# Linking metacentric height to drive force

Discussion in 'Stability' started by jakaranda, Apr 30, 2010.

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### jakarandaNew Member

Hello,

Does anybody know of an empirical formula linking the metacentric height (KMT)to drive force on a sailing yacht?

i.e. If I increase my kmt by 1%, how much do I increase my drive force for a given angle of heel assuming a modern sailplan?

Thanks

J

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### Tim BSenior Member

It doesn't quite work like that. Initial stability isn't really very meaningful on a yacht hull, unless it is very beamy or a catamaran. Most monohull yachts operate at high (20 to 30 degree) angles of heel. Here the keel makes a significant difference, as does the section shape near the deck.

You can legitimately relate driving force to side force for a certain apparent wind angle.

Performance prediction (which you are getting towards) is not totally straight-forward. Potentially, if you have a real (or estimated) GZ curve then you could link heel angle to sail side force, and infer an approximate driving force, which you could use to estimate a vessel speed.

Hope this helps.

Tim B.

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### TeddyDiverGollywobbler

That's only true with monohulls build before ~1960..

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### Tim BSenior Member

It's indicative, not accurate! My point was that initial stability calcs don't really apply in this region.

Tim B.

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### terhohalmeBEng Boat Technology

Up to 30 degrees...approximately:

Righting arm GZ = GM * sin(f) * cos(f)

Righting moment RM = m * g * GZ

Heeling moment HM = F * HCE

In equlibrium RM = HM, so

Sail Force F = m * g * GM * sin(f) * cos(f) / HCE, where

m = mass of sailboat
g = acceleration of gravity
GM = metacentric high between center of gravity and metacentre
f = heel angle
HCE = distance between sail centre and underwater lateral centre

...or not?

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### jehardimanSenior Member

Not....

Tim B is correct, GM is irrelevent for sail carrying equations. The above analysis (when GZ is correct) is only true when f == 0 (i.e. zero heeling force) AND the hull is wall sided, always a very poor assumption for most sailing vessels. Basicaly, the heeling moment must be balanced by the actual righting arm at f, not GZ=GM sin (f), the apparent righting arm.

Last edited: May 3, 2010
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### terhohalmeBEng Boat Technology

Well, jehardiman, could you then explain this.

The first picture is a quite normal modern sailboat hull. The other is her righting arm curve both calculated as above (purple) and calculated by hydrostatic software (blue). Can't see any significant difference up to 30 degrees.

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### jehardimanSenior Member

Your boat is a poor shape...?

While it is close, it is not the same, and it only works for that limited hull form; a CCA hull or a skimming dish wouldn't have a RM curve like that. Theirs would be much different in 10-30 degree range the due to form stability differences.

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### terhohalmeBEng Boat Technology

Never said the same, reed my first post ...approximately. Still some examples:

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### jehardimanSenior Member

No, I kenned what I read. FWIW, you still haven't shown any different hullforms, all those you show are effectvely equivalent. All I see that you did was manipulate the hardness of the turn of the bilge below the 30 degree heel waterline, which has no effect on Ixx or TCB other than trigometrically with heel. Change the flare, the flam, the depth, and the height of the turn of the bilge. To see the true effect do a scow, a cylinder, and a plank-on edge hull.

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### JoakimSenior Member

It is a good enough assumption that most modern monohull will have a constant RM (Nm/deg) at normal sailing angles in static conditions. However in dynamic conditions RM will be easily 10% or more lower due to wave pattern. Forgetting this will often be a bigger error.

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### terhohalmeBEng Boat Technology

Jeridiman,
You really think that cylinder or plank-on-edge sailboat hulls are common?

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### jehardimanSenior Member

Based on reading these forums the last few years, they more common than apparenty you expect.

It is important to remember that many of the people who come to this forum to ask questions are not in marine professions or avocations, and often do not know the question they need to ask. Tossing off a limited case rule of thumb use only in the preliminary design of a particular hull form without defining the boundaries is not only unprofessional, but disingenuous and potentially harmful to the op.

Tim B answered the op's question correctly, there is no single formula, empirical or otherwise, that correctly links KM to driving force. Even the rule of thumb you give does not link to KM, it uses GM, which was not part of the original question.

Without knowning more about the actual situation with generated the original question, it is improper not to give the absolutely correct answer. Once you have given that, then you can ask, as I usually do at work,"Why do you want to know that?" Which will lead to a discussion which reveals that the actual question that needs to be answered is different than the first one posed.

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