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. KEYWORDS exponential stability, finite difference, indefinite damping, Timoshenko system MSC CLASSIFICATION35L53; 35B40; 93D20; 65N06 Here, t denotes the time variable, x is the distance until the beam's centerline in equilibrium, the function = (t, x) denotes the vertical displacement of the beam's centerline, and the function = (t, x) denotes the rotation of the vertical fibers in the beam. Moreover, the coefficients 1 , 2 , b, and k denote positive constants, and they depend on the density of the mass material, the area of the cross-section, the second moment of the cross-section area, the Young's model, the modulus of rigidity, and the shear factor.The system (1)-(2) is conservative. So, if we want to search about asymptotic behavior, we must add a damping term. In this direction, the main types of dissipative mechanisms considered are frictional, thermal, viscoelastic and their combinations.Recently, researches have shown that the exponential stability of the Timoshenko system is achieved regardless of any specific relations between the coefficients when there are a dissipative mechanism in both equations. We refer the reader to, eg, previous studies 1-4 and the reference therein.Math Meth Appl Sci. 2020;43:225-241.wileyonlinelibrary.com/journal/mma
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