Resonance oscillations in some systems with aftereffect

Abstract

Non-linear systems, described by Volterra integro-differential equations, with a characteristic equation possessing a pair of pure imaginary roots with other roots with negative real parts are considered. A small limit periodic perturbation, specified by a time function, which tends exponentially with time to periodic oscillations, the frequency of which is equal to the frequency of the oscillations in the linearized system, acts on the system. The problem of the rotational motions of a solid plate in an air flow, taking into account the nonstationarity of the flow in the framework of the model which takes into account the nonstationarity by introducing integral terms into the moments of the aerodynamic forces, is considered. Amplitude equations, which solve the problem of the existence in the system of limit periodic rotational oscillations, which occur due to the action of a small perturbation of the flow when there is resonance, are constructed from third-order terms. An assertion is presented which generalizes the result previously obtained regarding the existence in the system of limit periodic motions, when the perturbation is specified by a function of time, having a derivative of bounded variation. The motions, indicated earlier, approach periodic oscillations, represented by absolutely converging Fourier series. The generalization relates to a class of perturbations, specified by continuous or piecewise-continuous functions of time.