Asymptotic behavior of weakly dissipative Bresse-Timoshenko system on influence of the second spectrum of frequency

In the present work, we consider a dissipative Bresse-Timoshenko type system, which is free of physical anomaly know as second spectrum of frequency according important observations made by Elishakoff etal., and we establish a new result of exponential decay. We prove that frictional damping acting on vertical displacement of this system is exponentially stable regardless the equality between velocities of wave propagation. This result is new and completely different from ones obtained by Almeida Junior etal. for the Timoshenko system with the same damping acting on vertical displacement. Our approach is strongly inspired in a recent result due to Almeida Junior and Ramos where the authors justified, from physical point of view, a classical and pioneering result due to Soufyane. That is, the dissipative mechanism of frictional type acting on angular rotation of the Timoshenko system truncates the consequences of the second spectrum when the velocities of wave propagation are equal. Also, for dissipative Bresse-Timoshenko systems which are free of the second spectrum, the exponential decay occurs regardless of any condition on coefficients of system. Based on the new results, it is demonstrated that the classical Bresse-Timoshenko beam theory, as an overcomplicated theory, requires damping terms to eliminate the second spectrum. In the present work, we consider a dissipative Bresse-Timoshenko type system, which is free of physical anomaly know as second spectrum of frequency according important observations made by Elishakoff etal.,([11-13]) and we establish a new result of exponential decay. We prove that frictional damping acting on vertical displacement of this system is exponentially stable regardless the equality between velocities of wave propagation. This result is new and completely different from ones obtained by Almeida Junior etal.([4]) for the Timoshenko system with the same damping acting on vertical displacement. Our approach is strongly inspired in a recent result due to Almeida Junior and Ramos([2]) where the authors justified, from physical point of view, a classical and pioneering result due to Soufyane.([36]) That is, the dissipative mechanism of frictional type acting on angular rotation of the Timoshenko system truncates the consequences of the second spectrum when the velocities of wave propagation are equal. Also, for dissipative Bresse-Timoshenko systems which are free of the second spectrum, the exponential decay occurs regardless of any condition on coefficients of system. Based on the new results, it is demonstrated that the classical Bresse-Timoshenko beam theory, as an overcomplicated theory, requires damping terms to eliminate the second spectrum.