THE MOTION OF A SYSTEM CLOSE TO HAMILTONIAN WITH ONE DEGREE-OF-FREEDOM WHEN THERE IS RESONANCE IN FORCED VIBRATIONS

The motion of an autonomous mechanical system with one degree of freed om subject to small time-periodic perturbations and small dissipative forces in the vicinity of a stable position of equilibrium of the syst em is considered. It is assumed that resonance occurs in forced vibrat ions when the ratio of the frequency of small vibrations of the system to the frequency of the external periodic perturbation is close to an integer. The qualitative behaviour of an approximate system is studie d. Depending on the parameters of the problem, namely, the magnitude o f the dissipation and resonance detuning, a rigorous solution of the p roblem of the existence, number, and stability of periodic motions (th e period being equal to that of the perturbation) arising from the pos ition of equilibrium of the unperturbed system is given. As an example the motion of a pendulum with oscillating point of suspension is cons idered. Copyright (C) 1996 Elsevier Science Ltd.