I have an old cruising bike with 21 gears. A few months ago I was pedalling back home when my chain broke, and I was only able to get the chain replaced this week. After having it replaced I noticed it was 'skipping' and the problem seems to be more severe in certain gears. Now, the teeth are a 'bit' worn out (particularly in the middle gear at the front), and the 'skipping' didn't happen with the old chain (makes sense as the old one probably stretched a lot). Any idea what could be causing this?
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As you asked for an explanation, why this skipping is happening, here it is:
For the beginning let's consider an unused chain on an unused sprocket. In this case the distance between the teeth of the sprocket is the same as the length of the chain links. Therefore, if you apply a force to the sprocket via the chain, the load is equally distributed over all teeth that are in contact with the chain. Due to the force the chain lengthens over time (not much, about 1-2% is enough to make the chain unuseable) which means that the chain links no longer fit the sprocket's teeth distance and also the load distribution isn't equal any more.
Instead, the tooth that is closest to the direction where the force comes from (the last one on the sprocket where the chain leaves towards the chain-blade and the first one one the chain-blade where the chain comes from the sprockets) gets the most load and therefore is wearing out a bit more than the other ones. As everything goes round and round and every tooth is in the position to wear out more than the others, the teeth will slightly adapt the valleys between them to fit the length of the chain.
If you replace now the chain by a new one while keeping the cassette the same (it would be the same if you replaced the cassette keeping the chain), then the chain doesn't fit any more to the cassette that has adapted to the length of the used chain links. If the length difference between old and new chain links is too much (as already said, some 1 or 2% are enough) the chain will slide up the shoulders of the teeth and skip into the next (or even the next-nearest) valley as soon as the applied force gets too high. One can make the components last somewhat longer by changing the chain early enough, but as both, chain and cassette are wearable parts they have to be replaced sooner or later.
It depends on different factors how fast this wear-out process will be:
This means that a road bike, where you normally don't have much dust and try to pedal rather smoothly without any rapid changes in force, can go several thousands of kilometers on one chain. With mountainbikes where you have the exact opposite (lots of dirt, often sharp load peaks when you go to rough terrain) the rule of thumb that I learned some years ago is to replace the chain every 2000km. Even then you will need a new cassette every second or third chain.
The likely cause is a worn cassette if the skipping is in the rear. Since the gears you use the most will wear the fastest the skip may be worse on some of them. But also look at the jockey wheels in the rear derailleur. If they have alot of wear they may not be keeping the chain aligned. If the front is worn and the back has never been replaced you are likely due for a cassette.
If a chain gets sufficiently worn, it will start inflicting wear damage on the chainrings and sprockets (in extreme cases, causing the teeth to take on a hook shape). This wear will cause problems with gear changes and pedalling, and your only option at that point is an expensive replacement of a significant portion of the drive train (You'll probably need to replace the chainrings, sprockets and cassette, and possibly your deraileurs as well if you can't get parts compatible with the current ones).
If you're lucky, your deraileurs just need tuning. Badly aligned deraileurs can cause all kinds of issues, but they should go away if you can sort the alignment out. There's a couple of adjustment screws you can use to calibrate the gear shift.
Keeping your chain in good working order and replacing it quickly when it starts showing signs of wear is important, because chains are one of the cheapest parts of the drivetrain and can cause very expensive damage if worn or damaged chains aren't replaced quickly.
If they're dirty or rusty, clean the chains (using a scrubber like http://www.parktool.com/product/cyclone-chain-scrubber-CM-5-2) or just lubricate it with WD-40 if you're lazy. WD-40 isnt a proper lubricant, but will dissolve a lot of things.
If your chain is clean, check for chain wear (unlikely since it's new).
Another possibility is that you have the wrong chain. There's different chains for 7, 8, 9, and 10 speed rear gears/cassettes.
But MOST IMPORTANTLY, I'd say check your derailleur alignment. There might be a dial/twisty along the cable that can help with alignment, or there's two small screws on the rear derailleur to change the alignment. You'll want the chain to be centered on each ring of the cassette. Here's a guide that sorta helps: http://sheldonbrown.com/derailer-adjustment.html