# Regularization of singular problems in mechanics

Lead Research Organisation: University of Bristol
Department Name: Engineering Mathematics

### Abstract

When a piece of chalk is pushed across a blackboard, it can jump and judder, often accompanied by an unpleasant screeching noise. This unwanted phenomenon occurs in many engineering applications, including robotic manipulators
In 1905, Painlevé [5] showed that, if a slender rod slides along a rough surface, the governing rigid body equations can produce a paradox: namely, when the coefficient of friction exceeds a critical value, the frictional torque exceeds the torque due to the normal reaction and the rod appears to be driven into the rigid surface.
To resolve this paradox, the rigid body assumption in the neighbourhood of the contact point, is relaxed, by assuming compliance there
This regularization produces a singularly perturbed problem, which can be solved using geometric singular perturbation theory (GSPT); a branch of mathematics developed to tackle problems that have two (slow and fast) or more time scales [3].
Using GSPT, Kristiansen and Hogan [1] proved that the compliant dynamics has three major phases: the torque compresses the compliant surface until the rod stops sliding, then the rod sticks, before finally lifting off the surface.
GSPT must be extended near non-hyperbolic points of the system using the blowup method. Using blowup, Kristiansen and Hogan also showed that the problem has a canard [2].
Hogan's original idea of a marriage of GSPT, blowup and mechanics, with friction and unilateral constraints, has proved very successful for problems with sliding friction.
The current project is about extending this approach to problems with rolling friction, to a problem that is even more familiar than the rod.
Euler's disk: Spin a coin on a table. Initially it rotates about a point on its edge, then its rolling speed rapidly increases (noisily) until the coin stops rotating and lands flat on the table. Why does the coin stop spinning? Recent experiments have shown that the rolling friction is nonlinear and that there is a finite time singularity a the coin stops spinning [4].
The approach to this unsolved problem is to assume compliance at the rolling contact point. As the coin rotates, it transfers energy to the compliant surface. When the rotation stops, the coin sticks to the table. Then energy can be released, with lift-off of the coin.
But what about the finite time singularity? The hypothesis is that all motions converge to a unique forward trajectory: a canard. Instead of the coin lifting off the compliant surface, it remains on the table. Blowup will be central central to the study of this problem.
References:
1. S. J. Hogan and K. Uldall Kristiansen. On the regularization of impact without collision: the Painlevé paradox and compliance. Proc. Roy. Soc. Lond. A., 473:20160773, 2017.
2. K. Uldall Kristiansen and S. J. Hogan. Le canard de Painlevé. SIAM Journal on Applied Dynamical Systems, To appear 2018 (53 pages).
3. C. Kuehn. Multiple Time Scale Dynamics. Springer-Verlag, Berlin, 2015.
4. D. Ma and C. Liu. Dynamics of a spinning disk. ASME J. Applied Mechanics, 83:061003, 2016.
5. P. Painlevé. Sur les loi du frottement de glissement. Comptes Rendu des Séances de l'Academie des Sciences, 141:401--405, 1905.

### ORCID iD

Stephen John Hogan (Primary Supervisor)
Ben Collins (Student)

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### Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/R513179/1 01/10/2018 30/09/2023
2117974 Studentship EP/R513179/1 24/09/2018 31/03/2022 Ben Collins