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Power Series Solution to a Simple Pendulum with Oscillating Support

Mohammad B. Dadfar and James F. Geer
SIAM Journal on Applied Mathematics
Vol. 47, No. 4 (Aug., 1987), pp. 737-750
Stable URL: http://www.jstor.org/stable/2101566
Page Count: 14
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Power Series Solution to a Simple Pendulum with Oscillating Support
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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 ωo and ω 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 (ω > ωo), below resonance (ω < ωo), and at resonance (ω = ωo). In each case, the approximate location of the nearest singularities which limit the convergence of the power series are obtained by using Padé approximants. Using this information, a new expansion parameter δ is introduced, where the radius of convergence of the transformed series is greater than the original series. The effects of primary and higher order resonances on the convergence of the series solution is noted and discussed.

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