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Analysis of lumped models with contact and friction

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Abstract

We consider two mathematical models that describe the vibrations of spring-mass-damper systems with contact and friction. In the first model, both the contact and frictional boundary conditions are described with subdifferentials of nonconvex functions. In the second model, the contact is modeled with a Lipschitz continuous function, and the restitution force is described by a differential equation involving a Volterra integral term. The two models lead to second-order differential inclusions with and without an integral term, in which the unknowns are the positions of the masses. For each model, we prove the existence of a solution by using an abstract result for first-order differential inclusions in finite dimensional spaces. For the second model, in addition, we prove the uniqueness of the solution by using a fixed point argument. Finally, we provide examples of systems with contact and friction conditions for which our results are valid.

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Jagiellonian University Institute of Computer Science, ul Stanisława Łojasiewicza 6, 30348, Krakow, Poland

    Stanisław Migórski & Anna Ochal

  2. Laboratoire de Mathématiques, Physique et Systèmes, Université de Perpignan, 52 Avenue Paul Alduy, 66860, Perpignan, France

    Mircea Sofonea

Authors
  1. Stanisław Migórski
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  2. Anna Ochal
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  3. Mircea Sofonea
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Corresponding author

Correspondence to Stanisław Migórski.

Additional information

This work was done within the project Polonium ”Nonsmooth Analysis with Applications to Contact Mechanics” under contract no. 7817/R09/R10 between the Jagiellonian University and Université de Perpignan. The first two authors were supported in part by the Ministry of Science and Higher Education of Poland under grant no. N201 027 32/1449.

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Migórski, S., Ochal, A. & Sofonea, M. Analysis of lumped models with contact and friction. Z. Angew. Math. Phys. 62, 99–113 (2011). https://doi.org/10.1007/s00033-010-0081-4

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  • DOI: https://doi.org/10.1007/s00033-010-0081-4

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