Response of the beams on random Pasternak foundations subjected to harmonic moving loads

Transcription

1 Journal of Mechanical Science and Technology 3 (9) 33~33 Journal of Mechanical Science and Technology DOI.7/ Repone of the beam on random Paternak foundation ubjected to harmonic moving loa Davood Youneian,* and Mohammad Hoein Kargarnovin School of Railway Engineering, Iran Univerity of Science and Technology, Tehran Iran Department of Mechanical Engineering, Sharif Univerity of Technology, Tehran Iran (Manucript Received October 6, 8; Revied June 3, 9; Accepted Augut 3, 9) Abtract Dynamic repone of infinite beam upported by random vicoelatic Paternak foundation ubjected to harmonic moving loa i tudied. Vertical tiffne in the upport i aumed to follow a tochatic homogeneou field coniting of a mall random variation around a determinitic mean value. By employing the firt order perturbation theory and calculating appropriate Green function, the variance of the deflection and bending moment are obtained analytically in integral form. To imulate the induced uncertainty, two practical cae of coine and exponential covariance are utilized. A frequency analyi i performed and influence of the correlation length of the tiffne variation on the beam repone are invetigated. It i found that in each frequency repone there i a peak value of frequency, which behave a a decreaing function of the correlation length. Among two coefficient of variation of the beam deflection and the bending moment, the former i higher in the cae of exponential covariance and it i independent of the magnification of the correlation length. Keywor: Timohenko beam; Moving load; Random vibration; Vicoelatic foundation Introduction Thi paper wa recommended for publication in revied form by Aociate Editor Eung-Soo Shin * Correponding author. Tel.: , Fax.: addre: Youneian@iut.ac.ir KSME & Springer 9 The invetigation on the conequence of a force moving along an infinite beam reted on vicoelatic foundation i of great theoretical and practical ignificance, in particular it application in the modeling of railway track [, ]. In mot of the cae and for implicity, the track parameter have been aumed to have a determinitic characteritic equation, but recently mechanical propertie of the ballat have been found to vary ignificantly along the track. The uncertainty of the track parameter and urface irregularity of the rail are two important ource of random vibration in railway track and they lead to vibration of the track even if no other external excitation i applied. Many reaon uch a non-uniformity in packing the ballat and alo ome natural and operational ource can eaily caue random ditribution of the track propertie along it length. Becaue of thi random ditribution, coefficient of the differential equation of motion become random function of the poition and a the moving load i travelling on the foundation, the randomne of the foundation parameter caue an exceive vibration. Since thi exceive vibration arie from an internal ource it i called parametric excitation. In the literature, a few tudie have been carried out and ome related problem have been treated numerically and analytically in recent year. Dynamic repone of an Euler-Bernoulli beam reting on a Winkler random foundation ha been obtained analytically in [3] under an idealitic aumption of the white noie pectral denity for the uncertainty of the foundation tiffne. Behavior of an infinite Euler-Bernoulli beam on a Kelvin foundation

2 34 D. Youneian et al. / Journal of Mechanical Science and Technology 3 (9) 33~33 with randomly varying parameter along the beam i examined in [4]. The author ued finite element method and found that randomne of the foundation tiffne i of greater importance than the uncertainty in the damping. They alo tudied the influence of the load peed on the beam repone. Interaction of an imperfect track and a moving random load uing the method of integral pectral decompoition wa tudied in [5, 6] tudied. They howed that imperfection could not be modeled by white noie. The repone of a ingle-degree-of-freedom vehicle moving uniformly along an Euler-Bernoulli beam reting on a modified Kelvin foundation with randomly ditributed tiffne ha been tudied in [7, 8]. They howed that the vertical motion of a moving vehicle and the beam deflection increae with the vehicle peed and alo with the correlation length of the tiffne variation. A comprehenive invetigation on the literature indicate that there are few work publihed in thi area and lack of a frequency analyi i quite clear. Vibration of infinite beam on variou type of upport coniting of linear, nonlinear and uncertain vicoelatic foundation ubjected to harmonic moving loa wa tudied in [9-]. In the preent paper, uing the firt order perturbation method, the repone analyi of an infinite Timohenko beam on the vicoelatic foundation under a harmonic moving load i carried out. To imulate the behavior of the foundation, Paternak vicoelatic model i ued. Thi model include a Kelvin foundation in conjunction with a hear elatic layer which can model the track more realitically [7, 8, 3]. Baed on thi model, the coefficient of variation of the beam deflection and bending moment at the point of the application of the moving force are obtained. Uing the complex Fourier tranformation, appropriate Green function are preented, and the mean and variance part of the repone of the beam are calculated analytically in the integral form. Uing the reidue theorem, for two practical cae of random foundation, a parametric tudy i carried out and the effect of correlation length on the coefficient of variation are tudied. A frequency analyi i performed a well. The olution and alo the parametric tudy are directed to make the outcome of thi paper more applicable in different branche of railway engineering uch a noie analyi of track, paenger comfort analyi of railway vehicle, dynamic and fatigue deign of track in which the frequency repone ha an important role. Fig.. Timohenko beam on random Paternak foundation.. Theory and formulation Fig. illutrate a beam on a random vicoelatic foundation under a harmonic moving load. By uing Hamilton principle and employing the Timohenko beam theory, one can obtain the differential equation of the motion a ([9-]) wxt (,) φ(,) xt ρ A k AG( wxt (,) ) () P ( xt, ) = FeiΩtδ( x vt) f φ( x, t) w( x, t) EI k AG( φ( x, t) ) () φ( xt, ) = ρi in which A, E, G, I, k* and ρ are cro-ectional area of the beam, the modulu of elaticity, hear modulu, econd moment of area, ectional hear coefficient and beam material denity and v, Ω, w, and φ are the load peed and frequency, beam deflection and beam lope due to bending, repectively. Moreover, Pf repreent the force induced by the foundation per unit length of the beam. In the cae of a random vicoelatic foundation it can be calculated a: Pf ( x, t) = k( x) w( x, t) wxt (, ) wxt (, ) µ c (3) in which, k (x) and µ are the foundation tiffne and the hear layer tiffne and c i damping coefficient of the foundation. The foundation tiffne can be decompoed into it contant mean value, km, and correponding tochatic component, k (x): kx ( ) = k ε k ( x) = k k( x) (4) m m in which k * (x) i a random tationary ergodic func-

3 D. Youneian et al. / Journal of Mechanical Science and Technology 3 (9) 33~33 35 tion with zero mean value and ε i the mall parameter. Uing the regular perturbation method one can aume [, 4]: wxt (, ) = w( xt, ) ε w( xt, ) ε w ( x, t)... φ( x, t) = φ( x, t) εφ( x, t) ε φ ( xt, )... (5) (6) Subtituting Eq. (5) and (6) in Eq. () and () and comparing the term with the ame power of parameter ε, one can get: w (,) x t ( xt, ) ρ A kmw φ (,) x t w (,) x t k AG( ) w ( x, t) w ( x, t) µ c = Fe i Ωt ( ) δ x vt φ ( x, t) w ( x, t) EI k AG( φ ( x, t) ) = φ ( xt, ) ρi (7) w(,) x t ρ A k (, ) mw x t φ (,) x t w(,) x t k AG( ) w(,) xt w(,) xt µ c ) = k (, ) (, ) xtw xt φ (,) xt w (,) xt EI k AG( φ ( x, t) ) = (8) φ ( xt, ) ρi To calculate the teady tate repone of the beam we ue the Galilean co-ordinate tranformation uch a: = x vt (9) Since the beam length i conidered to be infinite, the boundary condition are dwt () d wt () Lim wt () = Lim = Lim = ± ± ± dφt () Lim φt () = Lim = ± ± () in which w t () and φ t () are teady tate decription of w (x,t) and φ (x,t) and conequently can be written a w () = w () w w φt () = φ t () φ () ε φ () ε... t t t () ε t () ε... t t () () The above boundary condition repreent the fact that at point far enough from the point of the load application the central line deflection, it lope, curvature, hear force and bending moment are all approaching to zero. According to the Timohenko beam theory, the bending moment along the beam i given by φ () M () = EI = M () t t t t () t ε() ε... M M Uing the tate variable tranformation of i Ω t wj St() = wj() e ; wj() = wja() e iω t φj t() = φj() e ; φj() = φja() e iψ j ( ) iθ j ( ) (3) (4) In which, for j=,,, w ja and ψ j are the amplitude and the phae of the teady tate repone, w j and imilarly φ ja and θ j are the amplitude and the phae of φ j. After utilizing the chain rule differentiation on the Eq. (7) and (8) and conidering Eq. (4), one can get d d ρ( v iωv Ω ) w ( ) dφ() d w() d w() k AG( ) µ d c( v iω ) w() kmw() = Fδ () d φ() dw() EI k AG( φ ( ) ) = d d ρiv ( iωv Ω ) φ ( ) (5)

4 36 D. Youneian et al. / Journal of Mechanical Science and Technology 3 (9) 33~33 d d ρ( v iωv Ω ) w ( ) dφ() d w() d w() k AG( ) µ d c( v iω ) w() kmw() = k() w() d φ() dw() EI k AG( φ ( ) ) = d d ρiv ( iωv Ω ) φ ( ) (6) After applying the following complex Fourier tranformation, i.e., Eq. (7) to Eq. (5) and impoing the boundary condition of Eq. (), Fourier tranform of w () and φ () i.e. W (q) and Φ (q) are obtained a: iq F( q) = f( ) e ( B q B q B3 ) F W ( q) = 4 3 B4 q B5 q B6 q B7 q B8 Φ ( q) = ( B q) F B4 q B5 q B6 q B7 q B8 (7) (8) (9) General definition of all the above coefficient are lited in Table. If an invere Fourier tranform i taken from both ide of Eq. (8) and (9) one obtain: w M iq F ( Bq Bq B3) e dq () = 4 3 π B4q B5q B6q B7q B8 () iq F ( Bq ) e dq () = 4 3 π B4q B5q B6q B7q B8 () Uing a imilar procedure for Eq. (6) and uing appropriate Green function and the convolution integral theorem, the cloed form olution are obtained for j=,, a wj() = k() ξ wj ( ξ) Gw( ξ) dξ M j( ) = k( ξ ) wj ( ξ) GM( ξ) dξ () in which Green function G w (ξ) and G M (ξ) are defined a iξ q Bq Bq B3 e dq ( ) Gw( ξ ) = π (3) B q B q B q B q B GM ( ξ ) = π (4) B q B q B q B q B iξ q Bq e dq All B i coefficient in the above Green function are defined in Table. It hould be noted that the reidue theorem i ued for calculating the Green function and alo the numerical value of the dynamic repone. In thi way G w can be calculated a n r Gw( ξ) = πi [Re ( Integrand ( Gw( ξ)))] z= z (5) n j = j r = πi [Re ( Integrand ( Gw( ξ))) ] z= z for ξ j n r Gw( ξ) = πi [Re ( Integrand ( Gw( ξ)))] z= z (6) n k = j r= πi [Re ( Integrand ( Gw( ξ))) ] z= z for ξ < in which, z j repreent the pole of the integrand of G w in the upper half and z k repreent it pole in the lower half of the complex plane and z jr repreent it real pole. Moreover, the G M i obtained in a imilar trend. From Eq. () and (), the mean value of the beam deflection and bending moment are: Table. Definition of the coefficient appeared in related Green function. Parameter Definition B EI ρiv B Ω vρi B 3 k AG ρiω B 4 ( EI ρiv )( k AG ρ Iv µ ) B 5 EI( Ωvρ A civ) Ωvρ I ( k AG µ ) 3 ρivc 4Ωvρ IA ρav ( ρiω k AG) B 6 Ω vρi( Ω vρa civ) ( k ciω Ω ρa)( EI ρiv ) µ ( k AG ρiω ) k AGρIΩ B 7 ( ΩvρA civ)( k AG ρiω ) Ω vρi( k ciω Ω ρa) B 8 ( k ciω Ω ρa)( k AG ρiω ) B 9 k AGi B k AGEI k j r j r

5 D. Youneian et al. / Journal of Mechanical Science and Technology 3 (9) 33~33 37 Ew [ ( )] = Ew [ ( )] Ew [ ( )] ε t t t Ew [ t ( )] ε... EM [ ( )] = EM [ ( )] EM [ ( )] ε t t t EM [ t ( )] ε... (7) (8) Since the mean value of the tochatic part of the tiffne i zero, Eq. () implie that the mean value of the higher perturbation term are all zero, i.e., o Ew [ j t( )] = j=,,... (9) EM [ j t( )] = j=,,... (3) Ew [ t ( )] = Ew [ t ( )] (3) EM [ ( )] = EM [ ( )] (3) t t The centered value of the deflection and bending moment are: w ˆ () = w () Ew [ ()] w () ε (33) t t t Mˆ () = M () E [ M ()] M () ε (34) t t t With regard to the definition of the covariance of a random function [5] and Eq. (-), the covariance of the deflection and the bending moment are obtained a Cov (, ) = E[ ε w ( ) ε w ( )] = (35) ww t t w( ξ ) w( ξ ) G ( ξ) G ( ξ ) Cov ( ξ, ξ ) dξξ w w kk Cov (, ) = E[ ε M ( ) ε M ( )] = (36) MM t t M ( ξ ) M ( ξ ) G ( ξ ) G ( ξ ) M M Cov ( ξ, ξ ) dξξ kk the reult are dicued in thi ection. Thee covariance function are in the form of: ξ ξ Cov kk ( ξ, ξ) = σ exp( ) D (39) ξ ξ Cov kk ( ξ, ξ) = σ co( π ) D (4) where σ i the tandard deviation of the random tiffne and D i the correlation length. The numerical reult are related to the coefficient of variation for the deflection (C w ()) and the bending moment (C M ()) at the point of application of the moving force that i (=) σ w() Cw() = (4) Ew [ ( )] = σ M () CM () = (4) EM [ ( )] = Before the main parametric tudy and in order to verify the numerical procedure, a pecial cae of white noie i aumed for the covariance of the tiffne. The numerical imulation i carried out in the vicinity of the reonance (f=9 Hz) for the nondamped foundation. Analytical reult for the Euler- Bernoulli beam ubjected to the harmonic moving point load wa obtained in [3]. The obtained reult for the Timohenko beam are compared with thoe found in [3] in Fig.. A illutrated, very good correlation can be een between the two approache. in which Cov kk ( ξ,ξ ) i the covariance of the foundation tiffne. Then, the variance of the deflection and the bending moment are σ w() = Covww(,) (37) σ () = Cov (,) (38) M MM 3. Numerical reult For two practical type of foundation tiffne expreed in the form of exponential and coine covariance function, a parametric tudy i carried out and Fig.. Coefficient of variation veru the load peed in the vicinity of reonance (pecial cae of the white noie ha been aumed for covariance of the tiffne).

6 38 D. Youneian et al. / Journal of Mechanical Science and Technology 3 (9) 33~33 Table. Propertie of the UIC6 rail, track and load [9,, 6]. Item Notation Value Rail (UIC6) Young modulu (teel) E GPa Shear modulu (teel) G 77 GPa Ma denity ρ 785 kg/m 3 Cro ectional area A m Second moment of area I m 4 Shear coefficient k*.4 Foundation Mean tiffne k m.66 MN/m Vicou damping c 4.65 kn/m Stiffne variance σ.8 4 N /m 4 Moving Load Load F 65 kn Fig. 4. Frequency effect on the ditribution of the tandard deviation of the bending moment (v= km/hr) (Coine covariance). Fig. 3. Frequency effect on the ditribution of the tandard deviation of the deflection (v= km/hr) (Coine covariance). Fig. 5. Frequency effect on the ditribution of the tandard deviation of the deflection (v= km/hr) (Exponential covariance). For a real track [7], a frequency analyi i carried out uing the above two precribed covariance function. The phyical and geometrical propertie of the track are lited in Table. The effect of the load frequency on the ditribution of the tandard deviation of the beam deflection and bending moment are illutrated in Fig 3-6 for the two abovementioned cae of coine and exponential covariance. A can be een, by increaing the load frequency the maximum tandard deviation increae up to a point and then decreae. Moreover, it make the ditribution wider along the beam and caue the poition of the peak value move farther back with repect to the point of application of the moving load. Fig. 6. Frequency effect on the ditribution of the tandard deviation of the bending moment (v= km/hr) (Exponential covariance).

7 D. Youneian et al. / Journal of Mechanical Science and Technology 3 (9) 33~33 39 To compare two theorie of the Timohenko and Euler beam, the ame procedure ha been done for the Euler beam with the governing equation of 4 wxt (,) ρ A EI wxt (,) kw(,) x t 4 wxt (,) wxt (,) C µ = FeiΩtδ( x vt) (43) Numerical reult for the Euler beam are alo included in Fig 3-6. A een, one can generally ay that the magnitude of the maximum deflection of a Timohenko beam i larger than that of an Euler beam, and inverely the magnitude of the maximum bending moment of a Timohenko beam i lower than that of an Euler beam. Having a viewpoint focued on the energy tranfer, one can conclude that for a track modeled by a Timohenko beam on Paternak foundation larger fraction of the input. The frequency repone of the coefficient of variation of the deflection and bending moment are illutrated in Fig. 7 and 8. There are two frequencie at which C w () and C M () are at their highet level imilar to the frequency repone of an S.D.F ytem. The peak frequency of the coefficient of variation of the deflection i higher than the bending moment and it decreae with increaing of correlation length a well a the coefficient of variation of the bending moment. In the cae of exponential covariance, the correlation length ha much lower effect on the frequency repone (Fig. 9 and ). Alo can be een, for the cae of exponential covariance the coefficient of variance of the deflection i higher than that of the Fig. 8. Frequency repone of the beam bending moment at = for variou correlation length (D) (v= km/hr) (Coine covariance). Fig. 9. Frequency repone of the beam deflection at = for variou correlation length (D) (v= km/hr) (Exponential covariance). Fig. 7. Frequency repone of the beam deflection at = for variou correlation length (D) (v= km/hr) (Coine covariance). Fig.. Frequency repone of the beam bending moment at = for variou correlation length (D) (v= km/hr) (Exponential covariance).

8 3 D. Youneian et al. / Journal of Mechanical Science and Technology 3 (9) 33~33 Fig.. Relationhip between the coefficient of variation of the deflection at = and the correlation length (v= km/hr) (Coine covariance). Fig.. Relationhip between the coefficient of variation of the bending moment at = and the correlation length (v= km/hr) (Coine covariance). Fig. 3. Relationhip between the coefficient of variation of the deflection at = and the correlation length (v= km/hr) (Exponential covariance). Fig. 4. Relationhip between the coefficient of variation of the bending moment at = and the correlation length (v= km/hr) (Exponential covariance). bending moment, but in the cae of coine covariance it depen on the magnification of the correlation length. For the cae of coine covariance the influence of the correlation length on the C w () and C M () i illutrated in Fig. and for three different value of frequencie around the peak frequency. Saturation phenomenon that i getting independence from correlation length at high value of D, can be een for both C w () and C M (). In the cae of C M (), there i a peak value for D, which i a decreaing function of the frequency value and ha a value between to meter. Similar reult can be een for the cae of exponential covariance in Fig 3 and 4. In thi cae there i no peak value of D and alo the aturation phenomenon i taking place at much lower value of correlation length. Energy i tranferred to the foundation rather than the beam becaue of the larger deflection. But in the cae of Euler beam on the Paternak foundation, due to larger value of the bending tree, larger fraction of the input energy i tranferred into the beam (againt the Timohenko beam theory). On the other hand, Euler theory ha afer margin for deigning of the beam, and Timohenko theory ha afer margin for deigning of the foundation. To give more phyical explanation and application of the numerical reult, it hould be noted that the obtained reult may be eaily ued for real track ubjected to moving train. Since all of the term of the repone (w j ) are linear function with repect to the magnitude of the moving load, i.e., F (Eq. ()), the reult can be eaily extended for the number of moving force. It mean that the uperpoition principle here i till acceptable and therefore ucceive moving loa which well model the train loading

9 D. Youneian et al. / Journal of Mechanical Science and Technology 3 (9) 33~33 3 condition can alo be imulated with the ame approach. The ame comparion wa performed for the coefficient of variation of the two beam theorie, Timohenko and Euler beam. Since the coefficient of variation i a dimenionle parameter and i defined by Eq. (4-4), the type of the elected theory ha no ignificant effect on the coefficient of variation. Thi i becaue any change of the theory lea to the ame magnitude of variation both in denominator and numerator of the Eq. (4) and (4). 4. Concluion The repone of an infinite beam upported by Paternak foundation with randomly ditributed tiffne ubjected to harmonic moving loa wa invetigated. The mean value and variance of the beam deflection and bending moment were calculated by uing the firt order perturbation method. Two practical cae of coine and exponential covariance were ued to imulate the uncertainty of the foundation tiffne and a frequency analyi wa carried out for the firt time in thi paper. The influence of the correlation length on the repone were tudied a well. Concluion are a follow: () Uing Green function approach, a cloed-form olution in integral form applicable in the frequency analyi of track wa obtained. The preented method and alo the reult are applicable in ome related fiel in railway engineering uch a noie analyi of track, ride comfort of railway vehicle and alo dynamic and fatigue deign of railway track. () Uing the olution method preented, the ditribution of the tandard deviation of the deflection and bending moment along the beam were calculated and the influence of the load frequency on both of them were invetigated. Increaing the load frequency caue the poition of the maximum value to move farther back with repect to the load poition and alo make them to be ditributed in a wider area along the beam. (3) The parametric tudy howed that in each frequency repone there i a peak value for the frequency which depen on the correlation length. The peak frequency i a decreaing function of the correlation length. Dependency of the peak frequency on the correlation length i baed on the covariance type of the foundation tiffne and for the cae of coine covariance it i much tronger than exponential covariance. Moreover, it ha a higher value for the deflection than the bending moment. (4) For the cae of exponential covariance, a comparion between two dimenionle parameter of the coefficient of variation of the beam deflection and bending moment howed that the former one i higher. Regarding the unavailability of a device for direct meaurement of the bending moment, thi reult doe have a ignificant advantage in deign of railway track becaue it how that uing a typical meaured coefficient of variation of the beam deflection i a afe criterion in deign of track baed on the beam trength. (5) With increaing the correlation length, the coefficient of variation of the beam deflection and bending moment remain contant; thi phenomenon, i.e., aturation, doe occur for the two elected cae of exponential and coine covariance of the tiffne. For the exponential covariance, the aturation occur in lower correlation length other than the coine covariance. In the latter cae, there i a peak value of correlation length between to meter at which coefficient of variation of the bending moment i at it highet level and i a decreaing function of the load frequency. (6) Numerical reult obtained for the two theorie of Euler and Timohenko beam were compared. It wa found that the magnitude of the maximum deflection of a Timohenko beam i larger than that of an Euler beam and inverely the magnitude of the maximum bending moment of a Timohenko beam i lower with repect to the Euler beam. One can conclude that Timohenko and Euler theorie provide afer margin, repectively, for deign of the foundation and the beam. Nomenclature A : Cro-ectional area C : Foundation damping Cov : Covariance D : Correlation length E : Modulu of elaticity F : Moving force G : Shear modulu I : Cro-ectional moment of inertia k : Foundation tiffne k* : Sectional hear coefficient of the beam k m : Mean tiffne M : Bending moment

10 3 D. Youneian et al. / Journal of Mechanical Science and Technology 3 (9) 33~33 P f : Foundation force per unit length : Ditance from the moving load t : Time v : Load peed w : Beam deflection x : Longitudinal poition δ : Dirac Delta function µ : Shear layer tiffne ρ : Beam material denity σ : Standard deviation φ : Beam lope due to bending Ω : Load frequency Reference [] Y. Y. Chen and Y. H. Huang, Dynamic characteritic of an infinite and finite railway to moving loa, Journal of Engineering Mechanic, 9 (9) (997) [] O. R. Jaiwal and R. N. Iyengar, Dynamic repone of railway track to ocillatory moving mae, Journal of Engineering Mechanic, 3 (7) (997) [3] L. Fryba, Vibration of oli and tructure under moving loa, Thoma Telford, London, (999). [4] L. Fryba, S. Nakagiri and N. Yohikawa, Stochatic finite element for a beam on a random foundation with uncertain damping under a moving force, Journal of Sound and Vibration, 63 () (993) [5] J. Naprtk and L. Fryba, Interaction of a long beam on tochatic foundation with a moving random load, Proc. of the Conference: Structural Dynamic: Recent advance, Intitute of Sound and Vibration Reearch, Southampton, (993) [6] J. Naprtk and L. Fryba, Stochatic modelling of track and it ubtructure In: interaction of railway vehicle with the track and it ubtructure, Vehicle Sytem Dynamic, 4 (995) [7] L. Anderen and R. K. Nielen, Vehicle moving along a beam on a random modified Kelvin foundation, Proc. of Eighth Int. Congre on Sound and Vibration ICSV8, Hong Kong (). [8] L. Anderen and R. K. Nielen, Vibration of track caued by variation of the foundation tiffne, Probabilitic Engineering Mechanic, 8 (3) [9] M. H. Kargarnovin and D. Youneian, Dynamic repone analyi of Timohenko beam on vicoelatic foundation under an arbitrary ditributed harmonic moving load, Proc. of Fifth Int Conference on Structural Dynamic, Munich (). [] M. H. Kargarnovin and D. Youneian, Dynamic of Timohenko beam on Paternak foundation under moving load, Mechanic Reearch Communication, 3 (4) [] M. H. Kargarnovin, D. Youneian, D. J. Thompon and C. J. C. Jone, Repone of beam on nonlinear vicoelatic foundation to harmonic moving loa, Computer and Structure, 83 (5) [] M. H. Kargarnovin, D. Youneian, D. J. Thompon and C. J. C. Jone, Parametrically excited vibration of a Timohenko beam on vicoealtic foundation ubjected to a harmonic moving load, Nonlinear Dynamic, 45 (-) (6) [3] J. Ocaron, Dynamic train/track interactionlinear and non-linear track model with property catter, Ph.D thei, Department of Solid Mechanic, Chalmer Univerity of Technology, Goteborg (). [4] A. H. Nayfeh and D. Mook, Nonlinear Ocillation. John Wiley & Son, New York (995). [5] Solne J., Stochatic Procee and Random Vibration, John Wiley & Son, New York (997). [6] J. A. Zakeri and H. Xia, Senitivity Analyi of Track Parameter on Train- Track Dynamic Interaction, Journal of Mechanical Science and Technology, (7) (8) [7] R. George, Railway ballat quality monitoring, Ph.D thei, Intitute of Sound and Vibration Reearch ISVR, Southampton (3). Davood Youneian received hi MSc and PhD in Mechanical Engineering, both from Sharif Univerity of Technology, Iran. He joined Iran Univerity of Science and Technology in 5 a an aitant profeor in the School of Railway Engineering. Dr Youneian reearch area i mainly focued on non-linear and random vibration, optimal control of vibration, dynamic and vibration of tructure and railway vehicle ytem. He ha publihed more than 75 paper in international journal and conference proceeding in the area of hi reearch.

11 D. Youneian et al. / Journal of Mechanical Science and Technology 3 (9) 33~33 33 Mohammad. H Kargarnovin received hi MSc from Maachuett Intitute of Technology (MIT) and hi PhD from Renelaer Polytechnic Intitute (U.S.A), both in Mechanical Engineering. He currently act a a full profeor in Sharif Univerity of Technology, Department of Mechanical Engineering. Profeor Kargarnovin reearch area i mainly focued on olid mechanic (elaticity, platicity) and vibration of continuou ytem. He ha publihed more than paper o far in international journal and conference proceeding in hi reearch field.