Axis Parameterization and Conventions
The body reference-point position vector Ro is best given by its components along the inertial Earth axes, shown as xe, ye, ze in Figure 9.2. In contrast, the local position vector r, and the body velocity and rotation
rate vectors U, Q are best given in the body axes xb, yb, zb, since these are also used to describe the flow about the body, e. g. via the velocity field V(r, t). To translate the individual components of any vector, such as U, from body to Earth axes, we apply the general vector transformation (F.1) derived in Appendix F.
The direction-cosine transformation matrix Tb is now formed as the product of three simple rotation matrices for the individual Euler angles in the standard sequence —ф, —в, —ф:
cos в cos ф sin ф sin в cos ф—cos ф sin ф cos ф sin в cos ф+sin ф sin ф
cos в sin ф sin ф sin в sin ф+cos ф cos ф cos ф sin в sin ф — sin ф cos ф
— sin в sin ф cos в cos ф cos в
The reciprocal conversion matrix T is composed of the reverse rotation sequence ф, в,ф. But this is also
the inverse of Tb, which is simply its transpose as derived in Appendix F for the general case.
As an application example, consider equations (9.1) and (9.2) used to obtain the gust velocity at an aircraft point P. The aircraft position Ro and the Vgust(R, t) function are typically provided in Earth axes, i. e. as RO and Vgust(Re, t), while the point P vector rp is known in body axes, as rp. Expressions (9.1) and (9.2) would therefore need to be evaluated as
RP = RO + Te rp (9.6)
Vgbustp = T eVusAR.*) (9.7)
with the final result Vgustp being in the body axes. This would then be usable for calculation of body forces and moments which are typically performed in the body axes.