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Double Bass Trouble with a binding D tuning key

Like so many things mechanical, you've got to get in there and try stuff.

Can you remove the worm, just leaving the shaft with its gear in place, and then run it in its place to see whence the binding? (Seems to me that you could pull down on the shaft with your finger to partially simulate the string load.) Is the shaft properly constrained in the through hole on the tuner side and the blind hole on the other side, or are those holes worn into ellipses? What does the surface of the shaft look like at each bearing surface? Is it smooth and polished, or rough and galled up? Is the shaft bent? Is the gear square on the shaft?

I would have answers to all these questions before I start squirting lubricants around. Unless the shaft is properly constrained in two good round holes just a wee bit larger than the shaft ("close running fit" is what I would expect), all the bearing surfaces are smooth and polished, the shaft is straight and the gear square on the shaft (and there are probably some other considerations too) adding lubricant won't fix your issue and might make it worse. What is the material of the shaft, is it brass, wood, or something else?

[edit, I just took a look at my inexpensive individual-plate tuners of recent manufacture]

If you have ordinary individual-plate tuners like mine it looks like you could undo the gear from the shaft, pull the side plate (which holds the worm in place) then put the gear back on the shaft and see who it runs in its "bearings". Another thing that occurred to me is the total shaft length could be too long and its end bearing on the bottom of the blind hole on the far side of the pegbox. Is the stiffness of the tuner something that has recently developed, or has it always been like this?
 
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Two more reasons for stiffness -
(1) The way the string is wound could now have it jammed hard up against the inside of the peg box cheek
(2) If the other end of the peg from the gear goes through the cheek and is secured by a screw and washer on the outside then the screw might tighten as you wind the string if the peg is too short. The screw and washer should turn with the peg if all is correct.

On many, if not most basses the ends of the pegs may not go right through the cheeks or may be covered over by the plate on the other side

If your machines are individually mounted on 1/4 plates then it is not difficult to remove the string and machine. Use a DRY lubricant such as graphite powder of a soft lead pencil on the inside of the holes in the cheeks and make sure that the peg is clean before re-assembling.
Try not to lose, bruise or over tighten the screws. A complete set of the same screws in good condition looks best. If two machines are mounted on the same plate you will have take the entire side apart to go through the same procedure. In each case I would let a bit of tension off the remaining strings (don't drop the sound post!!)

Last step is to lubricate the big gear and worm gear with a smear of petroleum jelly (called Vaseline here). Not too much and wipe off the excess.

Incidently, the greatest wear occurs when brass turns against brass. Steel turning against brass gives less wear so I am told.
 
Incidently, the greatest wear occurs when brass turns against brass. Steel turning against brass gives less wear so I am told.

Dissimilar metals are generally better for sliding joints, but it depends on the relative hardness of the metal's oxide and the base metal. For example, aluminum is a very soft material with a very hard oxide. Aluminum running on aluminum is almost always a no-go, because the tiny particles of Al oxide disturbed by sliding friction tear up the much softer substrate; the bits of substrate torn up oxidize very quickly, making essentially a very hard abrasive material in between the two sliding parts; the process continues till the two parts are locked together by galling. Lubricating can only reduce the degree to which this happens. Steel running on brass is much better because the oxides on the brass are much softer than those on the steel so they won't chew up the steel and result in galling. Galling requires two participants. Thus the steel worm and brass gear of most upright bass machines. Stainless steel fasteners in aluminum is OK but still subject to galling especially if assembled without lubrication, because stainless while harder than aluminum, is still considerably softer than aluminum oxide. High carbon steel fasteners on aluminum are almost always good because the iron oxide surface of the steel fastener is hard enough that the aluminum oxide doesn't chew it up.

There's a lot more to the science of surfaces in sliding friction than this, including the characteristics of different lubricants and the relative surface energies of the metals; but that's kind of a most basic introduction to the concept.