Pendulum attached to a vibrating point: Semi-analytical solution by optimal and modified homotopy perturbation method

Abstract

A simple pendulum of length $\left(b\right)$ and bob mass $\left(m\right)$ attached to point $\left(O\right)$ is considered and investigated. The point $\left(O\right)$ is oscillating vertically according to the relation $\left(q_o\cos \Omega t\right)$, where $q_o$ and $\Omega$are amplitude and angular frequency of the external agent, respectively. The presence of time dependent oscillating term makes the governing equation is not solvable analytically. An attempt was to explore the application of optimal and modified homotopy perturbation method (OM-HPM) as a powerful semi-analytical tool for solving the oscillatory problem which exhibiting regular and irregular oscillation for some parameter set. Furthermore, the analytical expressions in series form, which is very close to the numerical solution of Runge-Kutta method is obtained. In addition, the analytical expression for the amplitude and the frequency of the oscillations for two cases: simple regular oscillation and the irregular oscillation is computed. Finally, the simplicity of the obtained solutions facilities a clear understanding, and the OM-HPM offer a robust and efficient analytical tool to obtain series based analytical solution for such kind of problems.