Identification of Damping from Experimental Dynamic Stiffness Matrix: Error Analysis

Gokhan O. Ozgen and Jay H. Kim

Structural Dynamics Research Laboratory Mechanical Engineering Department

University of Cincinnati, PO box 210072 Cincinnati, OH, 45221

Nomenclature

d.o.f.(s) : Degree(s) of freedom )]([H : Frequency response function matrix

DSM(s) : Dynamic stiffness matrix(ces) ][ DK : Dynamic stiffness matrix

FRM(s) : Frequency response function matrix(ces) r

: Modal vector for rth mode

FRF(s) : Frequency response function(s) rQ : Modal scaling factor for rth mode

SVD : Singular value decomposition r : Complex modal frequency for rth mode

: Frequency (rad/s) : Singular value matrix of SVD of a matrix

N : Total number of degrees of freedom U : Left hand side vector matrix of SVD of a matrix

][I : Identity matrix V : Right hand side vector matrix of SVD of a matrix

)]([L : Frequency dependent damping matrix U : Left hand side vector of SVD of a matrix

][M : Mass matrix V : Right hand side vector of SVD of a matrix

][K : Stiffness matrix r : Modal damping ratio for rth mode

][C : Viscous damping matrix nw

kH /)( : FRF contaminated with noise

j : 1 okH )( : FRF without noise

Abstract

Some practical issues related to application of the dynamic stiffness matrix (DSM) based damping identification method to real life mechanical systems are studied in this paper. The DSM based direct damping matrix identification method was proposed to identify both the mechanism and spatial distribution of damping as a matrix of general function of frequency. The method obtains the damping matrix from the imaginary part of the DSM, which is computed by inverting the measured frequency response function matrix (FRM). The major issue regarding the feasibility of this method is how the errors in measured frequency response functions (FRF) affect the DSM, because it is known that the DSM is highly sensitive to errors that are present in the FRFs. In this paper, a detailed analytical and computational study is performed, which leads to a sound physical explanation of the high sensitivity of the DSM to measurement errors that was observed in earlier studies. A new and also important conclusion is that the leakage error drastically affects the accuracy of the DSM in addition to other common measurement errors. Based on the findings of this study, recommendations are made for future experimental implementations of the damping identification method.

1 Introduction

Development of a method to identify spatial distribution of damping in mechanical systems has been attempted by many investigators. A common approach for this purpose has been to obtain a damping matrix that will give modal damping loss factors close to experimentally obtained values[1,2,3], which requires identification of modal parameters. A direct method, which eliminates the need for modal parameters, was proposed by Lee and Kim[4,5] to identify damping matrices from frequency response functions (FRF). This method uses the dynamic stiffness matrix (DSM) to estimate damping matrices, which is obtained by inverting the experimentally obtained frequency response function matrix (FRM). The damping matrix )]([L , a general function of frequency, is found

by taking the imaginary part of the DSM )]([ DK , i.e.;

12 )()()()( HimagLjKMimagKimagL D (1)

Magnitude of each element in the damping matrix )]([L represents damping distributed at the node point

associated with the element. Damping matrix )]([L identified as such can be used for any further analyses in the frequency domain. If it is necessary, the viscous damping matrix ][C can be found by a pseudo-inverse procedure as shown:

kNxNkkNxNk

NxN

Himag

Himag

Himag

I

I

I

C

))](([

.

.

))](([

))](([

][

.

.

][

][

1

12

11

2

1

(2)

where “+” is the pseudo-inverse of the matrix, ][I is the identity matrix, and N is the number of degrees of freedom (d.o.f.). The ][C matrix found as such will represent the best spatial description of damping when the damping mechanism is assumed viscous.

One problem that plagues the DSM based approach is that the procedure is very sensitive to measurement errors. Since most experimental modal parameter estimation methods are based on measured frequency response functions (FRFs), the characteristics of measurement errors and their effect on the FRFs are widely studied and well understood. On the other hand, their effect on the DSM is a relatively untouched area since the DSM is seldom used for dynamics parameter identification purposes. Following some earlier development efforts of the DSM based damping identification method[4,5,6], we now investigate effects of measurement errors on the DSM in this paper. Measurement errors being considered include both variance and bias type of errors. Variance type errors have random characteristics, while bias type errors are systematic errors which require special signal processing techniques other than statistical averaging. One interesting outcome of this study is that the leakage error, which is a bias type error, has a quite drastic effect on the accuracy of the DSM, especially on its imaginary part.

In the following sections, respective and combined effects of variance type random measurement error and bias type leakage error on the DSM are demonstrated both experimentally and numerically. The effect of variance type random measurement error is also analyzed from a theoretical perspective.

2 Preliminary Study of Measurement Errors

Before carrying out detailed error analyses, two preliminary studies are conducted. In the first study a numerical simulation is conducted using a simple mathematical model in order to see the effect of variance type errors. In the second study a test is performed on a real structure to compute the DSM. As will be discussed in Section 2.2, this study reveals not only a somewhat obvious effect of variance type errors but also significant effect of the leakage error on the accuracy of the computed DSM.

2.1 Numerical Case Study on the Effect of Variance type Random Error

A four degree of freedom (d.o.f.) lumped parameter system shown in Figure 1 is used to conduct numerical simulations. The system is viscously damped with 1% damping ratio at all four modes. In order to simulate the effect of variance type measurement error, different levels of random noise are introduced to the analytically calculated FRFs. The DSM is then calculated by inverting the FRM composed of FRFs contaminated by random noise. The noise is introduced to the analytically calculated FRFs using the following formula:

100

_))))((max()))((max(()()( /

noisePercentHimagjHrealHH o

kko

kko

knw

k , (3)

where nwkH /)( is the FRF contaminated with noise, o

kH )( is the original FRF without noise, k is the frequency

index, and k and k are random number vectors (-1 <…< 1). DSM and FRF plots for three different signal-to-noise (S/N) ratios are given in Figure 2. For the first case, the

S/N ratio is 0.01%, which is a very small ratio and hard to achieve in actual testing conditions. As seen in Figure 2b, the effect of noise on the FRF is insignificant in both the phase and the magnitude. Imaginary part of the DSM obtained by inverting the FRM, however, is drastically affected as seen in Figure 2a. Yet, damping matrix identified using Equation (2) is still in good agreement with the original damping matrix as seen in Table 1.

The S/N ratio of 0.1%, a more typical ratio in actual tests, is considered next. The DSM and FRF for this case are shown in Figure 2c and Figure 2d. The increased noise in FRFs causes larger errors in identified damping matrix as shown in Table 1.

The third case is when the S/N ratio is 1%. This is the case when noise is roughly the same level as the damping that has to be found. The effect of noise on the imaginary part of the DSM is rather significant as shown in Figure 6e. The identified damping matrix becomes completely different from the original damping matrix as seen in Table 1.

Figure 1) Four d.o.f. lumped parameter system

Table 1) Original and identified damping matrices Original Damping Matrix

(N.s/m) Identified Damping Matrices

(N.s/m) (0.01% noise in FRFs) (0.1% noise in FRFs) (1% noise in FRFs)

8.58 -2.47 -0.44 -0.21 8.54 -2.430 -0.39 -0.22 6.46 -0.29 -1.35 0.72 -17.78 -25.65 28.01 -28.80

-2.47 8.14 -2.69 -0.66 -2.27 7.99 -2.52 -0.53 -0.96 5.94 -2.55 -1.53 -36.41 -33.45 -41.45 -59.65

-0.44 -2.69 7.92 -3.13 -0.37 -2.85 8.17 -3.17 -2.33 -0.68 8.29 -3.22 18.91 -1.28 -61.54 5.39

-0.21 -0.66 -3.13 5.45 -0.05 -0.78 -2.98 5.50 0.71 -0.73 -2.729 5.56 -49.94 -14.63 -24.70 -29.43

m1

x1(t)f1(t)

c1

k1

m2

x2(t)f2(t)

c2

k2

m3

x3(t)f3(t)

c3

k3

m4

x4(t)

f4(t)

c4

k4

0 10 20 30 40 50 60 70 80-50

0

50

100

Phase(degrees)

DSM plot - (Imaginary part)

With NoiseNo Noise

0 10 20 30 40 50 60 70 80-2000

-1000

0

1000

2000

3000

4000

5000

Frequency (Hz)

Imaginarypart

0 10 20 30 40 50 60 70 80-100

-50

0

50

Phase(degrees)

Sample FRF plot - (Magnitude)

With NoiseNo Noise

0 10 20 30 40 50 60 70 8010

-7

10-6

10-5

10-4

10-3

Frequency (Hz)

M agnitude

(a) (b)

0 10 20 30 40 50 60 70 80-50

0

50

100

Phase(degrees)

DSM plot - (Imaginary part)

With NoiseNo Noise

0 10 20 30 40 50 60 70 80-3

-2

-1

0

1

2

3x 10

4

Frequency (Hz)

Imaginarypart

0 10 20 30 40 50 60 70 80-100

-50

0

50

Phase(degrees)

Sample FRF plot - (Magnitude)

With NoiseNo Noise

0 10 20 30 40 50 60 70 8010

-7

10-6

10-5

10-4

10-3

Frequency (Hz)

M agnitude

(c) (d)

0 10 20 30 40 50 60 70 80-100

0

100

Phase(degrees)

DSM plot - (Imaginary part)

With NoiseNo Noise

0 10 20 30 40 50 60 70 80-8

-6

-4

-2

0

2

4

6

8x 10

5

Frequency (Hz)

Imaginarypart

0 10 20 30 40 50 60 70 80-100

0

100

Phase(degrees)

Sample FRF plot - (M agnitude)

0 10 20 30 40 50 60 70 8010

-7

10-6

10-5

10-4

10-3

Frequency (Hz)

M agnitude

With NoiseNo Noise

(e) (f) Figure 2) FRF and DSM plots for three different signal to noise (S/N) ratios (a) Imaginary part of 1st diagonal element of the DSM with S/N ratio of 0.01% (b) Magnitude of the 1st diagonal element of the FRM with S/N ratio of 0.01% ratio (c) Imaginary part of 1st diagonal element of the DSM with S/N ratio of 0.1% (d) Magnitude of the 1st diagonal element of the FRM with S/N ratio of 0.1% (e) Imaginary part of 1st diagonal element of the DSM with S/N ratio of 1% (f) Magnitude of

the 1st diagonal element of the FRM with S/N ratio of 1%

2.2 Experimental Case Study

Results presented in this section are based on the experimental data obtained by Hylok[6]. The test object is an aluminum beam which is suspended by bungee cords to simulate free-free beam case (Figure 3). The beam is 408 mm long and has a rectangular cross-section with a thickness of 7.6 mm and a width of 43 mm. Six measurement points are selected along the longitudinal mid-axis of the beam. Single axis accelerometers are super-glued at these locations in order to measure the transverse response of the suspended beam. The test structure is excited using an impact hammer at measurement points through the opposite face of the beam.

A 6 x 6 FRM is obtained utilizing the Multi Reference Impact Testing (MRIT)[7] method. Measured FRFs seem to be fairly clean (Figure 4). But when the imaginary part of the DSM, which is calculated by inverting the measured FRM, is examined in Figure 5, effects of the measurement errors can be clearly seen. The effect of measurement errors of random nature becomes more evident in the off-diagonal elements. Note that the imaginary part of the DSM given in Figure 5 represents the spatial distribution of damping in the structure as functions of frequency.

An interesting observation is that in almost the entire frequency range, diagonal elements of the imaginary part of the DSM (Figure 5) are negative valued. Negative diagonal elements in the damping matrix indicate that the damping mechanism of the structure is creating energy rather than dissipating, which is obviously impossible. This is a rather unexpected result, which is not seen in numerical simulation results of random measurement errors shown in Figure 2. This observation made us to look for other types of errors than random errors. Impact testing, inherently contains leakage error[8,9]. An exponential window, which reduces the leakage error, cannot be used in this particular test, where the main purpose of the test is to identify damping, because it introduces artificial damping in the system[8]. Therefore, it is expected that the measured FRFs are highly contaminated by leakage error, in addition to the variance type random measurement error.

To better understand the effects of the variance type random error and the leakage error on the damping estimation, theoretical and numerical analyses are performed and summarized in Section 3 and Section 4 respectively.

Figure 3) Experimental set-up – A suspended aluminum beam[6].

Figure 4) Spatial plot of the experimentally obtained FRM of the aluminum beam

Figure 5) Spatial plot of the imaginary part of the experimentally obtained DSM of the aluminum beam

3 Analysis of Variance Type Measurement Error

Although it has been demonstrated through numerical and experimental examples that the DSM is highly sensitive to variance type measurement errors that are present in the FRFs[6],how the errors actually affect the results are not well known. A theoretical analysis is conducted to explain this. The modal superposition property is utilized to explicitly relate the FRM and the DSM, which explains how errors that are present in the FRFs influence the estimation of the DSM.

3.1 Explicit Relation of FRM to DSM (A Simplified Case)

Let us start by defining the FRM of a linear, time-invariant and viscously damped system in terms of its modal parameters:

)()()(

1 r

Trrrn

r r

Trrr

j

Q

j

QH , (4)

where r is the modal vector for rth mode (can be complex), rQ is the modal scaling factor for rth mode, is

the frequency (rad/s) and r is the complex modal frequency for rth mode. Equation (4) can be also written in matrix form as follows:

TNNx

NNxr

r

r

rNNxNxN

j

Q

j

Q

j

Q

H 2

22*

*

1

1

2

)(00

0)(

0

00)(

)( (5)

The square matrix for the N d.o.f. system defined in equation (5) has a total of 2N diagonal elements while only N physical modes contribute to the system response. This is because each mode has a conjugate pair of modal frequencies and mode shapes.

An alternative form of the FRM can be written utilizing the singular value decomposition (SVD) as shown:

HNxNNxNNxNNxN VUH )( , (6)

where is a diagonal matrix called “the singular value” matrix, and matrices U and V contain vectors that are orthogonal. The physical significance of comes from the fact that each diagonal element is associated with a different set of vectors U and V , and it represents the contribution of these vectors to the overall

system characteristics. Comparing equation (5) to equation (6), U and V can be described as linear combinations of the modal vectors of the system of interest.

The DSM is simply the inverse of the FRM and it can be defined in terms of the SVD parameters of the FRM as follows:

HD UVK

1 (7)

For a general system with non-proportional viscous damping, there is no direct relation between the SVD parameters and the modal parameters. In order to obtain such an explicit relation, a proportionally dampedsystem with a diagonal and uniform mass matrix will be considered; i.e.

IaM . (8)

Since modal vectors of such a system are real valued and orthogonal with respect to each other, a direct comparison of the SVD and the modal superposition form of the FRM becomes possible. In order to do this, equation (4) has to be rewritten as

N

r r

r

r

rTrr

j

Q

j

QH

1 )()()( , (9)

or in matrix form

TNxN

NxN

r

r

r

rNNxN

j

Q

j

QH

00

0)()(

0

00

)(*

*. (10)

Based on equation (10) and equation (6), a direct comparison of SVD and modal parameters can be made as follows:

00

0)()(

0

00

*

*

r

r

r

r

j

Q

j

Q, (11)

and,

VU , . (12)

Examining equations (6) and (7) in conjunction with equations (11) and (12), it can be concluded that at a certain frequency, the mode that has the lowest contribution to the system response characteristics has the lowest singular value associated with it. Lowest singular value for the FRM becomes the largest singular value of the DSM (equation (7)).

Based on the discussions above, the effect of measurement noise on computed DSM can now be investigated using the Complex Mode Indicator Function (CMIF). The CMIF is commonly used in experimental modal analysis studies to identify the number of modes present in the frequency range of interest. Number of modes is decided by examining the plot of singular values obtained from the SVD of the FRM. Common name for this plot is the “CMIF plot”. Peaks observed in the CMIF plot indicate the existence of modes at corresponding frequencies. Overall distribution of the singular value plot shows the contributions of different modes to the overall system response characteristics.

The CMIF plot of the FRM obtained for the four d.o.f. system (Figure 1) defined in Section 2.1 is given in Figure 6a. Looking at this plot, it can be easily seen that there exist only two modes within the given frequency range. Lowest two singular values are associated with the modes that are in the higher frequency range and they are contaminated by the measurement errors. Note also that the contribution of those modes is quite small when compared to the modes present in the frequency range. However, as noted before, lowest singular value associated with the FRM becomes the highest singular value for the DSM. This can also be seen in the CMIF plot for the DSM given in Figure 6b. Singular values associated with the modes outside the frequency range become dominant in the CMIF plot of the DSM. Since they are already highly contaminated by noise present in the FRFs, error in the DSM is amplified drastically.

0 10 20 30 40 50 60 7010

-6

10-4

10-2

CMIF plot of the FRM

Frequency (Hz)

1st mode

2nd mode

Modes outside the frequency range

0 10 20 30 40 50 60 70

102

104

106 CMIF plot of the DSM

Frequency (Hz)

Modes outside the frequency range

1st mode

2nd mode

(a) (b) Figure 6) CMIF plot of the (a) FRM and (b) DSM

3.2 Effect of Residual Modes

One possible way to attenuate the random measurement errors present in the FRFs and the DSM is to utilize modal curve fitting techniques. However, for most modal parameter estimation methods, there are no constraints imposed on the identified modal parameters to be representative of a second order system. Although synthesized FRFs can be well-correlated with experimentally measured ones, accuracy of the DSM obtained from the synthesized FRM can not be guaranteed.

Another problem associated with the use of modal curve fitting techniques is the incompleteness of the modal information. Only the modes in the frequency range of FRF measurements can be identified, while the contribution of modes outside the frequency range are represented as higher and lower residual terms. These lower and higher terms are commonly defined as mass and stiffness terms for each individual FRF. Since contribution of these modes to the FRFs is relatively small, damping information associated with these modes is generally not considered. It is not possible to say the same thing for the synthesis of the DSM. For example, the imaginary part of the DSM of the proportionally damped system defined in the Section 3.1 can be derived in terms of the modal parameters as follows:

N

rr

r

rN

rNN

rr

r

rrN

N

rr

r

rN

rN

rr

r

rr

QQ

QQDSMimag

11

1

1

1

1

11

)( , (13)

where

CDSMimag )( . (14)

Comparing Equations (13) and (14), viscous damping matrix C can be written in terms of the modal parameters as follows:

N

rr

r

rN

rNN

rr

r

rrN

N

rr

r

rN

rN

rr

r

rr

QQ

QQC

11

1

1

1

1

11

(15)

Note that Equation (15) indicates that each mode has a constant contribution to the viscous damping matrix, and this in fact necessitates the inclusion of all the modes of the system of interest if the purpose is to identify a complete damping matrix.

4 Analysis of Leakage Error

Complex nature of the leakage error requires a novel numerical study to understand its effects on the DSM. For this purpose, an impact testing procedure is simulated using a simple model in order to obtain FRFs that are contaminated by leakage error. The procedure for this numerical simulation is composed of several steps. In the first step, analytical FRFs of the chosen discrete mathematical model are computed. Following this, a force vector at a selected d.o.f. is defined as the time history of the impulsive input force. The linear spectrum of the input force vector is then computed using the Fast Fourier Transform technique. Having the input force spectrum, the linear spectrums of the response at all d.o.f.s are computed using the analytical FRFs. In order to be consistent with common FRF measurement procedures, acceleration responses are used rather than displacement responses. Once the response spectrums are available at all d.o.f.s, the time histories of the acceleration response vectors are computed by inverse Fourier transforming the analytically obtained response spectrum vectors. After this, the input force and acceleration response time histories are resampled using a smaller block size than the original one. The column of the FRF matrix corresponding to the selected input d.o.f. is estimated using the resampled time histories of input force and acceleration responses. All these steps are repeated for each d.o.f. to finally obtain the complete FRF matrix of the selected model. Since the time sampling increment is not changed during the resampling of time histories, maximum frequency of the input and response vectors does not change either (no aliasing error is induced through resampling). But since the block size is reduced, the frequency increment is automatically increased and this is actually what causes the leakage error in the FRFs estimated using the resampled time vectors. Random input testing can also be simulated by replacing the impulsive force vector by a random force vector.

The first model used for leakage error simulation is the four d.o.f. system given in Figure 1. The frequency range is chosen to include all four modes of the system analyzed. Spatial plot of the imaginary part of the DSM obtained using the leakage contaminated FRFs are given in Figure 7. As it can be clearly seen from this plot, the effect of the simulated leakage error on the imaginary part of the DSM, which is making diagonal elements negative in a wide frequency range, resembles what is seen in the experimental case presented in Section 2.2. In order to further support this conclusion, the test performed on the aluminum beam is re-processed using the same approach explained in the previous paragraph with the inclusion of variance type of random measurement error. A correlated finite element analysis (FEA) model is used to obtain the 6 x 6 FRM of the beam structure. Damping matrix of the FEA model is constructed using the modal damping ratios identified from the experimental FRFs. Spatial plot of the imaginary part of the DSM of the FEA model, which is contaminated with both leakage and random errors, is given in Figure 8. Comparing the simulation and experimental results (Figure 5 and Figure 8), it can be concluded that the actual test results are closely reproduced. This confirms the initial explanations made in Section 2.1 and Section 2.2 for the errors observed in the experimental DSM.

An exponential window cannot be used with the impact test because it introduces artificial damping effect. Therefore, impact testing induces quite a large leakage error in addition to variance type error. An obvious conclusion from this study is that impact testing is not a good choice to implement the DSM-based direct damping identification. Other testing methods, which can better compensate for leakage error and have less sensitivity to random errors, will have to be pursued because the proposed DSM based direct damping identification method requires high accuracy. For example, random input testing methods[10,11] will be a good option considering possible applications of various windows, cyclic averaging, special input force functions like burst random, pseudo random, which can minimize leakage errors efficiently, and the spectral averaging that largely eliminates the variance type random measurement errors.

Figure 7) Spatial plot of the imaginary part of the DSM of the four d.o.f. system (contaminated with leakage error)

Figure 8) Spatial plot of the imaginary part of the DSM of the FEA model of the aluminum beam (contaminated with both leakage and random errors)

5 Conclusion

The DSM based direct damping identification method is a very attractive and promising approach to identify the spatial information of damping in dynamic systems. Because the method identifies the damping matrix as a general function of frequency, it will enable building an analytical-experimental hybrid model by combining analytically formulated mass and stiffness matrices with an experimentally identified damping matrix. However, there still exist several issues that have to be resolved to make the method practical enough. As previously reported[6], an obvious issue is the effect of measurement errors which includes both variance and bias type of errors. In the previous study[6], it was shown by numerical and experimental examples that the DSM obtained by inverting the FRM was very sensitive to errors present in the FRFs. Experimental results also showed presence of bias errors in addition to variance type errors. This study is conducted to understand the way these errors influence the damping matrix identified by the DSM based approach. First a theoretical analysis to simulate the effect of variance type random errors is conducted using a simplified mathematical model and modal superposition property. It is shown that for a fixed frequency range, modes outside this range have a low contribution to the FRFs, while they have the highest contributions to the DSM. These higher modes are relatively more susceptible to random errors, which amplifies errors in the identified DSM. The use of modal curve fitting techniques is also discussed as a means to attenuate the error in the DSM. Following this, a numerical simulation is performed in order to understand how the leakage error affects the computed DSM. The simulation shows that the leakage error has a quite significant effect on the accuracy of the DSM. The pattern of the effect of leakage error also partially explains previous experimental results of direct damping matrix identification that showed negative diagonal damping elements. Considering the nature of impact testing that introduces relatively large leakage error, other testing methods will have to be pursued to obtain accurate damping matrix from the DSM-based damping identification method. The random input testing method is believed to be the most promising method to overcome the leakage problem associated with the impact testing method.

6 References

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2 PILKEY, D. F., GYUHAE, P. and INMAN, D.J., “Damping Matrix Identification and Experimental Verification”, Smart Structures and Materials, SPIE Conference on Passive Damping and Isolation, Newport Beach, California, pp.350-357, 1999.

3 ADHIKARI, S. and WOODHOUSE, J., “Towards Identification of a General Model of Damping”, Proceedings of the 18th International Modal Analysis Conference, San Antonio, TX, pp. 377-383, 2000.

4 LEE, J. H. and J. Kim, “Identification of Damping Matrices from Measured Frequency Response Functions”, Journal of Sound and Vibration 240(3), pp 545-565, 2001.

5 LEE, J. H. and J. Kim, “Development and Validation of a New Experimental Method to Identify Damping Matrices of a Dynamic System”, Journal of Sound and Vibration 246 (3), pp. 505-524, 2001.

6 HYLOK, J. E., “Experimental identification of distributed damping matrices using the dynamic stiffness matrix”, Thesis (M.S.), University of Cincinnati, 2002.

7 FLADUNG, W. A. Jr., BROWN, D. L., “Multiple Reference Impact Testing”, Proceedings of the 11th

International Modal Analysis Conference (IMAC), v.2, 1993, p.1221-1229. 8 FLADUNG, B., “Windows used for impact testing”, Proceedings of the 15th International Modal Analysis

Conference (IMAC), v.2, 1997, p.1662-1666. 9 BROWN, D., “Weaknesses of impact testing”, Proceedings of the 15th International Modal Analysis

Conference (IMAC), v.2, 1997, p.1672-1676. 10 ALLEMANG, R. J., Modal Analysis lecture notes, UC-SDRL-CN-20-263-662, Structural Dynamics Research

Laboratory, University of Cincinnati, 1999. 11 ALLEMANG, R. J., Vibrations: Experimental Modal Analysis, UC-SDRL-CN-20-263-663/664, Structural

Dynamics Research Laboratory, University of Cincinnati, 1999.