# Optimal Piezoelectric Potential Distribution for Controlling Multimode Vibrations
## 3 Experimental and Numerical Tests  
In order to test the model, numerical and experimental investigations were carried out. An experimental workbench was developed and two arrays of 13 PZT plates (PPs) were attached by means of an epoxy glue on both sides of an aluminium alloy cantilever beam. The dimensions of beam and PZT elements are summarized in Table 1, while a picture of the beam coupled with the PPs is shown in Figure 6.  
By means of an _ad hoc_ developed power supply circuit, the \(V(t)\) voltage of Equation (11) is provided to each piezo plate. Every PP is wired to a mechanical relay that is actuated by a micro controller. The relays can switch independently the sign of each PP supply voltage \(V(t)\) in order to execute a specific PZT activation pattern (i.e., the potential distribution of Table 2. The voltage \(V(t)\) is provided by an arbitrary function generator (Yokogawa FG420) and properly amplified and inverted by two amplifier circuits specifically developed by the authors. The circuit diagram of the amplifiers is shown in Figure 7 and provides appropriate voltages to actuate the piezoelectric plates. It basically consists of two pairs of complementary transistors (_MJE340_, _MJE350_), 10 resistors, and three fast \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Material** & **Length** & **Width** & **Thickness** \\ \hline Beam & Aluminium & 185 & 36 & 1.5 \\ PZT plate & PIC 255 (PI Ceramic) & 40 & 14 & 0.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Cantilever beam and lead zirconate titanate (PZT) plate specifications, and lengths are expressed in mm. Figure 4: \(f_{1,2}(r,\tilde{x})\)—the optimal potential distribution for coupling between the first two modes. Figure 5: \(f_{1,2}(r,\tilde{x})\)—the optimal potential distribution for \(r=0.2\) and \(r=0.8\). diodes; such a circuit is able to achieve an amplification factor of 10 up to 300 \(V_{pp}\). These features, along with their inexpensiveness and their ease of implementation, make these amplifiers particularly suitable for piezoelectric plate actuation. An ICP micro-accelerometer (_PCB_ 352A56) was placed at the free end of the beam and a data acquisition board (_NI_ USB 6251), and a _Labview_ subroutine was used for collecting and processing the measured data. The sampling rate was set at 15 kHz and 4 s of acceleration was acquired for each PP activation pattern. The FFT analysis of the measured data was performed for each PP activation pattern, so that the amplitudes of the acceleration at \(\omega_{i_{1}}\) and \(\omega_{i_{2}}\) could be assessed. Finally, the total tip displacement was calculated for each PP activation pattern in order to identify the optimal experimental potential distributions of the PPs. The overall measurement chain scheme is illustrated in Figure 8.  
The experimental trials were repeated several times for every PP activation pattern, and four combinations of \(\omega_{i_{1}}\) and \(\omega_{i_{2}}\) were tested. The \(r\) ratio (the parameter \(r\) takes into account the distribution of the load between the two eigenfrequencies (see Equation (10))) was varied by step of 0.1 from 0 to 1. The actual experimental setup is shown in Figure 9, and its uncertainty (i.e., the accelerometer, the acquisition board, the power supply system, and the data processing) is estimated to be 5% according to previous studies [23; 24]. Figure 6: A detail of the aluminium beam coupled with two arrays of 13 PZT plates (PPs). Figure 7: Measurement chain and experimental workbench functional scheme. A COMSOL(r) FEM code was chosen to conduct the numerical simulations. The first four modes, and their pairings, were considered. For every combination (Table 2), and every value of \(r\), the amplitude displacement at the free end of the beam was calculated by the frequency response function. Among all of these, the potential distribution, which implied the maximum amplitude, was considered optimal. The characteristics of the beam and of the piezoelectric elements are reported in Table 1.