A general solution procedure for nonlinear single degree of freedom systems including fractional derivatives

Abstract

This paper considers oscillations of systems with a single-degree-of-freedom (SDOF) including fractional derivatives. The system is assumed to be an unforced condition. A general solution procedure that can be effectively applied to various types of fractionally damped models, where damping is defined by a fractional derivative, in engineering and physics is proposed. The nonlinearity of the mentioned models contains not only damping but can also consist of acceleration or displacement. This study proposed a new general model that includes but not limited to modified fractional versions of the well-known linear, quadratic, Coulomb and negative damped models. The method of multiple time scales is performed to obtain approximate analytical solutions. The solution, the amplitude, and the phase in the applications are plotted for various fractional derivative parameter values. In order to confirm their validity, our results for the case of the fractional derivative parameter equal to one are compared with others available in the literature.