Steady-state dynamic response of various hysteretic systems endowed with fractional derivative elements

In this paper, the steady-state dynamic response of hysteretic oscillators comprising fractional derivative elements and subjected to harmonic excitation is examined. Notably, this problem may arise in several circumstances, as for instance, when structures which inherently exhibit hysteretic behavior are supplemented with dampers or isolators often modeled by employing fractional terms. The amplitude of the steady-state response is determined analytically by using an equivalent linearization approach. The procedure yields an equivalent linear system with stiffness and damping coefficients which are related to the amplitude of the response, but also, to the order of the fractional derivative. Various models of hysteresis, well established in the literature, are considered. Specifically, applications to oscillators with bilinear, Bouc-Wen, and Preisach hysteretic models are reported, and related parameter studies are presented. The derived results are juxtaposed with pertinent numerical data obtained by integrating the original nonlinear fractional order equation of motion. The analytical and the numerical results are found in good agreement, establishing, thus, the accuracy and efficiency of the considered approach.