区间和未确知参数结构（机构）分析方法研究及应用
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摘要
本论文首先以区间参数、随机区间混合参数和未确知参数结构为研究对象,探索性地研究了当结构参数和外载荷为区间变量或未确知变量时结构的静力响应、动力特性、动力响应以及非概率可靠性指标;然后以在研的国家863高技术研究发展计划项目《大型星载可展开天线结构系统多状态全过程的可靠性综合分析研究》为工程背景,针对小样本、贫信息事件,基于未确知理论对星载天线展开机构中的旋转关节运动可靠性进行了计算、预测和分析。主要内容如下:
1、基于区间模型的结构静力分析。
将结构系统中的不确定性参数用区间数表示,建立系统的区间有限元控制方程。对该方程组的求解提出一种基于导数信息的仿射算法。此方法通过令独立的不确定性参数转换成仿射型,将区间线性方程组的求解转化为相应的确定性问题,再搜索各方程解中的最大最小值得到位移响应的区间范围。
2、基于区间模型的结构动力特性和动力响应分析方法研究。
对多自由度区间参数结构的广义特征值问题和在区间荷载激励下的动力响应分析问题进行研究。将不确定结构系统中的区间参数用仿射型来表示,对获得的动力方程的求解方法进行研究,提出了一种基于区间离散的改进的仿射算法。此方法考虑到广义特征值及响应方程中各元素的相关性,通过独立的区间参数在子区间上转为仿射型,将特征值及响应方程的求解转化为相应的确定性问题,再利用常规的仿射算法,搜索方程解中的最大最小值来确定各特征值和响应的范围。
3、区间参数杆系结构非概率可靠性的仿射算法。
将杆系结构中的不确定性参数用区间数表示,建立结构的非概率可靠性指标求解方程。将矩阵形式的仿射算术和递归导数信息结合,提出了改进的二元区间多项式矩阵形式的仿射上下界公式。给出了有界不确定性变量的仿射型和区间形式的相互转化,将有界不确定性变量的仿射型及结合导数信息的矩阵形式的仿射运算引入到杆系结构基于区间模型的非概率可靠性指标计算中。分别以某外伸梁和十杆桁架结构为例对文中方法的可行性和有效性进行了验证。
4、基于可信度约束的区间参数天线结构力学分析。
构建了物理参数、几何参数和载荷同时具有不确定性的桁架结构有限元分析模型。基于不确定性的区间模型描述,将可信度约束引入区间分析,提出了基于可信度约束的区间因子法的结构分析方法;定义了具有可信度约束的区间运算规则;利用区间因子的数学表述,推导出结构位移、应力响应区间及动力特征值响应的计算表达式。
5、区间参数天线结构动力特性分析的概率处理。
将不确定桁架结构中的区间参数用随机变量来表示,对获得的广义区间特征值方程的求解方法进行研究。假定各区间参量在允许取值区间内为具有熵最大的矩形分布,各随机变量在所定义区域内均匀分布并假设它们彼此独立,将区间特征值方程的求解采用概率理论来处理,再利用随机因子算法,来确定结构动力特征值边界,并与用随机正态分布变量描述的区间求解做比较,得出相应的结论。
6、随机区间混合模型天线结构的有限元及可靠性分析。
对随机区间型天线结构有限元及可靠性分析方法进行研究,提出结构保精度和保强度两工况的概率描述。同时考虑结构的物理参数、几何参数的随机性和作用风载荷的区间性,首先将随机变量固定,利用区间因子法求得结构位移和应力响应的区间范围,然后在区间内任意点处利用随机因子法求结构响应的随机分布范围。通过推导,构造天线反射面位移响应和结构单元应力响应不确定变量的数字特征计算公式;进而得到结构各响应量的可靠性指标。
7、基于未确知理论的天线展开机构运动可靠性分析。
对某周边桁架式大型星载天线的展开运动机理和伞状天线展开机构中的旋转关节运动机理进行研究,建立展开机构的力学分析和未确知运动可靠性的分析模型。综合考虑尺寸误差和太空环境因素的影响,将运动功能函数视为未确知变量函数,利用未确知有理数计算的法则推导出可靠性计算公式,对机构在整个展开过程中的运动可靠性进行预测。与成熟的随机方法想比,该方法简单易行,且能在缺乏足够数据或信息不完整的情况下,获得更安全、可信度更高的机构可靠性计算结果。算例给出了未确知性天线展开机构功能函数的可能值及其可信度的计算结果,表明该方法的合理性和可行性。
8、基于未确知理论的板梁组合结构静力、动力特性分析。
为了克服随机方法对于小样本难以处理的缺陷,充分利用客观的不确定性信息,构建了物理参数和载荷同时具有未确知性的空间板梁组合结构有限元分析模型,并提出了基于未确知因子法的板梁组合结构分析方法;利用未确知因子的数学表述和未确知有理数的运算规则,推导出板梁组合结构静力响应、动力特征值的计算表达式。算例给出了未确知信息板梁组合结构的静力响应和固有频率可能值及其可信度的计算结果,表明该方法的可行性和有效性。
Firstly, structures with interval parameters, random-interval parameters or unascertained parameters are taken as research objects in this paper. Structural static responses, dynamic characteristics, dynamic responses and non-probability reliability index are derived under the conditions that physical parameters of materials, structural geometric dimensions and applied loads are all interval, random-interval or unascertained variables. Secondly, based on the engineering background of national 863 project, reliability computation, prediction and analysis on deployment mechanism system of satellite antenna are derived by means of unascertained theory. The main research works can be described as follows:
1. The static analysis of interval truss structures.
By representing the uncertain parameters as interval numbers, the governed equations of the structural system are obtained by means of the finite element method. Some solution methods for these equations are discussed and affine arithmetic polynomial evaluation method plus recursive derivative information is proposed. In this method, the independent uncertain parameters are transferred to affine forms, and the linear interval equations are changed to the corresponding certain ones. Then the bounds of every interval solution components are determined by searching for the maximums and the minimums.
2. The study on dynamic characteristic and dynamic responses of interval truss structures.
Not only considering the interval characteristics of structural physical parameters and geometric dimension, but also considering interval characteristics of applied load simultaneously, uncertainty of the MDOF structural dynamic response is studied. By describing the interval parameters of uncertain structure with affine forms, the interval structural dynamic equation is researched, and an improved affine arithmetic based on interval division is presented, where correlations between the interval elements in eigenvalue and responses equations are considered, independent uncertain parameters are transformed to affine forms, and the solution of eigenvalue and response equations are transformed into the corresponding certain ones. With general affine arithmetic, the eigenvalue of each order and response bounds are determined by searching for the maximum and minimum in the solutions. Some numerical examples were provided to illustrate the validity and feasibility of the present method.
3. Non-probabilistic reliability index of bar structures with interval parameters based on modified affine arithmetic.
By representing the uncertain parameters as interval numbers, the reliability index equations of bar structures are obtained. A modified matrix affine arithmetic polynomial evaluation method plus recursive derivative information is proposed in this paper, which keeps all powers of noise symbols without approximation. Based on the nature that affine forms and intervals variables can transform each other, affine forms of bounded uncertain variables and modified affine arithmetic including derivative information for univariate interval polynomial evaluation are introduced into modeling and calculating non-probabilistic reliability index. An extended beam example and a ten-bar truss structure example are provided to illustrate the validity and feasibility of the presented procedures.
4. Interval method with faith degree constraints for structures analysis.
The finite element analysis model of uncertain truss structures is built, in which the structural physical parameters, geometrical dimensions and the loads are all considered as unascertained variables. And a structural analysis method based on the interval factor method with faith degree constraint is given. The arithmetic operation rules of interval analysis with faith degree constraint are defined. By the mathematics expression of interval factor, the computational expressions of structural static responses, dynamic eigenvalue and dynamic responses are developed.
5. Dynamic eigenvalues analysis of structures with interval parameters based on probabilistic theory.
By describing the interval parameters of uncertain structure with random variables, a generalized eigenvalues interval equation was researched, and a simple arithmetic was presented. The interval variables were supposed to be rectangle distribution with maximum entropy in allowable range, and random variables obeyed uniform distribution in definition region on the assumption that they were independent each other. The solution of interval eigenvalues equations are tackled by using the probabilistic theory, then the random factor method is applied to obtain the bounds of eigenvalues. For comparisons, the interval variables are also supposed to be random normal distribution and the corresponding eigenvalues ranges are obtained. Finally an engineering application was applied to confirm the feasibility and validity of this approach.
6. Finite element and reliability analyses for antenna structures with the mixture of random and interval variables.
A model for finite element and reliability analyses for antenna structures with random parameters under interval loads was constructed, a new method of finite element analysis for dealing with structural uncertainty factors was presented, and the structural probability descriptions in the cases of preserved-precision and preserved-intensity were given. The stochastic property of physical parameters and geometry dimensions and the interval property of wind loads applied on antenna structures were considered simultaneously. Firstly the stochastic variables were fixed to obtain the ranges of structural displacement and stress by using the interval factor method, and then the random distribution ranges of structural responses for any points in the interval were gained based on the random factor method. The computational expressions for the numerical characteristic of antenna reflector responses including displacements and structural element stresses were constructed; thereby the reliability indexes of the structural responses were obtained. Finally, the rationality and the feasibility of the method were confirmed by the analysis of an antenna structure with an 8-meter caliber.
7. Reliability analysis for antenna deployment mechanism based on unascertained theory.
The deployment principium of a large hoop-truss satellite antenna was studied and the mechanical analysis model and the unascertained reliability model of its deployment mechanism were presented. Synthetically considering the effect of dimension errors and the space environment factors, we treat the mechanism movement as a function of some unascertained rational numbers, and derive the reliability formula by using computational theorem of unascertained rational numbers. The movement reliability of the mechanism of a large satellite antenna in the whole spreading process is predicted. Compared to the mature random method, the proposed method can obtain reliability result of safer and higher faith degree in the case of inadequate data or insufficient information; moreover, it is simple and easy to apply.
8. Static and dynamic eigenvalue analysis for beam-plates composite structures based on unascertained theory.
Random method doesn’t suit the case of small sample. To overcome this limitation, the objective unascertained information was made full use of, and the static finite element and dynamic characteristic analysis models of space beam-plate composite structure were built, in which the structural physical parameters and the applied loads were all considered as unascertained variables. And a structure analysis method based on the unascertained factor method was given. By the mathematics expression of unascertained factor and the arithmetic operation rules of unascertained rational numbers, the computational expressions of the structural displacement response, the element stress response and dynamic eigenvalue are developed. At last, an example is given, in which the possible values and faith degrees of the unascertained structure static responses and natural frequency are obtained. The rationality and validity of the presented method are demonstrated.
1、基于区间模型的结构静力分析。
将结构系统中的不确定性参数用区间数表示,建立系统的区间有限元控制方程。对该方程组的求解提出一种基于导数信息的仿射算法。此方法通过令独立的不确定性参数转换成仿射型,将区间线性方程组的求解转化为相应的确定性问题,再搜索各方程解中的最大最小值得到位移响应的区间范围。
2、基于区间模型的结构动力特性和动力响应分析方法研究。
对多自由度区间参数结构的广义特征值问题和在区间荷载激励下的动力响应分析问题进行研究。将不确定结构系统中的区间参数用仿射型来表示,对获得的动力方程的求解方法进行研究,提出了一种基于区间离散的改进的仿射算法。此方法考虑到广义特征值及响应方程中各元素的相关性,通过独立的区间参数在子区间上转为仿射型,将特征值及响应方程的求解转化为相应的确定性问题,再利用常规的仿射算法,搜索方程解中的最大最小值来确定各特征值和响应的范围。
3、区间参数杆系结构非概率可靠性的仿射算法。
将杆系结构中的不确定性参数用区间数表示,建立结构的非概率可靠性指标求解方程。将矩阵形式的仿射算术和递归导数信息结合,提出了改进的二元区间多项式矩阵形式的仿射上下界公式。给出了有界不确定性变量的仿射型和区间形式的相互转化,将有界不确定性变量的仿射型及结合导数信息的矩阵形式的仿射运算引入到杆系结构基于区间模型的非概率可靠性指标计算中。分别以某外伸梁和十杆桁架结构为例对文中方法的可行性和有效性进行了验证。
4、基于可信度约束的区间参数天线结构力学分析。
构建了物理参数、几何参数和载荷同时具有不确定性的桁架结构有限元分析模型。基于不确定性的区间模型描述,将可信度约束引入区间分析,提出了基于可信度约束的区间因子法的结构分析方法;定义了具有可信度约束的区间运算规则;利用区间因子的数学表述,推导出结构位移、应力响应区间及动力特征值响应的计算表达式。
5、区间参数天线结构动力特性分析的概率处理。
将不确定桁架结构中的区间参数用随机变量来表示,对获得的广义区间特征值方程的求解方法进行研究。假定各区间参量在允许取值区间内为具有熵最大的矩形分布,各随机变量在所定义区域内均匀分布并假设它们彼此独立,将区间特征值方程的求解采用概率理论来处理,再利用随机因子算法,来确定结构动力特征值边界,并与用随机正态分布变量描述的区间求解做比较,得出相应的结论。
6、随机区间混合模型天线结构的有限元及可靠性分析。
对随机区间型天线结构有限元及可靠性分析方法进行研究,提出结构保精度和保强度两工况的概率描述。同时考虑结构的物理参数、几何参数的随机性和作用风载荷的区间性,首先将随机变量固定,利用区间因子法求得结构位移和应力响应的区间范围,然后在区间内任意点处利用随机因子法求结构响应的随机分布范围。通过推导,构造天线反射面位移响应和结构单元应力响应不确定变量的数字特征计算公式;进而得到结构各响应量的可靠性指标。
7、基于未确知理论的天线展开机构运动可靠性分析。
对某周边桁架式大型星载天线的展开运动机理和伞状天线展开机构中的旋转关节运动机理进行研究,建立展开机构的力学分析和未确知运动可靠性的分析模型。综合考虑尺寸误差和太空环境因素的影响,将运动功能函数视为未确知变量函数,利用未确知有理数计算的法则推导出可靠性计算公式,对机构在整个展开过程中的运动可靠性进行预测。与成熟的随机方法想比,该方法简单易行,且能在缺乏足够数据或信息不完整的情况下,获得更安全、可信度更高的机构可靠性计算结果。算例给出了未确知性天线展开机构功能函数的可能值及其可信度的计算结果,表明该方法的合理性和可行性。
8、基于未确知理论的板梁组合结构静力、动力特性分析。
为了克服随机方法对于小样本难以处理的缺陷,充分利用客观的不确定性信息,构建了物理参数和载荷同时具有未确知性的空间板梁组合结构有限元分析模型,并提出了基于未确知因子法的板梁组合结构分析方法;利用未确知因子的数学表述和未确知有理数的运算规则,推导出板梁组合结构静力响应、动力特征值的计算表达式。算例给出了未确知信息板梁组合结构的静力响应和固有频率可能值及其可信度的计算结果,表明该方法的可行性和有效性。
Firstly, structures with interval parameters, random-interval parameters or unascertained parameters are taken as research objects in this paper. Structural static responses, dynamic characteristics, dynamic responses and non-probability reliability index are derived under the conditions that physical parameters of materials, structural geometric dimensions and applied loads are all interval, random-interval or unascertained variables. Secondly, based on the engineering background of national 863 project, reliability computation, prediction and analysis on deployment mechanism system of satellite antenna are derived by means of unascertained theory. The main research works can be described as follows:
1. The static analysis of interval truss structures.
By representing the uncertain parameters as interval numbers, the governed equations of the structural system are obtained by means of the finite element method. Some solution methods for these equations are discussed and affine arithmetic polynomial evaluation method plus recursive derivative information is proposed. In this method, the independent uncertain parameters are transferred to affine forms, and the linear interval equations are changed to the corresponding certain ones. Then the bounds of every interval solution components are determined by searching for the maximums and the minimums.
2. The study on dynamic characteristic and dynamic responses of interval truss structures.
Not only considering the interval characteristics of structural physical parameters and geometric dimension, but also considering interval characteristics of applied load simultaneously, uncertainty of the MDOF structural dynamic response is studied. By describing the interval parameters of uncertain structure with affine forms, the interval structural dynamic equation is researched, and an improved affine arithmetic based on interval division is presented, where correlations between the interval elements in eigenvalue and responses equations are considered, independent uncertain parameters are transformed to affine forms, and the solution of eigenvalue and response equations are transformed into the corresponding certain ones. With general affine arithmetic, the eigenvalue of each order and response bounds are determined by searching for the maximum and minimum in the solutions. Some numerical examples were provided to illustrate the validity and feasibility of the present method.
3. Non-probabilistic reliability index of bar structures with interval parameters based on modified affine arithmetic.
By representing the uncertain parameters as interval numbers, the reliability index equations of bar structures are obtained. A modified matrix affine arithmetic polynomial evaluation method plus recursive derivative information is proposed in this paper, which keeps all powers of noise symbols without approximation. Based on the nature that affine forms and intervals variables can transform each other, affine forms of bounded uncertain variables and modified affine arithmetic including derivative information for univariate interval polynomial evaluation are introduced into modeling and calculating non-probabilistic reliability index. An extended beam example and a ten-bar truss structure example are provided to illustrate the validity and feasibility of the presented procedures.
4. Interval method with faith degree constraints for structures analysis.
The finite element analysis model of uncertain truss structures is built, in which the structural physical parameters, geometrical dimensions and the loads are all considered as unascertained variables. And a structural analysis method based on the interval factor method with faith degree constraint is given. The arithmetic operation rules of interval analysis with faith degree constraint are defined. By the mathematics expression of interval factor, the computational expressions of structural static responses, dynamic eigenvalue and dynamic responses are developed.
5. Dynamic eigenvalues analysis of structures with interval parameters based on probabilistic theory.
By describing the interval parameters of uncertain structure with random variables, a generalized eigenvalues interval equation was researched, and a simple arithmetic was presented. The interval variables were supposed to be rectangle distribution with maximum entropy in allowable range, and random variables obeyed uniform distribution in definition region on the assumption that they were independent each other. The solution of interval eigenvalues equations are tackled by using the probabilistic theory, then the random factor method is applied to obtain the bounds of eigenvalues. For comparisons, the interval variables are also supposed to be random normal distribution and the corresponding eigenvalues ranges are obtained. Finally an engineering application was applied to confirm the feasibility and validity of this approach.
6. Finite element and reliability analyses for antenna structures with the mixture of random and interval variables.
A model for finite element and reliability analyses for antenna structures with random parameters under interval loads was constructed, a new method of finite element analysis for dealing with structural uncertainty factors was presented, and the structural probability descriptions in the cases of preserved-precision and preserved-intensity were given. The stochastic property of physical parameters and geometry dimensions and the interval property of wind loads applied on antenna structures were considered simultaneously. Firstly the stochastic variables were fixed to obtain the ranges of structural displacement and stress by using the interval factor method, and then the random distribution ranges of structural responses for any points in the interval were gained based on the random factor method. The computational expressions for the numerical characteristic of antenna reflector responses including displacements and structural element stresses were constructed; thereby the reliability indexes of the structural responses were obtained. Finally, the rationality and the feasibility of the method were confirmed by the analysis of an antenna structure with an 8-meter caliber.
7. Reliability analysis for antenna deployment mechanism based on unascertained theory.
The deployment principium of a large hoop-truss satellite antenna was studied and the mechanical analysis model and the unascertained reliability model of its deployment mechanism were presented. Synthetically considering the effect of dimension errors and the space environment factors, we treat the mechanism movement as a function of some unascertained rational numbers, and derive the reliability formula by using computational theorem of unascertained rational numbers. The movement reliability of the mechanism of a large satellite antenna in the whole spreading process is predicted. Compared to the mature random method, the proposed method can obtain reliability result of safer and higher faith degree in the case of inadequate data or insufficient information; moreover, it is simple and easy to apply.
8. Static and dynamic eigenvalue analysis for beam-plates composite structures based on unascertained theory.
Random method doesn’t suit the case of small sample. To overcome this limitation, the objective unascertained information was made full use of, and the static finite element and dynamic characteristic analysis models of space beam-plate composite structure were built, in which the structural physical parameters and the applied loads were all considered as unascertained variables. And a structure analysis method based on the unascertained factor method was given. By the mathematics expression of unascertained factor and the arithmetic operation rules of unascertained rational numbers, the computational expressions of the structural displacement response, the element stress response and dynamic eigenvalue are developed. At last, an example is given, in which the possible values and faith degrees of the unascertained structure static responses and natural frequency are obtained. The rationality and validity of the presented method are demonstrated.
引文
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