The rolling elliptical cylinder

Abstract

Solving the equations of motion for a cylinder with elliptical cross section rolling down an inclined plane constitutes a rather challenging problem. Using Lagrangian mechanics, we have derived the equations of motion analytically and the resulting differential equations are solved numerically. Using a CNC milling machine, we manufactured a solid elliptic cylinder and recorded its dynamics using a high-speed video camera for various inclinations of the plane. We compared theory and experiment by extracting the center of mass coordinates and the angular velocity from the video recording. Limits imposed by the support force reaching zero and insufficient friction to prevent slipping were examined both experimentally and theoretically.