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| Hlavní autoři: | , |
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| Médium: | Preprint |
| Vydáno: |
1999
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| Témata: | |
| On-line přístup: | https://arxiv.org/abs/math/9910013 |
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| _version_ | 1866908600175165440 |
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| author | Paoli, Laetitia Schatzman, Michelle |
| author_facet | Paoli, Laetitia Schatzman, Michelle |
| contents | We consider a mechanical system with impact and n degrees of freedom, written in generalized coordinates. The system is not necessarily Lagrangian. The representative point of the system must remain inside a set of constraints K; the boundary of K is three times differentiable.
At impact, the tangential component of the impulsion is conserved, while its normal coordinate is reflected and multiplied by a given coefficient of restitution e between 0 and 1. The orthognality is taken with respect to the natural metric in the space of impulsions.
We define a numerical scheme which enables us to approximate the solutions of the Cauchy problem: this is an ad hoc scheme which does not require a systematic search for the times of impact. We prove the convergence of this numerical scheme to a solution, which yields also an existence result.
Without any a priori estimates, the convergence and the existence are local; with some a priori estimates, the convergence and the existence are proved on intervals depending exclusively on these estimates.
This scheme has been implemented with a trivial and a non trivial mass matrix. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_math_9910013 |
| institution | arXiv |
| publishDate | 1999 |
| record_format | arxiv |
| spellingShingle | A numerical scheme for impact problems Paoli, Laetitia Schatzman, Michelle Numerical Analysis Primary: 65J10, 65M20, 65B05; secondary: 17B09, 46N20, 47D03 We consider a mechanical system with impact and n degrees of freedom, written in generalized coordinates. The system is not necessarily Lagrangian. The representative point of the system must remain inside a set of constraints K; the boundary of K is three times differentiable. At impact, the tangential component of the impulsion is conserved, while its normal coordinate is reflected and multiplied by a given coefficient of restitution e between 0 and 1. The orthognality is taken with respect to the natural metric in the space of impulsions. We define a numerical scheme which enables us to approximate the solutions of the Cauchy problem: this is an ad hoc scheme which does not require a systematic search for the times of impact. We prove the convergence of this numerical scheme to a solution, which yields also an existence result. Without any a priori estimates, the convergence and the existence are local; with some a priori estimates, the convergence and the existence are proved on intervals depending exclusively on these estimates. This scheme has been implemented with a trivial and a non trivial mass matrix. |
| title | A numerical scheme for impact problems |
| topic | Numerical Analysis Primary: 65J10, 65M20, 65B05; secondary: 17B09, 46N20, 47D03 |
| url | https://arxiv.org/abs/math/9910013 |