The classical Kapitsa’s problem of stability of an inverted pendulum subjected to vertical vibration of the support and some generalizations are investigated. The asymptotic method of two-scale expansions allows one to determine the level of vibrations that stabilizes the vertical position of the rod. The cases of inextensible and extensible rod are studied, and a benchmark of results is carried out. An attraction basin of the stable vertical position of the rod is found. Both harmonic and random stationary vibrations of the support are considered, too. Stability and attraction basin of the vertical position for a flexible rod are studied in detail. A single-mode approximation is used for the approximate solution to the nonlinear problem of stability of a flexible rod.

研究了在支撑的垂直振动下倒立摆的经典Kapitsa稳定性问题和一些推广。两尺度展开的渐近方法使人们能够确定使杆的垂直位置稳定的振动水平。研究了不可伸长杆和可伸长杆的情况，并对结果进行了基准测试。找到了杆的垂直位置稳定的吸引盆。支架的谐波振动和随机静止振动也都考虑在内。详细研究了柔性杆垂直位置的稳定性和吸引力。单模逼近用于柔性杆的非线性非线性问题的逼近解。