Power Series Solution to a Simple Pendulum with Oscillating Support

Abstract

The problem of determining some of the effects of a small forcing term on a regular perturbation solution to a nonlinear oscillation problem is studied via a simple example. In particular, we investigate the periodic solution of a simple pendulum with an oscillating support. A power series solution is constructed in terms of $\epsilon = ( \frac{\omega }{\omega _o } )^2 \frac{a}{L}$, where $\omega _o $ and $\omega $ are the natural and driving frequencies respectively, a is the amplitude of the support oscillation, and L is the length of the pendulum. These solutions are analyzed for three cases: above resonance $( \omega > \omega _o )$, below resonance $( \omega < \omega _o )$, and at resonance $( \omega = \omega _o )$. In each case, the approximate location of the nearest singularities which limit the convergence of the power series are obtained by using Pade approximants. Using this information, a new expansion parameter $\delta $ is introduced, where the radius of convergence of the transformed series ...