# A Flutter-Based Electromagnetic Wind Energy Harvester: Theory and Experiments
## 3 Flutter Analysis
\[U_{\sigma}=\left[1+\left(\frac{\omega_{a1}}{\omega_{h1}}-0.5\right)\sqrt{ \frac{0.72\mu r}{\pi\rho_{a}b^{3}}}\right]\omega_{h1}b \tag{36}\]  
\[U_{\sigma}=7.44 bf_{\alpha}\left\{\frac{\mu r}{\rho_{a}(2b)^{3}}\bigg{[}1- \left(\frac{f_{h1}}{f_{a1}}\right)^{2}\bigg{]}\right\}^{1/2} \tag{37}\]  
where \(r\) is the cross-section radius of gyration. For Equation (35), it is impossible to consider all of the bending and torsional modes. Figure 2(a) shows the relationship between the diverging rate and the wind speed, when all the first three bending and torsional modes are involved in the calculation (the magnets are located in the middle of the PET membrane). The diverging rate judges whether the flutter occurs or not. When the diverging rate is negative, the vibration is attenuated and no flutter occurs; when the diverging rate is positive, the vibration is divergent and the flutter takes place; when the diverging rate is zero, the corresponding wind speed is the critical wind speed. There are three curves corresponding to three sets of solutions for Equation (35). It indicates that the vibration of the energy harvester is divergent when wind speed is between 6.46 m/s and 9.39 m/s, and above 20.34 m/s, there will be large amplitudes when the flutter happens. The critical wind speed is 6.46 m/s, which is the most important parameter of the flutter, and the flutter phenomenon can occur only when wind speed is higher than the critical wind speed. Figure 2(b) shows the relationship between the diverging rate and the wind speed, when the first, second, and third bending and torsional modes are involved in the calculation, respectively. The results are similar to the three curves in Figure 2(a). Critical wind speed is 6.45 m/s when only the first bending and torsional modes are considered. We can infer that the flutter at low wind speed is mainly caused by the coupling of the first bending and torsional modes. The high order modes participate in the flutter as the wind speed increases and the state of motion of the wind belt is more chaotic. To verify the flutter analysis, we calculated the critical wind speed of the wind belt when there is no magnet attached to the PET membrane (shown in Figure 3c), and compared it with the results using Equations (36) and (37) (shown in Table 3); the calculated critical wind speed is in good agreement with previous empirical formulae, which can confirm the effectiveness of the presented analytical model. Figure 3d shows the relationship between the critical wind speed and the structural damping ratio. In a small range, the structural damping has a neglectable effect on the critical wind speed of the flutter, while aerodynamic damping is more dominating than structural damping. Therefore, it is reasonable to set the structural damping ratio as 0.02 in the flutter calculations. To investigate the influence of magnets' positions on the critical wind speed of flutter, the critical wind speeds of the wind belt with different positions of magnets was calculated. To ensure accuracy of the results, all the first three modes were involved in the flutter calculation process. As shown in Table 4, magnets placed near the middle of the PET membrane can result in a lower critical wind speed. However, the output value needs further investigation. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Equations** & **[PERSON]** & **Selberg** & **This Work** \\ \hline Critical wind speed (m/s) & 8.60 & 7.43 & 7.67 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The comparison of critical wind speed by different equations when there is no magnet attached to the PET membrane. Figure 3: **(a)** The diverging rate as a function of wind speed when all the first three bending and torsional modes are involved in the calculation. (**b**) The diverging rate as a function of wind speed when the first, second, and third bending and torsional modes are involved in the calculation, respectively. (**c**) The diverging rate as a function of wind speed when there is no magnet attached to the PET membrane.