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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.
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Keywords:
- an exact Cosserat elastic rod /
- analytical dynamics /
- variational principle /
- Lagrange equation Hamilton principle
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[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]
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[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 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 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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[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 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 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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