结构动力学逐步积分算法稳定性讨论

美国著名计算结构力学专家、地震工程专家Ray W Clough和Anil K Chopra分别写有在全世界范围内有很大影响的专著《DYNAMICS OF STRUCTURES》.两部著作在关于逐步积分算法的论述中都曾指出加速度由当前位移和速度代入运动方程求解将能得到更好的结果.但这一点存在可商榷之处,为避免读者误用,从算法的稳定性出发,经分析指出:专著观点应用于Newmark法增量格式时确实使得算法的谱半径有所减小;然而,应用于Newmark法全量格式时,算法特性并未发生改变,结果得不到改善;此外,书中给出了增量格式Wilsonθ法的推导步骤,按此推导的算法相对于全量格式稳定界限发生了改变——只有当θ≥1.5时算法才是无条件稳定的.如果误将加速度由当前位移和速度代入运动方程求解推广到Wilsonθ法全量格式,算法的稳定性将变得非常差.

A discussion to the stability of step-by-step integration methods in some textbooks of structure dynamics

Ray W. Clough and Anil K. Chopra, the famous experts of computer structural mechanics and earthquake engineering, have written two monographs with the same name: DYNAMICS OF STRUCTURES, which have a large influence around the world. A viewpoint about the step-by-step integration method, whose result comes out better when acceleration is calculated by the equations of motion according to displacement and velocity, is emphasized in both books. However, this point of view needs to be discussed carefully. After some analyses to the stability of the methods mentioned in the books, we point out that the application of this method to the increment format of Newmark method reduces the spectral radius as expected; while its application to the whole quantity format of Newmark method indicates that characteristics of the algorithm are not changed, and there’s no improvement to the result. This paper also provides the steps to deduce the increment format of Wilson θ method. The algorithm, which is whole quantity format, deduced by these steps changes the stability limit comparing with the algorithm discussed in Article 1. And the algorithm is unconditional stable only when θ is greater than or equals to 1.5. The stability of the algorithm becomes very bad when acceleration is replaced by current displacement and velocity to be used in motion equation and further proceeded to the whole quantity format of Wilson θ method.