Regular and chaotic motions of the Chaplygin sleigh with periodically switched location of nonholonomic constraint

Institute of Mathematics, Information Technologies and Physics, Udmurt State University - Universitetskaya 1, Izhevsk, 426034, Russia and Kotelnikov's Institute of Radio-Engineering and Electronics of RAS, Saratov Branch - Zelenaya 38, Saratov, 410019, Russia

Received: 25 March 2017
Accepted: 19 May 2017

Abstract

We consider motions of the Chaplygin sleigh on a plane supposing that the nonholonomic constraint is located periodically turn by turn at each of three legs supporting the sleigh. We assume that at switching on the constraint the respective element (“knife-edge”) is directed along the local velocity vector and becomes fixed relatively to the sleigh for a certain time interval till the next switch. Differential equations of the mathematical model are formulated and analytical derivation of a 2D map for the state transformation on the switching period is provided. The dynamics takes place with conservation of the mechanical energy. Numerical simulations show phenomena characteristic to nonholonomic systems with complex dynamics. In particular, on the energy surface attractors may occur responsible for regular sustained motions settling in domains of prevalent area compression by the map. In addition, chaotic and quasi-periodic regimes take place similar to those observed in conservative nonlinear dynamics.