计算具有区间参数结构特征值范围的一种新方法

基于区间效学的包含单调性和区间函效所表述的实际物理意义，把广义区间特征值问题转化为两个以非确定参效为优化变量，以关心的特征值为目标函效的全局优化问题，并采用遗传算法对优化问题求解，计算得到结构特征值的区间范围。通过效值算例对本文方法的有效性进行了验证，并和区间摄动法的计算结果进行了比较。     

A new method for computing the upper and lower bounds on frequencies of structures with interval parameters

In this paper, we will investigate the method for computing the upper and lower bounds on frequencies of structures with bounded uncertain (or interval) parameters. The stiffness matrix and the mass matrix of structures, whose elements have the initial errors, are unknown except for the fact that they belong to given bounded matrix sets. The set of possible matrices can be described by the interval matrix. By means of the stationary condition of Rayleigh Qutient and the minimax theorem of eigenvalues, the generalized eigenvalue problem of structures with bounded uncertain parameters can be transformed into two different generalized eigenvalue problems. The numerical results indicate that the proposed method is effective.     

有界参数结构特征值的上下界定理

与方法近似性的结构特征值包含定理不同,给出参数近似性的结构的特征值上下界定理．在结构刚度矩阵和质量矩阵可以利用结构参数进行非员分解的条件下,通过区间分析,将特征值的上下界分解成两个广义特征值问题进行求解．结果可以看成是胡海昌教授的特征值质量包含定理和刚度包含定理在结构参数近似性特征值问题中的一种推广和应用．     

Several solution methods for the generalized complex eigenvalue problem with bounded uncertainties

The aim of this paper is to evaluate the effects of uncertain-but-bounded parameters on the complex eigenvalues of the non-proportional damping structures. By combining the interval mathematics and the finite element analysis, the mass matrix, the damping matrix and the stiffness matrix were represented as the interval matrices. Firstly, with the help of the optimization theory, we presented an exact solution—the vertex solution theorem, for determining the exact upper bounds or maximum values and exact lower bounds or minimum values of complex eigenvalues of structures, where the extreme values are reached on the boundary of the interval mass, damping and stiffness matrices. Then, an interval perturbation method was proposed, which needs less computational efforts. A numerical example of a seven degree-of-freedom spring-damping-mass system was used to illustrate the computational aspects of the presented vertex solution theorem and the interval perturbation method in comparison with Deif’s method.     

不确定参数闭环控制系统特征值的区间分析

运用区间理论,讨论了区间参数的结构振动控制问题,给出了求解闭环系统区间特征值的一种方法．基于区间参数导出了区间刚度矩阵和质量矩阵,然后利用矩阵摄动理论和区间扩张理论,推导了复区间特征值上下界估计的算法．这些结果是从二阶系统的左右特征向量出发得到的．将该文方法应用到悬臂梁的控制问题,数值结果表明它是有效的．     

Upper and lower bounds to eigenvalues by weighting function approximations

A method is developed making use of variational principles and Rayleigh's quotient which yields lower bounds to eigenvalues. The method is the counterpart of the Rayleigh-Ritz method in the sense that the results obtained from both methods will improve, i.e. approach to the exact value, as more and more terms are considered, both rely on variational principles, they are similar systematically and conceptually, and this method yields lower bounds to eigenvalues which cannot be obtained from the Rayleigh-Ritz method. Therefore, with the results from both methods the eigenvalues can be bracketed into a small region. The most important advantage of the method is the lower bounds to all eigenvalues can be obtained from the solution of one transcendental equation.     

A new method for estimating bounds of eigenvalues

A new method for estimating the bounds of eigenvalues is presented. In order to show that the method proposed is as effective as Qiu's, an undamping spring-mass system with 5 nodes and 5 degrees of freedom is given. To illustrate that the present method can be applied to structures which cannot be treated by non-negative decomposition, a plane frame with 202 nodes and 357 beam elements is given. The results show that the present method is effective for estimating the bounds of eigenvalues and is more common than Qiu's. Project supported by the National Natural Science Foundation (No. 19872028) and the Mechanical Technology Development Foundation of China.     

Static response analysis of structures with interval parameters using the second-order Taylor series expansion and the DCA for QB

In this paper, based on the second-order Taylor series expansion and the difference of convex functions algorithm for quadratic problems with box constraints(the DCA for QB), a new method is proposed to solve the static response problem of structures with fairly large uncertainties in interval parameters. Although current methods are effective for solving the static response problem of structures with interval parameters with small uncertainties, these methods may fail to estimate the region of the static response of uncertain structures if the uncertainties in the parameters are fairly large. To resolve this problem, first, the general expression of the static response of structures in terms of structural parameters is derived based on the second-order Taylor series expansion. Then the problem of determining the bounds of the static response of uncertain structures is transformed into a series of quadratic problems with box constraints. These quadratic problems with box constraints can be solved using the DCA approach effectively. The numerical examples are given to illustrate the accuracy and the efficiency of the proposed method when comparing with other existing methods.     

区间参数振动系统的动力优化

对具有区间参数的多自由度振动系统的不确定性优化问题，提出一种新的区间优化方法．利用泰勒展开和函数的区间扩张，将区间优化问题转化为近似的确定性优化问题．该方法应用于多自由度线性扭振系统，并把区间设计变量的中值和不确定性半径取作优化参数．算例表明该方法是有效的．     

离散变量框架结构的可靠性优化

以框架结构系统的可靠性分析为基础,给出离散变量框架结构可靠性优化的相对差商方法及迭代格式,并用算例证明了该法的有效性及实用性。     

DYNAMIC CHARACTERISTIC ANALYSIS OF FUZZY- STOCHASTIC TRUSS STRUCTURES BASED ON FUZZY FACTOR METHOD AND RANDOM FACTOR METHOD

A new fuzzy stochastic finite element method based on the fuzzy factor method and random factor method is given and the analysis of structural dynamic characteristic for fuzzy stochastic truss structures is presented. Considering the fuzzy randomness of the structural physical parameters and geometric dimensions simultaneously, the structural stiffness and mass matrices are constructed based on the fuzzy factor method and random factor method; from the Rayleigh's quotient of structural vibration, the structural fuzzy random dynamic characteristic is obtained by means of the interval arithmetic; the fuzzy numeric characteristics of dynamic characteristic are then derived by using the random variable's moment function method and algebra synthesis method. Two examples are used to illustrate the validity and rationality of the method given. The advantage of this method is that the effect of the fuzzy randomness of one of the structural parameters on the fuzzy randomness of the dynamic characteristic can be reflected expediently and objectively.     

粘弹性阻尼结构系统振动特性的迭代分析

含粘弹性阻尼材料结构系统的振动特性分析是相当复杂的,求解比较困难,本文给出了一种有效的迭代解法。文中首先定义了广义Rayleigh商,指出求广义Rayleigh商的驻值问题等价于原特征问题的求解;由近似估计得到的特征向量,广义Rayleigh商给出一个较好的特征值近似估计。据此构造了迭代方法,并成功地计算了一些例子。     

频率约束板结构拓扑优化

本文利用ICM（独立、连续、映射）方法建立了频率约束下平板结构重量最轻的拓扑优化模型。采用指数函数作为单元重量、单元质量及单元刚度的过滤函数。通过瑞利商对刚度过滤函数倒变量的泰勒一阶展式,将频率约束近似显式化。利用对偶理论将含有大量设计变量的约束优化模型转化为易于求解的少设计变量拟无约束优化模型,通过序列二次规划将转化模型进行求解,提高了求解的效率。本文选择MSC.Patran&Nastran软件及PCL二次开发语言构架了平板结构频率约束拓扑优化问题的软件。数值算例表明：本文的方法具有迭代稳定性和收敛高效性。     

THE UNCERTAIN EIGENVALUES ANALYSIS OF STRUCTURES WITH UNCERTAIN PARAMETERS

This paper presents a method for estimating the upper and lower bounds of eigenvalues of structures with uncertainties. The uncertain parameters are described by the convex model. A numerical example of the frame structure is given to illustrate the efficiency of the method. This work is supported by the National Natural Science Foundation of China(No. 19872028).     

基于二阶摄动法求解区间参数结构动力响应

在处理区间参数结构动力响应问题时,现有的分析方法大多局限于一阶区间分析方法. 如果参数的不确定量稍大,采用一阶区间分析方法对结构动力响应范围进行估计可能会失效,所以需要考虑二阶区间分析方法.但是采用基于区间运算的二阶区间分析方法得到的结果将会对动力响应范围过分高估. 为了克服以上缺点,首先基于二阶摄动法得到结构动力响应广义函数. 然后通过求解此动力响应函数的最大和最小值,将结构动力响应区间的问题转化为序列低维箱型约束下的二次规划问题. 最后采用DC 算法(di erence of convex functionsalgorithm) 对这些箱型约束下的二次规划问题进行求解. 这样可以在不引入过多计算量的情况下,避免了对动力响应范围的过分估计. 通过数值算例,将该方法和其他区间分析方法进行比较,验证了该方法的有效性与精确性.      

一种新的有限元模型移频动力缩聚法

将矩阵幂迭代法与移频技术相结合,建立了一种新的结构动力缩聚方法.该方法首先应用矩阵幂迭代法对结构的初始有限元模型进行一次缩聚,计算初始缩聚模型的特征值,然后通过判断低阶特征值的收敛情况确定移频位置,选择合适的移频值,建立移频后的广义特征方程;再根据矩阵幂迭代法迭代计算新的广义特征方程的动力缩聚矩阵,经迭代收敛后得到精确...     

智能结构密频系统振动控制及其摄动分析

提出一种智能结构密频控制规律的设计方法。其主要工作为：将密频子空间转化为重频子空间,把密频系统作为重频系统上的摄动;为了解决重频密频子空间相对应的特征向量选取的敏感性问题,通过摄动分析得到与重频密频子空间相对应特征向量的线性组合并利用闭环系统特征值极点配置的方法得到重频密频系统的振动规律;把所设计的振动控制规律作用于原系统和摄动系统上,讨论了结构参数改变后对系统的动态特性的影响;最后通过算例证明该方法的有效性。     

考虑频率依赖性的涂层复合结构固有特性求解

粘弹性阻尼材料的力学特性参数会随着频率的变化而改变,即具有频率依赖性,因而传统的动力学建模及分析方法不能满足实际涂层结构优化设计的需要。在简要介绍粘弹性阻尼材料频率依赖性的基础上,本文提出用特征向量增值法来求解涂层复合结构的固有特性,并详细推导了特征向量增值法的求解原理。由此,提出了特征向量增值法的计算流程,包括计算无阻尼系统的固有特性,用Fox and Kapoor或者Nelson方法计算复特征向量增量;用Rayleigh熵法求解复特征值。最后,以涂敷粘弹性阻尼材料的钛基薄板为例,求解了该复合结构的固有特性,并与经典的模态应变能法进行了比较,证明了所提方法的正确性。     

结构鲁棒优化的非概率集合理论凸方法

以传统的优化理论为基础，考虑含不确定结构参数的情况，提出了非概率凸集合理论的 结构优化方法. 将结构优化列式中的目标函数与约束条件所含有的不确定参数用凸集合 定量化，只需知道其所在范围的边界，降低了以往处理不确定性问题概率方法需要知道不确 定参数的均值、方差或概率分布密度等详细统计信息的要求. 提出的鲁棒优化方法在使 目标函数达到设计要求的同时，结构还能承受结构参数在其所在范围内变化引起结构性能的 变异. 通过优化问题中普遍使用的10杆平面桁架和一个72杆空间桁架实例，给出了当 结构参数为名义值时结构的优化结果，以及结构参数具有不确定性时的优化结果，力求表明 所介绍的方法的可行性和优越性.