## 4 Semi-Active Vibration Control of the Thin Plate Using the ERF Embedded ATVA  
### Experimental Setup for the Thin Plate with Varying Harmonic Vibrations  
In our previous research, we have designed a semi-active controller for the cantilever beam equipped with the ERF embedded ATVA for the vibration at specified frequencies [17]. However, the thin plate's vibration mode shape is different from the cantilever beam. Figure 10 shows the plate's vibration modes, where the locations with the maximum displacement vary with the vibration modes. As shown in Figure 10a, the first mode shape of the plate is similar to the one of the cantilever beam; the maximum displacement appears at the free end. Therefore, to measure the vibration of the first mode, the displacement sensor can be located at the edge of the free end.  
However, as shown in Figure 10b, the maximum displacement of the second mode appears at the both side of the free end. If we need to measure the vibrations of both the first and second modes, the middle point of the free end is not a good choice to locate the displacement sensor. Even though the middle point of the free end can measure the vibration of the first mode, this location is not sensitive to the vibration of the second mode. As a result, to measure the vibration of the second mode, the displacement sensor should be located at both sides of the free end. With observing the mode shape of the third mode in Figure 10c, we found that the location with the maximum displacement appears at the middle point of the free end for the third vibration mode. Therefore, to measure the vibration of the third mode, we need to locate the sensor at the middle point to obtain the best sensitivity to the vibrations. Obviously, the choice of displacement sensor's location of the third mode is in conflict with the ones of the first and second modes.  
With consideration of measuring the multi-mode vibrations, we proposed to use two displacement sensors to measure the response of the plate for the different vibration modes. On the one hand, an eddy-current displacement sensor is located at the upper side of the free end to measure the vibrations of the first and second modes; on the other hand, the other displacement sensor is located at the middle of the free end to measure the vibration of the third mode. Figure 12 shows the schematic diagram of semi-active vibration control architecture for the ERF embedded ATVA. Two eddy-current displacement sensors are located at the middle and the upper of the free end to measure the vibration of the plate; the embedded controller (cRIO-9074) is used to implement the real-time control system. The analog input module (NI 9234) is employed to capture the feedback signals of the displacement sensors and the analog output module (NI 9263) is used to provide the analog signal according to the proposed semi-active vibration controller. Then, the output signal of NI 9263 is amplified via the high-voltage power amplifier (Trek 609A) to produce the required electric field to the ERF material.  
In this paper, we proposed a type-2 fuzzy controller, which has two inputs and one output as shown in Figure 12. The first input is the vibration frequency, which is obtained by fast-Fourier transformation (FFT) of the measured displacement signals; the second input is the root-mean-square value (RMS) of the displacement of the plate's free end. The RMS value of displacements and the vibration frequency are fed back to a fuzzy controller, which produces the required output voltage to the high-voltage power amplifier. Consequently, we will discuss how to design the type-2 fuzzy controller to establish the input-output mapping between the vibration signal and the required electric field of ERF material.