I've got a lathe that runs on 3-phase power but my home obviously has single phase. The lathe came with a power supply separate from it that looks home made. It has a 220 volt motor hooked via v-belt to another motor which is 3-phase 440 volt. How do I use this setup? There are no wires connecting this power supply to the lathe motor wires. Can anyone help?
You likely have what is known as a three-phase rotary converter. A quick google search will present uncounted returns for various designs. A friend of mine had one that required a kick-start in order to operate. He had a flywheel attached to the 220v motor which when switched on would not rotate until he struck it with his foot. Once up to speed, it then provided to the lathe motor the necessary power configuration to operate.
I would not begin to suggest that your configuration matches his, as his was home-spun and created without access to the internet! The links I've found have variations which may be challenging to match to yours.
One option you have is to replace the rotary converter with what is called a VFD converter. It is going to be a simpler device, with the added benefit of speed control as well as likely a more reliable bundle for your application. Unfortunately, it is an additional purchase, while you already have the devices on hand for the rotary phase converter.