Comparing dynamics initiated by an attached oscillating particle for the nonholonomic model of a Chaplygin sleigh and for a model with strong transverse and weak longitudinal viscous friction applied at a fixed point on the body. (English) Zbl 1412.37065
Summary: This paper addresses the problem of a rigid body moving on a plane (a platform) whose motion is initiated by oscillations of a point mass relative to the body in the presence of the viscous friction force applied at a fixed point of the platform and having in one direction a small (or even zero) value and a large value in the transverse direction. This problem is analogous to that of a Chaplygin sleigh when the nonholonomic constraint prohibiting motions of the fixed point on the platform across the direction prescribed on it is replaced by viscous friction. We present numerical results which confirm correspondence between the phenomenology of complex dynamics of the model with a nonholonomic constraint and a system with viscous friction – phase portraits of attractors, bifurcation diagram, and Lyapunov exponents. In particular, we show the possibility of the platform’s motion being accelerated by oscillations of the internal mass, although, in contrast to the nonholonomic model, the effect of acceleration tends to saturation. We also show the possibility of chaotic dynamics related to strange attractors of equations for generalized velocities, which is accompanied by a two-dimensional random walk of the platform in a laboratory reference system. The results obtained may be of interest to applications in the context of the problem of developing robotic mechanisms for motion in a fluid under the action of the motions of internal masses.
MSC:
| 37J60 | Nonholonomic dynamical systems |
| 70F25 | Nonholonomic systems related to the dynamics of a system of particles |
| 70E50 | Stability problems in rigid body dynamics |
Keywords:
Chaplygin sleigh; friction; parametric oscillator; strange attractor; Lyapunov exponents; chaotic dynamics; fish-like robotReferences:
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