[4] num integration
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Numerical IntegrationPengantar Teknologi Kelautan
Adi Wirawan Husodo
Why numerical integration?
◦ Ship is complex and its shape cannot usually be presented by mathematical equation.
◦ Numerical scheme, therefore, should be used to calculate the ship’s geometrical properties.
Which numerical method ?
◦ Trapezoidal rule◦ Simpson’s 1st rule◦ Simpson’s 2nd rule
Contoh-contoh
Contoh-contoh
Contoh-contoh
Trapezoidal rule (skip)
- uses 2 data points - assume linear curve
x1 x2 x3 x4s s s
y1 y2 y3
y4
A1 A2 A3
: y=ax+b
Total Area = A1+A2+A3 = s/2 (y1+2y2+2y3+y4)
A1=s/2 (y1+y2)A2=s/2 (y2+y3)A3=s/2 (y3+y4)
Simpson’s 1st Rule
- uses 3 data points - assume 2nd order polynomial curve
Area : )4(3
321
3
1yyy
sdxydAA
x
x
x1 x3
y(x)=ax²+bx+c
x
y
A
dx
x1 x2 x3s
y1 y2 y3
x
y
AdA
Mathematical Integration Numerical Integration
x2s
y(x)=ax²+bx+c
Simpson’s 1st Rule (cont)
x1 x2 x3
s
y1 y2 y3
x
y
x4 x5 x6 x7 x8 x9
y4y5
y6 y7y8 y9
Gen. Eqn.
Odd number
)y4y2y...2y4y(y3
sA n1n2n321
)4242424(3
)4(3
)4(3
)4(3
)4(3
987654321
987765
543321
yyyyyyyyys
yyys
yyys
yyys
yyys
A
- uses 4 data points - assume 3rd order polynomial curve
x1 x2 x3s s
y1 y2 y3 y(x)=ax³+bx²+cx+d
x
y
Area : )33(8
34321 yyyy
sA
A
x4
y4
Simpson’s 2nd Rule (skip)
Application of Numerical Integration
• Application
- Waterplane Area
- Sectional Area
- Submerged Volume
- LCF
- VCB
- LCB• Scheme
- Simpson’s 1st Rule
Numerical Calculation• Calculation Steps
1. Start with a picture of what you are about to integrate.
2. Show the differential element you are using.
3. Properly label your axis and drawing.
4. Write out the generalized calculus equation written in
the same symbols you used to label your picture .
5. Write out Simpson’s equation in generalized form.
6. Substitute each number into the generalized Simpson’s
equation.
7. Calculate final answer.
Not optional ! Always follow the above steps!
Waterplane Area
y
x
dxFPAP
y(x)
area
LppWP dxxydAA
0 )( 2 2
) width(aldifferenti
)(at breadth)-foffset(hal )(
)area( aldifferenti
)area( planewater 2
2
ftdx
ftxyxy
ftdA
ftAWP
Factor for Symmetric W.A.
Waterplane Area(cont.)
• Generalized Simpson’s Equation
..24y 3
1 2 210 yyxAWP
stations between distancex
y
x
FP AP0 1 2 3 4 5 6
x
Sectional Area
• Sectional Area : Numerical integration of half-breadth as a function of draft
WL
z
y
dz
y(z)T
area
Tt dzzydAA
0sec )( 2 2
) width(aldifferenti
)z(at breadth)-foffset(hal )(
)area( aldifferenti
)( toup area sectional2
2sec
ftdz
ftyzy
ftdA
ftzA t
Sectional Area(cont.)• Generalized Simpson’s equation
s waterlinebetween distancez
nn
area
T
t
yyyyz
dzzydAA
1210
0sec
4..24y 3
1 2
)( 2 2
z
y
WL
T
0
24
68
z
Submerged Volume : Longitudinal Integration
• Submerged Volume : Integration of sectional area over the length of ship
• Scheme z
x
y)(xAs
Submerged Volume
• Sectional Area Curve
• Calculus equation
volume
L
tssubmerged
pp
dxxAdVV0
sec )(
x
As
FP AP
dx
)(sec xA t
• Generalized equation
nns yyyyx 1210 4..24y 3
1
stations between distancex
Longitudinal Center of Floatation (LCF)
• LCF - Centroid of waterplane area - Distance from reference point to center of floatation - Referenced to amidships or FP - Sign convention of LCF
+
+-
FP
WL
Center of Flotation Merupakan titik berat dari luas bidang garis
air (water plane area). Suatu titik dimana kapal mengalami heel
atau trim. Titik ini terletak pada centre line (dalam
arah memanjang), disekitar midship (bisa di depan atau dibelakang midship).
contoh
Disebut juga dengan KB (Keel to Buoyancy)
Merupakan titik berat dari volume displacement kapal
KB atau VCB =
Vertical center of buoyancy (VCB)
ntdisplaceme vol.
keel about themoment total
Contoh KB