Understanding Roller Pressure: Effect of Angle on Sheave Force Distribution

[barvas11]

Hi Guys,
I have a sheave with 7 rollers spread evenly on 180° radius. There is an equal amount of force on each end of the sheave pulling down.
My question is will the force acting on rollers be the same for all rollers or will it differ depending on the angle?
I have attached the picture for better illustration.
Thanks for your advice

 

[jbriggs444]

Hi Guys,
I have a sheave with 7 rollers spread evenly on 180° radius. There is an equal amount of force on each end of the sheave pulling down.
My question is will the force acting on rollers be the same for all rollers or will it differ depending on the angle?
I have attached the picture for better illustration.
Thanks for your advice

Start by assuming that the individual rollers are friction-free and that the cable mass is negligible. If the cable mass is not negligible then the calculation becomes a little bit messier.

Given the above assumptions, the tension throughout the length of the cable must be equal to the amount of force on each end of the sheave. Call this F.

Consider one of the interior rollers (roller 2 through roller 6). Draw a free body diagram. There are three forces on this roller: Tension pulling rightward, tension pulling leftward and the outward support force from the sheave. Since the roller is not moving, these forces must sum to zero.

What angle do the cables make with the horizontal?
What is the vertical component of the tension on the left hand side? Can you write it as a formula involving force F and angle theta?
Repeat for the right hand side? What is the vertical component of tension?
What is the total of these two tensions in the vertical direction?
[For compleness] What is the total of the two tensions in the horizontal direction?

Now consider roller 1 or roller 7. Draw your free body diagram again. Rotate it 90 degrees so that the roller's support is vertical. What angle does the left hand cable make with the horizontal? What angle does the right hand cable make with the horizontal? What is the net vertical component of tension? What about the horizontal?