Journal of Software Engineering and Application, 011, 4, 67-681 doi:10.436/jea.011.41079 Publihed Online December 011 (http://www.scirp.org/journal/jea) Modelling and Wavelet-Baed Identification of 3-DOF Vehicle Supenion Sytem Khaled Nouri, Hichem Louifi, Naceur Benhadj Braie Laboratoire d Etude et Commande Automatique de Proceu (LECAP), Ecole Polytechnique de Tuniie, Univerité de Carthage, Tuni, Tuniia. Email: {haled.nouri, naceur.benhadj}@ept.rnu.tn, hichem.louifi@gmail.com Received October 1 t, 011; revied November 9 th, 011; accepted December 10 th, 011. ABSTRACT In thi paper, a three Degree of Freedom (DOF) model of a quarter vehicle upenion ytem i propoed including the eat driver ma. The modal parameter of thi ytem, which indicate the comfort and the afety of the upenion, are identified uing Wavelet analyi. Two application of wavelet analyi are preented: ignal denoiing baed on the Dicrete Wavelet Tranform (DWT) and modal identification baed on the Continuou Wavelet Tranform (CWT). It i hown that the CWT analyi of the ytem repone, initially denoied uing DWT, allow the etimation of the natural pulation and the damping ratio. The uefulne of the DWT in denoiing and the accuracy of the CWT in modal identification are teted and confirmed by applying them to the propoed model. The complete modeling and identification of a 3-DOF vehicle upenion ytem i developed and the imulation reult verify thee tatement and are atifactory. Keyword: Supenion Sytem, Dicrete and Continuou Wavelet Tranform, Signal Denoiing, Modal Identification 1. Introduction The importance of the upenion ytem of a vehicle i that it i reponible of the comfort of the driver and the afety of the vehicle ince it i the part of the vehicle which carrie the body of the vehicle and which tranmit all the force between thi body and the road. For that, the complete phyical model of the upenion i often reduced under invetigation of the vertical dynamic etablihed on a quarter-vehicle model. The tudy of thi model, and lie any mechanical ytem, preent challenge for engineer and reearcher. Noie reduction and parameter identification from ytem repone are common problem in ignal proceing and vibrational application [1-3]. That why numerou approache have been developed recently and applied to denoie ignal and to extract modal parameter of ytem and tructure in time and frequency domain [1]. The wavelet analyi i one of thee new approache that were revealed mainly becaue it repreent an eay way of extracting time varying frequency component. Indeed, the Wavelet Tranform technique i hown to be more effective than other denoiing method uch the Fourier Tranform (FT) or Compreed Sening technique [1], and more ueful than other identification method uch a Hilbert-Huang Tranform [4-6]. In noie reduction, wavelet ha been uccefully ued by Giaouri et al. [] in order to propoe a new peudoadaptive denoiing method baed on the Wavelet Tranform, thi method adjut the level of ignal decompoition and wa applied effectively to an electric drive. In ytem identification, reonant pulation and damping ratio are the mot difficult quantity to determine and require dynamic tet [3,4]. Numerou approache have been developed and applied to identify modal parameter of ytem and tructure in time-frequency domain uch a the WT [5,6] and the Hilbert-Huang Tranform (HHT) [7,8]. The objective of thi tudy i focue firt to propoe a model of a quarter vehicle upene ytem and then to identify the parameter of the propoed model of the conidered ytem uing WT. The different ection of thi paper are organized a follow: in ection II a complete modeling of a 3-DOF quarter upenion ytem i propoed with a dynamical tudy followed by a Simulin diagram of the conidered vehicle model. Beginning with ome mathematical fundament of the dicrete and the continuou WT, denoiing and the identification method are preented in ection III. The imulation reult are illutrated and dicued in Copyright 011 SciRe.
Modelling and Wavelet-Baed Identification of 3-DOF Vehicle Supenion Sytem 673 ection IV, and a concluion i drawn in ection V.. Propoed 3-DOF Model for the Vehicle Supenion The upenion ytem ha a big role in deign and dynamic of the vehicle. Indeed it goal i to maintain the comfort of the paenger and the tability of the vehicle by upporting the weight of the vehicle, by reducing the diturbance of road and by avoiding the road excitation experienced by the tire from being tranmitted to the paenger. The upenion ytem i generally reduced to a quarter vehicle model and the majority of the upenion tudie ue a model with two DOF. In our tudy, we propoe a three DOF model including the eat and drive ma ( m ), chai ma ( m ) and the tire ma ( c mt ). The propoed upenion ytem i modeled by interconnecting three ma-pring-damper ytem a hown in Figure 1. Auming that each ub-ytem i repreented by a ma connected to parallel arrangement of a linear pring and a linear vicou damper, the three DOF are the vertical motion of: the eat, the chai and the tire. The model hown in Figure 1 include for each ma a tiffne contant, a damping contant c and a vertical diplacement z. The vertical diplacement z r denote the change of road urface elevation. Auming that the tire doe not brea away from the road urface and that the diplacement are meaured form the tatic equilibrium poition, the linear equation of motion for the eat ma i: mz c z z z z (1) 0 c c Similarly, the equation of motion of the chai ma i given a: Figure 1. Propoed 3-DOF model of a quarter-vehicle upenion ytem. mc zccc z cz t c zczt c z z c zzc () and that of the tire ma i : mz c z z z z c z z z z t t t t r t t r c c t c c t (3) Thee three differential equation can be written in the matrix form a follow: MZ DZ S Z LU (4) where: m 0 0 c c 0 M 0 mc 0 ; D c c cc cc ; 0 0 m t 0 cc cc c t 0 S c c 0 c c t and z 0 0 z Z z ; 0 0 ; r c L U z z r t t c t M i the inertia matrix, D the damping matrix and S the tiffne matrix. From thee dynamical driving equation, we can derive the tate pace repreentation of the ytem by auming the tate pace variable are a follow: x1 z x zc x3 z X t x 4 z (5) x5 z c x6 z t A a reult, the tate pace repreentation of the propoed model i: X AX BU (6) Y CX DU where A i the tate matrix, Bthe input matrix, C the output matrix, D the tranmiion matrix, U the input of the ytem and Y it output. 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 A c c 0 0 m m m m c c c c cc c c mc mc mc mc m mc c c t cc cc ct 0 0 mt mt mt mt Copyright 011 SciRe.
674 Modelling and Wavelet-Baed Identification of 3-DOF Vehicle Supenion Sytem 0 0 0 0 0 0 B 0 0 ; 0 0 t c t mt m t 0 0 D 0 0 ; 0 0 z U r z r 1 0 0 0 0 0 C 0 1 0 0 0 0 0 0 1 0 0 0 From the equation of motion (1), () and (3) we can alo deduce the following diagram bloc of the propoed model (Figure ). 3. Wavelet Tranform Analyi Wavelet Tranform (WT) i conducted in a manner imilar to Short-Time Fourier Tranform (STFT). However, Wavelet Tranform i uperior in the ene that it window function, the analyzing wavelet, i caled (or dilated) in addition to being tranlated in time. Thi analyzing wavelet function i often called the mother wavelet becaue it give rie to a family of wavelet through the dilation and tranlation. A generalized wavelet family, decribed in the normalized form i: 1 t, t (7) where repreent the cale or the dilation parameter and repreent the time or the tranlation parameter. Otherwie, the wavelet function are able to adjut themelve a the tranform i computed for each component of the analyzed ignal: the cale parameter indicate the level of analyi. Small value of provide a local (or high frequency) analyi while large value correpond to large cale (or low frequency) analyi. Changing the parameter move the time localization center of each wavelet. Thee cale and tranlation parameter can vary in a dicrete or continuou way: it i what mae the difference between the Continue Wavelet Tranform (CWT) and Dicrete Wavelet Tranform (DWT). In thi tudy, we tated only ome ey equation and concept of each approach, more rigorou mathematical treatment of thi ubject can be found in [9,10]. 3.1. DWT for Denoiing For WT in the dicrete form, the widely ued cale and tranlation parameter etting that create an orthonormal bae are: j and j ( j, ). The wavelet family (7) then become: Figure. Diagram Bloc of the propoed 3-DOF Supenion Sytem. Copyright 011 SciRe.
Modelling and Wavelet-Baed Identification of 3-DOF Vehicle Supenion Sytem 675 j j j, x x (8) Analogou to the FT, the DWT calculate wavelet coefficient by taing the inner product of an input ignal f x with a function, that i in thi cae the wavelet family j,. DWT i deigned to give good time reolution and poor frequency reolution at high frequencie and good frequency reolution and poor time reolution at low frequencie. A ignal i decompoed uing the DWT into two et of coefficient called approximation coefficient repreent low-frequency ignal component defined by Vetterli et al. [11]: j j,, j, j d A f x f x x x (9) and detail coefficient repreent high-frequency ignal component defined by: j j,, j, j d D f x f x x j, m j1, m m j1, m h x(10) Thi decompoition, recurively repeated through filtering and down-ampling operation uing low-pa and high-pa filter (Figure 3), brea the ignal into lower level coefficient et. Without manipulating thee coefficient, the original ignal can be recontructed exactly through the invere DWT defined by: A h A g D m (11) where g and are repectively the high-pa and the low-pa filter (Figure 3). Manipulating wavelet coefficient prior to ignal recontruction change the original ignal. The original ignal can be modified, enhanced or denoied through variou coefficient manipulation operation. In our wor, the DWT will be ued to denoie the repone of the ytem on the road to identify hi modal parameter uing the CWT method. Figure 3. Signal decompoition uing DWT and recontruction uing invere DWT. 3.. CWT for Modal Identification The Continuou Wavelet Tranform (CWT) i defined a the convolution of the ignal x t and the caled-hifted verion ab, of an analyzing function called the mother wavelet:,,, *, * CWT x x t, t x t t dt (1) where i the complex conjugate of the analyzing function,. One of the mot widely ued mother wavelet for parameter identification i the well-nown modified Morlet wavelet defined by [1-14] : 0 t e e j t t N (13) where 0 i the central pulation of the Morlet mother wavelet and N i parameter introduced to control the hape of the baic wavelet intend to offer a better compromie in term of localization, in both time and frequency for a ignal. The optimal value of N i determined by minimization of the wavelet coefficient entropy [13] defined by: where Shannon m j log j WE N WC WC (14) m WC wc wc j j i i1 j1 and wc ; i 1, m i the et of wavelet coefficient energie. For a linear damped multi-dof ytem with real mode, the free-decay repone and impule repone can be ued for modal identification. Hence, a imulated freedecay repone with p mode i employed a the analytical ignal to invetigate the CWT baed time-frequency decompoition. Suppoe the ignal i given by: p p t 0 0 1 1 (15) xt () x A e co 1 t where A 0 i the amplitude of the th mode, 0 i the phae lag, i the damping ratio and i the natural pulation. Since the wavelet tranform are linear repreentation of the ignal, it follow that the CWT of thi multi-component ignal i: p p CWT ( x) CWT x CWT x,,, 1 1 p 1 * 0, j 10 1 A e e i (16) Thu, the CWT wor a a time-frequency filter and for a fixed value of the cale parameter, which maxi- Copyright 011 SciRe.
676 * mize, 1 Modelling and Wavelet-Baed Identification of 3-DOF Vehicle Supenion Sytem, only the mode aociated with give a ignificant contribution in the CWT, while all other are negligible: CWT, x A e e * 0 1 j 10 (17) Thee region where the amplitude of the CWT i maximum are called the ridge. By extracting the value of the wavelet coefficient along the ridge yield the eleton of the CWT. So, the CWT of each eparated mode i defined by a eleton expreed a follow: i CWT x A e (18), which implie that the CWT i able to decompoe a multicomponent ignal into eparated mode and to repreent them of a complex-valued ignal each one defined by amplitude A and a phae angle. Note that the phae angle of thi eleton i: 1 a b (19) where a i the lope of thi linear curve, and by applying logarithmic operator to the amplitude of the eleton, we obtain: ln A a b * ln A 0 A A 0 (0) Conequently, the natural pulation and the damping ratio can be defined by combining the lope of ln A a follow: and a a a A A (1) A pecialized program ha been developed on the MATLAB numeric computing environment for etimating thee parameter from the ytem repone. Thi program include the following procedure: Step 1: Calculate the wavelet entropy in order to determinate the optimal value of N correponding to the optimal Morlet wavelet. Step : Tranforming the time ignal (repone of the ytem) into time-cale domain uing the CWT. Step 3: Detecting ridge and extracting correponding eleton. Step 4: Calculate modal parameter according to equation (1) by leat-quare interpolation 4. Simulation In order to how the uefulne the WT in upenion tudy, the propoed 3-DOF upenion ytem wa created uing Simcape blocet library of Matlab oftware and the correpondent Simulin model hown on Figure 4 wa implemented with the imple configuration given in Table 1. In the imulation tudy, the propoed model i exited Figure 4. Simulin model of the propoed 3-DOF upenion ytem. Copyright 011 SciRe.
Modelling and Wavelet-Baed Identification of 3-DOF Vehicle Supenion Sytem 677 Table 1. Parameter configuration of the imulated model. Symbol Decription Value m ma of eat 00 Kg m c ma of chai 00 Kg m t ma of tire 00 Kg pring contant of eat 00 10 3 N m 1 c pring contant of chai 400 10 3 N m 1 t pring contant of tire 800 10 3 N m 1 c damping contant of eat 30 N m 1 c c damping contant of chai 0 N m 1 c t damping contant of tire 10 N m 1 by an impule ignal and three cae of identification parameter are developed and illutrated. 4.1. Cae 1: Identification from Original Repone The original repone ignal i hown in Figure 5. For thi repone he CWT of wa firtly calculated for N 1 a hown in Figure 6 and Figure 6 give a croection of the CWT at different time. According to thi time-cale repreentation, we cannot ee clearly the three mode of the conidered ytem. For thi reaon, we have to calculate the variation of the wavelet entropy. By increaing N from 1 to 50, the minimum value i obtained for N = (Figure 6(c)), and the optimized CWT of the repone correponding to thi value i hown in Figure 7 in which the three mode of the conidered ytem can be eaily oberved. According to the lat tudy, each mode can be iolated by ridge extraction and ued to etimate natural pulation and damping ratio. Figure 7 give the three related ridge repreentation at different time, from thi plot we can deduce the cale parameter correponding to each mode: the firt mode i localized in the neighborhood of 1 6, the econd mode neighbor to 44 and the third mode neighbor to 3 104. The retriction of wavelet coefficient to each obtained ridge give the correponding eleton. Figure 8 how the plot of the real part (Figure 8) and the imaginary part (Figure 8) of the eleton related to the firt mode. And a previouly etablihed by Equation (1), an interpolation of the phae (Figure 8(c)) and the modulu (Figure 8(d)) of thi firt eleton allow the etimated natural pulation and the damping ratio of the firt mode uing a linear leat-quare fit procedure. Similarly, Figure 9 and Figure 10 preent the identification of the other modal parameter. The identification reult are lited in Table and how that CWT method correctly identifie the natural pulation and the damping ratio correponding to the three Figure 5. Original repone of the upenion ytem. (c) Figure 6. CWT of the original repone ignal. a) CWT for N = 1; b) Cro ection at different intant for N = 1; c) Minimization of the wavelet entropy. Copyright 011 SciRe.
678 Modelling and Wavelet-Baed Identification of 3-DOF Vehicle Supenion Sytem Figure 10. Identification of the eleton correponding to the 3 rd mode. a) Real part; b) Imaginary part; c) Phae; d) Envelop. Figure 7. CWT of the original repone ignal. a) CWT for N = ; b) Cro ection at different intant. Figure 8. Identification of the eleton correponding to the 1 t mode. a) Real part; b) Imaginary part; c) Phae; d) Envelop. Figure 9. Identification of the eleton correponding to the nd mode. a) Real part; b) Imaginary part; c) Phae; d) Envelop. mode. We note that the damping ratio i lightly le accurate than the natural pulation becaue it value i etimated from that previouly etimated for the pulation which caue a propagation of error. 4.. Cae : Identification from Noied Repone In thi cae, two white Gauian noie with Signal-to- Noie Ratio (SNR) equal to 0 db and db were added to the original repone in order to imulate CWT performance under noiy condition. Signal with the two level of noie are hown on Figure 11 and Figure 1. The ame developed identification procedure i applied (Figure 13). By minimization of the wavelet entropy, the optimal value of N i the ame a in the firt cae. The CWT ha been applied uccefully at the two noie level. Figure 14 give the identification of the firt mode in the cae of SNR equal to db. The reult are not a good a in the cae without noie, but they are acceptable a hown in Table. 4.3. Cae 3: Identification from Denoied Repone In thi cae, the DWT-baed denoiing proce wa applied to the previou two noied ignal uing Symlet wavelet and the trouble component affecting the ignal were deleted a hown in Figure 15. Alo in thi cae the identification wa made following the ame procedure. The reult are imilar to the previou and that i why only the identification of the firt mode i repreented in Figure 16. Table preent a comparion between actual and etimated parameter. In order to highlight the reult obtained in each cae, it ueful to introduce a imple meaure of the error etimation for a given modal parameter etimated etimated via the CWT and compared to the exact value actual. Therefore, the Mean Abolute Error (MAE) which i an average of the abolute error i actual etimated deduced in a cae with the three modal parameter of the ytem. Thi MAE i given by () and i introduced at the lat column Table. Copyright 011 SciRe.
Modelling and Wavelet-Baed Identification of 3-DOF Vehicle Supenion Sytem 679 Figure 13. CWT of the noied repone (SNR = db). a) CWT; b) Cro ection at different intant. Figure 11. Noied repone of the upenion ytem (SNR = 0 db). Figure 14. Identification of the eleton correponding to the firt mode (SNR = db). a) Real part; b) Imaginary part; c) Phae; d) Envelop. Table. Identification reult of the 3-Dof Supenion Sytem. Figure 1. Noied repone of the upenion ytem (SNR = db). Modal 1 t mode nd mode 3 rd mode parameter Natural ω (rad 1 ) 83.67 50.09 1.6 pulation MAE Damping ζ (%) 1.55.5 0.79 Ratio Original ω (rad 1 ) 83.673 50.095 1.686 4.77E-03 repone ζ (%) 1.5568.593 0.7905 5.53E-03 Noied ω (rad 1 ) 83.6701 50.091 1.700 3.77E-03 repone SNR = 0 db ζ (%) 1.560.5188 0.7797 1.03E-0 Noied ω (rad 1 ) 83.6697 50.075 1.698 8.30E-03 repone ζ (%) 1.5749.519 0.8013 1.40E-0 SNR = db Denoied ω (rad 1 ) 83.6744 50.0976 1.669 6.30E-03 repone SNR = 0 db ζ (%) 1.5564.5198 0.7905 4.60E-03 Denoied ω (rad 1 ) 83.676 50.0981 1.688 7.70E-03 repone SNR = db ζ (%) 1.5568.587 0.7919 5.87E-03 Copyright 011 SciRe.
680 Modelling and Wavelet-Baed Identification of 3-DOF Vehicle Supenion Sytem Figure 15. Denoiing of the noiy repone (SNR = ) uing DWT with Symlet wavelet. a) Noiy ignal; b) Denoied ignal; c) Deleted noie. Figure 16. Identification of the eleton correponding to the firt mode of the denoied repone (SNR = db). a) Real part; b) Imaginary part; c) Phae; d) Envelop. 3 actual etimated () 1 1 MAE 3 The imulation reult obtained give a very accurate etimation of the actual parameter. Thi verifie the effectivene of the denoiing and identification method baed on DWT and CWT for the propoed model of upenion ytem. 5. Concluion We have propoed in thi paper a 3-DOF model of a quarter-vehicle upenion. After a dynamic tudy and the modeling of the ytem, we have invetigated the performance of the Wavelet analyi in the identification of pulation and damping parameter of the ytem. Due to it time-cale repreentation of ignal, and the location of a mode by a ridge and the correponding eleton, the CWT wa able to identify the three mode of upenion ytem in all cae. In the cae of poor reolution, the firt two mode were not ditinguihable, for thi a compromie between frequency and temporal reolution ha been etablihed by calculating the entropy of wavelet coefficient, and wa able to optimize the CWT to better locate thee particular mode. The important advantage of thi approach i that it i not too enitive to noie. Indeed, for all different noie level, the modal identification wa made with low error rate. The imulation reult obtained underlined the accuracy and the efficiency of the developed method in the cae where the repone ignal are denoied by the DWT, and even in the preence of noie. Finally, we can conclude that wavelet are a powerful tool for the modal analyi in vibration application epecially for upenion ytem. REFERENCES [1] L. Zhu, Y. Zhu, M. Mao and M. Gu, A New Method for Spare Signal Denoiing Baed on Compreed Sening, Second International Sympoium on Knowledge Acquiition and Modeling, Second International Sympoium on Knowledge Acquiition and Modeling, Wuhan, 30 November-1 December 009, pp. 35-38. [] D. Giaouri, J. W. Finch, O. C. Ferreira, R. M. Kennel and G. M. El-Murr, Wavelet Denoiing for Electric Drive, IEEE Tranaction on Indutrial Electronic, Vol. 5, No., 008, pp. 543-550. doi:10.1109/tie.007.911943 [3] H. S. Hu, J. Wang, S. X. Qian and X. Z. Jiang, Tet Modeling and Parameter Identification of a Gun Magnetorheological Recoil Damper, International Conference on Mechatronic and Automation, Changchun, 9-1 Augut 009, pp. 3431-3436. [4] N. Amann, J. Böcer and F. Prenner, Active Damping of Drive Train Ocillation for an Electrically Driven Vehicle, IEEE/ASME Tranaction on Mechatronic, Vol. 9, No. 4, 004, pp. 697-700. doi:10.1109/tmech.004.839036 [5] S.-L. Chen, J.-J. Liua and H.-C. Laia, Wavelet Analyi for Identification of Damping Ratio and Natural Frequencie, Journal of Sound and Vibration, Vol. 33, No. 1-, 009, pp. 130-147. doi:10.1016/j.jv.009.01.09 [6] J. Slavic, I. Simonovi and M. Boltezar, Damping Identification Uing a Continuou Wavelet Tranform: Application to Real Data, Journal of Sound and Vibration, Vol. 6, No., 003, pp. 91-307. doi:10.1016/s00-460x(0)0103-5 [7] D. S. Laila, A. R. Meina and B. C. Pal, A Refined Hilbert-Huang Tranform with Application to Interarea Copyright 011 SciRe.
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