# Electro-Aero-Mechanical Model of Piezoelectric Direct-Driven Flapping-Wing Actuator
## 2 Electro-Aero-Mechanical Model
### Mechanical Model  
Here, we describe the derivation of \(T\) and \(U\) from the material and dimensional parameters of the system. Figure 2 shows the dimensions and coordinates of the system (details of the wing geometry are explained in a later section). The unimorph is designed as a trapezoid. It can reduce the stress concentration at the fixed end against inertial load and the aerodynamic force of the wing, and it can also reduce the risk of brittle fracture [24]. The cross point of the extended line of the unimorph side edges is designed to coincide with the center of the wing (\(x=L_{A}+R/2\), where \(R\) and \(L_{A}\) are the lengths of the wing and unimorph, respectively). We assumed that the rotational axis of the stroke motion is placed at \(x=L_{A}/2\). This assumption can be valid when the inertial and aerial forces of the wing are considered as concentrated forces at \(x=L_{A}+R/2\). Of course, this is a very rough simplification, and it is not always suitable. Thus, we check the position of the rotational center of the wing experimentally; the result is described in the Result section. By using this assumption, the curvature of the unimorph \(\kappa\) is simply expressed by \(\phi\) as  
\[\kappa=\frac{\phi}{L_{A}}. \tag{4}\]  
The deflection of the unimorph \(\delta\) can be given by  
\[\delta=\frac{\kappa}{2}x^{2}. \tag{5}\]  
Thus, \(T_{A}\) is calculated from the following integral with respect to \(x\):  
\[T_{A}=\int_{0}^{L_{A}}\frac{1}{2}\Bigg{(}\sum_{i}\rho_{i}t_{i}\Bigg{)}w(x) \dot{\delta}^{2}\mathrm{d}x=\frac{1}{240}\Bigg{(}\sum_{i}\rho_{i}t_{i}\Bigg{)} (w_{r}+5w_{t})L_{A}^{3}\dot{\phi}^{2}, \tag{6}\]  
where \(\rho_{i}\) and \(t_{i}\) are the density and thickness of the \(i\)-th layer of the unimorph, respectively, as shown in Figure 2. The bending moment caused by the piezoelectric effect is given by  
\[M_{p}=E_{0}d_{pz}w(x)V(z_{0}-z_{c}), \tag{7}\]  
where \(E_{0}\) and \(d_{pz}\) are [PERSON]'s modulus and the piezoelectric coefficient of the piezoelectric plate along the x-axis, respectively [25]. \(V\) is the applied voltage to the piezoelectric material. \(z_{i}\) are the z-positions of the layers of the unimorph, and they are expressed as \(z_{i}=\sum_{j}t_{j}-t_{i}/2\). \(z_{c}\) is the neutral axis about the bending of the unimorph, and it is expressed as  
\[z_{c}=\sum_{i}E_{i}t_{i}z_{i}/\sum_{i}E_{i}t_{i}. \tag{8}\]  
Assuming the unimorph as an Euler-Bernoulli beam [26], the curvature generated by the piezoelectric actuation is given by  
\[\kappa_{pz}=\frac{M_{p}}{S_{A}w(x)}=\frac{E_{0}d_{pz}V}{S_{A}}(z_{0}-z_{c}), \tag{9}\]  
where \(S_{A}\) is the bending stiffness of the unimorph per unit width, and it is given by  
\[S_{A}=\sum_{i}E_{i}\Bigg{(}\frac{t_{i}^{3}}{12}+t_{i}(z_{i}-z_{c})^{2}\Bigg{)}. \tag{10}\]  
By using \(\kappa\) and \(\kappa_{pz}\), \(U_{A}\) is calculated as  
\[U_{A}=\int_{0}^{L_{A}}\frac{1}{2}S_{A}w(x)\big{(}\kappa-\kappa_{pz}\big{)}^{2} \mathrm{d}x=\frac{1}{2}S_{A}\frac{w_{r}+w_{t}}{L_{A}}\phi^{2}-\frac{E_{0}d_{pz }V}{2}(z_{0}-z_{c})(w_{r}+w_{t}). \tag{11}\]Next, we describe the kinematic and potential energy of the wing. Figure 3 shows the coordinates and dimensional parameters. We newly define a coordinate \(r\) as the distance from the wing root along the x-axis. \(T_{B}\) is obtained as  
\[T_{B}=\frac{1}{2}J_{B}\dot{\phi}^{2}, \tag{12}\]  
where \(J_{B}\) is the inertia of the leading edge part around the stroke rotation axis. \(T_{w}\) is calculated as  
\[\delta=\frac{\kappa}{2}x^{2}. \tag{5}\]0  
where \(M_{w}\) and \(\mathbf{J}_{\mathbf{w}}\) are the mass and inertia tensor of the wing, respectively. The origin of \(\mathbf{J}_{\mathbf{w}}\) is at the center of mass (COM) of the wing. \(\mathbf{v}\) and \(\mathbf{\omega}\) are the velocity of the COM and angular velocity of the wing, respectively. Upon defining these variables based on the wing-fixed x\({}^{\prime}\)y\({}^{\prime\prime}\)z\({}^{\prime\prime}\) frame, \(\mathbf{v}\) and \(\mathbf{\omega}\) can be expressed by \(\phi\) and \(\psi\) as  
\[\mathbf{v}=\left[