搜索

x

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

三自由度含间隙碰撞振动系统Neimark-Sacker分岔的反控制

伍新 文桂林 徐慧东 何莉萍

引用本文:
Citation:

三自由度含间隙碰撞振动系统Neimark-Sacker分岔的反控制

伍新, 文桂林, 徐慧东, 何莉萍

Anti-controlling Neimark-Sacker bifurcation of a three-degree-of-freedom vibration system with clearance

Wu Xin, Wen Gui-Lin, Xu Hui-Dong, He Li-Ping
PDF
导出引用
  • 分岔反控制作为传统分岔控制的逆问题, 其目的是在预先指定的系统参数点通过控制主动设计出具有所期望特性的分岔解. 以一类三自由度含间隙双面碰撞振动系统为研究对象, 在不改变原系统平衡解结构的前提下, 考虑到在碰撞振动系统反控制过程中由Poincaré映射的隐式特点和传统的映射Neimark-Sacker分岔临界准则带来的困难, 通过对原系统施加线性反馈控制器并利用不直接依赖于特征值计算的Neimark-Sacker分岔显式临界准则研究了此系统的分岔反控制问题. 首先对原系统施加线性反馈控制, 建立闭环控制系统的六维Poincaré映射. 由于六维映射的雅克比矩阵的特征值没有解析的表达式, 利用高维映射Neimark-Sacker分岔的显式临界准则, 获得了系统出现拟周期碰撞振动运动的控制参数区域. 然后采用中心流形-正则形方法分析了拟周期分岔解的稳定性. 数值仿真结果表明本文方法可以在指定的系统参数点通过控制设计出稳定的拟周期碰撞运动.
    Anti-control of bifurcation, as an inverse problem of conventional bifurcation analysis, is aimed at creating a certain bifurcation with desired dynamic properties at a pre-specified system parameter location via control. The main purpose of this paper is to address the problem of anti-control of Neimark-Sacker bifurcation of a three-degree-of-freedom vibro-impact system with clearance (i.e., the second Hopf bifurcation of the original system), which may be viewed as a design approach to creating a quasi-periodic impact motion (or torus solution) at a specified system parameter location via control. Firstly, in the premise of no change of periodic solutions of the original system, when the difficulties that are brought about by the implicit Poincaré map of the vibro-impact system are considered, a linear feedback controller is added to the original system and a six-dimensional Poincaré map of the close-loop control system is established. In order to design a desired bifurcation solution by control, the multiple control gains are used to tune the existence of this bifurcation based on the corresponding critical criterion. However, for six-dimensional map of the vibro-impact system in the paper, the analytical expressions of all eigenvalues of Jacobi matrix with respect to parameters are unavailable. This implies that when the classical critical criterion described by the properties of eigenvalues is used, we have to numerically compute eigenvalues point by point and check their properties to search for the control gains. So, the numerical calculation is a laborious job in the process of determining the control gains. To overcome the difficulty originating from the classical bifurcation criterion, the explicit critical criterion without using eigenvalue calculation of high-dimensional map is used to obtain the controlling parameters area when quasi-periodic impact motion occurs. Then, the stability of quasi-periodic bifurcation solution is analyzed by utilizing the center manifold and normal formal theory. Finally the numerical experiments verify that the stable quasi-periodic impact motion can be generated at a designated system parameter point by the proposed control.
    • 基金项目: 国家杰出青年科学基金(批准号: 11225212)、国家自然科学基金(批准号: 11002052, 11372101) 和湖南省教育厅科研基金(批准号: 12C0627)资助的课题.
    • Funds: Project supported by the National Science Fund for Distinguished Young Scholars of China (Grant No. 11225212), the National Natural Science Foundation of China (Grant Nos. 11002052, 11372101), and the Scientific Research Fundation of the Education Department of Hunan Province, China (Grant No. 12C0627).
    [1]

    Shaw S W, Rand R H 1989 Int. J. Non-linear Mech. 24 41

    [2]

    Hang W H, Chen B, Wang Z L 1994 Recent Developments of General Mechanics (Dynamic, Vibration and Control) (Beijing: Science Press) pp198-204 (in Chinese)[黄文虎, 陈滨, 王照林1994 一般力学(动力学、振动 与控制) 最新进展(北京: 科学出版社) 第198—204 页]

    [3]

    Luo G W, Lv X H, Zhu X F 2008 Int. J. Mech. Sci. 50 214

    [4]

    Jin L, Lu Q S, Wang Q 2005 Appl. Math. Mech. 26 810 (in Chinese) [金俐, 陆启韶, 王琪2005 应用数学和力学26 810]

    [5]

    Zhang H, Ding W C, Li F 2011 Eng. Mech. 28 209 (in Chinese) [张惠, 丁旺才, 李飞2011 工程力学28 209]

    [6]

    Yue Y, Xie J H 2008 J. Sound Vib. 314 228

    [7]

    Li Q H, Tan J Y 2011 Chin. Phys. B 20 040505

    [8]

    Feng J Q, Liu J L 2015 Chaos Soliton. Fract. 73 10

    [9]

    Chai L, Wu X M 2014 J. Xiamen Univ. (Natural Science) 53 508 (in Chinese) [柴林, 吴晓明2014 厦门大学学 报(自然科学版) 53 508]

    [10]

    Chen D S, Wang H O, Chen G R 2001 IEEE Trans. Circ. Syst. I 48 661

    [11]

    Chen Z, Yu P 2005 Chaos Soliton. Fract. 26 1231

    [12]

    Yu S M, Chen G R 2012 Commun. Nonlinear Sci. Numer. Simulat. 17 2617

    [13]

    Liu S H, Tang J S 2008 Acta Phys. Sin. 57 6162 (in Chinese) [刘素华, 唐驾时2008 物理学报57 6162]

    [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]

    Cheng Z S 2010 Neurocomputing 73 3139

    [16]

    Wu Z Q, Sun L M 2011 Acta Phys. Sin. 60 050504 (in Chinese) [吴志强, 孙立明2011 物理学报60 050504]

    [17]

    Wen G L 2005 Phys. Rev. E 72 026201

    [18]

    Luo G W, Xie J H 2004 Periodic Motion and Bifurcation of Vibro-impact System (Beijing: Science Press) pp118-119 (in Chinese) [罗冠炜, 谢建华2004 碰撞振动系 统的周期运动和分岔(北京: 科学出版社) 第118—119 页]

    [19]

    D'Amico M B, Moiola J L, Paolini E E 2003 Dynam. Conti. Dis. Ser. B 10 781

    [20]

    Kuznetsov Y A 1998 Elements of Applied Bifurcation Theory (2nd Ed.) (New York: Springer-Verlag) pp185-187

  • [1]

    Shaw S W, Rand R H 1989 Int. J. Non-linear Mech. 24 41

    [2]

    Hang W H, Chen B, Wang Z L 1994 Recent Developments of General Mechanics (Dynamic, Vibration and Control) (Beijing: Science Press) pp198-204 (in Chinese)[黄文虎, 陈滨, 王照林1994 一般力学(动力学、振动 与控制) 最新进展(北京: 科学出版社) 第198—204 页]

    [3]

    Luo G W, Lv X H, Zhu X F 2008 Int. J. Mech. Sci. 50 214

    [4]

    Jin L, Lu Q S, Wang Q 2005 Appl. Math. Mech. 26 810 (in Chinese) [金俐, 陆启韶, 王琪2005 应用数学和力学26 810]

    [5]

    Zhang H, Ding W C, Li F 2011 Eng. Mech. 28 209 (in Chinese) [张惠, 丁旺才, 李飞2011 工程力学28 209]

    [6]

    Yue Y, Xie J H 2008 J. Sound Vib. 314 228

    [7]

    Li Q H, Tan J Y 2011 Chin. Phys. B 20 040505

    [8]

    Feng J Q, Liu J L 2015 Chaos Soliton. Fract. 73 10

    [9]

    Chai L, Wu X M 2014 J. Xiamen Univ. (Natural Science) 53 508 (in Chinese) [柴林, 吴晓明2014 厦门大学学 报(自然科学版) 53 508]

    [10]

    Chen D S, Wang H O, Chen G R 2001 IEEE Trans. Circ. Syst. I 48 661

    [11]

    Chen Z, Yu P 2005 Chaos Soliton. Fract. 26 1231

    [12]

    Yu S M, Chen G R 2012 Commun. Nonlinear Sci. Numer. Simulat. 17 2617

    [13]

    Liu S H, Tang J S 2008 Acta Phys. Sin. 57 6162 (in Chinese) [刘素华, 唐驾时2008 物理学报57 6162]

    [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]

    Cheng Z S 2010 Neurocomputing 73 3139

    [16]

    Wu Z Q, Sun L M 2011 Acta Phys. Sin. 60 050504 (in Chinese) [吴志强, 孙立明2011 物理学报60 050504]

    [17]

    Wen G L 2005 Phys. Rev. E 72 026201

    [18]

    Luo G W, Xie J H 2004 Periodic Motion and Bifurcation of Vibro-impact System (Beijing: Science Press) pp118-119 (in Chinese) [罗冠炜, 谢建华2004 碰撞振动系 统的周期运动和分岔(北京: 科学出版社) 第118—119 页]

    [19]

    D'Amico M B, Moiola J L, Paolini E E 2003 Dynam. Conti. Dis. Ser. B 10 781

    [20]

    Kuznetsov Y A 1998 Elements of Applied Bifurcation Theory (2nd Ed.) (New York: Springer-Verlag) pp185-187

  • [1] 王海峰, 李旺, 顾国彪, 沈俊, 滕启治. 风力发电机自循环蒸发内冷系统稳定性的研究. 物理学报, 2016, 65(3): 030501. doi: 10.7498/aps.65.030501
    [2] 徐伟, 杨贵东, 岳晓乐. 随机参激下Duffing-Rayleigh碰撞振动系统的P-分岔分析. 物理学报, 2016, 65(21): 210501. doi: 10.7498/aps.65.210501
    [3] 朱霖河, 赵洪涌. 时滞惯性神经网络的稳定性和分岔控制. 物理学报, 2014, 63(9): 090203. doi: 10.7498/aps.63.090203
    [4] 鲁延玲, 蒋国平, 宋玉蓉. 自适应网络中病毒传播的稳定性和分岔行为研究. 物理学报, 2013, 62(13): 130202. doi: 10.7498/aps.62.130202
    [5] 张希, 包伯成, 王金平, 马正华, 许建平. 固定关断时间控制Buck变换器输出电容等效串联电阻的稳定性分析. 物理学报, 2012, 61(16): 160503. doi: 10.7498/aps.61.160503
    [6] 赵洪涌, 陈凌, 于小红. 一类惯性神经网络的分岔与控制. 物理学报, 2011, 60(7): 070202. doi: 10.7498/aps.60.070202
    [7] 陈海军, 李高清, 薛具奎. 变分法研究一维Bose-Fermi系统的稳定性. 物理学报, 2011, 60(4): 040304. doi: 10.7498/aps.60.040304.1
    [8] 唐春森, 孙跃, 戴欣, 王智慧, 苏玉刚, 呼爱国. 感应电能传输系统多谐振点及其自治振荡稳定性分析. 物理学报, 2011, 60(4): 048401. doi: 10.7498/aps.60.048401
    [9] 黄丽莲, 何少杰. 分数阶状态空间系统的稳定性分析及其在分数阶混沌控制中的应用. 物理学报, 2011, 60(4): 044703. doi: 10.7498/aps.60.044703
    [10] 张琪昌, 李伟义, 王炜. 斜拉索风雨振的静态分岔研究. 物理学报, 2010, 59(2): 729-734. doi: 10.7498/aps.59.729
    [11] 刘浩然, 朱占龙, 时培明. 一类相对转动时滞非线性动力系统的稳定性分析. 物理学报, 2010, 59(10): 6770-6777. doi: 10.7498/aps.59.6770
    [12] 刘爽, 刘彬, 张业宽, 闻岩. 一类时滞非线性相对转动系统的Hopf分岔与周期解的稳定性. 物理学报, 2010, 59(1): 38-43. doi: 10.7498/aps.59.38
    [13] 何学军, 张良欣, 任爱娣. 横向补给系统高架索的稳定性与分岔研究. 物理学报, 2010, 59(5): 3088-3092. doi: 10.7498/aps.59.3088
    [14] 时培明, 蒋金水, 刘彬. 耦合相对转动非线性动力系统的稳定性与近似解. 物理学报, 2009, 58(4): 2147-2154. doi: 10.7498/aps.58.2147
    [15] 孟 宗, 刘 彬. 一类非线性相对转动动力系统的平衡稳定性及组合谐波近似解. 物理学报, 2008, 57(3): 1329-1334. doi: 10.7498/aps.57.1329
    [16] 时培明, 刘 彬, 刘 爽. 一类谐波激励相对转动非线性动力系统的稳定性与近似解. 物理学报, 2008, 57(8): 4675-4684. doi: 10.7498/aps.57.4675
    [17] 刘素华, 唐驾时. 四维Qi系统零平衡点的Hopf分岔反控制. 物理学报, 2008, 57(10): 6162-6168. doi: 10.7498/aps.57.6162
    [18] 王作雷. 一类简化Lang-Kobayashi方程的Hopf分岔及其稳定性. 物理学报, 2008, 57(8): 4771-4776. doi: 10.7498/aps.57.4771
    [19] 王 坤. 二端面转轴相对转动非线性动力学系统的稳定性与近似解. 物理学报, 2005, 54(12): 5530-5533. doi: 10.7498/aps.54.5530
    [20] 张 凯, 冯 俊. 相对论Birkhoff系统的对称性与稳定性. 物理学报, 2005, 54(7): 2985-2989. doi: 10.7498/aps.54.2985
计量
  • 文章访问数:  6110
  • PDF下载量:  256
  • 被引次数: 0
出版历程
  • 收稿日期:  2015-04-13
  • 修回日期:  2015-05-22
  • 刊出日期:  2015-10-05

/

返回文章
返回