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
