Simple Rack and Pinion Zero Backlash (no springs)

[Muzzer]

Mechanically there's no significant difference between this and a split pinion apart from the fact that you get a "gearing up" of the spring force due to a) the wide angle of the belt between the main gears and the tensioner and b) the ratio of the belt diameter to the gear diameter. This results in not needing such a beefy spring. However you can also vary the preload tension more easily than with a split pinion as the spring is external. It's rather like a spilt pinion with the pinions moved apart on the driven gear

It wouldn't be difficult to calculate the equivalent torsional preload effect at the driven gears. Yes, the green pulley removes the backlash between the big gears and the red one tensions the motor pulley.

[deemoss]

Mechanically there's no significant difference between this and a split pinion apart from the fact that you get a "gearing up" of the spring force due to a) the wide angle of the belt between the main gears and the tensioner and b) the ratio of the belt diameter to the gear diameter. This results in not needing such a beefy spring. However you can also vary the preload tension more easily than with a split pinion as the spring is external. It's rather like a spilt pinion with the pinions moved apart on the driven gear

It wouldn't be difficult to calculate the equivalent torsional preload effect at the driven gears. Yes, the green pulley removes the backlash between the big gears and the red one tensions the motor pulley.

Actually you don't need to calculate the tension force. This is because the tooth-to-tooth force is not going through a spring. The belt tensioning force need only be such that facilitate the "touching" of the teeth of the respective pinions in each direction.

[Muzzer]

Actually you don't need to calculate the tension force. This is because the tooth-to-tooth force is not going through a spring. The belt tensioning force need only be such that facilitate the "touching" of the teeth of the respective pinions in each direction.

If you have no preload tension, then by definition you have some backlash in your system. There are 2 approaches to this - either a spring tensioner to maintain a tight belt / chain and take up any wear or unevenness - or a fixed tensioner. With the latter, you will always end up with some backlash. With the former, you must ensure the preload tension exceeds the maximum drive tension. You seem to have the latter but either means could be used here.

You get the same result with a dual pinion setup if you have an adjustment screw between the pinions, rather than a spring. That's essentially what is used on horizontal milling machines that have a backlash adjustment.

[m_c]

The issue with most racks, is the teeth are rarely perfect for the entire length, especially once you get some wear, so you need some kind of sprung tensioning system to minimise backlash, while avoiding excessive binding.

The system here, once adjusted correctly will likely have enough spring due to the belt flex to deal with some backlash.
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However, you have to put things in context. If you want precision with minimal/no backlash, rack and pinion is not the ideal choice.