Basics of Structural VibrationTesting and Analysis

APPLICATION NOTEAN011

IntroductionStructural vibration testing and analysis contributes toprogress in many industries, including aerospace,auto-making, manufacturing, wood and paper produc-tion, power generation, defense, consumer electron-ics, telecommunications and transportation. Themost common application is identification and suppres-sion of unwanted vibration to improve product quality.

This application note provides an introduction to thebasic concepts of structural vibration. It presents thefundamentals and definitions in terms of the basic con-cepts. It also emphasizes practical applications andreal world examples.

This application note covers the following topics:Basic terminologyModels in single and multiple degrees of freedom Continuous structure modelsMeasurement techniques and instrumentationVibration suppression methodsModal analysis Operating deflection shape analysis.

Basic Terminology of StructuralVibration The term vibration describes repetitive motion thatcan be measured and observed in a structure.Unwanted vibration can cause fatigue or degrade theperformance of the structure. Therefore it is desir-able to eliminate or reduce the effects of vibration. Inother cases, vibration is unavoidable or even desirable.In this case, the goal may be to understand the effecton the structure, or to control or modify the vibration,or to isolate it from the structure and minimize struc-tural response.

Vibration analysis is divided into sub-categories suchas free vs. forced vibration, sinusoidal vs. randomvibration, and linear vs. rotation-induced vibration.

Free vibration is the natural response of a structureto some impact or displacement. The response iscompletely determined by the properties of the struc-ture, and its vibration can be understood by examiningthe structure's mechanical properties. For example,when you pluck a string of a guitar, it vibrates at thetuned frequency and generates the desired sound.The frequency of the tone is a function of the tensionin the string and is not related to the plucking tech-nique.

Forced vibration is the response of a structure to arepetitive forcing function that causes the structure tovibrate at the frequency of the excitation. For exam-ple, the rear view mirror on a car will always vibrate atthe frequency associated with the engine's RPMs. Inforced vibration, there is a relationship between theamplitude of the forcing function and the correspon-ding vibration level. The relationship is dictated by theproperties of the structure.

Sinusoidal vibration is a special class of vibration.The structure is excited by a forcing function that is apure tone with a single frequency. Sinusoidal vibrationis not very common in nature, but it provides an excel-lent engineering tool that enables us to understandcomplex vibrations by breaking them down into simple,one-tone vibrations. The motion of any point on thestructure can be described as a sinusoidal function oftime as shown in Figure 1 (top).

Random vibration is very common in nature. Thevibration you feel when driving a car result from a com-plex combination of the rough road surface, enginevibration, wind buffeting the car's exterior, etc.Instead of trying to quantify each of these effects, theyare commonly described by using statistical parame-ters. Random vibration quantifies the average vibra-tion level over time across a frequency spectrum. Figure 1 (bottom) shows a typical random vibrationversus time plot.

Figure 1. Sinusoidal vibration (top) and random vibration (bottom).

Rotating imbalance is another common source ofvibration. The rotation of an unbalanced machine partcan cause the entire structure to vibrate. The imbal-ance generates the forcing function that affects thestructure. Examples include a washing machine, anautomobile engine, shafts, steam or gas turbines, andcomputer disk drives. Rotation vibration is usuallyunwanted, and the goal is to eliminate or minimize itby properly balancing the machine.

The common element in all these categories of vibra-tion is that the structure responds with some repeti-tive motion that is related to its mechanical proper-ties. By understanding some basic structural models,measures and analysis techniques, it is possible tosuccessfully characterize and treat vibration in struc-tures.

Time and Frequency AnalysisStructural vibration can be measured by electronicsensors that convert vibration motion into electricalsignals. By analyzing the electrical signals, you canlearn about the nature of the vibration. Signal analy-sis is generally divided into time and frequencydomains; each domain provides a different view andinsight into the nature of the vibration.

Time domain analysis starts by analyzing the signal asa function of time. An oscilloscope, data acquisitiondevice, or signal analyzer can be used to acquire thesignal. Figure 2 shows a structure (such as a single-story building) that responds to an impact with a vibra-tion that is measured at point A and plotted versustime. The dashed lines indicate the motion of thestructure as it vibrates about its equilibrium point.

Figure 2. Mechanical structure responds with vibration plotted ver-sus time.

The plot of vibration versus time provides informationthat helps characterize the behavior of the structure.It behavior can be characterized by measuring themaximum vibration (or peak) level, or finding the peri-od, (time between zero crossings), or estimating thedecay rate (the amount of time for the envelope todecay to near zero). These parameters are the typicalresults of time domain analysis.

Frequency analysis also provides valuable informationabout structural vibration. Any time history signal canbe transformed into the frequency domain. The mostcommon mathematical technique for transformingtime signals into the frequency domain is called theFourier Transform, after the French MathematicianJean Baptiste Fourier. The math is complex, buttoday's signal analyzers race through it automatically,in real-time.

Fourier Transform theory says that any periodic signalcan be represented by a series of pure sine tones.Figure 3 shows how a square wave can be construct-ed by adding up a series of sine waves; each of thesine waves has a frequency that is a multiple of the fre-quency of the square wave. The amplitude and phaseof each sine tone must be carefully chosen to get justthe right waveform shape. When using a limited num-ber of sine waves as in Figure 3, the result resemblesa square wave, but the composite waveform is stillragged. As more and more sine waves are added inFigure 4, the result looks more and more like a squarewave.

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Figure 3. A square wave can be constructed by adding pure sinetones.

Figure 4. As more sine tones are added the square waveformshape is improved.

In Figures 3 and 4, the third graph shows the ampli-tude of each of the sine tones. In Figure 3, there arethree sine tones, and they are represented by threepeaks in the third plot. The frequency of each tone isrepresented by the location of each peak on the fre-quency coordinate in the horizontal axis. The ampli-tude of each sine tone is represented by the height ofeach peak on the vertical axis. In Figure 4, there aremore peaks because there are more sine tones addedtogether to form the square wave. This third plot canbe interpreted as the Fourier Transform of the squarewave. There is also a fourth plot not shown that indi-cates the phase of each sine tone relative to the firstsine tone.

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In structural analysis, usually time waveforms aremeasured and their Fourier Transforms computed.The Fast Fourier Transform (FFT) is a computationallyoptimized version of the Fourier Transform. The thirdplot in Figure 4 also shows the measurement of thesquare wave with a signal analyzer that computes itsFast Fourier Transform. With test experience, you willgain the ability to understand structural vibration bystudying frequency data.

The Decibel dB ScaleVibration data is often displayed in a logarithmic scalecalled the Decibel (dB) scale. This scale is usefulbecause vibration levels can vary from very small tovery large values. When plotting the full data rangeon most scales, the small signals become virtuallyinvisible. The dB scale solves this problem because itcompresses large numbers and expands small num-bers. A dB value can be computed from a linear valueby the equation:

where xref is a reference number that depends on the

type of measurement. Comparing the motion of amass to the motion of the base, base measurement isused as the reference in the denominator and themass as the measurement in the numerator.

In the dB scale, if the numerator and denominator areequal, the level is zero dB. A level of +6 dB means thenumerator is about a factor of two times the refer-ence value and +20 dB means the numerator is a fac-tor of 10 times the reference.

Figure 5 shows an FFT in linear scale on the top anddB scale on the bottom. Notice that the peak near200 Hz is nearly indistinguishable on the linear scale,but very pronounced with the dB scale.

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Figure 5. The dB scale allows us to see both large and small num-bers on the same scale as shown for the FFT with linear scale onthe top and dB scale on the bottom.

Structural VibrationStructural vibration can be complex, so let's start witha simple model to derive some basic concepts andthen build up to more advanced systems. The sim-plest vibration model is the single-degree-of-freedom,or mass-spring-damper model. It consists of a sim-ple mass (M) that is suspended by an ideal spring witha known stiffness (K) and a dashpot damper from afixed support. A dashpot damper is like a shockabsorber in a car. It produces an opposing force thatis proportional to the velocity of the mass.

Figure 6. Simple Mass-Spring-Damper Vibration Model

In this model, the factors that affect vibration arecompletely characterized by the parameters M, K andC. Knowing these values, it be can predicted exactlyhow the structure will respond to vibration.

The mass is a measure of the density and amount ofthe material. A marble has a small mass and a bowl-ing ball has a relatively larger mass. The stiffness isa measure of how much force the spring will pull whenstretched a given amount. A rubber band has a smallstiffness and a car leaf spring has a relatively largerstiffness. A sports car with a tight suspension hasmore damping than a touring car with a soft suspen-sion. When the touring car hits a bump, it oscillatesup and down for a longer time than the sports car.Different materials have different damping qualities.Rubber, for example, has much more damping thansteel.

If you displace the mass by pulling it down and releas-ing it, the mass will respond with motion similar toFigure 7. The mass will oscillate about the equilibriumpoint and after every oscillation, the maximum dis-placement will decrease due to the damper, until themotion becomes so small that it is undetectable.Eventually the mass stops moving.

Figure 7. Free Vibration of Mass-Spring-Damper Model

Figure 7 shows that the time between every oscillationis the same. The time plot crosses zero at regularintervals. The time for the displacement to cross zerowith a positive slope to the next zero crossing withpositive slope is named the period. It is also relatedto the frequency of oscillation.

Frequency can be computed by dividing one by theperiod. For the mass-spring-damper model subject tofree vibration, the frequency of oscillation is complete-ly determined by the parameters M, K, and C. It is

called the natural frequency denoted by the symbol ωn.

Frequency is measured in cycles per second withunits of Hertz (Hz). Assuming the damping is small,then the mathematical relationship is given by

A larger stiffness will result in a higher ωn, and a larg-

er mass will result in a lower ωn.

Figure 7. also reveals something about damping.Theory tells us that the amplitude of each oscillationwill diminish at a predictable rate. The rate is relatedto the damping factor C. Usually damping is described

in terms of the damping ratio--ζ. That ratio is relatedto C by

The damping ratio can vary from zero to infinity.When it is small (less than about 0.1), the system islightly damped. When excited, it will oscillate, or ring,for a long time as shown in Figure 8. When the damp-ing ratio is large, the system is 'over damped.' It willnot oscillate at all and it may take a long time to returnto its equilibrium position. When the damping ratio =1, the system is 'critically damped' and will return tothe equilibrium position in the shortest possible time.

Figure 8. Free vibration of mass-spring-damper for three cases: A - under damped, B - critically damped, C - over damped.

Vibration theory also describes how the mass-spring-damper model will respond to forced vibration.Imagine that the base of the mass-spring-damper isnot fixed, but instead forced to move up and down asmall distance. If this base motion is sinusoidal, it isstraightforward to predict how the mass will respondto the forced vibration. The mass will begin to movein sinusoidal motion at the same frequency as thebase; the amplitude of the mass displacement will varydepending on the frequency of the base motion. Thisis shown in Figure 9 where the mass displacement islarger than the base displacement. There will also bea phase difference between the base and the massdisplacement.

Figure 9. Mass-Spring-Damper Model responds to forced vibrationwith change in amplitude and phase: A-mass vibration, B-base dis-placement.

The relative amplitude and phase of the mass displace-ment will vary with the frequency of the base excita-tion. Varying the base frequency and recording thecorresponding mass displacement amplitude (dividedby the base displacement amplitude) and the phase,results in a plot of the vibration amplitude and phaseversus excitation frequency.

Figure 10 shows the results for three different values

of the damping ratio (ζ). In all cases, the amplituderatio (Magnitude) is zero dB for low frequencies. ZerodB is equivalent to one. This means that the ampli-tudes are equal at low frequencies. As the frequencyincreases, the magnitude rises to some maximum

value. The frequency of this peak is ωn, the natural

frequency. At frequencies above ωn, the magnitude

falls of at a constant rate. This plot is known as aBode Diagram. Note the horizontal axis is a log scalewith units rad/sec (2*3.141 radians = 1 cycle persecond, Hz). Note that the peak is higher for lightdamping. For critical damping and over damping (notshown), the magnitude does not increase above 0 dB.

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Figure 10. Bode Diagram of the vibration of the mass-spring-

damper system with A: ζ = 0.01, B: ζ = 0.025, C: ζ = 0.1

The lower plot in Figure 10 shows the phase relationbetween the base and the mass for different frequen-

cies. At low frequencies below ωn, the phase is zero

degrees meaning that the mass is in phase with thebase. When the base vibration frequencies coincide

with ωn (at resonance), the phase is minus 90

degrees. At high frequencies, the phase is minus 180degrees; the mass is out of phase with the base andthey move in opposite directions. Note that the differ-ent damping ratios affect the slope of the phasechange.

The Q Factor is a common technique used to estimatethe damping from the Bode Diagram. Vibration theo-ry shows that the damping ratio is related to thesharpness of the peak of the magnitude plot. The

damping ratio, ζ, can be determined by computing Q,defined as the resonant frequency divided by the band-width. Q is computed as:

and

where fn is the resonant frequency at the peak in Hz;

f1 and f2 are the half power points measured -6 dB

down from the peak as shown in Figure 11.

Another way to measure damping is to simply recordthe peak amplitude ratio. However this should not beconfused with the Q factor as they are not equivalent.

Figure 11. The Q factor is computed by dividing the peak frequen-cy by the bandwidth.

The single-degree-of-freedom, mass-spring-dampermodel is an over simplification of most real structures.But the concepts and terminology introduced in analyz-ing how free and forced vibration affect this simplemodel also apply to analyzing more complex struc-tures.

Multi-Degree of Freedom ModelThe first model introduced in this paper was a single-story building. This model can be extended to a twoor three story building resulting in a two or threedegree of freedom system. The building can be rep-resented by interconnecting simple mass-spring-damper systems as shown in Figure 12.

Figure 12. A two story building can be modeled as a two degree offreedom model and simplified into two coupled mass-spring-damper

systems.

The coupled mass-spring-damper system will have tworesonant frequencies. This system is characterized bya Bode Diagram similar to Figure 13. The damping ofeach resonance can be determined using the Q factortechnique. Note that the first resonance is more light-ly damped than the second resonance judging fromthe sharpness of the peaks.

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Figure 13. Bode diagram of 2 mass-spring-damper system.

In the models considered so far, mass is lumped intoone point. A continuous structure such as a beam orstring where the mass is distributed over volume pro-vides another type of model. Figure 14 shows a beampinned at both ends so that it can rotate but cannotbe displaced. Subject to base excitation, the beam willdeform, vibrate and take on different shapes depend-ing on the frequency of the excitation and the mount-ing of the beam ends. The beam will have a first res-onant frequency at which all its points will move in uni-son; at the first resonant frequency, the beam willtake the shape shown to the right in Figure 14 labeledFirst Mode Shape. At a higher frequency, the beamwill have a second resonant frequency and modeshape, and a third, and fourth, etc. Theoreticallythere are an infinite number of resonant frequenciesand mode shapes.

However at higher frequencies, the structure acts likea low-pass filter and the vibration levels get smallerand smaller. The higher modes are harder to detectand have less effect on the overall vibration of thestructure.

Figure 14. A beam fixed at both ends is an example of a continu-

ous structure model.

Figure 15 shows an experimentally measured BodeDiagram for a 1 x 0.25 inch steel beam that is 8 inch-es long. There are many resonant frequencies, begin-ning as low as 800 Hz and located at approximatelyequal intervals. At every resonant peak, the phasegoes through a 180 degree phase shift. Notice thatwhen the resonances are located at separate frequen-cies, the frequency

response of each resonance is similar to that of a sim-ple spring-mass-damper system.

Figure 15 A typical Bode Diagram for a beam.

This overview of structural vibration analysis showsthat the field has many complexities and considerabledepth. Fortunately for technical and non-technicalpeople alike, the fundamental phenomena and con-cepts apply to models both simple and complex,whether they represent single-degree-of-freedom, mul-tiple-degree-of-freedom, or continuous structures.

Vibration Measurements

Vibration SensorsStructural vibration is commonly measured with elec-tronic sensors called accelerometers. These sensorsconvert an acceleration signal to an electronic voltagesignal that can then be measured, analyzed andrecorded with electronic hardware. There are manytypes of accelerometers. The most common requiresa power supply connected by a wire to the accelerom-eter as shown in Figure 16. Some accelerometershave internal circuitry that allows the analyzer to pro-vide the power, eliminating the need for a power sup-ply. The signal analyzer includes a calibration settingthat allows the voltage signal to be converted back intoa measurement of acceleration.

Figure 16. Typical instrumentation for accelerometer.

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Manufacturers calibrate each accelerometer and giveit a sensitivity value. For example, you can order a100 mV/gn accelerometer with a calibration value of

102.3 mV/gn. Accurate measurements depend on

using the correct sensitivity value in the signal analyz-er and on using an accelerometer with the right sen-sitivity for the application. A high sensitivity such as1000 mV/gn may not be appropriate for an applica-

tion with a high acceleration level. In this case toohigh a voltage could saturate the input circuitry on thesignal analyzer. If the acceleration level is very low, asmall sensitivity, such as 10 mV/gn, may produce a

signal that is too weak to accurately measure.

Sensitivity also has an important impact on the signalto noise ratio. Signal to noise is the ratio of the sig-nal level divided by the noise floor level and measuredin the dB scale as

All sensors and measurement hardware are subject toelectronic noise. Even if you know that the accelerom-eter is not moving, electronic noise may cause meas-urements that suggest some small acceleration. Thisis due to the sensor cables picking up electronic noisefrom stray signals in the air, from noise in the powersupply or from internal noise in the analyzer electron-ics. High quality hardware is designed to minimize theinternal noise making low signal measurements possi-ble. The signal to noise ratio limits the lowest meas-urement that can be made.

For example, if the noise floor is 1 mV, then with a typ-ical 100 mV/gn accelerometer, the smallest level

that can be read, can be computed as

1 mV divided by 100 mV/gn = 0.01 gn

If you measure a 20 gn acceleration with this

accelerometer, then the signal to noise ratio can becomputed as

In this case, the analyzer will always show a level of atleast 0.01 gn because of the noise floor. It is a good

practice to not trust a measurement with a signal tonoise ratio that is below 3:1 or 4:1. When the signalto noise ratio is too small, one solution is to use anaccelerometer with a higher sensitivity.

For example, with 1 mV of noise, the smallest levelthat can be read by a 1000 mV/gn accelerometer

can be computed as

1 mV divided by 1000 mV/gn = 0.001 gn

Some accelerometer power supplies include a gainsetting that multiplies the signal by 1, 10, or 100.Unfortunately, this setting also multiplies the noisethat is picked up by the wires and inherent in the sen-sor and power supply. In most cases, increasing thepower supply gain will not solve signal to noise prob-lems.

Excitation MethodsIn some applications, vibration measurements aremade during normal operation of a structure ormachine. They can instrument an automobile withmany accelerometers and drive it on roads or a testtrack while analyzing or recording signals. Many othercases require a more controlled excitation to yieldreproducible and predictable results. The two mostcommon methods are the impact hammer and elec-trodynamic shaker.

An impact hammer, as shown in Figure 17, is a spe-cialized measurement tool that produces short dura-tion vibration levels by striking the structure at somepoint. The hammer incorporates a sensor (called aload cell) that produces a signal proportional to theforce of impact. This enables precise measurementof the excitation force. An impact hammer is oftenused for modal analysis of structures where use of amechanical vibrator is not convenient; examples are inthe field or with very large structures. Differentimpact tip materials allow tailoring of the frequencycontent of the impact force. For low frequency meas-urements, a soft rubber tip concentrates the excita-tion energy in a narrow frequency range. A hardmetal tip gives good excitation energy content out to

high frequencies.Figure 17. Impact hammer instrumented with a load cell to meas-ure the excitation force and different hardness tips.

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For laboratory vibration measurements, shakers arethe instrument of choice. Shakers are rated by theforce they produce. Shakers vary in size and forcefrom baseball-sized systems with six pounds of forceto SUV-sized tables which are capable of generating65,000 pounds force and are used to vibrate satel-lites and other large structures.

Figure 18. Mechanical vibrators are used in laboratory measure-

ments and vary in size from small to very large.

Shakers connect to structures in three ways: bymeans of a thin metal rod named a stinger; by placingthe structure on a table mounted on the top of theshaker; or by a slip table that is built onto the shakerand vibrates in the horizontal direction. In mostcases, an accelerometer is mounted on the structure,near the attachment point to the shaker, to measurethe driving acceleration levels. In some cases, a loadcell on the stinger measures the excitation force.

A computer-based controller usually drives a shaker bygenerating carefully controlled voltage signals that arethen amplified and sent to the shaker to vibrate thetest article.

Dynamic Signal AnalyzersThe most common equipment for analyzing vibrationsignals is a computer based data acquisition systemcalled a dynamic signal analyzer (DSA) (see Figure19). The first generation DSAs used analog trackingfilters to measure frequency response. Modern ana-lyzers use digital technology and are far faster andmore versatile. Using disk drives to store large vol-umes of data for post processing, the can record fre-quency, amplitude and statistical data.

Figure 19. Dactron FocusTM (left) and the Dactron PhotonTM (right)Dynamic Signal Analyzers

A modern DSA consists of many electronic modulesas shown in Figure 20. First, the analyzer measureselectronic signals with an analog front end that mayinclude special signal conditioning such as sensorpower supply, TEDS (transducer electronic datasheets that read the calibration and other informationfrom a chip embedded in the sensor), adjustable volt-age gains settings and analog filters. Next, the sys-tem converts the analog signal to a digital format viaan analog to digital converter (ADC). After the signalis digitized, the system processes it with it a digitalsignal processor (DSP), which is a mini-computer opti-mized to do rapid mathematical calculations. The DSPperforms all required calculations, including additionalfiltering, computation of time and frequency measure-ments, and management of multiple channel signalmeasurements.

Most modern signal analyzers connect with a PC forthe reporting and display function. The DSP interfaceswith the PC; a software user interface displays resultson the PC screen. Older analyzers did have a PC inter-face; they included buttons and a display screen onthe analyzer chassis. The PC interface speeds andsimplifies test setup and reporting.

Figure 20. Signal analyzer architecture

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Quantization: analog to digital con-version and effective bitsOne measure of DSA quality is the bit count of the ana-log to digital conversion (Demler, 1991). When ananalog signal is converted into a digital signal, it under-goes quantization, which means that a perfectlysmooth analog signal is converted into a signal repre-sented by stair steps as shown in Figure 21. To accu-rately represent an analog signal, the stair steps ofthe digital signal should be as small as possible. Thestep size depends on the number of "bits" in the ADCand the voltage range of the analog input. For exam-ple, if the voltage range is 10 volts and the ADC uses16 bits, the smallest step size can be computed as

Step size = 10 volts divided by 216= 0.15 millivolts

A 24-bit ADC, reduces the step size to

Step size = 10 volts divided by 224 = 0.0006 millivolts

The 24-bit ADC has a step size that is 256 timessmaller than the step size of the 16-bit ADC. Twentyfour bits is the highest bit count available in modernDSAs. A 24-bit ADC can accurately measure largesignals and small signals at the same time. ADCs withlower bit counts require the user to change lower thevoltage range when measuring a smaller signal inorder to achieve the same measurement accuracy.

Figure 21. When an analog signal is converted to digital thesmooth analog signal becomes a stair step signal with the size of

the step depending on the bit count of the ADC.

The bit count is also related to dynamic range of theDSA, another important measurement quality consid-eration. Dynamic range shows the ratio of the largestsignal to the smallest signal that a DSA can accurate-ly measure; dynamic range is reported in the dB scale.A typical 24-bit ADC with good low-noise performancewill have 110 - 120 dB of dynamic range. Dynamicrange is also affected by the noise floor of the device.

Measurement Types: time vs. fre-quency, FFT, PSD, FRF, CoherenceMost analyzers compute time and frequency measure-ments. Time measurements include capturing tran-sient signals, streaming long duration events to thecomputer disk drive, and statistical measures. Thesampling rate, yet another measure of DSA quality, isrelated to time measurements. High speed ADCs canattain sampling rates of up to 100,000 samples persecond (100 kHz). Some analyzers incorporate multi-plexers that use one ADC sampling at a high rate andswitch it between the different input channels. Forexample, if the DSA has 8 channels and a multiplexing100 kHz ADC, it will only sample at 12.5 kHz perchannel. High-quality DSAs do not use multiplexersand provide high sampling rates regardless of thenumber of input channels.

Most DSAs can compute a variety of frequency meas-urements including Fast Fourier Transform, PowerSpectral Density, Frequency Response Functions,Coherence and many more. The DSP computes thesesignals from digitized time data. Time data is digitizedand sampled into the DSP block by block. A block is afixed number of data points in the digital time record.Most frequency functions are computed from oneblock of data at a time.

Fast Fourier Transform (FFT) is the discrete FourierTransform of a block of time signal. It represents thefrequency spectrum of the time signal. It is a complexsignal meaning that it has both magnitude and phaseinformation and is normally displayed in a BodeDiagram. Figure 22 shows the FFT of a square wavemeasured by a LDS-Dactron Photon. The horizontalaxis shows frequency ranging from zero to 1000 Hz;the vertical axis is volts ranging from 0 to 1.0 volts.A square wave is composed of many pure sine wavesindicated by the discrete peaks at even intervals in theFFT.

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Figure 22. FFT of a square wave computed with Dactron PhotonDSA.

Power Spectral Density (PSD) is computed from theFFT by multiplying the FFT by its complex conjugate.The result of this operation is a real signal with powerunits (squared values) know as the power spectrum.The phase information is lost, leaving only the magni-tude data. The final computation is the division of theresulting power spectrum by the frequency increment.This step normalizes the measurement to the FFT "fil-ter bandwidth" and converts the power spectrum intoa density function. The advantage over non-densityrepresentations of spectra is that the level will notchange when the frequency resolution is modified. Theunits of an acceleration signal PSD are gn

2/Hz. PSD

measurements are used to analyze "stationary" sig-nals such as random noise. Figure 23 shows the PSDof a broadband vibration signal displayed in dB ampli-tude units.

Figure 23. Power Spectral Density of broad band random vibration

Frequency Response Function (FRF) is computedfrom two signals. It is sometimes called a "transferfunction." The FRF describes the level of one signalrelative to another signal. It is commonly used inmodal analysis where the vibration response of thestructure is measured relative to the force input of theimpact hammer or shaker. FRF is a complex signalwith both magnitude and phase information.

Coherence is related to the FRF and shows thedegree of correlation of one signal with a second sig-nal. Coherence varies from zero to one and is a func-tion of frequency. In modal analysis, this functionshows the quality of a measurement. A good impactproduces a vibration response that is perfectly corre-lated with the impact, indicated by a coherence plotthat is near one over the entire frequency range. Ifthere is some other source of vibration, or noise, orthe hammer is not exciting the entire frequency range,the coherence plot will drop below one in someregions.

Most DSAs can compute many other time and fre-quency based signals including cross power spectrum,auto and cross correlation, impulse response, his-tograms, octave analysis, and order tracks.

Triggering, Averaging & WindowingGetting good measurements from an analyzerrequires careful selection of the measurement set-tings for the averaging, triggering, and windowingparameters.

Triggering is a technique for capturing an event whenyou do not know exactly when it will occur. A triggercan start data acquisition and processing when auser-specified voltage level is detected in an inputchannel. For example, you can set up a trigger to cap-ture a hammer impact. After you arm the trigger, theanalyzer will wait until the impact occurs before itstarts acquiring data.

Averaging improves the quality of the measurement.It applies to both the frequency and time domains.Frequency domain averaging uses multiple data blocksto "smooth" the measurements. You can average sig-nals with a linear average where all data blocks havethe same weight; or you can use exponential weight-ing. In this case, the last data block has the mostweight and the first has the least. Averaging acts toimprove the estimate of the mean value at each fre-quency point; it reduces the variance in the measure-ment. Time domain averaging is useful in measuringrepetitive signals to suppress background noise. Animpact test is good example of repetitive signals. Boththe force and acceleration signals are the same foreach measurement. This assumes that the triggerpoint is reliable. The presence of high backgroundnoise may adversely affect the reliability of the trigger.

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Windowing is a processing technique used when com-puting FFTs. Theoretically, the FFT can only be com-puted if the input signal is periodic in each data block(it repeats over and over again and is identical everytime). When the FFT of a non-periodic signal is com-puted, the FFT suffers from 'leakage.' Leakage is theeffect of the signal energy smearing out over a widefrequency range. If the signal were periodic, it wouldbe in a narrow frequency range. Since most signalsare not periodic in the data block time period, window-ing is applied to force them to be periodic. A window-ing function should be exactly zero at the beginningand end of the data block and have some specialshape in between. This function is then multiplied withthe time data block, and this forces the signal to beperiodic.

Figure 24 shows the effect of applying a Hanning win-dow to a pure sine tone. The left top graph is a sinetone that is perfectly periodic in the time window. TheFFT (left-bottom) shows no leakage; it is narrow andhas a peak magnitude of one, which represents themagnitude of the sine wave. The middle-top plotshows a sine tone that is not periodic in the time win-dow. This results in leakage in the FFT (middle-bot-tom). Applying the Hanning window (top-right),reduces the leakage in the FFT (bottom-right).

Figure 24. Hanning window (right) reduces the effect of leakage

(middle)

FFT Settings: dF, dT, time window,spanOther important settings on the DSA include time andfrequency resolution, and time and frequency span. Adata block consists of some fixed number of datapoints that represent the digitized time record. Thetime between the points is called the time resolution,or dT. The time span of the data block can be com-puted from dT multiplied by the number of points.These parameters are set on the DSA and they affectthe measurement results. If an event lasts a longtime, you must set the system for a long time span. Ifthe vibration levels change rapidly, use a smaller dT tocapture the details of the time history.

There are two main parameters in the frequencydomain: the frequency resolution, dF; and the frequen-cy span. All frequency measurements are plotted withfrequency on the horizontal axis. The dF defines thespacing between two points on the frequency axis.The frequency span defines the range between zeroand the highest value on the frequency axis. Theseparameters are also set in the DSA. If the PSD shapechanges dramatically over a short frequency range,use a small dF.

You can not select these four parameters independ-ently because of the interrelations between them. Thetable below summarizes the relationships.

Table 1. Interrelations between FFT Settings

Displays: scaling, waterfall, spectro-graphIn addition to two-dimensional plots, common displayformats include orbit plots, waterfall plots and spec-trographs. An orbit plot shows one time trace on thex axis and a second time trace on the y axis. A water-fall is a three dimensional plot made by stacking upconsecutive two-dimensional plots. Waterfall plotsshow how a signal changes over time, or how a signalmeasured from a rotating machine changes with vari-ations in the RPM. They are also useful for OrderAnalysis. Figure 25 shows a typical waterfall plot ofthe spectrum of the vibration measured on a rotatingmachine during a normal run-up. Often the waterfallplot includes an option to display one slice of thewaterfall in a separate pane. Waterfalls can also bepresented as a spectrograph as shown in Figure 26,a two dimensional format using color to representamplitude.

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Frequency ResolutiondF (Hz) = 1/Time Period (sec)0.5 second Time Period: dF = 1/0.5 = 2 Hz

Number of Frequency PointsNumber Frequency Points = ½ Number Time Points2048 time points yields 1024 frequency lines

Example:Time Record: 2048 points, dT=0.001 sec, T= 2.048 secFFT: 1024 frequency lines, 0.488 Hz - 500 Hz

Figure 25. Time waterfall plot of PSD measured from a rotating

machine during run-up with spectrum slice on top.

Figure 26. Spectrograph of rotating machine run-up shown in Figure 25.

Vibration SuppressionAfter taking vibration measurements and identifyingproblems, the next task is to fix them. This usuallymeans suppressing by modifying the structure to elim-inate the unwanted vibration effects. The many meth-ods available for vibration suppression include sourceisolation, absorbers, damping treatment and activesuppression. The examples that follow show tech-niques of vibration suppression in rotating equipment.

IsolationTreating the source of the vibration is the most effec-tive and often the most economical solution to vibra-tion problems. The rotating unbalance in Figure 27may cause the entire structure to vibrate. This can besolved by balancing the rotor and eliminating the vibra-tion source. Another method of isolation is to add ahighly damped material between the source and thestructure. For example, on automobile engines largerubber mounts isolate the chassis from the engine.

AbsorbersWhen you cannot isolate the vibration source from thestructure, another choice is to add a vibrationabsorber. The challenge is to absorb the vibrationwithout affecting the structure. A vibration absorberis simply a mass-spring-damper system added to thestructure and tuned to the same frequency as theoffending vibration. An example is shown in Figure 27where a vibration absorber is added to a rotatingmachine bracket to reduce the vibration of the brack-et and surrounding structure. The mass-spring-damper system resonates and vibrates with largeamplitudes, thus eliminating or reducing the vibrationfrom the rest of the structure and reducing the vibra-tion transmitted to other parts of the structure.

Figure 27. Vibration isolators (left) and absorbers (right) are meth-ods of passive vibration suppression.

Damping TreatmentThe most common form of vibration suppression is toadd viscoelastic damping treatment to structural ele-ments that are otherwise lightly damped. A thin stealbeam may be highly resonant and exhibit large vibra-tion that cause fatigue and failure, or transmit vibra-tion to other parts of the structure. Applying an elas-tic coating to the steel surface increases the dampingof the beam and thereby significantly reduces thevibration and its transmission. This method solves theproblem with a negligible increase to the mass of thestructure. Some structures use complex compositematerials to increase the damping. The layered sheetmetal used in automobile bodies is one example.

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Critical Speeds of RotatingEquipmentRotating machines such as turbines, compressorsand shafts are particularly subject to imbalances thatcause vibration. When the rotating speed corre-sponds to the resonance frequency of the first bend-ing mode of the shaft, large forces are generated andtransmitted to the bearings, eventually causing failure.This speed is called "critical speed," and the vibrationis called "synchronous whirl." Rotating machines nor-mally operate above the critical speed, so that therotor must go through the critical speed to come upto full speed. If the run-up is too slow, the resonanceat critical speed may build to unsafe levels. The rotorshould have sufficient damping in the first mode toavoid this problem.

Active Vibration SuppressionWhen no other means of vibration suppression is fea-sible, active vibration suppression may be the onlyanswer. Active suppression refers to using electron-ic controls to measure the vibration levels, processthe data and drive a mechanical actuator to counter-act the vibration levels as illustrated in Figure 28. Itis analogous to the anti-noise sound suppression sys-tems in airplanes and jets. Active suppression isexpensive to implement and requires careful designthat may be strongly dependent on the nature of thestructure. Off-the-shelf active vibration suppressionsystems are not available.

Figure 28. Damping treatment is the most common and active sup-pression is the most expensive technique for vibration suppression.

Vibration suppression is a broad topic. It is best tostart at the source and work down the transmissionpath when necessary. Most techniques add damping,or change the mass and stiffness to move the reso-nant frequencies away from fixed driving frequencies.However, changing the vibration response of one partof a structure is likely to change the vibrationresponse of other parts as well.

Modal AnalysisStructures vibrate in special shapes called modeshapes when excited at their resonant frequencies.Under normal operating conditions, the structure willvibrate in a complex combination of all the modeshapes. By understanding the mode shapes, all thepossible types of vibration can be predicted. ModalAnalysis refers to measuring and predicting the modeshapes and frequencies of a structure.

The mode shapes and resonant frequencies (themodal response) of a structure can be predicted usingmathematical models known as Finite Element Models(FEM). These models use points that are connectedby elements with the mathematical properties of thestructure's materials. Boundary conditions define themethod of fixing the structure to the ground and theforce loads applied. After defining the model andboundary conditions, the FEM software computes thestructure's mode shapes and resonant frequencies.This analytical model greatly aids in the design of astructure by predicting its vibration response before itis built. Figure 29 shows a Finite Element Model of apressurized storage tank with force loads and bound-ary conditions.

Figure 29. Finite Element Model of EADS Ariane 5 space vehicle.

After building the structure, good practice requiresverifying the FEM using experimental modal analysis.This identifies errors, particularity in the assumptionson the boundary conditions, to help improve the modelfor future designs. Experimental modal analysis isalso useful without FEM models because it can identi-fy the modal response of an existing structure to helpsolve a vibration problem.

Experimental Modal Analysis consists of exciting thestructure with an impact hammer or vibrator, meas-uring the frequency response functions between theexcitation and many points on the structure, and thenusing software to visualize the mode shapes.Accelerometers measure the vibration levels at sever-al points on the structure and a signal analyzer com-putes the FRFs (Frequency Response Functions).

Typically, the structure is divided into a grid patternwith enough points to cover the entire structure, or atleast the areas of interest. The size of the gridsdepends of the accuracy needed. More grid pointsrequire more measurements and take more time.

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One FRF measurement is made for every measure-ment location on the structure. The number of meas-urement points is determined by size and complexity ofthe structure and the highest resonant frequency ofinterest. High frequency resonances require a finegrid to fully determine the mode shape.

Each FRF identifies the resonant frequencies of thestructure and the modal amplitudes of the measure-ment grid point associated with the FRF. The modalamplitude indicates the ratio of the vibration accelera-tion divided by the force input. The mode shape isextracted by examining the vibration amplitudes of allthe grid points.

For example, the tuning fork shown in Figure 30 is avery simple structure that could reasonably be repre-sented by a very few points. Say that threeaccelerometers are placed on points one throughthree and then impact hammer is used to excite thestructure at point 4. A signal analyzer records theFRFs and produces the results shown in Figure 30.The resonant frequencies are the peaks that appearat every point at the same frequency. The amplitudeof the peak at each location describes the mode shapefor the associated resonant frequency. The resultsindicate that for the first mode, the base is fixed andthe end has maximum displacement as shown inFigure 31. The second mode has maximum deflectionat the middle of the fork as shown in Figure 31.

Figure 30. Modal analysis of a tuning fork results in FRFs for eachpoint on the structure with the amplitudes at each resonant fre-quency describing the mode shape.

Figure 31. The first mode shape of the tuning fork has the basefixed and the maximum deflection at the end and the second modeshape has the ends fixed and maximum deflection at the middle.

Specialized software applications such as VibrantTechnology's ME'scope or MTS's IDEAS® use FRF datato visualize the mode shapes. The software automat-ically identifies the resonant frequencies and com-putes the modal amplitudes. Then it animates themode shapes by drawing the deformed structureeither with a static image or a moving animation show-ing deformation from one extreme to the other. Thesoftware can generate color plots such as the bridgeshown in Figure 32 to represent the amount of defor-mation.

Figure 32. Modal analysis of a bridge with deformed geometry and

color shading.

The goal of modal analysis is to understand the modeshapes by visualizing the deformed geometry. Thisinformation may solve vibration problems. For exam-ple, a consumer product may be failing in the field. Itis suspected that vibration is causing the problem. Itis also known that an internal cooling fan rotates at3600 RPM, or 60 Hz. Modal analysis of the struc-ture shows that the cooling fan mounting bracket hasa resonant frequency with a bending mode at 62 Hz.This suggests that the fan is exciting the bendingmode of the bracket and causing the failure. Thebracket can be redesigned by stiffening the structureand increasing the resonant frequency to 75 Hz, thuseliminating the failure.

Operating Deflection Shape AnalysisOperating Deflection Shape Analysis (ODS) is similar tomodal analysis. There are two types of ODS analysis:time-based and frequency-based. Time-based ODS col-lects the time records of the motion of points on thestructure during normal operation. Then visualizationsoftware animates a geometric model of the structureso that moves in a manner proportional to the record-ed data. Time-based ODS is different from modalanalysis because there are no FRFs to record and nomathematical algorithms to apply.

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It is analogous to video taping the motion of the struc-ture during normal operation and then playing it back tounderstand the motion. The advantage is that somemeasured motions may be too small to see with thenaked eye. The software increases the scale of thesemotions so that they can be visualized.

Frequency-based ODS animates the vibration of a struc-ture at a specific frequency. Animated deformationshapes derive from frequency domain measurementssuch as auto-spectra or frequency response functions.Frequency-based ODS provides information on the rela-tive levels of structural deformation . It is useful forcases where a force is applied and dwells at a specificfrequency. It is a great tool for quickly spotting prob-lems due to loose parts or connections. It is also use-ful for determining whether vibration will excite a reso-nance.

ConclusionStructural vibration analysis is a multifaceted disciplinethat helps increase quality, reliability and cost efficiencyin many industries. Analyzing and addressing structuralvibration problems requires a basic understanding ofthe concepts of vibration, the basic theoretical models,time and frequency domain analysis, measurementtechniques and instrumentation, vibration suppressiontechniques, and modal analysis.

References

Demler, Michael J., "High-Speed Analog-To-DigitalConversion," Academic Press, Inc., San Diego,California 1991.

Hartmann, William M., "Signals, Sound, andSensation," American Institute of Physics, New York,1997.

Inman, Daniel J., "Engineering Vibrations, SecondEdition," Prentice Hall, New Jersey, 2001.

Ramirez, Robert W., "The FFT, Fundamentals andConcepts," Prentice-Hall, New Jersey, 1985.

Ziemer, Rodger E., et. al., "Signals & Systems,"Prentice Hall, New Jersey, 1998.

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Photon and Focus are the trademarks of LDS. All other trademarks are thetrademarks of their respective holders.

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