CRJ-200 Captain reported shuddering during flight that was more pronounced at higher speeds. The flight crew elected to request priority handling and divert to make a precautionary landing.

Narrative

While operating Aircraft X from ZZZ to ZZZ1; all parts of the flight regime were normal until we were climbing through approximately FL280. We were cleared up to and climbing to FL290 when we noticed the aircraft began an abnormal shuddering. My First Officer (FO) and I noticed it at the same time; so when he asked me what I thought it was I wasn't very confident when I said 'turbulence'. I had already begun scanning the instruments for any indication that would make sense of the shuddering. Shortly after leveling off at FL290 we were cleared to FL310 and we climbed. My hope was that we would get to a new pocket of air and the shuddering would stop. The shuddering did not stop. At this point I had checked the engine instruments; Primary; Status and Flight control pages for any indication that would explain what we were experiencing. I even checked the QRH but did not find an applicable checklist. After approximately 5 minutes; changing altitude and location the FO stated what I was already thinking; 'I don't think that's turbulence'. I asked the FO if he wanted a lower altitude to see if that changes it. He agreed. We requested a lower altitude and received FL290. At some point between level of and descending we had slowed to 240 knots. Previously we were flying 250 knots and we noticed that the shuddering wasn't as pronounced at the slower speed. We made the decision to remain at the lower speed. I had already begun conferring with ATC and Dispatch and my EFB to see our nearest practical landing airports. Dispatch told me ZZZ2 and I agreed. At FL290 the shuddering persisted. We asked for lower once again and while descending I noticed that the rudder trim was a little out of trim. I trimmed it up in hopes that the shuddering would subside. It got worse. At this point I was seriously concerned about getting the airplane on the ground safely. We untrimmed the rudder and made the decision to request priority handling and divert to our nearest practical airport. We got the weather; advised ATC and dispatch of our intentions; built; bugged; and briefed the approach; and completed all applicable checklists. I made the decision to take the landing for RWY XXR. It was a coupled ILS that I ensured minimal fight control abruptness. During the arrival and approach we experienced turbulence that 'muffled' the shuddering but we both agree that it was still present. The flight landed safely and we were happy to see emergency responders waiting on us. We canceled our urgent situation and continued to the gate without incident.