If condenser area is squeaky clean see if there is a way to reset the uit electronics. Check with Rudd for reset and then try again. If that does not do it then there may be an obstruction in the cooling system causing the high pressures and making it turn off for self preservation. You can also carefully push down on the contactor relay to see if the compressor starts at all - maybe there is an issue with the compressor.
#1 problem with AC units - dirty coils, this leads to many other problems and is caused by neglecting to change filters monthly. You can buy coil cleaner commercially, but it is not the coils you see that get dirty. There is another set inside that is usually not visible. If you look outside and your unit is "iced up" there is a 98% chance that is your problem. Sometimes it could be iced and you can't see it from the outside. Solution, turn AC off and blow fan to thaw. When thawed, clean coils with coil cleaner and coil comb.
Relay switches, you already pushed it so you may have a bad switch.
These are the two most common problems. A mechanic will be quickl to say it is your compressor for $. There are normally only half a dozen things that go wrong if your AC is 10yrs or less. If it is over that, it could possibly have a more serious problem. (Secret hint - I have known AC's to last 30+ years if the filters are changed regularly. Letting dust and dirt build up on coils invites a host of other problems due to the stress it puts on the other components.
Usually short cycling means that High side pressure is getting to high (due to not working condenser fan, or dirty condenser coil), or low side pressure is getting too low (might have low level of freon or restriction in sealed system). low side would drop dramatically if indoor blower is not running, so if air is not circulating thru evaporator coil, coil will freeze up and low side would be much lower then normal. Here is video how I found issue with package AC not coming on. Thermostat call for cool, but condenser would not come on, indoor blower runs ok.
First of all, clean your outside coils so that you are assured of good cooling air. The compressors have an overtemp device that is usually internal to protect them against heat. If you have an air compressor use it to blow out the coils good. If the compressor will start, the capacitor is good so it sounds like a cooling problem with the compressor
what type of system is it, if it's it Rheem the o/b needs to be set for B, if not check the freon charge, or thermostat, there will be a point where the aux. heat comes on when you turn it up, poss. only 4 degrees, is it a mercury stat or digital ?
More than likely, it is the hum of the coil in the contactor. Generally, residential uses 24 volts to energize the coil built into the contactor which pulls in a set of contacts (hence the name contactor) to send 220 volts to power the compressor & condenser fan motor. Contactors are relatively inexpensive... so if the noise bothers you, replace it or have it replaced. Put your mind @ ease, the humming noise will do no damage.
The evaporator coil is probably frozen too.The fan might of failed and it caused the evaporator coil to freeze along with the suction line. Based on what you said about hearing a noisy blower motor noise. I believe the blower motor failed and caused your problem.There's only a few things that cause freezing; 1. blocked or no airflow 2. low or loss of refrigerant. You have no airflow, most likely because of a defective blower motor.