A Minimalist Approach to Rolling Wheels

TL;DR

This paper derives equations for wheel-road pairs based solely on the no-slip rolling condition, enabling the analysis of wheels with complex, non-differentiable shapes.

Contribution

It introduces a minimalist derivation of wheel-road equations that accommodates non-differentiable wheel shapes, expanding the scope of rolling analysis.

Findings

Derived equations for arbitrary wheel shapes based on no-slip condition

Allowed analysis of non-differentiable, continuous wheel shapes

Extended classical rolling equations to more general scenarios

Abstract

In 1960, G. B. Robison discovered the general equations relating roads and wheels, where either can have an unusual shape (e.g., the square wheel rolls smoothly on a catenary). But he used some inobvious assumptions regarding the meaning of rolling. Here we derive the equations for the road appropriate for a given wheel using only the single assumption that rolling occurs with no slipping. We do not require that the wheel be differentiable, so this allows the construction of a wheel-road pair when the wheel is a continuous nowhere differentiable function.