振动方程的线性多步法数值求解

Linear Multistep Numerical Solutions for Vibration Equation

详细分析了工程中常用的振动问题数值解法Newmark及Wilson-θ法。并在此基础之上本文提出一种新的计算方法,即在计算t+△t时刻的状态时,不仅用到t时刻的各值,而且还将用到t-△t,t-2△t等各时刻的值。这样就可以提高数值积分时所用多项式的阶数,使得加速度在积分区间△t内可以为时间的二次及三次函数;从而大大提高了解法的数值精度及解的稳定性,并且基本上不增加计算量。该法在数值计算上讲属于线性多步法。

Two numerical solutions in common use for engineering vibration problems, Newmark and Wilson-θ Methods, are analyzed in detail. Based on the analysis,a new method is put forward. That is the calculation at time t ＋Δt concerns not only the results at time t ,but also at the times t-Δt and t-2Δt. Thus, in the time interval At,the polynomial assumption for acceleration can have higher order of second or third in time, so that the accuracy and stability of the numerical solution are enhanced greatly. The new method almost needs no more effort of calculation. It can be classified under the category of linear muhistep method in numerical analysis.