Dr. Daniel P. Schrage Georgia Institute of Technology Atlanta, GA 30332-0150 Rotorcraft Design I Day 3: Parametric Design Analysis Dr. Daniel P. Schrage

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


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


The Fully Articulated Rotor System


Lead-Lag Motion is least Damped and Often requires Dampers


A Bell Helicopter Bearingless Rotor – Typical of the Current State of the Art uses Elastomeric Dampers


Rotor Hub Moment Relations to Acceleration or Rate Type Response ( A Key for Ease of Operation)


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


Ideal Rotor (According to 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



• 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


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%



• 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


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


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


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


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.


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


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.


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


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.


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).


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.


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


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


2440

2420

2400

2380

600

625

650

675

700

Tip Speedft/sec

Gross Weight


Solution

Weight (lbs)


2460

2480

2500

Teetering Rotor CLro=.4



2440

2420

2400

2380

2360

2340

2320

2300

600

625

650

675

700

Tip Speed

ft/sec

Gross Weight


Solution

Weight (lbs)





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


2450

2400

2350

2500

600

625

650

675

700

Tip Speed(ft/sec)

Weight (lbs)


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


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


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


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


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)


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)


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)


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)


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)


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)


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)


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)


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)


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)


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


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:


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


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.


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


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


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)


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)


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)



Assume: Clro = .40 VT = 650 fpsw = 2.5 W = 2403A teetering rotor system__

Then: W = 2397 lbs


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)


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)




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)


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)


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)



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


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)


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


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)


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)


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)



• 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)



• 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)


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)


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)


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)


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



(3) Landing Flare

(2) Steady Autorotational Descent

(1) Autorotation Entry

descent of Rate dVdt

dh

Time, t

Altitude,

h

Segmentation of Autorotation Maneuver






i

i a1

Tip Path PlaneHorizontal

Flight Path

Vertical to Flight Path

Tip Path Plane AxisControl Axis

Rotor Axis Geometry in Autorotation


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


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


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)


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)


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


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)


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


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.


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’)


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)


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)


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)


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)


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.


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)


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)


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.


2440

2420

2400

2380

2360

2340

2320

2300

600

625

650

675

700

Tip Speed

ft/sec

Gross Weight


Solution

Weight (lbs)


2.4 2.5 2.6 2.7 2.8

Figure 5

Articulated Rotor CLro=.4


2440

2420

2400

2380

600

625

650

675

700

Tip Speed

ft/sec

Gross Weight


Solution

Weight (lbs)


2.2 2.3 2.4 2.5 2.6

2460

2480

2500

Figure 6



2440

2420

2400

2380

2360

2340

2320

2300

600

625

650

675

700

Tip Speed

ft/sec

Gross Weight


Solution

Weight (lbs)


2.4 2.5 2.6 2.7 2.8

Figure 7



2450

2400

2350

2300

600

625

650675700

Tip Speed

ft/sec

Weight (lbs)


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


2450

2400

2350

2500

600

625

650

675

700

Tip Speed(ft/sec)

Weight (lbs)


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


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


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


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


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


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


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

Documents

Dr. Daniel P. Schrage Georgia Institute of Technology Atlanta, GA 30332-0150 Rotorcraft Design I Day 3: Parametric Design Analysis Dr. Daniel P. Schrage