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# lecture_7 - 16.333: Lecture # 7 Approximate Longitudinal...

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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

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Fall 2004 16.333 6–1 More Stability Derivatives Recall from 6–2 that the derivative stability derivative terms Z w ˙ and M w ˙ ended up on the LHS as modiFcations 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 SigniFcant 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 U 0 , will take approximately Δ t = l t /U 0 to reach the tail. Instantaneous downwash at the tail ( t ) is due to the wing α at time t Δ t . ( t ) = ∂α α ( t Δ t ) Taylor series expansion α ( t Δ t ) α ( t ) α ˙ Δ t Note that Δ ( t ) = Δ α t . Change in the tail AOA can be com- puted as d d l t Δ ( t ) = α ˙ Δ t = α ˙ = Δ α t U 0
± ± Fall 2004 16.333 6–2 For the tail, we have that the lift increment due to the change in downwash is d l t Δ C L t = C L α t Δ α t = C L α t α ˙ dαU 0 The change in lift force is then 1 Δ L t = ρ ( U 0 2 ) t S t Δ C L t 2 In terms of the Z -force coeﬃcient Δ L t S t S t d l t Δ C Z = 1 = η Δ C L t = η C L α t α ˙ ρU 0 2 S S S dαU 0 2 c/ (2 U 0 ) to nondimensionalize time, so the appropriate stabil- We use ¯ ity coeﬃcient form is (note use C z to be general, but we are looking at Δ C z from before): ∂C Z 2 U 0 ∂C Z = = C Z α ˙ α ¯ 0 ( ˙ c/ 2 U 0 ) c ¯ ∂α ˙ 0 2 U 0 S t l t d = η c ¯ S U 0 C L α t d = 2 ηV H C L α t The pitching moment due to the lift increment is Δ M cg = l t Δ L t 1 ρ ( U 2 0 ) t S t Δ C L t 1 Δ C M cg = l t 2 ρU 0 2 Sc ¯ 2 d l t = ηV H Δ C L t = ηV H C L α t α ˙ dαU 0

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± ± Fall 2004 16.333 6–3 So that ∂C M 2 U 0 ∂C M = = C M α ˙ α ¯ 0 ( ˙ c/ 2 U 0 ) c ¯ ∂α ˙ 0 d l t 2 U 0 = ηV H C L α t dαU 0 c ¯ d l t = 2 ηV H C L α t c ¯ l t C Z α ˙ c ¯ Similarly, pitching motion of the aircraft changes the AOA of the tail. Nose
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## lecture_7 - 16.333: Lecture # 7 Approximate Longitudinal...

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