Rotating orbits of a parametrically-excited pendulum

Xu, X.Q.; Wiercigroch, Marian; Cartmell, M.P.

doi:10.1016/j.chaos.2004.06.053

Cited by 50 publications

(60 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We now consider the two pendulum systems (2) and (3), and estimate numerically the relative areas of the basins of attraction; previous study has be completed in [12,64,66] for the pendulum with oscillating support and in [16,18] for the pendulum with varying length. Throughout all this section, we fix the parameter values at α = 0.5 and δ = ε = 0.1.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…For this reason perturbed pendula are often used as toy models to study new phenomena in dynamics -for instance in [3,21,49,60,59,64] -or act as simplified models for more complex real world systems -see [46,57,62]. One of the most commonly studied perturbations is the pendulum with vertically oscillating support [6,12,13,30,64], often also referred to as the parametrically forced pendulum, see for example [21,29,32,44,66]. Another much studied perturbation of the simple pendulum is the pendulum with periodically varying length [8,16,58,18,56].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Comparisons between the pendulum with varying length and the pendulum with oscillating support

Wright

Bartuccelli

Gentile

2017

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

We consider two forced dissipative pendulum systems, the pendulum with vertically oscillating support and the pendulum with periodically varying length, with a view to draw comparisons between their behaviour. We study the two systems for values of the parameters for which the dynamics are non-chaotic. We focus our investigation on the persisting attractive periodic orbits and their basins of attraction, utilising both analytical and numerical techniques. Although in some respect the two systems have similar behaviour, we find that even within the perturbation regime they may exhibit different dynamics. In particular, for the same value of the amplitude of the forcing, the pendulum with varying length turns out to be perturbed to a greater extent. Furthermore the periodic attractors persist under larger values of the damping coefficient in the pendulum with varying length. Finally, unlike the pendulum with oscillating support, the pendulum with varying length cannot be stabilised around the upward position for any values of the parameters.

show abstract

Section: Numerical Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Comparisons between the pendulum with varying length and the pendulum with oscillating support

Wright

Bartuccelli

Gentile

2017

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

show abstract

“…We find that this Hamiltonian is isomorphic to that of a pendulum under a sinusoidal driving force (in the absence of gravity). The driven pendulum is a type of nonlinear Mathieu equation that is a subject of ongoing research in computational mathematics [19,20]. A canonical form for the driven pendulum may be parametrized as [19] …”

Section: Slip Stacking and The Driven Pendulummentioning

confidence: 99%

Dynamical stability of slip-stacking particles

Eldred

Zwaska

2014

Phys. Rev. ST Accel. Beams

View full text Add to dashboard Cite

We study the stability of particles in slip-stacking configuration, used to nearly double proton beam intensity at Fermilab. We introduce universal area factors to calculate the available phase space area for any set of beam parameters without individual simulation. We find perturbative solutions for stable particle trajectories. We establish Booster beam quality requirements to achieve 97% slip-stacking efficiency. We show that slip-stacking dynamics directly correspond to the driven pendulum and to the system of two standing-wave traps moving with respect to each other.

show abstract

“…This concept has been proposed by Wiercigroch [1] and is based on the conversion of the kinetic energy of sea waves into rotational motion of the pendulum, mounted on the floating pontoon. The rotational motion of the single pendulum regarding this application has been studied in [2], [3]. The working principle of the parametric pendulum has been illustrated at the pictures below (Fig.1).…”

Section: Introductionmentioning

confidence: 99%

“…A parametric pendulum experiences different types of motion, which can be represented on the phase plane (Fig.1b). The region of oscillatory solutions (closed loops denoted by (1)) is bounded by a critical motion described by the separatrices (2). The response outside this region is denoted by (3) and corresponds to the rotation, which is of main interest for the energy extraction purposes.…”

Section: Introductionmentioning

confidence: 99%

Synchronization control of parametric pendulums for wave energy extraction

Najdecka¹,

Hamaneh²,

Wiercigroch³

2011

REPQJ

Self Cite

View full text Add to dashboard Cite

Abstract.In this work the dynamics of a parametric pendulums system has been studied with a view to its application for sea wave's energy extraction. The idea is based on the conversion of the oscillatory motion of the waves into rotation of the pendulums. The system approximating a floating structure with two pendulums mounted on it has been modelled and analyzed. In the first stage of the study the general dynamics of the parametric pendulums has been investigated numerically and experimentally. The focus lies on synchronized rotational solutions, representing a most energetically favourable state of motion. The target state is to achieve a synchronized counter rotation of both pendulums. The controlling strategy, with the aim of initiating and maintaining the desired response, has been developed and verified numerically and experimentally. Different methods based on the delayed-feedback control have been suggested. The numerical and experimental results showing the difference in the system dynamics with and without control have been presented. Finally the energy extraction from the system has been simulated numerically and energy extraction control has been discussed.

show abstract

Rotating orbits of a parametrically-excited pendulum

Cited by 50 publications

References 14 publications

Comparisons between the pendulum with varying length and the pendulum with oscillating support

Comparisons between the pendulum with varying length and the pendulum with oscillating support

Dynamical stability of slip-stacking particles

Synchronization control of parametric pendulums for wave energy extraction

Contact Info

Product

Resources

About