# A Piezoelectric Energy Harvester with Bending-Torsion Vibration in Low-Speed Water
## 2 Modeling of the Harvester System
\begin{array}{l}EI_{\xi}\left.\frac{\partial^{2}\varphi}{\partial s^{2}} \right|_{s=L_{s}}+\frac{L_{c}}{2}M_{c}\left(\frac{\partial^{2}\varphi}{ \partial t^{2}}+\frac{\partial^{3}\varphi}{\partial\delta t^{2}}\frac{L_{c}} {2}\right)\bigg{|}_{s=L_{s}}\\ +\Big{(}L_{c}-\frac{L_{f}}{2}\Big{)}M_{f}\left[\frac{\partial^{2}\varphi}{ \partial\delta t^{2}}+\frac{\partial^{3}\varphi}{\partial\delta t^{2}}\Big{(} L_{c}-\frac{L_{f}}{2}\Big{)}\right]\bigg{|}_{s=L_{s}}=0\\ \end{array}  \tag{27}\]  
\[ \begin{array}{l}EI_{\xi}\left.\frac{\partial^{3}\varphi}{\partial s^{3}} \right|_{s=L_{s}}-M_{c}\left(\frac{\partial^{2}\varphi}{\partial t^{2}}+ \frac{\partial^{3}\varphi}{\partial\delta t^{2}}\frac{L_{c}}{2}\right)\bigg{|} _{s=L_{s}}\\ -M_{f}\left[\frac{\partial^{2}\varphi}{\partial t^{2}}+\frac{\partial^{3} \varphi}{\partial\delta t^{2}}\Big{(}L_{c}-\frac{L_{f}}{2}\Big{)}\right] \bigg{|}_{s=L_{s}}=0\\ \end{array}  \tag{28}\]  
\[ \begin{array}{l}GI_{p}\left.\frac{\partial\varphi}{\partial s}\right|_{s=L_ {s}}+J_{c}\frac{\partial^{2}\varphi}{\partial\delta t^{2}}\bigg{|}_{s=L_{s}}+M _{c}l_{d}\left(\frac{\partial^{2}\varphi}{\partial t^{2}}+\frac{\partial^{3} \varphi}{\partial\delta t^{2}}\frac{L_{c}}{2}\right)\bigg{|}_{s=L_{s}}=0\\ \end{array}  \tag{29}\]  
By substituting Equations (19)-(21) into Equations (24)-(29), the equations of boundary conditions expressed by mode shapes can be written as:  
\[ \begin{array}{l}\varphi_{vi}(0)=0\quad\varphi_{vi}^{\prime}(0)=0\quad \varphi_{wi}(0)=0\quad\varphi_{wi}^{\prime}(0)=0\quad\varphi_{\phi i}(0)=0\\ \end{array}  \tag{30}\]  
\[ \begin{array}{l}EI_{\eta}\varphi_{vi}^{\prime\prime}(L_{s})-\frac{L_{c}}{2}M _{c}\omega_{i}^{\prime 2}\Big{(}\varphi_{vi}(L_{s})+\varphi_{vi}^{\prime}(L_{s})\frac{L_{c}}{2} \Big{)}\\ -\Big{(}L_{c}-\frac{L_{f}}{2}\Big{)}M_{f}\omega_{i}^{\prime 2}\Big{[}\varphi_{vi}(L_{s})+ \varphi_{vi}^{\prime}(L_{s})\Big{(}L_{c}-\frac{L_{f}}{2}\Big{)}\Big{]}-M_{c} \omega_{i}^{\prime 2}l_{d}\frac{L_{c}}{2}\varphi_{\phi i}(L_{s})=0\\ \end{array}  \tag{31}\]  
\[ \begin{array}{l}EI_{\eta}\varphi_{vi}^{\prime\prime\prime}(L_{s})+M_{c} \omega_{i}^{\prime 2}\Big{(}\varphi_{vi}(L_{s})+\varphi_{vi}^{\prime}(L_{s})\frac{L_{c}}{2} \Big{)}\\ +M_{f}\omega_{i}^{\prime 2}\Big{[}\varphi_{vi}(L_{s})+\varphi_{vi}^{\prime}(L_{s}) \Big{(}L_{c}-\frac{L_{f}}{2}\Big{)}\Big{]}+M_{c}\omega_{i}^{\prime 2}l_{d}\varphi_{\phi i}(L_{s})=0\\ \end{array}  \tag{32}\]  
\[ \begin{array}{l}EI_{\xi}\varphi_{wi}^{\prime\prime}(L_{s})-\frac{L_{c}}{2}M _{c}\omega_{i}^{\prime 2}\Big{(}\varphi_{wi}(L_{s})+\varphi_{vi}^{\prime}(L_{s})\frac{L_{c}}{2} \Big{)}\\ -\Big{(}L_{c}-\frac{L_{f}}{2}\Big{)}M_{f}\omega_{i}^{\prime 2}\Big{[}\varphi_{wi}(L_{s})+ \varphi_{vi}^{\prime}(L_{s})\Big{(}L_{c}-\frac{L_{f}}{2}\Big{)}\Big{]}=0\\ \end{array}  \tag{33}\]  
\[ \begin{array}{l}EI_{\xi}\varphi_{wi}^{\prime\prime\prime}(L_{s})+M_{c} \omega_{i}^{\prime 2}\Big{(}\varphi_{wi}(L_{s})+\varphi_{vi}^{\prime}(L_{s})\frac{L_{c}}{2} \Big{)}\\ +M_{f}\omega_{i}^{\prime 2}\Big{[}\varphi_{wi}(L_{s})+\varphi_{wi}^{\prime}(L_{s}) \Big{(}L_{c}-\frac{L_{f}}{2}\Big{)}\Big{]}=0\\ \end{array}  \tag{34}\]  
\[ \begin{array}{l}GI_{p}\varphi_{\phi i}^{\prime}(L_{s})-J_{c\xi}\omega_{i}^{ \prime 2}\varphi_{\phi i}(L_{s})-M_{c}l_{d}\omega_{i}^{\prime 2}\Big{(}\varphi_{vi}(L_{s})+ \varphi_{vi}^{\prime}(L_{s})\frac{L_{c}}{2}\Big{)}=0\\ \end{array}  \tag{35}\]  
Normalizing the eigenfunctions using the following orthogonality conditions of mass and stiffness, the relations between different mode shapes are obtained. \[