Stability and oscillations of an inverted pendulum under polyharmonic excitation with the aliquant harmonic frequencies
Authors: Tushev O.N., Kondratyev E.K.
Published in issue: #3(159)/2025
DOI: 10.18698/2308-6033-2025-3-2428
Category: Mechanics | Chapter: Theoretical Mchanics, Machine Dynamics
Vertical high-frequency polyharmonic vibration of the suspension point is the external action on a pendulum. The harmonic component frequencies are aliquant, which differs fundamentally from the classical problem statement in the form of the Hill equation. Thus, the action is generally aliquant and aperiodic. The problem is solved by the well-known N.N. Bogolyubov method in two approximations with a small alteration. The pendulum motion is decomposed into two components: “slow” with a frequency of the order of the eigenfrequency and “fast with the external action frequencies. The process aperiodicity excludes a possibility of using an efficient method of averaging the solution over the rapid oscillations period. Therefore, the repeated motion segregation is used instead. As a result, an equation is obtained for a slow motion of the same type as in the case of the periodic external action. The pendulum stability regions are determined under the rapid vibrations. The paper shows that stability could be lost simultaneously in vicinity of the pendulum stable vertical position as a result of the parametric resonance or even several resonances (at least theoretically) on the external action combination frequencies. The results are illustrated with an example and their assessment is provided using the numerical simulation.
EDN UEZECH
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