The phenomenon of nonlinear coupling in an asymmetric pendulum

We investigate the nonlinear effect of a pendulum with the upper end fixed to an elastic rod which is only allowed to vibrate horizontally. The pendulum will start rotating while oscillating without initial angular momentum. We explain it as amplitude and phase modulation due to nonlinear coupling between the two dimensions. We build the theoretical model and obtain the pendulum's equations of motion. The pendulum's motion patterns are solved numerically and analytically using the method of multiple scales. In the analytical solution, the conversion period not only depends on the dynamical parameters, but also on the pendulum's initial releasing positions, which is a typical nonlinear behavior. The analytical approximate solutions are consistent with numerical solutions with good accuracy.