Tautochrone and Brachistochrone Shape Solutions for Rocking Rigid Bodies

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.