Sliding Bead on a Smooth Vertical Rotated Parabola: Stability Configuration

Galal M. Moatimid

Abstract


      The current article 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 of highly nonlinear second-order ordinary differential equation. An approximate solution of this equation is achieved via a coupling between the homotopy perturbation method (HPM) and Laplace transform ( ). On the other hand, a nonlinear frequency technique is utilized to obtain an approximate periodic solution. Therefore, the nonlinear frequency method is exercised to govern the stability criteria of the problem. An external excitation of the problem is examined through an oscillatory gravitational force. The multiple time-scales with the HPM is used to judge the stability criteria. The analyses reveal the resonance as well as the non-resonant cases. Numerical calculations are performed to illustrate, graphically, the perturbed solutions as well as the stability examination. It is found that the initial linear velocity as well as the angular velocity have a destabilizing influence. In contrast, the parameter that defines the reciprocal of the latus rectum has a stabilizing one.

 


Keywords


Homotopy perturbation methods, Laplace transform, Nonlinear frequency analysis, Multiple time-scales technique.

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