Evaluation of the stability of a two degrees-of-freedom dynamical system

Amer, T. S.; Ismail, A. I.; Amer, W. S.

doi:10.1177/14613484231177654

Cited by 7 publications

(2 citation statements)

References 58 publications

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“…If we can represent θk (k=1,2,3,4) and ak exponentially as in the forms qseλT (s=1,2,3,4,5,6,7,8), where the letters qs and λ stand for constants and the eigenvalues of the unidentified perturbations associated to them. If the steady-state solutions at θk0 and ak0 are asymptotically stable, the real parts of the roots of the below characteristic equations should have negative values 55 …”

Section: Solutions At the Scenario Of Steady-statementioning

confidence: 99%

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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: Solutions At the Scenario Of Steady-statementioning

confidence: 99%

“…The solutions of conservative nonlinear oscillators and micro-electromechanical systems are presented in References 47–49 using the modification of HPM. For more information about the solutions of nonlinear oscillators, dynamical systems, and the stability of such systems, see References 50–55.…”

Section: Introductionmentioning

confidence: 99%