渐变截面梁波透射反射系数求解的数值验证

摘要

The wave differential equation of prestressed box beam with any cross-section was deduced.For the purpose of calculating transmission and reflection coefficient matrix in segment with gradually varying cross section of the prestressed beam, the segment was divided into several pieces of beam with constant section.The transmission and reflection coefficient matrix of the wave amplitudes at the big end and the little end of the segment with non-uniform crosssection was calculated based on the transfer matrix method in wave analysis on sandwich structure, and the matrix accuracy which is affected by the division number and cross-section dimension in the beam segment was analyzed by performing several subroutines programmed based on Maple software.The accuracy of the calculation was verified by using the solution of wave differential equation according to the Galerkin method, in which the combination of first kind Bessel functions was employed as trial functions.What is worth to be mentioned is that the trial functions used in the paper can be used generally and accurately to obtain the approximate solutions of differential equation of higher order which contain coefficients of series functions with higher-order terms.%推导任意截面形状的预应力Timoshenko梁波动微分方程,对预应力箱板梁渐变截面段弹性波透(反)射系数矩阵进行计算,为求得小端传播到大端的透射波幅值,将渐变截面梁近似当作具有一系列截面间断的阶梯梁,按层状结构波传播的传递矩阵法求波幅在大小端间的透(反)射系数矩阵,通过基于Maple软件平台编写子程序对梁截面尺寸及渐变段分段数目对透(反)射系数矩阵精确性的影响进行数值分析.在波动微分方程伽辽金近似解基础上,对渐变截面段梁波透(反)射系数矩阵的求解法进行数值验证,验证中以第一类贝塞尔函数组合作为系数具有高次项幂级数函数的高阶常微分方程伽辽金法近似解的试函数,该试函数对这类方程具有普遍的适用性.