A computational model of the rolling coin using the pseudo-rigid body theory

Abstract

One of the classical problems of Rigid-Body mechanics is that of a coin rolling without sliding on a horizontal plane. The non-sliding condition gives rise to a so-called non-holonomic constraint, namely, a non-integrable differential form that places a restriction on the instantaneous velocity of the system. One of the intriguing challenges of the rolling coin problem is that of explaining the noise produced by the fast wobbling of a rotating dish on a table as it slowly approaches the final horizontal position of rest. It has been suggested that this phenomenon results from the coupling between the overall rigid body motion and the small elastic vibrations of the plate. To introduce this aspect into the formulation, we suggest the use of the Theory of Pseudo-Rigid Bodies, whereby a partial deformability of the coin is introduced by assuming that it can undergo homogeneous deformations only, that is, that the strain is uniform throughout the coin. In this study, however, only in-plane vibrations of the coin are considered, leaving the important issue of transverse vibrations for future development.

The shape of the deformed coin is an ellipse whose eccentricity and orientation keeps changing in time. As a result, the dynamical system is still a classical one governed by ordinary, rather than partial, differential equations.

Using the principles of Lagrangian Mechanics, the equations of motion of the system are formulated and solved numerically for different initial conditions. The solutions clearly show the coupling of the rolling with the deformation.