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Yes, I found that post on the Weightweenies forum but was put off by its internal inconsistencies. 'using 3.17x will result in the spokes being too long!' seems difficult to reconcile with 'When one laces a triplet wheel with 24 spokes (16 DS, and 8 NDS), the angle theta turns out to be.... the same angle as for... a 32 spoke wheel laced with 3.17 crosses'. The post seemed to be saying that 3.17x was wrong, you should be using a figure of 3.17x instead...
Brommers
@cycleclinic
Danstuff. This method is the correct way. The other methods give slightly different lengths i think.
Using 48 spokes cross 5 in the calculator:
resulting right side spoke length - ((3.14*right flange diameter)*0.0104).
I have used this a couple of times and it is spot on for 3x DS and in 16:8 pattern.
using 3.17x will result in the spokes being too long!
A full treatment is
For a normal build (equal # of spokes on each side), using the following notation:
E = ERD of rim
D = Diameter of flange
X = flange offset
N = Number of spokes
i = Number of crosses
Define:
Theta = 2 * Pi * i / (N / 2) = 4 * Pi * i / N = angle measured at the hub between a line passing through a hub spoke hole and wheel center and the line passing through rim spoke hole and wheel center for the same spoke
... then, aside from a correction of about 0.8mm from spoke stretch (~0.6mm) and extra spoke hole room (~0.2mm), the length of the spoke comes out to be:
Length = 0.5 * sqrt [ X^2 + E^2 + D^2 - 2 * E * D * cos( theta ) ]
[Note this matches with the results of United Bicycle Institute online calculator]
When one laces a triplet wheel with 24 spokes (16 DS, and 8 NDS), the angle theta turns out to be:
Theta_triplet = 2 * Pi * (19/4) / 24 = 4 * Pi * 4.75 / 48... ie the same angle as for a 48 spoke wheel laced with 4.75 crosses, or a 32 spoke wheel laced with 3.17 crosses
If one's calculator forces an integer for the number of crosses, and one made the calculation for a 5x wheel with 48 spokes, it would lead to a length that is slightly too long, because the angle theta would be off by Pi / 48 radians. If one imagines just the key spoke laced, then it'd be like rotating the hub by an extra quarter of the 24 hole spoke separation of the rim. Since the DS spokes are pretty much perpendicular to a radial line through the spoke hole, the effect of this is to change the length by roughly the angle of rotation multiplied by the radius of the hub flange, ie
Approx_length_error = Pi * (D / 2) / 48 = Pi * D / 96 = Pi * D * 0.010416666
There is a thread on Weight weenies a long time ago about this.