Mechanical energy and equivalent differential equations of motion for single-degree-of-freedom fractional oscillators

This paper addresses the total mechanical energy and equivalent differential equation of motion for single degree of freedom fractional oscillators. Based on the energy storage and dissipation properties of the Caputo fractional derivatives, the expression for total mechanical energy in the single degree of freedom fractional oscillators is firstly presented. The energy regeneration due to the external exciting force and the energy loss due to the fractional damping force during the vibratory motion are analyzed. Furthermore, based on the mean energy dissipation and storage in the fractional damping element in steady-state vibration, two new concepts, namely mean equivalent viscous damping and mean equivalent stiffness are suggested and the above coefficient values are evaluated. By this way, the fractional differential equations of motion for single-degree-of-freedom fractional oscillators are equivalently transformed into integer-order ordinary differential equations.