Damping optimization in mechanical systems with external force

We consider a mechanical system excited by an external force. Model of such a system is described by the system of ordinary differential equations: Mx¨(t) + Dx˙(t) + Kx(t) = ˆf(t), where matrices M, K (mass and stiffness) are positive definite and the vector ˆf corresponds to an external force. The damping matrix D is assumed to be positive semidefinite and has a small rank. The motivation for our approach is related to the harmonic response of the mechanical system under the influence of the harmonic force. Here we consider external function consisting of simple oscillating functions which is motivated by Fourier series which decomposes periodic functions into the sum of a set of simple oscillating functions. In this setting, we consider criterions average energy amplitude and average displacement amplitude 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 explicit formulas which have been used for efficient damping optimization. The efficiency of new formulas has been illustrated with a numerical examples.