4. Application for GMA in active control

#### 4.1 Active control model of vibration isolation system

The active control model of the vibration isolation system is shown in Figure 6, where M1, K1, C1, M2, K2, and C<sup>2</sup> are, respectively, the mass, stiffness, and damping of the upper isolated equipment and the middle vibration isolation platform. x, y, x\_, y\_, x€, and y€ are, respectively, displacement, velocity, and acceleration as above. f is the active control force of the GMA, and p is the external disturbance. The dynamics equations of the isolation system can be expressed as Eq. (13):

Figure 6. Active control model of vibration isolation system.

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

$$\begin{cases} M\_1 \ddot{\mathbf{x}} + \mathbf{C}\_1(\dot{\mathbf{x}} - \dot{\mathbf{y}}) + K\_1(\mathbf{x} - \mathbf{y}) = p - f \\ M\_2 \ddot{\mathbf{y}} + \mathbf{C}\_2 \dot{\mathbf{y}} + K\_2 \mathbf{y} + \mathbf{C}\_1(\dot{\mathbf{y}} - \dot{\mathbf{x}}) + K\_1(\mathbf{y} - \mathbf{x}) = f \end{cases} \tag{13}$$

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

$$\begin{cases} \dot{X}(\mathbf{t}) = AX(\mathbf{t}) + Bu(\mathbf{t}) + B\_1 w(\mathbf{t})\\ Y(\mathbf{t}) = \mathbf{C}X(\mathbf{t}) \end{cases} \tag{14}$$

$$\begin{aligned} \text{where } A &= \begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -K\_1/M\_1 & K\_1/M\_1 & -C\_1/M\_1 & C\_1/M\_1 \\ K\_1/M\_2 & -(K\_1+K\_2)/M\_2 & C\_1/M\_2 & -(C\_1+C\_2)/M\_2 \end{bmatrix}, \\ B &= \begin{bmatrix} 0 \\ 0 \\ -1/M\_1 \\ 1/M\_2 \end{bmatrix}, B\_1 = \begin{bmatrix} 0 \\ 0 \\ 1/M\_1 \\ 0 \end{bmatrix}, C = \begin{bmatrix} 0 & \mathbf{1} & \mathbf{0} & \mathbf{0} \end{bmatrix}, u = [f], w = [p], \text{and } Y = [y]. \text{ Let} \\ \mathbf{1} &= \begin{bmatrix} 0 & \mathbf{1} & \mathbf{0} \end{bmatrix}, \text{ and } \mathbf{1} = [y]. \text{ Let} \end{aligned}$$

<sup>U</sup> <sup>¼</sup> <sup>u</sup> <sup>þ</sup> <sup>B</sup>�<sup>1</sup> B1w. Eq. (14) can be further expressed as follows:

$$\begin{split} \dot{X}(\mathbf{t}) &= AX(\mathbf{t}) + Bu(\mathbf{t}) + B\_1 w(\mathbf{t}) \\ &= AX(\mathbf{t}) + BU(\mathbf{t}) \end{split} \tag{15}$$

#### 4.2 Sliding mode control design

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

The active control model of the vibration isolation system is shown in Figure 6, where M1, K1, C1, M2, K2, and C<sup>2</sup> are, respectively, the mass, stiffness, and damping of the upper isolated equipment and the middle vibration isolation platform. x, y, x\_, y\_, x€, and y€ are, respectively, displacement, velocity, and acceleration as above. f is the active control force of the GMA, and p is the external disturbance. The dynam-

selected in the study which can give the actuators better linearity.

ics equations of the isolation system can be expressed as Eq. (13):

4. Application for GMA in active control

Curves of displacement changes with the bias magnetic field intensity.

Noise and Vibration Control - From Theory to Practice

Figure 5.

Figure 6.

6

Active control model of vibration isolation system.

4.1 Active control model of vibration isolation system

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

$$S(t) = \Theta X \tag{16}$$

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:

$$Z(\mathbf{t}) = \Gamma X(\mathbf{t}) \tag{17}$$

where <sup>Γ</sup> <sup>¼</sup> <sup>I</sup><sup>3</sup> �B~1B~<sup>2</sup> 0 I<sup>1</sup> " #, <sup>B</sup> <sup>¼</sup> <sup>B</sup>~<sup>1</sup> B~2 " #, <sup>B</sup>~<sup>1</sup> <sup>∈</sup>R<sup>3</sup>�<sup>1</sup> , and <sup>B</sup>~<sup>2</sup> <sup>∈</sup>R<sup>1</sup>�<sup>1</sup> ; substituting

Eq. (17) into Eqs. (15) and (16), it gets the system canonical form and the switching surface as Eq. (18):

$$\begin{aligned} \dot{Z}(\mathbf{t}) &= \overline{A}Z(\mathbf{t}) + \overline{B}U(\mathbf{t}) \\ S(\mathbf{t}) &= \overline{\Theta}Z \end{aligned} \tag{18}$$

$$\text{where } \overline{A} = \Gamma A \Gamma^{-1}, \overline{B} = \begin{bmatrix} \mathbf{0} & \bar{B}\_2 \end{bmatrix}^T, \text{ and } \overline{\Theta} = \Theta I^{-1}; \text{ let } Z(\mathbf{t}) = \begin{bmatrix} Z\_1(\mathbf{t}) \\ Z\_2(\mathbf{t}) \end{bmatrix},$$

$$\overline{A} = \begin{bmatrix} \overline{A}\_{11} & \overline{A}\_{12} \\ \overline{A}\_{21} & \overline{A}\_{22} \end{bmatrix}, \text{and } \overline{\Theta} = \begin{bmatrix} \overline{\Theta}\_1 \\ \overline{\Theta}\_2 \end{bmatrix}^T, \text{ where } Z\_1(\mathbf{t}) \in \mathbb{R}^{3 \times 1}, Z\_2(\mathbf{t}) \in \mathbb{R}^{1 \times 1}, \overline{A}\_{11} \in \mathbb{R}^{3 \times 3},$$

$$\overline{A}\_{12} \in \mathbb{R}^{3 \times 1}, \overline{A}\_{21} \in \mathbb{R}^{1 \times 3}, \text{ and } \overline{A}\_{22} \in \mathbb{R}^{1 \times 1}; \text{ then Eq. (18) can be decomposed as Eq. (19):}$$

Noise and Vibration Control - From Theory to Practice

$$\begin{aligned} \overline{Z}\_1(\mathbf{t}) &= \overline{A}\_{11} Z\_1(\mathbf{t}) + \overline{A}\_{12} Z\_2(\mathbf{t})\\ \mathbf{S} &= \overline{\Theta}\_1 Z\_1(\mathbf{t}) + \overline{\Theta}\_2 Z\_2(\mathbf{t}) \end{aligned} \tag{19}$$

Let S ¼ 0 and Θ<sup>2</sup> ¼ I1; in Eq. (19), it gets:

$$\begin{aligned} Z\_2(\mathbf{t}) &= -\Theta\_1 Z\_1(\mathbf{t})\\ \dot{Z}\_1(\mathbf{t}) &= (\overline{A}\_{11} - \overline{A}\_{12}\overline{\Theta}\_1) Z\_1(\mathbf{t}) \end{aligned} \tag{20}$$

where the matrix Θ<sup>1</sup> can be designed by the optimal control method or pole assignment method, and then the sliding surface S tð Þ can also be determined. Finally, the saturation function of exponential reaching law is used as Eq. (21):

$$\dot{\mathbf{S}}(t) = -\beta \mathbf{S}(t) - \xi \text{sat}(\mathbf{S}(t)) \qquad \xi > 0, \beta > 0 \tag{21}$$

where α >0 and 1> β > 0; with the combination of Eqs. (15), (16), and (21), it can get active control force as Eq. (22) by ignoring the external disturbances:

$$f = -\left(\Theta B\right)^{-1} \left[\Theta AX(t) + \beta \mathbf{S}(t) + \xi sat(\mathbf{S}(t))\right] \tag{22}$$

#### 5. Experimental research

In this chapter, compared with passive vibration isolation, the active control experiment is studied to analyse the isolation effect on a double-layer vibration isolation system. The external disturbance is the force of shaker JZK-40, the hardware control system is designed by LabVIEW real time, and the isolation system experiment setup is shown in Figure 7.

When the external stimulus is applied to the system, the controllers can capture the acceleration signal of the upper and middle layers; then the signal is conveyed to a controller by an A/D converter. At the same time, D/A signals are given out by NI control calculations, passing through amplifier YE2706A, and the output finally is transmitted to GMA for carrying out the active control. The vibration isolation system control block diagram is shown in Figure 8.

In low frequencies, taking the middle platform displacement as the evaluation index, the isolation effect is compared between passive vibration isolation and active control. Experimental parameters are as follows:

<sup>n</sup> <sup>¼</sup> <sup>1200</sup>, Hb <sup>¼</sup> <sup>10</sup>kA=m, a <sup>¼</sup> <sup>7102</sup>A=m, <sup>α</sup><sup>~</sup> ¼ �0:01, Ms <sup>¼</sup> <sup>7</sup>:<sup>65</sup> � <sup>10</sup><sup>5</sup> A=m, <sup>λ</sup><sup>s</sup> <sup>¼</sup> <sup>1</sup>:<sup>005</sup> � <sup>10</sup>�<sup>6</sup>, k <sup>¼</sup> <sup>3283</sup>A=m, b <sup>¼</sup> <sup>0</sup>:18, SH <sup>¼</sup> <sup>1</sup>:<sup>3</sup> � <sup>10</sup>�<sup>11</sup>, d <sup>¼</sup> <sup>1</sup>:<sup>0</sup> � <sup>10</sup>�<sup>8</sup>m<sup>2</sup>=N, <sup>ξ</sup> <sup>¼</sup> <sup>7600</sup>Pa, C <sup>¼</sup> <sup>3000</sup>kNs=m<sup>2</sup>, <sup>ρ</sup> <sup>¼</sup> <sup>9250</sup>kg=m<sup>3</sup>, L <sup>¼</sup> <sup>8</sup>:<sup>6</sup> � <sup>10</sup>�<sup>4</sup>m, A <sup>¼</sup> <sup>78</sup>:<sup>5</sup> � <sup>10</sup>�<sup>6</sup>m<sup>2</sup>, M<sup>1</sup> ¼ 155:42kg, C<sup>1</sup> ¼ 302:7Ns=m, K<sup>1</sup> ¼ 57016:2N=m, M<sup>2</sup> ¼ 22:5kg, C<sup>2</sup> ¼ 290:544Ns=m, and K<sup>2</sup> ¼ 190000N=m:

(23)

active control makes the middle raft displacement decrease by 81.08, 78.21, 81.13, and 55.34% in a single frequency, multi-frequency, and random excitation, respectively. Therefore, the active control has obvious advantages and a good isolation effect. Figure 13 shows that the natural frequencies corresponding to the two peaks of the passive isolation system are fully consistent with the calculations. Moreover, the active control can change the system mode to eliminate the first- and second-order formants to achieve the purpose of isolation. It also shows the active control is poor when the excitation frequency is less than 2 Hz, and then the isolation effect increases with escalating excitation frequency. However, there are always crosspoints between active control and passive vibration isolation in the amplitudefrequency curve, which also shows the active control is more suitable for low- and middle-frequency vibration control. It can compensate for the lack of passive vibration isolation and effectively inhibit the transmission of vibration isolation and broaden isolation frequency band, all of which are of great significance for the study

Nonlinear Giant Magnetostrictive Actuator and Its Application in Active Control

of active vibration control in engineering applications.

Figure 8.

9

Figure 7.

The isolation system experiment setup.

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

Vibration isolation system control block diagram.

The system's natural frequency (f <sup>1</sup> ¼ 2:8Hz, f <sup>2</sup> ¼ 15:6Hz) can be calculated according to the above parameters, and the external disturbances were taken as single, multi-frequency, and random signals. The experimental results are in Figures 9–13 and in Table 1.

Figures 9–12 show that active control can effectively suppress the vibration generated by external incentives with a significant isolation effect and speed response. Table 1 shows that, compared with the passive vibration isolation, the Nonlinear Giant Magnetostrictive Actuator and Its Application in Active Control DOI: http://dx.doi.org/10.5772/intechopen.86463

#### Figure 7.

<sup>Z</sup>\_ <sup>1</sup>ðÞ¼ <sup>t</sup> <sup>A</sup>11Z1ð Þþ <sup>t</sup> <sup>A</sup>12Z2ð Þ<sup>t</sup>

Z2ðÞ¼� t Θ1Z1ð Þt <sup>Z</sup>\_ <sup>1</sup>ðÞ¼ <sup>t</sup> <sup>A</sup><sup>11</sup> � <sup>A</sup>12Θ<sup>1</sup>

Let S ¼ 0 and Θ<sup>2</sup> ¼ I1; in Eq. (19), it gets:

Noise and Vibration Control - From Theory to Practice

\_

5. Experimental research

and K<sup>2</sup> ¼ 190000N=m:

Figures 9–13 and in Table 1.

8

experiment setup is shown in Figure 7.

system control block diagram is shown in Figure 8.

active control. Experimental parameters are as follows:

<sup>n</sup> <sup>¼</sup> <sup>1200</sup>, Hb <sup>¼</sup> <sup>10</sup>kA=m, a <sup>¼</sup> <sup>7102</sup>A=m, <sup>α</sup><sup>~</sup> ¼ �0:01, Ms <sup>¼</sup> <sup>7</sup>:<sup>65</sup> � <sup>10</sup><sup>5</sup>

<sup>f</sup> ¼ �ð Þ <sup>Θ</sup><sup>B</sup> �<sup>1</sup>

S ¼ Θ1Z1ð Þþ t Θ2Z2ð Þt

<sup>Z</sup>1ð Þ<sup>t</sup>

S tðÞ¼�βS tðÞ� ξsat S t ð Þ ð Þ ξ>0, β >0 (21)

½ � ΘAXð Þþ t βS tðÞþ ξsat S t ð Þ ð Þ (22)

A=m,

(23)

where the matrix Θ<sup>1</sup> can be designed by the optimal control method or pole assignment method, and then the sliding surface S tð Þ can also be determined. Finally, the saturation function of exponential reaching law is used as Eq. (21):

where α >0 and 1> β > 0; with the combination of Eqs. (15), (16), and (21), it

In this chapter, compared with passive vibration isolation, the active control experiment is studied to analyse the isolation effect on a double-layer vibration isolation system. The external disturbance is the force of shaker JZK-40, the hardware control system is designed by LabVIEW real time, and the isolation system

When the external stimulus is applied to the system, the controllers can capture the acceleration signal of the upper and middle layers; then the signal is conveyed to a controller by an A/D converter. At the same time, D/A signals are given out by NI control calculations, passing through amplifier YE2706A, and the output finally is transmitted to GMA for carrying out the active control. The vibration isolation

In low frequencies, taking the middle platform displacement as the evaluation index, the isolation effect is compared between passive vibration isolation and

<sup>λ</sup><sup>s</sup> <sup>¼</sup> <sup>1</sup>:<sup>005</sup> � <sup>10</sup>�<sup>6</sup>, k <sup>¼</sup> <sup>3283</sup>A=m, b <sup>¼</sup> <sup>0</sup>:18, SH <sup>¼</sup> <sup>1</sup>:<sup>3</sup> � <sup>10</sup>�<sup>11</sup>, d <sup>¼</sup> <sup>1</sup>:<sup>0</sup> � <sup>10</sup>�<sup>8</sup>m<sup>2</sup>=N, <sup>ξ</sup> <sup>¼</sup> <sup>7600</sup>Pa, C <sup>¼</sup> <sup>3000</sup>kNs=m<sup>2</sup>, <sup>ρ</sup> <sup>¼</sup> <sup>9250</sup>kg=m<sup>3</sup>, L <sup>¼</sup> <sup>8</sup>:<sup>6</sup> � <sup>10</sup>�<sup>4</sup>m, A <sup>¼</sup> <sup>78</sup>:<sup>5</sup> � <sup>10</sup>�<sup>6</sup>m<sup>2</sup>, M<sup>1</sup> ¼ 155:42kg, C<sup>1</sup> ¼ 302:7Ns=m, K<sup>1</sup> ¼ 57016:2N=m, M<sup>2</sup> ¼ 22:5kg, C<sup>2</sup> ¼ 290:544Ns=m,

The system's natural frequency (f <sup>1</sup> ¼ 2:8Hz, f <sup>2</sup> ¼ 15:6Hz) can be calculated according to the above parameters, and the external disturbances were taken as single, multi-frequency, and random signals. The experimental results are in

Figures 9–12 show that active control can effectively suppress the vibration generated by external incentives with a significant isolation effect and speed response. Table 1 shows that, compared with the passive vibration isolation, the

can get active control force as Eq. (22) by ignoring the external disturbances:

(19)

(20)

The isolation system experiment setup.

active control makes the middle raft displacement decrease by 81.08, 78.21, 81.13, and 55.34% in a single frequency, multi-frequency, and random excitation, respectively. Therefore, the active control has obvious advantages and a good isolation effect.

Figure 13 shows that the natural frequencies corresponding to the two peaks of the passive isolation system are fully consistent with the calculations. Moreover, the active control can change the system mode to eliminate the first- and second-order formants to achieve the purpose of isolation. It also shows the active control is poor when the excitation frequency is less than 2 Hz, and then the isolation effect increases with escalating excitation frequency. However, there are always crosspoints between active control and passive vibration isolation in the amplitudefrequency curve, which also shows the active control is more suitable for low- and middle-frequency vibration control. It can compensate for the lack of passive vibration isolation and effectively inhibit the transmission of vibration isolation and broaden isolation frequency band, all of which are of great significance for the study of active vibration control in engineering applications.

Figure 8. Vibration isolation system control block diagram.

Figure 9.

The time history of middle-layer displacement with f <sup>1</sup> excitation.

Figure 10. The time history of middle-layer displacement with f <sup>2</sup> excitation.

6. Conclusions

Figure 12.

Figure 13.

Table 1.

11

The time history of middle-layer displacement with random excitation.

Nonlinear Giant Magnetostrictive Actuator and Its Application in Active Control

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

The amplitude-frequency curves of middle-layer displacement.

The middle-layer displacement RMS with excitations.

In this study, GMA linear model cannot fully describe the dynamic behaviour. Based on the magnetic domain theory, piezomagnetic theory, and with the consideration of the magnetic hysteresis, eddy current, and alternating stress, the GMA

f <sup>1</sup> f <sup>2</sup>

Passive vibration isolation 32.1970 0.6576 3.7067 0.1892 Active control 6.0899 0.1432 0.6996 0.0845

The middle-layer displacement RMS

ffiffiffi 2 <sup>p</sup> <sup>f</sup> <sup>1</sup> <sup>þ</sup> ffiffiffi 2

<sup>p</sup> <sup>f</sup> <sup>2</sup> <sup>þ</sup> <sup>50</sup> Random

Figure 11. The time history of middle-layer displacement with ffiffi 2 <sup>p</sup> <sup>f</sup> <sup>1</sup> <sup>þ</sup> ffiffi 2 <sup>p</sup> <sup>f</sup> <sup>2</sup> <sup>þ</sup> <sup>50</sup> excitation.

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

Figure 12. The time history of middle-layer displacement with random excitation.

Figure 13. The amplitude-frequency curves of middle-layer displacement.


#### Table 1.

Figure 9.

Figure 10.

Figure 11.

10

The time history of middle-layer displacement with f <sup>1</sup> excitation.

Noise and Vibration Control - From Theory to Practice

The time history of middle-layer displacement with f <sup>2</sup> excitation.

The time history of middle-layer displacement with ffiffi

2 <sup>p</sup> <sup>f</sup> <sup>1</sup> <sup>þ</sup> ffiffi 2

<sup>p</sup> <sup>f</sup> <sup>2</sup> <sup>þ</sup> <sup>50</sup> excitation.

The middle-layer displacement RMS with excitations.
