Sensitivity Analysis for Dynamic Stability Problems
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
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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: • 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. • Classification of systems and solutions for choosing the right tools to solve specific problems. • 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-offreedom 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. A. P. Seyranian et al. (eds.), Modern Problems of Structural Stability © Springer-Verlag Wien 2002
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P. Pedersen
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 Floquet 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 sp
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