Homogenized and classical expression for the response of free vibration of simply supported FGM Levinson beams
Wang Xuan2, Li Shirong1,2
1. School of Civil Science and Engineering, Yangzhou University, Yangzhou, Jiangsu 225127；
2. School of Hydraulic, Energy and Power Engineering, Yangzhou University, Yangzhou, Jiangsu 225127
An analytical transition relation between the natural frequencies of functionally graded material (FGM) beam based on the Levinson beam theory and that of the reference homogenous Euler-Bernoulli (E-B) beam is presented under simply supported boundary conditions. Material properties of the FGM beam are assumed to be varied continuously in the beam depth. By examining the mathematical similarity between the governing equations with boundary conditions of the two types of the beams for free vibrations, natural frequency of the FGM Levinson beam is expressed analytically in terms of that of the reference homogenous E-B beam. Consequently, solving the complex coupling differential equations with boundary conditions, or searching for the natural frequency of the FGM Levinson beam is simplified as the calculation of a serious of coefficients which can be easily determined by a specified gradient of the material properties and the geometry of the FGM beam. As a result, homogenized and classical expression for the response of free vibration of simply supported FGM Levinson beams is realized, which can provide a good convenience for engineering applications.
收稿日期: 2016-12-02
出版日期: 2017-09-15
引用本文:
王瑄1，李世荣1,2 . 功能梯度Levinson梁自由振动响应的均匀化和经典化表示[J]. 振动与冲击, 2017, 36(18): 70-77.
Wang Xuan2, Li Shirong1,2. Homogenized and classical expression for the response of free vibration of simply supported FGM Levinson beams. JOURNAL OF VIBRATION AND SHOCK, 2017, 36(18): 70-77.
[1] 仲政, 吴林志, 陈伟球. 功能梯度材料与结构的若干力学问题研究进展[J]. 力学进展, 2010, 40(5):528-541.
ZHONG Zheng, WU Lin-zhi, CHEN Wei-qiu. Research Progress on some mechanical problems of functionally graded materials and structures [J]. Advances In Mechanics, 2010, 40(5):528-541.
[2] Pradhan S C, Murmu T. Thermo-mechanical vibration of FGM sandwich beam under variable elastic foundations using differential quadrature method [J]. Journal of Sound and Vibration, 2009, 321(1–2):342-362.
[3] Li S-R, Su H-D, Cheng C-J. Free vibration of functionally graded material beams with surface-bonded piezoelectric layers in thermal environment [J]. Applied Mathematics and Mechanics (English Ed.), 2009, 30(8): 969-982
[4] Şimşek M, Kocatür T. Free and forced vibration of a functionally graded beam subjected to a concentrated moving harmonic load [J]. Composite Structures, 2009, 90(4): 465-473
[5] Khalili S M R, Jafari A A, Eftekhari S A. A mixed Ritz-DQ method for forced vibration of functionally graded beams carrying moving loads [J]. Composite structures, 2010, 92(10): 2497-2511.
[6] 李世荣,刘平. 功能梯度梁与均匀梁静动态解间的相似转换[J].力学与实践，2010, 32(5):45-49.
LI Shi-rong, LIU Ping. Similarity transformation between functionally graded beams and uniform Liang Jing dynamic solutions[J]. Mechanics in Engineering，2010, 32(5):45-49.
[7] 李清禄,李世荣. 功能梯度材料梁在后屈曲构形附近的自由振动[J].振动与冲击,2011, 30(9):76-78.
LI Qing-lu, LI Shi-rong. Free vibration of functionally graded material beam near the post buckling configuration[J]. Journal of vibration and shock, 2011, 30(9):76-78.
[8] Yang J, Chen Y. Free vibration and buckling analysis of functionally graded beams with edge cracks [J]. Composite Structures, 2008, 83(1): 48-60
[9] Taeprasartsit S. Nonlinear free vibration of thin functionally graded beams using the finite element method [J]. Journal of Vibration and Control, 2015, 21(1): 29–46
[10] Atmane H A, Tounsi A, Meftah S A, and Belhad H A. Free vibration behavior of exponential functionally graded beams with varying cross-section [J]. Journal of Vibration and Control, 2011, 17(2): 311–318.
[11] Cazzani A, Stochino F, Turco E. On the whole spectrum of Timoshenko beams. Part I: a theoretical revisitation [J]. Zeitschrift Für Angewandte Mathematik Und Physik Zamp, 2016, 67(24):1-30.
[12] Cazzani A, Stochino F, Turco E. On the whole spectrum of Timoshenko beams. Part II: further applications [J]. Zeitschrift Für Angewandte Mathematik Und Physik Zamp, 2016, 67(25):1-21.
[13] Li X F. A unified approach for analyzing static and dynamic behaviours of functionally graded Timoshenko and Euler-Bernoulli beams [J]. Journal of Sound and Vibration, 2008, 318(4-5):1210-1229
[14] Sina S A, Navazi H M, Haddadpour H. An analytical method for free vibration analysis of functionally graded beams [J]. Materials and Design, 2009, 30(3): 741–747
[15] 吴晓,罗佑新. 用Timoshenko梁修正理论研究功能梯度材料梁的动力响应[J].振动与冲击, 2011, 30(10):245-248.
WU Xiao, LUO You-xin. Dynamic responses of a beam with functionally graded Materials with Timoshenko beam correction theory [J]. Journal of vibration and shock, 2011, 30(10):245-248.
[16] 马连生,牛牧华. 基于物理中面FGM梁的非线性力学行为[J].兰州理工大学学报, 2010, 36(5):163-167.
MA Lian-sheng, NIU Mu-hua. Nonlinear mechanical behavior of FGM classic beams based on physical neutral surface [J]. Journal of Lanzhou University of Technology, 2010, 36(5):163-167.
[17] Pradhan K K, Chakarverty S. Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh-Ritz method [J]. Composites Part B, 2013, 51(4): 175-184
[18] Nguyen T K, Vo T P, Thai H T. Static and free vibration of axially loaded functionally graded beams based on the first-order-shear deformation theory [J]. Composites Part B, 2013, 55(55):147-157.
[19] Li S R, Wan Z Q, Zhang J H. Free vibration of functionally graded beams based on both classical and the first-order shear deformation beam theories [J]. Applied Mathematics and Mechanics, 2014, 35(5):591-606.
[20] Tossapanon P, Wattanasakulpong N. Stability and free vibration of functionally graded sandwich beams resting on two-parameter elastic foundation [J]. Composite Structures, 2016(142):215-225.
[21] Ebrahimi F, Mokhtari M. Free Vibration Analysis of a Rotating Mori-Tanaka-Based Functionally Graded Beam via Differential Transformation Method [J]. Arabian Journal Science and Engineering, 2016, 41(2):577-590.
[22] Aydogdu M, Taskin V. Free vibration analysis of functionally graded beams with simply supported edges [J]. Materials and Design, 2007, 28(5):1651-1656.
[23] Şimşek M. Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories [J]. Nuclear Engineering and Design, 2010, 240(4):697-705.
[24] Mahi A, Bedia E A A, Tounsi A, Mechab I. An analytical method for temperature-dependent free vibration analysis of functionally graded beams with general boundary conditions [J]. Composite Structures, 2010, 92(8): 1877-1887.
[25] Vo T P, Thai H T. Vibration and buckling of composite beams using refined shear deformation theory [J]. International Journal of Mechanical Sciences, 2012,62 (1):67-76.
[26] Thai H T, Vo T P. Bending and free vibration of functionally graded beams using various higher-order shear deformation beam theories [J]. International Journal of Mechanical Sciences, 2012, 62 (1): 57–66.
[27] Pradhan K K, Chakraverty S. Effects of different shear deformation theories on free vibration of functionally graded beams [J]. International Journal of Mechanical Science, 2014, 82(5):149-160.
[28] Vo T P, Thai H T, Nguyen T K, Inam F. Static and vibration analysis of functionally graded beams using refined shear deformation theory [J]. Meccanica, 2014, 49(1):155–168.
[29] Vo T P, Thai H T, Nguyen T K, Maheri A, Lee J. Finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory [J]. Engineering Structures, 2014, 64(4):12-22
[30] Shen H S, Wang Z X. Nonlinear analysis of shear deformable FGM beams resting on elastic foundations in thermal environments [J]. International Journal of Mechanical Sciences, 2014,81(4) :195-206
[31] Zoubida K, Daouadji T H, Hadji L, Tounsi A, Bedia A, Abbes E. A New Higher Order Shear Deformation Model of Functionally Graded Beams Based on Neutral Surface Position [J]. Transations of the Indian Institute of Metals, 2016,69(3):683–691
[32] Khan A A, Alam M N, Rahman N U, Wajid M. Finite Element Modelling for Static and Free Vibration Response of Functionally Graded Beam [J]. Latin American Journal of Solids and Structures, 2016, 13(4):690-714
[33] Wang X, Li S. Free vibration analysis of functionally graded material beams based on Levinson beam theory [J]. Applied Mathematics and Mechanics, 2016, 37(7):861-878.
