Bearing axial tolerance stack

A bearing axial tolerance stack expresses the effect of contributing dimensional deviations on the realized axial position of a bearing assembly. Each contribution enters with the sign and sensitivity imposed by the geometry. The resulting setting variation consumes bearing axial travel margin. The manufacturer's locating/non-locating bearing discussion identifies relative axial displacement and accumulated dimensional variation as design considerations.^[ref]

For an original linear stack, let shaft, housing and shim deviations be $\delta L_s$, $\delta L_h$ and $\delta t$, with setting errorIf their independent admissible ranges are respectively $\pm 0.02$, $\pm 0.03$ and $\pm 0.01\mathrm{mm}$, the worst-case setting range is $\pm 0.06\mathrm{mm}$. The positive extreme uses the upper shaft and shim deviations with the lower housing deviation. Here “independent admissible ranges” means that all combinations are allowed, rather than a probabilistic independence assertion.

A design with nominal margin0.10mm and required reserve0.05mm can accommodate at most0.05mm of bounded setting error. This stack is therefore too wide by0.01mm. Reducing the housing deviation bound from0.03 to0.02mm makes the worst-case sum0.05mm and meets that geometric reserve exactly. A bearing nominal axial setting can shift the range but cannot shrink its width.

Statistical propagation answers another question. With coefficient vector $a=(1,-1,1)$ and covariance matrix $\Sigma$, the setting variance isFor independent errors with standard uncertainties0.010,0.010 and0.005mm, this gives $u(\eta )=0.015\mathrm{mm}$. If the shaft and housing errors instead have correlation0.9 while the shim remains independent, the subtraction cancels much of their shared variation:giving about0.00671mm. Reversing a geometric sign or changing correlation changes that result. The GUM requires the covariance contribution when inputs are correlated.^[ref]

Neither statistical value replaces the0.06mm deterministic bound without a justified model relating probabilities and allowed ranges. Shared reference measurements or common thermal conditions can introduce dependence. Keeping the dimensional signs, bounds and covariance assumptions in the stack makes the resulting bearing axial setting tolerance reproducible.