Coordinate systems¶

In PyTornado there are two different coordinate systems, a body-fixed system and an aerodynamic system.

Body-fixed coordinate system¶

The body-fixed coordinate system can also be seen as a global system. The aircraft geometry, the centre of gravity and centre of rotation are defined in this system. Data in the aircraft input file is assumed to be defined in this system (see Aircraft file). The longitudinal aircraft axis is expected to be parallel to \(X\). The starboard wing span points in the \(Y\) direction and \(Z\) points upwards (right-handed coordinate system).

Fig. 9 Body fixed coordinate system

Note

CPACS uses the same convention. See also:

https://www.cpacs.de/pages/documentation.html

Aerodynamic coordinate system¶

The aerodynamic coordinate system is only relevant for the lift, drag and side force and the respective coefficients. The freestream direction expressed in the global (body-fixed) system is given as

\[\begin{split}\mathbf{V}_\infty = \begin{pmatrix} V_x \\ V_y \\ V_z \end{pmatrix}_\infty = V_\infty \cdot \begin{pmatrix} \cos \alpha \cdot \cos \beta \\ -\sin \beta \\ \sin \alpha \cdot \cos \beta \end{pmatrix}\end{split}\]

Fig. 10 Angle of attack \(\alpha\) and sideslip angle \(\beta\). Here \(\alpha\) is negative and \(\beta\) is positive.

where \(\alpha\) is the angle of attack and \(\beta\) is the sideslip angle. The transformation of the global forces \(F_x\), \(F_y\) and \(F_z\) into the aerodynamic system is given as

\[\begin{split}\begin{pmatrix} F_D \\ F_C \\ F_L \end{pmatrix} = \begin{bmatrix} \cos \beta \cdot \cos \alpha & -\sin \beta & \cos \beta \cdot \sin \alpha \\ \sin \beta \cdot \cos \alpha & \cos \beta & \sin \beta \cdot \sin \alpha \\ -\sin \alpha & 0 & \cos \alpha \end{bmatrix} \cdot \begin{pmatrix} F_x \\ F_y \\ F_z \end{pmatrix}\end{split}\]

where \(F_D\), \(F_C\) and \(F_L\) are drag, side force and lift, respectively.