机械结构有限元动力优化设计的逆摄动方法研究
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摘要
随着现代化的复杂机械或工程结构的发展，对其优化设计提出了更高的要
求。即需全面考虑强迫振动、频率禁区、刚度、强度等的影响因素，进行符合工
程实际要求的、基于有限元分析的动力优化设计，以满足现代化设计要求。目前
复杂机械或工程结构的动力有限元优化设计存在的主要问题：一是它的有限元分
析的结构重分析计算繁复，优化过程中重分析次数过多，优化无法进行；二是工
程技术人员要求该方法和软件容易掌握和使用，便于人机交互。这就需要研究新
的实用化的动力有限元优化设计的方法，这是急待解决的工程实际问题。
动力优化设计主要有两类问题：频率禁区（特征值问题）和动力响应（强迫振
动）。这两个问题既可以用“正求”方法求解，也可以归结为其反问题来求解。
频率禁区问题（带频率禁区约束的动力优化设计），目前常用方法，一是数学规
划法和优化准则法；二是结构矩阵摄动法。前者的局限性是结构重分析次数多，
计算时间长；后者虽可在一定程度上避免重分析，但其迭代步长难以确定。动力
响应问题（带动力响应约束的动力优化设计），目前研究尚少，因为它更复杂，
更为困难，但却是工程中急需解决的问题。
本文课题属国家自然科学基金资助项目：“可视逆摄动理论和方法及其实
现”（编号：59675041）。本文针对上述存在问题，研究机械和工程结构动力优化
设计的理论和方法，重点研究逆摄动法求解上述两类反问题。主要工作有以下三
方面：
1．提出了广义特征值反问题（频率禁区优化问题）求解的新的逆摄动法：将频
率禁区问题求解归结为广义特征值反问题求解。利用广义特征值反问题理论和有
限单元刚度阵和质量阵的分解，推导出特征值λ与逆摄动参数ε（与设计变量x
相对应）关系的显式表达的计算公式。通过给定的特征值增量逆求出ε值，解决
了结构矩阵摄动法的无法确定送代步长问题，避免或减少了重分析次数。广义特
征值反问题求解存在不适定问题，为解决此问题，给出了逆摄动参数ε若干种取
值方法，如基于平均概率的敏度分析方法。另外，方法简单直观，便于人机交互。
2．首次提出了动力响应振幅（或动柔度）反问题（动力响应振幅优化问题）求
解的逆摄动法：根据逆求思想和有限单元刚度阵和质量阵分解方法，推导出逆摄
动参数ε与动力响应振幅（或动柔度）关系的显式表达式。避免或减少了重分析。
用逆摄动法求解动力响应振幅问题，既可以将其作为反问题求解，也可以将其作
为正问题求解。有助于动力响应优化难题的求解。
3．逆摄动法在动力优化中的应用及其编程：给出了人机交互的基于约束的
n 大连理工大学博土学位论文 机械结构有限元动力优化设计的逆摄动方法研究
一
优化方法和边界映射的优化方法，推荐了暂时消除约束的分部（步）优化方法。编
制了本逆摄动法的软件：本软件采用MATLAB编程语言，并利用了MATLAB的平台，
完成了上述特征值问题和动力响应振幅问题逆摄动法的软件编程。实现了对杆单
元的轴向振动系统、析架系统，平面梁单元的转子系统等上述两类问题的求解。
算例或工程实例验证结果表明，该方法可有效地减少重分析次数，计算效
率高，为结构动力优化设计提供了一种有效方法。所给公式及求解方法具有工程
直观性，编程容易，便子工程设计人员应用，便于发挥其主观能动性。本方法可
与矩阵摄动法结合应用，相辅相成。根据本文逆摄动法的有限元矩阵分解组装方
法，可以推导出基于其它有限元分析模型的动力优化设计的逆摄动法，如板、壳
等单元的逆摄动参数。的有关计算公式。
With the development of modern complicated machine and engineering structures, an even higher demand is posed to the optimal design. That is the dynamic optimal design which concerns comprehensively influencing factors about forced vibration, unacceptable frequency band, stiffness, intensity etc., based on the analytical model of FEM, to satisfy the needs of practical engineering. The crux of the matter at present is to study a new and utility dynamic optimal design method, which strives to reduce reanalysis and is easy for engineers to master and use and convenient for manmachine interaction. This is the most urgent task for the problems of practical engineering.
There are two primary problems in dynamic optimal design: The unacceptable frequency bands(eigenvalue problems) and dynamic response(forced vibration). These two problems can be solved by 慸irect solution?method or be solved as inverse problems. The methods most in use to the solution of the problem with unacceptable frequency bands: one is a mathematical programming or optimum criterion method; the other is structure matrix perturbation. The former needs more structure reanalysis and more computational time, the later may reduce reanalysis to some extend, but the small-perturbed parameter s is hard to set. The problem with constraints of dynamic response amplitude hasn抰 been studied widely, because it is more complicated and more difficulty.
The question studied in the dissertation is supported by National Natural Science Fund(N259675041): Theory and Method of Visual Inverse Perturbation Method. The purpose of the dissertation is to study and advance a new effective and practical dynamic optimal design method梚nverse perturbation method based on the analytical model of FEM. The work to solve the problems mentioned above by inverse perturbation method includes 3 aspects as follows:
1 .The inverse perturbation method to the solution for the inverse problem of general eigenvalue: The problems with the constraints of unacceptable frequency bands are attributed to the solution of the inverse problem of general eignenvalue.
Taking advantage of the theories of the inverse problem of general eigenvalue and the decomposition of finite element matrices, the explicit expressions of eigenvalue 2 and perturbed parameter e (correspondence with design variables) and the computation equations of ~ are derived. The s 憇 value is determined by the difference of the increment of 2, so in this way reanalysis can be avoided or reduced. The characteristic of the inverse problem of general eignenvalue is multi-value and indetermination. By solving this problem several methods to determine the value of the perturbed parameter e are described in the dissertation, such as sensitivity ratio analytical method based on average of probability.
2.The inverse perturbation method to the solution for the inverse problem of dynamic response amplitude: According to the solution of inverse thought and the decomposition method of finite element stiffness and mass matrices, the explicit expressions of dynamic response amplitude and perturbed parameter s are also derived. The inverse perturbation method for the problem of dynamic response amplitude can be applied both for inverse and optimal problem of dynamic response amplitude.
3.The application of inverse perturbation method for dynamic optimum design and its software: The optimum method with constraints of fractional optimum design and man-machine synergy is introduced in the dissertation. The software of the inverse perturbation method is programmed in MATLAB. It has been successfully applied to solve the problems of truss structure system, axial vibration system, rotor system and plane beam system etc.
Illustrative examples demonstrate that the method is characterized in reducing reanalysis significantly, bears the merits in computing speed and precision. The inverse perturbation method can be as an approach for dynamic optimum design. Though illustrated with beam element system, in accordance with t
求。即需全面考虑强迫振动、频率禁区、刚度、强度等的影响因素，进行符合工
程实际要求的、基于有限元分析的动力优化设计，以满足现代化设计要求。目前
复杂机械或工程结构的动力有限元优化设计存在的主要问题：一是它的有限元分
析的结构重分析计算繁复，优化过程中重分析次数过多，优化无法进行；二是工
程技术人员要求该方法和软件容易掌握和使用，便于人机交互。这就需要研究新
的实用化的动力有限元优化设计的方法，这是急待解决的工程实际问题。
动力优化设计主要有两类问题：频率禁区（特征值问题）和动力响应（强迫振
动）。这两个问题既可以用“正求”方法求解，也可以归结为其反问题来求解。
频率禁区问题（带频率禁区约束的动力优化设计），目前常用方法，一是数学规
划法和优化准则法；二是结构矩阵摄动法。前者的局限性是结构重分析次数多，
计算时间长；后者虽可在一定程度上避免重分析，但其迭代步长难以确定。动力
响应问题（带动力响应约束的动力优化设计），目前研究尚少，因为它更复杂，
更为困难，但却是工程中急需解决的问题。
本文课题属国家自然科学基金资助项目：“可视逆摄动理论和方法及其实
现”（编号：59675041）。本文针对上述存在问题，研究机械和工程结构动力优化
设计的理论和方法，重点研究逆摄动法求解上述两类反问题。主要工作有以下三
方面：
1．提出了广义特征值反问题（频率禁区优化问题）求解的新的逆摄动法：将频
率禁区问题求解归结为广义特征值反问题求解。利用广义特征值反问题理论和有
限单元刚度阵和质量阵的分解，推导出特征值λ与逆摄动参数ε（与设计变量x
相对应）关系的显式表达的计算公式。通过给定的特征值增量逆求出ε值，解决
了结构矩阵摄动法的无法确定送代步长问题，避免或减少了重分析次数。广义特
征值反问题求解存在不适定问题，为解决此问题，给出了逆摄动参数ε若干种取
值方法，如基于平均概率的敏度分析方法。另外，方法简单直观，便于人机交互。
2．首次提出了动力响应振幅（或动柔度）反问题（动力响应振幅优化问题）求
解的逆摄动法：根据逆求思想和有限单元刚度阵和质量阵分解方法，推导出逆摄
动参数ε与动力响应振幅（或动柔度）关系的显式表达式。避免或减少了重分析。
用逆摄动法求解动力响应振幅问题，既可以将其作为反问题求解，也可以将其作
为正问题求解。有助于动力响应优化难题的求解。
3．逆摄动法在动力优化中的应用及其编程：给出了人机交互的基于约束的
n 大连理工大学博土学位论文 机械结构有限元动力优化设计的逆摄动方法研究
一
优化方法和边界映射的优化方法，推荐了暂时消除约束的分部（步）优化方法。编
制了本逆摄动法的软件：本软件采用MATLAB编程语言，并利用了MATLAB的平台，
完成了上述特征值问题和动力响应振幅问题逆摄动法的软件编程。实现了对杆单
元的轴向振动系统、析架系统，平面梁单元的转子系统等上述两类问题的求解。
算例或工程实例验证结果表明，该方法可有效地减少重分析次数，计算效
率高，为结构动力优化设计提供了一种有效方法。所给公式及求解方法具有工程
直观性，编程容易，便子工程设计人员应用，便于发挥其主观能动性。本方法可
与矩阵摄动法结合应用，相辅相成。根据本文逆摄动法的有限元矩阵分解组装方
法，可以推导出基于其它有限元分析模型的动力优化设计的逆摄动法，如板、壳
等单元的逆摄动参数。的有关计算公式。
With the development of modern complicated machine and engineering structures, an even higher demand is posed to the optimal design. That is the dynamic optimal design which concerns comprehensively influencing factors about forced vibration, unacceptable frequency band, stiffness, intensity etc., based on the analytical model of FEM, to satisfy the needs of practical engineering. The crux of the matter at present is to study a new and utility dynamic optimal design method, which strives to reduce reanalysis and is easy for engineers to master and use and convenient for manmachine interaction. This is the most urgent task for the problems of practical engineering.
There are two primary problems in dynamic optimal design: The unacceptable frequency bands(eigenvalue problems) and dynamic response(forced vibration). These two problems can be solved by 慸irect solution?method or be solved as inverse problems. The methods most in use to the solution of the problem with unacceptable frequency bands: one is a mathematical programming or optimum criterion method; the other is structure matrix perturbation. The former needs more structure reanalysis and more computational time, the later may reduce reanalysis to some extend, but the small-perturbed parameter s is hard to set. The problem with constraints of dynamic response amplitude hasn抰 been studied widely, because it is more complicated and more difficulty.
The question studied in the dissertation is supported by National Natural Science Fund(N259675041): Theory and Method of Visual Inverse Perturbation Method. The purpose of the dissertation is to study and advance a new effective and practical dynamic optimal design method梚nverse perturbation method based on the analytical model of FEM. The work to solve the problems mentioned above by inverse perturbation method includes 3 aspects as follows:
1 .The inverse perturbation method to the solution for the inverse problem of general eigenvalue: The problems with the constraints of unacceptable frequency bands are attributed to the solution of the inverse problem of general eignenvalue.
Taking advantage of the theories of the inverse problem of general eigenvalue and the decomposition of finite element matrices, the explicit expressions of eigenvalue 2 and perturbed parameter e (correspondence with design variables) and the computation equations of ~ are derived. The s 憇 value is determined by the difference of the increment of 2, so in this way reanalysis can be avoided or reduced. The characteristic of the inverse problem of general eignenvalue is multi-value and indetermination. By solving this problem several methods to determine the value of the perturbed parameter e are described in the dissertation, such as sensitivity ratio analytical method based on average of probability.
2.The inverse perturbation method to the solution for the inverse problem of dynamic response amplitude: According to the solution of inverse thought and the decomposition method of finite element stiffness and mass matrices, the explicit expressions of dynamic response amplitude and perturbed parameter s are also derived. The inverse perturbation method for the problem of dynamic response amplitude can be applied both for inverse and optimal problem of dynamic response amplitude.
3.The application of inverse perturbation method for dynamic optimum design and its software: The optimum method with constraints of fractional optimum design and man-machine synergy is introduced in the dissertation. The software of the inverse perturbation method is programmed in MATLAB. It has been successfully applied to solve the problems of truss structure system, axial vibration system, rotor system and plane beam system etc.
Illustrative examples demonstrate that the method is characterized in reducing reanalysis significantly, bears the merits in computing speed and precision. The inverse perturbation method can be as an approach for dynamic optimum design. Though illustrated with beam element system, in accordance with t
引文
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