Stability, bifurcation, and large-amplitude vibration analysis of a symmetric magnetic spherical pendulum

Abstract

Existing studies on the symmetric spherical pendulum are limited to small- and moderate-amplitude vibrations. This study was conducted to obtain accurate solutions for analysis of the large-amplitude vibration of a symmetric magnetic spherical pendulum using the continuous piecewise linearization method (CPLM). The stability conditions and bifurcation of the pendulum were derived based on the critical points, while the CPLM was used to estimate the frequency response and vibration histories to less than 0.1% and 1.0% relative error respectively when compared to numerical solutions. The CPLM was found to be significantly more accurate than the Laplace transform homotopy perturbation method and predicted the large-amplitude bi-stable vibrations accurately. The stability analysis that was conducted enabled the characterization of all bounded symmetric vibrations based on the relationship between the cyclotron frequency and azimuthal velocity, whereas the bifurcation analysis confirmed that the symmetric vibrations can undergo pitchfork bifurcation that results in transition from single-well to double-well (or bi-stable) vibrations and vice versa. Finally, a parametric analysis was conducted to study the effect of the cyclotron frequency and uniform azimuthal velocity on the frequency–amplitude response and vibration histories The parametric analysis showed that the frequency–amplitude response has a strong dependence on the cyclotron frequency and azimuthal velocity for all amplitudes. On the other hand, the oscillation profile only depends on the cyclotron frequency and azimuthal velocity for some amplitudes. The results of this study can be applied in the design of energy harvesters and elliptic tanks for liquid transport.