The complexity of a basic impact mapping for rigid bodies with impacts and friction
We consider two impact mappings, the Brach impact mapping and an energetic impact mapping, for rigid-body mechanisms with impacts and friction. The two impact mappings represent the opposite end of the spectrum from basic to advanced impact mappings. Both impact mappings are briefly derived and described. For the Brach impact mapping we will introduce the concept of impulse ratio and discuss how the kinetic energy changes during an impact as the impulse ratio is varied. This analysis is used to further extend the Brach impact mapping to cover situations that were previously omitted. Finally, we make comparisons between the two impact mappings and show how the Painlevé paradox appears in the two impact mappings. The conclusion of the comparisons is that while the basic impact mapping seems easy to implement in a computer simulator it may in the end be more complex and also introduce unnecessary complications that are completely artificial.
Keywordsrigid-body mechanics Coulomb friction impact law non-smooth Painlevé paradox
MSC2010 numbers34-XX 70-XX
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- 4.Keogh, P. S. and Cole, M.O.T., Rotor Vibration with Auxiliary Bearing Contact in Magnetic Bearing Systems: P. 1. Synchronous Dynamics, in Proc. of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 2003, vol. 217, no. 4, pp. 377–392.Google Scholar
- 7.Brach, R.M., Mechanical Impact Dynamics, Rigid Body Collisions, New York: Wiley & Sons, 2007.Google Scholar
- 11.Goldsmith, W., Impact: The Theory and Physical Behaviour of Colliding Solids, Mineola, NY: Dover, 2001.Google Scholar
- 16.Sahinkaya, M. N., Abdul, A.G., and Keogh, P. S., On the Modelling of Flexible Rotor/Magnetic Bearing Systems When in Contact with Retainer Bearings, in Proc. of the 9th Internat. Symp. on Magnetic Bearings, 2004.Google Scholar
- 17.Painlevé, P., Sur les lois de frottement de glissement, C. R. Acad. Sci. Paris, 1905, vol. 141, pp. 564–552.Google Scholar
- 21.Hall, B., Why Does Chalk Squeak?, Master’s Thesis, University of Bristol, Department of engineering Mathematics, 2009.Google Scholar
- 23.Routh, E. J., A Treatise on the Dynamics of a System of Rigid Bodies: P. 2. The Advanced Part, 6th ed., New York: Macmillan, 1905. See also: New York: Dover, 1955 (reprint).Google Scholar
- 24.Glocker, Ch., Set-Valued Force Laws: Dynamics of Non-Smooth Systems, Lecture Notes in Appl. Mech., vol. 1, Berlin: Springer, 2001.Google Scholar