Fa2004 16.33368 4. Solve to show that mUoM-ZmM mU0Mb。-Z6M ZuM AnMa -mom q ZuMu s2w/l Zo. Mu-zuMe d ZuMa- mom LuMg -mUmU 5. Substitute into the reduced equations to get full approximation S+n(mnM-nM)-9「u mOm ZuMy-zuMn 0 Xu muo Moe -se Mq m mzuMg-mUoM 6。Mu-uM6 ZuMg-mUoMy 6. Still a bit complicated. Typically get that Mn2<|MnZM(1.44) -|MnU0m>|M4Z|(10.13) Mu Xw/Ma< Xu small 7. With these approximations, the longitudinal dynamics reduce to the coarse approximation ph= AphIph+ Bphde where Se is the elevator input� � � � � � � � � � � Fall 2004 16.333 6–8 4. Solve to show that ⎡ ⎤ ⎡ ⎤ w q = ⎢ ⎢ ⎣ mU0Mu − ZuMq ZwMq − mU0Mw ZuMw − ZwMu ⎥ ⎥ ⎦ u + ⎢ ⎢ ⎣ mU0Mδe − ZδeMq ZwMq − mU0Mw ZδeMw − ZwMδe ⎥ ⎥ ⎦ δe ZwMq − mU0Mw ZwMq − mU0Mw 5. Substitute into the reduced equations to get full approximation: � ⎤ mU0Mu−ZuMq ZwMq−mU0Mw ⎡ Xu + Xw −g u˙ θ ˙ ⎢ ⎣ m m ⎥ ⎦ u = Z θ uMw−ZwMu 0 ZwMq−mU0Mw ⎡ ⎤ Xδe Xw mU0Mδe−ZδeMq ⎢ m + m ZwMq−mU0Mw ⎣ ⎥ + ⎦ δe ZδeMw−ZwMδe ZwMq−mU0Mw 6. Still a bit complicated. Typically get that – |MuZw| � |MwZu| (1.4:4) – |MwU0m| � |MqZw| (1:0.13) – |MuXw/Mw| � Xu small 7. With these approximations, the longitudinal dynamics reduce to the coarse approximation x˙ ph = Aphxph + Bphδe where δe is the elevator input