Statistics of a noise-driven elastic inverted pendulum

Abstract

Recently an elastic inverted pendulum structure was proposed as a means to make nonlinear energy harvesters. An effective dynamical model of this bi-stable system has an effective lumped mass that is dependent on the displacement, hence preventing direct application of previous analyses for nonlinear harvesters driven by random vibrations. We have set up a stationary Fokker-Planck equation for the inverted pendulum and solved it to obtain explicit expressions for the stationary probability densities of the system. We found that the marginal distribution of velocity is non-Gaussian, but numerically it differs little from a Gaussian when parameters for a recently published device are used. The conditional probability of position given velocity, has two peaks for low velocities. These merge into one upon increase of velocity.