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

分数阶时滞反馈对Duffing振子动力学特性的影响

温少芳 申永军 杨绍普

引用本文:
Citation:

分数阶时滞反馈对Duffing振子动力学特性的影响

温少芳, 申永军, 杨绍普

Dynamical analysis of Duffing oscillator with fractional-order feedback with time delay

Wen Shao-Fang, Shen Yong-Jun, Yang Shao-Pu
PDF
导出引用
  • 研究了含分数阶时滞耦合反馈的Duffing自治系统, 通过平均法得到了系统周期解的一阶近似解析形式, 定义了以反馈系数、分数阶阶次、时滞参数表示的等效刚度和等效阻尼系数, 发现分数阶时滞耦合反馈同时具有速度时滞反馈和位移时滞反馈的作用. 比较了三种参数条件下近似解析解与数值积分的结果, 二者的吻合精度都很高, 证明了近似解析解的正确性和准确性. 分析了反馈系数、分数阶阶次和非线性刚度系数等参数对系统分岔点、周期解稳定性、周期解的存在范围、零解的稳定性以及稳定性切换次数等系统动力学特性的影响.
    With increasingly strict requirements for control speed and system performance, the unavoidable time delay becomes a serious problem. Fractional-order feedback is constantly adopted in control engineering due to its advantages, such as robustness, strong de-noising ability and better control performance. In this paper, the dynamical characteristics of an autonomous Duffing oscillator under fractional-order feedback coupling with time delay are investigated. At first, the first-order approximate analytical solution is obtained by the averaging method. The equivalent stiffness and equivalent damping coefficients are defined by the feedback coefficient, fractional order and time delay. It is found that the fractional-order feedback coupling with time delay has the functions of both delayed velocity feedback and delayed displacement feedback simultaneously. Then, the comparison between the analytical solution and the numerical one verifies the correctness and satisfactory precision of the approximately analytical solution under three parameter conditions respectively. The effects of the feedback coefficient, fractional order and nonlinear stiffness coefficient on the complex dynamical behaviors are analyzed, including the locations of bifurcation points, the stabilities of the periodic solutions, the existence ranges of the periodic solutions, the stability of zero solution and the stability switch times. It is found that the increase of fractional order could make the delay-amplitude curves of periodic solutions shift rightwards, but the stabilities of the periodic solutions and the stability switch times of zero solution cannot be changed. The decrease of the feedback coefficient makes the amplitudes and ranges of the periodic solutions become larger, and induces the stability switch times of zero solution to decrease, but the stabilities of the periodic solutions keep unchanged. The sign of the nonlinear stiffness coefficient determines the stabilities and the bending directions of delay-amplitude curves of periodic solutions, but the bifurcation points, the stability of zero solution and the stability switch times are not changed. It could be concluded that the primary system parameters have important influences on the dynamical behavior of Duffing oscillator, and the results are very helpful to design, analyze or control this kind of system. The analysis procedure and conclusions could provide a reference for the study on the similar fractional-order dynamic systems with time delays.
      通信作者: 申永军, shenyongjun@126.com
    • 基金项目: 国家自然科学基金(批准号: 11372198)、河北省高等学校创新团队领军人才计划(批准号: LJRC018)、河北省高等学校高层次人才科学研究项目(批准号: GCC2014053)和河北省高层次人才资助项目(批准号: A201401001)资助的课题.
      Corresponding author: Shen Yong-Jun, shenyongjun@126.com
    • Funds: Project supported by National Natural Science Foundation of China (Grant No. 11372198), the Cultivation plan for Innovation Team and Leading Talent in Colleges and Universities of Hebei Province, China (Grant No. LJRC018), the Program for Advanced Talent in the Universities of Hebei Province, China (Grant No. GCC2014053), and the Program for Advanced Talent in Hebei Province, China (Grant No. A201401001).
    [1]

    Gorenflo R, Abdel-Rehim E A 2007 J. Comput. Appl. Math. 205 871

    [2]

    Jumarie G 2006 Comput. Math. Appl. 51 1367

    [3]

    Ishteva M, Scherer R, Boyadjiev L 2005 Math. Sci. Res. J. 2005 9 161

    [4]

    Agnieszka B M, Delfim F M T 2011 Fract. Calc. Appl. Anal. 14 523

    [5]

    Leung A Y T, Guo Z J, Yang H X 2012 J. Sound Vib. 331 1115

    [6]

    Yang J H, Zhu H 2013 Acta Phys. Sin. 62 024501 (in Chinese) [杨建华, 朱华 2013 物理学报 62 024501]

    [7]

    Zhou Y, Ionescu C, Machado J A T 2015 Nonlinear Dyn. 80 1661

    [8]

    Wang Z H, Hu H Y 2009 Sci. China G: Phys. Mech. Astron. 39 1495 (in Chinese) [王在华, 胡海岩 2009 中国科学G辑: 物理学 力学 天文学 39 1495]

    [9]

    Wang Z H, Du M L 2011 Shock Vib. 18 257

    [10]

    Shen Y J, Wei P, Yang S P 2014 Nonlinear Dyn. 77 1629

    [11]

    Shen Y J, Yang S P, Xing H J 2012 Acta Phys. Sin. 61 150503 (in Chinese) [申永军, 杨绍普, 邢海军 2012 物理学报 61 150503]

    [12]

    Shen Y J, Yang S P, Xing H J 2012 Acta Phys. Sin. 61 110505 (in Chinese) [申永军, 杨绍普, 邢海军 2012 物理学报 61 110505]

    [13]

    Shen Y J, Yang S P, Xing H J, Gao G S 2012 Commun. Nolinear Sci. 17 3092

    [14]

    Shen Y J, Yang S P, Xing H J, Ma H X 2012 Int. J. Nonlin. Mech. 47 975

    [15]

    Li C P, Deng W H 2007 Appl. Math. Comput. 187 777

    [16]

    Deng W H, Li C P 2008 Phys. Lett. 372 401

    [17]

    Li Q D, Chen S, Zhou P 2011 Chin. Phys. B 20 010502

    [18]

    Chen L C, Hu F, Zhu W Q 2013 Fract. Calc. Appl. Anal. 05 189

    [19]

    Wahi P, Chatterjee A 2004 Nonlinear Dyn. 38 3

    [20]

    Yin H, Chen N 2012 Chin. J. Comput. Mech. 29 966 (in Chinese) [银花, 陈宁 2012 计算力学学报 29 966]

    [21]

    Xu Y, Li Y G, Liu D 2013 Nonlinear Dyn. 74 745

    [22]

    Zhang R X, Yang S P 2009 Chin. Phys. B 18 3295

    [23]

    Zhang R R, Xu W, Yang G D 2015 Chin. Phys. B 24 020204

    [24]

    Hale J K, Lunel S M V 1993 Introduction to Functional Differential Equations (New York: Springer-Verlag) p6

    [25]

    Hu H Y, Wang Z H 2002 Dynamics of Controlled Mechanical Systems with Delayed Feedback (Berlin: Springer) p213

    [26]

    Wang Z H, Hu H Y 2000 J. Sound Vib. 233 215

    [27]

    Wang H L, Hu H Y 2003 Nonlinear Dyn. 33 379

    [28]

    Shi M, Wang Z H 2011 Automatica 47 2001

    [29]

    Babakhani A, Baleanu D, Khanbabaie R 2012 Nonlinear Dyn. 69 721

    [30]

    elik V, Demir Y 2014 Signal Image Video P. 8 65

    [31]

    Petras I 2011 Fractional-order Nonlinear Systems: Modeling, Analysis and Simulation (Beijing: Higher Education Press) p19

  • [1]

    Gorenflo R, Abdel-Rehim E A 2007 J. Comput. Appl. Math. 205 871

    [2]

    Jumarie G 2006 Comput. Math. Appl. 51 1367

    [3]

    Ishteva M, Scherer R, Boyadjiev L 2005 Math. Sci. Res. J. 2005 9 161

    [4]

    Agnieszka B M, Delfim F M T 2011 Fract. Calc. Appl. Anal. 14 523

    [5]

    Leung A Y T, Guo Z J, Yang H X 2012 J. Sound Vib. 331 1115

    [6]

    Yang J H, Zhu H 2013 Acta Phys. Sin. 62 024501 (in Chinese) [杨建华, 朱华 2013 物理学报 62 024501]

    [7]

    Zhou Y, Ionescu C, Machado J A T 2015 Nonlinear Dyn. 80 1661

    [8]

    Wang Z H, Hu H Y 2009 Sci. China G: Phys. Mech. Astron. 39 1495 (in Chinese) [王在华, 胡海岩 2009 中国科学G辑: 物理学 力学 天文学 39 1495]

    [9]

    Wang Z H, Du M L 2011 Shock Vib. 18 257

    [10]

    Shen Y J, Wei P, Yang S P 2014 Nonlinear Dyn. 77 1629

    [11]

    Shen Y J, Yang S P, Xing H J 2012 Acta Phys. Sin. 61 150503 (in Chinese) [申永军, 杨绍普, 邢海军 2012 物理学报 61 150503]

    [12]

    Shen Y J, Yang S P, Xing H J 2012 Acta Phys. Sin. 61 110505 (in Chinese) [申永军, 杨绍普, 邢海军 2012 物理学报 61 110505]

    [13]

    Shen Y J, Yang S P, Xing H J, Gao G S 2012 Commun. Nolinear Sci. 17 3092

    [14]

    Shen Y J, Yang S P, Xing H J, Ma H X 2012 Int. J. Nonlin. Mech. 47 975

    [15]

    Li C P, Deng W H 2007 Appl. Math. Comput. 187 777

    [16]

    Deng W H, Li C P 2008 Phys. Lett. 372 401

    [17]

    Li Q D, Chen S, Zhou P 2011 Chin. Phys. B 20 010502

    [18]

    Chen L C, Hu F, Zhu W Q 2013 Fract. Calc. Appl. Anal. 05 189

    [19]

    Wahi P, Chatterjee A 2004 Nonlinear Dyn. 38 3

    [20]

    Yin H, Chen N 2012 Chin. J. Comput. Mech. 29 966 (in Chinese) [银花, 陈宁 2012 计算力学学报 29 966]

    [21]

    Xu Y, Li Y G, Liu D 2013 Nonlinear Dyn. 74 745

    [22]

    Zhang R X, Yang S P 2009 Chin. Phys. B 18 3295

    [23]

    Zhang R R, Xu W, Yang G D 2015 Chin. Phys. B 24 020204

    [24]

    Hale J K, Lunel S M V 1993 Introduction to Functional Differential Equations (New York: Springer-Verlag) p6

    [25]

    Hu H Y, Wang Z H 2002 Dynamics of Controlled Mechanical Systems with Delayed Feedback (Berlin: Springer) p213

    [26]

    Wang Z H, Hu H Y 2000 J. Sound Vib. 233 215

    [27]

    Wang H L, Hu H Y 2003 Nonlinear Dyn. 33 379

    [28]

    Shi M, Wang Z H 2011 Automatica 47 2001

    [29]

    Babakhani A, Baleanu D, Khanbabaie R 2012 Nonlinear Dyn. 69 721

    [30]

    elik V, Demir Y 2014 Signal Image Video P. 8 65

    [31]

    Petras I 2011 Fractional-order Nonlinear Systems: Modeling, Analysis and Simulation (Beijing: Higher Education Press) p19

  • [1] 倪龙, 陈晓. 基于频散补偿和分数阶微分的多模式兰姆波分离. 物理学报, 2018, 67(20): 204301. doi: 10.7498/aps.67.20180561
    [2] 高飞, 胡道楠, 童恒庆, 王传美. 分数阶Willis环脑迟发性动脉瘤时滞系统混沌分析. 物理学报, 2018, 67(15): 150501. doi: 10.7498/aps.67.20180262
    [3] 陈晓, 汪陈龙. 基于赛利斯模型和分数阶微分的兰姆波信号消噪. 物理学报, 2014, 63(18): 184301. doi: 10.7498/aps.63.184301
    [4] 张路, 谢天婷, 罗懋康. 双频信号驱动含分数阶内、外阻尼Duffing振子的振动共振. 物理学报, 2014, 63(1): 010506. doi: 10.7498/aps.63.010506
    [5] 韦鹏, 申永军, 杨绍普. 分数阶van der Pol振子的超谐共振. 物理学报, 2014, 63(1): 010503. doi: 10.7498/aps.63.010503
    [6] 李晓静, 陈绚青, 严静. 一类具时滞的厄尔尼诺-南方涛动充电-放电振子模型的Hopf分岔与周期解问题. 物理学报, 2013, 62(16): 160202. doi: 10.7498/aps.62.160202
    [7] 赖志慧, 冷永刚, 孙建桥, 范胜波. 基于Duffing振子的变尺度微弱特征信号检测方法研究. 物理学报, 2012, 61(5): 050503. doi: 10.7498/aps.61.050503
    [8] 张勇. 时滞对一类单位负反馈二阶振荡系统的正面作用分析. 物理学报, 2012, 61(23): 230202. doi: 10.7498/aps.61.230202
    [9] 徐昌进. 厄尔尼诺-南方波涛动时滞海气振子耦合模型的分岔分析 . 物理学报, 2012, 61(22): 220203. doi: 10.7498/aps.61.220203
    [10] 申永军, 杨绍普, 邢海军. 含分数阶微分的线性单自由度振子的动力学分析 . 物理学报, 2012, 61(11): 110505. doi: 10.7498/aps.61.110505
    [11] 申永军, 杨绍普, 邢海军. 含分数阶微分的线性单自由度振子的动力学分析(Ⅱ). 物理学报, 2012, 61(15): 150503. doi: 10.7498/aps.61.150503
    [12] 吴勇峰, 张世平, 孙金玮, Peter Rolfe, 李智. 脉冲激励下环形耦合Duffing振子间的瞬态同步突变现象. 物理学报, 2011, 60(10): 100509. doi: 10.7498/aps.60.100509
    [13] 吴勇峰, 张世平, 孙金玮, Peter Rolfe. 环形耦合Duffing振子间的同步突变. 物理学报, 2011, 60(2): 020511. doi: 10.7498/aps.60.020511
    [14] 赵艳影, 杨如铭. 利用时滞反馈控制自参数振动系统饱和控制减振频带. 物理学报, 2011, 60(10): 104304. doi: 10.7498/aps.60.104304.2
    [15] 尚慧琳. 时滞位移反馈对Helmholtz振子系统的分形侵蚀安全域的控制. 物理学报, 2011, 60(7): 070501. doi: 10.7498/aps.60.070501
    [16] 戎海武, 王向东, 徐 伟, 方 同. 谐和与噪声联合作用下Duffing振子的安全盆分叉与混沌. 物理学报, 2007, 56(4): 2005-2011. doi: 10.7498/aps.56.2005
    [17] 莫嘉琪, 王 辉, 林万涛. 厄尔尼诺-南方涛动时滞海-气振子耦合模型. 物理学报, 2006, 55(7): 3229-3232. doi: 10.7498/aps.55.3229
    [18] 戎海武, 王向东, 徐 伟, 方 同. 有界随机噪声激励下软弹簧Duffing振子的安全盆分叉. 物理学报, 2005, 54(10): 4610-4613. doi: 10.7498/aps.54.4610
    [19] 戎海武, 王向东, 徐 伟, 孟 光, 方 同. 窄带随机噪声作用下Duffing振子的双峰稳态概率密度. 物理学报, 2005, 54(6): 2557-2561. doi: 10.7498/aps.54.2557
    [20] 张 强, 高 琳, 王 超, 袁 涛, 许 进. 具有时滞的一阶细胞神经网络动态行为研究. 物理学报, 2003, 52(7): 1606-1610. doi: 10.7498/aps.52.1606
计量
  • 文章访问数:  3548
  • PDF下载量:  347
  • 被引次数: 0
出版历程
  • 收稿日期:  2015-12-17
  • 修回日期:  2016-01-08
  • 刊出日期:  2016-05-05

分数阶时滞反馈对Duffing振子动力学特性的影响

  • 1. 石家庄铁道大学交通运输学院, 石家庄 050043;
  • 2. 石家庄铁道大学机械工程学院, 石家庄 050043
  • 通信作者: 申永军, shenyongjun@126.com
    基金项目: 国家自然科学基金(批准号: 11372198)、河北省高等学校创新团队领军人才计划(批准号: LJRC018)、河北省高等学校高层次人才科学研究项目(批准号: GCC2014053)和河北省高层次人才资助项目(批准号: A201401001)资助的课题.

摘要: 研究了含分数阶时滞耦合反馈的Duffing自治系统, 通过平均法得到了系统周期解的一阶近似解析形式, 定义了以反馈系数、分数阶阶次、时滞参数表示的等效刚度和等效阻尼系数, 发现分数阶时滞耦合反馈同时具有速度时滞反馈和位移时滞反馈的作用. 比较了三种参数条件下近似解析解与数值积分的结果, 二者的吻合精度都很高, 证明了近似解析解的正确性和准确性. 分析了反馈系数、分数阶阶次和非线性刚度系数等参数对系统分岔点、周期解稳定性、周期解的存在范围、零解的稳定性以及稳定性切换次数等系统动力学特性的影响.

English Abstract

参考文献 (31)

目录

    /

    返回文章
    返回