功能梯度材料可以提高结构的强度、改善质量分布和保证工程结构的完整性，因此轴向功能梯度变截面梁已广泛应用于土木、机械和航空工程。文章提出了用插值矩阵法计算轴向功能梯度Timoshenko梁自由振动固有频率，首先，基于Timoshenko梁理论，将轴向功能梯度Timoshenko梁自由振动固有频率的计算转化为一组非线性变系数常微分方程特征值问题，然后，运用插值矩阵法可一次性地计算出轴向功能梯度变截面梁各阶振动固有频率，并可同时获取相应的振型函数。论文方法对于材料梯度函数和截面几何轮廓的具体形式无任何限制条件，计算结果与现有结果对比，发现吻合良好，表明了论文方法的有效性。
Abstract
Non-uniform beams with varying axially material properties are widely used in civil, mechanical and aeronautical engineering, due to the fact that they can improve distribution of strength and weight, and guarantee the structural integrity. In this paper, an interpolating matrix method (IMM) for determining the natural frequencies orders of free transverse vibration of axially functionally graded Timoshenko beams is proposed. Firstly, based on the Timoshenko beam theory. the governing equations of free vibration analysis of axially functionally graded Timoshenko beams are transformed into a set of nonlinear characteristic ordinary differential equations with variable coefficients. Then, the interpolating matrix method (IMM) is adopted to solve the established equations, all the natural frequencies orders of free transverse vibration companying with the corresponding vibration mode functions of axially functionally graded beam were calculated at a time. Furthermore, the present methods do not pose any restrictions on both the type of material gradation and the variation of the cross section profile. and by comparing with the existing results of numerical examples, the validity of the present method is confirmed.
关键词
变截面梁 /
横向振动 /
固有频率 /
插值矩阵法 /
功能梯度材料
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Key words
variable cross-section beam /
transverse vibration /
natural frequency /
the interpolating matrix method, functionally graded material(FGM).
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参考文献
[1] Ichikawa K. Functionally graded materials in the 21 century: a workshop on trends and forecasts [M]. Japan: Kluwer Academic Publishers, 2000.
[2] B.V. Sankar, An elasticity solution for functionally graded beams [J].Compos. Sci. Technol. 61 (2001) 689–696.
[3] A Chakraborty, S. Gopalakrishnan, J.N. Reddy, A new beam finite element for the analysis of functionally graded materials [J]. Int. J. Mech. Sci. 45 (2003):519–539.
[4] A.J.Goupee, S.V. Senthil, Optimization of natural frequencies of bidirectional functionally graded beams [J].Struct. Multidiscip. Optim. 32 (2006) 473–484.
[5] M. Aydogdu, V. Taskin, Free vibration analysis of functionally graded beams with simply supported edges [J] Mater. Des. 28 (2007) 1651–1656.
[6] Tong X, Tabarrok B, Yeh KY (1995) Vibration analysis of Timoshenko beams with non-homogeneity and varying cross-section [J]. J Sound Vib 186:821–835.
[7] Sorrentino S, Fasana A, Marchesiello S (2007) Analysis of non-homogeneous Timoshenko beams with generalized damping distributions [J]. J Sound Vib 304:779–792.
[8] Sundaramoorthy Rajasekaran. Emad Norouzzadeh Tochaei. Free vibration analysis of axially functionally graded tapered Timoshenko beams using differential transformation element method and differential quadrature element method of lowest-order[J]. Meccanica (2014) 49:995–1009
[9] Shahba A, Attarnejad R, Tavanaie Marvi M, Hajilar S (2011), Free vibration and stability analysis of axially functionally graded tapered Timoshenko beams with classical and non-classical boundary conditions. Composites [J]. Part B, Eng 42:801–808.
[10] Huang Y, Li X F. A new approach for free vibration analysis of axially functionally gradedbeams with non-uniform cross-section [J].Journal of Sound and Vibration, 2010, 329:2291-2303.
[11] Elishakoff I, Candan S. Apparently first closed-form solutions for vibrating in homogeneous beams[J].International Journal of solids and structures,2001,38:3411-3441.
[12] Calio I, Elishakoff I. Closed-form trigonometric solutions for inhomogeneous beam columns on elastic foundation [J]. International Journal of structural Stability and Dynamics, 2004, 4:139-146.
[13] Hein H, Feklistova L. Free vibrations of non-uniform and axially functionally graded beams using Haar wavelets [J].Engineering Structures, 2011, 33:3696-3701.
[14] Mabie H H, Rogers C B. Transverse vibrations of tapered cantilever beams with end loads [J]. Journal of the Acoustical Society of America, 1964, 36:463-469.
[15] Niu Zhong-rong, Ge Da-li, Cheng Chang-zheng, Hu Zong-jun. Determining stress singularity exponents of plane V-notches in bonded bimaterial [J].Journal of University of Science and Technology of China, 2008, 38(3): 314- 319.
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