Sliding bead on a smooth vertical rotated parabola: stability configuration

This paper investigates the motion of a sliding bead on a smooth vertical parabola. The parabola is rotated about its vertical axis with a uniform angular frequency. The governing equation of motion is a highly nonlinear second-order ordinary differential equation. An approximate solution is achieved via the coupling of the homotopy perturbation method and Laplace transform. On the other hand, an expanded frequency concept is utilized in order to obtain an approximate periodic solution. Therefore, the expanded frequency method is applied to govern the stability criterion of the problem. An external excitation of the problem is examined through an additional oscillatory gravitational force. The multiple time-scales with the homotopy perturbation method are used to judge the stability criteria. The analyses reveal the resonance together with the non-resonant cases. Furthermore, the linearization techniques are utilized to check the stability of the linearized equation and to compare the findings with those obtained in the multiple time-scales. Numerical calculations are performed to graphically illustrate, the perturbed solutions as well as the stability examination. It is found that the initial and angular velocities have a destabilizing influence. In contrast, the parameter of the reciprocal of the latus rectum has a stabilizing effect.