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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.

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 [

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 [

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 [

Later Krasnosel’skii and Pokrovskii and their followers proposed a rigorous mathematical algorithm [

In 1967, Bouc suggested a solution to the problem of induced vibrations in a mechanical system whose restoring force hysteresis was displacement-dependent [

The differential Bouc-Wen “black box” model [

Let us consider a hysteretic system that transforms a time-dependent input signal

Equation (

Using (

Hysteretic trajectories within an enveloping cycle.

The identification method is useful for determining model parameters. The error (divergence) between the experimentally acquired output values and those calculated by model algorithms remain relatively small. Calculations are carried out for the reference input signal, after which hysteresis is modelled for other input signals.

There are known cases when the calculated Bouc-Wen model parameters do not agree with the experimental results for other input signals. This demonstrates the ambiguity of identification, which can lead to model instability, in relation to the input signal.

There are established conditions, to which the Bouc-Wen model must adhere [

In the present paper, we establish the system parameters describing a hysteresis process given by (

We use computational experiments to show that the proposed model features asymptotical stability and allows description of complex hysteretic trajectories that are adequate to a real process.

It is well known that dependencies between force and kinematic parameters in a hysteretic system are of cyclic nature. The deformation diagrams show that the path of each cycle is a loop shape formed by two curves (branches), which correspond to the increase and decrease of an external action (load) parameter with time

Experiments show that, in a broad range of hysteresis systems, single type branches of local cycles tend asymptotically to the relative curves of the envelope cycle during monotone process parameter change.

The physical model used to describe such hysteresis by the differential approach (

It is assumed that any physically possible hysteresis trajectory is bound within the space of the envelope cycle, which is experimentally constructed for the maximum range of process parameter changes

The whole hysteresis process is considered to be independent of frequency. The envelope cycle curves can be thus obtained from quasi-static experiments.

It is also assumed that all local forward process curves are asymptotically related to the

The suggested approach is using a standard first-order differential equation (

A standard first-order differential equation is used to model the two hysteresis trajectory branches, with the rhs being a polynomial of two variables

Equation (

The direct and reverse processes are described by (

Because of (

In most cases, the forward and reverse process curves are identical, differing only in the direction of

Cycle branches, symmetrical about the origin.

The reverse process can thus be described by the forward process equation

With variables

In this case, the equations of direct and reverse processes can be united and written as

As stated above,

Let us assume that experimental data gave us a sequence of

Instead, we use a reverse method, analytically approximating the hysteretic branch’s derivative, by use of a polynomial

Minimizing the square functional

Thus, from (

Let

In accordance with the ordinary least squares method, we construct a discrepancy function as quadratic functional

Minimization of (

Equation (

From (

One practical example of natural hysteresis is the galloping of the power lines, which is a major concern for both engineers and utilities [

To counteract the potentially dangerous phenomenon, a range of devices are used. Let us examine a pendulum Torsional Damper and Detuner (TDD) which dampens and mismatches vertical and torsional oscillations with energy dissipation hysteresis.

Figure

TDD designs: (a) two-phase power line (Lilien J.-L., Keutgen R., 1998) and (b) three-phase power line (Lilien J.-L., Vinogradov A. A., 2002).

TDD for a three-phase line: (a) main elements and (b) kinematic diagram.

The main elements, which make up the TDD, are the damping unit 1; curvilinear pendulum rods 4 with weights 2 and counterweights 3; and upper rods 5 to suspend the dampener from the wires, by anchor type fittings 6.

The damping unit (shown disassembled in Figure

Damping unit.

In spite of the device’s simple design, the energy dissipation mechanism is rather complex as it consists of several simultaneously operating independent processes. These include nonlinear elastomeric deformations, friction, and sliding displacement which generate heat. As said above, it is possible to avoid this laborious task of describing TDD’s hysteresis based on the fundamental physical processes involved by applying the phenomenological method, which we will do in the following chapter.

The experimental research of the TDD prototype was carried out at JSC “Elektrosetstroyproyekt” [

Experimentally, we have obtained a set of trajectories where the torsional moment

Experiments showed the following:

A selection of hysteretic dependencies, obtained from quasi-static TDD tests, is shown in Figure

An example of hysteretic dependencies of

From the experiments, we obtained data for the envelope cycle curves

It should be noted that the upper

A principle TDD schematic is shown in Figure

Interaction torques between damper elements.

The generated torque

From the above considerations, the vibration equations of the coupled discs can be written as

To model hysteresis and evaluate TDD efficiency in energy dissipation, it is necessary to define the motion of the driving disk. It is accepted that

We can now rewrite (

Damper efficiency can be evaluated in terms of energy dissipation power.

Differentiating (

The solution of (

Below are the TDD energy dissipation analysis results for two design concepts shown in Figure

Pendulum type dampers: I: inertial and G: gravitational.

For the I-damper, the differences between axis of rotation and weighs’ center of mass are

Figures

Hysteresis dependence and time dependence at

Hysteresis dependence and time dependence at

Hysteresis dependence and time dependence at

Hysteresis dependence and time dependence at

Dependence of power dissipation on time at

(a) The power dissipation on driving frequency under input amplitude and (b) transfer function.

Figures

Figure

Power calculations of

The results are illustrated in Figure

Figure

Two-pendulum damper diagram.

This damper was also built and tested at JSC “Elektrosetstroyproyekt.” Its theoretical analysis was based on the method outlined above. Also the free oscillations of the driven disc and pendulums, with the driving disc immobilized, were studied.

Figure

Free oscillations of a two-pendulum damper. (a) Rotation angle regarding time and (b) resistance moment to rotation angle.

The paper develops a kinematic approach to the description of hysteresis in a range of nonstationary processes. The approach is related to the famous Bouc-Wen method, where a mathematical model is based on a standard first-order differential equation. However, unlike the Bouc-Wen, the rhs is depicted as a two-variable polynomial, the time-dependent hysteresis parameter and a function of this parameter. These variables can be translation (angular) and corresponding force (torque) to real-life scenario. The polynomial coefficients are found by analytical approximation of the envelope cycle curves during stabilized oscillations.

The suggested model of physical dependencies is analytical, which makes it suitable for description of nonlinear mechanical systems.

The approach can be used to solve problems related to nonstationary oscillations in mechanisms with hysteretic type of energy dissipation. This is demonstrated on the example of a pendulum type low-vibration damper. An algorithm to analyze its effectiveness in counteracting wire galloping was also outlined.

The authors declare that they have no conflicts of interest.

This work was carried out with financial support from the Russian Ministry of Education and Science (Federal Programme no. 14.604.21.0188).