Given a conservative dynamical system we determine where, and in what quantity, dissipative mechanisms should be placed in order to best absorb an initial disturbance. We work in the context of linear vibrating systems described by constant mass, M, stiffness, K, and damping, D, matrices. We suppose that M and K are fixed and proceed to minimize, with respect to D, two measures of energy absorption: (i) the least exponent for which the energy exhibits exponential decay and (ii) the maximum over initial disturbances of unit energy of the infinite time integral of the energy. We construct a general setting and provide explicit solutions in the special case of normal mode systems.

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