  Common criteria used for measuring performance of vibrating systems have one
thing in common: they do not depend on initial conditions of the system. In
some cases it is assumed that the system has zero initial conditions, or some
kind of averaging is used to get rid of initial conditions. The aim of this
paper is to initiate rigorous study of the dependence of vibrating systems on
initial conditions in the setting of optimal damping problems. We show that,
based on the type of initial conditions, especially on the ratio of potential
and kinetic energy of the initial conditions, the vibrating system will have
quite different behavior and correspondingly the optimal damping coefficients
will be quite different. More precisely, for single degree of freedom systems
and the initial conditions with mostly potential energy, the optimal damping
coefficient will be in the under-damped regime, while in the case of the
predominant kinetic energy the optimal damping coefficient will be in the
over-damped regime. In fact, in the case of pure kinetic initial energy, the
optimal damping coefficient is $+\infty$! Qualitatively, we found the same
behavior in multi degree of freedom systems with mass proportional damping. We
also introduce a new method for determining the optimal damping of vibrating
systems, which takes into account the peculiarities of initial conditions and
the fact that, although in theory these systems asymptotically approach
equilibrium and never reach it exactly, in nature and in experiments they
effectively reach equilibrium in some finite time.
