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# lecture_6 - 16.333 Lecture 6 Aircraft Longitudinal Dynamics...

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16.333: Lecture # 6 Aircraft Longitudinal Dynamics Typical aircraft open-loop motions Longitudinal modes Impact of actuators Linear Algebra in Action!

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Fall 2004 16.333 5–1 Longitudinal Dynamics Recall: X denotes the force in the X -direction, and similarly for Y and Z , then (as on 4–13) ∂X , . . . X u ∂u 0 Longitudinal equations (see 4–13) can be rewritten as: mu ˙ = X u u + X w w mg cos Θ 0 θ + Δ X c m ( ˙ w qU 0 ) = Z u u + Z w w + Z w ˙ w ˙ + Z q q mg sin Θ 0 θ + Δ Z c I yy q ˙ = M u u + M w w + M w ˙ w ˙ + M q q + Δ M c There is no roll/yaw motion, so q = θ ˙ . Control commands Δ X c , Δ Z c , and Δ M c have not yet been specified.
Fall 2004 16.333 5–2 Rewrite in state space form as mu ˙ X u X w 0 mg cos Θ 0 u Δ X c Δ Z c ( m Z w ˙ ) ˙ w Z u Z w Z q + mU 0 mg sin Θ 0 M u M w M q 0 w q + = Δ M c M w ˙ w ˙ + I yy q ˙ θ ˙ 0 0 1 0 θ 0 m 0 0 0 u ˙ w ˙ q ˙ 0 m Z w ˙ 0 0 0 M w ˙ I yy 0 θ ˙ 0 0 0 1 X u X w 0 mg cos Θ 0 u Δ X c Δ Z c Z u Z w Z q + mU 0 mg sin Θ 0 M u M w M q 0 w q + = Δ M c 0 0 1 0 θ 0 ¯ E ˙ = A X + ˆ descriptor state space form c X ¯ = E 1 ( A X + ˆ c ) = A X + c ⇒ X ˙

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Fall 2004 16.333 5–3 Write out in state space form: X u X w 0 g cos Θ 0 m m Z q + mU 0 mg sin Θ 0 Z u Z w m Z w ˙ m Z w ˙ m Z w ˙ m Z w ˙ I 1 [ M u + Z u Γ] I 1 [ M w + Z w Γ] I 1 [ M q + ( Z q + mU 0 )Γ] yy mg sin Θ 0 Γ I 1 yy yy yy 0 0 1 0 A = M w ˙ Γ = m Z w ˙ Note: slight savings if we defined symbols to embed the mass/inertia ˆ ˆ X u = X u /m , Z ˆ u = Z u /m , and M q = M q /I yy then A matrix collapses to: ˆ ˆ 0 g cos Θ 0 X u X w A ˆ = Z ˆ u Z ˆ w g sin Θ 0 Z ˆ q + U 0 1 Z ˆ w ˙ 1 Z ˆ w ˙ 1 Z ˆ w ˙ 1 Z ˆ w ˙ ˆ ˆ ˆ ˆ ˆ M u + Z ˆ u Γ M w + Z ˆ w Γ M q + ( Z ˆ q + U 0 ) ˆ ˆ Γ g sin Θ 0 Γ 0 0 1 0 ˆ ˆ M w ˙ Γ = 1 Z ˆ w ˙ 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 5–4 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 ) , . . .

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