## Theoretical Modeling
### Coupling of Aeroelasticity and Flight Dynamics  
In the traditional flight dynamics, the aircraft is modeled as a rigid body with six degrees-of-freedom. However, due to the different velocities and orientation angles at different locations of the flexible aircraft, it is no longer proper to represent the aircraft as a rigid body. The Euler angles (roll, pitch and yaw angles) and airflow angles (angles of attack and sideslip) for a flexible aircraft should be redefined at the local element level.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Description** \\ \hline \(\rho A\) & \multicolumn{2}{c}{Sectional mass per unit length} \\ \(\rho_{y}^{\prime}\) & Bending inertia about principal axis \(y^{\prime}\) at G \\ \(\rho_{z}^{\prime}\) & Bending inertia about principal axis \(z^{\prime}\) at G \\ \((y_{c}^{\prime},z_{c}^{\prime})\) & Position of C referred to \(G_{xy^{\prime}z^{\prime}}\) \\ \(\rho_{t}\) & \(\rho_{y}^{\prime}+\rho_{z}^{\prime}+(y_{c}^{\prime 2}+z_{c}^{\prime 2})\rho A\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Variables defining mass matrix.  
In the present framework, given the spatial locations and orientations of every element, the ground, body and airflow axes for flight dynamics are re-defined analogously to the global and elemental reference frames for structural dynamics at the local element level.  
As illustrated in Figure 2, the ground axes \(\mathbf{U}_{g}\) for flight dynamics is defined to be identical to the global reference frame for aeroelasticity. For each element, there is a body-attached element frame \(\mathbf{U}_{e}\), which translates and rotates continuously with the varying generalized displacements of the element. The element frame \(\mathbf{U}_{e}\) in aeroelasticity is then taken as the body axes \(\mathbf{U}_{b}\) for flight dynamics. Note that, according to the most usual practice in flight dynamics, the \(x\) axis of \(\mathbf{U}_{b}\) is directed to \"forward\", \(y\) axis to \"rightward\", and \(z\) axis to \"downward\", while, in aeroelasticity, the \(x\) axis of the element frame is preferred to lie along the axial line of the beam, and \(y\) and \(z\) axes along the principal axes of inertia. Thus, the body axes \(\mathbf{U}_{b}\) of a wing element is defined as:  
\[\mathbf{U}_{b}=[\mathbf{U}_{ey},\mathbf{U}_{ex},-\mathbf{U}_{ez}] \tag{34}\]  
where \(\mathbf{U}_{ex}\), \(\mathbf{U}_{ey}\) and \(\mathbf{U}_{ez}\) are the base vectors of \(\mathbf{U}_{e}\).  
With ground axes \(\mathbf{U}_{g}\) and elemental body axes \(\mathbf{U}_{b}\) defined, the Euler angles (roll, pitch and yaw angles) of each element can be defined in the usual way. The translational velocity and acceleration of the element in body axes can be expressed as:  
\[\mathbf{V}_{b}=\mathbf{U}_{b}^{T}\mathbf{V}_{g};\quad\mathbf{a}_{b}=\mathbf{U }_{b}^{T}\mathbf{a}_{g} \tag{35}\]  
where \(\mathbf{V}_{g}\) and \(\mathbf{a}_{g}\) are, respectively, the inertial velocity and acceleration vectors expressed in the ground or global axes.  
The angular velocity and acceleration in body axes can be expressed as:  
\[\mathbf{\omega}=[w_{y},w_{x},-w_{z}]^{T};\quad\mathbf{\alpha}=[a_{y},a_{x},-a_{z}]^{T} \tag{36}\]  
where \(w_{x},w_{y},w_{z}\) and \(a_{x},a_{y},a_{z}\) are, respectively, the components of the elemental angular velocity and angular acceleration, which are defined in the element frame \(\mathbf{U}_{e}\), in light of the co-rotational approach.  
With the translational velocity in body axes, the angles of attack and sideslip of an element can be calculated as:  
\[\alpha=\arctan(V_{bz}/V_{bx});\quad\beta=\arcsin(V_{by}/V) \tag{37}\]  
Figure 2: Global (ground), elemental and body reference frames of the flexible wing.  
where \(V_{bx}\), \(V_{by}\) and \(V_{bz}\) are the components of \(\mathbf{V}_{b}\) and \(V\) is the norm of \(\mathbf{V}_{b}\).