Variation of Response Amplitude in Parametrically Driven Single Duffing Oscillator and Unidirectionally Coupled Duffing Oscillators
DOI:
https://doi.org/10.5890/JAND.2017.03.009Abstract
We present our investigation on the effect of parametric force on the response amplitude in the single Duffing oscillator and unidirectionally coupled n Duffing oscillators. In the single oscillator parametric perturbation is of the form f xsinωt. Parametric perturbation induced oscillatory motion is found for values of f above a critical value. In the oscillatory motion the dominant frequency is found to be ω/2. A(ω/2), the amplitude of oscillation at the frequency ω/2, is found to vary linearly with ω. We consider unidirectionally coupled n oscillators with first oscillator alone driven by a parametric force and the other oscillators are nonlinearly or linearly coupled but one-way only. Depending upon the values of the coupling strength δ the oscillators, after first several oscillators, exhibit damped or undamped signal propagation. In the nonlinearly coupled oscillators the dominant frequency of oscillation is ω/2. In the linearly coupled system the frequency ω/2 is absent. The oscillators other than the first oscillator exhibiting oscillatory motions have frequencies ω or 2ω or both depending upon the values of the coupling strength.References
[1] Jordan, D.W. and Smith, P. (2007), Nonlinear Ordinary Differential Equations, Oxford University Press: Oxford.
[2] Rajasekar, S. and Sanjuan, M.A.F. (2016), Nonlinear Resonances, Springer: Heidelberg.
[3] Landau, L.D. and Lifshitz, E.M. (1960), Mechanics, Pergamon: London.
[4] Fossen, T.I. and Nijmeijer, H. (2012), Parametric Resonance in Dynamical Systems, Springer: Berlin.
[5] Berthet, R., Petrosyan, A., and Roman, B. (2002), An analog experiment of the parametric instability, Am. J. Phys. 70, 744-749.
[6] Rowland, D.R. (2004), Parametric resonance and nonlinear string vibrations, Am. J. Phys., 72, 758-766.
[7] Butikov, E.I. (2004), Parametric excitation of a linear oscillator, Eur. J. Phys., 25, 535-554.
[8] Batista,A.A. and Moreira, R.S.N. (2011), Signal-to-noise ratio in parametrically driven oscillators, Phys. Rev. E, 84, 061121-8.
[9] Nayfeh, A.H. and Asfar, K.R. (1988), Non-stationary parametric oscillations, J. Sound Vib., 124, 529-537.
[10] Curzon, F.L., Loke, A.L.H., Lefrancois,M.E., and Novik, K.E. (1995), Parametric instability of a pendulum, Am. J. Phys., 63, 132-136.
[11] Porter, J., Tinao, I., Laveron-Simavilla, A, and Rodriguez, J. (2013), Onset patterns in a simple model of localized parametric forcing, Phys. Rev. E, 88, 042913-16.
[12] Zounes, R.S. and Rand, R.H. (1998), Transition curves for the quasi-periodic Mathieu equation, SIAM J. Appl. Math., 58, 1094-1115.
[13] Rand, R., Guennoun, K., and Belhaq, M. (2003), 2:2:1 resonance in the quasiperiodic Mathieu equation, Nonlinear Dyn., 31, 367-374.
[14] Requa, M.V. and Turner, K.I. (2007), Precise frequency estimation in a microelectromechanical parametric resonator, Appl. Phys. Lett., 90, 173508.
[15] Baskaran, R. and Turner, K.I. (2003), Mechanical domain coupled mode parametric resonance and amplification in a torsional mode micro electro mechanical oscillator, J. Micromech. Microeng., 13, 701-707.
[16] Gudkov, V., Shimizu, H.M., and Greene, G.L. (2011), Parametric resonance enhancement in neutron interferometry and application for the search for non-Newtonian gravity, Phys. Rev. C, 83, 025501-10.
[17] Cairncross, W. and Pelster, A. (2014), Parametric resonance in Bose-Einstein condensates with periodic modulation of attractive interaction, Eur. Phys. J. D, 68, 106-112.
[18] Wustmann, W. and Shumeiko, V. (2013), Parametric resonance in tunable superconducting cavities, Phys. Rev. B, 87, 184501-23.
[19] D’Ambroise, J., Malomed, B.A., and Kevrekidis, P.G. (2014), Quasi-energies, parametric resonances and stability limits in ac-driven PT-symmetric systems, Chaos, 24, 023136-10.
[20] Clerc, M.G., Falcon, C., Fernandez-Oto, C., and Tirapegui, E. (2012), Effective-parametric resonance in a non-oscillating system, Europhys. Lett., 98, 30006.
[21] Rajasekar, S. and Lakshmanan, M. (1994), Bifurcation, chaos and suppression of chaos in FitzHugh-Nagumo nerve conduction model equation, J. Theor. Biol., 166, 275-288.
[22] Jeyakumari, S., Chinnathambi, V., Rajasekar, S., and Sanjuan, M.A.F. (2011), Vibrational resonance in an asymmetric Duffing oscillator, Int. J. Bifurcation Chaos, 21, 275-286.
[23] Rajasekar, S. (1993), Controlling of chaos by weak periodic perturbations in Duffing-van der Pol oscillator, Pramana J. Phys., 41, 295-309.
[24] Rajasekar, S., Used, J., Wagemakers, A., and Sanjuan, M.A.F. (2012), Vibrational resonance in biological nonlinear maps, Commun. Nonlinear Sci. Numer. Simul., 17, 3435-3445.
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