Dynamical Stability of a 3-DOF Auto-Parametric Vibrating System

Abstract

Abstract Purpose This article concentrates on the oscillating movement of an auto-parametric dynamical system comprising of a damped Duffing oscillator and an associated simple pendulum in addition to a rigid body as main and secondary systems, respectively. Methods According to the system generalized coordinates, the controlling equations of motion are derived utilizing Lagrange's approach. These equations are solved applying the perturbation methodology of multiple scales up to higher orders of approximation to achieve further precise unique outcomes. The fourth-order Runge–Kutta algorithm is employed to obtain numerical outcomes of the governing system. Results The comparison between both solutions demonstrates their high level of consistency and highlights the great accuracy of the adopted analytical strategy. Despite the conventional nature of the applied methodology, the obtained results for the studied dynamical system are considered new. Conclusions In light of the solvability criteria, all resonance scenarios are classified, in which two of the fundamental exterior resonances are examined simultaneously with one of the interior resonances. Therefore, the modulation equations are achieved. The conditions of Routh–Hurwitz are employed to inspect the stability/instability regions and to analyze them in accordance with the solutions in the steady-state case. For various factors of the examined structure, the temporary history solutions, the curves of resonance in terms of the adjusted amplitudes and phases, and the stability zones are graphically presented and discussed. Applications The results of the current study will be of interest to wide range experts in the fields of mechanical and aerospace technology, as well as those working to reduce rotors dynamical vibrations and attenuate vibration caused by swinging structures.