# A Piezoelectric Energy Harvester with Bending-Torsion Vibration in Low-Speed Water
## 3 Numerical Analysis and Experimental Validation
### Numerical and Experimental Analysis  
Figure 6 illustrates the numerical and experimental values of average power versus external load resistance \(R\). The configurations of the harvester are \(D\) = 20, \(d\) = 10 and \(e\) = 5 mm. The powers are measured at flow velocities of 0.355 and 0.404 m/s, respectively. It is found that the power increases at first then decreases with the increase of the resistance value. There is an optimal resistance around 170 and 140 k\(\Omega\) for the output power when the widths of the beams are 20 and 25 mm, respectively. From Figure 6, one can find that the optimal resistance is not related to the flow velocities, and the optimal resistance of the numerical analysis is consistent with that of the experiment. Table 2 shows the configurations of eccentric cylinder used in experiment. To document the measured results, it is necessary to number the cylinders. The sequence numbers represent the diameter of the cylinder (\(D\)) and the diameter of the eccentric hole (\(d\)). For example, the number Con 2006 represents a cylinder with \(D\) = 20 and \(d\) = 6 mm. In addition, the number Con 2000 represents a solid cylinder with a diameter of 20 mm. Figure 5: Experimental test platform. Figure 6: Numerical and experimental values of average power versus load resistance: (**a**) \(b\) = 20 mm; (**b**) \(b\) = 25 mm. Figure 7: Values of average power numerically calculated and experimentally measured at various flow velocities with different cylinder configurations of energy harvesters with a beam width of 25 mm (**a**) \(D\) = 20 mm; (**b**) \(D\) = 25 mm; (**c**) \(D\) = 30 mm. Figure 7 illustrates the average power of harvesters with the configurations in Table 2, which are numerically calculated and experimentally measured at various flow velocity \(U_{0}\) with the same resistance of 80 k\(\Omega\). It can be found that the output power \(P\) first increases to \(P_{max}\) (maximum value of power) with \(U_{0}\), while \(U_{0}\) continues to increase, \(P\) decreases. The optimal velocity is about 0.339-0.366 m/s for the nine configurations. As the flow velocity \(U_{0}\) rises up, the vortex shedding frequency \(\omega/\) increases, which is gradually close to the natural frequency of the harvester. As a result, the lift force from the vortex drives the eccentric cylinder to resonance, and leads to a significant increase in amplitude of the cylinder rigidly connected with piezoelectric beam. At the same time, the optimal velocity becomes greater with the increase of \(d\), which causes a lighter mass of the cylinder. As a consequence, the natural frequency of the harvester increases. Meanwhile, there should be a positive correlation between \(P_{max}\) and \(l_{d}\) dominated by the eccentricity distance \(e\) and the diameter of cylinder eccentric hole \(d\). It can be concluded that \(P_{max}\) increases with the increase of eccentricity of mass \(l_{d}\), as shown in Figure 7a and Table 3. \(P_{max}\) is 0.3978 mW with a maximum value of \(l_{d}\) in configurations 2006, 2008, and 2010 in Figure 7a. Besides, the \(P_{max}\) of the solid cylinder is 0.1999 mW, and there is an improvement of 99%. The torque excited by vortex becomes bigger when \(l_{d}\) increases. It leads to a bigger twist angle. Further, more energy is harvested. In Figure 7, it also can be summarized that the cylinder diameter \(D\) has a negative influence on \(P_{max}\). \(P_{max}\) decreases from 0.3978 to 0.2482 mW, while \(D\) increases from 20 to 30 mm with a similar \(l_{d}\) in Con 2010, Con 2510, and Con 3010. In addition, the same conclusion can be obtained from Table 3. The lift force reduces with the increase of diameter \(D\), which weakens cylinder vibration response. By comparing the results of different experiments, the best configuration and working environment for the harvester is summarized. The optimal load resistance is 170 k\(\Omega\) when \(b\) = 20 mm, and 140 k\(\Omega\) when \(b\) = 25 mm. The greater the \(l_{d}\) is, the greater \(P_{max}\) will be.