this is an aircraft avionics mystery! it's made by Bendix, but i have no part number because the cover is missing (bought it this way from our local electronics flea market). as this video shows, i got it to work, and this thread explains how...
the electronics module is connected to the rest of the unit with a 15-pin D-sub connector, so it comes apart easily. on the other side, you can see a transformer on the far left, and next to it are two magnetic amplifiers!
i spend a few hours puzzling over the electronics module and reverse engineered a schematic. this is step 1, which matches closely with the physical layout.
closeup of the Digital Analog Converter. based on the patent numbers, it appears to use a system of brushes to convert the shaft position to a set of parallel bits.
on the other side of the unit you can see two more synchros as well as a "motor and rate generator" which is basically just a motor with an extra winding that generates a voltage proportional to the motor speed.
there are 3 26-pin connectors. there are four synchros inside, a motor, a "digital analog converter", and two large potentiometers. there's also a gear train and an electronics module on top.