Periodic solutions of equations of motion of a heavy solid applying Krylov–Bogoliubov–Mitropolski method

Abstract

Poincaré's small parameter method and Krylov–Bogoliubov asymptotic method are among the number of basic methods used for the study of nonlinear oscillations. Poincaré's method was developed in conformity with stationary (periodic) oscillations (Malkin, Certain problems in the theory of nonlinear oscillations, ABC-tr-3766, United States Atomic Energy Commission, Technical Information Service, 1959), although it may be extended to nonstationary oscillations (Bakhmutskii, On the application of Poincaré's method to the study of unsteady oscillations, Izv. Akad. Nauk SSSR OTN Mekh. Mashinostroenie (3) (1961)). The Krylov–Bogoliubov method may be used, first of all, for a study of nonstationary oscillations, but it is, of course, completely applicable to periodic oscillations (Bogoliubov and Mitropolski, Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordon and Breach, New York, 1961.). It is sometimes asserted that these methods are different in principle. In the present paper, the method of Krylov–Bogoliubov–Mitropolski (Krylov and Bogoliubov, Introduction to Non-linear Mechanics, Princeton University Press, Princeton, NJ, 1947, pp. 165–174) is modified to investigate periodic solutions, with nonzero basic amplitudes, for the equations of motion of a heavy solid, with one fixed point, rapidly spinning about the major or the minor axis of the ellipsoid of inertia. Numerical solutions for the reduced system of the equations of motion are obtained using fourth-order Runge–Kutta method (Rice, Numerical Methods, Software and Analysis, McGraw-Hill, New York, 1983). The analytical solutions are compared with the numerical ones in order to show the small deviation between them.