Abstract:
There are many approaches to study equations with time-periodic coefficients, in which a small parameter may be assigned, in particular the classical asymptotic methods. The present paper is concerned with the analysis of the applicability of these approaches for solving nonlinear equations, which don’t contain a small parameter explicitly. A new modifi-cation of the method of direct separation of motions (MDSM) [1,2], which may be employed to study such equations, is proposed. As an example a classical problem about the stability of a pendulum with vibrating suspension axis is considered in an unconventional case, when the frequency of external loading and the natural frequency of the pendulum in the absence of this loading are of the same order. As the result it is shown that in the considered range of parameters not only the effective “stiffness” of the system changes due to the external loading, but also its effective “mass”. It is noted that application of the classical asymptotic methods in this case leads to erroneous results. A correlation between the proposed modification of the MDSM and Ritz’s method of harmon-ic balance, Van der Pol’s method of slowly varying amplitudes, the classical asymptotic methods and other approaches is discussed.
