**Numerical Linear Algebra with Applications**

# Optimal damping of selected eigenfrequencies using dimension reduction

Zoran Tomljanović, Department of Mathematics, J.J. Strossmayer University in Osijek, Trg Lj. Gaja 6, 31000 Osijek, Croatia.

E-mail: ztomljan@mathos.hr

## SUMMARY

We consider linear vibrational systems described by a system of second-order differential equations of the form , where *M* and *K* are positive definite matrices, representing mass and stiffness, respectively. The damping matrix *D* is assumed to be positive semidefinite. We are interested in finding an optimal damping matrix that will damp a certain (critical) part of the eigenfrequencies. For this, we use an optimization criterion based on the minimization of the average total energy of the system. This is equivalent to the minimization of the trace of the solution of the corresponding Lyapunov equation *AX* + *XA*^{T} = −*GG*^{T}, where *A* is the matrix obtained from linearizing the second-order differential equation, and *G* depends on the critical part of the eigenfrequencies to be damped. The main result is the efficient approximation and the corresponding error bound for the trace of the solution of the Lyapunov equation obtained through dimension reduction, which includes the influence of the right-hand side *GG*^{T} and allows us to control the accuracy of the trace approximation. This trace approximation yields a very accelerated optimization algorithm for determining the optimal damping. Copyright © 2011 John Wiley & Sons, Ltd.