Vibrations Fundamentals and Practice ch11

Vibrations Fundamentals and Practice ch11 Maintaining the outstanding features and practical approach that led the bestselling first edition to become a standard textbook in engineering classrooms worldwide, Clarence de Silva''s Vibration: Fundamentals and Practice, Second Edition remains a solid instructional tool for modeling, analyzing, simulating, measuring, monitoring, testing, controlling, and designing for vibration in engineering systems. It condenses the author''s distinguished and extensive experience into an easy-to-use, highly practical text that prepares students for real problems in a variety of engineering fields.

de Silva, Clarence W “Experimental Modal Analysis”

Vibration: Fundamentals and Practice

Clarence W de Silva

Boca Raton: CRC Press LLC, 2000

11 Experimental Modal Analysis

Experimental modal analysis, basically, is a procedure of “experimental modeling.” The primarypurpose is to develop a dynamic model for a mechanical system using experimental data In thissense, experimental modal analysis is similar to “model identification” in control system practice,and can utilize somewhat related techniques of “parameter estimation.” It is the nature of thedeveloped model that can distinguish experimental modal analysis (EMA) from other conventionalprocedures of model identification Specifically, EMA produces a modal model that consists of

1 Natural frequencies

2 Modal damping ratios

3 Mode shape vectors

as the primary result Once a modal model is known, standard results of modal analysis can beused to extract an inertia (mass) matrix, a damping matrix, and a stiffness matrix, which constitute

a complete dynamic model for the experimental system, in the time domain The modal analysis

of lumped-parameter systems is covered in Chapter 5, and that of distributed-parameter systems

in Chapter 6 Vibration testing and signal analysis are studied in Chapters, 4, 8, 9, and 10 Thesechapters should be reviewed for the necessary background prior to reading the present chapter.Since experimental modal analysis produces a modal model (and in some cases, a completetime-domain dynamic model) for a mechanical system form test data of the system, its uses can

be extensive In particular, EMA is useful 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, may use modal control (see Chapter 12) In this approach, thesystem is first expressed as a modal model Then, control excitations, parameter adaptations, etc.are established that would result in a specified (derived) behavior in various modes of the system.

Of course, techniques of experimental modal analysis are commonly used here, both in obtaining

a modal model from test data, and in establishing modal excitations and parameter changes thatare needed to realize a prescribed behavior in the system

The standard steps of experimental modal analysis are:

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

2 Compute the frequency transfer functions (frequency response functions) of the pairs oftest data using Fourier analysis Digital Fourier analysis using fast Fourier transform(FFT) is the standard way of accomplishing this Either software-based (computer)equipment or hardware-based instrumentation can be used

3 Curve fit analytical transfer functions to the computed transfer functions Determinenatural frequencies, damping ratios, and residues for various modes in each transferfunction

4 Compute mode shape vectors

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

Some variations of these steps may be possible in practice, and step 5 is omitted in manysituations The present chapter focuses on some of the standard techniques and procedures associ-ated with the process of experimental modal analysis The first step in generating test data is notdiscussed here, as it is extensively covered elsewhere (see Chapters 8 and 10)

Frequency-domain analysis of vibrating systems is very useful in a wide variety of applications Theanalytical convenience of frequency domain methods results from the fact that differential equations

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 results without having totransform them back to the time domain through inverse Fourier transformation In the context ofthe present chapter, frequency-domain representation is particularly important because it is thefrequency-transfer functions that are used for extracting the necessary modal parameters

For the convenience of notation, the frequency-domain results are developed using the Laplacevariable s As usual, the straightforward substitution of s = jω, or s = j2πf, gives the correspondingfrequency–domain results

11.1.1 T RANSFER F UNCTION M ATRIX

Consider a linear mechanical system that is represented by

(11.1)where

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

y = displacement response vector (nth order column)

m = mass (inertia) matrix (n×n)

My˙˙+Cy˙+Ky= f( )t

C = damping (linear viscous) matrix (n×n)

K = stiffness matrix (n×n)

If the assumption of proportional damping is made, it is seen in Chapter 5 that the coordinatetransformation

(11.2)decouples equations (11.1) into the canonical form of modal equations

(11.3)where

Ψ = modal matrix (n×n) of n independent modal vector vectors [ψ1, ψ2, …, ψn]

M= diagonal matrix of modal masses M i

C = diagonal matrix of modal damping constants C i

K = diagonal matrix of modal stiffnesses K i

Specifically,

(11.4)(11.5)(11.6)

If the modal vectors are assumed to be M-normal, then

and, furthermore, one can express C i in the convenient form

where

ωi= undamped natural frequency

ζi = modal damping ratio

By Laplace transformation of the response canonical equations of modal motion (11.3), ing zero initial conditions, one obtains

Laplace transforms of the modal response (or generalized coordinate) vector q(t) and the forcingexcitation vector f(t) are denoted by the column vectors Q(s) and F(s), respectively The squarematrix on the left-hand side of equation (11.7) is a diagonal matrix Its inverse is obtained byinverting the diagonal elements Consequently, the following modal transfer relation results:

(11.8)

in which, the diagonal elements are the damped simple-oscillator transfer functions:

(11.9)

Note that ωi, the ith undamped natural frequency (in the time domain), is only approximately equal

to the frequency of the ith resonance of the transfer function (in the frequency domain) as given by

(11.10)

As discussed before, and clear from equation (11.10), the approximation improves for decreasingmodal damping Consequently, in most applications of experimental modal analysis, the resonantfrequency is taken equal (approximately) to the natural frequency for a given mode

From the time-domain coordinate transformation (11.2), the Laplace-domain coordinate formation relation is obtained as:

trans-(11.11)Substitute equation (11.8) into (11.11); thus,

G G

An alternative version of equation (11.13) that is extensively used in experimental modalanalysis can be obtained using the partitioned form (or assembled form) of the modal matrix inequation (11.13) Specifically,

(11.15)

On multiplying out the last two matrices on the RHS of equation (11.15), term by term, the followingintermediate result is obtained:

Note that G i are scalars, while ψi are column vectors The two matrices in this product can now

be multiplied out to obtain the matrix sum:

(11.16)

in which ψr is the rth modal vector that is normalized with respect to the mass matrix Notice that each

term ψrψr T in the summation (11.16) is an n×n matrix with the element corresponding to its ith row and kth column being (ψiψk)r The ikth element of the transfer matrix G(s) is the transfer function G ik (s), which determines the transfer characteristics between the response location i and the excitation location

k From equation (11.16), this is given by

(11.17)

where s = jω = j2πf in the frequency domain Note that (ψi)r is the ith element of the rth modal vector,

and is a scaler quantity Similarly, (ψiψk)r is the product of the ith element and the kth element of the rth modal vector, and is also a scalar quantity This is the numerator of each modal transfer function

within the RHS summation of equation (11.17), and is the “residue” of the pole (eignevalue) of that mode.Equation (11.17) is useful in experimental modal analysis Essentially, one starts by determining

the residues (ψiψk)r of the poles in an admissible set of measured transfer functions One candetermine the modal vectors in this manner In addition, by analyzing the measured transfer

G s

G G

G

n

n

T T

n T

r r r T r

n

i k r

r r r r

functions, modal damping ratios ζi and the natural frequencies ωi can be estimated From these results,

an estimate for the time-domain model (i.e., the matrices M, K, and C) can be determined.

auto-is applied at the kth degree of freedom with all the other excitations set to 0 The resulting response

at the ith degree of freedom is given by

(11.20)

Similarly, when a single excitation F i (s) is applied at the ith degree of freedom, the resulting response at the kth degree of freedom is given by

(11.21)

In view of the symmetry that is indicated by equation (11.18), it follows from (11.20) and (11.21)

that if the two separate excitations F k (s) and F i (s) are identical, then the corresponding responses

Y i (s) and Y k (s) also become identical In other words, the response at the ith degree of freedom due

to a single force at the kth degree of freedom is equal to the response at the kth degree of freedom when the same single force is applied at the ith degree of freedom This is the frequency-domain

version of the principle of reciprocity

E XAMPLE 11.1

Consider the two-degree-of-freedom system shown in Figure 11.1 Assume that the excitation forces

f1(t) and f2(t) act at the y1 and y2 degrees of freedom, respectively The equations of motion aregiven by

LLM

L ( ) ( )ss F s n

Y s i( )=G ik( ) ( )s F s k

Y s k( )=G s F s ki( ) ( )i

m m

c c

00

00

22

It has been noted in Chapter 5 that this system has proportional damping (specifically, it is clear

that C is proportional to M) and, hence, possesses the same real modal vectors as for the undamped

system First obtain the transfer matrix in the direct manner By taking the Laplace transform (with

zero initial conditions) of the equations of motion (i), one obtains

(partic-FIGURE 11.1 A vibrating system with proportional damping.

2

22

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

By comparing the residues (numerators) of these expressions with relation (11.17), one can

deter-mine the M-normal modal vectors Specifically,



It has been noted that the process of extracting modal data (natural frequencies, modal damping,and mode shapes) from measured excitation-response data is termed “experimental modal analysis.”Modal testing and the analysis of test data are the two main steps of experimental modal analysis.Information obtained through experimental modal analysis is useful in many applications, including

+ ++ +

1 22

1 22

12

12

12

12

ψψ

ψψ

m m

and

validation of analytical models for dynamic systems, fault diagnosis in machinery and equipment,

on-site testing for requalification to revised regulatory specifications, and design development of

mechanical systems

In the present development, it is assumed that the test data are available in the frequency domain

as a set 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

measured excitation-response (input-output) test data (in the time domain) is known as model identification in the 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 bediscussed in a later section

The basic result used in experimental modal analysis is equation (11.17) with s = jω or s = j2πf

for the frequency-transfer functions For convenience, however, the following notation is used:

(11.22)

or, equivalently,

(11.23)

where ω and f are used in place of jω and j2πf in the function notation G( ) As already observed

in Example 11.1, it is not necessary to measure all n2 transfer functions in the n×n transfer function

matrix G in order to determine the complete modal information Due to the symmetry of G, it

follows that, at most, only 1/2n(n + 1) transfer functions are needed In fact, it can be “shown by construction” (i.e., in the process of developing the method itself) that only n transfer functions are needed These n transfer functions cannot be chosen arbitrarily, however, although there is a wide choice for the admissible set of n transfer functions A convenient choice would be to measure any one row or any one column of the transfer-function matrix It should be clear from the following development that any set of transfer functions that spans all n degrees of freedom of the system

would be an admissible set, provided that only one auto-transfer function is included in the minimalset Hence, for example, all the transfer functions on the main diagonals, or on the main cross-

diagonal of G, do not form an admissible set.

Suppose that the kth column (G ik , i = 1, 2, …, n) of the transfer-function matrix is measured

by applying a single forcing excitation at the kth degree of freedom and measuring the corresponding responses at all n degrees of freedom in the system The main steps in extracting the modal

information from this data are given below

Step 1: Curve-fit the (measured) n transfer functions to expressions of the form given by

equation (11.22) In this manner, determine the natural frequencies ωr, the dampingratios ζr, and the residues (ψiψk)r for the set of modes r = 1, 2, …

Step 2: The residues of a diagonal transfer function (i.e., point transfer functions or auto-transfer

modal vectors are arbitrary up to a multiplier of –1 Hence, one can choose this row

to have all positive elements

Step 3: The residues of a nondiagonal transfer function (i.e., cross-transfer function) G k+i,k are

(ψk+iψk)1, (ψk+iψk)2, …, (ψk+iψk)n By substituting the values obtained in step 2 into

these values, determine the k+ith row of the modal matrix (ψk+i)1, (ψk+i)2, …, (ψk+i)n

( ) ( )1, 2, K, ( )

The complete modal matrix Ψ is obtained by repeating this step for i = 1, 2, …, n – k

and i = –1, –2, …, –k + 1 Note that the associated modal vectors are M-normal.

The procedure just outlined for determining the modal matrix verifies, by construction, that

only n transfer functions are needed to extract the complete modal information It further reveals

that it is not essential to perform the transfer function measurements in a row fashion or column

fashion A diagonal element (i.e., a point transfer function, or an auto-transfer function) should always be measured The remaining n – 1 transfer functions have to be off diagonal, but otherwise can be chosen arbitrarily, provided that all n degrees of freedom are spanned either as an excitation

point or 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

would be available in addition to the minimal set that is required Such redundant data are usefulfor checking the accuracy of the modal estimates Examples for an admissible (nonminimal) set,

a minimal set, and an inadmissible set of transfer function matrix elements are shown schematically

in Figure 11.2 Note that the inadmissible set in this example contains 8 transfer function ments, but the 6th degree of freedom is not covered by this set On the other hand, a minimal setrequires only six transfer functions

measure-11.2.1 E XTRACTION OF THE T IME -D OMAIN M ODEL

Once the complete modal information is extracted by modal analysis, it is possible — at least in

theory — to determine a time-domain model (M, K, and C matrices) for the system To obtain the

necessary equations, first premultiply by (ΨT)–1 and postmultiply by Ψ−1 the equations (11.4), (11.5),and (11.6) to get

Since the modal matrix Ψ is nonsingular, because M is assumed nonsingular in the dynamic models

that are used here (i.e., each degree of freedom has an associated mass, or the system does notpossess static modes), the inverse transformations given by equations (11.24) to (11.26) are feasible

It appears, however, that two matrix inversions are needed for each result Since ces are diagonal, their inverse is given by inverting the diagonal elements This fact can be used

matri-to obtain each result through just one matrix inversion

Equations (11.24) to (11.26) are written as

1

1 2

2 2

2

1

ωω

is fixed both by the signal record length (T) and the type of time window used in digital

Fourier analysis, but the resonant peaks are sharper for higher frequencies Frequencycoverage depends on the data sampling rate

3 Low signal-to-noise ratio (SNR) at high frequencies, in part due to noise and poordynamic range of equipment, and in part due to low signal levels, will result in datameasurement errors Signal levels are usually low at high frequencies because inertia in

a mechanical system acts as a low-pass filter

4 Computations involving high-order matrices (multiplication, inversion, etc.) in complexsystems with many degrees of freedom, will lead to numerical errors

It is customary, therefore, to extract modal information only for the first several modes In thatcase, it is not possible to accurately recover the mass, stiffness, and damping matrices Even ifthese matrices were computed, their accuracy would be questionable due to their sensitivity to thefactors listed above

11.3 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 often 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 on systems and control A parameter estimation procedure that involves transfer functions, and which is particularly useful in experimental modal analysis, is presented

mω



11.3.1 P ROBLEM I DENTIFICATION

Transfer functions that are computed from measured time histories using digital Fourier analysis(e.g., fast Fourier transform, or FFT) cannot be directly used in modal analysis computations Thedata must be available as analytical transfer functions Therefore, it is important to represent thecomputed transfer functions by suitable analytical expressions This is done, in practice, either bycurve-fitting a suitable transfer function model into the computed data or by simplified methodssuch as “peak picking.” Accordingly, this convention of data is an experimental modeling technique.Identification of transfer function models from measured data is an essential step in experimentalmodal analysis Apart from that, it has other important advantages In particular, analytical transferfunction plots clearly identify system resonances and generate numerical values for the correspond-

BOX 11.1 Main Steps of Experimental Modal Analysis

1 Measure an admissible set of excitation (u) and response (y) signals (Cover all dof;

one response measurement should be for the excitation location.)

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

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

4 Compute ordinary coherence functions:

and choose the accurate transfer functions on this basis (γuy close to 1 ⇒ accept; γuy close

to 0 ⇒ reject)

5 Curve-fit n admissible transfer functions to expressions:

Hence, extract:

Residues (ψiψk)r⇒ mode shapes vectors ψr , which are M-normal

Natural frequencies (undamped) ωr

Modal damping ratios (viscous) ζr

6 Form the modal matrix Ψ = [ψ1, ψ2, …, ψn]

Compute Ψ–1

7 Modal mass matrix M = I

Modal stiffness matrix

Modal damping matrix

8 Compute the system model:

C =diag[2ζ ω1 1, 2ζ ω2 2, K, 2ζ ωn n]

ing parameters (resonant frequencies, damping, phase angles, and magnitudes) in a convenientmanner This form represents a significant improvement over the crude transfer function plots thatare normally far less presentable and rather difficult to interpret.

11.3.2 S INGLE -D EGREE - OF -F REEDOM AND M ULTI -D EGREE - OF -F REEDOM

T ECHNIQUES

Several single degree-of-freedom (dof) techniques exist for extracting analytical parameters fromexperimental transfer functions In particular, the methods of curve fitting (circle fitting) and peakpicking are considered here In a single-dof method, only one resonance is considered at a time

Single-degree-of-freedom curve fitting or, more correctly, single-resonance curve fitting is the

term used to denote any curve-fitting procedure that fits a quadratic (second-order) transfer functioninto each resonance in the measured transfer function, one at a time In the case of closely spacedmodes (or closely spaced resonances), the associated error can be very large The accuracy isimproved if expressions of higher order than quadratic are used for this purpose, but unacceptableerrors can still exist In peak picking, each resonance of experimental transfer function data isexamined individually, and resonant frequency, and the damping constant corresponding to thatresonance, are determined by comparing with an analytical single-dof transfer function

In multi-degree-of-freedom curve fitting or, more appropriately, multiresonance curve-fitting,

all resonances (or modes) of importance are considered simultaneously and fitted into an analyticaltransfer function of suitable order This method is generally more accurate but computationallymore demanding than the single-resonance method In choosing between the single-resonance andmultiresonance methods, the required accuracy should be weighed against the cost and speed ofcomputation

11.3.3 S INGLE -D EGREE - OF -F REEDOM P ARAMETER E XTRACTION IN THE

imaginary axis for the frequency transfer function, is a circle Similarly, it can be shown that the

receptance or dynamic flexibility or compliance transfer function (displacement/force) of a dof system with hysteretic damping, when plotted on the Nyquist plane, is also a circle Note that

single-for hysteretic damping, the damping constant (in the time domain) is not actually a constant but

is inversely proportional to the frequency of motion (see Chapter 7) But, in the frequency domain,the damping term will be independent of frequency, in this case The fact that such circle repre-sentations are possible for transfer functions of a single-dof system, can be used in fitting a circle

to a transfer function that is computed from experimental data This will lead to determining theanalytical parameters for the transfer function This approach is illustrated now, through analyticaldevelopment

Case of Viscous Damping

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

(11.39)

my˙˙+cy˙+ky= f t( )

where m, k, and c are the mass, stiffness, and the damping constant of the system, respectively;

f(t) is the excitation force; and y is the displacement response Equation (11.39) can be expressed

in the standard form:

where the constant parameter m in equation (11.42) has been omitted, without loss of generality.

In the frequency domain (s = jω), then

(11.44)

Multiply the numerator and the denominator of G(jω) in equation (11.44) by the complex conjugate

of the denominator (i.e., ) Then, the denominator is converted to the square of

its original magnitude, as given by

(11.45)and the frequency-transfer function (11.44) is converted into the form

Hence, in view of equation (11.47), one obtains

It follows that the transfer function G(jω) represents a circle in the real-imaginary plane, with thefollowing properties:

A sketch of this circle is shown in Figure 11.3(a) As mentioned before, the plane formed by

the real and imaginary parts of G(jω) as the Cartesian x and y axes, respectively, is the Nyquist plane The plot of G(jω) on this plane is the Nyquist diagram It follows that the Nyquist diagram

of the mobility function (11.42) [or (11.44)] is a circle

Case of Hysteretic Damping

Consider a single-dof system with hysteretic damping The equation motion is given by

44

116

1

Note the frequency-dependent damping constant, with the hysteretic damping parameter h, in the time domain The receptance function G(jω) is given by

FIGURE 11.3 (a) Circle fit of a mobility function with viscous damping, and (b) circle fit of a receptance

function with hysteretic damping.

my˙˙+ h y˙+ky= f t( ) f t( )= f sin t

Note that the damping term jh is independent of frequency in the frequency domain, for this case.

As for the case of viscous damping, one can easily show that the Nyquist plot of this transferfunction is a circle with

(11.52)

A sketch of the resulting circle is shown in Figure 11.3(b)

In general, for a multi-dof viscous-damped system, one has the “mobility” function

Note: This method will lead to larger errors if the resonances are closely spaced and, consequently,

if significant modal interactions are present

Peak Picking Method

This is also a single-dof method in view of the fact that each resonance of an experimentallydetermined transfer function is considered separately The approach is to compare the resonanceregion with an analytical transfer function of a damped single-dof system One of three types oftransfer functions — receptance, mobility, or accelerance — as listed in Table 11.1, can be usedfor this purpose Note that when the level of damping is small, it can be assumed (approximately)that the resonance is at the undamped natural frequency Substituting this value for ω

in each of the frequency transfer functions, one can determine the transfer function value at

resonance, denoted by Gpeak(jω) It is noted from Table 11.1 that, in general, this function valuedepends on the damping constant and the natural frequency Because ωn is known directly from

the peak location of the transfer function, it is possible to compute the damping constant c (or

damping ratio ζ) by first determining the corresponding peak magnitude

Specifically, from Table 11.1, it is clear that one should pick the imaginary part of the transfer function for receptance or accelerance data, and the real part of the transfer function formobility data Then, one picks the peak value of the chosen part of the transfer function, and thefrequency at the peak

frequency-Peak picking is good for cases where modes are well separated and lightly damped It doesnot work when the system is highly damped (or overdamped), or when the damping is 0 (infinitepeak) It is a quick approach that is appropriate for preliminary evaluations and troubleshooting

Radius= 21 and Center= 0 

12

11.3.4 M ULTI -D EGREE - OF -F REEDOM C URVE F ITTING

A general multi-resonance curve-fitting method is now presented; the corresponding nance method should also be clear from this general procedure Note that many different versions

single-reso-of problem formulation and algorithm development are possible for least-squares curve fitting, butthe results should be essentially the same The method presented here is a frequency-domain method,

as one is dealing in this chapter with experimentally determined frequency-transfer functions In acomparable time-domain method, a suitable analytical expression of the complex exponential form

is fitted into the experimental impulse response function obtained by the inverse Fourier mation of a measured transfer function That method inherits additional error due to truncation

transfor-(leakage) and finite sampling rate (aliasing) during the inverse FFT (see Chapter 4 and Appendix D).

Formulation of the Method

The objective of the present multi-resonance (multi-dof) curve-fitting procedure is to fit the puted (measured) transfer function data into an analytical expression of the form:

com-(11.55)

The data for curve fitting would be the N complex transfer function values [G1, G2, …, G N] computed

at discrete frequencies [ω1, ω2, …, ωN] Typically, if 1024 samples of time history were used inthe FFT computations to determine the transfer function, one would have 512 valid spectral lines

of transfer function data But, near the high-frequency end, these data values become excessivelydistorted due to the aliasing error; only a part of the 512 spectral lines is usable, typically the first

400 lines In that case, one has N = 400 This value can be doubled by doubling the FFT block

c

r

n m