空气阻力对完全非弹性蹦球动力学行为的影响

姜泽辉; 韩红; 李翛然; 王福力

doi:10.7498/aps.61.240502

摘要

一个在振动台面上蹦跳的小球具有复杂的运动形式, 如倍周期分岔和混沌. 如果球与台面间的碰撞是完全非弹性的, 则球的运动是倍周期的, 不存在混沌. 在分岔相图中, 鞍-结不稳定性引入平台结构, 同时存在倍周期轨道的密集区. 这里将研究空气的黏滞阻力对完全非弹性蹦球动力学行为的影响. 分析表明, 空气阻力很弱时, 分岔序列不受影响, 但分岔点的数值变大, 平台和密集区加宽. 空气阻力较大时, 平台与密集区重叠. 重叠区内原有产生倍周期运动的机理被破坏, 球的运动是混沌的.

关键词:

倍周期分岔 /
蹦球 /
混沌 /
空气阻力

Abstract

A ball dropped on a vertically vibrating table exhibits intricate dynamical behaviors including period-doubling bifurcations and chaos. If the collision between the ball and the table is completely inelastic, the motion of the ball is always periodic, and the plateaus caused by saddle-node instability and clumping structures for periodic trajectories occur in the bifurcation diagram. Here the effect of air damping on the dynamics of the ball with zero elasticity is analyzed. The air damping is treated as linear viscous one. It is shown that a weak air damping does not change the sequence of bifurcations, but makes the bifurcation points shift to larger values and broadens the transverse dimensions of the plateaus and the clumping zones in the diagrams. However, when the air damping becomes larger, overlapping between the plateaus and clumping zones takes place. In the overlapping section, the mechanism originally leading to periodic motion is destroyed, and chaos is introduced.

Keywords:

作者及机构信息

1.
哈尔滨工业大学物理系, 哈尔滨 150001

基金项目: 国家自然科学基金(批准号: 10974038)资助的课题.

Authors and contacts

1.
Department of Physics, Harbin Institute of Technology, Harbin 150001, China

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 10974038).

参考文献

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[30]	Naylor M A, Swift M R, King P J 2003 Phys. Rev. E 68 012301
[31]	Yan X, Shi Q, Hou M, Lu K, Chan C K 2003 Phys. Rev. Lett. 91 14302
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[34]	Naylor M A, Sanchez P, Swift M R 2002 Phys. Rev. E 66 057201
[35]	Zhou Z G, Shi Y Y, Liu C B, Wang G H, Yang H J 2012 Acta. Phys. Sin. 61 200501 (in Chinese) [周志刚, 石玉仁, 刘丛波, 王光辉, 杨红娟 2012 61 200501]
[36]	Jiang Z H, Guo B, Zhang F, Wang F L 2010 Acta. Phys. Sin. 59 8444 (in Chinese) [姜泽辉, 郭波, 张峰, 王福力 2010 59 8444]

施引文献

[1]	Fermi E 1949 Phys. Rev. 15 1169
[2]	Lichtenberg A J, Lieberman M A 1983 Regular and Stochastic Motion (New York: Springer-Verlag) p190
[3]	Pustylnikov L D 1978 Trans. Moscow Math. Society 2 1
[4]	Tufillaro N B, Albano A M 1986 Am. J. Phys. 54 939
[5]	Tufillaro N B, Abbott T, Reilly J 1992 An experimental approach to nonlinear dynamics and chaos (Redwood: Addison-Wesley)
[6]	Pierański P, Kowalik Z, Franaszek M 1985 J. Phys. 46 681
[7]	Celaschi S, Zimmerman R L 1987 Phys. Lett. A 120 447
[8]	Holmes P 1981 J. Sound Vibration 84 173
[9]	Bapat C, Sankar S, Popplewell N 1986 J. Sound Vibration 108 99
[10]	Tufillaro N B 1994 Phys. Rev. E 50 4509
[11]	Luo A C J, Han R P S 1996 Nonlinear Dynamics 1996 10 1
[12]	Barroso J J, Carneiro M V, Macau E E N 2009 Phys. Rev. E 79 026206
[13]	Lichtenberg A J, Lieberman M A, Cohen R H 1980 Physica D 1 291
[14]	Mehta A, Luck J M 1990 Phys. Rev. Lett. 65 393
[15]	Luck J M, Mehta A 1993 Phys. Rev. E 48 3988
[16]	Giusepponi S, Marchesoni F 2003 Europhys. Lett. 64 36
[17]	Giusepponi S, Marchesoni F, Borromeo M 2005 Physica A 351 142
[18]	Jiang Z H, Zhao H F, Zheng R H 2009 Acta. Phys. Sin. 58 7579 (in Chinese) [姜泽辉, 赵海发, 郑瑞华 2009 58 7579]
[19]	Gilet T, Vandewalle N, Dorbolo S 2009 Phys. Rev. E 79 1539
[20]	Wassgren C R, Brennen C E, Hunt M L 1996 J. Appl. Mech. 63 712
[21]	Jiang Z H, Liu X Y, Peng Y J, Li J W 2005 Acta. Phys. Sin. 54 5692 (in Chinese) [姜泽辉, 刘新影, 彭亚晶, 李建伟 2005 54 5692]
[22]	Jiang Z H, Wang Y Y, Wu J 2006 Europhys. Lett. 43 417
[23]	Jiang Z H, Zheng R H, Zhao H F, Wu J 2007 Acta. Phys. Sin. 56 3727 (in Chinese) [姜泽辉, 郑瑞华, 赵海发, 吴晶 2007 56 3727]
[24]	Melo F, Umbanhowar P B, Swinney H L 1995 Phys. Rev. Lett. 75 3838
[25]	Moon S J, Shattuck M D, Bizon C, Goldman D I, Swift J B, Swinney H L 2001 Phys. Rev. E 65 011301
[26]	Pak H K, Van Doorn E, Behringer R P 1995 Phys. Rev. Lett. 74 4643
[27]	Aoki K M, Akiyama T, Yamamoto K, Yoshikawa T 1997 Europhys. Lett. 40 159
[28]	Paster J M, Maza D, Zuriguel I, Garcimartíin A, Boudet J F 2007 Physica D 232 128
[29]	Möbius M E, Lauderdale B E, Nagel S R, Jaeger H M 2001 Nature 414 270
[30]	Naylor M A, Swift M R, King P J 2003 Phys. Rev. E 68 012301
[31]	Yan X, Shi Q, Hou M, Lu K, Chan C K 2003 Phys. Rev. Lett. 91 14302
[32]	Klein M, Tsai L L, Rosen M S, Pavlin T, Candela D Walsworth R L 2006 Phys. Rev. E 74 010301
[33]	Liu C, Wang L, Wu P, Jia M 2010 Phys. Rev. Lett. 104 188001
[34]	Naylor M A, Sanchez P, Swift M R 2002 Phys. Rev. E 66 057201
[35]	Zhou Z G, Shi Y Y, Liu C B, Wang G H, Yang H J 2012 Acta. Phys. Sin. 61 200501 (in Chinese) [周志刚, 石玉仁, 刘丛波, 王光辉, 杨红娟 2012 61 200501]
[36]	Jiang Z H, Guo B, Zhang F, Wang F L 2010 Acta. Phys. Sin. 59 8444 (in Chinese) [姜泽辉, 郭波, 张峰, 王福力 2010 59 8444]

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