research article research of jiles-atherton dynamic model ...giant magnetostrictive materials (gmm)...

Research ArticleResearch of Jiles-Atherton Dynamic Model inGiant Magnetostrictive Actuator

Yongguang Liu, Xiaohui Gao, and Chunxu Chen

School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China

Correspondence should be addressed to Xiaohui Gao; [email protected]

Received 18 March 2016; Revised 29 July 2016; Accepted 9 August 2016

Academic Editor: Xiao-Qiao He

Copyright © 2016 Yongguang Liu et al.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.

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

1. Introduction

Giant magnetostrictive materials (GMM) are widely usedin transducer, precision actuator, and active vibration [1–3] because of large magnetostrictive coefficient, excellentdynamic characteristic, and high energy density. When theyare applied in high frequency, establishing accurate math-ematical model is crucial to improving controllability andstability [4, 5]. J-A model is established based on magne-tization principle by Jiles and Atherton, which has a wideapplication and fast development because of clear physicalconception, high stability, and accuracy [6–9]. In recent years,many experts established the mathematical models of hys-teresis materials including frequency, pressure, temperature,and other sensitive factors based on the J-A model thathad developed a relatively mature modeling theory [10–13].However, these modes of GMA are mainly studied frommaterial and ignore the structural factors such as load, discspring stiffness, and friction which have a strong impact onoutput 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 ofGMA, Ampere circuit law, and nonlinear piezomagneticequation through analyzing the working principle of GMA.Then, the correctness and validity of this model are provedby experiments in different conditions. Finally, some outputcharacteristics laws are revealed by analyzing key structuralparameters.

2. Working Principle of GMA

The GMM rod drives load under the action of the magneticfield generated by the excitation coil and permanent magnet(Figure 1).Themagnetostrictive coefficient of GMM is raisedby prepressurewhich is generated by compressing disc spring.Biasmagnetic fields are produced by permanentmagnets. Onthe one hand, the double frequency characteristic of GMMcould be removed; on the other hand, the high linearity partof hysteresis loop near the bias magnetic fields is appliedto reduce the nonlinearity. The temperature control systemprovides relatively constant temperature environment for theGMM rod.

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016, Article ID 2609069, 8 pageshttp://dx.doi.org/10.1155/2016/2609069

2 Mathematical Problems in Engineering

FoundationOriented bar

GMM rod

Temperature sensor

Permanent magnet

Inertia load

Output rod

Preload sensor

Exciting coil

Lower cover

Cooling chamber

Shell

Holder

Top cover

Load sensor

Screw

Linear bearing

Disc spring

Friction sensor

Figure 1: Structure of GMA.

3. Jiles-Atherton Dynamic Model

The J-A model is developed taking into account the phe-nomenology of domain wall translation through the so-called pinning center, which might be nonmagnetic inclu-sions and impurities, grain boundaries, voids, and so forth,causing residual stress that hinders the magnetization pro-cess [14]. The motion of magnetic domains can be dividedinto reversible or irreversible movement and rotation. Thus,according tomodel developers, total magnetization𝑀 can bedivided into reversible and irreversible terms [6]. Hence,

𝑀 = 𝑀rev +𝑀irr, (1)

where𝑀rev is reversiblemagnetization and𝑀irr is irreversiblemagnetization. Consider

𝑀rev = 𝑐 (𝑀an −𝑀irr) ,

𝑀irr =(𝑀 − 𝑐𝑀an)

(1 − 𝑐),

(2)

where 𝑐 is reversible loss coefficient.ThemodifiedLangevin equation is used to fit the isotropic

material and anhysteretic magnetization𝑀an is obtained in

𝑀an = 𝑀𝑠 (coth𝐻 + 𝛼𝑀an

𝑎−

𝑎

𝐻 + 𝛼𝑀an) , (3)

where 𝑀𝑠is saturation magnetization, 𝑎 is shape parameter

of the anhysteretic magnetization, 𝛼 is internal couplingfield coefficient among different magnetic domains, and𝐻 ismagnetic strength.

Equation (4), which takes into account dynamic effectsdue to eddy currents in thematerial, is derived by introducingadditional energy term in the basic J-A equation [15]. Hence,

𝑀 = 𝑀an − 𝑘𝛿 (1 − 𝑐)d𝑀irrd𝐻𝑒

− 𝑘1

d𝑀d𝑡

d𝑀d𝐻𝑒

− 𝑘2

d𝑀d𝑡

1/2 d𝑀d𝐻𝑒

,

(4)

where 𝑘1is eddy-current loss factor, 𝑘

2is anomalous loss

factor, 𝑘 is irreversible loss coefficient,𝐻𝑒is effectivemagnetic

field strength, and 𝛿 is direction coefficient, when d𝐻/d𝑡 > 0,𝛿 = 1 and when d𝐻/d𝑡 < 0, 𝛿 = −1.

Stress can change certain features of some magneticdomains to influence 𝐻

𝑒. When the stress is 𝜎, the effective

magnetic field 𝐻𝑒becomes formula (5) and parameter 𝛼 is

modified to be �̃� [16, 17]. Hence,

𝐻𝑒= 𝐻 + 𝛼𝑀 +

9𝜆𝑠𝜎

2𝜇0𝑀2𝑠

𝑀 = 𝐻 + �̃�𝑀, (5)

�̃� = 𝛼 +9𝜆𝑠𝜎

2𝜇0𝑀2𝑠

, (6)

where 𝜆𝑠is saturation magnetostrictive coefficient.

According to parameters characteristics of J-Amodel andthe difference between simulation and experiment, theminorloop J-Amodel can be got throughmodifying the parameters𝑎, 𝛼, 𝑐, and 𝑘 [18]. Hence,

𝑎minor = 𝑎𝑒𝛾𝑎(𝜆𝑠−𝜆𝑚)

,

𝛼minor = 𝛼𝑒𝛾𝛼(𝜆𝑠−𝜆𝑚)

,

𝑐minor = 𝑐𝑒𝛾𝑐(𝜆𝑠−𝜆𝑚)

,

𝑘minor = 𝑘𝑒𝛾𝑘(𝜆𝑠−𝜆𝑚)

,

(7)

Mathematical Problems in Engineering 3

GMM rod

Exciting current

Exciting force

Output displacement

Equivalent inertia force

RigidityDamping

Figure 2: Equivalent mechanical mode.

where 𝛾𝑘, 𝛾𝑎, 𝛾𝑐, and 𝛾

𝛼are, respectively, modified coefficient

of 𝑘, 𝑎, 𝑐, and 𝛼; 𝜆𝑚

is the maximum magnetostrictioncoefficient of minor loop.

We can get dynamic minor loop J-A model of GMMaccording to formulas (1)–(7).

4. Structural Dynamics Model

GMA can be considered as a single degree of freedom mass-spring-damping system. GMM rod drives the load under theaction of magnetic field. In this section, the dynamic mathe-matical model of GMA is established considering structuralcharacteristics such as load, disc spring, and friction and thefollowing assumptions should be accepted:

(1) GMM rod should be in uniform magnetic field.(2) The working temperature is 293K and remains con-

stant all the time.(3) The output displacement is 0 when biasmagnetic field

is 22.5 kA/m, and prepressure is 6Mpa.(4) The output displacement of one end is 0, while the

other end keeps the same output characteristics withload such as displacement, velocity, and acceleration.

(5) The output displacement 𝑥 of GMM rod is equal to 𝜀𝐿and the output force 𝐹 is equal to −𝜎𝐴.

Based on the above assumptions, the equivalentmechani-cal mode can be obtained according to the working principleof GMA (Figure 2). According to Newton’s second law, theoutput force 𝐹 of GMA is

𝐹 = −𝜎𝐴 = 𝑀𝑒�̈� + 𝐶

𝑀�̇� + 𝐹𝑓+ 𝐹𝑑,

𝑀𝑒=𝑀𝑀

3+𝑀𝐿,

(8)

D

d

t

H0

Figure 3: Structural parameters of disc spring.

where𝑀𝑒is equivalent mass, 𝐶

𝑀is the impedance factor of

GMA, 𝐹𝑓is friction, 𝐹

𝑑is the pressure of disc spring,𝑀

𝑀is

the quality of GMM bar that is equal to 1/3 of itself accordingto kinetic energy Lagrange function [19], and 𝑀

𝐿is mass of

the load.

4.1. The Output Force of GMM Rod. When the magneticfield intensity is relatively large, the magnetization process inthe material will produce serious hysteresis nonlinearity. Thequadraticmagnetic domain rotationmodel is introduced intothe linear piezomagnetic equation [20]. Hence,

𝜀 =𝜎

𝐸+ 𝛾1𝑀2

, (9)

where 𝛾1is the magnetic elastic coefficient of GMM rod and

𝐸 is the elasticity modulus of GMM rod.

4.2. Pressure of Disc Spring. The pressure generated by asingle disc spring [21] is given by

𝐹𝑑=(𝑥 + 𝑥

0) 𝑡3

𝜒𝐷2[(

ℎ0

𝑡−(𝑥 + 𝑥

0)

𝑡)

⋅ (ℎ0

𝑡−(𝑥 + 𝑥

0)

2𝑡) + 1] − 𝜎

0𝐴,

(10)

where 𝜎0is the prestress, 𝑥

0is the predisplacement in the

prestress, 𝜒 is shape factor of disc spring, and ℎ0is its

maximumdeformation.The geometric parameters are shownin Figure 3.

4.3. Friction. According to principle and characteristics offriction and working environment of GMA, Coulomb +Viscosity friction model (Figure 4) is selected [22].

Friction model is shown in

𝐹𝑓= 𝑓𝑐V + 𝑓max sgn (V) , (11)

where 𝑓max is the maximum static friction, 𝑓𝑐 is the Coulombfriction coefficient, and V is velocity of the GMM rod.

4.4. Ampere Circuital Theorem. According to Ampere cir-cuital theorem considering leakage in magnetic circuit [23],the magnetic field strength𝐻 is

𝐻 = 𝐻bias + 𝑘coil𝐼, (12)

where 𝐻bias is the bias magnetic field strength and 𝑘coil isexciting coefficient of the coil.


Velocity 0Fr

ictio

n fmax

fc

Figure 4: Friction model.

Table 1: Model parameters.

Parameter Value𝑀𝑠

380769A/m𝑘 22643A/m𝐶𝑀

3.02 × 104Ns/m𝜆𝑠

1500 ppm𝛾1

1.07 × 10−14 A2/m2

𝑎 47704A/m𝑐 0.1𝑘1

2.62 × 10−5

𝛾𝑎

−10𝛾𝑘

−30𝛼 0.417𝑘coil 16492m

−1

𝑘2

0.312𝛾𝛼

10𝛾𝑐

1500

The current-displacement mathematical model of GMAcould be obtained by connecting formulas (1)–(7) with (9)–(12) and its parameters (Table 1) are identified by adoptingmodified simulated annealing differential evolution algo-rithm which has a high convergence speed and accuracy.

5. Test Verification

GMA testbed is mainly composed of GMA, laser displace-ment sensor, and temperature control system (Figure 5).The measurement and control system applies RTX as lowercomputer system software and LabWindows as upper com-puter system software and sampling period is 0.5ms. Thetemperature sensor is installed near GMM rod (Figure 1) andtemperature control system can pass heat through circulatingwater which can be controlledwithin 1 K by heater and cooler.The V100-MS laser displacement sensor can measure the

Laser sensor

GMA

Temperature control system

Figure 5: GMA testbed.

Table 2: Structural parameters of disc spring.

𝐷 (mm) 𝑑 (mm) 𝑡 (mm) ℎ0(mm)

A 63 31 3.5 1.4B 63 31 2.5 1.75C 63 31 1.8 2.35

displacement, velocity, and acceleration and its displacementmeasurement accuracy is up to 5 𝜇m/V. The ISF20DA250servo driver can provide exciting current with feedbackcontrol and its frequency response is up to 15 kHz (seeFigure 6). The simulation and experiment curves in differentconditions are shown in Figure 7, which prove the validityand accuracy of this mathematical model.

6. Studies on Key Parameters of the Model

In this section, the output characteristics laws of GMA arerevealed by studying structural parameters such as disc springstiffness and friction.

6.1. Studies on Characteristics of Disc Spring. The disc spring,a key part that provides prepressure, has a serious impacton output characteristics of GMA because of its nonlinearity.This paper studies three series of disc springs (A, B, andC) whose inner and outer diameter are the same and theirstructural parameters and mechanical property are shown inTable 2 and Figure 8. Figure 9 shows the output displacementof GMA under 20Hz. After analyzing Figures 8 and 9, wecan conclude that the larger stiffness of disc spring can reducemagnetostriction coefficient.

6.2. Studies on Friction. This sectionmainly studies Coulombfriction coefficient 𝑓

𝑐and the maximum static friction 𝑓max

in friction model. When 𝑓𝑐takes 10Nm/s and 100Nm/s,

respectively, the output displacement of GMA under 20Hzis shown in Figure 10. The larger the 𝑓

𝑐, the more serious

the hysteresis energy loss. As a result, the Coulomb frictioncoefficient should be as small as possible.


Servo driver

Current sensor

High frequency

isolator

Figure 6: Servo driver.

SimulationExperiment

0−1−2−3 2 31Current (A)

Disp

lace

men

t (𝜇

m)

−30

−20

−10

0

10

20

30

(a) Different minor loops


0−1−2−3 2 31Current (A)

Disp

lace

men

t (𝜇

m)

−30

−20

−10

0

10

20

3020 Hz

50 Hz

200 Hz

(b) Different frequencies

20 Kg

1 Kg


0−1−2−3 2 31Current (A)

Disp

lace

men

t (𝜇

m)

−20

−10

0

10

20

(c) Different loads

Figure 7: Experiment and simulation.


Com

pres

sive f

orce

(kN

) A

B

C

0

2

4

6

8

10.4 0.6 0.80.20Deformation (mm)

Figure 8: Mechanical characteristics curve.

ABC

0

0

−1−2−3 2 31Current (A)

Disp

lace

men

t (𝜇

m)

−30

−20

−10

10

20

30

Figure 9: Hysteresis loop about springs.

0

0

−1−2−3 2 31Current (A)

Disp

lace

men

t (𝜇

m)

−30

−20

−10

10

20

30

fc = 100Nm/sfc = 10Nm/sFigure 10: Hysteresis loop about 𝑓

𝑐.


0

0

−1−2−3 2 31Current (A)

Disp

lace

men

t (𝜇

m)

−30

−20

−10

10

20

30

fmax = 5Nfmax = 100N

Figure 11: Hysteresis loop about 𝑓max.

When 𝑓max takes 5N and 100N, the output displacementof GMA in 20Hz is shown in Figure 11.With the increasing of𝑓max, serious low speed crawling and buffeting will appear inthe end of hysteresis loop. Therefore, 𝑓max should be as smallas possible.

7. Conclusion

(1) Structural parameters have a strong impact on outputcharacteristics of GMA in high frequency.

(2) The dynamic mathematical model of GMA includingstructural element such as load, disc spring, andfriction has a great adaptability and accuracy.

(3) The stiffness of disc spring influences the outputcharacteristics of GMA.

(4) Friction affects unfavourably the output characteris-tics.

Competing Interests

The authors declare that they have no competing interests.

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