##
**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).

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

### MSC:

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 |