Stability and analysis of the vibrating motion of a four degrees-of-freedom dynamical system near resonance

This article investigates the impact of three harmonically external moments on the motion of a four-degrees-of-freedom (DOF) nonlinear dynamical system composed of a double rigid pendulum connected to a nonlinear spring with linear damping. In light of the system's generalized coordinates, the governing system (GS) of motion is derived using Lagrange's equations. With the use of the multiple-scales method (MSM), the approximate solutions (AS) of the equations of this system are obtained up to a higher order of approximation, maybe the third order. Within the framework of the absence of secular terms, the conditions of solvability are obtained. Accordingly, the different resonance cases are categorized, and three of them are investigated simultaneously. Thus, these conditions are updated in preparation for achieving the modulation equations (ME) for the examined system. The numerical solutions (NS) of the GS are achieved using the algorithms of fourth-order Runge-Kutta (4RK), which are compared with the AS, which displays their high degree of consistency and demonstrates the precision of the MSM. The motion's time history, steady-state solutions, and resonance curves are graphed to demonstrate the influence of various system physical parameters. All relevant fixed points at steady-state situations are identified and graphed in accordance with the Routh-Hurwitz criteria (RHC). Therefore, the zones of stability/instability of are checked and analyzed. Numerous real-world applications in disciplines like engineering and physics attest to the significance of the nonlinear dynamical system under study such as in shipbuilding, automotive devices, structure vibration, developing robots, and analysis of human walking.