= Bearing end-stop engagement temperature
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.<Rolling bearings by SKF>[Section B.2, printed p.70, thermal displacement and support functions]{ref}
For an illustrative common temperature increase $\Delta T$, write free relative movement as
$$
\delta_{\rm free}=\delta_0+\beta\Delta T,\qquad
\beta=\alpha_sL_s-\alpha_hL_h,
$$
where $\delta_0$ is the assembly offset toward the stop. With positive remaining gap and $\beta>0$, engagement occurs at
$$
\Delta T_*=(g-\delta_0)/\beta.
$$
For hypothetical $\alpha_s=12\times10^{-6}\,\mathrm{K^{-1}}$, $\alpha_h=8\times10^{-6}\,\mathrm{K^{-1}}$ and $L_s=L_h=0.50\,\mathrm m$, $\beta=2\times10^{-6}\,\mathrm{m/K}$. 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 $C_b+C_s=1.25\times10^{-7}\,\mathrm{m/N}$, the engaged <bearing thermal reaction slope> is
$$
\frac{dF}{dT}=\frac\beta{C_b+C_s}=16\,\mathrm{N/K}.
$$
At60K 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.
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