The 3D-Printed Gearbox That Binds and Slips
A team is 3D printing a two-stage gearbox for a model wind turbine for the science fair. Stage one is a PLA spur gear of 40 teeth driving a 12-tooth PLA pinion, module 2 mm, 20° pressure angle, 8 mm face width. The pinion runs on a 5 mm silver steel shaft, located by a printed D-flat in the hub. The printer has a 0.4 mm nozzle and is set to 0.2 mm layers. After printing, the pinion tooth thickness measures 3.45 mm where 3.14 mm is nominal. When mounted at the calculated centre distance the gears bind and the drive motor stalls. Opened up by 0.5 mm they run, but after about 20 minutes under 0.4 Nm the pinion slips on the shaft and the D-flat rounds out. There are six days until the fair. The store has a bush kit and a 12-tooth acetal gear, module 2, with a 6 mm bore.
1. Calculate the nominal centre distance and the interference caused by the oversize teeth. Give two CAD or process remedies with numbers.
Read the worked answer
Centre distance. a = m(z₁ + z₂) / 2 = 2 × (40 + 12) / 2 = 52 mm.
Tooth thickness. Nominal circular tooth thickness at the pitch circle = πm / 2 = 3.14 mm. Measured 3.45 mm, so each tooth is 0.31 mm thick.
Interference. Two oversize teeth mesh, so the pair needs 2 × 0.31 = 0.62 mm more space than nominal. That is why the gears bind at 52 mm: there is negative backlash. Opening the centre distance by 0.5 mm relieved most of it, which matches the observation that the drive then ran.
Remedy 1 - CAD tooth thinning. Offset each tooth flank inward by 0.20 mm, giving 0.40 mm of backlash across the mesh, and set the centre distance to 52.2 mm. Leave 0.1–0.2 mm of backlash as a target; more than 0.3 mm makes the small pinion noisy and weak.
Remedy 2 - calibrate the printer. An over-extrusion of about 10% thickens every wall. Print a 20 mm test cube, measure the walls with calipers, and reduce flow (typically to 92–95%) and set horizontal expansion to −0.05 mm. Also check that the belt tension and the slicer's XY compensation are not adding material.
Practical sequence. Calibrate first, reprint the pinion only (12 teeth, quick), then set the centre distance in the housing with slotted motor mounts so backlash can be adjusted on assembly rather than guessed.
2. Explain quantitatively why the pinion slips on the 5 mm shaft, and recommend a fix that can be made in the time available.
Read the worked answer
Load at the pitch circle. Torque T = 0.4 Nm, pitch radius r = mz/2 = 2 × 12 / 2 = 12 mm = 0.012 m. Tangential force F = T / r = 0.4 / 0.012 = 33.3 N.
What carries that force. A printed D-flat on a 5 mm shaft gives a bearing area of roughly 5 mm × 2 mm = 10 mm². Nominal PLA shear strength is around 25–30 N/mm², so on paper the flat should survive - but the printed flat is a stack of layer lines, so the effective strength across layers is far lower, and PLA creeps. Thirty-three newtons acting on a 1 mm deep printed flat for 20 minutes is enough to deform and then round off, exactly as observed. The D-flat was also printed as a vertical wall, which is the weakest orientation.
Fix: put the torque through metal, not plastic.
- Drill a 2 mm cross-hole through the hub and the shaft and fit a 2 mm steel pin (or a cut-down nail). The pin carries the 33 N in shear over about 25 mm² of steel, with a huge margin. Cost: a few cents, time: 15 minutes.
- Add a metal collar or a printed hub of 12 mm length at 100% infill and 4 perimeters to spread the load.
- File a proper flat on the shaft with a file, not a printed flat, and add a grub screw if a metal hub is used.
Recommended action. Cross-pin the printed pinion now as a stopgap, then switch to the metal-bore solution below for the final build.
3. Should the team switch to the acetal 12-tooth gear with its 6 mm bore? Assess the benefits, the changes needed, and the risks.
Read the worked answer
Yes, for the pinion - with conditions.
Benefits. Acetal is moulded to nominal tooth thickness, so the binding cause disappears at that mesh; it has a low friction coefficient and good wear resistance against PLA; it is far tougher than PLA and will not creep or round out under 0.4 Nm. The 12-tooth pinion is the highly stressed, fast-running part, so it is the right one to replace.
Changes needed.
- The 5 mm shaft must become 6 mm silver steel to suit the 6 mm bore, using the bush kit for the bearing housing. That means reaming or reprinting the housing and re-checking the 52 mm centre distance.
- The 40-tooth PLA gear is still oversize, so the stage-one mesh still needs 0.3–0.4 mm of centre-distance adjustment; the acetal pinion alone does not fix that mesh.
- Acetal cannot be glued and does not take thread-forming screws well, so retain it with a pin or a grub screw on a filed flat.
Risks. A 12-tooth acetal pinion has a small hub around a 6 mm bore, so the wall is thin; check for cracks at the bore. Mixing acetal with an oversize PLA wheel may still bind if the PLA gear is not reprinted. Acetal and PLA have different thermal expansion, but at this size that is negligible.
Judgement. Use acetal for the pinion, reprint the 40-tooth wheel with 0.2 mm flank offset, and set both centre distances on slotted mounts. Six days is enough for one reprint and one housing revision, but not for a third iteration, so test the mesh by hand before powering the motor.