16.333: Lecture #1 Equilibrium States Aircraft performance Introduction to basic terms
16.333: Lecture #1 Equilibrium States Aircraft performance Introduction to basic terms
Fa2004 16.3331-1 Aircraft performance Accelerated horizontal flight - balance of forces Engine thrust t Lift l(⊥toV) Drag L(‖tov Weight w T-D=m dt 0 for steady fight and L-W=0 Define l= dev ScI where P-air density(standard tables) s- gross wing area=cx b, c= mean chord wIng span AR-wing aspect ratio=b/c namIc pressure V= speed relative to the air
Fall 2004 16.333 1–1 Aircraft Performance • Accelerated horizontal flight balance of forces – Engine thrust T – Lift L (⊥ to V ) – Drag D (� to V ) – Weight W dV D T mg L T − D = m = 0 for steady flight dt and L − W = 0 • Define L = 1 2ρV 2SCL where – ρ – air density (standard tables) – S – gross wing area = c¯× b, – c¯ = mean chord – b = wing span – AR – wing aspect ratio = b/c¯ ct c0 kc �k edge b = 2.s c Sweepback angle Trailing – Q = 1 2ρV 2 dynamic pressure – V = speed relative to the air
Fa2004 16.3331-2 Y lift coefficient -for low Mach number, CL=CI o a angle of incidence of wind to the wing o ao is the angle associated with zero lift Back to the performance 1v2SCi and L=mg which implies that v so that and we can relate the effect of speed to wing lift o a key number is stall speed which is the lowest speed that an aircraft can fly steadily 2 779 SC where typically get Clma at max= 10
� � Fall 2004 16.333 1–2 – CL lift coefficient – for low Mach number, CL = CLα(α − α0) 3 α angle of incidence of wind to the wing 3 α0 is the angle associated with zero lift • Back to the performance: 1 L = ρV 2 SCL and L = mg 2 2mg which implies that V = ρSCL so that V ∝ C−1/2 L and we can relate the effect of speed to wing lift • A key number is stall speed, which is the lowest speed that an aircraft can fly steadily 2mg Vs = ρSCLmax where typically get CLmax at αmax = 10◦
Fa2004 16.3331-3 Steady Gliding Flight Aircraft at a steady glide angle of y e Assume forces are in equilibrium L-mg cos y =0 D+mg sin y =0 Gives that tan DL >Minimum gliding angle obtained when Cp/Cl is a minimum High L/d gives a low gliding angle Note: typically Cp=CD:+ ARe where ' Dmin is the zero lift( friction/parasitic)drag ii gives the lift induced drag is Oswald's efficiency factor a 0.7-0.85
Fall 2004 16.333 1–3 Steady Gliding Flight • Aircraft at a steady glide angle of γ • Assume forces are in equilibrium L − mg cos γ = 0 (1) D + mg sin γ = 0 (2) Gives that D CD tan γ = L ≡ CL ⇒ Minimum gliding angle obtained when CD/CL is a minimum – High L/D gives a low gliding angle • Note: typically CD = CDmin + 2 CL πARe where – CD is the zero lift (friction/parasitic) drag min – C2 L gives the lift induced drag – e is Oswald’s efficiency factor ≈ 0.7 − 0.85
Fa2004 16.3331-4 Total drag then given by d= Spv SCp=pvS(CDmin+hCl v SCD_ +k (4 Total drag No-lift dra Lift-dependent drag So that the speed for minimum drag is 2m9/k 1/4 min drag
Fall 2004 16.333 1–4 • Total drag then given by 1 1 � � D = ρV 2 SCD = ρV 2 S L CD (3) min + kC2 2 2 1 (mg) 2 = 2 ρV 2 SCDmin + k 1ρV 2S (4) 2 D VE VS1 VEmd Total drag No-lift drag Lift-dependent drag • So that the speed for minimum drag is � � �1/4 2mg k Vmin drag = ρS CDmin
Fa2004 16.3331-5 Steady Climb T-D R/C D W Equations: T-D-wsin y=0 L-wcos → which gives L T-D oS SIn that T-D tan Consistent with 1-3 if T=0 since then y as defined above is negative Note that for small,tany≈y≈siny B/C= V siny≈V≈ (T-D)V L so that the rate of climb is approximately equal to the excess power available(above that needed to maintain level flight)
Fall 2004 16.333 1–5 Steady Climb � V R/C W L D T � T - D • Equations: T − D − W sin γ = 0 (5) L − W cos γ = 0 (6) � which gives L T − D − cos γ = 0 sin γ so that T − D tan γ = L • Consistent with 1–3 if T = 0 since then γ as defined above is negative • Note that for small γ, tan γ ≈ γ ≈ sin γ R/C = V sin γ ≈ V γ ≈ (T − D)V L so that the rate of climb is approximately equal to the excess power available (above that needed to maintain level flight)
Fa2004 16.3331-6 Steady Turn L sin o Lcosφ Centrifugal force R Radius of turn W quations: Lsin g centrifugal force L o=w=mg V=Rw an Note: obtain Rmin at C W Rmin(pv sclmax)sin o ImIn 1/2pg CLmax sin max where w/ S is the wing loading andφmax<30°
Fall 2004 16.333 1–6 Steady Turn R W L cos � force L L sin � � � Radius of turn Centrifugal • Equations: L sin φ = centrifugal force (7) mV 2 = R (8) L cos φ = W = mg (9) ⇒ tan φ = V 2 Rg V =Rω = V ω g (10) • Note: obtain Rmin at CLmax Rmin( 1 2 ρV 2 SCLmax)sin φ = WV 2 g ⇒ Rmin = W/S 1/2ρgCLmax sin φmax where W/S is the wing loading and φmax < 30◦
Fa2004 16.3331-7 Define load factor N=L/mg. i.e. ratio of lift in turn to weight secp=(1+tan2) 1/ tan g N g o 9VN2-1 For a given load factor(wing strength Compare straight level with turning flight If same light coefficien L m79 V2s 5PV2S so that V=VNV gives the speed increase(more lift Note that Constant→ Cp constant→D∝V2C T∝D∝VCn~ND so that must increase throttle or will descend in the turn
Fall 2004 16.333 1–7 • Define load factor N = L/mg. i.e. ratio of lift in turn to weight N = sec φ = (1 + tan2 φ) 1/2 � (11) tan φ = N2 − 1 (12) so that V 2 V 2 R = g tan φ = g √N2 − 1 • For a given load factor (wing strength) R ∝ V 2 • Compare straight level with turning flight – If same light coefficient L mg 1 CL = 1 = ρV 2S ≡ Nmg ρVt 2 S ρV 2S 1 2 2 2 so that Vt = √ NV gives the speed increase (more lift) • Note that CL constant CD constant ⇒ D ∝ V 2 ⇒ CD ⇒ Tt ∝ Dt ∝ Vt 2 CD ∼ ND so that must increase throttle or will descend in the turn