Autoparametric amplification of two nonlinear coupled mass–spring systems

Khirallah, Kareem

doi:10.1007/s11071-018-4068-6

Cited by 6 publications

(2 citation statements)

References 22 publications

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“…In Reference 41, an auto-parametric damped Duffing system coupled with an excited pendulum was examined using the MSM to obtain the resonance requirements. However, the solution of two coupled mass springs was obtained using this method, and new excitation conditions in the presence of auto-parametric resonance were discussed in Reference 42. In Reference 43, the authors shed their attention on the resonance of an auto-parametric vibrating system with a third-order nonlinearity.…”

Section: Introductionmentioning

confidence: 99%

Stability and analysis of the vibrating motion of a four degrees-of-freedom dynamical system near resonance

Amer,

Ismail,

Shaker

et al. 2024

Journal of Low Frequency Noise, Vibration and Active Control

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This article investigates the impact of three harmonically external moments on the motion of a four-degrees-of-freedom (DOF) nonlinear dynamical system composed of a double rigid pendulum connected to a nonlinear spring with linear damping. In light of the system’s generalized coordinates, the governing system (GS) of motion is derived using Lagrange’s equations. With the use of the multiple-scales method (MSM), the approximate solutions (AS) of the equations of this system are obtained up to a higher order of approximation, maybe the third order. Within the framework of the absence of secular terms, the conditions of solvability are obtained. Accordingly, the different resonance cases are categorized, and three of them are investigated simultaneously. Thus, these conditions are updated in preparation for achieving the modulation equations (ME) for the examined system. The numerical solutions (NS) of the GS are achieved using the algorithms of fourth-order Runge–Kutta (4RK), which are compared with the AS, which displays their high degree of consistency and demonstrates the precision of the MSM. The motion’s time history, steady-state solutions, and resonance curves are graphed to demonstrate the influence of various system physical parameters. All relevant fixed points at steady-state situations are identified and graphed in accordance with the Routh–Hurwitz criteria (RHC). Therefore, the zones of stability/instability of are checked and analyzed. Numerous real-world applications in disciplines like engineering and physics attest to the significance of the nonlinear dynamical system under study such as in shipbuilding, automotive devices, structure vibration, developing robots, and analysis of human walking.

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Section: Introductionmentioning

confidence: 99%

Stability and analysis of the vibrating motion of a four degrees-of-freedom dynamical system near resonance

Amer,

Ismail,

Shaker

et al. 2024

Journal of Low Frequency Noise, Vibration and Active Control

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“…The MMS was utilized in [15] to obtain the auto-parametric conditions of a damped Duffing system connected with a pendulum. By virtue of this technique, the solution of two coupled mass springs was obtained in [16]. The author discussed new excitation conditions in the presence of auto-parametric resonance.…”

Section: Introductionmentioning

confidence: 99%

Resonance in the Cart-Pendulum System—An Asymptotic Approach

Amer

Starosta

et al. 2021

Applied Sciences

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The major objective of this research is to study the planar dynamical motion of 2DOF of an auto-parametric pendulum attached with a damped system. Using Lagrange’s equations in terms of generalized coordinates, the fundamental equations of motion (EOM) are derived. The method of multiple scales (MMS) is applied to obtain the approximate solutions of these equations up to the second order of approximation. Resonance cases are classified, in which the primary external and internal resonance are investigated simultaneously to establish both the solvability conditions and the modulation equations. In the context of the stability conditions of these solutions, the equilibrium points are obtained and graphically displayed to derive the probable steady-state solutions near the resonances. The temporal histories of the attained results, the amplitude, and the phases of the dynamical system are depicted in graphs to describe the motion of the system at any instance. The stability and instability zones of the system are explored, and it is discovered that the system’s performance is stable for a significant number of its variables.

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