The Rolling Wheel Equations without Magic: A Combined Slip-Stiction Approach by the Udwadia-Kalaba Formulation with Constraints Relaxation for a Smooth Simulation
- 1 KDYNAMICS Logiciel Inc., Sherbrooke, Canada
Abstract
The Udwadia-Kalaba formulation is proposed to model the dynamics of the rolling wheel. A unified approach that addresses both the slip and the stiction in the contact section is considered. Purely rolling constraints are associated with stiction and are suitably lifted as slip occurs. An extended formulation for the Uwadia-Kalaba equations of motion is introduced for that matter. It resorts to the weighted minimum norm and the weighted semi-least-squares solutions of the constraints equations. This not only allows a bias on constraints, by an appropriate description of weight functions based on friction, it also leads to a smooth activation or deactivation of selected constraints without rewriting the equations of motion or upsetting their numerical integration.
- Caverley, R. (2001) Constrained or Unconstrained, That Is the Equation. USC News.
- Udwadia, F.E. and Kalaba, R. (1992) A New Perspective on Constrained Motion. Proceedings of the Royal Society, 439, 407-410.
- Soltakhanov, S.K., Yushkov, M.P. and Zeghzda, S. (2009) Mechanics of Nonholonomic Systems. Springer, Berlin Heidelberg. https://doi.org/10.1007/978-3-540-85847-8
- Neimark, J.I. and Fufaev, N.A. (1972) Dynamics of Nonholonomic Systems. American Mathematics Society.
- Rao, C.R. and Mitra, S.K. (1971) Generalized Inverse of Matrices and Its Applications. Wiley, New York, London.
- Mitra, S.K. and Rao, C.R. (1974) Projections under Seminorms and Generalized Moore Penrose Inverses. Linear Algebra and Its Applications, 9, 155-167. https://doi.org/10.1016/0024-3795(74)90034-2
- Gauss, C.F. (1829) über ein neues allgemeines Grundgesetz der Mechanik. Journal für die reine und angewandte Mathematik Journal für die reine und angewandte Mathematik, 1829, 232-235. https://doi.org/10.1515/crll.1829.4.232
- Lanczos, C. (1949) The Variational Principles of Mehanics. University of Toronto Press, Toronto. https://doi.org/10.3138/9781487583057
- Kalaba, R., Natsuyama, H.H. and Udwadia, F.E. (2004) An Extension of Gauss’ Principle of Least Constraint. International Journal of General Systems, 33, 63-69. https://doi.org/10.1080/0308107031000139996
- Popp, K. and Schiehlen, W. (2010) Ground Vehicle Dynamics. Springer, Berlin Heidelberg. https://doi.org/10.1007/978-3-540-68553-1
- Cardona, A. and Géradin, M. (2001) Flexible Multibody Dynamics. Wiley, New York.
- Oden, J. T. and Martins, J. (1985) Models and Computational Methods for Dynamic Friction Phenomena. Computer Methods in Applied Mechanics and Engineering, 52, 527-634. https://doi.org/10.1016/0045-7825(85)90009-X