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Discussion of d'Alembert's Principle for Non‐Smooth Unilateral Constraints
Author(s) -
Glocker C. H.
Publication year - 1999
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.19990791324
Subject(s) - tangent cone , cone (formal languages) , constraint (computer aided design) , tangent , mathematics , dual cone and polar cone , set (abstract data type) , mathematical analysis , geometry , computer science , algorithm , regular polygon , programming language
In optimization theory the tangent cone and the contingent cone are used to classify the regularity properties of sets at a given point. If both cones coincide the set is called tangentially regular, otherwise we speak of a reentrant corner. A third cone, the normal cone, is defined by the polarity with respect to the tangent cone, which is commonly egresses an terms of a variation inequality. This concept, directly applied to rigid body dynamics, allows an easy classification of possible Constraints. We define constraint forces to be the elements of the normal cone, whereas virtual displacements have to be taken from the contingent cone. A combination of both, the constraint forces and the virtual displacements, consitutes the Principle of d'Alembert as long as the constraints are tangentially regular. In the other case alternative formulations will be discussed.