本文之目的在探討一頂端攜帶一(偏心)集中質量與質量慣性矩之部分浸水均勻樑，於其底端固定與彈性支撐等兩種情況下之自然頻率與振態的正解(exact solutions)。為達此目的，吾人首先將樑分為浸水與不浸水兩部份，然後由各部份的運動方程式，求得各部份的側向位移函數，接著，利用整根樑之頂端及底端的邊界條件，連同該樑在浸水與不浸水交界處之位移與斜度的相容性，以及力及彎矩的平衡等條件，吾人便可求得該樑的自然頻率與振態。就底端固定的樑而言，將根據本文數學式及電算程式所得的結果，與現有文獻裡的結果相比較，吾人發現兩者非常接近，故本文中，相關的數學式及電算程式之可靠性，應可被接受。但就底端彈性支撐的樑而言，由於找不到現有文獻的資料供比較，吾人乃採用間接比較法：除了底端的支撐情況外，假設底端彈性支撐樑的所有條件，完全與底端固定樑者相同，然後，令彈性支撐樑底端的線性彈簧與旋轉彈簧之勁度，逐漸增大，此時吾人發現，底端彈性支撐樑的自然頻率逐漸趨近於底端固定樑的自然頻率，而當上述線性彈簧與旋轉彈簧的勁度，超過某值後，底端彈性支撐樑的自然頻率與底端固定樑的自然頻率，幾乎沒有差別，而且兩種樑的對應振態，亦幾乎重疊，故本文中，與底端彈性支撐樑相關的理論及電算程式之可靠性，亦應可被接受。除了上述驗證工作外，某些重要參數對底端固定樑及底端彈性支撐樑的自然頻率之影響，本文亦有所探討，這些重要參數包括：浸水長度、集中質量大小與質量慣性矩大小等。

　　The objective of this thesis is to determine the natural frequencies and mode shapes exact solutions of an immersed uniform beam carrying a (eccentricity)tip mass and mass moment of inertia which fixed and elastically supported end vibration. For the purpose , first to confer beam immersed and not immersed respectively , then to find out side of displacement function which use each part motion equation . Next use the boundary condition of beam which the top and the bottom , and force equilibrium can get the natural frequencies and mode shapes of beam . To the fixed beam , according to the result which computer program to compare with existing paper . We can detect the result very close with both . Therefore it is accept of relation computer program about fixed in this paper . It is can’t find any data to compare which elastically supported , so we can use indirect compare . Assumption the elastically supported with fixed is the same which other condition , just let the spring’s strength even more large to the fixed’s strength . Therefore we can find the both of mode shapes is superimposed , it is can accept of relation computer program about elastically supported in this paper . Besides other parameter influence with natural frequencies and mode shapes of fixed and elastically supported beam that to confer.

1. 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.
2. 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.
3. 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.
4. 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, 443-449, 1986.
5. 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.
6. W.H. Liu, J.R. Wu, and C.C. Huang, “Free Vibration of Beams with Elastically Restrained Edges and Intermediate Concentrated Masses,” Journal of Sound and Vibration, 122(2), 193-207, 1988.
7. M.N. Hamdan and B.A. Jubran, “Free and Forced Vibrations of a Restrained Uniform Beam Carrying an Intermediate Lumped Mass and a Rotary Inertia,” Journal of Sound and Vibration, 150(2), 203-216, 1991.
8. J.S. Wu and T.L. Lin, “Free Vibration Analysis of a Uniform Cantilever Beam with Point Masses by an Analytical-and-Numerical-Combined Method,” Journal of Sound and Vibration, 136(2), 201-213, 1990.
9. 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.
10. J.S. Wu and H.M. Chou, “A New Approach For Determining the Natural Frequencies and Mode Shape of a Uniform Beam Carrying Any Number of Spring Masses,”Journal of Sound and Vibration, 220(3), 451-468, 1999.
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. K. Nagaya, “Transient Response in Flexure to General Uni-Directional Loads of Variable Cross-Section Beam with Concentrated Tip Inertias Immersed in a Fluid,” Journal of Sound and Vibration, 99, 361-378, 1985.
13. K. Nagaya and Y. Hai, “Seismic Response of Underwater Members of Variable Cross Section,” Journal of Sound and Vibration, 103, 119-138, 1985.
14. J.Y. Chang and W.H. Liu, “Some Studies on the Natural Frequencies of Immersed Restrained Column,” Journal of Sound and Vibration, 130(3), 516-524, 1989.
15. J.T. Xing, W.G. Price and M.J. Pomfret, L.H. Yam, “Natural Vibration of a Beam-Water Interaction System,” Journal of Sound and Vibration, 199(3), 491-512, 1997.
16. 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.
17. J.S. Wu and K.W. Chen, “An Alternative Approach to the Structural Motion Analysis of Wedge-Beam Offshore Structures Supporting a Load,” Ocean Engineering, 30, 1791-1806, 2003.
18. J.D. Faires and R.L. Burden, Numerical Methods, PWS Publishing Company, 1993.
19. F.H. Todd, Ship Hull Vibration, Edward Arnold Ltd, London, 1961.
20. T. Sarpkaya, “Forces on cylinders and spheres in a sinusoidally oscillating fluid,” Journal of Applied Mechanics, American Society of Mechanical Engineers, E42, 32-37, 1975.