ILAS2016 — 11–15 July 2016 — KU Leuven, Belgium

## 20th ILAS Conference

In minisymposium: Matrix Polynomials, Matrix Functions and Applications

Thu 17:30–18:00, Room AV 03.12
On the eigenstructure of classically damped quadratic systems and phase synchronization
Ion Zaballa (Universidad del Pais Vasco UPV/EHU)

Classically damped quadratic systems are those whose coefficient matrices can be simultaneously diagonalized by strict equivalence. That is to say, there are non-singular square matrices $P$ and $Q$ such that $PL(\lambda)=\widetilde{L}(\lambda)Q$ where $L(\lambda)=M\lambda^2+D\lambda+ K$ is the given system and $\widetilde{L}(\lambda)=\widetilde{M}\lambda^2+ \widetilde{D}\lambda+\widetilde{K}$ is a diagonal quadratic matrix polynomial. If such matrices exist for $L(\lambda)$ then it is said that the system has been decoupled by modal analysis.

A well-known result by Caughey and O'Kelly gives a necessary and sufficient condition for decoupling when $L(\lambda)$ is symmetric and $M$ is positive definite. Ma and Caughey studied this problem for general systems and Lancaster and Zaballa provided a solution for symmetric systems when the pencil $\lambda M+K$ is semisimple and its eigenvalues are of definite type, and for general systems when $\lambda M+K$ has simple eigenvalues. In all these cases, the necessary and sufficient condition for reducing a given system to diagonal form by strict equivalence is the following commutative expression: $KM^{-1}D=DM^{-1}K$. This is quite a restrictive condition and so most systems cannot be decoupled by modal analysis.

Recently, Ma, Morzfeld and Imam ([2,3]) introduced the notion of phase synchronization. The goal of this method is, for a given system, to produce a new intermediate isospectral system that is classically damped yielding a physically implementable procedure to decouple the original one. The idea is to change, by means of a non-linear transformation, the eigenvectors of the systems so that the new vectors are eigenvectors of the intermediate classically damped system. However, it looks like there is no a thorough study that characterize the possible eigenstructures of the systems that can be decoupled by modal analysis. Such a characterization may help to improve the method of phase synchronization and/or to provide other methods to obtain isospectral classically damped systems out of the eigenvectors of the original one.

It will be shown that the notion of Filters connecting two isospectral quadratic systems ([1]) can be used to give a complete characterization of the possible eigenvectors of classically damped systems. Also, a method based on linear transformation will be presented that transform the eigenvectors of the original system into eigenvectors of a classically damped one.

1. S. Garvey, P. Lancaster, A. Popov, U. Prells and I. Zaballa, Filters connecting isospectral quadratic systems, Linear Algebra Appl. 438 (2013) pp.  1497–1516.
2. F. Ma, A. Imam and M. Morzfeld, The decoupling of damped linear systems in oscillatory free vibration, J. Sound Vibration 324 (2009), pp.  408–428.
3. F. Ma, M. Morzfeld and A. Imam, The decoupling of damped linear systems in free and forced vibration, J. Sound Vibration 329 (2010), pp.  3182–3202.