# Estimation of Stability Parameters for Wide Body Aircraft Using Computational Techniques
## 1 Introduction
[PERSON] [39] presented an overview of the simulated stability and control system of an aircraft. Flying and handling qualities were determined using integrated optimization techniques. [PERSON] [40] investigated the dynamic stability and control characteristics of a tilt-rotor aircraft using mathematical nonlinear flight dynamics techniques. The damping coefficients in longitudinal and lateral modes were estimated and their effects on overall stability characteristics were analyzed. The geometric parameters along with flight conditions are the basic and most important requirements for modeling the aircraft in any software. The data of a Boeing 747-200 are readily available in online sources to be used as the foundations of modeling and analysis of the aircraft. The geometric parameters along with control surface clearances were acquired from established literature and reliable online sources [41]. Based upon theoretical knowledge such as the vortex latex method (VLM), source code was generated in both computer programs. For modeling of aircraft to acquire stability and control derivatives, design flight conditions were used. Analytical results available in the literature for low-cruise-flight conditions were selected as a benchmark. Figure 2 shows the flow chart of the methodology adopted during the project. The parameters in both software applications were derived from basic equations of motion (Equations (1)-(6)) for an aircraft (six degrees of freedom system) readily available in the literature [1; 42]. Force equations:  
\[F_{x}=M(\frac{du}{dt}+qw-rv) \tag{1}\]  
\[F_{y}=M(\frac{dv}{dt}+ru-pw) \tag{2}\]  
\[F_{z}=M(\frac{dw}{dt}+pv-qu) \tag{3}\]  
Moment equations:  
\[l=I_{x}\frac{dp}{dt}-I_{xz}\frac{dr}{dt}+qr(I_{z}-I_{y})-I_{xz}pq \tag{4}\]\[m=I_{y}\frac{dq}{dt}+I_{xz}(p^{2}-r^{2})+pr(I_{x}-I_{z}) \tag{5}\]  
\[n=I_{z}\frac{dr}{dt}-I_{xz}\frac{dp}{dt}+pq(I_{y}-I_{x})+I_{xz}qr \tag{6}\]  
where \(l\), \(m\), and \(n\) represent the rolling, pitching, and yawing moments. \(F_{x}\), \(F_{y}\), and \(F_{z}\) represent the resultant forces in longitudinal, lateral, and vertical axes, respectively. The symbols \(M\) and \(t\) represent the mass of the body and the time domain, respectively. The symbols \(u\), \(v\), and \(w\) represent the axial, side, and normal velocity components, respectively. The symbols \(p\), \(q\), and \(r\) represent the roll, pitch, and yaw rates, respectively. \(I_{x}\), \(I_{y}\), and \(I_{z}\) represent the moment of inertia components in x, y, and z-axes. \(I_{yz}\), \(I_{zx}\), and \(I_{xy}\) represent the products of inertia in roll, pitch, and yaw axes. Figure 2: Methodology adopted during the research. The equations of motion are derived for a particular axis system specific to the aircraft. Equations for aerodynamic, gravitational, and propulsive are derived using [PERSON]'s equations of motion while applying Euler angle corrections to make the equations consistent in one frame of reference. Small-angle approximation and steady flight conditions are used to linearize the equations with the assumption that small perturbations occur in the motion of an aircraft during steady flight. This assumption holds good for the theoretical determination of stability for the small-amplitude motion of aircraft. Once employed, forces/moments in x, y, and z-directions are computed and observation is the coupling of lateral and directional motion. The stability and control of an aircraft can be troublesome if things are discussed in theoretical form but a representation using stability and control coefficients makes life easier. After the development of complete linearized equations of motion, [PERSON]'s series expansion is used to determine the relations of the rates of change of stability parameters. The expressions of stability/control coefficients and derivatives are used in software solutions to run the simulations of modeled aircraft. The results are obtained using visual tools along with insights into the aerodynamic behavior of aircraft during motion.