In:
ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Wiley, Vol. 91, No. 3 ( 2011-03), p. 179-191
Abstract:
We consider a mathematical model of a linear vibrational system described by the second‐order differential equation , where M and K are positive definite matrices, called mass, and stiffness, respectively. We consider the case where the damping matrix D is positive semidefinite. The main problem considered in the paper is the construction of an 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 AX + 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. Numerical results illustrate the effectiveness of our approach.
Type of Medium:
Online Resource
ISSN:
0044-2267
,
1521-4001
DOI:
10.1002/zamm.201000077
Language:
English
Publisher:
Wiley
Publication Date:
2011
detail.hit.zdb_id:
203011-1
detail.hit.zdb_id:
1474638-4
SSG:
17,1
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