Variable pulley distance - tensioner force help needed

[supernaught]

Greetings, lurked for a long time, but I need some help here so I joined up. Please have some patience with me if I miss any important information.

I am working on a belted pulley drive system where the driveN pulley will move relative to the driveR while in the stopped state.
An outside idler pulley is used to take up the slack as the distance between R&N changes.
I did not design this system, but I am tasked with making it work.

The idler force is managed using a proportional valved pneumatic cylinder, controlled by a PLC which also knows the distance between the R & N pulleys.

I need to know the force to apply to the cylinder in order to maintain correct tension on the belt to prevent skipping and excessive wear.

Here are the conditions

The belt is 1750mm long, 8mm pitch, synchronous(ribbed), 30mm wide
The pitch diameter of the R&N pulleys are the same ~180mm
The idler pulley(smooth) diameter is ~160mm
The distance between the R & N pulleys is variable between 485mm and 560mm

The next part that makes this more difficult (at least in my head) is that the idler does not apply force perpendicular to the line between the pulleys. The DriveN pulley and the idler move on a parallel axis, offset by ~185mm(from pulley centres).

To illustrate this, picture an XY plane with the driveR fixed at (X0 Y0). The driveN pulley can travel between (X-400 Y400) and (X-286 Y400). The idler then travels along a rail at Y215, parallel to the N pulley just to keep tension.

Power
The power and speed transmitted from driveR is variable. There is a 2 speed gearbox(4:1 & 1:1) where speed is limited (4:1 is 0-1500rpm), 1:1 is 1500-6000rpm. This is a speed controlled motor, so maximum power available is 45KW peak through a very wide rpm range.

So all I really need to know, is what is the ideal proportional curve/slope of the idler force given the geometry.

Feel free to ask for more information, I am interested in sorting this out. Also if there is a calculator somewhere that could help me figure this out(as well as forces acting on the pulley shafts at maximum power output in 4:1 gear), it would be greatly appreciated.

Many thanks,
Rob

 

[supernaught]

Also, I have some diagrams of the setup that might help, but I will post tomorrow if I can.

 

[billy_joule]

I'm having a hard time imagining a situation where this is the best solution.

Why can't the motor connect directly to the driven shaft?

I did not design this system, but I am tasked with making it work.

Ah, So you'll be the scapegoat when it fails ;-)

As for the problem at hand, the works been done already, the belt manufacturer will have all the tensioner data you require.

 

[supernaught]

I'm having a hard time imagining a situation where this is the best solution.

Why can't the motor connect directly to the driven shaft?

This motor is driving 3 separate shafts... 2 of the shafts will move independently, relative to the main drive. the 3rd driven shaft is fixed, and does not have an idler.

Ah, So you'll be the scapegoat when it fails ;-)

As for the problem at hand, the works been done already, the belt manufacturer will have all the tensioner data you require.

Having trouble getting good application support from the belt supplier right now, and my mech engineers are having a tricky time getting the range of forces needed and generated for this system to work reliably(we don't want damage belts or bearings by pushing the shaft side-load too high), considering how dynamic the system is(pulley and idler wrap amount, spans, pulley distances, rpms, etc.)

I will try to get some images up today of what I'm describing.