Contact the pet store and buy special crayons and mineral stone for hamsters. Put them in a cageand hang it better or reinforce the bars so they were always in sight of the animal. At the pet store you can buy special treats for hamsters promote grinding of the teeth, which they, like all rodents, grow throughout life.
Place in the cage a wooden block or twigs (all except cherry, plum or coniferous trees). Multiple pieces can be consolidated and on the bars of the cage. Periodically replenished supply of twigs.
You can't miss the bars of the cage sharp odorous substances (mustard, hot sauce, etc.), as this "treats" the hamster may die. The best thing you can do is to wipe the rods with a weak solution of chloramphenicol or lemon juice to the pesto, tasted once again the cage "by heart", I realized that it is extremely tasteless.
Buy at a pet store spray "Antigryzin", which can be sprayed on the bars of the cage. However, in many cases, the spray is useless, and the hamster continues to chew on his home as if nothing had happened.
Please note which side of the hamster chewing the cage. If he's eating her from the door side, you may just want to walk. Take it out of the cells and release the run or put in a special walking the ball to then not find it. After the animal frolic on freedom, again will put him in a cage, but hold it with the door to a blank wall. Perhaps, then, the interest of the hamster to the bars of the cage somewhat weaken.
Buy a new home for your hamster: aquarium tank or terrarium, put pieces of wood and stones. Chew everything from the hamster, of course, will not stop, but you almost will not hear.
If possible, move at night the cage with the hamster in another room or in the bathroom. In extreme cases, cover the cage with a dense blanket, to absorb the sounds.