I've got a mechanical design for a piece of catering equipment and I need to figure out a way to achieve some variable speed control for a slow RPM motor. This is a direct drive application (imagine the motor shaft connected to the center of a lazy susan - with the load supported by a bearing plate). I've been using a uxcell Synchronous Motor AC 110V 50/60Hz 6W 2.5/3RPM . What I really need to do is utilize a foot pedal that would allow the turntable to speed up to anywhere between 6-20 RPM. My background is the mechanical side of engineering so I'm lacking knowledge about the exact electrical components I could be using. Any suggestions are appreciated! I'd like to keep the cost at or below $200 if possible but I'm willing to consider anything at this point. Thanks!
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Your synchronous motor for that output RPM is already going to be an assembly with an integrated gear box. A synchronous motor is going to give you a normal constant no load RPM due to the fact that the motor design is based off the AC line frequency. For your variable speed application you will need to source a motor that you can vary the speed of (a small DC permanent magnet brushed motor will be the easiest to work with) and mate that up with a suitable gear box to reduce the speed to the range that you need.
I would suggest that you do some research into the drive motors and gearing systems used for small wheeled robot type projects. It should be possible to use your mechanical abilities to adapt such components to your application. The robot research will also expose you to the available electronic technologies for controlling the motor speed as well.
There is also a huge base of low cost hardware for motor speed control and components around Aruduino based home built 3D printers. You may want to look into that as well.
It may be possible to utilise a sewing-machine motor and speed control with the right gearbox. something would need to be arranged to hold the pedal part-way down. this will be much more bulky than your existing system, so you may need to use a flex-drive between the motor and the turntable.