关于Newmark-β法机理的一种解释  

An interpretation on Newmark beta methods in mechanism of numerical analysis

作　　者：李鸿晶[1] 王通[2] 廖旭[1] 

机构地区：[1]南京工业大学土木工程学院,江苏南京210009 [2]同济大学土木工程防灾国家重点实验室,上海200092

出　　处：《地震工程与工程振动》2011年第2期55-62,共8页Earthquake Engineering and Engineering Dynamics

基　　金：国家自然科学基金重大研究计划资助项目(90815013);地震行业科研专项经费资助项目(200808081)

摘　　要：一般认为,Newmark-β法属于积分类型的动力数值分析方法,和基于荷载分段插值类型的数值方法不是相同类型的方法。在本文中,研究了这两类方法之间的关系,以最常使用的两种Newmark方法——平均常加速度法和线性加速度法为例,从Newmark基本假定出发推导出这两种方法所具有的荷载分布模式。结果发现:平均常加速度法和线性加速度法与各时间步距内荷载分布模式分别取为二次函数和三次函数时的荷载分段精确法完全相同,但平均常加速度法在时步的始末端点处荷载是不连续的,且同真实荷载值不重合。因此,Newmark-β法亦可看作是一种基于荷载分段插值类型的数值分析方法,可以从荷载分布模式假定的角度解释Newmark-β法的数值机理。最后,通过一个单自由度体系实例阐释了本文结论的正确性。In general,Newmark beta methods are considered to be a type of integration method for numerical dynamic analysis,so they are different from methods based on interpolation of excitation or so called piecewise exact method in the mechanism of numerical analysis.In this paper,two special cases of the Newmark beta methods,the constant average acceleration method and the linear acceleration method,are selected and researched,and the distribution patterns of excitations over the time step are deduced based on the basic Newmark formulation.It is found that the constant average acceleration method is equivalent to the piecewise exact method,in which the distribution pattern of the excitation is assumed to be a quadratic function in each time step,and for the linear acceleration method the excitation is with the pattern of a cubic function.But for the constant average acceleration method,the assumed excitation function is not continuous at the initial point and the end of discrete time interval,and do not coincide with the existing excitation of the structure.Thus,the Newmark beta methods are actually a type of methods based on interpolation of excitation,and their numerical mechanism may be explained in another way.Finally,a single degree-of-freedom system is provided for numerical illustration.

关 键 词：Newmark-β方法 分段精确法 算法机理 

分 类 号：P315.96[天文地球—地震学]