Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Ideal characterizations of multiple impacts: A frame-independent approach by means of jet-bundle geometry


Author: Stefano Pasquero
Journal: Quart. Appl. Math. 76 (2018), 547-576
MSC (2010): Primary 70F35, 70E55, 70G45
DOI: https://doi.org/10.1090/qam/1494
Published electronically: November 17, 2017
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Abstract: We present, in the geometric setup given by the space-time bundle $ \mathcal {M}\/$ and its first jet-extension $ J_1\/(\mathcal {M}\/)$, an ideal constitutive characterization based on the conservation of kinetic energy for a general mechanical system with a finite number of degrees of freedom in contact/impact with a multiple unilateral constraint $ \mathcal {C}\/$ comprising a finite number of regular constraints of codimension $ 1$.

We prove that the geometric structures associated to the elements of $ \mathcal {C}\/$ determine a natural criterion to choose the simplest non-trivial constitutive characterization among the various possibilities preserving the kinetic energy in multiple impacts.

We put this specific choice at the core of an algorithm that determines the right-velocity of the system once the massive properties of the system, the elements of the multiple constraint and the left-velocity of the system are known, in cases of both single and multiple contact/impact.

We show the application of the algorithm in three significant examples: the Newton Cradle, the simultaneous impact of a disk with two disks at rest and in contact, the impact of a disk with a disk at rest and in contact with two other disks.


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Additional Information

Stefano Pasquero
Affiliation: Department of Mathematical, Physical and Computer Sciences, Parco Area delle Scienze 53/a (Campus), 43124 - Parma, Italy
Email: stefano.pasquero@unipr.it

DOI: https://doi.org/10.1090/qam/1494
Keywords: Multiple Impact, constitutive characterization, impulsive constraint
Received by editor(s): November 16, 2016
Received by editor(s) in revised form: October 9, 2017
Published electronically: November 17, 2017
Article copyright: © Copyright 2017 Brown University

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