On the periodic orbits of the perturbed Wilberforce pendulum

Bustos, M. Teresa de; López, Miguel A.; Martínez, Raquel Suriá

doi:10.1177/1077546314538299

Cited by 3 publications

(4 citation statements)

References 8 publications

(6 reference statements)

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“…Let us remark that averaging techniques have been used to study a different problem, the existence of periodic orbits in a perturbed weakly coupled Wilberforce pendulum, see De Bustos et al (2016). There, these authors parameterize the periodic solutions of that perturbed system by the simple zeros of an associated system of nonlinear equations.…”

Section: Introductionmentioning

confidence: 99%

Closed stable orbits in a strongly coupled resonant Wilberforce pendulum

Avendaño-Camacho

Torres-Manotas

Vallejo

2021

Journal of Vibration and Control

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Section: Introductionmentioning

confidence: 99%

Closed stable orbits in a strongly coupled resonant Wilberforce pendulum

Avendaño-Camacho

Torres-Manotas

Vallejo

2021

Journal of Vibration and Control

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“…Additionally, Plavčić et al formulated a Lagrangian of the system assuming linear relation of the azimuthal rotation of spring due to longitudinal displacement [5]. de Bustos et al also theoretically investigated the response to periodic dependent perturbation of the pendulum using the averaging theory of dynamical systems [6]. Note that even though some preliminary studies were reported in [2], it lacks exploration of the phenomenon in depth and breadth.…”

Section: Introductionmentioning

confidence: 99%

“…For example, it neither included any derivation of the motion equations, nor numerical solutions of the equations and their experimental verification. In addition, while the theories in [5,6] are simplistic and easy to implement, the use of a linear coupling constant does not have theoretical backing, remaining purely as an empirical relationship. Kopf attempted to generate a system of equations for the Wilberforce pendulum by examining the forces and moments on each spring element, but the extensive use of small angle approximation compromises the accuracy of the theory.…”

Section: Introductionmentioning

confidence: 99%

“…We then directly derive a dynamic model of the Wilberforce pendulum from the elastic deformation of helical springs using Lagrangian mechanics. The obtained model is thus much more comprehensive than the common use of a linear coupling constant between the two degrees of freedom in [6,7], yet the coupled differential equations can be numerically solved in any software with high computational efficiency. In section 3, we verify our model with experiments.…”

Section: Introductionmentioning

confidence: 99%

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Theoretical and experimental studies of the Wilberforce pendulum

Wen¹,

Yang²

2021

Eur. J. Phys.

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This work focuses on exploring the phenomenon of Wilberforce pendulum, a research problem for the International Young Physicists’ Tournament in 2021. A Wilberforce pendulum is a system which has a mass hanging vertically from the bottom of a long helical string. The mass is allowed to undergo both rotational motion about and translational motion along its vertical axis with the spring. Under appropriate initial conditions, the pendulum can exhibit coupled oscillations, where the mass switches between vertical oscillations (where it bobs up and down) and azimuthal oscillations (periodic rotations about the vertical axis). This paper aims to explore this phenomenon in three stages: firstly, an analysis on the mechanics of helical springs for an intuitive explanation for the onset of coupled oscillations; secondly, a comprehensive Lagrangian formulation which models the ‘beating’ motion path of the pendulum and its lightly damped oscillations; thirdly, an experimental verification of our model and the conditions required for clear coupling between the two modes of oscillations. The findings presented in this paper reveal some interesting features of the Wilberforce pendulum and help us gain some understanding of its physical properties. This phenomenon can also be regarded as an extremely insightful problem for introductory college physics courses focussed on the subject of oscillations and resonance, as its construction is easy and inexpensive, and it is a clear demonstration of coupled oscillatory motion.

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Initial release condition dependent decoupling motion of a Wilberforce pendulum in the case of near resonance

Zeng

xie

et al. 2023

Eur. J. Phys.

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The vertical and angular oscillation of Wilberforce pendulum was found to be coupled in the case of resonance. Many studies have discussed the effects of the physical properties of Wilberforce pendulum on coupled motion. In this article, we mainly study the influence of initial release conditions on the coupled motion of Wilberforce pendulum in the near resonance case. The experimental results clearly exhibit that the coupled motion phenomenon depends on the initial release condition. Theoretical calculation further demonstrates that the coupled strength between two normal modes is determined by the ratio of initial angular to vertical displacement in the case of near resonance. The experimental results also verify the calculated variation of decoupling coeffcient with this displacement ratio. Our study indicates that even in the case of resonance, the motion of Wilberforce pendulum is probably decoupled at a given initial release condition. Complete decoupled motion is found to appear only when the damping coeffcients in vertical and angular motion generate equal energy dissipation rates. The corresponding decoupling phenomenon in resonance case occurs when the initial release energy of vertical and angular motion is equal.

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On the periodic orbits of the perturbed Wilberforce pendulum

Cited by 3 publications

References 8 publications

Closed stable orbits in a strongly coupled resonant Wilberforce pendulum

Closed stable orbits in a strongly coupled resonant Wilberforce pendulum

Theoretical and experimental studies of the Wilberforce pendulum

Initial release condition dependent decoupling motion of a Wilberforce pendulum in the case of near resonance

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