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The behavior of a completely inelastic ball bouncing on a vertically vibrating table in the presence of frictional force is investigated. The frictional force is assumed to be constant. It is found that the sequence of bifurcation, controlled solely by the normalized vibration acceleration Γ, is the same as that in the absence of frictional force, but the value of each bifurcation point becomes larger. In the bifurcation diagram of ball flight time, the structure consisting of an infinity of bifurcation cascades in a narrow range of Γ is observed. Compared with that of no frictional force, it is longitudinally compressed and transversely stretched, and has a different fractal property. A comparison with the bifurcations observed in vertically vibrated granular beds is also made. When the fractional force is set to be 20%—30% of the whole weight of the particles, the results from the bouncing ball model are in good agreement with experimental observations.
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Keywords:
- bouncing ball /
- granular matter /
- period-doubling bifurcation /
- fractal
[1] Tufillaro N B, Abbott T, Reilly J 1992 An Experimental Approach to Nonlinear Dynamics and Chaos (New York: Addison-Wesley Publishing Company)
[2] Pierański P 1988 Phys. Rev. A 37 1782
[3] Holmes P J 1982 J. Sound Vib. 84 173
[4] Mehta A, Luck J M 1990 Phys. Rev. Lett. 65 393
[5] Luck J M, Mehta A 1993 Phys. Rev. E 48 3988
[6] 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]
[7] 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]
[8] Gilet T, Vandewalle N, Dorbolo S 2009 Phys. Rev. E 79 055201
[9] Melo F, Umbanhowar P B, Swinney H L 1995 Phys. Rev. Lett. 75 3838
[10] Moon S J, Shattuck M D, Bizon C, Goldman D I, Swift J B, Swinney H L 2001 Phys. Rev. E 65 11301
[11] Luding S, Clément E, Blumen A, Rajchenbach J, Duran J 1994 Phys. Rev. E 49 1634
[12] Douady S, Fauve S, Laroche C 1989 Europhys. Lett. 8 621
[13] Wassgren C R, Brennen C E, Hunt M L 1996 J. Appl. Mech. 63 712
[14] Aoki K M, Akiyama T, Yamamoto K, Yoshikawa T 1997 Europhys. Lett. 40 159
[15] Jiang Z H, Wang Y Y, Wu J 2006 Europhys. Lett. 74 417
[16] Jiang Z H, Li B, Zhao H F, Wang Y Y, Dai Z B 2005 Acta Phys. Sin. 54 1273 (in Chinese)[姜泽辉、李 斌、赵海发、 王运鹰、戴智斌 2005 54 1273] 〖17] 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、彭雅晶、李建伟 2005 54 5692]
[17] Pastor J M, Maza D, Zuriguel I, Garcimartín A, Boudet J F 2007 Physica D 232 128
[18] Ho C K, Webb S W 2006 Gas Transport in Porous Media (Dordrecht: Springer) pp5—26
[19] Nield D A, Bejan A 2006 Convection in Porous Media (3rd ed) (New York: Springer) Chap 1
[20] Pak H K, Doorn E V, Behringer R P 1995 Phys. Rev. Lett. 74 4643
[21] Yan X, Shi Q, Hou M, Lu K, Chan C K 2003 Phys. Rev. Lett. 91 14302
[22] Mbius M E, Cheng X, Eshuis P, Karczmar G S, Nagel S R, Jaeger H M 2005 Phys. Rev.E 72 011304
[23] Akiyama T, Kimura N, Iguchi T 1996 Powder Technol. 89 133
[24] Akiyama T, Yoshikawa T 1999 Powder Technol. 103 139
[25] Jiang Z H, Jing Y F, Zhao H F, Zheng R H 2009 Acta Phys. Sin. 58 5923 (in Chinese) [姜泽辉、荆亚芳、赵海发、郑瑞华 2009 58 5923]
[26] Evasque P, Szmatula E, Denis J P 1990 Europhys. Lett. 12 623
[27] Knight J B, Jaeger H M, Nagel S R 1993 Phys. Rev. Lett. 70 3728
[28] Jiang Z H, Wang Y Y, Wu J 2006 Acta Phys. Sin. 55 4748 (in Chinese) [姜泽辉、王运鹰、吴 晶 2006 55 4748]
[29] Zeilstra C, Collignon J G, van der Hoef M A, Deen N G, Kuipers J A M 2008 Powder Technol. 184 166
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[1] Tufillaro N B, Abbott T, Reilly J 1992 An Experimental Approach to Nonlinear Dynamics and Chaos (New York: Addison-Wesley Publishing Company)
[2] Pierański P 1988 Phys. Rev. A 37 1782
[3] Holmes P J 1982 J. Sound Vib. 84 173
[4] Mehta A, Luck J M 1990 Phys. Rev. Lett. 65 393
[5] Luck J M, Mehta A 1993 Phys. Rev. E 48 3988
[6] 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]
[7] 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]
[8] Gilet T, Vandewalle N, Dorbolo S 2009 Phys. Rev. E 79 055201
[9] Melo F, Umbanhowar P B, Swinney H L 1995 Phys. Rev. Lett. 75 3838
[10] Moon S J, Shattuck M D, Bizon C, Goldman D I, Swift J B, Swinney H L 2001 Phys. Rev. E 65 11301
[11] Luding S, Clément E, Blumen A, Rajchenbach J, Duran J 1994 Phys. Rev. E 49 1634
[12] Douady S, Fauve S, Laroche C 1989 Europhys. Lett. 8 621
[13] Wassgren C R, Brennen C E, Hunt M L 1996 J. Appl. Mech. 63 712
[14] Aoki K M, Akiyama T, Yamamoto K, Yoshikawa T 1997 Europhys. Lett. 40 159
[15] Jiang Z H, Wang Y Y, Wu J 2006 Europhys. Lett. 74 417
[16] Jiang Z H, Li B, Zhao H F, Wang Y Y, Dai Z B 2005 Acta Phys. Sin. 54 1273 (in Chinese)[姜泽辉、李 斌、赵海发、 王运鹰、戴智斌 2005 54 1273] 〖17] 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、彭雅晶、李建伟 2005 54 5692]
[17] Pastor J M, Maza D, Zuriguel I, Garcimartín A, Boudet J F 2007 Physica D 232 128
[18] Ho C K, Webb S W 2006 Gas Transport in Porous Media (Dordrecht: Springer) pp5—26
[19] Nield D A, Bejan A 2006 Convection in Porous Media (3rd ed) (New York: Springer) Chap 1
[20] Pak H K, Doorn E V, Behringer R P 1995 Phys. Rev. Lett. 74 4643
[21] Yan X, Shi Q, Hou M, Lu K, Chan C K 2003 Phys. Rev. Lett. 91 14302
[22] Mbius M E, Cheng X, Eshuis P, Karczmar G S, Nagel S R, Jaeger H M 2005 Phys. Rev.E 72 011304
[23] Akiyama T, Kimura N, Iguchi T 1996 Powder Technol. 89 133
[24] Akiyama T, Yoshikawa T 1999 Powder Technol. 103 139
[25] Jiang Z H, Jing Y F, Zhao H F, Zheng R H 2009 Acta Phys. Sin. 58 5923 (in Chinese) [姜泽辉、荆亚芳、赵海发、郑瑞华 2009 58 5923]
[26] Evasque P, Szmatula E, Denis J P 1990 Europhys. Lett. 12 623
[27] Knight J B, Jaeger H M, Nagel S R 1993 Phys. Rev. Lett. 70 3728
[28] Jiang Z H, Wang Y Y, Wu J 2006 Acta Phys. Sin. 55 4748 (in Chinese) [姜泽辉、王运鹰、吴 晶 2006 55 4748]
[29] Zeilstra C, Collignon J G, van der Hoef M A, Deen N G, Kuipers J A M 2008 Powder Technol. 184 166
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