A Two-Scale Analysis for a Spherical Pendulum with a Vertically Vibrating Pivot

In this article, we consider the behaviour of a simple undamped spherical pendulum subject to high-frequency small amplitude vertical oscillations of its pivot. We use the method of multiple scales to derive an autonomous ordinary differential equation describing the slow time behaviour of the polar angle which generalises the Kapitza equation for the plane problem. We analyse the phase plane structure of this equation and show that for a range of parameter values there are conical orbits which lie entirely above the horizontal. Going further, we identify a family of quasi-conical orbits some of which may lie entirely above the pivot and establish that initial conditions can be chosen so that precession is eliminated for these orbits. For the general initial value problem, we show that the leading order solutions for the polar and azimuthal angles diverge significantly from their exact counterparts. However, by consolidating the slow scale error term into the leading order structure we may construct extremely accurate solutions for the slow scale evolution of the system. These solutions, confirmed by exact numerical simulations, show that by suitable choice of initial data orbital precession can be eliminated.