Mine does that too and has since the get-go. On a cold start it's like a sewing maching but on a hot start it's very coarse and unpleasant. My car is a 1.8 with manual trans. I doubt mine is motor mounts because it starts so smoothly when cold. If they were broke I'd expect it to be rough all the time.
I had a theory it might be the dual mass flywheel but that is totally a guess.
I found another thread about this. One guy reports they changed the cam sprocket but I don't know if it's really the same issue.
http://www.cruzetalk.com/forum/25-service-issues/2973-vibration-during-start-up.html
Let me know if you get some kind of answer or fix.
I had a theory it might be the dual mass flywheel but that is totally a guess.
I found another thread about this. One guy reports they changed the cam sprocket but I don't know if it's really the same issue.
http://www.cruzetalk.com/forum/25-service-issues/2973-vibration-during-start-up.html
Let me know if you get some kind of answer or fix.