The starter looks like a typical geared Denso starter off some Japanese automobile.
Ah, great, now I know what I'm dealing with.
I feel, based on your questions you should learn more about electricity before proceeding.
Bless your little heart, I apologize for never having worked on an electric prerotation system previously, however I am quite well-versed in electricity as well as electronics. Now that you mention it, I actually am quite intimate with electricity. In 1972 I was electrocuted in the rain by one of the 600 amp, 14.4Kv primaries off a "Y" feed while rebuilding a distribution line on the northern coast of Nova Scotia. It's a great story, really, but I won't go into it here. I also have a patent on a photoelectric alignment device, and spent a good part of my professional life as manager of a huge data center at 56 Marietta Street, Atlanta, while at the same time being the Georgia Regional Manager of a multi-billion dollar telecom - meaning I had to know quite a lot about troubleshooting all types of electronic devices and 48v DC UPS power systems, etc.
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The owner knows very little to nothing about the system, as the gyro came to him with jury-rigged wiring. The soft-start switch never worked and the safety solenoid at the bottom of the mast was fried. It was by-passed by the original owner first with welding rod which lasted a handful of starts each, then a stainless steel bar which also burned up after a few uses, and finally a soft steel bar which is how it came to my shop in need of work. So far as we know it never had two batteries.
The wiring from the DPDT panel switch to the starter was melted along four feet of 16AWG wire, and had melted several other wires in the harness. He and I both think that one wire from the switch to the starter went through the relay coil, as the "Lo" energizer, using that solenoid coil as a resistor before going on up the mast to the starter where it activated the integral relay inside the starter+ reversing gear housing. A second wire from the DPDT switch - the "Hi" - then bypassed the solenoid coil (resistor) and ran directly up to the starter, while the "Lo" exiter was still engaged and keeping the relay on the solenoid closed. But that's just a guess I came up with this morning after reading the posts here to help us out, and talking with him on the phone.
The solenoid used was a golf-cart thing that was probably rated 80 Amps long-time use, and 300 amps brief use, like 5-6 seconds. Which woudl explain why the thing is burned out.
I found a 500A battery disconnect solenoid rated at 200A long time use, which I believe should hold up to the long burn required of a prerotator. Not sure if I will need to add resistors to the "Lo" side exitation or not, but no way am I putting coat hanger wire coils in the loom of a customer gyro, LOL!
I'm just trying to help him get it back in the air without spending too much $$$. I might bring it to Bensen Days if it's ready and the weather looks like it won't rain, since all 3 of our trailers are open.
I really have to buy an enclosed trailer soon.