On the west side of the ROOM WITH TWO HUGE GEARS, there are two doorways. Go through the one on the left. (The one on the right eventually leads you to the same place, but the left path is a little less complicated.) Just beyond the entrance turn left and advance cautiously.
The swinging chains will hurt Lara if they hit her. Stay to the left side of the doorway as you approach the first chain.
Wait for the chain to swing away to the right, then run past it and stop so you don't run into the second chain.
- OR -
Keep close to the left wall as you crawl under the first chain and it won't touch Lara.
Just beyond the first chain, turn left.
Take a few steps forward and turn right to face the second chain.
Wait for it to swing to the right and then quickly run past it and around the corner to the left to avoid the third chain (or crawl around the corner to avoid both chains).
Continue into the southeast corner room, which contains a spinning gear...
...and a metal trapdoor. Stand at the edge of the trapdoor with the gear on Lara's left and press Action to open the trapdoor.