# A Piezoelectric Energy Harvester with Bending-Torsion Vibration in Low-Speed Water
## 2 Modeling of the Harvester System
\begin{array}{l}\rho_{b}\int_{V_{b}}\big{(}\varphi_{vi}(s)\varphi_{vj}(s)+ \varphi_{wi}(s)\varphi_{wj}(s)\big{)}dV_{b}\\ +M_{c}\Big{(}\varphi_{vi}(L_{s})+\frac{L_{c}}{2}\varphi_{vi}^{\prime}(L_{s})\Big{)} \Big{(}\varphi_{vj}(L_{s})+\frac{L_{c}}{2}\varphi_{vj}^{\prime}(L_{s})\Big{)} \\ +M_{c}\Big{(}\varphi_{wi}(L_{s})+\frac{L_{c}}{2}\varphi_{vi}^{\prime}(L_{s}) \Big{)}\Big{(}\varphi_{wj}(L_{s})+\frac{L_{c}}{2}\varphi_{wj}^{\prime}(L_{s}) \Big{)}\\ +M_{f}\Big{(}\varphi_{vi}(L_{s})+\frac{(2L_{c}-L_{f})}{2}\varphi_{vi}^{\prime}(L_{s })\Big{)}\Big{(}\varphi_{vj}(L_{s})+\frac{(2L_{c}-L_{f})}{2}\varphi_{vj}^{ \prime}(L_{s})\Big{)}\\ +\Big{(}l_{c\xi}+J_{f\xi}\Big{)}\left[\varphi_{\phi i}(L_{s})+\varphi_{wi}^{ \prime}(L_{s})\varphi_{vi}^{\prime}(L_{s})\right]\left[\varphi_{\phi j}(L_{s})+ \varphi_{vj}^{\prime}(L_{s})\varphi_{vj}^{\prime}(L_{s})\right]\\ =\delta_{ij}\end{array}  \tag{36}\]\[\int_{0}^{L}\varphi_{v^{\prime\prime}}{}^{\prime}E_{\eta}\varphi_{v^{\prime\prime}} ds+\int_{0}^{L}\varphi_{\phi^{\prime}}{}^{\prime}G_{Ip}\varphi_{\phi\phi^{ \prime}}{}^{\prime}ds=\delta_{ij}\omega_{i}{}^{2} \tag{37}\]  
where \(\delta_{ij}\) is the Kronecker delta, defined as 1 when \(i\) is equal to \(j\) and 0 otherwise. As a consequence, combining the boundary conditions (30)-(35) and orthogonality conditions (36) and (37), as well as Equation (23), mode shapes and natural frequency of the beam can be obtained. As the energy harvesting system is designed to work in the low frequency environment, the first mode shape [33] is accurate enough to research the energy harvesting in this paper. By substituting Equations (19)-(21) and the orthogonality conditions (36) and (37) into the kinetic energy Equations (3)-(5), potential energy Equations (6), (9) and (10), and the virtual work Equations (11), (12), (16)-(18), the energy equations and virtual work equations are rewritten as:  
The kinetic energy equation \(T\):  
\[T=\dot{r}^{2}(t)\Big{/}2 \tag{38}\]  
The potential energy equation \(U\):  
\[U=H_{1}Vr(t)+\frac{1}{2}\omega_{i}{}^{2}r^{2}(t)-\frac{1}{2}H_{1}Vr(t)+H_{2}V^ {2}+H_{3}r^{2}(t) \tag{39}\]  
where \(H_{n}\) is the coefficient of the variables, given by:  
\[H_{1}=-\varepsilon_{31}\frac{b(h_{b}{}^{2}-h_{a}{}^{2})}{2h_{p}}\int_{0}^{L_{p }}\varphi_{v^{\prime\prime}}(s)ds\]  
\[H_{2}=-\frac{\varepsilon_{11}bL_{p}}{2h_{p}}\]  
\[H_{3}=M_{c}\xi\Big{[}\varphi_{v}(L_{s})\varphi_{v}{}^{\prime}(L_{s})+\frac{L_ {c}}{2}\varphi_{v}{}^{\prime}{}^{2}(L_{s})+\varphi_{w}(L_{s})\varphi_{w^{ \prime}}{}^{\prime}(L_{s})+\frac{L_{c}}{2}\varphi_{w}{}^{\prime}{}^{2}(L_{s} )\Big{]}\]  
\[+\rho_{b}S_{b}\xi\int_{0}^{L_{s}}(\varphi_{v}\varphi_{v}{}^{\prime}+\varphi_{w }\varphi_{w})ds\]  
The virtual work equation:  
\[\frac{\delta W}{\delta r(t)}=H_{4}F(t)+H_{5}r(t)+H_{6}\dot{r}(t) \tag{40}\]  
where the coefficient \(H_{n}\) are expressed as:  
\[H_{4}=\bigg{[}\varphi_{v}(L_{s})+\varphi_{v}^{\prime}(L_{s})\bigg{(}L_{c}- \frac{L_{t}}{2}\bigg{)}\bigg{]}+l_{d}\varphi_{\phi}(L_{s})\]  
\[H_{5}=\rho_{w}\xi\frac{\pi D^{2}}{4}\int_{0}^{L_{d}}\Big{[}\varphi_{v}(s) \varphi_{v^{\prime}}(L_{s})+\bigg{(}L_{c}-\frac{L_{t}}{2}\bigg{)}\varphi_{v^{ \prime}}{}^{2}(s)\Big{]}ds\]  
\[T=\dot{r}^{2}(t)\Big{/}2 \tag{38}\]0  
\[T=\dot{r}^{2}(t)\Big{/}2 \tag{38}\]1  
Now, the Lagrange function which is the subtraction of kinetic energy Equation (38) and potential energy Equation (39) is established and expressed as:  
\[ \begin{array}{l}L=T-U\\ =\frac{1}{2}\dot{r}^{2}(t)-\bigg{(}\frac{1}{2}\omega_{i}{}^{2}+H_{3}\bigg{)}r^ {2}(t)-\frac{1}{2}H_{1}Vr(t)-H_{2}V^{2}\end{array}  \tag{41}\]  
This occurs when substituting Equation (40) and Lagrange function (41) into [PERSON]'s Equation (2), and assigning variables \(r\) and \(V\), respectively.