Helicopter Sizing by Statistics - Faculty of Aerospace Engineering

AHS Forum 58, June 2002 

Helicopter Helicopter Sizing Sizing by by Statistics 

Statistics 

Omri Rand Vladimir Khromov 

Faculty of Aerospace Engineering 

Technion – Israel Institute of Technology 

Haifa 32000, Israel. 

Presented at the 

American Helicopter Society 58 th Annual Forum, 

Montreal, Canada, June 11-13, 2002. 

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

1


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. 


2


Multiple Regression Analysis (MRA): 

The Analysis Methodology 

• 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: 

Y = a Xα 1 Xβ 2 Xγ 

3... 

ε (%) = 100 

| EV - DBV| 

DBV 

where EV and DBV stand for the estimated value and the database value, respectively. For each case we 

shall present the averaged ε and maximum error obtained, defined by 

AVER ε MAX 

1 

N 

∑ 

i= 

1 

AVER MAX 

ε = ε and ε = max( ε 

N i i 

ε i 

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

. 


) 

3


The Analysis Methodology (cont.) 

yi yi 

R 

y y 

e 

N 2 

∑e 

− j 

2 = 1− 

i= 

1 

N 2 

∑e 

i − j 

i= 

1 

yi y ye i 

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 

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. 


4


The Analysis Methodology Scheme 

RAPID/RaTE 

Helicopter 

Configurations 

DATABASE 

Helicopter 

Configuration 

Model 

Generation of helicopter parameters 

combinations {X i , i=1,2,…} 

Filtering of Database configurations for selected 

parameters groups 

Calculation of the MRA parameters 

a , α , β , γ ,etc. 

and the multiple correlation measure R 

AVER MAX c h 

Evaluating error estimation ε …...…. , ε 

Identification of parameter combinations with 

high correlation measures 

Helicopter “Design trends” 


5


Vertical tail average chord, ± 12% 

Tail Rotor RPM, ± 6.6% 

Hover Tip Speed 170-250 m/sec 

Main Rotor RPM, ± 6.5% 

Hover Rotor RPM, ± 6% 

Overall length, rotor turning, ± 1.7% 

Fuselage Length, ± 6.2% 

Tail Rotor Arm, ± 3.3% 

Height to rotor head, ±7% 

Vertical tail Arm, ± 4.2% 

Width over landing gears, ± 10% 

Clearance Fuselage - Ground .3-.7 m 

Main Scheme of Helicopter Sizing 

Fenestron diameter, ± 7.7% 

Tail rotor diameter, ± 7.6% 

Tail Rotor Solidity 

Tail rotor Blade number 

Tail Rotor chord, ±10% 

Main rotor chord, ±10% 

Main rotor Blade number 

Main Rotor Solidity 

Main Rotor 

Diameter, ± 6% 

Long range speed, ± 6.5% 

Horizontal tail arm, ± 16% 

Horiz. tail area, ±29% 

Gross 

Weight 

Disc load 

Max speed, S/L 

Never exceed speed, ± 5.9% 

Range with 

standard fuel, S/L 

Fuel value (liters), ± 11.5% 

Empty Weight, ± 9.3% 

Useful load, ± 9.5% 

Total power T-O, ± 14.3% 

T-O Transmission rating, ± 8% 

Total power Max cont., ± 10.5% 

Max cont. Transm. rating, ± 9% 


6

FH = f ( D) 

c = f ( W0, N B) 

σ = f N D c 

B 

( , , ) 

Ω= f( D) 


FL = f ( D) 

FLf D 

RT = ( ) 

aMT = f ( D) 

aVT = f ( D) 

D & D f ( W , V ) 

LOAD 

a = f ( W ) 

HT 

= 0 

0 

max 

Helicopter Geometry Sizing 

Parameters 

cVT = f ( DTR 

) 

FW = f ( D) 

D = f ( W ) 

S = f ( W ) 


TR 

HT 

TR 

c = f ( W , N ) 

TR 0 B 

= ( , , 

) 

TR 

σ TR f NB DTR cTR 

ΩTR = f ( DTR) 

0 

0 

7

Main Rotor Diameter (m) 

40 

30 

20 

10 

0 


Main Rotor Diameter 

The " square− cube" scaling law: 

D ∝ W 13 

0 

D = f (W 0) ) 

D = f (W (GW 0, Vm & V ) max 

Estimation 

0 10 20 30 40 

Main Rotor Diameter Estimation (m) 

Error (%) 

40 

30 

20 

10 

0 

7 

D = f ( W0) 

6 

30 

21 

Average error Max error 

D = 0. 

980 W 0. 308, 

0 

D W 

AVER ε MAX ε 

where is in [ m] and is in [ kg] 

( = 7%, = 30%, R = . 9606) 

D = 9 . 133 W0. 380 / V0. 

515 

m , 

0 

Vm 

is in [ km / hr] 

S/ L 

AVER MAX 

ε ε R 

( = 6%, = 21%, = . 9744). 

0 

D = f ( W0, Vm) Faculty of Aerospace Eng., Technion - I.I.T. 

8

Disc Loading (kg/m 2 ) 


100 

90 

80 

70 

60 

50 

40 

30 

20 

10 

0 


Main Rotor Disc Loading 

0 10000 20000 30000 40000 50000 60000 

RAH-66 Comanche 


80 

70 

60 

50 

40 

30 

20 

10 

0 

CH-53E 

Mi-6 & Mi-22 

Mi-26 

Database configurations 

RAH-66 

Estimation 

Comanche 

ASI Ultrasport 254 


Estimation 

0 2000 4000 6000 8000 10000 12000 14000 16000 

Gross Weight (kg) 

Gross Weight 

(kg) 


9

Wing & Disc Loading (lb / f t 2 ) 

Wing/Disk Loading (lb / f t 2 ) 

60 

50 

40 

30 

20 

10 


0 

Wing & Disc Loading Comparison 

2.94 (W 1/3 - 6) 

[McCormick, 1995] 

Fixed-wing 

Helicopter Database configurations 

Disc Loading (rotory-wing) 

Wing Loading Upper boundary (fixed-wing) 

Wing Loading Lower boundary (fixed-wing) 

1.54 (W 1/3 - 6) 

[McCormick, 1995] 

0 10 20 30 40 50 60 

(Gross Weight, lb) 1/3 

[Current study] 

.334 (W 1/3 - .74) 


10

Wing & Disk Loading (kg/m 2 ) 

1000 

100 

10 


1 

Jet transport/bomber 

586 

30 

Fixed-wing 

typical loading 

[Raymer, 1999] 

Wing & Disc Loading 

Sailplane 

Some transport helicopters 

73 

20 

Rotary-wing 

typical loading 


Civil/utility low 

speed helicopters 

73 

7.5 

Rotary-wing 



11

Main Rotor Blade Chord (m) 

1 

0.8 

0.6 

0.4 

0.2 

0 


Main Rotor Blade Chord & Solidity 


Estimation 

0 0.2 0.4 0.6 0.8 1 

Main Rotor Blade Chord Estimation (m) 

c = . W N , 

0108 0540 . / 0. 

714 

0 b 

c W 

ε ε 

where is in [ m] and 0 is in [ kg] 

( AVER = 10%, MAX = 41%, R = . 9535). 


12

Rotor Angular Velocity (RPM) 

5000 

4500 

4000 

3500 

3000 

2500 

2000 

1500 

1000 

500 

0 


Main & Tail Rotor Angular Velocity 

Main Rotor Database 

Main Rotor Estimation 

Tail Rotor Database 

Tail Rotor Estimation 

FENESTRON Database 

0 5 10 15 20 25 30 35 

Rotor Diameter (m) 


13


Main Rotor: 

Ω = 2673. 

Ω 

ε ε 

/ D 0 

. 829 

D 

where is in [ RPM] and is in [ m] 

( AVER = 6%, MAX = 35%, R = . 9630). 

Ω RPM = 3475. 

/ D 

ε ε 

[ ] 

0. 828 

TR 

( AVER = 7%, MAX = 16%, R = . 9737) 

or 

Main & Tail Rotor Angular Velocity (cont.) 

Tail Rotor: 

Ω 

D 

[rad / sec] 

where is in [ m]. 

TR 

= 

364. / 

D 

0. 828 

TR 

Main Rotor Angular Velocity (RPM) 

800 

600 

400 

200 

0 


Estimation 

0 200 400 600 800 

Main Rotor Angular Velocity Estimation (RPM) 


14

Tip Speed (m/s) 

300 

250 

200 

150 

100 

50 

0 


Main & Tail Rotor Tip Speed 

Main Rotor Database configurations 

Tail Rotor Database configurations 

Fenestron Database configurations 

Main Rotor Estimation 

Tail Rotor Estimation 

0 5 10 15 20 25 30 35 

Main Rotor: 

Tail Rotor: 

Rotor Diameter (m) 

V TIP = 140. D 0 . 171, 

V TIP m sec 

where is in [ / ] 

V TIP = 182. D . 172, 

V TIP m sec 

0 where is in [ / ] 

TR 


15

Tail Rotor Diameter (m) 

8 

7 

6 

5 

4 

3 

2 

1 

0 


USAAMRDL Report 1974: 

RAPID/RaTE analysis: 

Tail Rotor Diameter 


Estimation 

FENESTRON Estimation 

FENESTRON configurations 

0 10000 20000 30000 40000 50000 60000 


2 

D/ D = [. 70÷ 73 .] − . 27DL{ lb/ ft } 

TR 

2 

D/ D = 688 . − . 19DL{ lb/ ft } 

TR 

D .0895 W 0 

TR = 

. 

D 

W 

ε ε 

where TR is in [ m] and 0 is in [ kg] 

( AVER = 8%, MAX = 25%, R = . 9754) 

.3081 W 

0 

0 154 . 


D 

TR 

F 

= 

0 391 

16

Tail Rotor Arm (m) 

25 

20 

15 

10 

5 

0 


Tail Rotor Arm 


RAPID+ RAPID/RaTE Estimation 

Ref. USAAMRDL 13 Estimation Report 1977 

0 5 10 15 20 25 

Tail Rotor Arm Estimation (m) 

USAAMRDL Report 1977: 

RAPID / RaTE analysis: 

a = . 5107 D 1061 . , 

aMT D 

AVER MAX 

ε ε 


MT 

where and is in [ m] 

( = 3%, = 14%, R = . 9907). 

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

MT TR 

17

Tail Rotor Blade Chord (m) 

0.5 

0.4 

0.3 

0.2 

0.1 

0 


Tail Rotor Blade Chord & Solidity 


(without FENESTRON) 

0 0.1 0.2 0.3 0.4 0.5 

Tail Rotor Blade Chord Estimation (m) 

c = . W N , 

TR 

/ 0 

b TR 

0058 0506 . . 720 

0 

c W 

ε ε 

where TR is in [ m] and 0 is in [ kg] 

( AVER = 10%, MAX = 30%, R = . 9437). 


18

Surface Area of 

Horizontal Tail (m 2 ) 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 


Horizontal Tail Surface Area 

S = . 0021 W 0. 758 

0 , 

HT 

S W 

ε ε 

where HT is in [ m2] and 0 is in [ kg] 

( AVER = 29%, MAX = 214%, R = .9117). 

Mi-28 


Estimation 

0 10000 20000 30000 40000 50000 60000 


Mi-26 


20

Vertical Tail Arm (m) 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

0 


Vertical Tail Surface Arm 


Estimation 

0 5 10 15 20 25 30 35 


a D 

= . 5914 0. 

995, 

a D 

ε ε 

where VT and are in [ m] 

( AVER = 4%, MAX = 20%, R = .9853). 


VT 

21

Average Chord of Vertical Tail (m) 

3.0 

2.5 

2.0 

1.5 

1.0 

0.5 


Vertical Tail Average Chord 

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) 

0.0 

0 1 2 3 4 5 VT = 6. 297 106 . 

TR 7 8TR > 35 . 9 

For Fenestron configurations 

Tail Rotor Diameter (m) 

cVT ≈ 

DTR 

c D 0 

VT = 909 . 927 

TR Fenestron 

cVT = . 161 D 1745 . 

TR DTR < 35 . m 

ε ε 

AVER MAX ( = 12%, = 43%, R = .9709) 

c D D m 

c D 

where and are in [ m] 


. 

VT 

TR 

22

Fuselage Length 

40 

35 

30 

25 

20 

15 

10 

5 

0 


Configuration Length 


Estimation 

0 5 10 15 20 25 30 35 40 


FL 1 D 

RT = . . 

AVER MAX ε 2%, ε 9%, R 

F D 

09 103 

( = = = .9982) 

where and are in [ m]. 

L RT 

FL = 0 . D . 

ε ε 

F D 

824 1 056 

AVER MAX ( = 6%, = 17%, R = .9807) 


L 

0 5 10 15 20 25 30 35 40 



50 

45 

40 

35 

30 

25 

20 

15 

10 

5 

0 

Overall Length, Rotor Turning (m) 

23

Height to Rotor Head (m) 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 


Fuselage Height and Width 

FH D 

ε ε 

F D 

( AVER = 7%, MAX = 25%, R = .9371) 



Estimation 

0 5 10 15 20 25 30 35 

F 0 D 0 

W = . 436 . 

ε ε 

F D 

AVER MAX ( = 10%, = 40%, R = .8818) 


697 


W 

H 

= 0. 642 0. 

677 

EH 101 

Mi-26 

Mi-6 & Mi-22 

0 5 10 15 20 25 30 35 40 



8 

7 

6 

5 

4 

3 

2 

1 

0 

Width Over Landing Gears/Skids (m) 

24

Clearance Fuselage - Ground (m) 

0.8 

0.7 

0.6 

0.5 

0.4 

0.3 

0.2 

0.1 

0 


Fuselage-Ground Clearance 

0 10000 20000 30000 40000 50000 60000 




25

Empty Weight (kg) 

35000 

30000 

25000 

20000 

15000 

10000 

5000 

0 


Empty Weight & Useful Load 

0 10000 20000 30000 40000 50000 60000 

W 0 0 

U = . 4709 W . , 0 99 

ε ε 

( AVER = 10%, MAX = 46%, 

R = .9870), where 

and are in [ kg] 

W W 

U 

0 

Helicopter Database configurations 

Rotor-wing Estimation 

Fixed-wing Estimation [McCormick, 1995] 

WE 

= 0 . 4854 W 1. 

, 0 015 

ε ε 

( AVER = 9%, MAX = 30%, 

R = .9932), where 

and are in [ kg] 

W W 

0 2000 4000 6000 8000 10000 12000 14000 



E 


0 

7000 

6000 

5000 

4000 

3000 

2000 

1000 

0 

Useful Load (kg) 

26

Weight Estimation (kg) 

14000 

12000 

10000 

8000 

6000 

4000 

2000 


0 

Empty Weight & Useful Load (continued) 

0 2000 4000 6000 8000 10000 12000 14000 


W = W + W + W + W 

0 

Useful Load Est. + Empty Weight Est. 

Empty Weight Estimation 

Useful Load Estimation 

�� PL �� 

F �C 

E 

W 

U 


+ 

. 4709 W 0. . 4854 W1. 015 W 

0 99 

0 

≅ 

0 

27

Empty Weight / Gross Weight, % 

100 

90 

80 

70 

60 

50 

40 

30 

20 

10 

0 


Empty Weight & Useful Load (continued) 

Fixed-wing 

[McCormick, 

1995] 

60 

50 

Fixed-wing 

[Raymer, 

1999] 

70 

30 

Rotary-wing 

[Raymer, 

1999] 

80 

* Group Weight Statement of the US military. 


45 

* 

Rotary-wing 

[current 

study] 

74 

42 

Upper 

bound 

Lower 

bound 

28

Fuel Value (liters) 


Fuel Value 

W = W + W + W + W 

0 

3000 

2500 

2000 

1500 

1000 

500 

0 

�� PL �� 

F �C 

E 

W 

U 


Estimation 

0 500 1000 1500 2000 2500 3000 

Fuel Values Estimation (liters) 

W 

F 

Rg 

0 0038 W 0976 Rg 0650, 

0 

= Range with standard fuel at sea level 

( AVER = 11%, MAX = 33%, R = .9942), 

where WFis fuel value in [ liters], W0is 

in [ kg] 

, 

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

. 

Rg 

ε ε 

= . . . 


29


8215 V 1. 

0565 

M 

AVER MAX 

( = 6%, = 20%, R = .9399), 

, 

Speed 

where VNE is never exceed speed (S/ L) in [ km / hr], VLR 

is long range speed (S/ L) in [ km / hr], 

is in [ km / hr] 

, 

V 

M 

VNE 

= . 

ε ε 

Speed; S/L (km/hr) 

400 

350 

300 

250 

200 

150 

100 

50 

0 

= . 

ε ε 

Long Range Speed Database configurations 

Long Range Speed Estimation 

Never Exceed Speed Database configurations 

Never Exceed Speed Estimation 

V M 

5475 V 0899 

M 

VLR 

1. 

( AVER = 6%, MAX = 31%, R = .9408), 

0 50 100 150 200 250 300 350 

Max Speed; S/L (km/hr) 


, 

30

Total Power T-O (kW) 

25000 

20000 

15000 

10000 

5000 

0 


Take-Off Total Power & Transmission Rating 

0 10000 20000 30000 40000 50000 60000 


Estimation 

= . 0366 W 2107 

0 , 

T 

1. 

TO 

AVER MAX 

( ε = 8%, ε = 22%, 

R = .9943), where 

TTO 

is the take- off 

transmission rating 

in [ kW] 

and W is in [ kg] 

0 


P 

1. 

TO 

AVER MAX 

( ε = 14%, ε = 37%, 

R = .9891), where 

PTO 

is the take-off total power 

in [ kW] 

and W is in [ kg] 

0 5000 10000 15000 20000 25000 30000 35000 


= . 0764 W0 

, 1455 

12000 

10000 

8000 

6000 

4000 

2000 


0 

0 

T-O Transmission Rating (kW) 

31


Max Continuous Total Power & Transmission Rating 

Max Continuous Total 

Power (kW) 

Max Continuous Total Power Estimation 

0 3000 6000 

Database 

Estimation 

9000 12000 

P 

0 0 

MC = . 0013 W . 9876 V . 9760 

0 M , 

AVER MAX 

( ε = 10%, ε = 37%, R = .9889), where 

PMC 

is the Max Cont. total power in [ kW], 

in [ kg] , is Max speed in [ km/ hr] 

12000 

9000 

6000 

3000 

0 

TMC 

= . . . 

ε ε 

000141 W 09771 V 13393 

0 M 

( AVER = 9%, MAX = 20%, R = .9870), where 

TMC 

is the Max Cont. transmission rating 

in [ kW], 

in [ kg] , is in [ km / hr] 

Wis V 

0 

EH 101 

M 

CH - 53E 

, 

Wis V 

0 

EH 101 


M 

0 3000 6000 9000 12000 

12000 

9000 

6000 

3000 

0 

Max Continuous 

Transmission rating (kW) 

Max cont. transmission rating estimation (kW) 

32


Power & Transmission Loading 

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. 

Power/Transmission Loading 

(kg/kW) 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

MINI-500 BRAVO 

Transmission Loading Database configurations 

Transmission Loading Estimation 

Power Loading Database configurations 

Power Loading Estimation 

0 10000 20000 30000 40000 50000 60000 


Mi - 26 


33

Power / Transmission 

Loading (kg/kW) 

10 

8 

6 

4 

2 

0 


Propeller powered aircrafts 

9 

6 

Fixed-wing 

typical power 

loading 


Power & Transmission Loading 

4.9 

1.8 

Rotary-wing 

typical power 

loading 


6.3 

2.6 

Rotary-wing 

power loading 


6.4 

3.5 

Rotary-wing 

transmission 

loading 



34


RAPID/RaTE Helicopter Sizing Software 


35


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. 


36

Helicopter Sizing by Statistics - Faculty of Aerospace Engineering

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