# A Flutter-Based Electromagnetic Wind Energy Harvester: Theory and Experiments
## 2 Device Structure and Modal Analysis
\begin{split}\phi_{1}(0)=0,\ \left.\frac{\partial\phi_{1}(x_{1})}{ \partial x_{1}}\right|_{x_{1}=0}=0,\ \phi_{3}(I_{3})=0,\ \left.\frac{\partial\phi_{3}(x_{3})}{ \partial x_{3}}\right|_{x_{3}=I_{3}}=0\\ \phi_{i}(l_{i})=\] \[\left.\left(EI\right)_{i}\frac{\partial^{3}\phi_{i}(x_{i})}{ \partial x_{i}^{3}}\right|_{x_{i}=I_{i}}=\left.\left(EI\right)_{i+1}\frac{ \partial^{3}\phi_{i+1}(x_{i+1})}{\partial x_{i+1}^{3}}\right|_{x_{i+1}=0}\\ \left.\left(EI\right)_{i}\frac{\partial^{3}\phi_{i}(x_{i})}{ \partial x_{i}^{3}}\right|_{x_{i}=I_{i}}=\left.\left(EI\right)_{i+1}\frac{ \partial^{3}\phi_{i+1}(x_{i+1})}{\partial x_{i+1}^{3}}\right|_{x_{i+1}=0}\\ \end{split}  \tag{4}\]  
\(l_{i}\) is the length of each section. By substituting Equation (3) into the boundary conditions (4), and considering the orthogonal normalization condition (to obtain the bending vibration modes):  
\[\sum_{i}\mu_{i}\int_{0}^{l_{i}}\phi_{ir}(x_{i})\phi_{is}(x_{i})dx_{i}=\delta_{ rs} \tag{5}\]  
where \(r\) and \(s\) represent the \(r\)th and \(s\)th bending mode, respectively, \(\delta_{rs}\) is the Kronecker delta (\(\delta_{rs}=0\) when \(r\
eq s\) and \(\delta_{rs}=1\) when \(r=s\)). We can obtain:  
\[\lambda_{i}=\lambda_{ir},\ f_{hr}=\frac{\omega_{hr}}{2\pi}=\frac{\lambda_{ir} ^{2}}{2\pi}\sqrt{\frac{(EI)_{i}}{\mu_{i}}},\ \phi_{i}(x_{i})=\phi_{ir}(x_{i}),\ p(t)=p_{r}(t),\ r=1,2,\cdots \tag{6}\]  
where \(f_{hr}\), \(\phi_{r}\), \(p_{r}\) represents the \(r\)th bending natural frequency, the \(r\)th bending vibration mode, and the \(r\)th bending modal, respectively. Bending displacement \(l_{i}\) can be written as:  
\[l_{i}(x_{i},t)=\sum_{r}\phi_{ir}(x_{i})p_{r}(t) \tag{7}\]  
Similar to the bending modal, the torsional modal can be obtained from the following equation:  
\[(EI_{\omega})_{i}\frac{\partial^{4}\alpha_{i}(x_{i},t)}{\partial{x_{i}}^{4}}- (GI_{t})_{i}\frac{\partial^{2}\alpha_{i}(x_{i},t)}{\partial{x_{i}}^{2}}+(\rho l _{m})_{i}\frac{\partial^{2}\alpha_{i}(x_{i},t)}{\partial t^{2}}=0,i=1,2,3 \tag{8}\]  
where \((EI_{\omega})_{i}\), \((GI_{t})_{i}\), \((\rho l_{m})_{i}\) and \(\alpha_{i}\) represent the warping stiffness, the torsional stiffness, the moment of inertial per unit length, and the twist angle of the \(i\)th section, respectively. Equation (8) can also be solved by separating variables as \(\alpha_{i}(x_{i},t)=\varphi_{i}(x_{i})q(t)\)[31], where:  
\[q(t)=S\sin\omega_{\alpha}t+T\cos\omega_{\alpha}t \tag{9}\]  
\[\varphi_{i}(x_{i})=S_{i}\sinh(m_{i}x_{i})+T_{i}\cosh(m_{i}x_{i})+U_{i}\sin(n_{ i}x_{i})+V_{i}\cos(n_{i}x_{i}) \tag{10}\]  
\[m_{i}=\frac{\sqrt{4\chi_{i}^{2}+k_{i}^{2}}+k_{i}}{2},\ n_{i}=\frac{\sqrt{4 \chi_{i}^{2}+k_{i}^{2}}-k_{i}}{2},\ k_{i}=\frac{(GI_{t})_{i}}{(EI_{\omega})_{i}} \tag{11}\]  
where \(\varphi_{i}\) and \(q\) represent variables related to coordinate and time, respectively; \(\omega_{\alpha}\) is the torsional angular frequency, \(\chi_{i}=\sqrt{\omega_{\alpha}^{2}(EI_{\omega})_{i}/(\rho l_{m})_{i}}\), \(S\), \(T\), \(S_{i}\), \(T_{i}\), \(U_{i}\), and \(V_{i}\) are undetermined coefficients. The boundary conditions are:  
\[\begin{split}\varphi_{1}(0)=0,\ \left.\frac{\partial\varphi_{1}(x_{1})}{ \partial x_{1}}\right|_{x_{1}=0}=0,\ \varphi_{3}(I_{3})=0,\ \left.\frac{\partial \varphi_{3}(x_{3})}{\partial x_{3}}\right|_{x_{3}=I_{3}}=0\\ \varphi_{i}(l_{i})= \tag{12}\]  
\[ \begin{split}(GI_{t})_{i}\frac{\partial\varphi_{i}(x_{i})}{ \partial x_{i}}\bigg{|}_{x_{i}=I_{i}}=\left.(GI_{t})_{i+1}\frac{\partial\varphi _{i+1}(x_{i+1})}{\partial x_{i+1}}\right|_{x_{i+1}=0}\\ \left.(EI_{\omega})_{i}\frac{\partial^{3}\varphi_{i}(x_{i})}{ \partial x_{i}^{3}}\bigg{|}_{x_{i}=I_{i}}=\left.(EI_{\omega})_{i}\frac{ \partial^{3}\varphi_{i+1}(x_{i+1})}{\partial x_{i+1}}\right|_{x_{i+1}=0}\end{split}