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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 Fall 2004 16.333 6–1 More Stability Derivatives • Recall from 6–2 that the derivative stability derivative terms Zẇ and Mẇ 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 ẇ 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) = α(t − Δt) ∂α – Taylor series expansion α(t − Δt) ≈ α(t) − α̇Δt – Note that Δ�(t) = −Δαt. Change in the tail AOA can be com­ puted as Δ�(t) = − d� d� lt α̇Δt = − α̇ = −Δαt dα dα U0 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 α̇ dα U0 The change in lift force is then 1 ΔLt = ρ(U02)tStΔCLt 2 • In terms of the Z­force coeﬃcient St d� lt ΔLt St ΔCZ = −1 2 = −η ΔCLt = −η CLαt α̇ S S dα U0 2 ρU0 S ¯ • We use c/(2U 0 ) 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 CZα̇ = = ˙ c/2U0) 0 ∂ (α¯ c̄ ∂α̇ 0 d� 2U0 St lt CLαt = −η c̄ S U0 dα d� = −2ηVH CLαt dα • The pitching moment due to the lift increment is ΔMcg = −ltΔLt 1 2 2 ρ(U0 )t St ΔCLt →ΔCMcg = −lt 1 2 2 ρU0 Sc̄ = −ηVH ΔCLt = −ηVH CLαt α̇ d� lt dα U0 Fall 2004 16.333 6–3 • So that � CMα̇ = ∂CM ˙ c/2U0) ∂ (α¯ � � ∂CM ∂α̇ 0 d� lt 2U0 = −ηVH CLαt dα U0 c̄ d� lt = −2ηVH CLαt dα c̄ lt ≡ CZα̇ c̄ = 0 2U0 c̄ � • 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 Δαt = qlt U0 which changes the lift at the tail (and the moment about the cg). • Following same analysis as above: Lift increment ΔLt = CLαt qlt 1 ρ(U02)tSt U0 2 ΔLt qlt St = −η C Lαt 2 )S S U0 ρ(U 0 2 ΔCZ = − 1 � CZq ≡ ∂CZ ∂(qc̄/2U0) � 0 2U0 = c̄ � ∂CZ ∂q • Can also show that CMq = CZq � lt c̄ 2U0 lt St CL c̄ U0 S αt = −2ηVH CLαt = −η 0 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: ⎡ ⎤ ⎡ X X ⎣ u̇ ẇ q̇ θ̇ m Zu m−Zw˙ [Mu +Zu Γ] Iyy m Zw m−Zẇ [Mw +Zw Γ] Iyy 0 −g cos Θ0 Zq +mU0 m−Zẇ [Mq +(Zq +mU0 )Γ] Iyy −mg sin Θ0 m−Zw ˙ Θ0 Γ − mg sin Iyy 0 0 1 0 u ⎦= ⎣ w ⎤� ⎦ u w q θ � � + ΔX c ΔZ c ΔM c 0 � Fall 2004 16.333 6–5 • For the Short Period approximation, 1. Since u ≈ 0 in this mode, then u̇ ≈ 0 and can eliminate the X­force equation. ⎤ ⎡ Zq +mU0 −mg sin Θ0 ⎡ ⎤ ⎡ ⎡ ⎤ ⎤ Zw c ẇ w ΔZ m−Zẇ m−Zẇ m−Zw˙ ⎥ ⎢ ⎣ q̇ ⎦ = ⎢ [Mw +Zw Γ] [Mq +(Zq +mU0)Γ] − mg sin Θ0Γ ⎥ ⎣ q ⎦+ ⎣ ΔM c ⎦ ⎦ ⎣ Iyy Iyy Iyy θ 0 θ̇ 1 0 0 2. Typically ﬁnd that Zẇ � m and Zq � mU0. Check for 747: –Zẇ = 1909 � m = 2.8866 × 105 –Zq = 4.5 × 105 � mU0 = 6.8 × 107 Γ= Mẇ Mẇ ⇒Γ≈ m m − Zẇ ⎡ Zw U0 m � � � � ẇ Mẇ Mẇ ⎢ Mw +Z Mq +(mU0 ) m w m ⎣ q̇ ⎦= ⎢ ⎣ Iyy Iyy ⎡ ⎤ θ̇ 0 1 −g sin Θ0 Θ0 Mẇ − mg Isin m yy 0 ⎤ ⎤ ⎡ ⎤ c w ΔZ ⎥ ⎥ ⎣ q ⎦+ ⎣ ΔM c ⎦ ⎦ θ 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 ẋsp = Aspxsp + Bspδe where δe is the elevator input, and � � � � Zw /m U0 w xsp = , Asp = −1 −1 (Mw + Mẇ Zw /m) Iyy (Mq + Mẇ U0) Iyy q � � Zδe /m Bsp = −1 Iyy (Mδe + Mẇ Zδe /m) Fall 2004 16.333 6–6 2 • Characteristic equation for this system: s2 + 2ζspωsps + ωsp = 0, where the full approximation gives: � � Zw Mq Mẇ 2ζspωsp = − + + U0 m Iyy Iyy Z w M q U0 M w 2 ωsp = − Iyy mIyy • Given approximate magnitude of the derivatives for a typical aircraft, can develop a coarse approximate: � ⎫ Mq Mq ζsp ≈ − 2 U0M−1w Iyy 2ζspωsp ≈ − Iyy ⎬ → � U0 Mw ⎭ 2 ωsp ≈ − Iyy ωsp ≈ −UI0yyMw • Numerical values for 747 Frequency Damping rad/sec Full model 0.962 0.387 0.963 0.385 Full Approximate 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) Fall 2004 16.333 6–7 • For the Phugoid approximation, start again with: ⎡ ⎤ ⎡ Xu Xw 0 −g cos Θ0 m m u̇ Zq +mU0 −mg sin Θ0 Zu Zw ⎢ ⎥ ⎢ ⎢ ẇ m−Z m−Z m−Z m−Zẇ ⎢ ⎥ ⎢ ẇ ⎢ ⎥ = ⎢ [Mu+ZẇuΓ] [Mw +Zẇw Γ] [Mq +(Zq +mU 0 )Γ] Θ0 Γ ⎣ q̇ ⎦ ⎣ − mg sin Iyy Iyy Iyy Iyy θ̇ 0 0 1 0 ⎤⎡ ⎤ ⎡ c ⎤ ⎥ ⎢ u ⎥ ⎢ ΔXc ⎥ ⎥ ⎢ w ⎥ ⎢ ΔZ ⎥ ⎥ ⎢ ⎥+ ⎢ ⎥ ⎣ q ⎦ ⎣ ΔM c ⎥ ⎦ ⎦ θ 0 1. Changes to w and q are very small compared to u, so we can – Set ẇ ≈ 0 and q̇ ≈ 0 – Set Θ0 = 0 ⎤ ⎡ ⎤ ⎡ ⎡ ⎤ ⎡ Xu ⎤ Xw 0 −g c m m u̇ Zq +mU0 ⎢ ⎥ ⎢ u ⎥ ⎢ ΔXc ⎥ Z Z w u ⎢ ⎥ ⎢ 0 ⎥ ⎢ w ⎥ ⎢ ΔZ ⎥ m−Zẇ m−Zẇ ⎢ 0 ⎥ ⎢ m−Zẇ ⎥ ⎢ ⎥+ ⎢ ⎢ ⎥ = ⎢ [Mu+ZuΓ] [Mw +Zw Γ] [Mq +(Zq +mU0)Γ] ⎥ ⎣ q ⎦ ⎣ ΔM c ⎥ ⎣ 0 ⎦ ⎣ ⎦ 0 ⎦ Iyy Iyy Iyy θ 0 θ̇ 0 0 1 0 2. Use what is left of the Z­equation to show that with these ap­ proximations (elevator inputs) ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ Z Zq +mU0 Zw δ Z e u � � m−Zẇ m−Zẇ m−Z ẇ ⎢ ⎥ w ⎢ m−Zẇ ⎥ ⎣ ⎦ =− u− ⎣ ⎣ ⎦ ⎦ δe [M +Z Γ] q M +(Z +mU )Γ M +Z Γ u u [ ] [ ] q q 0 [Mw +Zw Γ] δe δe Iyy Iyy Iyy Iyy 3. Use (Zẇ � m so Γ ≈ Mm ẇ ) and (Zq � mU0) so that: � � � � Z mU w 0 w � � Mw + Zw Mmẇ [Mq + U0Mẇ ] q � � � � Zδe Zu � u− � � δ = − � e Mẇ Mẇ Mδe + Zδe m M u + Zu m Fall 2004 16.333 6–8 4. Solve to show that ⎡ ⎤ ⎡ mU0Mδe − Zδe Mq mU0Mu − ZuMq � � ⎢ Zw Mq − mU0Mw ⎥ ⎢ Zw Mq − mU0Mw w ⎢ ⎥ = ⎣ u + ⎢ ⎣ Zδe Mw − Zw Mδe ⎦ ZuMw − Zw Mu q Zw Mq − mU0Mw Zw Mq − mU0Mw ⎤ ⎥ ⎥ δe ⎦ 5. Substitute into the reduced equations to get full approximation: � � ⎡ ⎤ mU M − Z M u u q X X 0 u � � + m w Zw Mq −mU0Mw −g � � m u̇ ⎢ ⎥ u = ⎣ ⎦ � � θ θ̇ Zu Mw −Zw Mu 0 Zw Mq −mU0 Mw ⎡ � � ⎤ Xδe Xw mU0 Mδe −Zδe Mq + m Zw Mq −mU0Mw ⎥ ⎢ m + ⎣ ⎦ δe Zδe Mw −Zw Mδe Zw Mq −mU0 Mw 6. Still a bit complicated. Typically get that – |MuZw | � |Mw Zu| (1.4:4) – |Mw U0m| � |Mq Zw | (1:0.13) – |MuXw /Mw | � Xu small 7. With these approximations, the longitudinal dynamics reduce to the coarse approximation ẋph = Aphxph + Bphδe where δe is the elevator input. Fall 2004 16.333 6–9 And ⎤ Xu � � ⎢ m −g ⎥ u ⎥ xph = Aph = ⎢ ⎣ ⎦ θ −Zu 0 mU0 ⎡ � � � � ⎤ Xw Xδe − Mw Mδe ⎥ ⎢ ⎥ ⎢ m ⎥ ⎢ Bph = ⎢ � � � �⎥ ⎥ ⎢ −Z + Zw M ⎦ ⎣ δe δ e Mw ⎡ mU0 8. Which gives 2ζphωph = −Xu/m gZu 2 ωph = − 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 Fall 2004 16.333 6–10 • Further insights: recall that � �� � � �� � U0 ∂Z U0 ∂L = − ≡ −(CLu + 2CL0 ) QS ∂u 0 QS ∂u 0 M2 CL − 2CL0 ≈ −2CL0 = − 1 − M2 0 so � � � � ∂Z ρUoS 2mg Zu ≡ = (−2CL0 ) = − ∂u 0 2 U0 • Then � ωph −gZu = = mU0 √ g = 2 U0 � mg 2 mU02 which is exactly what Lanchester’s approximation gave Ω ≈ √ 2 Ug0 • Note that � Xu ≡ ∂X ∂u � � = 0 ρUoS 2 � (−2CD0 ) = −ρUoSCD0 and 2mg = ρUo2SCL0 so XuU0 Xu =− √ 2mωph 2 2mg � 2 � 1 ρUo SCD0 = √ SCL0 2 � ρUo2� 1 CD0 = √ 2 CL0 ζph = − so the damping ratio of the approximate phugoid mode is inversely proportional to the lift to drag ratio. −2 10 −250 −200 −150 −100 −50 0 50 10 −2 10 0 10 1 2 10 10 3 4 10 Freq (rad/sec) −1 10 Freq (rad/sec) −1 10 0 10 0 Transfer function from elevator to flight variables γ de de |Gude| arg Gude |G | arg Gγ 10 −2 −2 10 −350 −300 −250 −200 −150 −100 −50 0 10 −1 10 0 10 1 10 10 2 10 3 4 10 −1 10 Freq (rad/sec) −1 10 Freq (rad/sec) 0 10 0 10 Fall 2004 16.333 6–11 Freq Comparison from elevator (Phugoid Model) – B747 at M=0.8. Blue– Full model, Black– Full approximate model, Magenta– Coarse approximate model −2 10 0 −2 10 50 100 150 200 250 300 10 −1 0 10 10 1 2 10 Freq (rad/sec) −1 10 Freq (rad/sec) −1 10 0 10 0 Transfer function from elevator to flight variables γ de de |Gα | de arg Gude |G | arg Gγ 10 −2 −2 10 −350 −300 −250 −200 −150 −100 −50 0 10 −1 10 0 10 1 10 2 10 −1 10 Freq (rad/sec) −1 10 Freq (rad/sec) 0 10 0 10 Fall 2004 16.333 6–12 Freq Comparison from elevator (Short Period Model) – B747 at M=0.8. Blue– Full model, Magenta– Approx­ imate model Fall 2004 16.333 6–13 Summary • Approximate longitudinal models are fairly accurate • Indicate that the aircraft responses are mainly determined by these stability derivatives: Property Stability derivative Damping of the short period Mq Frequency of the short period Mw Damping of the Phugoid Xu Frequency of the Phugoid Zu Fall 2004 16.333 6–14 • Given a change in α, expect changes in u as well. These will both impact the lift and drag of the aircraft, requiring that we re­trim throttle setting to maintain whatever aspects of the ﬂight condition might have changed (other than the ones we wanted to change). We have: � � � �� � ΔL Lu Lα u = ΔD Du D α Δα But to maintain L = W , want ΔL = 0, so u = − LLα u Δα � � Lα Giving ΔD = − Lu Du + Dα Δα 2CL0 CL → Dα = QSCDα πeAR α → Lα = QSCLα QS (4 − 16) Du = (2CD0 ) U0 QS Lu = (2CL0 ) (4 − 17) U0 � � � � 2CD0 CLα ΔD = QS − + CDα Δα 2CL0 /U0 U0 � � 2 2CL0 QS −CD0 + CLα Δα = CL0 πeAR CDα = (T0 + ΔT ) − (D0 + ΔD) −ΔD = L + Δ L L0 0 � � 2CL0 CLα Δα CD0 − = CL0 CL0 πeAR tan Δγ = For 747 (Reid 165 and Nelson 416), AR = 7.14, so πeAR ≈ 18, CL0 = 0.654 CD0 = 0.043, CLα = 5.5, for a Δα = −0.0185rad (6–7) Δγ = −0.0006rad. This is the opposite sign to the linear simulation results, but they are both very small numbers.