Drive chain goes loose and tight on back wheel turn

Any time a chain comes off it means there's slack in it or something's not tight enough. Not looking at your bike, I would have to guess that you need to loosen the bolts on the back wheels and then pull the wheel assemblies back until the chain is tight. This is probably a two person job, one to unbolt and bolt and one to apply tension. That being said, many years ago I remember bikes my friends used had a problem where if they were riding at a good speed and then stopped peddling to coast, then slightly reversed peddling, then started fast peddling again, that they would throw a chain. I think what was happening was they were creating slack in the chain by coasting, slightly reversing their peddling, then starting again. The chains were also slightly gummed up, allowing slack at this time.

Excessive drive chain noise is caused by the following, lack of proper maintenance, keep it clean and lubed, next would be correct adjustment, most bikes about 3/4" deflection at center run of chain up and down play, next would be a worn out chain and/or drive or driven sprockets, last but not least and very noisy is the chain being stretched, if you rotate the wheel the chain gets loose then tight. Always check the sprockets for excess wear when replacing a chain, the drive sprocket on the engine is hardened steel and wears the least.

chech to see that the chain has not stretched too much

Provided you have not been in an accident that has bent your frame,upper and lower triple tree, or forks, the only alignment you can make is to adjust your chain. If the the front wheel is not tracking with the back wheel you just have an improperly adjusted chain. On each side of the swing arm by the rear axle are two chain adjuster cam/wheels. The have numbered bump/stops. Both chain adjusters should be set in the same numbered bump/stop to align the front and rear chain sprockets and the front and rear wheel tracking. the adjusters are on the machine for the purpose of setting the amount of chain slack. Your chain should not be too tight or too loose. If you grab your chain in the middle and push it up and down, you should have a slack measurement of 1 1/4 inches from the bottom edge of the chain to the top edge of the chain. If the setting is off the best way to make the adjustment is to raise the rear wheel off of the ground. A race bike stand is very helpful for this and oiling and caring for your chain. Harbor freight tools sells a cheap rear stand for about $35 (buy the cheaper one piece non-collapseable stand). Pull out the cotter pin, loosen the axle bolt, turn both adjusters till you have the correct amount of slack and they are both set to the same number. Tighten the axle and recheck the slack. If you notice when spinning the rear wheel that the chain goes tight then slack, you have a stretched section of chain. If this is the case you should replace the chain and any worn sprockets at this time.

Thanks for the inquiry!
From your information I can't tell if this is a off-road bike or a street bike. With the extended swing arm I'm suspecting off-road. In either case it's very much same issues. You have two issues and we will deal with them separately.
Chain catching sound:
In low gear when you have maximum torque applied to your rear sprocket & wheel many times the primary sprocket (smallest one under the engine side cover) will slip and actually jump teeth. You indicated this is the third season, so it's seen a lot of wear. It would also explain that when you adjust/ tighten the chain tensioner the noise is gone as the front sprocket is not slipping in the chain. If this happens a couple of times the front sprocket is ruined. Pull the front sprocket cover and inspect the teeth. The teeth will be smooth and ground away if this has occurred. Most off-road riders go through 1 - 2 sprocket and chain sets per season with the abrasion form the sand and mud if they are serious about their riding. Replace front sprocket or both sprockets & chain.
Other possibilities are that you have lost your chain guide or the plastic wear plates inside the chain guide are gone and the chain is banging against the metallic sides or your swing arm making the chain jumping sound you have described. Replace the guide or wear plates to solve this.
Whooping sound on front of bike while coasting: This familiar sound appears to be related to your front wheel. The best way to test this is to set your bike on a bench, crate, or test stand so the front wheel/tire is off the floor/ground and spin the wheel. Your rim may be bent allowing the tire to rub against the fork tubes or front fender making the whooping sound. The engine noise drown's it out under power but noise is audible when coasting with the clutch in and engine idling. The simple fix is a spoke tightening and adjustment to properly true the wheel. You will hear if its a bearing or something else being rubbed against by a wooble or something out of round (run out) causing the whooping sound.
I hope I've provided you useful assistance and that this information allows you to make an informed decision as to how to proceed from here. Please be sure to vote on my response. I hope I've earned a good rating from you today.
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Happy cycling,
TF

It sounds as though the chain is too tight. Check to see when it is "freewheeled" if the chain gets tighter. Generally this will happen because the chain ring (front sprocket) was not perfectly aligned with the "spider" (thing it bolts to). You can loosen the chain a little and see if this helps. Be careful because you don't want it so loose the it comes off while riding. What I usually do is adjust the tension of the chain with the sprocket in the tight spot. Kind of tight but without the "grinding" feeling.
Good luck.

Get the rear wheel off the ground and adjust the chain tension. When checking the tension spin the rear wheel and check that the tension does not go slack or get tighter. If the chain is riding up and down the sprocket teeth then the chain is stretched and needs to be replaced. This is the grinding/clunking noise you are hearing. If the sprockets are worn consider a chain and sprocket set, it can be a less expensive option than buying the parts separately.

To replace the main drive chain,you need to remove the chain cover on the engine case which should have a couple of bolts or screws securing it. Loosen the back wheel by untightening the rear axle bolts and then loosen the rear adjusters either side. Make a note of their positions they need to be be the same position either side, although the new chain may need a different setting but as long as they are both the same tightness each side is what matters.Loosen the wheel and move it forward when the axle is loose and the pin holding the axle nut is removed.Unscrew the two 6mm bolts holding the front sprocket a the engine side of the chain.Turn the part on the front sprocket where the two bolts have just been removed so that it lines with the gearbox splines and pull it forward along with the front sprocket at the same time as you push the loosened rear wheel the chain should pull away from the teeth on both sprockets.
To put on the new chain just reverse the proceedure.the ajusters need to be tight but not too tight just enough play to move it up and down slightly remember both adjustes need to be the same position each side and check that the wheel is in line with the bike and straight as uneven adjustments will cause the wheel to be turned slightly off the required vertical position.

Tighting the bolt very tight is OK but over doing it can collaspe the spacers. It's more likly that if loosing the bolt corrects the problem it's been put to gether wrong. Look at the bike bandit .com for your bikes rear wheel and see if yours has all the right parts in the right order. Maybe the rear brakes are worn out or something in the brake came loose. The chain and sprocket can make alot of niose. Axel not the same on both sides. chain dry or too loose or too tight. Chain gaurd touching chain or the tire. With the wheel jacked up turn it and work the brake to see if the noise changes. The wheel should turn very easy cheak breakes and chain for probelems wheel bears do fail but mileage needs to be high or wet.

I noticed a repetetive whirring coming from somewhere near the front sprocket- Or at least I'm pretty sure that's where it's coming from. It's not quite a grinding but it sounds like some sort of contact problem. I jacked up the bike, and the rear wheel spins freely in neutral except for a small area which seems to need just a *little* more force to spin around. Not sure if it's from the rear wheel or from whatever might be wrong up towards the front. The noise has an accompanying gritty feel, which rather reminds me of a bent rotor rubbing lightly at a certain point once per cycle on the brake pad. No amount of variation of pressure or speed in braking seems to affect the problem. There's no obvious performance issues, and I only hear it when I've got the clutch pulled in (probably because it's not loud enough to hear over the exhaust, not because it isn't happening). I ran the bike downhill in neutral and it still makes this noise. I lubed and adjusted the chain a day or so ago, and the adjustment notches are at the same point on each side. I had also ridden over a set of train rails cut deeply into the road, but I doubt that had much impact on anything. Any ideas?,Sounds like your chain may not have stretched evenly. You also may have it adjusted too tightly. You should always adjust the chain so that you have the specified slack at its tightest point. Do you know how many miles/KMs are on on this chain/sprocket set?,,,