摘 要
本文之主旨是使用矩陣組合法，來求解一多跨距均勻樑與階梯樑，附帶多個任意位置的各類集中元件(如集中質量、集中質量慣性矩、線性彈簧、旋轉彈簧、彈簧-質量系統等)時的自然頻率及振動模態之正解。為達此目的，吾人首先由一均勻段樑的運動方程式，求得該段樑的橫向位移函數，再將上述橫向位移函數，代入各個簡支撐點及集中元件附著點之位移(與斜率)相容及力(與彎矩)平衡的方程式中，而各別得到一組聯立方程式，接著，又將上述橫向位移函數，分別代入樑的左端及右端的邉界條件方程式中，而各別得到另一組聯立方程式，並將上述全部聯立方程式寫成矩陣的形式。最後，利用類似傳統有限元素法的組合技巧，求得整個結構(振動)系統的特徵方程式 。上述齊次方程式若有非零之解，則其係數行列式須等於零，即 。本文利用半間距法，求解頻率方程式 的根，各個根代表上述結構(振動)系統的一個自然頻率，而對應於各個根(或自然頻率)，可由特徵方程式 ，求得一組係數 ，將這些係數代入各相關段樑的橫向位移函數中，即得整個結構(振動)系統的對應振動模態。本文也同時探討簡支撐及各類集中元件，對樑構件自由振動的影響。

Abstract

The purpose of this study is to determine the exact natural frequencies and mode shapes of multi-span uniform and multi-step Euler-Bernoulli beams with various concentrated elements (such as point masses, rotary inertias, linear springs, rotational springs, spring-mass systems, etc.) by using the matrix assembly method (MAM). To this end, the coefficient matrices for an intermediate pinned support, an intermediate concentrated elements, left-end support and right-end support of a beam are derived, first. Next, the overall coefficient matrix for the whole structural system is obtained by using the assembly technique of the finite element method. Finally, the natural frequencies and the associated mode shapes of the vibrating system are determined by equating the determinant of the last overall coefficient matrix to zero and substituting the corresponding values of integration constants into the associated eigenfunctions respectively. The effects of in-span pinned supports and various concentrated elements on the free vibration characteristics of the beam are also studied.

目 錄
目錄.............................i
圖目錄 ..iv
表目錄 vi
符號說明 viii
中文摘要 xi
英文摘要.... .. xiii
第一章 緒論………………………………………………..….....1
1-1研究背景………………………………………...…...…....1
1-2文獻回顧……………………………..…....…..……….....2
1-2-1附帶「集中元件」均勻樑的文獻回顧.........................2
1-2-2附帶「彈簧-質量系統」均勻樑的文獻回顧.......……....4
1-2-3附帶集中元件「階梯樑」的文獻回顧.................….....5
第二章 理論分析…………………………………...…...….…..7
2-1附帶多個各類「集中元件」多跨距均勻樑的自由振動分析....7
2-1-1各類集中元件及簡支座的係數矩陣...........……..……9
2-1-2不同邊界條件的係數矩陣............………….………...12
2-1-3整個振動系統的係數矩陣………….…..….........….15
2-2附帶多個「彈簧-質量系統」之多跨距均勻樑的自由振動分析….........................................................................17
2-2-1彈簧-質量系統及簡支座的係數矩陣…..…….........18
2-2-2不同邊界條件的係數矩陣…………….……….........21
2-2-3整個振動系統的係數矩陣…………….....…............21
2-3附帶多個集中元件之「階梯樑」的自由振動分析...................23
2-3-1階梯樑上之集中質量及集中質量慣性矩的係數矩陣…..25
2-3-2不同邊界條件的係數矩陣………….........……….…26
2-3-3整個振動系統的係數矩陣……….........……..……..29
第三章 數值分析結果與討論…………………….……..…….31
3-1附帶多個「各類集中元件」多跨距均勻樑的自然頻率及模態…………............................................…...….....................31
3-1-1本文所提理論及電腦程式可靠性的驗證……....…31
3-1-2附帶多個各類集中元件之「單跨距均勻樑」..............…35
3-1-3附帶多個各類集中元件之「多跨距均勻樑」...............41
3-2附帶多個「彈簧-質量系統」多跨距均勻樑的自然頻率及模態………..…..............................................................44
3-2-1本文所提理論及電腦程式可靠性的驗證….......…44
3-2-2附帶一個彈簧-質量系統「二跨距均勻樑」.................51
3-2-3附帶三個彈簧-質量系統「三跨距均勻樑」.................53
3-2-4附帶三個彈簧-質量系統「四跨距均勻樑」.................55
3-3附帶多個集中元件「階梯樑」的自然頻率及模態...............57
3-3-1本文所提理論及電腦程式可靠性的驗證….......…57
3-3-2附帶一個「集中質量」三階梯圓形樑...........…...……59
3-3-3附帶一個「集中質量慣性矩」三階梯圓形樑..................61
3-3-4附帶三個「集中質量與/或集中質量慣性矩」三階梯圓
形樑....................................................................61

第四章 結論………………………………………………...…..70
參考文獻………………….…………………………....……........…73

Balasubramanian, T.S. and Subramanian, G., 1985, “On the performance of a four-degree-of-freedom per node element for stepped beam analysis and higher frequency estimation,” Journal of Sound and Vibration, 99(4), 563-567.
Balasubramanian, T.S., Subramanian, G. and Ramani, T.S., 1990, “Significance of very high order derivatives as nodal degrees of freedom in stepped beam vibration analysis,” Journal of Sound and Vibration ,137(2), 353-356.
Bapat, C. N. and Bapat, C., 1987, “Natural frequencies of a beam with non-classical boundary conditions and concentrated masses,” Journal of Sound and Vibration, 112(1), 177-182.
Bathe, K. J., 1982, Finite Element Procedures in Engineering Analysis, Prentice-Hall, Inc., Englewood Cliffs, New Jersey.
Cha, P. D., 2001, “Natural frequencies of a linear elastica carrying any number of sprung masses,” Journal of Sound and Vibration, 247(1), 185-194.
Chen, D. W., 2001, “Exact solution for the natural frequencies and mode shapes of the uniform and non-uniform beams carrying multiple various concentrated elements by numerical assembly method,” Ph. D. Disertation, National Cheng-Kung University, Tainan, Taiwan.
Chen, D. W. and Wu, J. S., 2002, “The exact solutions for the natural frequencies and mode shapes of non-uniform beams with multiple spring-mass system,” Journal of Sound and Vibration, 255(2), 299-322.
Chen, D. W., 2003, “The exact solutions for the natural frequencies and mode shapes of non-uniform beams carrying multiple various concentrated elements,” Structural Engineering and Mechanics, 16(2), 153-176.
Clough, R. W., Penzien, J. 1975 “Dynamics of structures,” McGraw-Hill, Inc.
De Rosa, M.A., 1994, “Free vibrations of stepped beams with elastic ends,” Journal of Sound and Vibration, 173(4), 557-563.
De Rosa, M.A., Belles P.M. and Maurizi M.J., 1995, “Free vibrations of stepped beams with intermediate elastic supports,” Journal of Sound and Vibration, 181(5), 905-910.
Epperson, J. F., 2003, An introduction to Numerical Methods and Analysis, John Wiley & Son, Inc., New York.
Gupta, G. S., 1970, “Natural flexural waves and the normal modes of periodically supported beams and plates,” Journal of Sound and Vibration, 13(1), 89-101.
Gürgöze, M., 1984, “A note on the vibration of restrained beams and rods with point masses,” Journal of Sound and Vibration, 96(4), 461-468.
Gürgöze, M., 1986, “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.
Gürgöze, M., 1996, “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.
Gürgöze, M., 1998, “On the alternative formulation of the frequency equation of a Bernoulli-Euler beam to which several spring-mass systems are attached in-span,” Journal of Sound and Vibration 217 (3) 585-595.
Gürgöze, M. and BATAN, H., 1996, “On the effect of an attached spring-mass system on the frequency spectrum of a cantilever beam,” Journal of Sound and Vibration 195 (1) 163-168.
Gürgöze, M. and Erol, H., 2001, “Determination of the frequency response function of a cantilever beam simply supported in-span,” Journal of Sound and Vibration, 247(2), 372-378.
Gürgöze, M. and Erol, H., 2002, “On the frequency response function of a damped cantilever beam simply supported in-span and carrying a tip mass,” Journal of Sound and Vibration, 255(3), 489-500.
Hamdan, M. N. and Latif, L. Abdel, 1994, “On the numerical convergence of discretization methods for the free vibrations of beams with attached inertia elements,” Journal of Sound and Vibration, 169(4), 527-545.
Hamdan, M. N. and Jubran, B. A., 1991, “Free and forced vibration of restrained uniform beam carrying an intermediate lumped mass and a rotary inertia,” Journal of Sound and Vibration, 150(2), 203-216.
Jang, S.K., Bert, C.W., 1989a, “Free vibrations of stepped beams: exact and numerical solutions,” Journal of Sound and Vibration, 130(2), 342-346.
Jang, S.K. and Bert, C.W., 1989b, “Free vibrations of stepped beams: higher mode frequencies and effects of steps on frequency,” Journal of Sound and Vibration, 132(1), 164-168.
Ju, F., Lee, H.P. and Lee, K.H., 1994, “On the free vibration of stepped beams,” International Journal of Solids and Structures, 31, 3125-3137.
Karnovsky, I. A. and Lebed, O. I., 2001, Formulas for Structural Dynamics, Tables, Graphs and Solutions, McGraw Hill, New York.
Laura, P. A. A, Maurizi, M. J, Pombo, J.L., 1975, “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.
Laura, P. A. A, Pombo, J.L. and Susemihl, E. A., 1974, “A note on the vibrations of a clamped-free beam with a mass at the free end,” Journal of Sound and Vibration, 37(2), 161-168.
Laura, P.A.A., Rossi, R.E., Pombo, J.L. and Pasqua, D., 1994, “Dynamic stiffening of straight beams of rectangular cross-section: a comparison of finite element predictions and experimental results,” Journal of Sound and Vibration, 150(1), 174-178.
Lee, J. and Bergman, L.A., 1994, “Vibration of stepped beams and rectangular plates by an elemental dynamic flexibility method,” Journal of Sound and Vibration, 171(5), 617-640.
Lin, Y. K. and Yang, J. N., 1974, “Free vibration of a disordered periodic beam,” Journal of Applied Mechanics, 3, 383-391.
Liu, W. H. and Huang, C. C., 1988, “Vibration of a constrained beam carrying a heavy tip body,” Journal of Sound and Vibration, 123(1), 15-29.
Liu, W. H., Wu, J. R. and Huang, C. C., 1988, “Free vibration of beams with elastically restrained edges and intermediate concentrated masses,” Journal of Sound and Vibration, 122(2), 193-207.
Maurizi, M.J. and Belles, P.M., 1994, “Natural frequencies of one-span beams with stepwise variable cross-section,” Journal of Sound and Vibration, 168(1), 184-188.
Meirovitch, L., 1967, Analytical Methods in Vibrations, Macmillan Company, London.
Naguleswaran, S., 2002a, “Natural frequencies, sensitivity and mode shape details of an Euler-Bernoulli beam with one-step change in cross-section and with ends on classical supports,” Journal of Sound and Vibration 252(4), 751-767.
Naguleswaran, S., 2002b, “Vibration of an Euler-Bernoulli beam on elastic end supports and with up to three step changes in cross-section,” International Journal of Mechanical Sciences, 44, 2541-2555.
Naguleswaran, S., 2003, “Transverse vibration of an Euler-Bernoulli uniform beam on up to five resilient supports including ends,” Journal of Sound and Vibration, 261, 372-384.
Subramanian, G. and Balasubramanian, T.S., 1987, “Beneficial effects of steps on the free vibration characteristics of beams,” Journal of Sound and Vibration, 118(3):555-560.
Wu, J. S. and Chen, D. W., 2001, “Free vibration analysis of a Timoshenko beam carrying multiple spring-mass systems by using the numerical assembly technique,” International Journal for Numerical Methods in Engineering, 50, 1039-1058.
Wu, J. S. and Chou, H. M., 1998, “Free vibration analysis of a cantilever beams carrying any number of elastically mounted point masses with the analytical-and-numerical-combined method,” Journal of Sound and Vibration, 213(2), 317-332.
Wu, J. S. and Chou, H. M., 1999, “A new approach for determining the natural frequencies and mode shapes of a uniform beam carrying any number of sprung masses,” Journal of Sound and Vibration, 220(3), 451-468.
Wu, J. S. and Lin, T. L., 1990, “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.