Non-stationary Random Vibration Analysis of Vehicle with Fractional

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1 13 th National Conference on Mechanim and Machine (NaCoMM7), IISc, Bangalore, India, December 1-13, 7 NaCoMM-7-77 Non-tationary Random Vibration Analyi of Vehicle with Fractional Damping Yunqing Zhang 1, Wei Chen 1 Liping Chen 1, Wenbin Shangguan 1 Center for Computer-Aided Deign, Huazhong Univerity of Science & Technology, Wuhan, Hubei, China College of Automotive Engineering, South China Univerity of Technology,Guangzhou, Guangdong, China * Correponding author ( zhangyq@hut.edu.cn) Abtract The vicoelatic damping i one of key factor that affect vehicle vibration. It ha been demontrated that fractional derivative model can accurately model thi kind of frequency dependence damping very well. Generally, the excitation of rough road can be acted a a tationary random vibration proce in time domain when a vehicle i traveling at contant peed. But the road roughne hould be conidered a a non-tationary random vibration proce when traveling at variable peed. In thi paper, the vehicle non-tationary proce i invetigated by uing the fractional damping model. The road roughne i built uing PSD (power pectral denity) in pace domain by Laplace tranform. The propoed method i validated by imulating a vehicle running with variable peed in time domain. The vehicle non-tationary random repone in frequency domain are alo acquired by FFT( Fat Fourier Tranform) algorithm. Keyword: non-tationary, random vibration, fractional damping, PSD 1 Introduction The road profile, the only input of the paive upenion ytem, ha a ignificant effect on the vehicle vibration. Hence in order to generate a uitable road profile, it wa invetigated very carefully [1]. There are many poible way to decribe the road input. Generally, the road roughne i conidered a a random proce in pace domain and if the velocity i contant, it can alo be conidered a a tationary random proce in time domain. The tationary random vibration method wa adopted by many reearcher to invetigate the vehicle vibration repone [, 3]. The road roughne wa acquired by a white noie ignal with certain power pectral denity (PSD). While previou tudy all aumed that the vehicle were running at certain contant peed. However, the vehicle alway travel at variable peed, epecially when tarting, accelerating and breaking. Therefore, the vibration caued by rough road urface hould be conidered a a non-tationary random proce [4, 5]. Fang [6] ha contructed the model for two type of evolutionary random excitation: one may be obtained by filtering a tationary random proce through a linear time-dependent ytem, while another may reult from a nonlinear tranformation of the argument of a tationary random proce. Baed on the model, the unified expreion of the repone evolutionary power pectra are derived through the complex modal analyi. Hammond and White [7] tarted with a tationary proce and then ditorted the independent variable o that the proce i tretched out or compreed o a to create a non-tationary proce. Sobczyk [8] invetigated tationary repone to profile impoed excitation with randomly varying peed uing perturbation method. Sun [8] ued tate pace approach and the tranfer function approach to tudy the tranient phenomenon ubjected to a non-tationary vibration. Baed on tranient tranfer function, Zhang [4] invetigated a new time domain method by uing the non-tationary excitation model of a rough road. In [5], intead of the conventional FFT ( Fat Fourier Tranform) method, the continuou wavelet tranform a well a the dicrete wavelet tranform were applied to tudy the non-tationary input and repone of the vehicle vibration ytem. Quarter-car model i very widely ued to imulate the repone of the vehicle ubjected to non-tationary vibration by many reearcher. The damping force wa alway modeled a a force which i proportional to the firt derivative of the relative diplacement between the prung ma and the tire. However, in recent year, a number of paper have hown that vicoelatic damping can be modeled uing fractional calculu, which can more accurately model the vicoelatic damping behavior, and can take frequency dependency into account. Caputo and Mainardi [1] introduced a memory-baed damping model and howed that fractional derivative and Mittag-Leffler function could be ued to model vicoelatic phenomena. Roikhin [11] howed that fractional derivative could more accurately model the vicoelatic behavior of the damping. Bagley [1, 13] howed that a fractional damping relationhip could be ued to predicted tranient tructural repone. In thi paper, baed on tranfer function method and the work done by Zhang [4], a new way for vehicle nontationary random vibration reearch i propoed. A quarter-car model with fractional damping wa built to analyze 171

2 13 th National Conference on Mechanim and Machine (NaCoMM7), IISc, Bangalore, India, December 1-13, 7 NaCoMM-7-77 the repone of the vehicle ubjected to the excitation of the road profile. Baed on Laplace tranform, the fractional order calculu wa approximated by rational function. The paper i organized a follow: Section introduce the fractional order calculu. In Section 3, the quarter-car model with fractional damping i introduced. Section 4 decribe the contruction of the non-tationary road roughne. The imulation of non-tationary vibration repone i conducted and illutrated in Section 5. Section 6 offer our concluion and further tudy. Fractional Order Calculu The idea of fractional order calculu ha been known ince the development of regular calculu, with the firt reference probably being aociated with Leibniz and L Hopital in Even though the idea of fractional order operator i a old a the idea of the integer order. In the lat decade, the ue of fractional order operator and operation ha become more and more popular among many reearch area, uch a in phyical chemitry, electronic, mechanic, automatic control, robotic, ignal proceing, et al [14]. Fractional calculu i a generalization of ordinary differentiation and integration to arbitrary order (non-integer). The mot common definition of a fractional derivative i through Riemann-Liouville integral [15]: N 1 t d f( τ ) cdt f() t = dτ Γ N c N N dt (1) + ( t τ ) ( ) 1 order of the derivative, { 1,, 3, } Where N + 1, N. Here i the arbitrary = K and the Γ() de- n 1 t note the gamma function defined a ( n) t e dt for n >. Normally the lower integration limit c i choen to be zero. The fractional order derivative definition i conitent with the definition of integer derivative when. Modeling of vicoelatic behavior of damping normally reult in the order of the time derivative being between zero and one and enabling the above equation to be rewritten in a impler form: 1 t d f( ) cdt f() t = τ dτ, 1 ( 1 c Γ ) dt ( t τ) () For convenience, Laplace domain notion i uually ued to decribe the fractional integro-differential operation. The formula for the Laplace tranform of the Riemann- Liouville fractional derivative ha the form Γ = n 1 t k k 1 t = t k = e D f() t dt F( ) D f() (3) for ( n 1 n ) where F() = L[f(t)] i the normal Laplace tranform. 1 If k Dt f() =, k =,1,, K n 1 then L Dt f() t = F( ) (4) According to Outaloup [14], it i poible to obtain ueful approximation to calculate the fractional calculu by uing of a frequency-band real non-integer differentiator: H() + =, (5) 17 Where i a real non-integer. We give high and low tranitional frequencie ω h andω b, and the unit gain frequency and the central frequency of a band of frequencie ω u ( ωu = ωω h b ) geometrically ditributed around it. The above function (5) can be approximated by a rational function: N ˆ ω 1 / () ( u ) n + H = with n = ωh k= N1 + / (6) uing the following et of ynthei formula:.5 ω = ω (7) u ω = ω (8).5 u = = η > 1 (9) + 1 = η > (1) ω = > k (11) log( ωn / ω ) N = log( η) (1) log µ = log( η) (13) More detail about the approximated rational function can be referred to [14]. 3 Quarter-car Model with Fractional Damping Quarter-car model ha been widely ued for upenion analyi and deign by many reearcher, becaue it i imple and can capture many important characteritic of the full model, and provide a uitable framework to invetigate upenion control concept [16]. k m m t k t z c z t z r Figure 1: Quarter-Car Model The quarter-car model, Figure 1, conit of a prung ma upported on a upenion ytem which ha tiffne and damping characteritic and the upenion ytem i connected to the unprung ma. The tire i modeled a a pring. Traditionally, the tiffne characteritic wa modeled a a force which i proportional to the relative diplacement between the prung ma and the tire, and the damping characteritic wa modeled a a force which i proportional to the firt derivative of the relative diplacement between the prung ma and the tire. However, in

3 13 th National Conference on Mechanim and Machine (NaCoMM7), IISc, Bangalore, India, December 1-13, 7 NaCoMM-7-77 recent year, a number of paper have hown that vicoelatic damping can be modeled by uing fractional calculu more accurately. Generally, the dynamic equation of motion for the quarter model are given a follow: m && z + k ( z z ) + c ( z& z& ) = (14) t t mt && zt + kt( zt zr) k ( z zt) c ( z& z& t) = (15) Where m i the prung ma, which repreent the car chai, mt i the prung ma, which repreent the wheel aembly, k and c are tiffne and damping of the uncontrolled upenion ytem, repectively; k t i the vertical tiffne of the tire, z i the diplacement of the prung ma, z t i the diplacement of the tire, z r i the road diturbance. when fractional damping i introduced, the equation can be written a follow: m && z + k ( z zt) + c Dt ( z zt) = (16) mt && zt + ( zt zr) k ( z zt) c Dt ( z zt) = (17) Where D t repreent the fractional differentiation operator, i the calculu order, can be any real number. In practice, the value for hould be obtained by parameter identification method baed on material experiment data [17]. While in thi paper, the author jut want to illutrate a new procedure for non-tationary reearch baed on the fractional damping model and i et to /3. 4 Simulation of Road Roughne One of the mot ueful tool to decribe the tationary road roughne i the power pectral denity (PSD). When a car move at a contant velocity u, the road roughne can be viewed a a tationary proce in pace domain, and the PSD of the road diturbance input can be expreed by n - G ( ) ( )( ) w q n = Gq n (18) n Where Gq( n ) i the road PSD, n i the patial frequency. The reference patial frequency n can be defined by n =.1( cycle/ m), Gq( n ) i the road roughne coefficient, which i the value of PSD at the reference patial frequency n, and repreent different grade of road, w i called wavine and indicate whether the road ha more long wavelength ( w i large) or hort wavelength ( w i mall). The wavine w i found within the range 1.75 w.5, generally w =. When introduce patial angular frequency Ω : Ω= π n (19) The equation (18) can be written a: G W ( ) ( )( ) w q W= G - q W W () Where G ( W ) i the road PSD, Ω i the patial angular q frequency ( rad/m ). W i the reference patial angular frequency, and W= pn. Gq( W ) i alo the road roughne coefficient, which i the value of PSD at the reference patial angular frequency Ω. w ha been mentioned before, and ha the ame definition. When the vehicle drive at a contant velocity u, the relationhip between the time frequency f and the vehicle forward velocity u i defined by: f = u n (1) Derived from equation (1), the PSD of ground diplacement ha the following general form Gq( n) nu Gq( f) = () f From the equation (), we can get theoretic ground PSD under different frequencie. The relationhip between the patial angular frequency Ω ( rad/m ) and the angular velocity ω (rad/) can be defined a follow: ω = uω (3) So the road roughne in frequency domain can be decribed a the follow: Gq( ω) = Gq( Ω )/ u = Gq( Ω ) u/ ω (4) However, it may be trouble at ω =, Gq ( ω ) =. An improved equation for PSD of road roughne in frequency domain i hown a, Gq( ω) = Gq( Ω ) u/( ω + ω) (5) Where ω i the lowet cut-off angular frequency, ω = π f = πun. Equation (5) can be conidered a a repone of a firt order linear ytem to white noie excitation. Baed on the theory of random vibration, the following relationhip can be obtained, Gq( ω) = H( ω) Sw( ω) (6) Where H ( ω ) i the tranfer function, and S w i the PSD of white noie, where normally Sw( ω ) = 1. So from the equation (5) and (6) the tranfer function H ( ω ) can be decribed a: H ( ω) = Gq ( Ω ) u ω + jω (7) According to Laplace tranform, the above equation can be written a: H() = Gq ( Ω ) u ω + (8) The equation (8) can be viewed a the tranfer function from white noie ignal to road roughne. From the above equation, we can get the following equation: z& r( t) + ωzr( t) = Gq( Ω) uw( t) (9) Where zr () t i the road roughne, wt () i a white noie ignal whoe power pectral denity i 1. Becaue ω = π f = πun, the above equation can be hown a: z& r() t + πunzr() t = Gq( Ω) uw() t (3) From equation (3), the road roughne can be obtained. The numerical calculation for the road roughne wa car- 173

4 13 th National Conference on Mechanim and Machine (NaCoMM7), IISc, Bangalore, India, December 1-13, 7 NaCoMM-7-77 ried out by Matlab/Simulink, a can be een in Figure. Figure : Simulink Block to Generate the Stationary Road Roughne Figure 3 how the time coure of the tationary road roughne when vehicle drive at m/ on Grade C road. FFT method i ued to get the PSD from the time domain to the frequency domain. Figure 4 how that the imulation road PSD fit the theoretical value very well, which illutrate the effectivene of the method to get a tationary road roughne. Figure5b: Stationary Repone of the Supenion Figure 3: Time Coure of Stationary Road Figure 4: PSD of Stationary Road Roughne Figure5c: Stationary Repone of the Tire Figure5: Stationary repone (m/) A can be een from the vehicle tationary repone, figure 5, the body, the upenion and the tire repone to the tationary road are very regular and the maxim or minimum magnitude of all the repone almot remain contant. When a vehicle drive at variable peed, the equation (3) can be expreed a: z& r() t + πutnz () r() t = Gq( Ω) utwt () () (31) A ame a the way to get a tationary random proce, a non-tationary random proce can be obtained from equation (31). The initial peed of the vehicle i m/, and then the vehicle will accelerate with the acceleration m/ and 3 m/, repectively. Figure 6 and Figure 8 how a velocity coure of a non-tationary road. From Figure 6-9, we can ee the road and the PSD of the no-tationary road roughne under the ued acceleration exhibit imilar variation proce. Figure5a: Stationary Repone of the Body Figure 6: Velocity Coure of Non-Stationary Road ( m / ) 174

5 13 th National Conference on Mechanim and Machine (NaCoMM7), IISc, Bangalore, India, December 1-13, 7 NaCoMM-7-77 Figure 7: PSD of Non-Stationary Road Roughne ( m / ) Figure 1: The Flowchart of the Sytem in Matlab/Simulink 5.1 Time domain analyi Figure 8: Velocity Coure of Non-Stationary Road (3 m / ) Figure 9: PSD of Non-Stationary Road Roughne (3 m / ) The initial peed of the vehicle i m/. The vehicle accelerate to 3 m/ with acceleration m/ and 3 m/, repectively. Figure 11 and Figure 1 how repone to the non- tationary road roughne with vehicle acceleration m/ and 3 m/, repectively. From Figure 11 and Figure 1, Compare to the tationary repone, it can be een that the amplitude of the vehicle body acceleration, the upenion travel diplacement and the tire deflection all increae with peed. In fact, vehicle alway drive at variable peed, o it i not accurate to imulate the vehicle repone ubjected to tationary excitation, while auming the vehicle drive at a contant peed. Comparing Figure 1 to Figure 11, we can een the periodicity of all the repone under acceleration 3m/ i maller than the reult under acceleration m/. Namely, it take le time for a vehicle to reach ame velocity with a big acceleration than that with a mall acceleration. Furthermore, from the two imulation reult, the RMS (root mean quare) of the body acceleration i.149 for acceleration m/, and the RMS of the body acceleration i.165 for acceleration 3 m/. So we can conclude that, the vibration i more everely when the vehicle drive with a larger acceleration. 5 Simulation of Non-tationary Repone The quarter-car model with fractional damping wa built with Matlab/Simulink. The Figure 1 how the flowchart of the ytem. The parameter of the quarter-car model ued in the numerical calculation are given a follow: m = 8kg, m = 36kg, k = 16 N / m, t kt = 16 N / m, c = 98 N / m. The fractional order of the damper i et to /3. Figure11a: Non-Stationary Repone of the Body 175

6 13 th National Conference on Mechanim and Machine (NaCoMM7), IISc, Bangalore, India, December 1-13, 7 NaCoMM-7-77 Figure11b: Non-Stationary Repone of the Supenion Figure1c: Non-Stationary Repone of the Tire Figure 1: Non-Stationary Repone (3m/ ) 5. Frequency domain analyi Figure11c: Non-Stationary Repone of the Tire Figure 11: Non-Stationary Repone (m/ ) In general, the ride comfort i frequency-enitive. From the ISO361 [3], the human body i very enitive to vertical vibration in the frequency range of 4-8 Hz. Hence, it i neceary to evaluate the upenion ubjected to nontationary road excitation in frequency domain. The nontationary random repone in frequency domain of the vehicle are acquired by FFT( Fat Fourier Tranform) Algorithm. Alo, the vehicle initial peed i m/ and it will accelerate with acceleration m/, 3 m/ and 4 m/,repectively. Figure 13 how the frequency repone of the body acceleration z&&, the upenion travel z zu, and the tyre diplacement z u. From Figure 13, we can ee there are two repone peak for a quarter-car model and the maximum repone value will increae with acceleration increaing. Of coure, the quarter-car repone i inenitive to the frequency range of 4-8 Hz. Figure1a: Non-Stationary Repone of the Body Figure13a: Frequency repone of the Body Figure1b: Non-Stationary Repone of the Supenion Figure13b: Frequency Repone of the Supenion 176

7 13 th National Conference on Mechanim and Machine (NaCoMM7), IISc, Bangalore, India, December 1-13, 7 NaCoMM-7-77 Technology, Vol. 3, No. 3,, pp [5] Y. Wang, C. M. Lee and L. Zhang, Wavelet analyi of vehicle nontationary vibration under correlated fourwheel random excitation, International Journal of Automotive Technology, Vol. 5, No. 4, 4, pp Figure13c: Frequency Repone of the Tire Figure 13: Frequency Repone of Supenion 6 Concluion Thi paper propoed a new method for analyzing vehicle non-tationary random vibration. A quarter-car model with fractional damping wa built to analyze the repone of the vehicle ubjected to the non-tationary excitation of the road profile. Baed on Laplace tranform, the fractional order calculu wa approximated by rational function. The comparion of the no-tationary vibration imulation reult to the reult of tationary vibration imulation reult indicated the propoed method i effective. However, the model built in the work i a linear ytem. The nonlinear ytem can be conidered in the future. The model adopted in thi paper i a quart-car model with two degree of freedom. A much more complicated vehicle model with more freedom will be built to analyze the repone ubjected to non-tationary excitation. The parameter identification method can alo be ued to obtain the accurate fractional damping model in the future reearch. Reference [1] D. Hrovat, Survey of advanced upenion development and related optimal control application, Automatica, Vol. 33, No. 1, 1997, pp [] Y. Zhang, Y. Zhao, J. Yang and L. Chen, Dynamic Sliding Mode Controller with Fuzzy Adaptive Tuning for Active Supenion Sytem, (in pre) Proc. Intn. Mech. Engr., Part D: Journal of Automobile Engineering. [3] H. Chen, Z. Liu and P. Sun, Application of contrained H control to active upenion ytem on halfcar model, ASME, Journal of Dynamic Sytem, Meaurement, and Control, No. 17, 5, pp [4] L. Zhang, C. M. Lee and Y. Wang, A tudy on nontationary random vibration of a vehicle in time and frequency domain, International Journal of Automotive [6] T. Fang, M. Sun, A unified approach to two type of evolutionary random repone problem in engineering, Archive of Applied Mechanic, Vol. 67, No. 7, [7] J. K. Hammond, P. R. White, The analyi of nontationary ignal uing time-frequency method, Journal of Sound and Vibration, Vol. 19, No. 3, 1996, pp [8] K. Sobczyk, D. B. Macvean and J. D. Robon, Repone to profile impoed excitation with randomly varying traveral velocity, Journal of Sound and Vibration, Vol. 5, No. 1, 1977, pp [9] L. Sun, F. Luo, Nontationary Dynamic Pavement Load Generated by Vehicle Traveling at Varying Speed, Journal of Tranportation Engineering, Vol. 133, No.4, 7, pp [1] M. Caputo and F. Mainardi, A new diipation model baed on memory mechanim, Pure and Applied Geophyic, No. 91, 1971, pp [11] Y. A. Roikhin and M. V. Shitikova, Application of fractional calculu to dynamic problem of linear and nonlinear hereditary mechanic of olid, Applied Mechanic Review, Vol. 5, No. 1, 1997, pp [1] R. L. Bagley and P. J. Torvik, Fractional calculu in the tranient analyi of vicoelatically damped tructure, AIAA Journal, Vol.3, No. 6, 1985, pp [13] R. L. Bagley, R. A. Calico, Fractional order tate equation for the control of vicoelatically damped tructure, Journal of Guidance Control Dynamic, Vol. 14, No., 1991, pp [14] A. Outaloup, F. Levron and B. Mathieu, Frequencyband Complex Noninteger Differentiator: Characterization and Synthei. IEEE Tranaction on Circuit and Sytem Ⅰ:Fundamental Theory and Application, Vol. 47, No.1,, pp

8 13 th National Conference on Mechanim and Machine (NaCoMM7), IISc, Bangalore, India, December 1-13, 7 NaCoMM-7-77 [15] K. B. Oldham and J. Spanier, The Fractional Calculu, 1974, Academic Pre, New York. [16] M. E. Mohamed and S. A. Zuhair, Linear Quadratic Gauian Control of a Quarter-Car Supenion, Vehicle Sytem Dynamic, No.3, 1999, pp [17] Buggich, P. Mazilu and H. Weber, Parameter identification for vicoelatic material, Rheol Acta No.7,1998,pp