A modifiedbouc-wen model to describe the hysteresis of non-stationary processes

A number of known phenomenological models are considered, which are used to describe a variety of hysteresis effects in nature. In this case, the system is considered as a "black box" with known experimental values of input and output parameters. Correlations between them are established by mathematical functions, whose parameters are identified using experimental data. Amongthe phenomenological models there are marked the Bouc-Wen model andits analogsthat have been successfully usedin variousscientific and technical fieldsdue to the possibilityof the analytical description ofvarious hysteresis loopsof non-stationary processes. The conditions are formulatedwhich must be satisfied by the Bouc-Wen model. The main ones are the model adequacy of the physical process and stability. To describe the hysteresis, a mathematical model is suggested, according to which the force and kinematic parameters are bound by a special differential equation of the first order. In contrast to the Bouc-Wen model, the right side of this equation is chosen in the form of a polynomial of two variables determining the trajectory of a hysteresis in the process diagram. It is stated that this presentation provides the asymptotic approximation of the solution to the curves of the comprehending (including) hysteresis cycle.This cycle is formed by curves of direct and reverse processes ("loading-unloading" processes), which are based on experimental data for the maximum possible or permissible intervals of parameter changes during the steady vibrations. Coefficients in the right part are determined from experimental data for the comprehending hysteresis cycle under conditions of steady-state oscillations. Approximation curves of the comprehending cycle are constructed using the methods of minimizing the discrepancy of analytical representations to the number of experimental points. The proposed approach allows by one differential equation to describe the trajectory of hysteresis with a random starting point within the area of the comprehending cycle.