### 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

The overlapping phenomenon was discovered in the problems of
structural stability and optimization (Herrmann & Bungay 1964, Claudon
1975, Hanaoka & Washizu 1980, Langthjem & Sugiyama 1999).
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.

Calendar of seminars

*Last update: November 14, 2002 by
Anders Forsgren,
anders.forsgren@math.kth.se.
*