On Sun, 06 Jan 2019 22:23:58 -0500, Frank Krygowski wrote:

I built up a lovely kids bike with a Shimano 333 coaster brake 3 speed
hub. The young lad rode it on the ride we took today.

There's been a consistent problem keeping gear #2 adjusted. It acts as
if an internal spring responsible for pulling the hub into 2nd gear is
somehow weak. That is, the little circle with the letter N doesn't
settle down in its target hole. As a kludge, I used a bit of inner tube
as a rubber band on the bellcrank to help pull the hub into 2nd, but
that rubber decayed and I didn't have a spare. In any case, I've
readjusted it several times in the past couple months.

I'd suspect a dragging cable, except the hub shifts into the other gears
just fine. 3rd gear requires the hub to retract the cable even more, and
shows no problems. Of course, I lubricated the cable during
installation. I even spent time grinding away part of the kickstand
mount to give the straightest possible cable routing.

Unfortunately, the bike is an hour away and I normally see it at most
once per week; so some things are difficult to check, other things may
be difficult to remember. But are these hubs extra sensitive to cable
friction? Instead of the full housing, should I consider scrounging
parts to run bare cable over pulleys, etc?

But if anyone has tips, I'd be grateful.

Its a few decades snce I rode with a Sturney Archer 3 spped hub and my
understanding of how they worked was you set #2, which was direct drive
and adjusted for #1/#3.

As you mentioned, things like cable drag, etc can affect gears meshing.
How old is it? I basically wore mine out. Well, dad purchased it second
hand and it was the kids bike although it ended up being me who rode it,
every where & everyday.

Is the hand piece(?)/adjuster correct for the device? I vaguely remember
one hand piece not being suitable, aka either twist grip or thumb
shifter. They were "indexed".