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Piecewise nonlinear constraint exists in various fields and it always affects the stability of a system. In order to realize the dynamic characteristic of the system constrained by these nonlinearity, we consider two kinds of typical piecewise nonlinear constraints under the dynamic conditions, and establish a dynamic model with double piecewise nonlinear constraint of elasticity and damping, according to the generalized dissipation Lagrange principle. An average method is used to solve the amplitude and frequency response of the system under a periodic external incentive. By a numerical simulation, we compare the time domain responses under different piecewise nonlinear elastic constraints. The results show that the stronger the piecewise nonlinear elastic constraint, the more obvious the piecewise nonlinear damping constraint is. We also compare the bifurcation responses under different piecewise nonlinear damping constraints, the results show that the chaos state will emerge in an enlarged scope with the increase of the piecewise nonlinear damping coefficient, and threaten the stability of the system. The dynamic evolution process of the system is shown by the phase diagrams and Poincaré sections under the corresponding constraint conditions. By comparing the amplitude-frequency characteristics of the system under different constraint conditions, we obtain the response characteristic of the system and its change rule with the piecewise nonlinear constraints. By comparing and analyzing the amplitude-frequency characteristics under the piecewise nonlinear elastic and piecewise nonlinear damping constraint, we obtain the law of system stability influenced by different nonlinear factors, and the interaction relationship between the two piecewise nonlinear constraints.
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Keywords:
- piecewise nonlinear /
- double constraint system /
- cycle response /
- stability
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[2] Wang C J, Yang K L, Qu S X 2013 Chin. Phys. B 22 030502
[3] Lin P, Qin K Y, Wu H Y 2011 Chin. Phys. B 20 108701
[4] Quentin B, Tetsushi U, Danièle F P, Takuji K 2009 Chaos Solitons Fract. 42 187
[5] Jiang H B, Li T, Zeng X L, Zhang L P 2014 Chin. Phys. B 23 010501
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[8] Zhang C, Yu Y, Han X J, Bi Q S 2012 Chin. Phys. B 21 100501
[9] Xu L, Lu M W, Cao Q 2003 J. Sound Vib. 264 873
[10] Xu L, Lu M W, Cao Q 2002 Phys. Lett. A 301 65
[11] Jiang J, Gao W H 2013 Chin. J. Theor. Appl. Mech. 45 16 (in Chinese) [江俊, 高文辉 2013 力学学报 45 16]
[12] Akhavan A, Samsudin A, Akhshani A 2009 Chaos Solitons Fract. 42 1046
[13] Partha S D, Soma D, Soumitro B, Akhil R R 2009 Phys. Lett. A 373 4426
[14] Zhang L M, Zhang J W, Wu R H 2014 Acta Phys. Sin. 63 160505 (in Chinese) [张玲梅, 张建文, 吴润衡 2014 物理学报 63 160505]
[15] Zachary P K, Paul C B 2010 Physica D 239 1048
[16] Jia Q F, Yu W, Liu X J, Wang D J 2004 Chin. J. Theor. Appl. Mech. 36 373 (in Chinese) [贾启芬, 于雯, 刘习军, 王大钧 2004 力学学报 36 373]
[17] Ji J C, Hansen C H 2005 J. Sound Vib. 283 467
[18] Simpson D J W, Meiss J D 2012 Physica D 241 1861
[19] Li X J, Yan J, Chen X Q, Cao Y 2014 Acta Phys. Sin. 63 200202 (in Chinese) [李晓静, 严静, 陈绚青, 曹毅 2014 物理学报 63 200202]
[20] Xuan B T, Nur H, Hideki Y 2012 Mechatronics 22 65
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[1] Liang F, Han M A, Valery G R 2012 Nonlinear Anal.-Theor. 75 4355
[2] Wang C J, Yang K L, Qu S X 2013 Chin. Phys. B 22 030502
[3] Lin P, Qin K Y, Wu H Y 2011 Chin. Phys. B 20 108701
[4] Quentin B, Tetsushi U, Danièle F P, Takuji K 2009 Chaos Solitons Fract. 42 187
[5] Jiang H B, Li T, Zeng X L, Zhang L P 2014 Chin. Phys. B 23 010501
[6] Wang L Z, Zhao W L, Chen X 2012 Acta Phys. Sin. 61 160501 (in Chinese) [王林泽, 赵文礼, 陈旋 2012 物理学报 61 160501]
[7] Zhang Y, Bi Q S 2011 Chin. Phys. B 20 010504
[8] Zhang C, Yu Y, Han X J, Bi Q S 2012 Chin. Phys. B 21 100501
[9] Xu L, Lu M W, Cao Q 2003 J. Sound Vib. 264 873
[10] Xu L, Lu M W, Cao Q 2002 Phys. Lett. A 301 65
[11] Jiang J, Gao W H 2013 Chin. J. Theor. Appl. Mech. 45 16 (in Chinese) [江俊, 高文辉 2013 力学学报 45 16]
[12] Akhavan A, Samsudin A, Akhshani A 2009 Chaos Solitons Fract. 42 1046
[13] Partha S D, Soma D, Soumitro B, Akhil R R 2009 Phys. Lett. A 373 4426
[14] Zhang L M, Zhang J W, Wu R H 2014 Acta Phys. Sin. 63 160505 (in Chinese) [张玲梅, 张建文, 吴润衡 2014 物理学报 63 160505]
[15] Zachary P K, Paul C B 2010 Physica D 239 1048
[16] Jia Q F, Yu W, Liu X J, Wang D J 2004 Chin. J. Theor. Appl. Mech. 36 373 (in Chinese) [贾启芬, 于雯, 刘习军, 王大钧 2004 力学学报 36 373]
[17] Ji J C, Hansen C H 2005 J. Sound Vib. 283 467
[18] Simpson D J W, Meiss J D 2012 Physica D 241 1861
[19] Li X J, Yan J, Chen X Q, Cao Y 2014 Acta Phys. Sin. 63 200202 (in Chinese) [李晓静, 严静, 陈绚青, 曹毅 2014 物理学报 63 200202]
[20] Xuan B T, Nur H, Hideki Y 2012 Mechatronics 22 65
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