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Analysis of a generalized kinematic impact law for multibody-multicontact systems, with application to the planar rocking block and chains of balls

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Abstract

In this paper, we analyze the capabilities of a generalized kinematic (Newton’s like) restitution law for the modeling of a planar rigid block that impacts a rigid ground. This kinematic restitution law is based on a specific state transformation of the Lagrangian dynamics, using the kinetic metric on the configuration space. It allows one to easily derive a restitution rule for multiple impacts. The relationships with the classical angular velocity restitution coefficient r for rocking motion are examined in detail. In particular, it is shown that r has the interpretation of a tangential restitution coefficient. The case when Coulomb’s friction is introduced at the contact impulse level together with an angular velocity restitution is analyzed. A simple chain of aligned balls is also examined, illustrating that the impact law applies to various types of multibody systems.

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Notes

  1. The kinetic metric is the metric defined with the mass matrix M(q)=M T(q)>0, such that for two vectors x and y the inner product is x T M(q)y.

  2. The kinetic angle is obtained after substraction from π because the normal vectors point outside the admissible domain of the configuration space.

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Acknowledgements

This work was performed with the support of the NSFC/ANR project Multiple Impact, ANR-08-BLAN-0321-01. H. Zhang was funded by China Scholarship Council No. 2009601276 and by ANR project Multiple Impact ANR-08-BLAN-0321-01.

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Authors and Affiliations

  1. BIPOP project-team, ZIRST Montbonnot, INRIA, 655 avenue de l’Europe, 38334, Saint Ismier, France

    Bernard Brogliato & Hongjian Zhang

  2. State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing, 100871, People’s Republic of China

    Hongjian Zhang & Caishan Liu

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  1. Bernard Brogliato
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  2. Hongjian Zhang
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Correspondence to Bernard Brogliato.

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Brogliato, B., Zhang, H. & Liu, C. Analysis of a generalized kinematic impact law for multibody-multicontact systems, with application to the planar rocking block and chains of balls. Multibody Syst Dyn 27, 351–382 (2012). https://doi.org/10.1007/s11044-012-9301-3

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