Methods of Simulation of Railway Wheelset Dynamics taking into account Elasticity

Abstract

Two simulation techniques for analyzing flexible wheelset dynamics are presented. They are applied within multibody approach and implemented in "Universal mechanism" software. Equations of wheelset motion are derived using floating frame of reference and component mode synthesis methods. Modal analysis is carried out in external FEA software. Kinematics of a wheel profile is described taking into account flexible displacements of wheelset nodes. In the first techniques, Lagrangian approach is applied to obtain all terms of equation of motion including inertia forces. In the second one, Eulerian approach is simulated in the stage of integration of equation of motion. Non-rotating finite element mesh of the wheelset is considered using the interpolation of flexible displacements in the nodes. The first simulation results obtained using both approaches are presented. These results confirm correctness of the suggested techniques.

Full text

First International Conference on Rail Transportation
Chengdu, China, July 10-12, 2017
Methods of Simulation of Railway Wheelset Dynamics taking into
account Elasticity
Gennady Mikheev1*, Dmitry Pogorelov1 and Alexander Rodikov1
1.Laboratory of computational mechanics, Bryansk State Technical University, Bryansk,
241035, Russia
*Corresponding author email: mikheev@universalmechnism.com
Abstract: Two simulation techniques for analyzing flexible wheelset dynamics are presented. They
are applied within multibody approach and implemented in "Universal mechanism" software.
Equations of wheelset motion are derived using floating frame of reference and component mode
synthesis methods. Modal analysis is carried out in external FEA software. Kinematics of a wheel
profile is described taking into account flexible displacements of wheelset nodes. In the first
techniques, Lagrangian approach is applied to obtain all terms of equation of motion including inertia
forces. In the second one, Eulerian approach is simulated in the stage of integration of equation of
motion. Non-rotating finite element mesh of the wheelset is considered using the interpolation of
flexible displacements in the nodes. The first simulation results obtained using both approaches are
presented. These results confirm correctness of the suggested techniques.
Keywords: dynamics of flexible wheelset; wheel–track interaction; railway vehicle dynamics
1 Introduction
Multibody software gives an effective tool for
researches of mechanical systems which can be
presented by rigid or flexible bodies connected
via joints and force elements. Development of
the models of wheel-rail contact is one of basic
problems, which should be solved to implement
this approach for dynamic analysis of railway
vehicles. Simulation of a wheelset by rigid body
interacting with the massless rail is the
acceptable approach for analysis of vehicle
dynamics in the frequency range up to 30 Hz.
However, many studies are impossible without
taking into account the elasticity of wheelsets
and without detailed models of railway including
its inertial and flexible properties. Examples of
such researches are analysis of high-frequency
(including sound) vibrations of wheelsets and
rails, calculating stressed-deformed state and
durability of wheelsets, study of corrugation of
wheels and rails etc.
The model of the wheelset developed by
Morys for study of out-of-roundness
enlargement of high-speed-train wheels is
presented in (Morys, 1999). The wheelset is
mounted on ICE-1 coach. Its elasticity is
simulated using a rigid–body method. The
wheelset is divided onto several rigid bodies
interconnected by springs and dampers. The
model includes eight rigid bodies: four brake
disks, mounted on the axle, two wheels and two
axle ends. The connecting force elements
simulate bending and torsional flexibility. The
main problem of similar approaches is the
parameters identification. Only several lowest
natural frequencies and corresponding modes are
taken into account.
Another approach consists in use of
continuous models of wheelsets with additional
lumped elements. A two-dimensional model was
developed by Szolc (Szolc, 1998) to study the
coupled vibrations of the wheelset and track.
The model allows analysis of bending, torsional
and lateral vibration in the frequency range of

First International Conference on Rail Transportation 2017
30–300 Hz. It includes the axle as continuous
beams having bending and torsional flexibility
but rigid axially. The wheels and brake disks are
presented by rigid rings connecting to axle with
massless elastically isotropic membranes.
Application of such models in multibody
software requires developing specialized
procedures and algorithms.
Currently, the modal approach is the most
common and universal way for dynamic
simulation of the flexible wheelsets. The modes
of wheelsets are calculated using finite element
(FE) method. With the progress of computers,
detailed finite element models exactly describing
geometry, flexible and inertial characteristics of
wheelsets became available.
Wheelset kinematics is considered as the sum
of its rigid body motion and small elastic
displacements. In some papers mentioned below,
rigid body displacements are also supposed to be
small. Within the multibody approach, dynamics
of a flexible structure is usually simulated
applying floating frame of reference method for
description of its motion as absolutely rigid
body. This method is presented, for example, in
(Shabana, 1997).
The main difficulty in implementation this
approach to flexible wheelset dynamics is the
computation of the moving contact forces and
evaluation of the generalized forces. In order to
overcome this problem, the attempts to derive
the equation of motion for the wheelset with not
rotating finite element mesh are made. Similar
approaches are commonly called as Eulerian
approach as opposed to Lagrangian one that is
traditionally applied in multibody dynamics. The
Eulerian approach is proposed in recent
publications (Baeza et al. 2008; Vila et al. 2011;
Frigerio, 2010; Kaiser and Popp, 2006; Kaiser et
al. 2007; Kaiser, 2012). The main advantage of
the Eulerian approach (coordinates) consists in
small displacements of contact points between
the wheel and rail relative to the local coordinate
system of the wheelset. It allows applying the
contact forces to several predetermined nodes or
even to a single node and thus to increase the
effectiveness of simulation.
In papers (Fayos et al. 2007; Baeza et al.
2008; Vila et al. 2011), equations of motion of a
rotating flexible body are considered. They are
derived in Lagrangian coordinates using the
modal approach. These equations are reformed
to Eulerian coordinates with the help of a
transformation matrix of modal coordinates
introduced by the authors. As the result, the
system of ordinary differential equations with
the constant matrices is obtained. The equations
describe the straight–line motion of the flexible
wheelset with the constant angular speed in an
inertial coordinate system.
In paper (Frigerio, 2010), the similar
equations are derived directly in Eulerian
coordinates using the material derivative for
calculation of speed of wheelset particle.
In papers (Kaiser and Popp, 2006; Kaiser et
al. 2007; Kaiser, 2012), flexible displacements
of the wheelset are considered relative to the
attached coordinate system not rotating around
main axis. Cylindrical coordinates are
introduced. The expressions for flexible
displacements are written applying the modal
approach and finite Fourier series. Eigenmodes
are calculated in the local reference frame.
Applying symmetry properties of the wheelset
and taking into account the periodicity of the
eigenmodes, the expressions are transformed to
new modal coordinates which define flexible
displacements in the not rotating coordinate
system attached to the wheelset.
In paper (Guiral et al. 2015), equations of
motion of the flexible wheelset are derived in a
noninertial frame of reference moving with the
vehicle. On the one side, this approach uses the
assumption about small changes of all degrees of
freedom of the wheelset except the rotation
around the main axis; on the other side, it allows
analysis of motion in the curved section of a
railway track. Dynamics of the flexible wheelset
is studied separately from the vehicle using the
assumption that the wheelset elasticity does not
influence the displacements of a bogie. For
calculation of interconnection forces between
the wheelset and the bogie, the values of
displacements and velocities of an attachment
points on the bogie are required. In order to
compute these values, a preliminary simulation
in multibody software packages is carried out.
In this paper, the simulation methods of the
flexible wheelsets dynamics developed by
authors and implemented in «Universal
mechanism» (UM) software
(www.universalmechanism.com) are considered.
Kinematics of any flexible subsystem is
described in UM using the floating frame of
reference and the component mode synthesis
methods. The modal analysis of a flexible
subsystem is carried out in external FEA
programs such as ANSYS, MSC.NASTRAN.
The Lagrangian approach is used for derivation
of equations of motion. In order to apply this

First International Conference on Rail Transportation 2017
method to simulation of rotating wheel, the
following algorithms have been developed: 1)
calculation of kinematics of a wheel profile
using positions and speeds of rim nodes located
near to the contact area at the moment as well as
2) calculation of generalized forces for the
wheel-rail interaction in arbitrary points of the
rolling surface. The obtained equations include
inertial forces without simplifications. Thus,
dynamic simulation of a vehicle in curved
railway sections and with acceleration is
possible.
Another approach suggested in the paper
consists in numerical integration of equations of
motion, which excludes the rotation of finite
element mesh of the wheelset. This solution
resembles the Eulerian or complex approaches
2 Theoretical model
In accordance with the underlying approaches of
UM, the flexible wheelset can move arbitrary as
a rigid body, but displacements of its points
because of deformations are assumed to be
small.
2.1 Kinematics of flexible wheelset
Position of an arbitrary point K relative to the
global coordinate system SC0 is the sum of the
radius-vector to the origin of the local SC1 and
the radius-vector of the point relative to SC1
(Figure 1):
)1(
01
)0(
1
)0(
kk
pArr  , (1)
where A01 is the rotation matrix of SC1 relative
to SC0, the vectors are presented in the
coordinate systems marked by upper indices in
brackets.
Fig. 1 Floating frame of reference of
wheelset
Flexible displacements of wheelset points are
calculated applying the finite element method
and modal approach. Node displacements in the
local reference frame are the product of the
modal matrix and the matrix-column of modal
coordinates:
Hwhx 

 H
j
jj w
1
, (2)
where x is the N×1 matrix-column of nodal
coordinates, N is the number of degrees of
freedom of the FE model, hj is the j-th mode of
the flexible wheelset, wj is the j-th modal
coordinate, H is the number of used modes, H is
the N×H modal matrix.
If a node is placed in point K, its flexible
displacements can be presented by the product
wHd kk

, (3)
where Hk is the part of the modal matrix
corresponding to node K. Then, the radius-vector
of point K can be expressed as follows
(Figure 2):
)( )1(
01
)0(
1
)0( wHρArr kkk  , (4)
where
)1(
k
ρ is constant radius-vector of point K
in the undeformed state.
Fig. 2 Local coordinates of wheelset point
The flexible modes hj are calculated in
accordance with the Craig-Bampton method
(Craig and Bampton, 1968; Craig, 2000). At the
beginning, interface nodes are selected in joint
points and in points of force elements
attachment. Then the constraint static modes
from unit displacements in the interface nodes
and the fixed-interface normal modes are
calculated. The number of used normal modes is
chosen by a researcher depending on an
necessary frequency range.
SC1
K
dk
k
SC0
SC1
K
p
k
r1

First International Conference on Rail Transportation 2017
The examples of constraint and
fixed-interface modes are shown in Figure 3.
Two interface nodes are selected in the axle
ends.
Fig. 3 Examples of wheelset component
modes
In order to exclude rigid body motion relative
to the local reference frame, component modes
are transformed using solution of generalized
eigenvalue problem with the reduced matrices of
the wheelset.
2.2 Equations of motion
Equations of motion of flexible wheelsets are
derived using Lagrange's equations of the second
kind:
i
ii
Q
q
T
q
T
dt
d

























, (5)
where T is the kinetic energy, qi is the i-th
generalized coordinate, Qi is the i-th generalized
force. The lumped approach is applied for
calculation of kinetic energy:


k
k
T
kk
V
TmdVT vvvv 2
1
2
1

, (6)
where

is wheelset material density, mk is nodal
mass, vk is the node velocity that is obtained by
differentiation of radius-vector (1):
qHABwHρAE
rv











kkk
kk
01
)1()1(
01
)0()0(
)(
~, (7)
where B is Jacobian matrix of angular velocity
with relation to generalized velocities, symbol
“~” means a skew-symmetric matrix created by
vector, the operation is applied to the result in
brackets. The matrix-column of generalized
coordinates consists of radius-vector of origin of
the local reference frame r, 31 array of its
orientation angles  and H1 array of modal
coordinates w:











w
φ
r
q. (8)
The result equation can be written in the
following form:
qDCqffkqM

 ag , (9)
where M is the time-dependent mass matrix, k is
the matrix-column of inertia forces including
gyroscopic and Coriolis forces, fg is the gravity
force, fa is the applied forces, C is the stiffness
matrix, D is the matrix of structural damping.
Stiffness matrix has the single diagonal block
corresponding to the modal coordinates. This
block contains squares of cyclic frequencies

k
on the main diagonal because of modes are
M-normalized. Other blocks are zero. Matrix D
has the same structure. The nonzero diagonal
block is specified by ratios k of the critical
damping for each of the modes, corresponding
to the reduced equations:
02 2 kkkkk www


, k=1..H, (10)
where 2
kkk d


.
Roots of the characteristic equations for (10)
are the following:
22
2,1 kkk
ki

 , (11)
where i is the imaginary unit.
The critical damping is the value
kk


corresponded to conversion of the
harmonic process to the aperiodic one. Then the
diagonal block is defined as follows:
k
kkk
d


2

, k
k



,
where the value 0

k

should be inputted.
x
Constraint mode from unit
rotation around X-axis
Fixed interface normal mode.
x
z

First International Conference on Rail Transportation 2017
2.3 Model of Contact forces
Simulation of railway vehicles surely requires an
accurate evaluation of rail-to-wheel contact
forces. The multi-point non-elliptical contact
model is used to determine the contact forces,
see Figure 4. A semi-Hertzian contact model
based on virtual interpenetration is used for the
normal contact problem (Piotrowski and Kik,
2008). The calculation of tangential creep forces
is based on the FASTSIM algorithm by Kalker
(Kalker, 1982).
Fig. 4 Sample result of multi-point contact
model
2.4 Kinematics of wheel profile
One of the main problems of dynamic
simulation of a flexible wheelset is calculation
of contact forces. These are movable forces for
the wheelset, i.e. their application points are
changed relative to the local frame and in
general do not coincide with any node. In
addition, a description of profile kinematics
taking into account the wheel flexibility should
be proposed.
For rigid wheelset, a wheel profile position is
specified in the rail coordinate system SCR0
Y0Z0 with the origin in the central point on rail
surface (Figure 5). Axis Z0 is parallel to Z-axis
of track. The coordinate system of rail profile
YrZr is turned on the r0 angle relative to SCR0
due to the rail inclination. Position and
orientation of the wheel profile coordinate
system SCW YwZw relative to SCR0 are defined
by shifts Z and Y of the origin and by the
angle . The angles r0 and  are supposed to
be small.
The wheel profile is non-deformable. Position
of any profile point relative to the global SC0 is
defined by the position and orientation of the
wheelset local SC1 and by the position of SCW
origin relative to SC1.
Similar to the rigid wheelset, the profile of the
flexible wheelset is also assumed to be
non-deformable but its position and orientation
are computed taking into account displacements
of nodes on wheel rolling surface applying
ordinary least squares.
Fig. 5 Relative position and orientation of
wheel and rail profiles
Let us shortly consider this method. The
wheelset FE model is created by rotation of a
planar half-section mesh around the wheelset
axis with angle the step . This mesh must
contain some nodes locating exactly on the
wheel profile. Thus, there are nodes belonging to
the wheel profile in each i-th section turned on
the angle i. Position and orientation of the
wheel profile in contact with the rail are
calculated taking into account flexible
displacements of the intersection points of
profile with the mesh lines between two sections
(Figure 6). The point displacements are
computed by interpolation of the corresponding
values of the neighboring nodes.
Fig. 6 Intersection points of profile with the
mesh lines
Let us introduce the following designations:
0i
ρ is the position of i-th profile point without
deformations (Figure 7); i
r

is the flexible
displacement of the point; )( , R

ZX ,


is the unknown shift and rotation of the profile
due to deformations.
Points of contact
profile
Finite element nodes
Contact profile

First International Conference on Rail Transportation 2017
Positions of the points taking into account
their flexible displacements can be written as the
sum
iii rρρ

0. (12)
Fig. 7 Position of i-th profile point in SCW
On the other hand, the point positions can be
expressed in terms of the profile position and
rotation as follows:
α
xiii  eρRρR 00
~
, (13)
where x
e is the unit vector along the X–axis
orthogonal to the picture plane (the direction of
vehicle motion). Then, the unknown values R

and


can be found from the minimization of
the residual
)()(),( ii
T
i
T
iρRρRR 


. (14)
2.5 Calculation of contact forces
A FE-model of wheelset consists of eight-node
hexahedrons. Let us consider the basic ideas for
computation of generalized forces corresponding
to a force applied in an arbitrary point of an
element facet. In other words, the force acting in
any surface point should be transformed into
nodal forces and then generalized forces are
calculated using the common algorithm for
flexible bodies.
Two algorithms are implemented in UM to
compute nodal forces: 1) a simplified one
considering only the polygon geometry under
the acting force and 2) the distribution of applied
force using shape functions of the finite element.
If the simplified algorithm is applied, the
external force is distributed between the polygon
nodes in two stages inversely to the distances
from the application point to the nodes
(Figure 8).
To describe the algorithm of distribution force
using shape functions, the system of
dimensionless coordinates

,

,

with an origin
in the center of the hexahedron is introduced
(Zienkiewicz and Taylor, 1989) (Figure 9).
Fig. 8 Simplified algorithm of calculation of
nodal forces
The values of the dimensionless coordinates
in the element nodes are equal to 1. The
displacements of element points along axes x, y
and z are correspondingly designated as u, v and
w. Then, displacement fields are defined by the
following expressions:
e
uuN ),,(),,(


,
e
vvN ),,(),,(


, (15)
e
wwN ),,(),,(


,
where
T
euu ],...,[ 81
u, T
evv ],...,[ 81
v,
T
eww ],...,[ 81
w, ],...,[ 81 NN

N,
)1)(1)(1(81 iiii
N


, i=1..8,
for example,
)1)(1)(1(81
3


N.
Fig. 9 Distribution of applied force using
shape functions of finite element
The nodal forces from the force acting in
point K are calculated as follows:
kxKKK
T
ex F),,(

NF ,
kyKKK
T
ey F),,(

NF , (16)
kzKKK
T
ez F),,(

NF ,
where
T
xxex
FF ],...,[
81
F, T
yyey FF ],...,[ 81
F,
Z0
Y0
0i

i
1
L
2
L1
Fk
2
3
4
1
L
11
Fk12
Fk11
K
L12
L22
Fk22
1
2
Fk1
Fk2
4
3
2
K
2
K
1
Fk1
L
21
Fk2
K
Fk
F6
F7
F5
F8
2
1
3
7
8
6
4
y
,

x
,

z
,

5
Fk21

First International Conference on Rail Transportation 2017
;],...,[ 81
T
zzez FFFKKK

,, are the
dimensionless coordinates of point K;
kzkykx FFF , , are the projections of force Fk.
Coordinates KKK

,, are calculated for
given values of the Cartesian coordinates of
point K with an iterative algorithm like the
Newton method.
2.6 Simulation of wheelset dynamics without
rotation of finite element mesh
Equations of motion (9) are integrated using
implicit multistep methods like Park method
(Park, 1975; Pogorelov 1998). This method
implements predict-corrector calculation
algorithm.
Let us consider the modelling of the flexible
wheelset with not rotating finite element mesh.
This method simulates the Eulerian approach at
the stage of integration of equations of motion.
Before calculating predicted values of
generalized coordinates on a new integration
step, node coordinates are turned on the angle
dα. The predicted increment of wheelset rotation
around main axis is used.
Let


d, where


is the angle step
of the wheelset finite element mesh (Figure 10),
],0[ max


, 1
max


for linear interpolation
that is considered below.
Fig. 10 To description of Eulerian approach
The interpolation is applied to flexible
displacements in the nodes of mesh. Let k
r

is
the flexible displacement in node with index k
on the previous integration step, k
r

is the
displacement in the point k

after rotation of
the wheelset on dα (Figure 10). The following
interpolation expression can be written:
)(kpkk ba rArr

. (17)
Here p(k) is the previous node for node with
index k, i.e. the node that takes the place of node
k after rotation on


, )(


y
AA is the
matrix of rotation about wheelset axis on


.
For linear interpolation, the values of the
coefficients are


ba ,1 .
The flexible displacements depend on the
modal coordinates as follows:
Hww
H
H
r
r
r
























NN
......
11
. (18)
The matrix H consists of the transformed
constraint and fixed interface modes calculated
according to the Craig-Bampton method. They
are orthogonal and normalized in M-norm, i.e
EMHH 
T, where M is the mass matrix of the
wheelset FE model. Now, the expressions (17)
and (18) take the form:
wAHHr 




 )(kpkk ba , (19)
wHwHHr 





 p
ba . (20)
As far as the modal coordinates of wheelset
are the part of generalized coordinates of
vehicle, their values w must be calculated
after rotation on dα on the basis of minimization
of the residual:
wHwH 
q
min (21)
in the some norm. M-norm is reasonable choice
in this case, that is
)).((min
)()(min
w
wHwHMHwHw
f
q
TTTT
q


(22)
To find a solution, the extremum of function
)(wf should be computed:
.0)(2
)(2




wHMHw
wHwHMH
T
T
q
f
(23)
Finally, the expression for the modal
coordinates looks as follows:
.
)(
wXw
wHHMHwHMHw
p
p
TT
ba
ba

 (24)
k
k'


v
d

k+1

First International Conference on Rail Transportation 2017
The matrix p
X must be calculated before
the simulation starts


 N
k
kp
T
kkp
T
pm
1
)(
AHHMHHX . (25)
2.7 Track models
Universal Mechanism supports three track
models which consider track with different level
of details:
- massless rail;
- inertial rigid body rail;
- flexible track.
The massless rail track model treats rail as a
massless force element. Generalized rail
coordinates are not introduced for this model.
Rail deflections are calculated as a result of
solution of equilibrium equations. This model is
recommended for analysis/optimization of
running gears of railway vehicles since intrinsic
rail dynamics weakly influences on simulation
results of rail vehicles.
The inertial rail track model considers rails as
rigid bodies under each wheel. Every rigid body
has three degrees of freedom: two translations
d.o.f. corresponding the rail shift relative to the
lateral (Y) and vertical (Z) axes and one
rotational d.o.f. relative to longitudinal (X) axis.
The underrail base is modelled as a special force
element. The inertial rail model is recommended
to be used for simulation of a complex scenario
of wheel-to-rail contact: railway track evolution
in the switches and turnouts, flange-back and
conformal contacts, simulation of vehicle
derailment cases, prediction of wheel and rail
wear, etc.
The flexible track model is a detailed 3D track
model that includes flexible rails, fasteners,
sleepers and sleeper foundation (Figure 11).
Rails are considered as Timoshenko beams.
Fasteners are modelled as a nonlinear force
element. Sleepers are simulated as rigid bodies.
Sleeper foundation is simulated with the help of
the nonlinear force elements connecting
semi-sleepers with the rigid base or finite
element flexible foundation. Verification of the
flexible track model is presented in (Rodikov et
al. 2016).
Note that the frequency range for the massless
rail model is 0 to 20 Hz. The inertial rail
provides reliable simulation in the frequency
range up to 100 Hz, and up to 1000 Hz for the
flexible track.
Fig. 11 Model of flexible track: 1) rail,
2) fasteners, 3) rigid semi-sleepers, 4) semi-
sleeper pads, 5) rigid/flexible foundation
3 Simulation results
In this chapter, the test example is considered.
The simulation results for moving AS4 railcar
with the constant speed 80 km/h in the curved
railway section are presented.
The UM model of the railcar consists of the
following elements (see Figures 12, 13):
- flexible front wheelset;
- rigid rear wheelset;
- car body and four bodies as axle-boxes;
- one joint introducing car body coordinates,
and four rotational joints for
axle-box/wheelset pairs;
- 12 bipolar force elements: four inclined and
four lateral dampers, four traction rods.
- eight force elements of the spring type as
suspension springs.
Fig. 12 General view of railcar model
Fig. 13 Model of railcar suspension
1) wheelset, 2) suspension spring, 3) inclined
frictional damper, 4) lateral damper, 5)
axle-box, 5) traction rod
4
1
2
3
5
Wheel-rail
penetration
contact model
1
6
4
3
2
5

First International Conference on Rail Transportation 2017
Main parameters of the flexible wheelset
model are presented in table 1.
Table 1 Main parameters of the flexible
wheelset model
Parameter Value
Number of finite elements 46 442
Number of nodes 56 561
Number degrees of freedom
339 366
Angle step of FE mesh 2
Number of flexible modes 35
Lowest frequency 29.6 Hz
Highest frequency 5456.6 Hz
Ratio of critical structural
damping for each mode
0.01
The rail track model includes 1070 beam
finite elements modelling the rails and 1072
bodies modelling semi-sleepers. The length of
each finite element is 0.6 m corresponding to the
sleeper spacing. The rail track model has 9648
degrees of freedom. Linear force elements
represent the rail fasteners.
Track geometry is shown in Figure 14. The
track geometry includes an initial straight
section, a first transition curve, a steady curve, a
second transition curve and a final straight
section. The parameters shown in the figure have
the following values: L0 = 10 m, P1 = 50 m,
S = 200 m, R = 300 m, P2 = 50 m, L1 = 1000 m.
The vehicle motion is analyzed taking into
account track irregularities. They are generated
using PSD functions and correspond to a track of
bad quality according to experts of International
Union of Railways (Figure 15).
Fig. 14 Track geometry for test case
The simulation results are obtained from a
control node initially located on the vertical
radius (Figure 16). Two methods of result
representations are used. The first one is
calculation of the parameters in the control node
rotating together with the wheel. The second
method consists in the computation of the
parameters in the point constantly located on the
vertical radius. The first method is called bellow
as «rotating mesh». The second one is
«non-rotating mesh».
Fig. 15 Track irregularities: 1) left rail,
vertical; 2) right rail, vertical; 3) both of rails,
horizontal.
Fig. 16 Control node
The simulation results are presented in
Figures 17, 18, 19.
Fig. 17 Radial flexible displacements in con-
trol node: 1) rotating mesh, Lagrangian ap-
proach (in black), 2) rotating mesh, Eulerian
approach (in red), 3) non-rotating mesh,
Lagrangian approach (in green), 4) non-
rotating mesh, Eulerian approach (in blue).
Control node
3
2
4
1
1
2
3
y
x
R
S
L0
P
1
P
2
L1

First International Conference on Rail Transportation 2017
Fig. 18 Equivalent von Mises stress in
control node: 1) rotating mesh, Lagrangian
approach (in black), 2) rotating mesh,
Eulerian approach (in red), 3) non-rotating
mesh, Lagrange approach (in green), 4)
non-rotating mesh, Eulerian approach (in
blue).
Fig. 19 Spectral power density of control
node acceleration
5 Conclusions
Application of the described methods to
dynamic simulation of the flexible wheelset
leads to practically identical results for flexible
displacements and stresses.
The spectral power density of acceleration is
presented in the range up to 1300 Hz, which
could not be obtained by the rigid wheelset
model.
One numerical experiment takes 72 to 75
minutes CPU time using parallel computations
on the computer with Intel Core i7 3.5 GHz
processor. The simulation using the Lagrangian
approach is faster compared to the Eulerian one
by 8 to 10%. Improvement of Eulerian
calculation algorithm is the subject of the future
research.
Acknowledgement
The authors would like to acknowledge the
support of Russian Foundation for Basic
Research, grant 17-01-00815.
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