Dynamics of the Rolling Disk in the Presence of Dry Friction

Authors: Cédric Le Saux, Remco Leine

How can moving bodies be stopped in finite time? Many answers different from classical solutions are provided by nonsmooth dynamics, which deals with measure differential inclusions from time-evolution problems in mechanics. The most familiar examples of such systems are collisions of rigid bodies and Coulomb’s law of dry friction. Nonsmooth dynamics, however, is much more than friction and impacts. It provides an extended concept of classical analytical mechanics with all its subdisciplines and modern extensions, such as multibody dynamics and robotics, by allowing for set-valued constitutive laws and discontinuities in the state variables as functions of time. Set-valued constitutive laws provide a sound basis for switching on and off various kinds of constraints in a most consistent way, where the switching rules are embedded in the constitutive laws themselves. One important subclass of such constitutive laws are those expressed by normal cone inclusions from convex analysis, as used in this paper, because they generalize the dynamics on manifolds to manifolds with boundary. As one of the most demanding examples, the classical problem of a rolling disk is chosen with the main emphasis being the numerical study of the dissipation mechanisms generated by different kinds of set-valued constitutive laws at the contact point.

Model of the rolling disk.

This research project is concerned with the dynamics and numerical treatment of a rolling disk on a flat support. The objective is to develop a numerical model which is able to simulate the dynamics of a rolling disk taking into account various kinds a friction models (resistance against sliding, pivoting and rolling). A mechanical model of a rolling disk is presented in the framework of Non-smooth Dynamics and Convex Analysis. In an analytical study, approximations are derived for the energy decay of the system during the final stage of the motion for various kinds of frictional dissipation models. Finally, the numerical and analytical results are discussed and compared with experimental results available in literature.

Position of the center of mass of the rolling disk.

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