Spring 2003 16.61AC1-10 Qo= Ro=O. Note that the forces and moments are also perturbed Linearization for symmetric fight U= Uo +u, Vo= Wo=0 P+△F]=U+QW-RV≈i+q-r≈t 0+△F=V+RU-PW≈i+r(U+)-p≈t+r F+△F]=W+PV-QU≈t+p-q(o+) qUo △F 元+rUo △F 3 Attitude motion L IxzP+lxR +QR(I2x-Iwy+ pQl M Q+PR(Ix-12)+(R2-P2)l2 I2zR+lxP +PQ(Iyy-Ixz)-QRIzz △L ap+1, 4 →|△M △N 6 Key aerodynamic parameters are also perturbed Total Velocity VT =((Uo+u)2+v2+w2)1/2NUo+ Perturbed Sideslip angle B= sin(u/vr)av/U Perturbed Angle of Attack ar tan(W/U)Nw/Uo To understand these equations in detail, and the resulting impact on the vehicle dynamics, we must investigate the terms AF NSpring 2003 16.61 AC 1–10 • Linearization for symmetric flight U = U0 + u, V0 = W0 = 0, P0 = Q0 = R0 = 0. Note that the forces and moments are also perturbed. 1 m F0 x + ∆Fx = U˙ + QW − RV ≈ u˙ + qw − rv ≈ u˙ 1 m F0 y + ∆Fy = V˙ + RU − PW ≈ v˙ + r(U0 + u) − pw ≈ v˙ + rU0 1 m F0 z + ∆Fz = W˙ + P V − QU ≈ w˙ + pv − q(U0 + u) ≈ w˙ − qU0 ⇒ 1 m      ∆Fx ∆Fy ∆Fz      =      u˙ v˙ + rU0 w˙ − qU0      1 2 3 • Attitude motion:      L M N      =      IxxP˙ + IxzR˙ +QR(Izz − Iyy) + PQIxz IyyQ˙ +P R(Ixx − Izz)+(R2 − P2)Ixz IzzR˙ + IxzP˙ +P Q(Iyy − Ixx) − QRIxz      ⇒      ∆L ∆M ∆N      =      Ixxp˙ + Ixzr˙ Iyyq˙ Izzr˙ + Ixzp˙      4 5 6 Key aerodynamic parameters are also perturbed: Total Velocity VT = ((U0 + u) 2 + v2 + w2 ) 1/2 ≈ U0 + u Perturbed Sideslip angle β = sin−1 (v/VT ) ≈ v/U0 Perturbed Angle of Attack αx = tan−1 (w/U) ≈ w/U0 • To understand these equations in detail, and the resulting impact on the vehicle dynamics, we must investigate the terms ∆Fx ... ∆N