带圆开孔的功能梯度矩形板的自由振动分析
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摘要
为研究带圆开孔功能梯度矩形板的自由振动特性,基于一阶剪切变形理论及等几何分析建立了计算模型,采用精确几何建模的非均匀有理B样条(NURBS)基函数描述场变量,实现了计算机辅助设计(CAD)与有限元分析(FEA)的无缝结合。与相关文献对比验证了该方法的精确性,并讨论了边界条件、梯度指数、长宽比、板厚度及开孔半径对矩形板固有频率的影响。数值算例表明,等几何分析具有较高的精度,能有效求解带圆开孔功能梯度板的振动问题。
Based on first-order shear deformation theory and isogeometric analysis,a model of functionally graded rectangular plate with a circular cutout was built to study its free vibration characteristics. The non-uniform rational B-spline( NURBS) basis functions used to precisely model the geometry also described the solution field,which achieved seamless integration of computer-aided design and finite element analysis. The accuracy of the present method was validated through comparison with other solutions. The influence of boundary conditions,gradient index,aspect ratio,plate thickness,and radius of circular cutout on the natural frequency of the plate was investigated in detail. Numerical examples show that the isogeometric analysis has relatively high accuracy,and the method can effectively solve the free vibration of a functionally graded rectangular plate with a central circular cutout.
Based on first-order shear deformation theory and isogeometric analysis,a model of functionally graded rectangular plate with a circular cutout was built to study its free vibration characteristics. The non-uniform rational B-spline( NURBS) basis functions used to precisely model the geometry also described the solution field,which achieved seamless integration of computer-aided design and finite element analysis. The accuracy of the present method was validated through comparison with other solutions. The influence of boundary conditions,gradient index,aspect ratio,plate thickness,and radius of circular cutout on the natural frequency of the plate was investigated in detail. Numerical examples show that the isogeometric analysis has relatively high accuracy,and the method can effectively solve the free vibration of a functionally graded rectangular plate with a central circular cutout.
引文
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[11]曹志远,唐寿高,程国华.复杂形状及开孔功能梯度板的三维分析[J].应用数学和力学,2009,30(1):15-20.CAO Zhiyuan,TANG Shougao,CHENG Guohua.3D A-nalysis of the functionally graded material plates with complex shapes and various holes[J].Applied mathematics and mechanics,2009,30(1):15-20.
[12]LAL A,SINGH H N,SHEGOKAR N L.FEM model for stochastic mechanical and thermal postbuckling response of functionally graded material plates applied to panels with circular and square holes having material randomness[J].International journal of mechanical sciences,2012,62(1):18-33.
[13]HUGHES T J R,COTTRELL J A,BAZILEVS Y.Isogeometric analysis:CAD,finite elements,NURBS,exact geometry and mesh refinement[J].Computer methods in applied mechanics and engineering,2005,194(39/40/41):4135-4195.
[14]COTTRELL J A,REALI A,BAZILEVS Y,et al.Isogeometric analysis of structural vibrations[J].Computer methods in applied mechanics and engineering,2006,195(41/42/43):5257-5296.
[15]WALL W A,FRENZEL M A,CYRON C.Isogeometric structural shape optimization[J].Computer methods in applied mechanics and engineering,2008,197(33/34/35/36/37/38/39/40):2976-2988.
[16]NGUYEN V P,ANITESCU C,BORDAS S P A,et al.Isogeometric analysis:An overview and computer implementation aspects[J].Mathematics and computers in simulation,2015,117:89-116.
[17]NGUYEN-XUAN H,TRAN L V,THAI C H,et al.Isogeometric analysis of functionally graded plates using a refined plate theory[J].Composites part B:engineering,2014,64:222-234.
[18]DINACHANDRA M,RAJU S.Isogeometric analysis for acoustic fluid-structure interaction problems[J].International journal of mechanical sciences,2017,131-132:8-25.
[19]PIEGL L,TILLER W.The NURBS book[M].Berlin Heidelberg:Springer,1997.
[20]REDDY J N.Mechanics of laminated composite plates and shells:theory and analysis[M].2nd ed.New York:CRCPress,2003.
[2]PRAVEEN G N,REDDY J N.Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plates[J].International journal of solids and structures,1998,35(33):4457-4476.
[3]FERREIRA A J M,BATRA R C,ROQUE C M C,et al.Natural frequencies of functionally graded plates by a meshless method[J].Composite structures,2006,75(1/2/3/4):593-600.
[4]ZHAO X,LEE Y Y,LIEW K M.Free vibration analysis of functionally graded plates using the element-free kp-Ritz method[J].Journal of sound and vibration,2009,319(3/4/5):918-939.
[5]杨正光,仲政,戴瑛.功能梯度矩形板的三维弹性分析[J].力学季刊,2004,25(1):15-20.YANG Zhengguang,ZHONG Zheng,DAI Ying.Three dimensional elasticity analysis of a functionally graded rectangular plate[J].Chinese quarterly of mechanics,2004,25(1):15-20.
[6]HUANG M,SAKIYAMA T.Free vibration analysis of rectangular plates with variously-shaped holes[J].Journal of sound and vibration,1999,226(4):769-786.
[7]KWAK M K,HAN Sangbo.Free vibration analysis of rectangular plate with a hole by means of independent coordinate coupling method[J].Journal of sound and vibration,2007,306(1/2):12-30.
[8]CHO D S,VLADIMIR N,CHOI T M.Approximate natural vibration analysis of rectangular plates with openings using assumed mode method[J].International journal of naval architecture and ocean engineering,2013,5(3):478-491.
[9]AVALOS D R,LARRONDO H A,LAURA P A A,et al.Transverse vibrations of simply supported rectangular plates with rectangular cutouts carrying an elastically mounted concentrated mass[J].Journal of sound and vibration,1997,202(4):585-592.
[10]LAURA P A A,ROSSI R E,AVALOS D R,et al.Transverse vibrations of a simply supported rectangular orthotropic plate with a circular perforation with a free edge[J].Journal of sound and vibration,1998,212(4):753-757.
[11]曹志远,唐寿高,程国华.复杂形状及开孔功能梯度板的三维分析[J].应用数学和力学,2009,30(1):15-20.CAO Zhiyuan,TANG Shougao,CHENG Guohua.3D A-nalysis of the functionally graded material plates with complex shapes and various holes[J].Applied mathematics and mechanics,2009,30(1):15-20.
[12]LAL A,SINGH H N,SHEGOKAR N L.FEM model for stochastic mechanical and thermal postbuckling response of functionally graded material plates applied to panels with circular and square holes having material randomness[J].International journal of mechanical sciences,2012,62(1):18-33.
[13]HUGHES T J R,COTTRELL J A,BAZILEVS Y.Isogeometric analysis:CAD,finite elements,NURBS,exact geometry and mesh refinement[J].Computer methods in applied mechanics and engineering,2005,194(39/40/41):4135-4195.
[14]COTTRELL J A,REALI A,BAZILEVS Y,et al.Isogeometric analysis of structural vibrations[J].Computer methods in applied mechanics and engineering,2006,195(41/42/43):5257-5296.
[15]WALL W A,FRENZEL M A,CYRON C.Isogeometric structural shape optimization[J].Computer methods in applied mechanics and engineering,2008,197(33/34/35/36/37/38/39/40):2976-2988.
[16]NGUYEN V P,ANITESCU C,BORDAS S P A,et al.Isogeometric analysis:An overview and computer implementation aspects[J].Mathematics and computers in simulation,2015,117:89-116.
[17]NGUYEN-XUAN H,TRAN L V,THAI C H,et al.Isogeometric analysis of functionally graded plates using a refined plate theory[J].Composites part B:engineering,2014,64:222-234.
[18]DINACHANDRA M,RAJU S.Isogeometric analysis for acoustic fluid-structure interaction problems[J].International journal of mechanical sciences,2017,131-132:8-25.
[19]PIEGL L,TILLER W.The NURBS book[M].Berlin Heidelberg:Springer,1997.
[20]REDDY J N.Mechanics of laminated composite plates and shells:theory and analysis[M].2nd ed.New York:CRCPress,2003.