# A Flutter-Based Electromagnetic Wind Energy Harvester: Theory and Experiments
## 2 Device Structure and Modal Analysis  
The structure of the proposed wind energy harvester is illustrated in Figure 1; it mainly consists of four parts: a flexible polyethylene terephthalate (PET) membrane, two pieces of magnets, a set of copper coils, and an aluminum shell. The material and structural parameters of the wind energy harvester are given in Table 1. The PET membrane was doubly clamped by four screws. The N35 N6 Fe8 magnets were attached to the PET membrane, which were magnetized along the thickness direction. The coils were as close to the magnets as possible for higher voltage and electric power. The overall dimensions of the device are 142 mm \(\times\) 30 mm \(\times\) 25 mm. The wind direction is perpendicular to the axis of the PET membrane. When wind speed is higher than the flutter critical wind speed, the PET membrane combined with magnets oscillate with a large amplitude, and the energy harvester generates high electric power based on [PERSON]'s law of electromagnetic induction. The vibrational structure of the energy harvester only consists of a doubly-clamped PET membrane and two pieces of magnet, which is similar to a bridge. It vibrates with two degrees of freedom in the wind: bending and torsion. Modal analysis is the foundation of flutter analysis; the bending modal can be obtained from the Euler beam equation with undamped free vibration [17; 30]  
\[\left(EI\right)_{i}\frac{\partial^{4}h_{i}(x_{i},t)}{\partial{x_{i}}^{4}}+\mu_{ i}\frac{\partial^{2}h_{i}(x_{i},t)}{\partial{t}^{2}}=0,i=1,2,3 \tag{1}\]  
where \((EI)_{i}\), \(\mu_{i}\), \(x_{i}\) and \(h_{i}\) represent bending stiffness, the linear density, the coordinate, and the bending displacement of the \(i\)th section, respectively; \(t\) is the time. The method of separation of variables can be used to solve Equation (1) by separating the spatial and temporal functions as \(h_{i}(x_{i},t)=\phi_{i}(x_{i})p(t)\)[31]. Thus, we can obtain:  
\[p(t)=A\sin\omega_{h}t+B\cos\omega_{h}t \tag{2}\]  
\[\phi_{i}(x_{i})=A_{i}\mathrm{sinh}(\lambda_{i}x_{i})+B_{i}\cosh(\lambda_{i}x_ {i})+C_{i}\sin(\lambda_{i}x_{i})+D_{i}\cos(\lambda_{i}x_{i}) \tag{3}\]  
where \(\phi_{i}\) and \(p\) represent variables related to coordinate and time, respectively, \(\omega_{h}\) is the bending angular frequency, \(\lambda_{i}=\left[\omega_{h}^{2}\mu_{i}/\left(EI\right)_{i}\right]^{1/4}A\), \(B\), \(A_{i}\), \(B_{i}\), \(C_{i}\) and \(D_{i}\) are undetermined coefficients. The boundary conditions are: \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Description** & **Value** \\ \hline \(E_{p}\) & [PERSON]’s modulus of PET membrane & 2.4 GPa [28; 29] \\ \(\rho_{p}\) & Density of PET membrane & 1400 kg/m\({}^{3}\)[28; 29] \\ \(v_{p}\) & [PERSON]’s ratio of PET membrane & 0.36 [28; 29] \\ \(l\times 2b\times t_{p}\) & Dimension of PET membrane & 130 mm \(\times\) 20 mm \(\times\) 0.18 mm \\ \(E_{m}\) & [PERSON]’s modulus of magnet & 160 GPa \\ \(\rho_{m}\) & Density of magnet & 7500 kg/m\({}^{3}\) \\ \(v_{m}\) & [PERSON]’s ratio of magnet & 0.3 \\ \(l_{2}\times 2b\times t_{m}\) & Dimension of magnet & 20 mm \(\times\) 20 mm \(\times\) 1 mm \\ \(\Phi_{1}\times\Phi_{2}\times h\) & Inner diameter, external diameter, and thickness of coils & 3 mm \(\times\) 27 mm \(\times\) 6 mm \\ \(N\) & Turns of coils & 5310 \\ \(R_{c}\) & Resistance of coils & 580 \(\Omega\) \\ \(d_{0}\) & Gap between magnets and coils & 14 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material and structural parameters of the device. Figure 1: Schematic diagram of the flutter-based electromagnetic energy harvester. \[