3. Giant magnetostrictive actuator experiment

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.

#### Figure 3.

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 coeffi-

ible 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

<sup>s</sup> is the integrated magnetic domain coefficient, c is the revers-

cient, α is the molecular field parameter of interacting magnetic torque,

Noise and Vibration Control - From Theory to Practice

2.2 Magnetostrictive actuator model for wide frequency range

dM dt <sup>¼</sup> <sup>∂</sup><sup>M</sup> ∂H dH dt þ

Eqs. (2)–(5), the differential susceptibility can be expressed as such:

magnetisation intensity M can be expressed as follows:

∂M

change rate can be determined by Eq. (11):

magnetisation change rate can be obtained:

dt <sup>¼</sup> ð Þ <sup>1</sup> � <sup>c</sup>

SHσ

þ

dM

4

<sup>∂</sup><sup>H</sup> <sup>¼</sup> ð Þ <sup>1</sup> � <sup>c</sup>

∂M <sup>∂</sup><sup>σ</sup> <sup>¼</sup> <sup>S</sup><sup>H</sup><sup>σ</sup>

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

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

> Man � Mirr δk � α~ð Þ Man � Mirr

According to magneto-mechanical coupling model [16], the magnetisation stress

<sup>ξ</sup> ð Þþ Man � <sup>M</sup> <sup>c</sup>

where ξ is the energy coupling parameters of per unit volume and other param-

Man � Mirr δk � α~ð Þ Man � Mirr

dσ~

In summary, Eqs. (1)–(12) constitute the hysteresis nonlinear model of the giant magnetostrictive actuator, which can describe the displacement and force

dH

∂Man ∂σ

eters are the same as above. Substituting Eqs. (10) and (11) into Eq. (9), the

<sup>ξ</sup> ð Þþ Man � <sup>M</sup> <sup>c</sup>

þ c

∂Man

þ c

dt

∂Man ∂H

∂Man

∂M ∂σ dσ~

dt (9)

<sup>∂</sup><sup>H</sup> (10)

<sup>∂</sup><sup>σ</sup> (11)

dt

(12)

loss, eddy current losses, additional loss, and stress changes. In this case,

<sup>α</sup><sup>~</sup> <sup>¼</sup> <sup>α</sup> <sup>þ</sup> <sup>9</sup>λsσ0=2μ0M<sup>2</sup>

transform operator.

The giant magnetostrictive actuator experiment.

Figure 4. Curves of displacement changes with the preload.

M1x€ þ C1ð Þþ x\_ � y\_ K1ð Þ¼ x � y p � f

Nonlinear Giant Magnetostrictive Actuator and Its Application in Active Control

Y ¼ y as system output. Thus, the state space can be obtained using Eq. (14):

B1w. Eq. (14) can be further expressed as follows:

surface, assuming the system switching function as Eq. (4) [29, 30].

, <sup>B</sup> <sup>¼</sup> <sup>B</sup>~<sup>1</sup> B~2 " #

<sup>X</sup>\_ ðÞ¼ <sup>t</sup> AXð Þþ <sup>t</sup> Buð Þþ <sup>t</sup> <sup>B</sup>1wð Þ<sup>t</sup>

Based on full state feedback, the sliding mode can make the system reach the sliding mode surface and achieve sliding mode movement in a jump way. Therefore, it is essential for dynamic characteristics to rationally design the sliding mode

where S is the switching function, Θ is 1 � 4 dimension switching matrix, and <sup>X</sup> <sup>¼</sup> <sup>x</sup>; <sup>y</sup>; <sup>x</sup>\_; <sup>y</sup>\_ � �<sup>T</sup> are the state variables. The non-singular state transition matrix

Γ∈R<sup>4</sup>�<sup>4</sup> is taken to regulate Eq. (16), and the coordinate transformation is the following:

Eq. (17) into Eqs. (15) and (16), it gets the system canonical form and the switching

<sup>Z</sup>\_ðÞ¼ <sup>t</sup> AZð Þþ <sup>t</sup> BUð Þ<sup>t</sup>

S tðÞ¼ ΘZ

<sup>T</sup> h i<sup>T</sup>

, B ¼ 0 B~<sup>2</sup>

Θ<sup>2</sup>

, and A<sup>22</sup> ∈R<sup>1</sup>�<sup>1</sup>

" #<sup>T</sup>

, <sup>B</sup>~<sup>1</sup> <sup>∈</sup>R<sup>3</sup>�<sup>1</sup>

, and <sup>Θ</sup> <sup>¼</sup> ΘΓ�<sup>1</sup>

, where <sup>Z</sup>1ð Þ<sup>t</sup> <sup>∈</sup>R<sup>3</sup>�<sup>1</sup>

¼ AXð Þþ t BUð Þt

YðÞ¼ t CXð Þt

(

3 7 7

�

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

3 7 7 5, <sup>B</sup><sup>1</sup> <sup>¼</sup>

4.2 Sliding mode control design

where <sup>Γ</sup> <sup>¼</sup> <sup>I</sup><sup>3</sup> �B~1B~<sup>2</sup>

surface as Eq. (18):

where <sup>A</sup> <sup>¼</sup> <sup>Γ</sup>AΓ�<sup>1</sup>

<sup>A</sup><sup>21</sup> <sup>A</sup><sup>22</sup> " #, and <sup>Θ</sup> <sup>¼</sup> <sup>Θ</sup><sup>1</sup>

, A<sup>21</sup> ∈R<sup>1</sup>�<sup>3</sup>

<sup>A</sup> <sup>¼</sup> <sup>A</sup><sup>11</sup> <sup>A</sup><sup>12</sup>

A<sup>12</sup> ∈R<sup>3</sup>�<sup>1</sup>

7

0 I<sup>1</sup>

" #

where A ¼

0 0 �1=M<sup>1</sup> 1=M<sup>2</sup>

B ¼

<sup>U</sup> <sup>¼</sup> <sup>u</sup> <sup>þ</sup> <sup>B</sup>�<sup>1</sup>

<sup>M</sup>2y€ <sup>þ</sup> <sup>C</sup>2y\_ <sup>þ</sup> <sup>K</sup>2<sup>y</sup> <sup>þ</sup> <sup>C</sup><sup>1</sup> <sup>y</sup>\_ � <sup>x</sup>\_ � � <sup>þ</sup> <sup>K</sup><sup>1</sup> <sup>y</sup> � <sup>x</sup> � � <sup>¼</sup> <sup>f</sup>

Take X <sup>¼</sup> <sup>x</sup>; <sup>y</sup>; <sup>x</sup>\_; <sup>y</sup>\_ � �<sup>T</sup> as state variables and the middle platform displacement

<sup>X</sup>\_ ðÞ¼ <sup>t</sup> AXð Þþ <sup>t</sup> Buð Þþ <sup>t</sup> <sup>B</sup>1wð Þ<sup>t</sup>

0010 0001 �K1=M<sup>1</sup> K1=M<sup>1</sup> �C1=M<sup>1</sup> C1=M<sup>1</sup> K1=M<sup>2</sup> �ð Þ K<sup>1</sup> þ K<sup>2</sup> =M<sup>2</sup> C1=M<sup>2</sup> �ð Þ C<sup>1</sup> þ C<sup>2</sup> =M<sup>2</sup>

5, <sup>C</sup> <sup>¼</sup> ½ � <sup>0100</sup> , <sup>u</sup> <sup>¼</sup> ½ � <sup>f</sup> , <sup>w</sup> <sup>¼</sup> ½ � <sup>p</sup> , and <sup>Y</sup> <sup>¼</sup> ½ � <sup>y</sup> . Let

S tðÞ¼ ΘX (16)

ZðÞ¼ t ΓXð Þt (17)

, and <sup>B</sup>~<sup>2</sup> <sup>∈</sup>R<sup>1</sup>�<sup>1</sup>

; let ZðÞ¼ t

, <sup>Z</sup>2ð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>1</sup>�<sup>1</sup>

; then Eq. (18) can be decomposed as Eq. (19):

; substituting

Z1ð Þt Z2ð Þt � �,

, A<sup>11</sup> ∈R<sup>3</sup>�<sup>3</sup>

(18)

,

(13)

(14)

(15)

Figure 5. Curves of displacement changes with the bias magnetic field intensity.

Figure 4 shows that the actuator displacement increases with the preload, but when the preload is greater than 7 MPa, experimental curves began to appear asymmetrical, so the preload cannot be greater than 7 MPa. Figure 5 shows that without the paranoid magnetic field, actuator output is butterfly-shaped with a strong nonlinearity, but when the paranoid magnetic field increases, the output linearity also gradually increases. Finally, σ<sup>0</sup> ¼ 6:7MPa and Hb ¼ 14kA=mare selected in the study which can give the actuators better linearity.
