摘要A super thin elastic rod is modeled with a background of DNA super coiling structure, and its dynamics is discussed based on the Jourdain variation. The cross section of the rod is taken as the object of this study and two velocity spaces about arc coordinate and the time are obtained respectively. Virtual displacements of the section on the two velocity spaces are defined and can be expressed in terms of Jourdain variation. Jourdain principles of a super thin elastic rod dynamics on arc coordinate and the time velocity space are established, respectively, which show that there are two ways to realize the constraint conditions. If the constitutive relation of the rod is linear, the Jourdain principle takes the Euler-Lagrange form with generalized oordinates. The Kirchhoff equation, Lagrange equation and Appell equation can be derived from the present Jourdain principle. While the rod subjected to a surface constraint, Lagrange equation with undetermined multipliers may be derived.
Abstract:A super thin elastic rod is modeled with a background of DNA super coiling structure, and its dynamics is discussed based on the Jourdain variation. The cross section of the rod is taken as the object of this study and two velocity spaces about arc coordinate and the time are obtained respectively. Virtual displacements of the section on the two velocity spaces are defined and can be expressed in terms of Jourdain variation. Jourdain principles of a super thin elastic rod dynamics on arc coordinate and the time velocity space are established, respectively, which show that there are two ways to realize the constraint conditions. If the constitutive relation of the rod is linear, the Jourdain principle takes the Euler-Lagrange form with generalized oordinates. The Kirchhoff equation, Lagrange equation and Appell equation can be derived from the present Jourdain principle. While the rod subjected to a surface constraint, Lagrange equation with undetermined multipliers may be derived.
[1] Westcott T P, Tobias I and Olson WK 1997 J. Chem.Phys. 107 3967 [2] Travers A A and Thompson J M T 2004 PhilosophicalTransactions: Mathematical, Physical and Engineering Sciences 362 1265 [3] Bernard D C and David S 2004 PhilosophicalTransactions: Mathematical, Physical and Engineering Sciences 362 1281 [4] Westcott T P, Tobias I, Olson W K 1995 J. Phys.Chem. 99 17926 [5] Liu Y Z 2006 Nonlinear Mechanics of Thin ElasticRod-Theoretical Basis of Mechanical Model of DNA (Beijing: TsinhuaUniversity Press {\& Springer) (in Chinese) [6] Liu Y Z 2003 Mech. Engin. 25 1 (in Chinese) [7] Liu Y Z, Xue Y and Chen L Q 2004 Acta Phys. Sin. 53 2424 (in Chinese) [8] Xue Y, Liu Y Z and Chen L Q 2004 Chin. Phys. 13 794 [9] Xue Y, Chen L Q and Liu Y Z 2004 Acta Phys. Sin. 53 2040 (in Chinese) [10] Xue Y, Chen L Q and Liu Y Z 2004 Acta Phys. Sin. 53 4029 (in Chinese) [11] Zhao W J, Weng Y Q and Fu J L 2007 Chin. Phys.Lett. 24 2773 [12] Love A E H 1927 A Treatise on Mathematical Theory ofElasticity 4th edn (New York: Dover) [13] Liu Y Z and Xue Y 2004 Technische Mechanik 24206 [14] Langer J and Singer D A 1996 SIAM Rev. 38 605 [15] Pozo C L M 2000 Physica D 141 248 [16] Xue Y, Liu Y Z and Chen L Q 2005 Acta Mech. Sin. 37 485 (in Chinese) [17] Xue Y and Liu Y Z 2006 Chinese Quarterly ofMechanics 27 550 (in Chinese) [18] Xue Y and Liu Y Z 2006 Acta Phys. Sin. 553845 (in Chinese) [19] Xue Y and Weng D W 2009 Acta Phys. Sin. 58 34(in Chinese) [20] Mei F X, Xie J F and Gang T Q 2007 Chin. Phys. Lett. 24(5): 1133 [21] Jia L Q, Cui J C, Luo S K and Yang X F 2009 Chin.Phys. Lett. 26 030303 [22] Zheng S W, Xie J F and Chen W C 2008 Chin. Phys.Lett. 25 809