They don't call them window shakers for nothing. Knowing what the rattle is (window or unit itself) is important to know. It could be the front cover unit or, most likely the filter itself. Take it out the next time the rattle starts and see if the rattle goes away.
Over time, window units begin to lose their "tightness" and you'll notice various vibrations. One of the culprits might be the fan blades. No matter how careful you are with the filter, the inside blade can pick up dust. The outside blade most certainly picks up dust as well as getting corrosion on it. These things throw the fan out of balance just enough cause annoying vibrations. You can put foam strips wherever the unit actually touches the window sill, sides and upper portion which the window lowers down on, to help damper the unit from the window frame, if that is where the rattle might be.

Just an fyi...turning the unit on and then off and then back on can cause problems. From your description, it appears the noise starts when the compressor starts which in itself will cause vibrations. The mounts holding the compressor in may need to be snugged down. The compressor mounts are rubber and of course rubber hardens and is not as resilient as when new. But when you turn the unit on and then off and then back on, the noise may stop because the compressor is not running. There is a safety device which prevents it from starting right back up because that could damage the compressor. The overload protector trips in the compressor circuit (for 3-5 minutes) because the refrigerant pressures need to equalize and then trips back to let the compressor come back on. They are not designed for repetitive functioning and you'll wear them out faster. Some are external and can be replaced but, some newer units are built into the compressor and cannot be replaced. So, don't cycle the unit off and on to get rid of the noise.

The next time it makes the noise and you determine it's not the filter or the front cover rattling, try grabbing opposite corners at the front of the unit and putting pressure first one way and then another. If this stops the noise, you'll know which side you'll need to shim.

I hope this helps!!

I did the things you said and the noise lessened but did not quit. It was a bearable level so I forgot about it. Just a little while ago it tripped the circuit breaker. Not the one in the apartment but the one in the units plug. I tried the reset and test button and it would not restart. I pulled the plug from the wall and noticed that it was very very warm. It looked a little like it was melting slightly by the prong. I assume it would be dangerous to try to get it to run in this condition? What might have caused this?