GA7 pilot reported rejecting the takeoff after experiencing violent nose wheel vibrations due to shimmy damper failure.

Narrative

I was cleared for takeoff. During the takeoff roll; prior to my airspeed alive call out; the plane began to shake violently. I increased back pressure on the yoke to reduce nose wheel pressure but the shaking continued.At that moment; I made the decision to abort the takeoff. I smoothly reduced the throttles to idle; increased yoke back pressure for aerodynamic braking; and gently applied the brakes. The Pilot Not Flying advised Tower that we were aborting takeoff. The shaking continued and at a safe speed I initiated a left turn towards [the] taxiway. As soon as I started to turn; the left engine shutdown followed by the right engine.I continued to attempt to maneuver the aircraft off the runway but our momentum stopped with the aircraft's tail still not clear of the hold short. The Tower asked if we needed assistance as we were exiting the runway and the Pilot Not Flying replied that our engines shutdown and we were trying to restart. He also informed Tower that our tail was not fully past the hold short.We then attempted to restart but we were unsuccessful; at this point Tower told us to contact Ground; which the Pilot Not Flying complied with and informed Ground of the situation and our position not fully past of the hold short. Ground told us that they were sending assistance. We attempted to restart a second time and were successful. The Pilot Not Flying informed Ground that we were able to restart the aircraft. Ground then told us to shut down and wait for assistance. Before we could reply; the rescue team keyed on the frequency and told Ground that if the plane can move; that they should. At that time; a different Controller came on the frequency (assuming a Supervisor) and asked us if we needed any further assistance.The Pilot Not Flying replied no and that we would like to taxi back to the FBO. The Ground Controller then gave us taxi instructions back to the FBO.In summation; the violent shaking was attributed to a complete shimmy damper failure which has been repaired by the flight school. I feel that my decision to abort takeoff was the right one and that the procedures of doing so were the proper actions as per my training.