# Estimation of Stability Parameters for Wide Body Aircraft Using Computational Techniques
## 3 Xflr
### XFLR Results  
In order to obtain the lift, drag, and moment coefficients, aircraft stability analysis was performed in XFLR software. The variations of lift, drag, and moment coefficients with angle of attack are shown in Figure 12. For a range of angle of attack from \(-6\) to \(17\) deg, a linear relationship between lift coefficient and angle of attack is visible (Figure 12a). The choice of a range of angles was deliberate because, in this range, the stability of the aircraft is most prominent. An angle less than \(-6\) degrees would give a useless result and an angle more than \(17\) would be beyond the stall region. The variations of lift and drag coefficient (Figure 12b) obtained from XFLR agree with typical drag polarity drawn for conventional aircraft. For Boeing 747-200, the results are as per design requirements ensuring smooth aerodynamic performance. The graph (Figure 12c) shows a proportional relation between aerodynamic efficiency and angle of attack, indicating the promising aerodynamic performance of the aircraft in the given regime. The result is in concurrence with the theory of stable flight. Figure 12d shows the linear relationship between lift and moment coefficients with negative slope values, such as the variation of moment coefficient, with angle of attack. For a longitudinally stable aircraft, the value of Cm0 should be positive; i.e., the y-intercept of the Cm\(\alpha\) curve should be positive and the moment coefficient against a changing angle of attack must have a negative slope. The plot depicts the expected result establishing that the selected aircraft results satisfy this condition of static stability. The longitudinal stability aerodynamic coefficients results obtained at low-cruise-flight conditions are shown in Table A1 of Appendix A. CLadot and Cmadot could not be estimated by XFLR due to software limitations. The results from XFLR are close to the analytical results; however, small errors occurred due to approximations used in the software. Moreover, due to panel creation, some errors may also occur because the wing panel's wake interacts with the fin and elevator to generate unwanted numerical interactions. Similarly, the lateral stability aerodynamic coefficients results obtained at low-cruise-flight conditions are shown in Table A2 of Appendix A. The obtained results are close to the DATCOM results and analytical results, despite small absolute differences in some of the parameters. The results may vary because XFLR's wake modeling is insufficient to cater for flow behavior experienced by aircraft in real flight. Moreover, the effect of the fuselage is not added in the XFLR analysis of the entire aircraft. Eigenvalues are a special set of scalars associated with a linear system of equations also known as characteristic roots, characteristic values, proper values, or latent roots. These are used to determine the stability and the rate of decay/growth of the system. For the present case, eigenvalues for longitudinal (short period and Phugoid) and lateral modes (spiral, roll, and Dutch roll mode) were also estimated, as shown in Table 3. These values were obtained with a negative real part, which shows that the aircraft is dynamically stable, and if it is given an initial disturbance, the motion will decay sinusoidally. The frequency of oscillation would be governed by the imaginary part of the complex eigenvalues. The root locus is one of the popular graphical representations in control theory to pictorially read the stability of a system. When an aircraft is successfully designed in XFLR, \begin{table}
\begin{tabular}{c c c c} \hline \hline
**S No.** & **Eigen Value** & **Mode** & **Stability** \\ \hline
1 & \(-3.35\pm 6.28\)i & Short Period & Longitudinal \\
2 & \(-0.000133\pm 0.0304\)i & Phugoid & -”- \\
3 & \(-0.00498+0\)i & Spiral & Lateral \\
4 & \(-7.21+0\)i & Roll & -”- \\
5 & \(-0.589\pm 2.877\)i & Dutch Roll & -”- \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Eigenvalues obtained from XFLR for the Boeing 747’s dynamic stability analysis.