Approximate analytical solutions for oscillatory and rotational motion of a parametric pendulum

In this paper, the authors have studied dynamic responses of a parametric pendulum by means of analytical methods. The fundamental resonance structure was determined by looking at the undamped case. The two typical responses, oscillations and rotations, were investigated by applying perturbation methods. The primary resonance boundaries for oscillations and pure rotations were computed, and the approximate analytical solutions for small oscillations and period-one rotations were obtained. The solution for the rotations has been derived for the first time. Comparisons between the analytical and numerical results show good agreements.