Free and forced oscillations of a class of self-excited oscillators

Abstract

Free and forced oscillations in oscillators governed by the equation [MATHEMATICAL NOTATION GOES HERE. VIEW IT IN THE DOCUMENT] are studied with appropriate constraints on [MATHEMATICAL NOTATION GOES HERE. VIEW IT IN THE DOCUMENT]. Theorems are proved on the existence and uniqueness of stable periodic solutions for free oscillations using the Poincaré-Bendixson theory in the phase-plane. There follow several examples to illustrate the theorems and limit cycles are obtained for these examples by the Liénard construction. A result on the existence of periodic solutions in the forced case is obtained by use of Brouwer's fixed point theorem. The part on topological methods is concluded by applying Yoshizawa's results on ultimate boundedness of solutions to the forced case. Approximate analytical solutions are obtained for specific examples for different regions of validity of the parameter (. For free oscillations, the perturbation solution is obtained for small (. A Fourier series approximation is given for other values of (, and the limit cycle for the case ( (tm)co is obtained. Finally, the first order solution for forced oscillations is obtained by the method of slowly varying parameters and the stability of this solution is examined.