The document discusses numerical integration methods for calculating ship geometrical properties. It introduces trapezoidal rule, Simpson's 1st rule, and Simpson's 2nd rule for numerical integration. Simpson's 1st rule is recommended for calculating properties like waterplane area, sectional area, submerged volume, and centers of floatation and buoyancy which involve integrating curves related to the ship's shape. Detailed steps are provided for applying Simpson's 1st rule to calculate these properties numerically.

1. Adi Wirawan Husodo

2.  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

8. - uses 2 data points
- assume linear curve : y=ax+b
y4
y2 y3 A1=s/2 (y1+y2)
y1
A2=s/2 (y2+y3)
A1 A3=s/2 (y3+y4)
A2 A3
s s x3 s x4
x1 x2
Total Area = A1+A2+A3
= s/2 (y1+2y2+2y3+y4)

9. - uses 3 data points
- assume 2nd order polynomial curve y(x)=ax²+bx+c
Mathematical Integration Numerical Integration
y
y dx y(x)=ax²+bx+c y1 y2 y3
dA
A
A x
x x1 s x2 s x3
x1 x2 x3
x3 s
Area : A dA y dx ( y1 4 y2 y3 )
x1 3

10. y y6 y7 y8
y2 y5 y9
y1 y3 y4
s
x
x1 x2 x3 x4 x5 x6 x7 x8 x9
s s
A ( y1 4 y2 y3 ) ( y3 4 y4 y5 )
3 3 Odd number
s s
( y5 4 y6 y7 ) ( y7 4 y8 y9 )
3 3
s
( y1 4 y2 2 y3 4 y4 2 y5 4 y6 2 y7 4 y8 y9 )
3
s
Gen. Eqn. A (y1 4y2 2y3 ... 2yn 2 4yn 1 yn )
3

11. Simpson’s 2nd Rule (skip)
- uses 4 data points
- assume 3rd order polynomial curve
y y4
y2 y3
y1
y(x)=ax³+bx²+cx+d
A
x
s s x3 x4
x1 x2
3s
Area : A ( y1 3 y2 3 y3 y4 )
8

12. Application of Numerical Integration
• Application
- Waterplane Area
- Sectional Area
- Submerged Volume
- LCF
- VCB
- LCB
• Scheme
- Simpson’s 1st Rule

13. 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!

14. Waterplane Area
y
y(x)
x
FP dx
AP
Lpp
AWP 2 dA 2 y( x ) dx
0
area
AWP water plane area( ft2 )
Factor for Symmetric W.A. dA differential area( ft2 )
y ( x ) y offset(hal - breadth) at x( ft)
f
dx differential width( ft)

15. Waterplane Area(cont.)
• Generalized Simpson’s Equation
y
x
x
FP 0 1 2 3 4 5 6
AP
1
AWP 2 x y0 4 y1 2 y2 ..
3
x distance between stations

16. Sectional Area
• Sectional Area : Numerical integration of half-breadth
as a function of draft
z
T
WL Asec t 2 dA 2 y( z ) dz
0
y(z) area
T
dz Asec t sectional area up to z ( ft2 )
dA differential area( ft2 )
y y ( z ) y offset(hal - breadth) at z( ft)
f
dz differential width( ft)

17. Sectional Area(cont.)
• Generalized Simpson’s equation
z
WL
8
6
z
T 4 z distance between waterlines
2 y
0
T
Asect 2 dA 2 y ( z ) dz
0
area
1
2 z y0 4 y1 2 y2 .. 4 yn 1 yn
3

18. Submerged Volume : Longitudinal Integration
• Submerged Volume : Integration of sectional area over
the length of ship
• Scheme z
x
As (x )
y

19. Submerged Volume
• Sectional Area Curve
As
Asec t ( x )
dx x
FP AP
• Calculus equation Lpp
Vsubmerged s dV Asec t ( x )dx
volume 0
• Generalized equation
1
s x y0 4 y1 2 y2 .. 4 yn 1 yn
3
x distance between stations

20. 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
+ WL
-
+ FP

21.  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).

25.  Disebut juga dengan KB (Keel to Buoyancy)
 Merupakan titik berat dari volume
displacement kapal
total moment about the keel
 KB atau VCB =
vol. displaceme nt