Dynamics of a multi-stage pendulum when the suspension point is excited by high-frequency polyharmonic vibration with non-multiple frequencies

Аuthors

Tushev O. N.^*, Kondratiev E. K.^**

Baumann Moscow State Technical University, 105005, Moscow, 2nd Baumanskaya St., b. 5, c. 1

*e-mail: sm2aerospace@yandex.ru
**e-mail: kondratyev@eugenek.ru

Abstract

In the linear formulation, the planar problem of the dynamics of a system consisting of a series of pivotally connected pendulum elements is considered. The suspension point is subject to "oblique" (at different angles to the vertical) high-frequency polyharmonic vibration with non-multiple frequencies.
In the presence of high-frequency vibration, the effect of increasing stability is observed in the system. An example is the inverted pendulum considered by P.L. Kapitsa. The general results, which have now become classical, were obtained by V.N. Chelomey. Based on the asymptotic methods developed by N.N. Bogolyubov and developed by Y.L. Mitropolsky, he proved the fact of increasing the stability of any elastic system.
A related task for the studies described above is the analysis of the dynamics of the pendulum with high-frequency movement of the suspension point at a certain angle to the vertical direction. As a result of the action of the vibration moment, a deviation ("departure") of the pendulum from the vertical occurs. In practice, this effect is observed in the form of a false signal (error) of arrow devices (arrow-pendulum) and rotation of loose nuts.
A similar result in the form of a constant component in the solution, as well as fluctuations at a combinational frequency, as shown in this article, are characteristic of linear systems of a general type.
The results in these studies (and not only in them) are obtained for harmonic or at least periodic effects that can be decomposed into a Fourier series. If they are polyharmonics, then the frequencies of its elements must be multiples. This condition is necessary because averaging of the solution over a period of rapid fluctuations is used to isolate the slow part. In this case, this assumption is not necessary.
In accordance with the method of N.N. Bogolyubov, the solution is presented as a superposition of slow and fast components with frequencies of external influences and two approximations are considered. Since, in general, the external effect is practically aperiodic due to the frequency discrepancy, the averaging of high-frequency harmonics over a period in the second approximation is replaced by repeated segregation of motion.
As a result, a quasi-stationary component (the departure of the pendulum) is obtained in the solution obtained in an analytical vector form convenient for analysis. Low-frequency oscillations are excited at the combination frequencies of external influence, which, under appropriate conditions, can manifest themselves in the form of conventional and parametric multiple resonances.

Keywords:

Parametric and additive components, slow and fast movements, the departure of the pendulum

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