On the nature of dissipative Timoshenko systems at light of the second spectrum of frequency
The paper addresses the system derived by \textit{S. P. Timoshenko} in [``On the correction for shear of the differential equation for transverse vibrations of prismatic bars'', Philos. Mag. (6) 41, No. 245, 744--746 (1921; \url{doi:10.1080/14786442108636264})] as a generalization of the Euler--Bernoulli beam equation. This system has two branches of dispersion relation: one is physical and the other one is non-physical. \par The authors prove that there exists a relation between the non-physical spectrum and the exponential decay of a dissipative Timoshenko system where the damping mechanism acts on angle rotation. They show that the non-physical spectrum can be truncated by taking a damping mechanism. \par The authors also address the simplified Bresse-Timoshenko equation which is free of the non-physical spectrum. They prove that the dissipative version of this equation is exponentially stable for any values of parameters. They provide physical explanations why weakly dissipative systems decay exponentially in time.
scientific article; zbMATH DE number 6828465
scientific article