Stability Configuration of a Rocking Rigid Rod over a Circular Surface Using the Homotopy Perturbation Method and Laplace Transform

The current paper is concerned with the motion of a rocking uniform rigid rod, without slipping, over a rigid circular surface. The governing equation of motion resulted in a highly nonlinear second-order ordinary differential equation. This nonlinear equation has no natural frequency. A coupling the homotopy perturbation method and Laplace transform is adopted to obtain an approximate solution of the equation of motion. In addition, He's transformation method is used to obtain a periodic solution. Stability conditions are derived by making use of a nonlinear frequency analysis. Numerical calculations are achieved to investigate the governed perturbed solutions as well as the stability picture. It is found that the radius of the length of the rigid rod as well as the radius of the circular surface has a stabilizing influence. In contrast, the initial angular velocity has a destabilizing effect.