Non-locating bearing

A non-locating bearing support carries the required radial load while allowing the shaft's intended axial displacement relative to the housing. It complements a locating bearing, which establishes axial position. The movement may occur within the bearing or through a suitable ring-to-seat arrangement; its purpose is to avoid imposing an unwanted axial constraint when dimensions change.^[ref]

An important cause of relative movement is thermal expansion. For an illustrative shaft with constant linear thermal expansion coefficient $\alpha$, length $L$ and uniform temperature increase $\Delta T$, free extension isUsing hypothetical $\alpha =12\times 10^{-6}{\mathrm{K}}^{-1}$, $L=0.50\mathrm{m}$ and $\Delta T=50\mathrm{K}$ gives0.30mm. If the housing expands differently, the required bearing axial displacement allowance depends on the shaft's extension relative to that housing, as well as assembly tolerances.

For contrast within the same mechanical model, constrain both ends of an elastic bar against extension. Axial strain balance gives a compressive bearing thermal-expansion constraint load with magnitude $F=EA\alpha \Delta T$. With hypothetical Young's modulus $E=200\mathrm{GPa}$ and area $A=100\mathrm{m}{\mathrm{m}}^2$, the preceding temperature change gives12kN. Real supports and housings are compliant, so this rigid-constraint calculation illustrates the mechanism rather than predicting a universal bearing load.

A non-locating bearing is not generally a support with no useful constraint: it can still transmit substantial transverse forces. Its permitted axial movement must remain compatible with the shaft's other functional requirements, including gear positioning. The actual support arrangement determines whether thermal movement is accommodated without losing the required location.

Bearing axial displacement allowance is the axial movement available in a stated shaft/support arrangement before an unintended restraint is encountered. A non-locating bearing support accommodates relative shaft/housing displacement while a locating bearing establishes axial position. The required movement includes thermal changes and assembly variation, rather than shaft expansion alone.^[ref]

For a simple uniformly heated shaft and housing, estimate relative free displacement asusing the applicable linear thermal expansion coefficients and distances from their common axial reference. A shaft extension0.30mm and housing extension0.12mm give relative movement0.18mm. Assuming the housing stationary would overstate that thermal contribution in this example.

The complete bearing axial travel interval can be asymmetric. Suppose the operating states require relative motion between $-0.12$ and $+0.18\mathrm{mm}$, with an independent assembly offset between $-0.05$ and $+0.05\mathrm{mm}$. Adding the intervals gives required travel from $-0.17$ to $+0.23\mathrm{mm}$ about the nominal reference. Its total span is0.40mm, but centering a0.40mm travel range symmetrically at zero would miss the positive endpoint by0.03mm.

Once available movement is exhausted, additional expansion can generate a bearing thermal end-stop load. Movement may be accommodated inside a bearing or at a suitable seat, depending on the actual design. This interval model is bookkeeping for a specified support arrangement; the relevant travel limits must correspond to that physical mechanism and its permitted direction.

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.

A bearing thermal-expansion constraint load is an axial reaction generated when supports prevent the relative thermal expansion that a shaft/housing system would otherwise undergo. For a uniform elastic shaft of length $L$, area $A$, Young's modulus $E$ and free extension ${\delta }_{\mathrm{th}}=\alpha L\Delta T$, completely rigid axial restraint gives compressive force magnitudeA locating/non-locating support arrangement is intended to accommodate the required displacement rather than impose this unwanted extra axial constraint.^[ref]

Actual supports have finite mechanical compliance. Let shaft axial compliance be $C_b=L/(EA)$ and the combined restraining support/housing compliance be $C_s$. Compatibility for a fully engaged restraint givesFor hypothetical $L=0.50\mathrm{m}$, $E=200\mathrm{GPa}$ and $A=100\mathrm{m}{\mathrm{m}}^2$, $C_b=2.5\times 10^{-8}m/N$. Free extension0.30mm gives12kN against rigid supports. If $C_s=10^{-7}m/N$, the force falls to2.4kN because part of the mismatch is taken up by support displacement.

A gap changes the onset of restraint. If the system permits positive free travel $g$ before reaching an axial stop, a simple one-direction model gives bearing thermal end-stop loadWith the preceding compliant supports and $g=0.10\mathrm{mm}$, the result is1.6kN. Below the gap-closing displacement, this modeled stop contributes no force. Applied gear axial force, bearing preload, temperature gradients and other constraints require their own contributions; the simple expression represents one stated thermal compatibility problem.

A bearing thermal end-stop load is the axial reaction that appears when relative thermal expansion exhausts the free travel of a support arrangement. In a quasistatic one-direction model, let free thermal mismatch be ${\delta }_{\mathrm{th}}$, initial travel to the stop be $g$, shaft compliance be $C_b$ and restraining support compliance be $C_s$. The engaged reaction magnitude isThe gap first closes; additional mismatch is taken up by elastic deformation of the shaft and support. This explains why the bearing axial displacement allowance matters. The locating/non-locating arrangement is intended to accommodate relative motion rather than create an unintended thermal restraint.^[ref]

For hypothetical $C_b=2.5\times 10^{-8}m/N$, $C_s=10^{-7}m/N$, $g=0.10\mathrm{mm}$ and ${\delta }_{\mathrm{th}}=0.30\mathrm{mm}$, the effective stiffness is $1/(C_b+C_s)=8MN/m$ and the load is1.6kN. The stored elastic energy in this combined linear compliance isThis is stored energy; the frictionless elastic model does not itself predict heat loss during a complete reversible cycle.

If the housing is fixed and free mismatch is $\alpha L\Delta T$, the bearing end-stop engagement temperature increment is $\Delta T_{\ast }=g/(\alpha L)$. With hypothetical $\alpha =12\times 10^{-6}{\mathrm{K}}^{-1}$ and $L=0.50\mathrm{m}$, the0.10mm gap closes at about16.67K temperature increase. Further heating raises the load according to the engaged stiffness.

The positive-part relation is a simple instance of unilateral contact complementarity: the stop supplies a reaction only when the corresponding clearance is exhausted. Shaft inertia, impact, friction or bearing preload require a more complete model. In particular, this static compatibility calculation does not determine a collision force when a moving assembly first hits the stop.

The bearing end-stop engagement temperature is the temperature state at which free relative expansion first exhausts the clearance to an axial stop. It must specify the initial setting and how both shaft and housing temperatures change. A non-locating bearing arrangement normally accommodates relative expansion; an exhausted travel limit changes the axial constraint.^[ref]

For an illustrative common temperature increase $\Delta T$, write free relative movement aswhere ${\delta }_0$ is the assembly offset toward the stop. With positive remaining gap and $\beta >0$, engagement occurs atFor hypothetical ${\alpha }_s=12\times 10^{-6}{\mathrm{K}}^{-1}$, ${\alpha }_h=8\times 10^{-6}{\mathrm{K}}^{-1}$ and $L_s=L_h=0.50\mathrm{m}$, $\beta =2\times 10^{-6}m/K$. A0.10mm gap with no initial offset closes after50K common heating. Treating the housing as fixed would instead give16.67K, a different boundary-condition model.

An initial offset ${\delta }_0=0.02\mathrm{mm}$ reduces the engagement increase to40K. With combined axial compliance $C_b+C_s=1.25\times 10^{-7}m/N$, the engaged bearing thermal reaction slope isAt60K increase, the excess20K beyond engagement therefore gives320N of bearing thermal end-stop load in this linear quasistatic model.

If $\beta =0$, common uniform heating does not change the relative gap in this model. If $\beta <0$, heating moves away from the stated stop; cooling or another operating change can be relevant instead. Temperature gradients or independently heated components require their actual relative-displacement model rather than a universal scalar engagement temperature.

The bearing thermal reaction slope is the local rate of change of a bearing assembly's thermal constraint force with temperature along a specified thermal path. For constant combined axial mechanical compliance $C=C_b+C_s$, relative free-expansion rate $\beta$ and an engaged end stop, the linear model givesIts units are force per temperature, such asN/K. It combines thermal mismatch with elastic restraint; it is not a stiffness measured inN/m. The need to accommodate differential expansion motivates locating/non-locating arrangements in the bearing manufacturer's design guidance.^[ref]

For an original one-stop model with initial relative position ${\delta }_0$, gap $g$, uniform temperature change $\Delta T$ and $\beta >0$,The slope is zero while the gap is open and $\beta /C$ after engagement. At the ideal switching point, the two one-sided slopes differ. The stated slope therefore needs the contact state; it should not be applied throughout the open-gap regime.

Take $\beta =2\mu m/K$ and $C=0.125\mu m/N$. The engaged slope is16N/K. If the bearing end-stop engagement temperature corresponds to a40K increase, increases of60K and70K give forces320N and480N. Conversely, those two force measurements would imply a16N/K slope, and a separately established $\beta$ would imply $C=\beta /(dF/dT)=0.125\mu m/N$. This is a possible bearing thermal stiffness identification measurement, not a reported manufacturer's test.

A bench assessment could measure temperature and axial force at successive equilibrated states after engagement. Measuring both shaft and housing temperatures helps distinguish their actual thermal path from an assumed common temperature. The inferred combined mechanical compliance does not separately identify $C_b$ and $C_s$. Temperature-dependent elasticity, friction, changing contact or temperature gradients can make the observed slope vary, while transient heating can produce a path different from the equilibrated one.

As a local sensitivity coefficient, the engaged slope also propagates a temperature contribution to force standard uncertainty: for a known16N/K slope and $u(T)=0.5\mathrm{K}$, that contribution is8N. Uncertainty in the thermal coefficient, compliance and gap requires its own terms, with covariance included when inputs are correlated.^[ref]

Bearing thermal stiffness identification uses measured force changes along a specified thermal path, together with independently known expansion behavior, to infer the axial restraint of an assembly. The observed bearing thermal reaction slope is $q=dF/dT$ inN/K. Mechanical stiffness is force per displacement, so converting $q$ into stiffness requires the free relative-expansion rate $\beta$ in displacement per temperature. In the constant-compliance engaged model,Here $C_b$ and $C_s$ represent the bearing and supporting structure's axial mechanical compliance contributions.

For an original identification example, suppose equilibrated engaged measurements give forces160,320 and480N at temperature increases50,60 and70K. A linear regression gives slope16N/K and, within this model, an extrapolated zero-force engagement increase of40K. If an independent free-expansion measurement gives $\beta =2\mu m/K$, the total compliance is $0.125\mu m/N$ and total stiffness is $8N/\mu \mathrm{m}=8\times 10^{6}N/m$. A16N/K thermal slope is therefore not itself a16N/m spring stiffness.

The measurements have an identifiability limit: the slope determines $C_b+C_s$ when $\beta$ is known, rather than determining both contributions separately. A known supporting-structure compliance $C_s=0.100\mu m/N$ would imply $C_b=0.025\mu m/N$. Without that extra information, many pairs give the same total. If $\beta$ is also unknown, scaling both $\beta$ and total compliance by the same positive factor leaves the measured slope unchanged.

A bench experiment can measure axial force with a load cell, track shaft and housing temperatures, wait for equilibrated states and fit only measurements from the engaged regime. A separate low-restraint expansion measurement can estimate $\beta$. Repeating heating and cooling helps test whether friction, changing contact or thermal gradients invalidate the assumed single slope. These are proposed measurement steps, not a claimed manufacturer's experiment. The physical motivation is the need to accommodate relative thermal displacement in a locating/non-locating bearing arrangement.^[ref]

For independent $\beta$ and $q$, first-order propagation givesFor $\beta =2.0$ with standard uncertainty0.1µm/K and $q=16.0$ with standard uncertainty0.8N/K, the inferred total compliance has standard uncertainty about0.00884µm/N. Correlated temperature references require the corresponding covariance contribution rather than this independent-input expression.^[ref]