提出一种基于改进傅里叶级数的方法，对矩形薄板在任意边界条件下自由振动特性进行求解。通过将薄板振动的位移函数表示成二维傅里叶余弦级数和辅助级数的线性组合，克服传统傅里叶级数法中薄板位移函数边界处不连续的缺陷。基于位移函数列出矩形薄板拉格朗日方程，然后通过Hamilton原理求解得到矩形薄板自由振动频率与相应位移函数的系数。本文计算结果与文献及有限元解吻合良好，方法准确可靠。此外，通过改变边界约束弹簧刚度模拟任意边界条件；大量计算表明，固支边界条件与弹性边界条件组合中，随着固支边条界范围增大，矩形薄板无量纲频率参数呈增大趋势。简支及自由边界条件与弹性边界条件组合中，随着弹性边条界的增多，矩形薄板无量纲频率参数呈增大趋势。
Abstract
A method based on improved Fourier series method (IFSM) was proposed to solve free vibration characteristics of rectangular thin plates under arbitrary boundary conditions. The plate vibration displacement function was expressed as a linear combination of 2-D Fourier cosine series and auxiliary series to overcome the defect of vibration displacement function being dis-continuous at boundary using the traditional Fourier series method. A plate’s energy functional was established based on vibration displacement function. Using Hamilton principle, a rectangular thin plate’s natural frequencies and the corresponding displacement function’s coefficients were solved. The calculation results agreed well with those published in literature and using the finite element method to verify the correctness and reliability of the proposed method. Boundary restraint springs’ stiffness values were changed to simulate arbitrary boundary conditions. A lot of calculation results showed that in combination of fixed boundary conditions and elastic ones, dimensionless frequencies of a rectangular thin plate grow with increase in the range of fixed boundary conditions; in combination of simply supported and free boundary conditions and elastic ones, dimensionless frequencies of a rectangular thin plate grow with increase in the range of elastic boundary conditions.
关键词
矩形薄板 /
振动特性 /
改进傅里叶级数 /
弹性边界条件
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Key words
Rectangular Thin Plates /
Vibration Characteristic /
Improved Fourier Series Method /
Elastic boundary condition
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参考文献
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