连续梁瞬态振动离散时间精细传递矩阵法

DISCRETE-TIME PRECISE TRANSFER MATRIX METHOD FOR TRANSIENT VIBRATION ANALYSIS OF A CONTINUOUS BEAM

针对连续梁振动问题,将通常诸如中心差分法、Hewmark-β法等结构动力学差分方法和矩阵指数精细积分法相结合,提出了一种高效求解连续梁瞬态振动的方法,离散时间精细传递矩阵法。该方法基于连续梁振动偏微分方程,将惯性项中加速度通过差分法线性化为位移的线性函数,于是原系统的连续偏微分方程就转化为线性非齐次常微分方程组。通过精细积分法求解该系统从一端到系统另一端的传递矩阵,非齐次项可选用梯形积分、高斯积分等方法得到,结合边界条件既可进行系统动力学分析。系统固有振动分析可视为读方法的特例。通过数值算例证明提出的方法精度高、有效。

To solve transient vibration of a continuous beam,an efficient method based on precision integration method of matrix exponential and finite difference methods,such as center difference method and Newmark-β method,named as discrete-time precise transfer matrix method is described.In terms of the proposed method,the dynamic partial differential equation of the beam can be transformed into linear inhomogeneous ordinary differential equations when the four order partial differential equation of the continuoua beam is expressed as a set of first order partial differential equations,and the acceleration is written as the linear function of the displacement at time ti by the finite difference methods.If the precision integration method can be used to calculate the transfer matrix relating the two boundaries of the beam and the inhomogeneous items can be obtained through trapezoidal integration,Gauss integration etc.,the system dynamic analysis can be implemented with combination of the boundary conditions.The computation of the natural frequencies of the beam can be regarded as the special example.The numerical example validates that the precision of the proposed method is better than that of finite element method.