## 2 Modeling and Principle  
Piezoelectric materials have been widely used in many key technology fields, including microelectronics, signal processing, sensors and actuators. Among all piezoelectric materials, PVDF material can be fabricated in desired shapes of thin film easily and it is very suitable for replacing traditional macromolecule polymer film material considering its characteristics of wide frequency range, great sensitivity of power conversion and high strength of mechanical properties. When a piezoelectric material is mechanically deformed, electrical charges are induced on its surfaces according to the following equation:  
\[D=dT+\in\varepsilon \tag{1}\]  
where \(D\) is the electrical displacement, \(d\) is the piezoelectric coefficient, \(T\) is the stress, \(\in\) is the dielectric constant of the piezoelectric material, and \(\varepsilon\) is the electric field intensity. Regarding PVDF film as an airfoil for sensing aerodynamic force, we get the charge density \(D_{i}(i=x,y,z)\) by the following piezoelectric transformation relationship:  
\[\left[ \begin{array}{c}D_{x}\\ D_{y}\\ D_{z}\end{array} \right]=\left[ \begin{array}{cccccc}0&0&0&0&d_{15}&0\\ 0&0&0&d_{24}&0&0\\ d_{31}&d_{32}&d_{33}&0&0&0\end{array} \right]\times\left[ \begin{array}{cccccc} \sigma_{x}&\sigma_{y}&\sigma_{z}&\tau_{xy}&\tau_{yz}&\tau_{zx}\end{array}  \right]^{T} \tag{2}\]The piezoelectric constant \(d_{15}\) and \(d_{24}\) are zero respectively, and other \(d_{ij}\) of PVDF film are shown in the following equation. \[\left\{ \begin{array}{cc}d_{31}=17,&pC/N\\ d_{32}=5\sim 6,&pC/N\\ d_{33}=21,&pC/N\end{array} \right. \tag{3}\]  
The mathematical model for resultant force generated by wing is described as follows. We assume that there is a pressure difference distribution \(\Delta p(x,y,z)\) between the upper and the lower sides of the flapping wing. Resultant force \(F\) is calculated by taking the surface integral of \(\Delta p\) over the wing surface area \(S\):  
\[F=\iint\limits_{S}\Delta p(x,y,z)dA \tag{4}\]  
When the pressure difference \(\Delta p\) caused by the interaction of wings and air flow acts on the PVDF film, just like the case of giving a force on a shell or a plate, the PVDF film becomes deformed and stretched. We assume that the plane-stress distribution on the flapping wing as \(\sigma_{x}(x,y,z)\) and \(\sigma_{y}(x,y,z)\) which are related to \(\Delta p(x,y,z)\) by the following linear relationship:  
\[\Delta p(x,y,z)=k_{x}(x,y,z)\sigma_{x}(x,y,z)=k_{y}(x,y,z)\sigma_{y}(x,y,z) \tag{5}\]  
where \(k_{x}\) and \(k_{y}\) are stiffness functions assumed by the authors. Due to the plane-stress problem of the flapping wing, \(\sigma_{z}\) can be neglected (\(\sigma_{z}=0\)), The charge density along the \(Z\)-direction therefore can be derived as:  
\[D_{z}(x,y,z)=d_{31}\sigma_{x}(x,y,z)+d_{32}\sigma_{y}(x,y,z) \tag{6}\]  
Combining Equations (5) and (6), the total charge \(Q\) in terms of the pressure difference \(\Delta p\) and the stiffness function \(k\) as follows. \[Q=\iint\limits_{S}D_{z}(x,y,z)dA=\iint\limits_{S}\Delta p(x,y,z)\bigg{[}\frac {d_{31}}{k_{x}(x,y,z)}+\frac{d_{32}}{k_{y}(x,y,z)}\bigg{]}dA \tag{7}\]  
Because the stiff function \(k_{x}\) and \(k_{y}\) are not uniform over the wing skin area \(S\), the force \(F\) in Equation (4) cannot be directly separated from Equation (7). We temporarily and equivalently replace the bracket in Equation (7) with \(d^{*}\), and obtain the following expression:  
\[Q=d^{*}F \tag{8}\]  
In Equation (8), the total charge \(Q\) can be also expressed by a voltage \(U\) across the PVDF capacitor \(C\) by the relationship:  
\[Q=CU \tag{9}\]  
Combining Equations (8) and (9), the force can be determined linearly with the voltage U.  
\[F=\frac{Q}{d^{*}}=\left(\frac{C}{d^{*}}\right)\times U \tag{10}\]  
The simplified model described by Equation (10) gives us a guideline in principle that the information can be obtained electrically from the voltage given by the PVDF piezoelectric film. According to [PERSON]'s law, stress and strain are directly proportional.