# Modeling and Control of an Articulated Multibody Aircraft
## 4 Simulation and Results
### Effect of Abdomen Mass on Longitudinal Stability  
Static stability in aircraft design provides insight into the flying qualities of an aircraft. It is the initial tendency of an aircraft to return to its equilibrium state when disturbed [96]. Where the main focus is longitudinal motion, static stability theory can be applied to cruise conditions, as well as to quasi- steady maneuvers. A way to determine the static longitudinal stability of an aircraft is through the change in pitching moment with lift, \(\frac{dC_{m}}{dC_{L}}\). This derivative must be negative for a stable aircraft. In addition, the pitching moment at zero angle of attack must be positive. The same condition applies using the change in pitching moment with angle of attack as an indicator in some cases, i.e., \(\frac{dC_{m}}{d\alpha}<0\) is indicative of a stable aircraft [96; 135]. Another measure of static stability is the static margin (\(SM\)), which will be used to analyze the static stability of the aircraft in consideration in this study due to its simplistic and intuitive nature. The \(SM\) of an aircraft is given by [136]  
\[SM=\frac{X_{NP}-X_{cg_{m}}}{c_{ref}}, \tag{70}\]  
where \(X_{NP}\) is the neutral point (\(NP\)), and \(X_{cg_{m}}\) is the center of gravity (\(cg\)) position of the whole aircraft along \(x_{B}\) axis.  
Generally, if \(cg\) is ahead of the \(NP\), then \(SM\) is positive and the aircraft is stable. The larger the \(SM\), the more stable and less maneuverable the aircraft is. However, if the \(cg\) is behind the \(NP\), resulting in a negative \(SM\), the aircraft is unstable in pitch, making it difficult or impossible to fly manually [136]. The \(SM\) value is usually chosen based on the desired performance of the aircraft in terms of handling qualities and usually ranges from (\(10\%\leq SM\leq 5\%\)), for piloted aircraft. The neutral point of the DISWA was estimated as (\(X_{NP}=~{}0.046\) m) in AVL [137; 138; 139].  
The eigenvalues of the state matrix of a linearized model also provide insight into the dynamic stability of the system. The effect of abdominal mass of the aircraft of interest was investigated. Five abdominal mass values were considered to be a percentage of central body mass of the aircraft, namely 6%, 11%, 16%, 21% and 26%, such that they resulted in negative and positive \(SM\) values. Each time, the aircraft was trimmed and linearized about steady cruise flight condition for an airspeed of \(V_{0}=10\) m/s and height \(H_{0}=100\) m with an undeflected abdomen. The obtained eigenvalues of the various \(\mathbf{A}\) matrices are summarized in Table 4.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Percentage of Central** & **Static Margin (\%)** & **Eigenvalues** & **Dynamic Stability** \\ \hline
6 & 37.3 & \(\begin{array}{c}-20.80\pm 28.21\text{i}\\ -0.22\pm 0.95\text{i}\end{array}\) & Stable \\ \hline
11 & 24.6 & \(\begin{array}{c}-18.81\pm 23.02\text{i}\\ -0.20\pm 0.69\end{array}\) & Stable \\ \hline
16 & 9.5 & \(\begin{array}{c}-18.00\pm 20.39\\ -0.16\pm 0.41\end{array}\) & Stable \\ \hline
21 & \(\begin{array}{c}-2.7\end{array}\) & \(\begin{array}{c}-17.52\pm 18.91\text{i}\\ -0.34\end{array}\) & Unstable \\ \hline
26 & \(\begin{array}{c}-13.9\end{array}\) & \(\begin{array}{c}-17.24\pm 17.77\text{i}\\ -0.55\end{array}\) & Unstable \\ \hline \hline \end{tabular}
\end{table}
Table 4: Effect of abdominal mass as a percentage of central body mass on longitudinal stability.