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Calculate spoke lengths manually

Old 12-09-18, 07:28 PM
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speedevil 
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Calculate spoke lengths manually

This is an interesting math exercise that has been largely replaced by web-based spoke length calculators, linked to nice databases for hub dimensions and rim ERD values. I always measure rim ERD for myself and measure at at least 2 different points around the rim. And I always check the supplied hub dimensions for myself. Sometimes a bad value is stored, more likely for uncommon hubs (fewer examples and lower usage), so checking is worth the extra few minutes it will take. Some hubs and rims are rare enough that they aren’t in the databases at all, so once you’ve determined the measurements, sending them to the various calculator sites is a nice way to pay it forward. Good karma from that can’t hurt. Remember, you are the one that will pay for replacing spokes that are not the correct length, so measure at least twice. Personally, I consider the correct length to be where the top of the spoke is level with the bottom of the slot in the nipple.

2/3/4 cross lacing:

The basic formula is:

L = sqrt(R^2 + H^2 + F^2 – 2RHcos(360/h*X)) – shd/2

Where:

L = calculated spoke length
R = rim radius to nipple seat (ERD/2)
H = hub radius to spoke holes (spoke hole circle diameter/2)
F = flange offset from hub centerline
X = cross pattern (2, 3, 4…)
h = number of holes in one side of the hub
shd = diameter of spoke hole in the hub

Check it out, it’s not really that complicated. Most any calculator can handle this fairly easily. When you’re done, try one of the online spoke length calculators to see what lengths they calculate. You can try more than one spoke length calculator and average the results (they will likely have slightly different results, depending on any fudging done during the calculations).

Since spokes are generally available in 2mm increments and some in 1mm increments, you’ll likely need to round the calculated values. For me, I tend to round to the closest available length, but you’ll have to determine that on your own. Experience with calculated lengths on successful wheel builds will tell you which way to go with different length calculators. Definitely a math geek’s exercise, but sometimes it’s nice to see how the “behind the curtain” work is actually done.

Radial lacing:

For radial lacing it’s much simpler. it’s just a right triangle and you’re solving for the hypotenuse. The formula is:

L = sqrt((R-H)^2 + F^2) – shd/2

Where:

L = calculated spoke length
R = rim radius to nipple seat (ERD/2)
H = hub radius to spoke holes (spoke hole circle diameter/2)
F = flange offset from hub centerline
shd = diameter of spoke hole in the hub

Be aware that there are multiple places in the formula for rounding, and length calculators can and do round differently. This accounts for the differences between methods. All of them should deliver results within a mm +/-.

And you thought you’d never use trigonometry once you finished school.


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Last edited by speedevil; 12-09-18 at 07:43 PM.
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Old 12-09-18, 07:34 PM
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Radial Lacing Visualization

To visualize what it is your calculating, consider the case of radial lacing (0 cross). You are calculating the hypotenuse of a right triangle, and you know the length of the other two sides.

The short side is the distance from the spoke flange to the center of the hub.

The long side is the distance from the spoke holes in the flange to the nipple seat in the rim. To get this value we subtract the radius of the spoke holes in the flange from one-half of the rim's ERD.

The longest side length is the spoke length itself. We make an small adjustment for the diameter of the spoke holes in the flange.
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Old 12-09-18, 07:42 PM
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2-3-4 cross Visualization

Review the radial lacing visualization and add this twist to it. Literally, the hub is "twisted" and the spokes are no longer exiting the hub straight to the rim, the spokes are moving towards tangent to the spoke hole circle.

By "exiting the hub straight to the rim" I mean that the hub axle, the spoke hole in the hub, and the spoke hole in the rim are aligned. With 2/3/4 cross lacing patterns, there is an angle between a radial spoke and the cross pattern spokes. Visually, this angle is the angle between the hub axle and the spoke hole in the hub, and the spoke hole in the rim. The greater this angle, the more required length to reach the same depth in the nipple.

By using the angle that the spoke makes, we can calculate the additional length required. We know how many increments there are around the rim based on the number of spokes per side, which gives us the angle as compared with a radial spoke position. This angle "lengthens" the long side of the right triangle, which lengthens the hypotenuse (the spoke itself) as well.

If the spoke was perfectly tangent to the spoke hole circle, it would need to be increased in length equal to the radius of the spoke hole circle. That's an extreme example but it demonstrates how the spoke lengths change as the number of crosses increases from 0.
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Last edited by speedevil; 12-11-18 at 10:04 AM.
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Old 12-09-18, 08:26 PM
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Originally Posted by speedevil
This is an interesting math exercise that has been largely replaced by web-based spoke length calculators, linked to nice databases for hub dimensions and rim ERD values. I always measure rim ERD for myself and measure at at least 2 different points around the rim. And I always check the supplied hub dimensions for myself.
Nice! Granted, I haven't built a lot of wheels, but I always do all of my own measurements and calculations. This lets me be precise about how I want the spoke ends to end up relative to the nipples, and to confirm that all of the numbers are accurate.

I use what I think is the same formula, but in the form of a brief Python notebook, where I also keep all of my previous calculations.
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Old 12-10-18, 07:46 AM
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I believe this is the same spoke length formula Jobst Brandt gives in "The Bicycle Wheel", along with a lot of other arcane formulas for "torsional elasticity", "spoke elongation", etc. Any scientific calculator could do the calculation on a one-time basis but t would be fairly easy to set up this formula in a spread sheet for multiple use and to develop your own data base if you use various rims, hubs and lacing configurations.
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Old 12-10-18, 09:44 AM
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Looks to be the same formula used in Sutherlands:


Source: Sutherland's 4th Edition
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Old 12-10-18, 10:02 AM
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I have seen the formula where the cosign variable uses 720 divided by the total number of spokes. 360 divided by the number of spokes on one side seems better to me.

it allows for easier calculations of spoke lengths for alternate lacing patterns - the crow's foot pattern for example.
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Old 12-10-18, 01:26 PM
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Fortunately, these formulas don't require conversion between inches and mm.
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Old 12-10-18, 03:01 PM
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Originally Posted by AnkleWork
Fortunately, these formulas don't require conversion between inches and mm.
That depends on the ruler and caliper you use to get the rim and hub dimensions.
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