The asymptotic analysis and stability of 3DOF non-linear damped rigid body pendulum near resonance

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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“…According to PTMS, we can seek the approximate solutions of the variables ξ, ϕ, and χ in the following forms [34]:…”

Section: The Perturbation Techniquementioning

confidence: 99%

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

et al. 2022

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“…In this section, we focus our attention on investigating the solutions at the steady-state case of the considered dynamical models. This case corresponds to the zero values of dθ 1 dτ , dθ 2 dτ , da dτ , and db dτ [43]. Therefore, we can use Equation ( 39) to obtain the following form of algebraic equations…”

Section: Steady-state Solutionsmentioning

confidence: 99%

“…Considering that a 11 , θ 11 , b 11 , and θ 21 are the unknown perturbed functions of the linear system (39), we can express their solutions in the form k s e λτ in which k s (s = 1, 2, 3, 4) are constants, and λ is the eigenvalue of the unknown perturbation. Consequently, the fixed points in (43) are asymptotically stable when the real parts of the roots of the following characteristic equation of (43),…”

Section: Steady-state Solutionsmentioning

confidence: 99%

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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“…In this section, we will demonstrate the properties of nonlinear amplitude for the system of Equations (37) and investigate their stabilities [28][29][30]. Then, the transformations below are taken into account…”

Section: Non-linear Analysismentioning

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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The asymptotic analysis and stability of 3DOF non-linear damped rigid body pendulum near resonance

Cited by 31 publications

References 26 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

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

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