## 2 Experimental Conditions
\begin{table}
\begin{tabular}{c c} \hline \hline
**Properties** & **Values** \\ \hline Piezoelectric layer density, \(\varrho_{p}\) (kg/m\({}^{3}\)) & 7500 \\ Substrate layer density, \(\varrho_{s}\) (kg/m\({}^{3}\)) & 2700 \\ Cylinder density, \(\varrho_{c}\) (kg/m\({}^{3}\)) & 1200 \\ Thickness of the piezoelectric layer, \(h_{p}\) (mm) & 0.2 \\ Thickness of the substrate layer, \(h_{s}\) (mm) & 0.2 \\ Active width of the piezoelectric beam, \(b\) (mm) & 20 \\ Active width of the Substrate layer, \(b_{s}\) (mm) & 20 \\ Active length of the piezoelectric beam, \(l_{b}\) (mm) & 80 \\ Active length of the Substrate layer, \(l_{s}\) (mm) & 80 \\ Length of the cylinder, \(l_{c}\) (mm) & 80 \\ Diameter of the cylinder, \(D\) (mm) & 12 \\ Young modulus of the piezoelectric layer, \(E_{p}\) (GPa) & 56 \\ Young modulus of the Substrate layer, \(E_{s}\) (GPa) & 71.7 \\ Piezoelectric constant, \(d_{31}\) (PC/N) & \(-\)274 \\ Permittivity, \(\varepsilon^{T}\) (mF/m) & 41.78 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties of the piezoelectric energy harvesters. Figure 2: Schematic of the harvesting system with two tandem identical piezoelectric energy harvests: (**a**) Tandem configuration diagram; (**b**) Composition diagram of each harvester. Each of the two piezoelectric energy harvesters can be explained by a lumped-parameter model for the VIV-based energy harvesters, respectively. The governing equations are written as  
\[\left\{ \begin{array}{c}M_{eq}\ddot{x}+C\dot{x}+Kx+\Theta V=F_{L}\\ C_{p}\dot{V}+\frac{V}{R}-\Theta\dot{x}=0\end{array} \right. \tag{1}\]  
where \(x\) is the displacement of cylinder in the direction normal to the water flow (\(U\)); (\(\cdot\)) means derivative with respect to the dimensional time t. \(M_{eq}\) is the equivalent mass of the harvester, which can be worked out based on simple beam theory and conservation of kinetic energy. \(M_{eq}\) takes into account both the mass of the structure and the fluid-added mass, which models viscid inertial effects. \(K\) is the equivalent stiffness of the harvester, which can be worked out based on simple beam theory and Hooke's law; \(C\) is the damping coefficient of the harvester, which can be worked out based on a free vibration test. \(\Theta\), \(C_{p}\), and \(V\) are the electromechanical coupling coefficient, the capacitance of the piezoelectric sheet, and the generated voltage, respectively; \(R\) is the electrical load resistance. \(F_{L}\) is the vortex-induced force. Because of the interaction between the two tandem harvesters, \(F_{L}\) is especially complicated, and cannot be empirically established based on the previous wake oscillator model. Currently, the most useful methods for investigating the dynamic response and energy harvesting performances of the two coupled energy harvesters are numerical simulation and experimental test. Here, we choose the experimental method. According to the parameters in Table 1, the values of \(M_{eq}\) and \(K\) are 0.0154 kg and 9.366 N/m, respectively. Hence, the natural frequency \(f_{n}\) is 3.92 Hz worked out by \(M_{eq}\) and \(K\). The Damping ratio is 0.038. Furthermore, the frequency of the vortex shedding from the cylinder is calculated as follows  
\[f_{s}=S_{t}\frac{U}{D} \tag{2}\]  
where \(U\) is the water speed, \(D\) is the diameters of the cylinder, and \(S_{t}\) is the Strouhal number. When\(f_{s}\) is around \(f_{n}\), the harvester will resonate. Hence, the harvester can be designed with a right \(f_{n}\) targeting an intended flow rate.