'That first gear has seen better days, but it's not beyond repair. I'll get to work on that, while you look for the second gear.
'According to the old records, the second gear was hidden in a chest in some old ruins to the south in the Vale of Thráin. The ruins are on the west side of the vale, off the road and across the Vale from Noglond. The records indicate that the chest is on the south side of the ruins.
'One thing you should be aware of, though, is that goblins have overrun those ruins. Be careful.'
Bávor is an artificer tasked with repairing the great lock which opens access to the fabled Vault of the Mountain.
Find the missing second gear
The chest containing the second gear can be found on the south side of the goblin-infested ruins in the Vale of Thráin, south of Thorin's Halls and west of Noglond.
Bring the second gear to Bavor
Bavor is in Thorin's Hall.
Bavor will be pleased to begin working with the second gear.
I would recommend collecting this quest, the fellowship, and the Dourhand burgler quest; so as to complete them all at once. I was a lvl 11 dwarf guardian and was able to solo all three parts of this quest fairly easily.
Keep going thought the ruins of Oordost, and you will find the chest 20.6S 102.4W with the gear just to the left of the main common area at the top of the stairs You will see the Dourhand Burgler off to the right, avoid him (or kill him) and go through the left doors, and you should find the chest in the next area. If you keep going you will find the Orc captain in the proceeding chamber.