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We investigate the nonlinear effect of a pendulum with the upper end fixed toan elastic rod which is only allowed to vibrate horizontally. The pendulum willstart rotating and trace a delicate stationary pattern when released withoutinitial angular momentum. We explain it as amplitude modulation due tononlinear coupling between the two degrees of freedom. Though the phenomenon ofconversion between radial and azimuthal oscillations is common for asymmetricpendulums, nonlinear coupling between the two oscillations is usuallyoverlooked. In this paper, we build a theoretical model and obtain thependulum's equations of motion. The pendulum's motion patterns are solvednumerically and analytically using the method of multiple scales. In theanalytical solution, the modulation period not only depends on the dynamicalparameters, but also on the pendulum's initial releasing positions, which is atypical nonlinear behavior. The analytical approximate solutions are supportedby numerical results. This work provides a good demonstration as well as aresearch project of nonlinear dynamics on different levels from high school toundergraduate students.

Nonlinear coupling in an asymmetric pendulum