Application of variable- and distributed-order fractional operators to the dynamic analysis of nonlinear oscillators

Abstract

Many physical processes in nature exhibit complex dynamics that result from a combination of multiscale, nonlinear, non-local, and memory effects. Recent experimental measurements conducted in a variety of physical domains have shown that, at the macroscale level, these effects typically result in significant deviations from the behavior predicted by classical models. Notably, the underlying dynamics was often shown to be of non-integer order and possibly better captured by fractional-order models. Fractional operators are intrinsically multiscale; thus, they provide a natural approach to account for non-local and memory effects. In this study, we present the possible application of variable-order (VO) and distributed-order (DO) fractional operators to a few classes of nonlinear lumped parameter models that have great practical relevance in mechanics and dynamics. More specifically, we present a methodology to define VO and DO fractional operators that are capable of capturing various physical transitions characteristic of contact dynamics, nonlinear reversible systems, hysteretic systems, and nonlinear damped oscillator systems. Despite using simplified lumped parameters models to illustrate the application of VO and DO operators to mechanics, we show numerical evidence of their unique modeling capabilities as well as their connection to more complex systems at the continuum scale. Further, for a selected problem involving distributed nonlinear damping, we provide approximate analytical solutions that are helpful to better understand the underlying dynamics and to quantify the accuracy of our numerical models.