oil splashing gearbox 1

If the oil distribution (for lubrication of the bearing or gear) is dependent on the actual rotational speed of shaft or an attached element (bearing, oil ring, gear, etc.) then YES - using a drive to operate at speeds significantly below the designed operating point may result in inadequate lubrication.

How low a speed creates the "problem" is dependent on the specific lubricant, the amount required, and the duration of operation at the reduced speed.

Converting energy to motion for more than half a century

Thank you for the comment. I had seen some CT fan’s gear issues as a premature damage, then contractor will overhaul or replace it. That was a gear box with oil pump. The splash type is recently installed, and it is working. however, planning to install new VFD. As you pointed out, this is a concern in both case of oil pump and splash system. In oil pump, speed reduction might be diminished oil pressure. Anyway, I have seek advice from manufacturer.

The problem with the world is that intelligent people are full of doubts, while stupid ones are full of confidence.
-Charles Bukowski-

The only lubrication method I know that meet the zero-speed, no forced lubrication, are dip-lubricated gears mounted on horizontal shafts. Or completely filled with lubricant.

I see it in the gearboxes used in my industry, not sure if they are an option for cooling tower gearboxes.