# Electro-Aero-Mechanical Model of Piezoelectric Direct-Driven Flapping-Wing Actuator
## 2 Electro-Aero-Mechanical Model  
### Direct-Driven Flapping-Wing Actuator  
Figure 1 shows the actuation principle of a piezoelectric unimorph actuator and the structure and motion of the flapping-wing actuation system with the direct-driven mechanism. A unimorph actuator consists of a combination of a piezoelectric plate and a shim plate (elastic material), as shown in Figure 1a. Upon applying a voltage to the piezoelectric material, the actuator bends owing to the induced transverse piezoelectric stress. Figure 1b shows the structure of the flapping-wing actuation system. As described above, this system comprises only a unimorph piezoelectric actuator and a wing, and the wing is directly attached to the tip of the unimorph actuator. By actuating the unimorph, the wing is swung around the y-axis. This y-axis motion is simply called the \"stroke motion\", and the stroke angle is expressed by \(\phi\). When the voltage signal frequency matches the resonance frequency of the system, large stroke amplitude is obtained. We define the \(x^{\prime}y^{\prime}z^{\prime}\)-coordinate frame that rotates with the stroke motion. The wing is passively rotated around the x\({}^{\prime}\)-axis by air pressure associated with the stroke motion. This x\({}^{\prime}\)-axis rotation is called \"pitching motion\", and its rotation angle is defined by \(\psi\). We define the \(x^{\prime}y^{\prime\prime}z^{\prime\prime}\)-coordinate as a wing fixed frame. The pitching motion is crucial to generate lift force; the pitched wing gets a certain angle of attack against the stroke motion and it generates lift force along the negative y-axis. Therefore, the wing should be designed to smoothly rotate around the x\({}^{\prime}\)-axis and to be rigid against the other directions. For this purpose, hinge structures have often been equipped near the top edge of the wing. In the section below, we describe the analytical model of the actuation system. To derive this model, we employ several approximations for simplicity:  
1. The system has only two degrees of freedom (DOFs), \(\phi\) and \(\psi\). Specifically, the wing can not only rotate but also translate on the x-z plane, and therefore, this assumption is generally incorrect. However, under a cyclic resonant condition, the wing motion can be approximated as a simple rotation around a certain axis. This approximation is verified based on the experimental results reported in a later section. 2. Aerodynamics forces/moments are modeled as the sum of the translational force, force due to the added-mass effect, and rotational damping, and we use the blade element theory proposed in [5; 6; 7; 8; 9]. 3. We ignore the piezoelectric and other material losses and anchor-loss because their contribution is much smaller than that of aerial damping. We derive the model based on [PERSON]'s motion equation [23]. The total kinematic energy \(T\) of the system is expressed as  
\[T=T_{A}+T_{B}+T_{w}, \tag{1}\]  
where \(T_{A}\), \(T_{B}\), and \(T_{w}\) are the kinematic energy of the unimorph actuator, leading edge part (square bar on the top edge of the wing in Figure 1b), and wing, respectively. The total potential energy \(U\) is also expressed by the sum of the elastic stored energy of the unimorph, \(U_{A}\), and the pitching hinge, \(U_{w}\), as  
\[U=U_{A}+U_{w}. \tag{2}\]  
By using \(T\) and \(U\) and the aerodynamic moment along the \(\phi\) and \(\psi\) directions, \(\tau_{\phi}\) and \(\tau_{\psi}\), respectively, the motion equation of the system can be written as  
\[\left. \begin{array}{c}\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial\Lambda} {\partial\phi}-\frac{\partial\Lambda}{\partial\phi}=\tau_{\phi},\\ \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial\Lambda}{\partial\psi}-\frac{ \partial\Lambda}{\partial\psi}=\tau_{\psi},\\ \Lambda=T-U,\end{array} \right\} \tag{3}\]  
where \(\Lambda\) is the Lagrangian of the system.