一种无条件稳定的结构动力学显式算法

AN UNCONDITIONALLY STABLE EXPLICIT ALGORITHM FOR STRUCTURAL DYNAMICS

利用离散控制理论,针对结构动力学方程时间积分提出了一种新的无条件稳定的显式算法.新算法采用CR算法的速度和位移递推格式,同时利用Z变换获得算法对应的传递函数,进而根据极点条件推导了递推格式系数的具体表达式.然后,在其系数中引入了一个控制周期延长率的变量s,从而调节新算法的精度.理论分析表明无条件稳定显式新算法具有二阶精度、零振幅衰减率、无超调和自起步特性,且周期延长率可以用变量s控制,而CR算法只是本文新算法的特例.最后,确定了非线性刚度硬化系统的稳定性界限,并给出了使新算法精度达到较高的变量s的区间.算例分析表明,在此变量区间内取值时,新算法的精度要优于纽马克常平均加速度算法和CR算法.

This paper proposes an unconditionally stable explicit algorithm for time integration of structural dynamics by utilizing the discrete control theory.New algorithm adopts the recursive formula of velocity and displacement of CR algorithm,and obtains the respective transfer function based on Z transformation.Further,the specific expressions of coefficients of recursive formula are derived according to the pole condition.Then,a variable s in the coefficients to control the period elongation is introduced,which is applied to adjust the accuracy of new algorithm.Theoretical analysis indicate that the new proposed unconditionally stable explicit algorithm possesses the properties of second accuracy,zero amplitude decay,non-overshoot and self-starting,and its period elongation can be controlled by the variable s.Moreover,the CR algorithm is a special case of the proposed algorithm.Finally,the stability limit of nonlinear stiffening system is determined,and variable interval corresponding to the higher accuracy of new algorithm is presented.Numerical examples demonstrate that in this interval of variable s,the accuracy of new algorithm is superior to that of Newmark constant average acceleration and CR algorithm.