On Some Analytical Approximate Solutions of Fourth Order Damped Oscillatory and Near Critically Damped Nonlinear Systems

Abstract

At first Krylov and Bogoliubov prcseflted a perturbation method known as the asymptotic averaging method' in the theory of nonlinear oscillations. Primarily, the method was proposed only to get the periodic solutions of second order autonomous systems with small nonlinearities. Later, the method has been extended by Bogoliubov and Mitropolskii. At present the method is used to obtain the solutions of second and higher order nonlinear equations for damped oscillatory, over damped, near critically damped, critically damped, more critically damped systems under some special conditions. The unified Krvlov-13ooliubov-Mitropolskii (KBM) method is used to find approximate solutions of fourth order nonlinear systems with large damping. In this thesis. the KBM method has been modified and elaborated to find out the solutions of' fourth order damped oscillatory and near critically damped non-oscillatory nonlinear systems by imposing some restrictions on the cigen-values. For verification of the results obtained by the modified KI3M method, we have compared them with those obtained by the fourth order Runge-Kutta method and a nice matching is observed.