  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 and trace a delicate stationary pattern when released without
initial angular momentum. We explain it as amplitude modulation due to
nonlinear coupling between the two degrees of freedom. Though the phenomenon of
conversion between radial and azimuthal oscillations is common for asymmetric
pendulums, nonlinear coupling between the two oscillations is usually
overlooked. In this paper, we build a 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 modulation 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 supported
by numerical results. This work provides a good demonstration as well as a
research project of nonlinear dynamics on different levels from high school to
undergraduate students.
