Abstract

The giant magnetostrictive actuator has great use in vibration control, but the linear model cannot fully describe its dynamic characteristics. In this chapter, based on the domain wall theory and piezomagnetic theory, a hysteresis nonlinear model is established to fully describe the actuator dynamic characteristics. In combination with the regularisation method, a sliding mode controller has been designed, and the giant magnetostrictive actuator is also studied in the application of active control. Experimental results show that the hysteresis nonlinear model proposed in the chapter can fully describe the actuator's dynamic characteristics in a wider frequency band and the active control also has a much better isolation effect than the passive vibration; it can significantly attenuate the external incentives.

Keywords: giant magnetostrictive actuator, hysteresis nonlinearity, sliding mode control, active control

### 1. Introduction

Passive vibration isolation has been widely used as an effective isolation method; it can significantly reduce the vibration transmission between mechanical equipment and the base, while the isolation effect is limited in the micro-vibration and low frequencies. Therefore, the active control has been a focus of research at home and abroad [1–4]. With the development of smart materials, intelligent actuators manufactured by those materials including the magneto-rheological actuator, shape memory alloy actuator, giant magnetostrictive actuator (GMA), and others. These actuators have played a huge role in promoting the active control as the executing agency [5, 6].

With the advantages of high-positioning accuracy, fast response, and wide frequency band, among others, the GMA has a wide variety of applications in fields including vibration control and precision positioning [7–10]. Many scholars have studied the linear modelling method of GMA, which is only suitable for describing low-frequency dynamic characteristics [11, 12]. However, the hysteresis nonlinear model based on the domain walls theory can more clearly reveal the coupling relationship among the magnetization process, the stress magnetic machine effect, and stretching amount, and it can more fully describe the GMA's dynamic characteristics on a wide frequency band [13], which is suitable for the active control

actuator. Zhang established the GMA dynamics equations and studied the active control on the basis of the proportional-integral-derivative (PID) algorithm, the results showed that the GMA has some damping effect, but the system's adaptive capacity is a little weak [14]. Francesco calculated the actuator's amplitudefrequency curve and studied the active control in a single freedom degree vibration isolation system. The simulations showed the GMA can significantly reduce the force transmitted to the base [15]. Wang designed a magnetostrictive actuator rod and analysed the system structure vibration on the basis of the linear-quadratic regulator (LQR) algorithm, the results showed the GMA can effectively reduce the structure response of acceleration and displacement [16]. However, these studies mainly focus on the linear actuator model; the isolation frequency band is narrow relatively. Therefore, the application of the GMA hysteresis nonlinear model is urgently needed in the active control.

Currently, some of the more commonly active control strategies are the PID control [17], robust control [18], fuzzy control [19], optimal control [20], adaptive control [21], and sliding mode control [22]. In particular, the sliding mode control has no effect on system parameter perturbation and external disturbance when the system is in sliding mode, so it has much better robustness [23], which is more suitable for the active vibration control algorithm.

Based on the domain wall theory [24], this chapter studied the GMA broadband hysteresis nonlinear model and applied it to the active control of the vibration isolation system. By optimising the design of the hysteresis nonlinear model, the actuator linearity is much better than the linear model on a wider frequency band. The results showed that the GMA hysteresis nonlinear model on a wider frequency band can more fully describe its dynamic characteristics. Compared with passive vibration isolation, the active control has a better isolation effect and wider frequency band isolation, and the sliding mode control strategy also has stronger robustness.

#### 2. Hysteresis nonlinear magnetostrictive actuator model

#### 2.1 Magnetostrictive actuator model

The GMA structure is shown in Figure 1. It is mainly made of the push rod, preload spring, magnetostrictive rod, drive coil, and bias coil, where the preload mechanism is comprised of the preload spring, push rod, and preload screw. The polarised magnetic field of the bias coil can make the magnetostrictive rod deform in a linear manner and can prevent the multiplier phenomenon from affecting system control precision. When an alternating current is passed through the drive coil, the magnetostrictive rod can also generate an alternating magnetic field, which causes a dynamic redistribution of magnetic domains in the magnetostrictive rod. Then the telescopic length can be obtained microscopically to promote a displacement and thrust the output push rod, which achieves energy conversion by transferring electromagnetic energy into mechanical energy.

He <sup>¼</sup> <sup>H</sup> <sup>þ</sup> <sup>α</sup><sup>M</sup> <sup>þ</sup> <sup>H</sup><sup>σ</sup> <sup>¼</sup> <sup>H</sup> <sup>þ</sup> <sup>α</sup><sup>M</sup> <sup>þ</sup> <sup>3</sup> <sup>σ</sup><sup>~</sup>

dH <sup>¼</sup> Man � Mirr

The GMA structure. (1) The top cover screw, (2) the output push rod, (3) the top cover, (4) the preload spring, (5) the jacket, (6) the drive coil, (7) the bias coil, (8) the magnetostrictive rod, (9) the preload screw, (10) the

Nonlinear Giant Magnetostrictive Actuator and Its Application in Active Control

<sup>λ</sup> <sup>¼</sup> <sup>3</sup> 2 λs M<sup>2</sup> s

ð Þ m1 þ m<sup>2</sup> s<sup>2</sup> þ ð Þ c1 þ c<sup>2</sup> s þ ð Þ k1 þ k<sup>2</sup>

<sup>2</sup> <sup>þ</sup> <sup>c</sup>2<sup>s</sup> <sup>þ</sup> <sup>k</sup><sup>2</sup> ð Þ m<sup>1</sup> þ m<sup>2</sup> s<sup>2</sup> þ ð Þ c<sup>1</sup> þ c<sup>2</sup> s þ ð Þ k<sup>1</sup> þ k<sup>2</sup>

where He is the effective magnetic field strength, H is the sum of the drive magnetic field Hd ¼ nI (n is the number of turns per unit coil length, I is the drive

δk � α~ð Þ Man � Mirr

dMirr

Figure 1.

Figure 2.

3

The GMA schematic diagram.

bottom cover screws, and (11) the bottom cover.

DOI: http://dx.doi.org/10.5772/intechopen.86463

<sup>x</sup>ð Þ<sup>s</sup> <sup>¼</sup> <sup>1</sup>

<sup>f</sup> <sup>¼</sup> <sup>m</sup>2<sup>s</sup>

2μ<sup>0</sup>

Mrev ¼ c Mð Þ an � Mirr (4) M ¼ Mirr þ Mrev (5)

Man ¼ Ms coth ð Þþ He=a a=He ½ � (2)

dλ dM σ

M<sup>2</sup> (6)

A

A

<sup>S</sup><sup>H</sup> <sup>λ</sup> (7)

<sup>S</sup><sup>H</sup> <sup>λ</sup> (8)

(1)

(3)

Based on the Jiles-Atherton theory [25], many researchers explored the GMA characteristics of the nonlinear hysteresis model, and the structure optimisation and parameter configuration can make the theoretical calculations accurately match with experiments in a higher linearity. Generally, the GMA motion can be equivalent to a single freedom degree model, shown in Figure 2. Based on the magnetic domain theory and piezomagnetic theory [26–28], the partial differential equations of the magnetisation process are established. Then the GMA dynamics equation is also deduced with the specific equations as follows:

Nonlinear Giant Magnetostrictive Actuator and Its Application in Active Control DOI: http://dx.doi.org/10.5772/intechopen.86463

#### Figure 1.

actuator. Zhang established the GMA dynamics equations and studied the active control on the basis of the proportional-integral-derivative (PID) algorithm, the results showed that the GMA has some damping effect, but the system's adaptive capacity is a little weak [14]. Francesco calculated the actuator's amplitude-

frequency curve and studied the active control in a single freedom degree vibration isolation system. The simulations showed the GMA can significantly reduce the force transmitted to the base [15]. Wang designed a magnetostrictive actuator rod and analysed the system structure vibration on the basis of the linear-quadratic regulator (LQR) algorithm, the results showed the GMA can effectively reduce the structure response of acceleration and displacement [16]. However, these studies mainly focus on the linear actuator model; the isolation frequency band is narrow relatively. Therefore, the application of the GMA hysteresis nonlinear model is

Currently, some of the more commonly active control strategies are the PID control [17], robust control [18], fuzzy control [19], optimal control [20], adaptive control [21], and sliding mode control [22]. In particular, the sliding mode control has no effect on system parameter perturbation and external disturbance when the system is in sliding mode, so it has much better robustness [23], which is more

Based on the domain wall theory [24], this chapter studied the GMA broadband

hysteresis nonlinear model and applied it to the active control of the vibration isolation system. By optimising the design of the hysteresis nonlinear model, the actuator linearity is much better than the linear model on a wider frequency band. The results showed that the GMA hysteresis nonlinear model on a wider frequency band can more fully describe its dynamic characteristics. Compared with passive vibration isolation, the active control has a better isolation effect and wider frequency band isolation, and the sliding mode control strategy also has stronger

The GMA structure is shown in Figure 1. It is mainly made of the push rod, preload spring, magnetostrictive rod, drive coil, and bias coil, where the preload mechanism is comprised of the preload spring, push rod, and preload screw. The polarised magnetic field of the bias coil can make the magnetostrictive rod deform in a linear manner and can prevent the multiplier phenomenon from affecting system control precision. When an alternating current is passed through the drive coil, the magnetostrictive rod can also generate an alternating magnetic field, which causes a dynamic redistribution of magnetic domains in the magnetostrictive rod. Then the telescopic length can be obtained microscopically to promote a displacement and thrust the output push rod, which achieves energy conversion by trans-

Based on the Jiles-Atherton theory [25], many researchers explored the GMA characteristics of the nonlinear hysteresis model, and the structure optimisation and parameter configuration can make the theoretical calculations accurately match with experiments in a higher linearity. Generally, the GMA motion can be equivalent to a single freedom degree model, shown in Figure 2. Based on the magnetic domain theory and piezomagnetic theory [26–28], the partial differential equations of the magnetisation process are established. Then the GMA dynamics equation is

2. Hysteresis nonlinear magnetostrictive actuator model

ferring electromagnetic energy into mechanical energy.

also deduced with the specific equations as follows:

urgently needed in the active control.

Noise and Vibration Control - From Theory to Practice

2.1 Magnetostrictive actuator model

robustness.

2

suitable for the active vibration control algorithm.

The GMA structure. (1) The top cover screw, (2) the output push rod, (3) the top cover, (4) the preload spring, (5) the jacket, (6) the drive coil, (7) the bias coil, (8) the magnetostrictive rod, (9) the preload screw, (10) the bottom cover screws, and (11) the bottom cover.

Figure 2. The GMA schematic diagram.

$$H\_{\epsilon} = H + a\mathcal{M} + H\_{\sigma} = H + a\mathcal{M} + 3\frac{\tilde{\sigma}}{2\mu\_0} \left(\frac{d\lambda}{d\mathcal{M}}\right)\_{\sigma} \tag{1}$$

$$M\_{an} = M\_s[\coth\left(H\_e/a\right) + a/H\_e] \tag{2}$$

$$\frac{dM\_{irr}}{dH} = \frac{M\_{an} - M\_{irr}}{\delta k - \tilde{\alpha} (M\_{an} - M\_{irr})} \tag{3}$$

$$M\_{rev} = \varepsilon (M\_{an} - M\_{irr}) \tag{4}$$

$$\mathbf{M} = \mathbf{M}\_{irr} + \mathbf{M}\_{rev} \tag{5}$$

$$
\lambda = \frac{3}{2} \frac{\lambda\_s}{M\_s^2} M^2 \tag{6}
$$

$$\mathcal{X}\_{\text{(s)}} = \frac{1}{(\mathbf{m}\_1 + \mathbf{m}\_2)s^2 + (\mathbf{c}\_1 + \mathbf{c}\_2)s + (\mathbf{k}\_1 + \mathbf{k}\_2)} \frac{\mathbf{A}}{\mathbf{S}^H} \lambda \tag{7}$$

$$f = \frac{m\_2 s^2 + c\_2 s + k\_2}{(m\_1 + m\_2)s^2 + (c\_1 + c\_2)s + (k\_1 + k\_2)} \frac{A}{S^H} \lambda \tag{8}$$

where He is the effective magnetic field strength, H is the sum of the drive magnetic field Hd ¼ nI (n is the number of turns per unit coil length, I is the drive current) and bias magnetic field Hb, λ<sup>s</sup> is the saturation magnetostrictive coefficient, σ~ ¼ σ<sup>0</sup> þ σ is the external stress (σ<sup>0</sup> is the preload), μ<sup>0</sup> is the vacuum permeability, Ms is the saturation magnetisation, λ is the axial magnetostriction strain, a is the shape factor of non-hysteresis magnetisation, k is the irreversible loss coefficient, α is the molecular field parameter of interacting magnetic torque, <sup>α</sup><sup>~</sup> <sup>¼</sup> <sup>α</sup> <sup>þ</sup> <sup>9</sup>λsσ0=2μ0M<sup>2</sup> <sup>s</sup> is the integrated magnetic domain coefficient, c is the reversible factor, the parameter δ � þ1 when dH >0 and parameter δ � �1 when dH < 0, Man is the non-hysteresis magnetisation, Mirr is the irreversible magnetisation, Mrev is reversible magnetisation, and M is the total magnetisation. C, ρ, L, A, and S<sup>H</sup> are, respectively, the damping, density, length and cross-sectional area, and axial compliance coefficient of the magnetostrictive rod; m<sup>1</sup> ¼ ρLA=3, c<sup>1</sup> ¼ cA=L, and <sup>k</sup><sup>1</sup> <sup>¼</sup> <sup>A</sup>=SHL are, respectively, the equivalent mass, damping, and stiffness coefficient of the magnetostrictive rod; m2, c2, and k<sup>2</sup> are, respectively, the equivalent mass, damping, and stiffness coefficient of the end load; and s is the Laplace transform operator.

#### 2.2 Magnetostrictive actuator model for wide frequency range

When the driving frequency is low, temperature and eddy current can be ignored on the actuator linearity. However, to fully describe the actuator dynamic characteristics on a wide frequency range, it is necessary to consider the hysteresis loss, eddy current losses, additional loss, and stress changes. In this case, magnetisation intensity M can be expressed as follows:

$$\frac{d\mathbf{M}}{dt} = \frac{\partial \mathbf{M}}{\partial H}\frac{dH}{dt} + \frac{\partial \mathbf{M}}{\partial \sigma}\frac{d\bar{\sigma}}{dt} \tag{9}$$

output on a wide frequency range, and then the output displacement and force can

Nonlinear Giant Magnetostrictive Actuator and Its Application in Active Control

The experiments mainly include the giant magnetostrictive actuator, NI host controller, capture cards and displays, LabVIEW software, FN15150 power amplifier, MEL laser displacement sensor, and CHINT current transformer. NI equipment and software are used to capture and display the current and displacement signal, the amplifier amplifies NI weak signals to drive actuators, and the laser displacement sensor is used to accurately measure the actuator displacement. The GMA experiment setup is shown in Figure 3. This experiment studies the hysteresis nonlinear model displacement responses to the preload and the bias magnetic field intensity, as shown in Figures 4 and 5. By optimising the actuator preload and bias magnetic field, the GMA can eliminate the doubling phenomenon and work perfectly within a linear range, which can enhance the actuator output linearity.

be calculated by the input current.

DOI: http://dx.doi.org/10.5772/intechopen.86463

Figure 3.

Figure 4.

5

The giant magnetostrictive actuator experiment.

Curves of displacement changes with the preload.

3. Giant magnetostrictive actuator experiment

where magnetisation change rate is made up of differential susceptibility ∂M=∂H, the magnetic field change rate ∂H=∂t, magnetisation stress change rate ∂M=∂σ, and stress change rate ∂σ=∂t. The key is to solve for ∂M=∂H and ∂σ~=∂t. By Eqs. (2)–(5), the differential susceptibility can be expressed as such:

$$\frac{\partial \mathbf{M}}{\partial H} = (\mathbf{1} - \mathbf{c}) \frac{\mathbf{M}\_{an} - \mathbf{M}\_{irr}}{\delta k - \tilde{\alpha} (\mathbf{M}\_{an} - \mathbf{M}\_{irr})} + c \frac{\partial \mathbf{M}\_{an}}{\partial H} \tag{10}$$

According to magneto-mechanical coupling model [16], the magnetisation stress change rate can be determined by Eq. (11):

$$\frac{\partial \mathcal{M}}{\partial \sigma} = \frac{\mathbb{S}^{H}\sigma}{\xi} (\mathcal{M}\_{an} - \mathcal{M}) + c \frac{\partial \mathcal{M}\_{an}}{\partial \sigma} \tag{11}$$

where ξ is the energy coupling parameters of per unit volume and other parameters are the same as above. Substituting Eqs. (10) and (11) into Eq. (9), the magnetisation change rate can be obtained:

$$\begin{split} \frac{d\mathbf{M}}{dt} &= \left( (\mathbf{1} - \mathbf{c}) \frac{M\_{an} - M\_{irr}}{\delta k - \tilde{\alpha} (M\_{an} - M\_{irr})} + c \frac{\partial M\_{an}}{\partial H} \right) \frac{dH}{dt} \\ &+ \left( \frac{S^H \sigma}{\xi} (M\_{an} - M) + c \frac{\partial M\_{an}}{\partial \sigma} \right) \frac{d\bar{\sigma}}{dt} \end{split} \tag{12}$$

In summary, Eqs. (1)–(12) constitute the hysteresis nonlinear model of the giant magnetostrictive actuator, which can describe the displacement and force Nonlinear Giant Magnetostrictive Actuator and Its Application in Active Control DOI: http://dx.doi.org/10.5772/intechopen.86463

output on a wide frequency range, and then the output displacement and force can be calculated by the input current.
