Dampening coils (also referred to as "dampening rods" by the quest text) are necessary to prevent a complete meltdown of the nuclear reactor on board the submarine Yangtze.
The original dampening coils were damaged beyond repair some time between 2077 and 2287. After the Sole Survivor discovered the Yangtze in 2287, its captain, Zao, asked them to acquire new coils. Captain Zao had pre-war intelligence that led the Sole Survivor to Saugus Ironworks, where the new coils were found.
The Sole Survivor returned the coils to the Yangtze, where they were used with the nuclear fuel from an unfired warhead to restore power to the submarine.