Studying the influence of external torques on the dynamical motion and the stability of a 3DOF dynamic system

This article explores the planar motion of a novel dynamical system of three degrees of freedom (DOF) triple pendulum, consisting of a double rigid pendulum attached to an un-stretched one, in which its suspension point is considered to be fixed. Applying Lagrange's equations, the controlling system of the equations of motion (EOM) is derived. The multiple scales method (MSM) is used to achieve the analytic solutions of these equations up to higher order of approximation as novel approximate solutions. The accuracy of these solutions is demonstrated by comparing them with the numerical results of the EOM. The system's resonance cases are characterized, and its modulation equations are established. In light of solvability conditions, the steady-state solutions are analyzed. A graphic depiction of the dynamical behavior regarding the motion's time histories and resonance curves is presented. The stability zones are discussed by analyzing their graphs to evaluate the positive effect of different parameters on the dynamic behavior. The importance of the studied dynamical system is evident from its many applications in various fields such as engineering and physics. However, the vibrational motion can be taken into account in engineering applications like shipbuilding and ships motion, structure vibration, automotive devices, designing robots, and human walking analysis. (c) 2021 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).

Automated summary

This article investigates the planar motion of a novel three degrees of freedom dynamic system and derives analytic solutions using the multiple scales method. The accuracy of the solutions is demonstrated by comparing them with numerical results. Resonance cases, modulation equations, and steady-state solutions are characterized and analyzed. The significance of the studied dynamical system in engineering and physics, as well as the impact of different parameters on dynamic behavior, is discussed.