A freely damped oscillating fractional dynamic system modeled by fractional Euler–Lagrange equations

Abstract

The behaviors of some vibrating dynamic systems cannot be modeled precisely by means of integer representation models. Fractional representation looks like it is more accurate to model such systems. In this study, the fractional Euler–Lagrange equations model is introduced to model a fractional damped oscillating system. In this model, the fractional inertia force and the fractional damping force are proportional to the fractional derivative of the displacement. The fractional derivative orders in both forces are considered to be variable fractional orders. A numerical approximation technique is utilized to obtain the system responses. The discretization of the Coimbra fractional derivative and the finite difference technique are used to accomplish this approximation. The response of the system is verified by a comparison to a classical integer representation and is obtained based on different values of system parameters.