Characterization of the Elliptical Pendulum

The relationship between the oscillation period and the length of a simple pendulum is well known. Other more complex pendulums such as physical, double, and inverted pendulums have been thoroughly examined as well. We present here, for the first time to the best of our knowledge, a systematic modeling and experimental analysis of the elliptical pendulum. An elliptical pendulum is a mass that is constrained to oscillate along the arc of an ellipse. Our dynamical modeling indicates that the period is dependent on both the initial displacement of the pendulum and the eccentricity of the ellipse. We have measured the period as a function of initial displacement and eccentricity in order to test our dynamical model. The period was measured using a rotary motion sensor and was compared to the period predicted from a Lagrangian dynamics analysis of the motion with the displacement angle from the vertical as the generalized coordinate. The Lagrangian analysis generated an equation of the motion which was used to predict the period of oscillation for a given ellipse eccentricity and initial displacement angle. Good agreement of the model with the data was obtained for ellipse eccentricities in the range of 0.1-0.3. For higher eccentricities, however, the model deviated appreciably from the data. There are particular indications in the data that suggest these deviations originate with the apparatus and not with the model. Future directions include building a new apparatus that addresses the faults of the original apparatus and continued refinement of the model to better account for small deviations of the actual path of the pendulum from that of a perfect ellipse.