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Systems with double-loop hysteresis are used increasingly in engineering, but few studies on their dynamics are reported. In this study, the bifurcation characteristics of the primary resonance of a system with double-loop bilinear hysteresis are investigated on the background of a shape memory alloy damper. First, the frequency-amplitude response equation is obtained by using the averaging methods. Then, the influences of the temperature and the amplitude of excitation on amplitude-frequency responses are analyzed by the constrained bifurcation singularity analysis method of non-smooth systems. The calculation results show that the parameter space composed of the temperature and the amplitude of excitation can be divided into 11 regions, which suggest that there are 11 qualitatively different kinds of amplitude-frequency responses to the variation of two parameters. In order to describe and compare the frequency-amplitude response curves conveniently, an encoding rule is proposed to describe their jump phenomena as the frequency sweeps. The above results can guide directly the design of frequency response mode of the system.
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Keywords:
- bilinearity /
- hysteresis /
- constrained bifurcation
[1] Zhang W, Hu H Y 2009 New Development of Nonlinear Dynamic Theory and Applications (Beijing: Science Press) p142 (in Chinese)[张 伟、胡海岩 2009 非线性动力学理论和应用的新进展 (北京:科学出版社) 第142页]
[2] [3] Li G J, Xu W, Wang L, Feng J Q 2008 Acta Phys. Sin. 57 2107 (in Chinese)[李高杰、徐 伟、王 亮、冯进钤 2008 物理学报 57 2107]
[4] Li S H, Yang S P 2006 J. Dynam. Contr. 4 8 (in Chinese) [李韶华、杨绍普 2006 动力学与控制学报 4 8]
[5] [6] Wu Z Q, Yu P, Wang K Q 2004 Int. J. Bifur. Chaos 14 2825
[7] [8] Chen Z, Wu Z Q, Yu P 2005 J. Sound Vib. 284 783
[9] [10] Huang C T, Kuo S Y 2006 Int. J. Nonlin. Mech. 41 888
[11] [12] [13] Fragiacomo M, Rajgelj S, Cimadom F 2003 Earthq. Eng. Struct. Dyn. 32 1333
[14] Katsaras C P, Panagiotakos T B, Kolias B 2008 Earthq. Eng. Struct. Dyn. 37 557
[15] [16] Christopoulos C 2004 J. Eng. Mech. ASCE 130 894
[17] [18] Lau S M L, Lau K T, Yin Y S, Li L, Wong M, Chan K, Chen W 2010 Mater. Manuf. Process. 25 281
[19] [20] [21] Williams K A, Chiu G T C, Bernhard R J 2005 J. Sound Vib. 280 211
[22] Rustighi E, Brennan M J, Mace B R 2005 Smart Mater. Struct. 4 19
[23] [24] Chou C C, Tsai K C, Yang W C 2009 Earthq. Eng. Struct. Dyn. 38 403
[25] [26] Stanton J F, Stone W C, Cheok G S 1997 PCI J. 42 20
[27] [28] Priestley M J N, Sritharan S, Conley J R, Pampanin S 1999 PCI J. 44 42
[29] [30] [31] Christopoulos C, Tremblay R, Kim H J, Lacerte M 2008 J. Struct. Eng. ASCE 134 96
[32] [33] Li H G, Zhang J W, Wen B C 2002 Mech. Res. Commun. 29 283
[34] [35] Motahari S A, Ghassemieh M 2007 Eng. Struct. 29 904
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[1] Zhang W, Hu H Y 2009 New Development of Nonlinear Dynamic Theory and Applications (Beijing: Science Press) p142 (in Chinese)[张 伟、胡海岩 2009 非线性动力学理论和应用的新进展 (北京:科学出版社) 第142页]
[2] [3] Li G J, Xu W, Wang L, Feng J Q 2008 Acta Phys. Sin. 57 2107 (in Chinese)[李高杰、徐 伟、王 亮、冯进钤 2008 物理学报 57 2107]
[4] Li S H, Yang S P 2006 J. Dynam. Contr. 4 8 (in Chinese) [李韶华、杨绍普 2006 动力学与控制学报 4 8]
[5] [6] Wu Z Q, Yu P, Wang K Q 2004 Int. J. Bifur. Chaos 14 2825
[7] [8] Chen Z, Wu Z Q, Yu P 2005 J. Sound Vib. 284 783
[9] [10] Huang C T, Kuo S Y 2006 Int. J. Nonlin. Mech. 41 888
[11] [12] [13] Fragiacomo M, Rajgelj S, Cimadom F 2003 Earthq. Eng. Struct. Dyn. 32 1333
[14] Katsaras C P, Panagiotakos T B, Kolias B 2008 Earthq. Eng. Struct. Dyn. 37 557
[15] [16] Christopoulos C 2004 J. Eng. Mech. ASCE 130 894
[17] [18] Lau S M L, Lau K T, Yin Y S, Li L, Wong M, Chan K, Chen W 2010 Mater. Manuf. Process. 25 281
[19] [20] [21] Williams K A, Chiu G T C, Bernhard R J 2005 J. Sound Vib. 280 211
[22] Rustighi E, Brennan M J, Mace B R 2005 Smart Mater. Struct. 4 19
[23] [24] Chou C C, Tsai K C, Yang W C 2009 Earthq. Eng. Struct. Dyn. 38 403
[25] [26] Stanton J F, Stone W C, Cheok G S 1997 PCI J. 42 20
[27] [28] Priestley M J N, Sritharan S, Conley J R, Pampanin S 1999 PCI J. 44 42
[29] [30] [31] Christopoulos C, Tremblay R, Kim H J, Lacerte M 2008 J. Struct. Eng. ASCE 134 96
[32] [33] Li H G, Zhang J W, Wen B C 2002 Mech. Res. Commun. 29 283
[34] [35] Motahari S A, Ghassemieh M 2007 Eng. Struct. 29 904
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