本文之目的在使用有限元素法，來求解一均勻Timoshenko曲樑(圓弧)或圓環未附帶及附帶多個集中質量(大小為 )、線性彈簧(勁度係數為 )、螺旋彈簧(勁度係數為 )及扭轉彈簧(勁度係數為 )時，在各種邊界條件下的自然頻率與振態。首先，吾人將一連續的均勻曲樑細分為許多直樑元素，將每組相鄰的兩根直樑元素以一節點連接之，並將上述各種集中元素附著於各節點上。因此，吾人只須調整曲樑的剖面積、中心角與半徑，以及附加於各節點上的各種集中元素之大小，便可輕易建立一附帶多個集中質量、線性彈簧、螺旋彈簧及扭轉彈簧之均勻Timoshenko曲樑(圓弧)或圓環的數學模型。根據此數學模型，吾人便可進行任意邊界條件下攜帶任意集中元素之Timoshenko曲樑(圓弧)或圓環的自由振動分析。剪切變形、旋轉慣量及細長比，在各種情況下，對曲樑或圓環自然頻率與振態的影響是本文探討的重點。

The purpose of this thesis is to determine the natural frequencies and mode shapes of the Timoshenko curved beams (or arcs) and rings carrying various concentrated elements in various boundary conditions by using the finite element method (FEM). The above-mentioned concentrated elements include lumped masses, translational springs, rotational (bending) springs and torsional springs. First of all, a uniform continuous curved beam (or ring) is replaced by many straight beam elements and any two adjacent straight beam elements is connected by a node. Next, the aforesaid concentrated elements are attached to each of the nodes, so that the mathematical model for a curved beam or ring carrying any number of lumped masses, translational springs, rotational springs and or torsional springs with various supporting conditions can easily be obtained. Based on the last mathematical model, the natural frequencies and mode shapes of a ring or a curved beam with any cross-sections, subtended angles and radii of curvature carrying various concentrated elements are determined. The influence of shear deformation, rotary inertia and slenderness ratio on the free vibration characteristics of the curved beams and rings are studied.

1.S. S. Rao, “Effects of Transverse Shear and Rotatory Inertia on The Coupled Twist-Bending Vibrations of Circular Rings,” Journal of Sound and Vibration, 16(4), 551-566, 1971.
2.J. Kirkhope, “Out-of-Plane Vibration of Thick Circular Ring,” Journal of the Engineering Mechanics Division, ASCE, 102, EM2, 239-247, April 1976.
3.J.M.M. Silva and A.P.V. Urgueira, “Out-of-Plane Dynamic Response of Curved Beams ─An Analytical Model,” Int. J. Solid Structures, 24(3), 271-284, 1988.
4.T.M. Wang, R.H. Nettleton and B. Keita, “Natural Frequencies for Out-of-Plane Vibrations of Continuous Curved Beams,” J. of Sound and Vibration, 68(3), 427-436, 1980.
5.W.B. Bickford and S.P. Maganty, “On the Out-of-Plane Vibrations of Thick Rings,” J. of Sound and Vibration, 108(3), 503-507, 1986.
6.M. Kawakami, T. Sakiyama, H. Matsuda and C Morita, “In-Plane and Out-of-Plane Free Vibrations of Curved Beams With Variable Sections,” J. of Sound and Vibration, 187(3), 381-401, 1995.
7.Y.B. Yang and C.M. Wu, “Dynamic Response of A Horizontally Curved Beam Subjected to Vertical and Horizontal Moving Loads,” J. of Sound and Vibration, 242(3), 519-537, 2001.
8.B.K. Lee, S.J. Oh and K.K. Park, “Free Vibrations of Shear Deformable Circular Curved Beams Resting on Elastic Foundation,” Int. J. of Structural Stability and Dynamics, 2(1), 77-97, 2002.
9.A.O. Lebeck and J.S. Knowlton, ‘A finite element for the three-dimensional deformation of a circular ring,’ Int. J. for Numerical Methods in Engineering, 21, 421-435, 1985
10.R. Palaninathan and P.S. Chandrasekharan, ‘Curved beam element stiffness matrix formulation,’ Computers & Structures, 21(4), 663-669, 1985
11.C.H. Yoo and J.P. Fehrenbach, ‘Natural frequencies of curved girders,’ Journal of the Engineering Mechanics Division, ASCE, Vol. 107, No. EM2, Apr., 339-354, 1981
12.S.K. Chaudhuri and S. Shore, ‘Dynamic analysis of horizontally curved I-girder bridges,’ Journal of the Structural Division, ASCE, Vol. 103, No. ST8, Proc. Paper 13121, Aug., 1589-1604, 1977
13.R. Davis, R. D. Henshell and G. B. Warburton, “Curved Beam Finite Elements for Coupled Bending and Torsional Vibration,” Earthquake engineering and structural dynamics, Vol.1, 165-175, 1972.
14.W.P. Howson and A.K. Jemah, “Exact Out-of-Plane Natural Frequencies of Curved Timoshenko Beams,” Journal of Engineering Mechanics, ASCE, 125(1), 19-25, Jan. 1999.
15.江列光﹐”曲形樑的自由與強迫振動分析”,國立成功大學系統及船舶機電工程研究所博士論文﹐民國92年.
16.J.S. Wu and L.K. Chiang, "Out-of-plane responses of a circular curved Timoshenko beam due to a moving load" Int. J. of Solids and Structures, Vol.40, p.7425-7448, 2003.
17.J.S. Wu and L.K. Chiang, "A new approach for free vibration analysis of arches with effects of shear deformation and rotary inertia considered" Journal of Sound and Vibration , Vol.64, p.1375-1399, 2005.
18.J.S. Wu and L.K. Chiang, "Free vibration of a circularly curved Timoshenko beam normal to its Initial plane using finite curved beam elements" computers and structures , Vol.82, p.2525-2540, 2004.
19.S.S. Rao. 1971 Aeronaut. J.﹐”Three-dimensional vibrations of a ring on an elastic foundation.
20.張柏豪﹐”應用分布質量轉移矩陣法於攜帶各種集中元素之Timoshenko樑的自由振動分析”﹐國立成功大學系統及船舶機電工程研究所碩士論文﹐民國95年.
21.R. Blevins, Formulas for natural frequency and mode shape, p205, 1979.