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首页> 外文会议>Conference on Modern Problems of Structural Stability Jul 16-20, 2001 Udine >Sensitivity Analysis for Dynamic Stability Problems
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Sensitivity Analysis for Dynamic Stability Problems

机译：动态稳定性问题的灵敏度分析

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0.1 Goal of the notes These notes are neither chapters in a textbook, nor short research papers. They are something in between and they include results known for some years as well as just published results. The six notes are written to be fairly independent without extensive mutual reference and the page limit for each note is set to 10. This page limit means that extensive examples are not included. The main goal is to communicate to the course participants: 1. Knowledge about sensitivity analysis, i.e. gradient information obtained without re-analysis. Especially for eigenvalue problems. The simplicity and the practical use of sensitivity analysis is today not widely known. 2. Classification of systems and solutions for choosing the right tools to solve specific prob-lems. 3. The versatility in the application of sensitivity analysis, which however, has a tendency to make these notes somewhat disconnected. 0.2 Comments to the individual notes/lectures 1) Note number 1 is rather general and abstract, hopefully not too theoretical. Restricted to linear systems we classify the systems as well as the possible instability behaviour. Then general variational analysis for non-selfadjoint systems is presented with focus on energy interpretation. Specific treatment is given to finite element discretization and to global Galerkin discretization. The necessary procedure for sensitivity analysis with multiple eigenvalues is derived for the selfadjoint case. 2) In note number 2 we are more specific, dealing with two rather simple two-degrees-of-freedom problems to exemplify the case of double eigenvalues with only one eigenvector. Analytical analysis of isolated eigenvalues explains in a more complete way the earlier reported results. The goal of this note is also to point out the importance of the Routh-Hurwitz stability criteria. 3) In note number 3 we concentrate on linear periodic systems and describe two possible methods of stability analysis for the solutions. For the case of a single linear differential equation with periodic coefficients, it is suggested to set up general stability diagram from which a more complete overview can be obtained. Actual examples of relevant physical problems are listed and classifications are described. For the case of multi-degrees-of-freedom, the Floquet method is suggested and therefore this methods is described in more details. This description also gives the background for note number 4. 4) In note number 4 we primarily establish the sensitivity analysis which is necessary to obtain the derivatives of the Floquet matrix, the reason being that the eigenvalues of the Flo-quet matrix give the information about the stability of solutions. The Floquet matrix is real but non-symmetric. Thus the eigenvalues of the Floquet matrix may be complex and the sensitivity analysis of these eigenvalues must include the possibility for both multi-modal solutions with full eigenvector space as well as not complete eigenvector space. 5) Applications of sensitivity analysis are described in note number 5 and we have concentrated on three specific examples. (1) The results of numerical studies of the influence of external damping are explained analytically by sensitivity analysis. (2) Sensitivity analysis is applied to laminate inverse problems and a possible iteration procedure is presented, although this example is not related to an eigenvalue problem. (3) In material identification problems, eigenfrequencies are often used as response quantity, here sensitivities to material parameters are involved. 6) Although sensitivity analysis has many domains of application, the domain of optimal design still shows the major interest. In note number 6 we as an introduction shortly give the background for the two major optimal design solution procedures in terms of sensitivities. Then four specific examples are presented: (1) Beam design with several eigenvalue constraints. (2) Material orientation of composite plates for o
机译：0.1注释的目的这些注释既不是教科书中的章节，也不是简短的研究论文。它们介于两者之间，其中包括多年来已知的结果以及刚刚发布的结果。这六个注释的编写是相当独立的，没有广泛的相互参考，并且每个注释的页数限制都设置为10。此页数限制意味着不包括大量示例。主要目标是与课程参与者进行交流：1.了解敏感性分析，即无需重新分析即可获得的梯度信息。特别是对于特征值问题。如今，灵敏度分析的简单性和实际用途尚未广为人知。 2.为选择解决特定问题的正确工具的系统和解决方案的分类。 3.敏感性分析应用的多功能性使这些注释有些脱节。 0.2对单个注释/讲座的注释1）注释1相当笼统和抽象，希望不要太理论化。限于线性系统，我们对系统以及可能的不稳定性行为进行分类。然后介绍了非自伴系统的一般变分分析，重点是能量解释。对有限元离散化和全局Galerkin离散化进行了特殊处理。对于自伴情况，导出了具有多个特征值的敏感性分析的必要过程。 2）在注解2中，我们更加具体，处理两个相当简单的两自由度问题，以仅具有一个特征向量的情况来说明双特征值的情况。对孤立特征值的分析分析以更完整的方式解释了先前报告的结果。本说明的目的还在于指出Routh-Hurwitz稳定性标准的重要性。 3）在注解3中，我们专注于线性周期系统，并描述了两种可能的稳定性分析方法。对于具有周期系数的单个线性微分方程，建议建立一般的稳定性图，从中可以获得更完整的概述。列出了相关物理问题的实际示例，并描述了分类。对于多自由度的情况，建议使用Floquet方法，因此将更详细地描述此方法。该说明还提供了音符4的背景。4）在音符4中，我们主要建立灵敏度分析，这对于获得Floquet矩阵的导数是必要的，原因是Flo-quet矩阵的特征值提供了信息。关于解决方案的稳定性。 Floquet矩阵是实数，但不是对称的。因此，Floquet矩阵的特征值可能很复杂，并且这些特征值的灵敏度分析必须包括具有完整特征向量空间以及不完整特征向量空间的多模态解的可能性。 5）注释5中描述了灵敏度分析的应用，我们集中在三个具体示例上。 （1）通过灵敏度分析来分析解释外部阻尼影响的数值研究结果。 （2）尽管本例与特征值问题无关，但对叠层反问题进行了灵敏度分析，并提出了可能的迭代程序。 （3）在材料识别问题中，本征频率通常用作响应量，这里涉及对材料参数的敏感性。 6）尽管灵敏度分析有许多应用领域，但最佳设计领域仍然显示出人们的主要兴趣。在注解6中，我们作为引言部分简要介绍了两种主要的最佳设计解决方案程序的敏感性。然后给出了四个具体示例：（1）具有多个特征值约束的梁设计。 （2）复合材料的材料取向

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