Bearing Preload Deflection⚠ unverified
Mechanical / Bearings · Compute the bearing deflection under a preload
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| F_pre | Fpre | N | 1.0 | Preload force |
| k | k | N/m | 1.0 | Bearing stiffness |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | δ | m | Deflection under preload, in metres (m). Returns 0.0 when the stiffness is non-positive |
The science & history
Understanding the Parameters
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Preload force $F_{\text{pre}}$ — the internal force locked into the assembly. Too little and clearance remains, letting the shaft rattle and rolling elements skid; too much and the extra contact stress and friction heat shorten fatigue life and can cause thermal runaway. Preload is a Goldilocks quantity, chosen from manufacturer light/medium/heavy classes.
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Bearing stiffness $k$ — the load–deflection slope from Bearing Stiffness Radial. Because bearings stiffen under load, the operative $k$ here is the stiffness at the preloaded state, which is already higher than the unloaded value — one reason preload is so effective at raising stiffness.
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Preload deflection $\delta$ — the resulting set-down. Its practical role is subtle: preload deflection removes the "dead band" of clearance, so that when an external load later arrives the shaft responds along the already-stiff part of the load–deflection curve instead of first crossing a soft, sloppy clearance zone. This is why a preloaded pair is far stiffer near zero external load than a single clearance-fit bearing.
Derivation (Approaching a Proof)
Preload deflection is simply Hooke's law applied to the bearing treated as an elastic spring of stiffness $k$. By definition of stiffness ($k = F/\delta$ for a linear element), the deflection caused by a preload force is
$$\delta = \frac{F_{\text{pre}}}{k}.$$
The deeper content is why preload helps. Model two opposed bearings, each following the nonlinear Hertzian law $F \propto \delta^{3/2}$. Preloading sets both to a compressed state $\delta_0$ with force $F_{\text{pre}}$. When an external axial load $F_e$ is applied, one bearing loads up and the other unloads; differentiating the pair's combined response about $\delta_0$ shows the system's tangent stiffness near zero external load equals the sum of the two preloaded bearing stiffnesses — far larger than a single un-preloaded bearing, which starts from zero contact and near-zero tangent stiffness. The single-spring formula here gives the set-down; the pair analysis explains the payoff.
Dimensional check. $[\delta] = \text{N} \div (\text{N/m}) = \text{m}$. ✓
History and Development
Preloading grew out of the precision machine-tool and instrument industries of the early 20th century, where spindle stiffness and repeatability were paramount. Matched angular-contact ball-bearing pairs (back-to-back "DB", face-to-face "DF", tandem "DT") became a catalogue standard from makers such as SKF, FAG, and NSK, each supplying defined preload classes. The nonlinear-spring understanding of why preload raises near-zero-load stiffness follows from the same Hertzian bearing mechanics that A. B. Jones formalised for rotordynamics in the 1960s.
Related Concepts: Bearing Stiffness Radial, Hertzian Contact Pressure, Preload For Backlash, Bolt Preload, Deflection and Stiffness Modeling
Notes: Uses a linear stiffness. Since real bearing stiffness rises with load, evaluate $k$ at the preloaded operating state for accuracy. Excessive preload trades stiffness for reduced life and higher running temperature.