Naslov
Dimension reduction for damping optimization in linear vibrating system

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

Izvornik
Zeitschrift für angewandte Mathematik und Mechanik (0044-2267) 91 (2011), 3; 179-191

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

Ključne riječi
Linear vibrational system; Damping optimization; Lyapunov equation; Dimension reduction

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, called mass and stiffness, espectively. We consider the case where the damping matrix $D$ is positive semidefinite. The main problem considered in the paper is the construction of efficient algorithm for calculating an optimal damping. As optimization criterion we use the minimization of the average total energy of the system which is equivalent to the minimization of the trace of the solution of the corresponding Lyapunov equation $A X+ X A^T =-I$, where $A$ is the matrix obtained from linearizing the second-order differential equation. Finding the optimal $D$ such that the trace of $X$ is minimal is a very demanding problem, caused by the large number of trace calculations, which are required for bigger matrix dimensions. We propose a dimension reduction to accelerate the optimization process. We will present an approximation of the solution of the structured Lyapunov equation and a corresponding error bound for the approximation. Our algorithm for efficient approximation of the optimal damping is based on this approximation.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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