Optimization of material with modal damping Ivana Kuzmanovic ´ ⇑ , Zoran Tomljanovic ´ , Ninoslav Truhar Department of Mathematics, J.J. Strossmayer University of Osijek, Trg Lj. Gaja 6, 31000 Osijek, Croatia article info Keywords: Modal damping Proportional damping Rayleigh damping Optimal parameters abstract This paper considers optimal parameters for modal damping D ¼ Mf 1 ðM 1 K; a 1 ; ... ; a k Þþ Kf 2 ðK 1 M; a 1 ; ... ; a k Þ in mechanical systems described by the equation M€ x þ D _ x þ Kx ¼ 0, where matrices M and K are mass and stiffness matrices, respectively. Different models of proportional and gen- eralized proportional damping are considered and optimal parameters with respect to dif- ferent optimization criteria related to the solution of the corresponding Lyapunov equation are given. Also, some specific example problems are compared with respect to the optimal and estimated parameters. Ó 2012 Elsevier Inc. All rights reserved. 1. Introduction There are many physical and engineering problems (e.g. free oscillations of some construction) which can be described by the system of ordinary differential equations: M€ x þ D _ x þ Kx ¼ 0; ð1Þ where matrices M and K are positive definite matrices, called mass and stiffness, respectively, and D is a damping matrix. There are two well-known problems related with construction or determination of the damping matrix D. The first problem is how one can construct the damping matrix D such that the considered model (1) describes some spe- cific properties in the best possible way. If we assume that the damping matrix D can be presented as D ¼ f ðM; K; a 1 ; ... ; a k Þ; ð2Þ where f is a known function, then the above problem can be reduced to the problem of determination of parameters a 1 ; ... ; a k such that model (1) describes some specific properties in the best possible way. The following damping models of the above type are widely used in various problems: a Rayleigh damping (also known as proportional damping) model (see e.g. [1]) D ¼ a 1 M þ a 2 K ð3Þ and a generalized proportional damping model, D ¼ Mf 1 ðM 1 KÞþ Kf 2 ðK 1 MÞ; ð4Þ 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2012.01.011 ⇑ Corresponding author. E-mail addresses: ikuzmano@mathos.hr (I. Kuzmanovic ´), ztomljan@mathos.hr (Z. Tomljanovic ´), ntruhar@mathos.hr (N. Truhar). Applied Mathematics and Computation 218 (2012) 7326–7338 Contents lists available at SciVerse ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc