Mathematical Problems in Engineering

Volume 2018 (2018), Article ID 7102796, 15 pages

https://doi.org/10.1155/2018/7102796

## Hysteresis Modelling of Mechanical Systems at Nonstationary Vibrations

^{1}Institute of Applied Mechanics, RAS, Moscow, Russia^{2}Moscow 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.

#### Abstract

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.

#### 1. Introduction

The hysteresis features of a given process (shape of the loop trajectories, asymptotical resemblance, symmetry of forward and reverse processes, etc.) are determined by physical nature [1–11] of the process. Often the dynamics of complex mechanical systems is not straightforward and not easy to model and may depend on a large number of interactions.

Alternative approach is to view the system as a “black box,” with known input and output parameters. Then the interactions between them can be established phenomenologically and their values can be identified experimentally [12–14]. Phenomenological approach is effective in creating a general mathematical model, which can be “migrated” from one field to another. Not only does this show the “depth” of mathematics per se, but more specifically, the overall similarity of various hysteretic processes in nature.

Among widespread phenomenological models that are used to describe hysteresis in various fields of science and technology, the models based on the spectral factorization by relay nonlinearities are particularly important. Originally such approach was proposed in 1935 by German physicist Preisach [15–17], who treated the magnetization process as statistical result of alternating magnetization of individual elementary regions, domains. It is believed that a domain can be in a state of saturation with the direction of magnetization along or against the external field. Accordingly, the magnetization of each domain is described by the switch functions determining the rectangular hysteresis loop. An important component in the model is the particle distribution function, which determines values of magnetization in an arbitrary field.

Later Krasnosel’skii and Pokrovskii and their followers proposed a rigorous mathematical algorithm [18] based on Preisach’s ideas. Similar phenomenological concepts were drawn up and are being developed in different fields of physics and mechanics [19–22]. A common drawback of these models is their parameter identification which requires complex experiments and data analysis.

In 1967, Bouc suggested a solution to the problem of induced vibrations in a mechanical system whose restoring force hysteresis was displacement-dependent [23]. Hysteretic trajectory is described by standard nonlinear first-order differential equation, whose coefficients are identified experimentally. In 1971, he proposed a model for an abstract physical system which he treated as a “black box” with known input and output parameters [24]. In 1976, the model was generalized by Wen [25, 26] and it is known as the Bouc-Wen model since that.

The differential Bouc-Wen “black box” model [12] became very popular due to the fact that it allows analytical description of various hysteretic loops of a damped system [27]. The model has been successfully applied to piezoelectric elements [28], magnetorheological shock absorbers [29], wooden junctions [30], isolation of structure foundations [31], and many more cases.

Let us consider a hysteretic system that transforms a time-dependent input signal into an output signal . In accordance with the Bouc-Wen model, a first-order nonlinear differential equation with switch functions is used to depict . The general form that links input and output signals is [32]where is the selected piecewise-smooth function, which is experimentally identified by a reference signal*.*

Equation (1) is a part of a general system of dynamic equations, where and are the unknown functions, and it can be used for description of a mechanical system with hysteretic energy dissipation. Let us consider a one-dimensional oscillator as an illustration. In this case, the dynamic equations can be written as [33, 34]The sum is damping force , where is viscous damping coefficient, is stiffness coefficient, is variable describing the hysteretic trajectory, and is constant components of . The values of , and are all identified experimentally. Various modifications of (2) are known, which allow the description of many physical hysteretic processes with relative accuracy [12, 27–37].

Using (1) and (2), it becomes possible to build a dependency that determines the piecewise-smooth continuous hysteretic trajectory. The value of *, *when , forms the continuity of points , where is the sequence number that grows with time* t* from the starting point. As it crosses these points, the derivative successively changes sign and the hysteresis branches shift, as it is depicted in Figure 1. Bold arrows in the 2nd and 4th quadrants represent the forward and reverse hysteresis directions, corresponding to the rise and fall of , when, respectively, or .