrumented with four

accelerometers located at the support, the end of the beam, and equally

spaced along the beam. The system is then excited at the base with broadband

random vibration and the response of the beam is measured.

Accelerometer (Accel.) corresponds to

channels on analyzer

Accel. 1

_TL

Accel. 4

D-

Accel. 3

D-

Small

Support

Accel. 2

Pi

Small Modal

Shaker

Aluminum Beam2"x9.5"x1/8"

Figure 23: Experimental Setup

7.1 Equipment

A small 50 Ibf modal shaker was used to excite the beam. A random

noise generator supplied the signal to the shaker amplifier. Four DJB Industry

accelerometers, model 28 ,were selected due to their high charge output with

low mass. This would minimize mass loading effect on the beam's dynamics.

A Hewlett-Packard (HP) 3566A analyzer was selected to make all

measurements. This analyzer is capable of time, frequency, and amplitude

66

domain measurements. The analyzer has a usable frequency range from DC to

12.8 kHz, which is ample for the beam investigation.

All measurements were tape recorded with a Kyowa Dengyo 21 channel

Beta tape recorder for future analysis. The tape recorder has a 46 dB signal to

noise ratio and was set to have a viable frequency range of 2.5 kHz.

7.2 Damping Analysis

The damping ratio of the beam was determined using log decrement. The

beam was supported, and the accelerometer attached to the free end monitored

while the beam was struck. The motion was allowed to decay while being

recorded. Using the following equation for log decrement:

xillog decrement 5 .

= ln

\X2J

See Figure 24 for damping decay result. The log decrement and damping ratio

are calculated in Table 2.

Table 2: Determination of Log Decrement and Damping Ratio

Vibration Peak Time (msec) Level ( g ) Log Decrement Period (msec) Damping Ratio

1 26.855 33.605

2 55.604 20.8 0.479 28.749 0.379

3 84.961 15.84 0.272 29.357 0.210

4 109.863 10.21 0.439 24.902 0.400

5 139.648 7.48 0.311 29.785 0.237

6 168.945 5.373 0.330 29.297 0.256

7 197.754 3.516 0.424 28.809 0.334

8 227.051 2.636 0.288 29.297 0.223

0.291 Average Damping Ratio

67

To determine c from the experimental data, the decay rate was

determined by using the data in table 2. An exponential discrete least squares

approximation technique was used (See table 3).

Table 3: Determination of Damping, C

Vibration Peak Time (sec) Level ( g ) In(Level) Time A 2 time*ln(Level)

1 0.026855 33.605 3.5156 0.0007212 0.0944

2 0.055604 20.8 3.0349 0.0030918 0.1688

3 0.084961 15.84 2.7625 0.0072184 0.2347

4 0.109863 10.21 2.3234 0.0120699 0.2553

5 0.139648 7.48 2.0122 0.0195016 0.2810

6 0.168945 5.373 1.6814 0.0285424 0.2841

7 0.197754 3.516 1.2573 0.0391066 0.2486

8 0.227051 2.636 0.9693 0.0515522 0.2201

Sum 1.010681 99.46 17.5557 0.1618040 1.7869

a= -12.6328084023769953

In b= 3.7904348874763876

y=44.276eA(-12.633)t

Using the relationship for determiningc on the aluminum beam:

qi(t) :=K-e2-H

c= 1.602*

10A-3 lb-sec/inA2

where:

p:= 63.41-

10-6

Ib-sec^/in^

68

L_

J

r

i

L L L

L

_J

L r r

r r i

L

J

r

n

r

i

1 ! ! 1 f

05.*

13

CO

CD

rr

c

CD

E(Dk-

JD

CD

Q

o

CO

D)

O

I ^J CM

= CD

u_

69

7.3 Spectral Analysis and Correlation

Power spectral measurements of all accelerometers were performed to

obtain the overall vibration environment of the positions. Analysis was carried

out between 0-1600 Hz. Be=4 Hz and Averaging = 50 counts was used. See

Figure 25 for the power spectra. The overall grms level of each position is

identified by RMS:. Using the formula for random error percent,

-/"dRandomError -

^u j

where,

11^= number of averages

Performing a stable average of 50 counts, random error = 14 %.

Next, frequency response is examined. Accelerometer 1 is designated

the control or input, and the accelerometers 2, 3, and 4 are responses or output.

See Figure 26 for results.

Some conclusions about the system can be derived from the power

spectra and frequency responses. A peak exists in all plots between 0 and 80

Hz. This is impart due to system limitations; the small modal shaker has limited

displacement and cannot excite low frequencies with high displacement. The

first natural frequency of the beam is also enclosed in this frequency range

causing some of the peak. Threeother peaks are noted in the spectra, at 240,

680, and 1356 Hz. These are identified as natural frequencies of the beam. In

channel 1, these appear as anomalies to the flat random background the

shaker is attempting to run. Natural frequency responses are often difficult to

control, resulting in irregular spikes and notches. In channel 2, the beam

70

>- 1 ^-~-r-<^ ,.

j. j .j.

.__ ir>'__

' ' > ' I

e r ^ '~S t r

cr*

t i \ 3 ! !r I t -S I I<-o l I i c? I Icn-| , [-> T r-a- ' I -t^ t i

Ln| |

_sr ] ]1

-j \~P^^] j T

^-Tirirtr^r^^-L^ ! 1 1

1 1 jJj i i

J 1 S i 1 1

1 1 -W^ 1 1

T-J

- ^i-u

^

1

1{.

1 S-^ 1

DO I I<^

1 I

o-j 1 I -^^I l

Cn, | _3 | |

^j. 1^- I !

_|7_ tr^c____i 1 l___

J- L 1 j- I 1

j l i>^-- ' '

4- -I

J^X^' '

E

i_*-*

o

CDQ.

CO

k_

CD

O

a.

16l_

<d>

O

LO

CVJ

CDi_

Zl

D)

U_

71

Freq Response Ch 2 : 1 Avg=50

LogMag

Functn Lin Hz g^arkerg RCLD 1.6k

Freq Response Ch 3 : 1 Avg=50

LogMag

0 Functn Lin Hz [^Markerg RCLD 1.6k

Freq Response Ch 4 : 1 Avg=50

LogMag

Functn Lin Hz [^Vlarker[g RCLD 1.6k

Figure 26: Frequency Response Spectra For Beam

72

natural frequencies appear as peaks. The greatest changes occur at

accelerometer position 2, which is the end of the beam. This is due to the lack

of constraints. This channel will be focused on for further analysis. The gain,

phase, and coherence are all examined (See Figure 27 a,b, and c). The

coherence from 300 to 1600 Hz is near unity, indicating a near ideal system

analysis. Data below this and at the identified resonances requires further

investigation. Between 0-30 Hz, the gain is at a minimum. System limitation is

one reason for this. Examining the gain and coherence leads to identification of

random and bias errors. Both graphs are at a minimum in the analyzed data,

suggesting that there is extraneous noise at the output. From 30-200 Hz,

coherence deviance from 1 could be attributed to noise at the input. This is

suspected, since the gain is not near a minimum. At the beam natural

frequencies, the coherence dips from unity corresponding to sharp peaks in the

gain. This is attributed to the signal to noise ratio being the highest at these

points. The phase plot also aids in the identification of the beam system

response. Phase shifts of 1 80 degrees are noted at the natural frequencies.

The above analysis was repeated to distinguish bias errors from possible

system nonlinearities. Measurements were obtained with finer spectral

resolution, Be = 2 Hz (Figures 28 a, b, and c) and Be = 1 Hz (Figures 29 a, b,

and c). The coherence should increase as spectral resolution becomes finer.

The coherence at 1356 is greatly reduced as resolution is increased, indicating

nonlinearities.

73

Freq Response Ch 2 : 1 Avg=S0

360

Phase

Deg

Functn Lin Hz [^rtarkerg RCLD

Freq Response Ch 2 : 1 Avg=50

Mag

1.6k

\ ! ! i I I ! iV_L_IlL__i i i I I

\i Ip::^TS

j j | jI

\ i i i ~i l l i0 Functn Lin Hz g^arkerg RCLD 1.6k

Coherence Ch 2 : 1 Avg=50

Functn Lin Hz gJMarkerg RCLD 1.6k

Figure 27: Frequency Response for Position 2

400 Line Resolution, Be = 4 Hz

a.) gain

b.) phase

c.) coherence

74

Freq Response Ch 2 : 1 Avg=40

360

Phase

Functn Lin Hz g)arker[g RCLD

Freq Response Ch 2 : 1 Avg=40

1.6k

Deg

0 Functn Lin Hz g^arkerg RCLD 1.6k

Coherence Ch 2 : 1 Avg=40

MagFjfH*

W 1 ! CL-,"*

"T

Functn Lin Hz [^larkerg RCLD 1.6k

Figure 28: Frequency Response for Position 2

800 Line Resolution, Be = 2 Hz

a.) gain

b.) phase

c.) coherence

75

Freq Response Ch 2 : 1 Avg=20

LogMag

2m

360

Functn Lin Hz [^ilarkerg RCLD

Freq Response Ch 2 : 1 Avg=20

1.6k

0 Functn Lin Hz g^flarker[g RCLD

Coherence Ch 2 : 1 Avg=20

1.6k

Mag

M^^VM

Functn Lin Hz Qiflarkerfg RCLD 1.6k

Figure 29: Frequency Response for Position 2

1600 Line Resolution, Be = 1 Hz

a.) gain

b.) phase

c.) coherence

76

7.4 Mode Shapes

Once the data has been deemed viable, it can be used to determine the

dynamics of the unit under investigation. Two type of analysis were performed

with the tape recorded data, time domain, and frequency domain. Time domain

analysis indicates how the beam deflects under excitation. The accelerometer

responses are used in Vibrant Technology's ME Scope to identify beam

dynamics in the time domain (See Figure 30) This shows how several modes

are excited simultaneously; their interactions can be identified.

Frequency domain analysis allows for the identifications of natural

frequencies of the beam. Aligning the frequency response of the beam at the

four analyzed locations with ME Scope, the dynamics of the beam are identified

at 36, 240, 680, and 1356 Hz (See Figures 31,32,33, and 34).

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77

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78

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Figure 33: Beam Response at Natural Frequency at 680 Hz

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Figure 34: Beam Response at Natural Frequency at 1356 Hz

79

8.0 Conclusions

Data from the experimental analysis are used to complement and finalize

theoretical data. The data is then compared, and discrepancies explored.

General guidelines extracted from this analysis are then summarized.

8.1 Theoretical Analysis Application

Material properties of the aluminum test beam are used in the theoretical

analysis. The following values are used:

[i := 63.41-io~6lb-secA2/inA2

E := io- io6

psi

I := 3.255- io"4 inA4

c := 1.602-io~3

lb-sec/inA2

L := 9.5in

Theoretical results and experiment results for natural frequency are presented

in Table 4.

Table 4: Comparison of Natural Frequencies

Natural Frequencies Analytical Results(Hz) Experimental Results(Hz) % Difference

1

2

3

4

5

6

The analytical results were derived without damping considerations, so values

80

44.475 36 19

277.966 240 13.66

779.569 680 12.77

1528 1356 11.26

2525

3772

for experimental data are lower. The first natural frequency has the greatest

error at 19 %. This is due to the difficulty of the system to control at such a low

frequency, noted by the large peak in the power spectra graphs on channel 1.

All other differences are less than 14 % which is within the random error

percentage calculated. Theoretical values are calculated beyond the

experimental results. The frequency analysis range was limited to 1600 Hz.

Mode shapes five and six are useful in determining how the structure behaves

at higher frequencies where testing was not recorded. Many situations occur in

which measurement of high frequency data is impossible due to test system

limitations, so the results of lower frequency analysis are used to confirm the

model and aid credibility of high frequency analysis. If a -14 % error is allowed

predicted from the trend of the first four mode shapes for the experimental

natural frequencies, then the fifth mode would occur in the 2171-2525 Hz

frequency range, and for the sixth mode a range of 3244-3772 Hz range.

The mode shape results are similar for the exact solutions and

experimental results, displaying the same general form, except for mode 4

(compare Figure 1 5 results to 31,1 6 to 32, 1 7 to 33, and Figure 1 8 to 34). This

can be attributed to the crude model of the beam used being insufficient to

model the mode deflection. There is also indication that the mode exhibits

nonlinear characteristics which influences the differences. More resolution of

the structure is possible with the theoretical exact solution, since transducer size

and limitation do not plague analytical analysis.

The frequency response of the system is determined for mode 1 using 1,

2, and 6 function assumed mode methods (See Figure 35 for calculation of 1

81

function, 36 for two function, and 37 for 6 function).

i--f^[n:=63.4iio~6

c:=1.602-10~3

z:=o,-..i

fiLj:= 1.875 co

t:= 279.445 kj:= 4.9402 Q j

:=.782989

i]> !(z) :=(cosh(PL1z)-cos(pL1z))- ( sinh( pL j-z)-

sin( pL j-z) )

Hjdo ) :=

CO - CO

co ~-q!(z)-

V i(z)

|*-Q

0 0

0.333 0.331

0.667 1.094

1 2

(-ton+iwc+kj)

H^co)

+ 1

1

0.927 - 2.865i

0.758 - 9.467i

0.557 - 17.309i

Figure 35: One Function Approximation for Mode 1

PL2:= 4.694 co 2

:= 1746.512 k2:= 194.0285 Q 2

:=.433931

o|>2(z):=(cosh(PL2z)- cos(PL2z))- 1.018467-

( sinh( PL 2z)-

sin(PL2z))

-co 2-o^ jCz)^-!*^!) -co2-il)2(z)(-nQ2)

H2(co ) :=i + -7j \+T2 V

I _co |i + i(o -c+k J I ~co -ji + ico c+ k 2)

1>l(z) ,>l'2(z)

0 0

0.333 0.331

0.667 1.094

1 2

0

1.179

0.845

-2

H2(co )

1

0.94- 2.865i

0.767 - - 9.467i

0.534 - -

17.308i

Note: This approximation uses values presented in single function

approximation.

Figure 36 Two Function Approximation for Mode 1

82

PL3:= 7.855 co 3

:= 14898.176 k3:= 1521.2379 Q3

:=.254435

PL4:= 10.996 co 4:= 9600 k4

:= 5841.3942 Q4:=. 181906

PL5

= 14.137 co 5:= 15865.043 k

5:= 15962.7052 Q5'=

.141471

PL6:= 17.279 co 6

:= 23700.175 kg:= 35620.5672 Q g

:=.1 15749

oj) 3( z):=

cosh( PL3-

z)-

cos( PL3-

z)-

( sinh( PL 3z)-

sin( PL3-

z) )

r|> 4(z) :=(cosh(PL4-z)-

cos(PL4z))-

(sinh(PL4-z)-

sin(PL4z))

t}> 5( z):=

( cosh( PL5-

z)-

cos( PL5-

z) )-

( sinh( PL5-

z)-

sin( PL5-

z) )

\|> 6( z):= ( cosh( PL

6-

z)-

cos( PLg-

z) )-

( sinh( PLg-

z)-

sin( PLg-

z) )

Ha^co) :=

Ha 2(co ) '=

Ha3( co ) :=

-co -Tj) i(z)-("n-Q1)

("co2n + icoc+ k,)

-co -il)2(z)-(-tx-Q2)

+ icoc+ k2)("%

-co Tl3(z)-(-n-Q3)

("co n + icoc+ k3J

Ha4(co) :-

Ha5(co) =

Ha6(co) :=

-co -aJ>4(z)-(-(i-Q4)

("co \i + icoc+ k4J

-co -^5(z)(-liQ5)

^-co2li + icoc+ k

-ip6(z)-(-ti-Q6)

(~co2|i + icoc+ k,

H6(oo):=l + Ha1(co) + Ha2(co) + Ha3(co)-I-Ha4(co)-I-Ha5(co)-|-Ha6(co)

z i|>i(z) n2(z) ^3(z) ^4(z) ^5(z) ^6(z) Hg(co)

0 0 0

0.333 0.331 1.179

0.667 1.094 0.845

1 2 -2

0 0

1.439 0.391

-1.361 0.367

1.001 -1

0 0

-0.991-

1.363

1-

1.366

1 -1

1

0.941 - 2.865i

0.766 - 9.467i

0.535 -

17.308i

Note: This approximation utilizes values presented in the one and two function

approximations.

Figure 37: Six Function Approximation of Mode 1

Similarly, results can be approximated for modes 2-6, See Tables 5-9

83

Table 5: Frequency Response Estimates for Mode 2

Utilizing 1, 2, and 6 Function Approximations

CO .- CO

Hjdo)

0

0.333

0.667

1

0.734-0.004i

0.121 0.013i

-0.607--

0.024i

H2(co)

1

8.082 - 33.78i

5.39-24.23i

-

13.068 + 57.254i

H6(co)

1

8.136- 33 .78i

5.343 - 24.23i

13.036 + 57.254i

Table 6: Frequency Response Estimates for Mode 3

Utilizing 1, 2, and 6 Function Approximations

CO --co 3

H^co)

0

0.333

0.667

H2(co)

1 1

0.741-4.399i-10~4

0.222- O.OOli

0.143-O.OOli

-

0.229 - 0.002i

-0.566- 0.003i0.313 - O.OOli

H6(co)

1

-1.458+ 0.01 li

0.996 - 0.015i

1.317 0.014i

Table 7: Frequency Response Estimates for Mode 4

Utilizing 1, 2, and 6 Function Approximations

CO --co 4

H^co) H2(co)

0

0.333

0.667

1 1

0.7410.211 -0.002i

-0.237- 0.003i0.143 -0.002i

0.33 - 0.002i-0.567- 0.004i

H6(co)

1

-4.634- 26.379i

-3.692- 24.735i

10.877 + 67.406i

84

Table 8: Frequency Response Estimates for Mode 5

Utilizing 1, 2, and 6 Function Approximations

CO - CO

0

0.333

0.667

1

5

H^co)

1

0.741 - 4.13i- 4

0.143- O.OOli

-0.566- 0.002i

H2(co)

1

0.223 - O.OOli

"0.228-- 0.002i

0.312--O.OOli

H6(co)

1

-8.777 +87.234i

8.354- 88.05i

8.653- 88.018i

Table 9: Frequency Response Estimates for Mode 6

Utilizing 1, 2, and 6 Function Approximations

co .-co 6

z

0

0.333

0.667

1

H^co) H2(co) Hg(to)

1 1 1

0.741- 2.764i-

10~4

0.226-

12.67 + 146.886i

-

12.913 + 147.234i0.143-9.132M0"4

-0.225- O.OOli

-9.307+ 107.805i

0.306 - 7.345i-10_4

-0.566 -0.002i

Values for the Frequency Response Function are in Table 10.

Table 10: Frequency Response Gains From Experimentation.

Location

Freq(Hz) 0.333 0.666 1

36 -0.301-.252i -0.508-.399i -0.736-.577i

240 -.933-6.07i -0.985-3.59i 1.407+8.91 01 i

680 -4.41-13. 5i -2.07+1 2. 59i 5.28-1 5.43i

1356 -0.262-.747i .444-5. 5i -10+1 9.81 i

The values are noted at peaks of the imaginary axis. This typically identifies

85

resonances, while the real portion should cross zero. The values do not match

the theoretical values. This is due to several inefficiencies in the calculation,

such as calculation of damping, resolution problems, approximation for

modeling, and other errors noted in chapter 7 . Damping is assumed to be

uniform in the model, when in fact it is not. Varying constraints on the beam

( cantilever being fixed at one end, and free at the other ) alter the damping

along the beam. Damping was calculated for the first natural frequency of the

beam, and assumed to be the same for all modes. Damping varies for each

mode; the damping constant assumed introduced error in the calculations. The

first natural frequency greatly influences the logarithmic decrement, and

analyzer limitations prevent a high enough frequency resolution to calculate

the higher mode damping. The approximation used to model the beam is

simple, with few terms, more sophisticated methods would produce more

accurate results. The technique is viable; it consists of superposing the

modes and base excitation to predict mode shapes. At natural frequencies the

output is high resulting in the mode shape being heavily influenced by the

corresponding natural frequency mode shape, all other shapes are a forced

vibration and typically would not have a great influence. The greater the

number of functions, the better the prediction, as shown in Figure 38. Figure 38

displays the frequency response function versus frequency for one, two, and six

function approximations for the first mode (compare to experimental results in

Figures 27, 28, and 29). For the frequency range displayed, the greater the

number of functions, the truer the frequency response.

The weakest portion of the calculation is the damping estimate. As stated

previously, damping is difficult to estimate and obtain good results. If the

damping is increased ten times, the frequency response more closely

86

approaches the experimental data as shown in Table 1 1 . One hundred time

multiplication of the damping constant improves results shown in Table 12.

|Hi(ip -

|H2(co)|

100

8000 1'10

100

|H6(a.)

2000 4000 6000 80001'104

CD

Figure 38: Frequency Response Function for Approximation

of Mode One at position 2

87

Table 1 1 : Frequency Response of First Mode Approximation with Six Functions

at Ten times the Damping Constant

z h 6( co )

0.333

0.667

1

1

1.014- 0.287i

1.006 - 0.948i

0.974- 1.73 li

Table 12: Frequency Response of First Mode Approximation with Six Functions

at One Hundred times the Damping Constant

z H6(co)

0.333

0.667

1

1

1.014- 0.032i

1.008-0.097i

0.979 - 0.168i

When using random vibration signals, precise values are important, but

indicate trends. The mode shape results are often the goal of the analysis. The

mode shapes of the bar are presented in Figure 39. These are a crude

approximation of the theoretical results due to the small number of transducers

used.

8.2 Conclusions and Recommendations

Random vibration signal analysis is a powerful tool for structural

dynamics analysis. Dynamics of structures can quickly and efficiently be

characterized through its use. Care must be taken to insure this type of analysis

is performed with viable data, since the random nature makes it difficult to

recognize errors.

To minimize errors, the following guidelines should be observed:

88

1. Have a general knowledge of the structure before proceeding with

experimentation. This will aid in the placement of transducers and help

distinguish between good and bad data.

CO - CO

CO .-co

H6() H6(<) 0 -

CO - CO CO .- CO

H6(co)

-10

20

Figure 39: Mode Shape Results Using the Six Function Approximation

2. Understand measurement equipment and limitations. Random signal

analysis noise is prevalent in any measurement, and must be examined to

make sound engineering judgment on data.

3. Maximize analysis parameters whenever feasible. Better frequency

resolution or averaging can exponentially improve data, but not without the

89

sacrifice of another parameter. Weight the results of each.

4. Always examine coherence and phase as well as gain in any

frequency response measurement. Trends in this data can indicate errors and

reveal necessary action to improve measurements.

Future studies should be performed to determine better estimates of the

damping. Filtering techniques can be investigated to limit the signal analysis

band and damping experimentation performed to improve the model. More

analysis can be performed changing the frequency span to obtain estimates of

the model at higher frequencies. The same analysis can be performed using

transient excitation instead of random and the results compared to determine

advantages and disadvantages of each technique. Other modeling techniques

can be used to enhance the model such as Galerkin approximation or finite

difference techniques.

90

References

1. Bendat, Julius S. and Allen G. Piersol., Engineering Applications of

Correlation and Spectral Analysis. New York: John Wiley and

Sons, 1980.

2. Bendat, Julius S. and Allen Piersol.,Random Data

,2nd edition,

New York: John Wiley and Sons, 1986.

3. Burden, Richard L. and J. Douglas Faires, Numerical Analysis, 4th ed.,

Boston: PWS-Kent Publishing Company, 1989.

4. Freund, John E.,Modern Elementary Statistics, 7th ed. New Jersey,

Prentice-Hall, 1988.

5. Harris,Cyril M. and Charles C. Crede, Shock and Vibration

Handbook, New York: McGraw-Hill Book Company, 1961.

6. Meirovitch, Leonard, Elements of Vibration Analysis, New York:

McGraw-Hill Book Company, 1986.

7. Newland, D. E., An Introduction to Random Vibration and Spectral

Analysis, Longman Group Ltd., London, 1975.

8. Randall, R. B. and B. Tech, Freouencv Analysis, Bruel and Kjaer,

Denmark, 1987

9. Thomson, William T., Theory of Vibration with Applications, 3rd ed.,

New Jersey, Prentice Hall, 1988.

10. Wylie, C. Ray and Louis C. Barrett, Advanced Engineering

Mathematics, McGraw-Hill Book Company, New York, 1982.

91