Rarely you are challenged by somehow exotic lacing. Being rim or spoke design. Paired spoke hole hub design is just one case that requires a specific calculation correction for the correct spoke length. To see how this type of hub looks, take a look at the drawing below, or refer to some real life product, such as Novatec’s A271SB or F372SB hubs.
In the past, Damon Rinard made contribution to this subject, while stating the following in his spokeCalc.xls:
“If paired hub spoke holes are 15 degrees apart, then: For 24 paired spokes laced 2x, enter 2.25 cross. For 24 paired spokes laced 1x, enter 1.25 cross. For 20 paired spokes laced 2x, enter 2.29 cross. For 20 paired spokes laced 1x, enter 1.29 cross. For 16 paired spokes laced 1x, enter 1.33 cross.”
Trying to understand the logic behind these numbers and to add a paired holes subcalculator to my SpokeCalc, I dived into some studying. So, let us look how to properly calculate spoke length for hubs that do not have equally spaced holes on its flanges.
For those in haste, there is a short answer. Lacing pattern input has to be modified to include a fracture number. You can simply follow Damon Rinard’s calculations, stated above, and write them as you go into the lacing pattern input field. Since SpokeCalc doesn’t allow a fracture lacing pattern number input, read further to see why this is even better as you can use it for even greater range of paired spoke hole angles. So, as for long answer - the method of getting the correct spoke length calculation based on your own hub measurements is being discussed below.
IDENTIFYING THE PROBLEM
To identify the problem, we should first visualize it. Below you will find normal hub geometry with 10 holes on each side. For this example, we are using 1 cross lacing pattern. Since the angle between nearest spoke holes in the flange of the hub is only determined by spoke count (N), we can calculate it:
∝= 360 / N
In our case, being 360 degrees divided by 10 spoke holes. Therefore, there is 36 degrees between centres of each hole in our hub’s flange. That number will serve us as a reference and a guiding point for further calculations.
Having calculated the angle for equally spaced hub spoke holes, we are now returning to paired spoke hole hub design. Let’s assume that holes in the flange are not equally spaced anymore but instead they are paired together with 20 degrees in between the paired holes as shown in the drawing below. You will notice the position of equally spaced holes a little darkened.
If you remember our reference angle, we are now moving from 36 to 20 degrees. In other words, we have subtracted original angle for the amount of 16 degrees. But we also did it by moving two holes of a pair of trailing and leading spoke further apart. But the thing to remember here is that each hole was moved only for one half of the total change, therefore 8 degrees (16 divided by 2). Again, take a close look and compare new holes to the original ones (equally spaced ones), which are lower opacity circles in the drawing.
Based on conclusions from drawings above, we can now implement a lacing fracture number with the following equation:
X = K / L
Where X is the lacing fracture number, a decimal number between integers. Since we are operating with one cross lacing in our example, we will add X to 1 cross and get 1.xx lacing pattern.
On the other side of equation, K is the numerator of the fraction X and calculated by angle change between our reference angle (equally spaced holes) and the new angle. That number is further divided by two, to account for moving pair of trailing and leading spoke simultaneously apart in geometry of paired hub design. To apply this to our example:
K = (angle if equally spaced holes - angle between paired holes) / 2 = (36 degrees – 20 degrees) / 2 = 8 degrees
And L being the denominator of the fraction X, the angular distance from one spoke hole to the next in the imaginary equally spaced holes’ hub's flange. Basically it is our reference angle we started with, 36 degrees.
In our case the cross fraction number (X) is 8 degrees divided by 36 degrees which makes is 0.22. Since we are still staying with 1 cross lacing, we just sum it up with our primary lacing pattern: 1 cross + 0.22 = 1.22. To put simply, we just divide the change in angle (16/2) by the original design to get the proportional change in angle. This calculation should properly adjust the spoke length for a given paired hub design.
Let’s now return back to Damon’s statement: “… If paired hub spoke holes are 15 degrees apart, then for 20 paired spokes laced 2x, enter 2.29 cross… “
X = (((36 degrees -15 degrees) / 2) / 36 degrees ) + 2 cross = 2.29 cross
You can see that the calculation works. Nice!
Let’s now look at two imaginary situations. What happens if we increase X by moving the spoke holes in the hub further apart?
One extreme is met when we set X to 0.5 + an integer, for example X = 1.5, which would mean that two spoke holes of a pair would actually overlap each other like shown in the drawing below. Again, original holes’ position is drawn with lowered opacity. In real world that situation would be impossible, as both leading and trailing spoke would start from the same position in the flange. Even though some spoke calculators use this exact coefficient as a constant when calculating spoke length for straight pull hubs which are of similar geometry.
On the other hand, if we continue to increase X past 0.5 + integer, we get somewhat different extreme, shown in the drawing below this paragraph. Note that it looks very similar to the second drawing with the angle between two nearest spoke holes being exactly 20 degrees. However, lacing pattern is now changed. We started with 1 cross lacing for our example, but here suddenly spokes start to cross two times. Except that they don’t and that’s the trick. Our total cross number is actually greater than 1.5 but smaller than 2.
Basically we have moved each hole 28 degrees apart from its original place in equally spaced holes’ hub (subtract 8 degrees from 36 degrees). Therefore, following the calculation above, X will be 0.77. Adding this number to our original lacing pattern (1 cross) it means we are looking at the case of 1.77 cross lacing pattern. Again, like we expected it to be between 1.5 and 2 cross. Below is the upclose drawing of how we moved original equally spaced holes apart.
The important thing here is to take caution when defining what your actual cross pattern is like. Put it simple, is it more in the realm of 1 – 1.5 or on the other side, meaning between 1.5 or above. Calculation works also for 2 or three cross lacing pattern.
APPENDIX: DEFINING THE PAIRED SPOKE HOLES ANGLE
In situation where technical drawing with full specs about the hub is not available, one has to rely on measuring the hub itself to define the angle between paired spoke holes. That is why SpokeCalc offers input of whatever angle between paired spoke holes. But how to define this angle?
Imagine two lines from the center of the hub through the center of the nearest pair of spoke holes in the flange. And then add another line from one spoke hole to its nearest. If we connect all the crossings, we get isosceles triangle, meaning the triangle has two legs of equal length. And we also know these legs’ length is exactly PCD (Pitch Circle Diameter) divided by 2. In real life we can only measure the distance between centres of two nearest spoke holes in the hub, marked as Y in the drawing below.
If you measure the distance between two centres of the nearest spoke holes in the hub, you will get Y. Having also measured PCD of the hub, you can use this as a parameter in the next equation.
Or to make it easier to get the full angle on the fly, use the bottom equation. You should be good either way. Maybe some day I add this as an automatic calculation in my SpokeCalc. It should make things somehow easier.
Now you know how to make use of more exotic hub designs. But what about rims? Should the same concept apply? Till next time... Happy wheelbuilding everybody!