A generalization of the classical Kapitza pendulum is considered: an inverted planar mathematical pendulum with a vertically vibrating pivot point in a time-periodic horizontal force field. We study the existence of forced oscillations in the system. It is shown that there always exists a periodic solution along which the rod of the pendulum never becomes horizontal, i. e., the pendulum never falls, provided the period of vibration and the period of horizontal force are commensurable. We also present a sufficient condition for the existence of at least two different periodic solutions without falling. We show numerically that there exist stable periodic solutions without falling.

中文翻译：

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Kapitza – Whitney摆的平均方法

考虑了经典的Kapitza摆的一般化：在时间周期的水平力场中具有垂直振动的枢轴点的倒置平面数学摆。我们研究了系统中强迫振荡的存在。结果表明，只要振动周期和水平力的周期是可比较的，则始终存在周期解，摆锤的杆永远不会变为水平，即，摆锤永远不会掉落。我们还为至少两个不同的周期解的存在提供了充分条件而不会下降。我们用数字表明存在稳定的周期解而不会下降。