The current paper investigates the dynamical property of a pendulum attached to a rotating rigid frame with a constant angular velocity about the vertical axis passing to the pivot point of the pendulum. He’s homotopy perturbation method is used to obtain the analytic solution of the governing nonlinear differential equation of motion. The fourth-order Runge-Kutta method (RKM) and He’s frequency formulation are used to verify the high accuracy of the obtained solution. The stability condition of the motion is examined and discussed. Some plots of the time histories of the gained solutions are portrayed graphically to reveal the impact of the distinct parameters on the dynamical motion.

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The response of a nonlinear multidegrees of freedom (M-DOF) for a nature dynamical system represented by a spring pendulum which moves in an elliptic path is investigated. Lagrange’s equations are used in order to derive the governing equations of motion. One of the important perturbation techniques MS (multiple scales) is utilized to achieve the approximate analytical solutions of these equations and to identify the resonances of the system. Besides, the amplitude and the phase variables are renowned to study the steady-state solutions and to recognize their stability conditions. The time history for the attained solutions and the projections of the phase plane are presented to interpret the behavior of the dynamical system. The mentioned model is considered one of the important scientific applications like in instrumentation, addressing the oscillations occurring in sawing buildings and the most of various applications of pendulum dampers.