The asymptotic analysis for the motion of 3DOF dynamical system close to resonances

Bek, M. A.; Amer, T. S.; Almahalawy, Ashraf; Elameer, Abdelkarim S.

doi:10.1016/j.aej.2021.02.017

Cited by 23 publications

(12 citation statements)

References 22 publications

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“…The authors verified the attained results by the comparison with the numerical results. On the other hand, the perturbation technique of multiple scales (PTMS) [6] has been used in several scientific works, such as [13][14][15][16][17][18][19][20][21], to obtain the approximate solutions of the controlling EOM of the corresponding dynamical systems. In refs.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, the motion of a three degrees of freedom (DOF) double pendulum was examined in [20] under the existence of a harmonic excitation force and two moments. The first pendulum was considered to be rigid, in which its first point was constrained to move only in an elliptical path, while its second end was connected with a damped spring pendulum.…”

Section: Introductionmentioning

confidence: 99%

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The Stability Analysis of a Vibrating Auto-Parametric Dynamical System Near Resonance

et al. 2022

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This paper examines a new vibrating dynamical motion of a novel auto-parametric system with three degrees of freedom. It consists of a damped Duffing oscillator as a primary system attached to a damped spring pendulum as a secondary system. Lagrange’s equations are utilized to acquire the equations of motion according to the number of the system’s generalized coordinates. The perturbation technique of multiple scales is applied to provide the solutions to these equations up to a higher order of approximations, with the aim of obtaining more accurate novel results. The categorizations of resonance cases are presented, in which the case of primary external resonance is examined to demonstrate the conditions of solvability of the steady-state solutions and the equations of modulation. The time histories of the achieved solutions, the resonance curves in terms of the modified amplitudes and phases, and the regions of stability are outlined for various parameters of the considered system. The non-linear stability, in view of both the attained stable fixed points and the criterion of Routh–Hurwitz, is investigated. The results of this paper will be of interest for specialized research that deals with the vibration of swaying buildings and the reduction in the vibration of rotor dynamics, as well as studies in the fields of mechanics and space engineering.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Stability Analysis of a Vibrating Auto-Parametric Dynamical System Near Resonance

et al. 2022

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“…The dynamical motion of a damped spring pendulum with two or three DOF in the presence of excitation forces and moments is investigated in several works, e.g., [33][34][35][36][37][38][39][40]. The damped motion of a spring pendulum, in which its suspended point moves in a circular trajectory, is investigated in [33].…”

Section: Introductionmentioning

confidence: 99%

“…It is observed that the approximated system has a bifurcation that leads to chaotic motion via a series of period-doubling bifurcations. The nonlinear motion of a double pendulum system with 3DOF in the presence of one external force and two moments is examined in [38]. The effects of absorber motion on the 3DOF dynamical system is studied in [39].…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, nonlinear movements of an elastic pendulum under the influence of two perpendicular excitation forces are investigated in [40]. The pivot points of the pendulums considered in [38][39][40] are considered to move in elliptic trajectories. The asymptotic solutions are obtained up to higher order of approximation using the approach of multiple scales (AMS).…”

Section: Introductionmentioning

confidence: 99%

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Influence of the Motion of a Spring Pendulum on Energy-Harvesting Devices

et al. 2021

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Energy harvesting is becoming more and more essential in the mechanical vibration application of many devices. Appropriate devices can convert the vibrations into electrical energy, which can be used as a power supply instead of ordinary ones. This study investigated a dynamical system that correlates with two devices, namely a piezoelectric device and an electromagnetic one, to produce two novel models. These devices are connected to a nonlinear damping spring pendulum with two degrees of freedom. The damping spring pendulum is supported by a point moving in a circular orbit. Lagrange’s equations of the second kind were utilized to obtain the equations of motion. The asymptotic solutions of these equations were acquired up to the third approximation using the approach of multiple scales. The comparison between the approximate and the numerical solutions reveals high consistency between them. The steady-state solutions were investigated, and their stabilities were checked. The influences of excitation amplitudes, damping coefficients, and the different frequencies on energy-harvesting device outputs are examined and discussed. Finally, the nonlinear stability analysis of the modulation equations is discussed through the stability and instability ranges of the frequency response curves. The work is significant due to its real-life applications, such as a power supply of sensors, charging electronic devices, and medical applications.

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The dynamical motion of a rolling cylinder and its stability analysis: analytical and numerical investigation

Amer

2022

Arch Appl Mech

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The present paper addresses the dynamical motion of two degrees-of-freedom (DOF) auto-parametric system consisting of a connected rolling cylinder with a damped spring. This motion has been considered under the action of an excitation force. Lagrange's equations from second kind are utilized to obtain the governing system of motion. The uniform approximate solutions of this system are acquired up to higher order of approximation using the technique of multiple scales in view of the abolition of emerging secular terms. All resonance cases are characterized, and the primary and internal resonances are examined simultaneously to set up the corresponding modulation equations and the solvability conditions. The time histories of the amplitudes, modified phases, and the obtained solutions are graphed to illustrate the system's motion at any given time. The nonlinear stability approach of Routh–Hurwitz is used to examine the stability of the system, and the different zones of stability and instability are drawn and discussed. The characteristics of the nonlinear amplitude for the modulation equations are investigated and described, as well as their stabilities. The gained results can be considered novel and original, where the methodology was applied to a specific dynamical system.

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The asymptotic analysis for the motion of 3DOF dynamical system close to resonances

Cited by 23 publications

References 22 publications

The Stability Analysis of a Vibrating Auto-Parametric Dynamical System Near Resonance

The Stability Analysis of a Vibrating Auto-Parametric Dynamical System Near Resonance

Influence of the Motion of a Spring Pendulum on Energy-Harvesting Devices

The dynamical motion of a rolling cylinder and its stability analysis: analytical and numerical investigation

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