Vibrations of a track caused by variation of the foundation stiffness
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Transcript of 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).
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.
L. Andersen, S.R.K. Nielsen / Probabilistic Engineering Mechanics 18 (2003) 171–184174
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
L. Andersen, S.R.K. Nielsen / Probabilistic Engineering Mechanics 18 (2003) 171–184 175
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
L. Andersen, S.R.K. Nielsen / Probabilistic Engineering Mechanics 18 (2003) 171–184176
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,
L. Andersen, S.R.K. Nielsen / Probabilistic Engineering Mechanics 18 (2003) 171–184 177
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.
L. Andersen, S.R.K. Nielsen / Probabilistic Engineering Mechanics 18 (2003) 171–184178
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.
L. Andersen, S.R.K. Nielsen / Probabilistic Engineering Mechanics 18 (2003) 171–184 179
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.
Fig. 4. Response spectra for a model with the parameters given in Eqs.
(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.
(46)–(48), v ¼ 75 m=s and various amounts of support stiffness variation.
L. Andersen, S.R.K. Nielsen / Probabilistic Engineering Mechanics 18 (2003) 171–184180
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
Fig. 6. Response spectra for a model with the parameters given in Eqs.
(46)–(48), v ¼ 100 m=s and various amounts of support stiffness variation.
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
Fig. 7. Response spectra for a model with the parameters given in Eqs.
(46)–(48), v ¼ 93 m=s and various amounts of support stiffness variation.
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
L. Andersen, S.R.K. Nielsen / Probabilistic Engineering Mechanics 18 (2003) 171–184 181
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:
L. Andersen, S.R.K. Nielsen / Probabilistic Engineering Mechanics 18 (2003) 171–184182
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:
L. Andersen, S.R.K. Nielsen / Probabilistic Engineering Mechanics 18 (2003) 171–184 183
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’.
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