Wheels Again, and Math

snoops lowrider trike

So I've been thinking about wheels again.  Sort of "again."  I mean, I guess I never really stop thinking about wheels, but recently they've been jacking a higher percentage of my brain processing power than normal.  I think it's because, due to increased boredom and indifference to my current project at work, I've been spending too much time on the internet reading equipment news and reviews from places like bikerumor.com.  So much of the content centers around new wheels and "wheel systems," and so many of the articles leave me with my Hanes all in a twist, because there's just so much bad information being perpetuated!

But rather than complain, I'd like to make something positive and constructive from my frustration.  That said, prepare for a quick engineering class that will hopefully help us all gain the power to see through the claims of companies who want you to buy their things, and make sense of what's right and what's pure fiction...

I had to dig back a ways to find an article that I remember really striking me as wonky. This is a thing to click on to see that article. I kind of feel like an internet god when I make a link thing that works.  My grandma would be so impressed with me!  Anyways, if you read that article, you'll see things said like, the wheels with more spokes (24) were more comfortable because the tension was lower, and that the ones with 20 spokes were "harder, stiffer, and faster."  Claims like these, where people comment on wheel properties and "feel" based on spoke tension, or say counter-intuitive things like "fewer spoke and more stiffness" are just plain nuts.  There are things that affect the properties of a wheel, and there are things that don't, and there are variables that result in changes you'd expect, and variables that make almost no difference at all, and it seems like those things get all mixed up all the time.  Let's rap about those things a bit, and see what gets clarified.

First off, for the purpose of this post, I'm going to bow out of talking about rims.  Huge topic, not simple at all, and ultimately the information required for informed discussion isn't readily available from manufacturers.  So let's assume in every scenario from here on that the rims remain the same.  Hubs, too, are a whole 'nuther can of worms, so let's leave them alone as well.  Basically, what I'm saying is, I only want to talk about spokes, and this whole lengthy intro was just an excuse for me to do that.  Confessions out of the way, let's move on!

So to start - let's call it PART ONE - what makes a spoke "stretchy" or "stiff?"  It's a combination of things.  Let's imagine a whole bunch front wheels, all identical, like we said above - say, 24 holes and radially laced, same rims and hubs, and the same tires and tire pressure - and the only variable among them is the type of spoke used.    If you could corner exactly the same on any of those wheels, it would produce the same forces within the system, and ask the same thing of the spokes.  That is to say, if a spoke had to take on 10 lbs of force (in addition to the force it sees all the time from the wheel build) to resist wheel flex on one wheel with one type of spoke, the same would be true for the others, regardless of the spoke used.  The forces involved are, at least for the sake of this somewhat simplified blog post, independent of the spoke type.  What the spoke type does affect, though, is how each spoke reacts to the forces it sees.  When a spoke is subjected to additional tension from the forces of riding and cornering, that spoke stretches, and that stretch is what allows the rim to move away from center, flexing the wheel laterally.  Too much flex, and you have a wheel that isn't terribly laterally stiff (and lateral stiffness is the best, says marketing), but a wheel with no stretch in the spokes would concentrate more of the stresses from riding in the hub flanges and the rim.  Anyway, let's take a closer look at that stretch.

Actually, before we do, I should note that because a spoke can pivot at both ends and is much longer than it is wide, it essentially acts like a string, in that it can only support pure tension.  That simplifies things, because we don't really have to think about bending or compression or buckling or anything like that, only tension, which makes life easier.

OK, so we've got a whole array of different types of spokes, all seeing the same force, and we want to know which stretches the most and lets the wheel flex the farthest.  From here on out, we're going to call "stretch" by its more official name, "strain," which is measured in inches per inch, or mm per mm (or unitless), meaning that for a given length of material, the length changes x amount.  If you had a strain of .5, then a foot long section of spoke would actually measure 18", you dig?  The spokes will all have some initial strain from the wheel build, so from here on, we'll just be looking at additional strain.

The amount of strain you see in a material is proportional to the amount of stress you have in that material, which in this case is the force the spoke must resist, divided by the cross sectional area of that spoke.  So a spoke with more cross sectional area will have less stress from a given force, and therefore also less strain.  The opposite is also true.  That's all very intuitive, and I hope it's not boring.

Also intuitive is that a stiffer material should have less strain for a given stress.  There's a measurement of a material's stiffness, called Young's modulus, and it defines that relationship between stress and strain.  A high Young's Modulus means a material deforms less at a given stress than a low Young's modulus material would.  So a spoke made out of a material with a higher Young's modulus will be stiffer, exhibit less strain for a given load, and will act much like a spoke with a bigger cross section.  All of our quality spokes are stainless steel, so at first glance it seems like the Young's modulus will be a constant, and the comparative strains in our different spokes will be determined by only their cross sectional areas.  Things unfortunately aren't quite that simple, though, as the Young's modulus of the different spokes' alloys can vary somewhat.  DT Swiss doesn't advertise their material properties, and I couldn't find anything for Wheelsmith either.  Sapim lists the yield strength of the various steel alloys they use, but that doesn't tell us anything about how a spoke stretches, only how much stress it can withstand.  Left with no specifics, the best we can do is assume a consistent Young's modulus, and note that any calculations aren't 100% real life accurate.  For blog-level math, I think that's OK.

So if we know the Young's Modulus, and we're assuming the same force is being seen by each spoke, the only variable left is the cross sectional area.  For now, let's look at 3 types of spokes; cheap straight 14 gauge jobs, skinny butted round spokes (like the DT Revolution), and ritzy aero bladed $$$ spokes like Aerolites or CX-Rays.  Based on info from the manufacturers' websites, we can figure out the following cross sectional areas:

- DT straight 14 gauge - 2.0mm diameter, yields 3.14 mm^2

- DT Revolutions - 1.5mm diameter, yields 1.77 mm^2

- DT Aerolites - .9mm x 2.3mm, yields 2.07 mm^2 (assumed rectangular area)

- Sapim CX-Rays - .9mm x 2.2mm, yields 1.98 mm^2 (assumed rectangular area)

Slight variations in material stiffness aside, these cross sectional areas tell us how "stretchy" these spokes are in proportion to each other.  I calculated the x-section of the bladed spokes with rectangles, so in reality the area of both would be a little less.  Even with the benefit of that assumption, you can see that the aero spokes don't beat out the skinny round spokes by much.  If you hear somebody giving DT Revolutions or Sapim Lasers grief for being whippy or flimsy, but complementing the stiff ride characteristics of the CX-Ray or the Aerolite, it's likely that they're repeating some well-executed marketing (I hope that doesn't sound judgey.  If you don't do the more in depth look, like this, the marketing stuff is all you have to go on).  It's also pretty obvious that if a stout, stiff wheel is your goal, you're not going to beat a big, thick spoke, and you'll save money too since they're cheap.

*All of that is based on the assumption that you want a really stiff wheel, of course.  I personally don't think that everybody always benefits from the stiffness that they seek, but I'm also grumpy about new tech and promises of performance gains as a rule, so feel free to ignore me.

OK, moving on to PART TWO!  Spoke tension.  What do we know about spoke tension?  We know that spoke tension decreases on the side of the wheel that the wheel bends towards.  We know that if the tension gets too low, or is relieved entirely, problems with the wheel's integrity can arise, since the strength of the wheel is a result of that spoke tension distributed all the way around.  We know that if you build a wheel with more tension, then as the tension is relaxed when a wheel deforms, you have a lower risk of that tension dropping too much.

We also know that there is some value of initial tension that exists, hopefully, between one too low that could potentially allow a spoke to de-tension entirely, and one too high that would damage a nipple or rim.  Finding that sweet spot isn't too tricky, luckily, because the grunt work has been done for us, and there are several ways to get recommended safe tension numbers for various rim and spoke combinations.

We know a lot about spoke tension!

Here's what we think we know about spoke tension, that is not correct:  more is better, and a really tight wheel offers something more in terms of performance.

If you go back up to the beginning of this post and see what variables affect the lateral stiffness of a wheel under a certain load, you'll see that spoke tension doesn't show up there.  Also, if you look at the stuff about the stretch of a spoke due to additional force inputs, you'll see again that the initial tension doesn't change anything.  Once you've got enough initial tension to ensure that the in-use tension never gets to dangerous lows, then you have enough.  There are no gains to be had from increasing the tension beyond that point.  The spokes don't stretch more or less with additional inputs, and the wheel doesn't feel any different.  If a wheel genuinely feels different from one spoke tension to the next, then one of those wheels wasn't up to the minimum to begin with.  Going back to that Mavic article, you can see why the claim that the wheel was more comfortable with lower tension was likely incorrect.  Spokes don't change their behavior based on the starting tension; they do not get stiffer the harder you pull on them.

PART THREE.  Everything we've done up to this point assumed identical wheels with the same number of spokes under the same loading situations, resulting in the same amount of load being carried in a given spoke from wheel to wheel.  Now, if you add spokes to one of those wheels, but don't change the loading scenario on the entire wheel, each spoke ends up carrying less load itself.  Remember that the strain in a spoke was directly proportional to the load through it, and remember that the lateral wheel stiffness was dependent on the strain in the spokes.  When you put all of that together, you see that more spokes yields a stiffer wheel.

So looking back at all of those words I typed (and I really hope they both made some sense and were all more or less correct...), it paints a useful picture of what a wheel designer can and can't do with spokes, and what claims are legitimate, and what claims are bogus.

-You can build a stiffer wheel by using more spokes, or using spokes with a bigger cross section.

-You can't build a stiffer wheel by reducing the number of spokes.

-You can build a perfectly reasonable wheel with DT Revolutions, as long as you agree that you can build a reasonable wheel with little aero spokes.

-You can't build a stiffer wheel by increasing the spoke tension.

-You can build a crappy wheel by not tensioning the spokes enough, but you can build an equally crappy wheel by going too high and starting closer to the breaking point of all your wheel parts.

-If you are a wheel company and you find yourself adding weight to your rims to reinforce them just so they can handle the proper amount of tension necessary to allow silly low spoke counts, you are weird.

-There is no magic in wheels, and the design principles are fairly simple, so if you're wondering why a "wheel system" that isn't terribly light, using normal aluminum rims and steel spokes, can still cost upwards of $1000, the answer is because the wheel pimps can sell it for that much.  Them companies ain't got access to different math or concepts than any of the rest of us do, so if it seems fishy, it probably is.

I really love to think about the smaller details that make up the machine that I'm passionate about, and I also love pretending that my engineering school stuff stuck with me somewhat.  I hope it's enjoyable for y'all, too.  Was it?  Was that cool at all?  Interesting?  Helpful?  Annoying, boring, or stupid?  I want to write about what people like to read about, so let me know if this geek crap is big-time-fun, or lame.


The Tiniest Sprinter

PS:  Mat Barton, grade A+ off-road shredder, had just about the worst luck anyone could have, and ended up with a severe spinal cord injury at the PIR short track race this past Monday.  Big life changing stuff ensued, and if anybody is feeling particularly generous, there is a donation site set up for him and his family to help them through the crisis.  It's at http://www.bartonpdx.com/ .

PPS:  As someone who appreciates a quality tan and also full fingered gloves, I find myself internally conflicted a lot.  Which to chose?  Life, man, it's tough.

PPPS:  I just realized last night that when I put my rear derailleur back together after servicing the pulleys, I put the back plate on upside down.  And then I rode it like that for a month and a half.  Maybe nobody should trust anything I say about bikes, because I'm an idiot a lot of the time.

PPPPS:  I am totally stuck on revisiting old videos.  Can't stop.  I just watched about a dozen Timbaland ones (lord knows why), and I'll spare you those.  Instead I have to include this one:


PPPPS:  And this one isn't old, but it's hilarious.  If I could be any rapper, it would be Method Man, no question about.


Posted on June 15, 2012 and filed under Uncategorized.