Fa2004 16.3334-9 Linearization for symmetric fight U=Uo+u, Vo=Wo=0, Po= Qo= Ro=0 Note that the forces and moments are also perturbed AX0+△Ⅺ=U+QW-F 1Y0+△Y]=V+RU-PW +T(U0+)-p≈t+r z0+△Z]=W+PV-QU≈t (U0+ △X △Y 0+rU0 △Z ● Attitude motion L IrrP+IzR +QR(I2z-Iuy)+ PQI, M Iyy Q +PR(Ixz-I2)+(R IzaR+Ix p+PQ(Iwu-Ixx)-QR △L cp+ →△M 5 △N I2+Ix2p」6Fall 2004 16.333 4–9 • Linearization for symmetric flight U = U0 + u, V0 = W0 = 0, P0 = Q0 = R0 = 0. Note that the forces and moments are also perturbed. 1 [X0 + ΔX] = U˙ + QW − RV u˙ + qw − rv ≈ u˙ m ≈ 1 [Y0 + ΔY ] = V˙ + RU − PW m ≈ v˙ + r(U0 + u) − pw ≈ v˙ + rU0 1 ˙ [Z0 + ΔZ] = W + PV − QU w˙ + pv − q(U0 + u) m ≈ ≈ w˙ − qU0 ⎡ ⎤ ⎡ ⎤ ΔX u˙ 1 1 ⎣ ΔY ⎦ = ⎣ v˙ + rU0 ⇒ ⎦ 2 m ΔZ w˙ − qU0 3 • Attitude motion: ⎡ ⎤ ⎡ ⎤ L ˙ IxxP˙ + IxzR +QR(Izz − Iyy) + PQIxz ⎣ M ⎦ = ⎣ I ⎦ yyQ˙ +PR(Ixx − Izz) + (R2 − P2)Ixz N ˙ IzzR + IxzP˙ +PQ(Iyy − Ixx) − QRIxz ⎡ ⎤ ⎡ ⎤ ΔL Ixxp˙ + Ixzr˙ 4 ⇒ ⎣ ΔM ⎦ = ⎣ Iyyq˙ ⎦ 5 ΔN Izzr˙ + Ixzp˙ 6