Asymptotic Analysis of a Nonlinear Autonomous Vibratory System

In a recent paper Senator analyzed a system consisting of a spring, dashpot, and mass upon which is mounted a rotating eccentric weight driven by a motor with a linear torque-speed characteristic. 1 This system has been analyzed by several authors, under different assumptions on the values of the parameters of the system (see Refs. 5 through 9), and Senator discusses their results. The system is a model for the interaction between the rotational motion of a motor driving an eccentric and the translational vibratory motion of the frame, which is caused by this rotation. In Ref. 1, Senator constructed periodic (rotational) solutions by means of a perturbation technique, based on small reciprocal of rotational inertia. However, he did not analyze the stability of the periodic solutions directly, but proceeded in a somewhat different manner. Thus, he introduced a van der Pol type transformation, but imposed a subsidiary condition on the slowly varying functions of time which differs from the one usually imposed in the method of averaging. He then 73