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Realization of non-holonomic constraints and singular perturbation theory for plane dumbbells

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Abstract

We study the dynamics of pairs of connected masses in the plane when non-holonomic (knife-edge) constraints are realized by forces of viscous friction, in particular its relation to constrained dynamics, and its approximation by the method of matching asymptotics of singular perturbation theory when the mass-to-friction ratio is taken as the small parameter. It turns out that long-term behaviors of the frictional and constrained systems may differ dramatically no matter how small the perturbation is, and when this happens is not determined by any transparent feature of the equations of motion. The choice of effective time scales for matching asymptotics is also subtle and non-obvious, and secular terms appearing in them cannot be dealt with by the classical methods. Our analysis is based on comparison to analytic solutions, and we present a reduction procedure for plane dumbbells that leads to them in some cases.

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Acknowledgements

The authors are grateful to Jaap Eldering for multiple suggestions and corrections during the preparation of the paper.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Houston-Downtown, 1 Main Street, Houston, TX, 77002, USA

    Sergiy Koshkin

  2. Systems, Implementation & Integration, Smith Bits, A Schlumberger Co., 1310 Rankin Road, Houston, TX, 77032, USA

    Vojin Jovanovic

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  1. Sergiy Koshkin
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  2. Vojin Jovanovic
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Correspondence to Vojin Jovanovic.

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Koshkin, S., Jovanovic, V. Realization of non-holonomic constraints and singular perturbation theory for plane dumbbells. J Eng Math 106, 123–141 (2017). https://doi.org/10.1007/s10665-017-9896-8

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  • DOI: https://doi.org/10.1007/s10665-017-9896-8

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