Chaos in a Magnetic Pendulum Subjected to Tilted Excitation and Parametric Damping

Mathematical Problems in Engineering, Nov 2012

The effect of tilted harmonic excitation and parametric damping on the chaotic dynamics in an asymmetric magnetic pendulum is investigated in this paper. The Melnikov method is used to derive a criterion for transition to nonperiodic motion in terms of the Gauss hypergeometric function. The regular and fractal shapes of the basin of attraction are used to validate the Melnikov predictions. In the absence of parametric damping, the results show that an increase of the tilt angle of the excitation causes the lower bound for chaotic domain to increase and produces a singularity at the vertical position of the excitation. It is also shown that the presence of parametric damping without a periodic fluctuation can enhance or suppress chaos while a parametric damping with a periodic fluctuation can increase the region of regular motions significantly.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

http://downloads.hindawi.com/journals/mpe/2012/546364.pdf

Chaos in a Magnetic Pendulum Subjected to Tilted Excitation and Parametric Damping

Chaos in a Magnetic Pendulum Subjected to Tilted Excitation and Parametric Damping C. A. Kitio Kwuimy,1 C. Nataraj,1 and M. Belhaq2 1Center for Nonlinear Dynamics and Control, Department of Mechanical Engineering, Villanova University, 800 Lancaster Avenue, Villanova, PA 19085, USA 2Laboratory of Mechanics, University Hassan II, Casablanca, Morocco Received 18 May 2012; Revised 3 August 2012; Accepted 3 August 2012 Academic Editor: Stefano Lenci Copyright © 2012 C. A. Kitio Kwuimy et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The effect of tilted harmonic excitation and parametric damping on the chaotic dynamics in an asymmetric magnetic pendulum is investigated in this paper. The Melnikov method is used to derive a criterion for transition to nonperiodic motion in terms of the Gauss hypergeometric function. The regular and fractal shapes of the basin of attraction are used to validate the Melnikov predictions. In the absence of parametric damping, the results show that an increase of the tilt angle of the excitation causes the lower bound for chaotic domain to increase and produces a singularity at the vertical position of the excitation. It is also shown that the presence of parametric damping without a periodic fluctuation can enhance or suppress chaos while a parametric damping with a periodic fluctuation can increase the region of regular motions significantly. 1. Introduction Various nonlinear phenomena have been found in physical systems and chaotic behavior has been reported in various engineering systems with applications in microelectromechanical [1–3], electromechanical [4–6], mechanical [7–10], electronic [11–13], and others. Usually, numerical indicators such as the Lyapunov exponent and bifurcation diagram are used to determine and study the occurrence of chaos. The Melnikov method [14], on the other hand, predicts analytically the lower bound in parameter space separating regular and chaotic dynamics. The Melnikov method has been recently applied in experimental and theoretical research in various fields of science, including epidemiology [15], biology [16], and engineering systems [2, 7, 8]. Along these lines, Cicogna and Papoff [17] considered a Duffing type potential with an additional linear term and estimated the threshold condition for the appearance of chaos by using a Taylor expansion with respect to the asymmetric parameter. The optimal control of chaos was studied by Lenci and Rega [18] for the Helmholtz-Duffing oscillator. Litak et al. [19] revisited the Melnikov criteria for a driven system under a single and double well asymmetric potential and expressed the integrals to be evaluated for the appearance of chaos in terms of logarithm function. Cao et al. [20] applied the Melnikov theory to a driven Helmholtz-Duffing oscillator and derived the condition for appearance of fractal basin boundaries. Recently, a magnetic pendulum driven by a high-frequency excitation under a magnetic potential was considered [21]. This paper aims to apply the well-known Melnikov theory to a fundamental physical device used in several engineering systems, namely, a magnetic pendulum, and discuss the possibility of chaos suppression in the system. Current literature examines various nontrivial phenomena caused by a high-frequency excitation in physical systems. Thomsen [22] considered the stiffening, biasing, and smoothening in such systems, Bartuccelli et al. [23] and Schmitt and Bayly [24] showed that a high-frequency excitation of a horizontally or vertically shaken pendulum results in oscillations about a nonzero mean angle. Yabuno et al. [25] considered an inverted pendulum and showed that a tilt angle of the excitation produces stable equilibrium states different from the direction of the gravity and the excitation. The symmetry breaking bifurcation due to the tilt angle was also investigated qualitatively and through experiments by Mann and Koplow [26]. In a related experimental work, Mann investigated the energy criterion for snap-through instability and nonperiodic motion. The effect of a fast parametric excitation on self-excited vibrations in a delayed van der Pol oscillator was reported in [27, 28]. Fidlen and Juel Thomsen [29] analyzed this effect on the equilibrium of a strongly damped system comparing to the case of a slightly damped one. Mann and Koplow [26] showed that a small deviation from either a perfectly vertical or horizontal excitation will result in symmetry breaking bifurcations opposed to pitchfork bifurcations obtained for vertical or horizontal excitation. Also, the condition for well escapes in a bistable configuration of the potential energy has been studied [21]. An earlier work on magnetic pendulum was done by Moon et al. [30] who showed evidence of homoclinic orbit and horsesho (...truncated)


This is a preview of a remote PDF: http://downloads.hindawi.com/journals/mpe/2012/546364.pdf

C. A. Kitio Kwuimy, C. Nataraj, M. Belhaq. Chaos in a Magnetic Pendulum Subjected to Tilted Excitation and Parametric Damping, Mathematical Problems in Engineering, 2012, 2012, DOI: 10.1155/2012/546364