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
Select parametric variables
W
W
Design Iteration
Optimum Configuration
Select ranges for parametric variables
Synthesis and Analysis Phase
4
Fixed Wing Aircraft Vehicle Synthesis
5
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
  Customers Mission Performance Requirements
• THEN
• The Concept Is FEASIBLE!!

7
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)(1e-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/da,
rad-1,or speed of sound (?gRT)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 Ap 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 (? Ktr)ahp-RhpH-hpacc ihp
Rotor induced horsepower php Parasite
horsepower rhp Total horsepower required 1/
(? 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/?)(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 ?V2,
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, RO, 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) af Fuselage
angle of attack (angle between fuselage W.L. and
horizontal), deg ar Blade section angle of
attack, degs or radians d Blade section drag
coefficient ? Mechanical efficiency ?T Blade
twist (referenced to centerline rotation),
degrees
18
List of Symbols
? Inflow velocity ratio u/VT (ui uR
up)/VT ? Induced power correction factor due to
ground effect µ Tip speed ratio V/VT ?
Density, slugs/ft3 s Rotor solidity bc/pR ?
Rotor azimuth angle, degrees O 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 Newtons second law of motion, Fma,
  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
P2
P3
P4
P1
V bV
V aV
V

• 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)

22
(No Transcript)
23
(No Transcript)
24
(No Transcript)
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)
27
(No Transcript)
28
Apply Bernoullis Principle

• It is applied ahead of the disk and behind the
  disk

(4)
(5)
29
Apply Bernoullis Principle

• Equating Equations (4) and (5)

(6)

• Substituting Equation (6) in (3)

(7)
30
Apply Bernoullis 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)
33
(No Transcript)
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
(15)

• Substituting equation (11)

(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)
47
(No Transcript)
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 ui
V
49
Thrust Calculation

• The thrust considering uniform velocity
  distribution

(22)
50
Induced Velocity Calculation

• Substituting (BR)2 for R2 gives

(23)

• Substituting equation (14) in equation (23)

(24)
51
Velocity Impact of Thrust Tilting
a
a
Tip Path Plane
ui sin a
V
ui
ui cos a
Rotor
V

• For triangular induced velocity in forward flight
  it may be seen from figure above that

(25)
52
The Induced Velocity Correction Factor

• Defining the Induced Velocity Factor

(26)
(27)

• Setting equation (27) equal to equation (24)

(28)
53
Calculation of the Correction Factor

• Substituting equations (25) and (27) in equation
  (28)

54
Calculation of the Correction Factor

• The Ku3 term for steady level flight is negligible

(29)

• Solving the quadratic equation (29) becomes

(30)
55
Induced Velocity Factor as a Function of
Forward Velocity
1
Induced Velocity Factor Ku
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
V/uH
56
Induced Horsepower in Forward flight

• Considering uniform induced velocity distribution

(31)

• As mentioned previously the distribution is more
  nearly triangular therefore

(32)
57
(No Transcript)
58
(No Transcript)
59
(No Transcript)
60
Determination of Rotor Profile Power

• From Blade Element Theory
• The resultant velocity on a blade element is

(33)
(34)
(35)
61
Rotor Blade Element Theory
Blade element _at_ Station rxR
Y0
270
a
Q
90
dr
f
u V sina
Wr V siny
Tip Path Plane
V
a
180
Dy
62
Determination of Rotor Profile Power

• Substituting equation (35) in (34) and expanding

(36)
(37)

• Profile power at any angle, y, is

(38)
63
Determination of Rotor Profile Power

• Substituting the identities

• Integrating equation (38)

(39)
64
Determination of Rotor Profile Power

• Substituting the expression VmWR in equation (39)

(41)
65
Determination of Rotor Profile Power

• The average profile power over the entire azimuth
  considering the reverse flow region is

(42)

• The limits of r0 to r-mRsiny is the region of
  reverse flow and is established by determining
  the point on the blade radius at any azimuth
  where the resultant velocity is zero.

66
Determination of Rotor Profile Power

• The radius where VR0 is locus of reverse flow
  boundary and occurs only between yp and y2p

67
Determination of Rotor Profile Power

• Introducing the term Kµ and defining it as

(54)

• Then

(55)
68
Determination of Rotor Profile Power

• Defining

(43)
(44)

• Then

(45)
69
Determination of Rotor Profile Power

• Substituting equation (41) in (43) and evaluating
  the limits for r

(46)

• Integrating with respect to y and evaluating

(47)
70
Determination of Rotor Profile Power

• Substituting equation (41) in (43), evaluating
  the limits for r and combining terms

• Substituting the identities

(49)
71
Determination of Rotor Profile Power
(50)

• Integrating with respect to y and evaluating

(51)
72
Determination of Rotor Profile Power

• Combining equations (47) and (51)

(52)

• The rotor profile power then becomes

(53)
73
General form to include Compounds

• Past flight tests have shown a need to account
  for stalling of the retreating blade. Equation
  (54) has been modified to read

(56)

• C4 accounts for the stalled region of the blade
  and varies with mean lift coefficient
• For the normal design range of lift coefficient
  C4 has been set to 30 for pure helicopters and 5
  for compound helicopters.

74
Profile Power Factor vs Forward Velocity
1.5
Tip Speed (ft/sec)
600
1.4
650
1.3
700
Profile Power Factor Km
1.2
1.1
1
0
20
40
60
80
100
Forward Velocity (Knots)
75
Design Problem VTOL Configuration Sensitivity
Analysis (Use of Nomographs)

• Consider the requirement for a VTOL aircraft
  which has the following operational
  characteristics
• Crew (4) _at_ 200 lbs each 800 lbs
• Passengers (30 _at_ 200lbs 30 lbs gear 6900 lbs
• Vcruise 250 knots
• Vrange (275 NM w/30 min reserve_at_VCr) 400 nm
• Engine Requirement Multiple
• Hover Reqts (All engines, 6000,95oF
  day) OGE(5min rating)
• Environmental Conditions Dictate wlt 50lbs/ft2
• The design issue is which VTOL concepts are
  competitors for such a specification? It is
  immediately evident that the pure helicopter is
  not a competitor by virture of the cruise
  requirement. Further, the competitive
  configurations are
• The Compound Helicopter (Cmpd)
• The Tilt-Rotor Aircraft (TR)
• The Tilt-Wing Aircraft (TW)

76
Design Problem VTOL Configuration Sensitivity
Analysis (Use of Nomographs)

• Depending on the particular manufacturer, or
  procuring agency, certain a priori knowledge
  exists regarding the following characteristics of
  these VTOL configurations
• Configuration Informaton Cmpd TR TW
• Cruise L/D5.0 9.00 10.5
• ?C 0.84 0.75 0.80
• C (Specific Fuel Consumption _at_ VCr
  0.55 0.55 0.55
• F (Empty-to-Gross Weight Ratio)
  0.64 0.70 0.68
• w (Disk Loading) 9.0 15.0 50.0
• Student Exercise
• Considering these vehicle characteristics and
  using the design charts provided, Figure A-1 and
  A-2, compare the different configurations on
  gross weight and installed power. Which has the
  lowest gross weight? Which has the highest
  installed power?