Fin Surface Area Equation

[tap]

I posted this last night, but it appears to have vanished. It's never as good the second time around...

 

I generated a fin surface area equation based on depth and length measurements. I used a 2016 Radar fin since I had one laying around and it seemed like a good industry standard shape. The equation obviously only holds for this fin shape, but it should hold good for the trends for any similarly shaped fin. I digitized the fin and used CAD to make the length and depth adjustments. The fin holes are accounted for. The equation is based on a simple linear regression model, which fit the measured data extremely well.

 

Surface Area = (2.697 * Length) + (4.841 * Depth) - 18.924

 

The equation fits the measured data with an R-Squared value of 0.999.

 

Here's the data out of CAD:

 

Here's the same data in graphical form along with the predicted values from the equation. The measured values are the points, the predicted values are the lines.

 

 

 

[Adam Caldwell]

I find it interesting that the generically accepted 'surface area' is very similar yet the overall feel and 'performance' of ski is significantly different when you run a deep short setup versus long shallow. The mechanics that result in the skis characteristics are so different.

 

I am of the opinion that a deep/short/fwd fin configuration is a more complimentary setting for longer line lengths, slower speeds, and heavier weights....

 

I skied on a few 7.5" 2.5" 4 hole fins with no wings this summer/fall. They actually had very similar deceleration rates into apex as a generic fin setting with a 7/8 deg wing. There were some amazing things about the turns, but the lack of the wing made it feel far more unstable behind the boat.

 

Good work @TAP

[Horton]

@adamhcaldwell I have come to believe that short/deep/long/shallow is relative and different from ski to ski.

 

@tap Super cools stuff. Is there and XLS or Google sheets that you can share?

 

Wondering about things like what is the percent surface area difference between 2.500 x 6.850 and 2.510 x 6.850. It is clearly more than enough to feel but on paper does is seem insignificant.

[thager]

Didn't Paul Jager do some fin area work previously?

[tap]

@Horton Just use the equation, that's why I posted it.

 

2.500 x 6.850

SA = (2.697 * 6.850) + (4.841 * 2.500) - 18.924 = 11.653 in^2

 

2.510 x 6.850

SA = (2.697 * 6.850) + (4.841 * 2.510) - 18.924 = 11.701 in^2

 

Therefore 2.510 x 6.850 is 0.4% larger than 2.500 x 6.850.

 

You can also rearrange the equation to solve for a depth or a length given a desired SA and the other known variable.

[Adam Caldwell]

@Horton

 

Its not that black and white. The OVERALL setup is what matters. Long/shallow setups tend to work better with bindings 3/8 to 1/2" forward of where the short/deep settings would feel good. Most people jumping between those settings never move the DFT or Boots enough to get the optimal performance out of it. Unless your making those appropriate boot and DFT adjustments, then I can see why you make that argument. As skis have more tail rocker, the long shallow settings will require a more significant shift in DFT and boot position, but can still 'work' fairly well.

 

Deep/Short settings tend to have for a less stable YAW during all Roll and Pitch combinations/conditions. If the boots are too far forward, the ski will tend to be slow to accelerate and bite at the finish of the turns.

 

Conversely, with long/shallow numbers, if the boots are too far back then the ski will tend to be slow to accelerate and track a long straight path down course and not want to turn well. But, move the boots forward enough to create adequate tip pressure to overcome the YAW stability of a long/shallow fin, then it can be very fast and turn on time at the ball.

 

For example, at 30" on my 66" Denali - Deep/Short is no good at all. But long/shallow rocks. Alternatively at 29.5" on the same ski, Deep/Short rocks, but long shallow is not so hot.

 

Typically, for me at least, in addition to moving the boots when using LongShallow vs DeepShort, the DFT also needs to shift anywhere from 30-75/1000s. Generically, you would move the fin forward and move boots back when shifting into a deep/short setup, or, alternatively, you would pull the fin back and move boots forward when shifting toward a long/shallow setup.

[DW]

Nice work @tap . Can the data also provide the Cp (enter of pressure?), or perhaps simply the mechanical moment center. I think that data would provide some insight on what @adamhcaldwell is referring to. Distance from ski surface and longitudinal distance from boot center would be interesting data to track.

[tap]

Equations for finding the Centroid of your Fin... turns out these are pretty linear as well.

 

Centroid, the geometric center of a shape.

 

New Term Definitions:

Cx, Centroid Distance From Back of Fin (X-Axis).

Cy, Centroid Distance From Bottom of Ski (Y-Axis).

 

Cx can be used with your DFT measurement and Boot measurement to calculate the distance of the fin's centroid along the x-axis to either the tail of the ski or to your boot placement.

 

CxDFT, Centroid Distance From Tail of Ski (X-Axis): CxDFT = Cx + DFT

 

CxDFB, Centroid Distance From Boot (X-Axis): CxDFB = Boot - CxDFT

 

Cx = (0.4300 * Length) + (-0.0783 * Depth) + 0.2119

 

Cy = (-0.0358 * Length) + (-0.3823 * Depth) + 0.1993

 

Both equations are based on linear regressions with R-Squared values of 0.9991 and 0.9997 for Cx and Cy respectively. In other words, really good fits.

 

Point data from CAD:

 

 

 

 

 

This is pretty much all of the data the spreadsheets contain, but if anyone wants a copy of the Excel file I'd be happy to share. But don't expect me to make it all pretty.

[DW]

@tap : Of course I would like a copy... thanks in advance and again great work.

[thager]

And how do we apply any of this? I like math but appears to be a waste of my time!

[eleeski]

Remember to factor in the fin stiffness!

 

Eric

[tap]

@eleeski I honestly can't tell if you're being serious or sarcastic.

[eleeski]

Both. Sarcastic in that too much is made of exact fin settings. @adamhcaldwell noted that different other variables can offset fin settings. There are so many variables that a focus on just the fin settings loses track of finding what actually works best for you.

 

Serious in that flex does make a significant difference in how a ski performs. If that can be factored in with real data and results, we would have better knowledge of where to start with our fin settings.

 

@Horton 's carbon fins worked best for me when soft (he gave me a bunch to play with and I experimented). I'm now making a ridiculously soft fin for my ski - and it might be too soft. I'd really like some other data points than just me.

 

Eric

[Horton]

@Tap you can just ignore @eleeski. He is like a labrador chasing a squirrel. He means well but always off the path.

[Horton]

@tap please send me the xls. Horton@BallOfSpray.com

 

I think the Center of Pressure stuff is especially interesting. Adding length the traditional way (not the way @SkiJay advocates ) adds surface area, leading edge AND moves the Center Of Pressure forward. It is really 3 factors. If I recall @JaySki's method correctly he adds surface but keeps the Center of Pressure from moving (as much) by keeping the moving the fin back at the same time. (I am aware this is a bad explanation - buy the Jay's book for the instructions)

 

The years I spent years playing with model rockets and then high performance amature rockets taught me the importance of Center Of Pressure. I have long thought that there was a chapter in "Handbook of model rocketry" (G. Harry Stine) that should be required reading for anyone who wrenches a fin.

 

I have wondered if thinking about Center of Pressure (forward and back plus up and down) could be a way to think about fin adjustments. Using model or sounding rocket logic - center of mass is a good analogy for the skis center of mass.

[Horton]

@adamhcaldwell I agree and/or am intreaged by your above text. Makes sense.

 

My point was more general. What I am saying is that what if there was a neutral (0,0) setting between long shallow and short deep it would not be the same on any two ski designs.

[Adam Caldwell]

For me, the difference in feel of the softer flex fin is a function of how much energy is getting into the ski.

 

If the fin is super stiff, there will be less loss and all energy will go toward torsional load in the ski. If the fin is softer and can flex more, it will put less energy into the ski and potentially make things more 'forgiving'.

 

All Stainless fins I have used I have hated. The unload, or rebound rate of the ski at the finish of the turn is almost 'harsh'.

 

On the other end of the spectrum, whenever I play with overly soft fins I can feel them almost oscillating under load when pulling into the first wake.

 

At slower speeds and longer lines a softer fin being coupled with softer tail torsion could be helpful and give a more natural feel to the skier in the turns.

 

 

[Horton]

I generally think that my fins (CarbonFins) were a success in an the beginning of the light ski era because the factories did not have a handle on torsion yet. As ski layups got better the softer flex fins had less and less positive impact. I am proud of the success of the product but in the last 15 years skis have evolved and I do not think softer fins benefit most skiers on current ski designs.

[Adam Caldwell]

@Horton

 

Would you say there is any correlation between ZO and the 'less positive impact' of the carbon fin?

 

I feel like that may be as much of a factor as the ski, if not more.

[Horton]

@adamhcaldwell as you alluded to above when talking about stainless fins, super stiff fins are fast but have less margin of error. Softer fins are slower but are more forgiving. I tend to think the perfect flex is NOT super different from flex of the standard Aluminum we all use.

 

As fins get softer there are attributes that can be felt from the hookup to the wakes but what I found is that skiers generally like a fin that is stiff enough so the change in flex is almost unnoticed after the hookup.

 

I had the best luck when skiers felt the difference only from apex to hookup. Those fins flexed just a bit more than aluminium without creating enough drag to be noticable. Some skiers reported more speed which I always attributed to more angle and smoother exits from the ball.

 

So to answer your original question. I am sure there is an interaction with ZO vs PP but the fins that were most successful were close enough to aluminum. I do not think they were much of a factor during the time that the skier was under much load from the boat. I will admit that I have never thought about that before and it is an interesting question.

[eleeski]

@Horton Sometimes labradors chasing squirrels come across "interesting" things.

 

Why wouldn't a spreadsheet about fin data include fin stiffness? It is a real effect and should be quantified.

 

I took a carbon fin that was too stiff and modified it like the popular HO Pac-Man fin. That helped a lot - was it the shape or the flex? Cutting a Pac-Man in the front made it much more flexible and skied better. But the proper shape and flex worked best.

 

A standard flex measuring protocol needs to be agreed upon to get useful data. Might put Carbonfins back in business as people seek the ultimate tune.

 

Eric

[tap]

@thager For the vast majority it's mostly just a thought exercise to help get through the winter. But for those few who just can't help themselves from trying to figure out exactly how something works, like fin tuning, it's another tool in the box to play with. I've heard many a time someone bring up the idea of tuning based on the centroid of the fin but have yet to see anyone actually post the math to enable such a thing. Whether or not there's value in tuning based on the centriod, I'll leave that to guys like @adamhcaldwell .

 

But just for example... lets say for some reason you've got it in your head that you want to add .020" of length to your fin but don't want to take out surface area. Your current fin is set to 6.850/2.490. You can use the surface area equation to know exactly how much depth to take out to maintain the same surface area, in this case .011" making your new fin settings 6.870/2.479.

 

Use of the centroid equations follow the same pattern, however the reasoning behind such a desire gets pretty deep. But you asked 'how', not 'why'.

 

The only reason we tune by length/depth/dft is because they are easy to measure. If surface area and centroid location were just as easy to measure I suspect some of the mystery of fin tuning my disappear. It may even allow you to start comparing fin numbers from one fin shape to another.

[thager]

And I am sure I will remember this by next spring!

[Horton]

@eleeski we should also log that astrological sign of the person who made the fin.

[DW]

Cp distance to COM line through ski intersection is the lever one is turning, Horton's old Ski is a Lever article may be a good reference to revisit.

[Adam Caldwell]

Lot easier to wrap your mind around thinking about torque in all three axis for yaw, pitch and roll, instead of one singular moment from Cp to COM.

[DW]

True, but if it was easy everybody could / would do it. :D

[Horton]

@dw that old Lever article was everything I knew 15 years ago. I guess it is not bad for someone reading on the subject for the first time but there are some pretty big holes in it.