Parametric excitation of an internally resonant double pendulum

Abstract

The response of two-degree-of-freedom systems with quadratic and quartic nonlinearities to a principal parametric resonance in the presence of two-to-one internal resonance is investigated. The method of multiple scales is used to construct a first-order uniform expansion yielding four first-order nonlinear ordinary differential (averaging) equations governing the modulation of the amplitudes and phases of the two modes. These equations are used to determine the steady state (fixed point) solutions and their stability. Bifurcations of the fixed points are investigated. Numerical solutions are carried out and graphical representations of the results are presented. The effect of the different parameters on the system response is studied. It is found that each mode has a single-valued curve and there exist zones of multivalued with the increasing and decreasing of some parameters. Both modes lose stability with the varying of some parameters. The region of definition decreases with the increasing and decreasing of some parameters.