功的互等法在简谐分布力矩下矩形板受迫振动问题的应用
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摘要
从功的互等法出发,研究了薄板在简谐分布力矩作用下的受迫振动问题。给出了三条边固定另一条边简支、三条边固定另一条自由矩形板受迫振动的挠曲线方程,并进行了数值运算,得到了一系列数据。通过与给出的Ansys值对比,其误差小于5%,表明了给出的挠曲线方程和计算结果的正确性。给出的计算矩形板受迫振动问题的方法,避免了传统方法庞大的高阶偏微分方程组和复杂的求解过程,不仅具有一定的理论价值,而且可以直接用于工程实际。
Based on the reciprocal method, the problem of forced vibration of thin plates under the harmonic distribution of moment is analyzed. The rectangular plates bending surface equation with four-sided fixed, three-sided fixed with another side of the simple supported and three sides fixed with another free is deduced. The numerical calculation is completed and a series of data is obtained. By comparison with the results via ANSYS, the error is less than 5%, which shows the correctness of the given deflection equation and calculation results. The method of calculating the forced vibration of rectangular plates avoids the huge high order partial differential equations and the complicated solving process of the traditional method. It is not only of a certain theoretical value, but also applicable directly to the engineering practice.
Based on the reciprocal method, the problem of forced vibration of thin plates under the harmonic distribution of moment is analyzed. The rectangular plates bending surface equation with four-sided fixed, three-sided fixed with another side of the simple supported and three sides fixed with another free is deduced. The numerical calculation is completed and a series of data is obtained. By comparison with the results via ANSYS, the error is less than 5%, which shows the correctness of the given deflection equation and calculation results. The method of calculating the forced vibration of rectangular plates avoids the huge high order partial differential equations and the complicated solving process of the traditional method. It is not only of a certain theoretical value, but also applicable directly to the engineering practice.
引文
[1]王淼,陈永强,李志敏.正交各向异性矩形薄板振动的一种半解析方法[J].振动与冲击,2016,35(14):13-18.(WANG Miao,CHEN Yongqiang,LI Zhimin.Semi-analytical approach for the vibration analysis of orthotropic thin rectangular plates[J].Journal of vibration and shock,2016,35(14):13-18(in Chinese)).
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[3]张福范.弹性薄板[M].北京:科学出版社,1984:5-11.(ZHANGFufan.Flexible sheet[M].Beijing:Science Press,1984:5-11(in Chinese)).
[4]覃霞,曾治平,彭林欣.弹性地基上矩形加肋板自由振动分析的无网格法[J].应用力学学报,2017,34(6):1027-1033.(QIN Xia,ZENG Zhiping,PENG Linxin.Meshless method for free vibration analysis of rectangular stiffened plates on elastic foundation[J].Chinese journal of applied mechanics,2017,34(6):1027-1033(in Chinese)).
[5]钟阳,张永山.弹性地基上四边自由矩形薄板的解析解[J].应用数学和力学,2006,27(6):735-740.(ZHONG Yang,ZHANGYongshan.Analytical solution of rectangular thin plates with four edges free on elastic foundation[J].Applied mathematics and mechanics,2006,27(6):735-740(in Chinese)).
[6]徐小明,杨迪雄.关于修正的功的互等定理的讨论[J].应用数学和力学,2016,37(9):993-998.(XU Xiaoming,YANG Dixiong.Discussion on the modified reciprocal theorem of works[J].Applied mathematics and mechanics,2016,37(9):993-998(in Chinese)).
[7]付宝连.弯曲薄板的修正的功的互等定理及其应用[J].应用数学和力学,2014,35(11):1197-1209.(FU Baolian.Corrected reciprocal theorem of works for bending thin plates and its application[J].Applied mathematics and mechanics,2014,35(11):1197-1209(in Chinese)).
[2]姚伟岸,蔡智宇,胡小飞.矩形正交各向异性薄板弯曲受迫振动问题的分析解[J].动力学与控制学报,2011,9(1):12-17.(YAOWeian,CAI Zhiyu,HU Xiaofei.Analytical solution for the forced vibration of orthotropic rectangular thin plates[J].Journal of dynamics and control,2011,9(1):12-17(in Chinese)).
[3]张福范.弹性薄板[M].北京:科学出版社,1984:5-11.(ZHANGFufan.Flexible sheet[M].Beijing:Science Press,1984:5-11(in Chinese)).
[4]覃霞,曾治平,彭林欣.弹性地基上矩形加肋板自由振动分析的无网格法[J].应用力学学报,2017,34(6):1027-1033.(QIN Xia,ZENG Zhiping,PENG Linxin.Meshless method for free vibration analysis of rectangular stiffened plates on elastic foundation[J].Chinese journal of applied mechanics,2017,34(6):1027-1033(in Chinese)).
[5]钟阳,张永山.弹性地基上四边自由矩形薄板的解析解[J].应用数学和力学,2006,27(6):735-740.(ZHONG Yang,ZHANGYongshan.Analytical solution of rectangular thin plates with four edges free on elastic foundation[J].Applied mathematics and mechanics,2006,27(6):735-740(in Chinese)).
[6]徐小明,杨迪雄.关于修正的功的互等定理的讨论[J].应用数学和力学,2016,37(9):993-998.(XU Xiaoming,YANG Dixiong.Discussion on the modified reciprocal theorem of works[J].Applied mathematics and mechanics,2016,37(9):993-998(in Chinese)).
[7]付宝连.弯曲薄板的修正的功的互等定理及其应用[J].应用数学和力学,2014,35(11):1197-1209.(FU Baolian.Corrected reciprocal theorem of works for bending thin plates and its application[J].Applied mathematics and mechanics,2014,35(11):1197-1209(in Chinese)).