16.333: Lecture #6 Aircraft longitudinal dynamics Typical aircraft open-loop motions · Longitudinal modes · Impact of actuators Linear Algebra in Action!
16.333: Lecture # 6 Aircraft Longitudinal Dynamics • Typical aircraft openloop motions • Longitudinal modes • Impact of actuators • Linear Algebra in Action!
Fa2004 16.3335-1 Longitudinal dynamics Recall: X denotes the force in the X-direction, and similarly for Y and Z, then(as on 4-13) OX Longitudinal equations(see 4-13 )can be rewritten as mu= Xuu+ Xww-mg cos 000+AX mla-qU0= Zuu+ Zww+ Zi+ Zaq-mg sin 000+AZ M2+Mn+ M,iwn+M0q+△Me · There is no roll/yaw motion,soq=θ Control commands△X,△ze,and△ M have not yet been specified
� � Fall 2004 16.333 5–1 Longitudinal Dynamics • Recall: X denotes the force in the Xdirection, and similarly for Y and Z, then (as on 4–13) ∂X Xu ≡ , . . . ∂u 0 • Longitudinal equations (see 4–13) can be rewritten as: mu˙ = Xuu + Xww − mg cos Θ0θ + ΔXc m(w˙ − qU0) = Zuu + Zww + Zw˙w˙ + Zqq − mg sin Θ0θ + ΔZc Iyyq˙ = Muu + Mww + Mw˙w˙ + Mqq + ΔMc There is no roll/yaw motion, so q = θ ˙ • . • Control commands ΔXc , ΔZc , and ΔMc have not yet been specified
Fa2004 16.3335-2 Rewrite in state space form as mg cos eo △XC (m-z) ∠n∠zq+mU0- ng sin6o △ZC Maiw+lug Mu Mw Mq △MC 0m-2t00 0 0 mg cos o △XC Lg mo -mg sin e △Zc M M 0 △Mc 0 Ex= Ar+c descriptor state space form →=E-(Ax+c)=A+c
Fall 2004 16.333 5–2 • Rewrite in state space form as ⎡ ⎡⎤ ⎤⎡ ⎡⎤ ⎤ mu˙ Xu Xw 0 −mg cos Θ0 u ΔXc ⎥ ⎥ ⎥ ⎦ ⎢ ⎢ ⎢ ⎣ ⎢ ⎢ ⎢ ⎣ ⎥ ⎥ ⎥ ⎦ ⎥ ⎥ ⎥ ⎦ ⎢ ⎢ ⎢ ⎣ ΔZc (m − Zw˙)w˙ Zu Zw Zq + mU0 −mg sin Θ0 Mu Mw Mq 0 w q = + −M ΔMc w˙w˙ + Iyyq˙ θ ˙ 0 0 1 0 θ 0 ⎡ ⎡⎤ ⎤ m 0 0 0 u˙ ⎢ ⎢ ⎢ ⎣ ⎢ ⎢ ⎢ ⎣ ⎥ ⎥ ⎥ ⎦ ⎥ ⎥ ⎥ ⎦ w˙ q˙ 0 m − Zw˙ 0 0 0 −Mw˙ Iyy 0 θ ˙ 0 0 0 1 ⎡ ⎤ ⎡⎤⎡ ⎤ Xu Xw 0 −mg cos Θ0 u ΔXc ⎢ ⎢ ⎢ ⎣ ⎢ ⎢ ⎢ ⎣ ⎥ ⎥ ⎥ ⎦ ⎥ ⎥ ⎥ ⎦ ⎢ ⎢ ⎢ ⎣ ⎥ ⎥ ⎥ ⎦ ΔZc Zu Zw Zq + mU0 −mg sin Θ0 Mu Mw Mq 0 w q = + ΔMc 0 0 1 0 θ 0 EX˙ = A¯X + ˆc descriptor state space form = E−1 ¯ ⇒ X (AX + ˆc) = AX + c ˙ ⎥ ⎥ ⎥ ⎦ ⎢ ⎢ ⎢ ⎣
Fa2004 16.3335-3 Write out in state space form 0 - g cos eo z Zg mlo m-2 -mg sin eo Iy[Mu+Zur Iw [M+ Zur[Ma+(Zg+mUo)r-IwyImg sin Oor Note: slight savings if we defined symbols to embed the mass/inertia X /m, Zu=Zu/m, and M / Iy then a matrix cola X COS Lq 9720 +么[+2[+2+ 1-Z, Check the notation that is being used very carefully To figure out the c vector, we have to say a little more about how the control inputs are applied to the system
� � � � � � Fall 2004 16.333 5–3 • Write out in state space form: ⎡ Xu Xw 0 −g cos Θ0 m m Zu Zw Zq + mU0 −mg sin Θ0 m − Zw˙ m − Zw˙ m − Zw˙ m − Zw˙ I−1 [Mu + ZuΓ] I−1 [Mw + ZwΓ] I−1 [Mq + (Zq + mU0)Γ] −Iyy mg sin Θ0Γ −1 yy yy yy 0 0 1 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ A = Mw˙ Γ = m − Zw˙ • Note: slight savings if we defined symbols to embed the mass/inertia Xˆ ˆ u = Xu/m, Zˆu = Zu/m, and Mq = Mq/Iyy then A matrix collapses to: ⎡ ⎤ ˆ ˆ Xu Xw 0 −g cos Θ0 Aˆ = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ Zˆu Zˆw Z −g sin Θ0 ˆq + U0 1 − Zˆw˙ 1 − Zˆw˙ 1 − Zˆw˙ 1 − Zˆw˙ Mˆ ˆ ˆ ˆ ˆ u + ZˆuΓ Mw + ZˆwΓ Mq + (Zˆq + U0) ˆ Γˆ −g sin Θ0Γ 0 0 1 0 ˆ ˆ Mw˙ Γ = 1 − Zˆw˙ • Check the notation that is being used very carefully • To figure out the c vector, we have to say a little more about how the control inputs are applied to the system
Fa2004 16.3335-4 Longitudinal Actuators Primary actuators in longitudinal direction are the elevators and thrust Clearly the thrusters/elevators play a key role in defining the steady-state/equilibrium flight condition Now interested in determining how they also influence the aircraft motion about this equilibrium condition deflect elevator u(t),w(t), q(t) Canard Rudde Recall that we defined AX as the perturbation in the total force in the x direction as a result of the actuator commands Force change due to an actuator deflection from trim Expand these aerodynamic terms using same perturbation approach △X=X66+X61b de is the deflection of the elevator from trim down positive on change in thrust Xs and Xs, are the control stability derivatives
Fall 2004 16.333 5–4 Longitudinal Actuators • Primary actuators in longitudinal direction are the elevators and thrust. – Clearly the thrusters/elevators play a key role in defining the steadystate/equilibrium flight condition – Now interested in determining how they also influence the aircraft motion about this equilibrium condition deflect elevator → u(t), w(t), q(t), . . . • Recall that we defined ΔXc as the perturbation in the total force in the X direction as a result of the actuator commands – Force change due to an actuator deflection from trim • Expand these aerodynamic terms using same perturbation approach ΔXc = Xδeδe + Xδpδp – δe is the deflection of the elevator from trim (down positive) – δp change in thrust – Xδe and Xδp are the control stability derivatives