The dynamical analysis for the motion of a harmonically two degrees of freedom damped spring pendulum in an elliptic trajectory

This article investigates the motion of two degrees of freedom non-linear dynamical system represented by a spring pendulum in which its suspension point moves in an elliptic trajectory. Lagrange's equations are utilized to derive the governing equations of motion in light of the generalized coordinates of this system. The asymptotic solutions for the governing equations of motion up to the third approximation are obtained using the multiple scales technique. The different resonance cases are classified according to the modulation equations. The steady-state solutions are verified in the presence of the solvability conditions. The time histories of the considered motion, the resonance curves, and steady-state solutions are presented graphically. In addition to the simulations of the non-linear equations of the system's evolution, the stability criteria are carried out. The stability and instability regions are investigated, in which it is found the behaviour of the system is stable for a large number of the system’s parameters. In various engineering applications, a better understanding of the vibrational motion near resonance is critical, in which it can be avoided continued exposure to events that can cause severe damage