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 responses
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 computed 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 stability 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