KBM Asymptotic Method for third order Nonlinear Oscillations

Abstract

In most treatments or nonlinear oscillations hy perturbation method. only periodic oscillations are treated; transients are not considered. Krylov and Bogoliubov have used a perturbation method to discuss transients in the second order autonomous systems with small nonlinearities. The method is well known as an 'averaging method' in the theory of nonlinear oscillations. Later the method has been amplified and justified by L3ogoliubov and Mitropolskii. In this dissertation, we invt:stigatc SOllll' third order nonlinear oscillations based on the work of Krylov-13ogoliubov-Mitroplskii (KBJ\:1). First, nonlinear oscillation described by a third order ordinary autonomous differential equation is considered and a new perturbation tcclmiquc is developed. Then a method has been dcwlopcd 1(1 find asymptotic solution or a damped nonlinear system. The method is a generalization of Bogoliubov's asymptotic method and covers both under-damped and overdamped systems. Later clamped oscillations including critically damped motion have also been investigated in presence or more significant damping forces. Third order nonlinear oscillations with clamping and time delay, and with varying coenicients have been investigated separately. Moreover. a simple overdamped solution has been found for the third order weakly nonlinear systems.