The dynamic analysis of the continuum with the fractional derivatives

This thesis consists of seven chapters.In the first chapter, the historical development and the definition of the vibration motion are given. Furthermore, free and forced vibrations, damped and undamped oscillations, linear and nonlinear vibrations and deterministic and random vibrations are expressed. Additionally, modeling of the vibration motion, the definition of resonance and viscoelasticity are given. In accordance with the fractional derivatives, the derivatives in the sense of Riemann-Liouville and Caputo are defined. Finally, the perturbation method is also given.In the second chapter, a general model representing linear vibrations of the continuum and involving many applications is proposed and the governing equation is represented by the arbitrary linear operators. Additionally, the analytical solutions are obtained by using the method of multiple time scales.In the third chapter, the presented solution algorithm is applied to three different problems of the vibration. The first model is the problem of the harmonic axial loaded beam with viscoelastic material, the second is a elastic beam placed on the ground modelled by Kelvin-Voight viscoelastic model and the latest is also the problem of the axially forced viscoelastic beam. The displacement of the system is determined for the different beam parameters. The cases of parametric and primary resonances are examined.In the fourth chapter, it is investigated the convergence of the general solution obtained from the first case. Considering the three different cases of the fractional order, the convergence is examined.In the fifth chapter, the non-homogeneous general model involving many applications in the continuum has been considered and the analytical solutions are obtained by the method of multiple time scales. Additionally, the parametric and the forced vibrations are also analyzed.In the sixth chapter, the solution algorithm obtained for the non-homogeneous general model is applied to the three types of the different problems. The first model is the problem giving longitudinal vibration of harmonic external forced beam and the second is the problem giving longitudinal vibration of harmonic loaded beam. The model obtained as the third application is considered as the problem of the viscoelastic axial loaded beam placed on the ground.In the seventh chapter, conclusions and recommendations of this study are proposed.