Theory of Differential Inclusions and Its Application in Mechanics

Abstract

The following chapter deals with systems of differential equations with discontinuous right-hand sides. The key question is how to define the solutions of such systems. The most adequate approach is to treat discontinuous systems as systems with multivalued right-hand sides (differential inclusions). In this work three well-known definitions of solution of discontinuous system are considered. We will demonstrate the difference between these definitions and their application to different mechanical problems. Mathematical models of drilling systems with discontinuous friction torque characteristics are considered. Here, opposite to classical Coulomb symmetric friction law, the friction torque characteristic is asymmetrical. Problem of sudden load chance is studied. Analytical methods of investigation of systems with such asymmetrical friction based on the use of Lyapunov functions are demonstrated. The Watt governor and Chua system are considered to show different aspects of computer modeling of discontinuous systems.

An algorithmic approach to the multiple impact of a disk in a corner

“…Moreau, 1988;Lötstedt, 1982), the DAE and HHT methods (see e.g. Haddouni et al, 2017;Negrut et al, 2007), the differential inclusions and measure derivative methods (Kiseleva et al, 2018;Monteiro Marques, 1993), analytical or numerical integration methods (Moreau, 1999;Paoli and Schatzman, 2002;Liu et al, 2008). The wide bibliographies of the books of Brogliato (2016) and Pfeiffer and Glocker (2000) give a clear idea of the hugeness of the state of the art about these methods.…”