Parametric stability of a charged pendulum with oscillating suspension point

Cabral, Hildeberto E.; Carvalho, Adecarlos C.

doi:10.1016/j.jde.2021.02.062

Cited by 10 publications

(8 citation statements)

References 6 publications

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“…In the articles [1,2,[5][6][7] we have used the expression parametric stability in the sense that the Hamiltonian depends on parameters, and studied the regions in the parameter space corresponding to the unstable systems of the parametric family (regions of parametric resonance) and the regions that correspond to the stable systems of the parametric family of Hamiltonians. In these articles we have not made a formal definition of parametric stability.…”

Section: Stability Of Periodic Linear Hamiltonian Systemsmentioning

confidence: 99%

Parametric Resonance in the Oscillations of a Charged Pendulum Inside a Uniformly Charged Circular Ring

Cabral¹,

Carvalho²

2022

Nelin. Dinam.

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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.

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Section: Stability Of Periodic Linear Hamiltonian Systemsmentioning

confidence: 99%

Parametric Resonance in the Oscillations of a Charged Pendulum Inside a Uniformly Charged Circular Ring

Cabral¹,

Carvalho²

2022

Nelin. Dinam.

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“…A strongly localized forcing of a pendulum without dissipation is also obtained by an electrostatic rather than magnetic interaction, e.g., when the bob and an external body are electrically charged. Such cases were recently studied for idealized setups [17,18].…”

Section: Introductionmentioning

confidence: 99%

“…Minimum values of B for instability as functions of C from the harmonic balancing method for (a) subharmonic and (b) harmonic cases. The approximate power laws(18) and (19) are shown as thin solid and dashed straight lines…”

mentioning

confidence: 99%

Parametric instability of a vertically driven magnetic pendulum with eddy-current braking by a flat plate

2022

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The vertically driven pendulum is one of the classical systems where parametric instability occurs. We study its behavior with an additional electromagnetic interaction caused by eddy currents in a nearby thick conducting plate that are induced when the bob is a magnetic dipole. The known analytical expressions of the induced electromagnetic force and torque acting on the dipole are valid in the quasistatic limit, i.e., when magnetic diffusivity of the plate is sufficiently high to ensure an equilibrium between magnetic field advection and diffusion. The equation of motion of the vertically driven pendulum is derived assuming that its magnetic dipole moment is aligned with the axis of rotation and that the conducting plate is horizontal. The vertical position of the pendulum remains an equilibrium with the electromagnetic interaction. Conditions for instability of this equilibrium are derived analytically by the harmonic balance method for the subharmonic and harmonic resonances in the limit of weak electromagnetic interaction. The analytical stability boundaries agree with the results of numerical Floquet analysis for these conditions but differ substantially when the electromagnetic interaction is strong. The numerical analysis demonstrates that the area of harmonic instability can become doubly connected. Bifurcation diagrams obtained numerically show the co-existence of stable periodic orbits in such conditions. For moderately strong driving, chaotic motions can be maintained for the subharmonic instability.

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“…Kapitza [3], [4]. Let us also mention some recent papers on the considering problem [5]- [8]. The comprehensive bibliography can hardly be listed here.…”

Section: Introductionmentioning

confidence: 99%

Asymptotically Stable Non-Falling Solutions of the Kapitza-Whitney Pendulum

Polekhin¹

2022

Preprint

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The planar inverted pendulum with a vibrating pivot point in the presence of an additional horizontal force field is studied. The horizontal force is not assumed to be small or rapidly oscillating. We assume that the pivot point of the pendulum rapidly oscillates in the vertical direction and the period of these oscillations is commensurable with the period of horizontal force. This system can be considered as a strong generalization of the Kapitza pendulum. Previously it was shown that for any horizontal force there always exists a non-falling periodic solution in the considered system. In particular, when there is no horizontal force, this periodic solution is the vertical upward position. In the paper we present analytical and numerical results concerning the existence of asymptotically stable non-falling periodic solutions in the system.

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Parametric stability of a charged pendulum with oscillating suspension point

Cited by 10 publications

References 6 publications

Parametric Resonance in the Oscillations of a Charged Pendulum Inside a Uniformly Charged Circular Ring

Parametric Resonance in the Oscillations of a Charged Pendulum Inside a Uniformly Charged Circular Ring

Parametric instability of a vertically driven magnetic pendulum with eddy-current braking by a flat plate

Asymptotically Stable Non-Falling Solutions of the Kapitza-Whitney Pendulum

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