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分数阶拟周期Mathieu方程的动力学分析

郭建斌 申永军 李航

郭建斌, 申永军, 李航. 分数阶拟周期Mathieu方程的动力学分析. 力学学报, 2021, 53(12): 1-10 doi: 10.6052/0459-1879-21-455
引用本文: 郭建斌, 申永军, 李航. 分数阶拟周期Mathieu方程的动力学分析. 力学学报, 2021, 53(12): 1-10 doi: 10.6052/0459-1879-21-455
Guo Jianbin, Shen Yongjun, Li Hang. Dynamic analysis of quasi-periodic mathieu equation with fractional-order derivative. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(12): 1-10 doi: 10.6052/0459-1879-21-455
Citation: Guo Jianbin, Shen Yongjun, Li Hang. Dynamic analysis of quasi-periodic mathieu equation with fractional-order derivative. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(12): 1-10 doi: 10.6052/0459-1879-21-455

分数阶拟周期Mathieu方程的动力学分析

doi: 10.6052/0459-1879-21-455
基金项目: 国家自然科学基金(U1934201,11772206)、石家庄铁道大学研究生创新项目(YC2021043)资助
详细信息
    作者简介:

    申永军, 教授, 主要研究方向: 机械系统动力学与振动控制. E-mail: shenyongjun@126.com

  • 中图分类号: O322,O313

DYNAMIC ANALYSIS OF QUASI-PERIODIC MATHIEU EQUATION WITH FRACTIONAL-ORDER DERIVATIVE

  • 摘要: 分数阶微积分有着诸多优异的特点, 目前在动力学领域主要用来提高非线性系统振动特性研究的准确性. 本文在拟周期Mathieu方程的基础上, 引入分数阶微积分理论, 研究了分数阶微分项参数对方程稳定性的影响. 首先, 采用摄动法得到方程稳定区和非稳定区分界线(即过渡曲线)近似表达式, 利用数值方法验证了解析结果的准确性, 图像显示两者吻合较好. 随后, 通过归纳总结不同情况下的过渡曲线近似表达式, 发现在系统中分数阶微分项以等效线性刚度和等效线性阻尼的方式存在. 根据这一特点, 得到了系统等效线性阻尼和等效线性刚度的一般形式, 并且定义了非稳定区域厚度. 最后, 通过数值仿真直观地分析了分数阶微分项参数对方程稳定区域大小和过渡曲线位置的影响. 结果发现, 分数阶微分项不仅具有阻尼特性还具有刚度特性, 并且以等效线性刚度和等效线性阻尼的方式影响着方程稳定区域大小和过渡曲线位置. 合理选择分数阶微分项参数可以使其呈现不同程度的刚度特性或阻尼特性, 方程稳定区域的大小和过渡曲线的位置也因此产生了不同程度的变化.

     

  • 图  1  分数阶拟周期Mathieu方程的过渡曲线

    Figure  1.  Transition curves of QP Mathieu equation with fractional-order derivative

    图  2  数值解和解析解的方程过渡曲线

    Figure  2.  Transition curves of numerical and analytical solutions

    图  3  分数阶微分项阶次$ p $对过渡曲线的影响

    Figure  3.  Effects of the fractional order$ p $on transition curves

    图  4  $p = 0.1$时分数阶微分项系数${K_1}$对过渡曲线的影响

    Figure  4.  Effects of the fractional coefficient$ {K_1} $on transition curves when$p = 0.1$

    图  5  $p = 0.5$时分数阶微分项系数对过渡曲线的影响

    Figure  5.  Effects of the fractional coefficient on transition curves when$p = 0.5$

    图  6  $p = 0.9$时分数阶微分项系数${K_1}$对过渡曲线的影响

    Figure  6.  Effects of the fractional coefficient${K_1}$on transition curves when$p = 0.9$

    图  7  线性阻尼系数$\zeta $对过渡曲线的影响

    Figure  7.  The evolutions of the transition curves due to the change of$\zeta $

    表  1  $ {\delta _0} $在不同情况下的等效线性阻尼和等效线性刚度

    Table  1.   Equivalent linear damping and equivalent linear stiffness of different $ {\delta _0} $

    $ {\delta _0} = 0 $$ {\delta _0} = \dfrac{1}{4}{\omega ^2} $$ {\delta _0} = \dfrac{1}{4} $$ {\delta _0} = \dfrac{1}{4}{\left( {1 + \omega } \right)^2} $$ {\delta _0} = \dfrac{1}{4}{\left( {1 - \omega } \right)^2} $
    $C(p)$ $ 2\zeta {\kern 1 pt} + \dfrac{{{K_1}}}{{{2^{p - 1}}}}{\omega ^{p - 1}}\sin \left( {\dfrac{{p{\text{π }}}}{2}} \right) $ $ 2\zeta + {K_1}\dfrac{1}{{{2^{p - 1}}}}\sin \left( {\dfrac{{p{\text{π }}}}{2}} \right) $ $ 2\zeta + \dfrac{{{K_1}}}{{{2^{p - 1}}}}{\left( {1 + \omega } \right)^{p - 1}}\sin \left( {\dfrac{{p{\text{π }}}}{2}} \right) $ $ 2\zeta + \dfrac{{{K_1}}}{{{2^{p - 1}}}}{\left( {1 - \omega } \right)^{p - 1}}\sin \left( {\dfrac{{p{\text{π }}}}{2}} \right)\; $
    $K(p)$ $ \delta + {\varepsilon ^2}\dfrac{{{K_1}\left( {{\omega ^4} + {\omega ^p}} \right)}}{{2{\omega ^4}}}\cos \left( {\dfrac{{p{\text{π }}}}{2}} \right) $ $ \delta + \dfrac{{{K_1}}}{{{2^p}}}{\omega ^p}\cos \left( {\dfrac{{p{\text{π }}}}{2}} \right) $ $ \delta + \dfrac{{{K_1}}}{{{2^p}}}\cos \left( {\dfrac{{p{\text{π }}}}{2}} \right) $ $ \delta + \dfrac{{{K_1}}}{{{2^p}}}{\left( {1 + \omega } \right)^p}\cos \left( {\dfrac{{p{\text{π }}}}{2}} \right) $ $ \delta + \dfrac{{{K_1}}}{{{2^p}}}{\left( {1 - \omega } \right)^p}\cos \left( {\dfrac{{p{\text{π }}}}{2}} \right) $
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  • [1] Caputo M, Mainardi F. A new dissipation model based on memory mechanism. Pure and Applied Geophysics, 1971, 91(1): 134-147 doi: 10.1007/BF00879562
    [2] Oldham KB, Spanier J. The Fractional Calculus. New York: Academic Press, 1974
    [3] Podlubny I. Fractional Differential Equations. London: Academic, 1999
    [4] Kilbas AA, Srivastava HM, Trujillo JJ. Theory and Applications of Fractional Differential Equations. Amsterdam: Elsevier, 2006
    [5] Petras I. Fractional-order Nonlinear Systems: Modeling, Analysis and Simulation. Beijing: Higher Education Press, 2011: 18-19
    [6] 林世敏, 许传炬. 分数阶微分方程的理论和数值方法研究. 计算数学, 2016, 38(1): 1-24 (Lin Shimin, Xu Chuanju. Theoretical and numerical investigation of fractional differential equations. Mathematica Numerica Sinica, 2016, 38(1): 1-24 (in Chinese) doi: 10.12286/jssx.2016.1.1
    [7] 杨建华, 朱华. 不同周期信号激励下分数阶线性系统的响应特性分析. 物理学报, 2013, 62(2): 374-380

    Yang Jianhua, Zhu Hua. The response property of one kind of factional-order linear system excited by different periodical signals, Acta Physica Sinica, 2013, 62 (2): 374-380 (in Chinese)
    [8] 蔡伟, 陈文. 复杂介质中任意阶频率依赖耗散声波的分数阶导数模型. 力学学报, 2016, 48(6): 1265-1280 (Cai Wei, Chen Wen. Fractional derivative modeling of frequency-dependent dissipative mechanism for wave in complex media. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(6): 1265-1280 (in Chinese) doi: 10.6052/0459-1879-16-186
    [9] 李占龙, 孙大刚, 韩斌慧. 基于分数阶导数的黏弹性减振系统时频特性. 应用基础与工程科学学报, 2017, 25(1): 187-198 (Li Zhanlong, Sun Dagang, Han Binhui. Time and frequency features of viscoelastic vibration damping system based on fractional derivative. Journal of Basic and Science and Engineering, 2017, 25(1): 187-198 (in Chinese)
    [10] 姜源, 申永军, 温少芳等. 分数阶达芬振子的超谐与亚谐联合共振. 力学学报, 2017, 49(5): 1008-1019 (Jiang Yuan, Shen Yongjun, Wen Shaofang, et al. Super-harmonic and subharmonic simultaneous resonances of fractional-order Duffing oscillator. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(5): 1008-1019 (in Chinese) doi: 10.6052/0459-1879-17-105
    [11] Shen YJ, Li H, Yang SP, et al. Primary and subharmonic simultaneous resonance of fractional-order Duffing oscillator. Nonlinear Dynamics, 2020, 102: 1485-1497 doi: 10.1007/s11071-020-06048-w
    [12] 薛定宇, 赵春娜. 分数阶系统的分数阶PID控制器设计. 控制理论与应用, 2007, 24(5): 771-776 (Xue Dingyu, Zhao Chunna. Fractional order PID controller design for fractional order system. Control Theory and Applications, 2007, 24(5): 771-776 (in Chinese) doi: 10.3969/j.issn.1000-8152.2007.05.015
    [13] 常宇健, 田沃沃, 金格. 基于分数阶PIλDμ的非线性分数阶主动控制悬架研究. 燕山大学学报, 2020, 44(6): 575-580

    Chang Yujian, Tian Wowo, Jin Ge. Research on nonlinear fractional active control suspension based on fractional order PIλDμ. Journal of Yanshan University, 2020, 44(6): 575-580 (in Chinese)
    [14] Cao J, Ma C, Xie H, et al. Nonlinear dynamics of duffing system with fractional order damping. Journal of Computational and Nonlinear Dynamics, 2010, 5(4): 041012-1-6 doi: 10.1115/1.4002092
    [15] 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): 158-163 (Shen Yongjun, Yang Shaopu, Xing Haijun. Dynamical analysis of linear single degree-of-freedom oscillator with fractional-order derivative. Acta Physica Sinica, 2012, 61(11): 158-163 (in Chinese)
    [16] 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): 47-58 (Wei Peng, Shen Yongjun, Yang Shaopu. Super-harmonic resonance of fractional-order van der Pol oscillator. Acta Physica Sinica, 2014, 63(1): 47-58 (in Chinese)
    [17] 王晓娜, 申永军, 张娜. 含分数阶微分项的Van del Pol振子的动力学分析. 振动与冲击, 2020, 39(20): 91-96 (Wang Xiaona, Shen Yongjun, Zhang Na. Dynamical analysis of a Van Del Pol oscillator with fractional-order derivative. Journal of Vibration and Shock, 2020, 39(20): 91-96 (in Chinese)
    [18] 韩东颖, 时培明, 赵东伟. 板带轧机机电传动系统参激非线性扭振鲁棒控制研究. 振动与冲击, 2016, 35(12): 1-6 (Han Dongying, Shi Peiming, Zhao Dongwei. Study on robust control for parametric excitation nonlinear torsional vibration of a strip-rolling mill's mechanical and electrical drive system. Journal of Vibration and Shock, 2016, 35(12): 1-6 (in Chinese)
    [19] 徐梅鹏, 李凌峰, 任双兴等. 多自由度参激系统稳定性分析的数值解法. 计算力学学报, 2020, 37(1): 48-52 (Xu Meipeng, Li Lingfeng, Ren Shuangxing, et al. Numerical method for stability analysis of multiple-degree-of-freedom parametric dynamic systems. Chinese Journal of Computational Mechanics, 2020, 37(1): 48-52 (in Chinese) doi: 10.7511/jslx20181121002
    [20] 丛戎飞, 叶友达, 赵忠良. 吸气式高超声速飞行器耦合运动数值模拟. https://doi.org/10.13700/j.bh.1001-5965.2020.0313. 2021-09-05

    Cong Rongfei, Ye Youda, Zhao Zhongliang. Numerical simulation of coupling motion of air-breathing hypersonic vehicle. https://doi.org/10.13700/j.bh.1001-5965.2020.0313. 2021-09-05 (in Chinese)
    [21] Qian CZ, Chen CP, Zhou GW. Nonlinear dynamical analysis for the cable excited with parametric and forced excitation. Journal of Applied Mathematics, 2014: 183257-1-6
    [22] 黄建亮, 王腾, 陈树辉. 含外激励van der Pol-Mathieu方程的非线性动力学特性分析. 力学学报, 2021, 53(2): 496-510 (Huang Jianliang, Wang Teng, Chen Shuhui. Nonlinear dyanamic analysis of a van der Pol-Mathieu equation with external excitation. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(2): 496-510 (in Chinese) doi: 10.6052/0459-1879-20-310
    [23] 温少芳. 分数阶参激系统的动力学与控制研究. [博士论文]. 石家庄铁道大学, 2018: 22-37

    Wen Shaofang. Dynamics and control of fractional-order parametrically excited system. [PhD Thesis]. Shijiazhuang: Shijiazhuang Tiedao University, 2018: 22-37 (in Chinese)
    [24] Kovacic I, Rand R, Mohamed SS. Mathieu's equation and its generalizations: overview of stability charts and their features. Applied Mechanics Reviews, 2018, 70: 020802-12-14 doi: 10.1115/1.4039144
    [25] Galeotti G, Toni P. Nonlinear modeling of a railway pantograph for high speed running. Transactions on Modelling and Simulation, 1993, 5: 422-436
    [26] Huan RH, Zhu WQ, Ma F, et al. The effect of high-frequency parametric excitation on a stochastically driven pantograph-catenary system. Shock and Vibration, 2014, 2014: 1-8
    [27] Rand R, Morrison T. 2: 1: 1 Resonance in the quasiperiodic Mathieu equation. Nonlinear Dynamics, 2005, 40: 195-203 doi: 10.1007/s11071-005-6005-8
    [28] Rand R, Kamar G. 2: 2: 1 Resonance in the quasiperiodic Mathieu equation. Nonlinear Dynamics, 2003, 31: 367-374 doi: 10.1023/A:1023216817293
    [29] Zounes RS, Rand RH. Transition curves in the quasi-periodic Mathieu equation. Siam Journal on Applied Mathematics, 1998, 58(4): 1094-1115 doi: 10.1137/S0036139996303877
    [30] Rossikhin YA, Shitikova MV. On fallacies in the decision between the Caputo and Riemann-Liouville fractional derivative for the analysis of the dynamic response of a nonlinear viscoelastic oscillator. Mechanics Research Communications, 2012, 45: 22-27 doi: 10.1016/j.mechrescom.2012.07.001
    [31] Nayfeh AH, Mook DT. Nonlinear Oscillations. New York: Wiley, 1973
    [32] 胡海岩. 应用非线性动力学. 北京: 航空工业出版社, 2000: 80-92

    Hu Haiyan. Applied Nonlinear Dynamics. Beijing: Aviation Industry Press, 2000: 80-92 (in Chinese)
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出版历程
  • 收稿日期:  2021-09-27
  • 录用日期:  2021-11-05
  • 网络出版日期:  2021-11-06

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