Airplane-Centered Coordinate Systems

Airplane-centered coordinate systems originate at a fixed point in the vehicle, called the reference point. The choice of the reference point is arbitrary, but it must be fixed point. In particular, the center of gravity (CG) cannot be the reference point because, over time, the CG does change in an airplane. Instead, a point such as the leading edge of the root chord, the tip of the nose, or the aerodynamic center of the wings serves as a reference point.

Body Axes.

Body axes are fixed with respect to the airplane. If the airplane moves, the body axes move with it.

The orientation of the body axes with respect to the vehicle frame is arbitrary. However, by convention, the x-axis points towards the front of the airplane, the y-axis points to the right, and the z-axis points to the bottom.

A special case of body axes is when they coincide with the vehicle's principal axes. This simplifies the equations of motion, because the airplane's products of inertia become zero. In most aircraft, the x-z-plane is a plane of symmetry, and so the y-axis is automatically a principal axis. While many simulators align the x- and z-axes with principal axes, some do not due to practical considerations. In many cases, it is simpler to make the x-axis parallel the horizontal reference axis from the blueprints, or some other line.

The x-z-plane in body axes is called the reference plane. This plane is useful for defining other vehicle coordinate systems.

Wind Axes.

In wind axes, the x-axis directly points into the relative wind. The z-axis remains in the reference plane, but rotates so that it remains perpendicular to the x-axis. The y-axis completes the right-handed system.

The transformation from body to wind axes consists of two rotations. First the body axes are rotated about the y-axis through the angle of attack $\alpha$; the axes are then rotated about the z-axis through the angle of sideslip $\beta$, yielding the wind axes. The angle of attack and the angle of sideslip are defined respectively by

  

$$\alpha \equiv \tan^{-1}(w_a/u_a)$$ (5)

$$\beta \equiv \sin^{-1}(v_a/V)$$ (6)

The main reason for the wind axis system is that it is more convenient for calculating aerodynamic forces. For instance, lift is, by definition, perpendicular to the relative wind, while drag is parallel. With wind axes, both lift and drag resolve into a force parallel to one of the axes.