亏损和接近亏损振动系统的若干问题研究
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摘要
系统的结构修改是一个重要的研究问题，在振动抑制以及系统的设计和控制等方面有着广泛的应用。在不同的设计和优化问题中，往往需要通过反复不断的改变结构设计以满足预定的需求。在结构动态优化过程中，反复的分析将会带来大规模的计算量。随着设计不断的复杂化，在现代结构设计中迫切需要有效并且准确的重分析技术。目前，结构振动重分析问题的研究主要集中在非亏损系统上。然而，在动态和对称结构中，系统对应的矩阵通常含有重频率。更特别的，当特征值的几何重数小于代数重数时，系统的特征向量系不完全即不能张满整个空间，此时称为亏损系统。亏损系统存在于实际工程问题中，例如飞机颤振、桥梁颤振和机轮的抖振等等，相应的系统都会出现亏损现象。
在工程优化问题中，例如模型修正、故障诊断等都会涉及灵敏度分析。因此，关于系统参数变化的灵敏度分析研究已经成为一个重要的研究领域。由于变化的不确定性，例如失调的参数选取或者不规则的几何参数，都会使一个动态模型和初始的设计大相径庭。出于这些原因，灵敏度分析的理论研究意义深远，而且将会为工程实际问题提供指导性的方案。关于亏损和接近亏损系统的灵敏度分析研究有两个难点：第一，如何克服特征矩阵奇异的困难；第二，如何建立一种计算上述两种系统的快速、有效的灵敏度分析算法。
本文结合2009年国家自然科学基金面上项目“基于重分析理论的简化车身多单元框架结构截面参数快速优化研究”（项目编号：50975121）、2009年教育部高等学校博士点基金项目“亏损振动系统快速自适应重分析算法研究”（项目编号：20090061110022）、2011年吉林大学研究生创新项目“基于组合近似法的亏损系统动力修改问题研究”（项目编号：20111056）、2012年吉林大学研究生创新项目“灵敏度分析的松弛移频组合近似方法研究”（项目编号：20121097）和2012年国家自然科学青年基金项目“概念车身框架结构有限元精细建模与截面几何形状优化设计”（项目编号：51205159），以亏损系统和接近亏损系统为研究对象，对结构设计修改后的系统进行了重分析和灵敏度分析研究。
本文的主要研究内容可以概述为：
针对亏损系统结构动力修改的重分析问题，本文提出RCA（Relaxation CombinedApproximations）重分析算法。该算法基于亏损振动系统特有的广义模态理论，针对N重亏损系统和一般重亏损系统，通过将松弛因子嵌入到CA方法中，并将初始方程化为等价方程的形式，从而克服了系数矩阵奇异性的困难。通过求解缩减的分析方程即可得到结构修改后的广义模态，同时，理论上证明了松弛组合近似方法的收敛性。通过经典的算例结合误差分析定义验证了本文提出的松弛组合近似方法的正确性和有效性。
基于本文提出的RCA重分析算法，求解亏损振动系统修改后结构动力响应的重分析问题，利用密集频率系统可以和重频率系统相互转化的紧密关系，对状态矩阵分块，将密集特征值部分与孤立特征值部分剥离出来，通过特征移频，将接近亏损系统的结构修改重分析问题转化成亏损系统的重分析问题。提出了一种SRCA（Shift RelaxationCombined Approximations）重分析方法，当结构参数发生变化时，将特征移频产生的误差矩阵和摄动矩阵的和整体考虑为相对于亏损系统的摄动矩阵，数值算例分析中的松弛因子与误差的关系图表明该算法的准确性。
在重分析算法研究的基础上，分别以亏损和接近亏损振动系统为研究对象，讨论了结构修改和结构优化过程中的灵敏度分析问题。基于RCA方法和SRCA方法，推导了灵敏度分析的计算公式，以亏损系统、密频系统和接近亏损系统中的典型算例，说明了基于RCA方法和SRCA方法的灵敏度算法不仅格式整齐而且精准、高效，适用于特殊系统更适用于普通系统。
Structural modification of systems is an important research issue and has a wide range ofapplications such as vibration suppression, system design and control. Changes of structuresare often necessary to satisfy predetermined demands in various design and optimizationproblems, the response of the numerous modified structures need to be evaluated repeatedly.In the structural dynamic optimization, the multiple repeated analyses are ones of the mostcostly computations. The need for efficient and accurate reanalysis technique in modernstructural design is crucial because the design becomes more complex and large. Until now,many researches of structural vibration reanalysis are concerned on non-defective systems. Inproblems such as those of dynamic and symmetric structures, however, the correspondingmatrices can have repeated eigenvalues. Very often indeed, the geometric multiplicity of theeigenvalues is less than the algebraic multiplicity, and so the system has an incomplete set ofeigenvectors, insufficient to form a base for the state space. Systems of this type are called thedefective systems. The defective systems can be encountered in actual engineering problems,such as flutters of airplane and missile wings or long blades of turbines which thecorresponding matrices are defective.
Many engineering optimization problems, for example, model updating or structuraldamage detections, lead to a sensitivity analysis of eigenproblems. As a result, the study ofthe sensitivity of eigensolutions due to variations in the system parameters has been animportant research area. A dynamic model can be far from the assumed prototype. Since thereis usually a variation, such as a mistuned parameter or a irregularity geometrical. For thesereasons, sensitivity analysis is meaningful to perform a theoretical study and give a guide for engineering practice. There are two main difficulties in computing the eigenvectorsderivatives. One of the main difficulties is how to change the irreversible state ofcharacteristic matrix. The other difficulty is how to establish a uniform efficient method forcomputing the eigenvectors derivatives.
The grant of the projects from the National Natural Science Foundation of China “FastOptimization of Cross-Sectional Parameters for Simplified Car Body Multi-Elements FrameStructure Based on Reanalysis Theory”(No.50975121) and Doctoral Program of HigherEducation “The Research of Adaptive Reanalysis Algorithm of the Defective VibrationSystem”(No.20090061110022) and2011Graduate Innovation Fund of Jilin University“Research on Dynamic Modification for Defective Systems Based on CombinedApproximations Method”(No.20111056) and2012Graduate Innovation Fund of JilinUniversity “Research on Relaxation Combined Approximations Method for SensitivityAnalysis”(No.20121097) is gratefully acknowledged for the financial support.
The main contents of this paper can be summarized as follows:
The Relaxation Combined Approximations (RCA) method is proposed which aims tosolve reanalysis problems of defective systems. A relaxation factor is introduced into the CAmethod based on the general mode theory, and it changes the characteristic matrix from beingsingular to non-singular. For N repeated eigenvalues and general defective systems, thismodification makes possible to the solution of generalized eigenvectors. An additionaladvantage of the method is that the generalized eigenvectors are expressed by a series ofbasic vectors and the dimension of basic vectors is usually much less than the dimension ofeigenvectors, so the computational cost is also reduced. Numerical examples show that theRCA method can lead to exact generalized eigenvectors, and is very easy to be used for thesame kind of these problems.
The RCA method gives a solution to the problem of defective vibration systems basedon the advantage of relaxation factor. During system optimization, some originally separatedfrequencies can approach closer and closer. In these cases, if the associated eigenvectors make groups of nearly parallel vectors the system can be classified as near defective. Fromthe view point of mathematics, the close eigenvalues of near defective systems are distinct,but the dynamic characteristic is still defective. By the frequency shift (Shift RelaxationCombined Approximations method), the reanalysis problems of near defective systems withclose eigenvalues can be transformed into one of the defective systems with repeated ones,which is equal to the average of the close ones. The relationship between the relaxation factorand error shows that the accuracy of the algorithm.
As the research basis of the reanalysis algorithms, the sensitivity analysis problems ofthe structural modification and structure optimization are discussed for the defective and neardefective systems. The formulae for calculating the sensitivity analysis are derived from theRCA and SRCA approach. Sensitivity algorithms are not only neat format but also accurate,efficient and more applicable to the particular system applicable to an ordinary system.
在工程优化问题中，例如模型修正、故障诊断等都会涉及灵敏度分析。因此，关于系统参数变化的灵敏度分析研究已经成为一个重要的研究领域。由于变化的不确定性，例如失调的参数选取或者不规则的几何参数，都会使一个动态模型和初始的设计大相径庭。出于这些原因，灵敏度分析的理论研究意义深远，而且将会为工程实际问题提供指导性的方案。关于亏损和接近亏损系统的灵敏度分析研究有两个难点：第一，如何克服特征矩阵奇异的困难；第二，如何建立一种计算上述两种系统的快速、有效的灵敏度分析算法。
本文结合2009年国家自然科学基金面上项目“基于重分析理论的简化车身多单元框架结构截面参数快速优化研究”（项目编号：50975121）、2009年教育部高等学校博士点基金项目“亏损振动系统快速自适应重分析算法研究”（项目编号：20090061110022）、2011年吉林大学研究生创新项目“基于组合近似法的亏损系统动力修改问题研究”（项目编号：20111056）、2012年吉林大学研究生创新项目“灵敏度分析的松弛移频组合近似方法研究”（项目编号：20121097）和2012年国家自然科学青年基金项目“概念车身框架结构有限元精细建模与截面几何形状优化设计”（项目编号：51205159），以亏损系统和接近亏损系统为研究对象，对结构设计修改后的系统进行了重分析和灵敏度分析研究。
本文的主要研究内容可以概述为：
针对亏损系统结构动力修改的重分析问题，本文提出RCA（Relaxation CombinedApproximations）重分析算法。该算法基于亏损振动系统特有的广义模态理论，针对N重亏损系统和一般重亏损系统，通过将松弛因子嵌入到CA方法中，并将初始方程化为等价方程的形式，从而克服了系数矩阵奇异性的困难。通过求解缩减的分析方程即可得到结构修改后的广义模态，同时，理论上证明了松弛组合近似方法的收敛性。通过经典的算例结合误差分析定义验证了本文提出的松弛组合近似方法的正确性和有效性。
基于本文提出的RCA重分析算法，求解亏损振动系统修改后结构动力响应的重分析问题，利用密集频率系统可以和重频率系统相互转化的紧密关系，对状态矩阵分块，将密集特征值部分与孤立特征值部分剥离出来，通过特征移频，将接近亏损系统的结构修改重分析问题转化成亏损系统的重分析问题。提出了一种SRCA（Shift RelaxationCombined Approximations）重分析方法，当结构参数发生变化时，将特征移频产生的误差矩阵和摄动矩阵的和整体考虑为相对于亏损系统的摄动矩阵，数值算例分析中的松弛因子与误差的关系图表明该算法的准确性。
在重分析算法研究的基础上，分别以亏损和接近亏损振动系统为研究对象，讨论了结构修改和结构优化过程中的灵敏度分析问题。基于RCA方法和SRCA方法，推导了灵敏度分析的计算公式，以亏损系统、密频系统和接近亏损系统中的典型算例，说明了基于RCA方法和SRCA方法的灵敏度算法不仅格式整齐而且精准、高效，适用于特殊系统更适用于普通系统。
Structural modification of systems is an important research issue and has a wide range ofapplications such as vibration suppression, system design and control. Changes of structuresare often necessary to satisfy predetermined demands in various design and optimizationproblems, the response of the numerous modified structures need to be evaluated repeatedly.In the structural dynamic optimization, the multiple repeated analyses are ones of the mostcostly computations. The need for efficient and accurate reanalysis technique in modernstructural design is crucial because the design becomes more complex and large. Until now,many researches of structural vibration reanalysis are concerned on non-defective systems. Inproblems such as those of dynamic and symmetric structures, however, the correspondingmatrices can have repeated eigenvalues. Very often indeed, the geometric multiplicity of theeigenvalues is less than the algebraic multiplicity, and so the system has an incomplete set ofeigenvectors, insufficient to form a base for the state space. Systems of this type are called thedefective systems. The defective systems can be encountered in actual engineering problems,such as flutters of airplane and missile wings or long blades of turbines which thecorresponding matrices are defective.
Many engineering optimization problems, for example, model updating or structuraldamage detections, lead to a sensitivity analysis of eigenproblems. As a result, the study ofthe sensitivity of eigensolutions due to variations in the system parameters has been animportant research area. A dynamic model can be far from the assumed prototype. Since thereis usually a variation, such as a mistuned parameter or a irregularity geometrical. For thesereasons, sensitivity analysis is meaningful to perform a theoretical study and give a guide for engineering practice. There are two main difficulties in computing the eigenvectorsderivatives. One of the main difficulties is how to change the irreversible state ofcharacteristic matrix. The other difficulty is how to establish a uniform efficient method forcomputing the eigenvectors derivatives.
The grant of the projects from the National Natural Science Foundation of China “FastOptimization of Cross-Sectional Parameters for Simplified Car Body Multi-Elements FrameStructure Based on Reanalysis Theory”(No.50975121) and Doctoral Program of HigherEducation “The Research of Adaptive Reanalysis Algorithm of the Defective VibrationSystem”(No.20090061110022) and2011Graduate Innovation Fund of Jilin University“Research on Dynamic Modification for Defective Systems Based on CombinedApproximations Method”(No.20111056) and2012Graduate Innovation Fund of JilinUniversity “Research on Relaxation Combined Approximations Method for SensitivityAnalysis”(No.20121097) is gratefully acknowledged for the financial support.
The main contents of this paper can be summarized as follows:
The Relaxation Combined Approximations (RCA) method is proposed which aims tosolve reanalysis problems of defective systems. A relaxation factor is introduced into the CAmethod based on the general mode theory, and it changes the characteristic matrix from beingsingular to non-singular. For N repeated eigenvalues and general defective systems, thismodification makes possible to the solution of generalized eigenvectors. An additionaladvantage of the method is that the generalized eigenvectors are expressed by a series ofbasic vectors and the dimension of basic vectors is usually much less than the dimension ofeigenvectors, so the computational cost is also reduced. Numerical examples show that theRCA method can lead to exact generalized eigenvectors, and is very easy to be used for thesame kind of these problems.
The RCA method gives a solution to the problem of defective vibration systems basedon the advantage of relaxation factor. During system optimization, some originally separatedfrequencies can approach closer and closer. In these cases, if the associated eigenvectors make groups of nearly parallel vectors the system can be classified as near defective. Fromthe view point of mathematics, the close eigenvalues of near defective systems are distinct,but the dynamic characteristic is still defective. By the frequency shift (Shift RelaxationCombined Approximations method), the reanalysis problems of near defective systems withclose eigenvalues can be transformed into one of the defective systems with repeated ones,which is equal to the average of the close ones. The relationship between the relaxation factorand error shows that the accuracy of the algorithm.
As the research basis of the reanalysis algorithms, the sensitivity analysis problems ofthe structural modification and structure optimization are discussed for the defective and neardefective systems. The formulae for calculating the sensitivity analysis are derived from theRCA and SRCA approach. Sensitivity algorithms are not only neat format but also accurate,efficient and more applicable to the particular system applicable to an ordinary system.
引文
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