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

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.

Dynamical Analysis of a Damped Harmonic Forced Duffing Oscillator with Time Delay

“…As opposed to that, the linearized stability can be investigated near the fixed points (equilibrium points). Therefore, one can use the following consistent assumptions [28]:…”

Dynamical analysis of a forced vibrating planar motion of a spring pendulum

“…It should be highlighted that many academics have become interested in the damped motion of linear or nonlinear elastic pendulums in various pathways [16][17][18][19][20][21][22][23][24][25][26]. In this regard, the planar motion of a harmonic oscillator close to resonance when the pivot point is assumed to move on trajectories of an ellipse and on a closed Lissajous curve is examined in [15] and [16], respectively.…”

Vibration Extraction for Melting Plastic Hydraulic Injection System with Stick Slip Vibration Analysis

“…Many different vibrational models have been investigated in the past few years [4][5][6][7][8][9][10][11][12] due to their significance in various engineering applications. The dynamics of the motion of a damped spring and rigid body pendulums are examined in [4][5][6][7][8] and [9][10][11][12], respectively.…”

“…Many different vibrational models have been investigated in the past few years [4][5][6][7][8][9][10][11][12] due to their significance in various engineering applications. The dynamics of the motion of a damped spring and rigid body pendulums are examined in [4][5][6][7][8] and [9][10][11][12], respectively. Lagrange's equations were used to gain the EOM (Table 1), and the approach of multiple scales [13] is used to achieve the solutions of the regulating system.…”