Stability for a composite of Timoshenko laminated beams

The present model describes the vibrations of a composite beam consisting of two interconnected beams. One component is a material that exhibits only structural dissipation, while the other is a material subjected to additional frictional dissipation. It is well known that the solutions for laminated Timoshenko beams with one component converge to the equilibrium point at rates that depend on the wave propagation velocity. Here, we show that the solution of the complete system decays with exponential or rational rates when the components share a transmission condition. Proofs of the main results are provided using the multiplier method in the frequency domain. Technically, the main ingredients are inequalities of observability type that serve as “propagators” of asymptotic properties from one component to another.