## 4 System Setup  
The proposed hybrid RC algorithm was implemented in Matlab at a sampling frequency of 1000 Hz for the active suppression of vibrations in a flexible structure. Figures 6 and 7, respectively, present a schematic diagram and photo of the hardware used in the vibration control system. Measurements were obtained using a data acquisition card (NI PCI-6259). After sending control commands to a power amplifier, they were respectively used to drive a servomotor (MHMD-042P1S from Panasonic, Osaka, Japan) and piezoelectric ceramic transducer (SB4020008 from SINOCERAMICS, Shanghai, China). Motor encoder values were sent back to another data acquisition card (MCC PCI QUADO4). Table 1 lists the properties of the thin flexible plate and Table 2 lists the properties of the piezoelectric actuator and piezoelectric thin film sensor used in this study. A piezoelectric amplifier (VP7206 from PiezoMaster, Marlboro, MA, USA) was used to amplify the piezoelectricity from the original piezoelectric actuator by 20 times, thereby generating sufficient bending force to suppress vibrations in the thin flexible plate.  
We conducted modal analysis using finite element software to optimize the placement of sensors for subsequent experiments on the suppression of vibration. Due to the large structural mass of the system, we considered only the first bending resonance. In simulations, we applied a 1 Hz sinusoidal moment to the root of the structure and excited the structure for 10 s. We recorded strain values at various locations (A-E in Figure 8) to determine the best location for the sensors used in feedback control. Figures 8 and 9 present the modal analysis associated with various time responses and Fast Fourier transform (FFT) results. The symmetric structure of the device allows us to present strain values on only one side of the thin plate. Our results clearly indicate that vibration was dominated by persistent disturbances with the first vibration mode appearing at approximately 1.5 Hz. Time plots obtained from every location present similar composite periodic responses, whereas the highest strain occurred at the root of the structure (location E), as confirmed by the peak values in the FFT results. We therefore placed the piezoelectric sensor (Model NO. LDT0-028K/L from MEAS, Hampton, VA, USA) at the fixed end of the thin plate, in order to obtain measurements of high sensitivity for use in vibration control. The frequency bandwidth of bandpass filter in the coordinating piezoelectric sensor amplifier (Piezo Film Lab Amplifier from MEAS) was set between 0.1 Hz and 10 Hz. The feedback signals of the piezoelectric vibration suppression system were amplified 10 times.  
Figure 6: Schematic diagram showing vibration control system for large flexible structures.  
Figure 8: Modal analysis of flexible structure under persistent excitation: **(a)** illustration of the structure used for modal analysis; and **(b)** strain distribution throughout entire structure. Capital letters refer to the five locations used in evaluating the placement of piezoelectric sensors.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Symbol** & **Description** & **Flexible Plate** & **Units** \\ \hline \(L_{\text{plate}}\) & Plate Length & 700 & mm \\ \(h_{\text{plate}}\) & Plate Thickness & 1.58 & mm \\ \(W_{\text{plate}}\) & Plate Width & 350 & mm \\ \(\rho_{\text{plate}}\) & Plate Density & 2700 & Kg/m\({}^{3}\) \\ \(E_{\text{plate}}\) & Young’s Modulus & \(7.0\times 10^{10}\) & N/m\({}^{2}\) \\ \(\
u_{\text{plate}}\) & Poisson Ratio & 0.359 & N/A \\ \hline \hline \end{tabular}
\end{table}
Table 1: Properties of the flexible plate.  
Figure 7: Hardware photograph representing a large flexible structure in experiments.