脉冲激励振动问题的高精度数值分析

High Precision Numerical Analysis of Vibration Problems under Pulse Excitation

利用数值稳定性好、计算精度高的重心Lagrange插值近似未知函数,得到未知函数各阶导数的微分矩阵,提出高精度数值分析任意激励下振动问题的重心插值配点法。采用附加法施加初始条件,得到一个n+2个方程n个未知量的代数方程组。利用最小二乘法求解线性方程组,得到振动位移,进而利用微分矩阵直接求得振动的速度和加速度。对于脉冲激励等复杂激励下的振动问题,将前一个时间区域末的位移和速度作为下一个时间区域的初始条件,分段进行数值分析。数值算例表明重心插值配点法在分析脉冲激励下的振动问题,具有非常高的计算精度。

The differential matrices of derivatives are constructed by using barycentric Lagrange interpolation which has good numerical stability and high computing precision. The barycentric interpolation collocation method for analyzing the vibration problems under arbitrary excitation is presented. The vibration differential equation and two initial conditions are transformed into a set of algebraic equation system and two algebraic equations respectively. Applying attached method to impose initial conditions, a new set of algebraic equation system which has n variables and n＋2 equations is obtained. The new algebraic equation system is solved by using least-square method. The velocity and acceleration of vibration is directly computed by using differential matrices. For vibration problems under pulse excitations, dividing the time domain into two intervals according to the characters of excitation, the displacements of vibration in the two intervals are computed respectively. The displacement and velocity at the end of first interval are taken as the initial conditions of vibration in the second interval. The numerical examples indicate that the proposed method has high computing precision in analysis of vibration under pulse excitations.