Ummm, a number of factors I suspect.
The bench I think is one of these?
It’s not very stiff, easily improved by cross-bracing, or firmly bolting plywood panels to sides and back. The panels resist front-back and side to side movement.
The worktop looks very thin, and may be acting as a drum or sounding board. I’d add a bit of kitchen worktop, screwed from underneath to provide weight and rigidity. Super cheap DIY store worktop will do, and you might get lucky if a neighbour is having their kitchen done up : skip full of old worktop for the having.
The feet are a little flimsy, but hey!
I’ve listened to the video too. Does sound slightly odd, bit clicky, but that could be the recording. Microphones tend to add colour. Belt tension too high, or perhaps the change gears are over-tightly meshed. Both adjustments. What does it sound like with:
- the leadscrew disengaged, and then
- the change gears disabled by removing the one at the top (connecting to the spindle,)
Another possibility is that the motor isn’t quite square with or positioned in line with the pulley. This twists the belt, maybe causing the rubber to visibly fray, as well as labouring / ticking sounds. Also an adjustment, bolts under the motor.
I see the lathe is a HobbyMat, so circa 1965 to 1980, and been round the block a few times. Is it new to you? If not, did it make that noise before? Older machines are more liable to faults like worn bearings, a set in the belt, or a motor full of swarf. Though I suspect the bench is acting as a sounding board, amplifying the noise, we might have to worry about the lathe itself.
Congratulations on a neat job setting up the workshop – you deserve a better result!
Dave