# A Piezoelectric Energy Harvester with Bending-Torsion Vibration in Low-Speed Water
## 2 Modeling of the Harvester System
\begin{array}{l}\omega(s,t)=\dot{\psi}\overset{\rightarrow}{Z}+\dot{\theta} \overset{\rightarrow}{Y_{1}}+\dot{\phi}\overset{\rightarrow}{\xi}\\ =(\dot{\phi}-\dot{\psi}\sin\theta)\overset{\rightarrow}{\xi}+(\dot{\psi}\cos \theta\sin\phi+\dot{\theta}\cos\phi)\overset{\rightarrow}{\eta}+(\dot{\psi} \cos\theta\cos\phi-\dot{\theta}\sin\phi)\overset{\rightarrow}{\zeta}\\ =\omega_{\xi}\overset{\rightarrow}{\xi}+\omega_{\eta}\overset{\rightarrow}{ \eta}+\omega_{\zeta}\overset{\rightarrow}{\zeta}\end{array}  \tag{1}\]  
where the over dot denotes the derivative with respect to time \(t\), and the vector arrow denotes the vector along the axes. \(\omega_{\xi}\), \(\omega_{\eta}\), \(\omega_{\zeta}\) are the angular velocity of the rotating beam with respect to the axes \(I\)-\(\xi\), \(I\)-\(\eta\) and \(I\)-\(\zeta\). According to the practical work environment of the coupled bending-torsion motions of the beam, it is necessary to keep the mathematical model as simple as possible. Compared with other displacements, the longitudinal vibration displacement \(u(s,\,t)\), and rotation angular displacements \(\psi(s,\,t)\) and \(\theta(s,\,t)\) are so small that they can be neglected. As a result, \(u(s,\,t)=0\), \(\psi(s,\,t)=\theta(s,\,t)=0\), and from Equation (1), \(\omega_{\xi}=\dot{\phi}\), \(\omega_{\eta}=\omega_{\zeta}=0\). Next, the [PERSON]'s equations are used to derive the governing equations of the motion of the harvesting system. \[\frac{\partial}{\partial t}\left(\frac{\partial L}{\partial\dot{\mu}}\right)- \left(\frac{\partial L}{\partial\mu}\right)=\delta W/\delta\mu \tag{2}\]  
The kinetic energy of the system is the sum of the kinetic energies of the beam (\(T_{b}\)), the eccentric cylinder (\(T_{c}\)) and the additional kinetic energy of fluid (\(T_{f}\)). \[T_{b}=\frac{1}{2}\rho_{b}\left[\int_{V_{b}}\left(\dot{v}^{2}+\dot{w}^{2}\right) dV_{b}+\int_{0}^{L}\left(I_{\xi}\omega_{\xi}^{2}\right)ds\right] \tag{3}\]  
where \(\rho_{b}\) and \(V_{b}\) are the density and volume of the beam, which concludes two parts: the PZT and substrate layer. \(I_{\xi}\) is the rotary inertia per unit length with respect to the axis \(I\)-\(\xi\), given by:  
\[I_{\xi}=\left(b^{2}+h^{2}\right)\Big{/}12\]  
where \(h\) is the thickness of the beam. \[T_{c}=\frac{1}{2}M_{c}\left[\left(\dot{v}+\dot{v}^{\prime}\frac{L_{c}}{2} \right)^{2}+\left(\dot{w}+\dot{w}^{\prime}\frac{L_{c}}{2}\right)^{2}\right] \Bigg{|}_{s=L}+\frac{1}{2}\,I_{\xi}\omega_{\xi}^{2}\Big{|}_{s=L} \tag{4}\]  
Figure 3: The coordinate transformation figure. where \(D\) and \(d\) are the diameter of the cylinder and the eccentric hole, respectively. \(M_{c}\) and \(J_{c\xi}\) are the mass and the rotary inertia with respect to the axis \(I\)-\(\xi\) of the eccentric cylinder, as follows:  
\[ \begin{array}{c}J_{c\xi}=\frac{1}{2}M_{c}\Big{(}\frac{D^{2}}{4}+\frac{d^{2}}{ 4}\Big{)}-\frac{1}{2}\pi\rho_{c}\frac{d^{2}}{4}L_{c}\varepsilon^{2}\\ T_{f}=\frac{1}{2}M_{f}\bigg{[}\Big{(}\dot{v}+\dot{v}^{\prime}\frac{2L_{c}-L_{f }}{2}\Big{|}_{s=L}\Big{)}^{2}+\Big{(}\dot{w}+\dot{w}^{\prime}\frac{2L_{c}-L_{f }}{2}\Big{|}_{s=L}\Big{)}^{2}\bigg{]}+\frac{1}{2}J_{f\xi}\omega_{\xi}{}^{2}|_{ s=L}\end{array}  \tag{5}\]  
where \(M_{f}\) and \(J_{f\xi}\) are the fluid-added mass and rotary inertia with respect to the axis \(I\)-\(\xi\) of the fluid due to the vibration of the eccentric cylinder, given by:  
\[M_{f}=\left.C_{M}\rho_{f}\pi D^{2}L_{f}\right/4\]  
where \(C_{M}=1\) is the fluid-added mass coefficient. \[J_{f\xi}=\left.M_{f}D^{2}\right/8\]  
The potential energies of the whole system include the potential energy (\(U_{i}\)), the electric potential energy (\(U_{V}\)) of the beam and the gravitational potential energy (\(U_{g}\)) of the system. \[U_{i}=\frac{1}{2}\int_{V_{s}}(\sigma^{s}\varepsilon^{s}+\tau^{s}\gamma^{s})dV_ {s}+\frac{1}{2}\int_{V_{p}}[\sigma^{p}\varepsilon^{p}+\tau^{p}\gamma^{p}]dV_{p} \tag{6}\]  
where the strains and stresses in the substrate and PZT layers are:  
\[