Numerical investigation of the parametric pendulum under filtered random phase excitation

The parametrically excited pendulum is a highly nonlinear system which has been thoroughly studied regarding the response's stability and the fundamental types of motion that could be established. The rotating potential of a pendulum having its suspension point vertically excited was numerically identified and the appropriate excitation characteristics were presented. In this paper, the excitation is modeled using the random phase modulation and the rotational motion is sought. The resulting stochastic system is analyzed by using a numerical Path Integration (PI) method solving the Chapman-Kolmogorov equation to construct parameter space plots. The Probability Density Function (PDF) is computed and the rotational regions are identified as well as the effect of noise intensity onto them. Previous studies have shown that for small values of noise intensity the stochastic response resembles the deterministic one. However numerical simulations showed that with increasing noise intensity the regions of rotational motion become narrower. In order to improve the system's response a linear singledegree-of-freedom (SDOF) system is intercepted to filter the noisy excitation, forming a base excited SDOF system acting on the pendulum suspension point. The interaction between the moving pendulum mass and the SDOF filter is investigated with the goal being for the former to establish rotational motion.