The math behind lengthwise flex

[NateW]

It's been a while since we had a really good math-geek thread, so I'm here to fix that.

I've been toying with the idea of writing a simple piece of software that would compute the amount of force along the edge of a snowboard, based on the stiffness of the board, the sidecut radius and the edge angle. The problem is, I'm not sure about the math, so that's where I need your help.

My probably-too-simple plan goes like this...

Inputs:

• edge running length
  
• waist width (assume no taper)
  
• sidecut radius (assume a circular sidecut, nothing fancy (at least at first))
  
• stance width (assume no setback)
  
• camber (assume a circular camber shape, centered, nothing fancy (yet))
  
• core thickness profile (actually a few of these, as I want to experiment)
  
• rider's weight (assume Earth gravity for now :) )
  

Output:

A curve showing the amount of force exerted on the snow at centimeter-spaced points along the edge of the board, from tip to tail.

How?

Create a function that will return the core thickness for any point along the length of the board. Actually I'll need a few such functions since the whole point is the experiment with core thickness profiles.

Create a function that will return the core width for any point along the length of the board, based on the waist width and sidecut radius (for now, assume no taper).

Find the stiffness of the board at any point along the length, based on the thickness and width (and a constant, which I will initially guesstimate but might eventually compute if I can find some good data about fiberglass and wood).

Create a function that will return the amount of flex required at any given point, based on the board specs and edge angle. Define "flex" over a 2-centimeter section, so for example a section might have 0.001mm of camber at rest, and need to bend -0.002mm under load, for 0.003mm of flex.

Now, starting at the position of the front binding, figure out how stiff the board is at that point, how much the board has to flex at that point, and from that info, figure out how much force the board will exert on the snow at the points in-front-of and behind the binding. Then repeat that process, working backward to the center of the board and working forward to the tip of the board one centimeter at a time. (assume a centered stance for now)

At best, that might tell me how much force it takes to decamber a board in a carve. I'm not even sure if that information is useful, but I gotta start somewhere.

Given that information, and the rider's weight, can I figure the load at each point on the edge? I'm not sure how.

Any help would be most appreciated!

[NateW]

Basically I want to create something like this, but specialized for snowboards:

http://www.geocities.com/richgetze/

And the more I read about this stuff, the more I feel like my proposal above is hopelessly oversimplified and the math is going to go over my head. :)

[Gleb]

that sounds pretty ambitous. The math is waaaay over my head but i think its a great idea.

what about an input for different binding stiffness?

[Justin A.]

I don't think that an input for binding stiffness would be nessicary. I think that where nate wants to go with this is to show the flex of the board itself. Perhaps addons to the program to map Boots and Binding would be fun to play with in the future once he gets the bugs worked out of the board program, but Im pretty sure that boots have MUCH more complex flexes in them than boards, and that bindings are probally on par in complexity with boots.

Nate: try contacting a Major Archery (RECURVE) Manufacturer, they use all kinds of advanced math and physics to construct better performing limbs, which only need to flex along one axis, like the board in your program. You will also need to incorporate torsional stiffness into your equation if you are to calculate the force on the edge of the board, but adding this in shouldnt be too difficult once you get the equation for the lengthwise flex of the board. I have some ideas in my head about the math, but its been 2 years since my last physics class...I'll need a few days if you want any help from me. Your best bet is to contact either Hoyt USA or Win&Win's R&D department. Good luck and let us know how it works out!

Edit: I think that you may find that the math will infact be too complex to handle for any rational amount of work. I know that aircraft specs CAN be computed on paper, but it takes far too much work for any manufacturer to spend the time and money to do, thus test flights to gather performance data. It may end up being the same with this edge loading thing.

After some short thinking, you will also need to add acceleration into your equation. F=MA, so the force of the snow holding the edge (wierd to think about, eh?) will need to be greater than the Mass of the Board system x Their rate of turning and change in speed. The change in velocity is easy enough to calculate, the mass of the rider is a constant, sooooo...

(you're pretty much calculating g-forces here)

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[fishrising]

...try contacting a Major Archery (RECURVE) Manufacturer, they use all kinds of advanced math and physics to construct better performing limbs, which only need to flex along one axis, like the board in your program...Hoyt USA or Win&Win's R&D department...

Good luck trying to get Hoyt USA on the phone to answer a question like that. Try Black Widow (http://www.blackwidowbows.com/), their bows are much better anway...

[Derf]

Justin A. & fishring

Sorry to threadjack, but are you guys in archery? I started archery last year (recurve) after taking a course a couple of years back. I have some basic equipment, but I like it a lot.

[Justin A.]

Good luck trying to get Hoyt USA on the phone to answer a question like that. Try Black Widow (http://www.blackwidowbows.com/), their bows are much better anway...

Or try Bear, or Matthews, or Martin, or PSE...wait...no, not PSE their bows suck ;)

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[jtslalom]

I think your problem lies in the Physics and Engineering part not necessarily the math part. I would contact a board manufacturer and ask for an engineering analysis/profile on board flex. I would think most companies have a pretty good idea of how exactly their boards flex and what kind of forces are involved in decambering the board. If not you may have to set up your own make shift lab and test different boards with bindings and boots. This would more likely fall under the categorie of an engineer or mechanical physicist. The math involved is not hard at all (depending on how theoretical you want). You are only dealing with at best torsion, momentum, and newtons first and second laws. None of the math is hard but the physics is.

As far as this function:

Create a function that will return the amount of flex required at any given point, based on the board specs and edge angle. Define "flex" over a 2-centimeter section, so for example a section might have 0.001mm of camber at rest, and need to bend -0.002mm under load, for 0.003mm of flex.

If you can get data on this and actually measure the decamber amount under different loads over a given length, I will definately give you a polynomial that will describe it in any dimension you need. The only problem is that you may have to create a new polynomial for each board you test. Once you test enough boards we can generlize the poly an add different variables as needed. First you can create a few board types based on their specs of stiffness. The function will recognize each different type and adjust the poly as per type. It can then run through a few hundred computations with different weights of boarders and maybe the angulation of the board considered. It can print a table of of flex patterns and even graph it. It can return an array filled with hundreds of "flexions" along the given length.

I have no problem with the math or even writing a chunk of code for you. I have an associates degree in engineering sciece, a batchelors degree in applied math and just finished my masters in math. My concentration was not in this type of area but I know a thing or two about it. I can also write in c, c++, or java. The physics part is another story. I don't think you should get to deep into the molecular structure or anything like that. First start simple and test as much as possible. Any data you get, I will help you with the math to develope equations that model the situation. Also look to see what is out there already. Maybe someone has calculated pressure points along a board using different board/boot/binding combos for different ranges of weight. I will also try to look.

Good Luck

[Donek]

Quite honestly there is no simple way to model every board. A lot of manufacturers use straight line tapers to develop flex patterns. You could make a generalized analysis to show how ineffective they are at distributing a load along the boards edge. The really good board manufacturers are going to take a different approach to generating a better result. Some will use stringers or butterfly shapes that liniarize the flex pattern. Others will take different approached to tapering the core to solve the problem. Your model would assume that every manufacturer uses the exact same approach and the exact same shaping concepts in developing their boards. This simply isn't the case.

You'd be better off approaching someone who doesn't really understand what they are doing and offering to generate an analysis of their designs in order to help them improve. Unfortunately, such individuals are few and far between in this industry.

[Justin A.]

Quite honestly there is no simple way to model every board. A lot of manufacturers use straight line tapers to develop flex patterns. You could make a generalized analysis to show how ineffective they are at distributing a load along the boards edge. The really good board manufacturers are going to take a different approach to generating a better result. Some will use stringers or butterfly shapes that liniarize the flex pattern. Others will take different approached to tapering the core to solve the problem. Your model would assume that every manufacturer uses the exact same approach and the exact same shaping concepts in developing their boards. This simply isn't the case.

You'd be better off approaching someone who doesn't really understand what they are doing and offering to generate an analysis of their designs in order to help them improve. Unfortunately, such individuals are few and far between in this industry.

pfffft...what do you know! ;)

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[jtslalom]

Well there you go. If most boards have flex patterns that are generally linear, there is no sense in trying to model them all. Assume a linear flex but since it seems like different board manufaturers go about this by totally different means, I guess it would be pretty hard to calculate board thicknesses along the edge. You could research the way each manufacturer goes about accomplishing linear flex and base the one function off the different ways to manufacture the boards. I guess this would entail a list of all different ways to make boards and how board thickness varies between each method. Well in any case I love to screw around with modeling equations. Let me know if I can help.

[Donek]

pfffft...what do you know! ;)

It seems like the older I get, the less I know.

[Donek]

Well there you go. If most boards have flex patterns that are generally linear, there is no sense in trying to model them all. Assume a linear flex but since it seems like different board manufaturers go about this by totally different means, I guess it would be pretty hard to calculate board thicknesses along the edge. You could research the way each manufacturer goes about accomplishing linear flex and base the one function off the different ways to manufacture the boards. I guess this would entail a list of all different ways to make boards and how board thickness varies between each method. Well in any case I love to screw around with modeling equations. Let me know if I can help.

Actually a straight line taper produces a flex that is very far from linear. Stiffness falls off exponentially. This is based on the equation for moment of inertia bh^3/12. Stiffness varies directly with the cube of the thickness.

[NateW]

Actually the whole point of this exercise is to predict how the rider's weight will be distributed over the edge for a variety of flex patterns. I realize that different manufacturers use different flex patterns - what got me started on this path was seeing two boards stacked and noticing that they had very different core shapes... that got me wondering about the theory behind flex patterns and force distribution.

If I can get a reasonable model of lengthwise flex, it would be fun to experiment with different flex profiles and see how the force distributions differ. Making the assumption that flex is primarily a function of width and thickness (i.e. assuming simple and uniform lamination) should make the math tractable (I hope).

That would of course raise the question of what sort of distribution is better, but that's a whole other discussion. :)

[NateW]

It seems like this kind of thing would be highly prized intellectual property for a company that makes bows or snowboards. Why let the rest of the world know how to mimic something that provides a competitive advantage? Or maybe the math itself isn't proprietary, just the actual flex patterns. But it seems like it would be straightforward to compute the latter with a good understanding of the former. (Although actually creating a board or bow that has the designed flex pattern is another matter.)

JTSlalom: I was hoping to avoid measurements as much as possible - I mean one or two measurements per board should be enought to sanity-check the math. At least I hope so, as I am allergic to real work. :)

More later...

[Donek]

You're moving into the relm of analysis and a resulting design. If you have a good understanding of calcululus and beam theory (the simplest approach), this is an afternoons work.

[DrCR]

It seems like this kind of thing would be highly prized intellectual property for a company that makes bows or snowboards....

Probably not IP but trade secret.

One is clearly defined and protected, the other on the opposite end of the spectrum, open to the world basicly but tight-lipped and the in-the-know held accountable.

Sorry, just had to point that out. :)