# A Piezoelectric Energy Harvester with Bending-Torsion Vibration in Low-Speed Water
## 2 Modeling of the Harvester System
\begin{array}{ll}\varepsilon_{11}=z\phi\eta^{\prime\prime}-y\eta^{\prime \prime}&\gamma_{12}=2\varepsilon_{12}=-z\phi^{\prime}&\gamma_{13}=2 \varepsilon_{13}=y\phi^{\prime}\\ \sigma_{11}{}^{s}=E_{s}\varepsilon_{11}&\sigma_{12}{}^{s}=2G_{s}\varepsilon_{ 12}&\sigma_{13}{}^{s}=2G_{s}\varepsilon_{13}\\ \sigma_{12}{}^{p}=2G_{p}\varepsilon_{12}&\sigma_{13}{}^{p}=2G_{p}\varepsilon_{ 13}\end{array}  \tag{7}\]  
Due to the poling of the PZT layer, the strain and electric displacement vector are governed by the constitutive equations, given by:  
\[ \begin{array}{ll}\sigma_{11}{}^{p}=E_{p}\varepsilon_{11}-\varepsilon_{31}E _{2}\\ D_{2}=e_{31}\varepsilon_{\eta}+\varepsilon_{11}\ E_{2}\end{array}  \tag{8}\]  
where \(E_{s}\) and \(E_{p}\) are the [PERSON]'s modulus at constant electric field, \(G_{s}\) and \(G_{p}\) are the shear modulus, \(E_{2}\) is the electric field intensity in the poling direction, and \(V(t)\) is the voltage between the two piezoelectric electrodes. \[ \begin{array}{l}E_{2}(t)=V(t)/h_{p}\\ U_{V}=-\frac{1}{2}\int_{V_{p}}E_{2}D_{2}dV_{p}\end{array}  \tag{9}\]  
where \(V_{p}\) is the volume of the PZT layer. \[U_{g}=M_{c}g\frac{\partial x(s,t)}{\partial s}\bigg{|}_{s=L}\left[x(s,t)+ \frac{L_{c}}{2}\left.\frac{\partial x(s,t)}{\partial s}\right|_{s=L}\right]+ \int_{V_{b}}\rho_{b}gx(s,t)\frac{\partial x(s,t)}{\partial s}dV_{b} \tag{10}\]  
where \(x(s,\,t)\) is the displacement of the beam in three axes direction of the reference coordinate system. The virtual work includes the virtual work of the external resistance \(\delta W_{R}\), lift force \(\delta W_{F}\), buoyancy of fluid \(\delta W_{b}\), mechanical damping \(\delta W_{c}\) and fluid-added damping \(\delta W_{c\!f}\). \[\delta W_{R}=-V(t)\delta Q(t) \tag{11}\]where \(Q(t)\) is the charge between the piezoelectric electrodes. \[\delta W_{F}=F(t)\delta\bigg{[}v(s,t)\big{|}_{s=L_{s}}+\left.v(s,t)^{\prime} \right|_{s=L_{s}}(L_{\text{c}}-\frac{L_{f}}{2})\bigg{]}+F(t)l_{d}\delta\phi \tag{12}\]  
where \(l_{d}\) is the eccentric distance of mass and given by:  
\[l_{d}=\frac{d^{2}}{D^{2}-d^{2}}e\]  
and \(F(t)\) is the lift force. It is generated when the eccentric cylinder is subjected to the incoming fluid [22], as:  
\[F(t)=\frac{C_{L}(t)\rho_{f}D{U_{0}}^{2}L_{f}}{2} \tag{13}\]  
where \(U_{0}\) is the speed of the incoming fluid. \(C_{L}\) is the vortex lift coefficient. The fluid variable \(q\) is interpreted as a reduced vortex lift coefficient, which is governed by Van [PERSON] equation [31], and \(\omega_{f}\) is the vortex shedding frequency. \[\ddot{q}+\varepsilon\omega_{f}\Big{(}q^{2}-1\Big{)}\dot{q}+\omega_{f}{}^{2}q=f \tag{14}\]  
where  
\[q(t)=2C_{L}(t)/C_{L0}\]  
\[\omega_{f}=2\pi S_{t}U_{0}/D\]  
where \(C_{L0}=0.3\)[32]. And \(f\) is the nondimensional force of vortex [22], given by:  
\[f=(\frac{A}{D})\frac{d^{2}\overline{x}}{dt^{2}}=(\frac{A}{D})\left[\left.\frac {\partial^{2}x(s,t)}{\partial t^{2}}\right|_{s=L_{s}}+\left(L_{c}-\frac{L_{f} }{2}\right)\frac{\partial^{3}x(s,t)}{\partial s\partial t^{2}}\right|_{s=L_{ s}}+\left.\frac{\partial^{2}\phi(s,t)}{\partial t^{2}}\right|_{s=L_{s}}\right] \tag{15}\]  
The virtual work of the buoyancy of fluid \(\delta W_{b}\) is:  
\[\delta W_{b}=\int\limits_{\
u_{w}}\rho_{f}\delta\left.\frac{\partial x(s,t)}{ \partial s}\right|_{s=L_{s}}\delta\bigg{[}x(s,t)+\left(L_{c}-\frac{L_{f}}{2} \right)\frac{\partial x(s,t)}{\partial s}\bigg{|}_{s=L_{s}}\bigg{]}dV_{f} \tag{16}\]  
The virtual work of fluid-added damping \(\delta W_{\sigma f}\) is:  
\[ \begin{array}{l}\delta W_{c_{f}}=c_{f}\left.\frac{\partial v(s,t)}{ \partial t}\right|_{s=L_{s}}\left(L_{c}-\frac{L_{f}}{2}\right)\delta\left[ \left.v(s,t)\right|_{s=L_{s}}\right]\\ +c_{f}\left.\frac{\partial w(s,t)}{\partial t}\right|_{s=L_{s}}\left(L_{c}- \frac{L_{f}}{2}\right)\delta\left[\left.w(s,t)\right|_{s=L_{s}}\right]+c_{f} \omega_{\zeta}\frac{\gamma}{2}\delta\phi(s,t)\big{|}_{s=L_{s}}\end{array}  \tag{17}\]  
where  
\[c_{f}=\gamma\omega_{f}\rho_{f}D^{2}\]  
where \(\gamma=0.8\)[22] is the fluid-added damping coefficient.