Compressor turns on but not the fan.

Question. Is the flapping at a very fast pace that would indicate the speed of a fan blade hitting something? If so, you may be on to something there such as a misplaced gasket or something has dropped/fallen into the blower box via a vent.

If it is not that speed then see the following..

Hope I can give you some direction. Speaking from a generic sense all ac/heaters these days have a door or doors in the blower box that control air flow from the main part of the box to different positions when you ask for vent, heater, mix, defrost etc..

Most these days are operated with a small electric servo/motor and unfortunately the mostly have plastic gears that wear and can "thump-thump" as the motor cycles trying to open or close the flapper door.

Try to change the air direction from Max AC to regular ac to heater, defroster, bi-level etc all with the AC switch in the on position if you have a separate on/off switch.

Listen to the sound and see if it comes and goes with a different location of the selector switch.

One last question. Does it happen at idle or driving down the road at any particular speed?

Lets start with that and see what happens. This is interesting.

Good luck!

if the fan slowly comes up to speed check the connections on the resister they over heat and melt then check the fan motor itself

If it's just not blowing the blower could have worn out or something such as a mouse could have gotten in the fan and stopped it.
If the line on the compressor is getting cold then the compressor is working.

A fan is on or off not slow do any fans in your house run slow because they are old?Your thermostat fan or temp sensor is bad

Check the relay box under the hood, and any other fuse box you have in it.

Sounds like you have either a bad fan motor or a bad fan motor capacitor. When you turn the unit on with the fan not working, the compressor (the humming noise) runs and gets real hot real quick resulting in the compressor cutting off ( the nothing after) on it's inside high temp switch. This is what the compressor is designed to do so that when this happens, it does not burn the compressor out. Replace you fan capacitor, give the compressor an hour to cool, then try it. In most cases and if I'm correct, the unit will run just like you want it to. Good luck.
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capacitor may be defective or the motor is shorted or open.

It will not function until it has a refrigerant charge. There is a low pressure cycling switch in the AC compressor that will not let it function with low pressure. The system should be filled with PAG oil and refrigerant as that is what lubricates the AC compressor keeping it from destroying itself.

this not really a problem as the fan speed is controlled by the high preasure switch in the ac system the hotter it is outside the higher the preassure will be in the system as hot as its been the preasure is high already when the compressor cycles it pulls suction on one side and outputs high preasure on the other the fans cool the condenser the higher the ambient temp the faster they go so all is fine...

is this a whole house air conditioner ,
if so check the fuse panel,
check to see if the outside unit runs at all , compressor or fan.
check the ac transformer power to the thermostat
check the switch in the ac closet
doe the internal fan run if you switch to fan only.