# Help with calculations using Simpson's rule

Discussion in 'Hydrodynamics and Aerodynamics' started by ADAM87, Oct 13, 2011.

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New to the site, but not new to boats or working with/on them. I have come a long ways, but I still have exponentially more to learn. I designed a small craft and modeled it in rhino. I can easily cut the hull at the desired waterline, find the volume and calculate the displacement. However I wanted to understand how to use Simpson's rule for this.

So I measured the area (in inches) of the half stations bellow the desired waterline, then solved the formula. The formula I used came form the Principles of Yacht Design. I understood the formula to read 'Volume = s/3 (sum of products)'. But, when I solved it, it yielded me the area under the curve (square inches) not the volume (cubic inches). Which puzzled me, I pulled up solidworks and sketched the curve based on measurements taken and verified that I actually did solve for the area. So I thought this formula was to find volume? What do I do next to find the volume? What did I miss?

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### Eric SponbergSenior Member

The ordinates of your area curve are in units of square area, square inches. The factor "s" is the station spacing, which should also be in inches. When you multiply the s x the sum of the area ordinates, you are getting cubic units, cubic inches.

Eric

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That is what I understood it to say when I read through the word part of the book. So I thought the formula would look more like V = s (sum of products), so why divide s by 3?

If I solve the formula without dividing s by 3, and treat the result as cubic inches, it works out that the boat should float at the desired waterline with about 323lbs. Which is about half the weight predicted by the 3d model, about 620lbs.

Maybe my calculations in finding the area of the stations are incorrect? I calculated the area of half the hull section below the waterline by selecting the lines surrounding the area and computing volume. This should be accurate, right?

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### DCockeySenior Member

The "area" under a curve of cross-sectional area vs length is the volume.

The "3" has nothing to do with calculating a volume vs an area. Rather it is inherent in Simpson's rule.

Are you multiplying each sectional area (ordinate value) by the S.M. factor (1,4,2,4,2,..,4,1) and then summing the products?

If you use the area of the half-sections then you need to multiply the result by 2 to get the total volume.

A check which finds some (but not all) mistakes. Use a constant value, say 100, for each section area. The resulting volume should then be the lenght multiplied by the constant section area. If you don't get that then there is a problems somewhere.

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### Eric SponbergSenior Member

That's right, the division by 3 is part of the definition of Simpson's Rule--it has to be there. This is calculus in its crudest form. And yes, if you are using the half areas below the waterline, then you have to multiply the whole by 2. So the equation becomes:

Vol = (2/3)s(sum of products)

Eric

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Ok, so I am on course with the way I calculate the formula and I ran through all numbers again to verify my calculations. I also reread the text carefully. I understand it to say that I must have an even number for my calculations, is this an even number of sections/stations, or an even number of spans so that I have an odd number of sections? I currently have an even number of sections... Possibly this is my error?

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### DCockeySenior Member

An even number of spans, odd number of stations:

Multipliers of the values at the sections are:
2 spans, 3 sections: 1 4 1
4 spans, 5 sections: 1 4 2 4 1
6 spans, 7 sections: 1 4 2 4 2 4 1
8 spans, 9 sections: 1 4 2 4 2 4 2 4 1
10 spans, 11 sections: 1 4 2 4 2 4 2 4 2 4 1
and so forth.

Not that "s" is the lenght of each section, not the overall length.

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### Eric SponbergSenior Member

David Cockey has this right. The stations must be of equal length, s, and the first and last multipliers must be 1. This follows Simpson's First Rule.

Just so that you know matters can get more complicated, there is also Simpson's Second Rule, the 6-ordinate rule, the Trapezoidal Rule, and the "5, 8, -1 Rule". And, to get really accurate at the ends of curves which may have a lot of curvature, such as the back end of a waterline or the ends of a sectional area curve, you can also use half stations and quarter stations (or even more finely divided stations) with the Simpson's Rules. You may want to do more reading on these rules. "Theoretical Naval Architecture" by Attwood and Pengelly, first written in 1899, is a particularly good source.

Eric

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Maybe this will be of some help-it's the Simpsons table from Skenes:

click on image:

#### Attached Files:

• ###### Simpsons Table from Skenes 001.jpg
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### DCockeySenior Member

If the curve isn't nice and smooth then the trapezoidal rule may be more accurate than Simpson's rule.

Many of what have been the "standard" proceedures in hull design are related to the use of Simpson's rule with 10 or 20 sections (11 or 21 stations).

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### PARYacht Designer/Builder

or the Durand's rule . . .

I also vaguely remember one called the prismatoid rule, but I think it needs parallel bases.

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Thanks for the replies guys, I indeed did have an even number of stations and odd spacing. I re-"divided" the hull for the current number of spans and stations, calculated areas again and worked the formula to be within 20lbs of the 3D model.

I do remember the trapezoidal rule from calc in college. Also, the name Simpson's rule sounded familiar but if I remember correctly it was applied in the form of a bar graph, then taking the center point of all bars, then 'creating' a curve and calculating the area underneath.

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### Wynand NRetired Steelboatbuilder

Email me and I will send you a simple Excel spreadsheet -10 and 20 station - that uses Simpson's rule to calculate most hydrostatics inclusive of displacement, LCB, VCG, Cf, immersion, moment to trim,waterplane area, etc which I wrote and used in the late 80's to mid 90's when I studied for my small boat design diploma.
At the time boat design software was scarce, moreso in my country and the internet non existent. I actually still uses this when doing a design on the drawing board, my preferred method still..

My email: wynand@in.com

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