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 Entry II ENAE 791 - aunch and Entry Vehicle esign
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 II ENAE 791 - aunch and Entry Vehicle esign
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 II ENAE 791 - aunch and Entry Vehicle esign
ifferential Elements As before, ifferentiating (6), dh dt = v sin γ ρ = ρ o e h dρ dt = ρ o h s e h dh dt = ρ dh h s dt dρ dt = ρ h s v sin γ (5) (6) (7) MARYAN 4 ifting Atmospheric Entry II ENAE 791 - aunch and Entry Vehicle esign
ifferential Elements () Solve (7) for v v = h s sin γ 1 ρ dρ dt Substitute (8) into (1) and rewrite as (8) dγ dt = ρv = ρ sin γ 1 ρ dρ dt dγ dt = h s sin γ dρ (9) MARYAN 5 ifting Atmospheric Entry II ENAE 791 - aunch and Entry Vehicle esign
Solving for Flight Path Angle γ MARYAN γ e sin γdγ = h s cos γ cos γ e = h s cos γ = h s 6 ρ 0 ρ ρ + cos γ e (11) γ = cos 1 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 II ENAE 791 - aunch and Entry Vehicle esign
Flight Path Angle and Velocity Equations γ = cos 1 ρ oe h 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 ρ oe h + cos γ e (14) MARYAN 7 ifting Atmospheric Entry II ENAE 791 - aunch and Entry Vehicle esign
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 1+ 1+ e h (15) ifting Atmospheric Entry II ENAE 791 - aunch and Entry Vehicle esign
Fiddling with Algebra n = ρ o Substitute in (14) n = ρ ov e X 1+ MARYAN exp / 9 1+ γ e + cos 1 n = ρ ov e e h v γ e + cos 1 1+ ρ oe h ρ oe h e X / + cos γ e + cos γ e (16) (17) ifting Atmospheric Entry II ENAE 791 - aunch and Entry Vehicle esign
More Algebra Set dn dh =0 0= ρ ov e 1+ e h e X / / dx dh + 1 h s e h e X / (18) Factoring out common terms, 1 = h s / dx m dh (19) MARYAN 10 ifting Atmospheric Entry II ENAE 791 - aunch and Entry Vehicle esign
Even More Algebra From (13), γ = cos 1 ρ oe h + 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 II ENAE 791 - aunch and Entry Vehicle esign
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 II ENAE 791 - aunch and Entry Vehicle esign
Back to the Algebra From (13), cos γ = h s dx dh = 1 1 cos γ dx dh = 1 sin γ d dh ρ oe h ρ o ρ o + cos γ e e h 1 h s + cos γ e e h () MARYAN dx dh = 13 ρ o sin γ e h (3) ifting Atmospheric Entry II ENAE 791 - aunch and Entry Vehicle esign
Maximum eceleration Case If we go back to the n max case, (19) gives 1 = dx m h s dh 1 h s = / MARYAN 14 / ρ o hm sin γ m e h s h m, γ m are values at n max 1 = ρ o hm h s β sin γ m e h s sin γ m = ρ oh s β hm e h s (4) (5) ifting Atmospheric Entry II ENAE 791 - aunch and Entry Vehicle esign
et s Go Back to Algebra et Φ ρ oh s cos γ m =Φe h m h s sin γ m =Φe h m h s + cos γ e Φe h m h s et H e h m h s + cos γ e + Φe h m h s =1 (ΦH + cos γ e ) +(ΦH) =1 MARYAN 15 ifting Atmospheric Entry II ENAE 791 - aunch and Entry Vehicle esign
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 II ENAE 791 - aunch and Entry Vehicle esign
Maximum eceleration Equations Skipping some painful algebra and trig, ρ o h s h m = h s 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 17 1+ vm ifting Atmospheric Entry II ENAE 791 - aunch and Entry Vehicle esign
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 II ENAE 791 - aunch and Entry Vehicle esign
Perturbation Analysis ḣ = vγ ḣ = ḣ1 + ḣ = γ 1v 1 + ḣ = ḣ ḣ =(v 1 + v 1 )(γ 1 + γ) =v 1 γ + v 1 γ v 1 g ḣ = v 1 γ γ = mg + 1 mg 1 v 1 v c MARYAN v 1 g γ = mg = ḧ g 19 ifting Atmospheric Entry II ENAE 791 - aunch and Entry Vehicle esign