Some types of calculation connected with averaging in the theory of non-linear vibrations

Abstract

Applications of the method of averaging to the calculation of non-linear vibrational systems are considered in the present review paper. In Section 1 are described the methods of N.N. Bogolyubov and D.N. Zubarev [1] relating to the averaging of systems with a rotating phase, and some generalizations of the averaging method. In Section 2 a scheme of applying averaging to non-linear vibrational systems, which contain slowly varying parameters, is given, and singularities of systems similar to Hamiltonian systems are considered. In Section 3 systems with one degree of freedom are studied, and methods are developed for calculating them. A number of specific examples illustrating the methods proposed are given in Section 4. In particular, the amplitude of vibrations of some real mechanical models is calculated. Attention is paid to the description of rules and the order of performing the calculations while applying the averaging method to the investigation of non-linear vibrations.