Analyzing the Stability for the Motion of an Unstretched Double Pendulum near Resonance

Amer, T. S.; Starosta, R.; Elameer, Abdelkarim S.; Bek, M. A.

doi:10.3390/app11209520

Cited by 22 publications

(13 citation statements)

References 24 publications

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“…The unknown functions A j ðj ¼ 1; 2; 3Þ can be determined by looking at the conditions ( 27)-( 29) 33)- (35). Therefore, the required AS can be obtained easily after substituting the solutions ( 24)-( 26), ( 30)-( 32), ( 36)- (38), and series (12) into hypothesis (11).…”

Section: The Recommended Methodsmentioning

confidence: 99%

“…The bifurcations and stability of high-frequency periodic motions with limited amplitude are also addressed, as well as the stability criteria. Recently, the stability analysis of un-stretched double pendulum and triple one when they are subjected to external harmonic moments and force are investigated in [35] and [36], respectively. The stability and instability zones for different parameters of the frequency responses are estimated.…”

Section: Introductionmentioning

confidence: 99%

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The stability of 3-DOF triple-rigid-body pendulum system near resonances

et al. 2022

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In this article, the motion of three degree-of-freedom (DOF) dynamical system consisting of a triple rigid body pendulum (TRBP) in the presence of three harmonically external moments is studied. In view of the generalized coordinates of the system, Lagrange's equations are used to obtain the governing system of equations of motion (EOM). The analytic approximate solutions are gained up to the third approximation utilizing the approach of multiple scales (AMS) as novel solutions. The solvability conditions are determined in accordance with the elimination of secular terms. Therefore, the arising various resonances cases have been categorized and the equations of modulation have been achieved. The temporal histories of the obtained approximate solutions, as well as the resonance curves, are visually displayed to reveal the positive effects of the various parameters on the dynamical motion. The numerical results of the governing system are achieved using the fourth-order Runge–Kutta method. The visually depicted comparison of asymptotic and numerical solutions demonstrates high accuracy of the employed perturbation approach. The criteria of Routh–Hurwitz are used to investigate the stability and instability zones, which are then analyzed in terms of steady-state solutions. The strength of this work stems from its uses in engineering vibrational control applications which carry the investigated system a huge amount of importance.

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Section: The Recommended Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The stability of 3-DOF triple-rigid-body pendulum system near resonances

et al. 2022

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“…The main objective of this section is to examine the oscillations of the considered system in the steady state. It is known that this case is generated when the transient processes disappear due to the presence of damping [37,38]. Then, we consider the zero value of the left hand side of Equation (46), i.e., dθ j dt = 0, dh j dt = 0 (j = 1, 2, 3).…”

Section: Solutions In the Steady Statementioning

confidence: 99%

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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“…The substitution of these solutions into the second set of Eqs. ( 14) and ( 15) produces undesired secular terms [34]. Elimination of these terms demands that…”

Section: The Approach Techniquementioning

confidence: 99%

On the solutions and stability for an auto-parametric dynamical system

2022

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The main goal of this study is to look at the motion of a damped two degrees-of-freedom (DOF) auto-parametric dynamical system. Lagrange’s equations are used to derive the governing equations of motion (EOM). Up to a good desired order, the approximate solutions are achieved utilizing the method of multiple scales (MMS). Two cases of resonance, namely; internal and primary external one are examined simultaneously to explore the solvability conditions of the motion and the corresponding modulation equations (ME). These equations are reduced to two algebraic equations, through the elimination of the modified phases, in terms of the detuning parameters and the amplitudes. The kind of stable or unstable fixed point is estimated. In certain plots, the time histories graphs of the achieved solutions, as well as the adjusted phases and amplitudes are used to depict the motion of the system at any instant. The conditions of Routh–Hurwitz are used to study the various stability zones and their analysis. The achieved outcomes are considered to be novel and original, in which the used strategy is applied on a particular dynamical system. The significance of the studied system can be observed in its applications in a number of disciplines, such as swaying structures and rotor dynamics.

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Analyzing the Stability for the Motion of an Unstretched Double Pendulum near Resonance

Cited by 22 publications

References 24 publications

The stability of 3-DOF triple-rigid-body pendulum system near resonances

The stability of 3-DOF triple-rigid-body pendulum system near resonances

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

On the solutions and stability for an auto-parametric dynamical system

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