In the multibody formulation of the contact problem, the kinematic contact constraint conditions are formulated in terms of the normal and tangents to the contact surfaces. Using the assumption of nonconformal contact, the second time derivatives of the contact constraints, which are required in the augmented Lagrangian formulation of the multibody equations, contain third derivatives of the position vectors of the contact points with respect to the surface parameters that describe the geometry of the contact surfaces. These derivatives must be accurately calculated in order to develop a robust numerical algorithm for solving the multibody differential and algebraic equations of the contact problem. An important application for the procedure developed in this paper is the wheel/rail interaction. In order to allow a general description for the wheel and rail profiles, the spline function representation is used. A multi-layer spline function algorithm is used in order to ensure accurate calculation of the third derivatives with respect to the surface parameters when a small number of nodal points is used. The problems of continuity of the derivatives and smoothness of these functions are addressed. The proposed method allows using wheel and rail profiles obtained from direct measurements. Numerical results show that this multibody approach, whic leads to accurate value of the normal force at the contact, can capture the coupling between the vertical and the lateral motion of the wheelset.

1.
Johnson, K. L., 1985, Contact Mechanics, Cambridge University Press, Cambridge, UK.
2.
Kalker, J. J., 1990, Three-dimensional elastic bodies in rolling contact, Kluwer Academic, Dordrecht/Boston/London.
3.
Vermeulen
,
P. J.
, and
Johnson
,
K. L.
,
1964
, “
Contact of nonspherical bodies transmitting tangential forces
,”
ASME J. Appl. Mech.
,
31
, pp.
338
340
.
4.
Kalker
,
J. J.
,
1982
, “
A fast algorithm for the simplified theory of rolling contact
,”
Veh. Syst. Dyn.
,
5
, pp.
317
358
.
5.
Kalker
,
J. J.
, 1996, “Book of Tables for the Hertzian Creep-Force Law,” Technical Report, Department of Applied Analysis. Delft University of Technology, Delft, The Netherlands.
6.
De Pater
,
A. D.
,
1988
, “
The geometric contact between track and wheel-set
,”
Veh. Syst. Dyn.
,
17
, pp.
127
140
.
7.
Fisette
,
P.
, and
Samin
,
J. C.
,
1994
, “
A wheel/rail contact model for independent wheels
,”
Arch. Appl. Mech.
,
64
, pp.
180
191
.
8.
Shabana, A. A., and Sany, J. R., 2001, “An Augmented Formulation for Mechanical Systems with Non-Generalized Coordinates: Application to Rigid Body Contact Problems,” Journal of Nonlinear Dynamics, Vol. 24, No. 2, pp. 183–204.
9.
Litvin, F. L., 1994, Gear Geometry and Applied Theory, Prentice-Hall, Englewood Cliffs, NJ.
10.
Shabana, A. A., 1998, Dynamics of Multibody Systems, Second Edition, Cambridge University Press, Cambridge, UK.
11.
Berzeri, M., 2000, Ph.D. thesis, Department of Mechanical Engineering, University of Illinois at Chicago, Chicago, IL.
12.
Shabana, A. A., 2001, Computational Dynamics, Second Edition, J. Wiley, New York.
13.
Ahlberg, J. H., Nilson, E. N., and Walsh, J. L., 1967, The Theory of Splines and Their Applications, Academic Press, New York.
14.
Shikin, E. V., and Plis, A. I., 1995, Handbook on Splines for the User, CRC Press, Boca Raton, FL.
15.
Garg, V. K., and Dukkipati, R. V., 1984, Dynamics of Railway Vehicle Systems, Academic Press, Orlando, FL.
16.
Love, A. E., 1927, A Treatise on the Mathematical Theory of Elasticity, 4th Edition, Dover.
17.
Knothe
,
K.
, and
Bo¨hm
,
F.
,
1999
, “
History of stability of railway and road vehicles
,”
Veh. Syst. Dyn.
,
31
, pp.
283
323
.
You do not currently have access to this content.