Instant BM

[terhohalme]

Hello,

Found one day a very simple equation for metacentric radius in modern sail boat:

BM = 0.1 B^2 / T, where B is the waterline width and T is the draught of canoe body.

This formula looks to be right in a few percent.


How did I get there?

Displacement volume D = L * B * T * Cb

Second moment of the water plane I = L * B^3 / 12 * k

k is a constant and depens on waterplane shape, approximation k = Cw^2

Just using algebra:

Metacentric radius BM = I / D = B^2 / (12 T) * Cw^2 / Cb

(found later, this was statet by Mc Cloghrie)

Approximation: In modern sailboat hull Cw is about 0.7 and Cb about 0.4, so Cw^2 / Cb is about 1.2 . Put this in BM equation, we have after reduce:

BM = 0.1 B^2 / T

Example a boat with B = 3.0 m and T = 0.6 m have BM = 0.1 * 3.0^2 / 0.6 = 1.5 m

This very simplistic, but tells quite clear what geometric factors influence on instant stability.

Any comments? Seen this before?

[TeddyDiver]

But.. but.. then just anybody can calculate BM

[terhohalme]

Well, anybody can calculate his/her money balance:

Property = Earnigs - Expencies

Still, only some are rich.

[Guillermo]

I got another approximation as:
(See: What is (BM*T)/B**2?)

(BM*T)/B^2 = 0.08*Cwp^2/Cp

Asuming Cwp = Cp ^ 2/3 (Schneekluth)

we come to (BM*T)/B^2 = 0.08*Cp^1/3
and so BM = 0.08*Cp^1/3*(B^2/T)

For an efficient motoring hull Cp = 0,63

so roughly BM = 0.07 B^2/T

Another approximate formula is:

BM = (0.008 +0.0745 * Cwl^2) * B^2/(Cb*T),

As we can estimate also the vertical centre of buoyancy using:

Munro-Smith: KB = T*Cwl/(Cb+Cwl), or
Normand-Morrish: KB = T*[5/6 - Cb/(3*Cw)]

(The M-S formula usually gives a value about 0.01*T lower than the N-M)


we can, as KM = KB + BM, get KM and then performing a weights/centers calculation estimate KG and so GM.


Note: L = length, B = beam, T = draft, Cb = block coefficient, Cwl = waterplane area coefficient

Cheers.

[terhohalme]

Originally Posted by Guillermo

I got another approximation as:

(BM*T)/B^2 = 0.08*Cwp^2/Cp

Cheers.

This is true only if Cm = 1 then Cp = Cb*Cm = Cb

[Taildragger]

Hello,

Would anyone know Terho Halme's email contact details as the email quoted on his website keeps rebounding?

Cheers

[terhohalme]

My website is outdated history... just foating somewhere in cyperspace and no controllable anymore

Just pick my name and send me an email.

Terho

[Taildragger]

Hi Terho,

Tried to email you without success.
Could you email me on: janie.50@bigpond.com
I would just like to canvass your opinion on designs.

Cheers

Chris