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Unilateral contact and dry friction in finite freedom dynamics. (English) Zbl 0703.73070
Nonsmooth mechanics and applications, CISM Courses Lect. 302, 1-82 (1988).
[For the entire collection see Zbl 0652.00016.]
This comprehensive paper presents some results and reflexions of the author concerning generalizations of Lagrange equations to the case of contact problems. The system of particles is assumed to obey some constraints, possibly nonsmooth. Only the scleronomic case is studied. Frictionless motions as well as motions with friction are carefully investigated. Motions may be smooth or nonsmooth; in the latter case velocity u is discontinuous and is an element of the space \(lbv(I,{\mathbb{R}}^ n)\). Here \(I\subset {\mathbb{R}}^ 1\) is a time interval and \(lbv(I,{\mathbb{R}}^ n)\) is the space of functions that have locally bounded variations on I.
If the motion is smooth then the velocity function satisfies a differential inclusion. Thus this differential inclusion represents a generalization of Lagrange equations. An approximation scheme for finding a solution is proposed. More complicated is the case of nonsmooth motions. In this case the motion of a system is described by a measure differential inclusion. An approximation method is proposed and some simple examples are studied.
This valuable paper should be read by anyone working on dynamic contact problems. As a completion to some mathematical aspects of the paper the reviewer strongly advises the reader the following contribution by the same author: “Bounded variation in time”, in: Topics in nonsmooth mechanics, 1-74 (1988; Zbl 0657.28008).
Reviewer: J.J.Telega

74A55 Theories of friction (tribology)
74M15 Contact in solid mechanics
74S30 Other numerical methods in solid mechanics (MSC2010)
74P10 Optimization of other properties in solid mechanics
28B05 Vector-valued set functions, measures and integrals
49Q99 Manifolds and measure-geometric topics
49J52 Nonsmooth analysis