Modal Analysis Using Time-Frequency Transform

Transcription

1 Moal Aalysis Usig Time-Frequecy Trasform Zhuag Li a Malcolm J. Crocker Departmet of Mechaical Egieerig, Aubur Uiversity Aubur, AL 36849, Uite States lizhua@eg.aubur.eu Abstract Traitioal moal aalysis methos use either time omai or frequecy omai approaches. Because vibratio sigals are geerally o-statioary, time a frequecy iformatio is eee simultaeously i may cases. This paper presets a overview of the applicatios of joit time-frequecy methos for moal aalysis. Sice a joit time-frequecy aalysis ca ecouple vibratio moes, it has a avatage, especially whe iformatio about the excitatio is ot available. I this paper the moal parameters are estimate usig Gabor aalysis. Numerical simulatios a experimets have also bee carrie out. 1 INTRODUCTION Three elemetary parameters use i the moal aalysis of a yamic system are its atural frequecy, moal ampig a magitue. A cosierable amout of attetio has bee evote to frequecy respose fuctio (FRF) aalysis [1,,3]. However, the covetioal Fourier aalysis approach is restricte to oly oe omai, because the elemetary fuctios use to ecompose the sigals exist from egative ifiity to positive ifiity i the time omai. I other wors, Fourier aalysis caot provie iformatio o how the frequecy cotet of a sigal chages with time. Therefore, Fourier aalysis is oly useful for statioary sigals. For o-statioary sigals, time-frequecy represetatios are eee. Cosier a simple case, the ecay of free vibratio. The ecayig vibratio sigal is o-statioary because its magitue ecreases expoetially. So it ca be viewe as a trasiet pheomeo. Although it is ot ifficult to calculate the loss factor by the ecay rate metho for lightly ampe systems, the metho oly works for a sigle moe at resoace, a the result is very sesitive to oise. Joit time-frequecy aalysis (JTAF) ca be use istea to separate the moal compoets cotaie i sigals a to reuce oise. The moal parameters of each moe separate thus ca be extracte. Joit time-frequecy aalysis algorithms fall ito two categories: the liear JTFA a the quaratic JTFA. I the liear JTFA, the short-time Fourier trasform (STFT) a the Gabor expasio, which ca be regare as the iverse of the STFT, are two algorithms. If we cosier the liear JTFA as the evolutio of the covetioal Fourier trasform, the quaratic JTFA is the couterpart of the staar power spectrum. Quaratic algorithms iclue the Gabor spectrogram, Cohe s Class a the aaptive spectrogram [4,5,6]. The ifferece betwee liear a quaratic JTFA methos is that the liear trasform ca be iverte to recostruct the time sigal. Thus, the liear trasform is suitable for sigal processig, such as time-variat filterig. However, the quaratic JTFA escribes the eergy istributio of the sigal i the joit time-frequecy omai, which is useful for sigal aalysis. Sice the phase iformatio is lost i the quaratic time-frequecy represetatio, the time histories caot be recostructe. I this paper both liear a quaratic JTFA approaches are use for ampig calculatios. I some applicatios, iformatio about the excitatio force is ot available. Schwarz a Richarso use a curve fittig techique to estimate the moal parameters from ambiet respose ata [7]. Boato et al. use Cohe s class to estimate the moal parameters from the o-statioary respose to a ukow excitatio [8]. However, the algorithms i Cohe s class, such as the Wiger-Ville Distributio, have cross-term iterfereces i the time-frequecy represetatio [5]. I this paper we stuy the moal parameter estimatio usig the Gabor expasio a the Gabor spectrogram. GABOR ANALYSIS.1 Gabor trasform For a give iscrete time sequece x [k], the Gabor trasform ca be compute from

2 C j πk = N m x[ k] γ * [ k mt ] e,, (1) where C m, is a matrix whose etries are calle the Gabor coefficiets, γ[k] is the aalysis wiow, a * eotes the complex cojugate. The parameters T a N represet the iscrete time samplig iterval a the total umber of frequecy lies [5]. The aalysis fuctio γ[k] is localize i the joit time-frequecy omai. So the Gabor coefficiets will epict the local time-frequecy properties of x [k]. Fig. 1. (a) The measure sigal, (b) the origial Gabor coefficiets, (c), () a (e) the three moes separate by Gabor-aalysis base time-variat filters. Raom oise is evely istribute over the etire joit time-frequecy omai because it is ot limite to a short time perio or a arrow frequecy ba. O the cotrary, the joit time-frequecy represetatio of a sigal is always cocetrate i a relatively small regio. After ietifyig the sigal compoet, a mask ca be applie to filter the sigal compoets a take the iverse trasform i orer to obtai the oise-free waveform sigal i the time omai. After computig the Gabor coefficiets by Eq. (1), a time-variat filter, which is actually a two-imesioal biary mask fuctio M m,, is use to moify the Gabor coefficiet as ˆ =. () C m, Mm, Cm,

3 The compoet of iterest ca the be extracte. As log as some requiremets are satisfie, the compoet i the time omai ca be recostructe as M 1 N 1 x ˆ [ k] = Cˆ h[ k mt ] e N,, (3) M = 0 = 0 where h[k] is calle the sythesis fuctio [5,9]. Qia ha show that if the fuctios h[k] a γ[k] are ietical, the Gabor coefficiets of the recostructe sigal x ˆ[ k] will be optimally close to C ˆ m,, i the sese of least square error. This process is calle orthogoal-like Gabor trasformatio [5]. I this proceure Eq. (1) is calle the Gabor trasform (or aalysis). A the iverse trasform Eq. (3) is calle the Gabor expasio (or sythesis). Figure 1 shows the proceure i Eqs. (1) to (3). I Fig. 1, (a) is a sigal obtaie from the free vibratio of a alumium catilever beam, a (b) is the origial Gabor coefficiets calculate usig Eq. (1). Three moal resposes at 34.5 Hz, 14.4 Hz a Hz ca be see clearly. The color itesity i the Gabor coefficiet plot represets the isplacemet magitue which is isplaye i ecibels. The oise is istribute i the etire timefrequecy omai. By usig three time-variat filters, or actually three mask matrices, the three moal resposes ca be ecouple as show i Figs. (c), () a (e). A 104-poit optimal Gaussia wiow is use to serve as the aalysis a sythesis fuctios i this calculatio. The three ecouple moes thus ca be recostructe usig Eq (3). Figure illustrates the recostructe waveforms a their spectra. It is see that the property of the sigal is improve sigificatly a the oise is ramatically reuce. Sice each recostructe waveform becomes a sigle-moe sigal, the atural frequecy, magitue, phase a ampig ratio ca be extracte easily. m j πk (a) (b) Fig.. (a) the three recostructe moes, (b) compariso of the spectra of the origial a recostructe sigals.. Gabor spectrogram As metioe before, the Gabor trasform is liear JTFA, but the Gabor spectrogram is quaratic. Base o the Gabor trasform, the Gabor spectrogram is efie as GS D [ i, k] = C C WVD [ i, ], (4) m m' + ' D m, m', ' h, h k ' where WVD h,h' [i,k] is the cross Wiger-Ville istributio of the frequecy-moulate Gaussia fuctios. The orer of the Gabor spectrogram, D, cotrols the egree of smoothig [6]. Figure 3 illustrates the Gabor spectrogram of the same sigal show i Fig. 1 (a). The eergy istributio of the three moes is clearly see i the spectrogram.

4 From this the atural frequecies, ampig ratios a magitue relatioships betwee these moes ca be extracte. However, sice the phase iformatio is lost, time histories caot be recostructe from the spectrogram. Fig. 3. The Gabor spectrogram of a free vibratio sigal obtaie from a catilever beam. 3 A GABOR ANALYSIS-BASED MODAL TESTING 3.1 Dampig Calculatio Basically there are four measures of ampig: loss factor, quality factor, ampig ratio a imagiary part of the complex moulus. However, these four measures are relate. The loss factor η a the ampig ratio ζ are those most commoly use i measuremets of ampig. There are may refereces i which reviews of methos to measure ampig are presete [10~16]. The free respose of a uerampe sigle-egree-of-freeom (DOF) system ue to a impact excitatio is give by ζω t y( t) = Ae cos( ω t φ), (5) where ζ is the ampig ratio, ω is the uampe atural agular frequecy a the ampe atural agular frequecy ω is ω = ω 1 ζ. (6) For a small value of ampig coefficiet ζ, ω ω. The ampig ratio ca thus be calculate by obtaiig the evelope. The traitioal ecay rate metho i which the ratio of successive peak amplitues is measure, is very sesitive to oise. Aother approach to obtai the evelope is to costruct the aalytic sigal by usig the Hilbert trasform. For a give real sigal y(t), its aalytic sigal (t) is y a y a (t) = y(t)+jh{y(t)}, (7) where the subscript a stas for aalytic, a the Hilbert trasform of y(t) is efie as Usig Parseval s formula, we ca show that the Fourier trasform of H{y(t)} is 1 y( τ ) H{ y( t)} = τ π. (8) t τ H( ω) = j sg( ω) Y( ω), (9) where sg(ω) is a sig fuctio, a Y(ω) is the Fourier trasform of y(t). So the Hilbert trasform ca be easily realize by takig the fast Fourier trasform (FFT) of y(t). The the magitue of the vector y a (t) is the evelope of the sigal y(t). The ampig ratio ζ associate with each moe ca be evaluate by expoetial curve fittig from Π e ζ =, (10) πf

5 where Π e is the power of the best expoetial fit, a f is the atural frequecy extracte from the recostructe sigal. (a) (b) (c) () (e) (f) Fig.4. (a) A simulate oisy free vibratio sigal, (b) the origial Gabor coefficiets, (c) a () the recostructe sigals, (e) a (f) evelopes a their best expoetial fits. Figure 4 (a) illustrates a simulate free vibratio sigal obtaie usig y( t) = e 0.01 π 00t si(π 00t) e π 350t si( π 350t) ( t), (11) There are two ampe siusois, 00 Hz a 350 Hz. Their ampig ratios are chose to be 0.01 a 0.008, respectively. The oise level (t) is 0.1. The samplig rate is 1000 Hz a the ata legth is 300 poits. Figure 4 (b) shows the Gabor coefficiets. By usig the Gabor aalysis-base time-variat filters, the two simulate moes ca be separate a recostructe, as show i Figs. 4 (c) a (). Figures 4 (e) a (f) show the evelopes selecte from the ecay parts a the correspoig best expoetial fits. The results a the mea square errors (MSE) are liste i Table 1. The umbers i brackets are the relative errors.

6 Moe Dampig ratio Calculate ampig ratio MSE of expoetial curve fittig first (0.00%) seco (0.375%) Table 1. Dampig ratios of two-moe ecay sigal calculate usig Gabor expasio. If oe is oly itereste i the ampig, a recostructio is ot ecessary, a alterative approach, the Gabor spectrogram ca also be utilize. Figure 5 (a) illustrates the same sigal simulate by (11). Figure 5 (b) is the Gabor spectrogram calculate usig Eq. (4). The two moes, their frequecies, a the ifferece betwee their magitues a ampig ca be istiguishe i this figure. By settig the frequecy zoom, we ca easily ecouple the moes as show i Figs. 5 (c) a (). If the riges are extracte from the two 3-D plots, the the expoetial ecay curves are recovere agai. Referece [17] escribes several algorithms for rige etectio. A avace wavelet applicatio, the so calle rigelet, has bee evelope i recet years [18]. I this paper, it is quite simple to recover the ecayig vibratio sigals because the riges are cocetrate at fixe frequecies. (a) (c) (b) () Fig. 5. (a) The origial sigal geerate by Eq. (1), (b) Gabor spectrogram, (c) a () the first a the seco moes separate by zoomig Gabor spectrogram. Ulike the expoetial evelope of the sigal which ca be recostructe usig the Gabor expasio, the moulatio term i Eq. (5) is square because the Gabor spectrogram calculate usig Eq. (4) is quaratic. So the term shoul be ivie oce more by two compare with Eq. (10). The the ampig ratio is Π es ζ =, (1) 4πf where Π es represets the expoetial power of the rige i the Gabor spectrogram. The results are liste i Table. The umbers i brackets are the relative errors. O comparig Tables 1 a, it ca be see that the error of the curve fittig obtaie with the Gabor spectrogram metho is smaller tha that obtaie with the Gabor expasio metho.

7 Moe Dampig ratio Calculate ampig ratio MSE of expoetial curve fittig first (1.00%) seco (0.375%) Table. Dampig results calculate usig the Gabor spectrogram metho. 3. Natural frequecy The atural frequecy of each recostructe sigal ca be calculate easily usig the FFT. Figure () shows the spectra of the three recostructe moes. The atural frequecies correspo to the three peaks i the spectra. 3.3 Moe shape For a N-egree-of-freeom (DOF) ampe system, the geeral equatios of motio writte i matrix form are [ M ]{ y& } + [ C]{ y& } + [ k]{ y} = { f ( t)}, (13) where [M], [C] a [K] are the mass, ampig a stiffess matrices, a {f(t)} is the force vector. For a passive system, the N N matrices [M], [C] a [K] are symmetric a positive efiite. The the moe shapes are ietical to the moe shapes for the uampe system [19]. For a uampe system, the atural frequecies are the eigevalues of the matrix [ M] 1 [ K], a the moe shape correspoig to oe atural frequecy is the eigevalue which satisfies 1 [ M] [ K]{ y i } = ωi { y i }, (14) where ω i is the i-th atural frequecy, a the moe shape {y i } is a N-imesioal colum vector i [ y, y, y ] T i 1 i { y } = L,. (15) Here the subscripts 1 through N iicate the gri poits which are evely istribute o the N-DOFs structure. The absolute values of these elemets are the magitues of the vibratio at the N gri poits. The sigs iicate the phase iffereces. The same sig for two elemets meas that the vibratio at these two poits is i phase. Differet sigs mea that the two poits are vibratig 180 out-of-phase. Although the actual values of the vector elemets are arbitrary, the ratios betwee them are uique. Therefore, eve without the iformatio of excitatio, the moe shapes ca be obtaie by simply measurig the magitues a phase agles of the resposes at all the gri poits. We ca choose oe of the gri poits as the referece poit a compare the magitues a phase agles measure at other poits to those measure at the referece poit for all the moes of iterest. After the moes i a vibratio sigal measure at a poit are ecouple a recostructe, the magitue a the phase agle for each moe ca be obtaie usig the FFT. Fourier trasformatio is a complex process, resultig i both magitue a phase iformatio. ςω Let t p( t) = Ae a q( t) = cos( ω t φ). The Eq. (5) becomes y( t) = p( t) q( t). The sigle-sie jφ spectrum of the pure cosie fuctio q(t) is Q ω) = πe δ( ω ). The Fourier trasform of p(t) is P( ω) = ( ςω t jωt ςω jω t A ( + ) ( ςω + jω) t Ae e t = A e t = e jω + ςω A =. (16) jω + ςω Usig the covolutio property of the Fourier trasform, Y ( ω) πa = ( ω ) = Y ( ω) e jφ cos φ ω jφ πae = P( ω) Q( ω) = jω + ςω [( ςω si φ) j( ω in jφ πae = ( ςω ) + ( ω ) ( ςω cos φ + ςω jω si φ)]. (17) )

8 It is easy to show that the calculate magitue Y (ω) a phase agle Φ of a sigle-moe free vibratio at its atural frequecy ω is πa Y( ω) =, a Φ = 90 φ. (18) ω The the moe shape ca be obtaie by the magitue ratios a phase iffereces which are y i1 A Y ( ω) i1 i1 = =, m =, LN, (19) y im Aim Yim ( ω) a φ φ = Φ Φ, m, LN. (0) i1 im i1 im = 4 EXPERIMENTS 4.1 Experimetal setup A three-dof moel of a alumium catilever beam was stuie experimetally as show i Fig. 6. For a catilever beam, the first three atural frequecies are 1,, 3 (1.194,.988,5 ) π κ c / L f = l 8, (1) where L is the legth of the catilever beam, κ is the raius of gyratio a c l is the logituial wave spee. The 3 thickess of the catilever beam is 6.43 mm. So κ = h / 1 = m. A the wave spee i alumium is 5055 m/s. The theoretical atural frequecies are liste i Table 3. A heavy steel block a two clamps were use to fix the beam at the left e. The free e, poit #1 was selecte as the referece poit. Two Polytech laser vibrometers were employe to measure the beam resposes. Sice the referece poit a oe of the other poits must be measure simultaeously, oe laser vibrometer was fixe at poit 1, a the other oe was use to measure the resposes at the other two poits. A Natioal Istrumets PCMCIA 6036E car was use for ata acquisitio a aalysis programs were evelope i LabVIEW. The samplig rate was 1500 Hz. Totally 0 measuremets were carrie out at poits # a #3, te measuremets at each poit. Each ata file also cotais the free respose acquire at the referece poit. y x 17 mm Poit #3 Poit # Poit #1 5 mm 381 mm Fig. 6. Experimetal setup for moal testig. 4. Natural frequecy a moe shape Figures 1 a show oe of the typical measuremets. Base o the Gabor trasform proceure i Eqs. (1) to (3), the moes which were overlappig i oe free vibratio sigal were separate a recostructe. The the atural frequecy, magitue a phase agle associate with each moe were extracte usig the FFT.

9 Cosequetly, the magitue ratios a phase iffereces at ifferet gri poits are etermie usig Eqs. (19) a (0). Table 3 lists the theoretical a calculate values. The measure atural frequecies are less tha the theoretical values ue to the o-ieal bouary coitios. However, the magitue ratios a phase iffereces are quite accurate. Figure 7 compares the theoretical a measure moe shapes for the three moes. Fuametal moe Seco moe Thir moe Theoretical value Calculate value Frequecy (Hz) (-3.3%) A 11 /A (-0.70%) A 11 /A (-3.69%) φ φ 1 φ11 φ Frequecy (Hz) (-4.5%) A 1 /A (9.98%) A 1 /A (0.34%) φ φ φ1 φ Frequecy (Hz) (-4.89%) A 31 /A (1.36%) A 31 /A (-1.37%) φ φ 3 φ31 φ Table 3. Error aalysis of the ew moal testig metho. Fig. 7. Compariso of the theoretical a measure moe shapes. 4.3 ampig ratio I the 0 measuremet ata files there are altogether 40 time histories. Both the Gabor expasio metho a the Gabor spectrogram metho were use to calculate the ampig ratios. Table 4 presets the averages. The umbers i the brackets are the correspoig staar eviatios. The staar eviatio of the ampig of the thir moe is the highest because it is relatively ifficult to excite the higher moe ito free vibratio. Sice the vibratio magitue of the thir moe is the smallest, the sigal-to-oise ratio of the thir moe is the lowest. Moe Gabor expasio metho Gabor spectrogram metho Fuametal % (.9 10 ) % ( ) Seco % (.3 10 ) % (.7 10 ) Thir % ( ) % ( ) Table 4. Dampig ratios calculate usig the Gabor expasio a the Gabor spectrogram.

10 5 CONCLUSIONS For a N-DOF system, i geeral a vibratio sigal cotais the yamic eflectio of N moes if these moes are all properly excite. The Gabor trasform a expasio ca be use ecouple a recostruct these moes to effectively make them ito sigle-moe sigals. The the atural frequecy, moal ampig, vibratio magitue a phase ca be extracte for each moe. The moe shape ca also be obtaie by comparig the magitues a phase agles at ifferet gri poits. A quaratic joit time-frequecy metho, the Gabor spectrogram, is a alterative approach which ca be use to calculate the atural frequecy a moal ampig of each moe excite. However, the Gabor spectrogram caot be use to etermie the moe shape sice the phase iformatio is lost i such a quaratic process. The ew Gabor aalysis-base moal testig metho is especially useful for o-statioary vibratio sigals, because the joit time-frequecy properties of a ostatioary sigal ca be represete by these methos. I this paper the moal parameters for a simulate free vibratio sigal were stuie. The metho prouce quite accurate results. The measuremets were mae o a catilever beam. Without ay iformatio o the excitatio, the moal parameters ca be obtaie very well usig the Gabor expasio approach. REFERENCES [1] D. Rormeti, M. Richarso 00 0 th IMAC Coferece. Parameter Estimatio from Frequecy respose Measuremets Usig Ratioal Fractio Polyomials. [] Mark. Richarso, Bria. Schwarz 003 Sou a Vibratio Magazie 37(1), Moal Parameter Estimatio from Operatig Data. [3] D. J. Ewis 000 Moal Testig: Theory, Practice a Applicatio. Balock, Hertforshire: Research stuies press LTD. [4] L. Cohe 1995 Time-Frequecy Aalysis. Eglewoo Cliffs: Pretice Hall PTR. [5] S. Qia 00 Itrouctio to Time-Frequecy a Wavelet Trasforms. Upper Sale River: Pretice Hall PTR. [6] Natioal Istrumets 001 Sigal Processig Toolset User Maual. [7] Bria. Schwarz, Mark. Richarso, th IMAC Coferece. Moal Parameter Estimatio from Ambiet Respose Data. [8] P. Boato, R. Ceravolo, A. De Stefao 003 Joural of Sou a Vibratio 37(5) Use of Cross- Time-Frequecy Estimators for Structural Ietificatio i No-Statioary Coitios a Uer Ukow Excitatio. [9] H. Shao, W. Ji, S. Qia 003 IEEE Trasactios o Istrumetatio a Measuremet 5(3), Orer Trackig by Discrete Gabor Expasio. [10] E. E. Ugar, 199 Structural Dampig. Chapter 1 i Noise a Vibratio Cotrol Egieerig: Priciples a Applicatios, Leo. L. Beraek, eitor. New York: Joh Wiley & Sos, Ic. [11] A. D. Nashif, D. I. G. Joes, J. P. Heerso 1985 Vibratio Dampig. New York: Joh Wiley & Sos, Ic. [1] S. Gae & H. Herlufse 1994 Dampig measuremet. Bruel a Kjaer Techical Review. [13] D. J. Ewis 000 Moal Testig: Theory, Practice a Applicatio. Balock, Hertforshire: Research stuies press LTD. [14] M. J. Crocker 1997 Ecyclopeia of Acoustics. New York: Joh Wiley & Sos, Ic. [15] R. H. Lyo, R. G. DeJog 1998 Theory a Applicatios of Statistical Eergy Aalysis. Cambrige, MA: RH Lyo Corp. [16] Zhuag Li, Malcolm J. Crocker, 004 Joural of Sou a Vibratio. New Data Aalysis Approach a Moelig of the Dampig i Sawich Structures. (submitte) [17] R. Carmoa, W. Hwag, B. Toeersai 1998 Practical Time-Frequecy Aalysis: Gabor a Wavelet Trasforms with a Implemetatio i S. Sa Diego: Acaemic Press. [18] G. V. Wella 003 Beyo Wavelets. Sa Diego: Acaemic Press. [19] T. K. Caughey 1960 Joural of Applie Mechaics Jue Classical Normal Moes i Dampe Liear Dyamic Systems.