Motion of a disk in contact with a parametric 2D curve and Painlevé’s paradox
Tóm tắt
This paper addresses the modeling and simulation of a homogeneous disk that undergoes plane motion constrained to be in permanent contact with and in the same plane as a two-dimensional curve described by a parametric equation. Except for the elementary cases in which the curve is a straight line or an arc of circumference, the investigated problem presents significant challenges to both modeling and simulation. In addition to the necessity of using differential geometry methods to calculate local geometrical properties of the motion, this system exhibits typical difficulties of non-smooth dynamics. Different modes of operation require the synthesis of suitable rules for switching systems of differential equations and, due to the discontinuities caused by the exchange of dynamic models, numerical integration methods suitable for solving stiff problems are necessary. It should also be emphasized that the dynamic model here developed shows the Painlevé’s paradox, caused by the application of a simplified law of friction (Coulomb’s law) to a rigid body subjected to an unilateral constraint. Among the results obtained in this work, we would like to highlight the following: (i) the realization that both the geometry of the contact curve and that followed by the center of mass of the disk must be considered to construct a correct dynamic model for the motion; (ii) a detailed procedure to identify the instant the disk stops sliding and initiates a pure rolling motion on the two-dimensional track represented by a $\mathfrak{C}_{2}$ class curve $C$ in parametric form; (iii) the realization that, for arbitrary geometries of flat two-dimensional curves of class $\mathfrak{C}_{2}$ and simplified models of friction, it is not possible to establish an arbitrary initial kinematic condition (a necessity for integrating the equations of motion) without incurring in a paradoxical result, known in the literature as Painlévé’s paradox. We consider that the approach adopted in this work might contribute to building dynamic models of hybrid systems, especially those with non-elementary geometric characteristics.
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