Full Terms & Conditions of access and use can be found athttp://www.tandfonline.com/action/journalInformation?journalCode=uths20

Download by: [University of Michigan] Date: 03 January 2018, At: 08:00

Journal of Thermal Stresses

ISSN: 0149-5739 (Print) 1521-074X (Online) Journal homepage: http://www.tandfonline.com/loi/uths20

FINITE ELEMENT ANALYSIS OF FRICTIONALLYEXCITED THERMOELASTIC INSTABILITY

Shuqin Du , P. Zagrodzki , J. R. Barber & G. M. Hulbert

To cite this article: Shuqin Du , P. Zagrodzki , J. R. Barber & G. M. Hulbert (1997) FINITEELEMENT ANALYSIS OF FRICTIONALLY EXCITED THERMOELASTIC INSTABILITY, Journal ofThermal Stresses, 20:2, 185-201, DOI: 10.1080/01495739708956098

To link to this article: https://doi.org/10.1080/01495739708956098

Published online: 27 Apr 2007.

Submit your article to this journal

Article views: 143

View related articles

Citing articles: 34 View citing articles

FINITE ELEMENT ANALYSIS OF FRICTIONALLYEXCITED THERMOELASTIC INSTABILITY

Shuqin Du, P. Zagrodzki, J. R. Barber, and G. M. HulbertDepartment ofMechanical Engineering and Applied Mechanics

University ofMichiganAnn Arbor, Michigan, USA

The frictional heat generated during braking causes thermoelastic distortion that modi­fies the contact pressure distribution. If the sliding speed is sufficiently high, this canlead to frictionally excited thermoelastic instability, characterized by major nonuniformi­ties in pressure and temperature. In automotive applications, a particular area ofconcern is the relation between thermoelastically induced hot spots in the brake disksand noise and vibration in the brake system. The critical sliding speed can be found byexamining the conditions under which a perturbation in the temperature and stressfields can grow in time. The growth has exponential character, and subject to certainrestrictions, the growth rate b is found to be real. The critical speed then corresponds toa condition at which b = 0 and hence at which there is a steady-state solution involvingnonuniform contact pressure. We first treat the heat sources Q at the contact nodes asgiven and use standard finite element analysis (FEA) to determine the correspondingnodal contact forces P. The heat balance equation Q = jVP, wheref is the coefficient offriction, then defines a linear eigenvalue problem for the critical speed V. The method isfound to give good estimates for the critical speed in test cases with a relatively coarsemesh. It is generally better conditioned and more computationally efficient than a directfinite element simulation of the system in time. Results are presented for severalexamples related to automotive practice and show that the flexural rigidityof the frictionpad assembly has a major effect on the critical speed.

It is well known that thermoelastic distortion due to frictional heating can initiate aform of instability in sliding systems known as thermoelastic instability (TEl).Briefly, the distortion affects the contact pressure distribution, which in turnaffects the distribution of frictional heating and hence the thermoelastic distortion[1-3]. This feedback process is unstable if the sliding speed is sufficiently high andcan lead to localization of the load in a small region of the nominal contact area.The resulting high local temperatures and thermal stresses have various undesir­able effects such as material transformations, thermal cracking, and brake fade.

The earliest rigorous investigation of TEl for a simple sliding system is due toDow and Burton [2], who introduced the idea of determining the conditions under

Received 30 July 1996; accepted 18 August 1996.The authors are pleased to acknowledge support from the Ford Motor Company and from the

National Science Foundation under contract number CMS-9322106.P. Zagrodzki is on leave from the Institute of Transport, Warsaw University of Technology,

Koszykowa 75, 00-662 Warsaw, Poland.Address correspondence to Professor James Barber, Department of Mechanical Engineering,

University of Michigan, Ann Arbor, MI 48109-2125. E-mail: jbarber@engin.umich.edu

Journal of Thermal Stresses, 20:185-201, 1997Copyright © 1997 Taylor & Francis

0149-5739/97 $12.00 + .00

185

Dow

nloa

ded

by [

Uni

vers

ity o

f M

ichi

gan]

at 0

8:00

03

Janu

ary

2018

186 S. DU ET AL.

which a perturbed thermoelastic field can grow exponentially in time. This methodhas since been used extensively to determine the critical sliding speed for a varietyof geometries, mostly involving bodies of infinite extent. The results generallyunderestimate the experimentally observed critical speed for practical systems, andthere is evidence that this is due to the influence of the finite dimensions of thesystem. Lee and Barber [4] considered the case of an infinite layer sliding betweentwo half-spaces and demonstrated that the finite thickness of the layer can raisethe critical speed significantly and have a major effect on the nature of thedominant perturbation. Thus, if practical brakes and clutches are to be designedagainst TEl-related failure, it is essential to find some way of taking into accountthe finite geometry of the sliding system.

An obvious approach to this problem would be to simulate the transientbehavior of the finite geometry system, using a finite element description. Aparametric investigation of the behavior would then enable stability boundaries tobe determined. However, such an effort would be extremely computer-intensivesince the solution of the transient heat conduction equation with reasonably finemeshing involves the use of a very small time increment for numerical stability andconvergence. This is manageable in problems with two spatial dimensions [5] butwould pose considerable difficulties in the three-dimensional geometry needed togive a realistic approximation to a practical brake system.

A preferable alternative is to apply the finite element method (FEM) directly toBurton's perturbation method. We first formalize the method by postulating theexistence of a small perturbation on the steady state that grows exponentially intime. This leads to an eigenvalue problem for the growth rate b, and we anticipatea denumerably infinite set of eigenvalues with corresponding eigenfunctions de­scribing the perturbation fields. An arbitrary initial perturbation could be ex­panded as an eigenfunction series, and hence instability is indicated if anyone ofthe eigenvalues is positive or has positive real part.

The problem is further simplified if we can assume that instability is governedby a real eigenvalue since in that case the stability boundary is determined by thepresence of an eigenvalue of zero. An exponential growth rate of zero correspondsto a nontrivial perturbation in equilibrium that satisfies the steady-state equations.In this case, the method is equivalent to the determination of critical loads forstructural stability by seeking conditions under which nontrivial equilibrium solu­tions exist.

This approach was first suggested by Yeo [6], who developed it in the contextof the related static contact problem, where instability results from the pressuredependence of an interfacial contact resistance.

It is usual to consider Burton's method as a means of determining the stabilityof a steady state of the system. However, it is equally applicable to transientprocesses as long as the contact area does not change with time, since in that casethe homogeneous perturbation problem has the same boundary conditions as itwould have for a steady process. This is not the case for the static thermoelasticcontact problem of Yeo and Barber [6], where changes in contact pressure affectstability through the local gradient of contact resistance.

Dow

nloa

ded

by [

Uni

vers

ity o

f M

ichi

gan]

at 0

8:00

03

Janu

ary

2018

FRICflONALLY EXCITED THERMOELASTIC INSTABILITY

STATEMENT OF THE PROBLEM

187

We consider the plane strain system shown in Figure 1, in which an elastic body 0,1

slides at constant speed V in the z-direction (normal to the plane of the figure)against a second elastic body 0,2' The boundary of o' y is denoted by T, ("Y = 1,2)and the (plane) contact surface by f e • Tractions are applied in the region T, U f 2 ­

f e such as to ensure contact over the entire region fe' i.e.,

where u y n is the outward normal displacement component of body "y.

Coulomb friction is assumed to occur at the contact surface, so that for eachbody

(2)

where f is the coefficient of friction; unl' unn are tangential and normal in-planestresses, respectively; and Un z is the out of plane tangential stress.

No additional constraints are imposed on the applied tractions, except theequilibrium condition implied by the requirement of constant velocity. Indeed, thetractions can be permitted to change in a fairly general way with time.

In the contact region fe' the local shear tractions will cause frictional heating,which flows into the bodies; so the local heat generated at the contact surface isdenoted

q =fVp on r; (3)

where we define the contact pressure p = -Unn on f e • Continuity of temperatureis assumed across the contact interface.

We supposed that the other boundaries are insulated, hence the local heatfluxes are

(4)

/

0,

Figure I. Two-body sliding system. Relative motion occursnormal to the plane of the figure.

Dow

nloa

ded

by [

Uni

vers

ity o

f M

ichi

gan]

at 0

8:00

03

Janu

ary

2018

188 S. DU ET AL.

The Transient Problem

The problem defined above is necessarily transient because there is a positive heatflow into the bodies in the contact region and no heat flow elsewhere, so thetemperature must rise with time. If the applied tractions are constant and have aresultant F in the negative x-direction, the problem might be expected to approacha quasi steady state where the temperature increases linearly with time, i.e.,

A fVFtT(x,y,t) -> T(x,y) + (C

1+ C

z) (5)

where c.y is the thermal capacity of body n~.

Fortunately, we do not need to solve this transient problem in order todetermine the stability of the system. Instead, we consider the conditions underwhich a small perturbation in the temperature field can grow exponentially in time,i.e.,

T(x,y,t) =eb'O(x,y) (6)

As long as contact is retained throughout fe' the perturbed problem remainslinear and superposition applies. In particular, the perturbation must satisfy theheat conduction and thermoelastic equations. Also, since the prescribed boundaryconditions are assumed to be satisfied by the unperturbed solution, it follows thatthe perturbation must satisfy a corresponding set of homogeneous boundaryconditions.

The problem for the perturbation therefore has a trivial solution in which allthe perturbed fields are zero (as we should expect since, ex hypothesi, theunperturbed solution is a solution of the problem), but for certain eigenvalues ofthe exponential growth rate b we anticipate nontrivial solutions. If any of these hasa positive real part, we conclude that the unperturbed solution will be unstable.

If we make the further assumption that instability is always associated with realeigenvalues of b, we can conclude that the stability boundary-i.e., the criticalspeed Va at which instability begins-is determined by the existence of an eigen­value at the origin b = O. Referring to Eq. (6), we see that this is equivalent to theperturbation being independent of time and hence to the condition that thehomogeneous steady state problem has a nontrivial solution. It is this eigenvalueproblem that we shall formulate and solve in this article.

The Eigenvalue Problem

We first develop a suitable continuous model of the problem and then discretize it.The perturbation problem is defined by the homogeneous boundary conditions

(1), (2) in the contact region and the requirement that all other surfaces betractionfree and insulated. The eigenvalue Va is introduced only in the boundarycondition (3), which has the form

q(y)=fVaP(Y)

Dow

nloa

ded

by [

Uni

vers

ity o

f M

ichi

gan]

at 0

8:00

03

Janu

ary

2018

FRICTIONALLY EXCITED THERMOELASTIC INSTABILITY 189

This condition also represents the only influence of the elastic solution on the heatconduction problem and, therefore, is convenient to impose at the end of thederivation.

We first treat the q(y) as known, in which case it becomes a straightforwardheat conduction problem to determine the temperature field, recalling that forb = 0 the temperature is independent of time and that for the perturbationproblem the heat flux is zero at all noncontacting surfaces. A sequential thermo­elastic problem then permits us to determine the stress and displacement fieldsand, in particular, the contact pressures p(y) needed to satisfy the contactboundary conditions (1)-(3), the remaining surfaces being traction-free.

Since the problem is linear, the resulting pressures are linear functions of theassumed heat fluxes, i.e.,

p =~q or q = '?J'p

where ~ is a linear operator and '?J' =~-l is its inverse.Substituting for q from Eq. (7), we obtain

p =fV~p or fVp = '?J'p

(8)

(9)

which is a linear eigenvalue problem for l/fV or the product fV in continuousform. We next use the FEM to obtain the corresponding discrete form of thisproblem.

FINITE ELEMENT FORMULATION

The Heat Conduction Problem

For a solution of the form (6) with zero growth rate b, the perturbation in thetemperature field 0 must satisfy the steady-state heat conduction equations

with the boundary conditions

(10)

in F,in f 1 U f 2 - F,

(11)

where n y denotes the unit outward normal vector to the boundary and K; is thethermal conductivity.

Dow

nloa

ded

by [

Uni

vers

ity o

f M

ichi

gan]

at 0

8:00

03

Janu

ary

2018

190 s. DU ET AL.

The heat fluxes qy in f e are defined implicitly by the conditions of tempera­ture continuity and energy conservation, i.e.,

(12)

(13)

in fe' where q is given by Eq. (3).Notice that the solution to the problem defined by Eqs. (10), (11) exists only if

y = 1,2 (14)

and in that case is nonunique, having the form O(x, y) + C, where C is an arbitraryconstant. The constant C corresponds to a uniform expansion and does notinfluence the stress and, in particular, the contact traction for linear elasticmaterials with the stated boundary conditions. Therefore, we can restrict attentionto the spatially variable part O(x, y) of the solution of Eq. (10).

To solve this problem, we use the standard Galerkin finite element procedure.We multiply Eq. (10) by an arbitrary weight function wand integrate over Oy

y=I,2 (15)

Using Green's theorem we obtain

Substituting boundary condition (11) into Eq. (16), we obtain

Adding the two equations (17) and using Eq. (13), we obtain

/,KVO.VwdO= /, qwdfn r,

(16)

(17)

(18)

where 0 = 0 1 U O 2 , 0 represents the temperature field in the whole domain 0;and K denotes the conductivity K; appropriate for the particular subdomain Oy,y= 1,2.

The unknown 0 is approximated by

O(x,y) =W(x,y)O (19)

Dow

nloa

ded

by [

Uni

vers

ity o

f M

ichi

gan]

at 0

8:00

03

Janu

ary

2018

FRICfIONALLY EXCITED THERMOELASTIC INSTABILITY 191

where W, 6 are vectors whose components are the piecewise shape functionsW;(x, y) and the nodal temperatures ()i' respectively.

In the Galerkin method, the shape functions W; are also used as weightfunctions w. Substituting Eq. (19) into Eq. (18) and replacing w by Uj leads to the(M X M) system of algebraic equations

L6=Q*

where M is the total number of nodes,

L= fnK(VW)TVWdfl

is the thermal conductivity matrix, and

(20)

(21)

(22)

is a vector of nodal heat sources.Notice that although the heat source vector Q* has M components, only N of

these are nonzero, corresponding to the N nodes in the contact area. For all othernodes, the shape functions Uj make zero contribution to the surface integral. Wecan therefore define a new vector Q of order N containing just these nonzeroelements of Q*.

The continuous distribution q(y) is approximated using the shape functionsUj, i.e.,

(23)

where We denotes the subset of N shape functions corresponding to the N contactnodes and q is a nodal heat flux vector of order N. Substituting Eq. (23) into Eq.(22), we obtain

Q=Wq

where

is a matrix of dimension N X N.

The Elastic Problem

The constitutive equation for the elastic problem with thermal distortion is

(J' = D(E - Eo)

(24)

(25)

(26)

Dow

nloa

ded

by [

Uni

vers

ity o

f M

ichi

gan]

at 0

8:00

03

Janu

ary

2018

192

where

S. DU ET AL.

(27)

is the initial strain vector resulting from thermal expansion; (J", E are stress andstrain vectors, respectively; and D, a are the elasticity matrix and thermal expan­sion coefficient, respectively.

We first treat the two subdomains 01' O 2 as separate problems. The standardfinite element treatment of the elastic problem, based on the variational principle[7], leads to the following equation of equilibrium for domain Oy

(28)

where u, is a vector of nodal displacements,

(29)

is the stiffness matrix,

(30)

is a vector of nodal forces, B is the strain matrix [7], and Py* is a vector of nodalforces resulting from the unknown contact pressure. Substituting Eq. (27) into Eq.(30), we obtain

(31)

where 'l'y is a matrix.The two problems (28) are coupled through the boundary condition (1), which

states that the displacement vectors u 1' u2 have common normal components inthe contact region. However, rather than eliminating the dependent degrees offreedom between the two subproblems, we simply impose a penalty function on therelative normal displacements of corresponding nodes at the contact interface,which is equivalent to the connection of the two subdomains through a set ofcontact elements.

This leads to an equation of the form

Ku=F

where u is a vector containing all the components of u" u2, and

F = '1'0

(32)

(33)

The forces P; become internal forces in the assembled problem and thereforecancel in Eq. (33). Notice that in our case the only contribution to F is thatresulting from thermal expansion.

Dow

nloa

ded

by [

Uni

vers

ity o

f M

ichi

gan]

at 0

8:00

03

Janu

ary

2018

FRIcnONALLY EXCITED THERMOELASTIC INSTABILITY 193

Equation (32) is solved with boundary conditions (2) and an additional condi­tion constraining rigid body motion, after which the vector P; can be computedfrom Eqs. (28), (31). .

In the vector Py*, only N components are nonzero, corresponding to thenormal forces at the N contact nodes. We therefore define a vector P of order Ncontaining just these nonzero elements of P;. The relation between P and thepiecewise-continuous contact pressure distribution p(y) is analogous to that be­tween the nodal heat sources and the heat flux defined in Eqs. (23), (24), i.e.,

and

P = «Ilp

where p is the vector of nodal pressures and «Il is defined by Eq. (25).

Stability Analysis

In the discrete representation Eqs. (7), (8) take the forms

Q=fVP

P=AQ

(34)

(35)

(36)

(37)

respectively, where vectors Q and P were defined in previous subsections and A is amatrix to be determined. The discrete form of the eigenvalue problem (9) is

P=fVAP (38)

The vectors Q and P have dimension N. However, the components of thesevectors are not independent because the heat flux has to satisfy condition (14) andthe contact pressure must satisfy the equilibrium relations

1p(y)df = 0r;

The discrete forms of these relations are

1p(y)ydf=Or;

(39)

(40)

N

LPj=Oi=1

N

L PiYi=Oi=1

(41)

Thus, Q has N - 1 independent components, P has N - 2 independentcomponents, and relation (37) can be reduced to

(42)

Dow

nloa

ded

by [

Uni

vers

ity o

f M

ichi

gan]

at 0

8:00

03

Janu

ary

2018

194 S. DU ET AL.

where Q and P are vectors with N - 1 and N - 2 components, respectively, i.e.,

(43)

and Ahas dimension (N - 2) X (N - 1).To find the coefficients of A, we first solve the heat conduction problem (20)

.for a set of N - 1 linearly independent vectors Qi, j = 1, ... , N - 1. We then solvethe corresponding N - 1 thermoelastic problems (31), (32) with the frictionlesscontact boundary condition (1) and all other boundaries tractionfree. Once thedisplacement vectors u i are determined, we follow the procedure described inprevious subsection to find the corresponding nodal contact forces pi.

From the applied vectors Qi and the resulting vectors pi we form the matrices[Q](N-I)X(N-I) and [P](N-2)X(N-I)" The matrix Ais then computed from the equation

- = --IA = [p][Q]

Equation (36) implies that Q= jVP; hence, using Eq. (42), we have

P=jVAP

(44)

(45)

Using Eqs. (41), we obtain an expression for the component PN - 1 in terms ofPi' i = 1,... , N - 2, which we then use to eliminate this component on theright-hand side of Eq. (45), obtaining

where Ais a new matrix of dimension (N - 2) X (N - 2).Finally, we rewrite Eq. (46) in the standard eigenvalue form as

cp- jVP=O

(46)

(47)

where C = A-I. This constitutes a linear eigenvalue problem for the critical slidingspeed Vo at the stability boundary, the eigenfunctions P defining the form of thecorresponding unstable perturbation in contact pressure.

Note that in the above formulation we do not use elemental stresses or heatfluxes but their discrete nodal counterparts, i.e., forces and heat sources. It is wellknown that the accuracy of FEM solutions for the latter is higher by one orderthan for the former [7]. Thus, the applied method ensures the highest accuracyavailable for the specific element type.

RESULTS

The method was first validated using the simple example of a rectangular blocksliding against an insulated rigid plane, as shown in Figure 2. The block is assumed

Dow

nloa

ded

by [

Uni

vers

ity o

f M

ichi

gan]

at 0

8:00

03

Janu

ary

2018

FRICIlONALLY EXCITED THERMOELASTIC INSTABILITY 195

a

bFigure 2. Rectangular block sliding against a rigid nonconductingwall.

to be homogeneous and isotropic and to have width a (defining the contact length)and length b. The matrix C was determined as described in the last section usingthe FEM, with quadrilateral bilinear elements.

Convergence of the algorithm was tested for the case of a square block (a = b)with an increasing number of nodes from 10 X 10 up to 100 X 100. Figure 3 showsthe leading (lowest) eigenvalue as a function of the number of nodes N along eachside. For this purpose, it is convenient to define a dimensionless eigenvalue as

(48)

7.2r-------,-----.----,--------,-----,------,

7.18

7.18

- FEAsoluUon- - - Converged soluUon

7.14

v;7.12

7.1

---------------~--~--==---~--~---------7.08

1207·06

0L--- - - L:-- - - .L-- - - .L...-- - - -<,-- - - -'--- - - ...J

Figure J. Convergence of the critical speed with increased mesh refinement.

Dow

nloa

ded

by [

Uni

vers

ity o

f M

ichi

gan]

at 0

8:00

03

Janu

ary

2018

196 S. DU ET AL.

where E is Young's modulus. The results converge monotonically on the valueVo* = 14.1816, no further change being obtained beyond N = 60. Taking this as theexact result, Figure 3 shows that 0.6% accuracy in the critical speed can beobtained for this configuration with a 10 X 10 mesh and 0.12% accuracy with a20 X 20 mesh. Thus, for practical engineering purposes, a quite coarse mesh issufficient to obtain good estimates of the critical speed.

Effect of Aspect Ratio

We define the aspect ratio as

br= ­

a(49)

so that large values of r correspond to contact of a strip on an end face and smallvalues to contact of a thin layer.

Figure 4 shows the first three eigenvalues as functions of the aspect ratio inthe range 0.2 < r < 1. All the eigenvalues tend to a constant as r --> co, and inpractice there is very little further change beyond r = 1. The explanation for thiscan be seen by considering the corresponding eigenfunctions. Figure 5 shows thecontact pressure distribution corresponding to the first three eigenvalues for thecase r = 1. Equilibrium considerations [Eqs. (41)J demand that the eigenfunctions

45r-----,r-----.--......,.--.....,---...,.----.....,----,----,

3

2

Figure 4. Effect of the aspect ratio on critical speed for the first three modes.

Dow

nloa

ded

by [

Uni

vers

ity o

f M

ichi

gan]

at 0

8:00

03

Janu

ary

2018

FRICflONALLY EXCITED THERMOELASTIC INSTABIUTY 197

-<1.1

-0.2

-<1.3

II

I-,'

,,,,,,

5

...10

y

. .. ,. ,. . ,. ..,\ J I" , ...-,

15

-lsI Mode

--- 2nd Mode

--- 3rd Mode

20 25

Figure 5. Eigenfunctions (contact pressure perturbations) for the first three modes.

have zero sum across the width and hence have approximately sinusoidal form. Theperturbed fields satisfy the steady-state heat conduction equation and hence tendto decay exponentially with x at a rate related to the wavenumber of the variationthrough the width. It follows that the addition of material distant from the contactarea has little effect on the eigenfunctions and hence on the eigenvalues.

An approximation to the lowest eigenvalue for large r can be obtained byassuming that the corresponding eigenfunction is exactly sinusoidal. In that case,Burton's half-plane solution [18] yields the result

47TKV. =-­

o faEA(50)

where A is the wavelength of the perturbation. Writing A= a and substituting intoEq. (50), we obtain

Vo* = 47T = 12.5664 (51)

The free boundaries of the block place a constraint on the mode shape, and it istherefore to be expected that the actual critical speed will be somewhat larger thanthis approximation. This is confirmed by the results of Figure 4, which show thefirst mode tending to a limiting value of 14.1982 at large r,

Dow

nloa

ded

by [

Uni

vers

ity o

f M

ichi

gan]

at 0

8:00

03

Janu

ary

2018

198 S. DU ET AL.

We note from Figure 5 that the lowest eigenvalue is characterized by theeigenfunction with the smallest number of zero crossings (two) and that eachsuccessive eigenfunction has one additional zero crossing, consistent with the set ofeigenfunctions comprising a complete set of orthogonal self-equilibrated functionson the contact region. When r < 0.38 a change in this behavior is observed, thelowest eigenvalue (and hence the critical speed of the system) being associated withan eigenfunction with three zero crossings. The corresponding curves in Figure 4cross near r = 0.38. This behavior is consistent with the results of Lee [4], whofound that the thermoelastic instability for a layer sliding between two half-planesis associated with a sinusoidal form whose wavelength is related to the layerthickness. Similar results were also obtained by Yeo [9] for the related problem ofthe stability of thermoelastic contact between a block and a plane across apressure-sensitive contact resistance.

Three-Body Contact

Figure 6 shows several geometries chosen to illustrate the application of themethod to geometries closer to the conditions in an automotive disk brake. InFigure 60 a layer of cast iron slides between two pads of composition frictionmaterial, in Figure 6b the backs of the pads are reinforced with steel layers, in

/ V/ V/ V/ V/ V/ V/ V/ V/ V/ V/ V/ V

(a)

r--

-(e)

(b)

(d)

Figure 6. Test geometries related toautomotive practice: (a) a layer be­tween two coextensive pads of frictionmaterial, (b) pads reinforced by steelback plates, (c) disk extends beyond thepad surfaces, and (d) back plate flexi­bility increased by grooves.

Dow

nloa

ded

by [

Uni

vers

ity o

f M

ichi

gan]

at 0

8:00

03

Janu

ary

2018

FRICfIONALLY EXCITED THERMOELASTIC INSTABILITY 199

Table 1 Material properties

Properties Disk Pad Back Plate

Elastic modulus E (N/m')Thermal expansion a (j°ClThermal conductivity K (W/mOC)Poisson's ratio v

1.25 X 1011l.05 X 10-5

49.250.29

1.0 X 10"l.OX 10- 5

5.00.29

2.0 X 10111.15 X 10- 5

42.00.30

Figure 6c the cast iron layer extends somewhat beyond the contact region, and inFigure 6d the reinforcement layer is interrupted by grooves in order to reduceoverall bending stiffness. The pertinent properties of the materials are summarizedin Table 1.

Figure 7 shows the pressure distributions and the corresponding critical speedsassociated with the first three modes for the system of Figure 6a. In each case, the

201510Y

5o

0.3,-----------,

-0.2

5o

-0.2

201510Y

5

-0.2 -0.2

0 5 10 15 20 0 5 10 15 20 0 5 10 15 20

Va =1.2144 Vo =1.2526 Vo =1.5159

0.3 0.3 0.3

0.2

0.1

0.2

0.3r-----.:----,

-0.2

o

Figure 7. Eigenfunctions for the first three modes of the system of Figure 6a. The upper and lowerfigures present the pressure perturbations on the left and right interfaces, PI' P" respectively.

Dow

nloa

ded

by [

Uni

vers

ity o

f M

ichi

gan]

at 0

8:00

03

Janu

ary

2018

200 S. DU ET AL.

upper and lower figures present the pressure perturbations on the left and rightinterfaces, Pt' P2' respectively.

The lowest critical speed (Vo* = 1.2144) corresponds to an antisyrnmetric mode(Pt = -P2)' consistent with the results of Lee [4]. The second mode (Vo* = 1.2526)is also antisymmetric, but the third mode (Vo* = 1.5159) is symmetric (PI = P2)'

The pressure distribution for the first mode of each of the systems of Figure 6is shown in Figure 8. In each case, the perturbation is antisymmetric (PI = -P2);hence only the curves for PI are presented. The addition of steel back plates as inFigure 6b has a major effect on the critical speed, reducing it by almost a factor oftwo to Vo* = 0.6129. The wavelength is increased by about 8%, but in the spirit ofBurton's equation (50), the critical speed is probably more influenced by theincrease in effective modulus due to the steel back plate.

Essentially similar results were obtained for the geometry of Figure 6c,indicating that the extension of one of the two bodies in the contact plane has verylittle effect on the stability behavior.

Clearly, the addition of a back plate increases the stiffness of the pad, enablingit to support an unstable perturbation at lower speeds. We might therefore expectthat structural changes designed to restore flexibility would increase the criticalspeed. This is confirmed by results for the geometry of Figure 6d, where theaddition of a groove increases bending flexibility in the direction parallel to thecontact area and increases the critical speed to Vo* = 0.90n-an increase of 47%over the value for Figure 6b.

Vo =0.61290.5 (b)YO =1.21440.5r.-:--------::';";"'"c-::-c:-:o

(a)

-0.50

-0.5 1

5 10 15 20 0 5 10 15 20

Y Y0.5 0.5 (d)(c) Vo=0.6161 YO =0.9072

1510Y

5-0.51--=---:::---:,=---::o 20

Figure 8. Eigenfunctions (contact pressure perturbations) for the first (dominant) mode of the systemsof Figures 6a-d. In each case, the perturbation is antisymmetric (PI = -P2)'

Dow

nloa

ded

by [

Uni

vers

ity o

f M

ichi

gan]

at 0

8:00

03

Janu

ary

2018

FRICTIONALLY EXCITED THERMOELASTIC INSTABILITY

CONCLUSIONS

201

These results show that the numerical implementation of Burton's perturbationanalysis for thermoelastic stability provides an extremely efficient method fordetermining the critical speed in sliding systems, for cases where instability isassociated with real eigenvalues of exponential growth rate. By contrast, a directnumerical simulation of such a system would be very computationally intensive andrequire several runs to establish the critical sliding speed.

A relatively coarse mesh is sufficient to obtain a good estimate of the firsteigenvalue and, hence, of the critical sliding speed for the system. The methodtherefore bodes well for use in three-dimensional systems of significant complexity.

The results show that the flexural rigidity of the friction pad assembly has amajor effect on the critical speed.

REFERENCES

1. J. R. Barber, Thermoelastic Instabilities in the Sliding of Conforming Solids, Proc. Roy. Soc. A, vol.312, pp. 381-394,1969.

2. T. A. Dow and R. A. Burton, Thermoelastic Instability of Sliding Contact in the Absence of Wear,Wear, vol. 19, pp. 315-328,1972.

3. F. E. Kennedy and F. F. Ling, A Thermal, Thermoelastic and Wear Simulation of a High EnergySliding Contact Problem, J. Lub. Tech., vol. 96, pp. 497-507, 1974.

4. Kwangjin Lee and J. R. Barber, Frictionally-Excited Thermoelastic Instability in Automotive DiskBrakes, ASME J. Tribology, vol. 115, pp. 607-614, 1993.

5. P. Zagrodzki, Analysis of Thermomechanical Phenomena in Multidisc Clutches and Brakes, Wear,vol. 140, pp. 291-308,1990.

6. Taein Yeo and J. R. Barber, Finite Element Analysis of Thermoelastic Contact Stability, ASME J.Appl. Mech., vol. 61, pp. 919-922, 1994.

7. O. C. Zienkiewicz, The Finite Element Method, 3rd ed., McGraw-Hill, New York, 1977.8. R. A. Burton, V. Nerlikar, and S. R. Kilaparti, Thermoelastic Instability in a Seal-Like Configuration,

Wear, vol. 24, pp. 177-188, 1973.9. Taein Yeo and J. R. Barber, Finite Element Analysis of the Stability of Static Thermoelastic Contact,

J. Thermal Stresses, vol. 19, pp. 169-184, 1996.

Dow

nloa

ded

by [

Uni

vers

ity o

f M

ichi

gan]

at 0

8:00

03

Janu

ary

2018