All else being equal, a wider rim will make it worse because the tire will bulge out more.
Lets say you got a wider hub with your wider wheel.... or just got a wider hub. You have to fit it between the forks in the back, that is the potenital issue. You may be able to "cold set" the frame if it is steel. If it is alloy or carbon, you are probably screwed. If you cold set it (bend on it) be sure to do a string test to make sure the same thing happened on both sides.
Hubs are getting wider all the time on multispeed bikes and the new frames have to be built wider to match the newer wider hubs. Chainline usually moves out too with each new change. A wider hub would help a lot, but you have to get it in the frame....
Lets have an extremely exaggerated example. Lets say you widened the rear of the frame an inch (no I dont think you could widen it anywhere near that much without ruining it, but maybe you could widen a little bit in the real world).
Now, that we have an inch more to fill in the frame, so lets add a one inch spacer to the ARM SIDE of the coaster brake between the cone and the locknut. We have widened the bike an inch, (1/2 inch per side) and moved the wheel an inch to the right (chain side). Now the tire is 1/2 inch to the right of center. Now we re-dish the wheel to get the tire centered in the frame again (move the tire 1/2 inch left).
In doing all this, we have gained 1/2 inch of chainline at the rear, but had to widen the frame at the rear forks a whole inch to do it.
If we now make a change in the front to increase the chainline 1/2 inch so it matches the rear, then the chain will run straight again. Now we have the chain 1/2 inch further from the tire than it was before.
So coming back to reality, yes you can probably buy some different wheel that solves the chainline issues in the back, but that part is easy. It is the bike itself that has to change. That wheel will be wider at the locknuts, and you will have to get it in somehow, and then you will have to change the chainline in the front to get it to match.
As long as the chain is on that bike, you could probably get away with a mismatched chainline front to rear if you had to, but it works against you. Another example: Suppose you widened the back 10mm, and gained 5mm in chainline at the back, and did nothing at the front. The tire looks like it is about 2/3 of the way back on the chain, so you would only get 2/3 of the 5mm (3.3mm) for extra tire clearance. If you make the chainline match in the front you get the whole 5mm tire clearance.
Some modern coaster brakes have 3 notches and a ring to attach the sprocket. I think thats what you meant you have. If you can find a sprocket that is offset, and install it offset it to the outside of the wheel, that gives you more rear chainline with no downside and no other changes. Then shim the crank (put a washer between the right cone and the chainwheel) if the chainwheel is also too far in. Make the chainlines match. It might be enough to fix your problem.
Lets say you got a wider hub with your wider wheel.... or just got a wider hub. You have to fit it between the forks in the back, that is the potenital issue. You may be able to "cold set" the frame if it is steel. If it is alloy or carbon, you are probably screwed. If you cold set it (bend on it) be sure to do a string test to make sure the same thing happened on both sides.
Hubs are getting wider all the time on multispeed bikes and the new frames have to be built wider to match the newer wider hubs. Chainline usually moves out too with each new change. A wider hub would help a lot, but you have to get it in the frame....
Lets have an extremely exaggerated example. Lets say you widened the rear of the frame an inch (no I dont think you could widen it anywhere near that much without ruining it, but maybe you could widen a little bit in the real world).
Now, that we have an inch more to fill in the frame, so lets add a one inch spacer to the ARM SIDE of the coaster brake between the cone and the locknut. We have widened the bike an inch, (1/2 inch per side) and moved the wheel an inch to the right (chain side). Now the tire is 1/2 inch to the right of center. Now we re-dish the wheel to get the tire centered in the frame again (move the tire 1/2 inch left).
In doing all this, we have gained 1/2 inch of chainline at the rear, but had to widen the frame at the rear forks a whole inch to do it.
If we now make a change in the front to increase the chainline 1/2 inch so it matches the rear, then the chain will run straight again. Now we have the chain 1/2 inch further from the tire than it was before.
So coming back to reality, yes you can probably buy some different wheel that solves the chainline issues in the back, but that part is easy. It is the bike itself that has to change. That wheel will be wider at the locknuts, and you will have to get it in somehow, and then you will have to change the chainline in the front to get it to match.
As long as the chain is on that bike, you could probably get away with a mismatched chainline front to rear if you had to, but it works against you. Another example: Suppose you widened the back 10mm, and gained 5mm in chainline at the back, and did nothing at the front. The tire looks like it is about 2/3 of the way back on the chain, so you would only get 2/3 of the 5mm (3.3mm) for extra tire clearance. If you make the chainline match in the front you get the whole 5mm tire clearance.
Some modern coaster brakes have 3 notches and a ring to attach the sprocket. I think thats what you meant you have. If you can find a sprocket that is offset, and install it offset it to the outside of the wheel, that gives you more rear chainline with no downside and no other changes. Then shim the crank (put a washer between the right cone and the chainwheel) if the chainwheel is also too far in. Make the chainlines match. It might be enough to fix your problem.
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