Figure 1 shows one of the plates of a typical multidisk wet clutch
after a period of normal service.
The dark areas correspond to regions in which high local temperatures
have been experienced. Evidence of surface melting
can be found in extreme cases. In addition, transfer of
friction material components and the products of overheated
transmission fluid may be involved. The pattern seen in
Figure 1 represents the effect of multiple engagements
of the clutch and shows several series of hot spots
at different locations overlayed on each other.

A better
picture of the phenomenon is obtained if the clutch is
examined after a single engagement, in which case the
regular pattern shown in Figure 2 is obtained. The complete
disk in this particular case exhibits 12 equally spaced hot spots
on each side and they are arranged antisymmetrically.
In other words, the hot spots on the opposite side of the
disk are located in the gaps between those shown in the
figure.

Despite the considerable idealizations involved in Lee's
theory, it provides plausible predictions for the critical
speed and the mode shape in typical brake assemblies and
is therefore quite widely used in the brake and clutch
industry for TEI analysis. However, there is a clear
need for a method that will account for other features of
the system geometry, such as the finite width of the sliding
surface, the axisymmetric geometry of the disk and the `hat'
section used to attach the disk to its support. One
approach is to use the finite element method to solve the
coupled transient thermoelastic contact problem in time
(Zagrodzki 1990, Johannson 1993, Zagrodzki * et al.* 1999).
This method is extremely flexible, in that it can
accommodate non-linear or temperature-dependent constitutive
behaviour, more realistic friction laws and practical loading
cycles. However, it is also extremely computer-intensive and
appears unlikely to be a practical design tool for three-dimensional
problems in the foreseeable future.

Du's method rests on the assumption that the dominant
perturbation has a real growth rate. The limited range of problems
that have been solved analytically suggest that this
assumption is justified if one of the
two sliding bodies is a thermal insulator, or if the
dominant perturbation is independent of the coordinate in
the sliding direction, as in `banding' instabilities in
axisymmetric systems, where the frictional heating is
concentrated in an axisymmetric annular band within the
contact area. However, a
rigorous proof of this result has never been advanced.
When both materials are thermally conducting, the stability
boundary is generally determined by a disturbance that
migrates with respect to both bodies in or opposed to the direction of
sliding (Burton * et al.* 1973).
In a stationary frame of reference, the perturbation
will then appear to oscillate in time, corresponding to
a complex exponential growth rate.
The migration speed is smaller relative to the better
thermal conductor and this relative motion approaches zero when the other
body tends to the limit of thermal insulation.
Practical systems such as brakes and clutches usually
involve a steel or cast iron disk sliding against a
composite friction material of significantly lower
conductivity (typically two orders of magnitude lower
than that of steel). As a result, the dominant
perturbation migrates only very slowly relative to the metal
disk. However, this migration plays an important part in the
process, because it reduces the thermal expansion due to
a given perturbation in heat input and hence increases
the critical speed.

We first assume that the temperature, stress and displacement fields can be written as the product of a function of the spatial coordinates and an exponential function of time When these expressions are substituted into the governing equations and boundary conditions of the problem, the exponential factor cancels and we are left with a modified system of equations in which the growth rate appears as a linear parameter. A finite element discretization of this modified problem then yields a linear eigenvalue problem for the growth rate.

In order to adapt this method to the sliding contact problem,
we need to choose a suitable frame of reference, relative to which
at least one of the bodies will necessarily be moving. This
introduces convective terms into the heat conduction equation and
can present numerical problems when the convective term is large
(Christie * et al.* 1976). The relative magnitude of convective
and diffusive terms can be assessed by calculating the Peclet number
Pe = Va/k,
where V is the convective velocity, k is the thermal
diffusivity and a is a representative length scale.
Peclet numbers in tribological applications are typically
very large. For example, a steel clutch disk of mean
diameter 0.2 m rotating at 2000 rpm corresponds to
a Peclet number of about 35,000 using the mean diameter for a
and even the element Peclet number will be large compared
with unity for a realistic discretization. Thus, the
convective terms will tend to dominate the finite element
solution.

Fortunately, difficulties with convective terms can be avoided by using Fourier reduction in the sliding direction as long as (i) no material points on either sliding body experience intermittent contact and (ii) periodic boundary conditions apply in the sliding direction. These conditions are satisfied for multidisk brakes and clutches, which have an axisymmetric geometry, but which often exhibit signs of damage attributable to TEI with a non-axisymmetric eigenmode, as shown in the figures above. In this case, orthogonality arguments show that all the eigenmodes must have Fourier form in the circumferential direction and the sinusoidal function can be factored out of the equations, leading to an eigenvalue problem in radial and axial coordinates only, for given values of wavenumber and rotational speed. A finite element description of this problem leads to a linear eigenvalue problem for the growth rate.

The critical speed for the five disk clutch system is
shown in Figure 4 as a function of wavenumber n.
Instability first occurs in the banding mode (n=0) at a
rotational speed of 218 rad/s, but there is also a local
minimum of 237 rad/s at n=10. This clutch is designed
for initial engagement speeds of 628 rad/s,
so we also calculated the exponential growth rate
at this speed. The maximum value of b=39.6 (1/s)
corresponds to the Fourier mode with 10 hot spots
per revolution, whilst the banding mode grows only at
a rate of 12.2 (1/s). Thus, the non-axisymmetric
mode would be expected to be dominant in this application.

Notice that the axisymmetric mode in this example
has the lowest critical speed but that a non-axisymmetric
mode becomes dominant at larger speeds. The general
solution of the transient thermoelastic contact problem
at constant sliding speed can be written down as an eigenfunction
expansion. Furthermore, the fact that some of
these terms grow exponentially but that most decay suggests that
a severely truncated series (a * reduced order model*) might
give a highly efficient numerical approximation to the transient
behaviour, whilst retaining adequate accuracy. This is the subject
of an ongoing investigation.

The corresponding temperature contours in the (r,z) plane predicted for the dominant eigenmode are shown on the right. Notice that the eigenmode is antisymmetric with respect to the central stator, in which the greatest temperature perturbations are recorded. The temperatures in the rotors are close to zero except in a thermal boundary layer that is too thin to be visible in the figure. The two rotors exhibit a `qualitative' symmetry in the sense that hot regions occur at the same locations on the two sides, but the maximum temperatures are lower on those surfaces nearest to the piston and the end plate. Both these predictions were confirmed by the experimental observations. The most severe damage was observed on the central steel disk and the location of hot spots on the other two stators indicated a mode symmetric with respect to the rotors. The attenuation of the disturbance near the ends of the disk stack is probably attributable to the extra rigidity provided to the terminal stators by the piston and end plate. In fact, a simpler model in which the piston and end plate were replaced by rigid non-conducting surfaces predicted a critical speed within 1% of the more exact value.

This explanation also suggests that a more exact sequence of antisymmetric and symmetric perturbations in the stators and rotors respectively would be observed in a clutch with a larger number of disks. This was confirmed by additional finite element calculations for clutches of the same form, but with odd numbers of disks between 3 and 13. The critical speed decreases towards a limit as the number of disks increases and the dominant mode approaches a state in which the perturbation is strictly antisymmetric in the steel disks and symmetric in the composite disks, except for those near to the ends of the stack. This limiting condition was also obtained independently by modelling half of one rotor and one stator, using symmetric/antisymmetric boundary conditions at the respective mid-planes. The number of hot spots in the dominant eigenmode increases slightly with the number of disks, but the solution has essentially converged on the limit for clutches with 11 disks or more.

In conventional clutch systems with alternating steel and composite disks, the dominant unstable mode is usually antisymmetric with respect to the steel disks, symmetric with respect to the composite disks and involves an integer number of focal hot spots around the circumference. This prediction is confirmed by experimental observations of thermal damage in a 5 disk clutch.

The method is easily applied to other examples and can therefore be used to assess the effect of design modifications such as changes in geometry and material properties on the thermoelastic stability of multidisk brakes and clutches.

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