# Resonance-Driven Passive Folding/Unfolding Flapping Wing Actuator
## 4 Results and Discussion  
In this section, we first present the observed folding/unfolding behavior, and then compare the measurement and theoretical solutions. Figure 5 shows the measured values of \(\Phi\) and \(\theta\) with sweeping of the applied voltage amplitude, \(V\). Here, \(\omega\) is fixed at 95 Hz. The figure also presents images captured via the cameras for easy understanding. First, we consider the case of increasing \(V\). For \(V<193\) V, \(\Phi\) was small and the wing remained in the folded state (\(\theta\) was largely negative). When \(V\) exceeded 193 V, \(\Phi\) and \(\theta\) showed a sudden increase, and the wing was unfolded. In contrast, for a decreasing \(V\), for \(V>193\) V, although \(\Phi\) and \(\theta\) were almost equal to those in the case of increasing \(V\); they do not show a sudden increase for \(V=193\) V. In addition, the wing remained in the unfolded state even when \(V\) exceeds the jump-up voltage and finally drops to \(V=121\) V. This hysteresis behavior indicates that the fabricated actuator is a bistable system. We measured the \(V\)-\(\Phi\) and \(V\)-\(\theta\) curves at different frequencies, as shown in Figure 6. Figure 6a,b show \(\Phi\) and \(\theta\) with increasing \(V\), respectively, and Figure 6c,d show \(\Phi\) and \(\theta\) with decreasing \(V\). The jump-up/down voltages of \(\Phi\) and \(\theta\) were identical with the all frequencies. We determined that a lower frequency resulted in higher jump-up and jump-down voltages. In addition, \(\Phi\) and \(\theta\) in the unfolded state increase with decreasing frequency. This trend can be explained in a qualitative manner as follows. To unfold the wing, a large torque exerted by the centrifugal or lift force is needed, thus suggesting the requirement of a large \(\Phi\) because these forces are proportional to \(\Phi^{2}\). \(\Phi\) depends on \(V\) and the closeness of \(\omega\) to the resonant frequency; this helps determine the magnitude of the  
Figure 4: Typical captured image showing (**a**) stroke amplitude and (**b**) folding angle. Figure 3: Measurement setup. resonant amplification. When considered together, either a sufficiently high \(V\) or \(\omega\) close enough to the resonant frequency is required to unfold the wing. The system in the folded state has a relatively higher resonant frequency than that in the unfolded state owing to its smaller inertia around the \(z\)-axis. Thus, the jump-up voltage should decrease with a higher \(\omega\); a higher \(\omega\) value matches the resonant frequency of the folded state more than that of the unfolded state and will result in a large \(\Phi\) and lower torque (or \(V\)). After unfolding (jump-up), the inertia increases, while the resonant frequency of the system decreases. This results in a mismatch between \(\omega\) and the resonance. Thus, \(\Phi\) in the unfolded state with a higher \(\omega\) is considered to be smaller than that for a lower \(\omega\). Next, \(\theta\) in the unfolded state with a higher \(\omega\) also decreases because smaller value of \(\Phi\) results in smaller values of centrifugal and lift forces. Conversely, a smaller \(\omega\) results in a larger jump-up voltage and larger \(\Phi\) and \(\theta\). This is because a smaller \(\omega\) matches the resonant frequency of the unfolded state more than that of the folded state. Next, we compare the experimental results with the simplified theoretical model. In Figure 7, the solution of Equation (19) is represented by blue lines (the parameters used in solving the equation are summarized in the Appendix A), and the measured values are plotted as red points and crosses. The theoretical solutions show S-shape curves; the overhanging region represents a set of unstable solutions, and the other region represents stable solutions. The top and bottom lines of the S-shape indicate two stable solutions at \(V\), i.e., bistability.