Pendulum Analogy for Nonlinear Fluid Oscillations in Spherical Containers

Abstract

The nonlinear response of a pendulum model is compared to that of fluid motion in a spherical container undergoing lateral oscillations. It is observed that the pendulum does not duplicate the nonlinear response of the fluid except at a particular fluid height. The strong boundary curvature of the spherical container is blamed for this disagreement. Therefore, a cubic spring is included to provide sufficient elasticity to compensate for the boundary effects. The proper values for the coefficient of the cubic spring are found from an experimental observation of planar fluid motion in a spherical tank. It is also noted that, if the pendulum and the fluid are not constrained to a planar motion, at a certain excitation amplitude and frequency, both depart from the plane of excitation and rotate about a vertical axis. The upper and lower boundaries of this rotational motion of the fluid are obtained by a mathematical analysis of its analog pendulum model and compared to those measured experimentally. Mathematical predictions agreed more favorably with the experimental results after the inclusion of the cubic spring in the pendulum model.