1 1/8" forks in a 1" headttube, have a look at my IS41 adaptor and let me know what you think!

Not need to be an engineer to see 0.1mm clearance have very little chance to spin freely

I've always wondered why headsets don't use bushings rather than bearings. An Igus bushing would be fine for that purpose

Too much friction in the lower bearing, which has a lot of axial thrust on it. You can ride with a lot of friction (see friction steering "dampers" on some old BMW motorcycles), but it's not fun as you end up thrupenny bitting every curve.

Lol

Not need to be an engineer to see 0.1mm clearance have very little chance to spin freely

Depends on the application. If the shaft and bore are both accurately cylindrical, 50μm radial clearance is about right for this size if you want to treat it as a plain bearing with either hydrostatic or hydrodynamic lubrication.
Of course, a 1⅛" steerer, especially carbon fibre, is unlikely to be any of circular, straight or nominal size even without load, and it can get a long way from straight under load in comparison with a clearance that small.

I was wondering whether part of the solution to this would be to sleeve the entire headtube with a very thin bushing of low-friction material and then polish the steerer or coat it with something similarly low-friction but, TBH, at that point you may as well just get someone to custom manufacture you the fork you want with a 1" steerer.

The only way I can see it working well is to make the headset cups totally external

That's what I meant with "external cups". So, how about this (see below). Would be easier anyway, as they could be made from (easier to machine) aluminium then. When the paint has been cleaned off at the headtube contact surfaces and the ID of the cups are matching the headtube, the assembly would be a press-fit, like a normal cup, even if the headtube isn't perfectly circular the press-fit would probably work well enough.
Any thoughts?

I'm not sure this will work either, they'll be a load of lateral force on the fork steerer that might walk the cups off of the head tube.

I admire your persistence but nothing you make work, is going to be as effective as just using a 1" steerer.

they'll be a load of lateral force on the fork steerer that might walk the cups off of the head tube

No offense but this is really not going to happen. Remember that there is the whole ahead force clamping it down. If the lateral forces would actually stretch the material of the cups so they get loose then this would happen with every cup in the world. But it doesn't.

But you're not applying the force in the same way.

In the traditional way the headset cup is supported on both sides, internally by the steerer tube and externally by the head tube. You have no support outboard of your cup because it goes over the head tube. I'm not an engineer but I can tell you your way is going to be significantly less effective.

Either way, good luck. Let us know how it goes.

In the traditional way the headset cup is supported on both sides, internally by the steerer tube and externally by the head tube.

That's not right, in the traditional way the press-fit section of the cup is supported externally by the head tube (yes), but internally the steerer does not support it, it supports (if you can even call it that) only the cup itself which sits further up. With the press-fit section outside, forces are justed turned inside-out but the support is about the same. And the wall thicknesses (see my updated drawing) are higher too when it's external.

Of course it's always ideal if you use a 1" fork in a 1" steerer etc., but I am proposing a solution for a specific case that sometimes comes up here and there and I hope that people can take advantage from that, and the effort needed is actually quite low as it's really only about making the parts on the lathe. I will try the external cup now, thanks anyway for the input.

I will try the external cup now

Yes! Excited for this despite not at all being interested in bikes old enough to not have a tapered steerer.

If I understand it correctly the problem is that the bearings exert an outward radial force on the cups (and a corresponding inward force on the steerer) in addition to the axial force. In a normal headset the cups are braced against that radial force by the headtube, but in your design they're not. Instead you're relying on the strength of the cup itself to resist any non-axial forces that might deform it. If the cup does deform then it might unseat slightly from the headtube and, even if it can't walk off it, it might fret and wear both itself and the headtube.

I assume that's why previous efforts at this have actually brazed or bonded the cups to the headtube.

bearings exert an outward radial force on the cups

I understand that point and I am still thinking about it. But, if deformation exerted by the bearings on the cups would be a problem, then even with normal cups in a standard 1" setup for example, the bracing support by the headtube would never be sufficient to prevent a gradual deformation of the cup by the bearings over time, because the headtube is sitting quite a few mm below, not in the same "plane" as the cup itself).
But, apparently reality shows that this is not happening?

One more thing to consider: the bearings itself in my concept are IS41 industry bearings, which usually have a closed stainless steel body. Unlike an open BB ring like it is used in 1" headsets, these bearings offer a certain structural integrity in themselves, helping to distribute any forces coming from the steerer much more evenly onto the cup

the bracing support by the headtube would never be sufficient to prevent a gradual deformation of the cup by the bearings over time, because the headtube is sitting quite a few mm below, not in the same "plane" as the cup itself).

I'm not sure about that. The effect of the force over the whole cup would be to expand it slightly. With a normal cup that would tighten the cup against the inside of the headtube. In your design it loosens it from the headtube, which could lead to fretting

If the force is not consistent around the cup i.e., it's acting to push just one point of the cup outwards then, in a normal cup, that force is dealt with locally by the cup pushing against the inside of the headtube at that point. In your design that force acts against the outside of the headtube on the opposite side, which stretches the cup overall.

I don't think it's a bad solution, it had occurred to me that you could do this. I just think there's a reason that previous versions have actually been brazed or bonded to the headtube.

I get your point. But:

in a normal cup, that force is dealt with locally by the cup pushing against the inside of the headtube at that point.

Why should the headtube deal with that force to the full extent, it would be more likely that the cup main section which is sticking out of the heattube and has quite thin walls would be stretched, too.

In the end, there is no way getting around trying it I guess ;)
At least nothing can go horribly wrong, i.e. there's no danger involved if the external cups should gradually go loose.
Of course brazing them to the headtube would be better but then I can't take them off anymore if I have another idea and also I would have to make them out of steel instead of aluminium which takes much more time and effort on the lathe, and theres the extra effort of brazing involved.
Just making two cups out of aluminium would be super easy and fast.

there is no way getting around trying it

Well, there's always not trying it.
If 1" forks were impossible to obtain and there was a plethora of options in 1⅛", it might be worth pursuing, but neither of those things is the case.

I'm keen to learn the results now. Push on @jetski

If 1" forks were impossible to obtain

Not as easy. The forks shall be disc forks, but disc forks in 1" are actually almost impossible to obtain

I’ve considered this for years but I’ve yet to attempt a 1 1/8 in a 1 inch fork.
Any success?

If you were going to all this trouble, could you swap out the head tube?

Easy to get made though.

I'm all for clever solutions but bicycle tolerances being what they are, I'm really not so sure about this

Why should the headtube deal with that force to the full extent, it would be more likely that the cup main section which is sticking out of the heattube and has quite thin walls would be stretched, too.

But even if that were to happen in a normal headset the only effects would be that the whole fork would move minutely upwards as the (effectively) conical crown race moves upwards relative to the cup (because the ring of bearings in the lower race would have expanded slightly) and that the top cup-cone interface would loosen minutely. It wouldn't act to loosen either cup from the headtube, which is the risk in your design.

my gut feeling is to agree with the naysayers. i'm not sure how tight a fit you can get on the outside of a (presumably painted) headtube. i'd try and bond the top cup at least.

but please, please conduct the trial because i'm interested now.

It wouldn't act to loosen either cup from the headtube, which is the risk in your design.

We'll see. I get your point, but I could also imagine that it will only loosen after many thousands of km, or not at all, that's why I need to run this test.

Also because..

If you were going to all this trouble, could you swap out the head tube?

.."all this trouble" is actually just 20 minutes of lathe work, compared to the massive surgery of swapping the head tube.