Damping optimization in mechanical systems with external force

We consider a mechanical system excited by external force. Model of such a system is described by the system of ordinary differential equations: Mx¨(t)+Dẋ(t)+Kx(t)=fˆ(t), where matrices M,K (mass and stiffness) are positive definite and the vector fˆ corresponds to an external force. The damping matrix D is assumed to be positive semidefinite and has a small rank. We introduce two criteria that allow damping optimization of mechanical system excited by an external force. Since in general a damping optimization is a very demanding problem, we provide a new formulas which have been used for efficient damping optimization. The efficiency of new formulas is illustrated with a numerical experiment.