For fixed-wing aircraft longitudinal motions, it is often desirable to use the stability-frame instead of the body-fixed frame, as shown in the figure, where Vr = √²+w² is the true air speed, which is the magnitude of the body-frame velocity under no wind condition, and a = tan-¹. Note that u Vtcosa, w = Vtsina. 

r

Use the body-frame longitudinal translational EOM 

 

 to derive the stability-frame translational EOM 

 

u = V cos a w = Vr sin a ZS W U Zb Longitudinal Axis Flight Path Vector VT a Angle of Attack Flight Path Angle Y Xs Xb Pitch Angle 0