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Fin Surface Area Equation

taptap Posts: 72 Solid Baller
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.



Comments

  • adamhcaldwelladamhcaldwell Posts: 390 Open or 55K Rated Skier
    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
  • HortonHorton Posts: 25,182 Administrator
    @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.

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  • thagerthager Posts: 4,178 Mega Baller
    Didn't Paul Jager do some fin area work previously?
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  • taptap Posts: 72 Solid Baller
    @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.
  • DWDW Posts: 1,787 Crazy Baller
    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.
    adamhcaldwellHorton
  • taptap Posts: 72 Solid Baller
    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.
    GloersenDW
  • GloersenGloersen Posts: 766 Crazy Baller
    winter weather + NWB = Cool Math!

    thanks @tap
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  • DWDW Posts: 1,787 Crazy Baller
    @tap : Of course I would like a copy... thanks in advance and again great work.
  • thagerthager Posts: 4,178 Mega Baller
    And how do we apply any of this? I like math but appears to be a waste of my time!
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    Drago
  • eleeskieleeski Posts: 3,459 MM Trick Skier / Eccentric Person
    Remember to factor in the fin stiffness!

    Eric
    Horton
  • taptap Posts: 72 Solid Baller
    @eleeski I honestly can't tell if you're being serious or sarcastic.
  • eleeskieleeski Posts: 3,459 MM Trick Skier / Eccentric Person
    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
  • HortonHorton Posts: 25,182 Administrator
    @Tap you can just ignore @eleeski. He is like a labrador chasing a squirrel. He means well but always off the path.

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  • HortonHorton Posts: 25,182 Administrator
    edited January 9
    @tap please send me the xls. [email protected]

    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.

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  • HortonHorton Posts: 25,182 Administrator
    @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.

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    adamhcaldwell
  • adamhcaldwelladamhcaldwell Posts: 390 Open or 55K Rated Skier
    edited January 9
    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.

    eleeskiDW
  • HortonHorton Posts: 25,182 Administrator
    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.

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    Than_Boganadamhcaldwelleleeski
  • adamhcaldwelladamhcaldwell Posts: 390 Open or 55K Rated Skier
    edited January 9
    @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.
  • HortonHorton Posts: 25,182 Administrator
    edited January 9
    @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.

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  • eleeskieleeski Posts: 3,459 MM Trick Skier / Eccentric Person
    @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
    Horton
  • thagerthager Posts: 4,178 Mega Baller
    And I am sure I will remember this by next spring!
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  • HortonHorton Posts: 25,182 Administrator
    @eleeski we should also log that astrological sign of the person who made the fin.

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    adamhcaldwell
  • DWDW Posts: 1,787 Crazy Baller
    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.
  • adamhcaldwelladamhcaldwell Posts: 390 Open or 55K Rated Skier
    edited January 10
    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.
  • DWDW Posts: 1,787 Crazy Baller
    True, but if it was easy everybody could / would do it. :D
  • HortonHorton Posts: 25,182 Administrator
    edited January 11
    @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.

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