 Bearing thermal reaction slope

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]