The chatter remains, even when the piece was relatively short and hold close to chuck. I have tried may tricks I read on the Internet, slow feeding, deeper cuts, etc. The vibration will still come. The only way I have found that I can turn anything on this lathe is to hold the end of a piece with a center from tailstock.
After playing with it for some time, I finally got a lathe dog so I can turn between centers. Finally I was able to turn a piece and it cuts smoothly. What a wonderful feeling when the cutter just glide through steel like butter.
I have tried to measure the vibration frequency, and it is somewhere around 100 Hz. My lathe at the setting I used most is turning at around 700 RPM. Interesting at one of the pieces I can see the chatter marks that goes about 8 cycles per turn. For 700 RPM and 8 dimples per turn, it turns out at right about 100 Hz. That's what I am seeing.
I have tried to tighten down my tool post, and it did not help. And the disappearance of chatter when turning between centers confirmed that the vibration is probably in the headstock and chuck.
Then I realized that the problem is probably with this size of the chuck and the lathe. The chuck is quite heavy, and it sticks out quite a bit. At the same time, the spindle that supports it is relatively small 0.5" steel rod. That is an ideal case for some good oscillation. This chuck does not fit this lathe. I will need a lighter chuck that sits closer to the headstock. It turns out there are many choices.