## 3 Numerical Validation
### Cantilevered Goland Wing with Inertial Coupling  
The cantilevered Goland wing was analyzed as a method-verification study for the present time-marching aeroelastic approach and the coupling of bending and torsional inertia. The properties of the Goland wing are given in Table 3. With the effects of gravity neglected, the wing starts from rest with a very small AOA of 0.05 degree. The flutter speed was estimated in the time domain with different freestream velocities evaluated until the flutter onset was found.  
Time histories of the wing-tip deflection and the corresponding torsional angle at three distinctive freestream velocities of 131.4, 136.4 and 141.4 m/s are plotted in Figures 3 and 4, which, respectively, correspond to the damped, neutral and divergent states.  
At the air speed of 131.4 m/s, the amplitude of oscillation decays and finally approaches to a steady state. As the air speed goes up to 136.4 m/s, a periodic motion with unchanged amplitude occurs, indicating the flutter speed is 136.4 m/s. The flutter frequency was calculated to be 69.2 rad/s, with a time-domain peak-finding technique. Above this speed, the wing oscillation tends to grow.  
Table 4 compares the flutter speed and frequency obtained in the present framework to the results found in the literature. It can be seen that the flutter frequencies calculated with different models are almost the same, but the flutter speeds calculated with strip theory are approximately 18% lower than those with unsteady vortex lattice method (UVLM). This is because the two-dimensional strip theory cannot simulate the loss of lift near the tip of a wing with finite span, while the three-dimensional vortex  
Figure 3: Time history of Goland wing-tip deflection at various freestream velocities.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Property** & **Value** & **Property** & **Value** \\ \hline Span & 6.096 m & Torsional stiffness & \(9.87581\times 10^{5}\) Nm\({}^{2}\) \\ Chord & 1.8288 m & Bending stiffness & \(9.77221\times 10^{6}\) Nm\({}^{2}\) \\ Elastic axis & 33\% chord from L.E. & Chordwise bending stiffness & \(9.77221\times 10^{6}\) Nm\({}^{2}\) \\ Center of gravity & 43\% chord from L.E. & Mass per unit length & 35.709121 kg/m \\ Air density & 1.020 kg/m\({}^{3}\) & Mass moment of inertia about \(x\) axis & 8.6405832 kgm \\ \hline \hline \end{tabular}
\end{table}
Table 3: Properties of Goland wing.  
lattice method can. Different from high aspect-ratio wings, the Goland wing has an aspect-ratio of only 3.33, in which the 3D effect cannot be neglected. Using the strip theory to calculate the flutter speed of 3D finite-span wings may result in underestimation.  
To use 2D theory for a 3D wing, a spanwise distribution coefficient was applied to correct for the tip loss of lift:  
\[C_{corr}=1-e^{-\tau s}\quad(0\leq s\leq 1,\ \tau>1) \tag{51}\]  
where \(\tau\) is a user-defined parameter that controls the spanwise lift distribution and \(s\) is the distance between wingtip and the spanwise location of interest.  
Despite the difference from 3D UVLM, excellent agreement with other models based on strip theory is reached, from which the sufficient accuracy of the established framework and its corresponding computing codes can be observed.