EMB ERJ 170 Captain reported engine failure and diversion to nearby airport.

Narrative

We departed ZZZ; and we were climbing up to our cruising altitude of FL330. Climbing through FL290; we felt the plane shudder; and we saw the low pressure vibrations on the number 2 engine went amber. Following this; N1 dropped to about 18%. After this; we got an ENG 2 FAIL EICAS; and a fail indication on the engine 2 N1 indicator. My FO called out identify cancel; and I responded with engine 2 fail. My FO then called for the ENG 2 FAIL QRH. During this time; the engine tried to auto relight; but it was unsuccessful. We began running the QRH; and advised ATC . We were over ZZZ1; so I made the decision to divert there. My flight attendants called me; asking what was going on. I told them that we had an engine issue we were working through; and I would call them back. Following the QRH; we made the decision to not restart the engine due to the vibrations prior to the failure. Upon reaching the appropriate section of the QRH; I emergency called the flight attendants and gave them a briefing. I followed this by making a quick PA advising the passengers of an engine issue; that we'd be diverting to ZZZ1; and that we'd be landing shortly. I briefed the approach; and we ran the engine out descent checklist. We were vectored onto the approach and ran the engine out landing checklist. We landed without incident and taxied to the gate.