Naslov
Optimal Damping of Selected Eigenfrequencies Using Dimension Reduction

Autori
Benner, Peter ; Tomljanović, Zoran ; Truhar, Ninoslav

Izvornik
Numerical linear algebra with applications (1070-5325) 20 (2013), 1; 1-17

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
Vibrating system; Lyapunov equation; Energy minimization; Dimension reduction; Error bound; Partial spectra

Sažetak
We consider a mathematical model of a linear vibrational system described by the second-order differential equation $M \ddot{; ; ; ; x}; ; ; ; + D \dot{; ; ; ; x}; ; ; ; + Kx = 0$, where $M$ and $K$ are positive definite matrices, representing mass and stiffness, respectively. The damping matrix $D$ is positive semidefinite. We are interested in finding an optimal damping matrix which will damp a certain (critical) part of the undamped eigenfrequencies. For this we use an optimization criterion based on 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 $A X+ X A^T =-GG^T$, 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 corresponding error bound for the trace of the solution of the Lyapunov equation obtained by dimension reduction, which includes the influence of the right-hand side $G G^T$ and allows us to control the accuracy of the trace approximation. This trace approximation yields a much accelerated optimization algorithm for determining the optimal damping.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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