On the nature of dissipative Timoshenko systems at light of the second spectrum of frequency

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

“…In that direction, Almeida Júnior and Ramos [2] analyzed the dispersion properties of the dissipative Bresse-Timoshenko system given by and they conclude that second spectrum is truncated when 1 = 2 (similar results hold in the general case, when 1 ≠ 2 ), which constitutes similar conclusions those obtained by Manevich and Kolakowski. [24] Therefore, we are conduced to conclude that damping effects truncate the second spectrum and, in that direction, the first spectrum is the only spectrum resulting all dissipative system.…”

“…The authors in [24] concluded that 'internal friction damping eliminates the second branch for sufficiently short wavelengths, and there remains only the first branch' (see Figure 3 on page 12). Thus, from the point of view of the work of Manevich and Kolakowski, we are conduced to conclude that damping effects on Bresse-Timoshenko model are able of truncating the consequences of the second spectrum (see also Almeida Júnior and Ramos [2] ). This has a notable consequence because we can investigate, from the dispersion analysis, if other damping mechanisms have the same property.…”

“…In this paper we consider the truncated version of the Bresse-Timoshenko beam model with a damping term acting on vertical displacement and we prove a new result of exponential decay. Our study is strongly inspired by works of Elishakoff et al [11][12][13] and also by recent result on stabilization of truncated version of the dissipative Bresse-Timoshenko due to Almeida Júnior and Ramos [2].…”

“…At the level of higher frequency (for large wave number) these modes converge to horizontal asymptotes given by 1 and 2 . However, at level of the lower frequency (for lower wave number) these modes are inversely proportionals and the second spectrum blows-up when the wave number goes to zero (see red plots in Figure 1 builded from the dispersion analysis of harmonic waves [2] ). By supposing that the Equations 1.5-1.6 admit harmonic solutions given by ( + ) where is the wave number, is the phase velocity and = √ −1, it can be shown that (see [2])…”

“…This result is new and completely different from ones obtained by Almeida Júnior et al [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 Júnior 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.…”