Novel Asymptotic Solutions for the Planar Dynamical Motion of a Double-Rigid-Body Pendulum System Near Resonance

Purpose The planar dynamical motion of a double-rigid-body pendulum with two degrees-of-freedom close to resonance, in which its pivot point moves in a Lissajous curve has been addressed. In light of the generalized coordinates, equations of Lagrange have been used to construct the controlling equations of motion. Methods New innovative analytic approximate solutions of the governing equations have been accomplished up to higher order of approximation utilizing the multiple scales method. Resonance cases have been classified and the solvability conditions of the steady-state solutions have been obtained. The fourth-order Runge-Kutta method has been utilized to gain the numerical solutions for the equations of the governing system. Results The history timeline of the acquired solutions as well as the resonance curves have been graphically displayed to demonstrate the positive impact of the various parameters on the motion. The comparison between the analytical and numerical solutions revealed great consistency, which confirms and reinforces the accuracy of the achieved analytic solutions. Conclusions The non-linear stability analysis of these solutions have been examined and discussed, in which the stability and instability areas have been portrayed. All resonance cases and a combination of them have been examined. The archived results are considered as generalization of some previous works that are related to one rigid body and for fixed pendulum's pivot point.

Automated summary

This study investigated the planar dynamical motion of a double-rigid-body pendulum near resonance, where its pivot point moves in a Lissajous curve. The controlling equations of motion were constructed using Lagrange equations. New approximate analytical solutions were obtained using the multiple scales method, and their accuracy was confirmed through comparison with numerical solutions. The stability of these solutions was analyzed, and the results provide a generalization of previous works.