= Bearing thermal stiffness identification
<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,
$$
q=\frac\beta{C_b+C_s},\qquad
C_{\rm total}=\frac\beta q,\qquad
k_{\rm total}=\frac q\beta.
$$
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 $\beta=2\,\mathrm{\mu m/K}$, the total compliance is $0.125\,\mathrm{\mu m/N}$ and total stiffness is $8\,\mathrm{N/\mu m}=8\times10^6\,\mathrm{N/m}$. 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 $C_s=0.100\,\mathrm{\mu m/N}$ would imply $C_b=0.025\,\mathrm{\mu m/N}$. 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.<Rolling bearings by SKF>[Section B.2, printed p.70, thermal expansion and relative axial displacement]{ref}
For independent $\beta$ and $q$, first-order propagation gives
$$
u^2(C_{\rm total})\approx\frac{u^2(\beta)}{q^2}+\frac{\beta^2u^2(q)}{q^4}.
$$
For $\beta=2.0$ with <standard uncertainty>0.1µm/K and $q=16.0$ with <standard uncertainty>0.8N/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.<Guide to the expression of uncertainty in measurement by JCGM>[§5.1.3, sensitivity coefficients; §5.2, correlated inputs]{ref}
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