A modified perturbation method for treating non-linear oscillation problems

Abstract

In this investigation a method that uses multiple time scales for the purpose of obtaining uniform asymptotic solutions of non-linear ordinary differential equations is modified through the introduction of a new small parameter, μ, defined by μ=ɛ/(1+ɛ), where ɛ denotes the non-linearity parameter given in the differential equation under consideration. Since the value of μ is positive and less than unity for any positive value of ɛ, asymptotic expansions in terms of μ are attempted in the hope of enlarging the radius of convergence of these expansions. The technique is applied to a differential equation which arises in the nonlinear theory of transverse vibrations of hinged-hinged thin cylindrical shells. A series of numerical calculations reveal that the μ-expansion proposed here yields an approximation for the frequency of free periodic motion of the shell which represents an improvement over the corresponding ɛ-expansion for a significant range of the parameters studied.