Resonance of Non-Linear Systems Subjected to Multi-Parametrically Excited Structures: (Comparison Between two Methods, Response and Stability)

El-Bassiouny, A F; Eissa, M.

doi:10.1088/0031-8949/70/2-3/006

Cited by 16 publications

(17 citation statements)

References 21 publications

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“…The effect of damping, magnitudes of non-linear excitation and self-excitation are analyzed. El-Bassiouny and Eissa [21] analyzed the behavior of two-degrees-of-freedom vibrating mechanical structure, which is described by two nonlinear differential equations with quadratic and cubic non-linearity's, subjected to multi-frequency parametric excitations in the presence of two-to-one internal resonance. Two approximate methods (the multiple scales and the generalized synchronization) are used to obtain a uniform firstorder expansion.…”

Section: Introductionmentioning

confidence: 99%

Periodic Solutions of Modified Duffing Equation Subjected to Abi-Harmonic Parametric and External Excitations

El-Naggar

El-Bassiouny

Khalil

et al. 2016

Benha Journal of Applied Sciences

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In this paper, we investigated the periodic solutions of modified Duffing's equation subjected to a biharmonic parametric and external excitations. The method of multiple scales is used to construct a first order uniform expansion yielding two first-order non-linear ordinary differential equations are derived for the evolution of the amplitude and phase. These oscillations involve a super-harmonic and sub-superharmonic oscillations. Steady state responses and their stability are computed for selected values of the system parameters. The effects of these (quadratic and cubic) non-linearities on these oscillations are specifically investigated. With this study, it has been verified that the qualitative effects of these non-linearities are different. Regions of hardening (softening) behavior of the system are exist for the case of sub-superharmonic oscillation. Numerical solutions are presented which illustrate the behavior of the steady-state response amplitude as a function of the detuning parameter.

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Section: Introductionmentioning

confidence: 99%

Periodic Solutions of Modified Duffing Equation Subjected to Abi-Harmonic Parametric and External Excitations

El-Naggar

El-Bassiouny

Khalil

et al. 2016

Benha Journal of Applied Sciences

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“…El-Bassiouny et al [14] studied two-to-one internal resonance in a nonlinear two-degrees-of-freedom system with parametric and external excitations. El-Bassiouny and Eissa [15] analysed the behaviour of a two-degrees-offreedom vibrating mechanical structure, which is described by two nonlinear differential equations with quadratic and cubic nonlinearities, subjected to multi-frequency parametric excitations in the presence of two-to-one internal resonance. Two approximate methods (the multiple scales and the generalized synchronization) are used to obtain a uniform firstorder expansion.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear interfacial Rayleigh–Taylor instability of two-layers flow with an ac electric field

Moatimid¹,

El-Bassiouny²

2007

Phys. Scr.

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This paper deals with the stability of a plane interface between two different superposed fluids moving through two different porous media. The system is acted upon by an alternating electric field parallel to the interface. In addition, the influence of mass and heat transfer is taken into consideration. The boundary-value problem contains the linearized equation of motion together with nonlinear boundary conditions. It leads to a nonlinear characteristic differential equation governing the surface evolution. This characteristic equation is accomplished up to a third-order. It has periodic coefficients. The method of the multiple time scales is applied to a one-degree-of freedom system with quadratic and cubic nonlinearties subjected to parametric excitations. This method is used to determine the modulation of the amplitude as well as the phase. The steady state solutions and their stability are investigated. Previous relevant works are references. New and interesting properties of the solutions are derived. For example, we found that a softening behaviour causes a significant shift in the primary resonance frequency and the solution loses stability. Effects of different parameters such as: the damping factor, the coefficients of nonlinear terms and the amplitudes of the parametric excitations of the system are numerically investigated. It is also found that the response peak decreases with the increase and decrease of some parameters. The response amplitude will be contracted and becomes oval for increasing and decreasing parameters. The response amplitude loses stability with the varying of the parameters. In the case of sub-superharmonic resonance, the steady state response of the system shows a hardening behaviour in most cases for the chosen parameters although the physical nonlinearity of the system is of the softening type. The response amplitude loses stability with the decreasing and increasing of some parameters.

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“…Arafat and Nayfeh [33] studied the nonlinear responses of suspended cables to primary resonance excitations. El-Bassiouny and Eissa [34] investigated the resonances of nonlinear systems subjected to multi-parametrically excited structures. Bi [35] studied the dynamical analysis of two coupled parametrically excited van der pol oscillators.…”

Section: Introductionmentioning

confidence: 99%

Parametric excitation of an internally resonant double pendulum

El-Bassiouny¹

2007

Phys. Scr.

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The response of two-degree-of-freedom systems with quadratic and quartic nonlinearities to a principal parametric resonance in the presence of two-to-one internal resonance is investigated. The method of multiple scales is used to construct a first-order uniform expansion yielding four first-order nonlinear ordinary differential (averaging) equations governing the modulation of the amplitudes and phases of the two modes. These equations are used to determine the steady state (fixed point) solutions and their stability. Bifurcations of the fixed points are investigated. Numerical solutions are carried out and graphical representations of the results are presented. The effect of the different parameters on the system response is studied. It is found that each mode has a single-valued curve and there exist zones of multivalued with the increasing and decreasing of some parameters. Both modes lose stability with the varying of some parameters. The region of definition decreases with the increasing and decreasing of some parameters.

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Resonance of Non-Linear Systems Subjected to Multi-Parametrically Excited Structures: (Comparison Between two Methods, Response and Stability)

Cited by 16 publications

References 21 publications

Periodic Solutions of Modified Duffing Equation Subjected to Abi-Harmonic Parametric and External Excitations

Periodic Solutions of Modified Duffing Equation Subjected to Abi-Harmonic Parametric and External Excitations

Nonlinear interfacial Rayleigh–Taylor instability of two-layers flow with an ac electric field

Parametric excitation of an internally resonant double pendulum

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