# Enhancing Wind Energy Harvesting Using Passive Turbulence Control Devices
## 2 Physical Mathematical Model
### Solution and Verification of the Model  
The influence of the load at the interface of the circuit on the natural frequency, the system damping and the critical wind speed of the harvester can be determined by linear analysis of electromechanical coupling problems. The following state variables are introduced:  
\[X=\left[ \begin{array}{c}x_{1}\\ x_{2}\\ x_{3}\end{array} \right]=\left[ \begin{array}{c}y(t)\\ \dot{y}(t)\\ V(t)\end{array} \right] \tag{11}\]  
Therefore, the matrix form of Equations (1) and (2) can be expressed as:\[\dot{X}=\left\{ \begin{array}{c}x_{2}\\ -\frac{C_{eff}}{M_{eff}}x_{2}-\frac{K_{eff}}{M_{eff}}x_{1}-\frac{\theta}{M_{eff}} x_{3}+\frac{\beta SU^{2}}{2M_{eff}}\\ \left(a_{1}\big{(}\frac{x_{2}}{U}+\lambda x_{1}\big{)}+a_{3}\big{(}\frac{x_{1}}{U }+\lambda x_{1}\big{)}^{3}\right)\\ \frac{\theta}{C_{p}}x_{2}-\frac{x_{3}}{RC_{p}}\end{array} \right. \tag{12}\]  
MATLAB program solver \"ODE45.m\" was used to obtain the analytical solution of Equations (9) and (12). The aerodynamic parameters and the fitting results of linear and nonlinear coefficients \(a_{1}\) and \(a_{3}\) in the matrix equation are shown in Table 1. Positive nonlinear mechanical damping is introduced into the harvester. The amplitude is self-limited, so that the harvester undergoes steady-state limit cycle oscillation at each wind speed. In addition, the resistance value ranges from 100 K\(\Omega\) to 10 M\(\Omega\). The electromechanical coupling coefficient is \(\theta\) = 2.0 \(\times\) 10\({}^{-5}\) N\(\cdot\)V\({}^{-1}\). The piezoelectric plates are made of fiber MFC (Macro Fiber Composite). Capacitance is \(C_{p}\) = 15.7 nF. The other values of the relevant parameters are listed in Table 2.  
In order to verify the theoretical model of the harvester, a T-shaped device was designed, as show in Figure 2. The device was installed in a circular wind tunnel with a diameter of 400 mm and was connected to a linear resistance load with wires. The honeycomb region in the middle of the wind tunnel is the stable wind section. The piezoelectric plate was 35 \(\times\) 10 \(\times\) 0.3 mm\({}^{3}\) in size and was placed on the top of the aluminum cantilever beam. The width of the PTC was 3 mm and the length was the same as the bluff body. Simple harmonics produced by the bluff body can be captured by a digital oscilloscope (ISDS220B) and the vibration is measured by a laser sensor (Panasonic, hg-c1400, Kad \begin{table}
\begin{tabular}{c c c c} \hline \hline
**U** & **\(\lambda\)/K\({}_{\mathit{eff}}\)** & **(\(-\)1/2)\(\rho\)SU\({}^{2}\)** & **\(a_{1}\)** & **\(a_{3}\)** \\ \hline
1.276 & 0.62 & \(-\)0.0038 & 2.4 & \(-\)35 \\
1.413 & 0.62 & \(-\)0.0046 & 2.4 & \(-\)28 \\
1.550 & 0.62 & \(-\)0.0055 & 2.3 & \(-\)21 \\
1.687 & 0.62 & \(-\)0.0066 & 2.4 & \(-\)24 \\
1.824 & 0.62 & \(-\)0.0077 & 2.6 & \(-\)24 \\
1.961 & 0.62 & \(-\)0.0089 & 2.8 & \(-\)23 \\
2.098 & 0.62 & \(-\)0.0101 & 2.9 & \(-\)20 \\
2.235 & 0.62 & \(-\)0.0115 & 3.0 & \(-\)20 \\
2.372 & 0.62 & \(-\)0.0130 & 3.0 & \(-\)19 \\
2.509 & 0.62 & \(-\)0.0145 & 3.0 & \(-\)19 \\
2.646 & 0.62 & \(-\)0.0161 & 3.0 & \(-\)19 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Values of system aerodynamic parameters and \(a_{1}\), \(a_{3}\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Material and Physical Properties** & **Symbol** & **Numerical Value** \\ \hline The effective length of the & & \\ cantilever beam/m & \(L_{a}\) & 0.14 \\ Bluff body diameter/m & \(d\) & 0.032 \\ The length of the bluff/m & \(L\) & 0.12 \\ Equivalent mass/kg & \(M_{\mathit{eff}}\) & \(5.15\times 10^{-3}\) \\ width of the PTC /m & \(L_{b}\) & 0.004 \\ The system damping/N\(\cdot\)s\(\cdot\)m\({}^{-1}\) & \(C_{\mathit{eff}}\) & \(7.6\times 10^{-3}\) \\ The system stiffness/N\(\cdot\)m\({}^{-1}\) & \(K_{\mathit{eff}}\) & 17.21 \\ Damping ratio & \(\zeta\) & 0.013 \\ Natural frequency/Hz & \(f_{n}\) & 9.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of the harvester.