Dynamics of a rolling disc with an inclined sliding support 

Evgeniya A. Mikishanina, Candidate of physical and mathematical sciences, associate professor of the sub-department of actuarial and financial mathematics, Chuvash State University (15 Moscovsky avenue, Cheboksary, Russia), E-mail: evaeva_84@mail.ru 

531.384 

10.21685/2072-3040-2021-3-4 

Background. Nonholonomic mechanical systems can be models of various technical and robotic devices. Taking into account modern trends, the relevance of the study of nonholonomic systems is obvious and indisputable. Nonholonomic models include, among other things, rolling models of various bodies: balls, disks, ellipsoids, and others. This article is devoted to the study of another nonholonomic model of rolling along the horizontal plane of a heavy homogeneous axisymmetric disk connected to a weightless support. And at the point of contact of the disk with the plane of rolling is no slippage, and the contact point of the support plane rolling on the contrary, there is a perfect slide. This determines the imposition of additional links on the system: nonholonomic in the first case and holonomic in the second. Materials and methods. To analyze the dynamics of the considered system, the Poincare-Suslov equations of motion are formulated, the first integrals of motion and the invariant measure where it exists are determined. In earlier works, scientists have shown the integrability of the classical problem of rolling a disk on a horizontal surface. This statement also shows the integrability of a system of differential equations. Results and conclusions. Analytical solutions are obtained in the form of periodic functions of time in the case of a dynamically symmetric disk. In the case of a dynamically asymmetric disk, the analytical solution is reduced to elliptic quadratures. In the latter case, on the basis of numerical calculations, a phase portrait of the system reduced to the level of integrals and graphs of the desired mechanical parameters are also constructed. To build phase portraits and graphs of parameters, the software package “Computer Dynamics: Chaos” was used. 

nonholonomic model, dynamical system, disk, phase portrait 

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