Equivalent Dynamic Load (Bearing)⚠ unverified

Understanding the Parameters

• Radial load $F_r$ — the force component perpendicular to the shaft axis (weight of a rotor, belt pull, gear separating force). It is reacted by only the loaded arc of rolling elements, so the most heavily stressed ball or roller carries several times the average.

• Axial load $F_a$ — the thrust component along the shaft axis (helical-gear thrust, fan pressure, a shaft pushed by a coupling). In an angular-contact or tapered bearing it is carried through the contact angle; in a deep-groove ball bearing a modest thrust simply shifts the contact ellipse up the groove shoulder.

• Limiting ratio $e$ — the value of $F_a/F_r$ at which the internal load path changes character. Below $e$ the thrust is small enough that the radial elements absorb it with little penalty, so $Y$ is effectively zero and $P \approx F_r$. Above $e$ the thrust genuinely redistributes the load and the full two-term formula applies. $e$ itself depends on $F_a/C_0$ (the static capacity) and on the contact angle, which is why manufacturers tabulate it.

• Factors $X$ and $Y$ — weighting coefficients from the bearing catalogue that convert each load component into its fatigue-equivalent radial share. They encode the contact angle and internal geometry: a high-contact-angle bearing carries thrust efficiently and has a small $Y$; a deep-groove bearing handling the same thrust needs a larger $Y$.

Registry note (honest): the calculator applies the bare product $P = X F_r + Y F_a$ with the $X,Y$ you supply. The input $e$ is documentation of the threshold — it is not used to switch factors automatically. You must look up the correct $X,Y$ for your $F_a/F_r$ regime yourself. Flagged in Known Issues as an unused-in-formula input.

Derivation (Approaching a Proof)

Rolling-contact fatigue life depends on the amplitude of the sub-surface alternating shear stress, which in turn scales with the maximum contact load on the most heavily stressed rolling element, $Q_{\max}$. Lundberg and Palmgren showed that for a bearing under a pure radial load $F_r$, the life scales as $L \propto (C/F_r)^a$, i.e. life is governed by a single characteristic load.

For combined loading, $Q_{\max}$ is a nonlinear function of both $F_r$ and $F_a$ and of the contact angle. Rather than re-solve the internal load distribution for every duty point, the standards replace the true combined load with an equivalent pure-radial load $P$ defined so that

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

Expanding $Q_{\max}(F_r, F_a)$ and matching the resulting life to leading order produces a relation that is very nearly linear in the two load components over the practical range, which is exactly why the standard writes it as a weighted sum:

$$P = X F_r + Y F_a.$$

The coefficients $X$ and $Y$ are the fitted slopes of that linearisation for a given bearing type, and the kink at $F_a/F_r = e$ marks where the dominant term in $Q_{\max}$ switches from the radial to the axial contribution. So the formula is a piecewise-linear surrogate for a genuinely nonlinear contact-mechanics result — accurate because the standards fit $X$, $Y$, and $e$ to test data for each bearing series.

Dimensional check. $X$ and $Y$ are dimensionless, so every term carries the dimension of force: $[P] = \text{N}$.

History and Development

Arvid Palmgren, working at SKF from the 1920s, was the first to treat bearing failure statistically as rolling-contact fatigue rather than as a deterministic strength limit. With Gustaf Lundberg he published the Lundberg–Palmgren theory (1947, 1952), which underlies both the $(C/P)^a$ life law and the equivalent-load concept. The weighted-sum form with tabulated $X$, $Y$, $e$ was codified in the ABMA/AFBMA standards and internationally in ISO 281. Every major manufacturer (SKF, Timken, NSK, FAG) publishes series-specific $X,Y,e$ tables that trace back to this framework.

Related Concepts: Bearing Rating Life L10, Static Equivalent Load, Dynamic Load Rating, Minimum Required Dynamic Capacity, Bearing Life hours

Notes: Supply $X,Y$ from the bearing manufacturer's catalogue for your $F_a/F_r$ regime; the built-in defaults ($X=1$, $Y=1.5$) are placeholders, not universal values. For a static (non-rotating or shock) check use Static Equivalent Load instead.