**Stiffness, dish, and tension**
rruff said:

I can believe that the angle doesn't matter... I was imagining the limiting case of 0 degrees and 10 degrees and how that wouldn't work... but it is only because all the spokes are at zero tension... so as you say, it is the spokes going slack that is the problem, not the dish.

But... the *square* of the sine of the angle? I'm not seeing that... How is it derived?

That's easy. First to define a few terms:

L = Spoke length

R = Spoke radius (i.e. the length of the spoke projected along the radius)

O = Lateral distance from flange to rim

The lengths L, R, O form a right triangle, where R and O are at a right angle, and L is hypotenuse.

A = Angle of spoke (from radial)

T = Tension of spoke

Ks = Spring constant of the spoke (i.e longitudinal stiffness)

Kl = Component of lateral stiffness contributed by the spoke.

Fl = Portion of lateral force distributed to spoke

Kl = dFl / d O ( change in lateral force for change in lateral deflection )

dFl / dO = ( dFl / dT ) * ( dT / dL ) * ( dL / dO )

dFl / dT = sine( A ) ( change of lateral force for change in spoke tension)

dT / dL = Ks (change in spoke tension for change in change in spoke length)

dL / dO = sine( A ) ( change in spoke length for lateral offset )

Kl = ( sine( A ) ) * ( Ks ) * ( sine( A ) ) = Ks * sine^2 ( A )

Of course, the stiffness of the rim will also affect how much the spoke deflects for a given lateral deflection, but none the less, the lateral stiffness of the wheel will be greater with one set of spokes at a shallow angle, then with two sets of spokes at a steeper angle.

rruff said:

And if that is true, we could improve rear wheels by moving the left flange over some more... *and* using an OC rim

Yup. This part of the fallacy of narrow flanged rear hubs to reduce dish - although the dish is reduced (by making the left spokes steeper), the wheel stiffness is also decreased, as well as reducing overall wheel strength. That's why "traditional" hubs (like Shimano and Campagnolo) have stayed with the standard rear hub flange spacing as dish increased, instead of trying to reduce dishing by moving the left flange inward.

If you look at the HPW hubs used on Campagnolo pre-built wheels, they actually move the left flange

*outward* compared to standard hubs, increasing dish, but also increasing wheel stiffness. Of course, this only works if you make sure that the spoke tensions are very high, to prevent the left spokes from completely slackening in use. I wonder if the makers of narrow flange hubs realize that they are attacking a symptom, and not the problem - that the real problem with their wheels is not that they have too much dish, but that the spoke tensions are too low.