The Response of Two-Degree of Freedom Self-Sustained Systems with Quadratic Nonlinearities to a Parametric Excitation (RESEARCH NOTE)

In this study the interaction between self-excited and paramet rically excited oscillations in two-degree-of-freedom systems with quadratic nonlinearities is investigated. The fundamental parametric resonance of the first mode and 3:1 internal resonance is considered, followed by 1:2 internal and parametric resonances of the second mode. The method of multiple time scales is applied to derive four first-order non-linear ordinary differential equations that describe the modulation of the amplitudes and phases of both modes caused by resonance. These equations are used to determine steady state amplitudes. To determine stability of the steady state solutions, small disturbances in the amplitudes and phases are superposed on the steady state solutions and the resulting equations are linearized. The eigenvalues of the corresponding system of first-order differential equations determine the stability of the steady state solutions. The instability modes are discussed and the amplitude and frequency response curves are presented by varying parameters of the system.