# A Flutter-Based Electromagnetic Wind Energy Harvester: Theory and Experiments
## 2 Device Structure and Modal Analysis
 \tag{13}\]By substituting Equation (10) into the boundary conditions (12), and considering the orthogonal normalization condition (to obtain the torsional vibration modes):  
\[\sum_{i}{(\rho I_{m})_{i}}{\int_{0}^{i_{i}}{\varphi_{iu}(x_{i})\varphi_{iv}(x_{i} )dx_{i}}}=\delta_{uv} \tag{13}\]  
where \(u\) and \(v\) represent the \(u\)th and \(v\)th torsional mode, respectively; \(\delta_{uv}\) is the Kronecker delta (\(\delta_{uv}=0\) when \(u\
eq v\) and \(\delta_{uv}=1\) when \(u=v\)). We can obtain:  
\[ \begin{array}{c}\chi_{i}=\chi_{iu},\ f_{au}=\frac{\omega_{au}}{2\pi}=\frac{ \chi_{iu}}{2\pi}\sqrt{\frac{(\rho I_{m})_{i}}{(EI_{u})_{i}}},\ \varphi_{i}(x_{i})=\varphi_{iu}(x_{i}),\ q(t)=q_{u}(t),\\ u=1,2,\cdots\end{array}  \tag{14}\]  
where \(f_{au}\), \(\varphi_{u}\), \(q_{u}\) represents the \(u\)th torsional natural frequency, the \(u\)th torsional vibration mode and the \(u\)th torsional modal, respectively. The torsion angle \(\alpha_{i}\) can be written as:  
\[\alpha_{i}(x_{i},t)=\sum_{u}{\varphi_{iu}(x_{i})q_{u}(t)} \tag{15}\]  
The modal of the structure can be obtained by finite element simulation as well, which has been used for verification of modal analysis by the analytical model. Figure 2 and Table 2 show the first three bending and torsional vibration modes and the corresponding natural frequencies by finite element simulation (ANSYS workbench) and the derived analytical model, respectively (the magnets are located in the middle of the PET membrane). We can see that the results of the derived analytical model and the finite element simulation are consistent, which demonstrates that the derived analytical model is reliable. The errors of natural frequencies between the analytical model and the finite element results are less than 5%. The errors mainly come from the constraint of the derived analytical model on the freedom of vibration. In addition, the first and third bending and torsional vibration modes are symmetric to the perpendicular bisector of magnets, and the second bending and torsional vibration modes are anti-symmetric around the midpoint of the magnets. Therefore, it is easy to produce coupled flutter due to similarity between the bending and torsional vibration modes of the same order. There is no coupling between the adjacent bending and torsional modes (refer to the definition of \(\Lambda_{ru}\) in Equation (34)). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Natural** & \multicolumn{3}{c}{**Bending Mode**} & \multicolumn{3}{c}{**Torsional Mode**} \\ \cline{2-7}
**Frequency (Hz)** & **1 st Mode** & **2 nd Mode** & **3 rd Mode** & **1 st Mode** & **2 nd Mode** & **3 rd Mode** \\ \hline ANSYS FEM & 3.83 & 23.32 & 84.11 & 13.46 & 164.71 & 166.04 \\ Analytical model & 3.71 & 22.77 & 80.71 & 13.54 & 162.50 & 163.83 \\ Error (\%) & 3.13 & 2.36 & 4.04 & 0.59 & 1.34 & 1.33 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The bending and torsional natural frequencies by finite element simulation and the derived analytical model. Figure 2: **(a–c)** The first three bending vibration modes, and **(d–f)** the first three torsional vibration modes by finite element simulation. The **(g)** bending and **(h)** torsional vibration modes by the derived analytical model.