挠曲线复位的微分方程解法求梁的位移

梁位移求解的方法主要有两种：积分法和叠加法。积分法的困难是当多个荷载同时作用时，要按控制面分段列弯矩方程，这样，两次积分时带来的积分常数较多，导致依边界条件、连续条件定这些常数时要解多个联立方程，计算繁杂。梁位移求解的叠加法是借用典型荷载下梁已知积分结果来进行对应位置位移的叠加，具体是查转角方程、挠曲线方程表格。从表中知，梁任一位置处的位移方程大多是多次多项式，很难记忆，故离开表格便无法采用。本文找到了一种新的方法求梁的位移，通过设置比拟梁，使比拟梁的挠曲线复位于原梁，从而建立了相应的微分方程，求解此方程后便可得梁的位移，避开了分段多次积分或查表的弱点。

A method of differential equation using in restoration to deflection curve for solution of displacement of beams

Mechanics of materials generally introduces the integration method and the superposition method for computation of the slope and deflection of beams. The integration method has its own difficulty which lies in the integration when loads existing are at different places at the same time. Due to too many integration constants appearing, it is not simple. The superposition method is based on the deflection curve equations diagram and table where the deflection led by one load is given. Usually, the deflection curve equations have many items and their variables have many high powers. So this method requires the ability of memory too high to average people. This paper advances a new theorem for computation of the slope and deflection of beams. At first, we set up an analogy beam of the original beam. Then we restore its deflection curve contrasting with the original one's. So deduce a differential equation of the deflection curve. Finding the solution of the differential equation we can get the displacements of the original beam. It avoids the defects of the integration method and the superposition method.