Energy Considerations for Mechanical Fractional-Order Elements

This paper considers the energy aspects of fractional-order elements defined by the equation: force is proportional to the fractional-order derivative of displacement, with order varying from zero to two. In contrast to the typically conservative assumption of classical physics that leads to the potential and kinetic energy expressions, a number of important nonconservative differences are exposed. Firstly, the considerations must be time-based rather than displacement or momentum based variables. Time based equations for energy behavior of fractional elements are presented and example applications are considered. The effect of fractional order on the energy input and energy return of these systems is shown. Importantly, it is shown that the history, or initialization, has a significant effect on energy response. Finally, compact expressions for the work or energy, are developed.