一类非线性相对转动系统的组合谐波分岔行为研究

孟宗; 付立元; 宋明厚

doi:10.7498/aps.62.054501

摘要

针对一类具有非线性刚度、非线性阻尼的非线性相对转动系统, 应用耗散系统的拉格朗日原理建立在组合谐波激励作用下非线性相对转动系统的动力学方程. 构造李雅普诺夫函数, 分析相对转动系统的稳定性, 研究自治系统的分岔特性. 应用多尺度法求解相对转动系统的非自治系统在组合激励作用下的分岔响应方程. 最后采用数值仿真方法, 通过分岔图、时域波形、相平面图、Poincaré截面图等研究外扰激励、系统阻尼、非线性刚度对相对转动系统经历倍周期分岔进入混沌运动的影响.

关键词:

相对转动 /
组合激励 /
分岔 /
混沌

Abstract

Using the Lagrange principle of dissipative system, the nonlinear dynamic equation of a relative rotation with combined harmonic excitation is established, which contains nonlinear stiffness and nonlinear damping. The stability and bifurcation characteristics of autonomous system are analyzed by constructing Lyapunov function. Bifurcation response equation of non-autonomous system under the combined harmonic excitation is obtained by the method of multiple scale. Finally, numerical method is employed to analyze the effects of external excitation, system damping and nonlinear stiffness on the process that the system enter into chaos motion via period-doubling bifurcation by bifurcation diagram, time domain waveform, phase trajectory and Poincaré map.

Keywords:

作者及机构信息

1.
河北省测试计量技术及仪器重点实验室(燕山大学), 秦皇岛 066004;

2.
河北省工业计算机控制工程重点实验室(燕山大学), 秦皇岛 066004

基金项目: 国家自然科学基金 (批准号: 51105323) 和河北省自然科学基金 (批准号: E2012203166) 资助的课题.

Authors and contacts

1.
Key Laboratory of Measurement Technology and Instrumentation of Hebei Province, Yanshan University, Qinhuangdao 066004, China;

2.
Key Laboratory of Industrial Computer Control Engineering of Hebei Province, Yanshan University, Qinhuangdao 066004, China

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 51105323), and the Natural Science Foundation of Hebei Province, China (Grant No. E2012203166).

参考文献

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施引文献

[1]	Carmeli M 1985 Found. Phys. 15 175
[2]	Carmeli M 1986 Inter. J. Theor. Phys. 25 89
[3]	Luo S K 1996 J. Beijing Inst. Technol. 16 (S1) 154 (in Chinese) [罗绍凯 1996 北京理工大学学报 16 (S1) 154]
[4]	Luo S K 1996 Appl. Math. Mech. 17 683
[5]	Luo S K 1998 Appl. Math. Mech. 19 45
[6]	Luo S K 2003 Acta Phys. Sin. 52 2941 (in Chinese) [罗绍凯 2003 textbf52 2941]
[7]	Luo S K 2004 Acta Phys. Sin. 53 5 (in Chinese) [罗绍凯 2004 53 5]
[8]	Jiang W A, Luo S K 2011 Acta Phys. Sin. 60 060201 (in Chinese) [姜文安, 罗绍凯 2011 60 060201]
[9]	Luo S K, Fu J L, Chen X W 2001 Acta Phys. Sin. 50 383 (in Chinese) [罗绍凯, 傅景礼, 陈向炜 2001 50 383]
[10]	Luo S K, Guo Y X, Chen X W, Fu J L 2001 Acta Phys. Sin. 50 2049 (in Chinese) [罗绍凯, 郭永新, 陈向炜, 傅景礼 2001 50 2049]
[11]	Fu J L, Chen L Q, Luo S K, Chen X W, Wang X M 2001 Acta Phys. Sin. 50 2289 (in Chinese) [傅景礼, 陈立群, 罗绍凯, 陈向炜, 王新民 2001 50 2289]
[12]	Fu J L, Chen L Q, Xue Y, Luo S K 2002 Acta Phys. Sin. 51 2683 (in Chinese) [傅景礼, 陈立群, 薛纭, 罗绍凯 2002 51 2683]
[13]	Chen X W, Luo S K, Mei F X 2002 Appl. Math. Mech. 23 47
[14]	Guo Y X, Shang M, Luo S K 2003 Appl. Math. Mech. 24 62
[15]	Luo S K 2003 Appl. Math. Mech. 24 414
[16]	Chen X W, Luo S K, Mei F X 2000 Acta Mech. Solid. Sin. 21 251 (in Chinese) [陈向炜, 罗绍凯, 梅凤翔 2000 固体力学学报 21 251]
[17]	Ma C, Wang X Y 2011 Communications in Nonlinear Science and Numerical Simulation 17 721
[18]	Wang X Y, Liang Q Y, Meng J 2008 International Journal of Modern Physics C 19 1389
[19]	Di G H, Xu Y, Xu W, Gu R C 2011 Acta Phys. Sin. 60 020504 (in Chinese) [狄根虎, 许勇, 徐伟, 顾仁财 2011 60 020504]
[20]	Li Q H, Yan Y L, Yang D 2012 Acta Phys. Sin. 61 200505 (in Chinese) [李群宏, 闫玉龙, 杨丹 2012 61 200505]
[21]	Liu Y Z, Lin C S, Li X C 2011 Acta Phys. Sin. 60 060507 (in Chinese) [刘扬正, 林长圣, 李心朝 2011 60 060507]
[22]	Zhang Z X, Yu S M 2010 Acta Phys. Sin. 59 3017 (in Chinese) [张朝霞, 禹思敏 2010 59 3017]
[23]	Ding L, Xia G Q, Shen J T, Li N Y, Wu Z M 2011 Acta Phys. Sin. 60 014210 (in Chinese) [丁灵, 吴加贵, 夏光琼, 李能尧, 吴正茂 2011 60 014210]
[24]	Wang M J, Wang X Y 2009 Acta Phys. Sin. 58 1467 (in Chinese) [王明军, 王兴元 2009 58 1467]
[25]	Yan S L 2008 Acta Phys. Sin. 57 2819 (in Chinese) [颜森林 2008 57 2819]
[26]	Wang X Y, Liang Q Y 2008 Communications in Nonlinear Science and Numerical Simulation 13 913
[27]	Liu Z H, Zhang Y J, Zhang J, Wu J H 2011 Acta Phys. Sin. 60 019501 (in Chinese) [刘朝华, 张英杰, 章兢, 吴建辉 2011 60 019501]
[28]	Gu H G, Zhu Z, Jia B 2011 Acta Phys. Sin. 60 100505 (in Chinese) [古华光, 朱洲, 贾冰 2011 60 100505]
[29]	Zhong M, Tang G N 2010 Acta Phys. Sin. 59 3070 (in Chinese) [钟敏, 唐国宁 2010 59 3070]
[30]	Luo S K, Guo Y X, Chen X W 2001 Acta Phys. Sin. 50 2053 (in Chinese) [罗绍凯, 郭永新, 陈向炜 2001 50 2053]
[31]	Luo S K 2007 Acta Phys. Sin. 56 5580 (in Chinese) [罗绍凯 2007 56 5580]
[32]	Jia L Q, Xie Y L, Luo S K 2011 Acta Phys. Sin. 60 040201 (in Chinese) [贾利群, 解银丽, 罗绍凯 2011 60 040201]
[33]	Dong Q L, Liu B 2002 Acta Phys. Sin. 51 2191 (in Chinese) [董全林, 刘彬 2002 51 2191]
[34]	Zhao W, Liu B, Shi P M, Jiang J S 2005 Acta Phys. Sin. 54 3852 (in Chinese) [赵武, 刘彬, 时培明, 蒋金水 2005 54 3852]
[35]	Liu S, Liu B, Zhang Y K, Wen Y 2010 Acta Phys. Sin. 59 38 (in Chinese) [刘爽, 刘彬, 张业宽, 闻岩 2010 59 38]
[36]	Shi P M, Li J Z, Liu B, Han D Y 2011 Acta Phys. Sin. 60 094501 (in Chinese) [时培明, 李纪召, 刘彬, 韩东颖 2011 60 094501]
[37]	Wang Y Z, Zhou Y Z 2011 Chin. Phys. B 20 040501
[38]	Meng Z, Liu B 2007 Acta Phys. Sin. 56 6194 (in Chinese) [孟宗, 刘彬 2007 56 6194]
[39]	Liu H R, Zhu Z L, Shi P M 2010 Acta Phys. Sin. 59 6770 (in Chinese) [刘浩然, 朱占龙, 时培明 2010 59 6770]
[40]	Wang K, Guan X P, Qiao J M 2010 Acta Phys. Sin. 59 3648 (in Chinese) [王坤, 关新平, 乔杰敏 2010 59 3648]
[41]	Mo J Q, Cheng R J, Ge H X 2011 Acta Phys. Sin. 60 040203 (in Chinese) [莫嘉琪, 程荣军, 葛红霞 2011 60 040203]
[42]	Li X J, Chen X Q 2012 Acta Phys. Sin. 61 210201 (in Chinese) [李晓静, 陈绚青 2012 61 210201]
[43]	Meng Z, Liu B 2008 Acta Phys. Sin. 57 1329 (in Chinese) [孟宗, 刘彬 2008 57 1329]
[44]	Li H B, Wang B H, Zhang Z Q, Liu S, Li Y S 2012 Acta Phys. Sin. 61 094501 (in Chinese) [李海滨, 王博华, 张志强, 刘爽, 李延树 2012 61 094501]
[45]	Tang R R 2012 Acta Phys. Sin. 61 200201 (in Chinese) [唐荣荣 2012 61 200201]

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