区间参数结构的动力优化
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摘要
工程结构中的分析和设计方法一般是基于确定性的结构参数和确定性的数学模型。然而，实际上经常存在着与材料性质、几何特性、外力、初始条件、边界条件及与结构部件接头有关的误差或不确定性。虽然在多数情况下，误差或不确定性可能很小，但这些误差或不确定性结合在一起却可能使结构响应产生大的、意想不到的偏差或不可预知性，特别是在多部件系统中。
近几十年来，研究不确定性问题最常用的方法是概率统计方法。在方法中，人们常把结构参数看成随机变量或随机场，关于结构参数的所有信息均由结构参数的联合概率密度函数或分布函数提供。尽管概率统计方法在描述结构力学中的不确定现象方面获得了很大成功，但是在没有足够的实验数据证明由联合概率密度(联合概率密度函数)所给出的假设正确的情况下，这种方法并不能给出满足某种精度的可靠结果。因此，概率统计方法不应该是研究不确定性问题的唯一方法。
目前，除了概率统计方法外，用于研究不确定性问题的方法还有凸分析法、模糊集合法和区间分析法。考虑到凸分析法和模糊集合法中的主观信息的限制，同时也考虑到数学上线性区间方程组、非线性区间方程和区间特征值问题解法的复杂性，陈塑寰、邱志平等人区间分析模型用于研究具有不确定性参数结构的静、动态问题中。利用区间分析和矩阵摄动法，他们得到了几个重要的结果。但他们的结果是以下述假设为基础的：ΔK和Δf在方程K(α)U=f(α)中是预先设定的；ΔK和ΔM在方程K(α)U=λM(α)中也是预先设定的。一般来说，这种假设是有局限性的。因为ΔK、ΔM和Δf都是结
吉林大学博士学位论文
构参数的函数，它们的不确定性可根据结构参数的不确定性来计算，这已经
由杨晓伟、连华东等人给出了不错的结果。
区间分析方法是自二十世纪六十年代以来出现的，Moore和他的合作者
Alefeld和Herzberger己经作了许多开创性的工作。Hansen在他的书中基于
区间分析讨论了全局优化问题。但是由于这些区间算法的复杂性，很难用它
们来解决实际的工程问题。最近，区间分析方法被用于具有区间参数的不确
定性结构的静态位移和特征值的分析。但是，很少有文献涉及到具有区间参
数结构的结构优化问题。因此，很有必要提出一种有效的方法，用于解决具
有区间参数结构的结构优化问题。本文基于区间分析提出了一种区间优化方
法，这种方法的思想是:首先对具有区间参数结构的动力响应进行区间分析，
然后在此基础上用标准的优化算法对所得的区间的上界进行优化，使得结构
某点的响应值落在一个最窄的区间内。
在本论文中，作者在杨晓伟、连华东等人工作的基础上，主要做了下面
的一些工作:
1.第四章针对具有区间参数的离散系统提出一种动力区间优化方法。
2.第五章针对具有区间参数的连续系统提出一种动力响应的区间优化
方法。
3，第六章针对区间参数振动结构提出了一种具有频率约束的动力响应
的区间优化方法。
4.第七章针对区间参数振动结构提出了一种改进的动力响应的区间优
化方法。
In many practical engineering problems, the structural parameters are uncertain, for example, the inaccuracy of measurements, errors in manufacture, etc. Therefore, the concept of uncertainty plays an important role in the investigation of various engineering problems.
The most common approach to uncertain problems is to model the structural parameters as random variables or fields. In this case, all information about the structural parameters is provided by the joint probability density function(or distribution function) of them. Unfortunately, the probabilistic modeling is not the only way we can use to describe the uncertainty, and uncertainty is not tantamount to randomness. In many cases, the uncertainty phenomena do not have a stochastic nature. The reason why many researchers studying uncertain problems utilize stochastic modeling is that this randomization is the result of an established scientific stereotype. Indeed, probabilistic approaches are not able to deliver reliable results without sufficient experiment data.
Since the mid-1960's, a new method called the interval
analysis has appeared. Moore and his co-workers, Alefeld andHerzberger have done the pioneering work. Mathematically, the linear interval equations and nonlinear interval equations have been resolved. But because of the complexity of the algorithm, it is difficult to apply these results to practical engineering problems. Recently, Chen, Qiu ,etc. have used interval method in the study of the static response and eigenvalue problems of structures with bounded uncertain parameters. In their studies, several important results have been obtained, using interval analysis and matrix perturbation techniques. However, these results are based on the assumption that K, f are pre-selected in the equation K(a)U = f(a) and K, M are also pre-selected in the equation K(a)U = M(a). In general, K , M and Af are functions of the structural parameters, so they must can be calculated according to the uncertainties of the structural parameters. Yang Xiaowei and Lian Huadong have presented some effective interval methods for structures with interval parameters. Hansen in his book discussed the global optimization using interval analysis. Because of the complexity of the interval algorithm, it is difficult to deal with practical engineering problems. Recently, the interval analysis method has been used to deal with the static displacement and eigenvalue analysis of the uncertain structures with interval parameters. However, few papers can be found about the optimization of structures with interval parameters in engineering. Hence, it is necessary to develop an effective
method to solve the optimal problems of structures with interval parameters. This paper presents an interval optimization method based on the interval analysis.
In this paper, on the basis of the work of Yang Xiaowei and Lian Huadong, some problems are discussed:
1. In chapter 4, a dynamic interval optimization method for discrete systems with interval parameters is presented.
2. In chapter 5, a dynamic interval optimization method for continuous systems with interval parameters is presented.
3. In chapter 6, an interval optimization method of dynamic response for uncertain structures with natural frequency constraints is presented.
4. In chapter 7, an interval dynamic optimization method for uncertain structures using the improved 1st-order Taylor expansion is presented.
近几十年来，研究不确定性问题最常用的方法是概率统计方法。在方法中，人们常把结构参数看成随机变量或随机场，关于结构参数的所有信息均由结构参数的联合概率密度函数或分布函数提供。尽管概率统计方法在描述结构力学中的不确定现象方面获得了很大成功，但是在没有足够的实验数据证明由联合概率密度(联合概率密度函数)所给出的假设正确的情况下，这种方法并不能给出满足某种精度的可靠结果。因此，概率统计方法不应该是研究不确定性问题的唯一方法。
目前，除了概率统计方法外，用于研究不确定性问题的方法还有凸分析法、模糊集合法和区间分析法。考虑到凸分析法和模糊集合法中的主观信息的限制，同时也考虑到数学上线性区间方程组、非线性区间方程和区间特征值问题解法的复杂性，陈塑寰、邱志平等人区间分析模型用于研究具有不确定性参数结构的静、动态问题中。利用区间分析和矩阵摄动法，他们得到了几个重要的结果。但他们的结果是以下述假设为基础的：ΔK和Δf在方程K(α)U=f(α)中是预先设定的；ΔK和ΔM在方程K(α)U=λM(α)中也是预先设定的。一般来说，这种假设是有局限性的。因为ΔK、ΔM和Δf都是结
吉林大学博士学位论文
构参数的函数，它们的不确定性可根据结构参数的不确定性来计算，这已经
由杨晓伟、连华东等人给出了不错的结果。
区间分析方法是自二十世纪六十年代以来出现的，Moore和他的合作者
Alefeld和Herzberger己经作了许多开创性的工作。Hansen在他的书中基于
区间分析讨论了全局优化问题。但是由于这些区间算法的复杂性，很难用它
们来解决实际的工程问题。最近，区间分析方法被用于具有区间参数的不确
定性结构的静态位移和特征值的分析。但是，很少有文献涉及到具有区间参
数结构的结构优化问题。因此，很有必要提出一种有效的方法，用于解决具
有区间参数结构的结构优化问题。本文基于区间分析提出了一种区间优化方
法，这种方法的思想是:首先对具有区间参数结构的动力响应进行区间分析，
然后在此基础上用标准的优化算法对所得的区间的上界进行优化，使得结构
某点的响应值落在一个最窄的区间内。
在本论文中，作者在杨晓伟、连华东等人工作的基础上，主要做了下面
的一些工作:
1.第四章针对具有区间参数的离散系统提出一种动力区间优化方法。
2.第五章针对具有区间参数的连续系统提出一种动力响应的区间优化
方法。
3，第六章针对区间参数振动结构提出了一种具有频率约束的动力响应
的区间优化方法。
4.第七章针对区间参数振动结构提出了一种改进的动力响应的区间优
化方法。
In many practical engineering problems, the structural parameters are uncertain, for example, the inaccuracy of measurements, errors in manufacture, etc. Therefore, the concept of uncertainty plays an important role in the investigation of various engineering problems.
The most common approach to uncertain problems is to model the structural parameters as random variables or fields. In this case, all information about the structural parameters is provided by the joint probability density function(or distribution function) of them. Unfortunately, the probabilistic modeling is not the only way we can use to describe the uncertainty, and uncertainty is not tantamount to randomness. In many cases, the uncertainty phenomena do not have a stochastic nature. The reason why many researchers studying uncertain problems utilize stochastic modeling is that this randomization is the result of an established scientific stereotype. Indeed, probabilistic approaches are not able to deliver reliable results without sufficient experiment data.
Since the mid-1960's, a new method called the interval
analysis has appeared. Moore and his co-workers, Alefeld andHerzberger have done the pioneering work. Mathematically, the linear interval equations and nonlinear interval equations have been resolved. But because of the complexity of the algorithm, it is difficult to apply these results to practical engineering problems. Recently, Chen, Qiu ,etc. have used interval method in the study of the static response and eigenvalue problems of structures with bounded uncertain parameters. In their studies, several important results have been obtained, using interval analysis and matrix perturbation techniques. However, these results are based on the assumption that K, f are pre-selected in the equation K(a)U = f(a) and K, M are also pre-selected in the equation K(a)U = M(a). In general, K , M and Af are functions of the structural parameters, so they must can be calculated according to the uncertainties of the structural parameters. Yang Xiaowei and Lian Huadong have presented some effective interval methods for structures with interval parameters. Hansen in his book discussed the global optimization using interval analysis. Because of the complexity of the interval algorithm, it is difficult to deal with practical engineering problems. Recently, the interval analysis method has been used to deal with the static displacement and eigenvalue analysis of the uncertain structures with interval parameters. However, few papers can be found about the optimization of structures with interval parameters in engineering. Hence, it is necessary to develop an effective
method to solve the optimal problems of structures with interval parameters. This paper presents an interval optimization method based on the interval analysis.
In this paper, on the basis of the work of Yang Xiaowei and Lian Huadong, some problems are discussed:
1. In chapter 4, a dynamic interval optimization method for discrete systems with interval parameters is presented.
2. In chapter 5, a dynamic interval optimization method for continuous systems with interval parameters is presented.
3. In chapter 6, an interval optimization method of dynamic response for uncertain structures with natural frequency constraints is presented.
4. In chapter 7, an interval dynamic optimization method for uncertain structures using the improved 1st-order Taylor expansion is presented.
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