Mechanical systems with set-valued force laws

Abstract

Some theoretical background on non-smooth systems has been discussed in the previous chapter. Mechanical multibody systems form a special and important class of non-smooth systems, because they can be cast in an elegant structured form. The special structure of mechanical systems is due to the fact that the dynamics is described by the Lagrangian formalism, which links dynamics to variational calculus. Moreover, contact forces are incorporated in the equation of motion by using the Lagrange multiplier theorem. But most importantly, contact forces are (mostly) derived from (pseudo-)potentials or dissipation functions. In this chapter we will discuss the mathematical formulation of Lagrangian mechanical systems with unilateral contact and friction modelled with setvalued force laws. It is important to note that (finite-dimensional) Lagrangian mechanical systems encompass rigid multibody systems as well as discretized continuous systems (e.g. through a Ritz approach or a nite-element discretization) with possible frictional unilateral contacts. First, we discuss how set-valued force laws can be derived from non-smooth potentials. Subsequently, we treat the contact laws for unilateral contact and various types of friction within the setting of non-smooth potential theory. This leads to a unified approach with which all set-valued forces can be formulated. Finally, we incorporate the set-valued forces as Lagrangian multipliers in the Newton-Euler equations. The notation in this chapter is kept as close as possible to the notation of Glocker [63].