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: Mẍ(t) + D(v)ẋ(t) + Kx(t) = f̂(t), where matrices M,K (mass and stiffness) are positive definite and the vector f̂ corresponds to an external force. The damping matrix D(v) is a positive semidefinite matrix which depends on damping positions and parameter v, called viscosity. 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 functions into the sum of a set of simple oscillating functions. In order to determine optimal damping matrix D(v), we consider criteria average energy amplitude and average displacement amplitude that allow damping optimization of mechanical system excited by an external force. We consider criteria based on time which is determined from period of the excitation function, but also additionally we consider the case with the arbitrary time. Since in general a damping optimization is a very demanding problem, we provide a new explicit formulas for objective functions which have been used for efficient damping optimization. The efficiency of new formulas has been illustrated with a numerical examples.