statistics example

Helicopter Sizing by StatisticsHelicopter Sizing by StatisticsHelicopter Sizing by Statistics

Omri Rand Vladimir Khromov

Faculty of Aerospace Engineering

Technion – Israel Institute of Technology Haifa 32000, Israel.

Presented at the American Helicopter Society 58th Annual Forum,

Montreal, Canada, June 11-13, 2002.

Faculty of Aerospace Eng., Technion - I.I.T.

AHS Forum 58, June 2002

1



2

Introduction

• Sizing is the first and an important stage in helicopter preliminary design process.

• Preliminary design tools are relatively simple and were developed for fast design cycles.

• “Design trends” analysis is a well known technique in which flying configurations are analyzed in order to conclude or identify a trend which is common to many configurations, and therefore, it may represent physical constrains which are not clear and evident at the early stages.

• “Design trends” analysis is useful for the sizing stage in its broad sense: geometrical sizing and preliminary “sizing” of performance, power required, etc.

• The present study is based on a (partial) database for more than 180 conventional single rotor helicopter configurations. The analysis has been carried out using advanced computerized correlation technique which is based on Multiple Regression Analysis.



3

The Analysis MethodologyMultiple Regression Analysis (MRA):

• A computerized algorithm has been coded to generate and select hundreds of combinations of independent variables, in order to identify the groups that provide high correlation measure.

• The purpose was to find design trends that contain minimal number of independent unknowns(preferably one or two) that exhibit high correlation indicators.

• Error definition:

where EV and DBV stand for the estimated value and the database value, respectively. For each case we shall present the averaged and maximum error obtained, defined by

where N is the number of configurations involved, is the error calculated for the i-th configuration.

.Y a X X X= 1 2 3α β γ ...

ε (%) = | - |100 EV DBVDBV

ε AVER ε MAX

ε ε ε εAVER MAX

N i ii

N= = max(1 and

=∑

1)

ε i



4

The Analysis Methodology (cont.)

Multiple Regression Correlation Measure:

where are the data base values, is the averaged data base value, and are the estimated values.

Hence, R = 1 stands for a perfect correlation, while in most cases the minimal value of R was set to be above 0.9 in order to conclude that a correlation is of a value and represents a genuine trend.

The database used is the one stored in RAPID/RaTE (Rotorcraft Analysis for Preliminary Design / Rand Technologies & Engineering) - a desktop rotorcraft analysis package .

RAPID/RaTE is designed to model general rotorcraft configurations, conventional helicopters and tilt-rotors. RAPID/RaTE performs trim response, mission analysis, vibration analysis, stability analysis, and both flight mechanics and aeroelastic simulations.

Ry y

y y

i ie

i

N

ii

N2

2

12

1

1 = −−

−=

=

∑

∑

e j

e j

yi y yie

The Database



5

The Analysis Methodology SchemeRAPID/RaTE

Helicopter Configurations

DATABASE

Generation of helicopter parameters combinations Xi, i=1,2,…

Filtering of Database configurations for selected parameters groups

Calculation of the MRA parameters

and the multiple correlation measure Ra , , , , e t c .α β γ

Evaluating error estimation …...…. ε ε ,AVER MAXc h

Identification of parameter combinations with high correlation measures

Helicopter “Design trends”Helicopter

Configuration Model

Main rotor Blade number



6

Gross Weight

T-O Transmission rating, ± 8%

Vertical tail Arm, ± 4.2%

Fuselage Length, ± 6.2%

Empty Weight, ± 9.3%

Overall length, rotor turning, ± 1.7%

Long range speed, ± 6.5%

Disc load

Max speed, S/L

Fenestron diameter, ± 7.7%

Tail rotor diameter, ± 7.6%

Main Rotor Diameter, ± 6%

Never exceed speed, ± 5.9%

Height to rotor head, ±7%

Useful load, ± 9.5%

Total power T-O, ± 14.3%

Horizontal tail arm, ± 16%

Width over landing gears, ± 10%

Main rotor chord, ±10%

Main Rotor Solidity

Max cont. Transm. rating, ± 9%

Total power Max cont., ± 10.5%

Fuel value (liters), ± 11.5%

Range with standard fuel, S/L

Clearance Fuselage - Ground .3-.7 m

Tail Rotor Arm, ± 3.3%

Hover Tip Speed 170-250 m/sec

Hover Rotor RPM, ± 6%

Main Rotor RPM, ± 6.5%

Tail Rotor RPM, ± 6.6%

Tail Rotor chord, ±10%

Tail rotor Blade number

Tail Rotor Solidity

Vertical tail average chord, ± 12%

Horiz. tail area, ±29%

Main Scheme of Helicopter Sizing

F f DW = ( )



7

Helicopter Geometry Sizing Parameters

F f DLRT = ( )

F f DL = ( )

a f DMT = ( )a f DVT = ( )

F f DH = ( )

c f W NB= ( , )0

a f WHT = ( )0

D D f W VLOAD & ( , )max= 0

Ω= f D( )

σ = f N D cB( , , )

S f WHT = ( )0

c f W NTR BTR= ( , )0

D f WTR = ( )0

c f DVT TR= ( )ΩTR TRf D= ( )

σ TR TRf N D cBTR

TR= ( ), ,

7

30

6

21

0

10

20

30

40

Average error Max error

Erro

r (%

)

D WD W

D W VV

m kgR

km hrR

AVER MAX

AVER MAX

m

m

0

9 /

where is in [ ] and is in [ ]( = = = . )

is in [ ] S/ L

( = = = . ).

=

=

. ,

. ,

.

. .

/

980

133

00 308

0

00 380 0 515

7%, 30%, 9606

6%, 21%, 9744

ε ε

ε ε

D f W= ( )0

D f W Vm= ( , )0

The square cube scaling lawD W

" " :−∝ 0

1 3



8

0

10

20

30

40

0 10 20 30 40

D = f (W ) D = f (GW & V max Estimation

Main Rotor Diameter Estimation (m)

Mai

n R

otor

Dia

met

er (m

)

f (W0)f (W0, Vm )

Main Rotor Diameter

0

10

20

30

40

50

60

70

80

90

1000 10000 20000 30000 40000 50000 60000

Database configurations Estimation

RAH-66 Comanche

Mi-6 & Mi-22

Mi-26CH-53E

Dis

c Lo

adin

g (k

g/m

2)

0

10

20

30

40

50

60

70

80

0 2000 4000 6000 8000 10000 12000 14000 16000


Gross Weight (kg)

RAH-66 Comanche

ASI Ultrasport 254

Dis

c Lo

adin

g (k

g/m

2 )



9

Main Rotor Disc LoadingGross Weight

(kg)

Dis

c Lo

adin

g (k

g/m

2 )

Wing & Disc Loading Comparison

0

10

20

30

40

50

60

0 10 20 30 40 50 60

Helicopter Database configurations Disc Loading (rotory-wing)Wing Loading Upper boundary (fixed-wing)Wing Loading Lower boundary (fixed-wing)

(Gross Weight, lb) 1/3

Win

g/Dis

k Lo

adin

g (lb

/ f t

2 )

.334 (W1/3 - .74)1.54 (W1/3 - 6)

2.94 (W1/3 - 6) Fixed-wing



10

[McCormick, 1995]

[McCormick, 1995]

[Current study]

Wi n

g &

Dis

c Lo

a din

g (l b

/ f t

2 )

7373

586

7.5

2030

1

10

100

1000



11

Win

g &

Dis

k Lo

adin

g (k

g/m

2 )

Fixed-wing typical loading [Raymer, 1999]

Rotary-wing typical loading [Raymer, 1999]

Rotary-wing [Current study]

Sailplane

Jet transport/bomber

Civil/utility low speed helicopters

Some transport helicopters

Wing & Disc Loading

Main Rotor Blade Chord & Solidity



12

where is in [ ] and is in [ ]( = = = . ).

/ 0

c W N

c W

b

m kgRAVER MAX

= . ,. .0108 00540 714

010%, 41%, 9535ε ε

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Database configurations

Estimation

Main Rotor Blade Chord Estimation (m)

Mai

n R

otor

Bla

de C

hord

(m)

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 5 10 15 20 25 30 35

Main Rotor Database Main Rotor Estimation Tail Rotor Database Tail Rotor Estimation FENESTRON Database

Rotor Diameter (m)

Rot

or A

ngul

ar V

eloc

ity (R

PM)

Main & Tail Rotor Angular Velocity



13



14


/ D 0

ΩΩ

= 2673 829

6%, 35%, 9630

. .

RPM mR

DAVER MAXε ε

0

200

400

600

800

0 200 400 600 800


Main Rotor Angular Velocity Estimation (RPM) Mai

n R

otor

Ang

ular

Vel

ocity

(RPM

)

[ ] ( = = = . )

[rad / sec]

where is in [ ].

. /

364. /

0.

or 0.

Ω

Ω

RPM D

D

D

TR

TR

TR

AVER MAX R

m

=

=

3475 828

828

7%, 16%, 9737ε ε

Main & Tail Rotor Angular Velocity (cont.)

Main Rotor:

Tail Rotor:

Main & Tail Rotor Tip Speed



15

0

50

100

150

200

250

300

0 5 10 15 20 25 30 35

Main Rotor Database configurations Tail Rotor Database configurations Fenestron Database configurations Main Rotor Estimation Tail Rotor Estimation

Rotor Diameter (m)

Tip

Spe

ed (m

/s)

140. D 0 where is in [ / ]V VTIP TIP m= . , sec171

182. D 0 where is in [ / ]V VTIPTR

TIP m= . , sec172

Main Rotor:Tail Rotor:



16

Tail Rotor Diameter

USAAMRDL Report 1974:

RAPID/RaTE analysis:

D D DL lb ftTR/ . = ÷ −[ . . ] /7 0 7 3 27 2

D D DL lb ftTR/ . = −6 88 19 2. /

0

1

2

3

4

5

6

7

8

0 10000 20000 30000 40000 50000 60000


Estimation

FENESTRON Estimation FENESTRON configurations

Gross Weight (kg)

Tail

Rot

or D

iam

eter

(m)

where is in [ ] and is in [ ]( = = = . )

.0895 WW

.3081 W

0

0

DD

D

TR

TR m kgRAVER MAX

TRF

=

=

0391

0

0154

8%, 25%, 9754

.

.

ε ε

USAAMRDL Report 1977:

Tail Rotor Arm



17

RAPID RaTE analysisa D

a D

MT

MT mRAVER MAX

/

where and is in [ ]( = = = . ).

:. ,.= 5107 1061

3%, 14%, 9907ε ε

a D DMT TR ( + feet = +) / .2 5

0

5

10

15

20

25

0 5 10 15 20 25

Database configurations RAPID+ Estimation Ref. 13 Estimation

Tail Rotor Arm Estimation (m)

Tail

Rot

or A

rm (m

)

USAAMRDL Report 1977RAPID/RaTE Estimation

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5

Database configurations(without FENESTRON)

Tail Rotor Blade Chord Estimation (m) Tail

Rot

or B

lade

Cho

rd (m

)

Tail Rotor Blade Chord & Solidity



18


/ 0

c W N

c W

TR

TR

bTR

m kgRAVER MAX

= . ,. .0058 00506 720

010%, 30%, 9437ε ε

0

1

2

3

4

5

6

7

8

9

0 10000 20000 30000 40000 50000 60000


Gross Weight (kg)

Surf

ace

Are

a of

H

oriz

onta

l Tai

l (m

2 )

Mi-28

Mi-26

Horizontal Tail Surface Area



20

where is in [ ] and is in [ ] ( = = 214%, = .9117).

S W

S W

HT

HT m kgRAVER MAX

= . ,.0021 00 758

02

29%,ε ε

0

2

4

6

8

10

12

14

16

18

20

0 5 10 15 20 25 30 35


Estimation

Main Rotor Diameter (m)

Ver

tical

Tai

l Arm

(m)

Vertical Tail Surface Arm



21

where and are in [ ] ( = = = .9853).

0

a D

a D

VT

VT mRAVER MAX

= . ,.5914 995

4%, 20%,ε ε

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 1 2 3 4 5 6 7 8 9

Database configurations (Diameter < 3.5 m)Database configurations (Diameter > 3.5 m)Database configurations (FENESTRON)Estimation (Diameter < 3.5 m)Estimation (Diameter > 3.5 m)Estimation (FENESTRON)

Tail Rotor Diameter (m)

Ave

rage

Cho

rd o

f Ver

tical

Tai

l (m

)

Vertical Tail Average Chord



22

( = = = .9709)

[ ]

< .

> .

0

c D Fenestron

c D D m

c D D m

c D

VT

VT

VT

VT

TR

TR TR

TR TR

TR

AVER MAX R

where and are in m

=

=

=

.

.

.

.

.

.

909

161 35

297 35

927

1745

106

12%, 43%,ε ε

For Fenestron configurations c DVT TR≈

0

5

10

15

20

25

30

35

40

45

50

0 5 10 15 20 25 30 35 40


Estimation0

5

10

15

20

25

30

35

40

0 5 10 15 20 25 30 35 40


Fuse

lage

Len

gth

Configuration Length



23

1 ( = = = .9982)

F DLRT

AVER MAX R= . .09 103

2%, 9%,ε ε

( = = = .9807)

0

F DLAVER MAX R

= . .824 1 056

6%, 17%,ε ε [ ]. where and are in mF DL

[ ]. where and are in mF DLRT

Main Rotor Diameter (m) Ove

rall

Leng

th, R

otor

Tur

ning

(m

)

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40 Main Rotor Diameter (m)

Mi-6 & Mi-22

EH 101

0

1

2

3

4

5

6

7

8

9

0 5 10 15 20 25 30 35



Hei

ght t

o R

otor

Hea

d (m

)

Mi-26

Fuselage Height and Width



24

( = = = .8818)

0 0

F DWAVER MAX R

= . .436 697

10%, 40%,ε εwhere and are in mF DW [ ].

Wid

th O

ver L

andi

ng G

ears

/Ski

ds (m

)

( = = = .9371)

0 0

F DHAVER MAX R

= . .642 677

7%, 25%,ε ε [ ]. where and are in mF DH

Fuselage-Ground Clearance



25

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 10000 20000 30000 40000 50000 60000


Gross Weight (kg) Cle

aran

ce F

usel

age

- Gro

und

(m)



26

Empty Weight & Useful Load

0

1000

2000

3000

4000

5000

6000

7000

0 2000 4000 6000 8000 10000 12000 14000

Gross Weight (kg)

Use

ful L

oad

(kg)

0

5000

10000

15000

20000

25000

30000

350000 10000 20000 30000 40000 50000 60000

Helicopter Database configurations Rotor-wing Estimation Fixed-wing Estimation [McCormick, 1995]

Em

pty

Wei

ght (

kg)

( = = = .9932),

0 ,

1

[ ]

W

R where

E

E

W

W W

AVER MAX

and are in kg

= . .4854 0015

9%, 30%,

0

ε ε

( = =

= .9870),

0 ,

0

[ ]

W

R where

U

U

W

W W

AVER MAX

and are in kg

= . .4709 099

10%, 46%,

0

ε ε

Gross Weight (kg)

0

2000

4000

6000

8000

10000

12000

14000

0 2000 4000 6000 8000 10000 12000 14000

Useful Load Est. + Empty Weight Est.Empty Weight EstimationUseful Load Estimation

Gross Weight (kg)

Wei

ght E

stim

atio

n (k

g)



27

Empty Weight & Useful Load (continued)

+ . .0. .4709 4854099

01 015

0W W W≅

+ +

W W W W WPL F C E

WU

0 = +

Empty Weight & Useful Load (continued)



28

7480

70

60

4245

30

50

0

10

20

30

40

50

60

70

80

90

100

Em

pty

Wei

ght /

Gro

ss W

eigh

t, %

Upperbound

Lowerbound

Fixed-wing [McCormick,

1995]

Fixed-wing [Raymer,

1999]

Rotary-wing [Raymer,

1999]

Rotary-wing [current study]

* Group Weight Statement of the US military.

*



29

Fuel Value

( = = = .9942),

is

0 ,

0 0

= Range with standard fuel at sea level

where is fuel value in [ ], in [ ]and is range with standard fuel, S / L in [ ]

, .

W

R

km

F

F

W RgRg

W WRg

AVER MAX

liters kg

= . . .0038 0976 650

11%, 33%,

0

ε ε

0

500

1000

1500

2000

2500

3000

0 500 1000 1500 2000 2500 3000


Fuel Values Estimation (liters)

Fuel

Val

ue (l

iters

)

+ +

W W W W WPL F C E

WU

0 = +



30

Speed

( = = = .9399),

is

,

1

where is never exceed speed (S/ L) in [ / ], is long range speed (S / L) in [ / ], in [ / ]

,

VR

V V

NE

NE LR

V

V

M

M

AVER MAX

km hr km hrkm hr

= . .8215 0565

6%, 20%,ε ε ( = = = .9408),

,1

VR

LR VMAVER MAX

= . .5475 0899

6%, 31%,ε ε

0

50

100

150

200

250

300

350

400

0 50 100 150 200 250 300 350

Long Range Speed Database configurations Long Range Speed Estimation Never Exceed Speed Database configurations Never Exceed Speed Estimation

Max Speed; S/L (km/hr)

Spee

d; S

/L (k

m/h

r)

VM



31

Take-Off Total Power & Transmission Rating

0

2000

4000

6000

8000

10000

12000

0 5000 10000 15000 20000 25000 30000 35000

Gross Weight (kg)

0

5000

10000

15000

20000

250000 10000 20000 30000 40000 50000 60000


Tota

l Pow

er T

-O (k

W)

T-O

Tra

nsm

issi

on R

atin

g (k

W)

( = =

= .9891),

, 1

P

R where

TO WAVER MAX

= . .0764 01455

14%, 37%,ε ε

PW

TO is the take off total powerin and in kgkW

- [ ] [ ] is0

( = =

= .9943),

, 1

T

R where

TO WAVER MAX

= . .0366 02107

8%, 22%,ε ε

T

W

TO is the take offtransmission ratingin and in kgkW

-

[ ] [ ] is0

Gross Weight (kg)



32

Max Continuous Total Power & Transmission Rating

( = = = .9889),

[ ],[ ] [ ]

,

0 0

. , /

PR where

kW

MC

MC

W V

PW is V

M

M

AVER MAX

is the Max Cont total power inin kg is Max speed in km hr

= . . .0013 09876 9760

10%, 37%,

0

ε ε

0

3000

6000

9000

120000 3000 6000 9000 12000

Database Estimation

Max Continuous Total Power Estimation

Max

Con

tinuo

us T

otal

Po

wer

(kW

)

0

3000

6000

9000

12000

0 3000 6000 9000 12000

Max

Con

tinuo

us

Tran

smis

sion

rat

ing

(kW

)

EH 101

( = = = .9870),

[ ], [ ] [ ]

,

0 1

. , /

TR where

kW

MC

MC

W V

TW is V

M

M

AVER MAX

is the Max Cont transmission ratingin in kg is in km hr

= . . .000141 09771 3393

9%, 20%,

0

ε ε

Max cont. transmission rating estimation (kW)

EH 101

CH - 53E



33

Power & Transmission Loading

0123456789

10

0 10000 20000 30000 40000 50000 60000

Transmission Loading Database configurations

Transmission Loading Estimation

Power Loading Database configurations Power Loading Estimation

Gross Weight (kg)

Pow

er/T

rans

mis

sion

Loa

ding

(k

g/kW

)

MINI-500 BRAVO

Mi - 26

Power loading = the ratio of the take off gross weight over the maximum engine power.Transmission loading = the ratio of the take off gross weight over the take-off transmission rating.



34

Power & Transmission Loading

6.46.3

4.9

9

6

1.82.6

3.5

0

2

4

6

8

10

Fixed-wing typical power

loading [Raymer, 1999]

Rotary-wing typical power

loading [Raymer, 1999]

Rotary-wing power loading [Current study]

Prop

elle

r po

wer

ed a

ircr

afts

Rotary-wing transmission

loading [Current study]

Pow

er /

Tra

nsm

issi

on

Load

ing

(kg/

kW)



35

RAPID/RaTE Helicopter Sizing Software



36



36

Concluding Remarks

• A database for conventional helicopter configurations has been established and studied using advanced computerized correlation technique which is based on multiple regression analysis. Design trends were obtained and demonstrated. Currently, such data can not be found in the open literature.

• The study presented in this paper is expected to give designers a perspective of the existing flying designs and their inter-correlation. This is extremely important in the early preliminary stages where sizing issues are discussed in order to activate the preliminary design process.

• The collection of design trends presented in this paper contains also valuable information when comparison of performance of various configurations is under discussion.

• The present study results have been implemented as an autonomous component of RAPID/RaTE package.

statistics example

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