Kinematics of rolling contact: Derivation, misconceptions, and generalisations

Abstract

Rolling contact kinematics plays a foundational role in understanding the dynamic behaviour of a wide range of mechanical systems, from pneumatic tyres and railway wheels to tribological interfaces and robotic joints. Classical derivations of rolling contact kinematics often rely on vague or implicit assumptions about body elasticity or adopt non-rigorous arguments that can lead to incorrect or incomplete interpretations of slip and spin phenomena. This paper presents a unified and rigorous treatment of rolling contact, systematically addressing the cases of incommensurably elastic and similarly elastic bodies. Particular attention is dedicated to correcting several misconceptions widespread in the literature, including the misdefinition of rolling velocity and its pathological consequences. Based on the two limiting elastic cases, the paper further introduces a generalised elasto-kinematic framework combining kinematic constraints with local linear elastic equations. The resulting model is shown to be a symmetric hyperbolic system, and the novel theory is applied to an illustrative example to emphasise the role played by the bodies’ elasticity in the correct definition of slip and spin variables and in the calculation of the frictional forces and moments.