How to reduce the vibration of a bench lathe ( Emco Compact 8 )

Another thing is the tensioner is on the drive side not the slack side , designed like that but maybe try using a turnbuckle to tighten the tensioner instead of the spring to at least for a test
 
Yes, check the spindle bearing preload. Every chuck is unbalanced to some degree, so if the preload is loose, then the chuck will amplify it even if the imbalance in the chuck is not that bad. Since it is the same with both chucks, I would not expect the imbalance of the chucks to be too high that you would need to balance them.

To picture what is happening with loose preload, the spindle is like a lever with the belt acting like a spring with the chuck end bouncing up and down from the imbalance.
 
check the spindle bearing preload.
This is likely to be a stupid question but may I know what spindle bearing preload is ? Mine is just a simple hobby lathe so it doesn't have all the features of big lathes.
 
It is the amount of tension you place on the tapered bearings in the head stock. There is a adjustable collar on the pulley end, try cinching it up a little more after you unscrew the locking Allen screw, just a little a check the vibration again.
 
1) The original Emco 3-jaw chuck is unbalanced. Not sure if it indicates anything but I have tried mouting a four-jaw chuck with all the jaws touching each other at the center and the result is the same.

2) The additional mass of the chuck changes the resonating frequency of the whole structure causing it to resonate at 850 rpm.

What would be the sensible next step under such circumstance ?
I'd start with checking balance to address Q1. I've seen a few videos on YT about simple balancing jigs and techniques. Four bearings, some metal angle, a smooth (ideally ground...) steel rod and some patience can get you pretty far. Sounds like a therapeutic day in the shop, to me.
 
This is likely to be a stupid question but may I know what spindle bearing preload is ? Mine is just a simple hobby lathe so it doesn't have all the features of big lathes.
Doesn't look like your lathe has tapered roller bearings, so I don't think there's much to do about preload/bearing slop. Replacing the stock bearings, converting to tapered roller bearings or angular contact bearing$ would be logical next steps. Have you measured for bearing play with an indicator?

My next steps start with an indicator on the spindle to check for play, and then move on to checking balance on the chuck.

Probably not the root of your issue, but a heavy, bolted base like this couldn't hurt if you're looking for detour project.

This Old Tony did a video on bearing options:
 
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Have you measured for bearing play with an indicator?
Although it's a pretty basic lathe but it is equipped with tapered roller bearings. Just checked the play at the spindle end and there is none so I have some hesitation to adjust the bearing preload
 
They come stock with precision tapered bearings, I own one too.
 
Although it's a pretty basic lathe but it is equipped with tapered roller bearings. Just checked the play at the spindle end and there is none so I have some hesitation to adjust the bearing preload
That makes things easier, so might as well go for it. In addition to checking side to side play, also check end-play. That's your best confirmation of preload.

Tapered roller bearings are usually cheap, so no need to be hesitant about adjusting preload.
 
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