Optimization and Systems Theory Seminar
Friday, November 29, 2002, 11.00-12.00, Room 3721, Lindstedtsvägen 25

Dr. Oleg N. Kirillov
Institute of Mechanics
Moscow State Lomonosov University
Moscow, Russia
E-mail: kirillov@imec.msu.ru

Overlapping of the characteristic curves and optimization of nonconservative systems

We consider a non-conservative system governed by the equation y''+Ay=0, where A is a real non-symmetric m-by-m matrix smoothly dependent on a vector of n real parameters p=(p1,...,pn). At the fixed p this system is stable if and only if all the eigenvalues of the matrix A are positive and semi-simple. If the spectrum of A contains a complex conjugate pair then the system loses stability dynamically (flutter). Characteristic curve is a dependence of an eigenvalue of the matrix A on a chosen parameter, say on p1, while other n-1 parameters remain fixed. Characteristic curves of the stable system lie on the real plane. We define a functional of the critical load F as a minimal value of the parameter p1>=0 at which the flutter instability occurs. The optimization problem is to maximize F due to change of parameters p2,...,pn.

It turns out that due to change of parameters p2,...,pn any two characteristic curves of the stable system corresponding to positive simple eigenvalues may come together, merge at some point, and then overlap forming a closed curve of the complex eigenvalues (flutter instability) on some range of the parameter p1 In our work explicit formulae describing phenomenon of overlapping of characteristic curves in n-parameter non-conservative systems are derived. These formulae use information on a system only at the merging point and allow qualitative as well as quantitative analysis of behavior of characteristic curves near that point. It is shown that the overlapping of characteristic curves is closely connected with the convexity properties of the flutter instability boundary. Application of the developed theory to structural optimization problems is shown and mechanical examples are considered.

This work was done jointly with Alexander P. Seyranian.

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