simpson’s rules and it’s application in ship...

22 Mrs. Archana Ashish Bhange

International Journal of Computer & Mathematical Sciences

IJCMS

ISSN 2347 – 8527

Volume 6, Issue 2

February 2017

Simpson’s Rules and It’s Application in Ship Stability

Mrs. Archana Ashish Bhange,

B.Sc. Nautical Science Department,

Maharashtra Academy of Naval Education and Training (MANET)

Lonikalbhor, Pune

ABSTRACT:

Finding area of water plane is most important topic in ship stability. There are several ways by which this can be found,

one of them is Simpson’s Rules. Simpson’s Rules can be used to find areas of curved figures using numerical integration

technique. As the water plane is symmetrical about the center line, we need to calculate only half the area. For that

simpson’s rules are used. Simpson’s rules gives the better approximations to the areas. Simpson’s Rules are used to

calculate area of the space enclosed by a straight line and a curve. Aim of the paper is to derive simpson’s first rule,

second rule and third rule for different cases.

Keywords:

Water plane area, Simpson’s rules, Simpson’s multipliers, ordinates , intervals.

INTRODUCTION:

The process of evaluating a definite integral from the set of tabulated values of function is a numerical

Integration. There are basic Simpson’s rules ,Trapezoidal rule derived from Newton Cote’s formula. The aim

of the paper is to use numerical integration for deriving Simpson’s rules for ship stability.

Ship Stability is the area that deals with how a ship behaves at sea,both in still water and waves. Stability

calculations focuses on water plane area and center of gravity.

Simpson’s rules is used to find area of irregular figures .The rule is based on assumption that the

boundaries of such figures are curves which follow the definite mathematical law.The more the spacing the

more accurate answers .There three simpson’s rules depending on the number of ordinates.

Mathematical Model:

Simpson’s first rule:

Simpson’s first rule is used when number of ordinates are odd .Considering the parabola of third order with

equation

𝑦 = 𝑎0 + 𝑎1𝑥 + 𝑎2𝑥2 + 𝑎3𝑥

3,

Where 𝑎0 , 𝑎1 , 𝑎2𝑎𝑛𝑑 𝑎3 are constants.and𝑦1 , 𝑦2 and 𝑦3be three ordinates equally spaced at h units apart.



IJCMS

ISSN 2347 – 8527

Volume 6, Issue 2

February 2017

Fig 1.Simpson’s First Rule

The area of the elementary strip is ydx.Then the area enclosed by the curve is given by

Area Of the figure= 𝑦𝑑𝑥2ℎ

0

Area of the figure= (𝑎0 + 𝑎1𝑥 + 𝑎2𝑥2 + 𝑎3𝑥

3)𝑑𝑥2ℎ

0

= 2𝑎0ℎ + 2𝑎1ℎ2 +

8

3𝑎2ℎ

3 + 4𝑎3ℎ4…..(1)

Assume that the

area of the figure =𝐴𝑦1 + 𝐵𝑦2 + 𝐶𝑦3

Using the equation of the curve and substituting 0,h,2h respectively on the place of x.

Area of the figure

=𝐴𝑎0 + 𝐵 𝑎0 + 𝑎1ℎ + 𝑎2ℎ2 + 𝑎3ℎ

3 + 𝐶(𝑎0 + 2𝑎1ℎ + 4𝑎2ℎ2 + 8𝑎3ℎ

3)……(2)

Equating (1) and (2) we get

A+B+C=2h,

B+2C=2h,

B+4C=8

3ℎ

B+8C=4h

Which on solving gives

𝐴 =ℎ

3 , 𝐵 =

4ℎ

3 , 𝐶 =

ℎ

3

Therefore

Area of the figure=ℎ

3(𝑦1 + 4𝑦2 + 𝑦3)

This is Simpson’s First rule for ship stability.And the co-efficient of 𝑦1 , 𝑦2 and 𝑦3 are called Simpson’s

Multipliers.

This rule can be generalized according to number of ordinates as follows



IJCMS

ISSN 2347 – 8527

Volume 6, Issue 2

February 2017

1 4 1 if there are three ordinates,

1 4 2 4 1 if there are five ordinates,

1 4 2 4 2 4 1 if the ordinates are seven

1 4 2 4 2 4 2 4 1 for nine ordinates,

1 4 2 4…………………2 4 1 for any further odd number of ordinates.

Simpson’s Second Rule:

Simpson’s Second rule is used to find the area under the curve where number of ordinates are

(n-1) divisible by 3

Fig.2 Simpson’s second rule

Considering the parabola of third order with equation

𝑦 = 𝑎0 + 𝑎1𝑥 + 𝑎2𝑥2 + 𝑎3𝑥

3,

where𝑎0 , 𝑎1 ,𝑎2𝑎𝑛𝑑 𝑎3 are constants. and𝑦1 , 𝑦2 and 𝑦3be three ordinates equally spaced at h units apart.

The area of the elementary strip is ydx. Then the area enclosed by the curve is given by

Area Of the figure= 𝑦𝑑𝑥3ℎ

0

Area of the figure= (𝑎0 + 𝑎1𝑥 + 𝑎2𝑥2 + 𝑎3𝑥

3)𝑑𝑥3ℎ

0

= 3𝑎0ℎ +9

2𝑎1ℎ

2 + 9𝑎2ℎ3 +

81

4𝑎3ℎ

4…..(1)

Assume that the

Area of the figure =𝐴𝑦1 + 𝐵𝑦2 + 𝐶𝑦3 + 𝐷𝑦4

Using the equation of the curve and substituting 0, h, 2h , 3h respectively on the place of x.

Area of the figure

=𝐴𝑎0 + 𝐵 𝑎0 + 𝑎1𝑥ℎ + 𝑎2ℎ2 + 𝑎3ℎ

3 + 𝐶 𝑎0 + 2𝑎1ℎ + 4𝑎2ℎ2 + 8𝑎3ℎ

3 + 𝐷 𝑎0 + 3𝑎1ℎ + 9𝑎2ℎ2 +

27𝑎3ℎ3……(2)



IJCMS

ISSN 2347 – 8527

Volume 6, Issue 2

February 2017


A+B+C+D=3h,

B+2C+3D=9

2h,

B+4C+9D=9ℎ

B+8C+27D=81

4h


𝐴 =3ℎ

8 , 𝐵 =

9ℎ

8 , 𝐶 =

9ℎ

8,𝐷 =

3ℎ

8

Therefore

Area of the figure=3ℎ

8(𝑦1 + 3𝑦2 + 3𝑦3 + 𝑦4)

This is Simpson’s second rule for ship stability. Here Simpson’s multipliers can be generalized as follows

1 3 3 1 if there are four ordinates,

1 3 3 2 3 3 1 if there are seven ordinates,

1 3 3 2 3 3 2 3 3 1 if the ordinates are ten

Simpson’s Third Rule:

Simpson’s third rule is used to find the area between to consecutive ordinates when three consecutive

ordinates are known.

Consider the equation of the form

𝑦 = 𝑎0 + 𝑎1𝑥 + 𝑎2𝑥2

where𝑎0 , 𝑎1 𝑎𝑛𝑑 𝑎2 are constants. and𝑦1 ,𝑦2 ,𝑦3be three ordinates equally spaced at h units apart.

Fig 3.Simpson’s third rule



IJCMS

ISSN 2347 – 8527

Volume 6, Issue 2

February 2017

The area of the elementary strip is ydx. Then the area enclosed by the curve is given by

Area Of the figure= 𝑦𝑑𝑥ℎ

0

Area of the figure= (𝑎0 + 𝑎1𝑥 + 𝑎2𝑥2)𝑑𝑥

ℎ

0

= 𝑎0ℎ +1

2𝑎1ℎ

2 +1

3𝑎2ℎ

3…..(1)

Assume that the

Area of the figure =𝐴𝑦1 + 𝐵𝑦2 + 𝐶𝑦3

Using the equation of the curve and substituting 0, h, 2h respectively on the place of x.

Area of the figure

=𝐴𝑎0 + 𝐵 𝑎0 + 𝑎1ℎ + 𝑎2ℎ2 + 𝐶(𝑎0 + 2𝑎1ℎ + 4𝑎2ℎ

2)……(2)


A+B+C=h,

B+2C=ℎ

2,

B+4C=ℎ

3


𝐴 =5ℎ

12 , 𝐵 =

8ℎ

12 , 𝐶 = −

ℎ

12

Therefore

Area of the figure=ℎ

12(5𝑦1 + 8𝑦2 − 𝑦3)

This is Simpson’s Third rule for ship stability.Also known as five eight minus one rule.

Numerical Examples:

Ex.1The length of a ship's water-plane area is 70 m. The lengths of the equidistantly spaced half ordinates

commencing from forward are as follows: 0, 5.2, 6.4, 7.0, 6.0, 4.9, 0.3 Find the area of water plane.

Sol. h= 70 / 6 =11.7 m



IJCMS

ISSN 2347 – 8527

Volume 6, Issue 2

February 2017

Half-

ordinates

Simpson's

Multiplier

Area

Function

0 1 0

5.2 4 20.8

6.4 2 12.8

7.0 4 28.0

6.0 2 12.0

4.9 4 19.6

0.3 1 0.3

Total 93.23

Table.1

Area of the water-plane = 2× h /3 × Σ 1

=2× 11.7 m 3 ×93.23 m

=727.2 m 2

Ex.2Find the area of a water plane 12 m long .The half breadths at equal intervals from aft are: 7, 4.8

, 2.95 , 2 , 1.65 ,1.3 and 0 m

Sol .h=12/6=2m

By simpson’s second rule,

Half ordinates SM product

7 1 7

4.8 3 14.4

2.95 3 8.85

2 2 4

1.65 3 4.95

1.3 3 3.9

0 1 0

43.1

Table.2



IJCMS

ISSN 2347 – 8527

Volume 6, Issue 2

February 2017

Area=3×2

8× 2 × 43.1 = 64.65 𝑚2

Ex.3Find the area of a water plane 12 m long .The half breadths at equal intervals from aft are: 7, 4.8 , 2.95 , 2

, 1.65 ,1.3 and 0 m

Sol .h=12/6=2m

By simpson’s second rule,

Table.3

Area=3×2

8× 2 × 43.1 = 64.65 𝑚2

CONCLUSION:

In this paper three Simpson’s rules are derived . And further Simpson rules are used to find area of water

plane which is the essential thing for ship stability.

REFERENCES:

[1] Ship Stability for Masters and Mates (7th Edition) byC. B. Barrass , D.R. Derrett , Elsevier publication

[2] ShipStability I, II, III by Capt. H. Subramaniam , Vijaya Publications

Half ordinates SM product

7 1 7

4.8 3 14.4

2.95 3 8.85

2 2 4

1.65 3 4.95

1.3 3 3.9

0 1 0

43.1

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