非概率集合理论方法和概率方法的相容性问题研究

项目名称： 非概率集合理论方法和概率方法的相容性问题研究

项目编号： No.10872017

项目类型： 面上项目

立项/批准年度： 2009

项目学科： 武器工业

项目作者： 邱志平

作者单位： 北京航空航天大学

项目金额： 40万元

中文摘要： 结构中的不确定性是客观存在的，只是程度和形式不同而已，而且用不同的数学理论和方法进行研究所获得的结果应该是互相验证的，并且具有一定的数量关系，这就是不同数学理论和方法对同一结构力学现象研究结果的相容性问题。本项目将综合运用概率论与数理统计，区间数学等理论和方法，研究不确定结构动力学问题的非概率集合理论方法和概率方法的数量关系；研究如何根据少量的实验或样本数据确定出非概率集合理论方法所需要的不确定变量的上下界；研究不确定变量的各阶矩和其上下界的数量关系；在现有研究成果的基础上，研究如何根据结构动力响应的二阶运动微分方程直接确定结构响应的各阶矩的微分方程和数值计算方法；运用随机样本试验验证不确定变量区间和非概率集合理论方法所确定的结构响应区间的正确性；开发出能用于实际工程需要的结构不确定分析软件，并将其应用于航空航天工程实际。为航空航天工程结构的安全可靠分析和设计提供新的理论基础和技术储备。

中文关键词： 比较研究; 相容性; 集合理论方法; 概率方法; 不确定性

英文摘要： Uncertainty in structures exists objectively, which has different degree and form of uncertainty. The different results obtained by the different mathematical theory and method should be able to mutually verify, and there exists certain numerical relationship between them. It is the compatibility problem of results that obtained by different mathematical theory and method for the same structural mechanics phenomenon. In this project, the probabilistic theory and mathematical statistics and interval mathematics will be comprehensively applied to study the quantity relation of uncertain structural dynamics problem between them, to study how to determine the upper bound and lower bound of non-probabilistic set-theoretic method according to few experimental samples, to study the different order moment of uncertain variables and their quantity relation of their bounds. On the basis of present research results, we will study how to determine directly the differential equations for different order moment of the structural response and their numerical solution methods, and verify the bounds of uncertain variables by virtue of random sample experiments and correctness of the structural response bounds obtained by the non-probabilistic set-theoretic method, and develop the uncertain analysis software that can be used for the real practical needs and apply them to the aeronautical and astronautical engineering problems. It will supply the new theory and techniques for the reliability analysis and design of the aeronautical and astronautical structures.

英文关键词： comparative study; compatibility; set-theoretic method; probabilistic method; uncertainty