Relative Periodic Motion of a Rigid Body Pendulum on an Ellipse

In this paper, the properties of the relative periodic motions of a rigid body suspended on an elastic string in a vertical plane are considered. The supported point of this pendulum moves on an elliptic path with the rigid body suspended at the end point of the pendulum. Applying Lagrange's equation, the equations of motion are obtained in terms of small parameter (epsilon). These equations represent a quasi-linear system of second order, which can be solved in terms of generalized coordinates Theta, Phi, and beta using the method of small parameters. At the end, a discussion of the motion and a conclusion of the results are considered to show the orientation of the body and the geometric interpretation of this motion at any instant of time. Also, computerized data with graphical representations of the solutions are given for describing the behavior of the body in some periods of time.