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標題: 複合材料曲樑振動特性之探討
Vibration Analysis of Curved Composite Beams
作者: 劉大成
Liu, Da-Cheng
關鍵字: 複合材料;Vibration Analysis;曲樑;振動特性;Curved;Composite;Beams
出版社: 機械工程學系所
引用: [1] R. F. Gibson, Principles of Composite Material Mechanics, McGraw-Hill, New York, 1994 [2] 張榮明,含有壓電感測器與致動器之脫層複合材料樑動態特性之 探討,碩士論文,中興大學機械工程研究所, 1997 [3] J. N. Reddy, An Introduction to the Finite Element Method, McGraw-Hill, New York, 1984 [4] S.-K. Lee, B. R. Mace, and M. J. Brennan, “Wave propagation, reflection and transmission in curved beams,” Journal of Sound and Vibration 306, 636-656, 2007 [5] T. P. Philippidis and P. S. Theocaris, “The Transverse Poisson''s Ratio in Fiber Reinforced Laminase by Means of a Hybrid Experimental Approach,” Journal of Composite Materials, Vol. 28, No. 3, 1994. [6] 吳煌顏,樑理論應用於分析板狀複合疊層樑之探討,碩士論文,中興大學機械工程研究所, 1992 [7] 林高旭,含壓電片複合材料旋轉樑動態特性之探討,碩士論文,中興大學機械工程研究所, 1999 [8] B. Kang, C. H. Riedel, and C. A. Tan, “Free Vibration Analysis of Planar Curved Beams by Wave Propagation,” Journal of Sound and Vibration, 260, 19–44, 2003 [9] S. Timoshenko and D. H. Young, Vibration Problems in Engineering, Wiley, New York, 1974 [10] S.-Y. Yang and H.-C. Sin, “Curvature-Based Beam Elements for The Analysis of Timoshenko and Shear-Deformable Curved Beams,” Journal of Sound and Vibration, 187(4), 569-584, 1995 [11] A. Krishnan and Y. J. Suresh, “A Simple Cubic Linear Element for Static and Free Vibration Analyses of Curve Beams,” Computers and Structures, 68, 473-489, 1998 [12] M. Ganapathi, B. P. Patel, J. Saravanan, and M. Touratier, “Application of Spline Element for Large-Amplitude Free Vibrations of Laminated Orthotropic Straight/Curved Beams,” Composites Part B, 29B,1-8, 1998 [13] L.-W. Chen and G.-S. Shen, “Vibration and Buckling of Initially Stressed Curved Beams,” Journal of Sound and Vibration, 215(3), 511-526, 1998 [14] M. T. Piovan and V. H. Cortidnez, “Out-of-Plane Vibrations of Shear Deformable Continuous Horizontally Curved Thin-Walled Beams,” Journal of Sound and Vibration, 237(1), 101-118, 2000 [15] M.-Y. Kim and N.-I. Kim, “Analytical and Numerical Studyon Spatial Free Vibration of Non-Symmetric Thin-Walled Curved Beams,” Journal of Sound and Vibration, 258(4), 595-618, 2002 [16] N.-I. Kim and M.-Y. Kim, “Spatial Free Vibration of Shear Deformable Circular Curved Beams with Non-Symmetric Thin-Walled Sections,” Journal of Sound and Vibration, 276, 245–271, 2004 [17] N.-I. Kim and M.-Y. Kim, “Exact Dynamic Stiffness Matrix of Non-Symmetric Thin-Walled Curved Beams Subjected to Initial Axial Force,” Journal of Sound and Vibration, 284, 851–878, 2005 [18] C. H. Riedel and B. Kang, “Free Vibration of Elastically Coupled Dual-Span Curved Beams,” Journal of Sound and Vibration, 290, 820–838, 2006 [19] F. Yang, R. Sedaghati, and E. Esmailzadeh, “Free In-Plane Vibration of General Curved Beams Using Finite Element Method,” Journal of Sound and Vibration, 318, 850–867, 2008 [20] Z. H. Zhu and S. A. Meguid, “Vibration Analysis of A New Curved Beam Element,” Journal of Sound and Vibration, 309, 86–95, 2008 [21] A. M. Yu, C. J. Yang, and G. H. Nie, “Analytical Formulation and Evaluation for Free Vibration of Naturally Curved and Twisted Beams,” Journal of Sound and Vibration, 329, 1376–1389, 2010 [22] H. Ozturk, “In-Plane Free Vibration of A Pre-Stressed Curved Beam Obtained from A Large Deflected Cantilever Beam,” Finite Elements in Analysis and Design, 47, 229–236, 2011 [23] D. A. Saravanos and D. A. Hopkins, “Effects of Delaminations on The Damped Dynamic Characteristics of Composite Laminates: Analysis and Experiments,” Journal of Sound and Vibration, 192(5), 977-993, 1996
推導運動方程時,首先將採用文獻[2]之直樑元素的位移場加入文獻[7]位移場中的弦向曲率κ項及截面側向轉角 項。其次,求得直樑的動能,以及配合本構方程式,求出其應變能,再藉由漢米爾頓原理,推導出直樑元素的質量與勁度矩陣。為了求得曲樑系統的全域質量與勁度矩陣,考慮兩個相鄰直樑元素之間共同節點位移的連續性,找出位移轉換矩陣,再利用此轉換矩陣以及直樑元素的質量與勁度矩陣,推導出模擬的曲樑振動有限元素方程的全域質量與勁度矩陣。
採用本文模式首先分析等向性材密閉環樑,並與解析公式[9]和Ansys比較,結果還蠻一致。其次,本文也針對等向性材開口圓弧形曲樑的進行分析,發現與文獻所分析模態的頻率值接近。除此之外,本文亦分析無拘束密閉單層和疊層複材環樑,以及兩端為固定端(cc)和兩端為自由端(ff)開口圓弧單層和疊層複材曲樑,探討不同的半徑厚度比(R/h)、寬厚比(b/h)、纖維角、圓弧角對系統自然振動的頻率及模態之影響。由分析結果發現上述有些曲樑結構,隨著 b/h或R/h變化會出現模態交錯的現象。並且開口曲樑之頻率會隨著圓弧角變大而減少,而單層複材曲樑之頻率亦隨著纖維角的增加而變小,但其間模態的變化則無規律性。

In this thesis, a finite element model based on two-noded straight beam finite elements, with each node nine degrees of freedom, is developed to simulate the vibration of isotropic and composite closed and open curved beams. The straight beam element being considered contains the transverse shear deformation, torsion, cross-section warping, chordwise curvature, and lateral displacement effects in order to model more accurately the deformation of beam.
To develop the finite element model, first, the displacement fields used in [2] adding the chordwise curvature κ and lateral rotation of cross sections β_y considered in [7] are adopted. Next, the kinetic energy and the strain energy of the straight composite beam are found. Then, by employing Hamilton principle, the mass and stiffness matrices of the straight beam element are obtained. These element matrices are used together with the displacement transformation matrix, which is obtained from the displacement continuity condition between the common nodes of two adjacent straight beam elements, to derive the global mass and stiffness matrices of curved beams.
In the examples, first, the unconstrained closed isotropic curved beams and open curved isotropic beams with both ends fixed (clamped-clamped) or both ends free (free-free) are analyzed. The results are compared with those obtained from Ansys or from literature. They are found in good agreement. Furthermore, the closed as well as open curved single-layered and laminated composite beams are studied, in which the influences of the radius to thickness ratio (R/h), width to thickness ratio (b/h), the fiber angle, and the arc angle of beam on the natural frequencies and modal shapes are investigated. It is found that there exists a switch over of the types of vibration mode at a certain R/h or b/h value. Also, as the fiber angle or the arc angle increases, the natural frequencies get smaller. Their influence on mode shapes, however, is more irregular.
其他識別: U0005-2808201222493600
Appears in Collections:機械工程學系所

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