Abstract: A fourth order partial differential equation which originates from the free transverse vibration of beams, was changed into a second order mixed initial boundary value problem by introducing a couple of auxiliary functions. An implicit scheme of difference was constructed by discretization of two auxiliary functions and the second order mixed initial boundary value problem in which intermediate variables were eliminated and the second order boundary value conditions derived from vanishing deflection at both ends of the beam were approximated. Numerical experiments showed that the developed implicit scheme was absolutely stable and guaranteed a very high accuracy.

Key words: vibration equation of beams, finite difference method, absolutely stable difference scheme

摘要： 以简支梁的自由横振动问题为背景，求解一个非稳态四阶线性偏微分方程的初边值问题。通过引进辅助函数组，将四阶问题转化为二阶混合初边值问题。对两个辅助函数和二阶混合初边值问题进行离散并消掉中间变量，对由简支梁两端挠度为零得来的二阶偏微分边界条件进行近似处理，构造出求解四阶非稳态线性偏微分方程的差分隐格式。数值实验表明构造的隐格式绝对稳定并且具有很高的精度阶。

关键词: 梁横振动方程, 有限差分方法, 差分格式的绝对稳定