Small oscillations of a disk in a moving elliptical hoop

The paper analyzes small oscillations of a rigid disk in a moving rigid elliptical hoop. To derive the equations of motion for the considered system, which has two degrees of freedom, the Lagrange equations of the second kind were used. Due to the nonlinear nature of the equations of motion, they were solved numerically. Various initial conditions of the problem were analyzed. The results of the solution are illustrated with figures.The subject of the paper is small oscillations of a rigid disk in a moving rigid elliptical hoop. Using the method of Lagrange equations of the second kind, nonlinear equations of motion were derived for the considered system, which has two degrees of freedom. Due to the nonlinear nature of the equations of motion, they were solved numerically. The results of the solution are presented in figures.
2. EQUATIONS OF MOTION OF A DISK IN A MOVING ELLIPTICAL HOOP
ON A RIGID SURFACE
We consider a system consisting of a thin elliptical ring with a shell thickness ݁, total mass ‫ ܯ‬, and semi-axes ܽ and ܾ. Inside the ring, a rigid disk with a total mass ݉ and radius ‫ݎ‬ moves without slipping. The hoop is in plane motion on a rough surface, and the disk is in complex motion. In the problem, we exclude slipping of the hoop and the disk. The system has two degrees of freedom. To solve the problem, we will use the method of Lagrange equations of the second kind. (…)

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