Lifting Entry (continued)

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ifting Entry (continued) Basic planar dynamics

Akin - All rights reserved http://spacecraft.

1 ifting Entry (continued) Basic planar dynamics of motion, again Yet another equilibrium glide Hypersonic phugoid motion Planar state equations MARYAN 1 01 avid. Akin - All rights reserved ifting Atmospheric Entry ENAE aunch and Entry Vehicle esign

2 Basic Equations of Motion From last time, mv v, s r v dγ γ dt = v m + r g cos γ horizontal dv v mg dt = r g sin γ m Assume (at entry velocities close to orbital) g = v r v dγ dt = m = ρv c A m = ρv (1) MARYAN dv dt = m = ρv () ifting Atmospheric Entry ENAE aunch and Entry Vehicle esign

3 Solving for Velocity ivide () by (1) dv dt v dγ dt = ρv ρv = dv v = dγ / (3) v v e dv v = 1 / γ γ e dγ ln v v e = (γ γ e) / = v = e γ γ e / v e (4) MARYAN 3 ifting Atmospheric Entry ENAE aunch and Entry Vehicle esign

4 ifferential Elements As before, ifferentiating (6), dh dt = v sin γ ρ = ρ o e h hs dρ dt = ρ o e h hs dh dt = ρ dh dt dρ dt = ρ v sin γ (5) (6) (7) MARYAN 4 ifting Atmospheric Entry ENAE aunch and Entry Vehicle esign

5 ifferential Elements () Solve (7) for v v = sin γ 1 ρ dρ dt Substitute (8) into (1) and rewrite as (8) dγ dt = ρv = ρ hs sin γ 1 ρ dρ dt dγ dt = sin γ dρ (9) MARYAN 5 ifting Atmospheric Entry ENAE aunch and Entry Vehicle esign

6 Solving for Flight Path Angle γ MARYAN γ e sin γdγ = cos γ cos γ e = cos γ = 6 ρ 0 ρ ρ + cos γ e (11) γ = cos 1 hs ρ + cos γ e (1) Note that the negative sign was inserted because cos -1 is ambiguous as to direction, and the flight path angle on entry should be >0. dρ (9) (10) ifting Atmospheric Entry ENAE aunch and Entry Vehicle esign

7 Flight Path Angle and Velocity Equations γ = cos 1 hs ρ oe h hs Rewrite (4) as v = v e exp γ γ e = v e exp / and substitute into (13) + cos γ e γe γ / (13) v = v e exp 1 / γ e + cos 1 hs ρ oe h hs + cos γ e (14) MARYAN 7 ifting Atmospheric Entry ENAE aunch and Entry Vehicle esign

8 eceleration Along the flight path, m = ρv Perpendicular to the flight path, m = ρv Total deceleration (not in g s) n = + m MARYAN = 1 m m n = ρ ov 8 + = ρv e h hs (15) ifting Atmospheric Entry ENAE aunch and Entry Vehicle esign

9 Fiddling with Algebra n = ρ o Substitute in (14) n = ρ ov e X 1+ MARYAN exp / 9 1+ γ e + cos 1 hs n = ρ ov e e h hs v γ e + cos 1 hs 1+ ρ oe h hs ρ oe h hs e X / + cos γ e + cos γ e (16) (17) ifting Atmospheric Entry ENAE aunch and Entry Vehicle esign

10 More Algebra Set dn dh =0 0= ρ ov e 1+ e h hs e X / / dx dh + 1 e h hs e X / (18) Factoring out common terms, 1 = / dx m dh (19) MARYAN 10 ifting Atmospheric Entry ENAE aunch and Entry Vehicle esign

11 Even More Algebra From (13), γ = cos 1 hs ρ oe h hs + cos γ e X = γ e γ = cos 1 (cos γ)+γ e Y cos γ X = γ e cos 1 Y (0) MARYAN 11 ifting Atmospheric Entry ENAE aunch and Entry Vehicle esign

12 Trigonometry, for a Change Trig identity - d(cos 1 u) dx = 1 1 u du dx (1) dx dh = Y ) d(cos 1 dh = 1 1 Y dy dh dx dh = 1 1 cos γ d(cos γ) dh MARYAN 1 ifting Atmospheric Entry ENAE aunch and Entry Vehicle esign

13 Back to the Algebra From (13), cos γ = dx dh = 1 1 cos γ dx dh = 1 sin γ d dh ρ oe h hs ρ o hs ρ o hs + cos γ e e h hs 1 + cos γ e e h hs () MARYAN dx dh = 13 ρ o sin γ e h hs (3) ifting Atmospheric Entry ENAE aunch and Entry Vehicle esign

14 Maximum eceleration Case If we go back to the n max case, (19) gives 1 = dx m dh 1 = / MARYAN 14 / ρ o hm sin γ m e h m, γ m are values at n max 1 = ρ o hm β sin γ m e sin γ m = ρ o β hm e (4) (5) ifting Atmospheric Entry ENAE aunch and Entry Vehicle esign

15 et s Go Back to Algebra et Φ ρ o cos γ m =Φe h m sin γ m =Φe h m + cos γ e Φe h m et H e h m + cos γ e + Φe h m =1 (ΦH + cos γ e ) +(ΦH) =1 MARYAN 15 ifting Atmospheric Entry ENAE aunch and Entry Vehicle esign

16 Algebra is Fun, on t You Think? Φ H + ΦH cos γ e + cos γ e 1=0 Φ H + ΦH cos γ e sin γ e =0 H = Φ cos γ e ± 4Φ cos γ e + 4(Φ )sin γ e 4Φ cos γ e ± cos γ e +sin γ e H = 1 Φ H = 1 Φ MARYAN cos γ e ± 16 1+sin γ e ifting Atmospheric Entry ENAE aunch and Entry Vehicle esign

17 Maximum eceleration Equations Skipping some painful algebra and trig, ρ o h m = ln sin γ e 4+ csc γ e cot γ e cos γ m = cos γ e (/)sinγ e 4+(/) csc γ e (/) cot γ e MARYAN v m = v e e γ m γe / n max = ρ ov e vm ifting Atmospheric Entry ENAE aunch and Entry Vehicle esign

18 Phugoid Oscillations Assume a shallow, linear entry trajectory: γ 1 0 ḣ = v sin γ = vγ v γ g = 1 mg v v c γ o 0 Small perturbations = h = h 1 + h γ = γ 1 + γ γ MARYAN 18 ifting Atmospheric Entry ENAE aunch and Entry Vehicle esign

19 Perturbation Analysis MARYAN 19 ḣ = vγ ḣ = ḣ1 + ḣ = γ 1v 1 + ḣ = ḣ ḣ =(v 1 + v 1 )(γ 1 + γ) =v 1 γ + v 1 γ m = 1 + m ḣ = v 1 γ Neglecting drag = v = constant = v Ac m ρ oe = v Ac m ρ oe h 1 + h hs h 1 hs e h hs = 1 h m e hs ifting Atmospheric Entry ENAE aunch and Entry Vehicle esign

20 Perturbation Analysis Using Taylor s series expansion, m = 1 1 h m v 1 g m = 1 m γ = mg + 1 mg h 1 v 1 v c MARYAN v 1 γ = m = ḧ 0 ifting Atmospheric Entry ENAE aunch and Entry Vehicle esign

21 Perturbed ift On an equilibrium glide, ḧ = m = 1 m MARYAN γ =0= m = g v r 1 mg =1 v 1 gr =1 v 1 1 h ḧ + 1 v 1 v c = v c 1 v 1 v c g h =0 g h = Simple harmonic motion (undamped) ifting Atmospheric Entry ENAE aunch and Entry Vehicle esign

22 Phugoid Parameters Frequency: ω = Period: P = π ω = MARYAN 1 v 1 v c π 1 v 1 v c 8000 m sec = v 1 g v c g = 169 sec 1 v 1 v c m sec P m1s m15s m15s 000 m55s ifting Atmospheric Entry ENAE aunch and Entry Vehicle esign

23 Free-Body iagram with Spherical Planet r v γ ω m m θ MARYAN 3 ifting Atmospheric Entry ENAE aunch and Entry Vehicle esign

24 Planar State Equations r v γ ω m m ω = γ θ Sum of transverse accelerations m g cos γ = ωv Sum of parallel accelerations θ m g sin γ = v MARYAN 4 ifting Atmospheric Entry ENAE aunch and Entry Vehicle esign

25 Planar State Equations () MARYAN ṙ = ḣ = v sin γ r θ = v cos γ ω = γ θ = γ v r cos γ m g cos γ = γ v r cos γ v g m v cos γ = γv r 1 m v g cos γ = γv rg 5 ifting Atmospheric Entry ENAE aunch and Entry Vehicle esign

26 The Canonical Planar State Equations v γ = m 1 v v c v = m g sin γ ṙ = ḣ = v sin γ r θ = v cos γ g cos γ Coupled first-order OEs MARYAN 6 ifting Atmospheric Entry ENAE aunch and Entry Vehicle esign

27 Associated Parameters to State Eqns m = 1 ρv Ac m = ρv Ac m c = ρv c m = 1 ρv Ac m = ρv ρ = ρ o e h hs MARYAN h = r r o g = g o ro r 7 ifting Atmospheric Entry ENAE aunch and Entry Vehicle esign

28 Numerical Integration - 4th Order R-K Given a series of equations ȳ = f(t, x) k 1 = t f (t n, y n ) k = t f t n + t, y n + k 1 k 3 = t f t n + t, y n + k k 4 = t f t n + t, y n + k 3 y n+1 = y n + k k 3 + k k O( t5 ) MARYAN 8 ifting Atmospheric Entry ENAE aunch and Entry Vehicle esign

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Lifting Entry 2. Basic planar dynamics of motion, again Yet another equilibrium glide Hypersonic phugoid motion MARYLAND U N I V E R S I T Y O F

Lifting Entry 2. Basic planar dynamics of motion, again Yet another equilibrium glide Hypersonic phugoid motion MARYLAND U N I V E R S I T Y O F ifting Entry Basic planar dynamics of motion, again Yet another equilibrium glide Hypersonic phugoid motion MARYAN 1 010 avid. Akin - All rights reserved http://spacecraft.ssl.umd.edu ifting Atmospheric