A compound-lead differential screw combines two screw motions to produce a net displacement equal to their lead difference. One threaded section passes through a fixed nut. The other engages a moving nut constrained against rotation. Because the screw itself advances, the moving nut travels relative to the frame by p₁ − p₂ per revolution for the same-hand arrangement shown.
Differential Screw Interactive Calculator
A rotating stepped screw advances through a fixed nut while its second thread moves a guided nut in the opposite relative direction. Their lead difference sets the moving nut’s net travel. Adjust both leads and the knob rotation to see the two motions combine.
Equation Used
- Both nuts are prevented from rotating. The first nut is fixed to the frame.
- One rigid screw carries two same-hand, single-start threaded sections.
- Rigid, backlash-free and perfectly made threads; no elastic or thermal error.
- The moving nut is guided along the frame; the screw translates while it rotates.
- Equal leads give zero net output motion even though the screw moves.
Both threads have the same hand and are treated as single-start, so pitch equals lead. Opposite-handed threads would add their leads instead. Ideal travel per degree is not measured resolution or accuracy.
Two motions on one rotating screw
Turning the knob advances the first threaded section through the fixed frame nut. The entire stepped screw, including the knob and second thread, therefore translates while rotating.
The second nut cannot rotate because its carriage runs on a guide. Relative to the translating screw, that nut moves by the second lead in the opposite direction. Its absolute motion is the first advance minus the second relative movement. The larger drawing keeps the fixed nut stationary and connects the moving nut to its guide.
The cutaway nut sections expose the threaded portions. Their visible thread phase follows the same rotation and translation as the displacement equations. A full forward stroke followed by a return makes direction changes visible without accumulating unbounded travel.
Fine adjustment from a lead difference
Hiscox’s mechanism 1378 presents differential-pitch movement as a way to measure small motions. The principle also appears in differential micrometer adjusters, although commercial products can package the threaded parts differently from this exposed two-nut arrangement.
A small lead difference can give fine nominal travel without requiring either individual thread to have that very small lead. It does not remove manufacturing error, backlash, compliance or temperature effects.
Signed displacement from two leads
Let p₁ be the lead at the fixed nut and p₂ the lead at the moving nut, each in millimetres per revolution. For forward rotation θ in degrees, the screw advances xₛ = p₁θ/360. The moving nut’s displacement relative to that screw is xᵣ = −p₂θ/360.
Adding the two gives the moving nut’s frame displacement x = (p₁ − p₂)θ/360. Multiplying by 1000 expresses it in micrometres. Ideal displacement per degree is 1000(p₁ − p₂)/360, while turns per millimetre of travel is 1/|p₁ − p₂| when the leads differ.
Equal leads cancel exactly in the ideal model: the moving nut stays still while the screw rotates and translates through it. The calculator reports “No motion” because turns per millimetre is undefined when the net lead is zero.
0.25 mm and 0.20 mm leads
A full turn advances the screw 0.25 mm and moves the output nut −0.20 mm relative to the screw. The resulting output travel is 0.05 mm, or 50 µm. One degree produces about 0.139 µm of ideal output travel.
For 90° of rotation, the output moves 12.5 µm. Swapping the two leads reverses that travel. Setting them equal cancels output displacement; setting the angle to zero stops the demonstration and gives zero selected-angle travel.
Small nominal travel is not guaranteed precision
The per-degree figure is a geometric sensitivity. It does not establish how small a motion a user can set repeatedly, or how accurately the output reaches that setting. Thread error, clearance, preload, friction and structural deformation need separate assessment.
The illustrated same-hand threads subtract. An opposite-handed arrangement has a different sign relationship and adds the lead magnitudes. Multi-start threads must be described by lead, not by adjacent-crest pitch alone.
Differential screw questions
Why does the screw itself move?
Its first thread is turning through a nut fixed to the frame, so its rotation creates axial advance.
Why does the moving nut not rotate?
The carriage guide constrains it to axial translation.
What if the second lead is larger?
The output moves in the negative direction for the defined forward knob rotation. The calculator preserves that sign.
What if the leads are equal?
The two axial contributions cancel, so net output travel is zero.
Is ideal travel per degree a resolution rating?
No. It is calculated motion sensitivity. Actual resolution, repeatability and accuracy require a real mechanism and measurements.
Reference
Gardner D. Hiscox, Mechanical Movements, Powers, Devices and Appliances, mechanism 1378, printed page 333. The diagram shows differential-pitch movement with a fixed support and a guided traversing part. The exposed two-nut illustration here states its thread-hand and lead assumptions explicitly.
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