# A Piezoelectric Energy Harvester with Bending-Torsion Vibration in Low-Speed Water
## 2 Modeling of the Harvester System  
Figure 1a shows the energy harvester using piezoelectric cantilever beam (PZT) and its working environment. The energy harvester is composed of a piezoelectric cantilever beam and an eccentric cylinder tip. Figure 1b shows the schematic of energy harvester. Table 1 lists the geometric and physical properties of the energy harvesting system. The symbolic variables \(e\) and \(d\) are the eccentricity distance and diameter of cylinder eccentric hole, respectively, which determine the value of the eccentric distance of mass \(l_{d}\). Figure 1: Energy harvester: (**a**) Harvester and its working environment; (**b**) Schematic of harvester. To model the harvester, the following assumptions are adopted:  
1. The piezoelectric beam is assumed to be inextensible. 2. The attachments of the tip cylinder, beam and fixed end are assumed to be rigid. 3. The beam is an Euler-Bernoulli beam. Figure 2 shows a schematic diagram of the cantilever beam. \(L\) and \(b\) are the length and width of the beam (Baoding Hengsheng Acoustics Electron Apparatus Co., Ltd\({}^{\circledoplus}\), Baoding, China), respectively. The thickness of the PZT layer made of PZT 5H defined in Table 1 is \(h_{p}\), and that of the substrate layer is \(h_{s}\). Two coordinate systems are used to describe the coupled bending-torsion motions of the beam. The reference coordinate system (\(O\)-\(XYZ\)) is fixed to the clamped side of the beam, which is adopted to describe the bending motion. \(s\) is used as the arc-coordinate in the length direction of the beam. In the fixed end, \(s=0\), and \(s=L\) in the tip. \(s+u(s,\,t)\), \(v(s,\,t)\) and \(w(s,\,t)\) are the displacements of the infinitesimal element at three directions in the coordinate system \(O\)-\(XYZ\). Meanwhile, a local coordinate system (\(I\)-\(\xi\eta\zeta\)) is adopted to describe the torsion motion. The axis \(I\)-\(\xi\), \(I\)-\(\eta\) and \(I\)-\(\zeta\) are the central principal axes of the cross section in three directions. The local coordinate system \(I\)-\(\xi\eta\zeta\) can be obtained by coordinating the rotation of the reference coordinate system, as shown in Figure 3. Firstly, \(O\)-\(X_{1}\), \(O\)-\(Y_{1}\) are obtained by a counter-clockwise rotation of \(O\)-\(X\), \(O\)-\(Y\) with respect to the axis \(O\)-\(Z\) by the angle \(\psi\). Secondly, \(I\)-\(\xi\), \(O\)-\(Z_{1}\) are obtained by a rotation of \(O\)-\(X_{1}\), \(O\)-\(Z\) with respect to the axis \(O\)-\(Y_{1}\) by the angle \(\theta\). Finally, \(I\)-\(\eta\), \(I\)-\(\zeta\) are obtained by a \begin{table}
\begin{tabular}{c l c} \hline \(\rho_{p}\) & Density of the piezoelectric layer (kg/m\({}^{3}\)) & 7800 \\ \(\rho_{s}\) & Density of the substrate layer (aluminum; kg/m\({}^{3}\)) & 2700 \\ \(\rho_{f}\) & Fluid density (kg/m\({}^{3}\)) & 1000 \\ \(\rho_{c}\) & Density of the cylinder (acrylic; kg/m\({}^{3}\)) & 1200 \\ \(E_{p}\) & Young modulus of the piezoelectric layer (Gpa) & 59.77 \\ \(E_{s}\) & Young modulus of the substrate layer (Gpa) & 70 \\ \(L\) & Length of the beam (mm) & 85 \\ \(b\) & Width of the beam (mm) & 25 \\ \(h_{p}\) & Thickness of the piezoelectric layer (mm) & 0.2 \\ \(h_{s}\) & Thickness of the substrate layer (mm) & 0.2 \\ \(e_{31}\) & Piezoelectric constant (C/m\({}^{2}\)) & \(-\)16.6 \\ \(e_{11}\) & Piezoelectric permittivity (nF/m) & 41.78 \\ \(L_{c}\) & Diameter of the cylinder (mm) & 75 \\ \(L_{f}\) & Length of the cylinder submerged in fluid (mm) & 60 \\ \hline \end{tabular}
\end{table} 
Table 1: The geometric and physical properties of energy harvesting systems. Figure 2: The cantilever beam. rotation of \(O\)-\(Y_{1}\), \(O\)-\(Z_{1}\) with respect to the axis \(O\)-\(\xi\) by the angle \(\phi\). As a result, the angular velocity of the three rotation processes can be derived by the coordinate transformation, as shown in Equation (1). \[