Common transmission failures result from the glazing of clutch surfaces by
deposition of oil oxidation and decomposition products. Both the oxidation and
decomposition processes depend on temperature histories of the surfaces.
Because it is difficult to measure these surface temperatures during
operation, it is useful to develop models to predict clutch-interface thermal
behavior. The relative velocity between mating clutch plates, from the initial
engagement to the zero-sliding (lock-up) condition, generates heat that is
transferred to the plates. The relative velocity decreases at a constant rate
during this period, producing a heat flux that is initially very large and
decreases linearly with time, until lock-up occurs. Accordingly,
\(q_{f}^{\prime \prime}=q_{o}^{\prime \prime}=\left[1-\left(t /
t_{\mathrm{lu}}\right)\right]\), where \(q_{o}^{\prime \prime}=1.6 \times 10^{7}
\mathrm{~W} / \mathrm{m}^{2}\) and \(t_{1 \mathrm{u}}=100 \mathrm{~ms}\) is the
lock-up time. The plates have an initial uniform temperature of
\(T_{i}=40^{\circ} \mathrm{C}\), when the prescribed frictional heat flux is
suddenly applied to the surfaces. The reaction plate is fabricated from steel,
while the composite plate has a thinner steel center section bonded to low-
conductivity friction material layers. The thermophysical properties are
\(\rho_{s}=\) \(7800 \mathrm{~kg} / \mathrm{m}^{3}, c_{\mathrm{s}}=500
\mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), and \(k_{s}=40 \mathrm{~W} /
\mathrm{m} \cdot \mathrm{K}\) for the steel and \(\rho_{\mathrm{im}}=1150
\mathrm{~kg} / \mathrm{m}^{3}, c_{\mathrm{fm}}=1650 \mathrm{~J} / \mathrm{kg}
\cdot \mathrm{K}\), and \(k_{\mathrm{fm}}=4 \mathrm{~W} / \mathrm{m} \cdot
\mathrm{K}\) for the friction material.
(a) On \(T-t\) coordinates, sketch the temperature history at the midplane of
the reaction plate, at the interface between the clutch pair, and at the
midplane of the composite plate. Identify key features.
(b) Perform an energy balance on the clutch pair over the time interval
\(\Delta t=t_{\mathrm{lu}}\) to determine the steadystate temperature resulting
from clutch engagement. Assume negligible heat transfer from the plates to the
surroundings.
(c) Compute and plot the three temperature histories of interest using the
finite-element method of FEHT or the finite-difference method of IHT (with
\(\Delta x=0.1 \mathrm{~mm}\) and \(\Delta t=1 \mathrm{~ms}\) ). Calculate and
plot the frictional heat fluxes to the reaction and composite plates,
\(q_{\mathrm{rp}}^{\prime \prime}\) and \(q_{\mathrm{cp}}^{\prime \prime}\),
respectively, as a function of time. Comment on features of the temperature
and heat flux histories. Validate your model by comparing predictions with the
results from part (b). Note: Use of both \(F E H T\) and \(I H T\) requires
creation of a look-up data table for prescribing the heat flux as a function
of time.
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