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I am really challenged as I need to fit a bunch of motors in a 6" box acting as linear actuators. I was looking for linear actuators (most I found were brushed DC) at first but the ones I found that would fit in 6" didn't have a suitable speed torque curve for my application. I started looking at steppers and continuous spin servos, combined with a rack and pinion to transfer the rotational movement into a linear motion.

I need to calculate a linear force resulting from a torque applied on a rack from the pinion gear attached to the spinning motor. Let's say I have a servo applying a torque of 29 kg-cm on the rack, how is this force going to be transferred on the linear rack? (I do not know the pitch and size of rack and pinion yet, but I want to know how big the gear could be to make the rack move faster at a given motor RPM).

Thank you so much!

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The linear force of the rack is equal to the tangential force on your pinion teeth (less power lost due to friction, typically 2-3% for spur tooth and racks). Tangential force = torque/(pitch diameter/2).

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