任意随机激励下结构随机振动分析的一种数值方法

应用复化Cotes数值积分方法改进精细积分方法，建立一种新的高效的精细积分方法：C-PTSIM，并基于有限元理论讨论了此方法在任意随机激励下线性结构随机动力响应的应用。采用复化Cotes积分方法计算结构动力响应状态方程一般解的积分项，推导出随机激励下结构动力响应的显式表达式，利用一阶矩和二阶矩运算规律计算结构响应的均值和方差。C-PTSIM方法避免了精细积分过程中系数矩阵求逆问题，有效改善了精细积分在时间步长内载荷线性化假设带来的误差，在不改变时间步长时采用高次数复化积分时获得与更精细步长时同样精度的结果，表明该方法对时间步长的弱敏感性，并能节省大量的计算时间。基于此方法给出结构随机振动响应分析算例，并与其他方法对比，说明了该方法的高效率和高精度。

A Numerical Method for the Random Vibration Analysis of Structures Under Arbitrary Random Excitations

With the complex Cotes integral method improving the precise integration method，a new efficient precise integration method: C-PTSIM is developed. Based on finite element theory, the application of the method is discussed in random vibration dynamic response of linear structures under arbitrary random excitations. The complex Cotes integration method is used to calculate the integral term of the general solution of state equation of structural dynamic response, and the explicit expression of structural dynamic response is deduced, and then the mean and variance of structural response are calculated with the first moment and second moment operation laws. The C-PTSIM method avoided the problem to solve the coefficient matrix inversion in the process of precise integration, and efficiently reduced the errors by assuming linear loads within the time step. With the time step keeping unchanged, the high number complex Cotes integration can obtain the same precision of the results as integration of finer time step, showing weak sensitivity to the time step, and large amounts of time can be saved. The example of the random vibration response of structures is analyzed based on C-PTSIM, and the high efficiency and precision of this method are illustrated compared with other methods.