The Stability Analysis of a Vibrating Auto-Parametric Dynamical System 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.