Periodic Motion of a Simple Pendulum With Periodic Disturbance

Struble, Raimond A.; Marlin, J.A.

doi:10.1093/qjmam/18.4.405

Cited by 8 publications

(3 citation statements)

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“…[18]) may be used to excite discrete breather formation in an electrical lattice [19]. More recently, this idea has been examined further in the context of a horizontally shaken pendulum (which has long been known to display a variety of subharmonic resonances [20]), and the possibility of mixed-frequency breathers was identified in a pendulum chain [21]. These breathers exhibit the remarkable response that while energy is localized on a few pendula responding at a sub-harmonic of the driving force, the pendula in the tails of the breather are oscillating with the driving frequency.…”

Section: Introductionmentioning

confidence: 99%

Multifrequency and edge breathers in the discrete sine-Gordon system via subharmonic driving: Theory, computation and experiment

et al. 2016

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We consider a chain of torsionally-coupled, planar pendula shaken horizontally by an external sinusoidal driver. It has been known that in such a system, theoretically modeled by the discrete sineGordon equation, intrinsic localized modes, also known as discrete breathers, can exist. Recently, the existence of multifrequency breathers via subharmonic driving has been theoretically proposed and numerically illustrated by Xu et al. in Phys. Rev. E 90, 042921 (2014). In this paper, we verify this prediction experimentally. Comparison of the experimental results to numerical simulations with realistic system parameters (including a Floquet stability analysis), and wherever possible to analytical results (e.g. for the subharmonic response of the single driven-damped pendulum), yields good agreement. Finally, we report on period-1 and multifrequency edge breathers which are localized at the open boundaries of the chain, for which we have again found good agreement between experiments and numerical computations.

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Section: Introductionmentioning

confidence: 99%

Multifrequency and edge breathers in the discrete sine-Gordon system via subharmonic driving: Theory, computation and experiment

et al. 2016

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“…[2,3], and references therein), where the well-known stabilization of the inverted pendulum may be observed [4]. Horizontal driving has received less attention, with early works considering the nature of the periodic solutions [5] and the appearance of subharmonic excitations [6]. More recently, interest in this problem has been revived, with recent results exploring the appearance of chaotic motion [7], dynamic stabilization of two off-center equilibrium points [8], and complex bifurcation behavior of the period-1 oscillation [9].…”

Section: Introductionmentioning

confidence: 99%

Instability dynamics and breather formation in a horizontally shaken pendulum chain

Alexander

Sidhu

et al. 2014

Phys. Rev. E

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Inspired by the experimental results of Cuevas et al. [Phys. Rev. Lett. 102, 224101 (2009)], we consider theoretically the behavior of a chain of planar rigid pendulums suspended in a uniform gravitational field and subjected to a horizontal periodic driving force applied to the pendulum pivots. We characterize the motion of a single pendulum, finding bistability near the fundamental resonance and near the period-3 subharmonic resonance. We examine the development of modulational instability in a driven pendulum chain and find both a critical chain length and a critical frequency for the appearance of the instability. We study the breather solutions and show their connection to the single-pendulum dynamics and extend our analysis to consider multifrequency breathers connected to the period-3 periodic solution, showing also the possibility of stability in these breather states. Finally we examine the problem of breather generation and demonstrate a robust scheme for generation of on-site and off-site breathers.

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“…The pendulum with vibrating suspension is a classical example of a problem in which a parametric resonance can be observed. A large number of publications (see, e.g., [8,9]) are devoted to this problem. Other problems of this sort include the bending oscillations of straight rod under a periodic longitudinal force [10], the motion of a charged particle (electron) in the field of two running waves [11], etc.…”

Section: Examplementioning

confidence: 99%

Periodic Perturbations: Parametric Systems

Morozov¹

2018

Perturbation Methods With Applications in Science and Engineering

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We are not going to present the classical results on linear parametric systems, since they are widely discussed in literature. Instead, we shall consider nonlinear parametric systems and discuss the conditions of new motion existence in the resonance zones: the regular ones (on an invariant torus) and the irregular ones (on a quasi-attractor). On the basis of the self-oscillatory shortened system which determines the topology of resonance zones, we study the transition from a resonance to a non-resonance case under a change of the detuning. We then apply our results to some concrete examples. It is interesting to study the behavior of a parametric system when the ring-like resonance zone is contracted into a point, i.e., to describe the bifurcations which occur in the course of transition from the plain nonlinear resonance to the parametric one. We are based on article, and we follow a material from the book.

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Periodic Motion of a Simple Pendulum With Periodic Disturbance

Cited by 8 publications

References 0 publications

Multifrequency and edge breathers in the discrete sine-Gordon system via subharmonic driving: Theory, computation and experiment

Multifrequency and edge breathers in the discrete sine-Gordon system via subharmonic driving: Theory, computation and experiment

Instability dynamics and breather formation in a horizontally shaken pendulum chain

Periodic Perturbations: Parametric Systems

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