本研究主要在探討非線性波浪作用下纜繩結構的動力行為。在本研究中波浪作用下流體的運動包括速度與加速度乃根據勢能波浪理論來計算，對於波浪非線性效應的呈現則引用Stokes二階理論解。至於纜繩運動的描述，本研究主要沿用Lo(1982)以虛功原理為基礎所發展出來的有限元素模式，該模式足以呈現纜繩結構的非線性特性，包括大位移變形、非線性的應力應變關係與非保守力(nonconservative load)。另外，纜繩結構在波浪作用下所承受的流體作用力則利用Morison公式來計算。在求解程序中，對於高度非線性的方程式，則運用了增量法與Newton-Raphson疊代法等數值技巧來處理，進而得到二階的常微分運動方程式，並應用於探討非線性的靜態分析問題與動態分析問題。在靜態分析中藉由黏滯鬆弛法來處理纜繩結構在開始增量程序時勁度矩陣奇異的問題，因而建立纜繩結構的初始平衡形態。而在動態分析中，對於二階的常微分方程式則採用隱式的Newmark法來積分，可得到一組線性方程式進行求解。
在模式的驗證方面，由靜態分析所得到的纜繩結構靜平衡形態與懸垂方程式所得到的解析形態有良好的一致性；在動態分析中，由具繫纜浮球在靜止水域中釋放問題的動態模擬結果，與前人不同模式之計算結果有相當一致的趨勢。本研究進一步將模式運用在波浪作用下纜繩結構運動問題的計算，包括兩端點固定之纜繩在波浪作用下之運動，以及具繫纜之單浮球與多浮球系統在波浪作用下的運動。計算結果顯示纜繩結構在波浪作用下亦呈現週期性的運動反應，波浪的非線性效應可在纜繩的運動與受力上突顯出來，另外非保守力造成結構運動的平均位置偏離初始靜平衡位置。本研究所呈現的成果可應用在計算纜繩結構物在非線性波浪下動力分析之問題。

In this study, the dynamic problem of mooring cables subject to nonlinear wave forces is investigated. The governing equation for cable structure is derived from the principle of virtual work. A finite element method is then used to obtain the discrete equations for numerical computation. Due to high nonlinearity of the discrete equations of motion for cables, further manipulations, including incremental and iterative schemes, are used in the solution procedures. Before performing dynamic analysis, the initial equilibrium configuration of cable structure is calculated by the viscous relaxation technique in static analysis. In this study, the implicit Newmark’s method is chosen for integration of the equations of motion in nonlinear dynamic analysis. To describe the wave forces, the Morison equation is used to calculate the drag forces on cables, and the fluid motion generated by water wave is based on potential wave theory, in which the nonlinear Stokes waves is applied up to the second-order solution. Good agreements between numerical results and analytic solutions are shown in calculation of equilibrium configurations, which can be used to start the dynamic simulations. From the example problems, the present numerical model is shown to simulate dynamic motions of mooring cables reasonably and can be used in dynamic analysis of cable systems.

1. Ablow, C. M. and Schechter, S., “Numerical simulation of undersea cable dynamics,” Ocean Engineering, Vol. 10, pp. 443–457, 1983.
2. Bajer, C. I. and Dyniewicz B., “Virtual functions of the space-time finite element method in moving mass problems,” Computers and Structures, Vol. 87, pp. 444-455, 2009.
3. Bathe, K-J. and Wilson, E. L., Numerical Methods in Finite Element Analysis, Prentice Hall, 1976.
4. Buckham, B. and Nahon, M., “Formulation and validation of a lumped mass model for low-tension ROV tethers,” International Journal of Offshore and Polar Engineering, Vol. 11, pp. 282-289, 2001.
5. Burgess, J. J., “Bending stiffness in a simulation of undersea cable deployment,” International Journal of Offshore and Polar Engineering, Vol. 3, pp. 197-204, 1993.
6. Chakrabarti, S. K., Hydrodynamics of Offshore Structure, Computational Mechanics Publications, London, Great Britain, 1987.
7. Chatjigeorgiou, I. K. and Mavrakos, S. A., “Comparison of numerical methods for predicting the dynamic behavior of mooring lines,” Proceedings of the Ninth International Offshore and Polar Engineering Conference, Brest, France, 1999.
8. Chiou, R. B., Nonlinear hydrodynamic response of curved singly-connected cables, PhD Dissertation, Oregon State University, 1990.
9. Chiou, R. B. and Leonard, J. W., “Nonlinear hydrodynamic response of curved singly-connected cables,” Computer Modelling in Ocean Engineering, Balkema, Rotterdam, 1991.
10. Clough, R. W. and Penzien, J., Dynamics of Structures, McGraw Hill Book Company, New York, 1975.
11. Courant, R. and Hilbert, D., Methods of Mathematical Physics, Vol. II, Interscience, N. Y., 1962.
12. Culla, A. and Carcaterra, A., “Statistical moments predictions for a moored floating body oscillating in random waves,” Journal of Sound and Vibration, Vol. 308, Issue: 1-2, pp. 44-66, 2007.
13. Dean, R. G., “Stream function representation of nonlinear ocean wave,” Journal of Geophysical Research, Vol. 70, pp. 4561-4572, 1965.
14. Dean, R. G. and Dalrymple, R. A., Water Wave Mechanics for Engineers and Scientists, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1984.
15. Delmer, T. N., Stephens, T. C. and Coe, J. M., “Numerical simulation of towed cables,” Ocean Engineering Vol. 10, pp. 119-132, 1983.
16. Engseth, A., Bech, A. and Larsen, C. M., “Efficient method for analysis of flexible risers,” Proceedings of Behavior of Offshore Structures, pp. 1357-1371, 1988.
17. Gobat, J. I. and Grosenbaugh, M. A., “Applications of the generalized-a method to the time integration of the cable dynamics equations,” Computer Methods in Applied Mechanics and Engineering Vol. 190, pp. 4817-4829, 2001a.
18. Gobat, J. I. and Grosenbaugh, M. A., “Time-domain numerical simulation of ocean cable structures,” Ocean Engineering, Vol. 33, No. 10, pp. 1373-1400, 2006.
19. Hann, M., “Statics and dynamics of multi-cable systems for submersibles,” Marine Structures, Vol. 8, pp. 555-583, 1995.
20. Hedges T. S., “Regions of validity of analytical wave theories,” Proc. Instn Civ. Engrs Wat., Marit. And Energy, 112, June, pp. 111-114, 1995.
21. Hoff, C. and Pahl, P. J., “Development of an implicit method with numerical dissipation from a generalized single-step algorithm for structural dynamics,” Computer Methods in Applied Mechanics and Engineering, Vol. 67, pp. 367-385, 1988.
22. Howell, C. T., Investigation of the dynamics of low-tension cables, PhD Dissertation, Massachusetts Institute of Technology and Woods Hole Oceanographic Institution Joint Program, Woods Hole, MA, 1992.
23. Huang, S., “Dynamic analysis of three-dimensional marine cables,” Ocean Engineering, Vol. 21, pp. 587-605, 1994.
24. Hu, H. Y. and Jin, D. P., “Non-linear dynamics of a suspended travelling cable subject to transverse fluid excitation,” Journal of Sound and Vibration, Vol. 239, Issue: 3, pp. 515-529, 2001.
25. Irons, B. and Ahmad, S., Techniques of Finite Elements, Ellis Horwood, London, 1980.
26. Jain, A. K., “Nonlinear coupled response of offshore tension leg platforms to regular wave forces,” Ocean Engineering, Vol. 24, Issue: 7, pp. 577-592, 1997.
27. Larsen, C. M., “Flexible riser analysis—comparison of results from computer programs,” Marine Structures 5, pp. 107-119, 1992.
28. Le Méhauté, B., An introduction to hydrodynamics and water waves, Springer, ISBN 0387072322, 1976.
29. Leonard, J. W., Behavior and analysis of tension structures, McGraw Hill Book Company, New York, 1988.
30. Leonard, J. W. and Karnoski, S. R., “Simulation of tension controlled cable deployment”, Applied Ocean Research, Vol. 12(1), pp. 34-42, 1990.
31. Leonard, J. W. and Nath, J. H., “Comparison of Finite Element and Lumped Parameter Methods for Ocean Cables,” Engineering Structures, Vol. 3, pp. 15B-167, 1981.
32. Lo, A., Nonlinear dynamic analysis of cable and membrane structure, PhD Dissertation, Oregon State University, 1982.
33. Massel, S. R., “Harmonic generation by waves propagating over a submerged step,” Coastal Engineering, Vol. 7, pp. 357-380, 1983.
34. McNamara, J. F., O’Brien, P. J. and Gilroy, S. G., “Nonlinear analysis of flexible risers using hybrid finite elements,” Proceedings of Fifth International Conference on Offshore Mechanics and Arctic Engineering, Tokyo, Vol. 3, pp. 371-377, 1986.
35. Meggit, D. J., Palo, P. A. and Buck, E. F., “Small-size laboratory experiments on the large-displacement dynamics of cable systems, Vol. I-V,” TM No. M-44-78-11, Civil Engineering Lab., Port Hueneme, CA, 1978.
36. Milinazzo, F., Wilkie, M. and Latchman, S. A., “An efficient algorithm for simulating the dynamics of towed cable systems,” Ocean Engineering, Vol. 14, pp. 513-526, 1987.
37. Morison, J. R., O'Brien, M. P., Johnson, J. W. and Schaaf, S. A., “The force exerted by surface waves on piles,” Petroleum Transactions (American Institute of Mining Engineers) 189, pp. 149-154, 1950.
38. Nath, J. H. and Felix, M. P., “Dynamics of single point mooring in deep water,” Journal of Waterways, Harbors and Coastal Engineering Division, ASCE, Vol. 96, WW4, pp. 815-833, 1970.
39. Newmark, N. M., “A method of computation for structural dynamics,” ASCE Journal of the Engineering Mechanics Division, Vol. 85, pp. 67-94, 1959.
40. Norris, H. C. and Wilbur, J. B., Elementary structural analysis, McGraw-Hill, N. Y., 1960.
41. Oden, J. T., Finite element of nonlinear continua, McGraw-Hill, N. Y., 1972.
42. Oden, J. T., Becker, E. B. and Carey, G. F., Finite elements, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1984.
43. Palo, P. A., “Comparisons between small-scale cable dynamics experimental results and simulations using SEADYN and SNAPLG computer models,” TM No. M-44-79-5, Civil Engineering Lab., Port Hueneme, CA, 1979.
44. Paulling, J. R. and Webster, W. C., “A consistent, large-amplitude analysis of the coupled response of a TLP and tendon system,” In: Proceedings of Fifth International Conference on Offshore Mechanics and Arctic Engineering, Tokyo, Vol. 3, pp. 126-133, 1986.
45. Polachek, H., Walton, T. S., Mejia, R. and Dawson, C., “Transient motion of an elastic cable immersed in a fluid,” Mathematics of Computation, Vol. 17, pp. 60-63, 1963.
46. Prpić-Oršić, J. and Nabergoj, R., “Nonlinear dynamics of an elastic cable during laying operations in rough sea,” Applied Ocean Research, Vol. 27, Issue: 6, pp. 255-264, 2005.
47. Rahman, M. A., Mizutani, N. and Kawasaki, K., “Numerical modeling of dynamic responses and mooring forces of submerged floating breakwater,” Coastal Engineering, Vol. 53, pp. 799-815, 2006.
48. Rayleigh, J. W. S., Theory of sound, Second edition, Vol. I, Dover Publication, New York, 1945.
49. Rienecker, M. M. and Fenton, J. D. “A Fourier approximation method for steady water waves,” Journal of Fluid Mechanics, Vol. 101, pp. 108-137, 1981.
50. Sanders, J. V., “A three-dimensional dynamic analysis of a towed system,” Ocean Engineering, Vol. 9, pp. 483-499, 1982.
51. Sarpakaya, T., “Force on cylinder and spheres in a sinusoidal fluid,” Journal of Applied Mechanics, Vol. 42, Issue: 1, ASME, 1976.
52. Sarpakaya, T. and Isaacson, M., Mechanics of wave forces on offshore structures, Van Nostrand Reihold Co., 1981.
53. Sneddon, I. N., Elements of partial differential equations, McGraw-Hill, London, 1957.
54. Sun, Y., Leonard, J. W. and Chiou, R. B, “Simulation of unsteady oceanic cable deployment by direct integration with suppression,” Ocean Engineering, Vol. 21, pp. 243-256, 1994.
55. Svendsen, I. A. and Jonsson, I. G., Hydrodynamics of Coastal Regions. Den Private Ingeniørfond, DTH, Lyngby, 1980.
56. Thomas, D. O., A numerical investigation of time integration schemes applied to the dynamic solution of mooring lines, PhD dissertation, The University of Newcastle upon Tyne, Newcastle, UK, 1993.
57. Thomas, D. O. and Hearn, G. E., “Deepwater mooring line dynamics with emphasis on seabed interference effects,” Proceedings of the 26th Offshore Technology Conference, Houston TX, pp. 203-214, 1994.
58. Tjavaras, A. A., Dynamics of highly extensible cables, PhD dissertation, Massachusetts Institute of Technology Cambridge, MA, 1996.
59. Tjavaras, A. A., Zhu, Q., Liu, Y., Triantafyllou, M. S. and Yue, D. K. P., “The mechanics of highly-extensible cables,” Journal of Sound and Vibration, Vol. 213, Issue: 4, pp. 709-737, 1998.
60. Vaz, M. A. and Patel, M. H., “Three-dimensional behaviour of elastic marine cables in sheared currents,” Applied Ocean Research, Vol. 22, pp. 45-53, 2000.
61. Walton, T. S. and Polachek, H., “Calculation of transient motion of submerged cables,” Mathematics of Computation, Vol. 14, pp. 27-46, 1960.
62. Webster, R. L., “Nonlinear static and dynamic response of underwater cable structures using the finite element method,” OTC 2322, Offshore Technology Conference, Dallas, Texas, pp. 75B-764, 1975.
63. Webster, R. L. and Palo, P. A., “Experiences in the development of a general-purpose mooring analysis capability,” Proceedings of Ocean Structural Dynamics Symposium, Oregon State University, Corvallis, Oregon, pp. 421-435, 1986.
64. Webster, R. L., “On the static analysis of structures with strong geometric nonlinearity,” Computer Structures, Vol. 11, pp. 137-145, 1980.
65. Wilson, B. W. and Garbaccio, D. H., “Dynamics of ship anchor-lines in waves and current,” Journal of the Waterways and Harbors Division, ASCE, Vol. 95, WW4, pp. 449-465, 1969.
66. Wood, W. L., Practical time-stepping schemes, Clarendon Press, Oxford, 1990.
67. Yu, L. and Tan, J. H., “Numerical investigation of seabed interaction in time domain analysis of mooring cables,” Journal of Hydrodynamics, Vol. 18, Issue: 4, pp. 424-430., 2006.
68. Zienkiewicz, O. C., The finite element method, McGraw-Hill, London, 1977.
69. Zienkiewicz, O. C., Wood, W. L. and Hine, N. W., “A unified set of single step algorithms part 1: general formulation and applications,” International Journal for Numerical Methods in Engineering 20, pp. 1529-1552, 1984.
70. Zueck, R. F., Local/global approach to nonlinear simulation of compliant marine structures, Technical Report TR-2073-OCN, Naval Facilities Engineering Service Center, Port Hueneme, CA, 1997.
71. 李兆芳、藍元志，“波浪與可透水結構互相作用之二階解分析”，第十四屆海洋工程研討會論文集，316頁~329頁，1992。
72. 李兆芳、劉正琪、藍元志，“直擺式造波二階波浪特性”，第十六屆海洋工程研討會論文集， A60頁~A83頁，1994。
73. 郭一羽主編，海岸工程學，文山書局發行，台灣，2001年。
74. 楊輔邦，“繫纜式輪胎浮堤消波特性模型試驗研究”， 國立中山大學海洋地質研究所，碩士論文，台灣，1992年。
75. 黃材成、王柏升，“單錨式箱網現場實驗研究”，第29屆海洋工程研討會論文集，第661-666頁，2007年。