## Reply To: 2012 Hub Review: Information overload?

All good stuff, Dave… hope I can explain things as well.

You are missing part of the equation for the spokes. The lateral component of the spoke’s force (tension) will be proportional to the bracing angle. But in addition the change in spoke tension for a given lateral displacement will also increase proportional to the bracing angle… resulting in stiffness being proportional to the square of bracing angle.

It might help to show an example… calculating the lateral component of force if the rim end of the spoke is laterally displaced 1mm for two different initial offsets of 20mm and 40mm:

Radial hub to rim= 280mm

Offset 1= 20mm

Offset 2= 40mm

Spoke stiffness= 100kg/mm

Lateral displacement= +1mm

20mm offset:

Spoke length initial L1a= (280^2+20^2)^.5= 280.713mm

Spoke length final L1b= (280^2+21^2)^.5= 280.786mm

Change in spoke tension= (280.786-280.713)*100kg/mm= 7.30kg

Lateral component~ 7.3*(21/280.786)= .55kg

40mm offset:

L2a= 282.843mm

L2b= 282.986mm

Change in spoke tension= 14.30kg

Lateral component~ 2.07kg

Doubling of the offset results in a ~4x increase in spoke lateral stiffness.

In either case the force vs displacement will be nearly linear. For instance, you could calculate the lateral force for 3mm displacement and it will be about 3x as high… and still 4x difference between 20mm and 40mm.

Many years ago Damon Rinard did some tests of wheel stiffness (http://sheldonbrown.com/rinard/wheel/data.htm), and several of them were front/rear sets with the same type and number of spokes. I counted 10 road sets that meet this criteria. As a ballpark value I’d expect the spokes’ contribution to lateral stiffness to be ~60% higher for a front wheel vs a rear wheel, if the number and type of spoke are identical. This is based on a typical 36mm x2 offset on the front wheel vs 36 and 18mm offsets on the rear. In Rinard’s tests the ratio of deflection between the front and rear ranged from 1.35 to 1.61 with an average of 1.48. 48% is high enough compared to 60%, that I feel confident in saying that the spokes were the greatest contributor to lateral stiffness for these wheels.

The Gavin paper is online if anyone is interested: http://www.duke.edu/~hpgavin/papers/HPGavin-Wheel-Paper.pdf

As you mentioned, figure 6 shows the modeled increase in lateral stiffness vs rim radial stiffness. In his example increasing the rim radial stiffness 3x resulted in a ~40% increase in lateral stiffness. IIRC, this is with 36 2mm spokes. This is in the same ballpark with your experience, where you increased rim radial stiffness by 8x and removed half the spokes to end up about where you started. An 8x increase in rim radial stiffness is a very large amount… and a 700g 40mm deep rim is certainly on the uber high end for stiffness. I’m also not surprised that you experienced good reliability with the 24 DH13- 700g 40mm deep set. Even that one is a beefy build. I’ve built tandem sets with 28f 32r CX-Rays that weigh about half as much (and are half as stiff) as the DH13s, and rims that are only 445g… and even those hold up under modestly sized riders.

You stated that rim depth is a more important factor than hub bracing angle… for the typical ranges encountered. But that last part is key. As I showed from Rinard’s tests, the difference in bracing angle between a front and rear wheel makes a large difference…48% on average. Based on Gavin you’d need a >3x increase in rim stiffness to equal that… which is certainly exceeded if you compare the least and greatest stiffness rims available. But typically riders limit their choices to a much smaller range, because they are also interested in other criteria… like weight, aerodynamics, and ride quality.

If you compare all 100mm front hubs to each other, and all 130mm rear hubs to each other, you won’t see a big range in bracing angles. The reason is *because* it is so significant. The hub manufacturers use about all the space they can… and the good ones use every small fraction of a mm on the rear hub. If it didn’t matter so much, there would be good reasons to have hubs and axle spacings narrower than they currently are. In fact bracing angle is the driving force behind the steady increase in rear dropout spacing as cassettes get wider, and why tandems have 145 and 160mm wide rear axles.

I agree that the rim stiffness is a substantial contributor to lateral wheel stiffness. That is one reason why I tend to favor rims with some depth to them even if the wheel ends up a little heavier, since you can also reduce the number of spokes and improve aerodynamics. But the bracing angle of the spokes is important as well… and there is everything to gain and nothing to lose by optimizing those angles within the constraints of the frame and drivetrain. Not just for lateral stiffness but for keeping tension on the NDS spokes, which I believe is more important.