Fa2004 16.3336-5 For the Short Period approximation 1. Since u 0 in this mode, then i 0 and can eliminate the X-force equation Zambo -mg sin eo m-li m-Z m-Z △Z IMo+Zu r IMo+(zg+mUorI_mg sin 6or q+|△Mc 2. Typically find that Zi m and lg muo. Check for 747 Zin=1909<m=2.86×105 z=4.5×105≤mU0=6.8×107 M →I≈ gsin eo △Z Mutz m Mil ma+(mUo mi mg sin eo Mri +△M 0 3. Set 00=0 and remove 0 from the model (it can be derived from With these approximations, the longitudinal dynamics reduce to +B where Se is the elevator input, and Uo w(mw+Mizu /) Iw(M+Mi Uo) B Iyy(Mse+Mize/m)� � � � � � � � � � Fall 2004 16.333 6–5 • For the Short Period approximation, 1. Since u ≈ 0 in this mode, then u˙ ≈ 0 and can eliminate the X­force equation. ⎡ ⎤ ⎡ ⎤ Zq+mU0 −mg sin Θ0 ⎡⎤⎡ ⎤ m−Z ˙ Zw ΔZ w c ˙ m−Zw˙ m−Zw˙ w [Mq+(Zq+mU0)Γ] ⎢ ⎢ ⎣ ⎥ ⎥ ⎦ w ⎣ ⎦ = [Mw+ZwΓ] ⎣ ⎦+⎣ ΔMc −mg sin Θ0Γ Iyy q˙ q Iyy Iyy θ ˙ θ 0 0 1 0 2. Typically find that Zw˙ � m and Zq � mU0. Check for 747: – Zw˙ = 1909 � m = 2.8866 × 105 – Zq = 4.5 × 105 � mU0 = 6.8 × 107 Mw˙ Mw˙ Γ = m − Zw˙ ⇒ Γ ≈ m ⎡ ⎤ ⎡ ⎤ Zw U ⎡⎤⎡ ⎤ 0 −g sin Θ0 ΔZ w c ˙ m w ⎢ ⎢ ⎣ ⎥ ⎥ ⎦ Mw+Zw Mw˙ Mq+(mU0) Mw˙ ⎣ ⎦ = m ⎣ ⎦+⎣ ΔMc −mg sin Θ0 Mw˙ Iyy q˙ m q Iyy Iyy m θ ˙ θ 0 0 1 0 3. Set Θ0 = 0 and remove θ from the model (it can be derived from q) • With these approximations, the longitudinal dynamics reduce to x˙ sp = Aspxsp + Bspδe where δe is the elevator input, and w Zw/m U0 xsp = q , Asp = I−1 (Mw + Mw˙Zw/m) I−1 (Mq + Mw˙U0) yy yy ⎦ ⎦ Zδe/m Bsp = I−1 (Mδe + Mw˙Zδe/m) yy