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In this paper, we consider damping optimization in mechanical system excited by an external force. We use optimization criteria based on minimizing average energy amplitude and average displacement amplitude over the arbitrary time. As the main result we derive explicit formulas for objective functions. These formulas can be implemented efficiently and accelerate optimization process significantly, which is illustrated in a numerical example. [ABSTRACT FROM AUTHOR]

In this paper, we present the Sherman–Morrison–Woodbury-type formula for the solution of the Sylvester equation of the formas well as for the solution of theT-Sylvester equation of the formwhereU1,U2,V1,V2are low-rank matrices. Although the matrix version of this formula for the Sylvester equation has been used in several different applications (but not for the case of aT-Sylvester equation), we present a novel approach using a proper operator representation. This novel approach allows us to derive a matrix version of the Sherman–Morrison–Woodbury-type formula for the Sylvester equation as well as for theT-Sylvester equation which seems to be new. We also present algorithms for the efficient calculation of the solution of structured Sylvester andT-Sylvester equations by using these formulas and illustrate their application in several examples. [ABSTRACT FROM PUBLISHER]