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Energy decay to Timoshenko system with indefinite damping

We consider the classical Timoshenko system for vibrations of thin rods. The system has an indefinite damping mechanism, ie, it has a damping function a=a(x) possibly changing sign, present only in the equation for the vertical displacement. We shall prove that exponential stability depends on conditions regarding of the indefinite damping function a and a nice relationship between the coefficient of the system. Finally, we give some numerical result to verify our analytical results.