Dr. Daniel P. Schrage Georgia Institute of Technology Atlanta, GA 30332-0150 Rotorcraft Design I Day...
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Transcript of Dr. Daniel P. Schrage Georgia Institute of Technology Atlanta, GA 30332-0150 Rotorcraft Design I Day...
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Rotorcraft Design IDay 3: Parametric Design Analysis
Dr. Daniel P. Schrage
Professor and Director, CERT and CASA
Georgia Institute of Technology
Atlanta, GA 30332-0150
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Presentation Outline• Review of Different Rotor Types• Ideal Rotor Characteristics• Development of a Parametric Design
Analysis for the Hiller Model 1100• Some Other Example Parametric Design
Analyses
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
The Fully Articulated Rotor System
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Lead-Lag Motion is least Damped and Often requires Dampers
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
A Bell Helicopter Bearingless Rotor – Typical of the Current State of the Art uses Elastomeric Dampers
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Rotor Hub Moment Relations to Acceleration or Rate Type Response ( A Key for Ease of Operation)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Characteristics of an Ideal Rotor (Tom Hanson)
• Simplication through elimination of the following:– All hub bearings and lubrication– Blade dampers– Droop stops– Gyroscopes and stabilizer bars– Powered control boost– Electronic stability augmentation systems– All structural joints except one per blade (to allow blade
removal and/or folding
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Ideal Rotor (According to Tom Hanson)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Characteristics of an Ideal Rotor (Tom Hanson)
• Structural safety improvements by providing:– Multiple load paths so that the failure of any one part could not result
in the loss of a blade
– A very high ratio of ultimate tensile strength to blade centrifugal force
– Stability about the feathering axis so that a blade would go to flat pitch in case of a broken pitch link
– Principle blade natural frequencies below their respective forcing frequencies so that structural damage would take the natural frequency further away from resonance and thus attentuate the remaining loads and stresses
– The absolute minimum number of structural joints and stress concentration points
– A capability to continue controlled flight after serious damage had been incurred
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Characteristics of an Ideal Rotor (Tom Hanson)
• Handling Quality Improvements by providing:– Force rather than displacement cockpit controls– High control power (& damping) about the pitch and
roll axes– Force feedback (feel) at the cyclic stick that was
proportional to the rotor moment being produced and to the rate of angular velocity of the aircraft
– High damping in pitch and roll– Zero sensitivity to gusts and other disturbances– Zero apparent time lag between control force
application– Force feedback (feel) at the collective lever that
was proportional to the rotor thrust being produced
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Some Ideal Traits can be QuantifiedUniform, effective, and immediate response to control inputs at all forward speeds including
zero
+
High pitch, roll, and yaw damping
Pitc
h ra
te d
ampi
ng
Pitch control power
0Unacceptable
Unsatisfactory (C-H = 6.5)
Optimum Line
Satisfactory (C-H = 3.5)
e=0
e=15%
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Characteristics of an Ideal Rotor (Tom Hanson)
• Some general improvements would be:– Minimum rotor noise by reducing tip speed– Minimum rotor and control system weight by
eliminating hydraulic boost– A design that is easy to operate, manufacture
and maintain by removal of bearings and required lubrication
– Minimum vibration generation by multiple blades (>2) and proper natural rotor blade frequency placement
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Benefits of a New Bearingless Rotor(Hanson’s Auto-trim Rotor) for the Hawk 4 Gyroplane
• Increase in low speed agility and maneuverability – thus enhancing safety during landing and takeoff– Current two bladed teetering rotor limited to positive thrust flight due to
potential for flapping limit excursions and “mast bumping”– Current two bladed teetering rotor has “acceleration” type response due
to low combination of pitch and roll control power & damping
• Increase in forward speed capability through compounding – thus providing a doorstop to destination speed advantage over the conventional helicopter– Excessive flapping of current two bladed teetering rotor limits high speed
flight, e.g. flapping limits of + or – 19 degrees could be exceeded– Collective pitch of 3 to 4 degrees required with current two bladed
teetering rotor to provide lift in absence of wing– Addition of a new bearingless rotor plus wing could add 40-50 kts
increase in cruise speed and unload the rotor to approximately 1 to 2 degrees, thus reducing rotor loads and vibration
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Benefits of a New Bearingless Rotor(Hanson’s Auto-trim Rotor) for the Hawk 4 Gyroplane
• Increase in Ease of Operation and Reduction in Training Requirements– Current two bladed teetering rotor behaves as an “acceleration-type” response
system and has a stiff feathering frequency which results in the pilot over reacting with control inputs and training requirements as experienced with the Robinson R-22/R-44
– New bearingless “auto-trim” rotor would have sufficient control power and damping in pitch and roll to provide a “rate type” response system
– By placing the feathering frequency of the new bearlingless “auto-trim” rotor at 1P( the rotor rotation speed) would require easy pilot inputs (without actuator augmentation) and provide direct force feedback (without artificial force-feel augmentation)
• Reduction in Rotor Blade Vibrations and Increase in Reliability– Current two bladed teetering rotor has natural frequencies that can respond to all
rotor harmonics in the rotating system; and airframe natural frequencies that can respond to 2P, 4P, 6P, 8P etc. in the fixed system
– New bearingless rotor (3 or 4 blades) would reduce the number of blade and airframe natural frequencies that could be excited
– The results of the new bearingless rotor should be reduced vibration and improved reliability, especially for dynamic components
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Parametric Design Study• Introduction
• Parametric Analysis Method (RF Method)
• Development of Performance and Weight Equations
• Autorotation Characteristics• Selection of Design Parameters• Discussion of Model 1100 Design
Parameters
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Parametric Design Study
• For any given payload and performance specification, an infinite number of helicopters satisfying the requirements are possible.
• The preliminary design problem is not which helicopter meets the specifications but which solution out of many will best meet the requirements.
• A secondary problem is presented by the question “What criteria are to be used in selecting the best solution?”
• These criteria must be compatible with any size and operating requirements imposed in addition to the payload and performance specifications.
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Parametric Design Study
• Before the design solution giving the best helicopter can be selected, an appropriate number of solutions satisfying the design specifications for payload and performance must be produced
• Since each solution is characterized by a different combination of design parameters, the selection can best be made through a parametric study which allows the optimization of many design parameters and the determination of the corresponding gross weight
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
The RF Parametric Analysis Method
• One type of parametric analysis is the RF method
• Based on graphical simultaneous solution of equations expressing the weight and performance characteristics of the helicopter.
• This method also enforces compatibility of resulting gross weight solution with both weight and performance predictions.
• Minimum gross weight is criterion by which optimum design parameters are selected.
• Design parameters include disk loading, power loading, tip speed and blade loading.
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Rotorcraft/VSTOL Aircraft Synthesis ( RF Method)
Engine Power
Available
Vehicle Power
Required
Vehicle
Power
Loading
Vehicle
Gross
Weight
Installed
Power
HP i
Requirements Modelsls SynthesisConfiguration
Solution
Mission Input
Payload
Block Range
Hover Time
Agility
Performance
Hover Alt.
Hover Temp.
Block Speed
Block Alt.itudes
ROC/Maneuver
Empty Fraction
Fuel Weight
Ratio Available
Mission Analysis
Fuel Weight
Ratio Required
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Parametric Analysis Method
• The objective of this parametric study is to select the design parameters for a four place light observation helicopter which meets the following requirements:– Turbo shaft engine ~ 250Hp Class– Hover (OGE) at 6000 feet altitude with air temperature of
95°F. Useful load at hover includes (a) 200 lb pilot (b) 400 lbs payload (c) 3 hour endurance at sea level and best range speed. Take-off (or military) power used.
– Maximum speed of at least 110 kts at sea level using Normal Rated Power
– Rotor diameter, overall length and gross weight should be equal to or less than 35.2 feet, 41.4 feet and 2450 lbs respectively.
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Selection of configuration and Parameters
• Chosen configuration is conventional single main rotor with tail rotor
• Tail rotor radius is assumed Rtr=.16 RMR
• Tail rotor moment arm is assumed ltr= 1.19RMR
• Two bladed, semi-rigid (teetering) and three bladed, fully articulated type of main rotor included
• Allison T-63 engine is used, due to derate at SL (larger %of SL Power available at 6000’/95o), ~ good SFC and light weight)
• Four parameters: gross weight (W), disk loading (w), tip speed (VT) and design mean blade lift coefficient (CLro).
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Parametric Study
• The RF method is modified to provide a method which establishes a family of solutions meeting hover and useful load requirements.
• The aerodynamic and weight equations are written in terms of four parameters.
• For selected values of VT and CLro the aerodynamic and weight equations are solved simultaneously to determine the values of gross weight and disk loading which establish a solution to the hover problem.
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Parametric Study
• The equations are solved graphically by first substituting a value for disk loading into the aerodynamic equation and solving for W
• This value of W and the related values of w, VT and CLro are utilized in the weight equation to establish a value for the empty weight plus useful load
• The intersection of the two weight curves, plotted versus disk loading, is the solution
• Figures 5 through 7 illustrate graphical solutions, while Figures 8 and 9 present families of solutions to the hover requirement for semi-rigid and articulated rotor systems, respectively
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
2440
2420
2400
2380
2360
2340
2320
2300
600
625
650
675
700
Tip Speed
ft/sec
Gross Weight
Empty Weight + Useful Load
Solution
Weight (lbs)
Disk Loading (lbs/ft2)2.4 2.5 2.6 2.7 2.8
Articulated Rotor CLro=.4
Figure 5: Graphical Solution of Hover & Useful Load Requirements
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
2440
2420
2400
2380
600
625
650
675
700
Tip Speedft/sec
Gross Weight
Empty Weight + Useful Load
Solution
Weight (lbs)
Disk Loading (lbs/ft2)2.2 2.3 2.4 2.5 2.6
2460
2480
2500
Teetering Rotor CLro=.4
Figure 6: Graphical Solution of Hover & Useful Load Requirements
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
2440
2420
2400
2380
2360
2340
2320
2300
600
625
650
675
700
Tip Speed
ft/sec
Gross Weight
Empty Weight + Useful Load
Solution
Weight (lbs)
Disk Loading (lbs/ft2)2.4 2.5 2.6 2.7 2.8
Teetering Rotor CLro=.5
Figure 7: Graphical Solution of Hover & Useful Load Requirements
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
2450
2400
2350
2300
600
625
650675700
Tip Speed
ft/sec
Weight (lbs)
Disk Loading (lbs/ft2)
2.0 2.2 2.4 2.6 2.8
.3
.4
.5.6
0rLC
Family of Solutions
Hover & Useful Load Requirements Articulated Rotor
Figure 8
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
2450
2400
2350
2500
600
625
650
675
700
Tip Speed(ft/sec)
Weight (lbs)
Disk Loading (lbs/ft2)
2.0 2.2 2.4 2.6 2.8
.3
.4
.5.6
0rLC
Family of Solutions
Hover & Useful Load Requirements Teetering Rotor
Figure 9
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Parametric Study• Figures 8 and 9 indicate that when only the hover & useful
load requirement is considered, increasing the rotor tip speed and mean blade lift coefficient results in a continually decreasing W
• Other requirements must be imposed to determine the upper limits on VT and Clro
• The remaining performance specification, a mini-mum cruise speed of 110 kts, has not been considered: but with it enters another parameter, the equivalent flat plate drag area, Aπ
• The forward speed potential of the family of solution helicopters can be obtained, however, and the range of values of Aπ which meet the cruise requirement can be determined
• The forward flight power required equation equation can be rewritten to yield an expression for Aπ in terms of V, power available, and the four selected design parameters
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Parametric Study• Expressions of Aπ in terms of the same variables, except for
power available, are also derived from the retreating blade stall and advancing blade drag divergence relationships
• Thus, for any solution to the hover and useful load requirement, a plot can be produced that shows rotor limited forward speed versus Aπ and power limited forward speeds versus Aπ , as illustrated in Figure 10
• Selecting hover solutions from lines of constant VT and Clro in a family of plots similar to Figure 10, with the rotor and power limited speed in terms of three variables, namely Aπ, Cl ro and VT
• Cross plots of the limiting forward speeds may be produced, such as those shown in Figures 18 thru 21
• These plots allow the selection of design parameters which yield the best speed potential
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Parametric Study
• Further requirements must still be investigated to determine the optimum design
• These include the dimensional restrictions on rotor radius and maximum W and structural limitations on rotor blade dimensions
• The selected configuration should also have a sufficient rotor autorotation and present a “safe” autorotative rate of descent
• The methods and criteria used to ensure compliance with these latter two requirements will be presented later,along with the optimum design configuration
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Hover Equations
TLL
H
L
L
T
HHH
WVCC
wW
rhp
and
aBIf
a
C
C
WVw
B
W
Rhpihprhp
r
r
FK
r
r
)80779.11()10(91971.
035479.
F)95 ft, (6000 749395.
002378. ,73.5 ,3. ,009. ,97.
4400
6
2550
13.1
25
020
2
20
0
0
0
95/'6
0
0
0
0
0
0
(3.1)
(3.3)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Tail Rotor Thrust
W
w
V
rhp
RV
rhp
A
Tw
RR
V
rhp
Vl
RrhpT
T
H
T
H
tr
trtr
tr
T
H
Ttr
Htr
)0256(.19.1
550
)16(.
1
19.1
550
is loadingdisk rotor tail the16.with
19.1
550550
2
(3.4)
(3.5)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Tail Rotor Power Required• By assuming tail rotor tip speed is equal to main
rotor tip speed and applying equation 3.1 to the tail rotor, tail rotor power required in hover is:
rtr
rtr
L
H
T
HtrH
trL
H
T
H
otrT
HtrH
C
rhp
V
rhp
W
wrhp
C
rhp
WV
wrhp
BV
rhprhp
012605.
)(7.2055
toreduces this90.B and 02. assuming
19.1)4400(
)550(6
)()0256(.19.1
550
219.1
13.1
23
0
0
trtr
(3.6)
(3.7)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Satisfactory Yaw Control• The tail rotor must provide adequate thrust for
satisfactory yaw control.• Achieved by designing tail rotor to counterbalance
a sea level main rotor torque equivalent to ninety percent of installed power with CLrotr= .4
• Using equation 3.5 the design tail rotor disk loading is
W
w
VV
w
VC
w
and
W
w
Vw
TT
tr
TL
tr
tr
Ttr
design
trr
design
design
)0256(.19.1
)250 (550
4.
66
)0256(.19.1
)250 x 9(.550
20
20 0
(3.8)
(3.9)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Satisfactory Yaw Control
• Utilizing equations (3.5) and (3.9) the tail rotor mean blade lift coefficient is given by:
)(5.562
)225)(550(6
)0256(.19.14.
)0256(.19.1
5506
6
)0256(.19.1
)250 x 9(.550
0
30
2
2
rhp
w
WV
WV
wrhp
V
V
wC
and
W
w
Vw
T
TT
Ttr
trL
Ttr
trr
design
(3.10)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Satisfactory Yaw Control
• Substituting equation (3.10) into equation (3.7) the tail rotor power required at 6000 feet altitude and 95°F is:
3134.5 7.23742
3
95/'6
T
HFKtrH V
rhp
W
wrhp
(3.11)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Total Brake Horsepower Required to Hover
• By imposing a gear loss of 3% and a cooling power loss of 1% for a turbine engine installation, the total brake horsepower required to hover is:
TLL
H
WVCC
wW
Bhp
r
r
FK
)80779.11()10(95803.
036957. 25
0
0
95/'6
96.trHH
H
rhprhpBhp
5348.5 6.24732
3
T
H
V
rhp
W
w
(3.12)
(3.13)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Some Simplifications for Bhpreq
• Partial substitution of equation (3.3) in equation (3.13) yields:
TLL
H
WVCC
wW
Bhp
r
r
FK
)80779.11()10(95803.
036957. 25
0
0
95/'6
6.24735348.52
3
43
H
T
rhpV
w
21
0
0
)80779.11()10(91971.
035479. 25
w
VC
CT
LL
r
r(3.14)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Some Simplifications for Bhpreq
• The last term in equation (3.14) which contains rhpH is comparatively small and the equation is simplified by noting from previous studies:
HHH
H
BhpBhprhp
rhptr
8832.)96(.92.
)hover tohorsepowerrotor total(08.
• If this equation is substituted in equation (3.14), the equation (3.14) can be solved for W as a function of the four design parameters
(3.15)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Solving for Gross Weight, W
)80779.11(95803.
7.3695
553480
)80779.11(00025929.
)10(
23
54
53
22
5
5
1
0
0
0
0
96/''6000
r
r
r
r
LL
LL
H
CC
K
KK
KK
CC
KK
BhpK
wKV
Kw
VK
V
wK
WT
T
T
4
321
21
23
43
151.4111
(3.16)
(3.17)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Sample Calculation for Demonstrating Utility of Equation (3.16)
Assume: CL ro = .40 VT = 650
w = 2.5 BhpH6000’/95o
Then: K5 = 3.0878 K4= 1196.9
K3 = 179250 K2 = .00083572
K1 = 6,671,4000
Gross Weight = W = 2403 lbs
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Obtaining Weight Equations• Statistical weight equations for the components which make
up the empty weight can be divided into three groups– Components having constant weight– Components whose weights depend only on installed
power– Components whose weights depend on gross weight or
on two or more of the following: gross weight(W), installed power, rotor tip speed(VT), rotor radius(R), rotor solidity(σ)
• Since a specific engine is being used the first two categories are constant
• By using the following relationships:
W/w=A=R2, W/lpm=MHP=250* hp, P= (A )1/2/VT *Military rating of Allison T-63 @SL
The component weights are reduced to the following:
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Constant Weights
Pylon and isolation
Lubrication and oil cooling
Engine
Communications
Engine controls
Engine accessories
Instruments
Starting System
Furnishings
Flight controls
Electrical system
Stabilizer
20 lbs
35 lbs
128 lbs
100 lbs
6 lbs
15 lbs
20 lbs
20 lbs
40 lbs
80 lbs
100 lbs
4 lbs
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Engine, Transmission, Drive Shaft Weight
Engine Section
Group
Main
Transmission
Rotor Drive Shaft
Fuel System capacitygallon per lb 4.
)(56.5
)(43.10
)/(053.
35.7.
05.1
432.863.
295.1
07.1
wVl
W
wVl
W
lW
Tp
Tp
p
m
m
m
FW
P
P
lbs
0615.
266
1221
5.19
7.
863.
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Tail Rotor Weights
785.57.
355.1
25.5.
75.
14.1
14.1
)(124.
)(7.3
)(2.32
wl
W
wVl
W
Vl
W
m
m
m
p
Tp
Tp
Tail rotor
Tail rotor
gear box
Tail rotor
drive shaft AP
P
VT
57.
14.1
886.2
47.58
17449
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Main Rotor and Fuel System Weights
21.
21.1
21.
21.1
205.
33.205.1
185.
33.185.1
00975.
0088.
77.19
15.35
w
W
w
W
wV
W
wV
W
T
T
Rotor Blades:
Teetering
Articulated
Rotor Hub:
Teetering
Articulated 21.
21.
33.205.
33.185.
00975.
0088.
77.19
15.35
WA
WA
AV
W
AV
W
T
T
Fuel System: .4 lb. per gallon capacity = .0615 WF
where WF = fuel weight
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Empty Weight, WE, Equation
• The component weights can now be combined into a single expression for the helicopter empty weight
weightshub and bladerotor eappropriat
0294.191.886.2
47.5817449
12210615.5.587
99.916.57.
14.17.
WWAP
PV
PWWT
FE
(3.18)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Useful Load Calculation
• Fuel weight for three hour endurance at 85% of Normal Rated Power = .85(212)=180.2 Hp
• Allison T-63, SFC at 180.2 Hp = .783 lbs fuel/Bhp-hr
• Assuming 3 minute warm-up and 5% SFC correction
WF= 3(180.2)(.822) + (3/60)(212)(.777) = 452.6 lbs
WU= 200 + 400+20 + 452.6 = 1072 lbs
WU= Pilot + Payload + WF + Trapped Oil and Fuel
(3.19)
(3.20)
Useful load consists of a pilot (200 lbs), payload (400 lbs) and the Required fuel weight (WF)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Operating Weight Empty Equation
• Operating weight empty includes empty weight plus useful load (summation of eqns. (3.18) & (3.20)
weightshub and bladerotor eappropriat
0294.191.886.2
47.5817449
26612219.1687
99.916.57.
14.17.863.
WWAP
PV
PPWT
(3.21)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Sample Calculation for Demonstrating Utility of Equation (3.21)
Assume: Clro = .40 VT = 650 fpsw = 2.5 W = 2403A teetering rotor system__
Then: W = 2397 lbs
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Forward Flight Equations
• The main rotor power required in forward flight may be written:
phpRhpKihpKphpRhpihprhp HHu
KWVC
C
VAK
wWrhp
TL
L
u
r
r
2
5
30
0
0
0
0
0
0
01524.11)10(22727.1
1100030713.
(3.22)
(3.23)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Forward Flight Equations• For forward velocities pertinent to the specification
requirements (greater than 100 knots) the following approximation is sufficiently accurate:
2
1 w
BVV
uK H
u
KWVCC
VAV
Wwrhp
TL
rL
r
01524.11)10(22727.1
)10(1618.2459124.
25
36
0
0
• Substitution of this approximation and the density ration equal to 1 for seal level:
(3.24)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Tail Rotor Power in Forward Flight
• Tail rotor power in forward flight is given by
• The approximation for the tail rotor is valid
• For sea level then
K
V
rhp
W
wrhp
Ttr )/(0903.7 K
)(7.2055 0u tr
23
0
2
1 tr
trtru
w
VBK
KV
rhp
WV
wrhp
Ttr 0903.7 )10(4502.4
2
6
(3.26)
(3.27)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Tail Rotor Power in Forward Flight• Substituting equations (3.24) and (3.27) in
equation(3.12)
2
36
25
26
25
36
)10(1618.2
01524.11)10(22727.1
459124.
)10(4502.40903.7
01524.11)10(22727.1
)10(1618.2459124.96.
0
0
0
0
VA
KWVCCV
Ww
WVV
wK
KWVCC
VAV
WwBhp
TL
T
TL
rL
r
rL
r
(3.28)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Tail Rotor Power in Forward Flight
• This equation can be written in the form
• where
KWVCCV
WwC
BhpKCVWV
wCC
VACWV
wVC
WV
wVC
TL
T
T
T
rL
r
01524.11)10(22727.1
459124.
96.0903.7)10(4502.4
)10(1618.2241.19
)10(0116.2
254
242
643
3642
2
2
2
55
1
0
0
0322
1 CACAC (3.29)
(3.21)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Sample Calculation for Demonstrating Utility of Equation (3.29)
Assume: Clro = .40 VT = 650 fps
w =2.5 lbs/ft2 W = 2400 lbs
V = 120 kts Bhp = 250 shp 202.68 fps
Then: μ = 202.68 = .3118 650Kμ = 1 + 3(.3118)2 + 30(.3118)4 = 1.575
C4 = 101.222, C3 = -127.06, C2 = 18.196, C1 = .016963Now,
.016963 Aπ2 + 18.196 Aπ – 127.06 = 0
or Aπ = 6.94 ft2
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Determining Rotor Limits
• From derivation section the tip angles of attack on retreating and advancing blades are
• The coefficients are functions only of tip speed ratio, and tip loss factor, B. The following figure plots these coefficients (B=.97) versus
TL AACAr
'3
''2
'1)90)(1( Tip Retreating
TL AACAr
3'
21)270)(1( Tip Advancing (3.31)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Figure 1. Constants in the Expression for Blade Tip Angle of Attack @ = 90° and = 270° vs Tip Speed Ratio
(B=.97)
.2
.1
0.5.3.2.10 .4
A2
A1
Tip Speed Ratio,
An & An
A3
A’ 3
A’ 2
A’ 1
.3
.4
.5
.6
.7
.8
.9
1.0
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Determining Rotor Limits• Previously shown that
21'
2
1'
''
'
2)(WV
550tan
2)(V
550H
drag parasite
tan
RhpW
qA
Rhp
D
whereW
HD
V
u
V
Vu
P
P
TT
(3.32)
(3.33)
(3.34)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Determining Rotor Limits• Since ’ is a small angle
• Substituting a=5.73, at sea level
0
0
0
0
0
0
013706.0135.
4400
6
WV
1100
'
2
22
0
2'
r
r
r
r
LL
L
L
T
CCW
qA
a
C
C
WV
W
qA
(3.35)
(3.36)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Determining Rotor Limits• For pertinent speed range
qB
w
VVB
w
V
u
Therefore
VB
wV
uuKu
Thus
w
BuV
uK
TT
HHu
HH
u
22
2
2
42
2
2
1
(3.37)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Determining Rotor Limits
• Substitution of equations (3.36) and (3.37) into equation (3.32) yields:
0
0
013706.0135.
42
2'
r
r
LL
CCW
qA
qB
w
(3.38)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Determining Rotor Limits
• By limiting forward speeds to advance ratios between .26 and .46 and assuming a blade twist of ten degrees (T= -.17453 radians) the A coefficients reduce to:
on dependent are ,,, where
7335.4966.
075.3726.
0
0
0
4321
4331
22131
'2
22
r
r
r
L
TL
TL
C
ACA
ACA
A
A
(3.39)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Determining Rotor Limits• The stall limit is considered to be at an angle of
attack of 12°. The operational stall limit is determined by substituting this value and equations (3.38) and (3.39) into equation (3.31) for (1)(270).
Wq
CCq
w
Wq
A
r
r
s
LL
2
2
221
0
0
013706.0135.
24.3
)075.3726(.
20944.
(3.40)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Determining Rotor Limits• Assuming the following form for drag divergence
Mach number effect on theoretical angle of attack
• This equation provides good agreement for angle 7 degrees or less which is sufficient for establishing advancing blade compressibility limits at sea level
• For the advancing blade tip at sea level
2595.1446.11178. dd MM
0.1117T
s
Td
VV
a
VVM
(3.41)
(3.42)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Determining Rotor Limits• Substitution of equation (3.42) in the equation for
advancing blade tip angle of attack yields
22
43
2
0
0
013706.0135.
24.3)7335.4966(.
1117595.1
1117446.11178.
X )7335.4966(.
r
r
c
LL
T
T
CCq
w
VV
VV
q
WA
(3.43)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Autorotation Characteristics• The complete autorotation maneuver can be
divided into three segments:– The interval between power failure and steady
autorotational descent called the autorotation entry which includes “delay” time”
– The interval of steady autorotational descent– The landing flare
• The most significant item is the available ”delay time” after power failure until collective pitch must be reduced in order to effect entry into steady autorotational descent
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
(3) Landing Flare
(2) Steady Autorotational Descent
(1) Autorotation Entry
descent of Rate dVdt
dh
Time, t
Altitude,
h
Segmentation of Autorotation Maneuver
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
i
i a1
Tip Path PlaneHorizontal
Flight Path
Vertical to Flight Path
Tip Path Plane AxisControl Axis
Rotor Axis Geometry in Autorotation
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Autorotation Characteristics• Rate of descent is important for two reasons
– The effect of descent velocity on the pilot’s judgment as to when to commence the landing flare
– The effect of descent velocity on the momentum change necessary in the flare to stop the descent
• In the landing flare the rotor must have the capability to stop the descent before the rotor its kinetic energy is expended
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Time Delay Aspect of Autorotation Entry
• After power failure, if collective pitch is not reduced immediately the rotor decelerates
• The limit on the time delay is the interval required for the rotor speed to decay to the minimum value from which autorotation entry can still be accomplished
• Theoretically this is difficult therefore assume it is rotor speed at which CLMax is reached if thrust remains the same during the interval.
2.100
0
min r
MAX
r L
L
L C
C
C
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Time Delay Aspect of Autorotation Entry• The equation of motion for the rotor immediately
after power failure is:
failurepower at (power) torqueengine
inertia ofmoment rotor
R
20
2
R
so
so
Q
Iwhere
Qdt
dI
• Integrating equation (4.1) applying condition = when t=0 and solving for t when = min
1
2.1
550t
0
2R
rLH
T
CWBhp
wVI
(4.1)
(4.2)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Time Delay Aspect of Autorotation Entry
• Assuming rhptr=.08rhpH, gear losses of 3%, and cooling losses of 1% (turbine) or 5% (reciprocating) the time delay equations reduce to:
12.1
8.206t
12.1
2.198t
0
0
2R
2R
r
r
LH
T
LH
T
CWrhp
wVI
CWrhp
wVI For turbine engines
For reciprocating engines
(4.3)
(4.4)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Rotor Inertia
• Rotor inertia can be closely approximated by considering only the blades
• With uniform spanwise mass distribution K=1/3 and study has shown this to yield good correlation for articulated rotor system
• For teetering system where a structural build-up is required to carry loads through the blade root attachment K+.3 is good
rotor for the weight blade totalis Wwhere
B
2Rg
WKI B
R
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Rotor Inertia
• With the previous K values and R2=W/w
w
WWI
w
WWI
BR
BR
0029682.
003298.
Articulated Rotors
Teetering Rotors
(4.6)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Autorotational Rate of Descent
• Autorotative performance is established through a plot of autorotative rate of descent against flight velocity.
W
R
T
HPtr+acc
A
= gross weight, lbs
= rotor radius, ft
= rotor angular velocity (or tip speed) during autorotation, rad/sec
= effective rotor solidity
= rotor blade twist, rad
= horsepower required for tail rotor and rotor driven accessories during autorotation
= helicopter equivalent flat plate drag area, ft2
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Thrust & Torque Equilibrium During Autorotation
• Thrust Equilibrium
• Torque Equilibrium
53
210,509,5
208,57,506,5
25,52
01,5
26,405,4
204,43,402,4
21,4
423,302,31,3
1100
)()()()()()(
)(
)()()()()()(
2)()()(
R
HP
tttttt
t
tttttta
Ra
Wttt
acctr
TTT
TTT
T
(4.8)
(4.9)
The (ti,j) coefficients are primarily a function of and are defined in “Experimental Investigation in the Langley Helicopter Test Tower of Compressibility Effects on a Rotor Having NACA 632-015 Airfoil Sections” by James P. Shivers and Paul J. Carpenter, NACA TN 3850, Dec 1956.
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Solving for Inflow Ratio and Collective Pitich
• For selected values of and , the thrust and torque equilibrium equations can be solved simultaneously for and 0
• Recommended method of solution is to solve thrust equation for 0 and substitute into torque equation yielding
1
31222
322
1
2
4
0
C
CCCC
CCC
(4.11)
(4.11’)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Solving for Inflow Ratio and Collective Pitch• The angle between the inclined flight path and the
horizontal is (+a1+i) with:
)(4.11' and (4.10) equations
from determined are and where
)()()(
and )(
5.tan
0
6,105,14,11
4222
T
TT
ttta
R
WC
C(4.12)
(4.13)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Solving for Inflow Ratio and Collective Pitch• The angle I is given by;
• The rate of descent is given by
cos
)(V and
2tan
2 R
W
AV
W
qA
lift
dragi
(4.14) and (4.13) (4.12), equations from
determined ,, where
)sin(
1
1
ia
iaVvd
(4.14)
(4.15)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Landing Flare
• The landing flare presents problem of converting rotor kinetic energy into additional rotor thrust to decrease the rate of descent.
• If the sinking speed is denoted Vs for a helicopter of mass, M then:
timeoffunction a asst rotor thru T(t) where
)(])([
tTMgtTdt
dvM s
(4.16)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Landing Flare
• Starting at vs=vd= steady autorotational rate of descent and time t=0 we can integrate the previous equation:
zero tospeed sinking
reduce torequired time twhere
)(
)(
1
0
0
0
1
1
t
d
t
v s
dttTMv
dttTdvMd
(4.17)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Landing Flare Discussion
• If T(t) is known then integral can be evaluated and t1 can be determined.
• At t=0, T(t) is assumed a maximum value.• As time increases, the thrust decreases
approximately as the square of the rotor RPM decrease.
• At some time t = t2 T(t) = Mg and T(t) = 0.
• At t > t2, T(t) is negative and R/D wil increase.
• Maximum flare capability could then be expressed as ratio t2/ t1.
• However T(t) can not be determined analytically.
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Relative Autorotative Landing Index
• For comparison of a new helicopter to a helicopter with known autorotational landing characteristics
• The ratio of rotor kinetic energy to gross weight is given by:
speed tiponalautorotati normal is V where
22
T
2
2
22
W
VWI
R
V
W
I
W
I
W
KE TR
TRR (4.18)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Relative Autorotative Landing Index
Ww
W
VI
W
CRALI
I
Ra
I
RabC
Ww
IRALI
TRL
e
R
e
R
e
R
r
2
54
1
1
0964.161
then5.73a and for ngsubstituti
where
1000
(4.20)
(4.19)
(4.21)
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Selection of Design Parameters
• The aerodynamic (hover) and weight equations were solved simultaneously for rotor tip speeds of 600, 625, 650, 675, and 700 feet per second in conjunction with design mean blade lift coefficients of .3, .4, .5 and .6
• Power available at 6000 ft/ 95°F was 206 Hp• Typical graphical solution are presented for both
articulated and teetering rotors illustrating the form of the aerodynamic and weight equations.
• Plots showing families of solutions are drawn for both rotor systems.
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
2440
2420
2400
2380
2360
2340
2320
2300
600
625
650
675
700
Tip Speed
ft/sec
Gross Weight
Empty Weight + Useful Load
Solution
Weight (lbs)
Disk Loading (lbs/ft2)
2.4 2.5 2.6 2.7 2.8
Figure 5
Articulated Rotor CLro=.4
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
2440
2420
2400
2380
600
625
650
675
700
Tip Speed
ft/sec
Gross Weight
Empty Weight + Useful Load
Solution
Weight (lbs)
Disk Loading (lbs/ft2)
2.2 2.3 2.4 2.5 2.6
2460
2480
2500
Figure 6
Teetering Rotor CLro=.4
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
2440
2420
2400
2380
2360
2340
2320
2300
600
625
650
675
700
Tip Speed
ft/sec
Gross Weight
Empty Weight + Useful Load
Solution
Weight (lbs)
Disk Loading (lbs/ft2)
2.4 2.5 2.6 2.7 2.8
Figure 7
Teetering Rotor CLro=.5
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
2450
2400
2350
2300
600
625
650675700
Tip Speed
ft/sec
Weight (lbs)
Disk Loading (lbs/ft2)
2.0 2.2 2.4 2.6 2.8
.3
.4
.5.6
0rLC
Figure 8
Family of Solutions
Hover & Useful Load Requirements Articulated Rotor
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
2450
2400
2350
2500
600
625
650
675
700
Tip Speed(ft/sec)
Weight (lbs)
Disk Loading (lbs/ft2)
2.0 2.2 2.4 2.6 2.8
.3
.4
.5.6
0rLC
Figure 9
Family of Solutions
Hover & Useful Load Requirements Teetering Rotor
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Selection of Design Parameters
• From inspection of graphs for family of solutions it is evident that additional requirements must be used to obtain upper limits on design mean lift coefficient and rotor tip speed
• The forward flight equations yield plots of power limited and rotor limited forward speeds (at sea level) versus A (fuselage drag coefficient)
• NRP = 212 Hp and MRP = 250 Hp
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Forward Speed Data
• Most useful when plotted versus design mean lift coefficient for a constant A and rotor tip speed
• A = 5.0, 6.0, 7.0, and 8.0 are used in this study
• A = 5.0 estimated minimum value in production of four place helicopter with fuel and avionics
• A = 8.0 approximate upper limit if VNRP=110 kts is to be satisfied
• Figure yields two boundaries based on forward flight performance that can be applied to Family of Solutions
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Boundaries from Forward Flight Performance
• 110 kt forward speed requirement– Maximum values of design mean lift coefficient
compatible with A = 5.0 and the appropriate tip speeds– These lift coefficients are seen in Figure as intersection
of the rotor limit portion of the 110 knot forward speed boundary and the line A = 5.0.
• Rotor limit on forward speed– Dashed boundary lines indicate contraction of right hand
rotor limit boundary to more stringent but desirable limit– Values of design mean lift coefficient at the junction of
the dashed rotor limit curves and the power limit potion of the 110 knot boundary for the appropriate tip speed
• Size restrictions on rotor diameter and gross weight can also be applied
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Aspect Ratio Limit
• A final restriction is the limitation on effective aspect ratio of the blades.
• AR<21 is lower limit on rotor solidity imposed in this study by structural and dynamic considerations
rotor bladed 3 000018022.
rotor bladed 2 000012015.126
621
2
22
0
20
0
0
0
0
TL
TLTL
TL
e
VC
VCVCb
w
or
w
VCbb
C
R
r
r
r
r
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Power Limit Boundary
Rotor Limit Boundary
(Sta
ll)
(Compressibility)
RLMRP VV
VT =
700
675650
625600
6.0
5.0.30
7.0
8.0
.35 .40 .45 .50 .55 .60
kts 110Vfor C vsNRPr0 MAX L A
A
Lr0C
Forward Flight Limits
Dr. Daniel P. SchrageGeorgia Institute of TechnologyAtlanta, GA 30332-0150
Final Carpet Plot for Teetering Rotor
DISK LOADING (lbs/sq. ft)
GR
OS
S W
EIG
HT (
lbs)
2300
2350
2400
2450
2500
2550
2.0 2.2 2.4 2.6 2.8 3.0
Final Design SolutionsTeetering Rotor
0.30
0.40
0.45
0.50
0.60
600
625
650
700
675
Gross Weight = 2450 lbs
VNRP = 110 kts
VMRP = VRL
VT, ft/sec
Aspect Ratio = 21
Rotor Diameter = 35.2 ft
Loci of Final Solution