I am trying to build a small 3D printed CNC machine but can't afford lead screws as of now. I need to 3D print a lead screw or use a rack and pinion gear system. I am a little skeptical able the rack and pinion because I don't know how big of a problem backlash will be. I need to know if I will get a +- 3mm accuracy. Is it possible to use 3D printed rack and pinion gears to make a CNC or should I 3D print a lead screw?
My experience-free book learning suggests that pre-loading the rack would eliminate backlash at the rack.
First, visualize where the backlash arises. When the pinion moves the rack in direction A, one side of the pinion teeth will engage one side of the rack teeth. Call these the A sides. But when the pinion goes to move the rack in direction B, first it has to rotate a little, until the B sides of the teeth engage, before the rack moves.
Now for the pre-loading: suppose you had a spring pulling the rack in direction B. When the pinion moves the rack in direction A, it has to stretch the spring, while again engaging the teeth on the A side. However, when the pinion goes to move the rack in direction B, the spring contracting actually does all the work, pulling the rack in direction B. The pinion's motion controls the rate of retraction, still fighting the spring. And all the while the A sides of the teeth remain engaged, eliminating the backlash.
This doesn't come for free, of course. Besides moving the rack and its load, the motor and geartrain driving the pinion will need to be strong enough to stretch the spring to its greatest extension. At its least extension, the spring will still need enough tension to pull the rack and whatever it is carrying. (One could fantasize the complication of replacing the spring with a separately controlled motor/geartrain whose job was solely to fight against the main one.)
Finally, backlash in the upstream geartrain won't be addressed by this scheme, unless the spring on the rack can somehow backdrive it all.