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Binary collision approximation for solitary wave in periodic dimer granular chains

Chen Qiong Yang Xian-Qing Zhao Xin-Yin Wang Zhen-Hui Zhao Yue-Min

Binary collision approximation for solitary wave in periodic dimer granular chains

Chen Qiong, Yang Xian-Qing, Zhao Xin-Yin, Wang Zhen-Hui, Zhao Yue-Min
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  • We study solitary wave propagation in periodic dimer granular chains of beads with the same material but different sizes by binary collision approximation. This kind of chain which is called N:1 dimer consists of pairs of N big beads and one small bead. First we present a method to map the actual chain into an effective chain, then use the binary collision approximation to obtain the transmitted solitary wave speed, the total time taken by the pulse to pass through the chain, and the frequency of oscillation of the small particle. Frequency of oscillation, which increases with the decrease of the radius of the small particle, is analytically obtained. And the results are in excellent agreement with numerical results. For the total time of the pulse passing through the chain, the results of theoretical analysis is in good agreement with numerical results when N 2. The relative error seems no change with the chain length but becomes larger with the increase of the value of N.
    • Funds: Project supported by the Science Fund for Creative Research Groups of the National Natural Science Foundation of China (Grant No. 50921002) and the Fundamental Research Funds for the Central Universities (Grant No. 2010LKWL09).
    [1]

    Huang J, Sun Q C 2007 Acta Phys. Sin. 56 6124 (in Chinese) [黄晋, 孙其诚 2007 物理学报 56 6124]

    [2]

    Wang P J, Xia J H, Liu C S, Liu H, Yan L 2011 Acta Phys. Sin. 60 014501 (in Chinese) [王平建, 夏继宏, 刘长松, 刘会, 闫龙 2011 物理学报 60 014501]

    [3]

    Sen S, Hong J, Bang J, Acalos E, Doney R 2008 Phy. Rep. 462 21

    [4]

    Carretero-Gonález R, Khatri D, Porter M A, Kevrekidis P G, Daraio C 2009 Phys. Rev. Lett. 102 024102

    [5]

    Hong J, Xu A 2001 Phys. Rev. E 63 061310

    [6]

    Hong J 2005 Phys. Rev. Lett. 94 108001

    [7]

    Daroio C, Nesterenko V F, Herbold E B, Jin S 2006 Phys. Rev. Lett. 96 058002

    [8]

    Daraio C, Ngo D, Nesterenko V F, Fraternali F 2010 Phys. Rev. E 82 036603

    [9]

    Job S, Santibanez F, Tapia F, Melo F 2009 Phys. Rev. E 80 025602(R)

    [10]

    Porter M A, Daraio C, Herbold E B, Szelengowicz I, Kevrekidis P G 2008 Phys. Rev. E 77 015601(R)

    [11]

    Vergara L 2006 Phys. Rev. E 73 066623

    [12]

    Wang P J, Xia J H, Li Y D, Liu C S 2007 Phys. Rev. E 76 041305

    [13]

    Wang P J, Li Y D, Xia J H, Liu C S 2008 Phys. Rev. E 77 060301(R)

    [14]

    Daraio C, Nesterenko V F, Herbold E B, Jin S 2005 Phys. Rev. E 72 016603

    [15]

    Nesterenko V F, Daraio C, Herbold E B, Jin S 2005 Phys. Rev. Lett. 95 158702

    [16]

    Hascoet E, Herrmann H J 2000 Eur. Phys. J. B 14 183

    [17]

    Daraio C, Nesterenko V F 2006 Phys. Rev. E 73 026612

    [18]

    Jayaprakash K R, Starosvetsky Y, Vakakis A F 2011 Phys. Rev. E 83 036606

    [19]

    Rosas A, Lindenberg K 2004 Phys. Rev. E 69 037601

    [20]

    Harbola U, Rosas A, Esposito M, Lindenberg K 2009 Phys. Rev. E 80 031303

    [21]

    Harbola U, Rosas A, Romero A H, Esposito M, Lindenberg K 2009 Phys. Rev. E 80 051302

    [22]

    Harbola U, Rosas A, Romero A H, Lindenberg K 2010 Phys. Rev. E 82 011306

    [23]

    Pinto I L D, Rosas A 2010 Phys. Rev. E 82 031308

    [24]

    Pinto I L D, Rosas A, Lindenberg K 2009 Phys. Rev. E 79 061307

  • [1]

    Huang J, Sun Q C 2007 Acta Phys. Sin. 56 6124 (in Chinese) [黄晋, 孙其诚 2007 物理学报 56 6124]

    [2]

    Wang P J, Xia J H, Liu C S, Liu H, Yan L 2011 Acta Phys. Sin. 60 014501 (in Chinese) [王平建, 夏继宏, 刘长松, 刘会, 闫龙 2011 物理学报 60 014501]

    [3]

    Sen S, Hong J, Bang J, Acalos E, Doney R 2008 Phy. Rep. 462 21

    [4]

    Carretero-Gonález R, Khatri D, Porter M A, Kevrekidis P G, Daraio C 2009 Phys. Rev. Lett. 102 024102

    [5]

    Hong J, Xu A 2001 Phys. Rev. E 63 061310

    [6]

    Hong J 2005 Phys. Rev. Lett. 94 108001

    [7]

    Daroio C, Nesterenko V F, Herbold E B, Jin S 2006 Phys. Rev. Lett. 96 058002

    [8]

    Daraio C, Ngo D, Nesterenko V F, Fraternali F 2010 Phys. Rev. E 82 036603

    [9]

    Job S, Santibanez F, Tapia F, Melo F 2009 Phys. Rev. E 80 025602(R)

    [10]

    Porter M A, Daraio C, Herbold E B, Szelengowicz I, Kevrekidis P G 2008 Phys. Rev. E 77 015601(R)

    [11]

    Vergara L 2006 Phys. Rev. E 73 066623

    [12]

    Wang P J, Xia J H, Li Y D, Liu C S 2007 Phys. Rev. E 76 041305

    [13]

    Wang P J, Li Y D, Xia J H, Liu C S 2008 Phys. Rev. E 77 060301(R)

    [14]

    Daraio C, Nesterenko V F, Herbold E B, Jin S 2005 Phys. Rev. E 72 016603

    [15]

    Nesterenko V F, Daraio C, Herbold E B, Jin S 2005 Phys. Rev. Lett. 95 158702

    [16]

    Hascoet E, Herrmann H J 2000 Eur. Phys. J. B 14 183

    [17]

    Daraio C, Nesterenko V F 2006 Phys. Rev. E 73 026612

    [18]

    Jayaprakash K R, Starosvetsky Y, Vakakis A F 2011 Phys. Rev. E 83 036606

    [19]

    Rosas A, Lindenberg K 2004 Phys. Rev. E 69 037601

    [20]

    Harbola U, Rosas A, Esposito M, Lindenberg K 2009 Phys. Rev. E 80 031303

    [21]

    Harbola U, Rosas A, Romero A H, Esposito M, Lindenberg K 2009 Phys. Rev. E 80 051302

    [22]

    Harbola U, Rosas A, Romero A H, Lindenberg K 2010 Phys. Rev. E 82 011306

    [23]

    Pinto I L D, Rosas A 2010 Phys. Rev. E 82 031308

    [24]

    Pinto I L D, Rosas A, Lindenberg K 2009 Phys. Rev. E 79 061307

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  • Received Date:  27 May 2011
  • Accepted Date:  01 September 2011
  • Published Online:  15 April 2012

Binary collision approximation for solitary wave in periodic dimer granular chains

  • 1. College of Science, China University of Mining and Technology, Xuzhou 221116, China;
  • 2. School of Chemical Engineering & Technology, China University of Mining and Technology, Xuzhou 221116, China
Fund Project:  Project supported by the Science Fund for Creative Research Groups of the National Natural Science Foundation of China (Grant No. 50921002) and the Fundamental Research Funds for the Central Universities (Grant No. 2010LKWL09).

Abstract: We study solitary wave propagation in periodic dimer granular chains of beads with the same material but different sizes by binary collision approximation. This kind of chain which is called N:1 dimer consists of pairs of N big beads and one small bead. First we present a method to map the actual chain into an effective chain, then use the binary collision approximation to obtain the transmitted solitary wave speed, the total time taken by the pulse to pass through the chain, and the frequency of oscillation of the small particle. Frequency of oscillation, which increases with the decrease of the radius of the small particle, is analytically obtained. And the results are in excellent agreement with numerical results. For the total time of the pulse passing through the chain, the results of theoretical analysis is in good agreement with numerical results when N 2. The relative error seems no change with the chain length but becomes larger with the increase of the value of N.

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