功能梯度材料圆柱壳的振动特性研究
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摘要
研究了具有指数型体积分数的功能梯度材料圆柱壳自由振动的固有频率。根据Love薄壳理论,确定功能梯度材料圆柱壳的内力、位移、应变和曲率的关系式;利用Rayleigh-Ritz方法建立了功能梯度材料圆柱壳自由振动固有频率的特征方程,推导出一端固定一端自由、两端简支两种基本边界条件下的固有频率参数表达式。最后通过两类材料组分的算例,分析了材料组分、体积分数、边界条件以及几何尺寸等因素对功能梯度材料圆柱壳的固有频率的影响。研究表明,构成功能梯度材料的材料组分对FG圆柱壳的频率特征有着明显的影响,体积分数所产生的影响则相对有限;而不同边界条件对FG圆柱壳固有频率的影响主要表现在壳体长度与半径比较小和周向波数较小的情况下。
The vibration of cylindrical shells composed of functionally graded material and with exponential volume fraction law is studied.Expressions of the internal force,displacement,strain and curvature of FG cylindrical shells are analysed by Love's thin shell theory.Based on the Rayleigh-Ritz method,the shell eigenfrequency equation is derived and the natural frequencies with simply supported ends and clamped-free ends are obtained.Then the influence of the configurations of constituent materials,volume fraction,boundary conditions and geometric dimensioning on the natural frequencies of FG cylindrical shells is studied through the examples.The results show that the configurations of constituent materials have considerable effect on the frequency characteristic of FG cylindrical shells,while the volume fraction has minor effect.The influence of various boundary conditions on natural frequencies is mainly reflected in the cases of small ratios of length to radius and small circumferential wave numbers.
The vibration of cylindrical shells composed of functionally graded material and with exponential volume fraction law is studied.Expressions of the internal force,displacement,strain and curvature of FG cylindrical shells are analysed by Love's thin shell theory.Based on the Rayleigh-Ritz method,the shell eigenfrequency equation is derived and the natural frequencies with simply supported ends and clamped-free ends are obtained.Then the influence of the configurations of constituent materials,volume fraction,boundary conditions and geometric dimensioning on the natural frequencies of FG cylindrical shells is studied through the examples.The results show that the configurations of constituent materials have considerable effect on the frequency characteristic of FG cylindrical shells,while the volume fraction has minor effect.The influence of various boundary conditions on natural frequencies is mainly reflected in the cases of small ratios of length to radius and small circumferential wave numbers.
引文
[1]Loy C T,Lam K Y,Reddy J N.Vibration of functionally graded cylindrical shells[J].International Journal of Mechanical Sciences,1999,41(3):309-324.
[2]Pradhan S C,Loy C T,Lama K Y,Reddy J N.Vibration characteristics of functionally graded cylindrical shells under various boundary conditions[J].Applied Acoustics,2000,61(1):111-129.
[3]沙哈A G,曼穆德T,那姆M N.指数型体积分数功能梯度材料的薄壁圆柱壳振动[J].应用数学和力学,2009,30(5):567-574.Shah A G,Mahmood T,Naeem M N.Vibrations of FGM thin cylindrical shells with exponential volume fraction law[J].Applied Mathematics and Mechanics,2009,30(5):567-574.(in Chinese)
[4]Arshad S H,Naeem M N,Sultana N.Frequency analysis of functionally graded material cylindrical shells with various volume fraction laws[J].Proc IMechE Part C:Journal of Mechanical Engineering Science,2007,221(12):1483-1495.
[5]曹志远.功能梯度复合材料圆柱壳固有频率解[J].地震工程与工程振动,2005,25(6):38-42.Cao Zhiyuan.Natural frequency solution for functionally graded material cylindrical shells[J].Journal of Earthquake Engineering and Engineering Vibration,2005,25(6):38-42.(in Chinese)
[6]Francesco Tornabene.Free vibration analysis of functionally graded conical,cylindrical shell and annular plate structures with a four-parameter power-law distribution[J].Comput Methods Appl.Mech.Engrg,2009,198(37-40):2911-2935.
[7]Patel B P,Gupta S S,Loknath M S,Kadu C P.Free vibration analysis of functionally graded elliptical cylindrical shells using higher-order theory[J].Composite Structures,2005,69(3):259-270.
[8]Haddadpour H,Mahmoudkhani S,Navazi H M.Free vibration analysis of functionally graded cylindrical shells including thermal effects[J].Thin-Walled Structures,2007,45(6):591-599.
[9]Sheng G G,Wang X.Thermoelastic vibration and buckling analysis of functionally graded piezoelectric cylindrical shells[J].Applied Mathematical Modelling,2010,34(9):2630-2643.
[10]伊克巴尔Z,纳伊姆M N,萨尔塔纳N,阿沙德S H,沙赫A.充液功能梯度材料圆柱壳振动特性的波动解[J].应用数学和力学,2009,30(11):1307-1317.Iqbal Z,az Naeem M N,Sultana N,in Arshad S H,Shah A.Vibration characteristics of FGM circular cylindrical shells containing fluid using wave propagation approach[J].Applied Mathematics and Mechanics,2009,30(11):1307-1317.(in Chinese)
[11]Hiroyuki Matsunaga.Free vibration and stability of functionally graded circular cylindrical shells according to a2D higher-order deformation theory[J].Composite Structures,2009,88(4):519-531.
[12]Zhao X,Lee Y Y,Liew K M.Thermoelastic and vibration analysis of functionally graded cylindrical shells[J].International Journal of Mechanical Sciences,2009,51(9-10):694-707.
[13]Kadoli R,Ganesan N.Buckling and free vibration analysis of functionally graded cylindrical shells subjected to a temperature-specified boundary condition[J].Journal of Sound and Vibration,2006,289(3):450-480.
[14]Bhangale R K,Ganesan N.Free vibration studies of simply supported non-homogeneous functionally graded magnetoelectro-elastic finite cylindrical shells[J].Journal of Sound and Vibration,2005,288(1-2):412-422.
[15]Loy C T,Lam K Y.Vibration of cylindrical shells with ring support[J].International Journal of Mechanical Sciences,1997,39(4):455-471.
[16]Love A E H.A treatise on the mathematical theory of elasticity[M].4th ed.,Cambridge:Cambridge University Press,1952.
[2]Pradhan S C,Loy C T,Lama K Y,Reddy J N.Vibration characteristics of functionally graded cylindrical shells under various boundary conditions[J].Applied Acoustics,2000,61(1):111-129.
[3]沙哈A G,曼穆德T,那姆M N.指数型体积分数功能梯度材料的薄壁圆柱壳振动[J].应用数学和力学,2009,30(5):567-574.Shah A G,Mahmood T,Naeem M N.Vibrations of FGM thin cylindrical shells with exponential volume fraction law[J].Applied Mathematics and Mechanics,2009,30(5):567-574.(in Chinese)
[4]Arshad S H,Naeem M N,Sultana N.Frequency analysis of functionally graded material cylindrical shells with various volume fraction laws[J].Proc IMechE Part C:Journal of Mechanical Engineering Science,2007,221(12):1483-1495.
[5]曹志远.功能梯度复合材料圆柱壳固有频率解[J].地震工程与工程振动,2005,25(6):38-42.Cao Zhiyuan.Natural frequency solution for functionally graded material cylindrical shells[J].Journal of Earthquake Engineering and Engineering Vibration,2005,25(6):38-42.(in Chinese)
[6]Francesco Tornabene.Free vibration analysis of functionally graded conical,cylindrical shell and annular plate structures with a four-parameter power-law distribution[J].Comput Methods Appl.Mech.Engrg,2009,198(37-40):2911-2935.
[7]Patel B P,Gupta S S,Loknath M S,Kadu C P.Free vibration analysis of functionally graded elliptical cylindrical shells using higher-order theory[J].Composite Structures,2005,69(3):259-270.
[8]Haddadpour H,Mahmoudkhani S,Navazi H M.Free vibration analysis of functionally graded cylindrical shells including thermal effects[J].Thin-Walled Structures,2007,45(6):591-599.
[9]Sheng G G,Wang X.Thermoelastic vibration and buckling analysis of functionally graded piezoelectric cylindrical shells[J].Applied Mathematical Modelling,2010,34(9):2630-2643.
[10]伊克巴尔Z,纳伊姆M N,萨尔塔纳N,阿沙德S H,沙赫A.充液功能梯度材料圆柱壳振动特性的波动解[J].应用数学和力学,2009,30(11):1307-1317.Iqbal Z,az Naeem M N,Sultana N,in Arshad S H,Shah A.Vibration characteristics of FGM circular cylindrical shells containing fluid using wave propagation approach[J].Applied Mathematics and Mechanics,2009,30(11):1307-1317.(in Chinese)
[11]Hiroyuki Matsunaga.Free vibration and stability of functionally graded circular cylindrical shells according to a2D higher-order deformation theory[J].Composite Structures,2009,88(4):519-531.
[12]Zhao X,Lee Y Y,Liew K M.Thermoelastic and vibration analysis of functionally graded cylindrical shells[J].International Journal of Mechanical Sciences,2009,51(9-10):694-707.
[13]Kadoli R,Ganesan N.Buckling and free vibration analysis of functionally graded cylindrical shells subjected to a temperature-specified boundary condition[J].Journal of Sound and Vibration,2006,289(3):450-480.
[14]Bhangale R K,Ganesan N.Free vibration studies of simply supported non-homogeneous functionally graded magnetoelectro-elastic finite cylindrical shells[J].Journal of Sound and Vibration,2005,288(1-2):412-422.
[15]Loy C T,Lam K Y.Vibration of cylindrical shells with ring support[J].International Journal of Mechanical Sciences,1997,39(4):455-471.
[16]Love A E H.A treatise on the mathematical theory of elasticity[M].4th ed.,Cambridge:Cambridge University Press,1952.
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