ON THE GENERALIZED AVERAGING METHOD OF A CLASS OF STRONGLY NONLINEAR FORCED OSCILLATORS

We present a modified version of the generalized averaging method for studying the periodic solutions of a class of strongly nonlinear forced oscillators of the form x + omega2x + epsilonf(x) P(OMEGAt) = 0, where f(x) is a nonlinear function of x, P(OMEGAt) is a periodic function of t, epsilon need not be small, and omega is a constant parameter. This equation can be used to describe, e.g., a pendulum with a vibrating length or the displacements of colliding particle beams in high energy accelerators. The new version is based on defining a new parameter alpha = alpha(epsilon) and a linear transformation of the time. This version is applied for the cases f(x) = x3 and f(x) = x4 with P(OMEGAt) = cost and excellent agreement is found with the results of numerical experiments, for large values of epsilon.