# Resonance-Driven Passive Folding/Unfolding Flapping Wing Actuator
## 2 Structure and Modelling
### Modelling
#### 2.2.1 Centrifugal Force  
Here, we define \(r\) as a coordinate along the wingspan, which originates from the wing root. Because the distance from the center of the stroke motion (\(z\)-axis) to a position on the wing (\(r\)) is expressed as \(r\cos\theta+R_{0}\), the centrifugal force acts on a small element of the wing, \(\mathrm{d}F_{C}\), which is defined as  
\[\mathrm{d}F_{C}=(r\cos\theta+R_{0})\dot{\phi}^{2}\gamma\ \mathrm{d}r, \tag{2}\]  
where \(\gamma(=M_{w}/L_{w})\) is the line density along the wingspan. Thus, the torque on the small element is formulated based on Equation (2) as  
\[\mathrm{d}T_{C}=r\sin\theta\mathrm{d}F_{C}=r\sin\theta(r\cos\theta+R_{0})\dot{ \phi}^{2}\gamma\ \mathrm{d}r. \tag{3}\]  
By integrating Equation (3) along the wingspan, \(T_{C}\) can be obtained as  
\[T_{C}=\int\limits_{0}^{L_{w}}r\sin\theta(r\cos\theta+R_{0})\dot{\phi}^{2} \gamma\mathrm{d}r=\gamma\ L_{w}^{2}\Big{(}\frac{L_{w}}{3}\cos\theta+\frac{R_{ 0}}{2}\Big{)}\sin\theta\dot{\phi}^{2}. \tag{4}\]  
Assuming the stroke motion as sinusoidal, a time-averaged centrifugal torque, \(\overline{T_{C}}\), is expressed as a function of \(\theta,\omega\), and \(\Phi\):  
\[\overline{T_{C}}(\theta,\omega,\Phi)=\gamma\ L_{w}^{2}\Big{(}\frac{L_{w}}{3} \cos\theta+\frac{R_{0}}{2}\Big{)}\sin\theta\frac{\Phi^{2}\omega^{2}}{2}, \tag{5}\]  
where \(\omega\) and \(\Phi\) are the angular frequency and amplitude of the stroke motion, respectively. Figure 2: Simplified model of the flapping-wing actuator. Stroke amplitude \(\Phi\) is determined by the resonant system about \(\phi\) and the applied driving signal. Because it is regarded as a second-order system, \(\Phi\) is obtained as [14]  
\[\Phi=\frac{1}{\sqrt{\left(1-\beta^{2}\right)^{2}+4\zeta^{2}\beta^{2}}}\frac{T_{a }}{K_{a}}, \tag{6}\]  
where \(\beta\) is the ratio of \(\omega\) to the resonant angular frequency of the system, \(\omega_{n}\) (\(\beta=\omega/\omega_{n}\)); \(\zeta\) is the damping ratio; and \(T_{a}\) is the amplitude of applied actuation torque. \(\omega_{n}\) and \(\zeta\) are defined using the inertia, damping factor, and spring constant of the system (\(J,\ B,\) and \(K_{a}\), respectively) as  
\[\zeta=\frac{B}{2\sqrt{JK_{a}}},\ \ \ \omega_{n}=\sqrt{\frac{K_{a}}{J}}. \tag{7}\]  
Because the torque generated by a piezoelectric actuator is linear to the applied voltage, it can be expressed as  
\[T_{a}=GV, \tag{8}\]  
where \(G\) and \(V\) are the proportional gain constant and applied voltage amplitude. By substituting Equations (6)-(8) into Equation (4), \(\overline{T_{C}}\) can be expressed as a function of \(\theta,\ \omega,\) and \(V\). Based on the rigid-bar approximation, the inertia of the wing around the \(z\)-axis can be given by \(\left(1/12\right)M_{w}\left(L_{w}\cos\theta\right)^{2}+M_{w}\left(L_{w}\cos \theta/2+R_{0}\right)^{2}\). Thus, \(J\) is expressed as a sum of the wing inertia, and \(J_{a}\) is expressed as  
\[J(\theta)=\frac{1}{12}M_{w}\left(L_{w}\cos\theta\right)^{2}+M_{w}\bigg{(} \frac{L_{w}}{2}\cos\theta+R_{0}\bigg{)}^{2}+J_{a}. \tag{9}\]  
Assuming the deformation of the piezoelectric actuator as parabolic, \(J_{a}\) is given by  
\[J_{a}=\frac{1}{120}\rho(5w_{t}+w_{r})L_{a}^{3}, \tag{10}\]  
where \(\rho\) is the area density of the actuator [13], and \(w_{t},\ w_{r},\) and \(L_{a}\) are the tip width, root width, and length of the actuator, respectively. Moreover, \(K_{a}\) is given by  
\[\mathrm{d}T_{C}=r\sin\theta\mathrm{d}F_{C}=r\sin\theta(r\cos\theta+R_{0})\dot{ \phi}^{2}\gamma\ \mathrm{d}r. \tag{3}\]0  
where \(S_{a}\) is the bending stiffness of the actuator per unit width [13]. We approximate the aerodynamic drag pressure as a concentrated damping force, \(F_{D}\), acting on the center of the wing; \(F_{D}=B_{0}v\), where \(B_{0}\) and \(v\) are a constant and the velocity at the center of the wing. Because \(v=(L_{w}\cos\theta/2+R_{0})\dot{\phi}\), a torque due to the aerodynamic damping force is expressed as \(T_{D}=B_{0}v(L_{w}\cos\theta/2+R_{0})\).