Analytical and differential reformulations of the Vaiana–Rosati model for complex rate-independent mechanical hysteresis phenomena

Abstract

A recent analytical model of hysteresis, developed by the authors to simulate a great variety of complex uniaxial rate-independent mechanical hysteresis phenomena, is reformulated in a twofold manner. First, it is proposed an analytical reformulation that, compared to the current version of the model, does not require the evaluation of any internal variable. Second, the closed form expressions provided by the analytical reformulation are expressed in rate form to foster its use especially in the field of nonlinear dynamics. To verify their accuracy, both formulations are first employed to reproduce four different complex hysteresis phenomena and the related results are compared in terms of generalized force–displacement hysteresis loops, tangent stiffness functions, and work-displacement relations. Subsequently, nonlinear time history analyses are carried out on four single degree of freedom mechanical systems for three different types of external generalized forces. To this end, two solution strategies are adopted. The former combines the model analytical reformulation with an explicit structure-dependent time integration method used to directly integrate the second-order ordinary differential equation of motion. The latter employs the model differential reformulation in conjunction with the Runge–Kutta method implemented to solve the equivalent system of three first-order ordinary differential equations. The comparison of the numerical results obtained by the two solution procedures shows an excellent agreement.