轴对称薄壁结构自由振动的边界元分析 - 周琪; 陈永强 (力学学报, 2019)
采用双互易法分析薄壁轴对称结构自由振动的特征频率以及特征模态.首先,采用径向基函数插值域积分里的位移,利用双互易法将域积分转化为子午面边界的积分.然后,将边界物理量、基本解和特解展开为傅里叶级数,沿环向积分后得到的边界积分方程可用于轴对称结构带体积力问题和受非对称载荷的动力学分析,其积分域为轴对称结构子午面边界上的线积分,进一步降低了问题的维度和离散的难度.文章详细探讨了源点处于对称轴的特殊情况,根据基本解和特解的退化形式,针对无体积力和有体积力分别给出了处理奇异矩阵的方案.对于薄壁结构,采用双曲正弦变换处理近奇异积分有效提高积分精度.最后将双互易法和双曲正弦变化应用于薄壁轴对称结构带体积力的静力学和自由振动分析.数值结果表明,文章提出的处理奇异矩阵的方法能够有效处理源点处于对称轴的情况;当圆筒厚高比为10~(-3),边界元计算的特征频率的相对误差为10~(-3),且优于有限元的结果.Free vibration analysis of thin-walled axisymmetric structures with Boundary Element Method The Dual Reciprocity Method(DRM) is extended to study the eigenvalue and eigenmode of thin-walled axisymmetric structures. First the displacement in the domain integral can be approximated by a set of radial basis functions and the domain integral can be converted to the boundary using DRM. Then the displacement and the traction can be expanded as Fourier series and integrate along the circumferential direction. The obtained boundary integral equation can be used for analysis of elastostatics of axisymmetric structure distributed body force and elastodynamics subject to asymmetric loading. The special case of the source point on the axis of symmetry is discussed in detail. New schemes are suggested for dealing with singular matrices for cases with and without body force respectively according to the degenerate form of the fundamental solution and the particular solution. For the thin walled structure, the sinh transformation is applied to improve the accuracy of evaluation of the nearly singular integrals. The developed project has been used to analyze elastostatics with body force and the free vibration of the thin axisymmetric structures. Numerical results indicate that the proposed method for dealing with singular matrices can effectively deal with the situation where the source point is on the axis of symmetry. and when the thickness ratio reaches 10^(-3), the relative error of the results can approach 10^(-3), which is better than those of FEM .
Keywords: 双互易法; 双曲正弦变换; 奇异矩阵; 自由振动