The asymptotic analysis and stability of 3DOF non-linear damped rigid body pendulum near resonance

Abstract

The pendulum's motion better understanding enhances the researcher's ability to solve many engineering challenges. This article explores the motion of three-degrees-of-freedom nonlinear dynamic mechanism described by a damped rigid body pendulum in which its suspended point travels along the Lissajous trajectory. In light of this system's generalized coordinates, we used Lagrange's equations to derive the governing system of motion. The asymptotic solutions of the equations of this system up to the third approximation are obtained using the multiple scale technique. The resonance cases are classified and the conditions of solvability are examined in view of the modulation equations in which the steady-state solutions are verified. The stability criteria were carried out in addition to simulations of the evolution of the non-linear equations of the considered system using the approach of nonlinear stability analysis. The time history of the considered motion, the resonance curves and steady-state solutions are graphically presented in accordance with the effect of various physical parameters on the motion. The numerical results of governing system are obtained using the fourth-order Runge–Kutta algorithms and compared with the asymptotic ones to reveal the high consistency between them and to clarify the accuracy of the used perturbation technique.