細菌鞭毛運動中，快速扭轉是最常見的一種運動行為。
近年來，物理學家藉由彈性理論成功解釋細菌鞭毛運動的力學行為與現象。本論文研究的目的是想探討一個細長桿子在旋轉下的行為。本文有三個主要部分。首先，我們推導出了一組可以描述彈性桿子一般性(general)運動方程式，並且對其自轉(spin rotation)與扭曲(twist)的動力行為與動態幾何關係做一個探討。第二部分是針對細長桿子在旋轉下研究其動態行為。我們採用了解析與數值方法求得一個臨界轉速，在這個轉速以下，棍子只會在旋轉軸自轉，超過這個轉速，棍子會轉出來，這是一個分歧現象。我們用數值方法將這個描述形狀的非線性邊界方程給求得。同時也找到這個臨界轉速與材料參數的關係。當馬達轉速超過第二個臨界轉速的時候，棍子會被彎曲回來並產生一個節點，其對應的形狀的非線性邊界方程和其材料參數關係也都做一個討論。最後，在實驗中發現若採用一長棍子會有旋轉轉速與馬達轉速不相同的現象。這種現象會使棍子旋轉速度一直停留在某個轉速上，即使我們增加了馬達轉速，但是對於短棍子並沒有這種現象發生。因此我們採用動態穩定性分析去了解其行為。

Motivated by the rare studies of the dynamical problems and the common characteristic in biophysics: twirling filaments, which includes the diverse bacterial filaments
motion. We consider a simple dynamical system which one end is clamped to a motor and the other is free to study the
dynamical behaviors of a slender elastic rod at different motor rotation rate. There are three main parts in this thesis. First, we derive the general equations of motion for a slender elastic rod subjected to the external force (gravity). The axial rotation dynamics and geometry constraint about the torsion rate and spinning rate are considered in detail. Second, analytical and numerical methods reveal that a critical twirling rate (we term it the first critical twirling rate) exists, which controls the bifurcation behavior of axial rotation (i.e., twirling) and the steady whirling. Above the first critical twirling rate, the governing nonlinear equation for the rod shape at different whirling rate is solved by numerical method and the results agree with experimental measurements. Our theory also shows that when the motor rotation rate is higher than the second critical twirling rate, the rod would be bent back. We find the relation between the first (the second) critical twirling rate and the material parameters, that agrees well with results observed in experiments. Third, in experiments rods with long length, the whirling rate would remain almost constant as we increase the motor rotation rate. Whereas rods with short length, the whirling rate would be the same with the motor rotation rate. Thus we develop the linear stability analysis to study dynamical stability of a twirling rod at different whirling rate.

Bibliography
[1] M. J. Kim, J. C. Bird, A. J. Van Parys, K. S. Breuer, and T. R. Powers, Proc.
Natl. Acad. Sci. 100, 15481 (2003).
[2] Stephan A. Koehler, Thomas R. Powers, Phys. Rev. Lett. 86, 4827 (2000).
[3] Bian Qian, Thomas R. Powers, Kenneth S. Breuer, Phys. Rev. Lett. 100, 078101
(2008).
[4] Alain Goriely and Sébastien Neukirch, Phys. Rev. Lett. 97, 194302 (2006).
[5] Leif E. Becker and Michael J. Shelley, Phys. Rev. Lett. 87, 198301 (2001).
[6] Zicong Zhou, Pik-Yin Lai, Béla Joós, Phys. Rev. E. 71, 052801 (2005).
[7] Raymond E. Goldstein and Alain Goriely, Phys. Rev. E. 74, 010901 (2006).
[8] Charles W. Wolgemuth, Thomas R. Powers, and Raymond E. Goldstein, Phys.
Rev. Lett. 84, 1623 (2000).
[9] Alain Goriely, J. Elasticity 84, 281 (2006).
[10] A. E. H. Love, A treatise on the mathematical theory of elasticity (Dover, New
York, 1994).
[11] Alain Goriely and Patrick Shipman, Phys. Rev. E. 61, 4508 (2000).
[12] Alain Goriely and M. Tabor, Physica D 105, 20 (1997).
43
[13] David A. Kessler and Yitzhak Rabin, Phys. Rev. Lett. 90, 024301 (2003).
[14] L. D. Landau and E. M. Lifshitz, Theory of Elasticity (Pergamon Press, Oxford,
1986), 3rd ed..
[15] F Tisseur and K Meerbergen, SIAM Review 43, 235 (2001).
[16] Yi-Ting Wang, and Peling Chen, Experiment of Dynamic Rotating Rod in Air,
M.S.Thesis, NCU, Taiwan (2009).
[17] Raymond E. Goldstein, Thomas R. Powers, and Chris H. Wiggins, Phys. Rev.
Lett. 80, 5232 (1998).
[18] Alain Goriely and Michael Tabor, Phys. Rev. Lett. 77, 3537 (1996).
[19] Raymond E. Goldstein and Alain Goriely, Phys. Rev. E. 74, 010901 (2006).
44