Aircraft Stability Analysis Including Unsteady Aerodynamic Effects

130A.2 Angle of attack and sideslip variation during rolling motion at fixedpitch angleIn a wind tunnel working in steady state conditions, the free stream velocity vector isconstant and horizontal. We therefore choose the X axis of the local horizontal referencesystem to be coincident with the velocity vector. When the airplane model rolls aroundthe sting, the effective angle of attack α( t)and sideslip ( t)to the model roll angle φ () t at that moment.β in a given time t are relatedThe definitions for the angle of attack and sideslip angle as a function of the body-axescomponents u, v, and w of the air velocity are [57], [59], [60]:wα () t = tan −1(A.2-1)u−1v −1vβ () t = sin = sin(A.2-2)V2 2 2u + v + wThe u, v, and w velocity body-axes components can be related to the air velocity througha local horizontal to the body axis reference systems transformation matrix T B,H , suchthat, in the particular case where the heading angle ψ is zero, we have⎧u⎫ ⎧V⎫⎪ ⎪ ⎪ ⎪⎨v⎬ = TB , H ⎨0⎬(A.2-3)⎪ ⎪ ⎪ ⎪⎩w⎭⎩0⎭⎡1⎢00⎤⎡cosθ0⎥⎢− sinθ⎤B ,=⎢0 cosφ() t sinφ()t⎥⎢0 1 0⎢ () () ⎥⎢⎥ ⎥⎥ ⎣0- sinφt cosφt ⎦⎣sinθ00 cosθ0⎦⎡ cosθ00 − sinθ0⎤⎢⎢sinθ0sinφ() t cosφ() t cosθ0sinφ()t⎢ () () ()⎥ ⎥⎥ sinθ0cosφt − sinφt cosθ0cosφt ⎦T H=⎣00(A.2-4)Therefore,