本文將採用模態法與有限元素法來探討三跨距之階梯複合層樑的動態響應；此結構中的第一和第三跨距為單層的Timoshenko 樑，第二跨距為三層的三明治複合層樑所組成。
在模態法方面，為瞭解階梯複合層樑之力學行為，則利用應力場、應變場與位移場的關係推導出應變能項和動能項，再以漢米爾頓原理求得階梯複合層樑之運動方程式，進而計算出模態頻率，並討論在不同的幾何參數下對模態頻率之影響。
在有限元素法方面，則使用靜態的運動方程式推導出其位移場之通解，再借由應變能項與動能項計算出結構的勁度矩陣和質量矩陣，進而利用Lagrange’s equation 解出系統的模態頻率，且選取不同數目之元素來堆疊此結構，並將結果與模態法之結果作比較。

In this thesis, the free vibration of stepped beams is investigated. The Timoshenko beam model is considered. The beam structure has one segment
of sandwich beam. The displacement fields are set up. The strains, stresses, stress resultants and stress-couple resultants, kinetic energy and strain energy of the entire beam are derived. The governing equations are formulated via the Hamilton’s principle.
An analytical method is presented to obtain the modal frequencies and the corresponding sets of mode shape functions of the stepped beam. Further, the shape functions of an element of core and an element of sandwich beam are derived, respectively. Then, the technique of finite element is employed to compute the modal frequencies of the entire beam.
The effects of length and location of the sandwich beam segment on the modal frequencies of the entire beam are studied. Further, the efficiency of the presented finite element computation also is investigated.

1. M. F. Rubinstein and W. C. Hurty, “ Effect of joint rotation on dynamics
of structures”, Journal of the Engineering Mechanics Division, ASCE,
Vol.87, No.6, pp.135-157, 1961.
2. K. W. Levien and B. J. Hartz, “Dynamics flexibility matrix analysis of
frames”, Journal of the Structural Division, ASCE, Vol.89, No.4,
pp.545-536, 1963.
3. L. Rayleigh, The theory of sound, Dover Publications, New York, Vol. I,
pp.293-294, 1945.
4. S. P. Timoshenko, “On the correction for shear of the differential equation
for transverse vibration prismatic bars”, Philosophical Magazine, Vol.41,
pp.744-746, 1921.
5. S. Timoshenko and D. H. Young, Vibration problems in engineering, Van
Nostrandi, Co., New York, N.Y., pp.329-335, 1955.
6. G. Herrmann, “Forced motions of Timoshenko beams”, Journal of
Applied Mechanics, Trans ASME, Vol.22, pp.53-56, 1955.
7. T. C. Huang, “The effect of rotatory inertial and of shear deformation on
the frequency and normal mode equations of uniform beam with simple
end conditions”, Journal of Applied Mechanics, Trans ASME, Vol.28,
No.4, pp.578-584, 1961.
8. J. Y. Shen, E. G. Abu-Saba, W. M. McGinley, L. Sharpe Jr., and L. W.
Taylor Jr., “Continuous dynamic model for tapered beam-like structures”,
Journal of Aerospace Engineering, Vol.47, No.4, pp.435-445, 1994.
9. S. Woinowsky-Krieger, “The effect of an axial force on the vibration of
hinged bars”, Journal of Applied Mechanics, Trans ASME, Vol.17,
pp.35-36, 1950.
10. G. Prathap and T. K. Varadan, “The large amplitude vibration of hinged
beams”, Computers and Structures, Vol.9, pp.219-222, 1978.
11. T. Wah, “The normal modes of vibration of certain nonlinear continuous
system ”, Journal of Applied Mechanics, Trans ASME, Vol.31, pp.139-140,
1964.
12. A. V. Srinivasan, “Large amplitude free oscillations of beams and plates”,
AIAA Journal, Vol.3, pp.1951-1953, 1965.
13. G. Venkateswara Rao, “A comparative study on the use of consistent and
lumped mass approach for the large amplitude free vibrations of slender
beams”, Journal of Structure Engineering, Vol.4, No.4, pp.243-246, 1979.
14. I. S. Raju, G. Venkateswara Rao and K. Kanaka Raju, “Effect of
longitudinal or in-plane deformation and inertia on the large amplitude
flexural vibrations of beams and thin plates”, Journal of Sound and
Vibration, Vol.49, pp.415-422, 1976.
15. Gajbir Singh, A. K. Sharma and G. Venkateswara Rao, “Large-amplitude
free vibrations of beams. —A discussion on various formulations and
assumptions”, Journal of Sound and Vibration, Vol.142, pp.77-85, 1990.
16. J. Hino, T. Yoshimura and N. Ananthanarayana, “Vibration analysis of
nonlinear beams subjected to a moving load using the finite element
method”, Journal of Sound and Vibration, Vol.100, No.4, pp.477-491,
1985.
17. J. Hino, T. Yoshimura and N. Ananthanarayana, “Vibration analysis of a
nonlinear beam subjected to moving loads by using the Galerkin method”,
Journal of Sound and Vibration, Vol.104, No.2, pp.179-186, 1986.
18. G. R. Buchanan, D. C. Huang and T. K. M. Cheng, “Effect of shear on
nonlinear behavior of elastic bars”, Journal of Applied Mechanics, Trans
ASME, Vol.91, pp.212-215, 1970.
19. G. Venkateswara Rao, I. S. Raju and K. Kanaka, “Nonlinear vibrations of
beams considering shear deformation and rotary inertia”, AIAA Journal,
Vol.14, No.5, pp.685-687, 1976.
20. P. N. Singh, V.Sundararajan and Y.C.Das, “Large amplitude vibration of
some moderately thick structural elements,” Journal of Sound and
Vibration, Vol.36, No.3, pp.375-387, 1974.
21. J. N. Reddy and I. R. Singh, “Large deflections and large-amplitude free
vibrations of straight and curved beams”, international Journal for
Numerical Methods in Engineering, Vol.17, pp.829-852, 1981.
22. S. Y. Lee, “On the finite deflection dynamics of thin elastic beams”,
Journal of Applied Mechanics, Trans ASME, pp.961, 1971.
23. J. S. Przemieniecki, “Theory of matrix structural analysis”, McGraw-Hill,
Inc.，pp.70-80, pp.292-297, 1968.
24. Kanwar K. Kapur, “Vibration of a Timoshenko beam, using finite-element
approach”, The Journal of the Acoustical Society of America, Vol.40, No.5,
pp.1058-1063, 1966.
25. D. L. Thomas, J. M. Wilson, R. R. Wilson，“Timoshenko beam finite
element”, Journal of Sound and Vibration, Vol.31, No.1, pp.315-330,
1973.
26. T. Yokoyama, “A reduced intergration Timoshenko beam elemet”, Journal
of Sound and Vibration, Vol.169, No.3, pp.411-418, 1994.
27. B. A. H. Abbas, J. Thomas, “Finite element model for dynamic analysis
of Timoshenko beam”, Journal of Sound and Vibration, Vol.41, No.3,
pp.291-299, 1975.
28. R. Davis, R. D. Henshell, G. B. Warburton, “A Timoshenko beam
element”, Journal of Sound and Vibration, Vol.22, No.4, pp.475-487,
1972.
29. D. J. Dawe, “A finite element for the vibration Timoshenko beams ”,
Journal of Sound and Vibration, Vol.60, No.1, pp.11-20, 1978.
30. M. Levinson, “Vibrations of steepped strings and beams”, Journal of
Sound and Vibration, Vol.49, No.2, pp.287-291, 1976.
31. H. Sato, “Non-linear Free vibrations of stepped thickness beams”, Journal
of Sound and Vibration, Vol.72, No.3, pp.415-422, 1980.
32. S. K. Jang, C. W. Bert, “Free vibration of stepped beams : higher mode
frequencies and effects of steps on frequency”, Journal of Sound and
Vibration, Vol.132, No.1, pp.164-168, 1989.
33. J. Lee.，L. A. Bergman，“The vibration of steeped beam and rectangular
plates elemental dynamic flexibility method”，Journal of Sound and
Vibration, Vol.171, No.5, pp.617-640, 1994.
34. A. P. Gupta，N. Shama，“Forced motion of a stepped semi-infinite
plate”，Journal of Sound and Vibration, Vol.203, No.4, pp.697-705, 1997.
35. A. P. Gupta, N. Shama, “Effect of transverse shear and rotatory inertia on
the force motion of a stepped rectangular beam ”，Journal of Sound and
Vibration, Vol.209, No.5, pp.811-820, 1998.
36. S. Naguleswaran，“vibration and stability of an Euler-Bernoulli beam with
up to three-step changes in cross-section and in axial force”，International
Journal of Mechanical Sciences Vol.45, pp.1563-1579, 2003.
37. Christopher Stuart Anderson，“Improving the structural dynamics of
slender beam-like structures”，Ph.D.dissertation，Victoria University of
Technology，2003.
38. Gere & Timoshenko, Mechanics of Materials, 2nd Ed., 1984.
39. G. R. Cowper, “The shear coefficient in Timoshenko’s beam theory”,
Journal of Applied Mechanics, Vol.33, pp.335-340, 1966.