For a hub of 120mm, what would the dropout spacing be on a typical frame?
Or for 130mm, or other sizes - in each case, how close to the hub width should one expect the frame spacing to be?
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You can find the whole gamut of tolerances, especially if you include very low-end bikes. All the numbers I have are my own estimations but they are based on taking rulers to plenty of frames and taking wheels in and out of thousands more.
There is no formal rule for what's acceptable, so any answer is likely to be viewed as too tight by some and too loose by others. However, a lot of framebuilders and industry types see +/-0.1mm as the gold standard for most of the key frame alignment tolerances on higher end bikes, including dropout spacing, rear end alignment as measured with a Park FAG or similar tool, and a bunch of other dimensions. Many decide upon .1mm as the number where if you try to do better you'll be chasing ghosts, struggling to reproduce measurements, etc. A lot of nicer bikes measured out of the box will conform to that for their dropout spacing.
Less premium but still "dealer-level" bikes these days are almost always +/-0.5mm or better for dropout spacing, although occasionally you see a bike where it's more like 1mm.
Low-end and historic bikes are way more all over the place with dropout spacing fidelity. On a contemporary steel department store bike, it wouldn't be surprising to see +/-3mm. For aluminum the dropouts can't be leaned on to flex into conformity the same way and things aren't quite as bad, maybe half that number.
You mention 120. Bike boom bikes had very sloppy tolerances for dropout spacing, probably something like +/-3mm. Good modern track and singlespeed frames are usually very close to dead-on, like +/-.2mm or better. Quickly dialing in optimal chain tension is a total nuisance when you've got gap at the dropouts.
ideally it would match 1-to-1. that is, a 120mm hub would fit in 120mm dropouts. hub width is measured as the distance between the outside of the locknuts. the locknuts are the parts of the hub that press against the inside of the frame. the axle length is a different measurement and depends on whether it's quick release or bolt-on axle (or thru-axle).
with steel frames you can get away with a hub that is narrower or wider by a few millimeters. for instance, lots of cyclocross frames come with a hub spacing that is between sizes, 132.5mm, so you can use either a road size (130mm) or MTB size (135mm). aluminum is less tolerant, so i would make sure it's the right size hub for the frame. some hubs can be modified to be wider or narrower.
Steel Bikes have tons of flex compared to aluminum so its not uncommon to see lots of slop. Some manufacturers even design their frames with 123mm wide dropouts so they can stretch to fit 126mm hubs or squeeze to fit 120mm hubs. There is also a process called cold setting where you can expand or contract dropout widths to fit your needs.
Aluminum has less flex so tolerances are tighter, but they can get quite loose of Walmart level bikes.
Carbon has very tight tolerances because very little flex and becasue carbon bikes are high end and it would be ridiculous to spend thousands on a bike and have sloppy dropouts. I also suspect its is easier to manufacture carbon to exact specs because it isn't welded which causes warping in steel and aluminum.
Not really sure about Ti. It can flex like steel but Ti bikes are high end and customers would expect tight tolerances.