Magnetic Circuit Optimization and Physical Modeling of Giant Magnetostrictive Actuator

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Introduction
With the advancement of active control on vibration, giant magnetostrictive actuator (GMA), which has the advantages of fast response, high reliability, and high precision, is applied in the high-precision active vibration isolation platform [1][2][3][4][5][6]. In order to design, evaluate, and control the performance of the GMA, it is necessary to establish its accurate mathematical model [3,[7][8][9][10].
Te earliest model of the GMA is a linear piezomagnetic equation without considering the efects of stress, temperature, and hysteresis [11]. Te linear piezomagnetic coefcient is used to measure the magnetic mechanical coupling characteristics of giant magnetostrictive materials. Experiments and applications show that it is feasible to use this model to describe the output performance of the GMA if a certain preload and bias magnetic feld are applied to a GMM rod under low frequency and weak magnetic drive [4]. However, because its core component GMM is the ferromagnetic material, the intrinsic hysteresis nonlinearity exists in its magnetization process, and when the driving frequency is slightly higher, it is also afected by the eddy current loss and other electromagnetic losses. Terefore, when the driving magnetic feld intensity and frequency are slightly higher, the accuracy of the model is not high. So, it is necessary to analyze the performance of the GMA based on the hysteresis nonlinear dynamic model considering the infuence of the eddy current.
At present, the models used to describe the hysteresis nonlinearity of a magnetostrictive actuator are mainly divided into three categories: Te frst category is the phenomenological model, which is a purely mathematical model without any physical mechanism, such as the Preisach hysteresis model [11]. Te second category is the physical model based on physical mechanisms, such as the Jiles-Atherton model based on the magnetization mechanism and the free energy hysteresis model based on the thermodynamic theory [12][13]. Te third category is the data-driven model based on the experimental data and the artifcial intelligence algorithm, such as the neural network model [7]. Te Preisach hysteresis model is a mathematical model without any physical mechanism, which has a strong prediction ability for the hysteresis nonlinearity. Clark et al. applied it to describe the hysteresis of the GMA, but this model can only describe the hysteresis characteristics under a quasistatic driving magnetic feld. With the increase in the driving frequency, the error of this model becomes large [11]. In order to study and control the dynamic nonlinearity of the GMA, Tan and Baras developed a dynamic Preisach hysteresis model based on the phenomenological theory of the electrocircuit and gave the dynamic compensation control method of the GMA based on the inverse model of this model [14]. Te results show that compared with the static Preisach hysteresis model, this model has higher tracking control performance, and its application range can reach 200 Hz. In order to control the GMA in real time, Yang used the Prandtl-ishlinskii model, which is an upgraded version of the Preisach model, for modelling and compensation control for the hysteresis of the GMA. Trough the experiment, the maximum prediction error of this model is 0.146 μm. Te accuracy of this compensation control can reach 0.309 μm [15]. Te Prandtl-ishlinskii model has the advantages of small computation, simple structure, and convenient inversion, so it especially meets the requirements of real-time control of the GMA.
Te Jiles-Atherton model is a hysteresis model based on the Weiss molecular feld theory and the domain wall motion mechanism. It has relatively few structural parameters and a clear magnetization physical mechanism. Sablik and Jiles developed this model to analyze the magnetic mechanical coupling efect caused by stress and the coupling efect of magnetization and magnetostriction. Later, Jiles added the eddy current loss and other loss terms to the Jiles-Atherton hysteresis model to describe the magnetization of ferromagnet driven by a medium or a low frequency alternating magnetic feld [11,[16][17][18][19][20][21][22]. Wang applied this modifed model to describe the performance of the GMA, established a GMA dynamic model considering the eddy current efect and the stress' change, and identifed the model parameters by a hybrid genetic algorithm [5]. Calkins et al. improved the Jiles-Atherton hysteresis model from the perspective of energy so that it can describe the hysteresis characteristics of the main loop and the symmetrical small loop of the GMA under diferent conditions. Chakrabarti and Dapino derived the relationship between the radial magnetic feld distribution of the GMM rod and the driving frequency from Maxwell equations, by substituting the average radial magnetic feld intensity of the GMM rod into the Jiles-Atherton hysteresis model. Te GMA dynamic hysteresis model considering the infuence of the driving frequency is given. Te application range of this model can reach 2 kHz [3].
Smith studied hysteresis from the perspective of thermodynamics and proposed the free energy hysteresis model. Because it is similar to the mathematical form of the Preisach model, it is also called the physical Preisach model.
Chakrabarti and Dapino applied the physical Preisach model to model the hysteresis of the GMA, which presented a good performance [3]. On the basis of this model and Tan Xiaobo's phenomenological theory of the electrocircuit, Tian et al. established a dynamic free energy hysteresis model, whose application range was 300 Hz [13]. Zheng et al. studied the coupling efects such as the preload, the temperature, and the alternating magnetic feld on the dynamic characteristics of the GMA. Based on the Gibbs free energy theory, they established a new model to describe magneticmechanical coupling characteristics of the GMM rod driven by a strong magnetic feld [16].
Because the neural network can ft any complex nonlinearity with high accuracy, it was used to accurately describe and control the dynamic nonlinear of the GMA. Cao et al. proposed a GMA control strategy based on the dynamic recurrent neural network model [7]. In addition, the RBF neural network and the Fuzzy-RBF neural network were also used to model and control the dynamic characteristics of the GMA by the online hysteresis compensation control method. Li et al. applied the small quadratic support vector machine theory to model the hysteresis of the GMA. Tis maximum relative error of the model was about 2.5%. Compared with the traditional neural network model, this model requires less experimental data [19]. Although the neural network hysteresis model has many advantages, it is only a nonlinear ftting of the experimental data and cannot be used to design and optimize the performance of the GMA.
To sum up, the frst and the third types of models cannot refect the physical mechanism of the GMA and the infuence of the physical parameters on the performance of the GMA cannot be studied through the model. Te second type of model is the implicit equation groups in diferential form. Te formulas are complex and need to be solved by programming, which is difcult for engineering practical applications. Terefore, an accurate and simple modeling method is very necessary for the engineering application of the GMA [20][21][22][23][24][25][26]. With the development of computer software, the physical modeling method based on simulation software is more and more welcomed by the engineers and technicians. Te physical modeling method only needs to connect and assemble the basic physical components to the module block diagram according to the schematic diagram, so as to establish a complex multidisciplinary system model and carry out simulation calculation and in-depth analysis on this basis, so as to avoid cumbersome mathematical modeling. In this work, the dynamic hysteresis nonlinear physical model of the GMA is given based on the Simulink Physical Modeling Toolbox (Simcape), and its performance is analyzed. In addition, the magnetic circuit physical model of the GMA is also developed based on Simcape, which includes all magnetic resistances in the magnetic circuit.
Te contents are organized as follows: Section 2 introduces the confguration and the working principle of the GMA, while Section 3 presents a physical model and optimization of the magnetic circuit. Section 4 shows the magnetomechanical model. Section 5 shows the kinetic models of the GMA. Section 6 discusses simulation and the experiment.

Configuration and Working Principle of the GMA
Te confguration of the GMA is shown in Figure 1. It mainly consists of an annular permanent magnet, a giant magnetostrictive material (GMM) rod, a coil, a spring, and an output rod. Te annular permanent magnet is used to provide a longitudinal bias magnetic feld to improve the sensitivity of the GMM rod and eliminate frequency doubling. In order to save cost, the annular permanent magnets are stacked with standard magnet rings, as shown in Figure 2. Te coil is used to generate the control magnetic feld to drive the GMA output displacement. Te spring is used to apply a prestress on GMM because a larger magnetostrictive strain can be obtained with a same magnetic feld when GMM is compressed. Figure 3 illustrates the prototype of the GMA. Te conversion from the current to the displacement of the GMA involves three energy conversion processes, namely, the conversion from electric energy to magnetization energy, the conversion from magnetization energy to strain energy, and the conversion from strain energy to mechanical energy. Terefore, the GMA's model includes the magnetization submodel describing the conversion from electric energy to magnetization energy, the magnetostriction submodel from magnetization energy to strain energy, and the mechanical submodel from strain energy to mechanical energy. In addition, when the driving frequency is high, the efect of the eddy current on magnetization should be included in the magnetization model.

Optimization of the Magnetic Circuit.
GMM is the core component of the GMA, which is driven by a magnetic feld. Both the magnetic feld uniformity and the magnetic energy utilization on the GMM rod seriously afect the performance of the GMA [27]. Under the same ampere-turns, the more uniform the magnetic feld is, the stronger the magnetic feld intensity is, the better the performance of GMM is, and the higher the efciency is. Assuming the magnetic fux density of the annular permanent magnet is 0.2 T, the length of the GMM rod is 80 mm, the diameter is 10 mm, the number of coil turns is 1200, the control current is 1 A, the coil's length is 70 mm, 80 mm, and 90 mm, respectively, and the inner diameter of the coil is 12 mm. Te linear "Hard" magnetic material is used for permanent magnets. Te nonlinear "Soft" magnetic material model is used for GMM. Te linear "Soft" magnetic material is used for other magnetic resistances in the magnetic circuit.
Based on the three-dimensional structure of the GMA, through the magnetic feld fnite element analysis with ANSYS/Maxwell, the three-dimensional electromagnetic feld distribution in the GMA can be obtained, as shown in Figure 4. Figure 4 shows that under the same number of ampereturns, the internal magnetic feld strength of the GMM rod changes with the coil's length. In order to clearly study the magnetic feld distribution in the GMM rod, only the magnetic feld on the GMM rod is displayed. Figure 5 is the magnetic feld distribution of the GMM rod in Figure 4(a). Figure 5 shows that the magnetic feld distributions in the GMM rod are uneven. Te magnetic feld strength is extracted on the GMM axis in Figure 4, and the distribution of the magnetic feld strength on the axis of the GMM rod under diferent lengths of the coil can be obtained as shown in Figure 6. It can be seen from Figure 6 that when the coil's length is 70 mm or 80 mm, the magnetic feld strength on the axis of the GMM rod is strong in the middle and weak at both ends. When the coil's length is 90 mm, the magnetic feld strength on the GMM axis is weak in the middle and strong at both ends.
Te magnetic feld intensity on the end face (Z � 0) and the middle section (Z � 40) of the GMM rod are extracted from Figure 4. Te distribution law of the radial magnetic feld intensity in the GMM rod is shown in Figures 7 and 8. Tey show that the radial distribution of the magnetic feld in the GMM is also uneven.
Te discretization form for the degree of the magnetic feld unevenness formula is defned as follows:

Shock and Vibration
where H is the average magnetic feld intensity, and it can be obtained as follows: where dU � dQ + dW is the number of the observation points.
Substituting the numerical simulation results of the magnetic feld intensity in Figures 6 to 8 into equations (1) and (2), the calculated results of the axial average magnetic feld intensity and the degree of the magnetic feld unevenness under diferent lengths of the coil are shown in Tables 1 and 2. It can be seen from Tables 1 and 2 that under the same number of ampere-turns, with the increase in coil length, the average magnetic feld intensity on the center line decreases, but the average magnetic feld intensity in the GMM rod increases and the degree of the magnetic feld unevenness decreases. Terefore, from the two aspects of magnetic energy utilization and magnetic feld uniformity, under the same ampere-turns, the longer the coil's length is, the better the performance is. So, the length of the coil should be longer than the length of the GMM rod. However, the improvement of GMA's performance is very small, as the coil's length is changing from 80 mm to 90 mm.
Assuming that the coil's length is 80 mm, the coil's radius is 6 mm, 7 mm, and 8 mm, respectively. According to the numerical simulation of the magnetic feld, the magnetic feld intensity on the center line, the end face, and the middle section of the GMM rod under the same ampere-turns are shown in Figures 9-11. It can be seen that when the coil's inner diameter is Figure 10 diferent, the magnetic feld distribution is also diferent. Te magnetic feld intensity on the center line is high in the middle and low at both ends, and the magnetic feld intensity on the middle section is high in the middle and low at both ends, but it is high in the middle and low at both ends at the coil's inner diameter of 6 mm. Figure 11 shows that the smaller the inner diameter of the coil is, the stronger the magnetic feld intensity in the middle section is.
By substituting the data of Figures 9-11 into equations (1) and (2), it can be seen that the average magnetic feld intensity and the unevenness of GMA with the coils having diferent inner diameters under the same ampere-turns are shown in Tables 3 and 4. It can be seen from Table 3 that with the increase in coil's inner diameter, the average magnetic feld intensity on the GMM decreases, but the magnetic feld distribution is more uniform. When the inner diameter of the coil is large, the coil's resistance and inductance will increase under the same number of turns, resulting in more heat and afecting the dynamic performance of the GMA. Terefore, considering the magnetic energy utilization and structural constraints, the inner diameter of the coil is chosen as 6 mm.
From the above analysis, it can be seen that when the number of ampere-turns is constant, the longer the coil's length is, the greater the average magnetic feld intensity is, while the smaller the nonuniformity is. Te smaller the coil's inner diameter is, the greater the magnetic feld intensity is, while the more uneven the magnetic feld distribution is. Terefore, from the aspects of the magnetic energy utilization and the magnetic feld uniformity, the design of the magnetic circuit in the GMA needs to meet the following requirements: (1) Te magnetic circuit is preferably closed; (2) Te length of the coil (excluding the coil frame) is greater than the length of the GMM rod, and the inner diameter is close to the diameter of the GMM rod; (3) Te permeability of other materials on the magnetic circuit should be much greater than the GMM rod.

Magnetization Physical Model Based on the Magnetic
Circuit. Figure 12 shows the magnetic circuit of the GMA, which can be obtained from Figure 1. Based on Figure 12, the magnetic circuit physical model of the GMA in Simulink/   Simcape can be given as shown in Figure 13. Te physical meanings of magnetic elements in Figure 13 are described in Table 5. Table 6 are substituted into the GMM magnetization simulation model shown in Figure 13. Assuming that the amplitude of the control current is 1 A, the driving frequency is 1 Hz and the magnetic fux of the bias magnetic feld is 0 and 0.5 T, respectively. Te relationship curves between the magnetization and the control current are shown in Figure 14 under diferent bias magnetic felds. Figure 14 shows that increasing the bias magnetic feld can greatly reduce the hysteresis nonlinearity, but the magnetic susceptibility is decreased. Te control current changes from −1 A to 1 A, and the magnetization changes by 152068 A/m under the     bias magnetic feld of 0, while the magnetization changes by 69450 A/m under the bias magnetic feld of 0.01 Wb. In addition, it can be seen from Figure 14(b) that the magnetization curve is no longer symmetrical about the center after the bias magnetic feld is applied.

Magnetostrictive Model of GMM
Te microphysical mechanism of the magnetostriction efect is very complex, but it can be explained macroscopically from the perspective of thermodynamics.
Te GMM rod satisfes the frst law of thermodynamics during magnetization.
where dU is the increment of internal energy of the GMM rod; dQ is the increment of heat absorbed by the GMM rod during magnetization; and dW is the increment of work performed by the outside world on the GMM rod. According to the second law of thermodynamics, where T is the thermodynamic temperature of the GMM rod and dS is the change of entropy of the GMM rod. Te increment of work done by the outside world to the GMM rod consists of the magnetization work done by the external magnetic feld to the GMM rod and the volume work done by the GMM rod due to volume change. Te magnetization work is also positive, and the volume work is also negative. Terefore, where H e is the efective magnetic feld applied inside the GMM rod and M is the magnetization. σ and λ are the stress and the strain caused by the magnetostrictive efect, respectively. Te efective magnetic feld in the GMM rod is not equal to the external magnetic feld applied to the GMM rod. Te function of the external magnetic feld is only to change the direction of the magnetic moment formed by spontaneous magnetization and make the magnetic moment rotate in a direction parallel to the external magnetic feld. Sablik and Jiles believe that under the action of axial stress, the efective magnetic feld inside the GMM rod is given as follows: where dU � TdS + μ 0 H e dM + σdλ is the preloading stress applied on the GMM rod; dU � TdS + μ 0 H e dM + σdλ is the molecular feld parameter of magnetic moment interaction; dU � TdS + μ 0 H e dM + σdλ and dU � TdS + μ 0 H e dM+ σdλ are the saturation magnetostriction and saturation magnetization of the GMM rod; dU � TdS + μ 0 H e dM + σdλ is vacuum permeability. Simultaneous equations (3)∼(5) can be obtained.
f (x) = 0 ER i + -   Gibbs free energy is written as follows: Hence, total diferential form of Gibbs free energy is written as follows: Substituting equation (7) into equation (9) yields And the total diferential of Gibbs free energy can be written as follows: Comparing equation (10) with equation (11) yields Since the second-order mixed partial derivatives are independent of the derivation order under continuous conditions, Subsisting equation (12) into equation (13) yields Tus, Equation (15) shows that the change of the magnetization state in the GMM rod will cause the change of the strain state. Te minus sign in the formula indicates that with the increase in magnetization, the GMM rod works externally and it is in an extended state.
Te relationship between stress and strain is given by the following equation: where E H is Young's modulus of the GMM rod. Subsisting equation (16) into equation (15) and taking the square root yield Integrating equation (17) yields where ε 0 is the strain caused by a preload stress and a thermal deformation. It can be obtained from Section 3.3 that the magnetic susceptibility of GMM is not a constant, so (zH/zM) is not a constant. Terefore, the relationship between the magnetostriction and the magnetization shown as equation (18) is not linear.
It can be seen from equation (18) that the magnetostriction is always positive no matter whether the magnetization is positive or negative. In engineering applications, in order to obtain a GMA with bidirectional output, the bias magnetic feld is always applied in the GMM rod. In addition, applying a suitable bias magnetic feld can improve the magnetostriction sensitivity of the GMM rod so that a large displacement can be controlled by a small magnetic feld.
After the bias magnetic feld is applied and the displacement generated by the bias magnetic feld is taken as the reference zero point, the magnetostriction model described by equation (18) can be rewritten as follows: where M b is the magnetization generated by the bias magnetic feld and λ b is the magnetostriction generated by the bias magnetic feld. χ(M) is the susceptibility of the GMM, which varies with the level of the driving magnetic feld. When the driving magnetic feld is small compared with the bias magnetic feld, equation (19) is reasonable. When the driving magnetic feld is large, the magnetic susceptibility changes greatly, and GMM will show a serious nonlinearity, so susceptibility of the GMM rod in equation (19) needs to be given as a variable in this condition.
Te relationship between magnetic feld strength and magnetization is given by the following equation: Hence, equation (19) can be written as follows:

Kinetic Models of the GMA
A large number of experiments and theoretical studies show that when the GMA system adopts the lumped parameter model, its equivalent physical model is the mass-springdamping system. So the kinetic model of the GMA can be equivalent to the system as shown in Figure 15, where m is the equivalent mass, C is the equivalent damping, and K is the equivalent stifness. Force generated by the magnetostriction is given by the following equation: Te physical model from magnetization to output force can be obtained by combining equations (21) and (22), as shown in Figure 16.
Based on the working principle of the GMA, the physical submodel of the magnetic circuit, the magnetostrictive output force submodel, the equivalent mechanical submodel of the GMA, and the physical model from the control current to the output displacement of the GMA can be established, as shown in Figure 17.

Simulation and Experiment
6.1. Simulation. Substituting simulation parameters shown in Table 7 (the susceptibility of the GMM with the bias magnetic feld of 0.2 T is 4) into the physical model in Figure 17 and assuming that the magnetic fux density of permanent magnet is 0, 0.5 T, and 0.2 T, respectively, the control current is a sinusoidal signal with the amplitude of 1 A and the frequency of 1 Hz, and the curves of GMA output displacement and the control current with time can be obtained, as shown in Figure 18. Figure 18(a) shows that when the bias magnetic feld is 0, the frequency of GMA output displacement is twice the frequency of the control current. Te reason is that when the bias magnetic feld is 0, GMA outputs a positive displacement under a positive or a negative control current, as shown in Figure 18(b). In order to realize the bidirectional control of the GMA, the method of applying the bias magnetic feld is usually used to eliminate this "frequency doubling" phenomenon shown in Figure 18(a). Figures 18(c) and 18(e) show that the "frequency doubling" phenomenon is gradually eliminated with the increase in the bias magnetic feld. Te bias magnetic feld can make the GMA obtain a large displacement under the same control current. Compared with Figures 18(b), 18(d), and 18(f ), it can be seen that the GMA can get a larger displacement under the same control current when the bias magnetic feld is 0.2 T. So, a proper bias magnetic feld can not only make the GMA bidirectional output displacement but also increase the sensitivity of the GMA.

Experiment.
Te experiment platform of GMA's output displacement is shown in Figure 19. Te experimental device consists of the GMA, a eddy current displacement sensor and its regulator, a constant current power amplifer, a signal generator, and an oscilloscope. Te signal generator generates a signal with adjustable amplitude and frequency to control the output current of the constant current power amplifer to drive the GMA. Its output displacement is measured by the eddy current sensor and fnally displayed on the oscilloscope.
Te control current is set as a sinusoidal signal with a frequency of 1 Hz and the amplitude of 0.25 A, 0.5 A, 0.8 A, and 1 A, respectively. Te comparison results of the experiment and model simulation for the GMA under diferent driving levels are shown in Figure 20.
GMA's physical model can simulate its hysteresis nonlinearity, as shown in Figure 20. Te error between this physical model and the experiment increases with the increase in the control current. Te model has high accuracy when the control current is not more than 0.8 A. Figure 20 shows that there is a signifcant error between the model and the experiment when the amplitude of the control current is 1 A.
When the amplitude of the control current is 0.8 A, the frequency is set as 50 Hz, 100 Hz, 150 Hz, and 200 Hz, respectively. By simulation and the experiment, the output displacement curves of the GMA under diferent driving frequencies are shown in Figure 21.
As shown in Figure 21, the physical model of the GMA has a good agreement with the experiment. Under the same control current, the maximum output displacement of the       GMA decreases with the increase in frequency, while the phase lag relative to the input control current increases. Setting the amplitude of the control current as 0.8 A and changing the frequency, the amplitude-frequency curve and the phase-frequency curve of the GMA can be obtained by simulation and experiments, as shown in Figure 22. Te result shows that the amplitude bandwidth of the GMA is about 150 Hz and the phase bandwidth can reach to 400 Hz.
Te experimental curve of the step response for the GMA can be obtained by setting the square wave signal with the control current amplitude of 0.5 A and the frequency of 1 Hz. Te comparison with the simulation curve is shown in Figure 23.
It can be seen from Figure 23 that the rise time of the measured curve for the GMA is less than 5 ms and the average value of the steady-state value is about 12 μm. Te rise time of simulation is less than 2 ms, and the steady-state value is 12.6 μm. So, the error is 0.6 μm. Te error of response time is large, which causes by slow response of the constant current power amplifer.

Conclusions
In order to make the GMA convenient to use in the practical engineering application, a nonlinear dynamic physical model of the GMA is developed. By simulation and the experiment, some conclusions are as follows: (1) In order to improve the magnetic feld uniformity and the magnetic energy utilization on the GMM rod, the magnetic circuit of the GMA must be closed, the length of the coil should be greater than the length of the GMM rod, and the inner diameter of the coil should be close to the diameter of the GMM rod. (2) Under a proper bias magnetic feld, the frequency doubling of the GMA can not only be eliminated but also a larger magnetostrictive strain can be obtained with the same magnetic feld. (3) When the magnetic susceptibility is large, the magnetostrictive strain is small under the same magnetic feld.
Under the zero bias magnetic feld, the output displacement of the GMA is insensitive at the initial section. (4) Te model is in good agreement with the experiment, when the GMA is driven by the low and medium magnetic felds. Te error of the output displacement of the GMA under step response is less than 0.6 μm. (5) Te efect of the eddy current is considered in the physical model. So, it can accurately describe the complex hysteresis behavior of the GMA not only under the quasistatic operating conditions but also under dynamic operating conditions of less than 200 Hz.

Data Availability
Te data are attached in the fgures and tables in the article.

Conflicts of Interest
Te author declares that there are no conficts of interest. Shock and Vibration 15