Torque Required to Drive Capstan and Bow

Question

I am trying to find the torque exerted on the shaft of a capstan and bow drive. In particular, the system I'm looking at has two wire rope pulleys, and two idler pulleys, and is used to drive a carriage as seen Below:

In this case the drive pulley consists of two separate pulleys tensioned against one another, with the end of two wire ropes terminated in the pulleys.

Naively, I would assume that the force F is equal to the Torque T * the radius r of the drive pulley.

Are there any additional effects due to a multiple number of wraps on the pulley (one side reels out, while the other side reels in, keeping the total angle of wraps equal throughout the motion)?

Does the capstan effect play a large part with this mechanism? From what I understand the capstan effect should cause less tension at the wire rope stops embedded in the drive pulley, but not effect F.

Thanks!

Answer 1

In theory, the force is just the torque, divided by the radius. In practice, wrapping the rope a few times around the pulley will rob a little torque from it, but it may even be unnoticeable. At the same time, friction between rope and pulley will increase, thus allowing you to transfer more force to the rope before it starts slipping over the pulley. It also helps to tension the rope to reduce slipping(much like a derailleur on mountainbikes), but again it'll cost you a little torque. So only consider it if the rope keeps slipping.

Answer 2

The force is actually F= Torque/ radius of drive pulleys.

F=T, tension in cable.

If we ignore the friction, the force is not changed by your configuration of pulleys.