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Based on the generalized principles of dynamics, the feature of Gauss principle of least constraint is that the motion law can be directly obtained by using the variation method of seeking the minimal value of the constraint function without establishing any dynamic differential equations. According to the Kirchhoff's dynamic analogy, the configuration of an elastic rod can be described by the rotation of rigid cross section of the rod along the centerline. Since the local small change of the attitude of cross section can be accumulated infinitely along the arc-coordinate, the Kirchhoff's model is suited to describe the super-large deformation of elastic rod. Therefore the analytical mechanics of elastic rod with arc-coordinate s and time t as double arguments has been developed. The Cosserat model of elastic rod takes into consideration the factors neglected by the Kirchhoff model, such as the shear deformation of cross section, the tensile deformation of centerline, and distributed load, so it is more suitable to modeling a real elastic rod. In this paper, the model of the Cosserat rod is established based on the Gauss principle, and the constraint function of the rod is derived in the general form. The plane motion of the rod is discussed as a special case. As regards the special problem that different parts of the rod in space are unable to self-invade each other, a constraint condition is derived to restrict the possible configurations in variation calculation so as to avoid the invading possibility.
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Keywords:
- Gauss principle /
- thin elastic rod /
- Kirchhoff' /
- s kinetic analogy
[1] Liu Y Z 2001 Advanced Dynamics (Beijing: High Education Press) (in Chinese) [刘延柱 2001 高等动力学 (北京: 高等教育出版社)]
[2] Popov E P, Vereshchagin A F, Zenkevich S A 1978 Manipulative Robots, Dynamics and Algorithm (Moscow: Science) (in Russian) [Попов ЕП, Берещагин АФ, Зенкевич С А 1978 Манипулядионные роботы, динамики и алгоритмы(Москва:Наука)]
[3] Lilov L, Lorer M 1982 Z. Angew. Math. Mech. 62 539
[4] Kalaba R E, Udwadia F E 1993 Trans. ASME J. Appl. Mech. 60 662
[5] Kalaba R, Natsuyama H, Udwadia F 2004 Int. J. General Syst. 33 63
[6] Dong L L, Yan G R, Du Y T, Yu J J, Niu B L, Li R L 2001 Acta Armament. 22 347 (in Chinese) [董龙雷, 闫桂荣, 杜彦亭, 余建军, 牛宝良, 李荣林 2001 兵工学报 22 347]
[7] Hao M W, Ye Z Y 2011 J. Guangxi Univ. (Nat. Sci. Ed.) 36 195 (in Chinese) [郝名望, 叶正寅2011广西大学学报(自然科学版) 36 195]
[8] Liu Y Z, Zu J W 2004 Acta Mech. 167 29
[9] Liu Y Z, Xue Y 2005 Chin. Quart. Mech. 26 1 (in Chinese) [刘延柱, 薛纭 2005力学季刊 26 1]
[10] Liu Y Z, Sheng L W 2007 Acta Mech. Sin. 23 215
[11] Liu Y Z, Xue Y 2011 Chin. J. Theor. Appl. Mech. 43 1151 (in Chinese) [刘延柱, 薛纭 2011 力学学报 43 1151]
[12] Liu Y Z 2009 Chin. Phys. B 18 1
[13] 13Liu Y Z, Xue Y 2011 Appl. Math. Mech. 32 570 (in Chinese) [刘延柱, 薛纭2011 应用数学和力学 32 570]
[14] Liu Y Z 2012 Chin. J. Theor. Appl. Mech. 44 832 (in Chinese) [刘延柱 2012 力学学报 44 832]
[15] Liu Y Z, Xue Y 2004 Tech. Mech. 24 206
[16] 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]
[17] Xue Y, Liu Y Z 2006 Acta Phys. Sin. 55 3845 (in Chinese) [薛纭, 刘延柱 2006 55 3845]
[18] Xue Y, Weng D W 2009 Acta Phys. Sin. 58 34 (in Chinese) [薛纭, 翁德玮 2009 58 34]
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[1] Liu Y Z 2001 Advanced Dynamics (Beijing: High Education Press) (in Chinese) [刘延柱 2001 高等动力学 (北京: 高等教育出版社)]
[2] Popov E P, Vereshchagin A F, Zenkevich S A 1978 Manipulative Robots, Dynamics and Algorithm (Moscow: Science) (in Russian) [Попов ЕП, Берещагин АФ, Зенкевич С А 1978 Манипулядионные роботы, динамики и алгоритмы(Москва:Наука)]
[3] Lilov L, Lorer M 1982 Z. Angew. Math. Mech. 62 539
[4] Kalaba R E, Udwadia F E 1993 Trans. ASME J. Appl. Mech. 60 662
[5] Kalaba R, Natsuyama H, Udwadia F 2004 Int. J. General Syst. 33 63
[6] Dong L L, Yan G R, Du Y T, Yu J J, Niu B L, Li R L 2001 Acta Armament. 22 347 (in Chinese) [董龙雷, 闫桂荣, 杜彦亭, 余建军, 牛宝良, 李荣林 2001 兵工学报 22 347]
[7] Hao M W, Ye Z Y 2011 J. Guangxi Univ. (Nat. Sci. Ed.) 36 195 (in Chinese) [郝名望, 叶正寅2011广西大学学报(自然科学版) 36 195]
[8] Liu Y Z, Zu J W 2004 Acta Mech. 167 29
[9] Liu Y Z, Xue Y 2005 Chin. Quart. Mech. 26 1 (in Chinese) [刘延柱, 薛纭 2005力学季刊 26 1]
[10] Liu Y Z, Sheng L W 2007 Acta Mech. Sin. 23 215
[11] Liu Y Z, Xue Y 2011 Chin. J. Theor. Appl. Mech. 43 1151 (in Chinese) [刘延柱, 薛纭 2011 力学学报 43 1151]
[12] Liu Y Z 2009 Chin. Phys. B 18 1
[13] 13Liu Y Z, Xue Y 2011 Appl. Math. Mech. 32 570 (in Chinese) [刘延柱, 薛纭2011 应用数学和力学 32 570]
[14] Liu Y Z 2012 Chin. J. Theor. Appl. Mech. 44 832 (in Chinese) [刘延柱 2012 力学学报 44 832]
[15] Liu Y Z, Xue Y 2004 Tech. Mech. 24 206
[16] 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]
[17] Xue Y, Liu Y Z 2006 Acta Phys. Sin. 55 3845 (in Chinese) [薛纭, 刘延柱 2006 55 3845]
[18] Xue Y, Weng D W 2009 Acta Phys. Sin. 58 34 (in Chinese) [薛纭, 翁德玮 2009 58 34]
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