《航空器的稳定与控制》（英文版）Lecture 7 More Stability Derivatives

16.333: Lecture #7 pproximate Longitudinal Dynamics Models A couple more stability derivatives Given mode shapes found identify simpler models that capture the main re sponses

16.333: Lecture # 7 Approximate Longitudinal Dynamics Models • A couple more stability derivatives • Given mode shapes found identify simpler models that capture the main re￾sponses

Fa2004 16.3336-1 More Stability Derivatives Recall from 6-2 that the derivative stability derivative terms zai, and Mi ended up on the lhs as modifications to the normal mass and Inertia terms displaced air is "entrained"and moves with the aircraf rounding These are the apparent mass effects- some of the sur Acceleration derivatives quantify this effect Significant for blimps, less so for aircraft e Main effect: rate of change of the normal velocity w causes a transient in the downwash e from the wing that creates a change in the angle of attack of the tail some time later -downwash lag effect If aircraft flying at Uo, will take approximately At= lt Uo to reach the tail Instantaneous downwash at the tail E(t) is due to the wing a at time t-△t Oe )=ba(t-△) Taylor series expat nsion (t-△)≈a(t)-a△t Note that Ae(t)=Aat. Change in the tail aoa can be com uted as △e(t) a△t

Fall 2004 16.333 6–1 More Stability Derivatives • Recall from 6–2 that the derivative stability derivative terms Zw˙ and Mw˙ ended up on the LHS as modifications to the normal mass and inertia terms – These are the apparent mass effects – some of the surrounding displaced air is “entrained” and moves with the aircraft – Acceleration derivatives quantify this effect – Significant for blimps, less so for aircraft. • Main effect: rate of change of the normal velocity w˙ causes a transient in the downwash � from the wing that creates a change in the angle of attack of the tail some time later – Downwash Lag effect • If aircraft flying at U0, will take approximately Δt = lt/U0 to reach the tail. – Instantaneous downwash at the tail �(t) is due to the wing α at time t − Δt. ∂� �(t) = ∂αα(t − Δt) – Taylor series expansion α(t − Δt) ≈ α(t) − α˙ Δt – Note that Δ�(t) = −Δαt. Change in the tail AOA can be com￾puted as d� d� lt Δ�(t) = − α˙ Δt = α˙ = −Δαt dα −dα U0

Fa2004 16.3336-2 For the tail, we have that the lift increment due to the change in downwash Is △CL=CL△a=Cn0 da Uo The change in lift force is then △L=5()S△CL In terms of the z-force coefficient △L △Cz= da u We use c/(2Uo) to nondimensionalize time, so the appropriate stabil- ity coefficient form is (note use C2 to be general, but we are looking at△C2 from before) aC, 2U0/C (ac/2U0) ca t de o The pitching moment due to the lift increment is △M lt△L p()S△CL △C apSo

� � � � Fall 2004 16.333 6–2 • For the tail, we have that the lift increment due to the change in downwash is d� lt ΔCLt = CLαt Δαt = CLαt α˙ dα U0 The change in lift force is then 1 ΔLt = ρ(U0 2 )tStΔCLt 2 • In terms of the Z­force coefficient ΔLt St St d� lt ΔCZ = 1 = −η ΔCLt = −η CLαt − α˙ ρU0 2S S S dα U0 2 • We use c/¯ (2U0) to nondimensionalize time, so the appropriate stabil￾ity coefficient form is (note use Cz to be general, but we are looking at ΔCz from before): ∂CZ 2U0 ∂CZ CZα˙ = = α¯ 0 ∂ ( ˙ c/2U0) c¯ ∂α˙ 0 2U0 St lt d� = −η c¯ S U0 CLαt dα d� = −2ηVHCLαt dα • The pitching moment due to the lift increment is ΔMcg = −ltΔLt 1ρ(U2 0 )tStΔCLt → ΔCMcg = − 1 lt 2 ρU0 2Sc¯ 2 d� lt = −ηVHΔCLt = −ηVHCLαt α˙ dα U0

Fa2004 16.3336-3 aCr 2Uo/aCM a(ac/200)/o 0 de lt 20 de lt de la Similarly, pitching motion of the aircraft changes the aoa of the tail Nose pitch up at rate g, increases apparent downwards velocity of tail y qlt, changing the aoa by which changes the lift at the tail (and the moment about the cg Following same analysis as above: Lift increment △Lt=Cn2CS △Cz 点p(U)S 0 aCz 20o/aCz Uo lt 9(qc/20)/0 0 2nVHCI ● Can also show that 1q Lg

• � � � � Fall 2004 16.333 6–3 So that ∂CM 2U0 ∂CM CMα˙ = = α¯ 0 ∂ ( ˙ c/2U0) c¯ ∂α˙ 0 d� lt 2U0 = −ηVHCLαt dα U0 c¯ d� lt = −2ηVHCLαt dα c¯ lt ≡ CZα˙ c¯ • Similarly, pitching motion of the aircraft changes the AOA of the tail. Nose pitch up at rate q, increases apparent downwards velocity of tail by qlt, changing the AOA by qlt Δαt = U0 which changes the lift at the tail (and the moment about the cg). • Following same analysis as above: Lift increment ΔLt = CLαt qlt U0 1 2 ρ(U2 0 )tSt ΔCZ = − ΔLt 1 2ρ(U2 0 )S = −η St S CLαt qlt U0 � � � � CZq ≡ ∂CZ ∂(qc/¯ 2U0) 0 = 2U0 c¯ ∂CZ ∂q 0 = −η 2U0 c¯ lt U0 St S CLαt = −2ηVHCLαt • Can also show that lt CMq = CZq c¯

Fa2004 16.3336-4 Approximate Aircraft Dynamic Models It is often good to develop simpler models of the full set of aircraft namIcs Provides insights on the role of the aerodynamic parameters on the frequency and damping of the two modes Useful for the control design work as well Basic approach is to recognize that the modes have very separate sets of states that participate in the response Short Period -primarily g and w in the same phase The u and q response is very small Phugoid -primarily 0 and u, and 0 lags by about 90 The w and g response is very smal Full equations from before △Xe △Zc q Mu+Zurl Mu+Zurl Ma+(gtmo)r q △Me

Fall 2004 16.333 6–4 Approximate Aircraft Dynamic Models • It is often good to develop simpler models of the full set of aircraft dynamics. – Provides insights on the role of the aerodynamic parameters on the frequency and damping of the two modes. – Useful for the control design work as well • Basic approach is to recognize that the modes have very separate sets of states that participate in the response. – Short Period – primarily θ and w in the same phase. The u and q response is very small. – Phugoid – primarily θ and u, and θ lags by about 90◦. The w and q response is very small. • Full equations from before: ⎡ ⎤ ⎡ Xu u˙ m ⎣ w˙ Zu w ⎦ = ⎣ [Mu+ZuΓ] m−Z ˙ q˙ θ ˙ Iyy 0 Xw m Zw m−Zw˙ [Mw+ZwΓ] Iyy 0 0 Zq+mU0 m−Zw˙ [Mq+(Zq+mU0)Γ] Iyy 1 −g cos Θ0 ⎤� � � � u ΔXc −mg sin Θ0 m−Z ˙ w ΔZc −mg sin Θ0Γ w ⎦ q + ΔMc Iyy 0 θ 0

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

Fa2004 16.3336-6 Characteristic equation for this system: s-+ 2Sspwsp s+Wsp=0, where the full approximation gives ZuMa Uom Given approximate magnitude of the derivatives for a typical aircraft can develop a coarse approximate 2 sas≈ 2 vUoMwlyy UoM Numerical values for 747 ency Damping rad/sec Full model 0.962 0.387 Full Approximate 0.963 0.385 Coarse Approximate 0.906 0.187 Both approximations give the frequency well, but full approximation gIves a much better damping estimate Approximations showed that short period mode frequency is de mined by Ma -measure of the aerodynamic stiffness in pitcher- Follows from discussion of CMa(see2- vard-changes sign Sign of m,n negative if cg sufficient far fory mode goes unstable) when cg at the stick fixed neutral point

� � � � Fall 2004 16.333 6–6 • Characteristic equation for this system: s2 + 2ζspωsps + ω2 = 0, sp where the full approximation gives: Zw Mq Mw˙ 2ζspωsp = − + + U0 m Iyy Iyy ω2 ZwMq U0Mw sp = mIyy − Iyy • Given approximate magnitude of the derivatives for a typical aircraft, can develop a coarse approximate: 2ζspωsp ≈ − Mq Iyy ⎫ ⎬ ζsp ≈ − Mq 2 −1 U0MwIyy ω2 U0Mw ⎭ → sp ≈ − ωsp ≈ −U0Mw Iyy Iyy • Numerical values for 747 Frequency Damping rad/sec Full model 0.962 0.387 Full Approximate 0.963 0.385 Coarse Approximate 0.906 0.187 Both approximations give the frequency well, but full approximation gives a much better damping estimate • Approximations showed that short period mode frequency is deter￾mined by Mw – measure of the aerodynamic stiffness in pitch. – Sign of Mw negative if cg sufficient far forward – changes sign (mode goes unstable) when cg at the stick fixed neutral point. Follows from discussion of CMα (see 2–11)

Fa2004 16.3336-7 For the Phugoid approximation, start again with X 0 m g cos ec △XC sin e m-∠iy △Z q Mu+Z,r Mo+Zur Mg+(zg+_mg sin eor △Mc yg 0 0 1. Changes to w and q are very small compared to u, so we can Set ai≈0andq≈0 Set 00=0 U 2n △X 0 △ZC [Mu+zury (Mu+zur)[Mg+(Zo+mor) 0 q △M 0 2. Use what is left of the Z-equation to show that with these ap- proximations(elevator inputs Lg+mlo Zain Mu+zurl Mg+(zq+mvo)r q LMu+Zurl [MBe+zer] 3.Use(∠Z<mso≈M)and(z≤mUo) so that Mu+Zu mil [mg+UoM Mu+Z,Mi lju

� � � � � � � � � � � � � � � � Fall 2004 16.333 6–7 • For the Phugoid approximation, start again with: ⎡ ⎤ ⎡ ⎤ Xu Xw m m 0 −g cos Θ0 −mg sin Θ0 ΔXc u˙ u ⎢ ⎢ ⎢ ⎢ ⎣ ⎥ ⎥ ⎥ ⎥ ⎦ ⎥ Zq+mU0 ⎥ ⎥ ⎦ Zu Zw ⎢ ⎢ ⎢ ⎣ ⎥ ⎥ ⎥ ⎦ ⎢ ⎢ ⎢ ⎣ ΔZc ΔMc w˙ q˙ w q m−Zw˙ m−Zw˙ m−Zw˙ m−Zw˙ = [Mq+(Zq+mU0)Γ] + [Mu+ZuΓ] [Mw+ZwΓ] −mg sin Θ0Γ Iyy Iyy Iyy Iyy θ ˙ θ 0 0 0 1 0 1. Changes to w and q are very small compared to u, so we can – Set w˙ ≈ 0 and q˙ ≈ 0 – Set Θ0 = 0 Xu Xw 0 −g m m ⎡ ⎤ ⎡ ⎤ ⎡⎤⎡ ⎤ ΔXc u˙ u ⎢ ⎢ ⎢ ⎢ ⎣ ⎥ ⎥ ⎥ ⎥ ⎦ ⎢ Zu Zw Zq+mU0 ⎢ ⎢ ⎣ ⎥ ⎥ ⎥ ⎦ ⎢ ⎢ ⎢ ⎣ ⎥ ⎥ ⎥ ⎦ ⎢ ⎢ ⎢ ⎣ ⎥ ⎥ ⎥ ⎦ 0 0 ΔZc ΔMc 0 0 w q m−Zw˙ m−Zw˙ m−Zw˙ = [Mq+(Zq+mU0)Γ] + [Mu+ZuΓ] [Mw+ZwΓ] Iyy Iyy Iyy θ ˙ θ 0 0 0 1 0 2. Use what is left of the Z­equation to show that with these ap￾proximations (elevator inputs) ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ Zu Zδe Zw Zq+mU0 ⎢ m−Zw˙ m−Zw˙ m−Zw˙ m−Zw˙ ⎣ ⎥ ⎦ ⎢ ⎣ ⎥ ⎦ w = − ⎣ ⎦u− δe [Mw+ZwΓ] [Mq+(Zq+mU0)Γ] [Mδe+ZδeΓ] q [Mu+ZuΓ] I Iyy yy Iyy Iyy 3. Use (Zw˙ � m so Γ ≈ Mw˙ ) and (Zq � mU0) so that: m Zw mU0 w Mw + Zw Mw˙ [Mq + U0Mw˙ ] q m ⎡⎤⎡ ⎤ ⎥ ⎥ ⎥ ⎦ ⎢ ⎢ ⎢ ⎣ Zu Zδe = Mu + Zu Mw˙ − u − Mw˙ m Mδe + Zδe m δe

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|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

Fa2004 16.33369 And X B (Z+[翻M 8. Which gives 2Sphwp X a/7 Z Numerical values for 747 Frequency Damping rad /sec Fu u mode 0.06730.0489 Fu| Appro× imate 0.06700.0419 Coarse Approximate.0611 0.0561

� � � � � � � � � � Fall 2004 16.333 6–9 And ⎡ ⎤ u xph = Aph = θ ⎢ ⎢ ⎣ Xu −g m −Zu 0 mU0 ⎥ ⎥ ⎦ ⎡ ⎤ Bph = ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ Xδe − Mw Mδe Xw m −Z + Zw δe Mw Mδe ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ mU0 8. Which gives 2ζphωph = −Xu/m ω2 gZu ph = −mU0 Numerical values for 747 Frequency Damping rad/sec Full model 0.0673 0.0489 Full Approximate 0.0670 0.0419 Coarse Approximate 0.0611 0.0561