Bearing Stiffness Radial⚠ unverified
Mechanical / Bearings · Compute the approximate radial stiffness of a bearing
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| C | C | N | 1.0 | Radial load |
| delta | δ | m | 1.0 | Radial deflection under load |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | k | N/m | Radial stiffness, in newtons per metre (N/m). Returns 0.0 when the deflection is non-positive |
The science & history
Understanding the Parameters
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Radial load $C$ — the force pressing the shaft against the bearing (rotor weight, belt or gear reaction). Note the symbol: here $C$ denotes the applied load, not the dynamic capacity of the same name in the life equations — a registry symbol reuse to be aware of. Flagged in Known Issues.
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Radial deflection $\delta$ — the elastic movement of the inner ring (and shaft) relative to the outer ring under that load. It is the sum of the Hertzian flattening at every loaded rolling-element contact, plus any clearance take-up. Typical magnitudes are micrometres, which is why bearing stiffness runs to $10^7$–$10^9$ N/m.
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Stiffness $k$ — the local slope of the load–deflection curve. The subtlety this simple ratio hides is that a real bearing is nonlinear and stiffening: because Hertzian contact deflection grows only as (load)$^{2/3}$, doubling the load less than doubles the deflection, so stiffness rises with load. The formula $k = C/\delta$ therefore gives a secant (average) stiffness at one operating point, not a universal constant.
Derivation (Approaching a Proof)
Stiffness is by definition the resistance to elastic displacement, $k = \dfrac{\mathrm{d}F}{\mathrm{d}\delta}$. For a linear elastic element $F = k\delta$, so the secant and tangent stiffness coincide and $k = F/\delta = C/\delta$ — the calculator's form.
For a real bearing the physics comes from Hertzian contact. A single ball–raceway contact deflects as
$$\delta_{\text{contact}} \propto Q^{2/3},$$
where $Q$ is the contact load. Summing over the loaded rolling elements gives an overall load–deflection law of the form $C \propto \delta^{3/2}$, whose tangent stiffness is
$$k_{\text{tangent}} = \frac{\mathrm{d}C}{\mathrm{d}\delta} \propto \delta^{1/2} \propto C^{1/3}.$$
The calculator's $k = C/\delta$ is the secant stiffness — exact for the idealised linear spring and a good local estimate for a bearing at a known operating load and its measured (or Hertz-computed) deflection. It is the number you feed into a rotordynamic or structural model that treats the bearing as a linear spring around its operating point.
Dimensional check. $[k] = \text{N}/\text{m}$. ✓
History and Development
Treating bearings as elastic springs became essential with high-speed rotating machinery in the 20th century, where getting critical speeds right is a safety issue. The nonlinear Hertzian foundation traces to Heinrich Hertz's 1882 contact theory; its application to full bearing load–deflection behaviour was developed by Palmgren and later systematised by A. B. Jones (1960s) into the "Jones–Harris" bearing stiffness matrices used in modern rotordynamics software. The single-number secant stiffness here is the entry-level version of that framework.
Related Concepts: Bearing Preload Deflection, Hertzian Contact Pressure, Contact Stress Basics, Deflection and Stiffness Modeling, Joint Stiffness
Notes: Result is a secant stiffness at the stated load/deflection point; real bearing stiffness increases with load and with preload. For rotordynamic models, evaluate $k$ at the actual operating load, or use a full Hertzian bearing-stiffness calculation.