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This article is mainly about reassemby of the original Shimano 3 speed hub. While it doesn't explain how a 3-speed hub works, the author's explantation of 3-speed-theory may be of interest. Understanding the theory may help with fixing the hub. But spare parts may be inpossibe to find and a quality repair by building up worn surfaces by welding may not be feasible.
If you took apart the Shimano 3-speed hub to overhaul it, you need to know how to adjust it when reassembling it. This is not the simple adjustment of the bell crank cable, which is adjusted so that a circle is centered when in 2nd gear and is visible to anyone standing on the ground next to the rear axle. In order to understand these instruction, you need to have a disassembled hub to inspect. It's easy to take apart, but not quite as easy to put back together correctly.
This adjustment is for the case where you took apart the hub starting from the brake arm side an is not for the case where you have started from the other side by first removing the sprocket (over which the chain goes). It assumes that you haven't repositioned the ball bearing cone for this sprocket.
On the brake-arm side of the hub there's an internal adjustment controls the bearing clearance. There can be problems (including problems braking) if it's not adjusted correctly. The adjustment of the bearing clearance is done by locating (with a wrench) a thin locknut on the axle inside the hub. You turn the locknut to the "correct" position, determined by trail and error. Then you put a non-turn washer on it and then a nut that looks just like the above locknut at one end, only it's much longer and most of it is cylindrical. You tighten down this long nut (which looks from the top like a short piece of pipe) onto the locknut but these two nuts have the thin, non-turn washer separating them, so turning one doesn't turn the other. Since these 2 nuts go onto the threaded axle, the axle needs to be kept from rotating using an adjustable wrench on one of the two flat spots on the axle. The flats of these two nuts should be close to each other when assembled, separated only by the thin non-turn washer.
To make this adjustment, you are supposed to use a special wrench since the two nuts are just flats on a cylinder, and you can't get very good access to them. But you can get by with just a 15mm or 19/32" open end wrench and be content to only grab onto only part of the flats of the nuts with your wrench.
The play (clearance) in the hub bearings is directly controlled by the position of the brake-cone/bearing-race (one part) that fits over the ball bearing cage (with little balls in it). This brake-cone not only spreads apart the brake shell (like brake shoes) when you apply the coaster brake but has a bearing race on it on which the ball bearings roll. This brake cone fits over the pipe-like long nut and is stopped from going in more by the fat end (larger diameter) of this long nut. Thus the position of the long nut sets the bearing clearance and since the thin locknut controls the position of the long nut, the locknut position determines the bearing clearance.
When you put the hub back together and are screwing on the thin locknut, screw it gently in as far as it will go and then back it off about 2 or 3 turns. Make sure that this nut is fully on the axle threads, since when the threads end the O.D of the axle decreases allowing the nut to go past the end of the threads. If the locknut is only partially on the axle threads, you can easily break off part of threads when you tighten down the other nut. Better yet, if the wheel bearing was adjusted correctly when you started to take apart the hub, check out the position of this thin locknut before removing it. To do this, when you are ready to unscrew it, do just the opposite and screw it on more as far as it will go. Count the number (or fractional number) of turns. Then when you put it back you'll know how many turns to back it off.
So to get this adjustment correct, you may want to experiment at first when you assemble this part of the hub by tightening things only hand tight while intentionally neglecting to put back the spring and brake shell. Just assemble the bare minimum (including the brake-cone/bearing-race and the ball bearings) and check the bearing for play while holding the parts together tightly. Be sure that the brake-cone is pushed in all the way which means that it must first be rotated to a position where the tab inside the hub is inside one of the 3 notches of the brake cone. There should be a very slight amount of play (looseness). When you tighten the long nut against it, there will be even less play.
Note that when it's adjusted correctly the end of the long nut will project out of the center hole of the brake cone by a fraction of a millimeter (just slightly). Then when you tighten the nut on the axle which holds the brake lever to the brake cone, it stops when it hits the end of the long nut, permitting the brake lever (and brake cone) to rotate somewhat freely with respect to the axle. The axle should also freely rotate with little (if any) play.
As was mentioned above, the brake cone (also serves as a ball-bearing-race) fits over the long nut. This brake cone has 3 notches in it: 2 large and one small. All 3 notches must fit in their corresponding tabs: 2 on the brake shell and a small tab of the friction (brake activating) spring (doesn't look like a spring). So when reassembling, rotate the brake shell until the small friction spring tab is centered between the two brake shell tabs. When installing the brake lever (connects to the bike frame), the brake cone might come out of its tabs (since a spring is trying to push it away from these tabs). So be sure that the brake cone is all the way in before wrench-tightening the nut for the brake lever. If it has fallen out of the friction spring tab, just rotate the brake lever (with the brake cone attached) until it will push easily into place.
To determine if the brake cone is all the way in, look at the position of the dust cover on the hub. If you see part of a groove of the hub shell, the brake cone is not in all the way, perhaps because the small notch is not aligned with the friction spring tab. Of course, you never want to wrench-tighten the brake lever nut to try to force the brake cone to move into a position it shouldn't move into due to a notch not being aligned with its tab.
When the wheel is put back on the bicycle, it should have little play. Wiggle the wheel rims very gently back and forth. They should only move about 1/2 mm or so. The wheel should spin freely, but not quite as freely as the front wheel since there's a friction spring inside which allows you to apply the brakes. If you spin the wheel when the bike is upside-down, the wheel should remain spinning for at least a half-minute before it comes to a halt. Mine spun for a minute using automobile ATF as a lubricant. The front wheel should spin like this for a few minutes before it stops.
There are two dog rings that are pressed or screwed into the hub shell. Thus the dog ring outside diameter (O.D.) is about the same as the inside diameter (I.D.) of the hub shell. Both are clearly visible on a fully assembled bicycle. They are extensions of the hub shell and rotate with the spokes, wheels, and hub shell. The location is just outboard of the spokes with one dog ring on each side. One has a groove on its circumference and the other has a small step so that the O.D. is slightly larger nearer the brake lever.
The dog ring for the high gear (sprocket side) screws into the hub shell. It has normal right-hand threads and there is a notch for a tool to remove it. If you don't have the tool, you can put a hose clamp (automotive) around it and then use large pliers to grip it and turn the clamp (along with the dog ring. This dog ring seldom comes loose by itself.
But the other dog ring was pressed into the hub shell at the factory (but not tight enough ??). The I.D> also serves as a brake drum and thus must take braking torque. It may break loose and start to rotate. At first it may only rotate a few degrees with respect to the hub shell. But as you ride the bike and use the brakes it is likely to get worse over time, eventually reaching a state where you have little or no coaster brakes and the bike's low and middle gears slip. For safety's sake don't let it get this bad.
How to repair this? I attempted to repair it with JB Weld epoxy, the type that claims to maintain it's strength to 650 deg F but it failed when I went down a steep grade and the hub got too hot to touch (but not likely 650 deg F). I had cleaned it as describe as follows. Perhaps soldering or brazing would work.
Before you attempt to reattach it you need to remove the dog ring and clean it say with lacquer thinner. Also clean the inside of the hub shell where it will fit in. But there's a problem because (at it not clear if the dog ring should not be driven all the way in since this might make the hub length too short. It should be pounded in just as far as it was when new, but I don't know what this was. Let me know about this
To remove the dog ring for cleaning, use a punch and hammer on it. The best punch would be say a large socket. I didn't have such a socket and instead used an old punch with a rectangular tip. I had to tap at numerous spots around the periphery to pound it off.
To replace a broken planet gear axle use drill rod obtained from a hardware store. Size 28 is 3.56 mm and replaces the original 3.50 mm axle. Drill rod is stronger than ordinary rod and hopefully will not break again.
Hope this is helpful. Let me know about any changes/corrections needed (my original instructions were wrong). I've only done this three times and didn't have any manual or other help so there are likely some shortcuts or faster ways to do this.