My A/C just went out today. When I turn the air on it's cold for maybe a minute, but then goes warm. The other day when I had the A/C on while idling, there was a shaking of some kind under the hood that lasted maybe 10 seconds, so I'm wondering if that's related.

 

Check whether clutch plate (end of ac compressor) is rotating when no cooling occurs.

If no clutch plate rotation, problem is likely electrical; open clutch coil, excess clutch gap, faulty AC relay, etc.

If clutch is rotating, the problem is low charge, faulty air mix, frozen evaporator, etc.

good luck