I keep my iPhone on silent pretty much all the time. When I plug it in to a charger, it goes "buzz-buzz". My iPad doesn't do this. Both behaviors are the same with iOS 6 and 7.
How do I make my phone not do that?
I have checked the activity-specific sound settings (like Lock Sounds [off], etc). The main suspicious ones are Vibrate on ring and vibrate on silent. I would still like to have it buzz (vibrate) upon receiving texts, calls, and calendar alerts.