Differential Screw and Gear Movement: How It Works, Diagram, Formula, and Precision Uses

Two gear ratios act on one threaded pair

The upper input shaft carries the long blue pinion F and the shorter pinion G. F meshes with D, which is fixed to the lower screw. G meshes with E, fixed to the nut. The nut is held axially in the right-hand standard but is free to rotate.

The lower screw both turns and slides. Its D gear therefore moves along the face of wide pinion F. When the screw and nut rotate together at equal speeds, their relative rotation is zero and the screw does not translate.

The illustrated gear pairs use module 2 and a fixed tooth-count sum of 80 in each pair. D=80−F and E=80−G keep both mesh center distances at 80 mm. Those are explicit model dimensions, not dimensions claimed for the historical machine.

Explore a fine differential feed

Choose nearby F and G counts to obtain a small difference in screw and nut speed. Equal counts give zero travel. Swapping the pinion counts reverses travel without reversing the input shaft.

The animation runs through the selected input turns and then reverses back to its starting position. A separate travel indicator enlarges the small axial result. Gear rotation, screw movement and the indicator all come from the same relative-rotation calculation.

Subtract screw and nut rotation

With input revolutions N, two external gear meshes give nD=−N×F/D and nE=−N×G/E. D rotates with the screw; E rotates with the nut. The signs use the same axial viewing direction.

For the selected screw-hand and positive-travel convention, x=L(nD−nE)=LN(G/E−F/D), where L is screw lead in mm per relative revolution. Travel per input turn is L(G/E−F/D).

This mechanism has one screw lead and two gear ratios. Subtract the screw and nut rotations, then multiply by the screw lead to find axial travel. Force and efficiency require additional load and friction information.

Example: F=20 and G=21

The matched gears are D=60 and E=59. One input turn gives −0.333333 screw turns and −0.355932 nut turns. With a 2 mm lead, their relative motion advances the screw by about +0.045198 mm.

Twenty input turns give about +0.903955 mm. Making F and G equal cancels the axial movement; reversing their values reverses its sign.

Small travel needs controlled clearance

This ideal calculation assumes rigid gears and an accurately made thread. Backlash, thread clearance, compliance and friction can be significant compared with the small calculated feed. No force, precision or self-locking rating is implied.

The cutaway shows shaft supports, the axially retained nut and a wide F pinion long enough for the permitted demonstration travel. Tooth bodies and the overview thread marks are illustrative. Actual tooth and thread profiles, bearing retention and production tolerances require separate design.

Differential screw-and-gear questions

Why is pinion F so wide?

Gear D slides with the screw. A wide mating face keeps D engaged throughout that axial movement.

Why do D and E change automatically?

Both gear pairs share the same two shaft axes. With the model’s common module, equal tooth-count sums keep their center distances equal.

What happens when F equals G?

The nut and screw rotate at equal speeds, giving zero relative turns and zero axial travel.

Is this a two-pitch differential screw?

No. It uses a single threaded pair and a differential produced by two gear ratios.