Given an incomplete set of natural frequencies and modes of a linear structure from a vibration test, the problem of computing the necessary changes in the structure to reach a desired spectra is considered. A criterion of optimality which gives the best approximation from a certain subspace is formulated. Then formulae for evaluating the family of all mathematical solutions are given, resulting in a class of mathematical incremental mass and stiffness matrices. Finally, we describe the problem of the realization of the mathematical solutions by constructing a model of a given type.

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