## 4 Simulation and Analysis  
In this section, numerical simulation of a smart plate is performed using COMSOL. The finite-element model for the smart plate with parameters listed in Tables 1 and 2 is shown in Figure 6.  
The external disturbance for the plate is placed at (0.33 m, 0.17 m) to excite as many modes as possible. Based on the maximum strain criteria [29], the PZT actuator and sensor are placed at (0.37 m, 0.50 m) and (0.13 m, 0.04 m), respectively. Thus, the controllability and observability are ensured.  
Making use of COMSOL yields the mode shapes of the smart plate. Based on theoretical analyses, several natural frequencies below 200 Hz are selected as the disturbance frequencies for AVC. The uncontrolled and controlled vibration responses at different natural frequency orders are shown in Table 3. Their corresponding values of \(\gamma\) calculated by Equation (15) are also shown in Table 3. The magenta triangle, the blue circle, and the cyan square represent the positions of the actuator, the disturbance, and the sensor, respectively. For comparison purposes, the range of the color bars in Table 3 has been set to the same. The darkest blue marks the same lowest value of the acceleration level, and the  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Thickness (m)** & **Piezoelectric Constant (pC/N)** & **Young’s** & **Density** & **Poisson’s** & **Surface** \\
**(m)** & **d\({}_{31}\)** & **d\({}_{22}\)** & **d\({}_{33}\)** & **Modulus (GPa)** & **(kg/m\({}^{3}\))** & **Ratio** & **Area (m\({}^{2}\))** \\ \hline
0.001 & 124 & 124 & 280 & 79 & 7500 & 0.3 & 0.04 \(\times\) 0.04 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Geometrical and material properties of PZT.  
Figure 6: Finite-element model for the smart plate.  
In Table 3, in the (2,3), (3,3), (3,4), and (3,6) modes, the values of \(\gamma\) are greater one, and their corresponding vibration responses are aggravated. These modes are uncontrollable modes, as mentioned above. Conversely, the residual modes listed in Table 3 are controllable. Among the residual modes, the vibration attenuations in the (1,4) and (2,4) modes are relatively small. The reason is that there are close relative modes near the concerned modes. They are difficult to distinguish in practical engineering. In general, only one of the close relative modes is dominant in the structural vibration response. Through AVC, the other close relative mode still contributes a great deal to the structural vibration response. Overall, the AVC effect and \(\gamma\) in the (1,4) and (2,4) modes are positive to verify the proposed criterion. With the results shown in Table 3, the controllable and uncontrollable modes can be effectively pre-estimated using the proposed criterion.  
For the uncontrollable modes mentioned above, the main factors of \(\gamma\) related with the location of PZT are listed in Table 4. As shown in Table 4, both \(v_{a}\) and \(v_{s}\) are greater than one. Therefore, the controlled vibration is aggravated because the adjacent mode dominates the actuating and sensing performance of PZT.  
From the simulation results, AVC is achieved when the controlled mode plays a dominant role in the performance of PZT. Thus, a series of controllable modes for AVC can be pre-specified according to the modal shapes of the controlled structure. In this way, the active vibration attenuation is ensured as much as possible. These results make contributions to promote the application of AVC with PZT actuators and sensors. In the following experiments, the pair of uncontrollable and controlled modes in Table 3 are randomly selected to further verify the theoretical results.