Prediction of bifurcations for external and parametric excited one-degree-of-freedom system with quadratic, cubic and quartic non-linearities

El-Bassiouny, A F; Abdelhafez, H. M.

doi:10.1016/s0378-4754(01)00292-0

Cited by 15 publications

(6 citation statements)

References 12 publications

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“…Single degree of freedom (SDOF) systems consisting nonlinear stiffness terms can exhibit a wide range of dynamical phenomena, as has already been reported by several researchers [7][8][9][10][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. Models describing continuous systems with multiple modes of vibration can still be adequately described as SDOF system with quadratic and cubic nonlinear terms, under specific operating conditions (e.g., Nayfeh and Lacabonara [14,15]).…”

Section: Introductionmentioning

confidence: 95%

Optimizing the dynamical behavior of a dual-frequency parametric amplifier with quadratic and cubic nonlinearities

Dolev

Bucher

2018

Nonlinear Dyn

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The paper describes a novel parametric excitation scheme that acts as a tunable amplifier by controlling two pumping signals and two nonlinear feedback terms. By modulating the stiffness of a mechanical oscillator with a digital signal processor, low frequency inputs are projected onto a higher resonance frequency, thus exploiting the natural selective filtering of such structures.Described is an optimized dual-term nonlinear stiffness resonator that enhances the input signal level and the sensitivity to changes in both amplitude and phase, while limiting the obtained response to desired levels. This amplifier is geared to cases when the frequency of the input is known or measurable, like in rotating structures, while the amplitude and phase are too weak to be detected without amplification. It is shown that by tuning the cubic and quadratic feedback terms, the amplifier benefits from a nearly linear response behavior, while exploiting the benefits of nonlinear and pumping signal enhancements.

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Section: Introductionmentioning

confidence: 95%

Optimizing the dynamical behavior of a dual-frequency parametric amplifier with quadratic and cubic nonlinearities

Dolev

Bucher

2018

Nonlinear Dyn

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“…Steady state solutions correspond to the fixed points (constant solutions) of equations ( 28)- (31). Thus, they correspond to a 1 = a 2 = 0 and 1 = 2 = 0.…”

Section: Primary Resonance Of the Pitch Modementioning

confidence: 99%

“…Bikdash et al [30] investigated the influence of a general roll-damping on the stability of ships using Melnikov criterion and characterized the effect of different damping models. El-Bassiouny and Abdelhafez [31] investigated predication of bifurcations for external and parametric excited one-degree-of-freedom system with quadratic, cubic, and quartic nonlinearities. Eissa and El-Bassiouny [32] investigated analytical and numerical solutions of a non-linear ship motion.…”

Section: Introductionmentioning

confidence: 99%

Coexistence of Stable Solutions in the Nonlinear Ship Motion

El-Bassiouny

2005

Phys. Scr.

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The dynamic stability and complicated motions of a vessel in regular sea are investigated when the frequency in the pitch is nearly twice the frequency in the roll. We use the Method of Multiple Scales to determine four first-order nonlinear ordinary differential equations that govern the modulation of the amplitudes and phases when the forcing frequency is near either the pitch or the roll natural frequency. These equations are used to determine the steady-state solutions and their stability. Force-response and frequency-response curves are generated. Coexistence of multiple solutions is found in the presence of quadratic nonlinearities. The linear solution is symmetric. The pitch mode has multi-valued solutions and the roll mode has a single-valued solution for the case of primary resonance of the pitch mode. Both pitch and roll modes have multivalued solutions in small intervals for the case of primary resonance of the roll mode.

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“…Oueini and Nayfeh [20] investigated the response of single-mode control of a cantilever beam under principal parametric excitation. El-Bassiouny and Abddelhafez [21] showed that single-mode discretization can lead to qualitatively erroneous frequency-response curves for the cases of primary resonance and one-half subharmonic resonance of a beam resting on a nonlinearelastic foundation with distributed quadratic and cubic nonlinearities. Nayfeh and Lacarbonara [22] analysed the prediction of bifurcations for an externally and parametrically excited one-degree-of-freedom system with quadratic, cubic and quartic nonlinearities.…”

Section: Introductionmentioning

confidence: 99%

Resonances in nonlinear structure vibrations under multifrequency excitations

El-Bassiouny

El-Latif

2006

Phys. Scr.

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The response of a single-degree-of-freedom system with quadratic, cubic and quartic nonlinearities subjected to a sinusoidal excitation that involves multiple frequencies is considered. The method of multiple scales is used to construct a first order uniform expansion yielding two first-order nonlinear ordinary differential equations that are derived for the evolution of the amplitude and phase. These oscillations involve a subharmonic oscillation of order one-fourth and superharmonic oscillation of order two. Steady state responses and their stability are computed for selected values of the system parameters. The effects of these (quadratic, cubic, and quartic) nonlinearities on these oscillations are specifically investigated. With this study, it has been verified that the qualitative effects of these nonlinearities are different. Regions of hardening (softening) behaviour of the system exist for the case of subharmonic resonance. The response curve is not affected by decreasing the damping factor for the case of superharmonic resonance. It is shown that the response curve contracts or expands as the parameters vary. The multivalued region increases or decreases when some parameters vary.

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Prediction of bifurcations for external and parametric excited one-degree-of-freedom system with quadratic, cubic and quartic non-linearities

Cited by 15 publications

References 12 publications

Optimizing the dynamical behavior of a dual-frequency parametric amplifier with quadratic and cubic nonlinearities

Optimizing the dynamical behavior of a dual-frequency parametric amplifier with quadratic and cubic nonlinearities

Coexistence of Stable Solutions in the Nonlinear Ship Motion

Resonances in nonlinear structure vibrations under multifrequency excitations

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