© L. Sankar Helicopt er Aerodynamics 1 Helicopter Aerodynamics and Performance Preliminary Remarks
Dec 29, 2015
© L. Sankar Helicopter Aerodynamics
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Helicopter Aerodynamics and Performance
Preliminary Remarks
© L. Sankar Helicopter Aerodynamics
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ThrustAeroelasticResponse
0
270 180
90
Dynamic Stall onRetreating Blade
Blade-Tip Vortexinteractions
UnsteadyAerodynamicsTransonic Flow on
Advancing Blade
Main Rotor / Tail Rotor/ Fuselage
Flow Interference
V
NoiseShock Waves
Tip Vortices
The problems are many..
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A systematic Approach is necessary
• A variety of tools are needed to understand, and predict these phenomena.• Tools needed include
– Simple back-of-the envelop tools for sizing helicopters, selecting engines, laying out configuration, and predicting performance
– Spreadsheets and MATLAB scripts for mapping out the blade loads over the entire rotor disk
– High end CFD tools for modeling• Airfoil and rotor aerodynamics and design• Rotor-airframe interactions• Aeroacoustic analyses
– Elastic and multi-body dynamics modeling tools– Trim analyses, Flight Simulation software
• In this work, we will cover most of the tools that we need, except for elastic analyses, multi-body dynamics analyses, and flight simulation software.
• We will cover both the basics, and the applications.• We will assume familiarity with classical low speed and high speed
aerodynamics, but nothing more.
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Plan for the Course
• PowerPoint presentations, interspersed with numerical calculations and spreadsheet applications.
• Part 1: Hover Prediction Methods• Part 2: Forward Flight Prediction Methods• Part 3: Helicopter Performance Prediction
Methods• Part 4: Introduction to Comprehensive Codes
and CFD tools• Part 5: Completion of CFD tools, Discussion of
Advanced Concepts
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Text Books
• Wayne Johnson: Helicopter Theory, Dover Publications,ISBN-0-486-68230-7
• References:– Gordon Leishman: Principles of Helicopter
Aerodynamics, Cambridge Aerospace Series, ISBN 0-521-66060-2
– Prouty: Helicopter Performance, Stability, and Control, Prindle, Weber & Schmidt, ISBN 0-534-06360-8
– Gessow and Myers– Stepniewski & Keys
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Grading
• 5 Homework Assignments (each worth 5%).• Two quizzes (each worth 25%)• One final examination (worth 25%)• All quizzes and exams will be take-home type.
They will require use of an Excel spreadsheet program, or optionally short computer programs you will write.
• All the material may be submitted electronically.
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Instructor Info.
• Lakshmi N. Sankar
• School of Aerospace Engineering, Georgia Tech, Atlanta, GA 30332-0150, USA.
• Web site: www.ae.gatech.edu/~lsankar/AE6070.Fall2002
• E-mail Address: [email protected]
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Earliest Helicopter..Chinese Top
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Leonardo da Vinci(1480? 1493?)
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Human Powered Flight?
HP 6.7 5.33/0.8
Merit of rePower/Figu Ideal Power Actual
33.5A2
W WPower Ideal
slugs. 0.00238Desnity
sq.ft 100 AreaRotor
6ft ~RadiusRotor
160lbfWeight
HP
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D’AmeCourt (1863)Steam-Propelled Helicopter
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Paul Cornu (1907)First man to fly in helicopter mode..
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De La Ciervainvented Autogyros (1923)
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Cierva introduced hinges at the rootthat allowed blades to freely flap
Hinges
Only the lifts were transferred to the fuselage, not unwanted moments.In later models, lead-lag hinges were also used toAlleviate root stresses from Coriolis forces
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Igor Sikorsky Started work in 1907, Patent in 1935
Used tail rotor to counter-act the reactive torque exerted by the rotor on the vehicle.
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Sikorsky’s R-4
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Ways of countering the Reactive Torque
Other possibilities: Tip jets, tip mounted engines
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Single Rotor Helicopter
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Tandem Rotors (Chinook)
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Coaxial rotorsKamov KA-52
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NOTAR Helicopter
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NOTAR Concept
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Tilt Rotor Vehicles
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Helicopters tend to grow in size..
AH-64A AH-64D
Length 58.17 ft (17.73 m) 58.17 ft (17.73 m)
Height 15.24 ft (4.64 m) 13.30 ft (4.05 m)
Wing Span 17.15 ft (5.227 m) 17.15 ft (5.227 m)
Primary Mission Gross Weight
15,075 lb (6838 kg)11,800 pounds Empty
16,027 lb (7270 kg) Lot 1 Weight
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AH-64A AH-64D
Length 58.17 ft (17.73 m) 58.17 ft (17.73 m)
Height 15.24 ft (4.64 m) 13.30 ft (4.05 m)
Wing Span 17.15 ft (5.227 m) 17.15 ft (5.227 m)
Primary Mission Gross Weight
15,075 lb (6838 kg)11,800 pounds Empty
16,027 lb (7270 kg) Lot 1 Weight
Hover In-Ground Effect (MRP)
15,895 ft (4845 m)[Standard Day]14,845 ft (4525 m)[Hot Day ISA + 15C]
14,650 ft (4465 m)[Standard Day]13,350 ft (4068 m)[Hot Day ISA + 15 C]
Hover Out-of-Ground Effect (MRP)
12,685 ft (3866 m)[Sea Level Standard Day]11,215 ft (3418 m)[Hot Day 2000 ft 70 F (21 C)]
10,520 ft (3206 m)[Standard Day]9,050 ft (2759 m)[Hot Day ISA + 15 C]
Vertical Rate of Climb (MRP)
2,175 fpm (663 mpm)[Sea Level Standard Day]2,050 fpm (625 mpm)[Hot Day 2000 ft 70 F (21 C)]
1,775 fpm (541 mpm)[Sea Level Standard Day]1,595 fpm (486 mpm)[Hot Day 2000 ft 70 F (21 C)]
Maximum Rate of Climb (IRP)
2,915 fpm (889 mpm)[Sea Level Standard Day]2,890 fpm (881 mpm)[Hot Day 2000 ft 70 F (21 C)]
2,635 fpm (803 mpm)[Sea Level Standard Day]2,600 fpm (793 mpm)[Hot Day 2000 ft 70 F (21 C)]
Maximum Level Flight Speed
150 kt (279 kph)[Sea Level Standard Day]153 kt (284 kph)[Hot Day 2000 ft 70 F (21 C)]
147 kt (273 kph)[Sea Level Standard Day]149 kt (276 kph)[Hot Day 2000 ft 70 F (21 C)]
Cruise Speed (MCP)
150 kt (279 kph)[Sea Level Standard Day]153 kt (284 kph)[Hot Day 2000 ft 70 F (21 C)]
147 kt (273 kph)[Sea Level Standard Day]149 kt (276 kph)[Hot Day 2000 ft 70 F (21 C)]
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Power Plant Limitations
• Helicopters use turbo shaft engines.
• Power available is the principal factor.
• An adequate power plant is important for carrying out the missions.
• We will look at ways of estimating power requirements for a variety of operating conditions.
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High Speed Forward Flight Limitations
• As the forward speed increases, advancing side experiences shock effects, retreating side stalls. This limits thrust available.
• Vibrations go up, because of the increased dynamic pressure, and increased harmonic content.
• Shock Noise goes up.• Fuselage drag increases, and parasite power
consumption goes up as V3.• We need to understand and accurately predict
the air loads in high speed forward flight.
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Concluding Remarks
• Helicopter aerodynamics is an interesting area.• There are a lot of problems, but there are also
opportunities for innovation.• This course is intended to be a starting point for
engineers and researchers to explore efficient (low power), safer, comfortable (low vibration), environmentally friendly (low noise) helicopters.
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Hover Performance Prediction Methods
I. Momentum Theory
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Background
• Developed for marine propellers by Rankine (1865), Froude (1885).
• Extended to include swirl in the slipstream by Betz (1920)
• This theory can predict performance in hover, and climb.
• We will look at the general case of climb, and extract hover as a special situation with zero climb velocity.
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Assumptions
• Momentum theory concerns itself with the global balance of mass, momentum, and energy.
• It does not concern itself with details of the flow around the blades.
• It gives a good representation of what is happening far away from the rotor.
• This theory makes a number of simplifying assumptions.
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Assumptions (Continued)
• Rotor is modeled as an actuator disk which adds momentum and energy to the flow.
• Flow is incompressible.
• Flow is steady, inviscid, irrotational.
• Flow is one-dimensional, and uniform through the rotor disk, and in the far wake.
• There is no swirl in the wake.
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Control Volume is a CylinderV
Disk area A
Total area S
Station1
2
3
4
V+v2
V+v3
V+v4
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Conservation of Mass
444
1
)(A-SV bottom he through tOutflow
m side he through tInflow
VS tophe through tInflow
AvV
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Conservation of Mass through the Rotor Disk
Flow through the rotor disk =
44
32
v
vv
VA
VAVAm
Thus v2=v3=v
There is no velocity jump across the rotor disk
The quantity v is called induced velocity at the rotor disk
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Global Conservation of Momentum
4444
42
42
4
44
1
2
vv)v(A T
in Rate Momentum
-out rate MomentumT,Thrust
.boundaries fieldfar the
allon catmospheri is Pressure
vA-S
bottom through outflow Momentum
vA
Vm side he through tinflow Momentum
V op through tinflow Momentum
mV
AVV
V
S
Mass flow rate through the rotor disk timesExcess velocity between stations 1 and 4
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Conservation of Momentum at the Rotor Disk
V+v
V+v
p2
p3
Due to conservation of mass across theRotor disk, there is no velocity jump.
Momentum inflow rate = Momentum outflow rate
Thus, Thrust T = A(p3-p2)
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Conservation of EnergyConsider a particle that traverses fromStation 1 to station 4
We can apply Bernoulli equation betweenStations 1 and 2, and between stations 3 and 4.Recall assumptions that the flow is steady, irrotational, inviscid.
1
2
3
4
V+v
V+v4
44
23
24
23
222
v2
v
v2
1v
2
12
1v
2
1
Vpp
VpVp
VpVp
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44
23
44
23
v2
v
v2
v
#38, slide previous theFrom
VAppAT
Vpp
From an earlier slide # 36, Thrust equals mass flow rate through the rotor disk times excess velocity between stations 1 and 4
4vv VAT Thus, v = v4/2
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Induced Velocities
V
V+v
V+2v
The excess velocity in theFar wake is twice the inducedVelocity at the rotor disk.
To accommodate this excessVelocity, the stream tube has to contract.
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Induced Velocity at the Rotor DiskNow we can compute the induced velocity at the rotor disk in terms of thrust T.
T = Mass flow rate through the rotor disk * (Excess velocity between 1 and 4).
T = 2 A (V+v) v
A
TV
222
V-v
2
There are two solutions. The – sign Corresponds to a wind turbine, where energy Is removed from the flow. v is negative.
The + sign corresponds to a rotor orPropeller where energy is added to the flow.In this case, v is positive.
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Induced velocity at the rotor disk
A
T
A
TV
2v
0V velocity climb Hover,In
222
V-v
2
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Ideal Power Consumed by the Rotor
A
TVVT
VT
Vm
mm
P
222
v
vv2
V2
12vV
2
1
in flowEnergy -out flowEnergy
2
22
In hover, ideal power
A
TT
2
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Summary• According to momentum theory, the downwash
in the far wake is twice the induced velocity at the rotor disk.
• Momentum theory gives an expression for induced velocity at the rotor disk.
• It also gives an expression for ideal power consumed by a rotor of specified dimensions.
• Actual power will be higher, because momentum theory neglected many sources of losses- viscous effects, compressibility (shocks), tip losses, swirl, non-uniform flows, etc.
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Figure of Merit
• Figure of merit is defined as the ratio of ideal power for a rotor in hover obtained from momentum theory and the actual power consumed by the rotor.
• For most rotors, it is between 0.7 and 0.8.
P
TT
C
CC
T
FM
2P
v
Hoverin Power Actual
Hoverin Power Ideal
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Some Observations on Figure of Merit
• Because a helicopter spends considerable portions of time in hover, designers attempt to optimize the rotor for hover (FM~0.8).
• We will discuss how to do this later.• A rotor with a lower figure of merit
(FM~0.6) is not necessarily a bad rotor.• It has simply been optimized for other
conditions (e.g. high speed forward flight).
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Example #1
• A tilt-rotor aircraft has a gross weight of 60,500 lb. (27500 kg).
• The rotor diameter is 38 feet (11.58 m).
• Assume FM=0.75, Transmission losses=5%
• Compute power needed to hover at sea level on a hot day.
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Example #1 (Continued)
HP 11528 1.05*10980shaft the toengine by the suppliedPower
lossion transmiss5% is There
HP 10980 power actual totalrotors, twoFor the
HP 5490 power Actual
4117/0.75Power/FM idealPower Actual
HP 4117Power Ideal
ft/sec lb 74.86 x 30250 Tv Power Ideal
! ft/sec 150 far wake in theDownwash
ft/sec 86.74v
A2
T v velocity,Induced
lbf 30250 T rotors. twoare There
feet cslugs/cubi 0.00238 Density
feet square 12.1134
19A AreaDisk 2
A
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Alternate scenarios
• What happens on a hot day, and/or high altitude?– Induced velocity is higher.– Power consumption is higher
• What happens if the rotor disk area A is smaller?– Induced velocity and power are higher.
• There are practical limits to how large A can be.
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Disk Loading
• The ratio T/A is called disk loading.• The higher the disk loading, the higher the
induced velocity, and the higher the power.• For helicopters, disk loading is between 5 and
10 lb/ft2 (24 to 48 kg/m2).• Tilt-rotor vehicles tend to have a disk loading of
20 to 40 lbf/ft2. They are less efficient in hover.• VTOL aircraft have very small fans, and have
very high disk loading (500 lb/ft2).
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Power Loading
• The ratio of thrust to power T/P is called the Power Loading.
• Pure helicopters have a power loading between 6 to 10 lb/HP.
• Tilt-rotors have lower power loading – 2 to 6 lb/HP.
• VTOL vehicles have the lowest power loading – less than 2 lb/HP.
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Non-Dimensional Forms
QC
P
2Q
3P
2T
C
QP
Torquelocity x Angualr ve Power hover,In
RAR
QtCoefficien TorqueC
RA
PtCoefficienPower C
RA
TtCoefficienThrust C
form. ldimensiona-nonin
expressedusually arePower and Torque, Thrust,
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Non-dimensional forms..
P
TT
i
C
CC
T
FM
2P
v
Hoverin Power Actual
Hoverin Power Ideal
2
C
A2
T
R
1
R
v inflow Induced T
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Tip Losses
R A portion of the rotor near theTip does not produce much liftDue to leakage of air fromThe bottom of the disk to the top.
One can crudely account for it byUsing a smaller, modified radius
BR, where
b
CB T2
1
BR
B = Number of blades.
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Power Consumption in HoverIncluding Tip Losses..
2
11 TTP
CC
BFMC
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Hover PerformancePrediction Methods
II. Blade Element Theory
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Preliminary Remarks
• Momentum theory gives rapid, back-of-the-envelope estimates of Power.
• This approach is sufficient to size a rotor (i.e. select the disk area) for a given power plant (engine), and a given gross weight.
• This approach is not adequate for designing the rotor.
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Drawbacks of Momentum Theory
• It does not take into account– Number of blades– Airfoil characteristics (lift, drag, angle of zero
lift)– Blade planform (taper, sweep, root cut-out)– Blade twist distribution– Compressibility effects
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Blade Element Theory
• Blade Element Theory rectifies many of these drawbacks. First proposed by Drzwiecki in 1892.
• It is a “strip” theory. The blade is divided into a number of strips, of width r.
• The lift generated by that strip, and the power consumed by that strip, are computed using 2-D airfoil aerodynamics.
• The contributions from all the strips from all the blades are summed up to get total thrust, and total power.
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Typical Blade Section (Strip)
R
dr
r
Tip
OutCut
Tip
OutCut
dPbP
dTbT
dT
Root Cut-out
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Typical Airfoil Section
r
V varctan
r
V+v
Line of Zero Lift
effective =
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Sectional Forces
Once the effective angle of attack is known, we can look-up the lift and drag coefficients for the airfoil section at that strip.
We can subsequently compute sectional lift and drag forces per foot (or meter) of span.
dPT
lPT
cCUUD
cCUUL
2
1
2
1
22
22
These forces will be normal to and along the total velocity vector.
UT=r
UP=V+v
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Rotation of Forces
r
V+v
L
D
T
Fx
XxT
ldPT
x
dlPT
rdFdFUdP
drCCcUU
drLDdF
drCCcUU
drDLdT
sincos2
1
sincos
sincos2
1
sincos
22
22
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Approximate Expressions
• The integration (or summation of forces) can only be done numerically.
• A spreadsheet may be designed. A sample spreadsheet is being provided as part of the course notes.
• In some simple cases, analytical expressions may be obtained.
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Closed Form Integrations• The chord c is constant. Simple linear twist.• The inflow velocity v and climb velocity V are small. Thus,
<< 1.
• We can approximate cos( ) by unity, and approximate sin( ) by ( ).
• The lift coefficient is a linear function of the effective angle of attack, that is, Cl=a() where a is the lift curve slope.
• For low speeds, a may be set equal to 5.7 per radian.
• Cd is small. So, Cd sin() may be neglected.
• The in-plane velocity r is much larger than the normal component V+v over most of the rotor.
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Closed Form Expressions
drrCrr
V
rr
VcbaP
drrrr
VcbaT
Rr
r
d
Rr
r
3
0
3
2
0
2
vv
2
1
v
2
1
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Linearly Twisted Rotor: ThrustHere, we assume that the pitch angle varies as
E Fr
R
vV
a
Rbc
where
a
R
abcC
RRcab
RR
vVFREca
bT
RT
R
Ratio Inflow
)2(~ slope CurveLift
/DiskAreaBladeArea/solidity
2/32
2/32
2/3224
3
3
1
2
75.75.
75.232
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Linearly Twisted Rotor
Notice that the thrust coefficient is linearly proportional to the pitch angle at the 75% Radius.
This is why the pitch angle is usually defined at 75% R in industry.The expression for power may be integrated in a similar manner, if the drag coefficient Cd is assumed to be a
constant, equal to Cd0.
80d
TP
CCC
Induced Power Profile Power
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Closed Form Expressions forIdeally Twisted Rotor
r
Rtip
tipT
aC
4
C CC
P Td
0
8Same as linearlyTwisted rotor!
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Figure of Merit according to Blade Element Theory
AreaArea/Disk Blade Solidity
Rv)/(V Ratio Inflow
,
8/0
where
CC
CFM
dT
T
High solidity (lot of blades, wide-chord, large blade area) leads to higherPower consumption, and lower figure of merit.
Figure of merit can be improved with the use of low drag airfoils.
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Average Lift Coefficient• Let us assume that
every section of the entire rotor is operating at an optimum lift coefficient.
• Let us assume the rotor is untapered.
T
T
R
C
R
bc
RR
TC
RbcdrrcbT
6C
6
C
6
C
6
CC
2
1
CtCoefficienLift Average
l
ll22
32l
l
0
2
l
Rotor will stall if average lift coefficient exceeds 1.2, or so.
Thus, in practice, CT/ is limited to 0.2 or so.
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Optimum Lift Coefficient in Hover
minimized. is
/C if maximized is FM
6/C If
82
2
2
C hover,In
8
2/3d0
T
02/3
2/3
T
0
l
l
dT
T
dT
T
C
C
CC
C
FM
CC
CFM
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Drawbacks of Blade Element Theory
• It does not handle tip losses.– Solution: Numerically integrate thrust from the cutout
to BR, where B is the tip loss factor. Integrate torque from cut-out all the way to the tip.
• It assumes that the induced velocity v is uniform.• It does not account for swirl losses.• The Predicted power is sometimes empirically
corrected for these losses.
15.18
0
dTP
CCC
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Example(From Leishman)
• Gross Weight = 16,000lb• Main rotor radius = 27 ft• Tail rotor radius 5.5 ft• Chord=1.7 ft (main), Tail rotor chord=0.8 ft• No. of blades =4 (Main rotor), 4 (tail rotor)• Tip speed= 725 ft/s (main), 685 ft/s (tail)• K=1.15, Cd0=0.008• Available HP =3000Transmission losses=10%• Estimate hover ceiling (as density altitude)
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Step I• Multiply 3000 HP by 550 ft.lb/sec.• Divide this by 1.10 to account for available
power to the two rotors (10% transmission loss).
• We will use non-dimensional form of power into dimensional forms, as shown below:
• P=Tv+(R)3A [Cd0/8]
• Find an empirical fit for variation of with altitude: 2553.4
16.288
00198.01
h
levelsea
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Step 2• Assume an altitude, h. Compute density, .• Do the following for main rotor:
– Find main rotor area A– Find v as [T/(2A)]1/2 Note T= Vehicle weight in lbf.– Insert supplied values of , Cd0, W to find main rotor P.– Divide this power by angular velocity W to get main rotor torque.– Divide this by the distance between the two rotor shafts to get tail
rotor thrust.• Now that the tail rotor thrust is known, find tail rotor power
in the same way as the main rotor.• Add main rotor and tail rotor powers. Compare with
available power from step 1.• Increase altitude, until required power = available power.• Answer = 10,500 ft
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Hover PerformancePrediction Methods
III. Combined Blade Element-Momentum (BEM) Theory
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Background
• Blade Element Theory has a number of assumptions.
• The biggest (and worst) assumption is that the inflow is uniform.
• In reality, the inflow is non-uniform.
• It may be shown from variational calculus that uniform inflow yields the lowest induced power consumption.
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Consider an Annulus of the rotor Disk
r
dr
Area = 2rdr
Mass flow rate =2rV+vdr
dT = (Mass flow rate) * (twice the induced velocity at the annulus) = 4r(V+v)vdr
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Blade Elements Captured by the Annulus
r
dr
Thrust generated by these blade elements:
drr
vVrabc
drCcrbdT l
2
2
2
12
1
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Equate the Thrust for the Elementsfrom the
Momentum and Blade Element Approaches
R
v
,
088
2
VR
V
where
R
raa
c
c
2168216
2
cc a
R
raa
Total Inflow Velocity from CombinedBlade Element-Momentum Theory
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Numerical Implementation of Combined BEM Theory
• The numerical implementation is identical to classical blade element theory.
• The only difference is the inflow is no longer uniform. It is computed using the formula given earlier, reproduced below:
2168216
2
cc a
R
raa
Note that inflow is uniform if = CR/r . This twist is therefore called the ideal twist.
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Effect of Inflow on Power in Hover
thrust!of level specified afor power, inducedleast produces inflow Uniform
constant. a bemust that vfollowsit ),multipliern (Lagrangeacontant a is Since
0v2v3 if is v s variationpossible allfor vanish willintegral heonly way t The
0vdrv2v34
0v4v4
0T-P .multiplier Lagrangean a is whereT-P minimize weTherefore,
T. of valuespecified afor power, induced minimize wish toWe
v4dTT
v4vdT
2
0
2
0
23
0
2
0
0
3
0
R
R
RR
RR
induced
r
drrr
drr
drrP
Variation of a functional
constraint
© L. Sankar Helicopter Aerodynamics
84
Ideal Rotor vs. Optimum Rotor
• Ideal rotor has a non-linear twist: = CR/r• This rotor will, according to the BEM theory, have a
uniform inflow, and the lowest induced power possible.• The rotor blade will have very high local pitch angles
near the root, which may cause the rotor to stall.• Ideally Twisted rotor is also hard to manufacture. • For these reasons, helicopter designers strive for
optimum rotors that minimize total power, and maximize figure of merit.
• This is done by a combination of twist, and taper, and the use of low drag airfoil sections.
© L. Sankar Helicopter Aerodynamics
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Optimum Rotor
• We try to minimize total power (Induced power + Profile Power) for a given T.
• In other words, an optimum rotor has the maximum figure of merit.
• From earlier work (see slide 72), figure of merit is maximized if is maximized.
• All the sections of the rotor will operate at the angle of attack where this value of Cl and Cd are produced.
• We will call this Cl the optimum lift coefficient Cl,optimum .
d
lC
C 23
© L. Sankar Helicopter Aerodynamics
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Optimum rotor (continued..)
twisted.bemust blade thehow determines This
2R
v and
r
varctan-
from find weselected, is attack of angle Once
maximum. is C
Cat which a optimuman at operate willstations radial All
d
23
l
TC
© L. Sankar Helicopter Aerodynamics
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Variation of Chord for the Optimum Rotor
drCcrbdT l 2
2
1
dT = (Mass flow rate) * (twice the induced velocity at the annulus) = 4r(v)vdr
Compare these two. Note that Cl is a constant (the optimum value).
It follows that
r
Const
rRCR
bcr
l
18v
2
2
Local solidity
© L. Sankar Helicopter Aerodynamics
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Planform of Optimum RotorRootCut out
Tip
Chord is proportional to 1/r
Such planforms and twist distributions are hard to manufacture, and are optimumonly at one thrust setting.
Manufacturers therefore use a combination of linear twist, and linear variation in chord (constant taper ratio) to achieve optimum performance.
r=R r
© L. Sankar Helicopter Aerodynamics
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Accounting for Tip Losses
• We have already accounted for two sources of performance loss-non-uniform inflow, and blade viscous drag.
• We can account for compressibility wave drag effects and associated losses, during the table look-up of drag coefficient.
• Two more sources of loss in performance are tip losses, and swirl.
• An elegant theory is available for tip losses from Prandtl.
© L. Sankar Helicopter Aerodynamics
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Prandtl’s Tip Loss Model
Prandtl suggests that we multiply the sectional inflow by a function F, which goes to zero at the tip, and unity in the interior.
rbf
where
earcCosF f
1
2
,
2
When there are infinite number of blades, F approaches unity, there is no tip loss.
© L. Sankar Helicopter Aerodynamics
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Incorporation of Tip Loss Model in BEM
All we need to do is multiply the lift due to inflow by F.
r
drThrust generated by the annulus:
dT = = 4rF(V+v)vdr
© L. Sankar Helicopter Aerodynamics
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Resulting Inflow (Hover)
132
116
16816
2
R
r
a
F
F
a
F
a
R
r
F
a
F
a
© L. Sankar Helicopter Aerodynamics
93
Hover Performance Prediction Methods
IV. Vortex Theory
© L. Sankar Helicopter Aerodynamics
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BACKGROUND
• Extension of Prandtl’s Lifting Line Theory• Uses a combination of
– Kutta-Joukowski Theorem– Biot-Savart Law– Empirical Prescribed Wake or Free Wake Representation of Tip
Vortices and Inner Wake• Robin Gray proposed the prescribed wake model in
1952.• Landgrebe generalzied Gray’s model with extensive
experimental data.• Vortex theory was the extensively used in the 1970s and
1980s for rotor performance calculations, and is slowly giving way to CFD methods.
© L. Sankar Helicopter Aerodynamics
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Background (Continued)
• Vortex theory addresses some of the drawbacks of combined blade element-momentum theory methods, at high thrust settings (high CT/).
• At these settings, the inflow velocity is affected by the contraction of the wake.
• Near the tip, there can be an upward directed inflow (rather than downward directed) due to this contraction, which increases the tip loading, and alters the tip power consumption.
© L. Sankar Helicopter Aerodynamics
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Kutta-Joukowsky Theorem
r
V+v
T
Fx
T (r)
Fx= (V+v)
: Bound Circulation surroundingthe airfoil section.
This circulation is physically stored As vorticity in the boundary Layerover the airfoil
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Representation of Bound and Trailing Vorticies
Since vorticity can not abruptly increase in space, trailing vortices develop. Some have clockwise rotation, others have counterclockwise rotation.
© L. Sankar Helicopter Aerodynamics
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Robin Gray’s Conceptual Model
Tip Vortex has a Contraction that can be fitted with an exponential curve fit.
Inner wake descends at a near constant velocity. It descends faster near the tip than at the root.
© L. Sankar Helicopter Aerodynamics
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Landgrebe’s Curve Fit for theTip Vortex Contraction
Rv
v 2v
RR
R 707.02
v
© L. Sankar Helicopter Aerodynamics
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Radial Contraction
blade thefrom measuredFilament
vortex theofPosition Azimuthal
AgeVortex
27145.0
78.0
)1(R
R
: vortex tip theofposition Radial
v
vortex v
TC
A
eAA
© L. Sankar Helicopter Aerodynamics
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Vertical Descent Rate
v
Zv
Initial descent is slow
Descent is fa
ster
After th
e first b
lade
Passes over the vorte
x
© L. Sankar Helicopter Aerodynamics
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Landgrebe’s Curve Fit forTip Vortex Descent Rate
degrees twist,2
degrees twist,1
21
1
01.0
001.025.0
2
2k
2
20
TT
T
VVV
VVV
CCk
Ck
bbbk
R
zb
kR
z
twist,degrees: Blade twist=Tip Pitch angle – Root Pitch AngleThis quantity is usually negative.
© L. Sankar Helicopter Aerodynamics
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Circulation Coupled Wake Model
• Landgrebe’s earlier curve fits (1972) were based on the thrust coefficient, blade twist (change in the pitch angle between tip and root, usually negative).
• He subsequently found (1977) that better curve fits are obtained if the tip vortex trajectory is fitted on the basis of peak bound circulation, rather than CT/.
© L. Sankar Helicopter Aerodynamics
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Tip Vortex Representation inComputational Analyses
• The tip vortex is a continuous helical structure.• This continuous structure is broken into
piecewise straight line segments, each representing 15 degrees to 30 degrees of vortex age.
• The tip vortex strength is assumed to be the maximum bound circulation. Some calculations assume it to be 80% of the peak circulation.
• The vortex is assumed to have a small core of an empirically prescribed radius, to keep induced velocities finite.
© L. Sankar Helicopter Aerodynamics
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Tip Vortex RepresentationControl Points on the Lifting Line where induced flow is calculated
15 degrees
The x,y,z positions of theEnd points of each segmentAre computed usingLandgrebe’s Prescribed Wake Model
Inne
r Wak
e
(Opt
iona
l)
Lifting Line
© L. Sankar Helicopter Aerodynamics
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Biot-Savart Law
1r
Segment
Control Point
2r
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Biot-Savart Law (Continued)
212
22
122
212
21
21
2121
212
1
4 rrrrrrrrr
rrrrrr
rrVc
induced
Core radius used to keepDenominator from going to zero.
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Overview of Vortex Theory Based Computations (Code supplied)
• Compute inflow using BEM first, using Biot-Savart law during subsequent iterations.
• Compute radial distribution of Loads.• Convert these loads into circulation strengths. Compute
the peak circulation strength. This is the strength of the tip vortex.
• Assume a prescribed vortex trajectory. • Discard the induced velocities from BEM, use induced
velocities from Biot-Savart law.• Repeat until everything converges. During each iteration,
adjust the blade pitch angle (trim it) if CT computed is too small or too large, compared to the supplied value.
© L. Sankar Helicopter Aerodynamics
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Free Wake Models• These models remove the need for empirical
prescription of the tip vortex structure.• We march in time, starting with an initial guess
for the wake.• The end points of the segments are allowed to
freely move in space, convected the self-induced velocity at these end points.
• Their positions are updated at the end of each time step.
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Free Wake Trajectories(Calculations by Leishman)
© L. Sankar Helicopter Aerodynamics
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Vertical Descent of Rotors
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Background
• We now discuss vertical descent operations, with and without power.
• Accurate prediction of performance is not done. (The engine selection is done for hover or climb considerations. Descent requires less power than these more demanding conditions).
• Discussions are qualitative.• We may use momentum theory to guide the
analysis.
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Phase I: Power Needed in Climb and Hover
Climb Velocity, V
Power
A
TVVT
VTP
222
v
2
Descent
© L. Sankar Helicopter Aerodynamics
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Non-Dimensional FormIt is convenient to non-dimensionalize these graphs, so that universal behavior of a variety of rotors can be studied.
h
h
Tvby
lizeddimensiona-non is v)T(VPower
A2
Tv velocity inflow
hoverby lizeddimensiona-non
islocity descent veor Climb
© L. Sankar Helicopter Aerodynamics
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Momentum Theory gives incorrect Estimates of Power in Descent
V/vh
(V+v)/vh
ClimbDescent
No matter how fast we descend, positive power is still required if we use the above formula.This is incorrect!
0222
v
2
A
TVVT
VTP
© L. Sankar Helicopter Aerodynamics
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The reason..
Climb or hoverPhysically acceptable Flow
V is down
V+v is down
V+2v is downV is down
V is down
V is up
V+v is down
V+2v is downV is up V is up
Descent: Everything insideSlipstream is downOutside flow is up
© L. Sankar Helicopter Aerodynamics
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In reality..
• The rotor in descent operates in a number of stages, depending on how fast the vertical descent is in comparison to hover induced velocity.– Vortex Ring State– Turbulent Wake State– Windmill Brake State
© L. Sankar Helicopter Aerodynamics
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Vortex Ring State(V is up, V+v is down, V+2v is down)
V is upV is up
V+v is down
The rotor pushes tip vortices down.
Oncoming air at the bottom pushes them up
Vortices get trapped in a donut-shaped ring.
The ring periodically grows and bursts.
Flow is highly unsteady.
Can only be empirically analyzed.
© L. Sankar Helicopter Aerodynamics
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Performance in Vortex Ring State
V/vh
ClimbDescent
Momentum TheoryVortex Ring State
Experimental data Has scatter
Cross-overAt V=-1.71vh
Power/TVh
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Turbulent Wake State(V is up, V+v is up, V+2v is down)
V is up
V is up
V+v is up
V+2v isdown
Rotor looks and behaves like a bluffBody (or disk). The vortices lookLike wake behind the bluff body.
Again, the flow is unsteady,Can not analyze using momentum theory
Need empirical data.
© L. Sankar Helicopter Aerodynamics
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Performance in Turbulent Wake State
V/vh
ClimbDescent
Mom
entu
m T
heor
y
Cross-overAt V=-1.71vh
Vortex Ring State
TurbulentWake State
Notice power is –veEngine need not supply power
Power/TVh
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Wind Mill Brake State(V is up, V+v is up, V+2v is up)
V is up
V is up
V+v is up
V+2v up
Flow is well behaved.
No trapped vortices, no wake.
Momentum theory can be used.
T = - 2Av(V+v)
Notice the minus sign. This is becausev (down) and V+v (up) have opposite signs. The product must be positive..
© L. Sankar Helicopter Aerodynamics
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Power is Extracted in Wind Mill Brake State
mill. windain as ,freestream thefrom
extracted ispower case, In this
extracted. ispower means 0 P
consumed ispower means 0 P
descent is 0 V climb, is 0 V
:conventionSign
)v(
222
Vv
get to
v)v(V-2T
:equation thesolvecan We
2
VTP
A
TV
A
© L. Sankar Helicopter Aerodynamics
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Physical Mechanism for Wind Mill Power Extraction
r
V+vTotal Velocity Vector
Lift
The airfoil experiences an induced thrust, rather than induced drag!This causes the rotor to rotate without any need for supplying power or torque. This is called autorotation.Pilots can take advantage of this if power is lost.
© L. Sankar Helicopter Aerodynamics
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Complete Performance Map
V/vh
ClimbDescent
Mom
entu
m T
heor
y
Cross-overAt V=-1.71vh
Vortex Ring State
Power/TVh
Turbulent WakeState
Wind Mill Brake State
© L. Sankar Helicopter Aerodynamics
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Consider the cross-over Point
h-1.7vV
extracted.nor
supplied,neither
ispower speed, at this
descents vehicle theIf
!!parachute! a as good As
A. area equivalent with parachute a as
tcoefficien drag same thehasrotor The
4.1
2 vUse
v7.12
1T
:follows asrotor theoft coefficien
drag theestimatecan We
h
2h
D
D
C
A
T
AC
© L. Sankar Helicopter Aerodynamics
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Hover Performance
Coning Angle Calculations
© L. Sankar Helicopter Aerodynamics
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Background
• Blades are usually hinged near the root, to alleviate high bending moments at the root.
• This allows the blades t flap up and down.• Aerodynamic forces cause the blades to flap up.• Centrifugal forces causes the blades to flap
down.• In hover, an equilibrium position is achieved,
where the net moments at the hinge due to the opposing forces (aerodynamic and centrifugal) cancel out and go to zero.
© L. Sankar Helicopter Aerodynamics
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Schematic of Forces and Moments
0
dL
dCentrifugalForce
r
We assume that the rotor is hinged at the root, for simplicity.This assumption is adequate for most aerodynamic calculations.Effects of hinge offset are discussed in many classical texts.
© L. Sankar Helicopter Aerodynamics
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Moment at the Hinge due toAerodynamic Forces
From blade element theory, the lift force dL =
drCrcdrCvrc ll222
2
1
2
1
Moment arm = r cos0 ~ r
Counterclockwise moment due to lift = drrCrc l2
2
1
Integrating over all such strips,Total counterclockwise moment =
Rr
r
ldrrCrc0
2
2
1
© L. Sankar Helicopter Aerodynamics
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Moment due to Centrifugal Forces
The centrifugal force acting on this strip = rdm
r
dmr 22
Where “dm” is the mass of this strip.This force acts horizontally. The moment arm = r sin0 ~ r0
Clockwise moment due to centrifugal forces = dmr 022
Integrating over all such strips, total clockwise moment =
02
0
022
IdmrRr
r
© L. Sankar Helicopter Aerodynamics
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At equilibrium..
Rr
r
ldrrCrcI0
20
2
2
1
Rr
r
effective
Rr
r
l
R
rd
R
r
I
acR
I
drCcr
0
340
3
0
21
Lock Number,
© L. Sankar Helicopter Aerodynamics
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Lock Number, • The quantity =acR4/I is called the Lock number. • It is a measure of the balance between the aerodynamic
forces and inertial forces on the rotor.• In general has a value between 8 and 10 for
articulated rotors (i.e. rotors with flapping and lead-lag hinges).
• It has a value between 5 and 7 for hingeless rotors. • We will later discuss optimum values of Lock number.