Looking to understand the math behind spoke length calculation. Help?

Does anyone here have insight into what I may be doing wrong or explaining incorrectly with this question on math.stackexchange regarding the math behind spoke length calculation?

https://math.stackexchange.com/questions/4632693/seeking-a-practical-layman-s-explanation-of-the-math-behind-spoke-length-selecti

• Does this answer your question? Deriving formula for spoke lengths Commented Feb 6, 2023 at 0:12
• Its asking the same question, for sure. Though it appears he's using Vector Algebra rather than standard trig formula I'm asking about here. Commented Feb 6, 2023 at 0:36
• Check out my answer for the trigonometric approach. If you're comfortable with a vectorial approach, I can upload that section of the document here if you'd like. Commented Feb 6, 2023 at 0:40
• Feel free to hop in the chat and ask me about specific parts of that answer if you'd like. The explanation is definitely not the greatest. Commented Feb 6, 2023 at 0:48
• If you just go through how the formulas work in spoccalc, which are all right there and easy to see in the cell values, you'll see how the math is done. Commented Feb 6, 2023 at 0:55

I can't speak to the maths side of it so replying here. Also, too long for comment, and images.

https://leonard.io/edd/ returns 261.1mm which agrees with your empirical value of 260mm based on the numbers given.

Playing with the calculator cannot generate the 231mm that the formula returns, so its not a matter of the formula representing radial spoking only.

The 231mm value is shorter than the `ERD less flangeheight` so even if the wheel was 2-dimensional and flat that spoke would be 246mm length for a radial design. For 2cross it has to be longer again.

It has to be an error in the formula - something is not being represented.

approximate sketch

This is a 3-dimensional geometry question at the core, and I wonder if the formula is not accounting for that, it is working in a flat plane only?

If we only look at the spokes on one side of the wheel, then that simplifies things.

R for Radial spoking, where the spoke would intersect with the center of the rim/hub.
1, 2, 3 for the number of spoke crossings

As the number of crossings increases, the spoke length has to increase because it has further around the rim to go before getting to the nipple.

I can get the calculator to say 230.4 mm if I double the flange diameter to 200, though this is more-likely coincidence.

I look forward to your results on math.se.

• Yeah, I agree we're missing something. The first thing is that we used the pitch circle diameter, instead of the radius, for r_1 in that formula. Using the radius, it comes out at 248.7mm. I don't know if the formula is correct, I got it from here: wheelfanatyk.com/blogs/blog/spoke-length Commented Feb 5, 2023 at 23:33
• My goal is to learn the math for this calculation to a level that lets me calculate spoke lengths myself, rather than relying on an online calculator. Commented Feb 5, 2023 at 23:35
• Using Damian Rinard's Spocalc.xls 2022 update off Sheldon Brown's site, the calculation for radial lacing is 248.9mm, so I suspect you are correct that we aren't accounting for the crossing angle correctly (or at all). Commented Feb 6, 2023 at 0:08
• My gut suspicion is that we're thinking in triangles, not pyramids. I guess that the simple case is getting a formula that works for a radial lacing correctly. And then work out how to calculate the "rise" of the subsequent deflection required for some number of crossings.
– Criggie
Commented Feb 6, 2023 at 0:25
• @Criggie That is precisely the approach I took when deriving the formula. Great minds think alike! Commented Feb 6, 2023 at 0:41

Not giving the mathematics, just what I think are the required steps and using previous answered illustrations.

1st step. Calculate the spoke length facing the wheel from the side, so I would think you would have to use some kind of vector calculation, Example spoke 2 (sketched in green). That would give you the height for your right angle triangle calculation in step 2.

2nd step. Use the calculated length from step 1 as the height of a right angle triangle.

As example, calculate spoke 2 length, for both sides of the hub, using L or R offset as your triangle base, a 90 degree angle and height from Step 1 (sketched in blue) to calculate your spoke length, using Pythagoras trigonometry... right?

Repeat the same process to calculate the other spokes, but take into account that the results from step 1 should be slightly different for each spoke, because the wheel rim is centered from the axle but off-centre from the flange holes, so each length from Step1, would be slightly different for spoke 1, 2 and 3.

I would think that with CAD (Computer Aided Design) software, you could sketch the setup and CAD would give you the length for each spoke for Step 1.