一自然懸掛的彈性棍在其固定端施於一旋轉之角速度時，觀察其形變對應不同角速度的改變。在低速時，即低於自然振盪頻率時，由彈性理論推之，此彈性棍應該會延著其旋轉軸做自旋。而所謂自然振盪頻率即是對一根懸掛的彈性棍施於一側向力使其偏離懸掛軸後，讓其自由振盪所對應之頻率。而此頻率亦對應到線性彈性力學所得到的最低本徵頻率。而本徵頻率是為一不連續離散頻率。然而從實驗中我們觀測到在低於自然頻率前依舊會有偏離旋轉軸的行為，我們推斷是因為棍子並非理想垂直體，故在彈性理論中假設其初始是為一傾斜的棍子。因此得到和實驗相近的趨勢行為。而隨著旋轉頻率大於自然頻率時，棍子開始依一固定頻率和形狀旋轉。直到接近另一本徵頻率。在施於的角速度介於最低本徵頻率和第二個本徵頻率時，旋轉的棍子會以一固定角速度旋轉。此時棍子折曲於一平面，以此平面對旋轉軸旋轉。此時棍子旋轉之角速度是為一小於馬達旋轉角速度之頻率。此頻率會隨著棍子長度縮短而以平方增加。但當棍子小於一特定長度時，此關係就會從平方反比變成立方反比。
隨著馬達頻率接近第二個本徵頻率時，此時棍子的軌跡會由兩個頻率對應兩個振幅決定。其中一個頻率是為馬達頻率，另一則為一固定頻率如上述所提:在最低本徵頻率和第二個本徵頻率，棍子旋轉的頻率。對應馬達頻率的振幅會隨著馬達頻率接近第二本徵頻率而增加。達到第二本徵頻率時，棍子即以此頻率和一折曲如匙狀於一平面上旋轉。另一頻率所對應的振幅即消失。我們透過兩台數位攝影機同時取像，並將其重建三維圖像，再於此觀察歸納。在高轉速時，重力的效應可以被忽略。而從實驗中觀察到，扭力亦可被忽略。所以此系統可簡化成一彈性力和旋轉產生的離心力系統。

This study is intended as an investigation of the dynamic process of a vertical hanging flexible rod rotating with motor. As the motor rotates, the rod will curve out of rotating axis and whirl. The purpose is to examine those phenomena by observing the shape of rod changing with time at different motor rate.
First, the critical frequency which predicted the rotating rod curves out of the rotating axis is larger than experimental results in our observation. The shape of rod is located in 2 dimension and fixed shape to rotate. This shape is described by beam theory for eigen-frequency . Furthermore, there exists another frequency which the rod will curve back to the rotating axis. It is like a spoon or a half wave. And it is same with eigen-frequency by beam theory. Between and in the rotating axis, the trajectory of rod could be associated with two frequencies effect. One is due to motor rotating rate; the other is probably due to non-linear effect. There exists a range in which the motor rotating rate increases but the rod selects a specific frequency to rotate with shape of . That is, whirling frequency of rod and magnitude of deflection of rod is almost constant, despite increasing motor rate. The shape of rod is also located in 2 dimension and fixed shape to rotate.
Before the motor rotating rate of rod forms shape of , the motor frequency will be involved into the motion of rod. Since there are two frequencies involved, the shape of rod will become 3D changing with time. We use two CCD cameras to rebuild the 3 dimensional figure of rod and illuminate this process by tracing the trajectory of rod. We depict the structure of rod in different motor rate. From the 3D reconstruction of rod, we see that as motor rate smaller than 574 rpm, the shape of rod is composed of two parts, one is shape of , and the other is with different ratio. Our system could neglect the gravity, the torsion, and the Magnus effect.

Bibliography
[1] E. M. Purcell. Life at low reynolds number. American Journal of Physics, 45(1), 1977.
[2] H. C. Berg. Motile behavior of bacteria. Phys. Today, 53(1):24–29, 2000.
[3] J. Teran, L. Fauci, and M. Shelley. Peristaltic pumping and irreversibility of a stokesian
viscoelastic fluid. Physics of Fluids, 20(073101), 2008.
[4] B. Behkam and M. Sitti. Modeling and testing of a biomimetic flagellar propulsion
method for microscale biomedical swimming robots. Proceeding of 2005 IEEE/ASME
International Conference on Advanced Intelligent Mechatronics, pages 37–42, 2005.
[5] B. Qian, T. R. Powers, and K. S. Breuer. Shape transition and propulsive force of an
elastic rod rotating in a viscous fluid. Physical Review Letters, 100(078101), 2008.
[6] C. W. Wolgemuth, T. R. Powers, and R. E. Goldstein. Twirling and whirling: Viscous
dynamcis of rotating elastic filaments. Physical Review Letters, 84(7), 2000.
[7] H. Wada and R. R. Netz. Non-equilibrium hydrodynamics of a rotating filament. Euro-
physics Letters, 75(4):645–651, 2006.
[8] T. S. Yu, E. Lauga, and A. E. Hosoi. Experimental investigations of elastic tail propulsion
at low reynolds number. Physics of Fluids, 18(091701), 2006.
[9] E. Lauga. Floppy swimming: Viscous locomotion of actuated elastica. Physical Review
E, 75(041916), 2007.
[10] M. Manghi, X. Schlagberger, and R. R. Netz. Propulsion with rotating elastic nanorod.
Physical Review Letters, 96(068101), 2006.
[11] M. Takatera, Y. Yazaki, T. Nakano, H. Kanai, S. Hosoya, and Y. Shimizu. Measurement
of fiber flexural-rigidity by rotating vertical cantilever method. Sen’i Gakkaishi, 59(12),
2003.
50
BIBLIOGRAPHY
51
[12] S. Lim and C. S. Peskin. Simulations of the whirling instability by the immersed boundary
method. SIAM J. Sci. Comput, 25(6):2066–2083, 2004.
[13] J. B. Keller and S. I. Rubinow. Slender-body theory for slow viscous flow. J. Fluid Mech.,
75(4):705–714, 1976.
[14] M. F. Carlier P. Venier, A.C. Maggs and D. Pantaloni. Analysis of microtubule rigidity
using hydrodynamic flow and thermal fluctuations. The Journal of Biological Chemistry,
269:13353–13360, 1994.
[15] B. Lin and K. Ravi-Chandar. An experimental investigation of the motion of flexible
strings: Whirling. ASME Journal of Applied Mechanics, 73:842–851, 2006.
[16] B. Lin and K. Ravi-Chandar. Steady-state whirling motions of thin filaments. Interna-
tional Journal of Solids and Structures, 44(9):3035–3048, 2007.
[17] R. W. Tucker D. Kershaw J. Coomer, M. Lazarus and A. Tegman. A non-linear eigenvalue
problem associated with inextensible whiring strings. Journal of Sound and Vibration,
239(5):969–982, 2001.
[18] E. H. Dill. Kirchhoff’s theory of rods. Archive for History of Exact Sciences, 44(1):1–23,
1991.
[19] Z. Y. Li and Peilong Chen. Nonlinear dynamics and dynamical instability of a rotating
rod. Master thesis, 2009.
[20] W. S. Yoo O. Dmitochenko and D. Pogorelov. Helicoseir as shape of a rotating string (i)
:2d theory and simulation using ancf. Multibody System Dynamics, 15(2):135–158, 2006.
[21] L. D. Landau and E. M. Lifshitz. Theory of Elasticity. 1970. 3rd ed.
[22] T. E. Faber. Fluid Dynamics for Physicists. 1995.