本文之主旨在提出兩種改善的轉移矩陣法(TMM)，來求解一攜帶
多種集中元素（包括具有偏心距ei 及轉動慣量a i J , 的集結質量a i m , 、線性彈簧t i k , 及螺旋彈簧r i k , ）的多階梯樑，在多種支撐情況（包括F-F、P-P、C-C、C-F 及C-P 支撐，其中縮寫字母F、P 及C 分別表示free、pinned 及clamped）下的自然頻率及振態。其中，第一種方法本文稱為集結質量轉移矩陣法(LTMM)，而第二種方法本文稱為分佈質量轉移矩陣法(CTMM)。在集結質量轉移矩陣法(LTMM)中，主要變數是每根段樑的橫向位移i Y、斜率i
ψ、彎矩i M 及剪力i V，利用各個站(station)及場(field)的轉移矩陣，吾人便可求得樑右端的狀態變數與其左端的狀態變數（即起始狀態變數， 0 Y 、0 ψ 、0 M 及0 V ）的關係式，集結質量轉移矩陣法(LTMM)的頻率方程式，便是令上述起始狀態變數，滿足實際的邊界條件而得。而在分佈質量轉移矩陣法(CTMM)中，主要變數是每根段樑的四個積分常數( i A 、i B 、i C 及i D )，利用各個節點(node)的轉移矩陣，吾人便可求得右端段樑的積分常數與其左端段樑
的積分常數（即起始積分常數， 1 A 、1 B 、1 C 及1 D ）之關係式，顯然地，分佈質量轉移矩陣法(CTMM)的頻率方程式，亦是令上述起始積分常數，滿足實際的邊界條件而得。
上述兩種改善的轉移矩陣法(TMM)之重點，在於考慮各個集中元素(包括a i m , , i e , a i J , , t i k , 及r i k , )的效應下，來推導一根F-F 樑之各個節點（或站）及各個段樑（或場）的轉移矩陣。因此，吾人只須令= = ∞ t,1 t,n+1 k k 便可獲得一P-P 樑，令= = ∞ t,1 r,1 k k 及= = ∞ t,n+1 r,n+1 k k 便可
獲得一C-C 樑，令= = ∞ t,1 r,1 k k 便可獲得一C-F 樑，…等等。根據以上的理論，在改善的集結質量轉移矩陣法(LTMM)裡，吾人便不必像傳統的集結質量轉移矩陣法那麼樣，必須根據各種不同的支撐條件，分別推導頻率方程式，並使用不同的起始狀態變數值。此外，吾人只須改變各個段樑的直徑及長度，並調整其所攜帶的各個集中元素（ a i m , ,i e , a i J , , t i k , 及r i k , ）的大小（從0 到∞ ），便可獲得許多種不同載荷情況的階梯樑。當然， 本文所提改善的分佈質量轉移矩陣法(CTMM)，亦具有類似的優點。在本研究中，吾人發現，本文所提出的兩種改善的轉移矩陣法(TMM)，可以很容易地解決出現在現有文獻裡的許多問題。為了驗證本文所提理論及所研發電算程式的可靠性，吾人曾將部分本文數值分析的結果，與由現有文獻或傳統有限元素法（FEM）所得的結果相比較，由於一致性良好，故其可靠性應可被接受。

The purpose of this thesis is to present two modified transfer matrix methods(TMM) to determine the natural frequencies and the associated mode shapes of a multi-step beam carrying various multiple concentrated elements (including lumped masses a i m , with eccentricities i e and rotary inertias a i J , , the translational springs t i k , and rotational springs r i k , ) with various supporting conditions (including F-F ,
P-P, C-C, C-F and C-P beams, with the abbreviations of F, P and C denoting the free,pinned and clamped ends, respectively). The first method is called the lumped-mass transfer matrix method (LTMM) and the second method is called the continuous-mass transfer matrix method (CTMM). In the LTMM, the active variables are the transverse
displacement i Y , slope i ψ , bending moment i M and shearing force i V of each beam segment. Use of the lumped-mass transfer matrices of each station and each field, one may determine the active variables of the right end in terms of the corresponding ones of the left end of the beam (i.e., the four initial state variables, 0 Y , 0 ψ , 0 M and 0 V ). The frequency equation of the LTMM is then obtained by setting the last state variables to satisfy the actual boundary conditions. In the CTMM, the active variables are the four integration constants ( i A , i B , i C and i D ) of each beam segment. Use of the consistent-mass transfer matrix of each segment, one may determine the active variables of the right-end segment in terms of the corresponding ones of the left-end segment (i.e., the four active variables, 1 A , 1 B , 1 C and 1 D ). It is evident that the frequency equation of the CTMM is obtained by setting the values of the last four integration constants of the left-end segment ( 1 A , 1 B , 1 C and 1 D ) to satisfy the actual boundary conditions.
The key point of the two modified methods is to derive the transfer matrices of each node (or station) and each beam segment (or field) by considering the effects of various attachments (including a i m , , i e , a i J , , t i k , and r i k , ) for a “F-F beam”, so that one may easily obtain a P-P beam by setting = = ∞ t,1 t,n+1 k k , a C-C beam by setting = = ∞ t,1 r,1 k k together with = = ∞ t,n+1 r,n+1 k k , a C-F beam by setting = = ∞ t,1 r,1 k k , … etc. For this reason, the conventional LTMM of deriving the frequency and using the associated initial state variables according to each specified supporting conditions is not necessary. Besides, one may obtain various stepped beams carrying various concentrated elements by only changing the diameter and length of each beam segment and adjusting the magnitudes (from zero to infinity) of the attached concentrated elements, a i m , , i e , a i J , , t i k , and r i k , . The last advantages for the LTMM are also true for the CTMM. It has been found that the two presented modified TMM can easily solve many problems appearing in the existing literature. The reliability of the presented theories and the developed computer programs have been confirmed by the good agreements between the numerical results of this thesis and those of the exiting literature and/or the conventional finite element methods (FEM).

1.J.S. Przemieniecki, Theory of Matrix Structural Analysis, New York, McGraw-Hill, 1968.
2.K.J. Bathe, Finite Element Procedures in Engineering Analysis, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1982.
3.N.O. Myklestad, 'A new method of calculating normal modes of uncoupled bending vibration of airplane wings and other types of beams,' Journal of Aeronautical Science, 11, 153-162, 1944.
4.W.D. Pilkey and P.Y. Chang, Modern Formulas for Statics and Dynamics, New York: McGraw-Hill, 1978.
5.J.S. Wu and P.Y. Shih, The dynamic analysis of a multispan fluid-conveying pipe subjected to external load, Journal of Sound and Vibration, 239(2), 201-215, 2001.
6.H. Holzer, Analysis of Torsional Vibration, Springer, Berlin, 1921.
7.A.H. Church, Mechanical Vibrations, Wiley, New York, 1957.
8.W.K. Wilson, Practical Solution of Torsional Vibration Problems, 3rd ed, Vol. 1, Chapman & Hall, London, 1956.
9.W.K. Wilson, Practical Solution of Torsional Vibration Problems, 3rd ed, Vol. 2, Chapman & Hall, London, 1963.
10.Calculation Sheets for the Torsional Vibration of Shafting System of a 42100 bhp Container, Ishikawajima-Harima Heavy Industry (IHI), Inc., Japan, 1993.
11.J.S Wu, M. Hsieh, Torsional vibration of a damped shaft system using the analytical-and-numerical-combine method, Marine Technology, Vol. 38(4), 250-260, 2001.
12.J.P. Hartog, J.P. Li, Forced torsional vibrations with damping: an extension of Holzer's method, Journal of Applied Mechanics, Trans. ASME, Vol. 68, p.A-276, 1946.
13.T.W. Spaetgens, B.C. Vancouver, Holzer method for forced-damped torsional vibrations, Journal of Applied Mechanics, Vol. 17, p.59, 1950.
14.J.S. Wu, W.H. Chen, Computer method for forced torsional vibration of propulsive shafting system of marine engineering with or without damping, Journal of Ship Research, Vol. 26, 176-189, 1982.
15.J.S. Wu, I.H. Yang, Computer method for torsion-and-flexure-coupled forced vibration of a shafting system with damping, Journal of Sound and Vibration, Vol. 180(3), 417-435, 1995.
16.M.A. Prohl, A general method for calculating critical speeds of flexible rotors, Journal of Applied Mechanics, 12(5), A-142 to A-148, 1945.
17.R.L. Urban, Extension of Holzer-Myklestad-Prohl calculation of turbo-rotor critical speeds, ASME Paper No. 58-A-246, 1958.
18.J.W. Lund, Stability and damped critical speeds of a flexible rotor in fluid-film bearings, Journal of Engineering for Industry, 509-517, 1974.
19.F. Leckie, E. Pestel, Transfer-matrix fundamental, International Journal of Mechanical Science, Vol. 2, 137-167, 1960.
20.L. Meirovitch, Analytical Methods in Vibrations, London: Macmillan, 1967.
21.C.W. Bert, A.D. Tran, Transient response of a thick beam of bi-modular material, Earthquake Engineering and Structural Dynamics, Vol. 10, 551-560, 1982.
22.A.D. Tran, C.W. Bert, Bending of thick beams of bi-modulus materials, Computers & Structures, Vol. 15(6), 627-642, 1982.
23.J.S. Wu, H.S. Jing, C.C. Cheng, Free vibration analysis of single-layer and two-layer bi-modular beams, Int. J. for Numerical Methods in Engineering, Vol. 28(4), 955-965, 1989.
24.J.S. Wu and C.W. Dai, Dynamic responses of multispan non-uniform beam due to moving loads, Journal of Structural Engineering, ASCE, 113(3), 458-474, 1987.
25.R. Firoozian, H. Zhu, A hybrid method for the vibration analysis of rotor-bearing systems, Proceedings of the Institute of Mechanical Engineers, part C, 205, 131-137, 1991.
26.A.C. Lee, Y. Kang, S.L. Liu, A modified transfer matrix for linear rotor-bearing systems, Journal of Applied Mechanics, Trans. ASME, 776-783, 1991.
27.O.S. Sener, H.N. Ozguven, Dynamic analysis of geared shaft systems by using a continuous system model, Journal of Sound and Vibration, 166(3), 539-556, 1993.
28.M. Aleyaasin, M. Ebrahimi, R. Whalley, Multivariable hybrid models for rotor-bearing systems, Journal of Sound and Vibration, 233, 835-856, 2000.
29.M. Aleyaasin, M. Ebrahimi, R. Whalley, Vibration analysis of distributed-lumped rotor systems, Computer Methods in Applied Mechanics and Engineering., Vol. 189, 545-558, 2000.
30.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(1), 117-182, 1987.
31.M.J. Maurizi, R.E. Rossi and J.A. Reyes,'Vibration frequencies for a uniform beam with one end spring-hinged and subjected to a translational restraint at the other end,'Journal of Sound and Vibration, 48(4), 565-568, 1976.
32.A. Rutenberg, 'Vibration frequencies for a uniform cantilever with a rotational constraint at a point,'American Society of Mechanical Engineers, Journal of Applied Mechanics, 45, 422-423, 1978.
33.K. Takahashi, 'Eigenvalue problem of a beam with a mass and spring at the end subjected to an axial force,' Journal of Sound and Vibration, 71(3), 453-457, 1980.
34.N.G. Stephen,'Vibration of a cantilevered beam carrying a tip heavy body by Dunkerley's method,' Journal of Sound and Vibration, 70(3), 463-465, 1980.
35.J.H. Lau,'Vibration frequencies and mode shapes for a constrained cantilever,'American Society of Mechanical Engineers, Journal of Applied Mechanics, 51, 182-187, 1984.
36.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.
37.M. Gurgoze,'On the vibrations of restrained beams and rods with heavy masses,'Journal of Sound and Vibration, 100(4), 588-589, 1985.
38.W.H. Liu and C.C. Huang,'Vibrations of a constrained beam carrying a heavy tip body,'Journal of Sound and Vibration, 123(1), 15-29, 1988.
39. M.F. Spotts, Design of Machine Elements, Prentice-Hall Inc., Englewood Cliffs, N.J., 4th ed, 1978.
40. S. Timoshenko, D.H. Young and W. Weaver, JR., Vibration Problems in Engineering, 4th Ed., John Wiley & Sons, Inc., 1974.
41. J.S. Wu and D.W. Chen, Free vibration analysis of a Timoshenko beam carrying multiple spring-mass systems by using the numerical assembly technique, Int. J. for Numerical Methods in Engineering, 50, 1039-1058, 2001.
42. J. Gere and S. Timoshenko, Mechanics of Materials, Wadsworth, Inc., Belmont, California, 2nd ed, 1984.