How to duplicate animated objects and offset the animation?

So I’ve animated a spinning cylinder that also moves up and down, kind of like a drill. I would like to have several spinning cylinders, but offset so they don’t move all in unison.

I’ve duplicated the objects with a group and now have an empty with which I can move them around, that’s working. But I can’t figure out how to offset the animations. Someone mentioned the NLA editor when I googled but I can’t seem to get separate tracks there or anything.

This is what it looks like:
I’d like to create some small animations, but this is one of the very basic things I’m not sure about at the moment. Would be cool if someone could help.