Minimum Required Dynamic Capacity⚠ unverified
Mechanical / Bearings · Compute the minimum dynamic load rating required to meet a target life
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| P | P | N | 1.0 | Equivalent dynamic bearing load |
| L_desired | Ldesired | — | 1.0 | Desired rating life, in millions of revolutions |
| a | a | — | 3.0 | Life equation exponent (3 for ball bearings, 10/3 for roller bearings). Default is 3.0 |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | C | N | Required basic dynamic load rating C, in newtons (N) |
The science & history
Understanding the Parameters
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Equivalent dynamic load $P$ — the single steady radial load representing the actual duty (from Equivalent Dynamic Load Bearing). It sets the scale of the required capacity: $C$ is directly proportional to $P$, so every extra newton of duty load demands proportionally more rated capacity.
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Desired life $L_{\text{desired}}$ — the target life in millions of revolutions. Because it enters through the reciprocal of the exponent, its influence is weak: for a ball bearing ($a=3$), demanding 8× the life needs only 2× the capacity ($8^{1/3}=2$). Life is cheap in capacity terms — a small oversizing buys a large life margin.
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Life exponent $a$ — the same load–life slope as in the rating-life law. A larger $a$ (roller bearings) makes $L^{1/a}$ closer to 1, so rollers need proportionally less extra capacity than ball bearings to hit the same life multiple.
Derivation (Approaching a Proof)
Start from the basic rating life (Bearing Rating Life L10):
$$L_{10} = \left(\frac{C}{P}\right)^{a}.$$
Set $L_{10}$ equal to the target $L_{\text{desired}}$ and solve for $C$. Raise both sides to the power $1/a$:
$$L_{\text{desired}}^{\,1/a} = \frac{C}{P} \quad\Longrightarrow\quad C = P\, L_{\text{desired}}^{\,1/a}.$$
This is an exact algebraic inversion — it inherits all the physics of the Lundberg–Palmgren theory and adds nothing new, but it reframes the law as a design tool: instead of "how long will this bearing last?" it answers "how big must the bearing be?"
Dimensional check. $L_{\text{desired}}$ is dimensionless (a pure count in millions), so $L_{\text{desired}}^{1/a}$ is dimensionless and $[C] = [P] = \text{N}$. ✓
History and Development
The inverse form is the everyday face of ISO 281 in machine-design practice — it appears in Shigley and every bearing-selection guide as the first sizing step, usually followed by a check of static capacity (Static Equivalent Load), speed limits, and life modifiers (Bearing Life Modifier). Its power lies in exposing the load–life asymmetry that Lundberg and Palmgren's exponent encodes: capacity scales with load one-for-one but with life only as a weak root, which is why designers size bearings generously against load spikes but rarely need dramatic oversizing to chase longer life.
Related Concepts: Bearing Rating Life L10, Equivalent Dynamic Load Bearing, Dynamic Load Rating, Bearing Life hours, Static Equivalent Load
Notes: Compare the resulting $C$ against catalogue values for candidate bearings. This gives the basic requirement; if reliability, material, or lubrication modifiers apply, size against the modified life or divide the target life by the appropriate factor before inverting.