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In this paper studied is the crises in the Duffing vibro-impact oscillator with non-viscously damping by the composite cell coordinate system method. It is assumed that the non-viscously damping depends on the past history of the velocities other than the instantaneous generalized velocities. The energy dissipation behaviors of real structural materials can be preferably represented in the non-viscously damping models. Numerical simulations show that as the damping coefficient or the relaxation parameter or the recovery coefficient is varied, there appear two kinds of crises: one is the interior crisis, which results from the collision between a chaotic attractor and a chaotic saddle on the basin boundary, and the other is the regular/chaotic boundary crisis, which is due to the collision of a chaotic attractor with a periodic/chaotic saddle on its basin boundary. All the crises result in a sudden change in size and shape of the attractor.
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Keywords:
- non-viscously damping /
- Duffing vibro-impact oscillator /
- crisis /
- the composite cell coordinate system method
[1] Rayleigh J W S 1877 Theory of Sound (Vol.2) (New York: Dover Publications) pp135-216
[2] Ruzziconi L, Litak G, Lenci S 2011 J. Vibroeng. 13 2238
[3] Rossikhin Y A, Shitikova M V 2010 Appl. Mech. Rev. 63 010801
[4] Sieber J, Wagg D J, Adhikari S 2008 J. Sound Vib. 314 1
[5] Jin D P, Hu H Y 1999 Adv. Mech. 29 155 (in Chinese) [金栋平, 胡海岩 1999 力学进展 29 155]
[6] Ding W C, Xie J H 2005 Adv. Mech. 35 513 (in Chinese) [丁旺才, 谢建华 2005 力学进展 35 513]
[7] Lei H, Gan C B, Xie C Y 2010 Eng. Mech. 27 105 (in Chinese) [雷华, 甘春标, 谢潮涌 2010 工程力学 27 105]
[8] Rong H W, Wang X D, Xu W, Fang T 2008 Acta Phys. Sin. 57 6888 (in Chinese) [戎海武, 王向东, 徐伟, 方同 2008 物理学报 57 6888]
[9] Su M B, Rong H W 2011 Chin. Phys. B 20 060501
[10] Grebogi C, Ott E, Yorke J A 1983 Physica D 7 181
[11] Grebogi C, Ott E, Yorke J A 1986 Phys. Rev. Lett. 57 1284
[12] Szemplińka-Stupnicka W, Zubrzycki A, Tyrkiel E 1999 Nonlinear Dynam. 19 1936
[13] Feng J X, Xu W 2011 Acta Phys. Sin. 60 080502 (in Chinese) [冯进钤, 徐伟 2011 物理学报 60 080502]
[14] Yue X L, Xu W, Wang L 2013 Commun. Nonlinear Sci. Numer. Simul. 18 3567
[15] Hsu C S 1980 J. Appl. Mech. 147 931
[16] Xu W, Sun C Y, Sun J Q 2013 Adv. Mech. 43 91 (in Chinese) [徐伟, 孙春艳, 孙建桥 2013 力学进展 43 91]
[17] Hong L, Xu J X 1999 Phys. Lett. A 262 361
[18] Xu W, He Q, Rong H W, Fang T 2003 Acta Phys. Sin. 52 1365 (in Chinese) [徐伟, 贺群, 戎海武, 方同 2003 物理学报 52 1365]
[19] Jiang J, Xu J X 1994 Phys. Lett. A 188 137
[20] He Q, Xu W, Li S 2008 Acta Phys. Sin. 57 743 (in Chinese) [贺群, 徐伟, 李爽 2008 物理学报 57 743]
[21] He Q, Xu W, Li S 2008 Acta Phys. Sin. 57 4021 (in Chinese) [贺群, 徐伟, 李爽 2008 物理学报 57 4021]
[22] Li S, He Q 2011 Chin. J. Theor. Appl. Mech. 43 579 (in Chinese) [李爽, 贺群 2011 力学学报 43 579]
[23] Yue X L, Xu W, Zhang Y 2012 Nonlinear Dynam. 69 437
[24] Biot M A 1955 Phys. Rev. 97 14
[25] Wagner N, Adhikari S 2003 AIAA J. 41 951
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[1] Rayleigh J W S 1877 Theory of Sound (Vol.2) (New York: Dover Publications) pp135-216
[2] Ruzziconi L, Litak G, Lenci S 2011 J. Vibroeng. 13 2238
[3] Rossikhin Y A, Shitikova M V 2010 Appl. Mech. Rev. 63 010801
[4] Sieber J, Wagg D J, Adhikari S 2008 J. Sound Vib. 314 1
[5] Jin D P, Hu H Y 1999 Adv. Mech. 29 155 (in Chinese) [金栋平, 胡海岩 1999 力学进展 29 155]
[6] Ding W C, Xie J H 2005 Adv. Mech. 35 513 (in Chinese) [丁旺才, 谢建华 2005 力学进展 35 513]
[7] Lei H, Gan C B, Xie C Y 2010 Eng. Mech. 27 105 (in Chinese) [雷华, 甘春标, 谢潮涌 2010 工程力学 27 105]
[8] Rong H W, Wang X D, Xu W, Fang T 2008 Acta Phys. Sin. 57 6888 (in Chinese) [戎海武, 王向东, 徐伟, 方同 2008 物理学报 57 6888]
[9] Su M B, Rong H W 2011 Chin. Phys. B 20 060501
[10] Grebogi C, Ott E, Yorke J A 1983 Physica D 7 181
[11] Grebogi C, Ott E, Yorke J A 1986 Phys. Rev. Lett. 57 1284
[12] Szemplińka-Stupnicka W, Zubrzycki A, Tyrkiel E 1999 Nonlinear Dynam. 19 1936
[13] Feng J X, Xu W 2011 Acta Phys. Sin. 60 080502 (in Chinese) [冯进钤, 徐伟 2011 物理学报 60 080502]
[14] Yue X L, Xu W, Wang L 2013 Commun. Nonlinear Sci. Numer. Simul. 18 3567
[15] Hsu C S 1980 J. Appl. Mech. 147 931
[16] Xu W, Sun C Y, Sun J Q 2013 Adv. Mech. 43 91 (in Chinese) [徐伟, 孙春艳, 孙建桥 2013 力学进展 43 91]
[17] Hong L, Xu J X 1999 Phys. Lett. A 262 361
[18] Xu W, He Q, Rong H W, Fang T 2003 Acta Phys. Sin. 52 1365 (in Chinese) [徐伟, 贺群, 戎海武, 方同 2003 物理学报 52 1365]
[19] Jiang J, Xu J X 1994 Phys. Lett. A 188 137
[20] He Q, Xu W, Li S 2008 Acta Phys. Sin. 57 743 (in Chinese) [贺群, 徐伟, 李爽 2008 物理学报 57 743]
[21] He Q, Xu W, Li S 2008 Acta Phys. Sin. 57 4021 (in Chinese) [贺群, 徐伟, 李爽 2008 物理学报 57 4021]
[22] Li S, He Q 2011 Chin. J. Theor. Appl. Mech. 43 579 (in Chinese) [李爽, 贺群 2011 力学学报 43 579]
[23] Yue X L, Xu W, Zhang Y 2012 Nonlinear Dynam. 69 437
[24] Biot M A 1955 Phys. Rev. 97 14
[25] Wagner N, Adhikari S 2003 AIAA J. 41 951
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