 Bearing thermal-expansion constraint load

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.