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Abstract
In this paper we show that there exists a unique weak solution for a model that was proposed for simulating the vibration of a damaged beam. The weak formulation of the problem is rewritten as an abstract differential equation and semigroup theory is then used to prove the results. Because of discontinuities in the spatial derivatives of the solution(due to the damage to the beam)it is necessary to use product spaces in the abstract formulation. All functions are represented as pairs of functions where the members of a pair are the restrictions of the functions to subintervals.