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 c-=( )2 L,where w0 and w 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 (w > wo), below resonance (w < wo), and at resonance (w = wo). In each case, the approximate location of the nearest singularities which limit the convergence of the power series are obtained by using Pad6 approximants. Using this information, a new expansion parameter 6 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.
Dadfar, Mohammad and Geer, James, "Power Series Solution to a Simple Pendulum with Oscillating Support" (1987). Computer Science Faculty Publications. 1.
SIAM Journal on Applied Mathematics