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Mathematical Problems in Engineering
Volume 2018 (2018), Article ID 7102796, 15 pages
Research Article

Hysteresis Modelling of Mechanical Systems at Nonstationary Vibrations

1Institute of Applied Mechanics, RAS, Moscow, Russia
2Moscow Aviation Institute, Moscow, Russia

Correspondence should be addressed to A. D. Shalashilin

Received 9 March 2017; Accepted 6 December 2017; Published 12 March 2018

Academic Editor: Salvatore Caddemi

Copyright © 2018 A. N. Danilin and A. D. Shalashilin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper considers and reviews a number of known phenomenological models, used to describe hysteretic effects of various natures. Such models consider hysteresis system as a “black box” with experimentally known input and output, related via formal mathematical dependence to parameters obtained from the best fit to experimental data. In particular, we focus on the broadly used Bouc-Wen and similar phenomenological models. The current paper shows the conditions which the Bouc-Wen model must meet. An alternative mathematical model is suggested where the force and kinematic parameters are related by a first-order differential equation. In contrast to the Bouc-Wen model, the right hand side is a polynomial with two variables representing hysteresis trajectories in the process diagram. This approach ensures correct asymptotic approximation of the solution to the enclosing hysteresis cycle curves. The coefficients in the right side are also determined experimentally from the hysteresis cycle data during stable oscillations. The proposed approach allows us to describe hysteretic trajectory with an arbitrary starting point within the enclosed cycle using only one differential equation. The model is applied to the description of forced vibrations of a low-frequency pendulum damper.