engleski

Optimal damping of infinitedimensional vibrational systems

We introduce the notion of an abstract vibrational system. Most mechanical vibrational systems can be written in this form. Under some natural conditions, we solve this equation by the use of the semigroup theory technique. An useful optimal damping criterion is \[ \min_{\gamma} \int_{\|u_0\|=1} \left(\int_0^{\infty} E(t ; u_0)\mathrm{d}t\right)\mathrm{d}\sigma, \] where $E(t ; u_0)$ is the energy of the system with initial state $u_0$ at the moment $t$, and $\sigma$ is some probability measure on the unit sphere. In other words, we minimize the average total energy of the system over all admissible damping forms. We give a precise mathematical formulation of this criterion and show how to choose an appropriate measure $\sigma$. Also, in the case of systems which posses an internal damping, we find the optimal damping forms.

damping; vibrational systems

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