Conveyor Belt Tension with Sliding Rollers

[squeakychair]

In this conveyor belt configuration, rollers P and R are horizontally aligned, while rollers R and S are vertically aligned, in a triangular arrangement. Roller Q then bends the belt between P and S.

Roller Q is free to slide vertically, while roller P slides horizontally. The remaining rollers R and S are fixed to their position. Of course all the rollers rotate with zero friction to allow the belt to move.

The system is in static equilibrium, held together by an applied horizontal force Fp at P and vertical force Fq at Q.

If I know all dimensions and geometrical relationships (angles, distances), how would I relate the belt tension T, and the external forces Fp & Fq?

 

[dvd]

It is a vector sum at each pulley, but the belt won't move if the tension is everywhere the same. Which pulley is the driveR? There will be a tightside tension and a slackside tension associated with the driveR. Where is the load applied? I ask myself if this is a sanding belt. The forces under no-load may be very different from the forces when the belt is loaded.

 

[squeakychair]

Hi dvd,

Thanks for your reply.

Just to check, you say it's a vector sum, so:

Fp = T (1 + sina)
Fq = T (sinb - cosa)

Leading up to

Fp = Fq (1 + sina) / (sinb - cosa)

Is this what you mean?

Take R as the driver rotating clockwise. That means T1 > T2 (new diagram).

From my studies, T1 and T2 are related (belt tension ratio) by the belt mass, velocity, friction coefficient, contact angle etc.... too many variables over many rollers, so I wonder if I can ignore it.

For now the load can be assumed to be thinly distributed across the whole belt - so I simply take it as part of the belt's inertia, that is, no load. Honestly, I'm not too sure how to account for the belt's behavior - in another method I even treated the blet as rigid and related Fq = Fp tana.

 

[dvd]

More wrap on the driveR would result in less slackside tension - why can't you drive S pulley?

You can solve all of your forces in CAD with proper scaling of the vectors. I am not going to review that aspect of your work.