CRJ900 flight crew experiences N1 vibration and poor engine performance during climb to FL380. After consulting the Flight Manual the decision is made to divert and shut down the engine.

Narrative

During climb out at approximately FL300 we started to notice a degraded climb performance. It was unusual because of our very light load (only 9 passengers). While scanning the engine instruments we noticed that the left (#1) engine N1 vibration was at 1.8 which was just in the yellow; caution; range. We leveled off at FL340 as assigned by ATC and then referred to the FOM. We were not in icing conditions nor did we encounter any in the climb. We did pass through a layer from approximately FL180 up to approximately FL260; but the ice light never came on. We did have the cowl heat on during that time. As per the FOM we adjusted the thrust which momentarily brought the vibration back into the green range. ATC issued our climb to our final altitude of FL380. Once again; during the climb and level-off; the N1 vibration moved into the caution range. However; this time we were unable to control the vibration. It would change unpredictability with either an increase or decrease of power. At one point the vibration reached 3.5. Also note that the entire time we could hear a loud hum and could feel a vibration throughout the whole aircraft. Afterward; both the flight attendant's and the passengers commented on it. Referring again to the FOM; if the vibration persisted or was uncontrollable; we were to proceed to the in-flight engine shut down checklist; which we did. At this time we were operating the engine at reduced thrust because of our altitude. We evaluated our position and situation and determined that ZZZ was our best option. At this point it was approximately 110 miles at our 11 o'clock. We notified ATC that we needed to change our destination to ZZZ and that we needed to descend. We also told them that we were most likely going to be shutting down an engine. I elected to hold off on shutting down the engine and running it idle for a few moments until we got lower (because our #2 pack was cycling and I did not want to shut down our only good pack) and the APU was running. Once we got lower and closer to our drift down altitude we completed the engine shut down and secured it according to the FOM. At this point the emergency had been declared and ATC was extremely helpful clearing us directly to ZZZ. The flight attendants were notified and were informed of the situation and the emergency. They did comment about the noise coming from the back of the aircraft. Approximately 30 miles out; we were cleared for the ILS approach. We verified that ARFF had all the information (souls on board; fuel and any hazmat). After landing we stopped on the runway and asked ARFF to verify the condition of our #1 engine. They advised that they saw no external damage or leaking of any fluids. I thanked them and we taxied to the gate under our own power.