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Tr. Mat. Inst. Steklova, 2018, Volume 300, Pages 168–175 (Mi tm3867)  

Nonlinear oscillations of a spring pendulum at the 1 : 1 : 2 resonance: theory, experiment, and physical analogies

A. G. Petrovab, V. V. Vanovskiyab

a Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, pr. Vernadskogo 101-1, Moscow, 119526 Russia
b Moscow Institute of Physics and Technology (State University), Institutskii per. 9, Dolgoprudnyi, Moscow oblast, 141701 Russia

Abstract: Nonlinear spatial oscillations of a material point on a weightless elastic suspension are considered. The frequency of vertical oscillations is assumed to be equal to the doubled swinging frequency (the 1 : 1 : 2 resonance). In this case, vertical oscillations are unstable, which leads to the transfer of the energy of vertical oscillations to the swinging energy of the pendulum. Vertical oscillations of the material point cease, and, after a certain period of time, the pendulum starts swinging in a vertical plane. This swinging is also unstable, which leads to the back transfer of energy to the vertical oscillation mode, and again vertical oscillations occur. However, after the second transfer of the energy of vertical oscillations to the pendulum swinging energy, the apparent plane of swinging is rotated through a certain angle. These phenomena are described analytically: the period of energy transfer, the time variations of the amplitudes of both modes, and the change of the angle of the apparent plane of oscillations are determined. The analytic dependence of the semiaxes of the ellipse and the angle of precession on time agrees with high degree of accuracy with numerical calculations and is confirmed experimentally. In addition, the problem of forced oscillations of a spring pendulum in the presence of friction is considered, for which an asymptotic solution is constructed by the averaging method. An analogy is established between the nonlinear problems for free and forced oscillations of a pendulum and for deformation oscillations of a gas bubble. The transfer of the energy of radial oscillations to a resonance deformation mode leads to an anomalous increase in its amplitude and, as a consequence, to the break-up of a bubble.

Funding Agency Grant Number
Russian Foundation for Basic Research 17-01-00901
Ministry of Education and Science of the Russian Federation АААА-А17-117021310382-5
This work was supported by the Russian Foundation for Basic Research (project no. 17-01-00901) and performed within the framework of the state task (project no. AAAA-A17-117021310382-5).


DOI: https://doi.org/10.1134/S0371968518010132

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English version:
Proceedings of the Steklov Institute of Mathematics, 2018, 300, 159–167

Bibliographic databases:

UDC: 534.1
Received: November 5, 2017

Citation: A. G. Petrov, V. V. Vanovskiy, “Nonlinear oscillations of a spring pendulum at the 1 : 1 : 2 resonance: theory, experiment, and physical analogies”, Modern problems and methods in mechanics, Collected papers. On the occasion of the 110th anniversary of the birth of Academician Leonid Ivanovich Sedov, Tr. Mat. Inst. Steklova, 300, MAIK Nauka/Interperiodica, Moscow, 2018, 168–175; Proc. Steklov Inst. Math., 300 (2018), 159–167

Citation in format AMSBIB
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\inbook Modern problems and methods in mechanics
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