弹性地基上矩形加肋板自由振动分析的无网格法
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摘要
应用移动最小二乘无网格法研究弹性地基上矩形加肋板的自由振动问题。假设弹性地基与加肋板紧密接触,以弹簧模拟弹性地基,将弹性地基上的加肋板视为板与肋条组合的结构。基于一阶剪切理论,用无网格伽辽金法推出了板和肋条各自的动能与势能;再通过位移协调条件将两者的能量叠加,得到了弹性地基上整个加肋板的动能与势能。由Hamilton原理导出了弹性地基上加肋板自由振动的控制方程。采用完全转换法引入边界条件,求解自由振动方程,并编制了计算程序,给出了算例。将算例与ABAQUS有限元解及已有文献结果进行了比较分析,其相对误差均在5%以内,验证了该方法计算弹性地基上矩形加肋板结构自振频率的有效性。
A moving least square meshfree method is introduced to study the free vibration of a rectangular ribbed plate which rests on elastic foundation. The Winkler foundation is assumed to closely contact with the ribbed plate, and the elastic foundation is simulated by springs. The ribbed plate on the elastic foundation is simulated as a composite structure of plate and ribs. Based on the first-order shear deformation theory, the moving least square approximation is used to derive the kinetic energy and potential energy of the plate and the ribs. By applying the displacement compatibility conditions, the superposition of the two kinds of energy is obtained, and the total kinetic energy and potential energy of the ribbed plate is derived. According to the Hamilton's principle, a governing equation that describes the free vibration of ribbed plates on Winkler foundation is deduced. The full transformation method is used to introduce the essential boundary conditions. And the corresponding computer program is compiled to solve the equation. The equation is solved, and corresponding computer program is compiled. Numerical examples are given to compare the solutions from the proposed method with those from ABAQUS and literatures, which proved the validity of the proposed method.
A moving least square meshfree method is introduced to study the free vibration of a rectangular ribbed plate which rests on elastic foundation. The Winkler foundation is assumed to closely contact with the ribbed plate, and the elastic foundation is simulated by springs. The ribbed plate on the elastic foundation is simulated as a composite structure of plate and ribs. Based on the first-order shear deformation theory, the moving least square approximation is used to derive the kinetic energy and potential energy of the plate and the ribs. By applying the displacement compatibility conditions, the superposition of the two kinds of energy is obtained, and the total kinetic energy and potential energy of the ribbed plate is derived. According to the Hamilton's principle, a governing equation that describes the free vibration of ribbed plates on Winkler foundation is deduced. The full transformation method is used to introduce the essential boundary conditions. And the corresponding computer program is compiled to solve the equation. The equation is solved, and corresponding computer program is compiled. Numerical examples are given to compare the solutions from the proposed method with those from ABAQUS and literatures, which proved the validity of the proposed method.
引文
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[9]BELYTSCHKO T,LU Y Y,GU L.Element-free Galerkin methods[J].International journal for numerical methods in engineering,1994,37(37):229-256.
[10]PENG L X,LIEW K M,KITIPORNCHAI S.Bending analysis of folded plates by the FSDT meshless method[J].Thin-walled structures,2006,44(11):1138-1160.
[11]REDDY J N.Theory and analysis of elastic plates[M].London:Taylor&Francis,1999.
[12]PENG L X,LIEW K M,KITIPORNCHAI S.Buckling and free vibration analyses of stiffened plates using the FSDT mesh-free method[J].Journal of sound and vibration,2005,289(3):421-449.
[13]彭林欣.加肋板自由振动的移动最小二乘无单元分析[J].振动与冲击,2011,30(6):68-73.(PENG Linxin.Moving-least-square meshless analysis on free vibration behavior of ribbed plates[J].Journal of vibration and shock,2011,30(6):68-73(in Chinese)).
[2]MOTAGHIAN S,MOFILD M,AKIN J E.On the free vibration response of rectangular plates,partially supported on elastic foundation[J].Applied mathematical modelling,2012,36(9):4473-4482.
[3]JOODAKY A,JOODAKY I.A semi-analytical study on static behavior of thin skew plates on Winkler and Pasternak foundations[J].International journal of mechanical sciences,2015,100:322-327.
[4]KUMAR Y,LAL R.Prediction of frequencies of free axisymmetric vibration of two-directional functionally graded annular plates on Winkler foundation[J].European journal of mechanics-A/solids,2013,42(6):219-228.
[5]滕兆春,丁树声,郑鹏君.弹性地基上变厚度矩形板自由振动的GDQ法求解[J].应用力学学报,2014(2):236-241.(TENG Zhaochun,DING Shusheng,ZHENG Pengjun.Free vibration analysis of rectangular plates with variable thickness on elastic foundation by using GDQ method[J].Chinese journal of applied mechanics,2014(2):236-241(in Chinese)).
[6]李善倾,袁鸿.Winkler地基上固支薄板自由振动问题的准Green函数方法[J].应用数学和力学,2011,32(3):253-262.(LI Shanqin,YUAN Hong.Quasi-Green’s function method for free vibration of clamped thin plates on Winkler foundation[J].Applied mathematics and mechanics,2011,32(3):253-262(in Chinese)).
[7]谭飞,张友良.弹性地基板弯曲的杂交边界点法[J].工程力学,2013,30(4):35-41.(TAN Fei,ZHANG Youliang.A hybrid boundary node method for the bending of plates on elastic foundation[J].Engineering mechanics,2013,30(4):35-41(in Chinese)).
[8]徐腾飞,邢誉峰.弹性地基上矩形薄板自由振动的精确解[J].工程力学,2014,31(5):43-48.(XU Tengfei,XING Yufeng.Exact solutions for free vibrations of rectangular thin plates on the Winkler foundation[J].Engineering mechanics,2014,31(5):43-48(in Chinese)).
[9]BELYTSCHKO T,LU Y Y,GU L.Element-free Galerkin methods[J].International journal for numerical methods in engineering,1994,37(37):229-256.
[10]PENG L X,LIEW K M,KITIPORNCHAI S.Bending analysis of folded plates by the FSDT meshless method[J].Thin-walled structures,2006,44(11):1138-1160.
[11]REDDY J N.Theory and analysis of elastic plates[M].London:Taylor&Francis,1999.
[12]PENG L X,LIEW K M,KITIPORNCHAI S.Buckling and free vibration analyses of stiffened plates using the FSDT mesh-free method[J].Journal of sound and vibration,2005,289(3):421-449.
[13]彭林欣.加肋板自由振动的移动最小二乘无单元分析[J].振动与冲击,2011,30(6):68-73.(PENG Linxin.Moving-least-square meshless analysis on free vibration behavior of ribbed plates[J].Journal of vibration and shock,2011,30(6):68-73(in Chinese)).