Vibration and Shock Handbook 18

Vibration and Shock Handbook 18 Every so often, a reference book appears that stands apart from all others, destined to become the definitive work in its field. The Vibration and Shock Handbook is just such a reference. From its ambitious scope to its impressive list of contributors, this handbook delivers all of the techniques, tools, instrumentation, and data needed to model, analyze, monitor, modify, and control vibration, shock, noise, and acoustics. Providing convenient, thorough, up-to-date, and authoritative coverage, the editor summarizes important and complex concepts and results into “snapshot” windows to make quick access to this critical information even easier. The Handbook’s nine sections encompass: fundamentals and analytical techniques; computer techniques, tools, and signal analysis; shock and vibration methodologies; instrumentation and testing; vibration suppression, damping, and control; monitoring and diagnosis; seismic vibration and related regulatory issues; system design, application, and control implementation; and acoustics and noise suppression. The book also features an extensive glossary and convenient cross-referencing, plus references at the end of each chapter. Brimming with illustrations, equations, examples, and case studies, the Vibration and Shock Handbook is the most extensive, practical, and comprehensive reference in the field. It is a must-have for anyone, beginner or expert, who is serious about investigating and controlling vibration and acoustics.

18 Experimental Modal

Analysis

Clarence W de Silva

The University of British Columbia

18.1 Introduction 18-118.2 Frequency-Domain Formulation 18-2

Transfer-Function Matrix † Principle of Reciprocity

18.3 Experimental Model Development 18-8

Extraction of the Time-Domain Model

18.4 Curve Fitting of Transfer Functions 18-10

Problem Identification † Single- and Multi-Degree-of-Freedom Techniques † Single-Degree-of-Freedom Parameter Extraction

in the Frequency Domain † Multi-Degree of Freedom Curve Fitting † A Comment on Static Modes and Rigid-Body Modes † Residue Extraction

18.5 Laboratory Experiments 18-18

Lumped-Parameter System † Distributed-Parameter System

18.6 Commercial EMA Systems 18-24

System Configuration

Summary

In experimental modal analysis (EMA), first the modal information (natural frequencies, modal damping ratios,and mode shapes) of a test object is determined through experimentation, and this information is then used todetermine a model for the test object Once an “experimental model” is obtained in this manner, it may be used in avariety of practical uses including system analysis, fault detection and diagnosis, design, and control This chapterpresents some standard techniques and procedures associated with EMA

18.1 Introduction

Experimental modal analysis (EMA) is basically a procedure of “experimental modeling.” The primarypurpose here is to develop a dynamic model for a mechanical system, using experimental data In thissense, EMA is similar to “model identification” in control system practice, and may utilize somewhatrelated techniques of “parameter estimation.” It is the nature of the developed model, which maydistinguish EMA from other conventional procedures of model identification Specifically, EMAproduces a modal model as the primary result, which consists of:

1 Natural frequencies

2 Modal damping ratios

3 Mode shape vectors

Once a modal model is known, standard results of modal analysis may be used to extract an inertia (mass)matrix, a damping matrix, and a stiffness matrix, which constitute a complete dynamic model for theexperimental system, in the time domain Since EMA produces a modal model (and in some cases acomplete time-domain dynamic model) for a mechanical system from test data of the system, its uses can be

18-1

extensive In particular, EMA is useful in mechanical systems, primarily with regard to vibration, in:

Control of a mechanical system may be based on modal analysis Standard and well-developedtechniques of modal control are widely used in mechanical system practice In particular, vibrationcontrol, both active and passive, can use modal control In this approach, the system is first expressed as amodal model, then control excitations, parameter adaptations, and so on are established that result in aspecified (derived) behavior in various modes of the system Of course, techniques of EMA arecommonly used here, both in obtaining a modal model from test data and in establishing modalexcitations and parameter changes that are needed to realize a prescribed behavior in the system.The standard steps of EMA are as follows:

1 Obtain a suitable (admissible) set of test data, consisting of forcing excitations and motionresponses for various pairs of DoF of the test object

2 Compute the frequency transfer functions (the frequency response functions) of the pairs of testdata, using Fourier analysis Digital Fourier analysis using Fast Fourier Transform (FFT) is thestandard way of accomplishing this Either software-based (computer) equipment or hardware-based instrumentation may be used

3 Curve fit analytical transfer functions to the computed transfer functions Determine naturalfrequencies, damping ratios, and residues for various modes in each transfer function

4 Compute mode shape vectors

5 Compute inertia (mass) matrix M, stiffness matrix K, and damping matrix C

Some variations of these steps is possible in practice, and Step 5 is omitted in some situations Inthe present chapter, we will study some standard techniques and procedures associated with theprocess of EMA

18.2 Frequency-Domain Formulation

Frequency-domain analysis of vibrating systems is very useful in a wide variety of applications.The analytical convenience of frequency-domain methods results from the fact that differentialequations in the time domain become algebraic equations in the frequency domain Once the

necessary analysis is performed in the frequency domain, it is often possible to interpret the resultswithout having to transform them back to the time domain through inverse Fourier transformation.

In the context of the present chapter, frequency-domain representation is particularly importantbecause it is the frequency transfer functions that are used for extracting the necessary modalparameters

For the convenience of notation, we shall develop the frequency-domain results using the Laplacevariable, s: As usual, the straightforward substitution of s ¼ jv or s ¼ j2pf gives the correspondingfrequency-domain results

18.2.1 Transfer-Function Matrix

Let us consider a linear mechanical system that is represented by

where

f(t) ¼ forcing excitation vector (nth order column)

y ¼ displacement response vector (nth order column)

M ¼ mass (inertia) matrix ðn £ nÞ

C ¼ damping (linear viscous) matrix ðn £ nÞ

C ¼ modal matrix ðn £ nÞ of n independent modal vector vectors ½c1; c2; …; cn

M¯ ¼ diagonal matrix of modal masses Mi

C¯ ¼ diagonal matrix of modal damping constants Ci

K¯ ¼ diagonal matrix of modal stiffnesses Ki

and furthermore, we can express Ciin the convenient form

Ci¼ 2zivi

where

vi¼ undamped natural frequency

zi¼ modal damping ratio

By Laplace transformation of the response canonical equations of modal motion (Equation 18.3),assuming zero initial conditions, we obtain

s2þ 2zv2s þv2

26666

3777

to be equal (approximately) to the natural frequency for a given mode

From the time-domain coordinate transformation (Equation 18.2), the Laplace domain coordinatetransformation relation is obtained as

26666

3777

26666

3777

7C

Notice in particular that GðsÞ is a symmetric matrix; specifically

which should be clear from the matrix transposition property, ðABCÞT¼ CTBTAT:

An alternative version of Equation 18.13 that is extensively used in EMA can be obtained by using thepartitioned form (or assembled form) of the modal matrix in Equation 18.13 Specifically, we have

GðsÞ ¼ ½c1; c2; …; cn

G2

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37777

cT 1

cT 2

cTn

26666

3777

G2cT2

GncT n

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37777

Note that Giare scalars while ciare column vectors The two matrices in this product can be multipliedout now to obtain the matrix sum

Equation 18.17 is useful in EMA Essentially, we start by determining the residues ðcickÞrof the poles

in an admissible set of measured transfer functions We can determine the modal vectors in this manner

In addition, by analyzing the measured transfer functions, the modal damping ratios, zi; and the naturalfrequencies, vi; can be estimated From these results, an estimate for the time-domain model (i.e., thematrices M, K, and C) can be determined

18.2.2 Principle of Reciprocity

By the symmetry of transfer matrix, as given by Equation 18.14, it follows that

This fact is further supported by Equation 18.17 This symmetry can be interpreted as Maxwell’s principle

of reciprocity To understand this further, consider the complete set of transfer relations given by Equation18.12 and Equation 18.13:

Example 18.1

Consider the two-DoF system shown in Figure 18.1 Assume that the excitation forces, f1ðtÞ and f2ðtÞ; act

at the y1and y2DoFs, respectively The equations of motion are given by

This system has proportional damping (specifically, it is clear that C is proportional to M) and hencepossesses the same real modal vectors as does the undamped system Let us first obtain the transfermatrix in the direct manner By taking the Laplace transform (with zero initial conditions) of theequations of motion (i), we have

ms2þ cs þ 2k 2k2k ms2þ cs þ 2k

m m

Hence, in the relation YðsÞ ¼ GðsÞFðsÞ; the transfer matrix G is given by

GðsÞ ¼ ms

2þ cs þ 2k 2k2k ms2þ cs þ 2k

We can put these transfer functions into the partial fraction form For example,

ms2þ cs þ 2kðms2þ cs þ kÞðms2þ cs þ 3kÞ ¼

A1s þ A2ðms2þ cs þ kÞ þ

A3s þ A4ðms2þ cs þ 3kÞ ðviiÞ

By comparing the numerator coefficients, we find that A1¼ A3¼ 0 (this is the case when the modes arereal; with complex modes, A1– 0 and A3– 0 in general) and A2¼ A4¼ 1=2: These results aresummarized below:

ðc1c2Þ1¼ 1

2m; ðc1c2Þ2¼ 2 1

2m

We need consider only two admissible transfer functions (e.g., G11and G12; or G11and G21; or G12and

G22; or G21and G22) in order to completely determine the modal vectors Specifically, we obtain

1=pffiffiffiffi2m

24

21=pffiffiffiffi2m

24

35

Note that the modal masses are unity for these modal vectors Also, there is an arbitrariness in the sign

As usual, we have overcome this problem by making the first element of each modal vector positive

18.3 Experimental Model Development

We have noted that the process of extracting modal data (natural frequencies, modal damping, and modeshapes) from measured excitation–response data is termed experimental modal analysis Modal testingand the analysis of test data are the two main steps of EMA Information obtained through EMA is useful

in many applications, including the validation of analytical models for dynamic systems, fault diagnosis

in machinery and equipment, in situ testing for requalification to revised regulatory specifications, anddesign development of mechanical systems

In the present development, it is assumed that the test data are available in the frequency domain as aset of transfer functions In particular, suppose that an admissible set of transfer functions is available.The actual process of constructing or computing these frequency transfer functions from measuredexcitation–response (input–output) test data (in the time domain) is known as model identification inthe frequency domain This step should precede the actual modal analysis in practice Numerical analysis(or curve fitting) is the basic tool used for this purpose, and it will be discussed in a later section.The basic result used in EMA is Equation 18.17 with s ¼ jv or s ¼ j2pf for the frequency-transferfunctions For convenience, however, the following notation is used:

G, in order to determine the complete modal information Owing to the symmetry of G it follows that atmost only 1=2nðn þ 1Þ transfer functions are needed In fact, it can be “shown by construction” (i.e., inthe process of developing the method itself) that only n transfer functions are needed These n transferfunctions cannot be chosen arbitrarily, however, even though there is a wide choice for the admissible set

of n transfer functions A convenient choice is to measure any one row or any one column of the transferfunction matrix It should be clear from the following development that any set of transfer functions thatspans all n DoF of the system would be an admissible set provided that only one autotransfer function isincluded in the minimal set Hence, for example, all the transfer functions on the main diagonals or onthe main cross diagonal of G, do not form an admissible set

Suppose that the kth column ðGik; i ¼ 1; 2; …; nÞ of the transfer function matrix is measured byapplying a single forcing excitation at the kth DoF and measuring the corresponding responses at all nDoF in the system The main steps in extracting the modal information from this data are given below:

1 Curve fit the (measured) n transfer functions to expressions of the form given by Equation 18.22

In this manner determine the natural frequencies vr; the damping ratioszr; and the residues

ðcickÞr; for the set of modes r ¼ 1; 2; and so on

2 The residues of a diagonal transfer function (i.e., point transfer functions or autotransferfunction), Gkk; are ðc2

kÞ1; ðc2

kÞ2; …; ðc2

kÞn: From these, determine the kth row of the modal matrix;

ðckÞ1; ðckÞ2; …; ðckÞn: Note that M-normality is assumed However, the modal vectors arearbitrary up to a multiplier of 21 Hence, we may choose this row to have all positive elements

3 The residues of a nondiagonal transfer function, that is, a cross-transfer function, Gkþi;k are

ðckþickÞ1; ðckþickÞ2; …; ðckþickÞn: By substituting the values obtained in Step 2 into these values,determine the k þ ith row of the modal matrix; ðckþiÞ1; ðckþiÞ2; …; ðckþiÞn: The complete modalmatrix C is obtained by repeating this step for i ¼ 1; 2; …; n 2 k and i ¼ 21; 22; …; 2k þ 1:Note that the associated modal vectors are M-normal

The procedure just outlined for determining the modal matrix verifies, by construction, that only ntransfer functions are needed to extract the complete modal information It further reveals that it is not

essential to perform the transfer function

measure-ments in a row fashion or column fashion A

diagonal element (i.e., a point transfer function, or

an autotransfer function) should always be

measured The remaining n 2 1 transfer functions

must be off diagonal but otherwise can be chosen

arbitrarily, provided that all n DoF are spanned

either as an excitation point or as a measurement

location (or both) This guarantees that no

symmetric transfer function elements are

included This defines a minimal set of transfer

function measurements An admissible set of more

than n transfer functions can be measured in

practice so that redundant measurements are

available in addition to the minimal set that is required Such redundant data are useful for checkingthe accuracy of the modal estimates Examples for an admissible (nonminimal) set, a minimal set, and aninadmissible set of transfer functions matrix elements are shown schematically in Figure 18.2 Note thatthe inadmissible set in this example contains 11 transfer function measurements but the sixth DoF is notcovered by this set On the other hand, a minimal set requires only six transfer functions

18.3.1 Extraction of the Time-Domain Model

Once the complete modal information is extracted by modal analysis, it is possible, at least in theory, todetermine a time-domain model (M, K, and C matrices) for the system To obtain the necessaryequations, first premultiply by ðCTÞ21 and postmultiply by C Equation 18.4, Equation 18.5, andEquation 18.6 to obtain

where M ¼ I ¼ identity matrix

Since the modal matrix C is nonsingular because M is assumed nonsingular in the dynamic models that

we use (i.e., each DoF has an associated mass, or the system does not possess static modes), the inversetransformations given by the equations from Equation 18.24 to Equation 18.26 are feasible It appears,however, that two matrix inversions are needed for each result Since M, K, and C matrices are diagonal,their inverse is given by inverting the diagonal elements This fact can be used to obtain each resultthrough just one matrix inversion

Equation 18.24, Equation 18.25, and Equation 18.26 are written as

By substituting the equations from Equation 18.30 to Equation 18.32 into the equations from Equation18.27 to Equation 18.29, we obtain the relations that can be used in computing the time-domain model:

26666

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7C

T

0BBBB

1CCCC

26666

3777

7C

T

0BBBB

1CCCC

on the data sampling rate

(3) Low signal-to-noise ratio (SNR) at high frequencies, in part due to noise and poor dynamicrange of equipment and in part due to low signal levels, will result in data measurement errors.Signal levels are usually low at high frequencies because inertia in a mechanical system acts as alow-pass filter 1=ðmv2Þ:

(4) Computations involving high order matrices (multiplication, inversion, etc.) will lead tonumerical errors in complex systems with many DoF

It is customary, therefore, to extract modal information only for the first several modes In that case, it

is not possible to recover the mass, stiffness, and damping matrices Even if these matrices are computed,their accuracy is questionable due to their sensitivity to the factors listed above

18.4 Curve Fitting of Transfer Functions

Parameter estimation in vibrating systems can be interpreted as a technique of experimental modeling.This process requires experimental data in a suitable form, preferably excitation–response data, and isoften represented as a set of transfer functions in the frequency domain Parameter estimation using

measured response data is termed model identification or, simply, identification in the literature onsystems and control We shall present a parameter estimation procedure that involves frequency transferfunctions, which is particularly useful in EMA.

2 Group the signals, assign windows, and filter the signals

3 Compute transfer functions using FFT and the spectral density method:

Residues ðcickÞr) mode shapes vectors crwhich are M-normal;

Natural frequencies (undamped),vr;

Modal damping ratios (viscous),zr:

6 Form the modal matrix C ¼ ½c1; c2; …; cn ;

Compute C21:

7 Modal mass matrix M ¼ I;

Modal stiffness matrix K ¼ diagv2;v2; …;v2;

Modal damping matrix C ¼ diag½2z1v1; 2z2v2; …; 2znvn ;

8 Compute the system model:

Mass matrix M ¼ ðC21ÞC21;

Stiffness matrix K ¼ ðC21ÞTKC21;

Damping matrix C ¼ ðC21ÞCC21

a suitable transfer function model into the computed data or by simplified methods such as “peakpicking.” Accordingly, this conversion of data is an experimental modeling technique.

Identification of transfer function models from measured data is an essential step in EMA Apart fromthat, it has other important advantages In particular, analytical transfer function plots clearly identifysystem resonances and generate numerical values for the corresponding parameters (resonantfrequencies, damping, phase angles, and magnitudes) in a convenient manner This form represents asignificant improvement over the crude transfer function plots, which are normally far less presentableand rather difficult to interpret

18.4.2 Single- and Multi-Degree-of-Freedom Techniques

Several single-DoF techniques exist for extracting analytical parameters from experimental transferfunctions In particular, the methods of curve fitting (circle fitting) and peak picking are considered here

In a single-DoF method, only one resonance is considered at a time

Single-DoF curve fitting, or more correctly, single-resonance curve fitting is the term used to denote anycurve fitting procedure that fits a quadratic (second-order) transfer function into each resonance in themeasured transfer function, one at a time In the case of closely spaced modes (or closely spacedresonances), the associated error can be very large The accuracy is improved if expressions of a higherorder than quadratic are used for this purpose, but unacceptable errors can still exist In peak picking,each resonance of experimental transfer function data is examined individually; the resonant frequencyand the damping constant corresponding to that resonance are determined by comparing with ananalytical single-DoF transfer function

In multi-DoF curve fitting, or more appropriately, multiresonance curve fitting, all resonances (ormodes) of importance are considered simultaneously and fitted into an analytical transfer function ofsuitable order This method is generally more accurate but computationally more demanding than thesingle-resonance method In choosing between the single-resonance and multiresonance methods, therequired accuracy should be weighted against the cost and speed of computation

18.4.3 Single-Degree-of-Freedom Parameter Extraction in

the Frequency Domain

The theory of curve fitting by a circle (i.e., circle fitting) for each resonance of an experimentallydetermined transfer function is presented first Next, the peak picking method will be described.18.4.3.1 Circle-Fit Method

It can be shown that the mobility transfer function (velocity/force) of a single-DoF system with linearviscous damping, when plotted on the Nyquist plane of real axis and imaginary axis for the frequencytransfer function, is a circle Similarly, it can be shown that the receptance or dynamic flexibility orcompliance transfer function (displacement/force) of a single-DoF system with hysteretic damping, whenplotted on the Nyquist plane, is also a circle Note that, for hysteretic damping, the damping constant (inthe time domain) is not actually a constant but is inversely proportional to the frequency of motion.However, in this case, in the frequency domain, the damping term will be independent of frequency Thefact that such circle representations are possible for transfer functions of a single-DoF system may be used

in fitting a circle to a transfer function that is computed from experimental data This will lead todetermining the analytical parameters for the transfer function This approach is illustrated now throughanalytical development

Case of Viscous Damping

Consider a single-DoF system with linear, viscous damping, as given by

where m; k; and c are the mass, stiffness, and damping constants of the system elements, respectively, f ðtÞ

is the excitation force, and y is the displacement response Equation 18.39 may be expressed in thestandard form:

€y þ 2zvn_y þv2y ¼ 1

Receptance

YðsÞFðsÞ ¼

1

m s2þ 2zvns þv2 with s ¼ jv ð18:41ÞMobility

VðsÞFðsÞ ¼

sYðsÞFðsÞ ¼

s

m s2þ 2zvns þv2 with s ¼ jv ð18:42ÞConsider the mobility (velocity/force) transfer function given by

D ¼ ðv22v2Þ2þ ð2zvnvÞ2 ð18:45Þand the frequency transfer function (Equation 18.44) is converted into the form

GðjvÞ ¼ jvD v22v22 2jzvnv ð18:46Þ

Gð jvÞ ¼ Re þ j Im where Re ¼ 2zvnv2

Dðv22v2Þ ð18:47ÞNow, we can write

Re 2 4zv1n ¼ 8z2v2v22 4z2v2v22 ðv22v2Þ2

4zvnD ¼ 4z2v2v22 ðv22v2Þ2

4zvnDHence, in view of Equation 18.47 we have