Hi,

I don’t know about Canadian cars if you have one or not but there is a wire underneath the hood that will pull in the starters solenoid and engage the bending to the flywheel.
You just need the switch in position two for it to run.
Once the ECU sees a signal that the engine is turning all the other stuff turns on!

The short wire is located near the oil dipper stick. It has a female spade end on it.
You jumper a wire from the positive side of the battery and it energizes the solenoid coil.
Under normal circumstances it would roll the engine.

The problem is inside the solenoid with the plunger bottoming out when retracted.
The contacts do not make up due to lack of stroke.
The rods inside the plunger are not pushing the spring far enough to put pressure on the contact bar that lays across the terminal studs.

The plunger needs close the contacts and push on past a little farther for the spring to apply that pressure.
That is why I say, check for that extra distance of plunger movement past the sound or reading on your ohmmeter.

I notice your calculations for the .003 shrinkage but that is not the issue. I used the term “Hare breath” in relation to a rabbit’s breath.. It’s something never seen or felt of because their short in length. A rabbit does everything thing fast, pretty much! (:)

In the cold, it’s more about the lack of cold cranking amperage! When there is higher amperage available and applied the current can jump a slight gap. The gap occurs when the previous burning and pitting of contacts surfaces increase throughout the earlier years of use.
Just guessing but I would say .015 per side or maybe all together could occur before it runs out of factory adjustment.
The modification restores that movement, lost, to make the contacts work properly again.
You can shorten the plungers body by .050 and deepen the 60 degree angle that much but requires a lathe to make it easy. Otherwise you are looking at a drill press for using a countersink and a hand held Angle grinder.
Moving the rod in the body is far easier as long as you lock that rod by a tack weld as the factory does or at least center punch it around the slotted rod to “Stake it” tighter.
I don’t think there is a way to lengthen the rod in the solenoids core.

Interesting that you brought up railroad rails!
The calculations that you use are correct for expansion and shrinking of steel lengths.
We used a similar equation (.0000065) for forged steel cylinders up to 40” in diameter by 8’ in length with two inch walls. We shrunk them onto solid steel cores as a water jacket in the Aluminum Continous Roll Casting business.
In my way younger days of the 70’s, I worked in making big machinery for ten years! It was a domestic business that exported world wide but by 1999, it was chopped up and it was exported!
The Assembly department heat soaked them for hours to like 1200 to 1500 F. A overhead crane dropped them over the cores that were standing up. You aligned it and lowered as fast as you could! That cold core sure could suck the heat right out!
You needed hot because you didn’t want it cooling in transport or have it get stuck, almost on all the way down!

On railroad rails they ever so often put in slip joints or side plates with a small gap allowed.
Today the rails are made even longer due to better equipment to move them and the steel is stronger.
Today many are thermite welded together on the site.
Thermite was used in WWll, as I was told by my Stepdad, that was a Ranger at Normandy’s Point Du Hocke and that they carried the chemicals on them!
They put it into the breaches of German artillery guns to disable them. It melted them shut!
Look up “thermite welding” it’s interesting stuff!

The technology of laying rails has removed lots of the clacking between them. The rail cars themselves are even made quieter.
The ties and spiking have all changed somewhat!
The physics remain the same! Railroads have always “demanded” pathways to be surveyed to use straight rail wherever possible due to that change in lengths.

Phil