斜削梁协调元素族及其在梁固有振动分析上的应用

本文给出了位移函数为三次、五次、七次幂多项式时斜削梁元素刚度与质量矩阵的显表达式以及计算任意奇次幂高阶斜削梁元素族刚度与质量矩阵显表达式的一般方法。然后应用高阶协调元素计算了六种线性斜削梁在悬臂支持、二端简支、二端固支等情况下的头五阶固有频率及振型。结果表明,在所有情况下,与低阶元素相比,高阶元素均显出有较好的收敛性。但若施加以力的边界条件,则在使用少量元素和梁的斜削度较大的情况下,高阶元素的结果反而不如低阶元素的结果好。

A FAMILY OF TAPERED BEAM CONFORMING ELEMENTS AND ITS APPLICATION TO BEAM NATURAL VIBRATION ANALYSIS

The explicit expressions for the stiffness and mass matrices of the tapered beam conforming elements with displacement functions based on the 3rd, 5th and 7th order polynomial respectively are given, and a general method for evaluating the stiffness and mass matrices of even higher order beam element with arbitrary odd power is also developed. Then by means of these elements are calculated the first five natural frequencies and corresponding modes of these elements are calculated the first five natural frequencies and corresponding modes of the linearly tapered beams with rectangular cross section in six manners under various boundary conditions （e. g. fixed-fixed, simply supported-simply supported, cantilever, etc. ） Numerical results are presented for several typical cases. The results indicate that the higher order element exhibits better convergence than the lower order element in all cases without imposition of natural boundary conditions. Otherwise the higher order element would lead to worse results than the lower order element for deeply tapered beams with a few elements used in calculation Hence, it may be concluded that the reason for the worse convergence of the higher order element in some cases is the imposition of the natural boundary conditions in advance, but it is not necessary for the displacement variational principle.