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This article was originally ublished in a journal ublished by Elsevier, and the attached coy is rovided by Elsevier for the author s benefit and for the benefit of the author s institution, for

1 This article was originally ublished in a journal ublished by Elsevier, and the attached coy is rovided by Elsevier for the author s benefit and for the benefit of the author s institution, for non-commercial research and educational use including without limitation use in instruction at your institution, sending it to secific colleagues that you know, and roviding a coy to your institution s administrator. All other uses, reroduction and distribution, including without limitation commercial rerints, selling or licensing coies or access, or osting on oen internet sites, your ersonal or institution s website or reository, are rohibited. For excetions, ermission may be sought for such use through Elsevier s ermissions site at: htt://

2 ARTICLE IN PRESS Mechanical Systems and Signal Processing 1 (7) Mechanical Systems and Signal Processing Abstract Estimation of modal arameters using bilinear joint time frequency distributions A. Roshan-Ghias a, M.B. Shamsollahi b,,1, M. Mobed b, M. Behzad a a Mechanical Engineering Deartment, Sharif University of Technology, Tehran, Iran b Electrical Engineering Deartment, Sharif University of Technology, Tehran, Iran Received 15 March 6; received in revised form 4 July 6; acceted 9 Setember 6 Available online 1 December 6 In this aer, a new method is roosed for modal arameter estimation using time frequency reresentations. Smoothed Pseudo Wigner Ville distribution which is a member of the Cohen s class distributions is used to decoule vibration modes comletely in order to study each mode searately. This distribution reduces cross-terms which are troublesome in Wigner Ville distribution and retains the resolution as well. The method was alied to highly damed systems, and results were suerior to those obtained via other conventional methods. r 6 Elsevier Ltd. All rights reserved. Keywords: Modal arameter estimation; Joint time frequency distributions; Highly damed systems 1. Introduction Numerous algorithms have been develoed for estimating modal arameters using time and frequency domains searately over the ast 3 years. In recent years, joint time frequency (JTF) and wavelet transform (WT) methods have attracted many researchers to use great caabilities of these transforms. Several aers have been ublished on alying WT for estimating modal arameters in recent years. Lardies and Gouttebroze [1] used WT and resented a secial form of the Morlet Wavelet which gave rewarding results. Le and Argoul [] studied three different wavelets and tackled the Edge Effect roblem. Haase and Widjajakusuma [3] resented a method based on the WT for fault detection using modal arameters. On the other hand, there are researches in Biomedical Magnetic Resonance Sectroscoy which are very similar to the modal analysis. Serrai et al. [4] used the Morlet Wavelet and established an algorithm with less aroximation in comarison with some other works. However, to the best of the authors knowledge, no work has yet been reorted on estimating modal arameters using bilinear JTF distributions. These distributions exand the energy of a signal concurrently in Corresonding author. Tel.: address: mbshams@sharif.edu (M.B. Shamsollahi). 1 Postal address: Electrical Engineering Deartment Sharif University of Technology Azadi street Tehran Iran /$ - see front matter r 6 Elsevier Ltd. All rights reserved. doi:1.116/j.ymss.6.9.9

3 16 ARTICLE IN PRESS A. Roshan-Ghias et al. / Mechanical Systems and Signal Processing 1 (7) time and frequency. This makes them a natural choice for fully decouling the modes of vibration which haen at distinct frequencies. Wigner ville distribution (WVD) which is a member of Cohen s class distributions has the best resolution in time frequency lane among all time frequency reresentations of this class. But it has a deficiency of socalled cross-terms. Cross-terms aear when a signal has two or more comonents in the time frequency lane. In a free decay resonse signal, these comonents are different modes of vibration. Thus, WVD is incaable of giving an aroriate discrimination of these comonents in the time frequency lane in general. In order to comensate this drawback, dozens of joint time frequency reresentations have been introduced during last decades [5 8]. Smoothed Pseudo Wigner Ville (SPWV) is a member of Cohen s class distributions which utilizes two different smoothing windows on WVD, searately in time and in frequency, and eliminates cross-terms considerably [6]. In this aer, we introduce a method for the estimation of modal arameters using the SPWV distribution. Free decay resonse of a linear mechanical system with roortional daming ratios is used to estimate modal arameters (natural frequency and daming) of the system. It can be alied to single-degree-offreedom (sdof) and multi-degree-of-freedom (mdof) linear systems. The exact analytical time-frequency distribution of a one-dof system is obtained and modal arameters are extracted from the formulation. Afterward, in view of the fact that SPWV decoules vibration modes in the time-frequency domain, results are exanded to a mdof system. In order to demonstrate the caability of the roosed method, the daming ratio estimation of a dof highly damed system is examined and comared to two other methods. The aer is organized as follows. Section is a brief introduction to Cohen s class distributions and SPWV characteristics. In Section 3, WVD of the free decay resonse of a sdof system is obtained. The SPWV of free decay resonse of a sdof system is derived in Section 4. Section 5 generalizes the results of the sdof system to mdof systems considering the fully decouling of vibration modes in time frequency domain. In Section 6, two examles are given and fully discussed. Finally concluding remarks are given in Section 7.. Cohen s class The reresentations that describe a signal s frequency behavior fall redominantly into two categories [5]: linear reresentations such as the Fourier transform, and quadratic reresentations such as the ower sectrum (PS). Quadratic reresentations can be viewed as distributing signal s energy into frequency, time frequency, or time-scale variables. In this section we introduce a counterart to the ower sectrum: the quadratic joint time frequency reresentation known as Cohen s class distributions. The main core of all TFR is the Wigner Ville Distribution (WVD) which is defined as: WVD x ðt; f Þ ¼ 1 xtþt= x t t= e jf t dt. (1) WVD can be seen as the instantaneous version of the ower sectrum [6]. Theoretically, the WVD has the best time frequency resolution among all time-frequency reresentations. But it does suffer from the serious roblem of cross-terms, which occurs when the signal has two or more distinct time frequency features [7]. In order to overcome this deficiency, other bilinear JTF distributions have been develoed over the last decades such as the Pseudo WVD (PWVD), Smoothed Pseudo WVD (SPWVD) [6], Choi-Williams distribution (CWD) and cone-shae distribution (AMD) [5]. It is interesting to note that all these bilinear reresentations can be written in a general form that was introduced by Cohen [5]. The discovery of the general form of bilinear TFR facilitates us with the design of the desired TFR. This general form can be written as T x ðt; f Þ ¼ ¼ Fðt s; f Þx sþ t x s t e jt dt ds d Fðt s; f ÞWVD x ðs; Þds d, ðþ

4 ARTICLE IN PRESS A. Roshan-Ghias et al. / Mechanical Systems and Signal Processing 1 (7) where F(t, f) is a two dimensional filter known as the kernel of the TFR. It can be seen that when F(t, f) ¼ d(t)d(f), WVD is obtained. Sometimes it is more convenient to use j(t, u) instead, which is the inverse -D Fourier transform of F(t, f) [5]. Smoothed seudo Wigner Ville is a member of Cohen s class with a searable kernel in the form: Fðt; f Þ¼gðtÞHðfÞ or jðt; nþ ¼hðtÞGðnÞ. (3) This characteristic enables us to determine each window searately to suit our requirements. Windowing reduces cross-terms considerably but at the cost of losing some resolution. 3. Alication of WVD to free decay resonse of sdof systems Consider a sdof mechanical system with viscous daming x. The free dislacement resonse (noise free) according to modal basis is exressed as xðtþ ¼F e x f t cos ð f t þ j ÞuðtÞ, (4) where f and f are the undamed and damed natural frequencies in Hz, F is the amlitude of vibration and u(t) is the Heaviside function. If x(t) is assumed asymtotic, that is if the hase of the signal varies much faster than the amlitude (which imlies that x 51= ffiffiffi ), it can be shown [] that the comlex signal x a ðtþ ¼F e x f t e jð f tþj Þ uðtþ, (5) is a good aroximation of the analytic signal. Analytic signal is more aroriate to use since the WVD of the analytical signal has less cross-terms than the WVD of the real signal [5], so Eq. (5) is used instead of Eq. (4). Using Eq. (1), the WVD of free decay resonse can be exressed as (see Aendix A) WVD x ðt; f Þ¼ F e 4f x t 4ðf f Þ sin 4ðf f Þt. (6) Although at f ¼ f, this function is undefined but its limit is lim WVD x ðt; f Þ¼F t e 4f x t. (7) f! f Fig. 1(a) shows a tyical WVD of a free decay resonse and Fig. 1(b) shows its value at the damed natural frequency. As it is seen in Fig. 1(a) and in Eq. (6), at the damed natural frequency, this function has a maximum. Thus f and the corresonding slice of the time frequency ma can be easily obtained without any Fig. 1. (a) WVD of a tyical free decay resonse, and (b) WVD value at the damed natural frequency.

18 ARTICLE IN PRESS A. Roshan-Ghias et al. / Mechanical Systems and Signal Processing 1 (7) 15 136 comutations. Then using Eq.

5 18 ARTICLE IN PRESS A. Roshan-Ghias et al. / Mechanical Systems and Signal Processing 1 (7) comutations. Then using Eq. (7) and a simle curve fitting algorithm, x can be calculated (the very beginning and the very end of the calculated WVD at the damed natural frequency should be omitted to avoid errors due to the edge-effect roblem). For any sdof system, this method is accurate and the cost of comutation is low. But in the case of mdof systems, this method fails to give reasonable results because of the above-mentioned cross-terms (Fig. ). 4. Alication of SPWVD to free decay resonse of sdof systems As it was stated before, knowing WVD of a signal, its SPWV distribution can be written as SPWV x ðt; f Þ ¼ 1 1 WVD x ðt s; f ÞgðsÞHðÞ ds d. (8) Looking at the WVD of a mdof free decay resonse reveals the nature of the cross-terms. Since every mode has a fixed frequency in the time-frequency lane, cross-terms occur at the middle of every two of them. In other words, cross-terms are at fixed frequencies too. It can be roved [5] that in this case, there is no need of the frequency smoothing window. So the function H in the kernel of SPWVD will be set to Kronecker delta function. The resulting reresentation is the counterart to Pseudo Wigner-Ville Distribution which has only the frequency smoothing window. Therefore Eq. (8) can be simlified to SPWV x ðt; f Þ ¼ Fig.. WVD of a tyical free decay resonse of a two degrees of freedom system. 1 WVD x ðt s; f ÞgðsÞ ds. (9) The Gaussian window is an aroriate candidate for the time smoothing window since it is a low-ass filter with the analytical form of gðtþ ¼ffiffiffiffiffi 1 e t =s, (1) s where s is the windowing arameter.

6 ARTICLE IN PRESS A. Roshan-Ghias et al. / Mechanical Systems and Signal Processing 1 (7) Substituting Eqs. (6) and (1) into Eq. (9), we get integrals of the form: 1 SPWV x ðt; f Þ ¼ A 1 s ex u t=s þ s 4x f þ j4ðf f Þ u du 1 A s ex u s 4x f þ j4ðf f Þ u du; t=s where A 1 ¼ F 1 ffiffiffiffiffi s j4ðf f Þ ex 4x f t þ j4ðf f Þt A ¼ F 1 ffiffiffiffiffi s j4ðf f Þ ex 4x f t j4ðf f Þt. ð1þ Using substitute variables a 1, a, D 1, D and a, Eq. (11) can be written as where SPWV x ðt; f Þ¼A 1 s e a 1 =þja 1D 1 I 1 A s e a =þja D I, (13) I 1 ¼ 1 a e t =þjd 1 t dt; I ¼ 1 a ð11þ e t =þjd t dt, (14) a ¼ t=s 4x f s; a ¼ a 1 ¼ a ¼ 4x f s; D ¼ D 1 ¼ D ¼ 4ðf f Þs. (15) Integrals of Eq. (14) do not have rimary functions, but they can be obtained with desired accuracy (see Aendix B). Substituting integral s value, we can write Eq. (13) as SPWV x ðt; f Þ¼Y B sin 4 f f a þ C cos 4 f f a, (16) where Y ¼ F 1 ffiffiffiffiffi ðf f Þ ex 4x f t þð4x f Þ =, (17) ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B ¼ e D = 1 e a, (18) C ¼ e D = ffiffiffiffiffiffiffiffiffiffiffiffiffiffi e D 1 D 1 e a =. (19) The signs in Eq. (18) and 7 in Eq. (19) are conditioned by the signs of a and D, resectively. Obviously, Eqs. (16) (19) are comlicated. However, similar to the WVD case, the exressions simlify when f! f : ( ) lim SPWV x ðt; f Þ¼F 1 se 4f x s aþ 4f x s a a þ a erf ffiffi þ e a, () f! f where x erf ðxþ ¼ ffiffiffi e t dt, (1) and finally using the same rocedure as in the WVD case, Eq. (1) can be used to find F and x.

7 13 ARTICLE IN PRESS A. Roshan-Ghias et al. / Mechanical Systems and Signal Processing 1 (7) Alication of SPWVD to free decay resonse of mdof systems In the case of mdof systems, free decay resonse can be exressed as the sum of all modes: x m ðtþ ¼ XN F mk e x kf t k cos ð f k t þ j k ÞuðtÞ, () k¼1 where k is the mode number. Assuming the signal to be asymtotic (all x k 51= ffiffi ), the analytic form can be written as x ma ðtþ ¼ Xn k¼1 F mk e x kf k t e jð f k tþj k Þ uðtþ. (3) Assuming the cross-terms to be negligible (because of the imlication of the time smoothing window, the cross-terms would be reduced significantly), the SPWVD of Eq. (3) can be written as SPWV x ðt; f Þ¼ XN k¼1 Y k B k sin 4 f f k ak þ Ck cos 4 f f k ak. (4) Similar to the sdof case, the SPWV distribution is maximum at damed natural frequencies. Hence using a eak-icking algorithm, the damed natural frequency of each mode can be estimated. Subsequently, a curvefitting algorithm with corresonding slices of the time frequency ma and Eq. () will yield the daming ratio of each mode. 6. Simulated results In order to demonstrate the caability of the SPWVD based method in modal analysis, a simulation is done with a two-dof system. The arameters of the system are given in Table 1. The free decay resonse and the frequency resonse of this system are shown in Fig. 3. The SPWVD of this signal is shown in Fig. 4(a), (b) and the ridges which corresond to damed natural frequencies are given in Figs. 4(c), (d). Using the rocedure described in Section 5, the natural frequencies and daming ratios of the two modes is extracted (Table ). In this case, the accuracy of the SPWV based method is the same as two other methods: Line-fit method (LFM) and rational fraction olynomial method (RFPM) [9]. However in some secial cases, such as in highly damed systems, this method shows a great imrovement with resect to other methods. Consider a dof system with initial modal arameters which is shown in Table 3. Daming ratios in both modes were increased stewise u to 15% and then the modal arameters were estimated using SPWV and other methods. Estimated natural frequencies were satisfactory in all methods, but estimating high damings becomes inaccurate Table 4. Figs. 5 and 6 show the ercentage of error for both modes in each method, when daming is increased. The accuracy of the SPWV based method is undoubtedly higher than those two other methods. This is due to the erfect decouling of the modes in the time-frequency domain using the SPWV distribution. 7. Conclusion A novel method in modal arameter identification using smoothed seudo Wigner Ville distribution was roosed. Time-frequency reresentations fully decoule vibration modes and it makes them an enhanced tool Table 1 Modal arameters of a dof system Frequency (Hz) Daming (%) 1st mode 1.5 nd mode 35 1

8 ARTICLE IN PRESS A. Roshan-Ghias et al. / Mechanical Systems and Signal Processing 1 (7) Fig. 3. (a) Time resonse of a dof system, and (b) frequency resonse of a dof system. Fig. 4. (a,b) SPWVD of a dof system, (c) ridge lot at the first natural frequency, and (d) ridge lot at the second natural frequency.

9 13 ARTICLE IN PRESS A. Roshan-Ghias et al. / Mechanical Systems and Signal Processing 1 (7) Table Estimated modal arameters for the first mode Frequency (Hz) Daming (%) SPWVD 1..5 LFM RPFM Table 3 Modal arameters of a dof system Frequency (Hz) Daming (%) 1st mode.5 nd mode 35.5 Table 4 Exact and estimated values of daming ratios Daming ratio of 1st mode (%) Daming ratio of nd mode (%) Exact LFM RFPM SPWV Exact LFM RFPM SPWV Error (%) LFM RFPM SPWV 1 st Mode Daming (%) Fig. 5. Amount of error in estimating daming of the first mode. in modal analysis. Because of this decouling, each natural frequency can be obtained with a simle ickeaking algorithm along the frequency axis. An analytic form of a free decay resonse of a sdof system was obtained in the time frequency lane using SPWVD, without any simlifying aroximation. Therefore modal daming can be obtained with a straightforward curve-fitting for each mode. The method shows its effectiveness in modal analysis of highly damed systems where most other methods encounter large errors.

10 ARTICLE IN PRESS A. Roshan-Ghias et al. / Mechanical Systems and Signal Processing 1 (7) LFM RFPM SPWV nd Mode Aendix A This aendix describes the calculation of the Wigner Ville distribution of the free decay resonse of a sdof system. Substituting Eq. (5) into Eq. (1) yields 1 WV x ðt; f Þ¼ 1 F e x f ðtþt=þ e j½ f ðtþt=þþj Š uðt þ t=þ 4 F e x f ðt t=þ e j½ f ðt t=þþj Š uðt t=þ e jf t dt, which is simlified to WV x ðt; f Þ¼ 1 t 4 F e 4x f t e jðf f Þt dt ¼ F e 4x f t h j8ðf f Þ ejðf f Þt and finally t i t, t¼ t WV x ðt; f Þ¼ F e 4x f t 4ðf f Þ sin 4ðf f Þt. (A.3) Aendix B This aendix rovides the comutation of the integral I ¼ R 1 a I may be written as I ¼ Error (%) 1 e½ ð t =ÞþjDt Š dt a ½ e ð t =ÞþjDt Š dt Daming (%) Fig. 6. Amount of error in estimating daming of the second mode. ða:1þ (A.) e½ ð t =ÞþjDtŠ dt, where a ¼ t=s 4x f s [4]. ¼ I 1 I 8a. ðb:1þ Consider the rectangle OABC in Fig. 7. Let f(z) be a comlex function on this rectangle, given by f ðzþ ¼e z =, where z ¼ t jd and t runs from to N. I 1 may be written as I 1 ¼ e D = 1 jd jd e z = dz. (B.)

11 134 ARTICLE IN PRESS A. Roshan-Ghias et al. / Mechanical Systems and Signal Processing 1 (7) The function e z = is analytical on and inside the rectangle OABCO. Using Cauchy theorem, we have I f ðzþ dz ¼ e z = dz þ e z = dz þ e z = dz þ e z = dz ¼. (B.3) OABCO OA AB BC (1) On thesegment ffiffiffi OA of the rectangle, z ¼ jy, where y runs from to D. Using the olar coordinate (r ½; DŠ; y ½; =Š) and the sign of D, f(z) is given by 1 h e z = dz ¼ je y = dy j i 1=. ðed 1Þ (B.4) OA () On the segment AB of the rectangle, z ¼ x jd; x ½; RŠ, so that R jd e z = dz ¼ e ðx jdþ = dx. AB jd Eq. () is I 1 u to the term e D =. (3) On the segment BC of the rectangle, z ¼ R jy; y ½D; Š. Thus, e z = dz ¼ j e ðr jyþ = dy ¼ je R = e ðjryþy =Þ dy. BC Eq. (B.6) may be estimated in the limit by e z = dz je R = e y = dy. BC D D Fig. 7. A rectangle in comlex lane. The integral R D = ey dy has a finite value; hence, Eq. (B.7) decreases to zero ffiffi when R-N. (4) On segment CO, z ¼ x, with x ½R; Š Using the olar coordinates (r ½; DŠ; y ½; =Š), we obtain R e z = dz ¼ e x = 1= dy 1 e R CO, (B.8) Eq. (B.8) aroaches ffiffiffiffiffiffiffiffi = when R-N. D Substituting the values of Eqs. (B.4) (B.7) into Eq. (B.8), we obtain 8 h ffiffiffiffiffiffiffiffiffiffiffiffiffiffii >< e D = 1 þ j e D 1 if D4; I 1 h ffiffiffiffiffiffiffiffiffiffiffiffiffiffii >: e D = 1 j e D 1 if Do: CO (B.5) (B.6) (B.7) (B.9)

12 ARTICLE IN PRESS A. Roshan-Ghias et al. / Mechanical Systems and Signal Processing 1 (7) For I ¼ R a = et e jdt dt, we use the Taylor series exansion of the term e jd. " # a X 1 ðjdtþ k I ¼ e t = dt ¼ X1 ðjdþ k a t k e t = dt ¼ X1 ðjdþ k U k, (B.1) k! k! k¼ k¼ k! k¼ where U k ¼ R a tk e t = dt. Ifk ¼ and the olar coordinates are used: 8 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a >< 1 e a if a4 U ¼ e t = dt ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (B.11) >: 1 e a if a4: If k ¼ 1, by changing variables, we obtain U 1 ¼ R h i a = te t dt ¼ 1 e a =. The general term for this series for kx, is given by and U k ¼ a t k e t = dt ¼ a t k ðe t = Þ dt a ¼ a k e a = þðk 1Þ t k e t = dt ¼ a k e a = þðk 1ÞU k. ðb:1þ The series P 1 k¼ ðjdþk =k! U k is recurrent and convergent. By combining Eqs. (B.1), (B.9) and (B.1), I is aroximated as ffiffiffiffiffiffiffiffiffiffiffiffiffiffi " # D = I e 1 j e D 1 X1 ðjdþ k U k 8a. (B.13) k! k¼ Eq. (B.13) may be described by I ¼½BþjCŠ, where B is " # ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B ¼ e D = 1 e X1 ðjdþ k a U k ðk evenþ, (B.14) k! C ¼ e D = k¼ " # ffiffiffiffiffiffiffiffiffiffiffiffiffiffi e D 1 D 1 e a = X1 ðjdþ k U k ðk oddþ. (B.15) k! k¼ The signs and 7 are determined by the signs of a and D, resectively. If we restrict k to unity, B and C become ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B ¼ e D = 1 e a, (B.16) ffiffiffiffiffiffiffiffiffiffiffiffiffiffi C ¼ e D = e D 1 D 1 e a =. (B.17) References [1] J. Lardies, S. Gouttebroze, Identification of modal arameters using the wavelet transform, International Journal of Mechanical Sciences () [] T.P. Le, P. Argoul, Continuous wavelet transform for modal identification using free decay resonse, Journal of Sound and Vibration 7 (4) [3] M. Haase, J. Widjajakusuma, Damage identification based on ridges and maxima lines of the wavelet transform, International Journal of Engineering Sciences 41 (3) [4] H. Serrai, L. Senhadji, J.D. De Certaines, J.L. Coartrieux, Time-domain quantification of amlitude, chemical shift, aarent relaxation time, and hase by wavelet-transform analysis, Alication to biomedical magnetic resonance sectroscoy, Journal of Magnetic Resonances 14 (1997) 34. [5] S. Qian, D. Chen, Joint Time-Frequency Analysis: Method and Alication, Prentice Hall, 1996.

13 136 ARTICLE IN PRESS A. Roshan-Ghias et al. / Mechanical Systems and Signal Processing 1 (7) [6] P. Flandrin, Time-frequency and Time-scale Analysis, Academic Press, [7] L. Cohen, Time Frequency Analysis, Prentice Hall, [8] F. Hlawatch, G.F. Boudreaux, Linear and quadratic time-frequency signal rocessing, IEEE Signal Processing Magazine 9 () (199) [9] M.H. Richardson, Global frequency and daming estimates from frequency resonse method, 4th IMEC conference, LA, 1986.

ME scope Application Note 16

ME scope Application Note 16 ME scoe Alication Note 16 Integration & Differentiation of FFs and Mode Shaes NOTE: The stes used in this Alication Note can be dulicated using any Package that includes the VES-36 Advanced Signal Processing