# Wind speed vs angle of heel

Discussion in 'Stability' started by rader, Dec 7, 2012.

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I have a righting arm vs heel angle curve of a 10m sailboat. Now, I want to find out for certain heel angles (say 2, 10, 20, 25, and 30 degrees), the wind speed that produced that angle of heel. How is this calculation done? Thanks.

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### TANSLSenior Member

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Heeling angle & wind speed

Thanks.
Is the cosine heeling angle raised to the power of 1.3?
I can estimate Center of Effort of sail plain but was is HLP?
Is As area of sail plan in square feet or square meters?
What is 'ev' in Mev mean?

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### TANSLSenior Member

You are right, "fi cosine raised to the power 1.3"
All values ​​are given in meters, kilograms and seconds
Mev, means (spanish): Momento escorante viento= Heeling moment due to wind
HLP is the vertical distance in meters from the center of gravity of the underwater area (area longitudinal, not cross section) to the waterline.
That's the heeling moment. To calculate the heeling arm, you have to divide by displacement. This will give you a heeling arm curve to be cut with the righting lever (GZ) at the equilibrium point.

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### Eric SponbergSenior Member

You can calculate what is known as the Dellenbaugh Angle, which is the angle of heel created under a wind wing pressure of 1 lb per square foot, which is about 16 miles per hour (statute miles). You can apply other wind pressures for different wind speeds (Pressure P = 0.004 x V^2) where V = wind speed in miles per hour. Plug those into the Dellenbaugh equation, and you will get different heel angles. Heel angle is dependent on the stability of the boat, so you need to know it's GM (metacentric height). If you don't know the GM, you can work it backwards from the righting arm curve.

You can read more about Dellenbaugh Angle and how to calculate it in my write-up that I did about 3 years ago on this forum, called The Design Ratios. It is posted below. See Chapter 9, page 32.

I hope that helps.

Eric

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In pursuit of the Dellenbaugh angle

Eric,
Thanks for the help and I have gone to your article. This all began when I tried to discern stability characteristics of my Tartan Ten from data on the ORR certificate and the graph of RM vs Heel Angle provided by US Sailing. I wanted to get a sense of wind strength causing somewhat larger heeling angles in the range of 15-30 deg, by which time we should have reefed. I did an eyeball plot of GM from the curve, projecting a line from one radian upward to intersect the slope (the intersection was beyone the limits of the original graph) and came up with an estimate of GM of 1.06 feet. The Dellenbaugh equation also calls for the value of the heeling arm but I do not see this on the ORR certificate and do not have a clue about this from any of the specs of the S&S Tartan Ten (although the certificate does have basic sail dimensions and foretriangle dimensions for this fractional rig). So, at this point, I am unable to calculate a DA. You state that the DA is quite accurate at small angles of heel. My projected line, which is tangent to the curve taking off from zero, seems to depart from the curve at around 12 deg (similar to Figure 15 in your article). But, I am also interested in predicting the wind speed that would cause whatever angles of heel in the range of 15-30 deg.

I have also looked at an article by Roger Long re Heeling Arm Curves (www.cruisingonstrider.us/HeelingArmCurves.htm), but seem to come up with impossibly small wind heeling arms (if I have done the calculation correctly).

The reply from Spain re the ISO 12217-2 calculation of Wind Heeling Moment also calls for hCE and hLP for the boat, which I don't have. The equation, translated from European notation is: M = 0.75*V^2*A*(hCE + hLP)*Cos theta^1.3, where V, wind speed, is in m/s and A (area of the sail plan is probably in m^2) and M (heeling moment) is also probably in m. But, again I lack the numbers for hCE and hLP.

So, I am still trying to relate a range of wind speeds to a range of heel angles.

Thanks.

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### Eric SponbergSenior Member

The heeling arm is the vertical distance between the geometrical center of the sail area and the geometrical center of the lateral plane of the underwater profile of the hull. If you have a full outboard profile/sail plan of the boat (can download one off the internet) you can calculate where these center are. If you find difficulty in calculating that, you can approximate it by estimating that the center of the sail area is at about 40% of the height of the mast above the waterline (to my eye, anyway), and estimating that the center of the lateral plane is at about half draft (again, to my eye). Add these two distances together for the heeling arm. Keep this constant for all your calculations.

In the Dellenbaugh Equation on page 33 of The Design Ratios, the number 1.0 at the right end of the equation is the wind pressure number. That is the number that will change for different wind pressures: less than 1.0 for low wind speeds, greater than 1.0 for higher wind speeds.

Eric

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### LPFlying Boatman

So, as heel angle increases, does it make sense to take the cosine value of the heel angle and use it as a modifier to both the sail area and the heeling arm?

i.e. At 60 deg. of heel, both the sail area and heeling arm are both effectively 1/2 of their non-heeled values.

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### Eric SponbergSenior Member

LP, yes, you can do that. Dellenbaugh angle is meant for small angles of heel, say below 10-15° where the slope of the righting arm curve is very nearly constant with heel angle. As I say in The Design Ratios: "Dellenbaugh Angle is quite accurate at small angles of heel, where the greeen slope line and the black GZ line come very close together. DA is less accurate at higher angles of heel where these lines start to separate."

A number of things are going on, however, as you go past 15° heel. Yes, the sail area and heeling arm are reduced by the cosine of the angle of heel. Also, the angle of attack and direction of wind flow over the sails is highly skewed, so you are not generating as much lift as if the boat was more upright and the air flow squarer to the sail leading edge. In addition, the righting moment of the boat (it's righting energy) starts to diminish--does not increase as fast--and by about 60° heel, is at its maximum. And then of course, drag due to heeling effects increases and the boat wants to slow down. So to predict performance at high angles of heel is really hard to do. And generally, people don't really like to sail at more than about 15-20° heel anyway, it's too uncomfortable. And if you are heeling that much, it really is time to pull in a reef.

And that brings up the next question: If you start to reef at about 15° heel, then the Dellenbaugh Angle comes back into play because the boat is standing more upright again. You tend to reef in discreet amounts, reef #1, reef #2, etc. and the boat can withstand higher wind pressures without heeling as much--sail area and heeling arm are reduced by reefing, not by heeling. So by this method, you can develop a reefing schedule for the boat.

Eric

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Wind heeling moment

Following this thread, Eric states that angle of heel increases as the cosine of the angle. However, the SI equation raises cosine of the angle by the power, 1.3. I just want to get a handle on this.

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### Eric SponbergSenior Member

I was not a participant in the creation of the standard, so I don't know specifically how this equation was determined, but I'll have a guess:

It's not that the angle of heel increases as the cosine of the angle. Rather, the equation states that as the boat heels, the heeling moment reduces in proportion to the cosine of the heel angle. In reality, the cosine of the heel angle affects both the sail area AND the heeling arm, so one is tempted to apply the cosine of the heel angle twice, which in this equation would be cosine^2. Cosine for any angle greater than zero degrees is always less than 1.0, and when you take a number less then 1.0 to any power, it makes that number even smaller. For example, cosine(15°) = 0.966. cosine(15°)^2 = 0.933. Cosine(15°)^1.3 = 0.956. So, using cosine to the power 1.3 reduces the heeling moment even further than just plain cosine alone, but not as much as cosine^2. But for some reason, whoever wrote the rule decided that cosine^2, as called for by proper math and physics in the ideal case, was too much of a reduction, and picked cosine^1.3 instead. This decision is probably based on some empirical data or other design experience which makes the equation more closely match what really happens, taking into account any number of unknown additional physical effects due to heel. So, my guess is that the choice of power is a bit arbitrary based on some kind of prior experience.

Eric

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Dellenbaugh

I am getting an implausibly high DA, using metacentric height of 1 foot:

DA = 57.3*486*21.5/(1*7005) = 85 ???

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Rudder efficiency

Analogous to the effect of heeling on sails, this is probably applicable to the rudder. Heeling to 45 deg (as might occur during a windward broach) would reduce the efficiency of the rudder by around 30% (cos 45).

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### Eric SponbergSenior Member

Rader, I would say that the only "iffy" number that you might have is the GM, which from your above discussion was an estimate "off the chart" so to speak. GM = 1.06' is actually a very small number, and I would bet that the actual GM is quite a bit more than that. Just checking my own designs that are nearest yours between 35' and 40', all have GMs well in excess of 3'. Even at 3', your DA would be very high, so I am guessing that it is somewhat higher, maybe between 4' and 5'. See if you can double-check that and come up with a corrected value for GM.

Eric

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