Definition of axes and angles

The aim of this topic is to explain the commonly used axes for aircraft simulation and aerodynamic analysis.

The aerodynamic forces and moments on an aircraft are produced by the relative motion with respect to the air and depend on the orientation of the aircraft with respect to the airflow. Two orientation angles (with respect to the relative wind $V$) are needed to specify the aerodynamic forces and moments. These angles are the angle of attack $\alpha$and the sideslip angle $\beta$. The image below shows the definition of $\alpha$and $\beta$with respect to the body-fixed coordinate system. The origin of the body-fixed coordinate system coincides with the aircraft's center of gravity and its $x$-axis is parallel to the fuselage reference line and its $z$-axis in the (conventional) aircraft plane of symmetry.

Body-fixed coordinate system and aerodynamic angles

The commonly adopted convention is to label rotations about the $x$-, $y$-, and$z$-axes by $\phi$, $\theta$, and$\psi$. The angles are referred to as the roll angle, the pitch angle, and the the yaw angle. Rotation rates about the axes are labeled by $p$, $q$, $r$and are usually measured in radians per second. Transnational velocities along the $x$-, $y$-, and$z$-axes are labelled by $u$, $v$, and$w$.

Force and moment coefficients

The force and moments acting on the aircraft are commonly defined in terms of dimensionless aerodynamic coefficients. The coefficients are functions of the two aerodynamic angles, the Mach number, the Reynolds number, the control surface deflection, and the aircraft thrust. The thrust generated by the aircraft engines can affect the aerodynamic coefficients.

$C_{()}=C_{()}\left(\alpha,\beta,M,Re,\delta,T\right)$

Other factors that affect the aerodynamic coefficient are geometry changes such as deployment of landing gears, addition of external fuel tanks, ground proximity effect, etc. The definition of the coefficients with respect to the body-fixed coordinate system is:

Force/Moment	Description
$F_{x}=qS_{ref}C_{X}$	X-force
$F_{y}=qS_{ref}C_{Y}$	Y-force
$F_{z}=qS_{ref}C_{Z}$	Z-force
$M_{x}=l=qS_{ref}b_{ref}C_{l}$	X-moment or rolling moment
$M_{y}=m=qS_{ref}c_{ref}C_{m}$	Y-moment or pitching moment
$M_{z}=n=qS_{ref}b_{ref}C_{n}$	Z-moment or yawing moment

In the above equations $S_{ref}$is the aircraft reference area, $b_{ref}$is the aircraft reference span, $c_{ref}$is the aircraft reference chord (mean aerodynamic chord), and $q$is the dynamic pressure ($q=\frac{1}{2}\rho|\mathbf{V}|^2$).

The Aviumtechnologies Panel Method uses the above definition of the aerodynamic coefficients.