Rotorcraft Design I Day Two : Rotorcraft Modeling for Hover and Forward Flight

1. Rotorcraft Design IDay Two: Rotorcraft Modeling for Hover and Forward Flight Dr. Daniel P. Schrage Professor and Director Center of Excellence in Rotorcraft Technology (CERT) Center for Aerospace Systems Analysis (CASA) Georgia Institute of Technology Atlanta, GA 30332-0150

2. Presentation Outline • Fundamental Concepts and Relationships • Induced Power Required • Hover • Forward Flight • Rotor Profile Power Required • Hover • Forward Flight • Parasite Power Required • Simplified Trim (Moment Trim) • Example Problems

3. The Iterative Nature of Aerospace Synthesis Initiation and Coordination Phase Change requirements Requirements Change concept Change technology assumptions Concepts/Tech Change methodology Select search techniques Change parametric variables Select Methodology Fuel Balance Sizing W W Select parametric variables Design Iteration Optimum Configuration Select ranges for parametric variables Synthesis and Analysis Phase

4. Fixed Wing Aircraft Vehicle Synthesis

5. Configuration Requirements Modelsls Synthesis Solution Engine Power Performance Available Hover Alt. Vehicle Hover Temp. Power Block Speed Loading Block Alt.itudes Vehicle Power ROC/Maneuver Required Installed Power HP i Empty Fraction Mission Input Fuel Weight Ratio Available Vehicle Payload Gross Block Range Weight Hover Time Mission Analysis Agility Fuel Weight Ratio Required Rotorcraft/VSTOL Aircraft Synthesis ( RF Method)

6. RF Method Key Relationships IF: Fuel Ratio Required = Fuel Ratio Available Horsepower Required = Horsepower Available A Configuration Solution can be found to meet the Customer’s Mission & Performance Requirements THEN: The Concept Is FEASIBLE!!

7. Configuration Requirements Modelsls Synthesis Solution Engine Power Performance Available Hover Alt. Vehicle Hover Temp. Power Block Speed Loading Block Alt.itudes Vehicle Power ROC/Maneuver Required Installed Power HP i Empty Fraction Mission Input Fuel Weight Ratio Available Vehicle Payload Gross Block Range Weight Hover Time Mission Analysis Agility Fuel Weight Ratio Required Rotorcraft/VSTOL Aircraft Synthesis ( RF Method)

8. Achieving a Fuel Balance and Initial Gross Weight

9. Achieving a Power Balance Total Horsepower Required(THP) for Generic Subsonic Fixed Wing and Rotorcraft: KL Horsepower Available(HPA) as a Function of Altitude, Temp, Time, and No. of Engines: HPA= HP*(N-1)(1-Kh1h1)(1-Ktdelta TS)(1+e-0.0173t) N 4 THP = HPA for Hover, Forward Speed, Maneuver Critical Power Loading(THP/GW) sizes the Engine

10. Power Required Comparison for Fixed Wing and Rotary Wing Aircraft Note: High, Hot Day (4000’, 95o F) increases HP required

11. Rotorcraft Sizing Issues • HP required is determined for either hover out-of ground effect (HOGE), forward speed, or manu requirement; In the notional example below . . . • if speed reqmt. is 100 kts, the rotor and engine will be sized for hover condition • if the speed reqmt. Is, say, 250 kts, the rotor/engine will be sized for fwd flight • Conditions (altitude and ambient temperature) also affect rotorcraft sizing HP required Army Hot Day (4000’, 95o F) Normal Day Velocity 250 kts 100 kts

12. Derivation of Power Required and Steady State Thrust Equations • Induced Velocity and Power of a Rotor in Hover and Forward Flight • Determination of Rotor Profile Power • Steady State Thrust and Equilibrium Cyclic Pitch Equations for Straight and Level Flight

13. List of Symbols a = Airfoil section lift-curve slope, dCL/dα, rad-1,or speed of sound = (γgRT)1/2, fps AB = Effective blade area of rotor (projected to centerline of rotation) = bcR, ft2 AD = Rotor disk area = .785D2, ft2 AR = Aspect ratio Aw = Fuselage wetted area, ft2 Aπ = Equivalent flat plate area, ft2 b = No. of rotor blades B = Blade tip loss factor BSFC = Brake specific fuel consumption, lbs/bhp-hr c = Airfoil chord, ft CD = Parasite drag coefficient based on frontal area Cdomin = Blade section drag coefficient at CL = 0 CLr = Rotor mean blade lift coefficient

14. List of Symbols D = Rotor diameter, ft, or Parasite Drag, lbs E = Endurance g = Acceleration due to gravity, ft/sec2 ahp = for Horsepower available at engine output shaft chp = Horsepower available for vertical climb = (η – Ktr)ahp-RhpH-hpacc ihp = Rotor induced horsepower php = Parasite horsepower rhp = Total horsepower required = [1/ (η – Ktr)][ihp + Rhp + php + hpacc] Rhp = Rotor profile horsepower hpacc = Accessory horsepower hpaux= Auxiliary horsepower hptr = Total tail rotor horsepower

15. List of Symbols i = Stabilzer incidence relative to fuselage W.L., degrees Ktr = Tail rotor power factor = hptr/(1/η)(ihp + Rhp + php + hptr + hpacc) Ku = Induced velocity factor Kμ = Profile power factor lst = horizontal stabilizer moment arm (distance between main rotor centerline and tail rotor centerline), ft ltr = Tail rotor moment arm (distance between main rotor centerline and tail rotor centerline), ft L = Lift, lbs M = Mach No. P = Absolute pressure, lbs/ft2 q = dynamic pressure = ½ ρV2, lbs/ft2 Q = Rotor torque, lb-ft

16. List of Symbols R = Rotor radius, ft, or Gas constant for dry air = 53.3 ft.lb/lboR R/Cmax = Maximum rate of climb, fpm R/CV = Vertical rate of climb, fpm S = Frontal area, ft2 T = Rotor thrust, lbs, or = Absolute temperature, oR u = Induced velocity, fpm uc = Induced velocity in climb, fpm uH = Induced velocity in hover, fpm ui = Induced velocity in forward flight, fpm up = Equivalent inflow velocity to overcome fuselage parasite drag, fpm uR = Equivalent inflow velocity to overcome rotor profile drag, fpm Uc = Total inflow velocity in climb, fpm

17. List of Symbols vd = Rate of descent in autorotation, fpm V = Forward velocity, fps or knots Vclimb = Velocity for best rate of climb, knots Vcr = Cruising velocity, knots VT = Rotor tip speed, RΩ, fps w = Rotor disk loading, lbs/ft2 W = Gross Weight, lbs z = Rotor height above the ground (reference to teetering point or top of hub) αf = Fuselage angle of attack (angle between fuselage W.L. and horizontal), deg αr = Blade section angle of attack, degs or radians δ = Blade section drag coefficient η = Mechanical efficiency θT = Blade twist (referenced to centerline rotation), degrees

18. List of Symbols λ’ = Inflow velocity ratio = u/VT = (ui + uR + up)/VT Λ = Induced power correction factor due to ground effect μ = Tip speed ratio = V/VT ρ = Density, slugs/ft3 σ = Rotor solidity = bc/πR Ψ = Rotor azimuth angle, degrees Ω = Rotor angular velocity, rad/sec Subscripts: mr = Main rotor st = Stabilizer tr = Tail rotor

19. Fundamental Concepts and Relationships • Rotor Theory may best be understood by beginning with the hover and vertical climb flight conditions. • No dissymmetry of velocity across the disk • Simple momentum theory (actuator disk theory) • The axial velocity of fluid through airscrew disk is higher than speed with which airscrew is advancing. • The increase of velocity at the airscrew arises from the production of thrust (T) and is called the induced velocity (u) • Thrust developed by airscrew is product of mass air flow through disk per unit time and the total increase in velocity.

20. Momentum Theory • Stems from Newton’s second law of motion, F=ma, and is developed on the basis that the axial velocity of the fluid through the airscrew disk is generally higher than the speed with which the airscrew is advancing through the air • The increase in velocity of the air from its initial value to its value at the airscrew disk, which arises from the production of thrust, is called the induced or downwash velocity, and is denoted by u • The thrust developed by the airscrew is then equal to the mass of air passing through the disk in unit time, multiplied by the total increase in velocity caused by the action of the airscrew

21. Momentum Theory Model • Because of the increase in velocity of air mass by the rotor there is gradual contraction of slipstream • Airscrew advancing to left with freestream velocity V • Velocity increase at disk (aV), downstream (bV) P2 P3 P4 P1 V + bV V V + aV

25. Simple Momentum Theory Assumptions • The power required to produce the thrust is represented only by the axial kinetic energy imparted to the air composing the slipstream • A frictionless fluid is assumed so that there is no blade friction or profile-drag losses • Rotational energy imparted to the slipstream is ignored • The disk is infinitely thin so that no discontinuities in velocities occur on the two sides of the disk

26. Generation of Thrust • From momentum theory the thrust is: (1) (2) • Thrust may also be expressed as: (3)

28. Apply Bernoulli’s Principle • It is applied ahead of the disk and behind the disk (4) (5)

29. Apply Bernoulli’s Principle • Equating Equations (4) and (5) (6) • Substituting Equation (6) in (3) (7)

30. Apply Bernoulli’s Principle • Equating Equations (2) and (7) (8) • Half of the increase in velocity produced by rotor occurs just above the disk and half occurs in the wake

31. Calculation of Induced Velocity • The velocity induced by the rotor in the hovering state is the total velocity through the disk. • Substituting in Equation (2) (9)

32. Induced Velocity in terms of Disk Loading • Since disk loading is equal to the thrust divided by the disk area, the induced velocity in hovering may be expressed in terms of disk loading as: (10)

34. Accounting for Blade Tip Losses • For a rotor with finite number of blades, a factor should be introduced which accounts for the reduction of thrust near the blade tips • In the production of lift there is a differential pressure between upper and lower surfaces of blade • Air at the tip tends to flow from bottom to top, destroying the pressure difference and thus the lift in the tip region. • Important variables in determining the losses are the number of blades and the total loadin on the blade.

35. Accounting for Blade Tip Losses • The empirical equation used to find B, the tip loss factor is: (11)

36. Relating Induced Velocity to Disk Loading • For preliminary analyses, it is sufficient to assume B a constant. • A value of B = .97 has been assumed to be reasonable for main rotors. • More highly loaded rotors such as propellers will have more tip losses • The tip loss factor is incorporated into the equations for induced velocity. (12)

37. # of Blades 4 3 2

38. Uniform Induced Velocity Distribution in Hover • The thrust developed in hovering, considering uniform induced velocity distribution by an annular section of actuator disk of radius r and width dr is given by:

39. Uniform Induced Velocity Distribution in Hover • Integrating with respect to r and evaluating (13) • Substituting (BR)2 for R2 gives: (14)

40. Induced Power Based on Uniform Inflow • Substituting equation (11) (15) (16)

41. Triangular Induced Velocity Distribution in Hover • The use of highly tapered and twisted blades theoretically tends to approach the ideal uniform distribution flow condition • Actual distribution is probably more nearly triangular • The induced velocity at any radius r is:

42. Thrust in Hover, Triangular Inflow

43. Integrating with respect to r and evaluating (17) • Substituting (BR)2 for R2 gives: (18)

44. Power in Hover, Triangular Inflow • Expression for ihp based on triangular distribution

45. Relationship of uniform and triangular distributions • Integrating with respect to r and evaluating (19) • The triangular induced velocity distribution may be expressed in terms of uniform distribution using equations (14) and (18) as follows: (20)

46. Induced Power Correction • Using Equations (15), (19) and (20) the ratio of ihp for triangular distribution to ihp for uniform induced velocity distribution in hovering is: (21)

48. Induced Velocity and Power in Forward Flight • The velocity induced by a rotor in forward flight may be represented by figure below. V + 2ui V V + ui

49. Thrust Calculation The thrust considering uniform velocity distribution (22)

50. Induced Velocity Calculation • Substituting equation (14) in equation (23) • Substituting (BR)2 for R2 gives (23) (24)