## 4 Results and Discussion  
The output power is one of the most significant indexes for the piezoelectric energy harvester, which mainly depends on the external resistance, the structural dimensions and the water velocity here. Firstly, the effects of external resistances on the output power were investigated. Figure 4 illustrates the variation of output power of the two piezoelectric beams with different load resistances, which the velocities of the water maintained at 0.25 m/s, 0.31 m/s, and 0.36 m/s. Also, the diameter of the cylinders is 30 mm here. It can be found that the output power initially increases with the increase of the resistance until reaching the peak output power, and then decreases while the resistance continues to increase. The optimal resistance corresponds to the peak output power. In other words, the maximum output power can be generated with a specific resistance. The optimal resistances of both piezoelectric beams are 190 k\(\Omega\), 170 k\(\Omega\), and 150 k\(\Omega\) when the velocities are 0.25 m/s, 0.31 m/s, and 0.36 m/s, respectively, as shown in Figure 4a,b. The optimal resistance \(R\) can be theoretically expressed as  
\[R=\frac{1}{2\pi fC_{p}} \tag{1}\]  
where, \(f\) is the vortex shedding frequency  
\[f=S_{t}\,\frac{U}{D} \tag{2}\]  
where, \(S_{t}\) is the Strouhal number, and \(U\) is the velocity of water flow. Hence, we can obtain the relationship \(R\propto 1/U\), \(R\propto D\). The optimal resistance \(R\) decreases with increasing velocity \(U\). However, the optimal resistance \(R\) increases when the diameter of the cylinder \(D\) increases. In addition, it can be found that the output power of the upstream beam is larger than that of the downstream beam, as shown in Figure 4. \begin{table}
\begin{tabular}{c c} \hline
**Parameters** & **Values** \\ \hline Piezolectric layer density, \(\rho_{\mathrm{p}}\) (kg/m\({}^{3}\)) & 5540 \\ Substrate layer density, \(\rho_{\mathrm{s}}\) (kg/m\({}^{3}\)) & 1400 \\ Fluid density, \(\rho_{\mathrm{f}}\) (kg/m\({}^{3}\)) & 1000 \\ Young modulus of the piezoelectric layer, \(E_{\mathrm{p}}\) (GPa) & 15.857 \\ Young modulus of the elastic beam, \(E_{\mathrm{s}}\) (GPa) & 3.5 \\ Piezolectric constant, \(d_{\mathrm{31}}\) (pC/N) & \(-\)210 \\ Capacitance, \(C_{\mathrm{p}}\) (nF) & 30.78 \\ Active width of the piezoelectric layer, \(b\) (mm) & 14 \\ Cylinder density, \(\rho_{\mathrm{c}}\) (kg/m\({}^{3}\)) & 1150 \\ \hline \end{tabular}
\end{table} 
Table 1: Material properties of the piezoelectric energy harvester. Here, the relationship between the output power of both piezoelectric beams (the upstream and downstream beams) was investigated. The diameter of the cylinder of 30 mm and the resistance of 170 k\(\Omega\) were set in the following tests. Figure 5a shows the output voltages of each piezoelectric beam _versus_ time at a water velocity of 0.225 m/s. Figure 5b,c is the FFT analysis result of the output voltages of the upstream and downstream beams, respectively. It can be found that the output peak-to-peak voltage \(V_{\text{p-p}}\) of the upstream piezoelectric beam is about 3.2 times as high as that of the downstream one. The average power \(P=V_{\text{\tiny{EMS}}}^{\text{z}}/R\) of the upstream beam is about 10 times as large as that of the downstream beam. In other words, the upstream piezoelectric beam plays a major role in the energy harvesting performance of the harvester. The reason is that, along with the \(x\) direction of water flow, the stress distribution of the piezoelectric beam varies with different bending deflections. Due to the fact that the strain near the fixed point is larger than that of the free end, the upstream piezoelectric beam is prone to producing more power than the downstream beam. Furthermore, it can be found that the vibration frequencies of the two cylinders are both 1.28 Hz, see Figure 5b,c. Also, the Strouhal number \(S_{\text{t}}\) (\(S_{\text{t}}=0.171\)) can be calculated by Equation (2), which is slightly smaller than the single-cylinder value (\(S_{\text{t}}=0.21\)).