Bearing thermal-expansion constraint load ($F=EA\alpha \Delta T$)

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]