线性、非线性响应计算的减缩误差分析

Analysis of reduction errors in computing the responses of linear and nonlinear systems

非线性动力学的研究方兴未艾，然而其理论应用于高维的实际工程问题还存在困难。自由度减缩技术对此应用可以取到桥梁作用。因此有必要研究减缩对计算结果的影响。本文研究Ｎｅｗｍａｒｋ－Ｎｅｗｔｏｎ－Ｒａｐｈｓｏｎ（ＮＮＲ）方法与自由度减缩技术求解非线性振动系统动态响应的过程，通过比较建立了相关的误差表达式。由此分析减缩对误差的影响，提出减小误差的途径。作为基础，在讨论非线性系统之前，分析了运用Ｎｅｗｍａｒｋ直接积分方法求解线性振动系统的减缩误差。研究表明，减缩误差不仅取决于减缩基，而且与外激励或系统响应有关。

? The research on nonlinear dynamics is just unfolding. But there are difficulties to apply nonlinear theory to high dimensional problems resulted from realistic engineering. The reduction technique might play the role of bridge for this application, so it is necessary to study the influence of reduction on the accuracy of computation. In this paper, the procedures of Newmark Newton Raphson (NNR) method and reduction techniques are investigated so as to solve the dynamic responses of nonlinear vibrational systems. By comparing the displacement responses obtained through NNR with and without reduction, the explicit formulas of displacement errors are derived. Based upon this, the contribution of reduction to the errors are analyzed, and some possible measures are put forward for decreasing the errors. As the base of the aforementioned discussions, the reduction errors of solving linear vibrational systems by Newmark directly integral method are analyzed at the first. The work shows that the reduction error is not only up to the base vector matrix used for reduction, but also dependent on the external forces or the response itself of the system.