# Can the contour surface of a ship be represented by a mathematical function?

Discussion in 'Stability' started by sun, May 31, 2022.

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### sunJunior Member

And，during the ship capsizing, is it possible to solve the function of the waterplane surface at any time?

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### jehardimanSenior Member

A ship? Yes.
A real ship? Not so much.

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### gonzoSenior Member

You could simplify a ship's shape for approximate calculations. However, it depends on the level of accuracy needed. Are you trying to comply with regulations or specifications?

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### sunJunior Member

The more accurate the better. Is there any method?

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### AlanXSenior Member

You will not find a simple formula. Best to use a boat design program (Carlson Design Hulls can do it, no doubt others can).

I have spent more time than I care to admit trying to writing code to do this.
Just to show I have been there, here is dinghy I did in Excel using macros:

Regards AlanX

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### philSweetSenior Member

A great deal of thought has gone into this by practical NAs. Rules and standards for stability have been produced from the results. Back when you had to hand-crank all the section properties and shell areas (for each heel angle and load case) some approximation methods were very handy in getting close, and revealing any hand-crank errors, and generally reducing the amount of squares-counting. If you express waterline offsets as a parametric polynomial in x and z, you can gather all the even power terms and have a fore-aft symmetrical shell. The collected odd powers represent the difference in symmetry fore-aft. You can now use calculus on these two function groups, which may be 90 percent of the volume, and only have to hand-crank the residuals from real sections. This greatly improves accuracy and speed.

I'm having trouble getting at this DTIC document at the moment, so here's a different one - one among dozens which look at the usefulness of different kinds of parameterizations.

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### jehardimanSenior Member

Just to restate your question so everyone understands "Is is possible to solve for a function to give the waterplane at any instant up to capsize?" Per my original answer this is possible for simple shapes of fixed weight; what would be returned is the location of the CG at the point of capsize or pitch pole. For a real ship, you could never have a simple function; it must be either measured from the plans or calculated from a panel model. Throw damage waterplane loss leading to capsize and you have too many conditions to ever hope for a discrete solution and the only method is piecewise calculation.

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### AlanXSenior Member

Hi @philSweet,

Your document is a bit of a find.

My current process is to:
• Import an offset table
• Interpolate (smoothed) chine lines for cross-section points
• Rotate the cross-section points
• Interpolate/slice the panel between sections.
• Collect the interpolated segments, calculate area and export DXF.
• Select the waterplane slice that has the closest required displacement.
Unfortunately my waterplane slices are not pretty as the panels are not curved like real stressed panels.
I am currently converting the waterplane segments into a polyline so I can smooth the waterplane slice.

Regards AlanX

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### jehardimanSenior Member

Just remember the Defense Technical Information Center (DTIC) is subdivided depending on the document and the IP domain address of the request, so a particular document may or may not be accessible. Most people worldwide can access "distribution unlimited"; the others, not so much (that W Arrangement again).

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### Alan CattelliotSenior Member

Hi Sun,

That's a question I still push myself, because it would be very handy to do so !!! As methods and technologie evolved, I find it very relevant to keep asking.

This is an old monohull ACC model, solely based on mathematical formulas. The user should make another file for the deck, the assembly of the two offset table forms a complete hull.

It can be used to do monohulls and multihulls. Here is the hull of the catamaran ORANGE2, which is now Vitalia 2.

I try as much as possible to find mathematical expression that could be used for "instantaneous stability calculations" for instance, but the truth is, as it was said earlier in the conversation, is that this approach tends to highly limit the design possibilities, since it is very very very difficult to handle complex mathematical expressions to perform your calculations to the end. For instance, the capsized stability would require a very detailed representation of the deck, which seems out of any pratical mathematical expression. I should say that, although we have pushed forward this approach in our design in the Gilles Ollier Design Team, this approach has always been very critized by a majority of architects.

It may be a solution that is pointing is nose to our affairs.

Last edited: Jun 2, 2022
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### AlanXSenior Member

A very simple hull can be defined as:
y=+/-0.5*Beam*(1-(z/Depth)^2)*(1-4*(x/Length)^2)​
where:
y is the half breath
z is the depth (positive!)
x is the length​
And:
Beam is the hull beam
Depth is the hull depth
Length is the hull length
The equation to a (water) plane:
y=(D-Ax+Cz)/B​
As "A" will be zero if the pitch is zero this simplifies to:
y=(D-Cz)/B​

So the intersection of the hull:
1. +0.5*Beam*(1-(z/Depth)^2)*(1-4*(x/Length)^2)=(D-Cz)/B
2. -0.5*Beam*(1-(z/Depth)^2)*(1-4*(x/Length)^2)=(D-Cz)/B
For "x" solve for "z", you will have to check for overflow (i.e. negative "z").

Now you could use each chine (y(x) and z(x)) equations in turn instead of a hull equation (i.e. y(x,z)).

You can then fit polynomials to these intersection points.
---
I should add, to solve the hull intersection equations above will require a numerical solver for even my simple hull model.

Regards AlanX

Last edited: Jun 2, 2022
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### HeimfriedSenior Member

I did a simple boat hydrostatics by numerical integration of the submerged part of the hull. Starting from the upright position with given cog vertically aligned with actual cob, you can disturb the equilibrium by putting a load in a certain place of the boat. To find the new equilibrium there has to be an iteration until the cob is newly aligned with the new cog. This means a new orientation of the hull (draft, heel, trim).
Try if you want.
Schwimmlagen eines Bootes https://www.bootsphysik.de/boot25m.php
(button "english" is top right)

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### rxcompositeSenior Member

Some 12 years ago, this was posted here in this forum. I am not sure if the site is still being maintained.

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### rxcompositeSenior Member

sun likes this.

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### sunJunior Member

For floating body, whatever its inclination angle or shape.

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