不确定结构（机构）分析和可靠性专题研究
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摘要
实际工程结构中存在大量的误差和不确定性，在结构分析中必须考虑这些不确定因素。研究不确定性问题的数学模型主要有三种：概率模型、模糊模型、凸集模型或区间分析。其中，区间分析具有简单实用的特点。由于机构形体可变、失效模式众多，机构可靠性问题较之结构可靠性问题难度要大得多。目前，关于机构可靠性的理论和方法尚不完善。
本文对不确定性结构的静力响应分析、动力特性分析和非概率可靠性计算，以及机构运动的可靠性等专题进行了研究。主要内容如下：
1、基于不确定性的区间模型描述，研究了不确定结构的区间有限元解法。提出了区间静力控制方程组的两种有效解法：区间逐步离散法和区间因子法。区间逐步离散法既考虑到区间线性方程组中各矩阵元素的相关性，又考察了区间元素取区间内点值的情况，结果趋近于真实解。用区间因子法求解不确定性问题方法简便，只需一次结构分析即可获得静力位移和应力响应的区间。
2、讨论了求解静力区间线性方程组的泛灰数方法。该方法将区间线性方程组转化为泛灰数方程组求解，方法简单易行，但在使用中有不完善的地方。本文对此提出一种新的泛灰数转换规则，通过算例验证了新规则的合理性。
3、基于区间有限元进行了不确定结构动力特性分析。针对广义区间特征方程的求解提出了区间离散方法和区间因子法，算例分析说明了所建模型和求解方法的可行性、有效性。区间离散算法不引入更多假设，适用面较广，收敛速度较快。区间因子法易于分析结构参数的区间不确定性对结构动力特性的影响。
4、基于传递矩阵法对不确定链式结构进行动力特性分析。对不确定链式结构导出关于系统固有频率的非线性区间方程，并用区间离散算法求解该方程。基于区间因子法推导了关于动力特性区间变量的计算表达式，并分析了结构参数的区间不确定性对链式结构动力特性的影响。
5、研究了大型星载天线的展开可靠性问题。分析了周边桁架式星载天线的展开运动机理，并基于随机模型，计算了展开机构的运动可靠性和同步齿轮防卡滞的可靠性。应用失效树分析方法建立起展开系统的失效树，通过各基本事件的失效概率和重要度计算，指出影响天线展开的关键因素。
6、基于非概率可靠性模型提出了结构可靠性的两种计算方法。两种算法对于任意功能函数都是可行的；对于线性功能函数，两者等效；对于非线性功能函数，两者略有差异。计算的差异来源于两种非概率可靠性定义的不同。将非概率可靠性模型应用到星载天线展开机构可靠性研究，并与概率可靠性相比较。结果表明：概率可靠性指标约为非概率可靠性指标的三倍，两者的变化趋势相同。
7、将灰色理论与方法引入机构的运动可靠性评估。提出了机构的灰色可靠性概念和分析模型。从机构运动条件和运动精度的极限状态方程出发，推导了灰色可靠度计算公式。研究结果表明：机构灰色可靠性模型是可行有效的。
本文的研究对于提高不确定结构的有限元求解能力，完善结构(机构)的可靠性理论，拓展不确定结构(机构)分析在航天工程领域的应用具有比较重要的理论和现实意义。
A large number of errors and uncertainties exist in the actual structure, and these uncertain factors should be considered in structural analysis. There are three mainly models to deal with uncertainty, which are probability model, fuzzy model and convex model or interval analysis. Among these models, the interval analysis is simple and practical to deal with uncertain structures. For changeable structural forms and sorts of failure types, the reliability analysis of a mechanism is more difficult than a fixed structure. Recently, the theory of mechanism reliability analysis is still imperfect.
Some researches on special topics are presented in this paper, which include the static response analysis, dynamic characteristic analysis, the non-probabilistic reliability calculating of an uncertain structure, the movement reliability of a mechanism.
The main research works can be described as follows:
1. Interval finite element methods (FEM) for uncertain structures are studied. Several efficient methods for solving the static governing equations of interval FEM are proposed, which include an interval-step-dividing method and an interval factor method. The interval-step-dividing method considers the correlation between matrix elements and the condition that each element obtains the interval's inner value. Therefore, it can get a result which is close to the exact one. The interval factor method is easy to solve the uncertain problem, which needs only once structural analysis to get the response intervals of the static displacement and stress.
2. An approach by using universal grey numbers to solve static interval linear equations is discussed. This method changes interval linear equations to universal grey linear equations to solve, which is simple, but has some disadvantages in application. A new rule of transforming an interval number into universal grey one is presented. Computing results show that the new transforming rule is reasonable.
3. Based on the interval FEM the dynamic characteristics of uncertain structures are analyzed. To solve the generalized interval eigenvalue equation, an interval-step-dividing method and an interval factor method are presented in this paper. The correctness and validity of the theory and method presented in this paper are inspected by several examples. The interval factor method is easy to analyze the effect of any parameter's uncertain on the dynamic characteristics.
4. Based on the transfer matrix method, the dynamic characteristics of uncertain chain-structures are analyzed. A nonlinear interval equation of a chain system's natural frequency is developed, which is solved by using the interval dividing method. Based on the interval factor method, some expressions of interval variables about the structure's dynamic characteristics are derived.
5. The deployment reliability analysis of a large satellite antenna is studied. The movement principium of a large satellite antenna is analyzed. Based on a random model, the movement reliability of the deployment mechanism and the seizure-preventing reliability of synchronous gears are calculated. The faulty tree of the deployment system is built by using the Faulty Tree Analysis method, and from the calculating result of the probability and importance degree of each base affair, some key factors to affect the deployment of the antenna are point out.
6. Based on a non-probabilistic reliability model, two methods for structural reliability computation are produced, which are practicable for any performance functions. These two methods are equivalent for linear functions but a little different for nonlinear ones. The computing difference comes from two definitions about non-probabilistic reliability. The non-probabilistic model is also used to give the reliability analysis of a satellite antenna's deployment mechanism, and the result is compared with another one based on the probabilistic model. The result shows that the trends of their variation are alike.
7. The grey theory and method is applied in the mechanism reliability research. The conception and model of grey reliability on a mechanism's movement is presented. From the limiting condition equations of movement requisition and accuracy, the computing formulae of the grey reliability are derived respectively. The research proves that the method is feasible and the result is reasonable.
The theories founded in the paper are important and meaningful in the following aspects: enhancing the ability of the FEM in solving uncertain structures, completing the structural (mechanism) reliability theory, improving the application of the structural (mechanism) analysis in the aerospace engineering.
本文对不确定性结构的静力响应分析、动力特性分析和非概率可靠性计算，以及机构运动的可靠性等专题进行了研究。主要内容如下：
1、基于不确定性的区间模型描述，研究了不确定结构的区间有限元解法。提出了区间静力控制方程组的两种有效解法：区间逐步离散法和区间因子法。区间逐步离散法既考虑到区间线性方程组中各矩阵元素的相关性，又考察了区间元素取区间内点值的情况，结果趋近于真实解。用区间因子法求解不确定性问题方法简便，只需一次结构分析即可获得静力位移和应力响应的区间。
2、讨论了求解静力区间线性方程组的泛灰数方法。该方法将区间线性方程组转化为泛灰数方程组求解，方法简单易行，但在使用中有不完善的地方。本文对此提出一种新的泛灰数转换规则，通过算例验证了新规则的合理性。
3、基于区间有限元进行了不确定结构动力特性分析。针对广义区间特征方程的求解提出了区间离散方法和区间因子法，算例分析说明了所建模型和求解方法的可行性、有效性。区间离散算法不引入更多假设，适用面较广，收敛速度较快。区间因子法易于分析结构参数的区间不确定性对结构动力特性的影响。
4、基于传递矩阵法对不确定链式结构进行动力特性分析。对不确定链式结构导出关于系统固有频率的非线性区间方程，并用区间离散算法求解该方程。基于区间因子法推导了关于动力特性区间变量的计算表达式，并分析了结构参数的区间不确定性对链式结构动力特性的影响。
5、研究了大型星载天线的展开可靠性问题。分析了周边桁架式星载天线的展开运动机理，并基于随机模型，计算了展开机构的运动可靠性和同步齿轮防卡滞的可靠性。应用失效树分析方法建立起展开系统的失效树，通过各基本事件的失效概率和重要度计算，指出影响天线展开的关键因素。
6、基于非概率可靠性模型提出了结构可靠性的两种计算方法。两种算法对于任意功能函数都是可行的；对于线性功能函数，两者等效；对于非线性功能函数，两者略有差异。计算的差异来源于两种非概率可靠性定义的不同。将非概率可靠性模型应用到星载天线展开机构可靠性研究，并与概率可靠性相比较。结果表明：概率可靠性指标约为非概率可靠性指标的三倍，两者的变化趋势相同。
7、将灰色理论与方法引入机构的运动可靠性评估。提出了机构的灰色可靠性概念和分析模型。从机构运动条件和运动精度的极限状态方程出发，推导了灰色可靠度计算公式。研究结果表明：机构灰色可靠性模型是可行有效的。
本文的研究对于提高不确定结构的有限元求解能力，完善结构(机构)的可靠性理论，拓展不确定结构(机构)分析在航天工程领域的应用具有比较重要的理论和现实意义。
A large number of errors and uncertainties exist in the actual structure, and these uncertain factors should be considered in structural analysis. There are three mainly models to deal with uncertainty, which are probability model, fuzzy model and convex model or interval analysis. Among these models, the interval analysis is simple and practical to deal with uncertain structures. For changeable structural forms and sorts of failure types, the reliability analysis of a mechanism is more difficult than a fixed structure. Recently, the theory of mechanism reliability analysis is still imperfect.
Some researches on special topics are presented in this paper, which include the static response analysis, dynamic characteristic analysis, the non-probabilistic reliability calculating of an uncertain structure, the movement reliability of a mechanism.
The main research works can be described as follows:
1. Interval finite element methods (FEM) for uncertain structures are studied. Several efficient methods for solving the static governing equations of interval FEM are proposed, which include an interval-step-dividing method and an interval factor method. The interval-step-dividing method considers the correlation between matrix elements and the condition that each element obtains the interval's inner value. Therefore, it can get a result which is close to the exact one. The interval factor method is easy to solve the uncertain problem, which needs only once structural analysis to get the response intervals of the static displacement and stress.
2. An approach by using universal grey numbers to solve static interval linear equations is discussed. This method changes interval linear equations to universal grey linear equations to solve, which is simple, but has some disadvantages in application. A new rule of transforming an interval number into universal grey one is presented. Computing results show that the new transforming rule is reasonable.
3. Based on the interval FEM the dynamic characteristics of uncertain structures are analyzed. To solve the generalized interval eigenvalue equation, an interval-step-dividing method and an interval factor method are presented in this paper. The correctness and validity of the theory and method presented in this paper are inspected by several examples. The interval factor method is easy to analyze the effect of any parameter's uncertain on the dynamic characteristics.
4. Based on the transfer matrix method, the dynamic characteristics of uncertain chain-structures are analyzed. A nonlinear interval equation of a chain system's natural frequency is developed, which is solved by using the interval dividing method. Based on the interval factor method, some expressions of interval variables about the structure's dynamic characteristics are derived.
5. The deployment reliability analysis of a large satellite antenna is studied. The movement principium of a large satellite antenna is analyzed. Based on a random model, the movement reliability of the deployment mechanism and the seizure-preventing reliability of synchronous gears are calculated. The faulty tree of the deployment system is built by using the Faulty Tree Analysis method, and from the calculating result of the probability and importance degree of each base affair, some key factors to affect the deployment of the antenna are point out.
6. Based on a non-probabilistic reliability model, two methods for structural reliability computation are produced, which are practicable for any performance functions. These two methods are equivalent for linear functions but a little different for nonlinear ones. The computing difference comes from two definitions about non-probabilistic reliability. The non-probabilistic model is also used to give the reliability analysis of a satellite antenna's deployment mechanism, and the result is compared with another one based on the probabilistic model. The result shows that the trends of their variation are alike.
7. The grey theory and method is applied in the mechanism reliability research. The conception and model of grey reliability on a mechanism's movement is presented. From the limiting condition equations of movement requisition and accuracy, the computing formulae of the grey reliability are derived respectively. The research proves that the method is feasible and the result is reasonable.
The theories founded in the paper are important and meaningful in the following aspects: enhancing the ability of the FEM in solving uncertain structures, completing the structural (mechanism) reliability theory, improving the application of the structural (mechanism) analysis in the aerospace engineering.
引文
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