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Numerical algorithm for dynamic problems of viscoelastic sandwich plates based on improved Fourier series

DOI：

 作者 单位 邮编 马乾瑛* 长安大学 建筑工程学院 710061 梅杨0 长安大学 建筑工程学院 孙正 长安大学建筑工程学院

本文提出了一种基于改进傅里叶级数的新型数值算法，用于求解任意边界条件下粘弹性夹层板的自由振动特性。该算法利用一阶剪切理论建立面外位移场，采用改进的傅里叶级数拟合位移容许函数，结合Rayleigh-Ritz法计算动能和势能，并利用Hamilton变分原理得到振动特征矩阵方程。在此基础上，引入粘弹性材料的复模量概念，得到夹层板结构的自振频率和损耗因子。结果表明，无论是梁式板结构还是各种边界条件下的夹层板结构，该算法得到的自振频率和损耗因子与解析解、2D有限元解、数值计算法、ABAQUS有限元软件等计算结果相比，误差均不超过1%，且计算效率提高了10倍，并且克服了传统方法只能求解特定边界下振动问题的局限性。此外，本文还分析了人工弹簧刚度和傅里叶级数项数等因素对结构自振特性的影响规律和现象。本文建立的方法具有收敛性好、计算精度高、计算效率高、适应性强等特点。

In this paper, a new numerical algorithm based on improved Fourier series is proposed to solve the free vibration characteristics of viscoelastic sandwich plates under arbitrary boundary conditions. The algorithm uses the first-order shear theory to establish the out-of-plane displacement field, uses the improved Fourier series to fit the displacement tolerance function, combines the Rayleigh-Ritz method to calculate the kinetic energy and potential energy, and uses the Hamilton variational principle to obtain the vibration characteristic matrix equation. On this basis, the concept of complex modulus of viscoelastic material is introduced to obtain the natural frequency and loss factor of sandwich plate structure. The results show that whether it is a beam-type plate structure or a sandwich plate structure under various boundary conditions, the natural frequency and loss factor obtained by the algorithm are compared with the analytical solution, 2D finite element solution, numerical calculation method, ABAQUS finite element software and other calculation results. The error does not exceed 1 %, and the calculation efficiency is increased by 10 times, and the limitation that the traditional method can only solve the vibration problem under specific boundaries is overcome. In addition, this paper also analyzes the influence of artificial spring stiffness and Fourier series on the natural vibration characteristics of the structure. The method established in this paper has the characteristics of good convergence, high calculation accuracy, high calculation efficiency and strong adaptability.
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