Content area
Full Text
Received Mar 9, 2017; Accepted Dec 6, 2017
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.
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...