Rates of decay to non homogeneous Timoshenko model with tip body

• We consider the Timoshenko beam Model with tip load.
• We prove the lack of exponential stability using the Weyl's Theorem.
• We show that the solution decays polynomially to zero.
• When the dissipative mechanism is effective only over the bending moment the decays is slower that the fully dissipative case.
• If the wave propagations speed are equals the partial dissipation behaves as fully dissipation.

We consider the uniform stabilization of a hybrid elastic model consisting of a Timoshenko beam and a tip load at the free end of the beam. Our main result proves that the semigroup eAteAt associated to this model is not exponentially stable. Moreover, we prove that the semigroup decays polynomially to zero as t−1/2t−1/2. When the damping mechanism is effective only on the boundary of the rotational angle, the solution also decays polynomially as t−1/2t−1/2 provided the wave speeds are equal. Otherwise it decays as t−1/4t−1/4 for any initial data taken in D(A)D(A).