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Aerospace Equation of the camber of a sail

  1. Feb 27, 2017 #1
    When sailing into the wind the lift creates a force on the sail that makes a cross section of the sail look like the Nike swoosh. For a given wind speed, sail size, and angle of attack, the equation of the slope should always look the same. I believe another assumption that needs to be made is the linear density of the sail, but I'm not sure.

    I looked at the differential equation for a telephone wire hanging between two telephone polls and tried to equate the relationship of the shape of a sail this way.

    One way I tried to do it was by showing the lift and drag vectors as two different forces acting at different angles. Another way was adding the Lift and Drag vectors and applying them at the combined angle.

    So far I haven't been able to derive a differential equation that works.

    What's bothering me is there is clearly a relationship which means there has to be an equation. Even if it's a messy one.

    I can provide more information if you need it, but any insight with the information that I've already given would probably be the most beneficial.

    Thanks in advance!
  2. jcsd
  3. Feb 27, 2017 #2


    Staff: Mentor


    Your sail is an airfoil. It is like the wing of an airplane, but a lot more complicated than a rigid wing. You can adjust the tension on all three sides, and the mast can bend in two axes.

    This wiki article can help you get started. To go deeper, use the links in the article's Reference section.

    Fair winds and following seas.
  4. Feb 27, 2017 #3
    Thanks for the tips. I know it's very complicated. I've been up and down those references. I am trying to over simplify it and say that the cross section was unaffected by the top and bottom and that the tension was applied at the beginning and end. And then start making adjustments from there. I was thinking like how a telephone wire hangs between two polls, but if the polls were different heights.
  5. Feb 27, 2017 #4


    Staff: Mentor

    That shape is called a catenary. I don't think it will help much in this case.

    There is no simple formula. it is very complex. Just look at this one figure. You won't find a simple formula for that.

  6. Feb 27, 2017 #5
    Considering the forces on a catenary all face the same direction and the forces on a sail rotate as they move about the sail should I scrap that idea? I thought I could use the combined force of drag and lift and equate that to the downward force of the catenary. It's quite possibly I have a conceptual misunderstanding.
  7. Feb 27, 2017 #6
    I've been familiarizing myself with how to work with catenary equations for this project.
  8. Feb 27, 2017 #7


    Staff: Mentor

    But the shape of an airfoil is nothing close to a catenary.
  9. Feb 27, 2017 #8
    The shape of a hanging uniform wire is called a catenary whose equation is well known. I do not think that this is related to a desirable shape for a sail since it is a symmetric curve and a sails profile is not. If you change the linear density of the wire you can change the shape and make it asymmetric but the curve will have a different equation although the technique for finding it would be the same as for the uniform wire. Sails are made usually of uniform density material and the panels are cut and sewn to produce a desirable shape.

    How much do you know about sail design and building?
  10. Feb 27, 2017 #9
    I know a little bit about the design. I'm starting off with a uniformly dense sail that has no batons, tell tales, or any of the like. I started to try and derive the equation from scratch, step by step like you would get a catenary, but I ran into some pretty irreducible and messy equations before I could create a ds/dx. Then I decided that maybe I could turn the combine force of lift and drag straight downward in the Y direction. This is all looking at it from the perspective of a cross section with an assumed camber, chord, angle of attack, and T on both ends.
  11. Feb 27, 2017 #10
    I'm picturing it as a sideways catenary, hanging in the direction of my lift and drag combined. Very ideal. Since a catenary assumes that the forces acting on it all face in the same direction, I think I might be at a fault. Even if I am, the shape should still be comprable. I plan on modeling it nonetheless (otherwise I'll get an F on my project) and saying that my conclusion does not support my hypothesis. Unless I find out that it matches some of the experimental data out there. This is a very backwards process, haha.
  12. Feb 27, 2017 #11


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    Gold Member

    Make an initial good guess at the deflected shape of the sail . Use simplified air flow analysis to get a pressure distribution over the sail . Calculate the deflected shape that this pressure distribution should cause . Compare with initial guess . Refine guess .
  13. Feb 27, 2017 #12
    I'm trying to do all of this without touching a sail (yet). What kind of computational program would I need to learn to be able to do this (I'm just a lowly, scum of the earth, undergrad)?
  14. Feb 27, 2017 #13


    Staff: Mentor

    It might help if you posted the problem statement of this project assignment.
  15. Feb 27, 2017 #14
    This is for a research project that I'm designing myself. So I can tweak it however I need to. By that I mean, I haven't written it down as a project statement, but my aim is to find the equation of the camber line. I do know the importance of writing down a specific goal and I have overlooked that part so I will fully articulate it as soon as I can, thanks!
  16. Feb 28, 2017 #15
    It has been pointed out by @anorlunda that a sail is a very complex object. let me elaborate as I think it is instructive. A sail is a deformable surface whose shape depends on the tension produced by the lines attached to the sail. The proper shape of the sail also depends on wind velocity and angle of incidence at the front of the sail which changes appreciable you rise above the deck. It depends on the interaction with other sails. It depends on the shape and design of the hull of the boat. I depends. on the expected sea conditions. It depends on the stretchiness of the sail material and how it is oriented on the sail. Try to find a copy of "Sail Power" by Wallace Ross for a more detail explanation of traditional cloth sail making .

    Most sail lofts today use some software to design sails. I think the simple programs are basically design on a semi empirical basis a combination of experience from centuries of building sails by hand and knowledge of the stresses within the sail that are required to best maintain the shape. More advanced programs can use wind tunnel simulations to produce a camber design appropriate for the boat and wind/sea conditions which is almost as good as you can get it.

    Sails are still made the way they have been for centuries i.e., with panels of woven cloth sewn/glued together but they are increasingly made from laminated sheets of plastic film as Mylar in which strings of very strong fibers as Kevlar or carbon are laid along stress lines. North Sails is the originator of this process and uses a solid male mold to actually form the sail shape. These sail do not stretch very much and maintain their original designed shape much longer.
  17. Feb 28, 2017 #16


    Staff: Mentor

    I asked because I see nothing you said that would eliminate a rigid flat triangle as a rough approximation of the sail. You have not stated any accuracy requirement, so there is no lower limit on how crude the approximation can be.

    Tsk tsk. You are mis-learning engineering. Always nail down the requirements before commencing the design. It is foolish to do otherwise.

    Experienced engineers told you there is no simple equation. You ignored them.

    You acknowledge that you're supposed to do requirements first, but you don't care. If you graduated today, I would not hire you.

    Sorry to be harsh, but you are wasting your time and our time on this thread.
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