The quick answer is yes, there will be a relationship from which you can estimate the RPM.
But this will depend upon the type of VFD, (vectored or not), the various loaded parameters and likely only valid for no load or constant load conditions.
The other question is WTF do you need to know lathe RPM to any degree of accuracy?- it tells you nothing.
The approx speed is usually more than adequate if you don't want to fit a spindle counter.
This was achieved in my case by using a linear control pot and setting the min and max frequency of the VFD to give the desired min and max motor RPM in line with manual belt speeds at min and max pot positions (usually 1400 to 1500 for a 4 pole).
Then print out a 0 to 100 dial for your control pot, so the approx spindle speed can be determined by the dial percentage times the max speed at any particular belt setting, plus or minus say 10%. Therefor if your pot setting is 40% and the full speed would be 600 RPM at the given belt setting, then 40% of 600 would be about 240 +/- 20 or 30 RPM.
As a caution, I would not drive the motor for high load operations at any less than about 20% of full pot deflection. If you have a fancy vectored thrust unit, you may be able to go to lower frequencies without cooking your motor.
I would add, that my experience has been the lower the frequency, the larger the range of speed variation under load. But then I am partial to the cheapest and nastiest VFDs money can buy, so your mileage may vary.
Edited By Martin of Wick on 17/03/2022 10:07:59
Edited By Martin of Wick on 17/03/2022 10:12:55