臺灣博碩士論文加值系統

一根楔形樑在空氣中(本文稱為乾樑)之自然頻率與其浸在水中(本文稱為濕樑)之自然頻率不一樣，係由於水的影響。因此，假如吾人在計算乾樑的自然頻率及相對應的振態時，有考慮到濕樑振動時所牽涉到的附加質量(added mass)效應，則所得之自然頻率及振態應該會與濕樑相等。根據上述的觀念，本文使用兩種方法來求楔形樑在水中振動的自然頻率及相對應之振態：有限元素法(finite element method；FEM)及點附加質量法(point added mass method ;PAM)。在使用有限元素法解題時，吾人將樑浸水部份的長度等分成 個樑元素，然後使用附加質量係數 來估算每一樑元素的附加質量 ，再將 平均分配到各相關樑元素之兩端節點(nodes)上，而以集中質量之方式附加到乾樑元素之相關節點上，以求得乾樑在水中振動時的自然頻率及相對應之振態。在使用點附加質量法解題時，吾人先使用Bessel函數來求楔形樑之頂端未附帶集中質量時在空氣中振動之自然頻率及正規化振態。接著，吾人將浸水部份的各個附加集中質量與樑頂端所附帶之實際集中質量所產生的慣性力(inertial forces)施加到上述的乾樑上，並利用擴展理論，來推導相關的特徵值方程式。後者之特徵值與特徵向量即為頂端附帶一集中質量之乾樑在水中振動之自然頻率及振態。

It is well-known that, due to effect of the surrounding water, the natural frequencies of a beam in air (or dry beam) are different from those of the same beam immersed in water (or wet beam). However, if the natural frequencies and the associated mode shapes of a dry beam are calculated by taking account of the “added mass” for the immersed beam, then the last natural frequencies and mode shaps will be equal to the corresponding ones of the wet beam. Based on the last concept, the closed form solutions for natural frequencies and the associated mode shapes of the dry beam were determined first, then the partial differential equation of motion for the wet beam was transformed into a matrix equation by using the expansion theorem and the foregoing closed form solutions of free vibration responses for the dry beam. Solving the last matrix equation will give the required natural frequencies and the associated mode shapes of the wet beam. The formulation of this paper is available for the fully or partially immersed double tapered beams with either circular, square or rectangular cross-sections. The taper ratio for width and that for depth may be equal or unequal. The numerical results of this paper were compared with the existing results or the finite-element-method results and good agreement was achieved.

Keywords: Dry beam; Wet beam; Added mass; Natural frequencies; Mode shapes; Circular or rectangular cross-sections

摘要…………………………………………………………I
誌謝………………………………………………………II
目錄………………………………………………………III
表目錄………………………………………………………V
圖目錄……………………………………………………VI
符號說明…………………………………………………VII
第一章 緒論……………………………………………… 1
1-1 研究動機………………………………………………1
1-2 文獻回顧………………………………………………2
1-3 研究方法………………………………………………5
第二章 有限元素分析…………………………………… 6
2-1 基本假設………………………………………………6
2-2 樑元素之質量矩陣與勁度矩陣………………………7
2-3 有限元素模型…………………………………………9
2-4 乾樑頂端附帶一集中質量之元素性質矩陣……… 10
2-5 濕樑頂端附帶一集中質量之元素性質矩陣……… 11
第三章 點附加質量法……………………………………13
3-1乾樑的運動方程式及其自然頻率與振態……………13
3-2 頂端附帶一集中質量之濕樑的自然頻率與振態… 18
第四章 數值分析結果與討論……………………………23
4-1 理論與電腦程式的可靠性………………………… 23
4-2 浸水的效應………………………………………… 27
4-3 點附加質量法(PAM)的適用性………………………30
第五章 結論………………………………………………33
參考文獻………………………………………………… 34
附錄A：乾樑振態之閉式解析解…………………………39
附錄B：Bessel函數的相關公式 …………………………42
附錄C：樑元素之質量矩陣與勁度矩陣…………………44
自述……………………………………………………… 47

1.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, Vol.41, p.p.397-405, 1975.
2.P.A.A. Laura, E.A. Susemihl, J.L. Pombo, L.E. Luisoni and R. Gelos, “On the dynamic behavior of structural elements carrying elastically mounted concentrated masses,” Applied Acoustic, Vol. 10, p.p.121-145, 1977.
3.M. G rg ze, “A note on the vibration of restrained beam and rods with point masses,” Journal of Sound and Vibration, Vol.96 (4), p.p.467-468, 1984.
4.P.A.A. Laura, C.P. Filipich and V.H. Cortinez, “Vibrations of beams and plates carrying concentrated masses,” Journal of Sound and vibration, Vol.112, p.p.177-182, 1987.
5.R.E. Rossi, P.A.A. Laura, D.R. Avalos and H.A. Larrondo, “Free vibration of Tiomshenko beams carrying elastically mounted concentrated masses,” Journal of Sound and Vibration, Vol.165, p.p.209-223, 1993.
6.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, Vol.136 (2), p.p.201-213, 1990.
7.J.S. Wu and C.G. Huang, “Free and forced vibration of a Timoshenko beam with any number of translation and rotational springs and lumped masses,” International Journal of Communications Numerical Method in Engineering, Vol.11, p.p.743-756, 1995.
8.J.S. Wu and S.S. Lou, “Use of the analytical-and-numerical-combined method to the free vibration analysis of a rectangular plate with any number of point masses and translational springs,” Journal of Sound and Vibration, Vol.200 (2), p.p.179-194, 1997.
9.J.S. Wu and H.M. Chou, “Free vibration analysis of a cantilever beam carrying any number of elastically mounted masses with the analytical-and-numerical- combined-method,” Journal of Sound and Vibration, Vol.213 (2), p.p.317-332, 1998.
10.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, Vol.229 (3), p.p.549-578, 2000.
11.B. Posiadala, “Free vibration of uniform Timoshenko beams with attachments,” Journal of Sound and Vibration, Vol.204 (2), p.p.359-369, 1997.
12.M.W.D. White and G.R. Heppler, “Vibration modes and frequencies of Tiomshenko beams with attached rigid bodies,” Journal of Applied Mechanics, Vol.62, p.p.193-199, 1995.
13.N. Popplewell and Daqing Chang, “Free vibration of a complex Euler-Bernoulli beam,” Journal of Sound and Vibration, Vol.190 (5), p.p.852-856, 1996.
14.M.A. De Rosa, S. Ascoli and S. Nicastro, “Exact dynamic analysis if beam mass systems,” Journal of Sound and Vibration, Vol.196 (4), p.p.529-533, 1996.
15.M.A. De Rosa and M.J. Maurizi, “The influence of concentrated masses and pasternak soil on the free vibrations of Euler beams-exact solution,” Journal of Sound and Vibration, Vol.212 (1), p.p.573-581, 1998.
16.J.S. Wu and H.M. Chou, “A new approach for determining the natural frequencies and mode shapes of a uniform beam carrying any number of spring masses,” Journal of Sound and Vibration, Vol.220 (3), p.p.451-468, 1999.
17.E.T. Cranch and Alfred A. Adler, “Bending vibration of variable section beams,” Journal of Applied Machines, ASME, March, 1956.
18.G.W. Housner, M. ASCE, and W.O. Keightley, “Vibration of linearly tapered cantilever beams,” Journal of Engineering Mechanics Division, Proceedings of Civil Engineers, Vol.88, EM2, p.p.95-123, April, 1962.
19.R.F. Rissone and J.J. Williams, “Vibration of linearly tapered beams,” Journal of Engineering Mechanics Division, Proceedings of the American Society Civil Engineers, EM5, p.p.191-196, October, 1962.
20.H.D. Conway, E.C.H. Becker, and J.F. Dubil, “Journal of Applied Mechanics, ASME,” p.p.329-331, 1964.
21.H.D. Conway and J.F. Dubil, “Vibration frequencies of truncated-cone and wedge beams,” Journal of Applied Mechanics, ASME,” p.p.932-934, December, 1965.
22.A.C. Heidebrecht, “Vibration of non-uniform simple-supported beams,” Journal of the engineering Mechanics Division, Proceedings of American Society of Civil Engineers, Vol.93, EM2, p.p.1-15, April, 1974.
23.Han-Chung Wang, “Generalized hypergeometric function solutions on the transverse vibration of a class of nonuniform beams,” Journal of Applied Mechanics, ASME,” p.p.702-708, September, 1967.
24.W. Carnegie and J. Thomas, “Natural frequencies of long taper cantilevers,” The Aeronautical Quarterly, Vol. XVIII, p.p.309-320, November, 1967.
25.A.V. Krishakaran, “Vibration of tapered cantilever beams and shats,” The Aeronautical Quarterly, p.p.171-177, May, 1969.
26.H.H. Mabie and C.B. Rogers, “Transverse vibration of double-tapered cantilever beams,” The Journal of the Acoustical Society of America, Vol.51, Number 5(part 2), p.p.1771-1775, 1972.
27.D.O. Banks and G.J. Kurowaki, “The transverse vibration of a doubly tapered beam,” Journal of Applied Mechanics, ASME,” p.p.123-126, March, 1977.
28.B. Downs, “Transverse vibration of cantilever beams having unequal breadth and depth tapers,” Journal of Applied Mechanics, ASME,” p.p.737-742, December, 1977.
29.C.D. Bailey, “Direct analytical solution to non-uniform beam problems,” Journal of Sound and Vibration, Vol.56 (4), p.p.501-507, 1978.
30.Arvind K. Gupta and M. ASCE, “Vibration of tapered beams,” Journal of Structural Engineering, Vol.111, No.1, January, 1985.
31.R.O. Grossi and R.B. Bhat, “A note on vibration tapered beams,” Journal of Sound and Vibration, Vol.147 (1), p.p.174-178, 1991.
32.S. Naguleswaran, “Vibration of an Euler-Bernoulli beam of constant depth and with linearly varying breadth,” Journal of Sound and Vibration, Vol.153 (3), p.p.509-522, 1992.
33.S. Naguleswaran, “A direct solution for the transverse vibration of Euler-Bernoulli wedge and cone beams,” Journal of Sound and Vibration, Vol.172 (3), p.p.289-304, 1994.
34.Serge Abrate, “vibration of non-uniform rods and beams,” Journal of Sound and Vibration, Vol.185 (4), p.p.703-716, 1995.
35.S. Naguleswaran, “Comments on Vibration of non-uniform rods and beams,” Journal of Sound and Vibration, Vol.195 (2), p.p.331-337, 1996.
36.P.A.A. Laura, R.H. Gutoerrez and R.E. Rossi, “Free vibration of beams of bilinearly varying thickness,” Ocean Engineering, Vol.23, No.1, p.p.1-6. 1996.
37.A.K. Datta and S.N. Sil, “An analysis of free undamped vibration of beams of varying cross-section,” Computers and Structures, Vol.59, No.3, p.p.479-483, 1996.
38.J.A. Hoffmann and T. Wertheimer, “Cantilever beam vibration,” Journal of Sound and Vibration, Vol.229 (5), p.p.1269-1276, 2000.
39.R.P. Goel, “Transverse vibration of tapered beams,” Journal of Sound and Vibration, Vol.47 (1), p.p.1-7, 1976.
40.S.Y. LEE, H.Y. Ke and Y.K. Kuo, “Analysis of non-uniform beam vibration,” Journal of Sound and Vibration, Vol.142 (1), p.p.15-29, 1990.
41.H. H. Mabie and C. B. Rogers, “Transverse vibrations of double-tapered beams with end support and with end mass,” Journal of the Acoustical Society of America Vol.55, p.p.986-991, 1974.
42.R. Fossman and A. J. Sorensen, “Influence of flexible connections on response characteristic of a beam Vol.129, p.p.829-834, 1980.
43.K. Takahashi, “Eigenvalue problem of a beam with a mass and spring at the end subject to a force.” Vol.71, p.p.453-457, 1980.
44.P. A. A. Laura and R.H. Gutierrez, “Vibration of an elastically restrained cantilever beam of varying cross-section with tip mass of finite length,” Journal of Sound and Vibration, Vol.108, p.p.123-131, 1986.
45.N. M. Auciello, “Transverse vibration of a linearly tapered cantilever beam with tip mass of rotatory inertia and eccentricity,” Journal of Sound and Vibration, Vol.194 (1), p.p.25-34, 1996.
46.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, Vol.99, p.p.361-378, 1985.
47.K. Nagaya and Y. Hai, “Seismic response of underwater members of variable cross section,” Journal of Sound and Vibration, Vol.103, p.p.119-138, 1985.
48.J. Y. Chang and W.H. Liu, “Some studies on the natural frequencies of immersed restrained column,” Journal of Sound and Vibration, Vol.130 (3), p.p.516-524, 1989.
49.A. Uscilowska and J. A. Kolodziej, “Free vibration of immersed column carrying a tip mass,” Journal of Sound and Vibration, Vol.216 (1), p.p.147-157, 1998.
50.G. D. Han and J. R. Sahglivi, “Dynamic model response of wave-excited offshore structures,” Journal of Engineering Mechanics, Transactions of the ASCE, Vol.120, p.p.893-908, 1994.
51.J. T. Xing, W.G. Price and M.J. Pomfret, L. H. Yan, “Natural vibration of a beam-water interaction system,” Journal of Sound and Vibration, Vol.199, p.p.491-512, 1997.
52.G. N. Watson, “A Treatise on the Theory of Bessel Functions,” Cambridge university press, 1995.
53.R. W. Clough and J. Penzien, “Dynamic of Structures,” McGraw-Hill book company, INC., 1975.
54.L. Meirovitch, “Analytical Methods in Vibration,” 1967.
55.J. S. Przemieniecki, “Theory of Matrix Structural Analysis,” Dover Publications, INC., 1968.
56.F. H. Todd, B. Sc., Ph. D., “Ship Hull Vibration.”
57.Klaus-Jurgen Bathe, “Numerical Methods in Finite Element Analysis,” Prentice, Inc., 1976.
58.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.
59.J.D. Faires and R.L Burden, Numerical Methods, PWS Publishing Company, 1993.