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.

Ideal characterizations of multiple impacts: A frame–independent approach by means of jet–bundle geometry / Pasquero, Stefano. - In: QUARTERLY OF APPLIED MATHEMATICS. - ISSN 0033-569X. - (2017). [10.1090/qam/1494]

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

#### 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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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11381/2836703
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