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This article is cited in 1 scientific paper (total in 1 paper)
Nonlinear physics and mechanics
Parametric Resonance in the Oscillations of a Charged
Pendulum Inside a Uniformly Charged Circular Ring
H. E. Cabrala, A. C. Carvalhob a Department of mathematics, Universidade Federal de Pernambuco,
CEP 50670-901 Recife-PE, Brazil
b Department of mathematics, Universidade Federal do Maranhão
Av. dos Portugueses, 1966, 65080-805, São Luís-MA, Brazil
Abstract:
We study the mechanical system consisting of the following variant of the planar pendulum.
The suspension point oscillates harmonically in the vertical direction, with small amplitude $\varepsilon$,
about the center of a circumference which is located in the plane of oscillations of the pendulum.
The circumference has a uniform distribution of electric charges with total charge $Q$ and the bob
of the pendulum, with mass $m$, carries an electric charge $q$. We study the motion of the pendulum
as a function of three parameters: $\varepsilon$, the ratio of charges $\mu=\frac{q}{Q}$ and a parameter $\alpha$ related to the
frequency of oscillations of the suspension point and the length of the pendulum. As the speed of
oscillations of the mass $m$ are small magnetic effects are disregarded and the motion is subjected
only to the gravity force and the electrostatic force. The electrostatic potential is determined
in terms of the Jacobi elliptic functions. We study the parametric resonance of the linearized
equations about the stable equilibrium finding the boundary surfaces of stability domains using
the Deprit – Hori method.
Keywords:
planar charged pendulum, Hamiltonian systems, parametric resonance, Deprit –
Hori method, Jacobi elliptic integrals.
Received: 04.11.2021 Revised: 12.07.2022
Citation:
H. E. Cabral, A. C. Carvalho, “Parametric Resonance in the Oscillations of a Charged
Pendulum Inside a Uniformly Charged Circular Ring”, Rus. J. Nonlin. Dyn., 18:4 (2022), 513–526
Linking options:
https://www.mathnet.ru/eng/nd808 https://www.mathnet.ru/eng/nd/v18/i4/p513
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