Bearing Rating Life (L10)⚠ unverified

Understanding the Parameters

• Dynamic load rating $C$ — a catalogue property of the bearing, defined as the constant radial load under which the basic rating life is exactly one million revolutions. It is not a load the bearing can carry "safely" indefinitely; it is a reference point on the load–life curve. Larger, more rolling elements and harder, cleaner steel all raise $C$.

• Equivalent dynamic load $P$ — the single steady radial load that represents the actual duty (see Equivalent Dynamic Load Bearing). $C$ and $P$ must be expressed in the same units so their ratio is dimensionless.

• Life exponent $a$ — the slope of the load–life line on log–log axes. Its value comes from the contact geometry: point contact in ball bearings gives $a = 3$; line contact in roller bearings distributes stress over a longer footprint and gives the gentler $a = 10/3 \approx 3.33$, so rollers lose life more slowly as load rises. The steep exponent is why halving the load roughly octuples ($2^3$) a ball bearing's life.

Derivation (Approaching a Proof)

Lundberg and Palmgren modelled a bearing raceway as a fatigue-critical volume in which the probability of survival $S$ after $N$ stress cycles follows a Weibull form driven by the maximum orthogonal shear stress $\tau_0$, the stressed depth $z_0$, and the stressed volume $V$:

$$\ln\frac{1}{S} \;\propto\; \frac{\tau_0^{\,c}\, N^{\,e}}{z_0^{\,h}}\, V.$$

Fixing survival at $S = 0.90$ (the definition of $L_{10}$) and using Hertzian contact relations — which tie $\tau_0$ to the contact load, and hence to the ratio of applied load $P$ to the rating load $C$ — the number of revolutions to 10 % failure collapses to a power law in $C/P$:

$$L_{10} = \left(\frac{C}{P}\right)^{a}.$$

The exponent $a$ emerges from how $\tau_0$ scales with load for the contact type: point contact (Hertz ellipse) yields $a = 3$; line contact (Hertz rectangle) yields $a = 10/3$. The rating load $C$ is the constant of integration, fixed by defining $L_{10}=1$ million revolutions when $P = C$ — which is why $C$ can be tabulated once per bearing and reused for any $P$.

Dimensional check. $C/P$ is dimensionless, so $L_{10}$ is a pure number, interpreted as millions of revolutions by the convention baked into the definition of $C$.

History and Development

The equation is the centrepiece of the Lundberg–Palmgren theory (Gustaf Lundberg and Arvid Palmgren, SKF, Dynamic Capacity of Rolling Bearings, 1947, and the 1952 sequel for roller bearings). It replaced earlier deterministic "static strength" thinking with a probabilistic fatigue model and became ISO 281. The 90 %-reliability $L_{10}$ benchmark is now universal in mechanical design, warranties, and machinery standards. Modern ISO 281 augments the basic life with a modified life $L_{nm} = a_1 a_{\text{ISO}} L_{10}$ that adds reliability, material, and lubrication/contamination corrections (see Bearing Life Modifier and Life Reliability Factor).

Related Concepts: Equivalent Dynamic Load Bearing, Dynamic Load Rating, Bearing Life hours, Minimum Required Dynamic Capacity, Bearing Life Modifier, Life Reliability Factor, Weibull Reliability

Notes: Output is in millions of revolutions; use Bearing Life hours to convert to operating hours at a given speed. Basic $L_{10}$ assumes good lubrication and clean operation; apply life modifiers for adverse conditions.