本文的主要目的是利用解析及數值混合法(analytical-and-numerical-combined method, ANCM)來求解一均勻水平Euler樑的左端彈性(或固定)支撐，右端攜帶一含有慣性矩及偏心距的集結質量，承受支座垂向激振時的動態反應。其步驟為先利用解析法來求出均勻樑自由振動時最低的數個自然頻率及對應的正規化振態的正解(exact solution)，然後再使用振態重疊法(mode superposition method)理論來推導出均勻樑強迫振動的運動方程式，最後利用數值方法來求出均勻樑的動態反應。除了探討均勻樑的強迫振動反應外，在自由振動分析方面，本文還探討與樑有關的各個參數(如集結質量、集結質量慣性矩及偏心距等)對樑最低的前六個自然頻率的影響。

The purpose of this paper is to determine the dynamic response of an elastically (or rigidly) supported horizontal uniform Euler beam carrying a tip mass with eccentricity and rotary inertia subjected to the vertical support excitation by means of the analytical-and-numerical-combined method (ANCM). Firstly, the exact solutions for the lowest several natural frequencies and the associated normal mode shapes of the vibrating system are determined. Next, the equation of motion for the forced vibration analysis of the vibrating system are derived by using mode superposition method. Finally, the dynamic responses of the vibrating system are calculated by using the conventional numerical method. Besides the dynamic responses, the influence of several parameters (such as tip mass, rotary inertia and tip-mass eccentricity) on the lowest six natural frequencies of the vibrating system is also studied.

1. L. Meirovitch, Analytical Methods in Vibrations , Macmillan Company, London, 1967.
2. R. W. Clough, J. Penzien, Dynamics of Structures , McGraw-Hill, Inc., 1993.
3. P. A. A. Laura, J. L. Pombo and E. A. Susemihl, “A note on the vibration of a clamped-free beam with a mass at the free end ”, Journal of Sound and
Vibration, 37(2), 161-168, 1974.
4. P. A. A. Laura, M. J. Maurizi and J. L. Pombo, “A note on the dynamic analysis of an elastically restrained-free beam with a mass at the free end ”, Journal of Sound and Vibration, 41(4), 397-405, 1975.
5. C. W. S. To, “Vibration of a cantilever beam with a base excitation and tip mass ”, Journal of Sound and Vibration, 83(4), 445-460, 1982.
6. M. Gurgoze, “A note on the vibrations of restrained beams and rods with point masses ”, Journal of Sound and Vibration, 96(4), 461-468, 1984.
7. M. Gurgoze, “On the approximate determination of the fundamental frequency of a restrained cantilever beam carrying a tip heavy body ”, Journal of Sound and Vibration, 105(3), 443-449, 1986.
8. M. Gurgoze, “On the eigenfrequencies of a cantilever beam with attached tip mass and a spring-mass system ”, Journal of Sound and Vibration, 190 (2), 149-162, 1996.
9. C. N. Bapat and C. Bapat, “Natural frequencies of a beam with non-classical boundary conditions and concentrated masses ”, Journal of Sound and Vibration, 112, 177-182, 1987.
10. N. Popplewell and Daqing Chang, “Free vibrations of a complex Euler- Bernoulli beam ”, Journal of Sound and Vibration, 190(5), 852-856, 1996.
11. N. M. Auciello, “Transverse vibrations of a linearly tapered cantilever beam with tip mass of rotatory inertia and eccentricity ”, Journal of Sound and Vibration, 194(1), 25-34, 1996
12. D. Zhou, “The vibrations of a cantilever beam carrying a heave tip mass with elastic supports ”, Journal of Sound and Vibration, 206(2), 275-279, 1997.
13. J. S. Wu and T. L. Lin, “Free vibration analysis of a uniform cantilever beam with point masses by an analytical-and-numerical-combined methed ”, Journal of Sound and Vibration, 136(2), 201-213, 1990.
14. J. S. Wu and C. G. Huang, “Free and forced vibrations of a timoshenko beam with any number of translational and rotational springs and lumped masses ”, Communications in Numerical Methods in Engineering , 11, 743-756, 1995.
15. J. S. Wu and S. S. Luo, “Use of the analytical-and-numerical-combined methed in the free vibration analysis of a rectangular plate with any number of point masses and translational springs ”, Journal of Sound and Vibration, 200(2), 179-194, 1997.
16. J. S. Wu and H. M. Chou, “Free vibration analysis of a cantilever beam carrying any number of elastically mounted point masses with the analytical-and-numerical-combined method ”, Journal of Sound and Vibration, 213(2), 317- 332, 1998.
17. J. S. Wu and D. W. Chen, “Dynamic analysis of a uniform cantilever beam carrying a number of elastically mounted point masses with dampers ”, Journal of Sound and Vibration, 229(3), 549-578, 2000.
18. 刁德勝,“頂端附帶一集結質量且彈性支撐之均勻樑在水中的自由振動分析”,國立成功大學系統及船舶機電工程研究所碩士論文,民國93年。
19. J. T. Xing, W. G. Price, M. J. Pomfret and L. H. Yam, “Natural vibration of a beam-water interaction system ”, Journal of Sound and Vibration, 199(3), 491-512, 1997.
20. A. Uscilowska and J. A. Kolodziej, “Free vibration of immersed column carrying a tip mass ”, Journal of Sound and Vibration, 216(1), 147-157, 1998.
21. J. S. Wu and S. H. Hsu, “A unified approach for the free vibration analysis of an elastically supported immersed uniform beam carrying an eccentric tip mass with rotary inertia ”, Journal of Sound and Vibration, 291, 1122-1147, 2006.
22. 廖為忠,“剪力結構物受地震作用時之決定性及統計動力分析”,國立成功大學土木工程研究所碩士論文,民國70年。
23. J. S. Wu and M. C. Tsai, “The dynamic analysis of an X-Braced lattice girder subjected to a support excitation ”, Journal of Sound and Vibration, 160(1), 123-136, 1993.
24. Chaojin Xu and C. C. Spyrakos, “Seismic analysis of towers including foundation uplift ”, Engineering Structures, 18(4), 271-278, 1996.
25. A. P. Wang and R. F. Fung, “Dynamic analysis of a tall building with a tuned-mass-damper device subjected to earthquake excitations ”, Journal of Sound and Vibration, 244(1), 123-136, 2001.
26. M. K. Shrimali and R. S. Jangid, “Earthquake response of isolated elevated liquid storage steel tanks ”, Journal of Constructional Steel Research, 59, 1267-1288, 2003.
27. K. J. bathe, Finite Element Procedure in Engineering Analysis , Prentice- Hall, Inc., 1982.
28. J. D. Faires, R. L Burden, Numerical Methods, PWS Publishing, Company, 1993.