Fractional mechanical oscillators

Abstract

In this contribution we propose a new fractional differential equation to describe the mechanical oscillations of a simple system. In particular, we analyze the systems mass-spring and spring-damper. The order of the derivatives is $0< \gamma \leq 1$. In order to be consistent with the physical equation a new parameter $\sigma$ is introduced. This parameter characterizes the existence of fractional structures in the system. A relation between the fractional order time derivative $\gamma$ and the new parameter $\sigma$ is found. Due to this relation the solutions of the corresponding fractional differential equations are given in terms of the Mittag-Leffler function depending only on the parameter $\gamma$. The classical cases are recovered by taking the limit when $\gamma=1$.