The Quadratically-Damped Mathieu Equation and Its Application to Submarine Dynamics

Rand, Richard H.; Ramani, D. V.; Keith, William L.; Cipolla, Kimberly

doi:10.1115/imece2000-1764

Cited by 7 publications

(4 citation statements)

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“…At order 1 3 of this resonance, we also see through the figures 13-17 that the three cubic nonlinear and σ and F parameter have the same effects as the case of order 1 2 . For subharmonic resonance, We also see through the figures (18)(19)(20)(21)(22)(23)(24)(25) that the cubic nonlinear parameters, µ, k 2 , scales the peak response, while both the quadratic nonlinear parameter and the direct excitation level affect the frequency value of the peak response respectively for two and first appearance of this resonance in order-two or order-three but λ affect the the peak amplitude and the frequency which his corresponds when the subharmonic resonance appear. The subharmonic resonance may not be critical to modified Rayleigh-Duffing oscillator.…”

Section: Discussionmentioning

confidence: 78%

Parametrics Resonances of a Forced Modified Rayleigh-Duffing Oscillator

Miwadinou,

Monwanou,

Orou

2013

Preprint

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We investigate in this paper the superharmonic and subharmonic resonances of forced modified Rayleigh-Duffing oscillator. We analyse this equation by method of multiple scales and we obtain superharmonic, subharmonic resonances order-two and order-three and primary resonance. We obtain also regions where steady-state subharmonic responses exist. We also use the amplitude-frequency curve for demonstrate the effect of various parameters on the response of the system. Finally, we focus our attention on chaotic motion of this oscillator by simulation and we obtain that this oscillator is chaotic for certains values for natural and excitation frequency but chaotic motion it is not the same in subharmonic and superharmonic cases.

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Section: Discussionmentioning

confidence: 78%

Parametrics Resonances of a Forced Modified Rayleigh-Duffing Oscillator

Miwadinou,

Monwanou,

Orou

2013

Preprint

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“…Phenomenological models describing some type of nonlinear dissipation have been used in some applied sciences such as ship dynamics [Bikdash et al, 1994;Falzarano et al, 1992], where a particular interest has deserved the role played by different damping mechanisms in the formulation of ship stability criteria, and vibration engineering [Ravindra and Mallik, 1994a;Ravindra and Mallik, 1994b]. Damping in certain applied systems plays an important role, since it may be used to suppress large amplitude oscillations or various instabilities, and it can be also used as a control mechanism [Litak et al, 2009;Miwadinou et al, 2015;Rand et al, 2000;Sanjuán, 1999;Soliman andThompson, 1992, Taylan, 2000].…”

Section: Introductionmentioning

confidence: 99%

Melnikov Chaos in a Modified Rayleigh–Duffing Oscillator with ϕ6 Potential

Miwadinou

Monwanou

Hinvi

et al. 2016

Int. J. Bifurcation Chaos

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The chaotic behavior of the modified Rayleigh-Duffing oscillator with φ 6 potential and external excitation is investigated both analytically and numerically. The so-called oscillator models, for example, ship rolling motions. The single well and triple well potential cases are considered. Melnikov method is applied and the conditions for the existence of homoclinic and heteroclinic chaos are obtained. The effects of nonlinear damping on roll motion of ships are analyzed in detail. As it is known, nonlinear roll damping is a very important parameter in estimating ship responses. It is noted that the pure and unpure quadratic damping parameters affect the Melnikov criterion in the heteroclinic and homoclinic cases respectively while the pure cubic parameter affects the amplitude in both cases. The predictions have been tested with numerical simulations based on the basin of attraction. It is pointed out that certain quadratic damping effects are contrary to cubic damping effect.

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“…Such damping can, in some systems, change the sign depending on velocity or displacement values, and provide excitation energy to the examined system. These, so called, self-excited damping terms are often used to describe systems with dry friction, bearings lubricated by a thin layer of oil, shimming in vehicle wheels or chatter in a cutting process [1,2], [5]- [8] and [16,17]. In Ref.…”

Section: Introductionmentioning

confidence: 99%

Regular and Chaotic Behaviors of Modified Rayleigh Duffing oscillator

Miwadinou,

Monwanou,

Ainamon

et al. 2014

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The regular and chaotic behavior of modified Rayleigh-Duffing oscillator is studied. We consider in this paper the dynamics of Modified Rayleigh-Duffing oscillator. The harmonic balance method are used to find the amplitudes of the oscillatory states, and analyze. The influence of system parameters are clearly found on the bifurcations in the response of this system is investigated. It is found also hysteresis and jump phenomenon are appered or desappered when certain parameters incrases or descrases. Various bifurcation structures, the variation of the Lyapunov exponent are obtained, using numerical simulations of the equations of motion. Various basin attraction are used to confirm the predictions of bifurcation structures and its corresponds Lyapunov exponent.

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The Quadratically-Damped Mathieu Equation and Its Application to Submarine Dynamics

Cited by 7 publications

References 0 publications

Parametrics Resonances of a Forced Modified Rayleigh-Duffing Oscillator

Parametrics Resonances of a Forced Modified Rayleigh-Duffing Oscillator

Melnikov Chaos in a Modified Rayleigh–Duffing Oscillator with ϕ6 Potential

Regular and Chaotic Behaviors of Modified Rayleigh Duffing oscillator

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