Periodic Solutions of Forced Systems Having Hysteresis

Abstract

In Section I, the motion of a mass on a material rod is discussed. The stress-strain law exhibits a hysteresis effect, so that the stress at any time is a function of the whole strain history. A "rupture restriction" which limits the magnitudes of the admissable displacements, velocities, and accelerations is also discussed. It is shown that the appropriate formulation of the phenomenon is in terms of function spaces and mappings of function spaces. The system is forced with a periodic forcing function of period\Omegaand the problem we set ourselves is to find solutions of period\Omega. In Section II, there is presented first a brief elementary introduction to "Banach Space for Engineers." Then, a purely mathematical theorem is proved concerning the convergence and the speed of convergence of an iterative method for solving a problem in mappings of a Banach Space, and the existence and uniqueness of solutions to that problem. In Section III, the problem posed in Section I is identified with the various concepts introduced in Section II, thus furnishing rigorous proof of the existence and uniqueness of the periodic solution sought in Section I, a computational method for numerically evaluating the solution, and bounds on the errors of approximation. The problem dealt with is quite general, including, in addition to the hysteresis effects, as special cases, the equations of Hill, Mathieu, Duffing, van der Pol (with a forcing term), and a quite general class of nonlinear ordinary differential equations. The method is also applicable to partial differential equations.