Possible Capstone Projects - Euler’s Disk

Project subject area – dynamics of rotating bodies including friction.

Introduction

The rolling of a coin is a well-known mechanical behaviour, featuring a coin precessing at ever shallower angle and higher frequency until it suddenly falls down flat on a table (try that again now!). A US company has designed a toy in which the duration of the rolling extends to more than a minute. [1]

A Cambridge academic, Prof Keith Moffatt suggested that the energy loss was due to air friction losses [2].

A follow-up communication queried that suggestion, indicating that the motion was largely the same in a near vacuum, and proposed that contact friction was the dominant energy dissipation mechanism [3]. Stronger criticism still can be read in a letter that Nature rejected [4].

A model of the rolling and sliding friction has been included in the analysis [5] and included in a simulation code which can be downloaded from [6].

Possible Capstone Project Topics

Capstone projects could be based on:

1)     a review of the above controversy;

2)     further theoretical analysis of the motion, including models of the contact friction;

3)     further simulation of the motion by computer code;

4)     experiments to parameterise rolling resistance (for various surfaces, disk edge radius, and force), to infer parameters for contact friction models, and to see if rolling resistance explains the energy loss.

Tackling any (or a mix) of these projects will require the complicated mathematics of solid body mechanics. Some lessons on rolling friction modelling will be learnt (if that is the dominant loss). Testing the disks, especially the commercial one, will be fun!

References

[1] http://www.eulersdisk.com/, Tangent Toy Company site, accessed on 2/4/04

[2] H.K.Moffatt (2000) “Euler’s disk and its finite-time singularity”, Nature 404, 833-834, which can be downloaded from http://www.nature.com/nature/journal/v404/n6780/pdf/404833.pdf (on 2/4/04).

[3] Ger Van Den Engh, Peter Nelson & Jared Roach  (2000) “Analytical dynamics: Numismatic gyrations”, Nature 408, 540, which can be downloaded from http://www.nature.com/nature/journal/v408/n6812/pdf/408540.pdf  (on 2/4/04).

[4] Runia A. (2002) “Comment on Moffat’s Disk”, which can be downloaded from http://tam.cornell.edu/~ruina/hplab/Rolling%20and%20sliding/Andy_on_Moffatt_Disk.pdf (on 2/4/04)

[5] P Kessler, O M O'Reilly (2002) "The ringing of Euler's disk", Reg. Chaot. Dyn., 7 (1), 49-60,  which can be downloaded from http://www.turpion.org/php/full/infoFT.phtml?journal_id=rd&paper_id=195 (on 2/4/04).

[6] Milan Batista (2003-4) Euler’s Disk Simulation Program”, which can be downloaded from http://www.fpp.edu/~milanb/euler/