Rotorcraft Design I
Day 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
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
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
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
The Iterative Nature of Aerospace Synthesis
Initiation and
Coordination Phase
Change requirements
Requirements
Change concept
Change technology assumptions
Concepts/Tech
Select search
techniques
Change methodology
Select Methodology
Change parametric variables
Select parametric
variables
W
Fuel Balance
W
Sizing
Design
Iteration
Select ranges for
parametric variables
Synthesis and Analysis Phase
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
Optimum
Configuration
Fixed Wing Aircraft Vehicle Synthesis
Design
Layout
(3-View)
Constraints
2
Aero
Aero
Data
Propulsion
1
Sizing
Fuel
Balance
3
Project
Manager
Fuel
Required
T/W
SFC
4
Weight
Control
Optimization
Fuel
Available
5
Cost
Cost
6
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
Rotorcraft/VSTOL Aircraft Synthesis ( RF Method)
Requirements
Performance
Modelsls
Synthesis
Configuration
Solution
Engine Power
Available
Hover Alt.
Vehicle
Hover Temp.
Power
Block Speed
Loading
Block Alt.itudes
ROC/Maneuver
Vehicle Power
Required
Installed
Power
HP i
Empty Fraction
Mission Input
Fuel Weight
Ratio Available
Payload
Gross
Block Range
Hover Time
Mission Analysis
Agility
Fuel Weight
Ratio Required
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
Vehicle
Weight
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!!
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
Rotorcraft/VSTOL Aircraft Synthesis ( RF Method)
Requirements
Performance
Modelsls
Synthesis
Configuration
Solution
Engine Power
Available
Hover Alt.
Vehicle
Hover Temp.
Power
Block Speed
Loading
Block Alt.itudes
ROC/Maneuver
Vehicle Power
Required
Installed
Power
HP i
Empty Fraction
Mission Input
Fuel Weight
Ratio Available
Payload
Gross
Block Range
Hover Time
Mission Analysis
Agility
Fuel Weight
Ratio Required
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
Vehicle
Weight
Achieving a Fuel Balance and Initial Gross Weight
R FR

 Wl

 AT 1  1.05

WG


AT 

  12
 W 
   AR 1  U 

 
 WG 

3
2
TH
4560 H
AR 
1
2
3
 
  1  e d  2 M
 C H
RE
   L 
650   
 c V  D V

1  WU
RFA 
 
1 
1  K  WG

Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
WG 
WU


1 

1  AR 1  K  AT 1  K  2 
 1

1  1.05
1  AR 1  K  1  AR 1  K  
W
  G

  WU










3
2




 



Achieving a Power Balance
Total Horsepower Required(THP) for Generic Subsonic Fixed Wing and Rotorcraft:
WG  1
ihp H

fV 3
0.332 
1  32


THP 

1

K

K
K

K

rhp
L
L
u

H


146000 p e p 
b  V  H 


2
Uo
V
K   1  3 2  C 4
Ku 
KL
3
ihp H 
rhp H 
0.0938 2 R 2
B 
AB
1.85
3
VTo 2  m
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
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
Power Required Comparison for Fixed Wing and
Rotary Wing Aircraft
Pr
Total
Parasite
Profile
Rotary Wing
Induced
V
Pr
Total
Parasite
Fixed Wing
Induced
V
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
Note: High, Hot Day (4000’, 95o F) increases HP required
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
HP required
• Conditions (altitude and ambient temperature) also affect rotorcraft sizing
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
Army Hot Day (4000’, 95o F)
Normal Day
100 kts
250 kts
Velocity
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
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
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
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
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
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
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
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
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
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
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
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
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
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
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.
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
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
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
Momentum Theory Model
P1
V
P2
P3
V + aV
P4
V + bV
• 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)
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
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
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
Generation of Thrust
• From momentum theory the thrust is:
T  M (V)
T  A(V  aV)(V  bV - V)
T  A(V  aV)(bV)
(1)
(2)
• Thrust may also be expressed as:
T  (P3 - P2 )A
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
(3)
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
Apply Bernoulli’s Principle
• It is applied ahead of the disk and behind the disk
P1  ρV  P2  ρV  aV 
1
2
2
2
1
2
P1  P2  12 ρV  aV   12 ρV 2
2
(4)
and
P3  ρV  aV   P4  ρV  bV 
2
1
2
1
2
2
P4  P3  ρV  aV   ρV  bV 
1
2
But
P1  P4
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
2
1
2
2
(5)
Apply Bernoulli’s Principle
• Equating Equations (4) and (5)
P2  12 ρV  aV   12 ρV 2 
2
P3  12 ρV  aV   12 ρV  bV 
2
2
P2  12 ρV 2  P3  12 ρV  bV 
2
P3  P2  ρbV 1  b 
2
1
2
2
(6)
• Substituting Equation (6) in (3)
T  bV 2 1 12 b A
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
(7)
Apply Bernoulli’s Principle
• Equating Equations (2) and (7)
ρAV  aV bV  ρAbV  ρAb V
2
2
1
2
2
ρAbV  AabV  ρAbV  ρAb V
2
2
2
1
2
2
2
a  12 b
2a  b
(8)
• Half of the increase in velocity produced by rotor
occurs just above the disk and half occurs in the
wake
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
Calculation of Induced Velocity
• The velocity induced by the rotor in the hovering
state is the total velocity through the disk.
aV  u and bV  2u
• Substituting in Equation (2)
T  A(V  u )( 2u )
T  2 R 2u 2 where A  R 2
T
2
or
u 
2 R 2
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
(9)
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:
u
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
w
2
(10)
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
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.
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
Accounting for Blade Tip Losses
• The empirical equation used to find B, the tip loss
factor is:
2CT
(11)
B  1
b
where
b is the number of blades per rotor
T
C T is the thrust coefficien t equal to
R 2VT2
where
VT is the blade tip speed equal to R
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
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.
u
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
w
2 B 2
(12)
Blade Tip Loss Factor vs. Thrust Coefficient
Balde Tip Loss Factor B
1
0.98
# of Blades
0.96
4
3
0.94
0.92
2
0.9
0
0.004
0.008
Thrust Coefficient Ct
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
0.012
0.016
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:
T  M (V )
dT   (2rdr )u H (2u H )
R
R
Y   dT    (2rdr )u H (2u H )
0
T  4u
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
0
2
H

R
0
rdr
Uniform Induced Velocity Distribution in Hover
• Integrating with respect to r and evaluating
T  2R 2 u H2
and
u 
2
H
T
2R
(13)
2
• Substituting (BR)2 for R2 gives:
w
T
uH 
where w  2
2
2 B
R
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
(14)
Induced Power Based on Uniform Inflow
Tu H
ihp H 
550
(15)
• Substituting equation (11)
T
w
ihp H 
2
550 2 B
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
(16)
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:
uH
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
r
 u H TIP
R
Thrust in Hover, Triangular Inflow
T  M (V )
r
r
dT   (2r u H TIP dr )( 2 u H TIP )
R
R
3
R r
2
dT  4u H TIP  2 dr
0 R
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
Integrating with respect to r and evaluating
T  R  u
2
and
u
2
H TIP
2
H TIP
T

2
R
(17)
• Substituting (BR)2 for R2 gives:
u H TIP 
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
w
T
where w 
2
B
R 2
(18)
Power in Hover, Triangular Inflow
• Expression for ihp based on triangular distribution
ihp  
Tu H 
550
1 R
r
r
r
ihp  
 (2r u H TIP dr )( 2 u H TIP )( u H TIP )

550 0
R
R
R
4
R r
4
ihp  
u 3HTIP  3 dr
0 R
550
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
Relationship of uniform and triangular distributions
• Integrating with respect to r and evaluating
1 4
ihp  
x Tu H TIP
550 5
(19)
• The triangular induced velocity distribution may be
expressed in terms of uniform distribution using
equations (14) and (18) as follows:
w
u H TIP
B 2

 2
uH
w
2 B 2
u H TIP  1.414u H
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
(20)
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:
ihp 
1
4
550
(
x T x 1.414u H )
ihp
550 5
Tu H
ihp  4
 (1.414)  1.13
ihp 5
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
(21)
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
Induced Velocity and Power in Forward Flight
• The velocity induced by a rotor in forward flight
may be represented by figure below.
V
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
V + ui
V + 2ui
Thrust Calculation
The thrust considering uniform velocity distribution
T  R 2 (V  ui )(V  2ui  V )
Let
V  ui  V
Then
T  R V 2ui
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
2
'
'
(22)
Induced Velocity Calculation
• Substituting (BR)2 for R2 gives
T
ui 
2 2 '
2 R B V
(23)
• Substituting equation (14) in equation (23)
2
H
'
u
ui 
V
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
(24)
Velocity Impact of Thrust Tilting
a’
ui sin a’
V
V’
a’
ui
ui cos a’
• For triangular induced velocity in forward flight it
may be seen from figure above that
(V ' ) 2  (V  ui sin a ' ) 2  (ui cos a ' ) 2
(V ' ) 2  V 2  2uiV sin a 'ui2 sin 2 a 'ui2 cos 2 a '
(V ' ) 2  V 2  2uiV sin a 'ui2
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
(25)
The Induced Velocity Correction Factor
• Defining the Induced Velocity Factor
ui
Ku 
uH
Then
ui  u H K u
(26)
(27)
• Setting equation (27) equal to equation (24)
u H2
uH Ku 
V'
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
(28)
Calculation of the Correction Factor
• Substituting equations (25) and (27) in equation (28)
Ku 
uH
(V  2u H K uV sin a 'u K )
2
2
H
2
u
1
2
2
u
2
h
Ku  2
V  2u H K uV sin a 'u H2 K u2
u  (V K  2u H V sin a ' K  u K )
2
h
V
1  
 uH
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
2
2
u
2
 2 V
 K u  2

 uH
3
u
2
H
4
u

 sin a ' K u3  K u4

Calculation of the Correction Factor
• The Ku3 term for steady level flight is negligible
V
K  
 uH
4
u
2
 2
 K u  1  0

(29)
• Solving the quadratic equation (29) becomes
1 V
K   
2  uH
2
u
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
2
 1 V
 

 2  uH
4

  4

(30)
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
V/uH
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
8
10
Induced Horsepower in Forward flight
• Considering uniform induced velocity distribution
ihp  ihp H K u
T
w
ihp 
Ku
2
550 2 B
(31)
• As mentioned previously the distribution is more
nearly triangular therefore:
ihp   1.13 ihp H K u
1.13
w
ihp  
T
Ku
2
550
2 B
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
(32)
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
Determination of Rotor Profile Power
• From Blade Element Theory
1
dD  VR2cdr
2
1
d ( RHP ) 
VR3cdr
1100
(33)
(34)
• The resultant velocity on a blade element is:
VR  r  V sin 
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
(35)
Rotor Blade Element Theory
Blade element @
Station r=xR
Y=0
270°
a

r + V sin
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
Q
u + V sina’
Tip Path Plane
90°
dr

180°
a’
V
Determination of Rotor Profile Power
• Substituting equation (35) in (34) and expanding:
1
d ( Rhp ) 
 (R  V sin  )3 cdr
(36)
1100
1

c(3 R 3  2 2 R 2V sin   RV 2 sin 2 
1100
(37)
2 2
2
2
3
3
  r V sin   2RV sin   V sin  )dr
• Profile power at any angle, , is
r2
Rhp ( )   d ( Rhp ) 
r1
c
r2
1100 r1
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
(38)
(3r 3  3 2 r 2V sin   3rV 2 sin 2   V 3 sin 3  )dr
Determination of Rotor Profile Power
• Substituting the identities:
sin 2   1  1 cos 2
2
2
sin 3   3 sin   1 sin 3
4
4
• Integrating equation (38):
RHP ( ) 
r2
 3r

3 2 2 3 2 2
2 3

  r V sin   r V  r V cos 2  (39)
c 
4
4
4


1100  3 3
1 3


V
r
sin


V
r
sin
3

 4
 r
4
1
4
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
Determination of Rotor Profile Power
• Substituting the expression V=mWR in equation
(39)
Rhp ( ) 
r2
r

3 2 2 2 3 2 2 2
3
 r R sin    r R   r R cos 2 
3 
c 4
4
4


1100  3 3 3
1 3 3



rR
sin



rR
sin
3

 4
 r
4
1
4
(41)
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
Determination of Rotor Profile Power
• The average profile power over the entire azimuth
considering the reverse flow region is:
R
b 2

Rhp 
Rhp ( ) d

0
2 0
 R sin
b 2
  Rhp ( )
d


(42)
0
• The limits of r=0 to r=-Rsin 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.
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
Determination of Rotor Profile Power

VR  r  V sin 
where
V  R
R
3/2
Therefore
VR  (r  R sin  )
Y
/2
Y=0
• The radius where VR=0 is locus of reverse flow
boundary and occurs only between = and =2
r  R sin   0
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
Determination of Rotor Profile Power
• Introducing the term Kμ and defining it as:
3 4
K   1  3  
8
2
• Then
Rhp 
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
(54)
A V K 
3
B T
4400
(55)
Determination of Rotor Profile Power
• Defining:
b 2
R

A1 
Rhp ( )0 d

2 0
b 2
 R sin
A2   Rhp ( )0
d

(43)
(44)

• Then
Rhp 
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
c3
1100
( A1  A2 )
(45)
Determination of Rotor Profile Power
• Substituting equation (41) in (43) and evaluating
the limits for r:
 R4

3 2 4 3 2 4
4
 R sin    R   R cos 2 
b 2  4
4
4
A1 

d

2 0  3 3 4
1 3 4

 4  R sin   4  R sin 3

• Integrating with respect to  and evaluating:
bR 4
A1 
(1  3 2 )
4
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
(46)
(47)
Determination of Rotor Profile Power
• Substituting equation (41) in (43), evaluating the
limits for r and combining terms:
3 4 4
 3 4 4 4

2
  R sin    R cos 2 sin  

2

b
4
A2    4
d


 1  4 R 4 sin 3 sin 

 4

• Substituting the identities:
cos 2  1  2 sin 2 
sin 3  3 sin   4 sin 3 
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
(49)
Determination of Rotor Profile Power
b
2
1 4 4
4
A2     R sin d
 
4
b  4 R 4 2
4

sin d

 4 
(50)
• Integrating with respect to  and evaluating
3b R
A2  
32
4
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
4
(51)
Determination of Rotor Profile Power
• Combining equations (47) and (51):
bR 4 3 2bR 4 3b 4bR 4
A1  A2 


4
4
32
bR 4
3 4
2

(1  3   )
4
8
(52)
• The rotor profile power then becomes:
c3  bR 4
3 4 
Rhp 
(1  3   )

1100  4
8

3
ABVT
3 4
2
Rhp 
(1  3   )
4400
8
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
2
(53)
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:
K   1  3  C4 
2
4
(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.
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
Profile Power Factor vs Forward Velocity
Profile Power Factor K
1.5
Tip Speed
(ft/sec) 600
1.4
650
1.3
700
1.2
1.1
1
0
20
40
60
Forward Velocity (Knots)
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
80
100
Design Problem: VTOL Configuration
“Sensitivity” Analysis (Use of Nomographs)
•
•
Consider the requirement for a VTOL aircraft which has the
following operational characteristics:
Crew (4) @ 200 lbs each
800 lbs
Passengers (30 @ 200lbs + 30 lbs gear 6900 lbs
Vcruise
250 knots
Vrange (275 NM w/30 min reserve@VCr) 400 nm
Engine Requirement
Multiple
Hover Reqts (All engines, 6000’,95oF day)
OGE(5min
rating)
Environmental Conditions Dictate
w< 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:
1.
2.
3.
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150
The Compound Helicopter (Cmpd)
The Tilt-Rotor Aircraft (TR)
The Tilt-Wing Aircraft (TW)
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:
1. Configuration Informaton:
Cmpd TR
TW
Cruise L/D……………………………………5.0
ηC…………………………………………… 0.84
C (Specific Fuel Consumption @ VCr…… 0.55
Φ (Empty-to-Gross Weight Ratio)………… 0.64
w (Disk Loading)…………………………… 9.0
•
9.00
0.75
0.55
0.70
15.0
10.5
0.80
0.55
0.68
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?
Dr. Daniel P. Schrage
Georgia Institute of Technology
Atlanta, GA 30332-0150