Effect of fast harmonic excitation on a self-excited motion in Van der Pol oscillator

This paper studies the effect of horizontal and vertical fast harmonic parametric excitation on periodic self-excited motion in Van der Pol pendulum. The method of direct partition of motion is first applied to derive the equation governing the slow motion of the oscillator which is the main equation of vibrational mechanics. Then, a perturbation method is used on the slow motion equation to obtain analytical approximation for the periodic solution and its frequency in the slow dynamic. A relationship between the frequency of the limit cycle and the frequency of the fast harmonic excitation is derived. It is shown that in the case where the point of suspension of the pendulum is subjected to horizontal fast harmonic excitation, the periodic self-excitation can disappear. In contrast, this periodic solution persists in the case of the vertical fast harmonic excitation. Numerical simulations on the slow flow and on the original equation confirms the analytical prediction