On the motion of a damped rigid body near resonances under the influence of harmonically external force and moments

El-Sabaa, F. M.; Amer, T. S.; Gad, H.M.; Bek, M. A.

doi:10.1016/j.rinp.2020.103352

Cited by 48 publications

(37 citation statements)

References 24 publications

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“…Therefore, asymptotic solutions have shed the interest of many scientists to deal with various nonlinear equations, such as the averaging method and the small parameter method for some weak nonlinear problems [1][2][3][4]. In addition, the multiple scales (MS) method and the Lindstedt-Poincaré (LP) method have great advantages in obtaining the solutions of vibratory systems [5,6]. However, these methods depend on a small parameter, and improper selection of this parameter leads to wrong solutions.…”

Section: Introductionmentioning

confidence: 99%

Periodic Property and Instability of a Rotating Pendulum System

Amer

Elnaggar

et al. 2021

Axioms

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The current paper investigates the dynamical property of a pendulum attached to a rotating rigid frame with a constant angular velocity about the vertical axis passing to the pivot point of the pendulum. He’s homotopy perturbation method is used to obtain the analytic solution of the governing nonlinear differential equation of motion. The fourth-order Runge-Kutta method (RKM) and He’s frequency formulation are used to verify the high accuracy of the obtained solution. The stability condition of the motion is examined and discussed. Some plots of the time histories of the gained solutions are portrayed graphically to reveal the impact of the distinct parameters on the dynamical motion.

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Section: Introductionmentioning

confidence: 99%

Periodic Property and Instability of a Rotating Pendulum System

Amer

Elnaggar

et al. 2021

Axioms

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“…On the other side, dynamical pendulum models with 2 or 3 DOF have been studied in many research works such as (Awrejcewicz et al, 2016;Starosta et al, 2011;Starosta et al, 2012;Amer et al, 2016;Nayfeh, 2004;Awrejcewicz et al, 2013;Amer et al, 2018;Amer et al, 2019;El-Sabaa et al, 2020). The plane motion of nonlinear spring pendulum with 2DOF is investigated in (Awrejcewicz et al, 2016) for a fixed supported point under the influence of two external forces in the direction of a spring arm and its perpendicular direction.…”

Section: Introductionmentioning

confidence: 99%

“…The vibrational motion of a damped spring rigid body pendulum is investigated in (Awrejcewicz et al, 2013) for the case of a fixed pivot point. The generalization of this problem is presented in (Amer et al, 2018), (Amer et al, 2019) when the supported point moves in an elliptic path for the cases of linear and nonlinear spring and in (El-Sabaa et al, 2020) when the supported point moves in a Lissajous curve. The conditions of solvability for the steady-state solutions are obtained and the time histories of the approximate and numerical solutions are plotted in some diagrams.…”

Section: Introductionmentioning

confidence: 99%

On the motion of a triple pendulum system under the influence of excitation force and torque

Amer

Galal

Abolila

2021

KJS publishes peer-review articles in Mathematics, Computer Sci

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In this article, a nonlinear dynamical system with three degrees of freedom (DOF) consisting of multiple pendulums (MP) is investigated. The motion of this system is restricted to be in a vertical plane, in which its pivot point moves in a circular path with constant angular velocity, under the action of an external harmonic force and a moment acting perpendicular to the direction of the last arm of MP and at the suspension point respectively. Multiple scales technique (MST) among other perturbation methods is used to obtain the approximate solutions of the equations of motion up to the third approximation because it is authorizing to execute a specific analysis of the system behaviour and to realize the solvability conditions given the resonance cases. The stability of the considered dynamical model utilizing the nonlinear stability analysis approach is examined. The solutions diagrams and resonance curves are drawn to illustrate the extent of the effect of various parameters on the solutions. The importance of this work is due to its uses in human or robotic walking analysis.

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“…Conversely, the plane motion of the 3-DOF rigid body pendulum with a fixed suspension point was investigated in [21]. The latter two works are generalized in [22] when the supported point of the damped rigid body moved in a path similar to the Lissajous curve. The authors discussed all the possible amplitudes of the steady-state solutions for different parameters of the considered models in light of the perturbation approach used.…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2021

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This work looks at the nonlinear dynamical motion of an unstretched two degrees of freedom double pendulum in which its pivot point follows an elliptic route with steady angular velocity. These pendulums have different lengths and are attached with different masses. Lagrange’s equations are employed to derive the governing kinematic system of motion. The multiple scales technique is utilized to find the desired approximate solutions up to the third order of approximation. Resonance cases have been classified, and modulation equations are formulated. Solvability requirements for the steady-state solutions are specified. The obtained solutions and resonance curves are represented graphically. The nonlinear stability approach is used to check the impact of the various parameters on the dynamical motion. The comparison between the attained analytic solutions and the numerical ones reveals a high degree of consistency between them and reflects an excellent accuracy of the used approach. The importance of the mentioned model points to its applications in a wide range of fields such as ships motion, swaying buildings, transportation devices and rotor dynamics.

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On the motion of a damped rigid body near resonances under the influence of harmonically external force and moments

Cited by 48 publications

References 24 publications

Periodic Property and Instability of a Rotating Pendulum System

Periodic Property and Instability of a Rotating Pendulum System

On the motion of a triple pendulum system under the influence of excitation force and torque

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

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