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.

中文翻译：

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具有垂直振动枢轴的球摆的两尺度分析

在本文中，我们考虑一个简单的无阻尼球摆在其枢轴的高频小振幅垂直振荡下的行为。我们使用多尺度方法推导出一个自治常微分方程，该方程描述了极角的慢时间行为，它概括了平面问题的 Kapitza 方程。我们分析了该方程的相平面结构，并表明对于一系列参数值，存在完全位于水平面上方的锥形轨道。更进一步，我们确定了一系列准圆锥轨道，其中一些可能完全位于枢轴上方，并确定可以选择初始条件，以便消除这些轨道的进动。对于一般的初始值问题，我们表明极角和方位角的领先阶解与它们的精确对应显着不同。然而，通过将慢尺度误差项合并到主阶结构中，我们可以为系统的慢尺度演化构建极其精确的解。These solutions, confirmed by exact numerical simulations, show that by suitable choice of initial data orbital precession can be eliminated.