Pilot reported aircraft making unusual vibrations which led to a diversion to another airport.

Narrative

Some point after cleanup; as the Pilot Flying; I noticed an unusual vibration in the airframe; like something was hanging. The vibration continued and at about 8;000 feet; I mentioned it to the FO (First Officer). Both of us have less than 400 hours in the Airbus fleet; and at that point; I was thinking it might still be something normal for an A319; though it was clear by then it wasn't turbulence-driven. We accelerated after 10;000 feet to 320 knots and the vibrations got slightly more noticeable. The FO pulled out his QRH and showed me the checklist for unusual vibrations. I slowed the aircraft to about 305 knots and elected to transfer aircraft control to the FO so that I could run the checklist. Upon attempting to diagnose the problem; it seemed that the vibrations were intermittent and not fully dependent on airspeed. The number 1 engine N1 vibration on first look was .9; which I felt was a bit high; but vibrations continued later with vibration indications all below .5; and the vibrations did not vary with thrust. The airframe vibration checklist is not a true checklist but appears to be a bunch of bullets for crew awareness. It explicitly states that aircraft vibrations due to flight controls 'do not represent handling or performance concerns.' The vibrations initially resembled the symptoms listed in the QRH as flight control vibrations; so we continued our climb. At 14;000 feet the Number 2 Flight Attendant called from the aft galley; to notify me of significant aircraft vibrations; unlike anything she had ever felt. I told her that the vibrations were a known problem in the Airbus fleet. I viewed her report as confirming the verbiage in the QRH; so we continued. But she also indicated that she only flew Airbus trips and her seniority number indicated over XX years as a Flight Attendant (FA); and she was insistent that she had never experienced these vibrations before. So at the least; that information suggested an atypical problem. I called the Number 1 FA to ask what she was experiencing. She said that she felt some minor vibrations but nothing that concerned her. She also suggested the Number 2 FA had a flair for drama; in so many words. We continued the climb; but I asked the FO to slow us a little more to 290; while we considered the QRH guidance. The way the QRH is written; it suggests troubleshooting the problem; which seems to conflict with the FOM section concerning troubleshooting. Upon discussion; we elected to put in 1.5 units of rudder trim for a short while to try to alleviate the vibrations; which appeared effective.Throughout the climb; the vibrations continued intermittently with varying intensity. At about 30;000 feet; the FO and I decided that the vibrations were not fitting any one specific problem listed in the QRH. I still didn't have enough information to decide upon whether to continue the flight; but I reasoned that the lower dynamic pressure at a higher altitude would be as safe or safer; while we continued to try to diagnose the problem. I had the FO slow to 260 KIAS while I prepared to call Dispatch to see if I could reach an Airbus specialist. I was interrupted by the Number 2 who called to repeat her concerns to me; before I could complete the call. I explained to her again that what she was experiencing was described in our checklist and that I could only go by the checklist. At this point; though; I still wasn't sure which specific problem we had. We were experiencing 'characteristics' that were listed under both elevator and rudder vibrations. On crew-phone; the Dispatcher was able to connect me to an Airbus maintenance specialist. By now; we had leveled off at a cruise altitude of 34;000 feet. I described the symptoms to him; and he didn't seem to understand any better than us what we were experiencing. I mentioned that the elevator vibe was supposed to appear above 280 knots; the rudder vibe was supposed to appear above 260 knots; and by this point; we were .77 Mach and 255knots and still experiencing the vibrations intermittently. The maintenance technician had no explanation and as ZZZ1 was 45 miles off the nose; I told the maintenance tech and the Dispatcher that we were diverting to ZZZ1. They both concurred; and I broke down the phone patch.The FO had heard me when I said we were diverting and concurred. I asked him to get a clearance to divert to ZZZ1 and to set the cockpit up for our arrival. We decided to keep the speed back at 250 KIAS for the descent; and I instructed the FO to not use speed brakes as a precaution; as we had decided we had a malfunction of unknown origin. The FO mentioned that we would be overweight by a few thousand pounds so I told him to [receive priority] and request ARFF (Airport Rescue and Firefighting) meet us after landing. ATC vectored us to manage our altitude loss. I discontinued the Airframe Vibration analysis and ran the Non-Routine Landing Considerations OD checklist as a Precautionary landing. After talking to the flight attendants and the passengers; I briefed the arrival and approach to the FO. Then I assumed pilot flying duties while the FO completed normal procedures checklists. ATC had offered an RNAV to [Runway] XXC; but I requested an ILS to XXR as I wanted to have the best possible indication of any pitch variations. I configured the aircraft and slowed early so we would have some altitude in case any controllability issues occurred in the landing configuration. I elected to land flaps full if things were normal up to CONF 3 as we were only a few thousand pounds above max landing weight; and we had plenty of go-around thrust. The aircraft handled fine; so we dropped flaps to full; though the FO and I agreed that there seemed to be slightly abnormal vibrations all the way to landing.Touchdown and landing rollout were normal. We asked the tower and fire chief if they saw anything unusual as we were ending the rollout. They said no; so I elected to taxi clear. I made a PA to the passengers reminding them that the emergency vehicles that we were expecting would be visible and that they would follow us as we taxied to the gate. I realize now in writing this account that I forgot to put the overweight landing in the logbook. I was also reminded by another pilot that the Airbus has an overweight landing checklist that we forgot to run; although the FO and I accomplished everything on the checklist in considering the overweight landing.That would depend upon what Maintenance discovers about this problem.There is some confusion in the way the relevant section in the QRH is written; and the direction in the QRH to troubleshoot the problem; which appears in various malfunctions; should be reconciled with FOM section which directs not to troubleshoot. Also; the Airframe Vibration checklist has zero guidance; if air frame vibrations don't fall into any of the listed categories. Obviously; there are too many possible causes of unusual vibrations to list them all; but some direction as to if/when to land based upon some generic criteria like vibe intensity; etc.; would have been nice. Additionally; we are trained in the school; over and over again to run V1 cuts; and we never or rarely run an overweight landing checklist. A simple bullet in the Non-Routine Landing Checklist suggesting to run the overweight landing checklist; time/conditions permitting; would have been helpful to us.