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精确Cosserat弹性杆动力学的分析力学方法

薛纭 翁德玮 陈立群

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精确Cosserat弹性杆动力学的分析力学方法

薛纭, 翁德玮, 陈立群

Methods of analytical mechanics for exact Cosserat elastic rod dynamics

Xue Yun, Weng De-Wei, Chen Li-Qun
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  • 以脱氧核糖核酸和工程中的细长结构为背景, 大变形大范围运动的弹性杆动力学受到关注. 将分析力学方法运用到精确Cosserat弹性杆动力学, 旨在为前者拓展新的应用领域, 为后者提供新的研究方法. 基于平面截面假定, 在弯扭基础上再计及拉压和剪切变形形成精确Cosserat弹性杆模型. 用刚体运动的概念描述弹性杆的变形, 导出弹性杆变形和运动的几何关系; 在定义截面虚位移及其变分法则的基础上, 建立用矢量表达的d’Alembert-Lagrange原理, 在线性本构关系下化作分析力学形式, 并导出Lagrange方程和Nielsen方程, 定义正则变量后化作Hamilton正则方程; 对于只在端部受力的弹性杆静力学, 导出了将守恒量预先嵌入的Lagrange方程, 并讨论了其首次积分. 从弹性杆的d’Alembert-Lagrange原理导出积分变分原理, 在线性本构关系下化作Hamilton原理. 形成的分析力学方法使弹性杆的全部动力学方程具有统一的形式, 为弹性杆动力学的对称性和守恒量的研究及其数值计算铺平道路.
    Thin elastic rod mechanics with background of a kind of single molecule such as DNA and other engineering object has entered into a new developing stage. In this paper the vector method of exact Cosserat elastic rod dynamics is transformed into the form of analytical mechanics with the arc length and time as its independent variables, whose aims are to find new tools for studying rod mechanics and to develop the area of applications of classical analytical mechanics. Based on the plane cross-section assumption, a cross-section of the rod is taken as an object. Basic formulas on deformation and motion of the section are given. After defining virtual displacement of a cross-section and its equivalent variation rule, a differential variational principle such as d’Alembert-Lagrange one is established, from which dynamical equations of thin elastic rod are expressed as Lagrange equations or Nielsen equations under the condition of linear elasticity of the rod. For the rod statics when there exist conserved quantities, Lagrange equation which makes use of these quantities is derived and its first integral is discussed. Finally integral variational principle is derived from differential one, and expressed as Hamilton principle under the condition of linear elasticity. Hamilton canonical equations in phase space with 3×6 dimensions are also derived. All of the results have formed the method of analytical mechanics of dynamics of an exact Cosserat elastic rod, so that the further problems such as symmetry and conserved quantities, and numerical simulation of the rod dynamics may be further studied.
    • 基金项目: 国家自然科学基金(批准号:10972143)、上海应用技术学院《化工过程机械》重点学科建设基金(批准号:1020Q101201)资助的课题.
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 10972143) and the Key Disciplines Chemical Process Machinery of Shanghai Institute of Technology, China (Grant No. 1020Q101201).
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    Xue Y, Liu Y Z 2006 Chinese Quarterly of Mechanics 27 550 (in Chinese) [薛纭, 刘延柱 2006 力学季刊 27 550]

    [11]

    Xue Y, Wen D W 2009 Acta Phys. Sin. 58 34 (in Chinese) [薛纭, 翁德玮 2009物理学报 58 34]

    [12]

    Xue Y, Shang H L 2009 Chin. Phys. Lett. 26 074501

    [13]

    Zhao W J, Weng Y Q, Fu J L 2007 Chin. Phys. Lett. 24 2773

    [14]

    Wang P, Xue Y, Liu Y L 2012 Chin. Phys. B 21 070203

    [15]

    Cao D Q, Tucker R W 2008 Int. J. Solids Struct. 45 460

    [16]

    Xue Y, Weng D W, Chen L Q 2009 Chinese Quarterly of Mechanics 30 116 (in Chinese) [薛纭, 翁德玮, 陈立群2009 力学季刊 30 116]

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  • [1]

    Liu Y Z 2006 Nonlinear Mechanics of Thin Elastic Rod–Theoritical Basis of Mechanical Model of DNA (Beijing: Tsinghua University Press, Springer) p1432 (in Chinese) [刘延柱 2006 弹性细杆的非线性力学–-DNA力学模型的理论基础 (北京: 清华大学出版社 Springer) 第1432页]

    [2]

    Liu Y Z 2003 Mech. Eng. 25 1 (in Chinese) [刘延柱 2003 力学与实践 25 1]

    [3]

    Ouyang Z C 2003 Physics 32 728 (in Chinese) [欧阳钟灿2003物理 32 728]

    [4]

    Li M, Ouyang Z C 2003 Science 55 15 (in Chinese) [黎明, 欧阳钟灿 2003 科学 55 15]

    [5]

    Malacinski G M (translated by Wei Q) 2005 Essentials of Molecular Biology (4th Ed.) (Bejing: Chemical Industry Press) p59 (in Chinese) [乔治 M 马拉森斯基著 (魏群 译) 2005分子生物学精要 (北京: 化学工业出版社)第59页]

    [6]

    Westcott T P, Tobias I, Olson W K 1995 J. Phys. Chem. 99 17926

    [7]

    Pozo Coronado L M 2000 Physica D 141 248

    [8]

    Xue Y, Liu Y Z, Chen L Q 2005 Chin. J. Theor. Appl. Mech. 37 485 (in Chinese) [薛纭, 刘延柱, 陈立群 2005 力学学报 37 485]

    [9]

    Xue Y, Liu Y Z 2006 Acta Phys. Sin. 55 3845 (in Chinese) [薛纭, 刘延柱 2006 物理学报 55 3845]

    [10]

    Xue Y, Liu Y Z 2006 Chinese Quarterly of Mechanics 27 550 (in Chinese) [薛纭, 刘延柱 2006 力学季刊 27 550]

    [11]

    Xue Y, Wen D W 2009 Acta Phys. Sin. 58 34 (in Chinese) [薛纭, 翁德玮 2009物理学报 58 34]

    [12]

    Xue Y, Shang H L 2009 Chin. Phys. Lett. 26 074501

    [13]

    Zhao W J, Weng Y Q, Fu J L 2007 Chin. Phys. Lett. 24 2773

    [14]

    Wang P, Xue Y, Liu Y L 2012 Chin. Phys. B 21 070203

    [15]

    Cao D Q, Tucker R W 2008 Int. J. Solids Struct. 45 460

    [16]

    Xue Y, Weng D W, Chen L Q 2009 Chinese Quarterly of Mechanics 30 116 (in Chinese) [薛纭, 翁德玮, 陈立群2009 力学季刊 30 116]

    [17]

    Xue Y, Weng D W 2011 Mech. Eng. 33 65 (in Chinese) [薛纭, 翁德玮 2011 力学与实践 33 65]

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出版历程
  • 收稿日期:  2012-07-11
  • 修回日期:  2012-08-29
  • 刊出日期:  2013-02-05

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