Preload For Backlash⚠ unverified
Mechanical / Power Screws · Compute the preload torque required to eliminate backlash
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| F | F | N | 1.0 | Axial load on the screw |
| mu | μ | — | 1.0 | Coefficient of friction between the screw and nut threads (dimensionless) |
| dm | dm | m | 1.0 | Mean (pitch) diameter of the thread |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | T | N*m | Preload torque to eliminate backlash, in newton-metres (N*m) |
The science & history
Understanding the Parameters
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Preload force $F$ — the axial force locked into the nut assembly to keep the flanks engaged. More preload gives stiffer, more repeatable positioning and eliminates backlash more firmly, but raises the drag torque proportionally — a Goldilocks quantity (too little leaves play, too much wastes power and wears the thread).
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Friction coefficient $\mu$ — governs how much drag the preload produces. This is also why ball screws are preferred for anti-backlash precision: their rolling friction ($\mu \approx 0.005$) makes the drag penalty of preload tiny, so they can be preloaded firmly for zero backlash with little efficiency loss (see Ball Screw Efficiency).
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Mean diameter $d_m$ — the effective radius at which the friction force acts, $d_m/2$.
Derivation (Approaching a Proof)
A preload force $F$ presses the screw and nut thread flanks together with normal force $\approx F$ (square- thread approximation). Coulomb friction opposes relative rotation with a tangential force $\mu F$, acting at the mean thread radius $d_m/2$. The resulting resisting torque is force times radius:
$$T = (\mu F)\,\frac{d_m}{2} = \frac{F \mu d_m}{2}.$$
This is precisely the thread-friction component of a screw's torque (the same term found inside the raising-torque formula Power Screw Torque raise and the bolt tightening torque Bolt Torque Preload), isolated here because for an anti-backlash preload the friction is the only effect of interest — the preload does no net lifting, it just holds the flanks. For an angled (Acme) thread, replace $\mu$ with $\mu\sec\alpha$. The drag adds directly to whatever working torque the screw carries.
Dimensional check. $[T] = \text{N} \cdot \text{m}$ ($\mu$ dimensionless). ✓
History and Development
Anti-backlash nuts (spring-loaded split nuts, preloaded double nuts) and preloaded ball screws are standard precision-motion elements in machine tools, measuring instruments, and servo actuators. The recognition that preload buys accuracy at the cost of friction drag — minimised by rolling-element (ball) screws — is a core precision-design trade-off covered in Shigley and machine-tool design references.
Related Concepts: Ball Screw Efficiency, Power Screw Torque raise, Bolt Torque Preload, Collar Torque, Bearing Preload Deflection
Notes: Returns the friction drag torque from a given preload force (not a preload-sizing formula). Use $\mu\sec\alpha$ for Acme threads. Ball screws minimise this drag, enabling firm anti-backlash preload.