MULTIBODY MODELS OF RAILWAY VEHICLES FOR REAL-TIME
SYSTEMS
Enrico Meli, Jury Auciello, Monica Malvezzi, Susanna Papini, Luca Pugi, Andrea Rindi
Dipartimento di Energetica, Sez.Meccanica Applicata, Università degli Studi di Firenze Via S.Marta 3, 50139 FIRENZE Italy
e-mail:, meli@mapp1.de.unifi.it, auciello@mapp1.de.unifi.it, malvezzi@mapp1.de.unifi.it susanna@mapp1.de.unifi.it, luca@mapp1.de.unifi.it, rindi@mapp1.de.unifi.it
http:// mapp1.de.unifi.it
Keywords:
Contact, dynamic, real-time application,Abstract.
Hardware and Software In the Loop (HIL & SIL) techniques are widely used for fast prototyping of control systems, electronic and mechatronic devices. In the railway field, several mechatronic on board subsystems (for example WSP, traction controls, active/semi-active suspensions, ATP,ATC, etc.) are often tested and calibrated following HIL and SIL approach.
The accuracy of HIL and SIL tests is deeply dependent on how the simulated virtual environment approxi-mates the real/physical experimental conditions. As the computational power available on real time hardware grows, the demand for more complex and realistic models of railway vehicles for real-time application in-creases.
In past experimental activities authors have worked with Trenitalia SPA to the implementation of simplified real time models in the Matlab-Simulink environment for several applications and in particular for the MI-6 Test rig [1a]. However in the last year the research activity has been focused on the development of a threedimensional dynamic model of a whole railway vehicle for the development of more complex applications. The fol -lowing sections describe briefly the features of the developed models.
1
INTRODUCTION
The proposed numerical model describes the dynamics of a railway vehicle running on a generic track, it has been developed with the objective of a real time implementation, in order use its results to control the actuators of Hardware In the Loop test rigs. The model has the following characteristics:
it is parametric;
it is able to simulate the train running on a generic track;
multiple contact points between wheels and rails can be managed; the model outputs are the dynamic parameters and the forces
In this paper, the main features of the developed numerical procedure are described. Then its application to a benchmark vehicle (the Manchester wagon) is summarized and the comparison between the obtained results and those obtained using a commercial multibody software (ADAMS™) are shown.
Figure 1: numerical procedure block diagram.
2
CONTACT POINT BETWEEN WHEEL AND RAIL
The wheel/rail interaction is fundamental to support and guide of the rail vehicles, but the wheel/rail contact analysis is a complex task and has been the subject of several investigations which presented different solutions. The problem can be divided in three separate but correlated parts: the contact geometry (to define the contact point the location), the contact kinematics (to characterize the creepages), and contact mechanics (to determine the tangential creep forces and spin moment on the basis of 3D rolling contact theories).
In order to find a solution for the problem of contact mechanics, detailed descriptions of the surfaces in contact, as well as the kinematics of the bodies, are required.
Because the wheel and the rail have profiled surfaces, the prediction of the location of the contact point on line is not easy, especially when the most general three-dimensional motion of a wheelset with respect to the rails is considered.
A simplified common method used in many existing computer algorithms to find the location of the contact point is to interpolate some precalculated table entries. The contact point location is given as function of some coordinates that measure the relative position of the wheelset with respect to the rails. The degree of accuracy of such algorithms depends on the number of coordinates that are used to define this relative position.
For solving the problem of wheel/rail contact in railroad dynamics two approaches can be used. The first is the commonly called constraint approach, in which non-linear kinematic contact constraint equations are introduced. The contact surfaces using the methods of differential geometry are represented in a parametric form. The coor-dinates of the contact points can be predicted on line during the dynamic simulation by introducing surface pa-rameters that describe the geometry of the contact surfaces. These surface papa-rameters are not generalized coordi-nates since there are no inertia forces associated with them. The contact constraints can then be formulated in terms of the system generalized coordinates and the surface parameters.
The second is elastic approach, the wheel set is considered to have six degrees of freedom with respect to the rails. The local deformation of the contact surface at the contact point is allowed and the normal contact forces are defined using Hertz’s contact theory or in terms of assumed stiffness and damping coefficients. This type of approach allows the separation between the wheel and the rail and to manage multiple contact points One of the main problems encountered when using the elastic approach is the definition of the contact point location on
line. In most elastic force models, the three-dimensional contact problem is reduced, for the sake of efficiency, to a two-dimensional problem when the location of the contact points is searched for.
Both of these approaches allows to define the component of the contact force normal to the surfaces.
In this paper the authors present a multibody model of a railway vehicle in which the contact is represented by means of an elastic approach. The normal component of the contact force at the contact point is calculated as function of the penetration between the surfaces.
The body shape of rail and wheel was is defined on the basis of real profiles, the wheel is modeled as a revolu -tion surface, while the rail as an extrusion in the longitudinal direc-tion as shown in Figure 2
. The proposed
model allows to perform simulations on a generic track, that can be defined as input by the
user in the pre-processing phase.
Figure 2: UIC 60 rail and ORES1002 wheel profiles.
Figure 3: wheel and rail three-dimensional geometry, configuration in which a double contact point is present. The contact point calculation is performed taking into account the complete wheelset three-dimensional motion, and it is based on a lookup table generated by an analytical procedure running offline.
The contact points are calculated off-line using a procedure based on the Simplex method (developed by Nelder e Mead in 1965) as the minimum values of the difference surface between the rail and the wheel. When the value of this difference minimum is positive (Figure 4 a)) there is not contact between the wheel and the rail and no forces are exchanged in the contact point (that is not defined in this case). When the minimum difference is negative there is a penetration between the bodies (Figure 4 b)).
a)
b)
Figure 4: wheel and rail relative position, a) the minimum value of the difference surface is positive, no contact
is present, b) the minimum of the difference surface is negative, the normal component of the contact force is positive.
Figure 5: track definition
The simulated vehicle can move on a generic track, defined by
s , a generic three dimensional curve definedby means of its generic abscissa s.
Figure 6: rail and wheel coordinate systems
Figure 6 shows the coordinate systems fixed on the rail and on the wheelset respectively. Concerning the rail co-ordinate system, x is parallel to the track direction, b y is on the plane defined by the rails and normal tob xb
, z is consequently defined. The origin b O is fixed on the track axis, in other terms it is on the plane identi-b
fied by the rails and is equally distant from both of them.
The wheel-set coordinate system origin is fixed at the wheel-set center of mass, the yr axis is directed as the
wheel-set axis, while the xr axis is parallel to the xbyb plane. The zr is consequently defined. The wheel
set coordinate system is not fixed on the wheel-set, because it is not constrained to rotate with the wheels. A third auxiliary coordinate system fixed on the track is defined, x is tangent to the track curve, b' y axis isb'
normal to the plane xbzb and the z axis is consequently defined. The origin b' O of the auxiliary coordi-b'
nate system is on the median railway curve
s
t
. The difference surface is then defined as:
b
b r b ry
u
D
,
P
P
k
Where Pb’ is the coordinates of a generic point on the rail and Prb’ is the coordinates of a generic point on the
wheel
So in order to find the contact points the local minima of this surface are searched.
4
,..,
2
,
1
,
,
min
, ,n
i
y
u
y
u
D
r ic irc y u rThe procedure used to find the minima is based on the Simplex method. This method is iterative, then needs a starting point and a stop criterion. The method is reliable, since it always converges to a local minima and it re -quires low computational resources and converges after a small number of iterations. In this application the con-tact point research starts from two points, as shown in Figure 7, if the research converges in the same point, the contact is unique, otherwise the contact is double.
Figure 7: local minima research starting points
3
CONTACT FORCES BETWEEN WHEEL AND RAIL
The normal component of the contact force, according to the elasto-viscous approach as sum of a term that de-pends on the normal penetration between the body (the elastic component), and a term proportional to the sur-face relative velocities in the contact point (the viscous term).
( ) 1 ( ) 1 2 2 b h v sign v sign p N k p k v
(2)
Where the penetration is defined as the difference between the contact point on the wheel and his projection on the rail. The constant values are calculated in accordance with the common models adopted in literature.
The viscous part of the normal contact force depends on the component of the relative speed in the normal direc-tion, given by:
b C b b C G ω (P G ) v
).
(
C b b Cv
v
n
P
Where n is the unitary vector normal to the rail in the contact point. The constant b' k depends on materialv
properties, its value was chosen on the basis of literature data ( 104Ns/m
v
k ).
The magnitude of the tangential component of the contact forces is calculated using Kalker and Hertz theory: the Hertz theory is used to define the shape and the dimension of contact area that depend on the normal force mag-nitude, the material properties and the local profile geometry, while the Kalker linear theory results are used to define the components of the creep forces:
11 22 23 23 33 x y sp
T
f
T
f
f
M
f
f
(3)
where f11, f22, f33, f23 are the linear creep coefficients, that depends on the contact ellipse semi-axis and on the
combined modulus of rigidity (their values are tabulated), and the values of the creepage components ξ, η and φ for the right and the left wheel are defined as:
G i v c G t v c G n ω
In the contact area during the slip the maximum value of tangential force is given by “adhesion limit”, so the cal-culated Kalker creep forces are then saturated at the product of kinematics coefficient and vertical load.
4
THE MULTIBODY MODEL
The multibody model of the railway vehicle was realized in Matlab-Simulink environment, and is shown in Fig-ure 8.
Figure 8: the multibody vehicle model.
The multibody complete model is composed of seven rigid bodies (the car body, two bogies, four wheelsets) connected by three-dimensional non linear elastic-viscous force elements (used to model the connection ele-ments between the bodies, for example the vehicle suspensions, dampers etc.). The wheel-rail interaction model considers a fully three dimensional rolling contact and can manage multiple contact points.
Figure 9: forces and torques acting on each wheelset
The forces acting on each wheel-set (Figure 9) are the creep forces in the contact area, the forces due to the in-teraction with the boogie, the external applied braking or traction torque and the weight. The dynamics of the wheel-set can be described by the following differential equa
tions:
1 1
( )
( )
( )
x y n j i ix iy j i n j iN iT iT isp s j iM G P
F
N
T
T
K G
G
K G
M
M
M
M
M
The numerical model was realized in the Matlab/Simulink environment, by means of SimMechanics, that allows to model multibody systems. The structure of the model is modular and different subsystems can be easily mod-eled and substituted in the main procedure. The developing environment allows to obtain a numerically efficient model, to test different types of algorithm in the integration, to manage singularities.
In order to evaluate the accuracy of the model and to verify the performance of the model in terms of computa-tional burden the results of the model were compared with those obtained with Adams Rail. The benchmark case is the Manchester wagon, whose features are known in the literature and whose Adams Rail model is available.
5
RESULTS
Two sets of tests were performed, when the vehicle travels along a curve with a constant speed and the stability analysis. In the first set of tests The curve radius values were 500, 1200 and 2400 m, the rail angle was 1/40 and 1/20, the simulated cant angles were 40/1435 rad and 90/1435 rad, the speed values were 15, 30 and 45 m/s, the wheel/rail adhesion was 0.2. In this set of tests the displacements of the wheel-set centers and the contact forces were analyzed and compared. Figure 10 shows the comparison between the results obtained by the developed multibody models, the time-history of the wheel-set center of mass displacements is analyzed in the presented example.
Table 1 summarizes the hunting frequencies calculated for different vehicle speed with the developed models, a quite good agreement between the results can be observed. The calculated critical speeds are 71 m/s with the Simulink model and 73 m/s with the Adams Rail model.
First wheel-set (front bogie, front wheel-set) Second wheel-set (front bogie, rear wheel-set)
Third wheel-set (rear bogie, front wheel-set) Fourth wheel-set (rear bogie, rear wheel-set) Figure 10: wheel-sets center of mass displacement: comparison between results obtained with Adams and
Simulink models
Speed (m/s) Hunting Frequencies (Hertz) ADAMS SIMULINK
25 1.7 1.6
40 2.5 2.6
55 3.3 3.5
70 4.6 4.4
Concerning the stability analysis, the vehicle response to a vertical and a horizontal track disturbance was ana -lyzed with the developed models. Figure 11 shows the frequency response of the vehicle in terms of vertical car-body acceleration and vertical contact force at a vertical track disturbance. Also in this case a good agreement between the different models can be observed.
Figure 11: Vertical carbody acceleration and vertical contact force for the vertical track disturbance. These sets of tests allow to conclude that the developed multibody models give substantially comparable results, even if they are obtained using different software environments and different modeling approaches. The differ-ences are particularly evident in the definition of the contact points and of the contact forces. The time history analysis shows that the Simulink model is generally characterized by an higher transient damping, due to the different method for the calculation of the normal force (in the Simulink model a non linear elastoviscous ap -proach is implemented).
Since the objective of the study is the definition of a numerical model that could be implemented in real time ap-plications, the computational load is an important parameter to be considered. In the presented results, the Simulink model was integrated with a ODE5 algorithm and a constant time step equal to 5e-4 s. For the compar-ison of the computational load, is considered a test with curve radius 1200 m, rail angle 1/40 rad, cant angle 40/1435 rad, speed 45 m/s.
The time necessary to perform the test with the Adams model was 461 s, while the time necessary to the Simulink model was 333 s.
6
CONCLUSIONS
The paper summarizes the features of a railway vehicle multibody model realized in the MatlabSimulink envi -ronment. The model is used to simulate the dynamical behavior of a benchmark railway vehicle, the results of the model are compared with those obtained with the Adams Rail multibody model of the same vehicle. The comparison shows a good agreement between the models and the relative errors are acceptable (it has to be highlighted that the models evaluate the deformation of the contact bodies by means of two completely different ap -proaches). With respect to the existing and available railway multibody models, the presented one features are mainly a more detailed modeling of the wheel/rail contact problems and the possibility to obtain an executable implementation that could run in real time conditions. In the developed model the aspects relative to the contact point definition and contact force calculation has been carefully investigated since they have a dominant effect on the vehicle dynamics. The Simulink model was developed in order to investigate the possibility of a real time implementation, for this reason a particular attention was focused on its numerical efficiency.
REFERENCES
[1] S. Papini, M.Malvezzi, L.Pugi, E.Meli Contact Wheel rail investigation for numerical integration of train motion dynamic equations , AITC-AIT 2006 International Conference on Tribology, 20-22 Sep-tember 2006, Parma.
[2] R. V. Dukkipati, J. R. Amyot, Computer aieded simulation in railway dynamics, Marcel Dekker, 1988W. Schiehlen. Multibody system dynamics: Roots and perspectives. Multibody System Dynamics, 1, 149-188, 1997.
[3] J.J.Kalker, Three-Dimensional Elastic Bodies in Rolling Contact, Kluwert Academic Publishers, 1990. [4] E.A.H. Vollebregt, J.J. Kalker, G. Wang, " CONTACT 93 Users Manual“, VORtech Computing,
In-dustrial and Scientific Computing, July 1992, revised March 1994. [5] K.L. Johnson, Contact mechanics, Cambridge University Press, 1985.
[6] P.J. Vermeulen., K.L. Johnson, Contact of Nonspherical Elastic Bodies Transmitting Tangential Forces, Journal of Applied Mechanics, Transactions of the ASME, June, 1964.
[7] M.P. do Carmo, Differential geometry of curves and surface, Prentice Hall, 1976.
[8] A. A. Shabana, J. R. Sany An augmented formulation for mechanical systems with non-generalized co-ordinates: application to rigid body contact problems. “Nonlinear dynamics”, 24, 2001.
[9] A. A. Shabana, K. E. Zaazaa, J. L. Escalona, J. R. Sany Development of elastic force model for wheel-rail contact problems. “Journal of sound and vibration”, 269, 2004.
[10] J. Pombo, J. Ambrosio, Dynamics analysis of a railway vehicle in real operation conditions using a new wheel-rail contact detection model, “Int. J. of Vehicle Systems Modelling and Testing”, 1, 2005 [11] Manchester Metropolitan University, The Manchester Benchmarks for rail vehicle simulation, Rail
Technology unit Editor, March 1998.
[12] A. A. Shabana, M. Berzeri, J. R. Sany, Numerical Procedure for the Simulation of Wheel/rail Contact Dynamics, Journal of Dynamic Systems, Measurement and Control, Transaction of the ASME, Vol. 123, June 2001, pp.168-178.
[13] De Pater, A. D., ‘The geometric contact between track and wheel-set’, Vehicle System Dynamics 17, 1988, 127–140.
[14] A. A. Shabana, M. Tobaa, H. Sugiyama, K. E. Zaazaa, On the Computer Formulation of the Wheel/Rail Contact Problem, Nonlinear Dynamics, Springer 2005, vol. 40, pp. 169-193.
[15] L. Baeza, A. Roda, J. Carballeira, E. Giner, Railway TrainTrack Dynamics for Wheelflats with Im -proved Contact Models, Nonlinear Dynamics, Springer 2006, vol. 45, pp. 385-397.
[16] J. L. Escalona, M. Gonzàlez, K. E. Zaazaa, A. A. Shabana, A technique for Validating a Multibody Wheel/Rail Contact Algorithm, Proceedings of DETC’03 ASME 2003 Design Engineering Technical Conference and Computers and Information in Engineering Conference, Chigago, Illinois, USA, Sep-tember 2-6 2003.
[17] Wolfgang Rulka, Eli Pankiewicz, MBS Approach to Generate Equations of Motions for HiL-Simula-tions in Vehicle Dynamics Multibody System Dynamics (2005) n.14, pp 367–386.
[18] A Bhaskar, K L Johnson, G DWood and J Woodhouse, Wheel/rail dynamics with closely conformal contact .Part 1: dynamic modelling and stability analysis, Proc Instn Mech Engrs Vol 211 Part F IMechE 1997, pp 11-25.
[19] A Bhaskar, K L Johnson, G DWood and J Woodhouse, Wheel/rail dynamics with closely conformal contact .Part 2: forced response, results and conclusions, Proc Instn Mech Engrs Vol 211 Part F IMechE 1997, pp 27-40.
[20] Sung-Soo Kim · Wanhee Jeong, Subsystem synthesis method with approximate function approach for a real-time multibody vehicle model Multibody Syst Dyn (2007) 17:141–156.
Multibody Models of Railway VEhicles for Real-Time SYSTEMS
Enrico Meli, Jury Auciello, Monica Malvezzi, Susanna Papini, Luca Pugi, Andrea Rindi
Dipartimento di Energetica, Sez.Meccanica Applicata, Università degli Studi di Firenze Via S.Marta 3, 50139 FIRENZE Italy
e-mail: malvezzi@mapp1.de.unifi.it, meli@mapp1.de.unifi.it,
susanna@mapp1.de.unifi.it, luca@mapp1.de.unifi.it, rindi@mapp1.de.unifi.it http:// mapp1.de.unifi.it
Keywords:
Contact, dynamic, real-time application,SUMMARY
Hardware and Software In the Loop (HIL & SIL) techniques are widely used for fast prototyping of control systems, electronic and mechatronic devices. In the railway field, several mechatronic on board subsystems (for example WSP, traction controls, active/semi-active suspensions, ATP,ATC, etc.) are often tested and calibrated following HIL and SIL approach.
The accuracy of HIL and SIL tests is deeply dependent on how the simulated virtual environment approxi-mates the real/physical experimental conditions. As the computational power available on real time hardware grows, the demand for more complex and realistic models of railway vehicles for real-time application in-creases.
In past experimental activities authors have worked with Trenitalia SPA to the implementation of simplified real time models in the Matlab-Simulink environment for several applications and in particular for the MI-6 Test rig. However in the last year the research activity has been focused on the development of a three-dimensional dynamic model of a whole railway vehicle for the development of more complex applications. The following sections describe briefly the features of the developed models.
The numerical model described in this paper simulates the dynamics of a railway vehicle running on a generic track, it has been developed with the objective of a real time implementation, in order use its results to control the actuators of Hardware In the Loop test rigs. The model has the following characteristics:
it is parametric;
it is able to simulate the train running on a generic track;
multiple contact points between wheels and rails can be managed; the model outputs are the dynamic parameters and the forces
In this paper, the main features of the developed numerical procedure are described. Then its application to a benchmark vehicle (the Manchester wagon) is summarized and the comparison between the obtained results and those obtained using a commercial multibody software (ADAMS™) are shown.
Figure 12: numerical procedure block diagram.
REFERENCES
[1] S. Papini, M.Malvezzi, L.Pugi, E.Meli Contact Wheel rail investigation for numerical integration of train motion dynamic equations , AITC-AIT 2006 International Conference on Tribology, 20-22 Sep-tember 2006, Parma.
[2] J.J.Kalker, Three-Dimensional Elastic Bodies in Rolling Contact, Kluwert Academic Publishers, 1990. [3] M.P. do Carmo, Differential geometry of curves and surface, Prentice Hall, 1976.
[4] A. A. Shabana, K. E. Zaazaa, J. L. Escalona, J. R. Sany Development of elastic force model for wheel-rail contact problems. “Journal of sound and vibration”, 269, 2004.
[5] J. Pombo, J. Ambrosio, Dynamics analysis of a railway vehicle in real operation conditions using a new wheel-rail contact detection model, “Int. J. of Vehicle Systems Modelling and Testing”, 1, 2005 [6] Manchester Metropolitan University, The Manchester Benchmarks for rail vehicle simulation, Rail
Technology unit Editor, March 1998.
[7] A. A. Shabana, M. Berzeri, J. R. Sany, Numerical Procedure for the Simulation of Wheel/rail Contact Dynamics, Journal of Dynamic Systems, Measurement and Control, Transaction of the ASME, Vol. 123, June 2001, pp.168-178.
[8] De Pater, A. D., ‘The geometric contact between track and wheel-set’, Vehicle System Dynamics 17, 1988, 127–140.
[9] A. A. Shabana, M. Tobaa, H. Sugiyama, K. E. Zaazaa, On the Computer Formulation of the Wheel/Rail Contact Problem, Nonlinear Dynamics, Springer 2005, vol. 40, pp. 169-193.
[10] Wolfgang Rulka, Eli Pankiewicz, MBS Approach to Generate Equations of Motions for HiL-Simula-tions in Vehicle Dynamics Multibody System Dynamics (2005) n.14, pp 367–386.
[11] A Bhaskar, K L Johnson, G DWood and J Woodhouse, Wheel/rail dynamics with closely conformal contact .Part 1: dynamic modelling and stability analysis, Proc Instn Mech Engrs Vol 211 Part F IMechE 1997, pp 11-25.
