XXV AIAS NATIONAL CONFERENCE
INTERNATIONAL CONFERENCE ON MATERIAL ENGINEERING
Gallipoli, 4 - 7 settembre 1996
MULTIBODY SIMULATION OF RAIL VEHICLE DYNAMIC
WITH NONLINEAR CONTACT
A. GUGLIOTTA, A. SOMA’
Dipartimento di Meccanica
Politecnico di Torino
Cso. Duca degli Abruzzi 24 - 10129 Torino - Italy
KEYWORDS: rail, dynamic, contact, multibody
ABSTRACT
In the present paper stability analysis and high speed transient simulation of a
railway vehicle has been studied. The system has been modelled by using the rail
oriented module of the multibody computer code ADAMS. Different wheel-rail contact
theories have been compared. At first an analysis according to the numerical benchmark
proposed by the European Rail Research Institute has been performed in order to check
the model. Thereafter the transient simulation is studied to evaluate the combined effect
of non-linear contact and measured track irregularities.
INTRODUCTION
The requirement to increase the speed of a railway vehicle may induce vibrations
and vehicle instability; as a consequence at minimum this leads to comfort reduction. In
the field of railway applications multibody computers codes are used as tools to
investigate whole vehicle dynamic, pantograph dynamic, noise reduction [1,2].
In the present work the dynamic of a railway vehicle is studied including different
wheel-rail contact models in order to evaluate the critical speed [3] and the track
irregularities interactions in transient simulations.
During the last decades several wheel-rail contact theories have been evolved and
widely used in railway dynamic simulation. An excellent survey of wheel-rail rolling
contact theories has been given by Kalker [4]. In the present paper different levels of
wheel-rail contact have been taken into account introducing first the linear Kalker
model and, as second step, the non-linear Kalker model [5-6].
A mathematical model for the simulation of the interaction between train, track and
supporting structure has been made by using a new module, oriented to the rail vehicle
simulation, of the multibody simulation code ADAMS [7,8]. It is possible to define a
user track layout taking into account the track irregularities, to model the vehicle
introducing the effect of body flexibility and the non linear formulation of the contact.
A. Gugliotta, A. Somà
As far as the code allows the possibility to write user subroutines, in the present
work an automatic procedure for critical speed evaluation has been developed. At first
the modal analysis solution allows to perform the extraction of the critical speed. In the
second part the transitory analysis is performed. The analysis has been compared taking
into account both the effect of wheel-rail contact and track irregularities.
SYMBOLS
a/b
h = ( hnx , hny , hnz )
contact area semiaxes
linear velocity of n-th body
ξx
ξy
longitudinal creepage
lateral creepage
W = (Wnx ,Wny ,Wnz )
angular velocity
ξsp
spin creepage
V
forward rolling velocity
Fcx
longitudinal creep force
R = ( Rnx , Rny , Rny )
principal radii of curvature
Fcy
lateral creep force
MATHEMATICAL MODEL OF WHEEL-RAIL CONTACT
In fig. 1 a general scheme of wheel-rail contact is given. The dynamic behaviour of a
railway vehicle is significantly affected by the interactive forces between the wheel and
rail.
Fig. 1: wheel-rail contact scheme
Fig. 2: contact forces scheme
These forces depends on the adhesion and creep. The phenomena of creep exist
when two elastic bodies are pressed against each other with force and allowed to roll
ever each other in presence of friction. A contact region is formed at the point at which
they touch (fig. 2).
Several wheel-rail rolling contact theories have been evolved. Basically the theories
may be divided in linear and non linear. The linear simplified theories starting from two
dimensional Carter’s theory (1926) to recent three dimensional Kalker theory (1967) [5]
calculate the creep forces with a constant contact area. The non linear theories [6] take
into account the variation of the contact area due to the different principal radii of
curvature of the bodies of revolution during rolling.
A complete three dimensional non linear exact theories is given by Kalker [6] based
on a minimum principle that satisfy the Coulomb inequality. Approximated non linear
A. Gugliotta, A. Somà
theory has been given by Kalker and may be used in numerical codes with the aids of
numerical tables [9].
In a general case, from a kinematics point of view, creep occurs when the
circumferential velocities of the two rolling bodies are not equal. The nondimensional
term creepage is used to define these deviation from a pure rolling motion of the two
bodies. The creepage are defined as follow:
• longitudinal creepage:
ξx =
(h
1x
− h2 x − R1x ⋅ W1 y − R2 x ⋅ W2 y
) = actual forward velocity - pure rolling forward velocity
V
forward velocity due to rolling
• lateral creepage:
h1 y − h2 y  W1x W2 x  actual lateral velocity - pure rolling lateral velocity
 =
+ 
−
ξy =
V
forward velocity due to rolling
 W1 y W2 y 
• spin creepage:
(W1z − W2 z ) = angular velocity of upper body - angular velocity of lower body
ξsp =
V
forward velocity due to rolling
The creep forces depends on the normal force trough the Coulomb inequality.
According to Hertz static theory the contact region is elliptical and the ratio of the
semiaxes (a/b) can be calculated from the principal radii of curvature of the bodies at
the contact point. The calculation of the actual dimension a and b depends from the
normal force. The area of contact will be divided as in scheme of fig. (3a) in a slip
region and adhesion region. This influences the creep force/creepage relation in a non
linear mode as schematically described in fig. (3b).
Fig 3a: slip and adhesion contact zone of the Fig 3b: creep force/creepage non linear
contact area
function
In a general case the non-linear three dimensional rolling contact creep forces are
derived as follows:
Fcx = − c( y )2 ⋅ G ⋅ C( y )11 ⋅ ξ x
Fcy = − c( y ) 2 ⋅ G ⋅ C( y ) 22 ⋅ ξ y
Mz = − c( y ) 3 ⋅ G ⋅ C( y ) 32 ⋅ ξ y − c( y ) 4 ⋅ G ⋅ C( y ) 33 ⋅ ξz
(2)
A. Gugliotta, A. Somà
Where the semiaxes (a/b) of the contact area and the C coefficients may be
calculated at each equilibrium position. In the linearized case the contact forces
coefficients depends in a linear mode trough the kinematics creepage and equation is
simplified as follows, where f ij are called Kalker’s coefficients:
Fcx = − f 11 ⋅ ξx
Fcy = − f 22 ⋅ ξy
(3)
Mz = − f 32 ⋅ ξy − f 33 ⋅ ξz
NUMERICAL MODEL
The dynamic simulation of a railway vehicle leads to the a set of non linear
equations. In order to perform the simulation of the vehicle multibody computer codes
are used. In the present paper has been investigated the performance of the rail module
of the ADAMS moltybody code.
The vehicle model used represent a European benchmark proposed by ERRI
(European Rail Research Institute) as tool for computer code improvements [10]. The
proposed vehicle is the Eurofima wagon and the basic data are listen in Table I.
Table I: mass and stiffness data of the vehicle Eurofima [10]
wagon mass
32000 [kg] wagon total length
boogie mass
2600 [kg] primary suspension stiffness
wheelset mass
1500 [kg] secondary suspension stiffness
Fig. 4a: scheme of the Eurofima bogie:
1-engine wheelset; 2- support
beam; 3-secondary suspension;
4- boogie; 5- primary suspension;
6- wheelset
21.5 [m]
617 [KN/m]
160 [KN/m]
Fig. 4b: ADAMS model of the bogie
Fig. 4 shows a view of Eurofima boogie (a) and the ADAMS model scheme (b). The
ADAMS View graphical simulation module allows the user to obtain a model closed to
the design scheme. Therefore the debug operations of the model is helped with a
powerful graphical interface.
A. Gugliotta, A. Somà
MODAL DATA AND CRITICAL SPEED
In a first part of the work the numerical model is used to perform a comparison in
terms of mode shape and frequencies between the benchmarks ERRI data and the
results obtained with ADAMS and linearized wheel-rail contact theory. (Table II)
Table II: mode shape description and frequencies comparison between present model
and ERRI benchmark models.
Mode description
ERRI Benchmark
ADAMS Rail
wagon lateral mode
0.47
0.55
yaw wagon
0.87
0.81
bounce wagon
1.16
1.10
pitch wagon
1.61
1.41
wagon vertical mode
1.55
1.36
1° wheelset yaw
2.23
2.38
2° wheelset yaw
2.27
2.438
bounce bogie (in phase)
6.34
6.42
bounce bogie (out phase.)
6.07
6.16
pitch bogie (in phase)
9.75
9.06
pitch bogie(out phase)]
9.74
9.07
The linear model has been also used in order to perform the stability analysis and the
evaluation of the vehicle critical speed. For this specific goal a user subroutine has been
written for the automatic critical speed vehicle identification. The critical speed
comparison are listed in Table III.
Table III: critical speed comparison
with non linear dampers
with anti - yaw dampers
ERRI benchmark
295 [Km/h]
334 [Km/h]
ADAMS Rail
311[Km/h]
342[Km/h]
TRANSIENT SIMULATION RESULTS
The transient simulation have been done by using a straight railway. The track is first
modelled by using an analytical description.
Thereafter the track layout has been defined taking into account the effect of
measured irregularities. The measured data are written in the code by using an input
files data with the track co-ordinates and the cant angle for discrete points.
Table IV: summary of simulation cases
analytical/measured
linear Kalker contact
track
wagon lateral displ. (fig.6a)
no dampers
wheelset lateral displ. (fig. 7a)
non-linear dampers wagon lateral displ. (fig. 6c)
wheelset lateral displ. (fig. 7c)
non linear Kalker contact
wagon lateral displ. (fig.6b)
wheelset lateral displ. (fig. 7b)
wagon lateral displ. (fig.6d)
wheelset lateral displ. (fig. 7d)
A. Gugliotta, A. Somà
0.06
0.06
0.04
0.04
lateral wagon displacement [m]
lateral wagon displacement [m]
The simulation correspond to a run with 250 km/h speed of a duration of 5 sec and
about 230 m length selected from the measured track data given by the ERRI
benchmark. Different simulations have been performed, as described in Table IV,
taking into account linear and non linear wheel-rail contact theories and linear and nonlinear dampers. In the graphs fig. 6-abcd the y displacement of the wagon are reported.
0.02
0.00
-0.02
0.00
-0.02
0
1
2
time [s]
3
4
5
a) case with no dampers and linearized
Kalker’s contact
0
1
2
time [s]
3
4
5
b) case with no dampers and non linear
Kalker’s contact
0.03
0.03
0.02
lateral wagon displacement [m]
lateral wagon displacement [m]
0.02
0.01
0.00
0.02
0.01
0.00
-0.01
-0.01
0
1
2
time [s]
3
4
5
0
1
2
time [s]
3
4
c) case with non linear dampers and
d) case with non linear dampers and non
linearized Kalker’s contact
linear Kalker’s contact
Fig. 6: Evaluation of the lateral wagon displacement. All the graphs present the
superposition of results in the case of analytical and measured track with
irregularities.
Figures 6-a,b refers to the case with no dampers, figures 6-c,d relates to the
simulation with the non linear dampers. In all the cases the irregularities effect create a
low disturbance superimposed to the numerical solution with analytical track. Due to
the motor eccentricity the lateral displacement of the wagon has a equilibrium position
5
A. Gugliotta, A. Somà
shifted, of about 0.015 m, from the starting position. In the more realistic damped case
the wagon, after a quick transient, oscillates around that equilibrium position.
The comparison of linear and non linear contact model may be obtained by a first
visual inspection of figures 6-a,c and figures 6-b,d. It is possible to observe that the two
different solutions have the same results if we consider the mean value and the
maximum amplitude.
0.002
lateral wheelset displacement [m]
lateral wheelset displacement [m]
0.002
0.000
-0.002
0.000
-0.002
0
1
2
time [s]
3
4
5
0
1
2
time [s]
3
4
5
a) case with linear dampers and linearized b) case with linear dampers and non linear
Kalker’s contact
Kalker’s contact
0.002
lateral wheelset displacement [m]
lateral wheelset displacement [m]
0.002
0.000
-0.002
0.000
-0.002
0
1
2
time [s]
3
4
5
0
1
2
time [s]
3
4
5
c) case with non linear dampers and
d) case with non linear dampers and non
linearized Kalker’s contact
linear Kalker’s contact
Fig. 7: Evaluation of the lateral wheelset displacement. All the graphs present the
superposition of results in the case of analytical and measured track with
irregularities.
A. Gugliotta, A. Somà
Fig. 7-abcd show the y lateral displacement of the wheelset. The difference of linear
and non-linear contact model is in these case more evident. In-fact in the case of the
linearized Kalker contact theory (fig. 7a-c) the wheelset oscillates around the zero
lateral displacement. In the case on non-linear theory (fig. 7b-d) the variation of the
contact point allows the wheelset to oscillates around a shifted displacement.
CONCLUSIONS
The dynamic numerical simulation of a rail vehicle has been performed by using the
ADAMS multibody computer code. Different transient simulation cases, taking into
account the linear and non linear wheel-rail contact model, are compared.
Measured track data of an European benchmark has been introduced to evaluate the
relative effect of the wheel-rail contact versus the track irregularities effect.
The results so obtained, even if as preliminary step of the research, allow to conclude
that the linearized wheel-rail contact gives good results for critical speed analysis and
wagon comfort studies also in presence of track irregularities.
The more realistic non-linear contact model allows to evaluate the wheelset and
bogie behaviour but does not give any improvement to vehicle comfort analysis.
REFERENCES
[1] G. Diana, F. Cheli, S. Bruni, A. Collina: Modelli matematici per lo studio della
interazione veicolo strutture - armamento. Ingegneria Ferroviaria, pp. 1066-1080
(1980).
[2] G. Diana, F. Cheli, S. Bruni, A. Collina: Dynamic Interaction between Rail
Vehicles and Track High Speed Train. Vehicle System Dynamic, vol. 24, pp. 15-30
, (1995).
[3] C. Casini, G. Tacci: Studio di un modello matematico per la previsione della
velocità critica di un veicolo ferroviario su un binario rettilineo senza difetti. Ing.
Ferroviaria, pp. 262-275, (1978).
[4] J.J. Kalker: Review of Wheel Rail contact theories. Applied Mechanical Division
Vol. 40, ASME PP. 77-92, (1980).
[5] J.J. Kalker: The computational of three dimensional contact with dry friction.
International Journal of Numerical Methods in Engineering, Vol. 14, p. 1293-307
(1979).
[6] J.J. Kalker: Three-Dimensional Elastic Bodies in Rolling Contact. Kluwer
Academic Publisher. 1990.
[7] ADAMS / SOLVER. Reference Manual. Version 8.0. Mechanical Dynamics.
1994.
[8] ADAMS / RAIL 2281.User’s Manual. Mechanical Dynamics GmbH ,Marburg.
1995.
[9] J.J. Kalker: Book of Tables for the Hertzian Creep-Force Law. Delft University
internal report. 1996.
A. Gugliotta, A. Somà
[10] B 176/3 Benchmark Problem Result and Assessment. European Rail Research
Institute (ERRI) . Utrecht , August 1993
0.004
vertical wagon displacement [m]
vertical wagon displacement [m]
0.004
0.002
0.000
-0.002
-0.004
0.002
0.000
-0.002
-0.004
0
1
2
time [s]
3
4
5
0
1
2
time [s]
3
4
5
b) case with linear dampers and linearized b) case with linear dampers and non linear
Kalker’s contact
Kalker’s contact
0.004
vertical wagon displacement [m]
vertical wagon displacement [m]
0.004
0.000
0.000
-0.004
-0.004
0
1
2
time [s]
3
4
c) case with non linear dampers
linearized Kalker’s contact
5
0
1
2
time [s]
3
and b) case with non linear dampers
linear Kalker’s contact
4
5
and non
Fig. 7: Evaluation of the vertical wagon displacement. All the graphs present the
superposition of results in the case of analytical and measured track with
irregularities.