A multibody loop constraints approach for modelling the wheel/ground rigid contact of rolling systems

The present paper follows a work dealing with wheel/rail contact constraints published by Samin and Fisette in 1994 [1] and cam/follower contact constraints published in 1999 [2]. According to the latter publications, mechanical system subjected to contact constraints is modelled using the multibody approach and relative coordinates. The proposed methodology is applied to the vehicle dynamics, for which the wheel/ground contact is fundamental. In a lot of cases, one can admit that some contacting elements are significantly elastic: this is especially obvious in the contact region. In spite of this, the rigid contact model still has an important meaning and raison d'etre for pedagogic purpose but also for the analysis of rolling systems for which the contact slip is sufficiently negligible to justify pure rolling conditions. The goal of the present work is to describe the permanent contact between a wheel and the ground in the normal, tangent and longitudinal directions. The proposed form of the constraints equations is not homogenous: whereas the holonomic normal contact constraint has a purely geometrical nature, the tangent and the longitudinal contact constraint can only be expressed as kinematic non holonomic constraints. As a consequence, a "mixed" integration scheme involving the first and second order differential equations becomes necessary for the system analyse. The illustrative example will be the well-known bicycle which, although being a very familiar system, is a very interesting and fruitful application in vehicle dynamics.