弹性地基上转动功能梯度材料Timoshenko梁自由振动的微分变换法求解
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摘要
基于Timoshenko梁理论研究弹性地基上转动功能梯度材料(FGM)梁的自由振动。首先确定功能梯度材料Timoshenko梁的物理中面,利用广义Hamilton原理推导出该梁在弹性地基上转动时横向自由振动的两个控制微分方程。其次采用微分变换法(DTM)对控制微分方程及其边界条件进行变换,计算了弹性地基上转动功能梯度材料Timoshenko梁在夹紧-夹紧、夹紧-简支和夹紧-自由三种不同边界条件下横向自由振动的量纲一固有频率,与已有文献的计算结果进行比较,退化后结果一致。最后讨论了不同边界条件、转速、弹性地基模量和梯度指数对功能梯度材料Timoshenko梁自振频率的影响。结果表明:功能梯度材料Timoshenko梁的量纲一固有频率随量纲一转速和量纲一弹性地基模量的增大而增大;在量纲一转速和量纲一弹性地基模量一定的情况下,梁的量纲一固有频率随着功能梯度材料梯度指数的增大而减小。
Free vibrations of a rotating FGM beam on elastic foundation was investigated based on Timoshenko beam theory. Firstly,the physical neutral surface position for the FGM Timoshenko beam was determined,and two motion governing differential equations of the transverse free vibrations of the rotating beam on elastic foundation were derived by using generalized Hamilton principle. Secondly,DTM was used to transform the differential equations and the boundary conditions. At the same time,the dimensionless natural frequencies of transverse free vibrations of rotating FGM Timoshenko beam on elastic foundation with clamped-clamped,clamped-simply supported and clamped-free three boundary conditions were solved,then the governing differential equation was degenerated and the good agreement among the results of this paper andthose available in the literatures validated the presented approach. Finally,the effects of different boundary conditions,different rotating speeds,different elastic modulus and different gradient indexs on the freevibration frequencies of FGM Timoshenko beam were discussed. The results show that the dimensionless natural frequencies of FGM Timoshenko beam increase with the growth of the dimensionless rotating speeds and the dimensionless elastic foundation modulus. Under a certain dimensionless rotating speeds and dimensionless elastic foundation modulus,the dimensionless natural frequencies decrease along with the growth of the FGM gradient indexes.
Free vibrations of a rotating FGM beam on elastic foundation was investigated based on Timoshenko beam theory. Firstly,the physical neutral surface position for the FGM Timoshenko beam was determined,and two motion governing differential equations of the transverse free vibrations of the rotating beam on elastic foundation were derived by using generalized Hamilton principle. Secondly,DTM was used to transform the differential equations and the boundary conditions. At the same time,the dimensionless natural frequencies of transverse free vibrations of rotating FGM Timoshenko beam on elastic foundation with clamped-clamped,clamped-simply supported and clamped-free three boundary conditions were solved,then the governing differential equation was degenerated and the good agreement among the results of this paper andthose available in the literatures validated the presented approach. Finally,the effects of different boundary conditions,different rotating speeds,different elastic modulus and different gradient indexs on the freevibration frequencies of FGM Timoshenko beam were discussed. The results show that the dimensionless natural frequencies of FGM Timoshenko beam increase with the growth of the dimensionless rotating speeds and the dimensionless elastic foundation modulus. Under a certain dimensionless rotating speeds and dimensionless elastic foundation modulus,the dimensionless natural frequencies decrease along with the growth of the FGM gradient indexes.
引文
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[2]ELISHAKOFF I,PENTARAS D,GENTILINI C.Mechanics of Functionally Graded Material Structures[M].Singapore:World Scientific,2016:13-22.
[3]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.
[4]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 Shirong,LIU Ping.Analogous Transformation of Static and Dynamic Solutions between Functionally Graded Material Beam and Uniform Beam[J].Mechanics in Engineering,2010,32(5):45-49.
[5]LI X F.A Unified Approach for Analyzing Static and Dynamic Behaviors of Functionally Graded Timoshenko and Euler-Bernoulli Beams[J].Journal of Sound and Vibration,2008,318(4):1210-1229.
[6]MOHANTY S C,DASH R R,ROUT T.Free Vibration of a Functionally Graded Rotating Timoshenko Beam Using FEM[J].Advances in Structural Engineering,2013,16(2):405-418.
[7]MESUT S.Fundamental Frequency Analysis of Functionally Graded Beams by Using Different Higher-order Beam Theories[J].Nuclear Engineering and Design,2010,240(4):697-705.
[8]赵家奎.微分变换及其在电路中的应用[M].武汉:华中理工大学出版社,1988:67-151.ZHAO Jiakui.Differential Transformation and Its Application for Electrical Circuits[M].Wuhan:Huazhong University of Science and Technology Press,1988:67-151.
[9]LIU Z F,YIN Y Y,WANG F.Study on Modified Differential Transform Method for Free Vibration Analysis of Uniform Euler-Bernoulli Beam[J].Structural Engineering and Mechanics,2013,48(5):697-709.
[10]KAYA,M O.Free Vibration Analysis of a Rotating Timoshenko Beam by Differential Transform Method[J].Aircraft Engineering and Aerospace Technology,2006,78(3):194-203.
[11]FARZAD E,MOHADESE M.Free Vibration Analysis of a Rotating Mori-Tanaka-based Functionally Graded Beam via Differential Transformation Method[J].Arabian Journal for Science and Engineering,2016,41(2):577-590.
[12]ZHANG D G,ZHOU Y H.A Theoretical Analysis of FGM Thin Plates Based on Physical Neutral Surface[J].Computational Materials Science,2008,44(2):716-720.
[13]GANGULI R.Finite Element Analysis of Rotating Beams[M].Singapore:Springer,2017.
[14]RAO S S.Vibration of Continuous Systems[M].Hoboken:John Wiley and Sons,2007:317-392.
[15]BANERJEE J R.Dynamic Stiffness Formulation and Free Vibration Analysis of Centrifugally Stiffened Timoshenko Beams[J].Journal of Sound and Vibration,2001,247(1):97-115.
[16]WATTANASAKULPONG N,MAO Q B.Dynamic Response of Timoshenko Functionally Graded Beams with Classical and Non-classical Boundary Conditions Using Chebyshev Collocation Method[J].Computers and Structures,2015,119:346-354.