I'm posting this as an answer only so I can include the charts.

I wrote Python code to do the "integration" simple-mindedly: I subtracted the mean acceleration from the acceleration data, then added acceleration/100 to the velocity at each point, then added velocity/100 to the position. The acceleration looks like:

The velocity looks like this:

and the position looks like this:

I don't see any periodicity in either the velocity or the position. Is this not what you were expecting?