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关于无碰撞冲击的正则化：庞加莱悖论与柔顺性。

On the regularization of impact without collision: the Painlevé paradox and compliance.

作者信息

Hogan S J, Kristiansen K Uldall

机构信息

Department of Engineering Mathematics, University of Bristol, Bristol BS8 1UB, UK.

Department of Applied Mathematics and Computer Science, Technical University of Denmark, 2800 Kongens Lyngby, Denmark.

出版信息

Proc Math Phys Eng Sci. 2017 Jun;473(2202):20160773. doi: 10.1098/rspa.2016.0773. Epub 2017 Jun 14.

DOI:10.1098/rspa.2016.0773
PMID:28690403
原文链接:https://pmc.ncbi.nlm.nih.gov/articles/PMC5493941/
Abstract

We consider the problem of a rigid body, subject to a unilateral constraint, in the presence of Coulomb friction. We regularize the problem by assuming compliance (with both stiffness and damping) at the point of contact, for a general class of normal reaction forces. Using a rigorous mathematical approach, we recover impact without collision (IWC) in both the inconsistent and the indeterminate Painlevé paradoxes, in the latter case giving an exact formula for conditions that separate IWC and lift-off. We solve the problem for arbitrary values of the compliance damping and give explicit asymptotic expressions in the limiting cases of small and large damping, all for a large class of rigid bodies.

摘要

我们考虑一个存在库仑摩擦的、受单侧约束的刚体问题。对于一般类别的法向反作用力，我们通过假设在接触点处具有柔顺性（包括刚度和阻尼）来对该问题进行正则化。使用严格的数学方法，我们在不一致和不确定的潘勒韦悖论中都恢复了无碰撞冲击（IWC），在后一种情况下给出了区分IWC和脱离的条件的精确公式。我们针对柔顺阻尼的任意值解决了该问题，并在小阻尼和大阻尼的极限情况下给出了明确的渐近表达式，所有这些都是针对一大类刚体的。

https://cdn.ncbi.nlm.nih.gov/pmc/blobs/f87b/5493941/6cbb96c7832a/rspa20160773-g7.jpg
https://cdn.ncbi.nlm.nih.gov/pmc/blobs/f87b/5493941/e3c76b86e674/rspa20160773-g1.jpg
https://cdn.ncbi.nlm.nih.gov/pmc/blobs/f87b/5493941/f2e0097b3dde/rspa20160773-g2.jpg
https://cdn.ncbi.nlm.nih.gov/pmc/blobs/f87b/5493941/0c819a9f7977/rspa20160773-g3.jpg
https://cdn.ncbi.nlm.nih.gov/pmc/blobs/f87b/5493941/beda72ec597a/rspa20160773-g4.jpg
https://cdn.ncbi.nlm.nih.gov/pmc/blobs/f87b/5493941/50a4083cd200/rspa20160773-g5.jpg
https://cdn.ncbi.nlm.nih.gov/pmc/blobs/f87b/5493941/625489843d73/rspa20160773-g6.jpg
https://cdn.ncbi.nlm.nih.gov/pmc/blobs/f87b/5493941/6cbb96c7832a/rspa20160773-g7.jpg
https://cdn.ncbi.nlm.nih.gov/pmc/blobs/f87b/5493941/e3c76b86e674/rspa20160773-g1.jpg
https://cdn.ncbi.nlm.nih.gov/pmc/blobs/f87b/5493941/f2e0097b3dde/rspa20160773-g2.jpg
https://cdn.ncbi.nlm.nih.gov/pmc/blobs/f87b/5493941/0c819a9f7977/rspa20160773-g3.jpg
https://cdn.ncbi.nlm.nih.gov/pmc/blobs/f87b/5493941/beda72ec597a/rspa20160773-g4.jpg
https://cdn.ncbi.nlm.nih.gov/pmc/blobs/f87b/5493941/50a4083cd200/rspa20160773-g5.jpg
https://cdn.ncbi.nlm.nih.gov/pmc/blobs/f87b/5493941/625489843d73/rspa20160773-g6.jpg
https://cdn.ncbi.nlm.nih.gov/pmc/blobs/f87b/5493941/6cbb96c7832a/rspa20160773-g7.jpg

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