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

Tarek S Amer,Roman Starosta,Ashraf Almahalawy,Abdelkarim S Elameer

doi:10.3390/app12031737

Tarek S Amer, Roman Starosta + Show 2 more

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https://doi.org/10.3390/app12031737

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Abstract

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.

Highlights

The dynamical motion of a pendulum in generality or the motion of any system containing a pendulum is considered as one of the oldest scientific subjects in non-linear dynamics
The dynamical motion of a tuned absorber was studied in [24], in which the perturbation technique of multiple scales (PTMS) was applied to obtain the approximate solutions up to the third order for the case of a movable pivot point on an ellipse with a stationary angular velocity
This paper focuses on the motion of a novel three degrees of freedom (DOF) auto-parametric dynamical system consisting of a damped spring pendulum as the secondary system suspended with a forced non-linear Duffing damping oscillator as the primary system

Summary

Introduction

The dynamical motion of a pendulum in generality or the motion of any system containing a pendulum is considered as one of the oldest scientific subjects in non-linear dynamics. The dynamical motion of a tuned absorber was studied in [24], in which the PTMS was applied to obtain the approximate solutions up to the third order for the case of a movable pivot point on an ellipse with a stationary angular velocity. The approximate results of another auto-parametric system consisting of a damped pendulum connected to a non-linear vibrating oscillator near the region of principal resonance were investigated in [26]. This paper focuses on the motion of a novel three DOF auto-parametric dynamical system consisting of a damped spring pendulum as the secondary system suspended with a forced non-linear Duffing damping oscillator as the primary system. These results have a significant impact in special dampers mounted in buildings, applied in the opposite directions of river vortices or earthquakes, and in the applications of pendulums that are mounted on bridge towers

Formulation of the Dynamical System

The Perturbation Technique

Non-Linear Analysis of Stability

Conclusions