FORCING INDUCED ASYMMETRY ON DYNAMICAL SYSTEMS WITH CUBIC NON-LINEARITIES

The present work investigates the dynamics of a class of two-degree-of-freedom oscillators with cubic non-linearity in the restoring forces. These oscillators are under the action of an external load including constant and harmonic components. Initially, a perturbation analysis is applied to the equations of motion, demonstrating the effect of the asymmetry induced by the constant loading component on the classical 1:1 and 1:3 internal resonances, as well as on the possibility of the appearance of a first order 1:2 internal resonance. Next, sets of slow-flow equations governing the amplitudes and phases of vibration are derived for the special case of no internal resonance and for the most complicated case corresponding to 1:1 internal resonance. The analytical findings are then complemented by numerical results, obtained by examining the dynamics of a two-degree-of-freedom mechanical system. First, the effect of certain system parameters on the existence and stability of constant and periodic solutions of the slow-flow equations is illustrated by presenting a sequence of response diagrams. Finally, the dynamics of the system used as an example is investigated further by direct integration of the slow-flow equations. This shows the existence of a period-doubling sequence culminating into a continual interchange between quasiperiodic and chaotic response. It also demonstrates a new transition scenario from phase-locked to phase-entrained and drift response.