Theoretical and experimental investigation of the dynamics and bifurcations of an impacting spherical pendulum with large deflection

The inverted spherical pendulum has been a common engineering paradigm for strong focusing mechanisms. The spherical pendulum has been used to damp irregular motions in helicopters and on space stations as well as for many other appHcations. An inverted impacting spherical pendulum with large deflection was investigated.

The model was designed to approximate an ideal pendulum, with the pendulum bob contributing the vast majority of the mass moment of inertia of the system. Two types of bearing mechanisms and tracking devices were designed for the system, one of which had a low Coulomb damping coefficient and the other with a high Coulomb damping coefficient.

In the development of the dynamics theory a thorough analysis of the air damping was performed. The analytical solution assumed that the velocity squared term and the Coulomb term dominated the system damping. This broke with conventional analysis which had linear velocity terms dominating the damping of the pendulum. The system description consisted of the equations of motion, steady state motions. Type I and II subharmonics, and inversion criteria, AU simulations were run at twenty times the natural frequency of the pendulum to maximize shaker output. Power outputs from 0 to 125 mm-htz were studied both numerically and experimentally. Additional numerical simulations in the region 0 to 600 mm-htz were performed to determine extended system dynamics. Numerical simulations were performed using Butcher's fifth-order Runge-Kutta approximations for speed and accuracy. Fractal dimensions and Lyapunov exponents were derived using a new computational scheme that reduced the time necessary to find these components by an order of magnitude.

This paper will discuss the characteristic response of the system for various coefficients of restitution, mass moment of inertias. Coulomb damping coefficients and forcing amphtudes. It will be shown that though the pendulum can move in three dimensions, it will act as a plane pendulum in its response. Furthermore, it will be shown that the system responded in a Type I response at the lower values of forcing amplitude, had a mixed Type I/II response and then exhibited a subharmonic Type II response at higher values of forcing amplitude. Numerical simulations showed that at large values of forcing amplitude the system exhibited a loss of stabihty after a harmonic region and then stably inverted.