HP-AN243-3_Fundamentals of Modal Testing

8/14/2019 HP-AN243-3_Fundamentals of Modal Testing

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The Fundamentals of Modal Testing

Application Note 243-3

() = n

r = 1

i j /m

( n - ) + (2 n) 2 2 2 2


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Modal ana lysis is defined as thestudy of the dynamic chara cteris-tics of a mechanical structure.

This application note emphasizesexperimental modal techniques,specifically the method knownas frequency response functiontesting. Other areas are treatedin a general sense to introducetheir elementary concepts andrelationships to one another.

Although modal techniquesare mathematical in nature, thediscussion is inclined towardpract ical application. Theory ispresented as needed to enhancethe logical development of ideas.The reader will gain a soundphysical understanding of modalanalysis and be able to carryout an effective modal surveywith confidence.

Chapter 1 provides a brief overview of structural dynamicstheor y. Chapter 2 and 3 whichis the bulk of the note describesthe measurement process foracquiring frequency responsedata. Chapter 4 describes theparameter estimation methodsfor extracting modal properties.Chapter 5 provides an overviewof analytical techniques of struc-tural analysis and their relationto experimental modal testing.

Preface


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Table o f Conte nts

Preface 2

Chapter 1 Structural Dynamics Background 4Introduction 4Struc tural Dynamic s of a Single Degree of Freedom (SDOF) System 5Pr esentation and Charac ter istic s of Fr equenc y Response Func tions 6St ruc tur al Dynamics for a Mult iple Degree of Fr eedom (M DOF) Syst em 9Damping Mechanism and Damping Model 11Frequency Response Func tion and Transfer Func tion Relationship 12System Assumptions 13

Chapter 2 Frequency Response M easurements 14Introduction 14General Test System Configurations 15Supporting the Structure 16Exciting the Structure 18Shaker Testing 19Impact Testing 22Transduction 25Measurement Interpretation 29

Chapter 3 Improving M easurement Accuracy 30Measurement Averaging 30Windowing Time Data 31Increasing Measurement Resolution 32

Complete Survey 34

Chapter 4 M odal Parameter Estimation 38Introduction 38Modal Parameters 39Curve Fitting Methods 40Single Mode Methods 41Concept of Residual Terms 43Multiple Mode-Methods 45Concept of Real and Complex Modes 47

Chapter 5 Structural Analysis M ethods 48Introduction 48Structural Modification 49Finite Element Correlation 50Substructure Coupling Analysis 52Forced Response Simulation 53

Bibliography 54


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response function are examinedto understand the trend s and theirusefulness in the measurementprocess. Finally, these conceptsare extended into MDOF systems,

since this is the type of behaviormost physical structures exhibit.Also, useful concepts associatedwith damping mechanisms andlinear system assumptionsare discussed.

Introduction

A basic understanding of struc-

tural dynamics is necessaryfor successful modal testing.Specifically, it is important tohave a good grasp of the relation-ships between frequency responsefunctions and their individualmodal parameters illustrated inFigure 1.1. This unde rstand ing isof value in both the measurementand analysis phases of the survey.Knowing the various forms andtrends of frequency responsefunctions will lead to more accu-racy during the measurementphase . During the analysis phase ,knowing how equations relateto frequency responses leads tomore accurate es timation of modal parameters.

The basic equations and theirvarious forms will be presentedconce ptually to give insight intothe relationships between thedynamic characteristics of thestructure and the correspondingfrequency response function mea-surements. Although practicalsystems are multiple degreeof freedom (MDOF) an d havesome degree of nonlinearity,they can generally be representedas a superposition of singledegree of freedom (SDOF) linearmodels and will be developed inthis manner.

First, the b asics of an SDOF

linear dynamic system are pre-sented to gain insight into thesingle mode concepts that arethe basis of some parameterestimation techniques. Second,the presentation and propertiesof various forms of the frequency

Chapter 1

Structural Dynamics Background

Figure 1.1Phases of amodal tes t

Test Structure

Frequency Response M easurements

M odal Parameters

Curve Fit Representation

() = ijn

r = 1

ir jrmr ( r - + j2 r)

2 2

Frequency

Damping{ } Mode Shape


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Structural Dynamics o f aSingle Degree o f Free dom( SDOF) System

Although most physical structuresare continuous, their behaviorcan usually be represented bya discrete parameter model asillustrated in Figure 1.2. The ide-alized elements are called mass,spring, damper and excitation.The first three elements describethe p hysical system. Energy isstored by the system in the massand the spring in the form of kinetic and potential energy,respectively. Energy enters thesystem through excitation andis dissipated through damping.

The idealized elements of thephysical system can be describedby the equation of motion shownin Figure 1.3. This equation re-lates the effects of the mass, stiff-ness and damping in a way thatleads to the calculation of naturalfrequency and damping factor of the system. This computation isoften facilitated by the use of thedefinitions shown in Figure 1.3that lead directly to the natu ralfrequency and damping factor.

The natural frequency, , isin units of radians per second(rad/s). The typical units dis-played on a digital signal ana-lyzer, however, a re in Hert z (Hz).The damping factor can also berepresented as a percent of criti-

cal damping the d amping levelat which the system experiencesno os cillation. This is the morecommon understan ding of modaldamping. Although there are threedistinct damping cases, onlythe underdamped case ( < 1) isimportant for structuraldynamics applications.

Figure 1.2SDOF discreteparameter model

Spring k

ResponseDisplacement (x) Mass m

Damper c

ExcitationForce (f)

Figure 1.3Equation of motion modal definitions

Figure 1.4Complex root sof SDOFequation

s = - + j d1,2

Damping Rate Damped Natural Frequency

mx + cx + kx = f(t).. .

2n = ,km

2 n =cm

or = c2km

Figure 1.5SDOFimpulseresponse/ free decay e- t

d


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When there is no excitation,the roots o f the equation areas shown in Figure 1.4. Each

root has two parts: the real partor decay rate, which definesdamping in the system and theimaginary part, or oscillatoryrate, which defines the dampednatural frequency, d. This freevibration response is illustratedin Figure 1.5.

When excitation is applied,the equation of motion leads tothe frequency response of thesystem. The frequency response isa complex quantity and containsboth real and imaginary parts(rectangular co ordinates). It canbe presented in polar coordinatesas magnitude and phase, as well.

Presentation andCharacteristics of Frequency Respons eFunctions

Because it is a complex quantity,the frequency response functioncanno t be fully displayed on asingle two-dimensional plot. Itcan, however, be presented inseveral formats, each of whichhas its own uses. Although theresponse variable for the previousdiscussion was displacement, itcould also be velocity or accelera-tion. Acceleration is c urren tly theaccepted method o f measuringmodal response.

One method of presenting thedata is to plot the polar coordi-nates, magnitude and phaseversus frequency as illustratedin Figure 1.6. At resona nce, when = n, the magnitude is a maxi-mum an d is limited only by theamount of damping in the system.The phase ranges from 0 to 180and the response lags the input by90 at resonance.

Figure 1.6Frequencyresponse polarcoordinates

Magnitude

Phase

d

d

H( ) =

( ) = tan-1

1/m

( n- ) + (2 n) 2 2 2 2

2 n

n-

2 2


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Another method of presentingthe data is to plot the rectangularcoordinates, the real part and the

imaginary part versus frequency.For a proportionally damped sys-tem, the imaginary part is maxi-mum at resonance and the realpart is 0, as shown in Figure 1.7.

A third method of presentingthe frequency response is to plotthe real part versus the imaginarypart. This is often called a Nyquistplot or a vector response plot.This display emphasizes thearea of frequency response atresonance and traces out acircle, as sho wn in F igure 1.8.

By plotting the magnitude indecibels vs logarithmic (log)frequency, it is pos sible to covera wider frequency range andconveniently display the rangeof amplitude. This type of plot,often known as a Bode plot,also has some useful parametercharacteristics which are de-scribed in t he following plots.

When > n the frequency responseis approximately equal to the as-ymptote also shown in Figure 1.9.

This asymptote is called the massline and has a slope of -2, -1 or 0for displacement velocity oracceleration responses,respectively.

Figure 1.7Frequencyresponse rectangularcoordinates

Real

Imaginary

H( ) = -2 n

( n- ) + (2 n)

2 2 2 2

H( ) =

2 2

2 2 2 2

n -

( n - ) + (2 n)

Figure 1.8Nyquist plotof frequencyresponse

( )

H( )


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The various forms of frequencyresponse function based on thetype of response variable are

also defined from a mechanicalengineering viewpo int. They aresomewhat intuitive and do notnecessarily correspond to electri-cal analogies. These forms aresummar ized in Table 1.1.

Figure 1.9Different fo rmsof frequencyresponse

Table 1.1Different formsof frequency response

Definition Response Variable

Compliance X DisplacementF Force

Mobility V VelocityF Force

Inertance A AccelerationF Force

Displacement

A

cceleration

Velocity

Frequency

Frequency

Frequency

1k

2

k

k

1m2

1m

1m


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Structural Dynamicsfor a Multiple Degree of Freedom ( MDOF) System

The extension of SDOF conceptsto a more general MDOF system,with n degrees of freedom, isa straightforward process. Thephysical system is simply com-prised of an interconnectionof idealized SDOF m odels, asillustrated in Figure 1.10, andis described by the matrix equa-tions of motion as illustrated inFigure 1.11.

The solution of the equationwith no excitation again leadsto the modal parameters (rootsof the equation) of the system.For the MDOF case, however, aunique displacement vector calledthe mode shape exists for eachdistinct frequency and dampingas illustrated in Figure 1.11.The free vibration response isillustrated in Figure 1.12.

The equations of motion for theforced vibration case also lead tofrequency response of the system.It can be written as a weightedsummation of SDOF systemsshow n in F igure 1.13.

The weighting, often c alled themodal participation factor, is afunction of excitation and modeshape coefficients at the inputand output degrees of freedom.

Figure 1.10MDOF discret eparameter model

k1

k3

k2

k4

c1

c3

c2

c4

m1 m2

m3

Figure 1.11Equations of motion modal definitions

Figure 1.12MDOF impuls eresponse/ free decay

Amplitude

0.0 Sec 6.0

[m]{x} + [c]{x} + [k]{x} = {f(t)}

{ }r , r = 1, n modes

.. .


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The participation factor identifiesthe amount each mode contrib-utes to the total response at a

particular point. An examplewith 3 degrees of freedomshowing the individual modalcontributions is shown inFigure 1.14.

The frequency response of anMDOF system can be presentedin the same forms as the SDOFcase. There are other definitionalforms and properties of frequencyresponse functions, such as adriving point measurement, thatare presented in the next chapter.These are related to specificlocations of frequency responsemeasurements and are introducedwhen appropriate.

Figure 1.13MDOF frequen cyresponse

dB Magnitude

0.01 2 3 Frequency

() = n

r = 1

i j /m

( n - ) + (2 n) 2 2 2 2

Figure 1.14SDOF modalcontributions

dB Magnitude

0.01 2 3 Frequency

Mode 1

Mode 2Mode 3


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measured to characterize the sys-

tem when using a linear model.

The equivalent viscous dampingcoefficient is obtained fromenergy considerations as illus-trated in the hysteresis loop inFigure 1.15. E is the energy

dissipated per cycle of vibration,

c eq is the equivalent viscous damp-ing coefficient an d X is the a mpli-tude of vibration. Note that thecriteria for equivalence are equalenergy distribution per cycle andthe same relative amplitude.

Figure 1.15Visco us dampingenergy dissipation

Damping Mechanismand Damping Model

Damping exists in all vibrato rysystems whenever there is energydissipation. This is true for me-chanical structures even thoughmost are inherently lightlydampe d. For free vibration, theloss of ene rgy from d amping inthe system results in the decayof the amplitude of motion. Inforced vibration, loss of ene rgy isbalanced by the energy suppliedby excitation. In either situation,the effect of damping is to removeenergy from the system.

In previous mathematical formu-lations the damping force wascalled viscous, since it waspropo rtional to velocity. How-ever, this does not imply thatthe physical damping mechanismis viscous in nature. It is simplya modeling method and it is im-portant to note that the physicaldamping mechanism and themathematical model of thatmechanism are two distinctlydifferent concepts.

Most structur es exhibit one ormore forms of damping mecha-nisms, such as coulomb or struc-tural, which result from loosenessof joints, internal strain and othercomplex causes. However, thesemechanisms can be modeled byan equivalent viscous dampingcomponent. It can be shown

that only the viscous compone ntactually accounts for energy lossfrom the system and the remain-ing portion of the dam ping is dueto nonlinearities that do n ot causeenergy dissipation. Therefore,only the viscous term needs to be

x

X E = ceq X

2

Force vs Displacement

cx.

Figure 1.16Systemblock diagram

InputExcitation

System

OutputResponseG(s)

Figure 1.17Definition of transfer function

Transfer Function =

G(s) =

OutputInput

Y(s)X(s)


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Frequency Respons eFunction and TransferFunction Relationship

The transfer function is a math-ematical mode l defining theinput-outpu t relationship of aphysical system. Figure 1.16shows a block diagram of a singleinput-output system. Systemresponse (output) is caused bysystem excitation (input). Thecasual relationship is looselydefined as s hown in Figure 1.17.Mathem atically, the trans ferfunction is defined as th e Laplacetransform of the output dividedby the Laplace transform of the input.

The frequency response functionis defined in a similar manner andis related to the transfer function.Mathem atically, the frequencyresponse function is defined asthe Fourier transform of the out-put divided by the Fourier trans-form of the input. These termsare often used interchangeablyand are occasionally a source of confusion.

Figure 1.18S-planerepresentation

i

n

s-plane

d

This relationship can be furtherexplained by the modal test pro-cess. The measurements takenduring a modal test are frequencyresponse function measurements.The parameter estimation rou-tines are , in general, curve fitsin the Laplace domain and resultin the transfer functions . Thecurve fit simply infers th e loca-tion of system po les in the s-planefrom the frequency response func-tions as illustrated in Figure 1.18.

The frequency response is simplythe transfer function measuredalong the j axis as illustratedin Figure 1.19.


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System Assumptions

The structural dynamics back-

ground theory and the modalparameter estimation theory arebased on two major assumptions:

Figure 1.193-D Laplac erepresentation

The system is linear. The system is stationary.

There are, of course, a numberof other system assumptions suchas obser vability, stability, andphysica l rea lizability. How ever,these assumptions tend to be ad-dressed in the inherent propertiesof mechan ical systems. As such,they do not present practical limi-tations when making frequencyresponse measurements as dothe assumptions of linearityand stationarity.

Real Part

Imaginary Part

Magnitude

Phase

j

j

j

Transfer Function surf aceFrequency Response dashed


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Introduction

This chapter investigates the

current instrumentation andtechn iques a vailable for ac quiringfrequency response measure-ments. The discussion beginswith the use of a dynamic signalanalyzer and associated peripher-als for making these measure-ments. The type of modal testingknown as the frequency responsefunction metho d, which measuresthe input excitation and output re-sponse simultaneously, as shownin the b lock diagram in F igure 2.1,is examined . The focus is on theuse of one input force, a tech-nique commonly known as single-point excitation, illustrated inFigure 2.2. By unde rstand ing thistechnique, it is easy to expand tothe multiple input technique.

With a dynamic signal analyzer,which is a Fourier transform-based instrument, many types of excitation sources can be imple-mented to measure a structuresfrequency response function.In fact, virtually any physicallyrealizable signal can be inpu t ormeasur ed. The selection andimplementation of the morecommon and useful types of signals for mod al testing arediscussed.

Chapter 2

Frequency Response Measurements

Figure 2.1System blockdiagram

Excitation ResponseH( )X( ) Y( )

Figure 2.2Structureunder tes t

Structure

Force Transducer

Shaker

Transducer selection andmounting methods for measuringthese signals along with systemcalibration methods, are alsoincluded. Techniques for improv-ing the quality and ac curac y of measurements are then explored.These include processes such asaveraging, windowing and zoom-ing, all of which reduce measure-ment errors . Finally, a section onmeasurement interpretation is in-cluded to aid in understanding thecomplete measurement process.


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Gene ral Test Syste mConfigurations

The basic test set-up requiredfor making frequency responsemeasurements depends on afew major factors . These includethe type of structure to be testedand the level of results desired.Other factors, including the sup-port fixture and the excitationmechanism, also affect theamount of hardware needed toperform the test. Figure 2.3shows a diagram of a basictest system configuration.

The heart of the test system is thecontroller, or computer, which isthe operators communicationlink to the analyzer. It can beconfigured with various levelsof memory, displays and datastora ge. The modal analysissoftware usually resides here, aswell as an y additional analysiscapabilities such as structura lmodification and forced response.

The analyzer provides the dataacquisition and signal processingopera tions. It can be configuredwith several input channels,for force and response measure-ments, and with one or moreexcitation sources for drivingshake rs. Measurem ent functionssuch as windowing, averaging andFast Fo urier Transforms (FFT)computation are usually pro-cessed within the analyzer.

Figure 2.3General tes tconfiguration

Structure

Transducers

Exciter

Controller

Analyzer

For making measurements onsimple struc tures, the excitermechanism can be as basic asan instrumented hammer. Thismechanism requires a minimumamount of hardware. An electro-dynamic shaker may be neededfor exciting more complicatedstructures. This shaker systemrequires a signal source, a poweramplifier and an attachmentdevice. The signal sourc e, asmentioned earlier, may be acomponent of the a nalyzer.

Transducers, along with a powersupply for signal con ditioning, areused to measure the desired forceand respo nses. The piezoelectrictypes, which measure force andacceleration, are the most widelyused for modal testing. Thepower supply for signal condition-ing may be voltage or ch argemode and is sometimes providedas a component of the analyzer,so care should be taken in settingup and matching this part of thetest system.


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Supporting The Structure

The first step in setting up a

structure for frequency responsemeasurements is to consider thefixturing mechanism necessaryto obtain the desired constraints(boun dary cond itions). This isa key step in the process as itaffects the overall structuralcharacteristics, particularly forsubsequent analyses such asstruct ural mo dification, finiteelement correlation a ndsubstructure coupling.

Analytically, b ounda ry c onditionscan be specified in a completelyfree or completely constrainedsense . In testing pract ice, how-ever, it is generally not po ssibleto fully achieve these conditions.The free condition means that thestruct ure is, in effect, floating inspace with no attachments toground and exhibits rigid bodybehavior at zero frequency. Theairplane sho wn in Figure 2.4a isan example of this free condition.Physically, this is n ot realizable,so the structure must be sup-ported in some manner. Theconstrained condition impliesthat the motion, (displacements/ rotations ) is set to zero. How-ever, in reality most structuresexhibit some degree o f flexibilityat the grounded connections. Thesatellite dish in Figure 2.4b is anexample of this condition.

In order to approximate thefree system, the structure canbe suspended from very softelastic cords or placed on a verysoft cushion. By doing this, thestructure will be constrained to adegree and the rigid body modeswill no longer h ave zero fre-quenc y. However, if a sufficient lysoft support system is used, therigid b ody frequencies will be

Figure 2.4aExample of free supportsituation

Figure 2.4bExample of constrainedsupportsituation

ConstrainedBoundary

much lower than the frequenciesof the flexible modes and thushave negligible effect. The ruleof thumb for free supports is thatthe highest rigid body modefrequency must be less than onetenth that o f the first flexiblemode. If this criterion is met,rigid bo dy modes will have

negligible effect on flexiblemodes. Figure 2.5 shows atypical frequency responsemeasurement of this type withnonzero rigid body modes.

The implementation of a con-strained system is much moredifficult to ac hieve in a te st en vi-ronme nt. To begin with, the baseto which the structure is attachedwill tend to have some motion of its own. Therefore, it is not goingto be purely grounded . Also, theattachment points will have some

degree o f flexibility due t o thebolted, riveted or welded connec-tions. One possible remed y forthese problems is to measure the

FreeBoundary


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frequency response of the baseat the attachment points overthe frequency range of interest.

Then, verify that this responseis significantly lower than thecorresponding response of thestructure, in which case it willhave a negligible effect. How-ever, the frequency response maynot be measurable, but can stillinfluence the test results.

There is not a best practical orappropriate method for support-ing a structure for frequencyresponse testing. Each situationhas its own characteristics. Froma practical standpoint, it wouldnot be feasible to support a largefactory machine weighing severaltons in a free test state. On theother hand, there may be noconvenient way to ground a verysmall, lightweight device for theconst rained test state . A situationcould occur, with satellite for ex-ample, where the results of bothtests are desired. The free test isrequired to analyze the s atellitesoperating environment in space.However, the constrained test isalso needed to assess the launch

Figure 2.5Frequencyresponseof freelysuspendedsystem

Hz 1.5625kFxdXY 0

-30.0

dB

10.0/Div

50.0

FREQ RESP

Rigid Body Mode

1st Flexible M ode

environment attached to the boostvehicle. Anothe r reaso n forchoosing the appropriatebound ary conditions is for finiteelement model correlation orsubstructure coupling analyses.At any ra te, it is cert ainly impor-tant during this phase of the testto ascertain all the conditions inwhich the results may be used.


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Periodic* Transientin analyzer w indow in analyzer w indow

Sine True Pseudo Random Fast Impact Burst Burststeady random random sine sine randomstate

Minimze leakage No No Yes Yes Yes Yes Yes Yes

Signal to noise Very Fair Fair Fair High Low High Fairhigh

RMS to peak ratio High Fair Fair Fair High Low High Fair

Test measurement time Very Good Very Fair Fair Very Very Verylong good good good good

Controlled frequency content Yes Yes* Yes* Yes* Yes* No Yes* Yes*

Controlled amplitude content Yes No Yes* No Yes* No Yes* No

Removes distortion No Yes No Yes No No No YesCharacterize nonlinearity Yes No No No Yes No Yes No

* Requires additional equipment or special hardware

Exciting the Structure

The next step in the measurement

process involves selecting anexcitation function (e.g., randomnoise) along with an excitationsystem (e.g., a shaker) that bestsuits the application. The choiceof excitation can make the differ-ence between a good measure-ment and a poor one. Excitationselection should be approachedfrom both th e type of functiondesired and the type of excitationsystem available because they areinterrelated. The excitation func-tion is the mathematical signalused for the input. The excitationsystem is the physical mechanismused to prove the signal. Gener-ally, the choice of the excitationfunction dictates the choice of theexcitation system, a true randomor burst random function requires

a shaker system for implementa-tion. In genera l, the reverse isalso true. Choosing a hammer

for the excitation system dictatesan impulsive type excitationfunction.

Excitation funct ions fall into fourgeneral categories: steady-state,random, periodic and transient.There are several papers thatgo into great det ail examining theapplications of the most commonexcitation functions . Table 2.1summarizes the basic characteris-tics of the ones that are mostuseful for modal testing. Truerandom, burst random and im-pulse types are considered in thecontext of this note since they arethe most widely implemented.The best choice of excitationfunction depends on several fac-tors: available signal proces sing

equipment, characteristics of thestructure, general measurementconsiderations and , of course, the

excitation system.

A full function dynamic signalanalyzer will have a s ignal sour cewith a sufficient number of func-tions for exciting the structure.With lower quality analyzers,it may be necessary to obtain asignal source as a se parate part of the signal processing equipment.These sources often provide fixedsine and true rando m functions assignals; however, these may notbe acceptable in applicationswhere high levels of accuracy aredesired. The types of functionsavailable have a significant influ-ence on measurement quality.

Table 2.1Excitationfunctions


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The dynamics of the structureare also important in choosing theexcitation function. The level of

nonlinearities can be measu redand characterized effectivelywith sine sweeps or chirps, but arandom function may be neededto estimate the best linearizedmodel of a nonlinear system.The amount of damping and thedensity of the modes within thestructure can also dictate the useof specific excitation functions. If modes are closely coupled and/orlightly damped, an excitationfunction that can be implementedin a leakage-free manner (burstrandom for example) is usuallythe most appropriate.

Excitation mechanisms fall intofour categories: shaker, impactor,step relaxation and self-operating.Step relaxa tion involvespreloading the structu re witha measured force through a

cable then releasing the cableand measuring the transients.Self-operating involves exciting

the structure through an actualopera ting load. This input cannotbe measured in many cases, thuslimiting its usefulness. Shakersand impactors are the most com-mon and are discussed in moredetail in the following sections.Another method of excitationmechanism classification is todivide them into attach ed andnonat tache d devices. A shake ris an attached device, while animpactor is not, (although it doesmake contact for a short per iodof time).

Shaker Testing

The most useful shakers for

modal testing are the electromag-netic sh own in Fig. 2.6 (oftencalled electrodynamic) and theelectro hydraulic (or, hydraulic)types. With the electromagneticshaker, (the more common of thetwo), force is generated by analternating current that drives amagnetic coil. The maximum fre-quency limit varies from appr oxi-mately 5 kHz to 20 kHz depend ingon the size; the smaller shakershaving the higher operating range.The maximum force rating isalso a function of the size of theshaker and varies from approxi-mately 2 lbf to 1000 lbf; thesmaller the shaker, the lowerthe force rating.

With hydraulic shake rs, forceis generated through the use of hydraulics, which can providemuch higher force levels someup to several thousand pounds.The maximum frequency rangeis much lower though about1 kHz and below. An advantageof the hydraulic shaker is itsability to a pply a large sta ticpreload to the structur e. This isuseful for massive structures suchas grinding machines that operateunder relatively high pre loadswhich may alter their structuralcharacteristics.


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There are several potentialproblem areas to consider whenusing a shaker system for excita-

tion. To begin with, the shakeris physically mounted to thestructure via the force transducer,thus creating the possibility of altering the dynamics of the struc-ture. With lightweight struc tures ,the mechanism used to mountthe load cell may add appreciablemass to the structure. Thiscauses the force measured bythe load cell to be greater thanthe force actually applied to thestruct ure. Figure 2.7 descr ibeshow this mass loading alters theinput force. Since the extra massis between the load cell and thestructure the load cell sensesthis extra mass as part of thestructure.

Since the frequency response is asingle input function, the shakershould transmit only one compo-nent o f force in line with th e mainaxis of the load cell. In practicalsituations, when a structure isdisplaced along a linear a xis italso tends to rotate about theother two axes. To minimize theproblem of forces being appliedin other directions, the shakershould be connected to the loadcell through a slender rod, calleda stinger, to allow the structureto move freely in the other direc-tions. This rod, shown in Figure2.8, has a stron g axial stiffness,but weak bending and shear

stiffnesse s. In effect, it acts likea truss member, carrying onlyaxial loads but no momen ts orshear loads.

Figure 2.6Electrodynamicshaker withpower amplifierand signal source

Power Amplifier

Figure 2.7Mass loadingfrom shakersetup

Ax

Fs

Fm

F1

Fs = Fm - MmAx

LoadingMass

LoadCell

Structure


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The method of supporting theshaker is another factor that canaffect the force imparted to the

struc ture. The main body of theshaker must be isolated from thestructure to p revent any reactionforces from being transmittedthrough the base of the shakerback to the structure. This canbe accomplished by mountingthe shaker on a solid floor andsuspending the structure fromabove. The shake r could alsobe supported on a mechanically-isolated foundation. Anothe rmethod is to suspend the shaker,in which case an inertial massusually needs to be attachedto the shaker body in order togenerate a measura ble force,particularly at lower frequencies.Figure 2.9 illustra tes the differenttypes of shaker set-ups.

Another potential problemassociated with electromagneticshakers is the impedance mis-match that can exist betweenthe structure and the shaker coil.The electrical impedance of theshaker varies with the amplitudeof motion of the coil. At a reso-nance with a small effective mass,very little force is required toproduce a response. This canresult in a drop in the forcespect rum in the vicinity of theresonance, causing the forcemeasurement to be susceptibleto noise. Figure 2.10 illustratesan example of this phenomenon.

The problem can usually becorrected by using shakers withdifferent size coils or d riving theshaker with a constant-currenttype amplifier. The shake r couldalso be moved to a point with alarger effective ma ss.

Figure 2.8Stingerattachmentto st ructure

Figure 2.9Test supportmechanisms

Figure 2.10Shaker/structureimpedancemismatch

POWER SPEC1 30 Avg 50%Ovlp Hann-20.0

dB

-100

Fxd Y 1.1k Hz 2.1k

FREQ RESP 30 Avg 50%Ovlp Hann40.0

dB

-40.0

Fxd Y 1.1k Hz 2.1k

rmsV2

FRF

Force Spectrum


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Impact Testing

Another common excitation

mechanism in modal testing isan impact device. Although it isa relatively simple tec hnique toimplement its difficult to obtainconsisten t results. The conve-nience of this technique is attrac-tive becaus e it requires very littlehardware and provides shortermeasurement times. The methodof applying the impulse, sh own inFigure 2.11, includes a hammer,an electric gun or a susp endedmass. The hammer, the mostcommon of these, is used in thefollowing discussion. However,this information also applies tothe other types of impact devices.

Since the force is an impulse,the amplitude level of the energyapplied to the structure is a func-tion of the mass and the velocityof the hammer . This is due to theconcept of linear momentum,which is defined as mass timesvelocity. The linear impulse isequal to the incremental changein the linear momen tum. It isdifficult though to co ntrol thevelocity of the hammer, so theforce level is usua lly contr olledby varying the mass. Impacthammers are available in weightsvarying from a few ounces toseveral poun ds. Also, mass canbe added to or removed frommost hammers, making themuseful for testing objects of

varying sizes and weights.

The frequency content of theenergy applied to the structure isa function of the stiffness of thecontacting surfaces and , to alesser extent, the mass of thehammer . The stiffness of the con-tacting surfaces affects the shapeof the force pulse, which in turndetermines the frequency content.

Figure 2.11Impact devicesfor test ing

Figure 2.12Frequencycontent of various pulse s

Hz 2.5kFxd X 0-130

dB

10.0/Div

-50.0

POWER SPEC1 1Avg 0%Ovlp Fr/Ex

rms

v 2

Sec 15.9mFxd Y

-46.9

-50.0m

Real

50.0m

/Div

350m

FILT TIIME1 0%Ovlp

v

t

Hard

Medium

Soft

Pulse

Soft

Medium

Hard


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23

It is not feasible to changethe stiffness of the test object,therefore the frequency content is

contr olled by varying the s tiffnessof the hammer tip. The harderthe tip, the shorter the pulseduration and thus the higher thefrequency conte nt. Figure 2.12illustrates this effect on the forcespect rum. The rule of thumb is tochoose a tip so tha t the amplitudeof the force spectrum is no morethan 10 dB to 20 dB down at th emaximum frequency of interestas show n in Figure 2.13. A disad-vantage to note h ere is that theforce spectrum of an impactexcitation cannot be band-limitedat lower frequencies when mak-ing zoom measureme nts, so thelower o ut-of-band modes will stillbe excited.

Impact testing has two pote ntialsignal processing problemsassoc iated with it. The first noise can be p resent in eitherthe force or response signal as aresult of a long time record. Thesecond leakage can be presentin the response signal as a resultof a short time record. Compen-sation for both these p roblemscan be accomplished withwindowing techniques.

Since the force pulse is usuallyvery short relative to the length of the time record, the portion of thesignal after the pulse is noise andcan be eliminated without affect-

ing the pulse itself. The windowdesigned to accomplish this,called a force window, is shownin Figure 2.14. The small amoun tof oscillation that occurs at theend of the pulse is actually partof the pulse. It is a result of signal processing and shouldnot be truncated.

Figure 2.13Useful frequencyrange o f pulse

spectrum

Hz 2.5kFxd Y 0

-130

dB

10.0/Div

-50.0

POWER SPEC1 1Avg 0%Ovlp Fr/Ex

rms

v 2

-72 dB

-88 dB

Figure 2.14Force pulsewith forcewindow applied

Sec 100mFxd X 250

-40.0m

Real

20.0m

/Div

120m

FILT TIIME1 0%Ovlp

v

Force Pulse

TimeForce W indow

Ampl

1.0


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24

The response signal is an expo-nential decaying function andmay decay out before or after the

end of the measurement. If thestructure is heavily damped, theresponse may decay out beforethe end of the time record. In thiscase, the response window can beused to eliminate the remainingnoise in the time record. If thestructure is lightly damped, theresponse may continue beyondthe end of the time record. In thiscase, it mus t be artificially forcedto decay out to minimize leakage.The window designed to accom-plish either result, called the ex-ponential window, is shown inFigure 2.15. The rule of thumbfor setting the time constant, (thetime required for the amplitude tobe red uced by a factor of 1/e), isabout one-fourth the time recordlength, T. The result of this isshown in Figure 2.15.

Unlike the force window, theexponential window can alterthe resulting frequency responsebecause it has the effect of addingartificial damping to the system.The added damping coefficientcan usually be backed out of themeasurement after signal process-ing, but numerical problems mayarise with lightly damped struc-tures. This can happen when theadded damping from the expone n-tial window is significantly morethan the true damping in thestructure. A better measurement

procedure in this case would beto zoom, thus utilizing a longertime record in order to capturethe entire response, insteadof relying on the exponentialwindow.

Figure 2.15Decayingresponse withexponentialwindow applied

Sec

Sec

512m

512m

Fxd Y

Fxd Y

0.0

0.0

-250m

-250m

Real

Real

62.5m

/Div

62.5m

/Div

250m

250m

FILT TIIME2

WIND TIIME2

No Data

0% Ovlp

v

v

Response

Exponential Decaying Response

Exponential W indow

Ampl

Time

Window ed Response


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25

Transduction

Now that an excitation system

has been set up to force the struc-ture into motion, the transducersfor sensing force and motion needto be selected. Although there arevarious types of transducers, thepiezoelectric type is the mostwidely used for mo dal testing. Ithas wide frequency and dynamicranges, good linearity and is rela-tively dura ble. The piezoelectrictransducer is an electromechani-cal sensor that generates an elec-trical output when subjected tovibration. This is acc omplishedwith a crystal element thatcreates a n electrical chargewhen mechanically strained.

The mechanism of the force trans-ducer, called a load cell, functionsin a fairly simple man ner. Whenthe crystal element is strained, ei-ther by tension or compression, itgenerates a charge proportionalto the applied force. In this case,the applied force is from theshaker. However, due to moun t-ing methods discussed earlier,this is not necessarily the forcetransmitted to the structure.

The mechanism of the responsetransducer, called an accelerom-eter, functions in a similar man-ner. When the accelerometervibrates, an internal mass in theassembly applies a force to thecrystal element which is propor-

tional to the acceleration. Thisrelationship is simply Newton sLaw: force equals mas s timesacceleration.

The properties to co nsider inselecting a load cell include boththe type of force sensor and itsperformance characteristics. Thetype of force sen sing for wh ichload cells are designed includecompression, tension, impact or

Figure 2.16Frequencyresponse of transducer

dB Mag

Frequency

Useable Frequency Range

Response Transducer (Accelerometer)

Figure 2.17Stud mountedload cell


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26

some combination thereof. Mostshaker tests require at least acompression and ten sion type. A

hammer test, for example, wouldrequire an impact transducer.

Some of the operating specifica-tions to consider are sensitivity,resonant frequency, temperaturerange and shock rating. Sensitiv-ity is measured in terms of volt-age/force with units of mV/1bor mV/N. Analyzers have a rangeof input voltage settings; there-fore, sensitivity should be chosenalong with a power supp lyamplification level to genera tea measurable voltage.

The resonant frequency of a loadcell is simply a function of itsphysical mass and stiffness char-acter istics. The frequency rangeof the test should fall within thelinear range below the resonantpeak of the frequency response of the load cell, as shown in Figure2.16. The rule of thumb for shock rating is that the maximum vibra-tion level expected during the testshould not exceed one third theshock rating.

The load cell should be mountedto the structure with a threadedstud for best results as shown inFigure 2.17. If this is not fea-sible, then an alternative methodof first fixing a spa cer to thestructure with some type of adhesive (such as dental cement)

and then stud mounting the loadcell to this spacer will usuallysuffice for low force levels.

The properties to consider inselecting an accelerometer arevery similar to tho se of the loadcell, although they are related toacceleration rather than force.The type of response is limited toacceleration as the term implies,since displacement and velocity

Figure 2.18Frequencyresponse of transducer




Stud

Cement, Wax

Magnet

20k-25k

15k-20k

5k-7k


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27

Figure 2.19Mass loadingfromaccelerometer

Hz 2.5kFxd Y 0

-30.0

dB

10.0/Div

50.0

FREQ RESP 3Avg 0%Ovlp FR/Ex

Frequency

Amplitude

transducers are not available inthe piezoelectric type. However,if displacement or velocityresponses are desired, the accel-eration response ca n be artifi-cially integrated once or twiceto give velocity and displacementresponses, respectively.

In general, the optimum acceler-ometer has h igh sen sitivity, widefrequency range and small mass.Trade-offs are usually made since

high sensitivity usua lly dictates alarger mass for all but the mostexpensive accelerometers. Thesensitivity, measure in mV/G,and the s hock rating should beselected in the same ma nner aswith the load cell.

Although the resonant frequencyof the acc elerometer (freelysuspended) is function of itsmass and stiffness characteris-tics, the actual natural frequency(when mo unted) is generallydictated by the stiffness of themounting method used. The effectof various mounting methods isshown in Figure 2.18. The ruleof thumb is to set the maximumfrequency of the test at no morethan one-tenth the mounted natu-ral frequency of the accelerom-eter. This is within the linear

range of the mounted frequencyresponse of the accelerometer.

Another important co nsiderationis the effect of mass loading fromthe accelerometer. This occursas a result of the mass of theaccelerometer being a significantfraction of the effective mass of a particular mode. A simpleprocedure to determine if thisloading is significant can bedone as follows:

If the two measurements differ

significantly, as illustrated inFigure 2.19, then ma ss loading isa problem and an accelerometerwith less mass should be used.On very small structures, it maybe necessary to measure theresponse w ith a non-contactingtransducer, such as an acousticalor optical sensor, in order toeliminate any mass loading.

Measure a typical frequency re-sponse function of the test ob jectusing the desired accelerometer.

Mount another accelerometer(in addition to the first) with thesame mass at the same point andrepeat the measurement.

Compare the two measurementsand look for frequency shifts andamplitude changes.


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Figure 2.20Example of the inputhalf ranging

Sec 8.0mFxd XY 0.0

-80.0m

Real

20.0m

/Div

80.0m

INST TIME2

FREQ RESP 50 Avg 50% Ovlp Hann40.0

dB

-40.0

Fxd XY 1.1k Hz 2.1k

COHERENCE 50 Avg 50% Ovlp Hann1.1

Mag

0.0

Fxd Y 1.1k Hz 2.1k

v

Figure 2.21Example of the inputunder ranging

Sec 8.0mFxd XY 0.0

-130m

Real

32.5m

/Div

130m

INST TIME2


dB

-40.0Fxd XY 1.1k Hz 2.1k


Mag

0.0

Fxd Y 1.1k Hz 2.1k

v

Figure 2.22Example of the inputover ranging

Sec 8.0mFxd X 0.0

-80.0m

Real

20.0m

/Div

80.0m

INST TIME2


dB

-40.0

Fxd XY 1.1k Hz 2.1k


Mag

0.0

Fxd Y 1.1k Hz 2.1k

v

Ov2

Ov2 Ov2


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29

MeasurementInterpretation

Having discussed the mechanicsof setting up a mo dal test, it isappropriate at this point to makesome trial measurements andexamine their trends beforeproceeding with data collection.Taking the time to investigatepreliminaries of the test, suchas exciter or response locations,various types of excitation func-tions and different signal process-ing parameters will lead to higherquality measur ement s. This sec-tion includes preliminary checkssuch as adequate signal levels,minimum leakage measurementsand linearity and reciprocitychecks. The concept and trendsof the driving point measurementand the combinations of measure-ments that constitute a c ompletemodal survey are discussed.

After the structure has beensupported and instrumented forthe test, the time domain signalsshould be examined before mak-ing measu rement s. The inputrange settings on the analyzershould be set at no more than twotimes the maximum signal levelas shown in Figure 2.20. Oftencalled half-ranging, this tak esadvantage of the dynamic rangeof the analog-to-digital c onverterwithout underranging oroverranging the s ignals.

The effect resulting from under-ranging a signal, where theresponse input level is severelylow relative to the analyzer set-ting, is illustrated in Figure 2.21.Notice the apparent noise be-tween the peak s in the frequencyresponse and th e resultingpoor coher ence function. In

Figure 2.23Randomtest signals

Figure 2.24Transienttest signals

FILT TIME 2 0%Ovlp700

m

Real

-700m

Fxd Y 0.0 Sec 512m

FILT TIME 1 0%Ovlp400

mReal

-400m

Fxd Y 0.0 Sec 512m

v

v

Typical Force

Exponential Decaying

Response

Figure 2.22, the resp onse isseverely overloading the an alyzerinput section and is being clipped.This results in poor frequencyresponse and, consequently,poor coherence since the actualresponse is not being measuredcorrectly.

It is also advisable to verify thatthe signals are indeed the typeexpected, (e.g., random noise).With a r andom signal, it is ad vis-able to measure th e histogram toverify that it is not contaminatedwith other signal components,

i.e., it has a Gaussian distributionas show n in Figure 2.23. This canbe visually checked as illustratedwith the transient signals inFigure 2.24.

HIST 1 90 Avg 50%Ovlp10.0

k

Real

0.0

Fxd XY -10.0m V 10.0m

FILT TIME 1 50%Ovlp

10.0m

Real

-10.0m

Fxd Y 0.0 Sec 799m

v

Histogram

Random Noise


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30

Measurement Averaging

In order to reduce the statistical

variance of a measurement w ith arandom ex citation function (suchas random noise) and also reducethe e ffects of nonlinearities, it isnecessary to employ an averagingproce ss. By averaging severaltime records together, statisticalreliability can be increased andrandom noise associated withnonlinearities can be reduced .One metho d to gain insight intothe variance of a measurement is

Chapter 3

Improving Measurement Accuracy

to observe the Nyquist displayof the frequency response. Thecircle appears very distorted for a

measurement with few averages,but begins to smooth out withmore and more averages. Thisprocess can be seen in Figure 3.1.With each data record acquired,the frequency spectrum has adifferent magnitude and phasedistribution. As these spectra areaveraged, the nonlinear termstend to cancel, thus resulting inthe best linear estimate.

Figure 3.1Measurementaveragingfrequencyresponse

Real 128Fxd Y -128

-100

Imag

100

FREQ RESP 5 Avg 50% Ovlp Hann

Real 128Fxd XY -128

-100

Imag

100


FREQ RESP40.0

dB

-40.0

Fxd Y 1.1k Hz 2.1k


Mag

0.0

Fxd Y 1.1k Hz 2.1k5 Avg 50% Ovlp Hann

FREQ RESP40.0

dB

-40.0

Fxd Y 1.1k Hz 2.1k


Mag

0.0

Fxd Y 1.1k Hz 2.1k30 Avg 50% Ovlp Hann

30 Averages

5 Averages


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Windowing Time Data

There is a property of the Fast

Fourier transform (FFT) thataffects the e nergy distribution inthe frequenc y spect rum. It is theresult of th e physical limitationof measu ring a finite length timerecord along with the periodicityassumption required of the timerecor d by the FFT. This does notpresent a prob lem when the sig-nal is exactly periodic in the timerecord or wh en a transient signalis completely captured withinthe time recor d. However, in thecase of true rand om excitationor in the transient case when theentire response is not captured,a phenomenon called leakageresults. This has the effect of smearing or leaking ener gy intoadjacent frequency lines of thespectrum, thus distorting it.Figure 3.2 illustrates an ex ampleof the effects of severe leakageproblems with true randomexcitation. The effect is tounderestimate the amplitudeand overestimate the dampingfactor.

One of the most common tech-niques for reducing the effects of leakage with a non-periodic signalis to artificially force the signal to0 at the beginning and end o f thetime record to make it appearperiodic to the analyzer. This isaccom plished by mu ltiplying thetime record by a mathematical

curve, known as a window func-tion, before processing the FFT.Another measurement is takenwith a Hann window applied tothe true random excitation signal,show n in Figure 3.3. This mea-surement is more accurate, butnotice that the coherence is stillless than unity at the resonance.

Figure 3.2Frequencyresponse withtrue randomsignal andno windows

Figure 3.3Frequencyresponse withburst randomsignal

Sec 799mFxd Y 0.0

-50.0m

Real

12.5m

/Div

50.0m

FILT TIME2

FREQ RESP 50 Avg 50% Ovlp Unif40.0

dB

-40.0

Fxd Y 1.1k Hz 2.1k

COHERENCE 50 Avg 50% Ovlp Unif1.1

Mag

0.0

Fxd Y 1.1k Hz 2.1k

v

True Random Noise

50%Ovlp


dB

-40.0

Fxd Y 1.1k Hz 2.1k


Mag

0.0

Fxd Y 1.1k Hz 2.1k


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The window does not eliminateleakage completely and it alsodistorts the measurement as a

result of eliminating some data.A better measurement techniqueis to use a n excitation that isperiodic within the time recordsuch as burst random, in order toeliminate the leakage problem asillustrate d in Figure 3.4.

Increasing MeasurementResolution

Another measurement capabilitythat is often needed, particularlyfor lightly damped structures, isto obtain more frequency resolu-tion in the vicinity of resonan cepeaks . It may not be possible ina baseband measurement toextract valid modal parameterswith inadequate information.Normally, the Fourier transformis calculated over a frequencyrange from 0 to some maximumfrequency. Zoom proce ssing isa technique in which the lowerand upper frequency limits areindependently selectable overfixed ranges within the analyzer.The capa bility to zoom allowsclosely spaced modes to be moreaccurately identified by concen-trating the measurement po intsover a narrower band. The resultof this increased measurementaccuracy is shown in Figure 3.5.Another result is that distortiondue to leakage is reduced,because the smearing of energy

is now within a narrower band-width, but not eliminated . An-other related process associatedwith zooming is the ability toband-limit the e xcitation toconcentrate the available energywithin the given frequencyrange of the test.

Figure 3.4Frequencyresponse withtrue randomsignal andHann window

Sec 799mFxd Y 0.0

-50.0m

Real

12.5m

/Div

50.0m

FILT TIME2

FREQ RESP 15 Avg 0% Ovlp Unif40.0

dB

-40.0

Fxd Y 1.1k Hz 2.1k

COHERENCE 15 Avg 0% Ovlp Unif1.1

Mag

0.0

Fxd Y 1.1k Hz 2.1k

v

Burst Random Signal

0%Ovlp


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Figure 3.5Effects o f increasingfrequencyresolution

Real 257Fxd Y -257

-200

Imag

200


Real

Real

128

128

Fxd XY

Fxd Y

-128

-128

-100

-100

Imag

Imag

100

100

FREQ RESP

FREQ RESP

50 Avg 5% Ovlp Hann

50 Avg 50% Ovlp Hann

FREQ RESP50.0

dB

-30.0

Fxd Y 0 Hz 4k


Mag

0.0

Fxd Y 0 Hz 4k50 Avg 0% Ovlp Hann

FREQ RESP

FREQ RESP

40.0

40.0

dB

dB

-40.0

-40.0

Fxd Y 745

Fxd Y 1.2k

Hz

Hz

2.345k

1.6k

COHERENCE

COHERENCE

50 Avg 5% Ovlp Hann


1.1

1.1

Mag

Mag

0.0

0.0

Fxd Y 745

Fxd Y 1.2k

Hz

Hz

2.345k

1.6k

50 Avg 5% Ovlp Hann


1st Zoom Span

2nd Zoom Span

Baseband M easurement


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Complete Survey

As frequency response functions

are being acquired and storedfor subsequent modal parameterestimation, an ad equate set of measurements must be collectedin order to arrive at a completeset of modal paramete rs. Thissection describes the numberand type of measurements thatconstitute complete modal sur-vey. Definitions and concept s,such as driving point measure-ment and a row or column of thefrequency response matrix arediscussed . Optimal shake r andaccelerometer locations arealso included.

A complete, although redundant,set of frequency response mea-surements would form a squarematrix of size N, where the rowcorresponds to response pointsand the columns correspond toexcitation points, as illustratedin Figure 3.6. It can be show n,however, that any particular rowor column contains sufficientinformation to compute the com-plete set of frequencies, damping,and mode shapes. In other words,if the ex citation is at point 3, andthe response is measured at allthe po ints, including point 3, thencolumn of the frequency responsematrix will be measure d. Thissituation would be the result of ashaker test. On the other hand, if an accelerometer is attached to

point 7, and a hammer is used toexcite the structure at all points,including point 7, then row 7 onthe matrix will be measured.This would be the result of animpact test.

The measurement where theresponse point and d irection arethe same as the excitation pointand d irection is called a driving

Figure 3.6Frequencyresponsematrix

point measurement. The beam in

Figure 3.7 illustrates a measure-ment of this type. Driving pointmeasurements form the diagonalof the frequency response matrixshown above. They also exhibitunique characteristics that are notonly useful for checking measure-ment quality, but necessary foraccomplishing a comprehensivemodal analysis, which includes

not only frequencies, damping

factors and scaled mode shapes,but modal mass and stiffness aswell. It is not necess ary to makea driving point measurement toobtain only frequencies, dampingfactors and unscaled modeshape s. However, a set of scaledmode shapes and consequently,modal mass and stiffness cannot

H11 H12 H1N

H21 H22

HN1 HNN

Figure 3.7Driving pointmeasurementsetup


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be extracted from a set of measurements that does notconta in a driving point.

Recall from Chapter 1, StructuralDynamics Background, that therespo nse o f a MDOF system issimply the weighted sum of anumber of SDOF systems. Thecharacteristics of the drivingpoint measurement can b e easilyexplained and prese nted as aconsequence of this property.Figure 3.8 shows a typical drivingpoint measurement displayed inrectangular and polar coordi-nates. As seen in the imaginarypart of the rectangular coordi-nates, all of the resonant peakslie in the same direction. In otherwords, they are in phase witheach other. This characteristicbecomes more intuitive whenillustrated with the beam modesin Figure 3.9. The respo nse p ointmoves in the same direction asthe excitation point at all themodes, since it is measured atthe same physical location asthe excitation.

By observing the tr ends o f thismeasurement in polar coordinatesin Figure 3.9, a further und er-standing of its characteristics canbe gained. When the magnitudeis displayed in log format (dB),anti-resonances occur be tweenevery resonance throughout thefrequency range. The individualSDOF systems sum to 0 at the

frequencies where the mass andstiffness lines of adjacent modesintersect since all the modes arein phase with each other. Thisresults in the near 0 magnitude of an anti-reson ance. Also noticethe phase lead as th e magnitudepasses thro ugh an an ti-resonanceand the opp osite phase lag asthe magnitude passes through aresonance. These trends of thedriving point measurement should

Figure 3.8Driving pointfrequencyresponse

FREQ RESP100

Imag

-100

Fxd Y 0 Hz 1.25k

FREQ RESP 3 Avg 0% Ovlp Fr/Ex56.0

Real

-56.0

Fxd Y 0 Hz 1.25k3 Avg 0% Ovlp Fr/Ex

FREQ RESP180

Phase

-180

Fxd Y 0 Hz 1.25k


dB

-40.0


Figure 3.9Typicalfree beammode shapes


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be observed and monitoredthroughout the measurementprocess as a check for maintain-

ing a consistent set of data.

The remaining measurements,where the response coordinatesare different from the excitationcoordinates, are called cross-point measu rement s. Figure 3.10illustrates a typical cross-pointmeasur ement . All the modesare not necessarily in phasewith each other, as seen in theimaginary display. Since theresponse points are not at thesame location as the excitationpoint, the response can moveeither in phase or out of phasewith the excitation. This motion,which defines the mo de shape,is a function of the measurementlocation, and w ill vary from m ea-surement to measurement. In thedB display, if any two adjacen tmodes are in phase at a particularpoint, then an anti-resonance willexist between them. If any twoadjacent modes are out of phase,then their mass and stiffness lineswill not cancel at the intersectionand a smooth curve will appearinstead, as seen in Figure 3.10.

In order to excite all themodes within the frequencyrange of interest, several shakeror accelerometer locations sho uldbe examined. A point or lineon the structure th at remainsstationary is called a node point

or node line. The node points of a cantilever beam are illustratedin Figure 3.11. The number an dlocation of these no des are afunction of the particular modeof vibration and increase as modenumber increases.

If the response is measured atthe end of the b eam at point 1 andexcitation is app lied at point 3, all

Figure 3.10Cross-pointfrequencyresponse

FREQ RESP100

Imag

-100

Fxd Y 0 Hz 1.25k


Real

-56.0


FREQ RESP180

Phase

-180

Fxd Y 0 Hz 1.25k


dB

-40.0


Figure 3.11Typicalcantileverbeam modeshapes

2 13


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modes will be excited and theresulting frequency response willcontain all the modes as shown

in Figure 3.10. However, if theresponse is at point 1 but theexcitation is moved to point 2,the second mode will not be ex-cited and the resulting frequencyresponse will appear as shown inFigure 3.12. Referring to Figure3.11, note that point 2 is a nodepoint for mode 2 and very near anode point for mode 3. Mode 2does not appear in the imaginarydisplay and mode 3 is barely dis-cernible. In dB coord inates, themode 3 appears to exist but it isstill difficult to ob serve mo de 2.It may be advisable or necessaryat times to gather more than oneset of data at different excitationlocations in order to measureall the modes. It should alsobe noted that the same ob serva-tions can be made in an impacttest where the response pointwould also be moved to variouslocations.

Another con cept associatedwith a linear structure concernsa property of the frequency re-spons e matrix. The frequencyresponse matrix for a linear sys-tem can be shown to be symmet-ric du e t o Maxwells Reciproc ityTheorem. Simply stated , a mea-surement with the excitation atpoint i and the response at point jis equal to the measurement withthe excitation at point j and the

respo nse at point i. This is illus-trated in Figure 3.13. A check can be made on the measurementprocess by comparing these tworeciprocal measurements at vari-ous pairs of points and observingany differences between them.This can be h elpful for no tingnonlinearities when applyingdifferent force levels.

Figure 3.12Frequencyresponse withexcitat ion atnode of vibration

FREQ RESP40.0

dB

-40.0

Fxd Y 0 Hz 1.25k

FREQ RESP 3 Avg 0% Ovlp Fr/Ex100

Imag

-100


FREQ RESP40.0

dB

-40.0

Fxd Y 0 Hz 1.25k


Imag

-56.0


Figure 3.13Symmetricfrequencyresponsematrix

H11 H12 H1N

H21 H22

HN1 HNN

Hji

Hij

Hij = H ji


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Introduction

The previous chapter presented

several techniques for makingfrequency response measure-ments for modal analysis. Havingacquired this data, the next majorstep of the process is the use of parameter estimation techniques curve fitting to identify themodal parame ters. A vast amoun tof literature exists on the subjectof curve fitting measured data toestimate the modal propertiesof a structure. However, thisinformation tends to be math-ematically vigorous and is gene r-ally biased toward a particulartype of algorithm. It is the intentof this chapter to categorize, in aconceptual manner, the differenttypes of curve fttters and discussthe applications and problemsassociated with those mostcommonly implemented.

It was discussed earlier that aminimum of one row or column of the frequency response matrix, orits equivalent, must be measuredin order to identify a complete setof modal parameter s. Althoughadditional data is, in pr inciple,redundant information, it can be

Chapter 4

Modal Parameter Estimation

used to verify and increase theconfidence level of the estimatedparame ters. The frequency and

damping for each mod e can beestimated from any combinationof these measurements. Theresidues and, consequently, themodal coefficients are thencomputed for each measurementpoint. The mode shapes arethen scaled and sorted for eachreson ant frequency. Finally,the modal mass and stiffnesscan be determined from thesescaled parameters as illustratedin Figure 4.1.

Figure 4.1Typical flowof modal tes t

Scale ModeShapel

l

Modal MassModalStiffness

Measurecompleteset offrequencyresponses

Identify Modall

l

l

FrequencyDampingMode Shape


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Modal Parameters

One of the most fundamental

assumptions of modal testing isthat a mode of vibration can beexcited at any point on the struc-ture, except at nodes of vibrationwhere it has no motion. This iswhy a single row or column of the frequency response matrixprovides su fficient informationto estimate modal parame ters. Asa result, the frequency and damp-ing of any mode in a structure areconstants that can b e estimatedfrom any one of the measure-ments as shown in Figure 4.2.In other words, the frequencyand damping of any mode areglobal properties of the structure.

In pract ical applications, it isimportant to include sufficientpoints in the test to completelydescribe all the modes of interest.If the excitation point has notbeen chosen carefully or if enough response points arenot measured, then a particularmode may not be adequately rep-resen ted. At times it may beco menecessary to include more thanone excitation location in orderto adequately describe all of themodes of interest. Frequencyresponses can be measuredindepend ently with single-pointexcitation or simultanously withmultiple point excitations.

The mode shapes as a whole

are also global properties of the structure, but have relativevalues depending on the point of excitation and scaling and sortingfactors. On the other hand, eachindividual modal co efficient th atmakes up the mode shape is alocal property in the sense that itis estimated from the particularmeasurement associated with thatpoint as shown in Figure 4.3.

Figure 4.2Concepts of modal paramet ers

F r e q u e n c y

D a m p i n g

Mode = 1Mode = 2

Mode = 3Vibrating Beam

MeasurementPoints

Damping, frequency same at each measurement pointMode shape obtained at same frequency from all measurement points

Figure 4.3Modalparameters

Frequency Damping

{ } Mode Shape

Global

Local


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Curve Fit ting Met hods

Due to the large amount of

literature and algorithmscurren tly available for curvefitting structural data, it has be-come difficult to determine theexact need for each method andwhich method is best. There is noideal solution and the commonmethods are only approximations.Also, many of the methods arevery similar to each other and,in some cases, simply extensionsof a few basic techniques.

Although there are severalways in which c urve fittingmethods can be categorized,the most straightforward issingle-mode versus multiple-mode classification. Besides theintuitive rea soning for single andmultiple mode approximations,there are some practical reasonsfor this classification. The majordifferenc e in th e level of sophisti-cation, or level of accuracy,among curve fitters is between asingle mode and a multiple modemethod . Also, the computingresources needed (computationspeed, memory size and I/0capab ility) for multiple-modemethods can increase tremen-dously. Other sub-catagoriesand extensions that fall mostlywithin multiple mode methodsare shown in Figure 4.4.

Users gener ally fall into one of three major group s. The firstgroup is primarily concerned

with troubleshooting existingmecha nical equipment. They areusually concerned with time andrequire a fast, me dium qualitycurve fitter. The second group ismore serious about quantitativeparameter estimates for use in amodal model. For example, they

require more accuracy andare willing to spend more timeobtaining results. The final group

is pushing the state of the art andis involved with de velopmentwork. Accura cy, rathe r thantime, is of paramount importance.

Figure 4.4Increasingaccuracy incurve fitmethods

Multi-Measurementl

l

Global AccuracyLarge Amount ofData Processing

Single-ModeMethodsl

l

Easy, fastLimited

Multi-ModeMethodsl

l

MoreAccurateSlower


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Single-Mode Met hods

As stated earlier, the general

procedure for estimating modalparameters is to estimate frequen-cies and damping factors, thenestimate modal coefficients.For most single-mode parameterestimation techniques, however,this is not always the case. Infact, it is not absolutely necessaryto estimate damping in order toobtain modal coefficients. Thisis typical in a troubleshootingenvironment where frequenciesand mode shapes are of primaryconcern.

The basic assumption for single-mode approximations is that inthe vicinity of a resonance, theresponse is due primarily to thatsingle mode. The reson ant fre-quency can be estimated from thefrequency response data (illus-trated in Figure 4.5) by observingthe frequency at which any of thefollowing trends occur:

Figure 4.5Frequencyresponse

n Frequency

Real

Imaginary

n Frequency

Magnitude

Phase

The magnitude of the frequencyresponse is a max imum.

The imaginary part of thefrequency response is amaximum or minimum.

The real part of the frequencyresponse is zero.

The response lags the input by90 phase .

It was discussed earlier thatthe height of the resonant peak is a function of damping. Thedamping factor can be estimatedby the half-power method or otherrelated mathematical or graphicalmethod . In the half-powermethod, the damping is estimatedby determining the sharpnessof the resonant peak. It can beshown from Figure 4.6 that damp-ing can be related to the width of the peak between the half-power


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points: points below and abovethe resonant peak at which therespo nse m agnitude is .7071 times

the resonant magnitude.

One of the simplest single-modemodal coefficient estimation tech-niques is the quadrature method,often called peak picking. Modalcoefficients are estimated fromthe imaginary (quadrature) partof the frequency response, sothe method is not a curve fit inthe strict sense of the term. Asmentioned earlier, the imaginarypart reaches a maximum at theresonant frequency and is 90out of phase with respect to theinput. The magnitude of themodal coe fficient is simply takenas the value of the imaginarypart at resonance as illustratedin Figure 4.7. The sign (pha se) istaken from the direction that thepeak lies along the imaginaryaxis, either positive or negative.This implies that the phase angleis either 0 or 180.

The quadrature response methodis one of the more popular tech-niques for estimating modalparameters because it is easy touse, very fast and requires mini-mum computing resou rces. It is,however, sensitive to noise on themeasurement and effects fromadjacent modes. This metho d isbest suited for structures w ithlight damping and well separatedmodes where modal coefficients

are essen tially real valued. It ismost useful for troubleshootingproblems, however, where it isnot necessary to create a modalmodel and time is limited.

Figure 4.6Damping facto rfrom half power

640m

80.0m

/Div

Mag

0.0

0 Hz 100

Damping

Figure 4.7Quadraturepeak pick

640m

160m

/Div

Imag

-640m

0 Hz 100

Modal Coefficient

Figure 4.8Circle fitmethod

Imaginary

Real

Modal Coefficient


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Anothe r single-mode techn ique,called the circle fit, was o riginallydeveloped for structural damping

but can be extended to theviscous damping case. Recallfrom Chapter 1 that the frequencyresponse of a mode traces out acircle in the imaginary plane.The method fits a circle to thereal and imaginary part of thefrequency response data byminimizing the error betweenthe radius of the fitted circle andthe measured data. The modalcoefficient is then determinedfrom the diameter of the circleas illustra ted in Figure 4.8. Thephase is determined from thepositive or negative half of theimaginary axis in wh ich thecircle lies.

Frequency and damping can beestimated by one of the methodsdiscussed earlier or by some of the MDOF methods to be dis-cusse d later. Damping can alsobe estimated from the spa cingof points along th e Nyquist plotfrom the circle.

The circle fit method is fairly fastand requires minimum computerresou rces. It usually results inbetter parameter estimates thanobtained by the quadraturemethod because it uses moreof the measurement informationand is not as sensitive to effectsfrom adjacent modes as illus-trated in Figure 4.9. It is also less

sensitive to noise and distortionon the measurement. However,it requires much more userinteraction than the quadraturemethod; consequently, it is proneto errors, particularly when fittingclosely spaced modes.

A SDOF meth od related to thecircle fit is a frequency domaincurve fit to a single-mode analyti-

cal expression of the frequencyrespo nse. This expre ssion isgenerally formulated as a secondorder polynomial with residualterms to take into account theeffects of out of band modes.Because of its similarities tothe circle fit, it possesses thesame basic advantages anddisadvantages.

Concept of Residual Terms

Before proceeding to multiple-mode methods, it is appropriateto discuss the residual effectsthat out-of-band modes have onestimated param eters. In general,structures possess a n infinitenumber of modes. However,there are only a limited numberthat are usually of concern.Figure 4.10 illustra tes the analyti-cal expression for the frequency

Figure 4.9Comparison o f quadrature andcircle fit methods

Imaginary

Real

PeakPick

PeakPick

Circle Fit

Circle Fit

Figure 4.10Analyticalexpression of frequencyresponse

H( ) = nr = 1

i j( n - ) + j(2 n) 2 2


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response of a structure takinginto account the total number of realizable modes. Unfortunately,

the measured frequency responseis limited to some frequencyrange of interest depending on thecapab ilities of th e ana lyzer andthe frequency resolution desired.This range may not necessarilyinclude several lower frequencymodes and most certainly will notinclude some higher frequencymodes. However, the residualeffects of the se o ut-of-bandmodes will be present in themeasurement and, consequently,affect the accuracy of parameterestimation.

Although parameters of theout-of-band modes cannot beidentified, their effects c an berepresented by two relativelysimple terms. It can be seenfrom Figure 4.11 that the effectsof the lower modes tend to havemass-like behavior and the effectsof the higher modes ten d to havestiffness -like beh avior. Theanalytical expression for theresidual terms can then be writtenas show n in Figure 4.12. Noticethat the residual terms are equiva-lent to the asymptotic behavior of the mass and stiffness of a SDOFsystem discussed in the ch apteron structural dynamics.

Useful information ca n often begained from the residual termsthat has some physical signifi-

cance . First, if the structu re isfreely supported during the test,then the low frequency residualterm can be a direct measure of rigid body mass properties of thestruct ure. The high frequencyterm, on the other hand, can be ameasur e of th e local flexibility of the d riving point.

Figure 4.11Single modesummation o f frequencyresponse

dB Magnitude

0.01 2 3 Frequency

dB Magnitude

0.01 2 3 Frequency

Mass Line

Stiffness Line

Figure 4.12Residual termsof frequencyresponse H( ) = i j

kHigher Modes

H( ) = i j-w m2

Lower Modes


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Multiple-Mode Methods

The single-mode method s

discussed earlier performreasonably well for structureswith lightly damped and wellseparated modes. These methodsare also satisfactory in situationswhere accuracy is of secondaryconcern. However, for structureswith closely spaced modes, par-ticularly when heavily damped,(as shown in Figure 4.13) theeffects of adjacent modes cancause significant approximations.In general, it will be necessaryto implement a multiple-modemethod to more accuratelyidentify the modal parametersof these types of structures.

The basic t ask o f all multiple-mode methods is to estimatethe c oefficients in a multiple-mode analytical expression forthe frequency response function.This is done b y curve fittinga multiple-mode form of thefrequency response function tofrequency domain measurements.An equivalent me thod is to curvefit a multiple response form of the impulse response functionto time domain data. In eitherprocess, all the modal parameters(frequency, damping and modalcoefficient) for all the modes areestimated simultaneously.

There are a number of multiple-mode methods currently available

for curve fitting measured data toestimate modal parameters.However, there are essentiallytwo different forms of the fre-quency response function whichare used for curve fitting. Theseare the partial fraction form and

Figure 4.13Damping andmodal coupling

64.0

64.0

80

/Div

2.0

/Div

Mag

Mag

0.0

0.0

433.7

443.7

Hz

Hz

2.4437k

2.4437k

Light Damping and Coupling

Heavy Damping and Coupling

Figure 4.14Frequencyresponserepresentation

H( ) =b sn + b, sn - 1 + . . .0

Polynomial

h(t) =n

k = 1 Complex Exponential

H( ) =n

k = 1

rk* j - pk*

rk j - pk

+ Partial Fraction

a s m + a, s m - 1 + . . .0

rk e- kt + rk*e

- *kt


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the polynomial form which areshown in Figure 4.14. They areequivalent an alytical forms a nd

can be shown to be re lated tothe structura l frequency responsedeveloped earlier in Chapter 1.The impulse response function,obtained by inverse Fourier trans-forming the frequency responsefunction into the time domain, isalso sho wn in F igure 4.14.

When th e par tial fraction formof the frequency response isused, the modal parameters canbe estimated directly from thecurve fitting proce ss. A leastsquares error approa ch yields aset of linear equations that mustbe solved for the modal coeffi-cients and a set of nonlinearequations that must be solved forfrequency and damping. Becausean iterative solution is requiredto solve these equations, there ispotential for convergence prob-lems and long computation times.

If the polynomial form o f thefrequency response is used, thecoefficients o f the po lynomialsare identified during the curvefitting process. A root finding

solution must then be used todetermine the modal parameters.The advantage of the polynomialform is that the equations arelinear and the coefficients can besolved by a noniterative process.Therefore, convergence problemsare minimal and computing time

is more reasonable.

The complex exponential methodis a time domain method that fitsdecaying exponentials to impulse

response data. The equationsare nonlinear, so an iterativeprocedure is necessary to obtain asolution. The method is relativelyinsensitive to noise on the data,but s uffers from sens itivity totime domain aliasing, as a resultof truncation in the frequencydomain from inverse Fouriertransforming the frequencyresponses.

In principle, it should not matterwhether frequency domain dataor time domain data is usedfor curve fitting since the sameinformation is contained in bothdomains. However, there aresome practical reasons, basedon frequency domain and timedomain operations, that seem tofavor the frequency domain. One,the measurement data can berestricted to some des ired fre-quency range and any noise ordistortion outside this range caneffectively be ignored . Anothe r,the cross spectra and autospectra

needed to compu te frequencyresponses can be formed fasterthan the corresponding time

domain correlation functions. Itis true that the time domain canbe used to select modes havingdifferent damping values, but thisis usually not as important as theability to select a frequency rangeof interest.

Each method has its advantagesand disadvantages, but thefundamental problems of noise,distortion and interference fromadjacent modes remain. As aresult, none of the methods work well in all situa tions . It is alsounlikely that som e magicmethod will be discovered thateliminates all of these problems.All of the methods work wellwith ideal data, but cannot beevaluated by analytical meansalone. The import ant factor ishow well they work, or gracefullyfail, with real experimentaldata complete with noise anddistortion.


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Concept of Realand Complex Modes

The structural model discussedso far is based on the concept of proportional viscous dampingwhich implies the existence of real, or normal, modes . Math-ematically, this implies tha tthe physical damping matrixcan be defined as linear combi-nation of the physical mass andstiffness matrices as shown inFigure 4.15. The mode shapes,are, in effect real valued, mean-ing the pha se a ngles differ by 0or 180. Physica lly, all thepoints reach th eir maximumexcursion at the same time asin a standing wave pattern. Oneof the consequences of thisassumption, discussed earlier,is that the imaginary part of thefrequency response re aches amaximum at resonance and thereal part is 0 valued a s illus-trated in Figure 4.16. Note alsothat the Nyquist circle lies alongthe imaginary axis.

However, physical structuresexhibit a more complicated formof damping which results innon-proportiona l damping. Themode shapes are, generally,complex valued, meaning thephase angles can have valuesother th an 0 or 180. Physi-cally, the points reach theirmaximum excursions at varioustimes as in a tra veling wave

pattern. With non-proportionaldamping, the imaginary partof the frequency response nolonger reaches a maximum atresonance nor is the real partnonzero valued as illustrated inFigure 4.17. Note also that theNyquist circle is rotated at anangle in the complex plane.

Physically, this means that thedamping is sufficiently small sothat coupling is a second-order

effect. It should be noted thatclosely-spaced modes oftenappear complex as a result of theeffects from adjacent modes asillust rat ed in Figure 4.18. Inreality, they may actu ally bemore real than they appear.

Figure 4.15Proportionaldampingrepresentation

[C] = a [M] + [K]

When damping is light, as isthe case in most mechan icalstructures, the proportional

damping assumption is generallyan accurate ap proximation.Although damping is not propor-tional to the mass and stiffness,the nonproportional couplingeffects may be small enoughnot to cause serious errors.

Figure 4.16Imaginary

Real

2 k

= k

Figure 4.17Imaginary

Real

= k

ak

r1k +r2k2 k

2 2

Figure 4.18Two closelyspaced realmodes

Imaginary

Real


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Introduction

The basic techniques for perform-

ing a modal test t o identify thedynamic properties of a structurehave been described in the previ-ous chapters. An introduction tothe applications of the resultingfrequency responses and modalparameters is the focus of thischapt er. The discussion is spe-cifically concerned with the usesof a response model or a modalmodel with structural analysismethods as shown in Figure 5.1.The intent is to bring together theexperimental and analytical toolsfor solving noise, vibration andfailure problems.

A response model is simply theset of frequency response mea-surements ac quired du ring themodal test. These measurementscontain all the dynamics of thestructure needed for subsequentanalyses. A modal model isderived from the resp onse modeland is a function of the parameterestimation techn ique used. Itnot only includes frequencies,damping factors, and modeshapes, but also modal mass andmodal stiffness. These masse sand stiffnesses depend on themethod that was used to scalethe mode shapes. A subset of themodal model consisting of onlythe frequencies an d unsca ledmode shapes ca n b

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HP-AN243-3_Fundamentals of Modal Testing