Nonlinear coupling in an asymmetric pendulum

Abstract

We investigate the nonlinear behavior of a simple pendulum fixed to an elastic rod, which can only vibrate horizontally. When released without initial angular momentum, the plane of the pendulum rotates, and the bob traces a delicate stationary pattern. We explain this effect as amplitude modulation due to the nonlinear coupling between the two degrees of freedom. We construct a theoretical model and approximately solve the equations of motion analytically (using the method of multiple scales); we also solve these equations numerically. In the analytical solution, the modulation period depends not only on the dynamical parameters but also on the pendulum's initial release position, which is typical of nonlinear systems. Finally, we build a simple apparatus and conduct a quantitative experiment. The approximate analytical solutions exhibit the same trends as the numerical results and experimental data. After adding linear dissipation, the numerical and experimental results match fairly well. This system can serve as an instructive demonstration as well as a nonlinear dynamics research project for undergraduate students.