# Bio-Inspired Morphing Tail for Flapping-Wings Aerial Robots Using Macro Fiber Composites
## 2 MFC Tail Actuator Characterization
### MFC Theoretical Displacement Model  
In this subsection, the theoretical model of MFC actuators is presented. This model is based on the classical lamination theory to calculate the properties of the MFC. Subse  
Figure 2: Representation of the Separated Layers Normally Present in the Macro Fiber Composites (MFC). quently, the equations for calculating the theoretical potential energy of the MFC are shown. This is described in more detail in [28]. MFC is composed of several layers that should be individually characterized to calculate the laminate stiffness quantities. The layers can be either isotropic or orthotropic. In the case of the orthotropic layers, as they have different properties depending on the direction they are measured, they need four independent short-circuit stiffness quantities \(E_{x}\), \(E_{y}\), \(v_{xy}\), and \(G_{xy}\) for describing their mechanical behavior, where \(E_{r}\) is the elastic modulus in direction \"\(r\)\", \(v_{xy}\) is the Poisson ratio, and \(G_{xy}\) is the shear modulus in the \(xy\) plane of the fiber. This is developed in greater detail in [41]. The isotropic layers can be analyzed as orthotropic ones by assuming that \(E_{x}=E_{y}\) and \(G_{xy}=G\). By using the Classical Lamination Theory (CLT), it is possible to obtain these variables starting from the individual layers and combining them for obtaining the whole MFC. To achieve this, some assumptions have to be made for transforming the problem from a third-dimensional problem to a two-dimensional one. These assumptions are the following: the displacement is assumed as continuous in the whole laminate; the Kirchhoff hypothesis is assumed; the strain-displacement relationship is linear and the material is linearly elastic, and the through-thickness stresses are small when compared with the in-plane ones. By doing so, it is possible to obtain the relation of the stress \(\sigma\) and strain \(\epsilon\) with the following equation:  
\[\sigma_{k}=Q_{k}\cdot\epsilon_{k} \tag{1}\]  
where \(Q\) is the stiffness matrix and \(k\) indicates the layer. Orthotropic layers have greater complexity while calculating their properties as isotropic layers, which produce reduced stiffness matrices. Some of the layers of the MFC are isotropic, like the Kapton or the Acrylic ones, which makes the calculations more simple. By applying these methods and taking the properties of the layers presented in Section 2.1 into account, it is possible to obtain the stiffness quantities of the whole MFC, as seen in [28]. These values shall be similar to the specifications of the manufacturer. The calculations to obtain the displacements equations are based on the Rayleigh-Ritz method. This method assumes that the potential energy, depending on the displacement variable, is constant in specific situations. This work focuses on the situation when the actuator is actuated. For a similar development, see [28]. The first step is to specify the displacement equations. It is assumed that the displacement in the mid-line of the actuator and the host lamina are:  
\[u^{0}(x,y) =c_{3}x-\frac{c_{1}^{2}x^{3}}{6}-\frac{c_{1}c_{2}xy^{2}}{4} \tag{2}\] \[v^{0}(x,y) =c_{4}y-\frac{c_{2}^{2}y^{3}}{6}-\frac{c_{1}c_{2}x^{2}y}{4}\] \[w^{0}(x,y) =\frac{1}{2}(c_{1}x^{2}+c_{2}y^{2})\]  
where \(u\), \(v\), and \(w\) are, respectively, the displacement in the \(x\), \(y\), and \(z\) directions. \(c_{1}\), \(c_{2}\), \(c_{3}\), and \(c_{4}\) are the unknown constants. The next step is to calculate the potential energy assuming the displacement equations. The state when the system is evaluated should be determined in order to calculate this potential energy. In this case, the potential energy is calculated in the actuated state. This is the most complex situation, including the potential energy of the actuated MFC and the potential energy of the host lamina.