The asymptotic analysis for the motion of 3DOF dynamical system close to resonances

This study explores the nonlinear motion of a double pendulum in which its pivot point travels on an elliptical trajectory with a constant angular speed, and it is attached to an unstretched arm. This arm is then fixed to a damped harmonic spring pendulum, with a linear force affecting on the arm of the pendulum. Multiple scales technique is utilized to solve the equations of motion. The conditions of solvability for the solutions at a steady-state are achieved in view of the investigated resonance cases. Therefore, the corresponding modulation equations are established. For the right physical parameters of the investigated model, the time histories of the attained solutions besides the resonances curves are outlined. The comparison between the achieved results and the numerical ones indicates a strong consistency between these results. These kinds of dynamical systems can be seen as a fine example which reflects the treatments of the seismic waves for the vibrations of the ground induced by seismic factors such as earthquakes and volcano activity.