Research of Jiles-Atherton Dynamic Model in Giant Magnetostrictive Actuator

Due to the existence ofmulticoupled nonlinear factors in the giantmagnetostrictive actuator (GMA), building precisemathematical model is highly important to study GMA’s characteristics and control strategies. Minor hysteresis loops near the bias magnetic field would be often applied because of its relatively good linearity. Load, friction, and disc spring stiffness seriously affect the output characteristics of the GMA in high frequency. Therefore, the current-displacement dynamic minor loops mathematical model coupling of electric-magnetic-machine is established according to Jiles-Atherton (J-A) dynamic model of hysteresis material, GMA structural dynamic equation, Ampere loop circuit law, and nonlinear piezomagnetic equation and demonstrates its correctness and effectiveness in the experiments. Finally, some laws are achieved between key structural parameters and output characteristics of GMA, which provides important theoretical foundation for structural design.


Introduction
Giant magnetostrictive materials (GMM) are widely used in transducer, precision actuator, and active vibration [1][2][3] because of large magnetostrictive coefficient, excellent dynamic characteristic, and high energy density.When they are applied in high frequency, establishing accurate mathematical model is crucial to improving controllability and stability [4,5].J-A model is established based on magnetization principle by Jiles and Atherton, which has a wide application and fast development because of clear physical conception, high stability, and accuracy [6][7][8][9].In recent years, many experts established the mathematical models of hysteresis materials including frequency, pressure, temperature, and other sensitive factors based on the J-A model that had developed a relatively mature modeling theory [10][11][12][13].However, these modes of GMA are mainly studied from material and ignore the structural factors such as load, disc spring stiffness, and friction which have a strong impact on output characteristics especially in high frequency.Therefore, a new dynamic model including load, disc spring stiffness, and friction is established based on combination of minor loops J-A model of GMM, structural dynamic equation of GMA, Ampere circuit law, and nonlinear piezomagnetic equation through analyzing the working principle of GMA.Then, the correctness and validity of this model are proved by experiments in different conditions.Finally, some output characteristics laws are revealed by analyzing key structural parameters.

Working Principle of GMA
The GMM rod drives load under the action of the magnetic field generated by the excitation coil and permanent magnet (Figure 1).The magnetostrictive coefficient of GMM is raised by prepressure which is generated by compressing disc spring.Bias magnetic fields are produced by permanent magnets.On the one hand, the double frequency characteristic of GMM could be removed; on the other hand, the high linearity part of hysteresis loop near the bias magnetic fields is applied to reduce the nonlinearity.The temperature control system provides relatively constant temperature environment for the GMM rod.

Jiles-Atherton Dynamic Model
The J-A model is developed taking into account the phenomenology of domain wall translation through the socalled pinning center, which might be nonmagnetic inclusions and impurities, grain boundaries, voids, and so forth, causing residual stress that hinders the magnetization process [14].The motion of magnetic domains can be divided into reversible or irreversible movement and rotation.Thus, according to model developers, total magnetization  can be divided into reversible and irreversible terms [6].Hence, where  rev is reversible magnetization and  irr is irreversible magnetization.Consider where  is reversible loss coefficient.
The modified Langevin equation is used to fit the isotropic material and anhysteretic magnetization  an is obtained in where   is saturation magnetization,  is shape parameter of the anhysteretic magnetization,  is internal coupling field coefficient among different magnetic domains, and  is magnetic strength.Equation ( 4), which takes into account dynamic effects due to eddy currents in the material, is derived by introducing additional energy term in the basic J-A equation [15].Hence, where  1 is eddy-current loss factor,  2 is anomalous loss factor,  is irreversible loss coefficient,   is effective magnetic field strength, and  is direction coefficient, when d/d > 0,  = 1 and when d/d < 0,  = −1.
Stress can change certain features of some magnetic domains to influence   .When the stress is , the effective magnetic field   becomes formula (5) and parameter  is modified to be α [16,17].Hence, where   is saturation magnetostrictive coefficient.
According to parameters characteristics of J-A model and the difference between simulation and experiment, the minor loop J-A model can be got through modifying the parameters , , , and  [18].Hence, where   ,   ,   , and   are, respectively, modified coefficient of , , , and ;   is the maximum magnetostriction coefficient of minor loop.
We can get dynamic minor loop J-A model of GMM according to formulas (1)-( 7).

Structural Dynamics Model
GMA can be considered as a single degree of freedom massspring-damping system.GMM rod drives the load under the action of magnetic field.In this section, the dynamic mathematical model of GMA is established considering structural characteristics such as load, disc spring, and friction and the following assumptions should be accepted: (1) GMM rod should be in uniform magnetic field.
(2) The working temperature is 293 K and remains constant all the time.(3) The output displacement is 0 when bias magnetic field is 22.5 kA/m, and prepressure is 6 Mpa.(4) The output displacement of one end is 0, while the other end keeps the same output characteristics with load such as displacement, velocity, and acceleration.(5) The output displacement  of GMM rod is equal to  and the output force  is equal to −.
Based on the above assumptions, the equivalent mechanical mode can be obtained according to the working principle of GMA (Figure 2).According to Newton's second law, the output force  of GMA is where   is equivalent mass,   is the impedance factor of GMA,   is friction,   is the pressure of disc spring,   is the quality of GMM bar that is equal to 1/3 of itself according to kinetic energy Lagrange function [19], and   is mass of the load.

4.1.
The Output Force of GMM Rod.When the magnetic field intensity is relatively large, the magnetization process in the material will produce serious hysteresis nonlinearity.The quadratic magnetic domain rotation model is introduced into the linear piezomagnetic equation [20].Hence, where  1 is the magnetic elastic coefficient of GMM rod and  is the elasticity modulus of GMM rod.

Pressure of Disc Spring.
The pressure generated by a single disc spring [21] is given by where  0 is the prestress,  0 is the predisplacement in the prestress,  is shape factor of disc spring, and ℎ 0 is its maximum deformation.The geometric parameters are shown in Figure 3.

Friction.
According to principle and characteristics of friction and working environment of GMA, Coulomb + Viscosity friction model (Figure 4) is selected [22].
Friction model is shown in where  max is the maximum static friction,   is the Coulomb friction coefficient, and V is velocity of the GMM rod.
4.4.Ampere Circuital Theorem.According to Ampere circuital theorem considering leakage in magnetic circuit [23], the magnetic field strength  is where  bias is the bias magnetic field strength and  coil is exciting coefficient of the coil.The current-displacement mathematical model of GMA could be obtained by connecting formulas ( 1)-( 7) with ( 9)-( 12) and its parameters (Table 1) are identified by adopting modified simulated annealing differential evolution algorithm which has a high convergence speed and accuracy.

Test Verification
GMA testbed is mainly composed of GMA, laser displacement sensor, and temperature control system (Figure 5).The measurement and control system applies RTX as lower computer system software and LabWindows as upper computer system software and sampling period is 0.5 ms.The temperature sensor is installed near GMM rod (Figure 1) and temperature control system can pass heat through circulating water which can be controlled within 1 K by heater and cooler.The V100-MS laser displacement sensor can measure the  displacement, velocity, and acceleration and its displacement measurement accuracy is up to 5 m/V.The ISF20DA250 servo driver can provide exciting current with feedback control and its frequency response is up to 15 kHz (see Figure 6).The simulation and experiment curves in different conditions are shown in Figure 7, which prove the validity and accuracy of this mathematical model.

Studies on Key Parameters of the Model
In this section, the output characteristics laws of GMA are revealed by studying structural parameters such as disc spring stiffness and friction.

Studies on Characteristics of Disc Spring.
The disc spring, a key part that provides prepressure, has a serious impact on output characteristics of GMA because of its nonlinearity.This paper studies three series of disc springs (A, B, and C) whose inner and outer diameter are the same and their structural parameters and mechanical property are shown in Table 2 and Figure 8. Figure 9 shows the output displacement of GMA under 20 Hz.After analyzing Figures 8 and 9, we can conclude that the larger stiffness of disc spring can reduce magnetostriction coefficient.

Studies on Friction.
This section mainly studies Coulomb friction coefficient   and the maximum static friction  max in friction model.When   takes 10 Nm/s and 100 Nm/s, respectively, the output displacement of GMA under 20 Hz is shown in Figure 10.The larger the   , the more serious the hysteresis energy loss.As a result, the Coulomb friction coefficient should be as small as possible.When  max takes 5 N and 100 N, the output displacement of GMA in 20 Hz is shown in Figure 11.With the increasing of  max , serious low speed crawling and buffeting will appear in the end of hysteresis loop.Therefore,  max should be as small as possible.

Conclusion
(1) Structural parameters have a strong impact on output characteristics of GMA in high frequency.
(2) The dynamic mathematical model of GMA including structural element such as load, disc spring, and friction has a great adaptability and accuracy.
(3) The stiffness of disc spring influences the output characteristics of GMA.

Figure 3 :
Figure 3: Structural parameters of disc spring.

Table 2 :
Structural parameters of disc spring.