CC BY
102
y^K 539.62
Hot bands and hot spots: Some direct solutions of continuous thermoelastic systems with friction
M. Graf, G.-P. Ostermeyer
Technische Universität Braunschweig, Braunschweig, 38106, Germany
Hot bands and hot spots are thermoelastic phenomena appearing in frictional systems with high energy dissipation like brake systems or clutches. These thermoelastic instabilities are driven by the interaction of friction-induced heat in the sliding plane and thermal expansion of the materials. Systems exposed to thermoelastic instabilities show a characteristic temperature pattern that can lead to local material damage and vibrations like judder or brake torque fluctuations. While hot bands are observable by a cut through the system normal to the direction of sliding, hot spots are described by a cut parallel to the direction of sliding. When an angle parameter is introduced in a model-based description, both types of thermoelastic instabilities can be described by one single model. Such a model is presented that comprises of layers corresponding to different mechanical parts (e.g. pad, disk, homogenized cooling channels). Every layer is described by field equations for thermoelastic behavior and heat conduction. All layers basically include the same set of solutions which can analytically be found by separation of complex variables. These solutions are scaled to satisfy the boundary conditions at the contact areas between the layers. No symmetry conditions are required, but if present, they can simplify the model. The stability of one thermoelastic phenomenon under investigation is determined by evaluating the characteristic equation of the system. The appearance of hot spots or bands, their spatial distribution and movement are discussed in terms of sliding velocity and other system parameters.
Keywords: brake, clutch, thermoelastic instability, hot banding, hot spotting, judder
1. Introduction
Thermoelastic instabilities are typically observed in frictional systems like clutches or brakes, which convert large amounts of kinetic energy into thermal energy. When reaching a critical sliding velocity, experiments show a nonho-mogeneous and often periodic temperature distribution on the sliding surface, e.g. [1-3]. The mechanism behind is driven by the interaction of heat generation and thermal expansion: A local rise in surface temperature results in a thermal expansion of the material nearby. Such a region of elevated temperature therefore is slightly higher than the surrounding topography and therefore carries a dominating part of the frictional load. A following rise in heat can destabilize the process [4]. Two basic forms of thermoelastic instabilities are commonly observed:
- Hot spots are the maxima in a periodic temperature field in circumferential direction [5] (Fig. 1).
- Hot bands describe a nonhomogeneous temperature field in radial direction with one or few hot rings of nearly constant radii [5] (Fig. 2).
One goal in development of frictional systems is to impede thermoelastic instabilities because their focused tempe-
rature can disrupt the material and they can raise unwanted vibrations and noise. Pioneering works in the modeling of thermoelastic instabilities were published in [6] for the case of two sliding half-planes and in [7], where a problem close to a brake system is solved: it comprises two elastic halfplanes with a third sliding body in between. Both include a direct solution of the coupled temperature and displacement field. The latter model was further developed by correcting the pad stiffness [8]. First approaches are available aiming for intermitted contact problems [9, 10]. It is observed that the appearance of thermoelastic instabilities can be described by linear stability criteria, where, as a first approximation, stability does not depend on global contact pressure, but on sliding velocity. Typically thermoelastic instabilities appear with a preferred wavelength in the temperature field. To cover more detailed geometries, numerous FE-models are available, e.g. [11], a review is given in [12].
Models describing any kind of thermoelastic instabilities must cover the coupled fields of interacting displacements and temperature. The structure of these fields depends on the configuration of involved bodies, which is typically a layered structure for brake systems or clutches. For layered
© Graf M., Ostermeyer G.-P., 2012
structures, a set of analytical solutions can be found that satisfied the boundary conditions and the coupled fields equations. The applied modeling approach includes the direct solution of the field equations, reduces the problem to a discussion of eigenvalues and therefore follows a similar way as e.g. [7]. The authors demonstrate the development of such a model for the example of a disk brake, where many types of boundary conditions appear. The same modeling technique can be applied for other sliding systems equally. Both effects, hot spots and hot bands, differ basically by their spatial orientation compared with the direction of sliding. Therefore one single two-dimensional model suffices to cover both phenomena, if the model includes a parameter that allows the definition of its spatial orientation.
2. Model development
2.1. One model for hot bands and hot spots
For model development a brake system is considered, which is defined in cylindrical coordinates as shown in Fig. 3 (depthy, radius r, angle Q). To investigate the region where the disk is covered with pads, a sectional plane is introduced, which is perpendicular to the sliding plane and which has an angle 9 to the sliding direction (Fig. 4). This plane is spanned by the Cartesian coordinates for depth y and one coordinate parallel to the sliding plane x. The intersection shows five bodies, representing the two brake pads and the disk. Oppositely to many analytical models, one body represents the region of the cooling channels within the disk. The bodies are labeled with numbers i = 1-5 and have a thickness h1. Note that the superscripts are not exponentials, but indices for the body number. The layered structure within the sectional plane is independent of the angle 9, if the finite length of the system in the x-direction is not taken into account.
When hot spots are under discussion, the periodicity of the temperature field is observed in circumferential direction on the disk, therefore 9 = 0° (Fig. 3): Coordinate x points in sliding direction (Fig. 5). A two-dimensional model on the sectional plane is reasonable to describe the basic interactions in the contact region. The assumptions are [13]:
- The cylindrical system can be replaced by the Cartesian system with x = rQ, because typically the width of the sliding path W is small compared to the average radius of the sliding path Rav: W< Rav.
- A constant temperature distribution in the radial direction is assumed that allows the application of a two-dimensional model.
When hot bands are investigated, similar assumptions as for the hot spots case apply. For hot bands, the periodicity of the temperature field, or at least, the nonuniformity, is observed in the radial direction, while the field is rather constant in the sliding direction. The sectional plane is defined at 9 = 90° (Fig. 3) and x = r. Coordinate x points perpendicular to the direction of sliding (Fig. 6).
Consequently one two-dimensional model suffices to cover both effects, if the model includes an angle parameter. This parameter ensures that the periodicity of the temperature field is always observed in the sectional plane. In the model developed in the following, both phenomena, hot spots and hot bands, are described separately. Interaction effects between both phenomena are not in the focus of the present investigation.
2.2. Field equation and required solutions
Every body i (Fig. 4) is described by interacting displacement and temperature fields: ux (x, y, t), uy (x, y, t) and Tl (x, y, t). Thermoelasticity requires two Lame’s parameter X1 and M1 for elasticity, thermal expansion a1 and thermal diffusivity k. For the two-dimensional field equations the equations under plane-strain conditions are chosen that well agree with the three-dimensional solution, if the normal width of the body is sufficiently high. This is the case for typical brake systems [13]. For the i-th body the coupled field equations yield [14]:
(X + 2M )u'x,xx + M'u'x,yy + (X + M-1 )u'y,xy - m Tx = 0
(X + 2M V yy +MUy xx + (X +M )u'x xy - m T'y = 0, (1)
k Tx + Fyy) - Tt - v cos 97,x = 0, herein
m = a1 (3X' + 2^!)
(2)
Fig. 1. Thermograph picture of hot spots on a disk brake [5]
Fig. 2. Thermograph picture of one hot band on a disk brake [5]
Sectional plane for 2D-model
Fig. 3. Brake system with disk and two pads in cylindrical coordinates. Two-dimensional sectional plane is inserted at angle 9 relative to sliding direction
is applied for simplicity. Partial derivatives are expressed in the subscript by a comma, for instance Tt := dT/dt or uxxy := d 2ux/(dxdy). The set of field equations (1) covers thermoelastic behavior and thermal conduction. The absolute in-plane velocity of a body is v1 that can lead to thermal convection. With respect to the x-coordinate, only the velocity component v' cos 9 is only felt in the time derivative of the temperature. This coordinate transformation is controlled by the angle parameter 9 and includes that for hot bands the convection does not play a role. An analytical solution of the linear system with constant coefficients can be found by separation of constants and by writing the variables in complex notation (imaginary unity j =-J-l). A solution can be derived and tested with support of computer algebra tools, mathematical details are omitted for brevity. One obtained solution for the coupled set of equation is
ux (x, y, t) = (Km keSy + K2lS1(X+Mi )es2y +
+ K3 (IsiM1 +1 (X +M1 ))es2 y )ejxll+Dt, uy (x, y, t) = (Km1 k ls{esy + K2(X +M1 )es2y + (3)
+ K3 (-(X + 2m1 ) + s2 y(X + M1 ))es2 y )ejx!l+Dt,
T1 (x, y, t) = K (X + 2m1 )(Dl + jv1 cos 9)eSy+jx/l+Dt with behavior in the y-direction
Fig. 5. Hot spots are observed on a sectional plane parallel to the sliding direction: 9 = 0°
s2 =±
(4)
This solution contains one complex eigenvalue D that determines the dynamics with respect to stability and periodicity. In the x-direction, parallel to the layered structure, a pure periodicity is assumed reflecting the periodicity of hot bands and hot spots. Its wavelength can be changed by l. The behavior in the y-direction is given by the variables sj and 4 One of the two possible signs is chosen to ensure that the modeled fields decay to zero for infinite distances from the coordinate system, corresponding to a half-plane solution. In this case three complex constants Kj, K2 and K are open to adapt the half-plane to boundary conditions:
Fig. 6. Hot bands are observed on a sectional plane perpendicular to the sliding direction: 9 = 90°
generally one condition for temperature or heat and two for displacements or stresses are required. For the present system, half-plane solutions are not of interest, because boundary conditions exist on both sides of each layer. Consequently the solutions in Eq. (3) are applied twice, once for positive signs of s1 and s2 and once with negative sign. The six constants K[, K2, K\, K4, K5 and K6 are available for boundary conditions on both sides of the layer. In this case the fields consist of two half-plane solutions. To adapt the solution to the boundary conditions, the normal and shear stresses are required and can be gained by derivatives:
, *\l I I rril
y,y + K ux,x - mT ,
x ).
(5)
xy x, y y,x
The solutions (3) cannot explicitly cover boundary conditions for limited width in the x-direction. For the case of hot spots, the boundary conditions are implicitly fulfilled, if an integer number n of hot spots appears on a circumference sliding length 2nRav, therefore In = Rav must be satisfied. From a rigorous point of view, for the case of hot bands, boundary conditions on inner radius and outer radius of the sliding path cannot be fulfilled. From an engineer’s experience it is plausible that the appearance of hot bands is dominated by materials involved and sliding conditions and is less strong influenced by the boundary conditions on the inner and outer region of the sliding path. Hence it is expected that main effects are included in the model and acceptable results will be obtained although the inner connection of the disk to the hat section and the free outer radius is not described.
2.3. Formulation of boundary conditions
Different classes of boundary conditions appear in the present system. They are formulated in terms of the structure shown in Fig. 4 and the disk-fixed coordinate system therein. Consequently the movement of the disk vanishes v2 = v3 = 0 while the pads move in the positive x-direction v1 > 0. The physical system state (temperatures, displacements and their derivatives) can be understood as the sum of two parts:
- A reference state that covers for example, how the brake system continuously heats up while braking or how the braking pressure is applied that avoids a loss of contact in the contact plane.
- A fluctuating disturbance about the reference state. This disturbance describes the dynamics of thermoelastic instabilities. The following boundary conditions are chosen to model only this periodic disturbance of the reference state:
(6)
ref
Ti Ti Ti
i ux = i ux - i ux
i uy / i uy / phys i uy /
2.3.1. Rigid clamping
At y = -h1 the pad is fixed to stiff and well-conducting backing plates. Therefore displacements and temperatures vanish:
u\ = 0, u\ = 0, T1 = 0.
(7)
2.3.2. Sliding plane with friction
One sliding plane appears at y = 0. No penetration or loss of contact in y-direction is allowed. Steady sliding in one direction and no stiction is assumed (v1 > v2). For sliding in the opposite direction, signs must be adapted. The frictional law determines the shear force that is observed in the sliding plane. Naturally the coefficient of friction fin brake systems is highly dynamic and depends on the state within the sliding boundary layer, e.g. [15]. For simplicity f is assumed as a constant value. The shear force on the intersectional plane is controlled by the angle parameter 9 and must vanish for the case of hot banding. Equality of stresses and temperatures between both sliding faces are to be satisfied. The heat balances includes thermal conductivity Ai and the source of friction-induced heat. This is no directional property and is therefore independent of parameter 9. The six conditions yield:
Uy - uy = 0
- f cos 9 a* = 0,
12
xy - xy = 0 CTyy -^ = 0,
(8)
T1 - T2 = 0,
A2T2 -ATy - fCTyy v1 = 0.
2.3.3. Transition conditions
At y = h2 two elastic bodies with different material properties are fixed to each other. The six transition conditions include equality in displacements, stresses, temperature and heat:
23
uy - uy = 0
23
ux - ux = 0,
23
xy - xy = 0,
CTyy - CTyy = 0,
T2 - T3 = 0,
a2j2 -aT, = 0.
(9)
2.3.4. Symmetry conditions
All further contacts required in the present system can be formulated as the contacts before, but the system can be simplified for the case of a symmetric system with respect
to material parameters and geometry (h1 = h5 and h2 = = h4). For temperature and displacement field that are symmetrical to y = h2 + 0.5h3, the three conditions are
Uy 0,
^ = 0>
ty = 0.
(10)
Alternatively, a shift of the fields along the x-direction of 180° is possible. In this antisymmetric case a temperature (or displacement) maximum directly is symmetric to a temperature (or displacement) minimum:
uy = 0,
CT yy = 0, Ty = 0.
(11)
2.4. Solution of the system
The set of solutions of the field equation (3) is valid for every body i. Every layer has six constants (K1i,..., K'6) open that can be adapted to the boundary conditions on both sides of the layer 3. For every boundary one thermal condition and two displacements or stress conditions must be fulfilled. The present case for i = 1-3 layers contains 18 constants
3 (12)
k = (K1,..., K6y)T
that parameterize the individual solutions of the field equation. They must fulfill the same number of boundary conditions, Eqs. (7)—(10) or (7)-(9), (11), which can be written in a row vector B = 0. After substituting the solution of the field equations (3) in B, the factor ejXl+Dt can be factored out. The system can be rewritten as
A(D, 9,...) K = 0, (13)
where matrix A equals the Jacobian
A =
d BT
d K
t '
(14)
A characteristic equation for the unknown eigenvalue D is obtained, if the determinant vanishes
det(A(D, ^,...)) = 0. (15)
When the null space K is computed, the eigen functions are available. The matrix A is analytically derived with support of computer algebra. The calculation of the determinant (15) is done numerically, as well as the root-finding algorithm for D and the computation of the null space K. Because matrix A has 18 x 18 entries, the numerical evalua-
tion can be performed efficiently within seconds. For practical evaluations Lame’s parameter of elasticity are replaced by the Young’s modulus E1 and Poisson’s ratio vi:
viEi
Xl =-
(1 + vl )(1 - 2vl )
El
(16)
2(1 + v1 )
The system is discussed in terms of dimensionless quantities. This increases generality of the solution and minimizes rounding errors in the root-finding algorithm. Different choices of dimensionless parameters are possible; for the present investigation, the authors used the set of transformations:
t =■
tv1
T
_ x _ y
x = —, y = —, l l'
H = h1 H y = *y h = h2 H = F' H = F' h = _•
- Dl _ v1h2
D = ^ v =~kT’
v k
E' = K,E y = El
E1 Ey
22 _1 a _y a
a =—, a = —, a1 a 3 k1 = k2 , k 3 = k2 , k = k1 , k = k 3 ,
A1 = — Ay = —
A1 ^ Ay
M =
l
2 2 2 a2k2E2
A2
(17)
From a numerical point of view, the low thermal conductivity of the pad will always lead to a nearly singular system. To overcome this difficulty, for the temperature field in the pad only one half-plane solution is applied: T1 (x, y = -h1, t) ~ T1 (x, y ^ -m, t) = 0. This is tolerable because the temperature field only enters a thin boundary layer on the pad surface. Therefore the constant in the temperature field equations (3), which scales the solution with Re(sJ) < 0, is set to zero. The dimension of A is consequently reduced to 17 x 17.
3. Model analysis
3.1. Parameters
Table 1 contains the applied set of parameters. Because of the sufficient distance of the sliding plane, the region with cooling channels within the disk can be approximated
Table 1
Material and geometry parameters for a gray cast iron disk and low-met brake pads
El, GPa vl a, 10-6/k Al ,W/(K • m) kl, mm2/s hl, mm
i = 1 : pad 1.5 0.1 11 0.64 0.23 10
i = 2: disk 125 0.26 11 50 14 5
i = 3: cooling channels 25 0 11 10 14 10
314
Wave length 2nl, mm 105 63
45
314
Wave length 2nl, mm 105 63
45
E
£
Dimensionless disk thickness h
Dimensionless disk thickness h
1 2 3 Re(D), 1/s -1.5 -1.0 -0.5 Im(D), 1/s
Fig. 7. Eigenvalues for hot spots with an antisymmetric field: real (a) and imaginary part (b). A practical instability is marked with a gray-scaled dot
by homogenization. The filling with gray cast iron is chosen as 20 %, which reduces stiffness and heat transition. For the sliding a constant coefficient of friction f = 0.4 is applied.
3.2. Criterion for practical instability of thermoelastic instabilities
For a linear system stability criteria are well-developed: A positive real part of an eigenvalue corresponds to an unstable solution. For the model under discussion, the root-finding algorithm detects many eigenvalues with a positive real part very close to zero for the hot banding case. They can arise due to a weak physical instability mechanism,
neglected stabilization effects (e.g. cooling air, dynamics of the coefficient of friction) or numerical rounding errors. From a practical point of view these weak instabilities are of minor interest because they cannot lead to significant rises within typical braking times. As a criterion for practical instability, a solution is called practically unstable if its amplitude grows by more than a given factor within a defined time span. Therefore the small positives threshold
eRe(D)10s > 10 ^ Re(D) >= 0.23 1 (18)
10 s s
must be reached for an instability of practical interest.
314
Wave length 2nl, mm 105 63
45
314
Wave length 2nl, mm 105 63
45
Dimensionless disk thickness h
Dimensionless disk thickness h
0.3 0.4 0.5 Re(D), 1/s -0.5 -0.45 -0.40 -0.35 Im(D), 1/s
Fig. 8. Eigenvalues for hot spots with a symmetric field: real (a) and imaginary part (b). A practical instability is marked with a gray-scaled dot
y
-0.4
1 2 --4 -2 0 2 4 x
-1.0 -0.8 ' -0.4 0.0 0.4 0.8 1.0
Temperature disturbance T, K
Fig. 9. Unstable antisymmetric temperature field at h = 0.2 and minimum disk, cooling channels, disk, pad
3.3. Evaluation of hot spots
By choosing 9 = 0°, the discussed model is applied on hot spots in circumferential direction. The eigenvalue D is investigated numerically, while the dimensionless sliding speed v and the dimensionless disk thickness h are varied. Additionally the diagrams show axes with dimensioned values for the present system (Table 1). This is the sliding velocity v1 in m/s and the distance between two temperature maxima 2nl in millimeters. The dynamics is discussed in terms of one chart for the real part and on for the imaginary part of the dimensionless eigenvalues. The results can be evaluated for both types of symmetry conditions: antisymmetric conditions or symmetric conditions. A comparison between the stability charts for both fields (Figs. 7 and 8) show that instabilities are stronger in the case of antisymmetric fields. This is in accordance with results of a simpler theoretical models and experimental investigations, e.g. [3].
The antisymmetric instability chart in Fig. 7 shows a minimum sliding velocity that is required for the development of hot spots. Obviously a preferred wavelength of the temperature field and consequently a preferred number of hot spots is typical for each system. In the present system the preferred wavelength appears at approximately 157 mm and a sliding velocity of 0.6 m/s. At the backing plate of the pad, the spatial integral of all stresses must vanish, therefore at least one full period of the temperature and displacement field must be in contact. For instance, it is not possible that only the maximum of the normal force is in contact, because in this case the pad will move backwards and the maximum will disappear [7]. In the present case, a contact length of at least 157 mm is required for the instability at 0.6 m/s. If the pad, for example, has a length of only 105 mm, the instability threshold is found at 0.7 m/s. The present investigation leads to stability limits appearing at clearly lower velocities than in other works. Reasons are the limited thickness of the pad material and the introduction of cooling channels. The imaginary part includes negative values. Vanishing imagi-
in Fig. 7. Layered structure according to Fig. 4. With increasing y: pad,
nary parts represent a temperature field which is stationary on the disk, where the coordinate system is fixed, while negative values correspond to a temperature field that slowly moves in the direction of the sliding pads. Therefore the velocity of the temperature field is between the velocities of the sliding bodies, but is closer to the velocity of the better conductor: the disk. Two mechanisms are available that lead to moving fields on the disk: The stronger effect is that heat is transported by the sliding pads, the weaker effect is the elastic coupling of stress and strains in normal and tangential direction.
Figure 9 shows the eigenform of the antisymmetric temperature field at the minimum unstable sliding velocity. In the antisymmetric temperature distribution a temperature maximum on one side of the disk exactly meets a temperature minimum on the opposite side of the disk. A flow of heat therefore passes the region of cooling channels, which is a stabilizing mechanism. This heat flow is strongly influenced by the thermal properties of the cooling channels. In the pad material, the temperature field quickly decays to y
-0.4
Temperature disturbance T, K
Fig. 10. Unstable symmetric temperature field at h = 0.2 and minimum v in Fig. 7. Layered structure according to Fig. 4. With increasing y : pad, disk, cooling channels, disk, pad
314
Wave length 2nl, mm 105 63
45
314
Wave length 2nl, mm 105 63
45
E
,ty
gnidi
E
,ty
gnidi
Dimensionless disk thickness h
Dimensionless disk thickness h
5 Re(D), 1/s
-6
- 4
-2
Im(D), 10-22 1/s
Fig. 11. Eigenvalues for hot bands with an antisymmetric field: real (a) and imaginary part (b). A practical instability is marked with a gray-scaled dot
zero; therefore for the temperature field a half-plane assumption is reasonable, if local disturbances within the temperature field are under investigation. The temperature only enters a thin boundary layer close to the pad surface. Here small temperature tails appear due to convection. Due to the moving hot spots on the disk, the regions of elevated temperature in the disk are not fully symmetric with respect to the y-axis. Figure 10 contains the temperature field for the symmetric case.
3.4. Evaluation of hot bands
For the case of 9 = 90°, hot bands are under investigation. For both cases, antisymmetric and symmetric temper-
ature field, unstable solutions can be observed (Figs. 11 and 12).
The antisymmetric field shows an instability which is slightly stronger than for the symmetric field and appears at a lower sliding velocity. From a geometric point of view, periodicities with wavelength of more that e.g. 63 mm are not possible on the contact width between typical sizes of pads and disks. Similarly to the case of hot spots, the left part of the diagram with low values of h is not of practical interest. For the case of 52 mm, instabilities appear for the first at approximately 0.8 m/s (antisymmetric case) or 1.2 m/s (symmetric case). For smaller wavelength the symmetry conditions loses its influences on the stability beha-
314
Wave length 2nl, mm 105 63
45
gni
idi
w
Dimensionless disk thickness h
314
Wave length 2nl, mm 105 63
45
400
mi
Di
* • * ****** * t • *■**«*• *•••••••••••■:•■• it*
:::::::::::::::::::::::::::::::::::::: 3.4
1.1
gni
idi
w
0.0
0.1
0.3 0.5 _
Dimensionless disk thickness h
0.7
0.4 0.6 0.8 1.0 Re(D), 1/s 0 5 10 Im(D), 10-24 1/s
Fig. 12. Eigenvalues for hot bands with a symmetric field: real (a) and imaginary part (b). A practical instability is marked with a gray-scaled dot
y
-1
0
1
2
3
-1.0 -0.8 - 0.4 0.0 0.4 0.8 1.0
Temperature disturbance T, K
Fig. 13. Unstable antisymmetric temperature field at h = 0.6 and minimum v in Fig. 11. Layered structure according to Fig. 4. With increasing y: pad, disk, cooling channels, disk, pad
-1.0 -0.8 -0.4 0.0 0.4 0.8 1.0
Temperature disturbance T, K
Fig. 14. Unstable symmetric temperature field at h = 0.6 and minimum v in Fig. 12. Layered structure according to Fig. 4. With increasing y : pad, disk, cooling channels, disk, pad
vior. Both effects are known from experiments: symmetric hot bands, antisymmetric hot bands or hot bands that do not satisfy any symmetry condition. The corresponding imaginary parts are zero within the range of numerical accuracy: The temperature field does not move in radial direction. Experiments showed that in some cases hot bands propagate between the inner and outer radius on a timescale of minutes [16]. This movement is driven by wear [17, 18] and local dynamics of the coefficient of friction. Because both effects are not yet included in the model, no mechanism is present that can lead to non-zero imaginary parts of the eigenvalues.
The unstable eigenforms of the temperature fields are shown in Figs. 13 and 14 for the antisymmetric and symmetric field respectively. Compared with the case for hot spots no convection is introduced because no relative movement is seen in the two-dimensional model. Consequently no temperature tails appear. For the same reason the periodicity in the temperature field enters the pad more clearly in the case of hot bands.
4. Conclusion
The discussed two-dimensional model can be applied on both types of thermoelastic instabilities: hot bands and hot spots. This is enabled by a parameter 9 describing the cutting angle of the model through the contact region with respect to sliding direction. By choosing 9 = 0° hot spots are observed while for 9 = 90° hot bands are under investigation. Values for 0° < 9 < 90° describe an inclined periodicity of the temperature field that has not yet been deeper investigated. Typically sliding systems like brakes or clutches show a layered structure in their intersection. Each layer is described by coupled fields for temperature and displacement for which a set of analytical solutions is available by separation of constants. These solutions can individually be scaled to satisfy the boundary conditions. Typical boundary condi-
tions are given for rigid clamping, frictional sliding, transition between materials with different materials and symmetry. Any arbitrary layered structure can therefore be adapted to an arbitrary combination of boundary conditions. In the present investigation three layers and all types of boundary conditions are applied for the case of a brake system: limited pad thickness, disk material and a homogenized region of cooling channels within the disk. Due to linearity of the continuous model, stability and periodicity can be discussed in terms of complex eigenvalues while eigenfunctions deliver the solutions of the coupled fields. Parameter studies confirm that for the case of hot bands an antisymmetric field is by far more unstable than a symmetric field. Hot spots move on the disk (better conductor) in the direction of the sliding pads. For the case of hot bands the stability behavior depends only weakly on the stability condition. Because the numerical effort of the model is reduced to a root-finding algorithm, stability diagrams are available within few seconds on an ordinary computer. This allows an efficient study of parameter influences.
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Graf Matthias, Dipl.-Ing., Technische Universitaet Braunschweig, Germany, mat.graf@tu-bs.de Ostermeyer Georg-Peter, Prof., Technische Universitaet Braunschweig, Germany, gp.ostermeyer@tu-bs.de
