Motor selection is surprisingly easy if you're using a 2" diameter capstan. A 2" diameter capstan (standard) has a 1" lever arm (the radius). If you allow a little bit of rounding in the formulas a 1" lever arm turns inch-pounds of torque into pounds of direct force.
In other words, a motor/hydraulics combination that gives 4000 in-lbs of torque into a 1" radius capstan will give 4000 pounds of theoretical pulling force. The theoretical here is because all motors have losses, the rope slipping [or hopefully not] on the capstan has loss, and the log dragging across the ground adds friction. If you add all that up you definitely don't get 1:1 log weight to pull force. I don't know what the real ratio is, but you seem to already have that idea and in my actual setup I've not hit any motor lack of torque problems. I sized my system for a max of 60fpm (that determines your RPMS) and 8000 lbs (that determines your motor's torque requirements).
Before we get to the actual motor sizing selections there's one more thing to consider. Most hydraulic motors are NOT designed for side torque (the rope pulling perpendicular to the output shaft). Luckily, most motors do specify the allowable torque in their spec sheets. Most of them specify that value at something like 80% of max RPMs. Since we won't be using these motors anywhere near the max RPM and we're not looking for thousands of hours of use, you can safely go up to max side torque and probably beyond. However, since I was looking for a very beefy system I chose a "drive wheel motor" since those are designed for heavy direct side torque.
Ok, here's the "nutshell" of hydraulic theory. I'm greatly over-simplifying here so real engineers in the house please don't crucify me.
PSI = torque, GPM = speed aka rpms
You can't really affect the max PSI of your tractor's system so you can almost* treat that as a constant.
*Two exceptions here:
a) your tractor won't actually generate max PSI unless your engine is near PTO RPMs AND the hydraulic system is actually under load.
b) the hydraulic motor's torque & rpm curves are not linear. The motor's datasheet will have tables for a series of max PSI values and RPMS. Don't overthink this.
You also can't really change your tractor's max GPMs. However, you can change how fast your target motor will spin based on that GPM. The value you will select for is the motor displacement, measured in cubic inches (cu in) in imperial measurements. Generally speaking, the larger the displacement the slower the motor will turn for a given GPM and the higher the torque. That is, until you drop below a minimum useful RPM and the motor actually stalls out.
There's a few different hydraulic motor designs out there that have different rpm/torque curves. I'm not an expert on them. I did enough research to determine that a "Gerotor" was the right style for me and gave the best torque-speed combination. I am very happy with my motor selection so I'd suggest you do the same.
So here was my process:
a) I knew I wanted 8000 lbs so I needed 8000 in-lbs of output torque.
b) I wanted 60fpm of pull rate at max RPM.
c) With a 2" diameter capstan that's a circumference of 6.28" or 0.523' (2*Pi*R) so I need 60' / 0.523' = ~115 rpm
d) my tractor can do ~3000psi @ ~11gpm of flow
e) 11gpm / 115rpm = 0.09 gallons/revolution
f) 1 gallon = ~231 cubic inches so 0.09 * 231 = 22.1
g) then I searched through all the drive motor on surpluscenter.com until I found one that was an affordable price, near enough to 22.1 cu-in and had at least the desired torque at 3000psi.
h) I ended up picking
https://www.surpluscenter.com/Hydra...355320A8321BAAAA-HYDRAULIC-MOTOR-9-8675-D.axd which yielded an expected 8231 in-lb of torque with 79 rpm at 11gpm/3000psi
For final plumbping, on advice from a thread on from TBN I also added a flow control valve. I *STRONGLY* recommend that you also add one as well. I don't actually use it much for flow control as I find that the tractor's throttle is far more useful for that. However, it has saved my butt several times as a quick way to cut off rotation.
As has been discussed before, my rope likes to climb over itself. If this happens with any load on the line the capstan's slip design no longer works and you can't release the load. If you can't stop the rotation it's going to continue until something breaks. I also found out this same condition can happen if you have a heavy load on the line. For example, if you're pulling a log up a hill, even a very small one. So definitely have some kind of quick cut-off in easy reach.
Other than that, it's a simple design and I'm fairly happy with mine now. I'll definitely try some other ropes in the future.