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Variational Analysis of Inequality Impact Laws for Perfect Unilateral Constraints

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Advanced Topics in Nonsmooth Dynamics

Abstract

This chapter deals with frictionless instantaneous impacts in rigid multibody dynamics. For autonomous multibody systems which are subjected to perfect unilateral constraints, a geometric description of the impacts on the respective tangent space to the configuration manifold is presented. The mass matrix of a mechanical system endows the configuration manifold with the structure of a Riemannian manifold and provides an isomorphism between the tangent space and the cotangent space at each point of the configuration manifold. Kinematic quantities (virtual displacements, velocities) are elements of the tangent space, while kinetic quantities (forces, impulsive forces) live in the cotangent space, the dual space of the tangent space. Impact laws, as constitutive laws relating primal and dual quantities, are introduced as set-valued mappings between these two spaces. Methods from Convex Analysis permit to study what the implications are if the impact law is maximal monotone. Finally, the generalized Newton’s and the generalized Poisson’s impact law are considered as illustrative examples.

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Notes

  1. 1.

    Sections 2–5 were written by the first author. Sections 2–4 gather a wealth of ideas and concepts of various authors, notably Aeberhard [3], Glocker [15, 16, 19], Ballard [5], and Moreau [34, 36] (in reverse chronological order). Sections 5 and 6 are based on the PhD thesis [6] of the second author.

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Acknowledgements

This research is supported by the Fonds National de la Recherche, Luxembourg (Proj. Ref. 8864427).

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Authors and Affiliations

  1. Institute for Nonlinear Mechanics, University of Stuttgart, Pfaffenwaldring 9, 70569, Stuttgart, Germany

    Tom Winandy, Michael Baumann & Remco I. Leine

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  1. Tom Winandy
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  2. Michael Baumann
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  3. Remco I. Leine
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Correspondence to Tom Winandy .

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Editors and Affiliations

  1. Institute for Nonlinear Mechanics, University of Stuttgart, Stuttgart, Germany

    Remco Leine

  2. Inria Grenoble—Rhône-Alpes Research Centre, Saint Ismier Cedex, France

    Vincent Acary

  3. Department of Aerospace and Mechanical Engineering, University of Liège, Liège, Belgium

    Olivier Brüls

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Winandy, T., Baumann, M., Leine, R.I. (2018). Variational Analysis of Inequality Impact Laws for Perfect Unilateral Constraints. In: Leine, R., Acary, V., Brüls, O. (eds) Advanced Topics in Nonsmooth Dynamics. Springer, Cham. https://doi.org/10.1007/978-3-319-75972-2_2

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  • DOI: https://doi.org/10.1007/978-3-319-75972-2_2

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