Approximate solutions and stability analysis of single and two degree of freedom systems with fractional damping

In this thesis, mathematical models, including damping terms, which are critical parameters in dynamic analyses of structures, are considered. Dynamic systems have been analyzed separately, considering both single-degree-of-freedom and two-degree-of-freedom systems, and their vibration behavior has been investigated. The models mainly focus on fractional-order damping to obtain more accurate results. Using fractional derivatives allows for a broader range of phenomena to be modeled compared to traditional integer-order derivatives, providing the ability to predict the dynamic responses of structures more precisely. In the thesis, the mathematical solutions of these damping models are obtained, and stability analyses are performed based on these solutions. Stability analyses reveal the system's behavior under various parameters and possible instabilities, and these analyses are critical for the safe design of structures. As a solution method, the method of multiple scales, which is a powerful tool for nonlinear systems and models containing fractional derivatives and a perturbation technique that can be used in cases where analytical solutions are difficult, is preferred. With the advantages provided by this method, the effects of both nonlinear effects and fractional damping terms on the behavior of the dynamic system are analyzed. As an application, the mathematical model of the two-degree-of-freedom system is examined. Free vibration and primary resonance general solutions and steady-state solutions are obtained. The regions where the steady-state solutions coincide with the general solutions are presented with graphics. The effects of 𝛼 parameter on the solutions are demonstrated with time-displacement and amplitude-time graphs.