区间方法研究及其在天线结构分析中的应用
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摘要
实际工程结构中存在大量的不确定性和误差,在工程结构分析和设计过程中不能简单忽略这些不确定性因素。目前有三种解决不确定性问题的方法:概率方法、模糊方法和区间方法。其中,区间方法只需要知道结构参数的上下限,是在无法获得概率分布和隶属函数情况下一种有效的分析方法。本文采用区间方法对复杂结构进行了静力响应分析及动力特性分析,主要内容如下:
应用区间因子法对天线结构进行分析。利用区间因子表征区间变量的不确定性,将区间变量表示为其均值和区间因子的乘积。构造具有区间参数的8米天线结构在区间荷载作用下的分析模型,根据区间位移和区间应力响应的区间变量表达式,计算出结构在不同区间载荷作用下应力和位移区间范围。通过构造几种区间模型考察了任一结构参数和荷载的不确定性对应力的影响。
应用区间因子法对天线结构动力特性进行了分析。在天线结构的物理和几何参数同时为区间变量的情况下,利用区间因子建立了结构的刚度矩阵和质量矩阵;从结构振动的瑞利商表达式出发,利用区间运算法则推导出结构区间特征值,并且考察了物理和几何参数的区间性对其动力特性的影响。并验证了采用此方法的合理性与可行性。利用该方法能够反映天线结构某一参数的不确定性对结构动力特性的影响。
运用随机因子法和区间因子法对随机—区间模型进行分析。将模型中的随机量和区间量分别表示为随机因子和区间因子与其均值的乘积形式。分别计算随机和区间部分:根据区间位移和区间应力响应的区间变量表达式,计算出结构在区间载荷作用下应力和位移区间范围;基于随机因子法,利用代数综合法推导出结构位移和应力响应的均值、方差的计算表达式。将随机部分与区间部分进行计算获得了具有区间特性的位移和应力响应的均值和方差。可以考察区间参数和随机参数对结构响应的不同影响。
A large number of errors and uncertainties exist in the actual structures. In the process of structural analysis and design for engineering, uncertainties can not be ignored simply. At present, there are three main methods for uncertain modeling, such as probability method, fuzzy method and interval method. With acquiring the upper and lower limit of structural parameter, interval method can be used, which is an effective method which probability density and membership function is not obtained. Static response analysis and the dynamic characteristic analysis of complex structure are made by using of interval method. The main research works can be described as follows:
Interval factor method is applied to analyze antenna structure. Uncertainties of the interval variables are represented by interval factors and interval variables are described as mean values multiplied by interval factors. The finite element analysis of the 8m-diameter antenna structure with interval parameters under different interval load is studied. According to the structural interval displacement and stress response, the structural displacement and the element stress were obtained under different load. The effects of any one of uncertain structural parameters or load on the displacement and stress are examined by some interval models.
Interval factor method is applied to analyze dynamic characteristics of antenna structure. Under the condition that the structural parameters are interval variables simultaneously, the structural stiffness matrix and mass matrix are constructed by using interval factor method. From the Rayleigh expression of structural vibration, the computation expressions of interval variables of structural eigen values are obtained on the basis of principle of interval arithmetic. The method presented in this paper is verified. By the method, the effect of uncertainty of any individual antenna structural parameters on the structural dynamic characteristics can be reflected.
Structural analysis of random and interval hybrid model is made by using of interval factor method and random factor method. Random quantities and interval quantities in the model are described by interval factors and random factors multiplied their mean values. Calculation of interval and random parts is carried on respectively: According to the structural interval displacement and stress response, the structural displacement and the element stress were obtained under the load; According to the random factor method, the computational expressions of the mean value and mean variance of the structural displacement response and stress response are deduced by algebra method. The mean value and mean variance of the structural displacement response and stress response with the interval character are obtained by the hybrid calculation of interval and random parts. The effects of interval or random parameters on the displacement and stress can be examined.
应用区间因子法对天线结构进行分析。利用区间因子表征区间变量的不确定性,将区间变量表示为其均值和区间因子的乘积。构造具有区间参数的8米天线结构在区间荷载作用下的分析模型,根据区间位移和区间应力响应的区间变量表达式,计算出结构在不同区间载荷作用下应力和位移区间范围。通过构造几种区间模型考察了任一结构参数和荷载的不确定性对应力的影响。
应用区间因子法对天线结构动力特性进行了分析。在天线结构的物理和几何参数同时为区间变量的情况下,利用区间因子建立了结构的刚度矩阵和质量矩阵;从结构振动的瑞利商表达式出发,利用区间运算法则推导出结构区间特征值,并且考察了物理和几何参数的区间性对其动力特性的影响。并验证了采用此方法的合理性与可行性。利用该方法能够反映天线结构某一参数的不确定性对结构动力特性的影响。
运用随机因子法和区间因子法对随机—区间模型进行分析。将模型中的随机量和区间量分别表示为随机因子和区间因子与其均值的乘积形式。分别计算随机和区间部分:根据区间位移和区间应力响应的区间变量表达式,计算出结构在区间载荷作用下应力和位移区间范围;基于随机因子法,利用代数综合法推导出结构位移和应力响应的均值、方差的计算表达式。将随机部分与区间部分进行计算获得了具有区间特性的位移和应力响应的均值和方差。可以考察区间参数和随机参数对结构响应的不同影响。
A large number of errors and uncertainties exist in the actual structures. In the process of structural analysis and design for engineering, uncertainties can not be ignored simply. At present, there are three main methods for uncertain modeling, such as probability method, fuzzy method and interval method. With acquiring the upper and lower limit of structural parameter, interval method can be used, which is an effective method which probability density and membership function is not obtained. Static response analysis and the dynamic characteristic analysis of complex structure are made by using of interval method. The main research works can be described as follows:
Interval factor method is applied to analyze antenna structure. Uncertainties of the interval variables are represented by interval factors and interval variables are described as mean values multiplied by interval factors. The finite element analysis of the 8m-diameter antenna structure with interval parameters under different interval load is studied. According to the structural interval displacement and stress response, the structural displacement and the element stress were obtained under different load. The effects of any one of uncertain structural parameters or load on the displacement and stress are examined by some interval models.
Interval factor method is applied to analyze dynamic characteristics of antenna structure. Under the condition that the structural parameters are interval variables simultaneously, the structural stiffness matrix and mass matrix are constructed by using interval factor method. From the Rayleigh expression of structural vibration, the computation expressions of interval variables of structural eigen values are obtained on the basis of principle of interval arithmetic. The method presented in this paper is verified. By the method, the effect of uncertainty of any individual antenna structural parameters on the structural dynamic characteristics can be reflected.
Structural analysis of random and interval hybrid model is made by using of interval factor method and random factor method. Random quantities and interval quantities in the model are described by interval factors and random factors multiplied their mean values. Calculation of interval and random parts is carried on respectively: According to the structural interval displacement and stress response, the structural displacement and the element stress were obtained under the load; According to the random factor method, the computational expressions of the mean value and mean variance of the structural displacement response and stress response are deduced by algebra method. The mean value and mean variance of the structural displacement response and stress response with the interval character are obtained by the hybrid calculation of interval and random parts. The effects of interval or random parameters on the displacement and stress can be examined.
引文
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[4]陈怀海.非确定结构系统区间分析的直接优化法.南京航空航天大学学报, 1999, 31(2):146-150.
[5] Moore R E. Methods and Applications of Interval Analysis. Philadelphia: Society for Industrial & Applied Methmatics,1979.
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[8]张乃良,王海鹰,刘蕴华.非线性规划的区间方法.河海大学学报, 1992, 20(3):97-103.
[9]曹德欣,沈祖和.一类非光滑优化问题的区间算法.高等学校计算数学学报, 1998, 3(1):23-33.
[10] Shen P P, Zhang K C. An Interval Method for Non-smooth Global Optimization Problem. or Transaction, 2002, 6(2):9-18.
[11]王海军,曹德欣,李苏北,邓喀中.非线性等式约束全局优化问题的区间算法.中国矿业大学学报, 2003, 32(2):204-208.
[12]蔡新,昊威,吴黎华等.土石坝结构区间优化设计.河海大学学报, 1998, 26(3):1-5.
[13]禹智涛,吕恩琳,王彩华.结构模糊有限元平衡方程的一种解法.重庆大学学报, 1996, 19(1):53-58.
[14] Rao S S,Berke L. Analysis of Uncertain Structural Systerms Using Interval Analysis. AIAA Journal, 1997, 35(4):725-735.
[15]吴晓,罗佑新,文会军等.非确定结构系统区间分析的泛灰求解方法.计算力学学报, 2003, 20(3):329-334.
[16]杨钊.区间数和泛灰数在区间分析中的比较,河北工业大学学报, 2001, 30(4):93-96.
[17] Markov S,An Iterative Method for Algebraic Solution to Interval Equations. Applied Numerical Mathematics, 1599, 30:225-239.
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