Parametric Pendulum

Although parametric resonance occurs in areas disparate as quantum mechanics, cosmology, and the mechanics of machinery, very few students in the physical sciences and engineering are ever exposed to the concept. The presence of time-varying coefficients in the differential equations describing a parametrically excited system leads to a rich set of dynamical behaviors, including counter-intuitive stability or instability. The study presented here describes a simple mechanical system consisting of a pendulum suspended on a pivot driven harmonically up and down by a slider-crank mechanism, and the mathematical model describing the angular position of the pendulum as a function of time. Floquet analysis is used to predict both the presence of stable and unstable parametric combinations of angular velocity and slider-crank length for the up and down pendulum equilibrium pendulum positions. These predictions are verified experimentally using a machine designed and built for this purpose