Tautochrone and Brachistochrone Shape Solutions for Rocking Rigid Bodies

TL;DR

This paper introduces a comprehensive analysis of rocking rigid bodies, deriving their equations of motion, and presents novel shapes called tautochrone and brachistochrone, which generalize classical curves and offer new dynamical properties.

Contribution

It provides a new inversion method for designing rocking rigid bodies with specific dynamics and introduces two novel shapes that extend classical tautochrone and brachistochrone curves.

Findings

Derived equations of motion for rocking rigid bodies

Developed a general shape construction method based on desired dynamics

Introduced novel tautochrone and brachistochrone shapes for rocking bodies

Abstract

Rocking rigid bodies appear in several shapes in everyday life: As furniture like rocking chairs and rocking cradles or as toys like rocking horses or tilting dolls. The familiar rocking motion of these objects, a non-linear combination of a rigid rotation and a translation of the center of mass, gives rise to a number of interesting dynamical properties. However, their study has received little attention in the literature. This work presents a comprehensive introduction to the dynamics of rocking rigid bodies, including a concise derivation of the equations of motion as well as a general inversion procedure to construct rocking rigid body shapes with specified dynamical properties. Moreover, two novel rigid body shapes are derived - the tautochrone shape and the brachistochrone shape - which represent an intriguing generalization of the well-know tautochrone and brachistochrone curves. In particular, tautochrone shapes offer an alternative construction of a tautochrone pendulum, in addition to Huygens' cycloid pendulum solution.