A PROBLEM OF EIGENFUNCTION CORRESPONDING TO THE MINIMAL EIGENVALUE
A Kolpakoff
34, Bld.12-A, A.Nevskogo str., Novosibirsk, 630065, Russia
Abstract: The aim of the paper is attract attention to a prospective (not solved completely for now) eigenvalue problem arising in structural engineering.
Let us consider a structure constructed by periodic repetition of a basic structural element (the so called periodicity cell [Bensoussan et al (1978)]). For many practical cases the periodicity cell is a structure made of beams, plates and so on. From the mathematical point of view it implies that both the periodicity cell and the structure can be described using the finite dimension model [Kalamkarov and Kolpakov (1997)].
The finite dimension problem has the form
Ax=pBx
where A,B are symmetric matrices, A is positive. The size of the matrices is large.
We are interested in the minimal eigenvalue pmin. At the same time we are very interesting in the type of the eigenvector xmin corresponding to pmin. From mechanical point of view the following types of the eigenvectors are expected:
1. corresponding to buckling in a global mode (homogenized eigenvalue [Bensoussan et al (1978)]);
2. corresponding to buckling in a global mode;
3. corresponding to buckling in a periodic mode.
These modes have mechanical interpretation and knowledge of the mode is important for structural safety and design.
Bensoussan, A., Lions, J.L. ? Papanicolaou, G. (1978) Asymptotic analysis for periodic structures. North - Holland Publ.Comp., Amsterdam.
Kalamkarov A.L., Kolpakov A.G. (1997) Analysis, design ? optimization of composite structures. J.Wiley ? Sons, Chichester, New York.