×

On the terminal motion of sliding spinning disks with uniform Coulomb friction. (English) Zbl 1188.70047

Summary: We review previous investigations concerning the terminal motion of disks sliding and spinning with uniform dry friction across a horizontal plane. Previous analyses show that a thin circular ring or uniform circular disk of radius \(R\) always stops sliding and spinning at the same instant. Moreover, under arbitrary nonzero initial values of translational speed \(v\) and angular rotation rate \(\omega\), the terminal value of the speed ratio \(\varepsilon _0=v/R\omega\) is always 1.0 for the ring and 0.653 for the uniform disk. In the current study we show that an annular disk of radius ratio \(\eta=R_1/R_2\) stops sliding and spinning at the same time, but with a terminal speed ratio dependent on \(\eta\). For a two-tier disk with lower tier of thickness \(H_1\) and radius \(R_1\) and upper tier of thickness \(H_2\) and radius \(R_2\), the motion depends on both \(\eta\) and the thickness ratio \(\lambda=H_1/H_2\). While translation and rotation stop simultaneously, their terminal ratio \(\varepsilon_0\) either vanishes when \(k>\sqrt{2/3}\), is a nonzero constant when \(1/2<k<\sqrt{2/3}\), or diverges when \(k<1/2\), where \(k\) is the normalized radius of gyration. These three regimes are in agreement with those found by S. Goyal, A. Ruina and J. Papadopoulos [Wear 143, No. 2, 331–352 (1991)] for generic axisymmetric bodies with varying radii of gyration using geometric methods. New experiments with PVC disks sliding on a nylon fabric stretched over a plexiglass plate only partially corroborate the three different types of terminal motions, suggesting more complexity in the description of friction.

MSC:

70E18 Motion of a rigid body in contact with a solid surface
70F40 Problems involving a system of particles with friction

Software:

Mathematica
Full Text: DOI

References:

[1] Zhukovskii, N. E., (Collected Works. Collected Works, General Mechanics, vol. 1 (1948), Gostekhizdat: Gostekhizdat Moscow-Leningrad), 339-354, (in Russian); Original paper published in Notices of the Society of Enthusiasts of Natural Science, Department of Physical Science, 9, 1897
[2] Goyal, S.; Ruina, A.; Papadopoulos, J., Wear, 143, 307 (1991)
[3] Goyal, S.; Ruina, A.; Papadopoulos, J., Wear, 143, 331 (1991)
[4] Ishlinskii, A. Yu.; Sokolov, B. N.; Chernous’ko, F. L., Izv. Akad. Nauk SSSR Mekh. Tverd. Tela, 16, 17 (1981)
[5] Voyenli, K.; Eriksen, E., Amer. J. Phys., 53, 1149 (1985)
[6] Mason, M. T., Mechanics of Robotic Manipulation (2001), MIT Press, pp. 130-139
[7] S. Goyal, Planar sliding of a rigid body with dry friction: Limit surfaces and dynamics of motion, Ph.D. Thesis, Cornell University, Ithaca, 1989; S. Goyal, Planar sliding of a rigid body with dry friction: Limit surfaces and dynamics of motion, Ph.D. Thesis, Cornell University, Ithaca, 1989
[8] Farkas, Z.; Bartels, G.; Unger, T.; Wolf, D. E., Phys. Rev. Lett., 90, 24, 248302 (2003)
[9] Halsey, T. C., Nature, 424, 1005 (2003)
[10] Kadau, D.; Bartels, G.; Brendel, L.; Wolf, D. E., Phase Transit., 76, 315 (2003)
[11] Knight, J. B.; Fandrich, G. C.; Lau, C. N.; Jaeger, H. M.; Nagel, S. R., Phys. Rev. E, 51, 3957 (1995)
[12] Nicodemi, M.; Coniglio, A.; Herrmann, H. J., Phys. Rev. E, 55, 3962 (1997)
[13] Weidman, P. D.; Malhotra, C. P., Phys. Rev. Lett., 95, 264303 (2005)
[14] Abramowitz, M.; Stegun, I., Handbook of Mathematical Functions (1972), US Government Printing Office: US Government Printing Office Washington, DC
[15] Wolfram, S., Mathematica: A System for Doing Mathematics by Computer (1991), Addison-Wesley: Addison-Wesley Redwood City · Zbl 0812.68063
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.