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The energy dissipation of a disc spinning on a horizontal plane is studied, as the angle α of the coin made with the horizontal plane decreases, while the angular velocity Ω of the point of contact increases. Effect of the ratio x between the thickness and diameter of an Euler disc and the α on the energy dissipation is studied. We find, by using numerical simulation, that when x is small enough, the lose of the kinetic energy and the gravitational potential energy of the mass center is dominant in energy dissipations; when x>0.4142, the rotational kinetic energy dissipation of the disc around the axis that is parallel to the disc surface, is the leading factor. The requirements in which thickness can be neglected are also obtained, and they can give some hints to the relevant theories and experiments. Our results show that when α≥10° and b/a[26] data very well. We also discuss the main energy dissipation distributed among different forms: variation of rolling friction and viscous shear of the air with x and α, also show their transition in the process of the motion. Furthermore, we find that the pure rolling friction is the unique dissipation as x=0.1733 and α>18°, which improves the results obtained before. We speculate that the dominant dissipation is the gliding friction in the final stage of the motion, because when the disc is motionless, one face of the disc lies absolutely in contact with the horizontal surface just before the disc halts. One can assume that they are in contact completely but the disc does not halt, thus axis 1 and axis Z are almost in the same direction. In this case, the energy dissipation of the Euler disc is due to the gliding friction. To some extent, this accounts for the disc final halt.
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Keywords:
- Euler disc /
- energy dissipation /
- thickness of the disc /
- angle
[1] Zhang H J, Brogliato B 2011 INRIA Research Report 7580
[2] Routh E J 1905 The Advanced Part of a Treatise on the Dynamics of Rigid Bodies, 6th ed (Cambridge: Cambridge University Press) pp196
[3] Milne E A 1948 Vectorial Mechanics 338
[4] Fowles G R, Cassiday G L 1999 Analytical Mechanics, 6th ed(Cambridge: Cambridge University Press) pp383
[5] Olsson M G 1972 Am. J. Phys. 40 1543
[6] Moffatt H K 2000 Nature 404 833
[7] van der Engh G, Nelson P, Roach J 2000 Nature 408 540
[8] Zhang H J, Liu C 2012 Program and Abstract Book of the sixth Asian Conference on Multibody Dynamics Shanghai, August 26-30, 2012 p129
[9] Liu C, Zhao Z, Brogliato B 2008 Proc. R. Soc. A 464 3193
[10] Liu C, Zhao Z, Brogliato B 2009 Proc. R. Soc. A 465 1
[11] Zhao Z, Liu C, Brogliato B 2009 Proc. R. Soc. A 465 2267
[12] Liu C, Zhang H, Zhao Z, Brogliato B 2013 Proc. R. Soc. A 469 20120741
[13] Ma D, Liu C, Zhao Z, Zhang H J 2014 Proc. R. Soc. A 470 20140191
[14] Zhao Z, Liu C, Ma D 2014 Proc. R. Soc. A 470 20140007
[15] Le Saux C, Leine R, Glocker C 2005 Sci. 15 27
[16] Borisov A V, Mamaev I S, Karavaev Y L 2015 Nonlinear Dynamics 79 2287
[17] Wang J, Liu C, Zhao Z 2014 Multibody System Dynamics 32 217
[18] Wang J, Liu C, Jia Y, Ma D 2014 The European Physical Journal E. 37 1
[19] Wang J, Liu C, Ma D 2014 Proc. R. Soc. A 470 20140439
[20] Zhang H, Brogliato B, Liu C 2014 Multibody System Dynamics 32 1
[21] Stanislavsky A A, Weron K 2011 Physica D 156 247
[22] Srinivasan M, Ruina A 2008 Phys. Rev. E 78 066609
[23] Weidman P, Malhotra C 2005 Phys. Rev. Lett 95 264303
[24] Leine R I 2009 Arch. Appl. Mech. 79 1063
[25] McDonald A J, McDonald K T 2001 arXiv:0008227 [physics]
[26] Petrie D, Hunt J L, Gray C G 2002 American Journal of Physics 70 1025
[27] Zhong H, Lee C, Su Z, Chen S, Zhou M, Wu J 2013 J. Fluid Mech. 716 228
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[1] Zhang H J, Brogliato B 2011 INRIA Research Report 7580
[2] Routh E J 1905 The Advanced Part of a Treatise on the Dynamics of Rigid Bodies, 6th ed (Cambridge: Cambridge University Press) pp196
[3] Milne E A 1948 Vectorial Mechanics 338
[4] Fowles G R, Cassiday G L 1999 Analytical Mechanics, 6th ed(Cambridge: Cambridge University Press) pp383
[5] Olsson M G 1972 Am. J. Phys. 40 1543
[6] Moffatt H K 2000 Nature 404 833
[7] van der Engh G, Nelson P, Roach J 2000 Nature 408 540
[8] Zhang H J, Liu C 2012 Program and Abstract Book of the sixth Asian Conference on Multibody Dynamics Shanghai, August 26-30, 2012 p129
[9] Liu C, Zhao Z, Brogliato B 2008 Proc. R. Soc. A 464 3193
[10] Liu C, Zhao Z, Brogliato B 2009 Proc. R. Soc. A 465 1
[11] Zhao Z, Liu C, Brogliato B 2009 Proc. R. Soc. A 465 2267
[12] Liu C, Zhang H, Zhao Z, Brogliato B 2013 Proc. R. Soc. A 469 20120741
[13] Ma D, Liu C, Zhao Z, Zhang H J 2014 Proc. R. Soc. A 470 20140191
[14] Zhao Z, Liu C, Ma D 2014 Proc. R. Soc. A 470 20140007
[15] Le Saux C, Leine R, Glocker C 2005 Sci. 15 27
[16] Borisov A V, Mamaev I S, Karavaev Y L 2015 Nonlinear Dynamics 79 2287
[17] Wang J, Liu C, Zhao Z 2014 Multibody System Dynamics 32 217
[18] Wang J, Liu C, Jia Y, Ma D 2014 The European Physical Journal E. 37 1
[19] Wang J, Liu C, Ma D 2014 Proc. R. Soc. A 470 20140439
[20] Zhang H, Brogliato B, Liu C 2014 Multibody System Dynamics 32 1
[21] Stanislavsky A A, Weron K 2011 Physica D 156 247
[22] Srinivasan M, Ruina A 2008 Phys. Rev. E 78 066609
[23] Weidman P, Malhotra C 2005 Phys. Rev. Lett 95 264303
[24] Leine R I 2009 Arch. Appl. Mech. 79 1063
[25] McDonald A J, McDonald K T 2001 arXiv:0008227 [physics]
[26] Petrie D, Hunt J L, Gray C G 2002 American Journal of Physics 70 1025
[27] Zhong H, Lee C, Su Z, Chen S, Zhou M, Wu J 2013 J. Fluid Mech. 716 228
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