請用此 Handle URI 來引用此文件: http://hdl.handle.net/11455/91292
標題: Studies of Coupled Free Vibration of a Spinning Flexible Disk and a Tapered Pre-twisted Beam attached to Its Rim
旋轉撓性圓盤與連接於其外緣漸縮預扭樑耦合自由振動之探討
作者: 江韋銘
Wei-Ming Jiang
關鍵字: disk
beam
vibration
frequency
model shape
圓盤

振動
頻率
模態
引用: 1. H. Lamb, and R. V. Southwell , 'The Vibration of Spinning Disk,' Proceedings of Royal Society, Vol. 99, pp. 272-280, 1921. 2. D. Lee, and A. M. Waas, 'Stability Analysis of a Rotating Multi-layer Annular Plate with a Stationary Friction Load,' International Journal of Mechanical Sciences, Vol. 39, No. 10, pp. 1117-1138 (1997). 3. S. C. Yu, and S. C. Huang, 'Vibration of a Three- layered Visco-elastic Sandwich Circular Plate,' International Joural of Mechanical Sciences, Vol. 43, pp. 2215-2236, 2001. 4. H. J. Wang, and L. W. Chen, 'Vibration and Damping Analysis of a Three-layered Composite Annular Plate with a Viscoelastic Mid-layer,' Composite Structures, Vol. 58, pp. 563-570, 2002. 5. 蔡家偉, 複合材料軸-圓盤系統振動特性之探討, 碩士論文, 中興 大學機械工程研究所, 2005. 6. 廖榮川, 承受空間中固定外力之複合材料軸-圓盤系統動態響應 分析, 碩士論文, 中興大學機械工程研究所, 2008. 7. B. Downs, Transverse Vibration of Cantilever Beams Having Unequal Breadth and Depth Tapers,' Journal of Applied Mechanics, ASME, pp. 737-742, December, 1977. 8. F. Sisto and A. T. Chang, 'A Finite Element for Vibration Analysis of Twisted Blades Based on Beam Theory,' AIAA Journal, Vol. 22, No. 11, pp. 1646-1651, 1984. 9. A. Rosen,'Structural and Dynamic Behavior of Pretwisted Rods and Beams,' American Society of Mechanical Engineers, Vol. 44, No. 12, Part I, pp. 483-514, 1911. 10. J. R. Banerjee, 'Free Vibration Analysis of a Twisted Beam Using the Dynamic Stiffness Method,' International Journal of Solids and Structures, Vol. 38, pp. 6703-6722, 2001. 11. S. M. Lin, J. F. Lee, S. Y. Lee and W. R. Wang, 'Prediction of Vibration of Rotating Damped Beams with Arbitrary Pretwist,' International Journal of Solids and Structure, Vol. 48, pp. 1494-1504, 2006. 12. O. Ozdemir and M. O. Kaya, 'Flapwise Bending Vibration Analysis of Rotating Tapered Cantilever Bernoulli-Euler Beam by Using the Differential Transform Method,' Journal of Sound and Vibration, Vol. 289, pp. 413-420, 2006. 13. O. Ozdemir and M. O. Kaya, 'Flapwise Bending Vibration Analysisof Double Tapered Rotating Euler-Bernoulli Beam by Using theDifferential Transform Method,' Meccanica, Vol. 41, pp. 661-670, 2006. 14. R. Ganesan and A. Zabihollah, 'Vibration Analysis of Tapered Composite Beams Using a Higher-Order Finite Element. Part I: Formulation,' Journal of Composite Structure, Vol. 77, pp. 306–318, 2007. 15. O. Ozdemir and M. O. Kaya, 'Vibration Analysis of a Rotating Tapered Timoshenko Beam Using DTM,' Archive of Applied Mechanics, Vol. 78, No. 5, 2010. 16. 林嘉慶, 含預扭角複合材料旋轉樑振動特性之探討, 碩士論文, 中興大學機械工程研究所, 2009. 17. 陳俊男, 固定於可移動圓盤旋轉預扭漸縮複合材料樑動態響應 之探討, 碩士論文, 中興大學機械工程研究所, 2011. 18. T. Tomioka, Y. Kobayashi and G. Yamada, 'Analysis of Free Vibraltion of Rotating Disk-Blade Coupled Systems by Using Artificial Springsand Orthogonal Polynomials,' Journal of Sound and Vibration, Vol. 191 , No. 1, pp. 53-73, 1996. 19. Y.J. Yan, P. L. Cui, H. N. Hao, 'Vibration Mechanism of a Mistuned Bladed-Disk,' Journal of Sound and Vibration, Vol. 317, pp. 294-307, 2008. 20. O. Luo, 'Free Transverse Vibration of Rotating Blades in a Bladed Disk assembly,' Acta Mech, Vol. 223, pp. 1385-1396, 2012. 21. R. F. Gibson, Principles of Composite Material Mechanics, McGraw-Hill, New York, 1994. 22. 蘇振文, 含壓電制動器與感測器之複合材料疊層板振動之數位控 制, 碩士論文,中興大學機械研究所,1995. 23. J. N. Reddy, An Introduction to Finite Element Method, McGraw-Hill , New York, 1984. 24. C. W. Bert and C. D. Kim, 'Whirling of Composite- Material Driveshafts Including Bending-Twisting Coupling and Transverse Shear Deformation, ' Journal of Vibration and Acoustics, Vol. 117, pp. 17-21, 1995. 25. 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, 中興大學機械研究所, 1999. 26. Gouri Dhatt and Gilbert Touzot, The Finite Element Method Displayed, 新智, 1984.
摘要: The main objective of this thesis is to develop a finite element model for the vibration analysis of a spinning flexible disk with a pre-twisted tapered beam attached to its rim, and use this model to investigate coupled vibration between the disk and the beam. In the modeling of the beam, the transverse shear deformation, torsion, chordwise curvature, warping, bending in both transverse and sideways directions, pre-twisted angle, and the width-tapered effects are considered. By adopting a rotating coordinate system, the assumed displacement fields of the disk and the beam along with constitutive equations of materials are then used to obtain the strain energy and kinetic energy of the coupled system. Next, the Hamilton's principle is employed together with the finite element method to derive the finite element equations of motion. These equations are then transformed into a first-order form for analyses of the natural frequencies and model shapes of systems. In the above finite element model, nine-node isoparametric quadrilateral elements with 7 degrees of freedom per node are used for the disk, while for the beam one-dimensional three-node elements, each having 23 degrees of freedom, are considered. Also, the Lagrangian and/or the mixed Lagrangian-Hermite interpolation functions are used to approximate the displacement fields of the disk and beam finite elements. In this thesis, the first ten mode shapes and natural frequencies of the single disk system or the single beam system (or single systems for short) and coupled disk-beam systems (or coupled systems for short) are studied. The influences on natural frequencies and model shapes of the factors such as materials being used, the thickness of disk, and the tapered width, stagger angle and precone angle of beams are analyzed. First in the numerical examples, three cases of thickness ratios of disk and beam, which are 1, 2, and 5, are analyzed. In these cases, the disk and beam are assumed made of isotropic material. The results are compared with those obtained from commercial software Ansys. It is shown that both natural frequencies and mode shapes obtained from the present model and those of Ansys are better in agreement when the thickness ratio is larger, i.e., 5. The results obtained from various examples being studied in this thesis indicate that mode shapes and natural frequencies of the coupling system and the single systems have many in common. However, when natural frequencies of the single disk system and the single beam system fall in the overlapping frequency range of both systems, modes corresponding to the coupled vibration of the disk and the beam appear. The coupling modes' frequencies are closer to those of the single disk system. For both single beam systems and coupled systems, in the case of lower beam-dominated modes, if the rotating speeds are the same the trends of the variation of frequencies with the tapered width, the stagger angle or the precone angle of beams of both systems are similar. While for higher modes, when coupling modes appear, both trends of variation of frequencies and mode shapes vary very differently with the above factors.
本論文主要目的在建立一有限元素模式分析旋轉撓性圓盤於其外 緣連接一撓性預扭漸縮樑之振動,並探討圓盤與樑振動耦合的影響。 所採用的樑模式中包含橫向剪力變形、扭轉、弦向曲率、截面翹曲、 橫向與側向彎曲、預扭、以及寬度漸縮等效應。在動座標的描述下, 利用圓盤和樑的假設位移場搭配本構方程式求出系統的動能和應變能 ,再利用漢米爾頓原理配合有限元素方法推導出圓盤和樑耦合系統之 有限元素運動方程式,並將此方程轉換為一階的系統方程用以分析系 統自然振動頻率和模態。有限元素模式中,圓盤採用九節點每節點有 七個自由度的等參四邊形元素,樑則採用一維三節點具23個自由度的 元素,搭配拉格朗治和拉格朗治-赫米特混合型內插函數,來近似圓 盤與樑有限元素的位移變形。 本文利用上述之有限元素模式分析圓盤-樑耦合系統與單純圓盤 或樑系統(通稱為單一系統)前10個模態及自然頻率,探討不同的圓盤 厚度、樑之寬度漸縮、使用之材料、樑之攻角與傾角等因素對系統自 然頻率與模態的影響。文中首先探討圓盤與樑之厚度比值分別為1、2 與5的情形,考慮材料為等向性,並與套裝軟體Ansys分析之結果比 較,發現比值較大時,兩者分析結果較為一致。由本文分析的各種結 果顯示,耦合系統與單一系統的模態與頻率有一定的相似度。然而於 單純圓盤與單純樑自然頻率重疊之頻域,耦合系統會有耦合模態出現 ,此耦合模態的頻率較為接近所對應之單純圓盤模態的頻率。對耦合 系統與單純樑系統而言,隨樑寬度漸縮幅度、攻角或傾角的改變,低 頻且以樑為主之模態,此二系統在相同轉速時的頻率變化趨勢較為一 致,但兩者之高頻模態頻率與模態的變化趨勢則變化相當大。
URI: http://hdl.handle.net/11455/91292
文章公開時間: 2016-11-27
顯示於類別:機械工程學系所

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