Pat,
According to ISMACS {
http://www.ismacs.net/singer_sewing_...s-400-499.html } the 413 series was made from 1972 to 1978. (the 1973 in my post above is a typo). The 413 from 72 to 73 and the 413K from 72 to 78.
I am NOT an X-spurt on these machine but I have done two vertical shaft top gear replacements. I'll try to describe what I did.
>Remove: the slide plate, needle plate, bobbin case, presser foot, then the top, the face plate, the bottom plate and the plastic trim on the front of the machine.
>Loosen the timing belt adjuster and slip the belt up off of the drive gear. Easier than taking the belt off.
(NOTE: At this point check the vertical movement of the shaft. You'll want to get it as close to this as you can when you put the new gear on. I should have checked mine with a gap gauge (feeler gauge) before I took the gear off, but I didn't think of it. )
>Using a long hollow ground screwdriver, slip it in from the outside of the machine through the "STITCH WIDTH" lever slot, loosen the two set screws on the gear. These screws will be VERY TIGHT. So make sure the screwdriver is straight into the slot.
>Then you'll need to reach under the bottom and pull the shaft down just enough to slip the old gear out.
>While the shaft is down and the old gear is out you'll need to clean all the debris and gear fragments out of the top of the machine and especially out of the teeth of the driving gear.
>Then slip the new gear in and push the shaft back up. Do not set the gear so far down on the shaft it is too tight. There needs to be just a tiny bit of play in it.
>Once you get it on and set so there isn't too much play, then you will have to tighten those set screws as tight as you can get them. If you don't they will loosen up. Tighten one, then the other, then repeat until you just cannot tighten them any more. Don't be timid with them, remember how tight they were when you took them off.
>For just because I put some Tri-Flow grease on the gears. I have read yes and no on this but I do know the Tri-Flow synthetic grease won't hurt and probably helps, so I use it.
At this point the machine is way out of time. Do not worry about where the vertical shaft is in relation to the timing belt, it makes no difference.
>Look carefully at the needle bar. You'll see two timing marks. Slowly turn the machine until the needle shaft is all the way down. The top mark should just be visible under the bottom of the aluminum piece that moves right to left as the machine ZZs.
NOTE if the machine binds as you turn it, you got the gear too tight. It should turn freely.
>Continue turning the hand wheel until the bottom timing mark is at the bottom of the aluminum piece as the needle bar is moving up.
>Now with the stitch width on "0", the needle position at the center point, and the flexi stitch "off", rotate the hook until it's point is centered behind the needle. ( The repair manual says use a size 18 needle. Probably so you can see through the eye. )
>Slip the timing belt back on the drive gear and return the adjuster back to were it was. Not too tight.
You'll need a hex key (Allen wrench) or a bit driver with the correct size hex bit for the driven gear on the bottom of the hook.
> Slowly rotate the machine several times to seat everything and then reset the needle bar like we did in the step above. The hook will be in the neighborhood, but not in time.
>Loosen the two set screws on the hook shaft so you can turn the hook. (Before you loosen the set screws check the vertical play in the hook shaft like you did the long shaft. You can get this one too tight or too loose too.)
>Rotate the hook until the point is centered behind the eye and gently lock down one of the set screws.
(There may be slack in the system. There was in both of my machines. You might be able to rock the hook back and forth a bit with your fingers. When I timed my two machines I tried to set the hook in the middle of this slack point with it's point behind the needle. I don't know if this is right wrong or what, but they work.)
>Turn the machine through several rotations again and double check the position of the hook. Adjust as needed.
>Once you get the hook timing where it needs to be, tighten those two set screws as tight as you can get them, just like you did the those on the top gear.
>Test run the machine under power and then check the timing one more time. If it's still were it should be reassemble the machine for a thread up test run.
That is pretty much what I did with the two 413s I had. I gave one away and still have one. My description above is as best as I can describe it. I would suggest you hunt down a service and repair manual. I did mine without one. Taking pics won't hurt either, but other than the external parts of the machine body there isn't anything that needs to be be removed.
The plastic bobbin case is referred to as the "Apollo hook". I don't know why. Both the plastic and metal bobbin cases use the same design bobbins, the class 66. But the metal bobbins can cause excessive wear on the plastic bobbin cases so they make the plastic ones for that. My wife used both metal and plastic bobbins in her 538 for years and we can't see any abnormal wear in the plastic bobbin case. So I doubt your grandmother's use of the metal ones hurt anything.
As for the motor belt, do not tighten it too tight. That will bind up the motor and slow the machine down. Just tight enough so it doesn't slip but no more.
Hope this helps a bit.
Joe