January 26, 2009 10 Comments

It’s so easy, these days, to establish the exact spoke length for any wheel. Just visit one of the Web’s many free calculation sites (DT’s, for example). But it wasn’t always so easy; in fact, the correct lengths were hard to discover, closely guarded secrets. For more than a century, wheel builders relied on charts and notebooks filled with observations. That’s the wheel world I entered in the early 1970’s, and here’s a glimpse.

For those of you who don’t build wheels, let me mention that the spoke length needs to be within a millimeter, less than 1/3 of one percent accurate. Otherwise you run the risk of too long (pops your tire) or too short (unsightly and weak) spokes. Compounding the task, manufacturing tolerances of rims, spokes, and hubs vary and they stretch and deform during building. With such a moving target, a tiny error in length calculation can become a time consuming failure. Here is the basic mathematical formula for spoke length determination:

where

- a = distance from the central point to the flange, for example 30 mm,
- r1 = spoke hole circle radius of the hub, for example 35 mm,
- r2 = nipple seat radius, equal to half the ERD of the rim, for example 301 mm,
- m = number of spokes to be used for one side of the wheel, for example 36/2=18,
- k = number of crossings per spoke, for example 3 and
- α = 360° k/m

For most bicycle wheel builders, this is an intimidating formula.

Up until the early 1970’s, builders relied on hand kept notes and published charts. Here’s a chart by Sturmey Archer, the famous English hub maker:

Schwinn, Raleigh, and other manufacturers created pages and posters with “our spoke lengths.” Unfortunately, they were often inaccurate. Craftsman builders kept their own notes. Wheelsmith’s book was kept by brother Jon and myself. It consisted of hand laid-out pages, holes reinforced with masking tape (how quaint), and entries according to observed results. Sections by rim maker, our log eventually contained 38 pages with about 35 lengths per page; more than 1,000 recorded lengths.

Here, for the very first time, is a page from the Wheelsmith log of lengths.

This is page three from the Super Champion tubular rim section. You can’t imagine the care we took with this book. It virtually defined our capacity to build custom wheels efficiently and it recorded, for our own satisfaction, how much territory we had experienced. These pages look like a travel-worn Passport book: dirty, filled with stamps from exotic places, almost a diary, rich with authenticity. Honestly, it was fun.

However, the future beckoned and everyone wanted change. Real change, right now. Howard Sutherland and his gang of East Bay bike nerds threw their efforts behind a comprehensive book of cycling standards. Howard gave us a spoke length determining system. You looked up your hub, spoke number, and cross pattern; then consulted a “rim correction factor,” and PRESTO - you had the length. This system worked great and we rarely needed our dog-eared spoke length notebook again. Here’s a page from an early Sutherland Manual. These are rim correction factors with our own notes.

By the late ‘70’s, the energy in the wheel scene was fierce. We had Spence Wolf at Cupertino Bike Shop importing every obscure length of Robergel spokes from France (the World’s best at the time) and also custom drilled Super Champion rims (48 spoke, 650B, etc.), Phil Wood in Campbell was making superb sealed bearing hubs in every drilling and configuration. Howard did his manuals and Jobst Brandt published The Bicycle Wheel, the seminal work on tensioned wire wheel physics. He also developed a wonderful spoke tension gauge.

Jon and I were motivated to make our own contribution. Besides collaborating with Specialized to design the first commercialized mountain bike wheels, we created the Wheelsmith tensiometer (many thanks to the late, great Norm Ogle). While admiring Jobst’s design, ours was cheaper to make and sell. It was also the dawn of handheld calculators and we figured one could be harnessed and dedicated to spoke length calculation. The Spoke Length System, with its rim measuring rods, was born.

Now, for the first time, non-mathematical builders could establish the correct length for wheels EVEN WHEN the hubs and rims had never been seen or used before. The consequence was huge. A host of innovators threw themselves at the opportunity. Phil introduced his spoke cutting machine, WTB began making extraordinary hubs with inspired dimensions and features, Keith Bontrager cut down Mavic rims for MTB use, Specialized came out with proprietary hubs and rims, and pretty soon we had a host of players from Mistral rims (later to become Bontrager), to Ringle hubs, to Ross Schafer’s Salsa quick releases, to Tom Ritchey’s hubs and rims, Don Millberger’s Nipple Driver, and many more.

The floodgates were opened between 1974 and 1984. A decade to remember, and a truly fun one for me.

November 02, 2021

Thanks for the equation!

It looks like a neat thing to use.

I came up with 113.9mm and 113.2mm spoke lengths for Sturmey Archer S2C hub specs with 36H Mavic Open Pro 700c rims and a three cross pattern. Feels like a fail as I’m pretty sure that’s one (or two) millimeters off.

Are only smart/math people posting on this site?

November 02, 2021

First off, thanks for showing that page from your bible of spoke lengths. It's a fun thing to see. However I did have some quick questions about it. In the column labeled rim spec, there were some abbreviations such as REC or ROUTE that I didn't quite understand. I also would like to know how, when looking back upon the book you knew which rim it was you used and its ERD.

November 02, 2021

@NeilP, spoke angle is related to number of spokes and number of crosses as follows:

Imagine a hub and rim with their spoke holes lined up. In a radially spoked wheel each spoke runs from its rim hole to the hub hole it is lined up with. For a 1 cross wheel each spoke runs from its rim hole to the adjacent hub hole on the same hub flange. For a k-cross wheel each spoke runs from its rim hole to the hub hole k spaces away, on the same flange, from the hub hole radially in line with its rim hole.

If it is not clear that the number of crosses is equal to the hub hole offset in this way, draw it out on paper and it will be easy to see that it is the case.

The spoke angle, α, in the formula is the angle between a spoke’s hub hole and the hub hole radially in line with its rim hole (angle measured from the centre of the hub flange to the 2 hub holes in question).

Now the angle between holes on each hub flange is 360°/m where m = number of spokes to be used for one side of the wheel, for example 36/2=18. Therefore the spoke angle α = 360°k/m.

November 02, 2021

Same formula as above but with an interesting hole diameter/tension fudge factor.

For sustained use, you'll need fudge factors. Like the nearly 1mm increase we make with aluminum nipples, to engage more of the thread, trying to minimize nipple fracture at the head to barrel transition.

Thanks, BikeBoy.

November 02, 2021

Great to see the formula published. Have found it difficult to find on line.

Only one thing, it is perfect if you are building a standard wheel, with constant spoke angles, constant spoke crosses etc.

The formula derives Spoke angle ( α ) from the number of spokes and number of crosses.

It would be nice for others in future if you could show how this is derived.

Why the hell should any one want to do that you may ask.

Well there are maniacs out here ( like myself) doing weird stuff like fitting 20 hole rims to 36 hole hubs etc, creating weird lacing patterns, where number of crosses changes around the wheel ( Crows foot pattern etc)

I did find this info on Sheldons site

http://sheldonbrown.com/mismatch/

but if you were to icnlude this derivation here it would be just great

Regards

Neil

November 02, 2021

Here's the formula I like to use:

L = sqrt(R^2 + H^2 + F^2 – 2RHcos(720/h*X))-phi/2-t

where:

L = Length of spoke

R = Rim radius to spoke ends (ERD/2)

H = Hub radius to spoke holes

F = Flange offset from centerline of hub

X = Cross pattern (2, 3, 4…)

h = Number of holes in rim

phi = Diameter of spoke hole in hub

t = Tension

November 02, 2021

Just a quick side note that no one seems to have taken into account: Nipple length. The longer ones will give you some room to play with (1-2mm) if your spokes are slightly short.

November 02, 2021

The formula might be intimidating to do by hand, but that’s what computers are for. I laugh when I see these huge spreadsheet programs or other graphical programs to calculate length. In 1985 I took the formula from Jobst’s book and wrote a very simple program in Pascal that that simply prompted for each hub/rim measurement and the # of crosses, then calculated the length when you hit “enter”. It compiled to less than 2KB (if my memory serves me right), fit on a floppy disc and ran instantaneously on an 8086-based computer. Sometimes “progress” really over-complicates things. :-)

November 02, 2021

I’ve put an Spoke Length Calculator from Lled.maps on my iPhone. Cheap, works great.

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## Aaron J

November 02, 2021

Have you used these online spoke calculators that supposedly make it easy? I’ve used four and none of them have been accurate. The spokes ended up too long or too short.