Damping efficiency of the fractional Duffing system and an assessment of its solution accuracy

Dynamical systems with high damping efficiency in a wide frequency band can be useful on a small scale - for harvesting energy from ambient vibrations, and on a large scale - for damping harmful vibrations of mechanical structures. In this paper we present an assessment of the quality of solutions and damping efficiency of systems with fractional order derivatives. To simulate the fractional system the fourth-order Runge-Kutta method and Gr & uuml;nwald-Letnikov methods are used. We propose a coefficient for assessing the quality of solutions to fractional systems by reference to the quality of the calculated energy balance. As an exemplary system we study the Duffing model with embedded additional fractional-order derivative terms. Based on this coefficient, intervals of key numerical simulation parameters are determined to ensure the appropriate quality for the calculations of energy flows and energy balance. The determined values of these parameters are then used in tests of the damping efficiency of the studied system. Our results show that by modifying the fractional terms it is possible to configure a system that exhibits a "broadband effect'', i.e. a system that is characterized by high-amplitude vibrations and, consequently, high energy efficiency in a wide range of excitation frequencies.