  We consider a beam and a wave equations coupled on an elastic beam through
transmission conditions. The damping which is locally distributed acts through
one of the two equations only; its effect is transmitted to the other equation
through the coupling. First we consider the case where the dissipation acts
through the beam equation. Using a recent result of Borichev and Tomilov on
polynomial decay characterization of bounded semigroups we provide a precise
decay estimates showing that the energy of this coupled system decays
polynomially as the time variable goes to infinity. Second, we discuss the case
where the damping acts through the wave equation. Proceeding as in the first
case, we prove that this system is also polynomially stable and we provide
precise polynomial decay estimates for its energy. Finally, we show the lack of
uniform exponential decay of solutions for both models.
