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

含分数阶微分的线性单自由度振子的动力学分析(Ⅱ)

申永军 杨绍普 邢海军

引用本文:
Citation:

含分数阶微分的线性单自由度振子的动力学分析(Ⅱ)

申永军, 杨绍普, 邢海军

Dynamical analysis of linear SDOF oscillator with fractional-order derivative (Ⅱ)

Shen Yong-Jun, Yang Shao-Pu, Xing Hai-Jun
PDF
导出引用
  • 研究了含两类分数阶微分项的线性单自由度振子, 通过平均法得到了系统的近似解析解. 在近似解中, 两类分数阶微分项的系数和阶次均以等效线性阻尼和等效线性刚度的形式影响着系统的动力学特性, 这一点和现有文献中大多数直接将分数阶微分项归类为阻尼进行处理是完全不同的. 对近似解析解和数值解进行了比较, 二者符合精度很高, 证明了该结果的准确性. 然后分析了两类分数阶微分项的系数和阶次对系统响应特性的影响, 发现两类分数阶微分项的系数和阶次都既可以影响系统的共振振幅, 又可以影响系统的共振频率. 最后研究了第二类分数阶微分项对共振频率的影响, 指出了在振动控制工程中如何通过选取合适的第二类分数阶微分项的系数达到满意的控制效果.
    A linear single degree-of-freedom (SDOF) oscillator with two kinds of fractional-order derivatives is investigated by the averaging method, and the approximately analytical solution is obtained. The effects of the parameters on the dynamical properties, including the fractional coefficients and the fractional orders in the two kinds of fractional-order derivatives, are characterized by the equivalent linear damping coefficient and the equivalent linear stiffness, and the results is entirely different from the results given in the existing literature. A comparison of the analytical solution with the numerical results is made, and their satisfactory agreement verifies the correctness of the approximately analytical results. The following analysis of the effects of the fractional parameters on the amplitude-frequency is presented, and it is found that the fractional coefficients and the fractional orders can affect not only the resonance amplitude through the equivalent linear damping coefficient, but also the resonance frequency by the equivalent linear stiffness. Finally, the effects of the fractional coefficient in the second fractional-order derivative on resonance frequency are analyzed, and the design rule for the fractional coefficient in the second fractional-order derivative to meet the satisfactory vibration control performance is pointed out.
    • 基金项目: 国家自然科学基金(批准号: 11072158,10932006), 河北省杰出青年科学基金(批准号: E2010002047), 教育部新世纪优秀人才项目和教育部创新团队项目(批准号: IRT0971)资助的课题.
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11072158, 10932006), the Natural Science Funds for Distinguished Young Scholar of Hebei Province (Grant No. E2010002047), the Program for New Century Excellent Talents in University and the Program for Changjiang Scholars and Innovative Research Team in University (Grant No. IRT0971).
    [1]

    Oldham K B, Spanier J 1974 The Fractional Calculus-Theory and Applications of Differentiation and Integration to Arbitrary Order (New York: Academic Press) p1

    [2]

    Podlubny I 1999 Fractional Differential Equations (London: Academic) p10

    [3]

    Petras I 2011 Fractional-Order Nonlinear System (China: Higher Education Press) p19

    [4]

    Rossikhin Y A, Shitikova M V 2010 Applied Mechanics Reviews 63 010801

    [5]

    Yang S P, Shen Y J 2009 Chao, Solitons and Fractals 40 1808

    [6]

    Wang Z Z, Hu H Y 2010 Science China: Physics, Mechanics & Astronomy 53 345

    [7]

    Wang Z Z, Du M L 2011 Shock and Vibration 18 257

    [8]

    Rossikhin Y A, Shitikova M V 1997 Acta Mechanica 120 109

    [9]

    Li G G, Zhu Z Y, Cheng C J 2011 Applied Mathematics and Mechanics 22 294

    [10]

    Cao J Y, Ma C B, Xie H, Jiang Z D 2010 ASME Journal of Computational and Nonlinear Dynamics 5 041012

    [11]

    Wu X J, Lu H T, Shen S L 2009 Physics Letters A 373 2329

    [12]

    Chen J H, Chen W C 2008 Chaos, Solitons and Fractals 35 188

    [13]

    Lu J G 2006 Physics Letters A 354 305

    [14]

    Zhang C F, Gao J F, Xu L 2007 Acta Phys. Sin. 56 5124 (in Chinese) [张成芬, 高金峰, 徐磊 2007 物理学报 56 5124]

    [15]

    Liu C X 2007 Acta Phys. Sin. 56 6865 (in Chinese) [刘崇新 2007 物理学报 56 6865]

    [16]

    Chen X R, Liu C X, Wang F Q, Li Q X 2008 Acta Phys. Sin. 57 1416 (in Chinese) [陈向荣, 刘崇新, 王发强, 李永勋 2008 物理学报 57 1416]

    [17]

    Zhang R X, Yang Y, Yang S P 2009 Acta Phys. Sin. 58 6039 (in Chinese) [张若洵, 杨洋, 杨世平 2009 物理学报 58 6039]

    [18]

    Hu J B, Xiao J, Zhao L D 2011 Acta Phys. Sin. 60 110515 (in Chinese) [胡建兵, 肖建, 赵灵冬 2011 物理学报 60 110515]

    [19]

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

    [20]

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

    [21]

    Qi D L, Wang Q, Yang J 2011 Chin. Phys. B 20 100505

    [22]

    Wu Z M, Xie J Y 2007 Chin. Phys. 16 1901

    [23]

    Zhou P 2007 Chin. Phys. 16 1263

    [24]

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

    [25]

    Deng W H. 2007 Journal of Computational Physics 227 1510

    [26]

    Chen L C, Zhu W Q 2009 Journal of Vibration and Control 15 1247

    [27]

    Wahi P, Chatterjee A 2004 Nonlinear Dynamics 38 3

    [28]

    Huang Z L, Jin X L 2009 Journal of Sound and Vibration 319 1121

    [29]

    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]

    [30]

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

    [31]

    Sanders J A, Verhulst F, Murdock J 2007 Averaging methods in nonlinear dynamical systems (New York: Springer) p150

    [32]

    Ni Z H 1988 Vibration Mechanics p79 (Xi'an: Xi'an Jiaotong University Press) (in Chinese) [倪振华 1988 振动力学 (西安: 西安交通大学出版社) 第79页]

    [33]

    Liu Y Z, Chen W L, Chen L Q 1998 Vibration Mechanics p36 (Beijing: Higher Education Press) (in Chinese) [刘延柱, 陈文良, 陈立群 1998 振动力学 (北京: 高等教育出版社) 第36页]

  • [1]

    Oldham K B, Spanier J 1974 The Fractional Calculus-Theory and Applications of Differentiation and Integration to Arbitrary Order (New York: Academic Press) p1

    [2]

    Podlubny I 1999 Fractional Differential Equations (London: Academic) p10

    [3]

    Petras I 2011 Fractional-Order Nonlinear System (China: Higher Education Press) p19

    [4]

    Rossikhin Y A, Shitikova M V 2010 Applied Mechanics Reviews 63 010801

    [5]

    Yang S P, Shen Y J 2009 Chao, Solitons and Fractals 40 1808

    [6]

    Wang Z Z, Hu H Y 2010 Science China: Physics, Mechanics & Astronomy 53 345

    [7]

    Wang Z Z, Du M L 2011 Shock and Vibration 18 257

    [8]

    Rossikhin Y A, Shitikova M V 1997 Acta Mechanica 120 109

    [9]

    Li G G, Zhu Z Y, Cheng C J 2011 Applied Mathematics and Mechanics 22 294

    [10]

    Cao J Y, Ma C B, Xie H, Jiang Z D 2010 ASME Journal of Computational and Nonlinear Dynamics 5 041012

    [11]

    Wu X J, Lu H T, Shen S L 2009 Physics Letters A 373 2329

    [12]

    Chen J H, Chen W C 2008 Chaos, Solitons and Fractals 35 188

    [13]

    Lu J G 2006 Physics Letters A 354 305

    [14]

    Zhang C F, Gao J F, Xu L 2007 Acta Phys. Sin. 56 5124 (in Chinese) [张成芬, 高金峰, 徐磊 2007 物理学报 56 5124]

    [15]

    Liu C X 2007 Acta Phys. Sin. 56 6865 (in Chinese) [刘崇新 2007 物理学报 56 6865]

    [16]

    Chen X R, Liu C X, Wang F Q, Li Q X 2008 Acta Phys. Sin. 57 1416 (in Chinese) [陈向荣, 刘崇新, 王发强, 李永勋 2008 物理学报 57 1416]

    [17]

    Zhang R X, Yang Y, Yang S P 2009 Acta Phys. Sin. 58 6039 (in Chinese) [张若洵, 杨洋, 杨世平 2009 物理学报 58 6039]

    [18]

    Hu J B, Xiao J, Zhao L D 2011 Acta Phys. Sin. 60 110515 (in Chinese) [胡建兵, 肖建, 赵灵冬 2011 物理学报 60 110515]

    [19]

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

    [20]

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

    [21]

    Qi D L, Wang Q, Yang J 2011 Chin. Phys. B 20 100505

    [22]

    Wu Z M, Xie J Y 2007 Chin. Phys. 16 1901

    [23]

    Zhou P 2007 Chin. Phys. 16 1263

    [24]

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

    [25]

    Deng W H. 2007 Journal of Computational Physics 227 1510

    [26]

    Chen L C, Zhu W Q 2009 Journal of Vibration and Control 15 1247

    [27]

    Wahi P, Chatterjee A 2004 Nonlinear Dynamics 38 3

    [28]

    Huang Z L, Jin X L 2009 Journal of Sound and Vibration 319 1121

    [29]

    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]

    [30]

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

    [31]

    Sanders J A, Verhulst F, Murdock J 2007 Averaging methods in nonlinear dynamical systems (New York: Springer) p150

    [32]

    Ni Z H 1988 Vibration Mechanics p79 (Xi'an: Xi'an Jiaotong University Press) (in Chinese) [倪振华 1988 振动力学 (西安: 西安交通大学出版社) 第79页]

    [33]

    Liu Y Z, Chen W L, Chen L Q 1998 Vibration Mechanics p36 (Beijing: Higher Education Press) (in Chinese) [刘延柱, 陈文良, 陈立群 1998 振动力学 (北京: 高等教育出版社) 第36页]

  • [1] 刘海平, 张世乘, 门玲鸰, 何振强. 面向激光跟踪仪宽频隔振器的理论分析及试验评价. 物理学报, 2022, 71(16): 160701. doi: 10.7498/aps.71.20220307
    [2] 杨建华, 马强, 吴呈锦, 刘后广. 分数阶双稳系统中的非周期振动共振. 物理学报, 2018, 67(5): 054501. doi: 10.7498/aps.67.20172046
    [3] 王世元, 史春芬, 钱国兵, 王万里. 基于分数阶最大相关熵算法的混沌时间序列预测. 物理学报, 2018, 67(1): 018401. doi: 10.7498/aps.67.20171803
    [4] 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
    [5] 王建立, 郭亮, 徐先凡, 倪中华, 陈云飞. 晶格振动的超快光谱调控. 物理学报, 2017, 66(1): 014203. doi: 10.7498/aps.66.014203
    [6] 温少芳, 申永军, 杨绍普. 分数阶时滞反馈对Duffing振子动力学特性的影响. 物理学报, 2016, 65(9): 094502. doi: 10.7498/aps.65.094502
    [7] 路永坤. 参数不确定统一混沌系统的鲁棒分数阶比例-微分控制. 物理学报, 2015, 64(5): 050503. doi: 10.7498/aps.64.050503
    [8] 李永强, 刘玲. 微重力下变内角毛细驱动流研究. 物理学报, 2014, 63(21): 214704. doi: 10.7498/aps.63.214704
    [9] 叶扬, 王树林. 铜微颗粒碰撞阻尼特性. 物理学报, 2014, 63(22): 224304. doi: 10.7498/aps.63.224304
    [10] 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
    [11] 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
    [12] 陆法林, 陈昌远, 尤源. 双环形Hulthn势束缚态的近似解析解. 物理学报, 2013, 62(20): 200301. doi: 10.7498/aps.62.200301
    [13] 王梦蛟, 曾以成, 谢常清, 朱高峰, 唐淑红. Chen系统在微弱信号检测中的应用. 物理学报, 2012, 61(18): 180502. doi: 10.7498/aps.61.180502
    [14] 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
    [15] 王梦蛟, 曾以成, 陈光辉, 贺娟. Chen系统的非共振参数控制. 物理学报, 2011, 60(1): 010509. doi: 10.7498/aps.60.010509
    [16] 张艳, 郑连存, 张欣欣. 边界耦合的Marangoni对流边界层问题的近似解析解. 物理学报, 2009, 58(8): 5501-5506. doi: 10.7498/aps.58.5501
    [17] 卫高峰, 龙超云, 秦水介, 张 欣. 具有Manning-Rosen标量势与矢量势的Klein-Gordon方程的任意l波束缚态近似解析解. 物理学报, 2008, 57(11): 6730-6735. doi: 10.7498/aps.57.6730
    [18] 郑连存, 冯志丰, 张欣欣. 一类非线性微分方程的近似解析解. 物理学报, 2007, 56(3): 1549-1554. doi: 10.7498/aps.56.1549
    [19] 杨鹏飞. 一类带限定变换的二阶耦合线性微分方程组的解析解. 物理学报, 2006, 55(11): 5579-5584. doi: 10.7498/aps.55.5579
    [20] 郑连存, 盛晓艳, 张欣欣. 一类Marangoni对流边界层方程的近似解析解. 物理学报, 2006, 55(10): 5298-5304. doi: 10.7498/aps.55.5298
计量
  • 文章访问数:  6057
  • PDF下载量:  798
  • 被引次数: 0
出版历程
  • 收稿日期:  2011-12-30
  • 修回日期:  2012-01-05
  • 刊出日期:  2012-08-05

/

返回文章
返回