Bearing axial travel interval ($[x_{\min },x_{\max }]$)

A bearing axial travel interval states the range of relative shaft/housing positions that must be accommodated, or the range physically available, using an explicit axial reference. It is more informative than one unsigned clearance number when heating, cooling and assembly offsets move the shaft in different directions. A non-locating bearing support accommodates such movement while a locating bearing determines the reference position.^[ref]

Suppose thermal operating states give displacement $d\in [-0.12,0.18]\mathrm{mm}$ and an independent assembly offset lies in $[-0.05,0.05]\mathrm{mm}$. Adding the endpoint bounds gives required displacement $d\in [-0.17,0.23]\mathrm{mm}$. This is a bound over the permitted combinations, not a standard deviation.

Let available physical travel be $[a,b]$ and choose bearing nominal axial setting $x_0$. Requiring every displacement in $[d_{\min },d_{\max }]$ to remain within the travel givesorIf physical travel is $[-0.20,0.20]\mathrm{mm}$ for the preceding displacement range, both endpoints force $x_0=-0.03\mathrm{mm}$. Setting $x_0=0$ would exceed the positive limit by0.03mm despite equal total required and available spans.

The example has no spare endpoint margin: requiring an additional positive margin at both ends makes the nominal-setting interval empty. Correlated thermal inputs can also make a naive independent-range bound conservative; a tighter result requires the permitted joint operating states. The interval must refer to the actual movement mechanism, because reaching its end can introduce a bearing thermal end-stop load.

A bearing nominal axial setting is the chosen reference position of a shaft/support assembly within its available bearing axial travel interval. It determines how much travel remains in each direction before thermal or assembly changes encounter a limit. The locating support establishes the axial reference, while the non-locating bearing arrangement accommodates the permitted movement.^[ref]

Let physical travel be $[a,b]$, nominal setting $x_0$ and all required changes be $d\in [d_{\min },d_{\max }]$. Feasibility requiresThe interval is nonempty exactly when the available span is at least the required span. This follows directly by requiring $a\le x_0+d_{\min }$ and $x_0+d_{\max }\le b$.

For an original example, physical travel $[-0.30,0.30]\mathrm{mm}$ and required changes $[-0.17,0.23]\mathrm{mm}$ permit settings from $-0.13$ to $+0.07\mathrm{mm}$. Setting $x_0=-0.03\mathrm{mm}$ gives operating endpoints $-0.20$ and $+0.20\mathrm{mm}$, leaving0.10mm at each limit. Setting $x_0=0$ instead leaves0.13mm on one side and0.07mm on the other.

The worst endpoint bearing axial travel margin isIn this symmetric-margin objective, its maximum occurs at $x_0=(a+b-d_{\min }-d_{\max })/2$, with $m_{\max }=[(b-a)-(d_{\max }-d_{\min })]/2$. The example gives0.10mm. A real setting can have additional positioning requirements; balancing these geometric margins alone does not determine every functional constraint.

The bearing axial travel margin is the smallest remaining distance between the required axial operating range and either limit of the available bearing axial travel interval. It measures how much extra displacement can be accommodated before the assembly reaches a limit. The physical purpose of the non-locating support is to accommodate relative axial displacement, including thermal changes and accumulated dimensional variation.^[ref]

Let physical limits be $[a,b]$, bearing nominal axial setting be $x_0$, and required changes range from $d_{\min }$ to $d_{\max }$. The two end margins and their worst value areA negative margin means that at least one required position is outside the available interval. Zero means touching a limit, with no reserve in that direction.

For an original example, $[a,b]=[-0.30,0.30]\mathrm{mm}$, changes $[-0.17,0.23]\mathrm{mm}$ and setting $x_0=-0.03\mathrm{mm}$ give $m_{-}=m_{+}=0.10\mathrm{mm}$. A bearing axial setting tolerance of $\eta \in [-0.04,0.04]\mathrm{mm}$ expands the possible operating range to $[-0.24,0.24]\mathrm{mm}$. The guaranteed margin is then0.06mm. Choosing $x_0=0$ would instead give a guaranteed margin of only0.03mm under the same uncertainty bound.

In general, a bounded setting error $|\eta |\le ϵ$ changes the guaranteed margin to $m-ϵ$. This follows by reducing the lower end margin with $\eta =-ϵ$ and the upper end margin with $\eta =+ϵ$. A required reserve $m_{\mathrm{req}}$ therefore needs $m\ge m_{\mathrm{req}}+ϵ$. The example can retain a0.05mm reserve with the centered setting and a0.04mm error bound.

A deterministic tolerance bound and a standard uncertainty are different specifications. Replacing the bound by one standard deviation would not preserve the guarantee without additional distributional assumptions. The margin calculation also concerns geometric travel: it does not by itself establish acceptable contact stress, friction or bearing thermal end-stop load after a limit is reached.

A bearing axial setting tolerance specifies the permitted difference between the realized axial reference position of an assembly and its bearing nominal axial setting. It consumes some of the available bearing axial travel margin. The setting tolerance must therefore be included when deciding whether all required thermal and operating changes remain inside the physical travel limits. The bearing manufacturer's locating/non-locating arrangement discussion explicitly includes relative displacement and accumulated dimensional variation.^[ref]

Let physical travel be $[a,b]$, required displacement changes be $[d_{\min },d_{\max }]$, and setting error be $\eta \in [{\eta }_{\min },{\eta }_{\max }]$. To retain a prescribed margin $m_{\mathrm{req}}\ge 0$ for every allowed combination, the nominal setting must satisfyThe lower bound follows from the smallest possible realized position $x_0+d_{\min }+{\eta }_{\min }$; the upper bound follows from the largest. This is a worst-case guarantee over the stated ranges and needs no assumption that the errors are statistically independent.

For an original example, take $[a,b]=[-0.30,0.30]\mathrm{mm}$, required changes $[-0.17,0.23]\mathrm{mm}$, setting error $[-0.02,0.04]\mathrm{mm}$ and required margin0.05mm. The admissible nominal-setting interval is $[-0.06,-0.02]\mathrm{mm}$. Choosing $x_0=-0.04\mathrm{mm}$ gives worst-case operating endpoints $-0.23$ and $+0.23\mathrm{mm}$, leaving0.07mm at each end. The asymmetric error range has shifted the balanced nominal setting relative to a symmetric tolerance assumption.

A bearing axial tolerance stack combines the contributing dimensions and their signs before applying this range calculation. Two separate bounded errors of $\pm 0.01$ and $\pm 0.02\mathrm{mm}$ can add to $\pm 0.03\mathrm{mm}$ in the worst case. Taking a square root of summed squares would answer a statistical uncertainty question under additional assumptions; it would not reproduce that deterministic bound. The distinction between a bound and standard uncertainty matters when a guaranteed clearance or reserve is specified.

The resulting interval can be empty. That outcome identifies incompatible travel, operating-change, tolerance and reserve requirements; adjusting the nominal setting alone cannot resolve it. A nonempty interval addresses axial geometry but does not establish acceptable friction, elastic loading or contact conditions after an end stop engages.

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.