CHOICE OF BODY AXES
The equations derived in the preceding sections are valid for any orthogonal axes fixed in the airplane, with origin at the CG, and known as body axes. Since most aircraft are very nearly symmetrical, it is usual to assume exact symmetry, and to let Cxz be the plane of symmetry. Then Cx points “forward,” Cz. “downward,” and Су to
the right. In this case, the two products of inertia, Ixy and lyz, are zero, and (4.5,9) are consequently simplified.
The directions of Cx and Cz in the plane of symmetry are conventionally fixed in one of three ways (see Fig. 4.3).
These are chosen to coincide with the principal axes of the vehicle, so that the remaining product of inertia I7X vanishes; (4.3,4 and 5) then yield
These are chosen so that Cx is aligned with V in a reference condition of steady symmetric flight. In this case, the reference values of v and w are zero, and the axes are termed stability axes. These axes are commonly used, owing to the simplifications that result in the equations of motion, and in the expressions for the aerodynamic forces.
With this choice, it should be noted that for different initial flight conditions the axes are differently oriented in the airplane, and hence the values of Ix, /,, and l7X will vary from problem to problem. The “stability axes,” just as the principal axes, are body axes that remain fixed to the airplane during the motion considered in any one problem.
The following formulas are convenient for computing Ix, Iz, I7X when the values IXp and / are known for principal axes (see Exercise 4.4).
Ix = I cos2 € + IZp sin2 € L = Ir sin2 e + L cos2 є
z Xp Zp
4* = 2 (4> – 4„) sin 2e
where є = angle between xp (principal axis) and xs (stability axis), positive as shown (see Fig. 4.3).
When the axes are neither principal axes nor stability axes, they are usually called simply body axes. In this case the x axis is usually fixed to a longitudinal reference line in the airplane. These axes may be the most convenient ones to use if the aerodynamic data have been measured by a wind-tunnel balance that resolves the forces and moments into body-fixed axes rather than tunnel-fixed axes.