# A Piezoelectric Energy Harvester with Bending-Torsion Vibration in Low-Speed Water
## 2 Modeling of the Harvester System
\[\delta W_{c}=-\int_{0}^{L}c_{m}\big{[}\dot{v}\delta v(s,t)+\dot{w}\delta w(s, t)\big{]}ds \tag{18}\]  
where \(c_{m}\) is the mechanical damping coefficient measured by experiment. In order to solve the above equations, the bending vibration displacements \(v(s,\,t)\), \(w(s,\,t)\) and torsion angular displacement \(\phi(s,\,t)\) are expressed by using Galerkin procedure [4] in the form of  
\[v(s,t)=\sum\limits_{i=1}^{n}\varphi_{vi}(s)r_{i}(t) \tag{19}\]\[w(s,t)=\sum_{i=1}^{n}\varphi_{wi}(s)r_{i}(t) \tag{20}\]  
\[\phi(s,t)=\sum_{i=1}^{n}\varphi_{\phi i}(s)r_{i}(t) \tag{21}\]  
where \(\phi\)(s) and \(r(t)\) are the mode shapes and the model coordinates, respectively. They can be expressed as:  
\[ \begin{array}{l}\varphi_{vi}(s)=A_{1}\sin\lambda_{1i}s+B_{1}\cos\lambda_{1i }s+C_{1}\text{sinh}\lambda_{1i}s+D_{1}\cosh\lambda_{1i}s\\ \varphi_{wi}(s)=A_{2}\sin\lambda_{2i}s+B_{2}\cos\lambda_{2i}s+C_{2}\text{sinh} \lambda_{2i}s+D_{2}\cosh\lambda_{2i}s\\ \varphi_{\phi i}(s)=A_{3}\sin\lambda_{3i}s+B_{3}\cos\lambda_{3i}s\end{array}  \tag{22}\]  
where the \(A_{n}\), \(B_{n}\), \(C_{n}\) and \(D_{n}\) are arbitrary constants and \(\lambda_{ni}\) are determined by the natural frequency of the piezoelectric beam system, given by:  
\[ \begin{array}{l}\lambda_{1i}=\sqrt[4]{\frac{\rho E_{I}g}{\rho E_{I}g}} \lambda_{2i}\\ \lambda_{1i}\frac{EI_{I}}{\rho S}={\lambda_{3i}}^{2}\sqrt{\frac{GI_{I}}{I_{c}^ {3}}}\end{array}  \tag{23}\]  
where \(EI_{\eta}\) and \(EI_{\xi}\) are the bending stiffness of the beam with respect to axes \(I\)-\(\eta\), \(I\)-\(\xi\). \(GI_{p}\) is the torsion stiffness of the beam with respect to axis \(I\)-\(\xi\), as follows:  
\[ \begin{array}{l}EI_{\eta}=\frac{b\left(E_{s}\left(h_{b}{}^{3}-h_{a}{}^{3} \right)+E_{p}\left(h_{c}{}^{3}-h_{b}{}^{3}\right)\right)}{3}\\ EI_{\xi}=\frac{b^{3}\left(E_{s}\left(h_{b}-h_{a}\right)+E_{p}\left(h_{c}-h_{b} \right)\right)}{12}\end{array} \]  
\[GI_{p}=G_{s}\bigg{[}\frac{h_{s}b^{3}}{12}+\frac{b(h_{b}{}^{3}-h_{a}{}^{3})}{3 }\bigg{]}+G_{p}\bigg{[}\frac{h_{p}b^{3}}{12}+\frac{b(h_{c}{}^{3}-h_{b}{}^{3}) }{3}\bigg{]}\]  
where \(h_{a}\), \(h_{b}\) and \(h_{c}\) are the positions of the layers given with respect to the neutral axis which are defined in Figure 4.  
\[h_{0}=\frac{\left(h_{s}+h_{p}\right)E_{p}h_{p}}{2\left(E_{p}h_{p}+E_{s}h_{s} \right)}+h_{s}/2\]  
as follows:  
\[h_{a}=-h_{0}\quad h_{b}=h_{s}-h_{0}\quad h_{c}=\left(h_{s}+h_{p}\right)-h_{0}\]  
According to the work condition, their associated boundary conditions are used to solve the mode shapes above. \[v(0,t)=0\quad\frac{\partial v}{\partial s}\bigg{|}_{(0,t)}=0\quad w(0,t)=0 \quad\frac{\partial w}{\partial s}\bigg{|}_{(0,t)}=0\quad\phi(0,t)=0 \tag{24}\]  
Figure 4: The neutral axis. \[ \begin{array}{l}EI_{\eta}\left.\frac{\partial^{2}\varphi}{\partial s^{2}}\right|_{s= L_{s}}+\frac{L_{c}}{2}M_{c}\left(\frac{\partial^{2}\varphi}{\partial t^{2}}+ \frac{\partial^{3}\varphi}{\partial\delta t^{2}}\frac{L_{c}}{2}\right)\bigg{|} _{s=L_{s}}\\ +\Big{(}L_{c}-\frac{L_{f}}{2}\Big{)}M_{f}\left[\frac{\partial^{2}\varphi}{ \partial t^{2}}+\frac{\partial^{3}\varphi}{\partial\delta t^{2}}\Big{(}L_{c}- \frac{L_{f}}{2}\Big{)}\right]\bigg{|}_{s=L_{s}}+M_{c}l_{d}\frac{L_{c}}{2} \left.\frac{\partial^{2}\varphi}{\partial\delta t^{2}}\right|_{s=L_{s}}=0\\ \end{array}  \tag{25}\]  
\[ \begin{array}{l}EI_{\eta}\left.\frac{\partial^{3}\varphi}{\partial s^{3}} \right|_{s=L_{s}}-M_{c}\left(\frac{\partial^{2}\varphi}{\partial t^{2}}+ \frac{\partial^{3}\varphi}{\partial\delta t^{2}}\frac{L_{c}}{2}\right)\bigg{|} _{s=L_{s}}\\ -M_{f}\left[\frac{\partial^{2}\varphi}{\partial t^{2}}+\frac{\partial^{3} \varphi}{\partial\delta t^{2}}\Big{(}L_{c}-\frac{L_{f}}{2}\Big{)}\right] \bigg{|}_{s=L_{s}}-M_{c}l_{d}\left.\frac{\partial^{2}\varphi}{\partial t^{2}} \right|_{s=L_{s}}=0\\ \end{array}  \tag{26}\]  
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