Modeling and simulation of magnetostrictive actuators based on ΔE effect Preisach model

Based on the classical Preisach model, combined with the magneto-mechanical coupling effect of Giant magnetostrictive materials (GMM), this paper introduces the Everett function to optimize the classical Preisach hysteresis model and improve its accuracy. By importing dynamic parameters, the dynamic Preisach hysteresis model is established. Combined with the constitutive model of GMM, a dynamic Preisach hysteretic nonlinear magnetostrictive actuator model based on the ΔE effect is established. Then, the finite element simulation is carried out by Comsol software and compared with the theoretical model simulation. The results show that the error between the magnetostrictive strain of the established hysteresis model and the finite element simulation results is less than 100 ppm, which verifies the accuracy of the proposed model. Finally, the curves of magnetic field intensity, magnetic induction intensity, magnetization, magnetostrictive strain and ΔE effect of GMA are obtained by using this model.


Introduction
In order to effectively control Giant Magnetostrictive Actuators (GMA) and promote their increased application in industrial settings, it is crucial to establish an accurate hysteresis model for electromagnetic-mechanical coupling.An ideal mathematical model should be capable of accurately predicting the relationship between input current and output displacement, providing essential insights for the design and enhancement of GMAs.However, the magnetostrictive effect and ΔE effect of GMMs further increase the nonlinearity and complexity of the model.Therefore, many researchers employ various methods to establish the nonlinear dynamic models of GMAs.
Kellogg and Flatau [1] established a dynamic model related to the longitudinal shape variable of Terfenol-D rods and the displacement transfer ratio.Chakrabarti and Dapino [2] used a free potential energy equipartition model to describe the ΔE effect of the Terfenol-D rod.Li et al. [3] studied the magnetization process and magnetostriction behavior of the Terfenol-D rod by combining the magnetoelastic effect with the Jiles-Atherton model.Fallah and Moghani [4] developed a new method for identifying and implementing an isotropic vector Preisach model.Zhu et al [5] presented an algorithm by a symmetric minor hysteresis loop; Manescu et al. [6] described a dynamic Preisach hysteresis model applicable in the case of thin non-oriented electrical steels; Yu et al. [7] proposed a new dynamic Preisach model based on the relationship between the Preisach function and the input change rate; Chen et al. [8] and Dang and Tan [9] proposed a neural Preisach model for modeling and control of piezoceramic actuator; Li et al. [10] used control technology to integrate with Preisach model to design a stable controller; Yu et al. [11][12] proposed a numerical implementation of a geometrically improved Preisach model; Li et al. [13] proposed a hysteresis model using probability.
But these modeling methods more or less have some problems, such as large calculation, low efficiency and low precision.In this paper, we propose a new method for identifying and implementing the isotropic vector Preisach model.The model takes full advantage of the large strain range, high energy density and significant mechanical thrust of giant magnetostrictive materials (GMM).The ΔE effect of the material, is fully considered.On this basis, combined with the magnetoelastic effect, this paper improves the Preisach hysteresis model and establishes the GMA nonlinear theoretical model based on the ΔE effect.

Classical Preisach model
The classical model utilizes a superposition of the certain number of elementary hysteresis operators with ideal rectangular hysteresis characteristics to characterize magnetic materials, as shown in Figure 1.The basic expression is as follows:

T B t h h h h H t dh dh
where B(t) is the magnetic induction value at time t; H(t) is the magnetic field intensity at time t; T is the integral domain; μ(h 1 , h 2 ) represents the Preisach distribution function;

Improved static Preisach model
In classical Preisach model equation calculations, there are complex double integral operations involving the distribution function.Therefore, some scholars have proposed an Everett function expression, defined as [14]: x h

T y y E x y h h dh dh h h dh dh
where the integral domain T is a right-angled triangle with its vertex at the current operating point A(x, y) and one of its sides along the line defined by the angle θ.In Figure 2, this triangle is known as triangle ABC.
where H m and B m represent the magnetic field intensity and magnetic flux density at a reversal point on the hysteresis loop, respectively.Assuming that the Preisach distribution function can be expressed as the product of two one-dimensional functions, you can write the distribution function as a sum of m terms: where ( ) i x ϕ is a one-dimensional function about the switching threshold.
In this paper, numerical simulations of the distribution function x b e e c c e a where a i represents the function amplitude; b i represents the function mean; c i represents the square of the function variance; α i , β i , and γ i are model parameters that satisfy the relationships given below the equation.The expression of the Preisach distribution function is: Substituting Equation ( 6) into Equation (1) yields the following Everett function expression: e e e E x y e e Compared with the mathematical model derived from other Everett functions, the above equation is a closed expression without integral form, and its advantage is that it only contains basic arithmetic operations, real numbers and logarithmic power operations, so that Everett functions can be calculated only through elementary function mathematics operations, improving the calculation speed of the model.The correct analytical expression of the Preisach hysteresis model can be obtained by combining the derived closed Everett expression (8) with the classical Preisach model expression (2), while considering the influence of reversible magnetization components on the magnetization process of magnetic materials.
where ( ) rev B t represents the magnetic flux density corresponding to the reversible magnetization component, and 1 2 3 k k k 、 、 represents the coefficient to be identified.In summary, the improved Preisach analytical model with universal applicability is as follows:

H t k H t k H H k
where m H indicates the magnetic field strength of the turning point.

Dynamic Preisach model
In the actual magnetization process of ferromagnetic materials, the relevant dynamic characteristics caused by eddy current and other factors should be considered.Therefore, this paper introduces the effective magnetic field intensity m H with 3 additional parameters m a , m b and m c to build a dynamic Preisach model: The derivative of the magnetization to the effective magnetic field strength is: After discretization of the above equation, the expression of the instantaneous effective magnetic field at i+1 can be obtained as follows: where The dynamic hysteresis model can realize the effective magnetic field strength of the model by replacing the magnetic field strength in static hysteresis.The calculation process is shown in Figure 3,

Nonlinear theoretical model based on ΔE effect
Combining the Preisach dynamic hysteresis model with the magnetoelastic effect, a nonlinear theoretical model of GMA based on the ΔE effect is established.In order to describe the relationship between the two more accurately, more higher-order terms can be retained in the Taylor series expansion of Gibbs free energy function, so as to obtain the nonlinear constitutive model of magnetostrictive materials [15]: where H is the foreign field, k represents the relaxation factor, s σ represents the saturation stress, s E is the saturation Young's modulus, s λ is the saturation strain, and s M is the saturation magnetization.In Equation 14, the slope of the strain-stress response of a material is the reciprocal of its elastic modulus.The ΔE effect curve can be calculated by Equation (14).

GMA finite element simulation analysis
In finite element analysis (FEA) of the GMM rod, the constant current density in the coil is set to 10 6 A/m 2 , the generated driving magnetic field strength is 60 kA/m, and the pre-pressure of the magnetostrictive rod is 0 MPa.The simulation results are shown in the figure below.Figure 4 shows the axial strain component of the GMM rod surface.In the parts outside the fixed end of the GMM rod, the strain caused by magnetostriction is obvious and uniform.Figure 5 shows the closed magnetic circuit generated by the coil excitation current.The magnetic conductive material of the shell makes the magnetic flux in the GMM rod dense.The magnetic flux density in the GMM rod is mostly uniform and relatively high, and most of the magnetic flux exists in the closed-loop composed of the shell and GMM rod.In Figure 6, the magnetostrictive curve of the GMM rod is obtained by simulating the parameterization of the quasi-static growth of the current density in the coil when the prepressure is 0 and 5 MPa.Since the direction of H is mainly axial, only the Z-axis component of the deformation vector is plotted.In Figure 6, the magnetostrictive curve has obvious nonlinear characteristics when the magnetic field intensity varies between 5 and 20 kA/m.When H is higher than 20 kA/m, the magnetostriction gradually tends to saturation.When the pre-pressure of 5 MPa is applied to the GMM rod, the magnetic field intensity between 0 and 10 kA/m shows better magnetostrictivity and better linearity, indicating that appropriate pre-pressure can improve the output and control performance of GMA.
Figure 7 shows the variation of E with H under the pre-pressure of 5 MPa, that is, the ΔE effect of GMM.In Figure 7, E decreases at a faster rate at the beginning as the magnetic field strength increases, and decreases to a certain extent, then increases at a slower rate.2) The hysteresis loop of GMM is calculated by H, B, M and λ, as shown in Figure 9, 10 and 11, respectively.
3) The Young's modulus changes greatly with the change of H and stress in Figure 12.
The results of the paper provide a reference of magnetostrictive models and the ΔE effect.
hysteresis operator and takes values of ±1.

Figure 2 .
Figure 2. The range of the Preisach distribution function.Therefore, the Preisach function model is: ( ( ), ( )) 0 and 0 ( ) 2 ( , ( )) 2 ( ( ), ) on the given magnetic hysteresis characteristics of the material.The results indicate that it follows an approximately exponential distribution.Three equivalent analytical expressions in the following equation are used to characterize the one-dimensional function(

H
+ expression; ε stands for floating point relative error limit.

Figure 4 .
Figure 4. Axial strain component and closed magnetic circuit distribution diagram.

Figure 8 (
a) shows the time-domain curve of B and its change frequency is consistent with that of H, which is 3 Hz.The obvious magnetic saturation phenomenon appears at the peak and valley values of the signal, and the amplitude is 1T.

Figure 8 (
b) shows the time-domain curve of H.In Figure8, the frequency of the excitation magnetic field signal is 3 Hz and the amplitude is 60 kA/m.

Figure 8 (
c) shows the timedomain curve of M, and its change frequency is also 3 Hz, which is consistent with H.A more obvious magnetic saturation phenomenon appears at the peak and valley values of the signal.

Figure 8 (
d) shows the time-domain curve of λ.It can be seen that magnetostrictive strain has an obvious frequency doubling effect, and its change frequency is 6 Hz, which is twice the change frequency of H. Obvious magnetostrictive saturation phenomenon occurs at the peak, which is consistent with the performance of B and M, and its amplitude is 1400 ppm.The saturated magnetostrictive strain is basically close to the finite element simulation results.The hysteresis loop of GMM is calculated by H, B, M and λ, as shown inFigures 9, 10 and 11, respectively.

Figure 9 .
Figure 9. Relation between H and B.

Figure 10 .
Figure 10.Relation between H and M.

Figure 12 (
Figure 12(a) shows the curve of E and H under the action of 5 MPa precompression stress.Figure 12(b) shows the curve of E and stress under the bias magnetic field of 40 kA/m.In the figure, E changes greatly with the change in H and stress.

Figure 12 (
Figure 12(a) shows the curve of E and H under the action of 5 MPa precompression stress.Figure 12(b) shows the curve of E and stress under the bias magnetic field of 40 kA/m.In the figure, E changes greatly with the change in H and stress.

Figure 12 .
Figure 12.ΔE effect varies with magnetic field intensity and stress.
Based on the classical Preisach model, the closed Everett function based on the magnetic-force coupling principle of GMM is introduced to optimize the static Preisach hysteresis model, and the calculation speed of the model is improved.Then, by introducing dynamic parameters, the corresponding dynamic Preisach hysteresis model is established.Combined with the constitutive model of GMM, the dynamic nonlinear Preisach hysteretic model based on ΔE effect is established.The finite element and theoretical model simulation were compared and analyzed, and the results were obtained: the error between the magnetostrictive output of the established hysteresis model and the finite element simulation results is about 100 ppm, which validates the accuracy of the established model.By the model, the H-M, H-B, H-λ and ΔE effect curves of the GMM rod are obtained: 1) When the external excitation magnetic field signal frequency is 3 Hz and amplitude is 60 kA/m, the time-domain curves of H, B, M and λ obtained by simulation are shown in Figure 8.