FRICTIONALLY-EXCITED THERMOELASTIC INSTABILITY (TEI)
When two bodies slide against each other, frictional heat is
generated and the resulting thermoelastic deformation alters
the contact pressure distribution. This coupled thermomechanical
process is susceptible to thermoelastic instability (TEI).
Above a certain critical speed, a nominally uniform pressure
distribution is unstable, giving way to localization of load
and heat generation and hence to hot spots at the sliding
interface (Barber, 1969, Kennedy & Ling, 1974,
Floquet & Dubourg 1994, Bryant et al., 1995,
Kao et al., 2000). The problem is particularly
prevalent in energy dissipation systems such as brakes and
clutches. Hot spots can cause material
damage and wear and are also a source of undesirable frictional
vibrations, known in the automotive disk brake community
as ``hot roughness'' or ``hot judder'' (Kreitlow et al., 1985,
Inoue, 1986, Zagrodzki, 1990, Anderson & Knapp, 1990,
Lee & Dinwiddie, 1998). For an animated view of infrared
experimental observations of the
development of hot spots during drag braking of an
automotive disk brake, click
A windows-based software package
for estimating the susceptibility of
brake and clutch systems to TEI is available for
purchase from the University of Michigan. For more
information, including sample input and output and
a demonstration that can be downloaded,
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.
Figure 1: Evidence of hot spotting on a clutch disk
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 2: Clutch disk after a single engagement
Burton et al. (1973) used a perturbation method to
investigate the stability of contact between two sliding
half planes. The system is linearized about the uniform
pressure state and perturbations are sought which can grow
exponentially with time. Their results provided useful
insight into the nature of the phenomenon, but there is
no inherent length scale in the problem as defined and
it was found that sufficiently long wavelengths are
always unstable. A length scale can be artificially
introduced into the analysis by restricting attention
to perturbations below a certain wavelength, estimated
as being comparable with the linear dimensions of the
practical system, but the resulting predictions for
critical speed do not generally show good agreement with
those observed experimentally (Dow & Stockwell, 1977,
Banerjee & Burton, 1979).
The effect of geometry
The first solution of a TEI problem involving a geometric
length scale was given by Lee & Barber (1993), who used
Burton's method to analyze the stability of a layer
sliding between two half planes. This geometry provided
a first step towards that of the typical disk brake
assembly, where a disk slides between two pads of a
friction material. Using typical material properties from
automotive applications, it was found that stability is
governed by a deformation mode that is antisymmetric with
respect to the mid-plane of the layer and that has a
wavelength proportional to the layer thickness.
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.
Numerical implementation of Burton's method
A promising alternative approach is to implement Burton's
perturbation method numerically, leading to an eigenvalue problem
to determine the stability boundary. If the exponential growth
rate of the dominant perturbation can be assumed to be real,
the critical sliding speed is defined by the condition that
there exists a steady-state equilibrium perturbation --- i.e. one with
zero growth rate. Du et al. (1997) used the finite
element method to develop the matrix defining this eigenvalue
problem. Yi et al. (1999) used Du's method
to explore the effect of disk geometry in an idealized disk
brake in which the brake pads are assumed to be rigid and
non-conducting. Their results showed that the critical speed is in many cases
quite close to that predicted by the considerably simpler
analysis of Lee & Barber (1993), which probably explains the
success of that analysis in practical applications.
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.
Eigenvalue formulation for the exponential growth rate
Burton's method can be implemented numerically for systems
of two thermal conductors by defining the eigenvalue problem
for the exponential growth rate. This method was first
suggested by Yeo & Barber (1996), who developed it in the
context of the related static thermoelastic contact problem,
where instability results from the pressure dependence
of an interfacial contact resistance.
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
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
This method has been implemented in a
windows-based software package
that is available for purchase from the University of Michigan.
Here we apply the method to the practical
clutch design shown in Figure 3. The design
parameters and material properties were chosen to be consistent
with the clutch whose
experimentally damaged disks were shown above.
The clutch has three steel stators and two composite rotors.
The rotors each have a steel core and a friction material layer
bonded onto each side. During operation, sliding occurs between the
friction material layers and the adjacent stators. A hydraulic
pressure, P is applied to the upper piston, causing the
stack of disks to be compressed against the lower reaction
plate. The corresponding boundary conditions on the
(homogeneous) perturbation problem are therefore that
the upper surface of the piston be traction-free
and the lower surface of the end plate be restrained from axial motion.
All exposed surfaces were assumed to be thermally
insulated, since practical heat transfer coefficients are so
small that they hardly affect thermoelastic instability.
Figure 3: Typical multi-disk clutch system. All
dimensions are in mm
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.
Figure 4: Critical speed as a function of wavenumber, n, for the five-disk clutch
The eigenfunction for temperature in the stator surface is
shown on the left for the n=10 mode.
Comparison of this figure with the experimental observations
of Figure 2 above shows that the perturbation analysis
correctly predicts an antisymmetric mode with focal hot
spots, but the dominant wavenumber is predicted to be 10
in contrast to the 12 hot spots observed experimentally.
Various explanations might be advanced for this relatively small
discrepancy. The initial speed for clutch engagement is well above
the predicted critical value and all wavenumbers between
4 and 14 are unstable at the beginning of the engagement.
However, the mode with wavenumber 10 has the highest
growth rate and would be expected to dominate the transient
process. A more plausible explanation is that clutch friction
materials exhibit quite complex constitutive behaviour and
it is difficult to select an appropriate incremental
elastic modulus for the analysis. The modulus used in the
finite element analysis was the incremental modulus obtained in compression
tests at the mean engagement pressure, but significant
stiffening may occur under service conditions. The critical
speed and the dominant eigenmode are both quite sensitive
to the modulus of the friction material and plausible values
could have been chosen to `fit' the theoretical predictions
to a wavenumber of 12. This highlights the fact that the
principal difficulty remaining in obtaining reliable theoretical
predictions for TEI performance lies in the accurate
characterization of the properties of the complex friction
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.
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.
The Fourier reduction method described here permits
a remarkably efficient solution of the frictional thermoelastic
stability problem for systems in which the geometry is
axisymmetric. The power of the method is demonstrated by the
multidisk clutch example, direct numerical simulation of which
would represent an extremely challenging computational problem.
Values are obtained for the critical sliding or rotational speed
and also for the exponential growth rate of each mode when
operating above the critical speed.
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
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.
Back to J.R.Barber's homepage
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