# Electro-Aero-Mechanical Model of Piezoelectric Direct-Driven Flapping-Wing Actuator
## 4 Results and Discussion
### Resonant Driving Characteristics  
In this experiment, we measured the resonant driving performance of the small- and large-type actuators with \(V_{0}\) of 15-30 and 20-50 V, respectively. Figures 11 and 12 show the motion of the small and large types, respectively. Given the simplicity of the analytical model, we found that the behaviors of the calculated \(\phi\) and \(\psi\) showed good agreement with the measured data. Figures 13 and 14 show the amplitudes of \(\phi\) and \(\psi\). The average absolute errors of \(\phi\) and \(\psi\) amplitude were 20% and 8.1%, respectively. Figures 15 and 16 show these tendencies of \(\phi\) with the lift force and power  
Figure 10: Measured and calculated resonant frequencies. Figure 9: Resonant frequency estimation by impulse response: (**a**) small type and (**b**) large type. consumption, respectively. The average absolute errors of the lift force and consumed power between the calculation and the experiment were 15% and 8.7%, respectively. These results suggest that the derived model can well simulate the performances of the flapping-wing actuation system. Figure 17 shows the lift-to-power efficiency of the actuation system; it is defined as the ratio of the lift force to the power consumption. Upon increasing the applied voltage, the efficiency of both types of actuators decreased. The efficiencies of the small and large types at their maximum applied voltages were 10.8 and 7.31 gf/W, respectively. Figure 11: Measured and calculated stroke and pitching angle of small-type actuator: blue circle and red line show experimental data and calculated result, respectively. Figure 12: Measured and calculated stroke and pitching angle of large-type actuator: blue circle and red line show experimental data and calculated result, respectively. Figure 14: Relationship between applied voltage amplitude and (**a**) stroke amplitude and (**b**) pitching amplitude of large-type actuator. The frequencies of the input signal in the experiment and calculation were 105 and 109 Hz, respectively. Figure 13: Relationship between applied voltage amplitude and (**a**) stroke amplitude and (**b**) pitching amplitude of small-type actuator. The frequencies of the input signal in the experiment and calculation were 210 and 209 Hz, respectively. Figure 15: Relationship between applied voltage amplitude and (**a**) lift force and (**b**) consumed power of small-type actuator. The frequencies of the input signal in the experiment and calculation were 210 and 209 Hz, respectively. There are two possible causes of lift force error: Wing motion (\(\phi\), \(\psi\)) and analytical model of lift force. As shown in Figures 13 and 14, the predicted \(\phi\) and \(\psi\) have nonnegligible error. In particular, the trend of the error in \(\phi\) showed a similarity to the lift force error. For example, for the small-type actuator, lower \(V_{0}\) resulted in underestimated \(\phi\) (calculated \(\phi\) was smaller than the measured data) and larger \(V_{0}\), in overestimated \(\phi\). On the other hand, for the large-type actuator, the calculated \(\phi\) was slightly smaller than the measured one for all tested \(V_{0}\) values. The lift forces showed the same trends (Figures 15 and 16). Thus, the error of the lift force was considered to be dominantly caused by the estimation error in the stroke motion. To verify this result, we performed additional calculations in which the lift force was estimated from the measured trajectory of \(\phi\) and \(\psi\) using Equation (35). The results are shown in Figures 18 and 19; in these figures, we compare the lift forces obtained using the experiment, calculation with the proposed aeromechanic model, and calculation with the prescribed measured trajectories. For both small- and large-type actuators, the lift force obtained from the measured trajectories resulted in smaller errors than that obtained from the fully analytical calculation. This result confirms that the prediction error of the wing motion is the main error source in the lift force estimation.