Strojniški vetnik - Journal of Mechanical Engineering 55(009)5, StartPage-EndPage Paper received: 09.0.009 UDC 53/533 Paper accepted: 00.00.00x Spatial-Mode-Shape Identification uing a Continuou Wavelet Tranform Martin Čenik - Janko Slavič - Miha Boltežar * Univerity of Ljubljana, Faculty of Mechanical Engineering, Ljubljana, Slovenia Thi paper preent an experimental modal analyi of a damped multi-degree-of-freedom mechanical ytem uing a continuou wavelet tranform. An approximation of the wavelet tranform of the impule repone function i deduced, which erve a a bai for the extraction of the natural frequencie, the damping ratio and the mode hape. Due to an approximation with a finite Taylor erie, a computational error in the identified ocillatory amplitude occur and i oberved for the imulated ytem repone. The preented approach of modal identification i applied to real mechanical ytem, uch a a teel beam and the horizontal tail of an ultralight aircraft. Uing the propoed meaurement methodology, it i poible to recontruct the patial mode hape of any dynamic linear ytem with an arbitrary geometry. 009 Journal of Mechanical Engineering. All right reerved. Keyword: modal parameter, continuou wavelet tranform, patial vibration 0 INTRODUCTION One of the key tep in the development of a numerical model i it validation, where the modal analyi of a numerical model and the experimental modal analyi (EMA) of the real ytem are compared. With the reult of the comparion it i poible to determine the adequatene of the numerical model. The mot common way to undertake a modal analyi i by meauring the frequency repone-function (FRF) [] and []. Thi i done by ignal tranformation from the time domain to the frequency domain uing a Fourier tranformation. With the ue of a fat Fourier tranform (FFT) algorithm, the calculation of the FRF i a imple and ueful method ued to perform an EMA [3] to [5]. The decribed method i uitable for analyzing tationary ignal, ince it average the ignal in time domain. For an analyi of non-tationary ignal it i convenient to ue tranformation in the time or time-frequency domain, for intance, leatquare complex exponential (LSCE) [], continuou wavelet tranform (CWT) [6], etc. The impule repone of a mechanical ytem i a typical non-tationary proce where the ocillation amplitude decreae exponentially with time. Generally, the modal identification, baed on non-tationary ignal, i uually done with time-domain method [], e.g. LSCE for impule repone function or Ibrahim time domain (ITD) for free decay repone. The idea of applying a CWT to a dynamic ytem repone for damping identification wa firt introduced by Stazewki [7]. While Stazewki ued the Morlet wavelet function, Slavič et al. [8, 9, 0] developed their idea uing the Gabor wavelet function and focuing it on the edge effect and the relatively hort ignal. The propertie of the Gabor wavelet have been tudied in detail by Simonovki and Boltežar [], who alo ued it for fault detection in DC electro-motor []. Le and Argoul [3] extended the work of precendent reearcher and identified all the modal parameter by applying a CWT to the theoretical impule repone of a patial 4-degreeof-freedom (DOF) model with the Morlet, Cauchy and harmonic wavelet function. They alo explored the problem of the edge-effect and time-frequency localization. Furthermore, Lardie and Gouttebroze [4] applied a CWT to analyze real ignal meaured on a tower excited by the wind and determined it natural frequencie, damping ratio and one-dimenional mode hape. They compared the reult, obtained with a CWT to thoe obtained by an autocorrelation method. Their work wa extended by Huang and Su [5], who applied a CWT to dicrete equation of motion in order to identify the modal parameter of a teel frame ubjected to an earthquake excitation. A a reult, they obtained the frame * Corr. Author' Addre: Univerity of Ljubljana, Faculty of Mechanical Engineering, Aškerčeva 6, Ljubljana, Slovenia, miha.boltezar@f.uni-lj.i
Strojniški vetnik - Journal of Mechanical Engineering 55(009)5, StartPage-EndPage mode hape by a determination of the planar diplacement in a horizontal plane. To continue their work, Čenik et al. [6] tried to extract the patial mode hape of a mechanical ytem with an arbitrary geometry. Thi work repreent a continuation and enhancement of that reearch. In thi paper, a procedure for the identification of the modal parameter i introduced, baed on a CWT of the impule repone function. Due to the low-order approximation, a computation error of the ocillatory amplitude occur when dealing with ytem with high damping, which i preented in a numerical example. The method i alo teted on a real ytem of a teel beam The method i alo teted on a real ytem of a teel beam where the etimated mode hape are compared with the theoretical one. With the identification of the patial mode hape of the horizontal tail of an ultralight aircraft, the feaibility of the developed method i proven. CONTINUOUS WAVELET TRANSFORM The introductory part of CWT theory will be ummarized according to Mallat [6] and Tchamitchian et al. [7]. The continuou wavelet tranformation of the time function f(t), that atifie f(t) L (R) i defined a + * ψ u, () Wf ( u, ) = f ( t) ( t)d t, where ψ(t) repreent the wavelet function, the upercript * denote the complex conjugate, u i the tranlation parameter related to time and i the cale parameter that erve a an invere of the frequency. Fig. how the influence of the parameter u and on the wavelet function. The function f(t) L (R)may be treated a a mother wavelet function when atifying ocillatory, energy preervation and admiibility condition [6]. A modified wavelet function ψ u, (t), tranlated in the time domain and dilated in the frequency domain alo ha to atify the energy-preervation condition, and o we obtain t u ψu, () t = ψ, () ˆ i ω u ψ, ( ) ˆ u ω = e ψ ( ω ), (3) where ψˆu, ( ω ) denote the integral Fourier tranformation of the wavelet function ψ u, (t) in the frequency domain. The impule repone of a multi-degreeof-freedom (MDOF) ytem i expected to be a ummation of N mutually independent function pair [7] N i i () t () = A () t e ϕ, (4) f t i= i where A i (t) denote the amplitude and ϕ i (t) the phae (9). Therefore, the linearity property of the CWT [6] can be ued N N W αi fi ( u, ) = αi( Wfi)( u, ). (5) i= i=. Gabor Wavelet Function The Gabor mother wavelet function i defined a [6] t i (6) ψgabor ( t, σ, η) = e σ e η t. 4 πσ ( ) By introducing the tranlation and dilation parameter into (6), we obtain a family of Gabor wavelet function, defined in the time and frequency domain a [] ( t u) t u iη σ ψ, (,, u t σ η) = e e, 4, ( πσ ) ( ) ( ) η ω σ 4 i (7) ω u ψˆ u ω, σ, η = 4 π σ e e. (8) Fig.. ( ) mother ψ Gabor ( tu, = 0, = ) and ( ) modified ψ ( = = ) Gabor, u, tu, 4, 0.5 Čenik, M. - Slavič, J - Boltežar, M.
Strojniški vetnik - Journal of Mechanical Engineering 55(009)5, Startpage-EndPage. Approximation of the CWT of the Aymptotic Function The impule repone of a vicouly damped ingle-degree-of-freedom (SDOF) ytem can be written a [8] ζω0 t f () t = Ae co ( ω0d t+ ϕ0 ), (9) where ζ denote the damping ratio, ϕ 0 i the phae hift, ω 0 i the undamped angular frequency and ω 0d = ω 0 ζ i the damped angular frequency. In order to eparate the amplitude and the phae information of the repone function it can be written in a more general form a an analytic function [7] i f () t fa() t = Af () t e ϕ. (0) Similarly, a wavelet function may alo be given in the form of an analytic function a i () t a () t A () t e ϕ ψ = ψ ψ. With regard to the relation Wf ( u, ) = Wf ( u, ) [6] we can deduce that a + t u Wf ( u, ) = Af () t Aψ e i ϕf () t t u ϕψ d. t () In order to olve the equation (), a few approximation have to be made: the function S f (t) i approximated with a zero-order Taylor erie (i.e., a contant value A), and the function ϕ f (t) i approximated with a firt-order Taylor erie. By uing thee approximation, a general olution of the CWT can be deduced [8] ( ) ( f ( ) f ( )) ( f '', f ' ) i ϕ f ( u) Wf ( u, ) = Af ( u) e ψˆ ϕf '( u), η, σ + Er A '' u, ϕ ' u, where Er A ( u) ( u) () ϕ denote the error function due to the ue of the finite Taylor erie. On the bai of the linearity property of the CWT (5) we can apply an approximation () to multicomponent function, uch a the impule repone of the MDOF ytem (4) in order to identify the modal parameter of a particular natural frequency..3 Modal Parameter Identification Due to the nature of a CWT it i poible to tranform the meaured ignal in form of a time erie to the frequency domain without the lo of time-domain information. Furthermore, a CWT offer very good reitance to the noie in the meaured ignal. Regarding the nature of the CWT, it i poible to oberve the development of natural frequencie with time. In order to apply the CWT to modal analyi it i eential to analytically decribe the ditinctive propertie of the impule repone wavelet tranformation. The deduction of the impule-repone CWT will be preented according to Stazewki [7] and Slavič [8], who analyzed the Morlet and Gabor wavelet, repectively. From equation () we can ee that the eential contribution to the magnitude i provided by the tationary point of the argument of the integrand [6], i.e., the point t, uch that t u ϕf '( t) = ϕ ψ '. (3) The canonical pair ( u, ), which atify the condition t (u,)=u define the ridge of the wavelet tranform [7]. The ridge can be detected with different approache, and Stazewki preented three: the cro-ection method, the amplitude method and the phae method [7]. In thi work, the phae method, which i baed on a frequency match (3), i ued. In order to determine the ridge an initial etimation of the intantaneou frequency ω ha to be made. It i then entered into the recurive equation φ ( ui+, ηωi) φ( ui, ηωi) ωi+ =, (4) ui+ ui which gradually converge to an intantaneou frequency in the ignal (Fig.). The value of the wavelet tranform retricted to it ridge are called the keleton of the wavelet tranform; they erve for the modalparameter identification (6, 7). When conidering an impule repone function (9) an aumption of contant frequency can be made. Spatial-Mode-Shape Identification uing a Continuou Wavelet Tranform 3
Strojniški vetnik - Journal of Mechanical Engineering 55(009)5, StartPage-EndPage Fig.. Ridge detection uing the phae method with the recurive equation (4) Due to the linear relation between frequency and cale for the Gabor wavelet φ'(u,)=η/ we can conclude that the cale variable on the ridge (u)i contant (u)= 0. By introducing the Gabor wavelet function into equation (), regarding the frequency match and neglecting the error function, it follow that [6] 4 ζω0d u Wf ( u, 0) = Ae ( 4 πσ 0). (5) Finally, we can deduce equation for the direct identification of the damping ratio ζ and the initial amplitude A ( ln Wf ( u, 0 )) ζ = u cont., (6) ω0d ζω0d u e Wf ( u, 0 ) A = cont. (7) 4 4πσ η ω ( ( 0d ) ) are coniderably fater than the amplitude variation A A f f '( t) () t () t << ϕ '. (8) f When conidering ignal by which the aymptotic condition i not entirely atified, e.g., a highly damped impule repone, an error occur during the amplitude A identification. The preented error repreent a deviation of the recontructed amplitude A recontr from the real amplitude in the ytem repone A real. It can be oberved from a numerical example of a highly damped impule repone and depend on the damping ratio ζ and the parameter σ (Fig.3 and Fig.4); however, it i independent of the real amplitude value A real. Conequently, the ratio between the real amplitude A real and the recontructed amplitude A recontr which erve for the mode-hape recontruction, i contant A real A real = = cont. (9) A recontr A recontr Thi finding i important when a recontruction of the mode hape i made only by comparing the amplitude of the diplacement at different meaurement poition (Section.). Fig. 3. Recontructed initial amplitude at variable parameter σ and A real (t=0)=..4 A( t) Variation Influence on the Identified Amplitude An approximation for the theoretical CWT of the impule repone i deduced by approximating an impule repone (9) a a finite Taylor erie. An approximation of A(t) = cont. i exact only for aymptotic ignal, like the undamped repone, where the phae variation Fig. 4. Recontructed initial amplitude at Areal ( t = 0) = and A real ( t = 0) =. 4 Čenik, M. - Slavič, J - Boltežar, M.
Strojniški vetnik - Journal of Mechanical Engineering 55(009)5, Startpage-EndPage.5 Spatial Mode Shape Geometry In order to recontruct the patial mode hape the patial vibration have to be meaured with accelerometer at pre-defined meaurement poition. When performing a modal analyi on imple geometry uch a a teel beam (Section.), the repone accelerometer can be oriented in the ame direction and no iue arie from the geometry point of view. When analyzing a ytem with three-dimenional geometry, the orientation of the repone accelerometer in the ame direction a the reference one i not alway poible (Fig. 5). To overcome thi problem, the local coordinate ytem of each accelerometer i referenced to the global coordinate ytem of the meaured ample. One poibility of decribing thi linkage i with the Euler angle (Eq. 0 to ). Fig. 5. Problem of patial vibration meaurement To recontruct a ingle patial mode hape, the identified diplacement need to be tranformed from the local coordinate ytem (a meaured) into the global coordinate ytem with the ue of the rotational matrice 0 0 Rx, α = 0 co( α ) in( α ), (0) 0 in( α) co( α) ( ) ( ) co β 0 in β Ry, β = 0 0, () in( β ) 0 co( β ) ( ) ( ) ( γ) ( γ) co γ in γ 0 Rz, γ = in co 0, () 0 0 where α, β and γ denote the Euler rotation angle about the x, y or z axi, repectively..6 Identification of the Spatial Mode Shape The procedure to perform the experiment and recontruct the ingle patial mode hape i preented in the following tep: a) Experiment. Definition of the excitation poition.. Definition of the global coordinate ytem, n repone poition with n local coordinate ytem on the meaured ample geometry. 3. Selection of the reference poition and the reference direction (thi hould be the poition/direction where high ocillation of the identified mode are expected). 4. Excitation of the ytem with the impule hammer, acceleration meaurement on the reference poition/direction and on the repone poition() in the choen direction(). b) Recontruction 5. Identification of the ridge (4) and of the initial amplitude A ref j and A rep i,j (7) for the reference and repone ignal, repectively; j denote the conecutive number of the impule excitation and i i the number of the repone poition (i=,,n). 6. Computation of the normalized repone amplitude A norm i with repect to the reference amplitude A ref j with the ratio A norm i = A rep i,j / A ref j. 7. Tranformation of the normalized diplacement A norm i from the local to the global coordinate ytem (Eq. 0 to ). 8. Graphical preentation of the patialmode hape. Spatial-Mode-Shape Identification uing a Continuou Wavelet Tranform 5
Strojniški vetnik - Journal of Mechanical Engineering 55(009)5, StartPage-EndPage EXPERIMENT For a relatively imple teel beam, the preented approach for modal identification i experimentally compared to the beam theoretical mode hape. Furthermore, to demontrate an experimental identification of the patial modal analyi, the horizontal tail of an ultralight aircraft i preented. identification of global mode hape only negligible error are expected at the validation experiment. Eleven meauring poition were et along the beam: poition wa defined a the "reference" poition, and the remaining poition were defined a the "repone" poition, a hown in Fig. 6.. Experimental Validation In order to perform a pecific numerical manipulation, cutom program and algorithm were developed by the author. Cutom-made program offer a higher flexibility and provide the author with an inight into the computation proce. To check the correctne of the developed program a imple experiment with well-defined theoretical mode hape wa carried out. With algorithm that are verified on imple model, it i poible to perform a reliable modal analyi on more complex patial tructure. A imple validation experiment for method verification wa performed on a free-free upported teel beam with dimenion of 5x40x000 mm, excited by an impule hammer at a pre-defined "excitation" poition. The experiment wa made with everal impule excitation; during each excitation the repone accelerometer wa at a different poition, a decribed in the following. For the ake of experimental implicity, only two accelerometer were ued for the meaurement. One accelerometer wa denoted a the "reference" enor and wa alway placed in the "reference" poition. The econd accelerometer, denoted a the "repone" accelerometer wa placed at a different "repone" poition during each impule excitation. When carrying out the decribed experimental procedure a precaution hould be taken; due to changing of poition of the repone accelerometer a ma matrix of the whole beam ytem i changing. A a reult ome error of the recontructed mode hape and belonging natural frequencie could occur, epecially when uing accelerometer with coniderable ma and when identifying high-order mode hape with local nature. With application of a lightweight accelerometer with a ma of g and at Fig. 6. Meauring-poition placement on the analyzed beam After the whole et of meaurement wa carried out, we could obtain the impule repone from any pre-defined poition along the beam. Fig. 7. The validation-experiment configuration et-up With the ue of a CWT on the meaured repone it i poible to identify the modal parameter according to the theoretical backgrounddecribed in Section.3. The equation (4) i ued to define the time development of an oberved natural frequency. Once the value of the natural frequency i known, equation (6) and (7) are ued to define the damping ratio ζ of the ytem and the initial amplitude A at the location of the meaurement. Baed on the ratio between the diplacement at the reference and elected poition (Section.6), the firt nine normalized mode hape were recontructed and compared to the theoretical mode hape by computing the Modal Aurance Criterion MAC (Φ X, Φ A ) [], hown in Figure 8 and tated in Table. Table how the aociated natural frequencie f i and the damping ratio ζ i. 6 Čenik, M. - Slavič, J - Boltežar, M.
Strojniški vetnik - Journal of Mechanical Engineering 55(009)5, Startpage-EndPage Table. MAC(Φ X, Φ A ) matrix Mode 3 4 5 6 7 8 9 0.9994 0.000 0.085 0.0000 0.0765 0.000 0.074 0.000 0.0553 0.000 0.9996 0.0000 0.0750 0.0000 0.075 0.0000 0.0708 0.0000 MAC (Φ X, Φ A ) 0.0695 0.000 0.9995 0.000 0.075 0.0000 0.075 0.0000 0.0588 0.000 0.0700 0.0000 0.999 0.0000 0.076 0.000 0.0758 0.0000 0.067 0.0000 0.0709 0.0000 0.9995 0.000 0.078 0.0000 0.0668 0.000 0.070 0.0000 0.078 0.000 0.999 0.000 0.0839 0.0000 0.0683 0.0000 0.0706 0.0000 0.0757 0.000 0.9997 0.000 0.0885 0.000 0.069 0.000 0.076 0.000 0.0807 0.0000 0.999 0.0000 0.0554 0.0000 0.058 0.0000 0.0639 0.0000 0.085 0.000 0.9997 Table. Beam natural frequencie and damping ratio i.. 3. 4. 5. 6. 7. 8. 9. f i [Hz] 6.08 7.60 40.63 3.73 347.88 486.5 647.93 83.6 040.7 ζ i.0 4 3.567.895 0.649.3.035 0.978.50 0.808 0.566 Fig. 8. Modal aurance criterion matrix. Horizontal Tail Experiment Fig. 9. Horizontal tail in iometric view An experiment imilar to the one decribed in Section.. wa carried out on the horizontal tail of an ultralight aircraft. Since the tail geometry i decribed uing patial curve (Fig. 9), a patial vibration meaurement had to be performed. In order to reduce the experimental complexity and the number of nonlinearitie in the ytem, the tail wa upported free-free. Twenty-eight meauring poition were defined on the tail urface, a hown in Fig.0. Altogether, 8 local coordinate ytem were defined in uch a way that the z axi wa normal to the urface and the x axi followed obviou line in the tail geometry. A a reference, meaurement poition wa choen (Fig. 0 and ), oriented in the local direction z. Fig. 0. Meauring poition placement and the global coordinate ytem of the horizontal tail In each meaurement two ignal were ampled: - the acceleration at the reference poition in the reference direction, - the acceleration at the repone poition in one of the defined local axe. Altogether, 84 eparate meaurement were made (8 meauring poition in 3 direction). Spatial-Mode-Shape Identification uing a Continuou Wavelet Tranform 7
Strojniški vetnik - Journal of Mechanical Engineering 55(009)5, StartPage-EndPage b) Second mode hape Fig.. A horizontal tail with accelerometer placed at the reference poition (left) and at the i-th repone poition (right) In order to tranform the identified diplacement from the local to the global coordinate ytem, pecial oftware wa developed by the author, to which an arbitrary geometry can be imported. The geometrical characteritic are imported into the oftware in the form of a finite-element-method (FEM) meh that conit of volume element. The uer define the poition and the orientation of the local coordinate ytem and the correponding identified diplacement of the ingle mode hape. The oftware generate a cript which can be run in ANSYS and give a recontructed mode hape in the form of a tatic problem olution. The identified natural frequencie and the damping ratio are hown in Table, and the experimentally recontructed normalized mode hape are hown in Fig.. Table. Horizontal tail natural frequencie and damping ratio i f i [Hz] ζ i.0 4..8 63.7. 43.4 83.89 3. 75.5 4.8 c) Third mode hape Fig.. Recontructed normalized horizontal tail mode hape 3 DISCUSSION From the validation experiment, which erved a an ideal, real ytem, it i obviou that the identified mode hape agree with theoretical mode hape to a high degree, although ome change in ma matrix occured due to the diplacement of a repone accelerometer. With the imple beam experiment the applicability of the continuou wavelet tranform to determine the modal parameter i confirmed. A meaurement approach with the introduction of a reference accelerometer wa hown to be uitable. According to the experiment realization and the reult, performed and obtained in Section., the preented approach provide a imple and feaible method for an experimental determination of the modal parameter. Although a mechanical ytem, uch a a horizontal tail, i not an ideal ytem without nonlinearitie, the extraction of it modal parameter with a CWT wa hown to be a robut and effective method. 4 CONCLUSIONS a) Firt mode hape In thi paper a modal parameter identification of a dynamic ytem uing a continuou wavelet tranformation of ytem impule repone wa preented. The computation error due to the finite Taylor erie wa indicated 8 Čenik, M. - Slavič, J - Boltežar, M.
Strojniški vetnik - Journal of Mechanical Engineering 55(009)5, Startpage-EndPage and it influence on the mode-hape recontruction wa hown. Steel beam mode hape were recontructed and compared to the theoretical mode hape. The patial mode hape of a free-free upported horizontal tail of an ultralight aircraft were recontructed. A comparion among the theoretical and experimental mode hape of the beam confirmed the uitability of the propoed method for modehape recontruction. The main contribution of thi paper i an enhancement of known method for EMA uing a CWT for one-dimenional cae with a new method for the recontruction of patial mode hape for an arbitrary geometry. In thi cae it i a horizontal tail. The experimental reult can be ued for the validation of the numerical model. 5 ACKNOWLEDGEMENT The author would like to thank Pipitrel d.o.o. for providing the horizontal tail of the ultralight aircraft. 6 REFERENCES [] Maia, N.M.M., Silva, J.M.M., Theoretical and experimental modal analyi, Reearch Studie Pre, Someret, 997. [] Ewin, D.J., Modal Teting: Theory and Practice, Reearch Studie Pre, Lechtworth, 986. [3] Čermelj, P., Boltežar, M., An indirect approach to invetigating the dynamic of a tructure containing ball bearing, Journal of Sound and Vibration, 004, vol. 76, no. /, p. 40-47. [4] Čermelj, P., Boltežar, M., Modelling localied nonlinearitie uing the harmonic nonlinear uper model, Journal of Sound and Vibration, 006, vol. 98, no. 4/5, p. 099-. [5] Boltežar, M., Čermelj, P., Dynamic of complex tructure - valid modeling of indutrial product, In: Korelc, J., Zupan, D. (Ed.): Kuhljevi dnevi 007, Snovik, p. 7-4, 007. [6] Mallat, S., A wavelet tour of ignal proceing, Academic Pre, 999, econd edition. [7] Stazewki, W.J., Identification of damping in mdof ytem uing time-cale decompoition, Journal of Sound and Vibration, 997, vol. 03, p. 83-305. [8] Slavič, J., Simonovki, I., Boltežar, M., Damping identification uing continuou wavelet tranform: application to real data, Journal of Sound and Vibration, 003, vol. 6, p. 9-307. [9] Slavič, J., Boltežar, M., Enhanced identification of damping uing continuou wavelet tranform, Journal of Mechanical Engineering, 00, vol. 48, no., p. 6-63. [0] Boltežar, M., Slavič, J., Enhancement to the continuou wavelet tranform for damping identification on hort ignal, Mechanical ytem and ignal proceing, 004, vol. 8, no. 5, p. 065-076. [] Simonovki, I., Boltežar, M., The norm and variance of the gabor, morlet and general harmonic wavelet function, Journal of Sound and Vibration, 003, vol. 64, p. 545-557. [] Boltežar, M., Simonovki, I., Furlan, M., Fault detection in DC electro motor uing the continuou wavelet tranform, Meccanica, 003, vol. 38, no., p. 5-64. [3] Le, T.P., Argoul, P., Continuou wavelet tranform for modal identification uing free decay repone, Journal of Sound and Vibration, 004, vol. 77, p. 73-00. [4] Lardie, J., Gouttebroze, S., Identification of modal parameter uing free decay repone, International Journal of Mechanical Science, 00, vol. 44, p. 63-83. [5] Huang, C.S., Su, W.C., Identification of modal parameter of a time invariant linear ytem by continuou wavelet tranform, Mechanical Sytem and Signal Proceing, 006, vol., p. 64-664. [6] Čenik, M., Slavič, J., Boltežar, M., Modal parameter identification uing continuou wavelet tranform, In: Boltežar, M., Slavič, J. (Ed.): Kuhljevi dnevi 008, Cerklje na Gorenjkem, p. 4-48, 008. [7] Tchamitcian, P., Torreani, B., Aymptotic wavelet and gabor analyi: Extraction of intantaneou frequencie, IEEE tranaction on information technology, 99, vol. 38, p. 644-664. Spatial-Mode-Shape Identification uing a Continuou Wavelet Tranform 9