Vibrations of a track caused by variation of the foundation stiffness

Vibrations of a track caused by variation of the foundation stiffness

Lars Andersen*, Søren R.K. Nielsen

Department of Civil Engineering, Aalborg University, Sohngaardsholmsvej 57, DK-9000 Aalborg, Denmark

Abstract

The paper deals with the stochastic analysis of a single-degree-of-freedom vehicle moving at constant velocity along a simple track

structure with randomly varying support stiffness. The track is modelled as an infinite Bernoulli–Euler beam resting on a Kelvin foundation,

which has been modified by the introduction of a shear layer. The vertical spring stiffness in the support is assumed to be a stochastic

homogeneous field consisting of a small random variation around a deterministic mean value. First, the equations of motion for the vehicle

and beam are formulated in a moving frame of reference following the vehicle. Next, a first-order perturbation method is proposed to

establish the relationship between the variation of the spring stiffness and the responses of the vehicle mass and the beam. Numerical

examples are given for various parameters of the track. The response spectra obtained from the perturbation analysis are compared with the

numerical solution, in which finite elements with transparent boundary conditions are used. The circumstances, under which the first-order

perturbation approach provides satisfactory results, are discussed.

q 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Uncertain parameters; Perturbation approach; Finite elements

1. Introduction

A road, runway or railway track is often modelled as a

beam structure on a Kelvin foundation. In itself this model

only weakly describes the real situation of a track resting on

a subsoil. However, a reasonable model may be obtained by

adjusting the stiffness of the Kelvin foundation as proposed

by Dieterman and Metrikine [1] and Metrikine and Popp [2],

respectively, for representation of a visco-elastic half-space

or a layer over a bedrock. Alternatively, Vallabhan and Das

[3] showed that an elastic layer under static load may be

approximated by a Winkler foundation modified by the

inclusion of a shear layer. Intuitively this is a better model

since the modified support is at least capable of transporting

energy in the along-beam direction. Analytic solutions for

shear and translational spring stiffnesses were derived by

Krenk [4].

Recently, the vertical stiffness of the ballast and sleepers

that are used to support railway tracks has been found to

vary significantly along the track with a correlation length

much smaller and a variation coefficient much larger than

the sub-soil [5]. This variation leads to vibrations of a

moving vehicle and the track itself, even when no external

excitation is applied. In the literature, similar problems have

been treated numerically using finite elements in a number

of papers. Yoshimura et al. [6] analysed a vehicle moving

along a simply supported beam with random surface

irregularities and varying cross-section. Fryba et al. [7]

examined the behaviour of an infinitely long Euler beam on

a Kelvin foundation with randomly varying parameters

along the beam. Muscolino et al. [8] formulated an

improved first-order perturbation method for the analysis

of multi-degree-of-freedom systems with uncertain mech-

anical properties. The method proposed by Muscolino et al.

appears to be convenient for problems involving stochastic

vectors, e.g. the uncertain stiffness variation over space

within a dynamic system described by the finite element

method. However, the method is not appropriate for the

analysis of problems involving stochastic processes, in

which an infinite amount of unknown parameters has to be

accounted for. Therefore, another means of carrying out

stochastic finite element analysis must be used for the

analysis of the present problem, in which the stiffness of the

system varies with time in the local moving frame of

reference following the vehicle.

In the present paper, a novel numerical method is

presented for the analysis of a single-degree-of-freedom

(SDOF) vehicle moving uniformly along a beam on a

random modified Kelvin foundation. The vertical support

stiffness is described by a weakly homogeneous random

process. The randomness is primarily due to the sleeper and

0266-8920/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved.

doi:10.1016/S0266-8920(03)00012-2

Probabilistic Engineering Mechanics 18 (2003) 171–184

www.elsevier.com/locate/probengmech

* Corresponding author. Tel.: þ45-9635-8080; fax: þ45-9814-2555.

E-mail address: [email protected] (L. Andersen).

http://www.elsevier.com/locate/probengmech

ballast stiffness variation, as mentioned above. Furthermore,

it should be noticed that the parameters of a sub-soil under,

e.g. a railway track usually are known beforehand from field

observations and/or laboratory tests. The problem is

formulated in a local coordinate frame, which follows the

vehicle, and the interaction between the vehicle and the

beam is taken into account.

When, for example, the surface of the track is varying

stochastically, a linear set of equations govern the vibration

problem, and an exact analytical solution can be found, see

[9]. However, a random variation of the support stiffness

gives rise to parametric excitation, thus preventing a direct

analytical solution. To compute the stationary responses of

the vehicle and beam due to the variation of the spring

stiffness, a first-order perturbation approach is proposed, in

which the parametric excitation is transformed into an

additive excitation. The perturbation technique is only

appropriate for relatively small levels of the stiffness

variation. In order to validate the results from the

perturbation analysis and to investigate, at which level of

stiffness variation it no longer provides satisfactory results,

a comparison with the results from a finite element (FE)

simulation is presented. The FE analysis is carried out in the

moving frame of reference following the vehicle, and it

includes the use of transmitting boundary conditions, which

are a generalization of the boundary conditions proposed by

Andersen et al. [10].

2. Theory

A vehicle modelled as an SDOF system with determi-

nistic mass m0; spring stiffness k0 and viscous damping c0 is

moving uniformly in permanent contact along the smooth

surface of a Bernoulli–Euler beam at the velocity v; thus

having along the beam position x ¼ vt at time t: The beam

has the deterministic bending stiffness EI and mass m per

unit length. Further, the beam axis forms a straight line in

the state of static equilibrium in the absence of the vehicle.

The beam rests on a modified Kelvin foundation with

deterministic shear stiffness G: Viscous damping is present

in the support, defined by the vertical damping constant g

and the shear damping constant D; both measured per unit

length of the beam, see Fig. 1.

The vertical stiffness of the support kðxÞ is assumed to be

described by a stochastic homogeneous field along the beam

with the mean value �k ¼ E½kðxÞ�: The auto-covariance

function Ckk and two-sided auto-spectral density SKK are

defined as

CkkðDxÞ ¼s2k e2lDx=xcl; Dx¼ x2 2 x1; ð1Þ

SKKðkÞ ¼1

2p

ð1

21CkkðDxÞe2ikDx dDx¼

xc

p

s2k

1þðxckÞ2: ð2Þ

Here s2k is the variance of kðxÞ; xc is the correlation

length and i¼ffiffiffiffi21

pis the imaginary unit. Further, k denotes

a wavenumber, k¼ ð2p=LÞ; where L is the corresponding

wavelength of each component of harmonic stiffness

variation. The auto-spectrum represents a first-order (Orn-

stein–Uhlenbech) filtration of Gaussian white noise. The

energy content in the low-wavenumber range is relatively

higher than the energy content in the high-wavenumber

range.

For the stationary analysis of the vehicle and beam

response due to the stiffness variation in the support, a

description in a local moving coordinate system following

the vehicle is convenient. Thus, a so-called Galilean

coordinate transformation is carried out in the form ~x ¼

x 2 vt: The mapping relates the fixed Cartesian coordinate x

to the co-directional convected coordinate ~x: Partial

derivatives in the two coordinate systems are related in

the following manner,

›

›x¼

›

›~x;

›

›t

��x¼

›

›t

��~x2v

›

›~x;

›2

›t2

��x

¼›2

›t2

��~x

22v›2

›~x›t

��~x

þv2 ›2

›~x2: ð3Þ

Let the vertical displacement of the vehicle mass relative to

the position in the state of static equilibrium and in the

absence of the beam be denoted z ¼ zðtÞ: Further, let u ¼

uð~x; tÞ be the vertical displacement of the beam relative to

the deformation in the state of static equilibrium and in the

absence of the vehicle. Introducing the notations

_z ¼dz

dt; €z ¼

d2z

dt2; _u ¼

›u

›t

��~x; €u ¼

›2u

›t2

��~x

; ð4Þ

for the local velocity and acceleration of the vehicle and

beam, respectively, the equations of motion for the vehicle

and beam become,

m0€z þ c0ð_z 2 _uð0; tÞÞ þ k0ðz 2 uð0; tÞÞ ¼ 0; ð5Þ

Fig. 1. SDOF vehicle on an Euler beam supported by a modified Kelvin

foundation with homogeneous shear stiffness and stochastically varying

vertical stiffness.

L. Andersen, S.R.K. Nielsen / Probabilistic Engineering Mechanics 18 (2003) 171–184172

EI›4u

›~x4þ m €u 2 2v

›_u

›~xþ v2 2

G

m

� �›2u

›~x2

!

2 D›2 _u

›~x22 v

›3u

›~x3

!þ g _u 2 v

›u

›~x

� �þ kð~x þ vtÞu

¼ f ðtÞdð~xÞ: ð6Þ

The additive force on the beam originates from the SDOF

vehicle, i.e. f ðtÞ ¼2m0ðg þ €zÞ: Here g is the gravitational

acceleration. However, the system is also subject to

parametric excitation due to the variation of the vertical

support stiffness with time. This precludes a direct

analytical solution. Expressions for k and G are derived in

Ref. [4] for an elastic layer of finite magnitude H; defined by

the mass density r and the Lame constants l;m; and

overlaying a bedrock.

2.1. First-order perturbation approach

As a first approach to the computation of the vehicle and

beam response due to the parametric excitation, a

perturbation analysis is proposed. The variation coefficient

V ¼ sk= �k of the vertical support spring stiffness is assumed

to be sufficiently small so that a first-order perturbation

method may be used. Hence, the displacements and the

spring stiffness are divided up into zero- and first-order

terms

zðtÞ ¼ �z þ zðtÞ; uð~x; tÞ ¼ �uð~xÞ þ uð~x; tÞ;

kð~x þ vtÞ ¼ �kþ kð~x þ vtÞ: ð7Þ

In Eq. (7) and below, the bar denotes mean values and the

hat denotes stochastic deviations. The mean value of the

vehicle displacement is constant, given as �z ¼ �uð0Þ: It has no

influence on the problem and will therefore be disregarded

in the further analysis.

A moving vehicle on a beam and Kelvin foundation with

constant stiffness causes no wave propagation in the moving

frame of reference, i.e. the wave pattern is locked in time.

The situation is analogous to the problem with a vehicle on

an elastic half-space. Hence, the zeroth-order, or quasi-

static, beam displacement term �u is the solution to the

equation

EId4 �u

d~x4þ m v2

2G

m

� �d2 �u

d~x2þ v D

d3 �u

d~x32 g

d�u

d~x

!þ �k�u

¼ 2m0gdð~xÞ; ð8Þ

which is independent of time. However, a note should be

made with respect to the material damping. In the case of

static load in a fixed frame of reference, the material

damping terms vanish completely. Contrary, in Galilean

coordinates the part of the material damping, which is due to

convection, exists even in the quasi-static case.

A fundamental solution set to the homogeneous part of

Eq. (8) may be found in the form

�unð~xÞ ¼ 2m0g �Un e�Kn ~x; n ¼ 1; 2; 3; 4: ð9Þ

The characteristic wavenumbers �Kn are found as the roots to

the characteristic polynomial

EI �K4n þ vD �K3

n þ m v22

G

m

� ��K2

n 2 vg �Kn þ �k ¼ 0;

n ¼ 1; 2; 3; 4; ð10Þ

which is obtained by assuming wave components of type (9)

in the homogeneous part of Eq. (8). Only solutions that are

decaying in the far-field are physically valid. Hence, only

wave components with Rð �KnÞ . 0 are acceptable for ~x , 0;

and only wave components with Rð �KnÞ , 0 can exist for

~x . 0: Here, Rð �KnÞ denotes the real part of �Kn: Due to this,

it turns out that two of the initial four components are

present on either side of the vehicle. The components, which

are present at ~x , 0; are assigned the subscripts n ¼ 1; 2;

whereas the components present at ~x . 0 are assigned the

subscripts n ¼ 3; 4:

In Eq. (9), �Un; n ¼ 1; 2; 3; 4; are the quasi-static beam

displacement amplitudes at ~x ¼ 0 due to a unit intensity

point force applied at ~x ¼ 0: Given a force of this kind,

continuity of the displacement, the rotation and the moment

must be assured across ~x ¼ 0; whereas a unit jump in the

shear force is present at ~x ¼ 0: Once �Kn; n ¼ 1; 2; 3; 4; have

been determined, and the wave components present on

either side of the vehicle have been identified, the

amplitudes �Un; n ¼ 1; 2; 3; 4; are found simultaneously as

the solutions to

21 21 1 1

2 �K1 2 �K2�K3

�K4

2 �K21 2 �K2

2�K2

3�K2

4

2 �K31 2 �K3

2�K3

3�K3

4

26666664

37777775

�U1

�U2

�U3

�U4

26666664

37777775 ¼

0

0

0

1=EI

26666664

37777775: ð11Þ

The full solution �uð~xÞ is then found as the sum of the two

components, which are present at the point ~x: Since the

amplitudes computed in Eq. (11) correspond to a point force

with unit intensity, but the influence stems from the gravity

on the vehicle mass, the solution should be multiplied by the

factor 2m0g as indicated in Eq. (9).

When �uð~xÞ has been determined, and neglecting the

second-order terms, the first-order terms zðtÞ and uð~x; tÞ can

be calculated by a simultaneous solution of the equations

m0€z þ c0ð_z 2 _uð0; tÞÞ þ k0ðzðtÞ2 uð0; tÞÞ ¼ 0; ð12Þ

L. Andersen, S.R.K. Nielsen / Probabilistic Engineering Mechanics 18 (2003) 171–184 173

EI›4u

›~x4þ m €u 2 2v

› _u

›~xþ v2 2

G

m

� �›2u

›~x2

!

2 D›2 _u

›~x22 v

›3u

›~x3

!þ g _u 2 v

›u

›~x

� �þ �kuð~x; tÞ

¼ fð~x; tÞ: ð13Þ

Here, fð~x; tÞ ¼ 2m0€zdð~xÞ2 kð~x þ vtÞ�uð~xÞ is the excitation of

the beam. Apparently the parametric excitation due to the

stiffness variation is transformed to an equivalent distrib-

uted additive excitation kð~x þ vtÞ�uð~xÞ in the differential

equation for the first-order beam displacement.

2.1.1. Discretization of the first-order spring stiffness term

In order to find a solution to Eqs. (12) and (13), the

influence of the distributed equivalent line load 2kð~x þ

vtÞ�uð~xÞ has to be evaluated. This is done numerically by a

discretization into J time-varying equivalent point loads fjðtÞ

acting on the beam at the fixed positions ~xj in the moving

frame of reference. The distance D~xj between each point

load must be sufficiently small so that both the bending

waves in the beam and the variation of the spring stiffness

are described satisfactorily. In practice this requires about

10 point loads per wavelength. The approximation may be

written

2 kð~x þ vtÞ�uð~xÞ <XJ

j¼1

fjðtÞdð~x 2 ~xjÞ;

fjðtÞ ¼ 2kð~xj þ vtÞ�uð~xjÞD~xj: ð14Þ

In order to obtain a good approximation, the summation has

to be carried out over the entire region, in which �uð~xÞ is not

very close to zero. The size of this region is easily

determined, once the quasi-static beam displacement is

known. The amounts of vertical and the shear damping, g

and D; are crucial to the length, LP; over which the support

stiffness variation has to be discretized in the perturbation

analysis. Furthermore, LP is highly dependent on the vehicle

velocity, v: The critical velocity

vcr ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4EIk

m2

sþ

G

m

vuutð15Þ

is identified as the convection velocity, at which the wave

components in the quasi-static beam displacement change

their qualitative behaviour, given that no material damping

is present in the system. For subsonic velocities, i.e. v , vcr;

the wave field consists of evanescent waves. In other words,

the quasi-static beam displacement is confined to a region in

the vicinity of the vehicle. However, for supersonic

velocities v . vcr; waves propagate into the far-field. As a

consequence of this, even the stiffness variation in the

support far from the vehicle will influence the vehicle and

beam response. However, for any practical applications,

material damping will be present in the support, ensuring

that all waves in the beam are to some extent dispersive.

Since the model is linear, the principle of superposition is

applicable and the total field uð~x; tÞ may be written as

uð~x; tÞ ¼ u0ð~x; tÞ þ uSð~x; tÞ; uSð~x; tÞ ¼XJ

j¼1

ujð~x; tÞ; ð16Þ

where u0ð~x; tÞ is the contribution from the interaction force

between the vehicle and the beam at ~x ¼ 0 and ujð~x; tÞ; j ¼

1; 2;…; J; are the displacement fields generated by the point

loads fjðtÞ: Eqs. (12) and (13) then lead to the formulation

m0€z þ c0ð_z 2 _u0ð0; tÞ2 _uSð0; tÞÞ þ k0ðzðtÞ2 u0ð0; tÞ

2 uSð0; tÞÞ

¼ 0; ð17Þ

EI›4uj

›~x4þ m €uj 2 2v

› _uj

›~xþ v2 2

G

m

� �›2uj

›~x2

!

2 D›2 _uj

›~x22 v

›3uj

›~x3

!þ g _uj 2 v

›uj

›~x

� �

þ �kujð~x; tÞ¼ fjðtÞdð~x 2 ~xjÞ; j ¼ 0; 1;…; J; ð18Þ

where f0ðtÞ ¼ 2m0€z; ~x0 ¼ 0; and fjðtÞ; j ¼ 1; 2;…; J; are

given in Eq. (14). The equations of motion (18) for j ¼

1; 2;…; J are decoupled. Hence ujð~x; tÞ; j ¼ 1; 2;…; J; may

be determined independently. Subsequently, Eqs. (17) and

(18) for j ¼ 0 must be solved simultaneously.

2.1.2. Frequency response functions for first-order terms

The origin of the vibrations is the first-order variation of

the spring stiffness, whereas the output is the displacement

of the vehicle and the beam. The input–output relations are

shown in Fig. 2, where HZKðvÞ and HUKð~x;vÞ are the

frequency response functions for zðtÞ and uð~x; tÞ for unit

harmonic forces at ~x0 ¼ 0 and ~x; respectively.

In the discretized system defined by Eqs. (17) and (18),

the input is however the discrete loads fjðtÞ: Hence, as a first

step in determining HZKðvÞ and HUKð~x;vÞ; a relationship

between the point loads and the stiffness variation must be

established. Assuming kðxÞ to vary harmonically with

wavenumber k and amplitude ~KðkÞ; the variation in the

moving frame of reference reads

kð~x þ vtÞ ¼ KðvÞe2iðv=vÞ~x e2ivt; KðvÞ ¼ ~Kðv=vÞ: ð19Þ

Fig. 2. Input–output relations for the vehicle and the beam response.


Here v ¼ kv is the apparent circular frequency of the

variation as seen by an observer following the moving

vehicle. KðvÞ is the complex amplitude of kð~x þ vtÞ at

~x ¼ 0: Eqs. (14) and (19) imply that a harmonic variation

of kð~x þ vtÞ leads to the following variation of the point

loads,

fjðtÞ ¼ FjðvÞe2ivt

; FjðvÞ ¼ HFjKðvÞKðvÞ;

HFjKðvÞ ¼ 2�uð~xjÞD~xj e2iðv=vÞ~xj ; ð20Þ

where FjðvÞ is the amplitude of fjðtÞ and HFjKðvÞ is the

frequency response function relating fjðtÞ to kð~x þ vtÞ:

With the point loads given by Eq. (20), four linearly

independent harmonic solutions exist to the equation of

motion, Eq. (18), for each j ¼ 1; 2;…; J: The solutions

take the form

uj;nð~x; tÞ ¼ FjðvÞ ~Uj;nð~x;vÞe2ivt

;

j ¼ 1; 2;…; J; n ¼ 1; 2; 3; 4: ð21Þ

Here, ~Uj;nð~x;vÞ are the amplitudes for a harmonically

varying force with unit amplitude at ~x ¼ ~xj; i.e. f ðtÞ ¼

e2ivtdð~x 2 ~xjÞ: Introducing ~UnðvÞ as the amplitudes at ~x ¼

~xj; ~Uj;nð~x;vÞ may be written

~Uj;nð~x;vÞ ¼ ~UnðvÞeiKnð~x2~xjÞ; ð22Þ

where the wavenumbers Kn are the roots to the

characteristic polynomial obtained by assuming solutions

of type (21) in the homogeneous version of Eq. (18), that

is with fðtÞ ; 0:

The vehicle displacement and the interaction terms of the

beam displacement are given as

zðtÞ ¼ ZðvÞe2ivt;

u0;nð~x; tÞ ¼ v2m0ZðvÞ ~U0;nð~x;vÞe2ivt

; n ¼ 1; 2; 3; 4: ð23Þ

Here, ZðvÞ is the first-order vehicle displacement amplitude

and ~U0;nð~x;vÞ ¼ ~UnðvÞeiKn ~x are the amplitudes of the

vehicle displacement for a harmonically varying force

with unit amplitude at ~x0 ¼ 0:

At any point along the beam, only two of the

fundamental solutions uj;nð~x; tÞ; n ¼ 1; 2; 3; 4; are present

for each j ¼ 0; 1;…; J; as it was also the case for �uð~xÞ: For

the sake of convenience, the two wave components existing

behind the respective loads fjðtÞ with reference to the

direction of the velocity v will be assigned the subscripts

n ¼ 1; 2; whereas the components existing in front of the

respective loads are assigned the subscripts n ¼ 3; 4: From

Eqs. (18) and (23), U0ð~x;vÞ; i.e. the amplitude of u0ð~x; tÞ;

may be expressed in terms of ZðvÞ: Inserting the result into

Eq. (17) and assuming that J0 of the point loads are applied

in front of the vehicle, the following frequency response

relation is obtained by isolating ZðvÞ on the left-hand side of

the resulting equation

ZðvÞ ¼ HZKðvÞKðvÞ; HZKðvÞ ¼ NZKðvÞ=DZKðvÞ; ð24Þ

where the numerator NZKðvÞ and denominator DZKðvÞ are

given as

NZKðvÞ ¼ ð2ivc0 þ k0ÞHUSKð0;vÞ; ð25Þ

DZKðvÞ ¼ ð2v2m0 2 ivc0 þ k0Þ

þ ðiv3m0c0 2 v2m0k0ÞX2

n¼1

~UnðvÞ; ð26Þ

respectively. HUSKð0;vÞ is a special case of the frequency

response function relating uSð~x; tÞ to kð~x þ vtÞ;

HUSKð~x;vÞ ¼XJ 0j¼1

HFjKðvÞ

X2

n¼1

~Uj;nð~x;vÞ

!

þXJ

j¼J0þ1

HFjKðvÞ

X4

n¼3

~Uj;nð~x;vÞ

!: ð27Þ

HFjKðvÞ and ~Uj;nð~x;vÞ are previously defined. It is noted that

the first sum is carried out over the point loads in front of the

observation point ~x: Here, the two wave components, which

are moving backwards relative to the convection velocity v;

are included. Behind the observation point, the sum is

carried out over the forward-propagating wave components,

since these are the components that will eventually arrive at

point ~x:

Inserting Eq. (23) into Eq. (18) and making use of Eq.

(24), the contribution from f0ðtÞ to the beam displacement

can be found. The contributions from the discrete loads fjðtÞ;

j ¼ 1; 2;…; J; are given by Eq. (27). Adding the respective

contributions, the frequency response function relating

uð~x; tÞ to kð~x þ vyÞ may eventually be written

Uð~x;vÞ ¼ HUKð~x;vÞKðvÞ;

HUKð~x;vÞ ¼ HUSKð~x;vÞ þ HU0Zð~x;vÞHZKðvÞ; ð28Þ

where

HU0Zð~x;vÞ ¼ m0v2Xj2j¼j1

~UjðvÞeiKj ~x;

{j1; j2} ¼ {1; 2} for ~x # 0;

{j1; j2} ¼ {3; 4} for ~x . 0:

8<: ð29Þ

2.1.3. Random stiffness variation

In the frequency domain, the two-sided auto-spectral

density for the stiffness variation becomes

~SKKðvÞ ¼xc

p

s2k

1 þ x2cðv=vÞ

2; ð30Þ

which is obtained from Eq. (2) by the transformation v ¼

kv: Notice that all the variation lies within the first-order


terms, i.e. ~SKKðvÞ ¼ ~SKKðvÞ: Next, the two-sided auto-

spectral density ~SZZðvÞ for the SDOF vehicle displacement

and the two-sided cross-spectral density ~SUUð~x1; ~x2;vÞ for

the beam displacement at two points ~x1 and ~x2 on the beam

axis may be found. Thus, see, e.g. Ref. [11]

~SZZðvÞ ¼ lHZKðvÞl2 ~SKKðvÞ; ð31Þ

~SUUð~x1; ~x2;vÞ ¼ Hp

UKð~x1;vÞHUKð~x2;vÞ~SKKðvÞ; ð32Þ

where HZKðvÞ and HUKð~x;vÞ are defined previously and

Hp

UKð~x1;vÞ is the complex conjugate of HUKð~x1;vÞ: Again it

should be noticed that the principle of superposition is valid,

because the governing equations are all linear, and that all

variation lies in the first-order terms. From the Wiener–

Khintchine relation, the auto-covariance function CzzðtÞ for

the vehicle displacement and the cross-covariance function

Cuuð~x1; ~x2; tÞ for the beam displacement may be expressed

as

CzzðtÞ ¼ð1

21cosðvtÞ~SZZðvÞdv; ð33Þ

Cuuð~x1; ~x2; tÞ ¼ 2ð1

21ðcosðvtÞ~SR

UU 2 sinðvtÞ~SIUUÞdv: ð34Þ

~SRUU and ~SI

UU are the real and imaginary parts of~SUUð~x1; ~x2;vÞ; respectively, and t ¼ t2 2 t1 is the time

difference between two observations, which in the case of

the beam displacement are related to the points ~x1 and ~x2:

2.2. Finite element simulation approach

To validate the perturbation analysis of Section 2.1, a

finite element analysis is carried out. The theory, which has

been implemented in the numerical examples, is described

in this section.

2.2.1. Derivation of the finite element system matrices

A finite part of the beam is considered. The artificial end

points, or boundaries, of the finite element model are located

at the positions ~x2b behind the vehicle and ~xþb in front of the

vehicle, respectively. To achieve the weak formulation of

the equation of motion for the beam, Eq. (6) is multiplied by

an arbitrary weight function, which is chosen as a virtual

displacement field, ~u ¼ ~uð~x; tÞ; and integration is carried out

by parts,

ð~xþb

~x2b

›2 ~u

›~x2EI

›2u

›~x2þ ~u kð~x þ vtÞu 2 gv

›u

›~x

"

þm v22

G

m

� �›2u

›~x2

!#d~x þ

ð~xþb

~x2b

~u g_u 2 2mv›_u

›~x

þ m€u 2 D›2 _u

›~x22 v

›3u

›~x3

!!d~x

¼ ~uð0; tÞf ðtÞ þ ~uQ 2›~u

›~xM

� �~xþb

~x2b

: ð35Þ

Here, Q ¼ Qð~x; tÞ is the shear force and M ¼ Mð~x; tÞ is the

moment. The moment and shear force as well as the additive

load, f ðtÞ; are discussed below.

Next, the displacement field as well as the virtual

velocity field is put in discrete form

uð~x; tÞ ¼ Nð~xÞaðtÞ; ~uð~x; tÞ ~Nð~xÞ~aðtÞ: ð36Þ

Vectors a and a store the nodal displacements and

rotations in the physical and the virtual field, respectively.

Thus there are two degrees of freedom per node. Accord-

ingly, Nð~xÞ and ~Nð~xÞ are global shape functions of the

dimension 1 £ 2n; where n is the number of nodal points.

The global shape functions are assembled from the local

Hermite shape functions for each element. Defining ~xe as a

local coordinate along an element, ranging between 0 at one

end and Le at the other end, these element shape functions

are given as [12]:

N1ð~xeÞ ¼ 1 2 3~x2

e

L2e

þ 2~x3

e

L3e

;

N2ð~xeÞ ¼ ~xe 1 2 2~xe

Le

þ~x2

e

L2e

!; ð37Þ

N3ð~xeÞ ¼~x2

e

L2e

32 2~xe

Le

� �; N4ð~xeÞ ¼

~x2e

Le

~xe

Le

2 1

� �: ð38Þ

The corresponding local degrees of freedom are a1ðtÞ ¼

uð0; tÞ; a2ðtÞ ¼ uð0; tÞ; a3ðtÞ ¼ uðLe; tÞ and a4ðtÞ ¼ uðLe; tÞ;

where u are the rotations, which in the global moving frame

are defined as uð~x; tÞ ¼ ›uð~x; tÞ=›~x:

Eq. (36) is inserted into Eq. (35). The stationarity

condition of the variation principle then leads to the finite

element method (FEM) formulation

M€a þ C_a þ Ka ¼ f þ fb; ð39Þ

where the system matrices and the boundary load vector are

defined as

M ¼ð~xþ

b

~x2b

~NTmN d~x;

C ¼ð~xþ

b

~x2b

~NT gN 2 2mv›N

›~x2 D

›2N

›~x2

!d~x; ð40Þ

K ¼ KðtÞ ¼ �K þ KðtÞ;

�K ¼ð~xþ

b

~x2b

›2 ~NT

›~x2EI

›2N

›~x2þ ~NT

�kN 2 gv›N

›~x

"

þm v2 2G

m

� �›2N

›~x2þ vD

›3N

›~x3

!#d~x; ð41Þ

KðtÞ ¼ð~xþ

b

~x2b

~NTkð~x þ vtÞN d~x;

fb ¼ fbðtÞ ¼ ~NTQð~x; tÞ2› ~NT

›~xMð~x; tÞ

" #xþb

~x2b

: ð42Þ


The system matrices are all sparse. If the elements are

numbered continuously from one end of the finite part of the

beam to the other, the bandwidth is reduced to a minimum

of four, i.e. the number of degrees of freedom in each finite

element. However, except for the mass matrix they are all

nonsymmetric due to the presence of convection, see

discussion below.

The mean part of the stiffness matrix, �K; must only be

assembled once. The time-varying part, KðtÞ; must on the

other hand be assembled for each time step in the time

integration. However, if the time step Dt in the integration is

chosen with care, the stiffness matrices need only be

computed for a single period of parametric excitation, T :

The rules for choosing the time step Dt and the beam

element length Le are addressed below.

The interaction force from the vehicle enters the additive

load vector f ¼ fðtÞ in the component corresponding to the

degree of freedom, at which the beam is coupled to

the vehicle. In practice, the coupling is described by the

introduction of an additional degree of freedom for

the vertical displacement of the vehicle mass. The vehicle

mass contributes only with a diagonal term in the global

mass matrix, whereas off-diagonal terms enter the damping

and stiffness matrices due to the interaction via the vehicle

spring and dashpot. Thus, no additive forces are applied to

the degrees of freedom associated with the beam. However,

it should be noted that the gravitation on the vehicle mass

must be included in the finite element analysis, i.e. the

vertical load 2m0g is applied to the node corresponding to

the vehicle mass. Otherwise, there will be no deformation of

the system, and hence no vibration.

To ensure that waves propagating from the vehicle

through the beam and support will not reenter the model,

transparent boundary conditions must be applied at ~x2b and

~xþb : The method proposed by Andersen et al. [10] is

implemented. This implies that the moment and shear force,

Mð~x; tÞ and Qð~x; tÞ; at the artificial boundaries are written in

terms of the displacement and the rotation at ~x2b and ~xþb :

Thus the interaction forces at the artificial boundaries are

automatically updated from the degrees of freedom

associated with the end nodes. As a consequence, the

boundary load vector fb is turned into an extra contribution

to the system matrices, so that the only remaining external

load in the model is the additive load due to gravitation on

the vehicle mass. The boundary conditions are derived on

the basis of the approximation that outside the region, which

is discretized with finite elements, there is no variation of

the support stiffness. In other words, the transparent

boundary conditions are kept constant through all time

steps in the integration. If ~x2b and ~xþb coincide with the end

points in the discretization of the equivalent additive line

load in the perturbation approach, i.e. LP ¼ LFE ¼ ~xþb 2 ~x2b ;

the same error will arise in the two methods due to

truncation of the model. In the numerical examples given

below, the same end points are used in the two models. It has

been tested by numerical experiment that the results

obtained with an FE model, in which the artificial

boundaries have been moved farther away from the vehicle,

does not provide results with significantly higher accuracy.

A note should be made regarding the convective term

related to the shear damping, i.e. the term ~NvDð›3N=›~x3Þ;

which appears in the integrand of the stiffness matrix, see

Eq. (41). Differentiation three times of the third-order shape

functions seems inconvenient. A higher accuracy in the

interior of the model would be achieved if one of the

derivatives is transferred to the virtual field by 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, but zero on the virtual displacement.

However, integration by parts of the convective terms

gives rise to additional boundary terms. This complicates

the matter of implementing transparent boundary con-

ditions. Hence, it has been chosen to use the formulation

given in Eqs. (39)–(42).

When a beam on a Kelvin foundation is discretized with

the FEM in the fixed frame of reference, the differential

operators in the integrals comprising the system matrices

become symmetric. However, this is not the case when

convection is present. To overcome this problem, asym-

metric shape functions for the virtual field may be used,

which are different from the shape functions used

to approximate the physical field. In the present

analysis, however, the standard Galerkin approach is

taken, i.e. ~Nð~xÞ ¼ Nð~xÞ: Previous work by Andersen et al.

[10] confirmed that this does not provide any problems of

instability for the kind of problem being investigated, as

long as the convection velocity is below approximately two

times the critical velocity. However, as discussed earlier, it

is generally a problem to carry out the analysis in case of

supersonic velocities. This is due to the fact that a large

region of the beam has to be considered in the perturbation,

and also in the FEM, approach, because the quasi-static

displacement of the beam does not decay exponentially

away from the vehicle. Instead it has a sinusoidal behaviour

in the far-field.

2.2.2. Time integration and discretization considerations

In the first-order perturbation method presented in

Section 2.1, the parametric excitation due to the variation

of the support stiffness is transformed into an additive

excitation. As a consequence, harmonic input with a single

frequency gives rise to harmonic output, or response, with

the same frequency. In contrast to this, in the finite element

method approach, multiple harmonic components will be

present in the response, even 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 problem. Hence, the higher-order

harmonics are included in the solution.

In principle, the standard deviations of the responses due

to the variation of the support stiffness may be computed by

means of traditional Monte Carlo simulation. Here,


a number of random time series of the foundation stiffness

variation are generated on the basis of the auto-spectrum

given by Eq. (2). The random process in this context is the

phase shift. Once a stationary condition has been reached in

each simulation, the standard deviations of the responses

can be found by ergodic sampling.

For the analysis of the present problem, the Monte Carlo

method (MCM) has the disadvantage that the frequencies in

the input are mixed. Since the input for each harmonic

component leads to response with multiple frequencies, it

provides no information about, where in the frequency

domain the relative error on the standard deviation has its

origin. Therefore, an alternative procedure is suggested, in

which the finite element simulation is carried out for one

harmonic component of the stiffness variation at a time.

When the stationary condition is reached, the rms values of

the responses for the individual frequencies are recorded.

Subsequently, Eqs. (33) and (34) are used to calculate the

standard deviations of the responses. For each frequency, or

wavenumber, in the input, the rms values obtained from the

FE simulation may be compared with the spectral

components SZZðkÞ ¼ ~SZZðvÞ and SUUð~x1; ~x2; kÞ ¼~SUUð~x1; ~x2;vÞ; hence providing an exact information

about, where in the wavenumber domain errors arise

between the perturbation method and the FEM results.

Since the rms values of the responses cannot directly be

interpreted as the amplitudes of harmonic displacement time

series, the present method may be understood as a pseudo-

spectral analysis method (P-SAM). It is noted that the

weighted sum of the rms values will still provide an estimate

of the total standard deviation, which on the other hand

would be the only information provided by the MCM.

Besides the fact that the P-SAM gives more information

about the nature of the relative errors than does the MCM,

the method has also some computational advantages over

the MCM. The boundary conditions proposed by Andersen

et al. [10] are limited to the analysis of excitation within a

relatively narrow frequency band and, in a Monte Carlo

simulation including numerous wavenumbers in the input,

many elements have to be used in the model, while the time

step must be very small, in order to obtain satisfactory

results. The discretization considerations are further dis-

cussed below. As a final remark it should be mentioned that

the standard deviations or the rms values of the responses

are in any case calculated with respect to the variation

around the deterministic quasi-static displacements, which

are computed correctly in the perturbation analysis and with

insignificant errors in the MCM or P-SAM. Hence, the

relative errors are found with respect to the first- and higher-

order contributions to the displacement, not with respect to

the total displacement, where the errors would be much

smaller.

The direct time integration scheme proposed by New-

mark [13] is used with the parameters d ¼ 1=2 and a ¼ 1=4

corresponding to constant acceleration over each time step.

Numerical experiment indicates that the time integration is

unstable for the present problem if linear interpolation of the

acceleration is assumed. This prohibits the use of a Wilson-u

integration scheme with u ¼ 1 as well as the Newmark

scheme with d ¼ 1=2 and a ¼ 1=6:

To achieve satisfactory results, the element length Le

must be small enough to ensure that all wave components in

the beam as well as the support stiffness variation are

described with sufficient accuracy. As already mentioned,

third-order interpolation is used for the beam displacement

field. Four elements per wavelength of the flexural waves

are known to provide accurate results for additive load on a

beam supported by a homogeneous Kelvin foundation [10].

In the present analysis, five elements will be used per

wavelength.

In principle, no interpolation of the stiffness variation is

necessary. Analytic expressions for KðtÞ are easily derived

for a sinusoidal variation of kð~x þ vtÞ in Eq. (41). However,

a more adaptable code is obtained, if linear shape functions

are used to interpolate the stiffness variation. Approximately

10 elements are required per wavelength of the stiffness

variation.

When direct time integration is used for the analysis of

wave propagation problems, the time step is determined

from the relation

C ¼lclDt

Le

; C # 1; ð43Þ

where C is the Courant number and c is the phase velocity of

the waves. The relation must be fulfilled for all wave

components in the system. In the present problem, this

includes the stiffness variation in the support. A Courant

number of C ¼ 0:5 is used in the numerical examples.

Furthermore, the time step must be small enough to ensure

that the vibration of the vehicle is described with sufficient

accuracy. Numerical experiment indicates that several time

steps per eigenvibration period are required for optimal

performance of the numerical scheme in terms of accuracy.

However, even with 100 time steps per eigenvibration

period, the vehicle vibrations are only the critical criterion at

low frequencies. At high frequencies, the length and speed

of the bending waves dictate Dt. It is noted that the

frequency of the parametric excitation dominates the

response, once the stationary condition is reached. How-

ever, during the transient part of the response history,

instability in the numerical integration may occur, if the

time step is not small enough to describe the eigenvibration

of the vehicle.

In terms of computation time, it is quite costly that KðtÞ

must be assembled for each time step. A faster code may be

obtained by adjusting the time step, so that a whole number

of time steps, NDt; are used over a single period of

parametric excitation. This way, KðtÞ only needs to be

assembled NDt times. Likewise, after the first period of

excitation, the system matrices produced in the Newmark

integration scheme can be reused from an earlier time step.


This is in particular advantageous, if the transient phase has

a duration of several excitation periods, which is the case for

excitation in the high-frequency range.

3. Numerical examples

An analysis will be carried out for the standard deviation

of the SDOF vehicle mass displacement response, sz ¼ffiffiffiffiffiffiffiffiCzzð0Þ

p; and the standard deviation of the beam displace-

ment response directly under the vehicle, su ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiCuuð0; 0; 0Þ

p: The standard deviations sz and su are

proportional to sk: Hence, they may conveniently be

described by the dynamic amplification factors,

sz ¼sz

sk

; su ¼su

sk

: ð44Þ

The dynamic amplification factors have the unit ½sz� ¼

½su� ¼ m3=N: In the perturbation analysis, sz and su are

linear functions of sk: Therefore, sz and su are independent

of sk: In the finite element analysis, this is not the case,

because the second-order, and higher-order, terms of the

response are not disregarded.

For convenience, the following parameters are intro-

duced:

v0 ¼

ffiffiffiffiffik0

m0

s; z0 ¼

c0

2ffiffiffiffiffiffim0k0

p ;

z ¼g

2ffiffiffiffim �k

p ; b ¼D

2ffiffiffiffiffimG

p : ð45Þ

Here v0 is the eigenfrequency of the SDOF vehicle. The

damping ratios z0 and z are related to the vertical

displacements of the vehicle mass and the beam, respect-

ively, whereas b is the damping ratio for the shear layer.

3.1. Verification of the model

In Section 3.2, a parameter study will be carried out, in

which the perturbation approach of Section 2.1 is used to

investigate the influence on the displacement responses

from the bending and foundation stiffness as well as the

correlation length of the stiffness variation. Firstly,

however, an analysis is performed with the aim of

determining the level of stiffness variation, beyond which

the perturbation analysis provides unsatisfactory results.

The response spectra for the vehicle and beam displace-

ment are computed with the perturbation method and the FE

model. In case of the FE approach, different levels of the

stiffness variation are considered, described in terms of the

variation coefficient V ¼ sk= �k: It is noted that the results of

the P-SAM analysis is not a true frequency spectrum, but the

rms values of the response, which are computed with

the method, may be compared directly with the results of the

perturbation method.

The SDOF vehicle has the parameters

m0 ¼ 1000 kg; v0 ¼ 2p s21; z0 ¼ 1:0; ð46Þ

i.e. the vehicle is critically damped. The parameters listed in

Eq. (46) are assumed to be typical for an automobile. The

beam has the mass per unit length and the damping ratios

m ¼ 1000 kg=m; z ¼ 0:1; b ¼ 0:1: ð47Þ

The properties listed in Eqs. (46) and (47) will be used for

all analyses in the examples. Here, the following parameters

are used in addition to those already listed,

EI ¼ 107 N m2; �k ¼ 107 N=m2

;

G ¼ 5 £ 106 N m=m; xc ¼ 5 m: ð48Þ

Figs. 3–6 show the response spectra and pseudo-spectra

obtained with the perturbation method and the P-SAM,

respectively, for the four vehicle velocities 25, 50, 75 and

100 m/s. For comparison it should be mentioned that the

critical velocity in the present case is 106.42 m/s, meaning

that all four velocities are sub-sonic. As discussed above,

only the part of the displacement, which is due to the stiffness

variation, has been included. In the P-SAM the results are

Fig. 3. Response spectra for a model with the parameters given in Eqs.

(46)–(48), v ¼ 25 m=s and various amounts of support stiffness variation.


obtained by a subtraction of the quasi-static displacement

from the total displacement. The quasi-static displacement is

found by solving the equation �a ¼ �K21f: Here, �a are the

nodal displacements and rotations, when no variation of the

stiffness has been applied. Alternatively, for the beam, Eq. (8)

may be used to find �uð0Þ: It is noted that �a has been used as

initial conditions for the time integration in order to reduce

the duration of the transient phase to a minimum.

Generally the qualitative behaviour of the response is the

same at all levels of stiffness variation for each individual

velocity. As expected, the results obtained with the FE model

for the variation coefficients 0.20 and below lie close to the

solution, which has been computed with the perturbation

approach. For V ¼ 0:30 the perturbation approach seems less

likely to provide satisfactory results. Clearly the relative

error is confined to the low-wavenumber range. For high

wavenumbers of the stiffness variation the relative error is

insignificant, regardless of the variation coefficient V : This

information would not be provided by the MCM, because no

means exist, by which the higher-order harmonics from low-

wavenumber excitation can be filtered from the first-order

harmonics of high-wavenumber excitation.

The relative errors on the standard deviations of the

responses are listed in Tables 1 and 2 for the vehicle and

beam displacement, respectively. It is observed that the

perturbation method does indeed provide satisfactory

results for variation coefficients up to 0.30 and including

with errors smaller than 10%; but only for velocities

below a certain value. For the velocity 100 m/s, the

relative error is approximately two times the error, which

is obtained at the lower velocities. Hence it may be

concluded that the perturbation method is better suited

for velocities well below vcr: Another interesting feature

concerning the relative error on the standard deviations

for the velocity 100 m/s is that the error has the opposite

sign of the errors, which are recorded for the velocities

75 m/s and below. This suggests that there may exist an

intermediate velocity, at which the error is close to zero.

To investigate this matter, an analysis is carried out, and

it has been found that the vehicle velocity 93 m/s gives

rise to very small relative errors, even for the variation

coefficient V ¼ 0:4: The response spectra for v ¼ 93 m=s

are plotted in Fig. 7, and the relative errors are listed in

Tables 1 and 2.


(46)–(48), v ¼ 50 m=s and various amounts of support stiffness variation. Fig. 5. Response spectra for a model with the parameters given in Eqs.



For the velocity v ¼ 25 m=s; Fig. 3 shows that the FE

model and the perturbation model do not provide the same

solution, even for the variation coefficient V ¼ 0:01: A

further analysis, the results of which are not included in the

paper, indicates that this is a general tendency for relatively

low velocities and bending stiffnesses and high shear

stiffnesses. Thus, when the quantity G 2 mv2 becomes

large in comparison to the bending stiffness, discrepancies

may arise between the results computed with the two

models, which are not due to shortcomings of the first-order

perturbation approach.

To further clarify the nature of the response at different

levels of the stiffness variation, an FE analysis is carried out

for the variation coefficients V ¼ 0:01 and 0.5. The vehicle

velocity is set to 50 m/s, and the analysis is carried out for a

single harmonic component of the stiffness variation with

arbitrary, but relatively low wavenumber k: Fig. 8 shows the

vehicle response and the beam response at ~x ¼ 0 during



Table 1

Relative error (in %) between the standard deviations of the vehicle

displacement response found by finite element and perturbation analysis

v V

0.01 0.10 0.20 0.30 0.40

25 0.58 1.15 2.95 6.20 11.40

50 20.17 0.49 2.56 6.32 12.37

75 0.19 0.85 2.91 6.62 12.39

100 20.39 22.38 28.12 216.64 225.86

93 0.90 0.79 0.38 20.70 23.08



Table 2

Relative error (in %) between the standard deviations of the beam

displacement response at ~x ¼ 0 found by finite element and perturbation

analysis

v V

0.01 0.10 0.20 0.30 0.40

25 0.58 1.15 2.92 6.13 11.28

50 20.36 0.20 1.97 5.18 10.36

75 0.07 0.60 2.27 5.25 9.91

100 20.31 22.02 26.83 213.77 221.17

93 0.80 0.70 0.35 20.58 22.59


the first five periods of parametric excitation, that is

including the transient phase. The response for V ¼ 0:5

has been divided by 50 in order to obtain response levels of

the same magnitude in the two cases.

The response obtained for the variation coefficient

V ¼ 0:01 varies harmonically with time, as it is assumed

in the first-order perturbation analysis. It is observed that

the stationary condition is reached almost immediately

(Fig. 8). For V ¼ 0:5 the response is however not

harmonic, in accordance with the discussion given in

Section 2.2.2. The negative displacements due to a local

softening of the support are much more pronounced than

the positive 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 be seen in Figs.

3–5, namely that the rms value of the response becomes

greater, when the stiffness variation is increased. For

high velocities, the reduction in the tips will be more

pronounced than the amplification of the dips in the

response, when the stiffness variation is increased, thus

leading to a reduction of the rms response, see Fig. 6.

3.2. Parameter study

A perturbation analysis is carried out for various

combinations of the bending stiffness EI and the correlation

length xc: Two different supports have been examined. In

Fig. 9 the results are given for the velocity range v [½1:50; 150� m=s and a beam resting on a foundation with

�k ¼ 107 N=m2 and G ¼ 106 N m=m: The parameters are

rather arbitrarily chosen but may to some extent represent a

soft elastic layer. Fig. 10 shows the results for an even softer

foundation with �k ¼ 106 N=m2 and G ¼ 105 N m=m: It is

noted that the critical velocities in the first case are 54.77,

144.91 and 448.33 m/s for the bending stiffnesses 105, 107

and 109 N m2, respectively. In the second case, the very soft

support, the critical velocities are 27.06, 80.15 and

251.69 m/s, respectively.

From the curves in Figs. 9 and 10 it may be concluded

that the amplification of the vehicle response (thick line) is

generally increasing with the correlation length, xc: In most

Fig. 8. Response of the vehicle mass and the beam at ~x ¼ 0 for the parameters

given in Eqs. (46)–(48) and for a wavenumber k < 0:2:The vehicle and beam

response for V ¼ 0:01 are plotted with the signatures ( ) and (—),

respectively, whereas the vehicle and beam response for V ¼ 0.5 are plotted

with the signatures ( ) and (- -) . The results for V ¼ 0.5 are divided by 50 in

order to get response levels similar to those for v ¼ 0.01. Further, all responses

are normalized with respect to vmax ¼ max(v(t)) for V ¼ 0.01.

Fig. 9. Dynamic amplification of vehicle response ( ) and beam response (—) �k ¼ 107 N=m2 and G ¼ 5 £ 107 N m=m: ½sz� ¼ ½su� ¼ m3=N and ½v� ¼ m=s:


cases this also applies to the beam displacement (thin line),

though a few exceptions exist. For subsonic velocities, the

amplification of both the vehicle and the beam response

grows significantly with the velocity. The vehicle response

increases less than the beam response. This phenomenon is

most pronounced for the stiffer support, i.e. Fig. 9. In the

velocity range near the critical velocity, a local dip occurs in

the response of the vehicle and, in most cases, also in the

response if the beam. After this local minimum, the

response level reaches a local, or even global, maximum.

When the velocity is further increased beyond vcr; the

response of the beam and in particular the response of the

vehicle decrease.

4. Conclusions

The response of a vehicle moving uniformly along a

track with random foundation stiffness has been analysed.

The vehicle is modelled as an SDOF oscillator and the track

is modelled as a Bernoulli–Euler beam resting on a Kelvin

foundation, which has been modified by the inclusion of a

shear layer with both stiffness and damping. The vertical

spring stiffness of the support is a random process along the

track with a variation given by an Ornstein–Uhlenbech

filtration of Gaussian white noise.

A first-order perturbation approach has been developed

for the analysis of the vehicle and track response to the

stiffness variation. The results obtained with the present

method have been compared with a numerical solution

achieved with a finite element simulation by means of a

pseudo-spectral 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 obtained with the perturbation approach is possible.

This is an advantage over traditional Monte Carlo

simulation, which may only be used to compute an estimate

of the total standard deviation of the response.

The quality of the perturbation method depends on the

velocity of the vehicle. For low velocities, a relative

error on the standard deviation of the responses of less

than 10% may be expected for a variation coefficient of

30% for the support stiffness. It should be emphasized

that the relative errors refer to the first- and higher-order

response only. The quasi-static displacement response is

computed accurately in either method. For high vel-

ocities, just below the critical velocity of the beam-

support system, the relative error is somewhat higher,

about 15%. For certain intermediate velocities, the results

obtained with the perturbation analysis have been found

to be more accurate. However, generally the perturbation

method breaks down for variation coefficients of the

stiffness variation beyond 30%.

The present perturbation method is advantageous to

finite element simulation in terms of computation time.

As such the method is a powerful tool for a parametric

study of the given problem. Furthermore, the proposed

perturbation analysis may be used to provide preliminary

results for the response, even when the variation

coefficient of the stiffness is larger than 30%. Based on

these results, an optimal distribution may be found for

the wavenumbers, at which the finite element analysis

should be carried out. Methods proposed in the literature

may be used to calibrate the model so that a description

of a realistic track on a half-space or elastic layer may

be achieved.

Fig. 10. Dynamic amplification of vehicle response ( ) and beam response (—) �k ¼ 106 N=m2 and G ¼ 5 £ 106 N m=m: ½sz� ¼ ½su� ¼ m3=N and ½v� ¼ m=s:


A parameter study indicates that the responses of the

moving vehicle and the beam underneath increases with the

velocity and the correlation length of the stiffness variation.

However, for vehicle velocities close to the critical velocity vcr

of the beam/support, the response of the beam and, in

particular, the vehicle mass due to the spring stiffness variation

drops dramatically. A local, or global, peak in the response may

be present for velocities just above the critical velocity, but for

velocities beyond vcr; the response is generally decreasing.

Acknowledgements

The authors would like to thank the Danish Technical

Research Council for financial support via the research

project: ‘Damping Mechanisms in Dynamics of Structures

and Materials’.

References

[1] Dieterman HA, Metrikine A. The equivalent stiffness of a half-space

interacting with a beam. Critical velocities of a moving load along the

beam. Eur J Mech, A/Solids 1996;15(1):67–90.

[2] Metrikine A, Popp K. Steady-state vibrations of an elastic beam on a

visco-elastic layer under a moving load. Arch Appl Mech 2000;70:

399–408.

[3] Vallabhan CVG, Das YC. Modified Vlasov model for beams on

elastic foundation. J Geotech Engng 1991;117:956–66.

[4] Krenk S. Mechanics and analysis of beams, columns and cables.

Copenhagen: Polyteknisk Press; 2000.

[5] Oscarsson J. Dynamic train/track interaction–linear and non-linear

track models with property scatter. PhD Thesis. Department of

Solid Mechanics, Chalmers University of Technology, Goteborg;

2001.

[6] Yoshimura T, Hino J, Katama T, Ananthanarayana N. Random

vibration of a non-linear beam subjected to a moving load: a finite

element analysis. J Sound Vib 1988;122(2):317–29.

[7] Fryba L, Nakagiri S, Yoshikawa N. Stochastic finite elements for a

beam on a random foundation with uncertain damping under a moving

force. J Sound Vib 1993;163(1):31–45.

[8] Muscolino G, Ricciardi G, Impollonia N. Improved dynamic analysis

of structures with mechanical uncertainties under deterministic input.

Probab Engng Mech 2000;15:199–212.

[9] Andersen L, Nielsen SRK, Iwankiewicz R. Vehicle moving along an

infinite Euler beam with random surface irregularities on a Kelvin

foundation. ASME J Appl Mech 2002;69:69–75.

[10] Andersen L, Nielsen SRK, Kirkegaard PH. Finite element

modelling of infinite Euler beams on Kelvin foundations exposed

to moving loads in convected co-ordinates. J Sound Vib 2001;

241(4):587–604.

[11] Lin YK. Probabilistic theory of structural dynamics. New York:

McGraw-Hill; 1967.

[12] Ottosen NS, Petersson H. Introduction to the finite element method.

London: Prentice Hall; 1992.

[13] Newmark NM. A method of computation for structural dynamics.

ASCE J Engng Mech Div 1959;85(EM3):67–94.


Vibrations of a track caused by variation of the foundation stiffness

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