## 2 Airfoil-Based Flutter Energy Harvester System Design and Control Equations
### Governing Equations  
Figure 2 illustrates a schematic diagram depicting the aerodynamic loads acting on the airfoil-based flutter piezoelectric energy harvester system. The motion model of this system is simplified to a \"spring-mass-damping\" system for descriptive purposes. The  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Parameters** & **Flexural Spring** & **Piezoelectric Patch** & **Holder** & **Spring Rod** \\ \hline Materials & Spring steel & PZT-5H & Aluminum alloy & Glass fiber \\ Density (kg/m\({}^{3}\)) & 7850 & 7500 & 2810 & 2500 \\ Young’s modulus (GPa) & 210 & 66 & 72 & 73 \\ Poisson’s ratio & 0.29 & 0.3 & 0.33 & 0.22 \\ Length (mm) & 180 & 40 & 90 & 50 \\ Width/Radius (mm) & 30 & 20 & 60 & 0.3 \\ Thickness (mm) & 0.8 & 0.2 & 4 & - \\ \hline \hline \end{tabular}
\end{table}
Table 1: Materials and dimensions parameters of the harvester system.  
Figure 1: Schematic diagram of airfoil-based flutter piezoelectric energy harvester system.  
control equations governing the two-degree-of-freedom dive-pitch flutter piezoelectric energy harvester are derived, taking into account the piezoelectric effect. These control equations are formulated as follows [25; 32]:  
\[\mathrm{m}_{T}\ddot{h}+S_{\alpha}\ddot{\alpha}+C_{h}\dot{h}+(k_{h0}+k_{h2}\mathrm{ h}^{2})\mathrm{h}+\theta V=L \tag{1}\]  
\[S_{\alpha}\ddot{h}+I_{\alpha}\ddot{\alpha}+C_{\alpha}\dot{\alpha}+(\mathrm{k}_ {\alpha 0}+\mathrm{k}_{\alpha 2}\alpha^{2})\alpha=M \tag{2}\]  
\[C_{p}\dot{V}+\frac{V}{R}-\theta\dot{h}=0 \tag{3}\]  
where \(L\) is the longitudinal force, positive in the upward direction, \(M\) is the moment, positive in the counterclockwise direction, and h is the sudden displacement, positive in the downward direction; \(\alpha\) represents the pitch displacement, positive in the clockwise direction; \(m_{t}\) represents the total mass, including the airfoil-based components, plunge spring, spring rod, and holder; \(m_{f}\) denotes solely the mass of the wing; \(I_{a}\) is the mass moment of inertia about the elastic axis. \(C_{h}\) and \(C_{\alpha}\) are the damping coefficients of the pitch and roll degrees of freedom, respectively; \(\mathrm{k}_{h0}\) and \(\mathrm{k}_{\alpha 0}\) represent the linear structural stiffness coefficients associated with the pitch and roll degrees of freedom, respectively; \(\mathrm{k}_{h2}\) and \(\mathrm{k}_{\alpha 0}\) represent the linear structural stiffness coefficients related to the cubic nonlinear pitch and roll degrees of freedom, respectively; \(\theta\) is the harvester electromechanical coupling term; \(R\) represents the external load resistance connected to the piezoelectric sheet of the collector; \(V\) is the output voltage generated by the external resistor; \(C_{p}\) represents the capacitance of the piezoelectric sheet; \(S_{\alpha}=m_{f}X_{a}b\) is the mass moment of the airfoil-based body about the rigid center; \(b\) represents the half chord of the airfoil-based -based; \(a\) denotes the dimensionless offset of the airfoil-based elastic axis measured from the half chord; \(X_{a}\) is the dimensionless distance from the center of gravity to the elastic axis.  
Table 2 presents a comprehensive summary of the parameters for the airfoil-based flutter piezoelectric energy harvesting system. These parameters can be classified into four distinct categories: (1) designated design parameters (\(s\), \(b\) and \(R\)); (2) directly measured parameters (\(m_{t}\), \(m_{f}\), \(a\) and \(X_{a}\)) obtained using standard measurement instruments; (3) dynamic parameters (\(\mathrm{k}_{h0}\), \(\mathrm{k}_{\alpha 0}\), \(C_{h}\), and \(C_{\alpha}\)) determined through free vibration experiments following [PERSON]'s methodology [33]; and (4) theoretical parameters (\(C_{p}\) and \(\theta\)) of the PZT-5H piezoelectric material as provided by the manufacturer.  
Figure 2: Schematic of a piezoelectric harvester.