Barrowman Equation

For Amraam Scale Level 3 Project

 

 

 

Definitions: a - Root Height

A = Area of Right Triangle Formula for Area of

a Right Triangle

b - Tip Height A = 1/2(a x s)

 

Cna - Force

 

d - Diameter

 

K - Interference Factor

 

l - Average Span

 

m - Top to Tip Height

 

r - Radius

 

s - Width

 

h - Hypotenuse

 

Center Fin:

 

s = 9.75 a = 5

 

 

Lower Fin:

 

 

a1 = 3.75 a2 = 9.25 a3 = 9.25

 

s1 = 9.75 s2 = 7 s3 = 2.75

 

 

ALf = (a1 x s1) + 1/2(a2 x s2) + (a3 x s3)

 

ALf = (3.75 x 9.75) + 1/2(9.25 x 7) + (9.25 x 2.75)

 

ALf = 36.5625 + .5(64.75) + 25.4375

 

ALf = 36.5625 + 32.375 + 25.4375

 

ALf = 94.375 - Total Area of Lower Fin

 


Center Fin Area:

 

AUf = 1/2(a x s)

 

AUf = 1/2(12 x 5)

 

AUf = .5(60)

 

AUf = 30 - Total Area of Center Fin

 

 

Lower Fin Tip Height Calculation for a Four Sided Fin Conversion:

 

Alfm = (1/2((a b) x s1)) + (b x s1)

 

94.375 = (.5(as1 - bs1)) + (b x s1)

 

94.375 = ((.5(13 x 9.75)) (b x 9.75)) + 9.75b

 

94.375 = (.5(126.75 9.75b)) + 9.75b

 

94.375 = (63.375 4.875b) + 9.75b

 

94.375 = 63.375 + 9.75b 4.875b

 

94.375 = 63.375 + 4.875b

 

94.375 63.375 = 4.875b

 

31 = 4.875b

 

4.875b = 31

 

b = 31/4.875

 

b=6.3589 - New Tip Height of Fin Conversion

 


 

Verify New Tip Height:

 

Alfm = (1/2((a b) x s1)) + (b x s1)

 

Alfm = (.5(6.6411 x 9.75) + (6.3589 x 9.75)

 

Alfm = (.5 x 64.7507) + 61.9992

 

Alfm = 32.375 + 61.9992

 

Alfm = 94.3745

 

Alfm <= Alf

 

94.3745 <= 94.375

 

Checked OK!

 

 

Nose Cone

 

Normal Force:

 

Cna = 2

 

Center of Pressure:

 

L = Length of Nosecone

 

Xn = .466L

 

Xn = .466 x 29

 

Xn = 13.514 inches

 


 

Center Fins

 

Definitions for l:

 

Sc = 5

 

bc = 0

 

ac = 1/2a - 1/2bm

 

ac = (.5 x 13) - (.5 x 0)

 

ac = 6.5 - 0

 

ac = 6.5

 

 

Calculating l :

 

l = (ac[2] + sc2)

 

l = (6.52 + 52)

 

l = (42.25 + 25)

 

l = (67.25)

 

I = 8.2006

 

 

Normal Force on Four Fins:

 

Cna = 4n (s / d)2

1 + 1 + [ 2l ]2

[a + b]

 

Cna = (4 x 4) (5 / 7.51)2

1 + 1 + [2 x 8.2006]2

[ 12 + 0 ]

 

Cna = 16 x (.6657)2

1 + 1 + [16.4012]2

[ 12 ]

 

Cna = 16 x .4432

1 + 1 + [1.3667]2

 

Cna = 7.0912

1 + 1 + 1.8680

 

Cna = 7.0912

1 + 2.8680

 

Cna = 7.0912

1 + 1.6935

 

Cna = 7.0912

2.6935

 

Cna = 2.6327 - Center Fin Normal Force

 

 

 

Radius of Body:

 

r = 1/2d

 

r = 1/2 x 7.51

 

r = 3.755

 

 

Interference Factor for Four Fins:

 

Kcf = 1 + r

(s + r)

 

Kcf = 1 + 3.755

(5 + 3.755)

 

Kcf = 1 + 3.755

8.755

 

Kcf = 1 + .4289

 

Kcf = 1.4289 - Center Fins Interference Factor

 

 

Normal Force on Four Fins in Presence of Body:

 

Cna cfb = Kcf x Cna

 

Cna cfb = 1.4289 x 2.6327

 

Cna cfb = 3.7618 - Center Fins Normal Force with Body

 


 

 

Center of Pressure for Center Fins:

 

Definitions:

 

Xcf = Distance from Nosecone Tip to Top of Center Fin

 

m = Top of Fin to Tip Height

 

a = Fin Height

 

b = Fin Tip Height

 

Xcf = 69 Inches

 

m = 12

 

a = 12

 

b = 0

 

Calculation for Center of Pressure:

 

Xcf = Xcf + Xcf

 

Xcf = Xcf + m(a +2b) + 1[a + b ab ]

3(a + b) 6[ a + b]

 

Xcf = 69 + 12(12 + (2 x 0)) + 1[12 + 0 (12 x 0)]

3(12 + 0) 6[ 12 + 0 ]

 

Xcf = 69 + 12(12 + 0) + 1[12 0 ]

3(12) 6[ 12 ]

 

Xcf = 69 + 12 x 12 + 1[12]

36 6[ 1 ]

 

Xcf = 69 + 144 + 2

36

 

Xcf = 69 + 4 + 2

 

Xcf = 69 + 6

 

Xcf = 75 inches - Center of Pressure for Lower Fins

 

 

 

 


 

Lower Fins

 

 

Definitions for l:

 

SL = s1

 

bL = 1/2b

 

aL = 1/2a 1/2bL

 

aL = (.5 x 13) (.5 x 6.3589)

 

aL = 6.5 3.1794

 

aL = 3.3205

 

 

Calculating l :

 

l = (aL2 + sL2)

 

l = (3.32052 + 9.752)

 

l = (11.0257 + 95.0625)

 

l = (106.0882)

 

I = 10.2999

 

 

Normal Force on Four Fins:

 

Cna = 4n (s / d)2

1 + 1 + [ 2l ]2

[a + b]

 

Cna = (4 x 4) (9.75 / 7.51)2

1 + 1 + [2 x 10.2999]2

[13 + 6.3589]

 

Cna = 16 x (1.2982)2

1 + 1 + [20.5998]2

[19.3589]

 

Cna = 16 x 1.6855

1 + 1 + [1.064]2


 

Cna = 26.9680

1 + 1 + 1.1320

 

Cna = 26.9680

1 + 2.1320

 

Cna = 26.9680

1 + 1.4601

 

Cna = 26.9680

2.4601

 

Cna = 10.9621 - Lower Fin Normal Force

 

 

 

Radius of Body:

 

r = 1/2d

 

r = 1/2 x 7.51

 

r = 3.755

 

 

Interference Factor for Four Fins:

 

KLf = 1 + r

(s + r)

 

KLf = 1 + 3.755

(9.75 + 3.755)

 

KLf = 1 + 3.755

13.505

 

KLf = 1 + .2780

 

KLf = 1.2780 - Lower Fins Interference Factor


 

 

Normal Force on Four Fins in Presence of Body:

 

Cna Lfb = KLf x Cna

 

Cna Lfb = 1.2780 x 10.9621

 

Cna Lfb = 1.2780 x 10.9621

 

Cna Lfb = 14.0095 - Lower Fins Normal Force with Body

 

 

 

Center of Pressure for Lower Fins:

 

Definitions:

 

XLf = Distance from Nosecone Tip to Top of Lower Fin

 

m = Top of Fin to Tip Height

 

a = Fin Height

 

b = Fin Tip Height

 

XLf = 130 Inches

 

m = 6.6411

 

a = 13

 

b = 6.3589

 

 

Calculation for Center of Pressure:

 

XLf = XLf + D XLf

 

XLf = XLf + m(a +2b) + 1[a + b ab ]

3(a + b) 6[ a + b]

 

XLf = 130 + 6.6411(13 + (2 x 6.3589)) + 1[13 + 6.3589 (13 x 6.3589)]

3(13 + 6.3589) 6[ 13 + 6.3589 ]

 

XLf = 130 + 6.6411(13 + (12.7178)) + 1[19.3589 (82.6657)]

3(19.3589) 6[ 19.3589 ]

 

XLf = 130 + 6.6411 x 25.7178 + 1[19.3589 4.2701]

58.0767 6[ 1 ]

 

XLf = 130 + 170.7945 + 15.0888

58.0767 6

 

XLf = 130 + 2.9408 + 2.5148

 

XLf = 130 + 2.9408 + 2.5148

 

XLf = 130 + 5.4556

 

XLf = 135.4556 inches - Center of Pressure for Lower Fins

 

 

Combination Calculations

 

 

Definitions:

 

Cnan = 2

 

Cnacfb = 2.6327

 

CnaLfb = 14.0095

 

Xn = 13.514

 

Xcf = 75

 

XLf = 135.4556

 

 

Calculations:

 

Cna = Cnan + Cnacfb + CnaLfb

 

Cna = 2 + 2.6327 + 14.0095

 

Cna = 18.6422

 


 

Center of Pressure for Rocket:

 

X = (Cnan x Xn) + (Cnacfb x Xcf) + (Cna Lfb x XLf)

Cna

 

X = (2 x 13.514) + (2.6327 x 75) + (14.0095 x 135.4556)

18.6422

 

X = 27.028 + 197.4525 + 1897.6652

18.6422

 

X = 27.028 + 197.4525 + 1897.6652

18.6422

 

X = 2122.1457

18.6422

 

X = 113.83558 Inches from Nose Tip

 

 

 

X = 113



[2]