Coulomb Friction Mathematical Background

In the preceding section we used a simple model containing Coulomb friction to demonstrate the problems when solving the equations of motion. We gave a refined model for this example. Now, we want to generalize this foregoing, i.e. we want to construct refined models for the general case of systems with jump discontinuities 

In the sequel we describe the problem of inconsistent switching and briefly introduce Filippov s approach. On this basis we construct an automatic treatment of this situation by a generalized three-valued switching logic. 

The problem of inconsistent switching is described for ordinary differential equa-tions of the form 

In the example given at the beginning of this section we have 

To overcome this difficulty, Filippov [Fil64] extended the definition of a solution for ordinary differential equations by allowing the right hand side to be any value of a set valued right hand side which consists of all convex combinations of / , /  

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