18.3 Experimental Model Development

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We have 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 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, 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 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-transfer

functions. For convenience, however, the following notation is used:

GikðvÞ ¼

Xn

r¼1

􀀑 ðcickÞr

v2r

2 v2 þ 2jzrvrv

􀀜 ð18:22Þ

Gikðf Þ ¼

Xn

r¼1

ðcickÞr

4p2

􀀑

f 2

1 2 f 2 þ 2jzr fr f

􀀜 ð18:23Þ

where v and f are used in place of jv and j2pf in the function notation Gð Þ: As already observed in

Example 18.1, 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. Owing to the symmetry of G it follows that at

most only 1=2nðn þ 1Þ 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, 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 transfer

function matrix. It should be clear from the following development that any set of transfer functions that

spans all n DoF of the system would be an admissible set provided that only one autotransfer function is

included in the minimal set. 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 ðGik; i ¼ 1; 2; …; nÞ of the transfer function matrix is measured by

applying a single forcing excitation at the kth DoF and measuring the corresponding responses at all n

DoF 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 ratios zr ; 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 autotransfer

function), Gkk; are ðc2k Þ1; ðc2k

Þ2; …; ðc2k

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

ðckÞ1; ðckÞ2; …; ðckÞn: Note that M-normality is assumed. However, the modal vectors are

arbitrary up to a multiplier of 2 1. 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; ðckþick Þ2; …; ðckþick Þn: 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 modal

matrix 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 n

transfer functions are needed to extract the complete modal information. It further reveals that it is not

18-8 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

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 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 checking

the accuracy of the modal estimates. Examples for an admissible (nonminimal) set, a minimal set, and an

inadmissible set of transfer functions matrix elements are shown schematically in Figure 18.2. Note that

the inadmissible set in this example contains 11 transfer function measurements but the sixth DoF is not

covered 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, to

determine a time-domain model (M, K, and C matrices) for the system. To obtain the necessary

equations, first premultiply by ðCTÞ21 and postmultiply by C Equation 18.4, Equation 18.5, and

Equation 18.6 to obtain

M ¼ ðCTÞ21M􀀊 C21 ð18:24Þ

where M􀀊 ¼ I ¼ identity matrix

K ¼ ðCTÞ21K􀀊C21 ð18:25Þ

C ¼ ðCTÞ21C􀀊C21 ð18:26Þ

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 inverse

transformations 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 result

through just one matrix inversion.

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

M ¼ ðCM􀀊 21CTÞ21 ð18:27Þ

K ¼ ðCK􀀊 21CTÞ21 ð18:28Þ

C ¼ ðCC􀀊 21CTÞ21 ð18:29Þ

Note that for the present M-normal case

M􀀊 21 ¼ I ð18:30Þ

K􀀊 21 ¼ diag½1=v21

; 1=v22

; …; 1=v2

n􀀉 ð18:31Þ

C􀀊 21 ¼ diag½1=ð2z1v1Þ; 1=ð2z2v2Þ; …; 1=ð2znvnÞ􀀉 ð18:32Þ

G =

An Admissible Set

A Minimal Set

An Inadmissible Set

FIGURE 18.2 A nonminimal admissible set, a minimal

set, and inadmissible set of possible transfer function

measurements.

Experimental Modal Analysis 18-9

© 2005 by Taylor & Francis Group, LLC

By substituting the equations from Equation 18.30 to Equation 18.32 into the equations from Equation

18.27 to Equation 18.29, we obtain the relations that can be used in computing the time-domain model:

M ¼ ðCCTÞ21 ð18:33Þ

K ¼ C

1=v21

0

1=v22

. .

.

0 1=v2

n

2

66666664

3

77777775

CT

0

BBBBBBB@

1

CCCCCCCA

21

ð18:34Þ

C ¼ C

1=ð2z1v1Þ 0

1=ð2z2v2Þ

. .

.

0 1=ð2znvnÞ

2

66666664

3

77777775

CT

0

BBBBBBB@

1

CCCCCCCA

21

ð18:35Þ

Alternatively, only one matrix inversion (that of C) is needed if we use the fact that

ðCTÞ21 ¼ ðC21ÞT

Then,

M ¼ ðC21ÞTM􀀊 C21 ð18:36Þ

K ¼ ðC21ÞTK􀀊C21 ð18:37Þ

C ¼ ðC21ÞTC􀀊C21 ð18:38Þ

The main steps of EMA are summarized in Box 18.1. In practice, frequency-response data are less

accurate at higher resonances. Some of the main sources of error are as follows:

(1) Aliasing distortion in the frequency domain, due to finite sampling rate of data, will distort highfrequency

results during digital computation.

(2) Inadequate spectral-line resolution (or frequency resolution) and frequency coverage

(bandwidth) can introduce errors at high-frequency resonances. The frequency resolution is

fixed both by the signal record length ðTÞ and the type of time window used in digital Fourier

analysis, but the resonant peaks are sharper for higher frequencies. Frequency coverage depends

on the data sampling rate.

(3) Low signal-to-noise ratio (SNR) at high frequencies, in part due to noise and poor dynamic

range 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 a

low-pass filter 1=ðmv2Þ:

(4) Computations involving high order matrices (multiplication, inversion, etc.) will lead to

numerical 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.

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