Vibrations of a track caused by variation of the foundation stiffness

Transcription

1 L. Andersen and S.R.K. Nielsen / Sumitted to Proailistic Engineering Mechanics Virations of a track caused y ariation of the foundation stiffness Lars Andersen and Søren R.K. Nielsen Department of Ciil Engineering, Aalorg Uniersity, Sohngaardsholmsej 57, DK-9 Aalorg, Denmark Astract The paper deals with the stochastic analysis of a single-degree-of-freedom ehicle moing at constant elocity along a simple track structure with randomly arying support stiffness. The track is modelled as an innite Bernoulli-Euler eam resting on a Kelin foundation, which has een modied y the introduction of a shear layer. The ertical spring stiffness in the support is assumed to e a stochastic homogeneous eld consisting of a small random ariation around a deterministic mean alue. First, the equations of motion for the ehicle and eam are formulated in a moing frame of reference following the ehicle. Next, a rst-order perturation method is proposed to estalish the relationship etween the ariation of the spring stiffness and the responses of the ehicle mass and the eam. Numerical examples are gien for arious parameters of the track. The response spectra otained from the perturation analysis are compared with the a numerical solution, in which nite elements with transparent oundary conditions are used. The circumstances, under which the rst-order perturation approach proides satisfactory results, are discussed. Keywords: Uncertain parameters; Perturation approach; Finite elements. Introduction A road, runway or railway track is often modelled as a eam structure on a Kelin foundation. In itself this model only weakly descries the real situation of a track resting on a susoil. Howeer, a reasonale model may e otained y adjusting the stiffness of the Kelin foundation as proposed y Dieterman and Metrikine [] and Metrikine and Popp [], respectiely, for representation of a isco-elastic half-space or a layer oer a edrock. Alternatiely Vallahan and Das [] showed that an elastic layer under static load may e approximated y a Winkler foundation modied y the inclusion of a shear layer. Intuitiely this is a etter model since the modied support is at least capale of transporting energy in the along-eam direction. Analytic solutions for shear and translational spring stiffnesses were deried y Krenk [4]. Recently, the ertical stiffness of the allast and sleepers that are used to support railway tracks has een found to ary signicantly along the track with a correlation length much smaller and a ariation coefcient much larger than the su-soil, see Ref. [5]. This ariation leads to irations of a moing ehicle and the track itself, een when no external excitation is applied. In the literature, similar prolems hae een treated numerically using nite elements in address: la@ciil.auc.dk (L. Andersen) a numer of papers. Yoshimura et al. [6] analysed a ehicle moing along a simply supported eam with random surface irregularities and arying cross-section. Frýa et al. [7] examined the ehaiour of an innitely long Euler eam on a Kelin foundation with randomly arying parameters along the eam. Muscolino et al. [8] formulated an improed rstorder perturation method for the analysis of multi-degreeof-freedom systems with uncertain mechanical properties. Howeer, the method proposed y Muscolino et al. is not appropriate for the analysis of the present prolem, in which the stiffness of the system aries with time in the local moing frame of reference following the ehicle. In the present paper, a noel numerical method is presented for the analysis of a single-degree-of-freedom (SDOF) ehicle moing uniformly along a eam on a random modied Kelin foundation. The ertical support stiffness is descried y a weakly homogeneous random process. The randomness is primarily due to the sleeper and allast stiffness ariation, as mentioned aoe. Furthermore it should e noticed that the parameters of a su-soil under e.g. a railway track usually are known eforehand from eld oserations and/or laoratory tests. The prolem is formulated in a local coordinate frame, which follows the ehicle, and the interaction etween the ehicle and the eam is taken into account. When, for example, the surface of the track is arying

2 L. Andersen and S.R.K. Nielsen / Sumitted to Proailistic Engineering Mechanics stochastically, a linear set of equations goern the iration prolem, and an exact analytical solution can e found, see [9]. Howeer, a random ariation of the support stiffness gies rise to parametric excitation, thus preenting a direct analytical solution. To compute the stationary responses of the ehicle and eam due to the ariation of the spring stiffness, a rst-order perturation approach is proposed, in which the parametric excitation is transformed into an additie excitation. The perturation technique is only appropriate for relatiely small leels of the stiffness ariation. To alidate the results from the perturation analysis and to inestigate, at which leel of stiffness ariation it no longer proides satisfactory results, a comparison with the results from a nite element (FE) simulation is presented. The FE analysis is carried out in the moing frame of reference following the ehicle, and it includes the use of transmitting oundary conditions, which are a generalization of the oundary conditions proposed y Andersen et al. [].. Theory A ehicle modelled as an SDOF system with deterministic mass m, spring stiffness k and iscous damping c is moing uniformly in permanent contact along the smooth surface of a Bernoulli-Euler eam at the elocity, thus haing the along eam position x = t at time t. The eam has the deterministic ending stiffness EI and mass m per unit length. Further, the eam axis forms a straight line in the state of static equilirium in the asence of the ehicle. The eam rests on a modied Kelin foundation with deterministic shear stiffness G. Viscous damping is present in the support, dened y the ertical damping constant fl and the shear damping constant D, oth measured per unit length of the eam, see Fig.. ( x) ~ uxt (,) t x zt () m k f() t c G x ~ D EI, m Fig.. Single-degree-of-freedom ehicle on an Euler eam supported y a modied Kelin foundation with homogeneous shear stiffness and stochastically arying ertical stiffness. The ertical stiffness of the support»(x) is assumed to e descried y a stochastic homogeneous eld along the eam with the mean alue μ» = E[»(x)]. The auto-coariance function C»» and two-sided auto-spectral density S KK are dened as C»» ( x) =ff» x e j j xc ; x = x x ; () Z S KK (k) = C»» ( x)e ik x d x ß = x c ß ff» (x c k) : () Here ff» is the ariance of»(x), x c is the correlation length and i = p is the imaginary unit. Further, k denotes a waenumer, k = ß, where L is the corresponding L waelength of each component of harmonic stiffness ariation. The auto-spectrum represents a rst-order (Ornstein- Uhlenech) ltration of Gaussian white noise. The energy content in the low-waenumer range is relatiely higher than the energy content in the high-waenumer range. For the stationary analysis of the ehicle and eam response due to the stiffness ariation in the support, a description in a local moing coordinate system following the ehicle is conenient. Thus, a so-called Galilean coordinate transformation is carried out in the form ~x = x t. The mapping relates the xed Cartesian coordinate x to the codirectional conected coordinate ~x. Partial deriaties in the two coordinate systems are related in the @ @ ~x ~x : () Let the ertical displacement of the ehicle mass relatie to the position in the state of static equilirium and in the asence of the eam e denoted z = z(t). Further, let u = u(~x; t) e the ertical displacement of the eam relatie to the deformation in the state of static equilirium and in the asence of the ehicle. Introducing the notations _z = dz dt ; z = d z dt _u ; ~x ü ~x ; (4) for the local elocity and acceleration of the ehicle and eam, respectiely, the equations of motion for the ehicle and eam ecome, m z c (_z _u(;t)) k (z u(;t)) = ; (5) ~x 4 m ~x u ~x ~x ~x»(~x t)u = f (t)f(~x); (6) The additie force on the eam originates from the singledegree-of-freedom ehicle, i.e. f (t) = m (g z). Here g

3 L. Andersen and S.R.K. Nielsen / Sumitted to Proailistic Engineering Mechanics is the graitational acceleration. Howeer, the system is also suject to parametric excitation due to the ariation of the ertical support stiffness with time. This precludes a direct analytical solution. Expressions for» and G are deried in Ref. [4] for an elastic layer of nite magnitude H, dened y the mass density ρ and the Lamé constants ; μ, and oerlaying a edrock.. First-order perturation approach As a rst approach to the computation of the ehicle and eam response due to the parametric excitation, a perturation analysis is proposed. The ariation coefcient V = ff» =μ» of the ertical support spring stiffness is assumed to e sufciently small so that a rst-order perturation method may e used. Hence, the displacements and the spring stiffness are diided up into zero- and rst-order terms, z(t) =μz ^z(t); u(~x; t) =μu(~x)^u(~x; t);»(~x t) =μ» ^»(~x t): (7) In Eq. (7) and elow, the ar denotes mean alues and the hat denotes stochastic deiations. The mean alue of the ehicle displacement is constant, gien as μz =μu(). It has no influence on the prolem and will therefore e disregarded in the further analysis. A moing ehicle on a eam and Kelin foundation with constant stiffness causes no wae propagation in the moing frame of reference, i.e. the wae pattern is locked in time. The situation is analogous to the prolem with a ehicle on an elastic half-space. Hence, the zeroth-order, or quasi-static, eam displacement term μu is the solution to the equation EI d4 μu d~x 4 m G m d μu d~x D d μu dμu fl d~x d~x μ»μu = m gf(~x); (8) which is independent of time. Howeer, a note should e made with respect to the material damping: In the case of static load in a xed frame of reference, the material damping terms anish completely. Contrary, in Galilean coordinates the part of the material damping, which is due to conection, exists een in the quasi-static case. A fundamental solution set to the homogeneous part of Eq. (8) may e found in the form μu n (~x) = m g μ U n e μ K n ~x ; n =; ; ; 4: (9) The characteristic waenumers K μ n are found as the roots to the characteristic polynomial EIK μ n 4 D K μ n m G μk n m fl μ K n μ» =; n =; ; ; 4; () which is otained y assuming wae components of type (9) in the homogeneous part of Eq. (8). Only solutions that are decaying in the far-eld are physically alid. Hence, only wae components with R( μ K n ) > are acceptale for ~x<, and only wae components with R( μ K n ) < can exist for ~x >. Here R( μ K n ) denotes the real part of μ K n. Due to this, it turns out that two of the initial four components are present on either side of the ehicle. The components, which are present at ~x <, are assigned the suscripts n =;, whereas the components present at ~x > are assigned the suscripts n =; 4. In Eq. (9), μ U n, n =; ; ; 4, are the quasi-static eam displacement amplitudes at ~x =due to a unit intensity point force applied at ~x =. Gien a force of this kind, continuity of the displacement, the rotation and the moment must e assured across ~x =, whereas a unit jump in the shear force is present at ~x =. Once μ K n, n =; ; ; 4,hae een determined, and the wae components present on either side of the ehicle hae een identied, the amplitudes μu n, n =; ; ; 4, are found simultaneously as the solutions to 6 4 μk K μ K μ K μ 4 μk μk K μ K μ 4 μk μk K μ K μ μu μu μu μu = 6 4 =EI 7 5 : () The full solution μu(~x) is then found as the sum of the two components, which are present at the point ~x. Since the amplitudes computed in Eq. () correspond to a point force with unit intensity, ut the influence stems from the graity on the ehicle mass, the solution should e multiplied y the factor m g as indicated in Eq. (9). When μu(~x) has een determined, and neglecting the second-order terms, the rst-order terms ^z(t) and ^u(~x; t) can e calculated y a simultaneous solution of the equations m ^z c _^z _^u(;t) k (^z(t) ^u(;t)) = ; () ψ ~x 4 m ~x ^u ~x D ~x fl ~x μ»^u(~x; t) = ^f (~x; t): () Here ^f (~x; t) = m ^zf(~x) ^»(~x t)μu(~x) is the excitation of the eam. Apparently the parametric excitation due to the stiffness ariation is transformed to an equialent distriuted additie excitation ^»(~x t)μu(~x) in the differential equation for the rst-order eam displacement.

4 4 L. Andersen and S.R.K. Nielsen / Sumitted to Proailistic Engineering Mechanics.. Discretization of the rst-order spring stiffness term In order to nd a solution to Eqs. () and (), the influence of the distriuted equialent line load ^»(~x t)μu(~x) has to e ealuated. This is done numerically y a discretization into J time-arying equialent point loads ^f j (t) acting on the eam at the xed positions ~x j in the moing frame of reference. The distance ~x j etween each point load must e sufciently small so that oth the ending waes in the eam and the ariation of the spring stiffness are descried satisfactorily. In practice this requires aout point loads per waelength. The approximation may e written Z JX ^»(~x t)μu(~x)d~x ß ^f j (t)f(~x ~x j ); j= ^f j (t) = ^»(~x j t)μu(~x j ) ~x j : (4) In order to otain a good approximation, the summation has to e carried out oer the entire region, in which μu(~x) is not ery close to zero. The size of this region is easily determined, once the quasi-static eam displacement is known. The amounts of ertical and the shear damping, fl and D, are crucial to the length, L P, oer which the support stiffness ariation has to e discretized in the perturation analysis. Furthermore, L P is highly dependent on the ehicle elocity,. The critical elocity cr = s r 4EI» m G m (5) is identied as the conection elocity, at which the wae components in the quasi-static eam displacement change their qualitatie ehaiour, gien that no material damping is present in the system. For susonic elocities, i.e. < cr, the wae eld consists of eanescent waes. In other words, the quasi-static eam displacement is conned to a region in the icinity of the ehicle. Howeer, for supersonic elocities > cr, waes propagate into the far-eld. As a consequence of this, een the stiffness ariation in the support far from the ehicle will influence the ehicle and eam response. Howeer, for any practical applications, material damping will e present in the support, ensuring that all waes in the eam are to some extent dispersie. Since the model is linear, the principle of superposition is applicale and the total eld ^u(~x; t) may e written as ^u(~x; t) =^u (~x; t)^u S (~x; t); ^u S (~x; t) = JX j= ^u j (~x; t); (6) where ^u (~x; t) is the contriution from the interaction force etween the ehicle and the eam at ~x =and ^u j (~x; t), j =; ;:::;J, are the displacement elds generated y the point loads ^f j (t). Equations () and () then lead to the formulation m ^z c _^z _^u (;t) _^u S (;t) k (^z(t) ^u (;t) ^u S (;t)) = ; (7) ^u ~x 4 D ψ m ^u _^u ~x ^u j m _^u ^u ~x fl _^u ^u ~x! μ»^u j (~x; t) = ^f j (t)f(~x ~x j ); j =; ; :::; J; (8) where ^f (t) = m ^z, ~x =, and ^f j (t), j =; ; :::; J, are gien in Eq. (4). The equations of motion (8) for j =; ; :::; J are decoupled. Hence ^u j (~x; t), j =; ; :::; J, may e determined independently. Susequently Eqs. (7) and (8) for j =must e soled simultaneously... Frequency response functions for rst-order terms The origin of the irations is the rst-order ariation of the spring stiffness, whereas the output is the displacement of the ehicle and the eam. The input-output relations are illustrated in Fig., where H ^Z ^K(!) and H ^U ^K(~x;!) are the frequency response functions for ^z(t) and ^u(~x; t) for unit harmonic forces at ~x =and ~x, respectiely. ^ ~ ( xt) ^ ( xt ~ ) H ^^( ) H ZK UK ^ ^ ~ (, x ) ^zt () ^ ux,t ( ~ ) Fig.. Input-output relations for the ehicle and the eam response. In the discretized system dened y Eqs. (7) and (8), the input is howeer the discrete loads ^f j (t). Hence, as a rst step in determining H ^Z ^K(!) and H ^U ^K(~x;!), a relationship etween the point loads and the stiffness ariation must e estalished. Assuming ^»(x) to ary harmonically with waenumer k and amplitude ~ K(k), the ariation in the moing frame of reference reads ^»(~x t) = ^K(!)e i(!=)~x e i!t ; ^K(!) = ~ K(!=): (9) Here! = k is the apparent circular frequency of the ariation as seen y an oserer following the moing ehicle. ^K(!) is the complex amplitude of ^»(~x t) at ~x =. Equations (4) and (9) imply that a harmonic ariation of ^»(~x t) leads to the following ariation of the point loads, ^f j (t) = ^F j (!)e i!t ; ^F j (!) =H ^F j ^K (!) ^K(!); H ^Fj ^K(!) = μu(~x j ) ~x j e i(!=)~xj ; () where ^F j (!) is the amplitude of ^f j (t) and H ^F j ^K (!) is the frequency response function relating ^f j (t) to ^»(~x t).

5 L. Andersen and S.R.K. Nielsen / Sumitted to Proailistic Engineering Mechanics 5 With the point loads gien y Eq. (), four linearly independent harmonic solutions exist to the equation of motion, Eq. (8), for each j =; ; :::; J. The solutions take the form ^u j;n (~x; t) = ^F j (!) ~ U j;n (~x;!)e i!t ; j =; ;:::;J; n =; ; ; 4: () Here U ~ j;n (~x;!) are the amplitudes for a harmonically arying force with unit amplitude at ~x = ~x j, i.e. f (t) = e i!t f(~x ~x j ). Introducing U ~ n (!) as the amplitudes at ~x =~x j, U ~ j;n (~x;!) may e written ~U j;n (~x;!) = ~ U n (!)e i ^K n(~x ~x j ) ; () where the waenumers ^K n are the roots to the characteristic polynomial otained y assuming solutions of type () in the homogeneous ersion of Eq. (8), that is with ^f (t). The ehicle displacement and the interaction terms of the eam displacement are gien as ^z(t) = ^Z(!)e i!t ; ^u ;n (~x; t) =! m ^Z(!) ~ U ;n (~x;!)e i!t ; n =; ; ; 4: () Here ^Z(!) is the rst-order ehicle displacement amplitude and ~ U ;n (~x;!) = ~ U n (!)e i ^K n ~x are the amplitudes of the ehicle displacement for a harmonically arying force with unit amplitude at ~x =. At any point along the eam, only two of the fundamental solutions ^u j;n (~x; t), n =; ; ; 4, are present for each j =; ;:::;J, as it was also the case for μu(~x). For the sake of conenience, the two wae components existing ehind the respectie loads ^f j (t) with reference to the direction of the elocity will e assigned the suscripts n =;, whereas the components existing in front of the respectie loads are assigned the suscripts n =; 4. From Eqs. (8) and (), ^U (~x;!), i.e. the amplitude of ^u (~x; t), may e expressed in terms of ^Z(!). Inserting the result into Eq. (7) and assuming that J of the point loads are applied in front of the ehicle, the following frequency response relation is otained y isolating ^Z(!) on the left-hand side of the resulting equation, ^Z(!) =H ^Z ^K(!) ^K(!); H ^Z ^K (!) =N ^Z ^K (!)=D ^Z ^K (!); (4) where the numerator N ^Z ^K(!) and denominator D ^Z ^K(!) are gien as N ^Z ^K (!) =( i!c k )H ^U S ^K (;!); (5) D ^Z ^K (!) =! m i!c k i! m c! m k X n= ~U n (!); (6) respectiely. H ^U S ^K (;!) is a special case of the frequency response function relating ^u S (~x; t) to ^»(~x t), ψ X J H ^U ^K (~x;!) = S JX j=j ψ j= X H ^F j ^K (!) n= 4X H ^Fj ^K(!) ~U j;n (~x;!) n= ~U j;n (~x;!)!! : (7) H ^F j ^K (!) and ~ U j;n (~x;!) are preiously dened. It is noted that the rst sum is carried out oer the point loads in front of the oseration point ~x. Here, the two wae components, which are moing ackwards relatie to the conection elocity, are included. Behind the oseration point, the sum is carried out oer the forward-propagating wae components, since these are the components that will eentually arrie at point ~x. Inserting Eq. () into Eq. (8) and making use of Eq. (4), the contriution from ^f (t) to the eam displacement can e found. The contriutions from the discrete loads ^f j (t), j =; ;:::;J, are gien y Eq. (7). Adding the respectie contriutions, the frequency response function relating ^u(~x; t) to ^»(~x y) may eentually e written ^U (~x;!) =H ^U ^K(~x;!) ^K(!); H ^U ^K(~x;!) =H ^US ^K(~x;!) H ^U ^Z(~x;!)H ^Z ^K(!); (8) where j X H ^U ^Z(~x;!) =m! U ~ j (!)e ikj ~x ; j=j ρ fj ;j g = f; g for ~x» fj ;j g = f; 4g for ~x > : (9).. Random stiffness ariation In the frequency domain, the two-sided auto-spectral density for the stiffness ariation ecomes ~S KK (!) = x c ß ff» ; () x c (!=) which is otained from Eq. () y the transformation! = k. Notice that all the ariation lies within the rst-order terms, i.e. S ~ KK (!) = S ~ ^K ^K (!). Next, the two-sided autospectral density S ~ ZZ (!) for the SDOF ehicle displacement and the two-sided cross-spectral density S ~ UU (~x ; ~x ;!) for the eam displacement at two points ~x and ~x on the eam axis may e found. Thus, see e.g. Ref. [], ~S ZZ (!) = H ^Z ^K(!) S ~ KK (!); () ~S UU (~x ; ~x ;!)=H Λ^U ^K (~x ;!)H ^U ^K(~x ;!) ~ S KK (!); ()

6 6 L. Andersen and S.R.K. Nielsen / Sumitted to Proailistic Engineering Mechanics where H ^Z ^K (!) and H ^U ^K (~x;!) are dened preiously and H Λ^U ^K (~x ;!) is the complex conjugate of H ^U ^K (~x ;!). Again it should e noticed that the principle of superposition is alid, ecause the goerning equations are all linear, and that all ariation lies in the rst-order terms. From the Wiener-Khintchine relation, the auto-coariance function C zz ( ) for the ehicle displacement and the crosscoariance function C uu (~x ; ~x ;) for the eam displacement may e expressed as Z C zz ( )= cos(!) S ~ ZZ (!)d!; () C uu (~x ; ~x ;) Z = cos(!) ~ S R UU sin(!) ~ S I UU d!: (4) ~S R UU and S ~ I UU are the real and imaginary parts of ~S UU (~x ; ~x ;!), respectiely, and = t t is the time difference etween two oserations, which in the case of the eam displacement are related to the points ~x and ~x. Next, the displacement eld as well as the irtual elocity eld is put in discrete form, u(~x; t) =N(~x) a(t) ; ~u (~x; t) = ~ N(~x) ~a(t) (6) where N(~x) and ~ N(~x) are gloal shape functions of the dimension n. Here n is the numer of nodal points. Vectors a and ~a store the nodal displacements and rotations in the physical and the irtual eld, respectiely. Equation (6) is inserted into Eq. (5). The stationarity condition of the ariation principle then leads to the nite element method (FEM) formulation Mä C _a Ka = f f ; (7) where the system matrices and the oundary load ector are dened as Z ~x M = Z ~x C = ~NmNd~x; ~N ~x N d~x; ~x. Finite element simulation approach To alidate the perturation analysis of Susection., a nite element analysis is carried out. The theory, which has een implemented in the numerical examples, is descried in this susection... Deriation of the nite element system matrices A nite part of the eam is considered. The articial end points, or oundaries, of the nite element model are located at the positions ehind the ehicle and ~x in front of the ehicle, respectiely. To achiee the weak formulation of the equation of motion for the eam, Eq. (6) is multiplied y an aritrary weight function, which is chosen as a irtual displacement eld, ~u =~u(~x; t), and integration is carried out y parts, Z ~x Z ~x Z ~x ~x ~u»(~x t) u d~x ~x m u d~x ~x ~u =~u(;t) f (t) fl _u ~x mü u» ~u ~x M ~x : (5) Here Q = Q(~x; t) is the shear force and M = M (~x; t) is the moment. The moment and shear force as well as the additie load, f (t), are discussed elow. K = K(t) = K μ ^K(t) ; Z ~x μk ~N = ~x ~x N ~ m G m ^K(t) = Z ~x f = f (t) = ~x D@ ~x d~x; ~N^»(~x t) Nd~x; (9) " # x ~NQ(~x; N ~x M (~x; t) : (4) The mean part of the stiffness matrix, μ K, must only e assemled once. The time-arying part, ^K(t), must on the other hand e assemled for each time step in the time integration. Howeer, if the time step t in the integration is chosen with care, the stiffness matrices need only e computed for a single period of parametric excitation, T. The rules for choosing the time step t and the eam element length ~x e are addressed elow. The interaction force from the ehicle enters the additie load ector f = f(t) in the component corresponding to the degree of freedom, at which the eam is coupled to the ehicle. In practice, the coupling is descried y the introduction of an additional degree of freedom for the ertical displacement of the ehicle mass. The ehicle mass contriutes only with a diagonal term in the gloal mass matrix, whereas offdiagonal terms enter the damping and stiffness matrices due to the interaction ia the ehicle spring and dashpot. Thus, no additie forces are applied to the degrees of freedom associated with the eam. Howeer, it should e noted that the

7 L. Andersen and S.R.K. Nielsen / Sumitted to Proailistic Engineering Mechanics 7 graitation on the ehicle mass must e included in the nite element analysis, i.e. the ertical load m g is applied to the node corresponding to the ehicle mass. Otherwise, there will e no deformation of the system, and hence no iration. To ensure that waes propagating from the ehicle through the eam and support will not reenter the model, transparent oundary conditions must e applied at and ~x. The method proposed y Andersen et al. [] is implemented. This implies that the moment and shear force, M (~x; t) and Q(~x; t), at the articial oundaries are written in terms of the displacement and the rotation at and ~x. These are the degrees of freedom associated with each node in the nite element model, in which standard third-order Hermite shape, or interpolation, functions are used oer each element. As a consequence, the oundary load ector f is turned into an extra contriution to the system matrices, so that the only remaining external load in the model is the additie load due to graitation on the ehicle mass. The oundary conditions are deried on the asis of the approximation that outside the region, which is discretized with nite elements, there is no ariation of the support stiffness. In other words, the transparent oundary conditions are kept constant through all time steps in the integration. If and ~x coincide with the end points in the discretization of the equialent additie line load in the perturation approach, i.e. L P = L FE =~x ~x, the same error will arise in the two methods due to truncation of the model. In the numerical examples gien elow, the same end points are used in the two models. It has een tested y numerical experiment that the results otained with an FE model, in which the articial oundaries hae een moed further away from the ehicle, does not proide results with signicantly higher accuracy. A note should e made regarding the conectie term related to the shear damping, i.e. the term ND N, ~x appears in the integrand of the stiffness matrix, see Eq. (9). Differentiation three times of the third-order shape functions seems inconenient. A higher accuracy in the interior of the model would e achieed if one of the deriaties is transferred to the irtual eld y means of partial integration. In principle this is also the case for terms, in which the order of differentiation is two on the physical displacement, ut zero on the irtual displacement. Howeer, integration y parts of the conectie terms gies rise to additional oundary terms. This complicates the matter of implementing transparent oundary conditions. Hence, it has een chosen to use the formulation gien in Eqs. (7) to (4). When a eam on a Kelin foundation is discretized with the FEM in the xed frame of reference, the differential operators in the integrals comprising the system matrices ecome symmetric. Howeer, this is not the case when conection and/or a shear layer is present. To oercome this prolem, asymmetric shape functions for the irtual eld may e used, which are different from the shape functions used to approximate the physical eld. In the present analysis, howeer, the standard Galerkin approach is taken, i.e. ~N(~x) =N(~x). Preious work y Andersen et al. [] conrmed that this does not proide any prolems of instaility for the kind of prolem eing inestigated, as long as the conection elocity is elow approximately two times the critical elocity. Howeer, as discussed earlier, it is generally a prolem to carry out the analysis in case of supersonic elocities. This is due to the fact that a large region of the eam has to e considered in the perturation, and also in the FEM, approach, ecause the quasi-static displacement of the eam does not decay exponentially away from the ehicle... Time integration and discretization considerations In the rst-order perturation method presented in Susection., the parametric excitation due to the ariation of the support stiffness is transformed into an additie excitation. As a consequence, harmonic input with a single frequency gies rise to harmonic output, or response, with the same frequency. In contrast to this, in the nite element method approach, multiple harmonic components will e present in the response, een if the input signal only contains a single frequency. This is due to the fact that the parametric nature of the excitation is kept in the FEM formulation of the prolem. Hence, the higher-order harmonics are included in the solution. In principle, the standard deiations of the responses due to the ariation of the support stiffness may e computed y means of traditional Monte Carlo simulation. Here, a numer of random time series of the foundation stiffness ariation are generated on the asis of the auto-spectrum as gien y Eq. (). The random process in this context is the phase shift. Once a stationary condition has een reached in each simulation, the standard deiations of the responses can e found y ergodic sampling. For the analysis of the present prolem, the Monte Carlo method (MCM) has the disadantage that the frequencies in the input are mixed. Since the input for each harmonic component leads to response with multiple frequencies, it proides no information aout, where in the frequency domain the relatie error on the standard deiation has its origin. Therefore, an alternatie procedure is suggested, in which the nite element simulation is carried out for one harmonic component of the stiffness ariation at a time. When the stationary condition is reached, the rms alues of the responses for the indiidual frequencies are recorded. Susequently, Eqs. () and (4) are used to calculate the standard deiations of the responses. For each frequency, or waenumer, in the input, the rms alues otained from the FE simulation may e compared with the spectral components S ZZ (k) = ~S ZZ (!) and S UU (~x ; ~x ;k)= ~ S UU (~x ; ~x ;!), hence proiding an exact information aout, where in the waenumer domain errors arise etween the perturation method and the FEM results. Since the rms alues of the responses cannot directly e interpreted as the amplitudes of harmonic dis-

8 8 L. Andersen and S.R.K. Nielsen / Sumitted to Proailistic Engineering Mechanics placement time series, the present method may e understood as a pseudo-spectral analysis method (P-SAM). It is noted that the weighted sum of the rms alues will still proides an estimate of the total standard deiation, which on the other hand would e the only information proided y the MCM. Besides the fact that the P-SAM gies more information aout the nature of the relatie errors than does the MCM, the method also has some computational adantages oer the MCM. The oundary conditions proposed y Andersen et al. [] are limited to the analysis of excitation within a relatiely narrow frequency and and, in a Monte Carlo simulation including numerous waenumers in the input, many elements hae to e used in the model, while the time step must e ery small, in order to otain satisfactory results. The discretization considerations are further discussed elow. As a nal remark it should e mentioned that the standard deiations or the rms alues of the responses are in any case calculated with respect to the ariation around the deterministic quasi-static displacements, which are computed correctly in the perturation analysis and with insignicant errors in the MCM or P-SAM. Hence, the relatie errors are found with respect to the rst- and higher-order contriutions to the displacement, not with respect to the total displacement, where the errors would e much smaller. The direct time integration scheme proposed y Newmark [] is used with the parameters f == and ff ==4 corresponding to constant acceleration oer each time step. Numerical experiment indicates that the time integration is unstale for the present prolem if linear interpolation of the acceleration is assumed. This prohiits the use of a Wilson- integration scheme with =as well as the Newmark scheme with f == and ff ==6. To achiee satisfactory results, the element length ~x e must e small enough to ensure that all wae components in the eam as well as the support stiffness ariation are descried with sufcient accuracy. As already mentioned, third-order interpolation is used for the eam displacement eld. Four elements per waelength of the flexural waes are known to proide accurate results for additie load on a eam supported y a homogeneous Kelin foundation []. In the present analysis, e elements will e used per waelength. In principle, no interpolation of the stiffness ariation is necessary. Analytic expressions for ^K(t) are easily deried for a sinesoidal ariation of ^»(~x t) in Eq. (9). Howeer, a more adaptale code is otained, if linear shape functions are used to interpolate the stiffness ariation. Approximately elements are required per wae length of the stiffness ariation. When direct time integration is used for the analysis of wae propagation prolems, the time step is determined from the relation, C = jcj t ~x e ; C» ; (4) where C is the Courant numer and c is the phase elocity of the waes. The relation must e fullled for all wae components in the system. In the present prolem, this includes the stiffness ariation in the support. A Courant numer of C = :5 is used in the numerical examples. Furthermore, the time step must e small enough to ensure that the iration of the ehicle is descried with sufcient accuracy. Numerical experiment indicates that seeral time steps per eigeniration period are required for optimal performance of the numerical scheme in terms of accuracy. Howeer, een with time steps per eigeniration period, the ehicle irations are only the critical criterion at low frequencies. At high frequencies, the length and speed of the ending waes ecome dominant. It is noted that the frequency of the parametric excitation dominates the response, once the stationary condition is reached. Howeer, during the transient part of the response history, instaility in the numerical integration may occur, if the time step is not small enough to descrie the eigeniration of the ehicle. In terms of computation time, it is quite costly that ^K(t) must e assemled for each time step. A faster code may e otained y adjusting the time step, so that a whole numer of time steps, N t, are used oer a single period of parametric excitation. This way, ^K(t) only needs to e assemled N t times. Likewise, after the rst period of excitation, the system matrices produced in the Newmark integration scheme can e reused from an earlier time step. This is in particular adantageous, if the transient phase has a duration of seeral excitation periods, which is the case for excitation in the high-frequency range.. Numerical examples An analysis will e carried out for the standard deiation of the SDOF ehicle mass displacement response, ff z = p Czz (), and the standard deiation of the eam displacement response directly under the ehicle, ff u = p Cuu (; ; ). The standard deiations ff z and ff u are proportional to ff». Hence, they may coneniently e descried y the dynamic amplication factors, s z = ff z ff» ; s u = ff u ff» : (4) The dynamic amplication factors hae the unit [s z ] = [s u ]=m =N. In the perturation analysis, ff z and ff u are linear functions of ff». Therefore s z and s u are independent of ff». In the nite element analysis, this is not the case, ecause the second-order terms of the response are not disregarded. For conenience, the following parameters are introduced: r k c! = ; = m p ; m k = fl p mμ» ; = p D ; (4) mg

9 L. Andersen and S.R.K. Nielsen / Sumitted to Proailistic Engineering Mechanics 9 Here! is the eigenfrequency of the SDOF ehicle. The damping ratios and are related to the ertical displacements of the ehicle mass and the eam, respectiely, whereas is the damping ratio for the shear layer.. Verication of the model In the next susection, a parameter study will e carried out, in which the perturation approach of Susection. is used to inestigate the influence on the displacement responses from the ending and foundation stiffness as well as the correlation length of the stiffness ariation. Firstly, howeer, an analysis is performed with the aim of determining the leel of stiffness ariation, eyond which the perturation analysis proides unsatisfactory results. The response spectra for the ehicle and eam displacement are computed with the perturation method and the FE model. In case of the FE approach, different leels of the stiffness ariation are considered, descried in terms of the ariation coefcient V = ff» =μ». It is noted that the results of the P-SAM analysis is not a true frequency spectrum, ut the rms alues of the response, which are computed with the method, may e compared directly with the results of the perturation method. The SDOF ehicle has the parameters S ZZ [m ] S UU [m ].5 x x FE V =. FE V =. FE V =. FE V =. FE V =.4 FE V =. FE V =. FE V =. FE V =. FE V =.4 m =kg;! =ß s ; =:; (44) i.e. the ehicle is critically damped. The parameters listed in Eq. (44) are assumed to e typical for an automoile. The eam has the mass per unit length and the damping ratios m =kg/m; =:; =:: (45) The properties listed in Eqs. (44) and (45) will e used for all analyses in the examples. Here, the following parameters are used in addition to those already listed, EI = 7 Nm ; μ» = 7 N/m ; G =5 m/m; x c =5m: (46) Figures to 6 show the response spectra and pseudospectra otained with the perturation method and the P- SAM, respectiely, for the four ehicle elocities 5, 5, 75 and m/s. For comparison it should e mentioned that the critical elocity in the present case is 6.4 m/s, meaning that all four elocities are su-sonic. As discussed aoe, only the part of the displacement, which is due to the stiffness ariation, has een included. In the P-SAM the results are otained y a sutraction of the quasi-static displacement from the total displacement. The quasi-static displacement is found y soling the equation μa = μ K f. Here μa are the nodal displacements and rotations, when no ariation of the stiffness has een applied. Alternatiely, for the eam, Eq. (8) may e used to nd μu(). It is noted that μa has een used as initial conditions for the time integration in order to reduce the duration of the transient phase to a minimum Fig.. Response spectra for a model with the parameters gien in Eqs. (44) to (46), =5m/s and arious amounts of support stiffness ariation. Generally the qualitatie ehaiour of the response is the same at all leels of stiffness ariation for each indiidual elocity. As expected, the results otained with the FE model for the ariation coefcients. and elow lie close to the solution, which has een computed with the perturation approach. For V = : the perturation approach seems less likely to proide satisfactory results. Clearly the relatie error is conned to the low-waenumer range. For high wae numers of the stiffness ariation the relatie error is insignicant, regardless of the ariation coefcient V. This information would not e proided y the MCM, ecause no means exist, y which the higher-order harmonmics from low-waenumer excitation can e ltered from the rst-order harmonics of high-waenumer excitation. The relatie errors on the standard deiations of the responses are listed in Tas. and for the ehicle and eam displacement, respectiely. It is osered that the perturation method does indeed proide satisfactory results for ariation coefcients up till. and including with errors smaller than %; ut only for elocities elow a certain alue. For the elocity m/s, the relatie error is approximately two times the error, which is otained at the lower elocities. Hence it may e concluded that the perturation

10 L. Andersen and S.R.K. Nielsen / Sumitted to Proailistic Engineering Mechanics S ZZ [m ].5 x FE V =. FE V =. FE V =. FE V =. FE V =.4 S ZZ [m ] 7 x FE V =. FE V =. FE V =. FE V =. FE V = S UU [m ] x FE V =. FE V =. FE V =. FE V =. FE V =.4 S UU [m ] 6 x FE V =. FE V =. FE V =. FE V =. FE V = Fig. 4. Response spectra for a model with the parameters gien in Eqs. (44) to (46), =5m/s and arious amounts of support stiffness ariation. Tale Relatie error (in percent) etween the standard deiations of the ehicle displacement response found y FE and perturation analysis. V Fig. 5. Response spectra for a model with the parameters gien in Eqs. (44) to (46), =75m/s and arious amounts of support stiffness ariation. Tale Relatie error (in percent) etween the standard deiations of the eam displacement response at ~x =found y FE and perturation analysis. V method is etter suited for elocities well elow cr. Another interesting feature concerning the relatie error on the standard deiations for the elocity m/s is that the error has the opposite sign of the errors, which are recorded for the elocities 75 m/s and elow. This suggests that there may exist an intermediate elocity, at which the error is close to zero. To inestigate this matter, an analysis is carried out, and it has een found that the ehicle elocity 9 m/s gies rise to ery small relatie errors, een for the ariation coefcient V =:4. The response spectra for =9m/s are plotted in Fig. 7, and the relatie errors are listed in Tas. and. For the elocity =5m/s, Fig. shows that the FE model and the perturation model do not proide the same solution, een for the ariation coefcient V =:. A further analysis, the results of which are not included in the paper, indicates that this is a general tendency for relatiely low elocities and ending stiffnesses and high shear stiffnesses. Thus, when the quantity G m ecomes large in comparison to the ending stiffness, discrepancies may arise etween the results computed with the two models, which are not due to shortcomings of the rst-order perturation approach. To further clarify the nature of the response at different leels of the stiffness ariation, an FE analysis is carried out for the ariation coefcients V =: and V =:5. The ehicle elocity is set to 5 m/s, and the analysis is carried out for a single harmonic component of the stiffness ariation

11 L. Andersen and S.R.K. Nielsen / Sumitted to Proailistic Engineering Mechanics S ZZ [m ].5 x FE V =. FE V =. FE V =. FE V =. FE V =.4 S ZZ [m ] x FE V =. FE V =. FE V =. FE V =. FE V = S UU [m ].5 x FE V =. FE V =. FE V =. FE V =. FE V =.4 S UU [m ] 8 x FE V =. FE V =. FE V =. FE V =. FE V = Fig. 6. Response spectra for a model with the parameters gien in Eqs. (44) to (46), =m/s and arious amounts of support stiffness ariation Fig. 7. Response spectra for a model with the parameters gien in Eqs. (44) to (46), =9m/s and arious amounts of support stiffness ariation. with aritrary, ut relatiely low waenumer k. Figure 8 shows the ehicle response and the eam response at ~x = during the rst e periods of parametric excitation, that is including the transient phase. The response for V =:5 has een diided y 5 in order to otain response leels of the same magnitude in the two cases. The response otained for the ariation coefcient V = : aries harmonically with time, as it is assumed in the rst-order perturation analysis. It is osered that the stationary condition is reached almost immediately, see Fig. 8. For V =:5 the response is howeer not harmonic, in accordance with the discussion gien in Susection... The negatie displacements due to a local softening of the support are much more pronounced than the positie displacements due to a local stiffening of the support. In other words, the local minima in the responses are changed more than the local maxima. This explains the tendency, which can e seen on Figs. to 5, namely that the rms alue of the response ecomes greater, when the stiffness ariation is increased. For high elocities, the reduction in the tips will e more pronounced than the amplication of the dips in the response, when the stiffness ariation is increased, thus leading to a reduction of the rms response, see Fig. 6.. Parameter study A perturation analysis is carried out for arious cominations of the ending stiffness EI and the correlation length x c. Two different supports hae een examined. In Fig. 9 the results are gien for the elocity range [:5; 5] m/s and a eam resting on a foundation with μ» = 7 N/m and G = m/m. The parameters are rather aritrarily chosen ut may to some extent represent a soft elastic layer. Figure shows the results for an een softer foundation with μ» = /m and G = 5 Nm/m. It is noted that the critical elocities in the rst case are 54.77, 44.9 and 448. m/s for the ending stiffnesses 5, 7 and 9 Nm, respectiely. In the second case, the ery soft support, the critical elocities are 7.6, 8.5 and 5.69 m/s, respectiely. From the cures on Figs. 9 and it may e concluded that the amplication of the ehicle response (thick line) is generally increasing with the correlation length, x c.in most cases this also applies to the eam displacement (thin line), though a few exceptions exist. For susonic elocities, the amplication of oth the ehicle and the eam response grows signicantly with the elocity. The ehicle response increases less than the eam response. This phenomenon is

12 L. Andersen and S.R.K. Nielsen / Sumitted to Proailistic Engineering Mechanics s z, s u EI = 5 Nm s z, s u EI = 7 Nm s z, s u EI = 9 Nm 8 x c =. m x c = m 5 5 x c = 5 m 5 5 Fig. 9. Dynamic amplication of ehicle response ( ) and eam response ( ). μ» = 7 N/m and G =5 7 Nm/m. [s z]=[s u]=m /N and [] =m/s. s z, s u EI = 5 Nm s z, s u EI = 7 Nm s z, s u EI = 9 Nm 6 x c =. m x c = m 5 5 x c = 5 m 5 5 Fig.. Dynamic amplication of ehicle response ( ) and eam response ( ). μ» = /m and G =5 m/m. [s z]=[s u]=m /N and [] =m/s. most pronounced for the stiffer support, i.e. Fig. 9. In the elocity range near the critical elocity, a local dip occurs in the response of the ehicle and, in most cases, also in the response if the eam. After this local minimum, the response leel reaches a local, or een gloal, maximum. When the elocity is further increased eyond cr, the response of the eam and in particular the response of the ehicle decrease.

13 L. Andersen and S.R.K. Nielsen / Sumitted to Proailistic Engineering Mechanics z(t), u(,t) 4 x t/t Fig. 8. Response of the ehicle mass and the eam at ~x =for the parameters gien in Eqs. (44)-(46). The ehicle and eam response for V =: are plotted with the signatures ( ) and ( ), respectiely, whereas the ehicle and eam response for V = :5 are plotted with the signatures ( ) and ( ). 4. Conclusions The response of a ehicle moing uniformly along a track with random foundation stiffness has een analysed. The ehicle is modelled as a single-degree-of-freedom oscillator and the track is modelled as a Bernoulli-Euler eam resting on a Kelin foundation, which has een modied y the inclusion of a shear layer with oth stiffness and damping. The ertical spring stiffness of the support is a random process along the track with a ariation gien y an Ornstein- Uhlenech ltration of Gaussian white noise. A rst-order perturation approach has een deeloped for the analysis of the ehicle and track response to the stiffness ariation. The results otained with the present method hae een compared with a numerical solution achieed with a nite element simulation y means of a pseudospectral analysis method. In this numerical method, the higher-order harmonics of the response are included, while at the same time a comparison with the response spectra otained with the perturation approach is possile. This is an adantage oer traditional Monte Carlo simulation, which may only e used to compute an estimate of the total standard deiation of the response. The quality of the perturation method depends on the elocity of the ehicle. For low elocities, a relatie error on the standard deiation of the responses of less than % may e expected for a ariation coefcient of % for the support stiffness. It should e emphasized that the relatie errors refer to the rst- and higher-order response only. The quasi-static displacement response is computed accurately in either method. For high elocities, just elow the critical elocity of the eam-support system, the relatie error is somewhat higher, aout 5%. For certain intermediate elocities, the results otained with the perturation analysis hae een found to e more accurate. Howeer, generally the perturation method reaks down for ariation coefcients of the stiffness ariation eyond %. The present perturation method is adantageous to nite element simulation in terms of computation time. As such the method is a powerful tool for a parametric study of the gien prolem. Furthermore the proposed perturation analysis may e used to proide preliminary results for the response, een when the ariation coefcient of the stiffness is larger than %. Based on these results, an optimal distriution may e found for the waenumers, at which the nite element analysis should e carried out. Methods proposed in the literature may e used to calirate the model so that a description of a realistic track on a half-space or elastic layer may e achieed. A parameter study indicates that the response of the moing ehicle and the eam underneath increases with elocity and the correlation length of the stiffness ariation. Howeer, for ehicle elocities close to the critical elocity cr of the eam/support, the response of the eam and, in particular, the ehicle mass due to the spring stiffness ariation drops dramatically. A local, or gloal, peak in the response may e present for elocities just aoe the critical elocity, ut for elocities eyond cr, the response is generally decreasing. Acknowledgements The authors would like to thank the Danish Technical Research Council for nancial support ia the research project: Damping Mechanisms in Dynamics of Structures and Materials. References [] HA Dieterman and A Metrikine. The equialent stiffness of a half-space interacting with a eam. Critical elocities of a moing load along the eam. European Journal of Mechanics, A/Solids, 5():67 9, 996. [] A Metrikine and K Popp. Steady-state irations of an elastic eam on a isco-elastic layer under a moing load. Archiee of Applied mechanics, 7:99 48,. [] CVG Vallahan and YC Das. Modied laso model for eams on elastic foundation. Journal of Geotechnical Engineering, 7: , 99. [4] S Krenk. Mechanics and Analysis of Beams, Columns and Cales. Polyteknisk Press, Copenhagen,. [5] J Oscarsson. Dynamic train/track interaction - linear and non-linear track models with property scatter. Ph.d. thesis, Department of Solid Mechanics, Chalmers Uniersity of Technology, Göteorg,. [6] T Yoshimura, J Hino, T Katama, and N Ananthanarayana. Random iration of a non-linear eam sujected to a moing load: a nite element analysis. Journal of Sound and Viration, ():7 9, 988. [7] L Frýa, S Nakagiri, and N Yoshikawa. Stochastic nite elements for a eam on a random foundation with uncertain damping under a moing force. Journal of Sound and Viration, 6(): 45, 99.