Arnold's Seminar, April 1, 2003

Kirillov O.N., Seyranian A.P.

E-mail: kirillov@imec.msu.ru, seyran@imec.msu.ru
http://www.imec.msu.ru/~kirillov/

Metamorphoses of characteristic curves in non-conservative systems dependent
on parameters.

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. 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.