# Estimation of Aircraft-Dependent Bumpiness Severity in Turbulent Flight
## 3 Experiments and Analysis
### Analysis of UVLM
#### 3.1.2 Verification of Pitching Moment  
There inevitably exists a gap between the aerodynamic performance of the whole aircraft and that of the wing-tail assembly. Fortunately, only the acceleration increment induced by turbulence is considered for aircraft bumpiness estimation. The pitching moment change, which has major effects on instantaneous acceleration change as shown in Equation (14), is mainly caused by the deflection of elevator or horizontal stabilizer. As a result, the pitching moment change based on wing-tail assembly should approximate the authentic aerodynamic data. The longitudinal pitching moment changes of B737-800 aircraft caused by elevator and stabilizer deflections are provided in [37; 38]. Figure 8 shows the results between UVLM results and the modeling data in three flight conditions, where \(c_{m\delta stab}\) represents the pitching moment change due to stabilizer deflection, and \(c_{m\delta e}\) stands for the moment change caused by elevator deflection. Taking Figure 8a as an example, it shows the comparison of \(c_{m\delta stab}\) between UVLM and modeling data in mach number \(M=0.2\), \(\alpha=3^{\circ}\) at sea level (\(H=0\)). It can be found that different flight conditions would change the dynamic response of the aircraft.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Grid No.** & \(N_{\text{CW}}\) & \(N_{\text{RW}}\) & \(N_{\text{CT}}\) & \(N_{\text{RT}}\) \\ \hline Grid 1 & 6 & 3 & 3 & 2 \\ Grid 2 & 12 & 6 & 3 & 2 \\ Grid 3 & 18 & 8 & 4 & 2 \\ Grid 4 & 20 & 10 & 4 & 2 \\ Grid 5 & 25 & 10 & 8 & 4 \\ Grid 6 & 30 & 12 & 8 & 4 \\ Grid 7 & 40 & 15 & 12 & 4 \\ Grid 8 & 40 & 20 & 12 & 6 \\ Grid 9 & 60 & 20 & 16 & 6 \\ Grid 10 & 80 & 20 & 16 & 8 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Number of cells included at each grid level used for convergence study.  
By comparing the computation results with the modeling data, the Mean Square Error (MSE) of \(c_{mistab}\) is 0.0045 and the MSE of \(c_{mise}\) is 0.0055. The fitting curves formed by the computation results show that the pitching moment change caused by the deflection of stabilizer or elevator approximates the modeling data under the above flight conditions.  
The above results are obtained by steady vortex lattice method because the test conditions remain unchanged. Several improvements on traditional VLM make the results approximate modeling data. Firstly, vortex rings are assigned on the mean camber surface rather than using the flat surface assumption. Secondly, the semi-circle division method is adopted to refine the lattices neighboring to wingtip, wing root, and leading edge. Thirdly, the divided lattices are as close as possible to the structural boundary of the control surface. As a result, the non-planar UVLM based on wing-tail assembly has satisfactory computing accuracy, which is beneficial to further analysis of vertical acceleration response. In addition, the accurate computation of pitching moment coefficient is also significant to separate aircraft maneuvers effects on bumpiness estimation.