- Email: [email protected]
Dynamic friction
Tribology Research: From Model Experiment to Industrial Problem G. Dalmaz et al. (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
37
DYNAMIC FRICTION K.L.Johnson 1 New Square, Cambridge CB 1 1EY, U.K. Dynamic friction, leading to frictionally excited vibrations, is a vast subject, coveting extreme ranges of space and time: from seismic faults stretching many kilometres to crystal lattice spacing in the Atomic Friction Microscope; from the years which separate earthquakes to the kilohertz frequencies of squealing brakes and railway wheels. In this paper it is hoped to give an overview of the field, to identify unanswered questions and to look to the future. 1. INTRODUCTION The subject of dynamic friction and frictionally excited vibrations is an old and extensive one. Most of the research in the past has started from the point of view of a particular application such as break squeal. In this paper the emphasis will be upon mechanism, with a view to bringing the field together, to identifying some of the unanswered questions and to look to the future. A useful review of the whole subject with a full bibliography has been presented by Ibrahim & Rivin (1994).
slider M I
I
|
1
C yI//////////f1:~;rl/////ll~
(a)
0T
driver
It is generally recognised that frictional vibrations arise from an interaction between the frictional characteristics of the interface and the dynamic characteristics of the system in which the interface exists. The textbook example is shown in Fig.la. A single degree-of-freedom mass-springdamper system rests on a flat surface which moves with a uniform velocity V. It is well known that, provided that the damping is sufficiently light, the mass will undergo 'stick-slip' oscillations if the force of static friction Fs is greater than sliding friction Fd, or if Fd(v) is a decreasing function of the slip velocity v. In the 'stick' phase the block moves with the velocity V of the moving surface; in the slip phase it moves as a spring-mass system under the action of variable friction (Fig.lb). It clearly would be desirable to separate the variations in friction from the dynamics of the system. Attempts to do this are discussed in the next section. 2. FRICTION CHARACTERISTICS 2.1 D y n a m i c m e a s u r e m e n t s
u
(bl
/"~
I~,/il,I,~i t
Effective measurements of the variation of friction during stick-slip oscillations were made by Bell & Burdekin (1969) using a machine tool slide, and by Ko & Brockley (1970) using a mass on the end of a cantilever leaf spring. The system in each case was equivalent to that represented in Fig.1. The equation of motion is: m ~ + cx + kx - F
Figure 1. (a) One degree-of-freedom system. (b) Stick slip trace.
(1)
Velocity and acceleration transducers were attached to the mass to measure the damping and inertia
38
forces on the LH side of equation (1), thereby determining the variation in the friction force F throughout the cyclic motion. The variation of friction F with slipping velocity v found by both researchers (and many others subsequently) was similar and is displayed in Fig.2. Significantly, this result shows that F is not a unique function of v; it is lower when v is decreasing than when increasing, leading to 'frictional hysteresis'.
0
in which B is generally greater than A. The state variable ~ evolves according to:
d~ = l_ VdPss dt D
(3)
o
The interface is envisaged as multi-asperity contacts in which the real area of contact is much less than the apparent area. Do is a characteristic sliding distance necessary to establish a new set of asperity contacts. Its magnitude is thus comparable with the mean asperity contact diameter. The state variable # is interpreted as the time which has elapsed since a particular set of contacts became established. In the steady state, from equation (3)
# =-#ss =Do/V Sliding speed
Substituting in (1) and taking
A possible explanation of this effect might be that the friction force decreases with slip distance s in addition to the velocity v. This hypothesis has found support from research into the mechanics of earthquakes, as discussed in the next section.
2.2 State- and rate-dependent friction During the slip phase of the system in Fig.1 the motion is close to that of a natural oscillation of frequency ~/k/m. A seismic fault, modelled as the interface between two elastic halfspaces, has no natural frequency and requires a different approach. Sliding friction measurements by Dieterich (1994) between nominally flat blocks of granite and other solids, gave rise to the characteristic and reproducible response shown in Fig.3. In this experiment steady sliding at velocity Vo is followed by a step change in speed (an alternate decrease and increase in Fig.3). The mean sliding speed is in the range 1-10 lxm/s, at which dynamic effects can be neglected. To model this behaviour Rice & Ruina (1983) introduced a 'state variable' # and proposed a 'state- and rate-dependent' friction law:
F( V, #) - Fo + Bln(# / #o ) + A ln(V / V )
(4)
(2)
F
ss
--- F - ( B - A ) I n ( V / V ) o
Vo=Do/#o (5)
With B>A equation (5) exhibits a friction characteristic which decreases with sliding speed. When combined with a simple system, such as that in Fig. 1, it gives rise to stick-slip vibrations During a sticking phase when V = 0, equation (2)gives d4gdt = 1, so that ~ - 9 § ~o and the static friction is given by
F-F s
o
+In
('+0;0) ....
(6)
We note that this relationship shows a growth of static friction with time, which has been attributed by many authors, from Kragelski (1939) onwards, to an increase in real contact area by creep of the asperities under the real contact pressure. Support for this hypothesis has been provided recently by Dieterich & Kilgore (1995) who measured the change in contact area with time between rough surfaces of transparent materials: quartz, calcite, glass and acrylic plastic. A thorough experimental investigation of the state-and rate-dependent friction law in both static and dynamic phases has been carried out by Baumberger et al. on nominally
39
V~ (V,=, V,)
V,
. . _v, _
_
Predicted re~oon$e Equation= (1) and (2)
J 75|
o~
=
p~ .
.
.
o~ .
Granite
70 L 9
'~
9
l~rn~ .....
10~Ivs
y-
.65
~
10atn~
'
~s
o =~M .
l~m~
~
9
~ .
I
62
70
15 MPa normal sires=
'
100 izm
67 r
.85 I
.o ,~
I 100 ~tm
,.
,~,v,
_ - -;
~--
t/
.
~ws .
.
o I.~ .
.
.
/ ~
_
--'--~- -
Granite #60 surface, Imm gouge I0 MPa normal s~reu
Soda-lime
glas=
S MPa normal stress
100 Izm
o,~m~, )
:r,-~ . .
'
,..~. --
100 ~tm
4
~
o,=~, ___
Lucite
plutic
. 6 0 .urfar 2 5 MPs normal s t r e ~
Figure 3. Theoretical (eq. 2 & 3) and experimental variations of friction following step changes in sliding speed
flat rough surfaces of PMMA in contact with PMMA and polystyrene on polystyrene (1999) Before leaving the state and rate law, it is relevant to note that the term in equation (2) and the growth of contact area given by equation (6) both imply rate dependence (e.g. viscoelasticity) in one or both contacting materials. The materials studied by Baumberger et al. (1999) are deliberately viscoelastic and it is conceivable that creep is important in the rock friction experiments where the sliding speed was very low and the stick times long. It is more difficult to accept that creep plays a significant role in the high frequency squeal of hard metal surfaces such as railway wheels in a fight curve. This would appear to pose an unanswered question.
Aln(V/Vo)
2. 3
T h e r m a l effects
A reduction of the force of friction with sliding speed V is not dependent on a change in contact area, but may be the result of a decrease in frictional stress x at the interface. One such cause could be frictional heating. In a concentrated contact (e.g. a square area 2a x 2a) in which one surface is stationary, the majority of the frictional heat is conducted into the moving surface. The rate of heat generation per unit area is q = zV. The resulting mean temperature rise in the contact- the 'flash temperature'-is given by
4 qa~v a 4~J2Va
(7)
40
where K = thermal conductivity, p = density, c = specific heat capacity and r = diffusivity = Kpc. Let fl = d , / d T be the rate of change of shear strength of the interface with temperature in the ambient sliding state. Then
"r
(8)
d T A T *' fl
For a reduction in frictional stress, fl must be negative, which is the usual situation. The effect becomes pronounced with a high sliding velocity and low thermal conductivity. (a)
-"
drive
Ul
lubricant~ I test discs
drive 20-
of a rolling contact disc machine for measuring lubricant traction in combined rolling and sliding contact. The discs were driven by independent motors through shafts of unequal length (Fig.4a). A typical variation of traction coefficient (friction coefft.) with sliding speed is shown in Fig.4b. At low sliding speed the traction increases with speed, as expected for a viscous lubricant. However at higher sliding speed, as demonstrated by Crook (1961), frictional heating in the contact so lowers the effective viscosity as to cause the traction to decrease with speed as shown. The effect was to excite torsional oscillations of the drive shafts. To eliminate this behaviour it was necessary to attach viscous torsional dampers to each shaft of sufficient strength to suppress the self-excited oscillations of the shafts. Stringed instruments such as the violin and cello provide classic examples of stick-slip vibrations excited by drawing the bow across the strings. The bow is coated with 'rosin', a natural resinous substance which shows rapid reduction of shear strength with temperature (negative/3) in the range 40-60 ~ and has low thermal conductivity. The velocity during the slip phase of the oscillation is of order 10m/s. Smith & Woodhouse (2000) have shown that the oscillations in a laboratory rig, equivalent to that in Fig.l, produced by a rosin covered rod are consistent with a thermal softening mechanism. 3. NORMAL MOTION
16
a...
Torque Ib tt 12
po= 176 0 0 0
Iblir 2
Ul= 153 in/sec h = e o ~in D = 0.075
.
.
Bearing /trict~on ,,...,.,,,
o
o.,
o.2
(b) u!
Figure 4. Rolling contact disc machine (b) Traction curve. A striking example of the thermal effect arose in our laboratory during the commissioning
So far we have only considered motion parallel to the sliding interface, but there has long been speculation about the role of motion normal to the interface. In a much quoted paper, Tolstoi (1967) reported observations of a relative normal displacement of the surfaces during the slip phase of a stick-slip oscillation in a system equivalent to that of Fig.1. Further the application of an externally excited high frequency vibration (~ 1 kHz) had the effect of suppressing the stick-slip motion and reducing the steady friction. These observations have received the support of other investigators. The separation of surfaces during sliding relative to their position at rest can be understood by the non-linear compliance of random rough surfaces. The Greenwood & Williamson model of a random rough surface, having an exponential distribution of asperity heights, in contact with a plane, relates a static load P to the separation d by (see Appendix)
41
It is often claimed that the separation between two sliding rough surface will increase with speed due to the impacts of colliding asperities. This intuitive supposition ignores the fact that, at sliding speeds much less than the shear wave speed of the solids, the interaction between contacting asperities is virtually quasi-static. It is, therefore, only responsive to sliding speed to the extent that the deformation of the solids is ratedependent, e.g. viscoelastic.
exp(-d / a ) (9) P where N is the asperity density, cr = roughness and Ip is definite integral, defined in the Appendix, which is independent of the separation d. The friction force can be expressed in the same way: P-CNI
(10)
F = CNI F exp(-d / a)
4. POINT CONTACS
Thus the coefficient of friction # = IF / 1p which is independent of d. Now assume that the separation varies sinusoidally according to:
4.1 The 'sprag effect' Studies of sliding point contacts have generally arisen from investigations of brake squeal. An early discussion of the 'sprag effect' was given by Spun" (1961-2) In its simplest form the system comprises a pin in point contact with a flat moving surface (e.g. brake disc) as shown in Fig.5a.
u
d = d + Asinrot
(11)
n
u
which gives: P - CNIpJexp( - d / a) m
m
F - CNIFJ exp(-d / a)
and
where J - - ~f~0~r exp( -
(12) (13)
AsinO ).dO. O"
e ~
Pin
~
If we compare the separation given by equation (12) with that do for an equal static load Po given by equation (9) we get:
~---- Disc
Vo = Disc Rotation
m
J exp(-d / or) = exp(-d / a ) 0 n
i.e.
d / a = d / cr + In(J) 0
(14)
This equation shows that the oscillation in separation of amplitude • A increases the mean separation d by amounts given in the Table below, but does not change the coefficient of friction which, by equations (12) and (13), remains equal to IF/Ip.
(a)
m=
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. . . . . . .
.
A/o" 0.5 1.0 1.5 2.0 2.5 In(J) "0.063 0.24 0.50 0.82 1.19 These conclusions are supported by Hess & Soom (1991). It seems as though a different explanation is necessary for the effect of an oscillating load or separation on the coefficient of friction. Presumably the frictional shear stress must be influenced directly.
(b)
: ~w
Figure 5. The spragg effect. (a) pin and disc. (b) 2 degree-of-freedom model.
42
The effect arises from a slight inclination 0 of the axis of the pin to the normal to the sliding surface. The bending stiffness of the pin is denoted by kI and the normal stiffness of the surface (disc) by kd as shown in Fig.5b. Damping can be included by dashpots cf and Cd in parallel with the springs. The equations of motion of pin and disc are then:
m/i~ +cfil + k f u = F c o s O - NsinO
(15)
m dfi, + Cd ~ + kdW = F sinO - N cosO
(16)
where w = u sin O Sliding velocity: V = V - ucosO
(17) (18)
Coulomb friction: F = # N s i g n ( V )
(19)
S
O
This 2 degree-of-freedom system has two solutions of the form Alexp(sl t) and A2exp(s2t ) , in which s1 and s2 can be real, positive or negative, or complex with positive or negative real parts., depending on the values of the angle 0, the friction # and the other parameters. The unstable case of a growing oscillation is of particular interest and is shown in Fig.6. 0 =
from disc drive. It is sometimes referred to as a 'flutter' oscillation from the analogy with the coupled bending and twisting oscillation of an aeroplane wing. The above solution applies while the oscillation is growing but, when the slip velocity given by equation (18) reaches zero, a series of high frequency stick-slips occur. This nonlinear behaviour will be referred to again below. 4.2
Disc vibrations
The compliance of the moving surface in the previous example was represented by a single spring, but a real rotating disc is a complex dynamic system. Linear stability analyses of a rotating flexible disc in frictional point contact have been made by Jarvis & Mills (1963) and by Earles & Badi (1984) where the spragg effect arises from the elastic rotation of the block holding the pin. Unstable combinations of parameters are calculated and supported by experiments with a laboratory rig.
Px
5.2
i,.____
Py Figure 7. Follower forces. _o > -00~
"~176
0.$
1
5
2.S
3
3.5 x 104
Time (s) Figure 6. Unstable oscillations of the system shown in Fig.4(b). The amplitude is limited by stick-slip. It is noteworthy that this unstable behaviour is predicted on the basis of Coulomb friction having no reduction with sliding speed. The instability arises from the coupling of the 2 degrees-of-freedom such that energy is extracted
Mottershead & Chan (1995) propose a further source of flutter instability in the case of thin rotating discs, which they attribute to 'follower loads' (Fig.7). In this model the inclination of the normal and tangential contact forces acting on the disc, which arises from bending of the disc, is taken into account. With the notation of Fig.7, P
=/aV
cosO + N sinO =--# N + NO
(20)
NVsinO = N - / a V O
(21)
X
P
= NcosO
-
Y where O = dy = l dy which introduces damping
dx
V dt '
terms (+ o r - ) into the equations of motion which
43
can promote instability. However these terms are generally very small. In an oscillation of amplitude Yo, frequency f and wavelength round the disc X, 0 - 2 . r t f y o / V = 2~r yo/~,.
4.3 Extended contact
Most practical brakes do not use a point contact, but make use of a flat flexible pad (disc brakes) or a flexible band (drum brakes). This introduces many more degrees of freedom and modes of vibration. Coupling of these modes through friction at the interface can lead to dynamic instabilities of the sort illustrated in Section 4.1 and frequently referred to as 'flutter vibrations' from the similarity of the mechanism to aeroplane wing flutter. A linear theory of frictional oscillations of both band and disc brakes is given by Nishiwaki (1993).
The linear stability of this system during sliding has been examined by Martins et al. (1990). They show that, with some combinations of parameters, including a coefficient of friction ~ = 0.6, unstable oscillations build up until the slip velocity vanishes, whereupon non-linear stick-slip vibrations take place. Note that these vibrations are at the contact resonance frequency of the block on the asperity stiffness, which greatly exceeds the frequency of the block on the stiffness of the restraining spring K. These high frequency (kHz) contact vibrations are flutter vibrations, which are genetically similar to those generated geometrically by the sprag effect and illustrated in Fig.6. What part they play in influencing friction characteristics and the stick-slip oscillations observed in practice is far from clear. 5.2 Dissimilar elastic solids
5. INTERFACIAL E F F E C T S 5.1 Rough surfaces
We now return to the sliding block shown in Fig.1. The block is effectively rigid, but the random roughness on one or both of the contacting surfaces is represented by a compliant layer with non-linear stiffnesses in the normal and tangential directions. Inelastic deformation of the asperities can be allowed for by adding damping to the normal elastic stiffness. We now have a 3 degreeof-freedom system: displacements in the tangential (x) and normal (y)directions, and rotation (pitching) about the z-axis. The modes are coupled through Coulomb friction at the contact surface (Fig.8).
Here we return to the earthquake problem: an extended sliding interface without asperities, where the elastic constants of the contacting solids are different. Following a disturbance at the surface of an elastic half-space, elastic waves known as Rayleigh waves are transmitted across the surface, which are held responsible for earthquake motions. They arise from the need for internal wave motion to satisfy the boundary conditions of a free surface. Similar wave motion occurs at an interface between dissimilar solids to satisfy continuity of stress and displacement across the interface. The generation of such waves by friction at an interface which transmits a steady mean pressure Po at a sliding velocity Vo as been studied by Adams (1995) and Martins et a1.(1995). It is shown that, under the action of friction at the interface, waves which propagate in the direction of sliding can increase in amplitude in an unstable way. At low sliding speed and high pressure the amplitude of the oscillation is limited by stick-slip motion but at high speed and low pressure a limit cycle is achieved by loss of contact. The change of regime is determined by a critical value of the parameter (vg/c2Po), where G is the shear modulus and c 2 the shear wave speed of the stationary solid. 6. CONCLUSION
Figure 8. The system in Fig.l(a) with a compliant rough interface which introduces normal and rotational degrees of freedom.
The expansive field of frictional vibration has been briefly reviewed from the point of view of
44
mechanism. Friction characteristics,
thermal and material characteristics, geometric (spragg) effects and self-excited 'flutter' vibrations have been identified as possible sources of unsteady sliding motion. But the list of possibilities is not exhausted: the non-linear dynamics of stick-slip oscillations is an open field. Parametric excitation and subharmonic resonance of rotating discs has been considered by Mottershead et al. (1996). Indeed stick-slip motion of a system like that in Fig.l, but with two spring coupled blocks, has been shown capable of chaotic motion! [Popp & Stelter (1990)]. Turning to the engineering scene, a first task is to decide which is the dominant mechanism of excitation in any particular practical case. This would appear to call for some form of mapping in terms of appropriately chosen non-dimensional parameters. Some such suggestions have been made in the text, for example the thermal parameter proposed in w
identical conditions, and the frequency can change for no apparent reason. Perhaps this capriciousness is itself an indication of 'chaos'. With these problems in the laboratory, the situation in the field with real engineering components seems even more problematical. With a remit to further study brake squeal, Duffour & Woodhouse (2000) have proposed a radically different approach. Recognising the difficulty of modelling a real brake system, they consider instead the general point contact with two degrees of freedom: normal and tangential, shown in Fig.8. The contacting bodies ('pin' & 'disc') are ascribed arbitrary linear transfer matrices [G] and [H]. A coefficient of friction which varies linearly with velocity is assumed. The genetic linear stability of this system can be analysed, so that those features of [Gl, [H] and /~ and their combination which lead to instability might be recognised and mapped. "Disc.
x'i
.
.
.
Brake"
.o V11
Ul
[H]
F N
:
~
_ V2
[G]
Figure 10. General dynamic system in sliding point contact. The field needs more novel approaches of this kind in order to impose some structure into its many manifestations.
ACKNOWLEDGEMENTS Figure 9. Chaotic motion in phase space caused by external vibration of the system of Fig. 1(a). So far nothing has been said about experiments. Experience in this regard is disappointing. Only in the case of a bowed violin string has detailed agreement been demonstrated between theoretical modelling and a real system (Schumacher & Woodhouse, 1995). Although some of the linear stability analyses of the pin and disc arrangement have been supported by laboratory tests on a laboratory rig designed to replicate the idealised theoretical model, that is far from general experience. The occurrence of self-excited oscillations is often very sporadic in apparently
The author wishes to acknowledge valuable help from Professors J.R.Rice, J.Woodhouse, J.H.Dieterich and Mr. Philippe Duffour in the preparation of this paper.
REFERENCES Adams G.G. (1995) ASME J.Appl.Mech. 62, 867872. Berthoud P., Baumberger T., G'Sell C & Hiver JM. (1999), Phys.Rev.B, 59, 14313-327. Baumberger T. Berthoud P.& Caroli C. (1999), Phys. Rev.B, 61), 3928-3939.
45
Bell R. & Burdekin M. (1969-70) Proc.I nst.Mech.Engrs. 184,543-551. Crook A.W. (1961) Phil.Tmns.Roy.Soc.Lond. A254, 223-258. Dieterich J.H. & Kilgrove B..D. (1994) Pure & Appl. Geophys. 143, 283-302. Dieterich J.H. & Kilgrove B.D. (1995) Techtonophysics 256, 219-239. Duffour P. & Woodhouse J. (2000) U.of Cambridge Engg. Dept. Tech.Rpt. Earles S.W.E. & Badi M.N.M. (1984) Proc.Inst.Mech.Engrs. 198 PtC, 43-50. Hess D.P. & Soom A. (1991) ASME J.Tribology, 113, 87-92. Ibrahim R.A. & Rivin E. (1994) ASME Appl.Mech Rev. 47, 209-253. Jarvis R.P. & Mills B. (1963) Proc.Inst.Mech.Engrs. 178 Pt.I, 847-866. Johnson K.L. Contact Mechanics, C.U.P. (1985). Ko P.L. & Brockley (1970) ASME J.Lub.Tech. 92, 543-549. Kragelski I.V. see Friction & Wear, 1965, Butterworths, London. p. 197. Martins J.A.C., Oden J.T. & Simoes F.M.F. (1990) Int.J.Engng.Sci. 28, 29-92. Martins J.A.C., Faria L.O. & Guimaraes J. (1995) ASME J.Vib.& Acoustics 117, 445-451. Mottershead J.E. & Chan S.N. (1995) ASME J.Vib.& Acoustics, 117, 161-163. Mottershead J.E., Outang H. & Cartmell M.P. (1996) in Advances in Automotive Braking Ed. D.C.Barton, Mech.Eng.Publ. Ltd. pp.49-65. Nishiwaki,M. (1993) Proc.Inst.Mech.Engrs. 207 Pt.D, J.Auto.Eng. 195-202. Popp K.& SteUter P. (1990) Phil.Trans.R.Soc.Lond. A332, 89-105. Rice J.R. & Ruina A.L. (1983) ASME J.Appl.Mech. 50, 343-349. Schumacher R.T. & Woodhouse J. (1995) Contemporary Physics, 36, 79-92. Smith J.H. & Woodhouse J. (2000) J.Mech.Phys.Solids, 48, 1633-81. Spurr R.T. (1961-62) Proc.Inst.Mech.Engrs. (Auto.Div.) 1, 33-52. Tolstoi D.M. (1967) Wear, 10 199-213.
APPENDIX Here we follow the presentation of the Greenwood & Williamson theory of rough surfaces contact in Johnson (1985) pp.411-414. The asperity heights will be assumed to be exponentially distributed, i.e. r
(C / cr)exp(z / cr)
If the force Pi to compress a single asperity through a displacement 6/is given by 1% = g( 6i ), the total load per unit nominal area is: P = CNI p e x p ( - d / o)
where I p ~,f o g(6 / o) exp( --6 / cr).d( 6 / or), We now express the asperity friction force Fi by Fi = NVi = fl6L), where # is not necessarily constant, but can vary with asperity contact pressure and hence dii. Thus the total friction force per unit nominal area is: F - CNI F exp(-d / or)
where I F ,, f o r ( 6 / o r ) exp(-6/cr).d(6 ~or). Note that: F/N = IF [ IF, = constant
even though the local friction at the asperity contacts is an arbitrary function of asperity pressure.
37
DYNAMIC FRICTION K.L.Johnson 1 New Square, Cambridge CB 1 1EY, U.K. Dynamic friction, leading to frictionally excited vibrations, is a vast subject, coveting extreme ranges of space and time: from seismic faults stretching many kilometres to crystal lattice spacing in the Atomic Friction Microscope; from the years which separate earthquakes to the kilohertz frequencies of squealing brakes and railway wheels. In this paper it is hoped to give an overview of the field, to identify unanswered questions and to look to the future. 1. INTRODUCTION The subject of dynamic friction and frictionally excited vibrations is an old and extensive one. Most of the research in the past has started from the point of view of a particular application such as break squeal. In this paper the emphasis will be upon mechanism, with a view to bringing the field together, to identifying some of the unanswered questions and to look to the future. A useful review of the whole subject with a full bibliography has been presented by Ibrahim & Rivin (1994).
slider M I
I
|
1
C yI//////////f1:~;rl/////ll~
(a)
0T
driver
It is generally recognised that frictional vibrations arise from an interaction between the frictional characteristics of the interface and the dynamic characteristics of the system in which the interface exists. The textbook example is shown in Fig.la. A single degree-of-freedom mass-springdamper system rests on a flat surface which moves with a uniform velocity V. It is well known that, provided that the damping is sufficiently light, the mass will undergo 'stick-slip' oscillations if the force of static friction Fs is greater than sliding friction Fd, or if Fd(v) is a decreasing function of the slip velocity v. In the 'stick' phase the block moves with the velocity V of the moving surface; in the slip phase it moves as a spring-mass system under the action of variable friction (Fig.lb). It clearly would be desirable to separate the variations in friction from the dynamics of the system. Attempts to do this are discussed in the next section. 2. FRICTION CHARACTERISTICS 2.1 D y n a m i c m e a s u r e m e n t s
u
(bl
/"~
I~,/il,I,~i t
Effective measurements of the variation of friction during stick-slip oscillations were made by Bell & Burdekin (1969) using a machine tool slide, and by Ko & Brockley (1970) using a mass on the end of a cantilever leaf spring. The system in each case was equivalent to that represented in Fig.1. The equation of motion is: m ~ + cx + kx - F
Figure 1. (a) One degree-of-freedom system. (b) Stick slip trace.
(1)
Velocity and acceleration transducers were attached to the mass to measure the damping and inertia
38
forces on the LH side of equation (1), thereby determining the variation in the friction force F throughout the cyclic motion. The variation of friction F with slipping velocity v found by both researchers (and many others subsequently) was similar and is displayed in Fig.2. Significantly, this result shows that F is not a unique function of v; it is lower when v is decreasing than when increasing, leading to 'frictional hysteresis'.
0
in which B is generally greater than A. The state variable ~ evolves according to:
d~ = l_ VdPss dt D
(3)
o
The interface is envisaged as multi-asperity contacts in which the real area of contact is much less than the apparent area. Do is a characteristic sliding distance necessary to establish a new set of asperity contacts. Its magnitude is thus comparable with the mean asperity contact diameter. The state variable # is interpreted as the time which has elapsed since a particular set of contacts became established. In the steady state, from equation (3)
# =-#ss =Do/V Sliding speed
Substituting in (1) and taking
A possible explanation of this effect might be that the friction force decreases with slip distance s in addition to the velocity v. This hypothesis has found support from research into the mechanics of earthquakes, as discussed in the next section.
2.2 State- and rate-dependent friction During the slip phase of the system in Fig.1 the motion is close to that of a natural oscillation of frequency ~/k/m. A seismic fault, modelled as the interface between two elastic halfspaces, has no natural frequency and requires a different approach. Sliding friction measurements by Dieterich (1994) between nominally flat blocks of granite and other solids, gave rise to the characteristic and reproducible response shown in Fig.3. In this experiment steady sliding at velocity Vo is followed by a step change in speed (an alternate decrease and increase in Fig.3). The mean sliding speed is in the range 1-10 lxm/s, at which dynamic effects can be neglected. To model this behaviour Rice & Ruina (1983) introduced a 'state variable' # and proposed a 'state- and rate-dependent' friction law:
F( V, #) - Fo + Bln(# / #o ) + A ln(V / V )
(4)
(2)
F
ss
--- F - ( B - A ) I n ( V / V ) o
Vo=Do/#o (5)
With B>A equation (5) exhibits a friction characteristic which decreases with sliding speed. When combined with a simple system, such as that in Fig. 1, it gives rise to stick-slip vibrations During a sticking phase when V = 0, equation (2)gives d4gdt = 1, so that ~ - 9 § ~o and the static friction is given by
F-F s
o
+In
('+0;0) ....
(6)
We note that this relationship shows a growth of static friction with time, which has been attributed by many authors, from Kragelski (1939) onwards, to an increase in real contact area by creep of the asperities under the real contact pressure. Support for this hypothesis has been provided recently by Dieterich & Kilgore (1995) who measured the change in contact area with time between rough surfaces of transparent materials: quartz, calcite, glass and acrylic plastic. A thorough experimental investigation of the state-and rate-dependent friction law in both static and dynamic phases has been carried out by Baumberger et al. on nominally
39
V~ (V,=, V,)
V,
. . _v, _
_
Predicted re~oon$e Equation= (1) and (2)
J 75|
o~
=
p~ .
.
.
o~ .
Granite
70 L 9
'~
9
l~rn~ .....
10~Ivs
y-
.65
~
10atn~
'
~s
o =~M .
l~m~
~
9
~ .
I
62
70
15 MPa normal sires=
'
100 izm
67 r
.85 I
.o ,~
I 100 ~tm
,.
,~,v,
_ - -;
~--
t/
.
~ws .
.
o I.~ .
.
.
/ ~
_
--'--~- -
Granite #60 surface, Imm gouge I0 MPa normal s~reu
Soda-lime
glas=
S MPa normal stress
100 Izm
o,~m~, )
:r,-~ . .
'
,..~. --
100 ~tm
4
~
o,=~, ___
Lucite
plutic
. 6 0 .urfar 2 5 MPs normal s t r e ~
Figure 3. Theoretical (eq. 2 & 3) and experimental variations of friction following step changes in sliding speed
flat rough surfaces of PMMA in contact with PMMA and polystyrene on polystyrene (1999) Before leaving the state and rate law, it is relevant to note that the term in equation (2) and the growth of contact area given by equation (6) both imply rate dependence (e.g. viscoelasticity) in one or both contacting materials. The materials studied by Baumberger et al. (1999) are deliberately viscoelastic and it is conceivable that creep is important in the rock friction experiments where the sliding speed was very low and the stick times long. It is more difficult to accept that creep plays a significant role in the high frequency squeal of hard metal surfaces such as railway wheels in a fight curve. This would appear to pose an unanswered question.
Aln(V/Vo)
2. 3
T h e r m a l effects
A reduction of the force of friction with sliding speed V is not dependent on a change in contact area, but may be the result of a decrease in frictional stress x at the interface. One such cause could be frictional heating. In a concentrated contact (e.g. a square area 2a x 2a) in which one surface is stationary, the majority of the frictional heat is conducted into the moving surface. The rate of heat generation per unit area is q = zV. The resulting mean temperature rise in the contact- the 'flash temperature'-is given by
4 qa~v a 4~J2Va
(7)
40
where K = thermal conductivity, p = density, c = specific heat capacity and r = diffusivity = Kpc. Let fl = d , / d T be the rate of change of shear strength of the interface with temperature in the ambient sliding state. Then
"r
(8)
d T A T *' fl
For a reduction in frictional stress, fl must be negative, which is the usual situation. The effect becomes pronounced with a high sliding velocity and low thermal conductivity. (a)
-"
drive
Ul
lubricant~ I test discs
drive 20-
of a rolling contact disc machine for measuring lubricant traction in combined rolling and sliding contact. The discs were driven by independent motors through shafts of unequal length (Fig.4a). A typical variation of traction coefficient (friction coefft.) with sliding speed is shown in Fig.4b. At low sliding speed the traction increases with speed, as expected for a viscous lubricant. However at higher sliding speed, as demonstrated by Crook (1961), frictional heating in the contact so lowers the effective viscosity as to cause the traction to decrease with speed as shown. The effect was to excite torsional oscillations of the drive shafts. To eliminate this behaviour it was necessary to attach viscous torsional dampers to each shaft of sufficient strength to suppress the self-excited oscillations of the shafts. Stringed instruments such as the violin and cello provide classic examples of stick-slip vibrations excited by drawing the bow across the strings. The bow is coated with 'rosin', a natural resinous substance which shows rapid reduction of shear strength with temperature (negative/3) in the range 40-60 ~ and has low thermal conductivity. The velocity during the slip phase of the oscillation is of order 10m/s. Smith & Woodhouse (2000) have shown that the oscillations in a laboratory rig, equivalent to that in Fig.l, produced by a rosin covered rod are consistent with a thermal softening mechanism. 3. NORMAL MOTION
16
a...
Torque Ib tt 12
po= 176 0 0 0
Iblir 2
Ul= 153 in/sec h = e o ~in D = 0.075
.
.
Bearing /trict~on ,,...,.,,,
o
o.,
o.2
(b) u!
Figure 4. Rolling contact disc machine (b) Traction curve. A striking example of the thermal effect arose in our laboratory during the commissioning
So far we have only considered motion parallel to the sliding interface, but there has long been speculation about the role of motion normal to the interface. In a much quoted paper, Tolstoi (1967) reported observations of a relative normal displacement of the surfaces during the slip phase of a stick-slip oscillation in a system equivalent to that of Fig.1. Further the application of an externally excited high frequency vibration (~ 1 kHz) had the effect of suppressing the stick-slip motion and reducing the steady friction. These observations have received the support of other investigators. The separation of surfaces during sliding relative to their position at rest can be understood by the non-linear compliance of random rough surfaces. The Greenwood & Williamson model of a random rough surface, having an exponential distribution of asperity heights, in contact with a plane, relates a static load P to the separation d by (see Appendix)
41
It is often claimed that the separation between two sliding rough surface will increase with speed due to the impacts of colliding asperities. This intuitive supposition ignores the fact that, at sliding speeds much less than the shear wave speed of the solids, the interaction between contacting asperities is virtually quasi-static. It is, therefore, only responsive to sliding speed to the extent that the deformation of the solids is ratedependent, e.g. viscoelastic.
exp(-d / a ) (9) P where N is the asperity density, cr = roughness and Ip is definite integral, defined in the Appendix, which is independent of the separation d. The friction force can be expressed in the same way: P-CNI
(10)
F = CNI F exp(-d / a)
4. POINT CONTACS
Thus the coefficient of friction # = IF / 1p which is independent of d. Now assume that the separation varies sinusoidally according to:
4.1 The 'sprag effect' Studies of sliding point contacts have generally arisen from investigations of brake squeal. An early discussion of the 'sprag effect' was given by Spun" (1961-2) In its simplest form the system comprises a pin in point contact with a flat moving surface (e.g. brake disc) as shown in Fig.5a.
u
d = d + Asinrot
(11)
n
u
which gives: P - CNIpJexp( - d / a) m
m
F - CNIFJ exp(-d / a)
and
where J - - ~f~0~r exp( -
(12) (13)
AsinO ).dO. O"
e ~
Pin
~
If we compare the separation given by equation (12) with that do for an equal static load Po given by equation (9) we get:
~---- Disc
Vo = Disc Rotation
m
J exp(-d / or) = exp(-d / a ) 0 n
i.e.
d / a = d / cr + In(J) 0
(14)
This equation shows that the oscillation in separation of amplitude • A increases the mean separation d by amounts given in the Table below, but does not change the coefficient of friction which, by equations (12) and (13), remains equal to IF/Ip.
(a)
m=
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. . . . . . .
.
A/o" 0.5 1.0 1.5 2.0 2.5 In(J) "0.063 0.24 0.50 0.82 1.19 These conclusions are supported by Hess & Soom (1991). It seems as though a different explanation is necessary for the effect of an oscillating load or separation on the coefficient of friction. Presumably the frictional shear stress must be influenced directly.
(b)
: ~w
Figure 5. The spragg effect. (a) pin and disc. (b) 2 degree-of-freedom model.
42
The effect arises from a slight inclination 0 of the axis of the pin to the normal to the sliding surface. The bending stiffness of the pin is denoted by kI and the normal stiffness of the surface (disc) by kd as shown in Fig.5b. Damping can be included by dashpots cf and Cd in parallel with the springs. The equations of motion of pin and disc are then:
m/i~ +cfil + k f u = F c o s O - NsinO
(15)
m dfi, + Cd ~ + kdW = F sinO - N cosO
(16)
where w = u sin O Sliding velocity: V = V - ucosO
(17) (18)
Coulomb friction: F = # N s i g n ( V )
(19)
S
O
This 2 degree-of-freedom system has two solutions of the form Alexp(sl t) and A2exp(s2t ) , in which s1 and s2 can be real, positive or negative, or complex with positive or negative real parts., depending on the values of the angle 0, the friction # and the other parameters. The unstable case of a growing oscillation is of particular interest and is shown in Fig.6. 0 =
from disc drive. It is sometimes referred to as a 'flutter' oscillation from the analogy with the coupled bending and twisting oscillation of an aeroplane wing. The above solution applies while the oscillation is growing but, when the slip velocity given by equation (18) reaches zero, a series of high frequency stick-slips occur. This nonlinear behaviour will be referred to again below. 4.2
Disc vibrations
The compliance of the moving surface in the previous example was represented by a single spring, but a real rotating disc is a complex dynamic system. Linear stability analyses of a rotating flexible disc in frictional point contact have been made by Jarvis & Mills (1963) and by Earles & Badi (1984) where the spragg effect arises from the elastic rotation of the block holding the pin. Unstable combinations of parameters are calculated and supported by experiments with a laboratory rig.
Px
5.2
i,.____
Py Figure 7. Follower forces. _o > -00~
"~176
0.$
1
5
2.S
3
3.5 x 104
Time (s) Figure 6. Unstable oscillations of the system shown in Fig.4(b). The amplitude is limited by stick-slip. It is noteworthy that this unstable behaviour is predicted on the basis of Coulomb friction having no reduction with sliding speed. The instability arises from the coupling of the 2 degrees-of-freedom such that energy is extracted
Mottershead & Chan (1995) propose a further source of flutter instability in the case of thin rotating discs, which they attribute to 'follower loads' (Fig.7). In this model the inclination of the normal and tangential contact forces acting on the disc, which arises from bending of the disc, is taken into account. With the notation of Fig.7, P
=/aV
cosO + N sinO =--# N + NO
(20)
NVsinO = N - / a V O
(21)
X
P
= NcosO
-
Y where O = dy = l dy which introduces damping
dx
V dt '
terms (+ o r - ) into the equations of motion which
43
can promote instability. However these terms are generally very small. In an oscillation of amplitude Yo, frequency f and wavelength round the disc X, 0 - 2 . r t f y o / V = 2~r yo/~,.
4.3 Extended contact
Most practical brakes do not use a point contact, but make use of a flat flexible pad (disc brakes) or a flexible band (drum brakes). This introduces many more degrees of freedom and modes of vibration. Coupling of these modes through friction at the interface can lead to dynamic instabilities of the sort illustrated in Section 4.1 and frequently referred to as 'flutter vibrations' from the similarity of the mechanism to aeroplane wing flutter. A linear theory of frictional oscillations of both band and disc brakes is given by Nishiwaki (1993).
The linear stability of this system during sliding has been examined by Martins et al. (1990). They show that, with some combinations of parameters, including a coefficient of friction ~ = 0.6, unstable oscillations build up until the slip velocity vanishes, whereupon non-linear stick-slip vibrations take place. Note that these vibrations are at the contact resonance frequency of the block on the asperity stiffness, which greatly exceeds the frequency of the block on the stiffness of the restraining spring K. These high frequency (kHz) contact vibrations are flutter vibrations, which are genetically similar to those generated geometrically by the sprag effect and illustrated in Fig.6. What part they play in influencing friction characteristics and the stick-slip oscillations observed in practice is far from clear. 5.2 Dissimilar elastic solids
5. INTERFACIAL E F F E C T S 5.1 Rough surfaces
We now return to the sliding block shown in Fig.1. The block is effectively rigid, but the random roughness on one or both of the contacting surfaces is represented by a compliant layer with non-linear stiffnesses in the normal and tangential directions. Inelastic deformation of the asperities can be allowed for by adding damping to the normal elastic stiffness. We now have a 3 degreeof-freedom system: displacements in the tangential (x) and normal (y)directions, and rotation (pitching) about the z-axis. The modes are coupled through Coulomb friction at the contact surface (Fig.8).
Here we return to the earthquake problem: an extended sliding interface without asperities, where the elastic constants of the contacting solids are different. Following a disturbance at the surface of an elastic half-space, elastic waves known as Rayleigh waves are transmitted across the surface, which are held responsible for earthquake motions. They arise from the need for internal wave motion to satisfy the boundary conditions of a free surface. Similar wave motion occurs at an interface between dissimilar solids to satisfy continuity of stress and displacement across the interface. The generation of such waves by friction at an interface which transmits a steady mean pressure Po at a sliding velocity Vo as been studied by Adams (1995) and Martins et a1.(1995). It is shown that, under the action of friction at the interface, waves which propagate in the direction of sliding can increase in amplitude in an unstable way. At low sliding speed and high pressure the amplitude of the oscillation is limited by stick-slip motion but at high speed and low pressure a limit cycle is achieved by loss of contact. The change of regime is determined by a critical value of the parameter (vg/c2Po), where G is the shear modulus and c 2 the shear wave speed of the stationary solid. 6. CONCLUSION
Figure 8. The system in Fig.l(a) with a compliant rough interface which introduces normal and rotational degrees of freedom.
The expansive field of frictional vibration has been briefly reviewed from the point of view of
44
mechanism. Friction characteristics,
thermal and material characteristics, geometric (spragg) effects and self-excited 'flutter' vibrations have been identified as possible sources of unsteady sliding motion. But the list of possibilities is not exhausted: the non-linear dynamics of stick-slip oscillations is an open field. Parametric excitation and subharmonic resonance of rotating discs has been considered by Mottershead et al. (1996). Indeed stick-slip motion of a system like that in Fig.l, but with two spring coupled blocks, has been shown capable of chaotic motion! [Popp & Stelter (1990)]. Turning to the engineering scene, a first task is to decide which is the dominant mechanism of excitation in any particular practical case. This would appear to call for some form of mapping in terms of appropriately chosen non-dimensional parameters. Some such suggestions have been made in the text, for example the thermal parameter proposed in w
identical conditions, and the frequency can change for no apparent reason. Perhaps this capriciousness is itself an indication of 'chaos'. With these problems in the laboratory, the situation in the field with real engineering components seems even more problematical. With a remit to further study brake squeal, Duffour & Woodhouse (2000) have proposed a radically different approach. Recognising the difficulty of modelling a real brake system, they consider instead the general point contact with two degrees of freedom: normal and tangential, shown in Fig.8. The contacting bodies ('pin' & 'disc') are ascribed arbitrary linear transfer matrices [G] and [H]. A coefficient of friction which varies linearly with velocity is assumed. The genetic linear stability of this system can be analysed, so that those features of [Gl, [H] and /~ and their combination which lead to instability might be recognised and mapped. "Disc.
x'i
.
.
.
Brake"
.o V11
Ul
[H]
F N
:
~
_ V2
[G]
Figure 10. General dynamic system in sliding point contact. The field needs more novel approaches of this kind in order to impose some structure into its many manifestations.
ACKNOWLEDGEMENTS Figure 9. Chaotic motion in phase space caused by external vibration of the system of Fig. 1(a). So far nothing has been said about experiments. Experience in this regard is disappointing. Only in the case of a bowed violin string has detailed agreement been demonstrated between theoretical modelling and a real system (Schumacher & Woodhouse, 1995). Although some of the linear stability analyses of the pin and disc arrangement have been supported by laboratory tests on a laboratory rig designed to replicate the idealised theoretical model, that is far from general experience. The occurrence of self-excited oscillations is often very sporadic in apparently
The author wishes to acknowledge valuable help from Professors J.R.Rice, J.Woodhouse, J.H.Dieterich and Mr. Philippe Duffour in the preparation of this paper.
REFERENCES Adams G.G. (1995) ASME J.Appl.Mech. 62, 867872. Berthoud P., Baumberger T., G'Sell C & Hiver JM. (1999), Phys.Rev.B, 59, 14313-327. Baumberger T. Berthoud P.& Caroli C. (1999), Phys. Rev.B, 61), 3928-3939.
45
Bell R. & Burdekin M. (1969-70) Proc.I nst.Mech.Engrs. 184,543-551. Crook A.W. (1961) Phil.Tmns.Roy.Soc.Lond. A254, 223-258. Dieterich J.H. & Kilgrove B..D. (1994) Pure & Appl. Geophys. 143, 283-302. Dieterich J.H. & Kilgrove B.D. (1995) Techtonophysics 256, 219-239. Duffour P. & Woodhouse J. (2000) U.of Cambridge Engg. Dept. Tech.Rpt. Earles S.W.E. & Badi M.N.M. (1984) Proc.Inst.Mech.Engrs. 198 PtC, 43-50. Hess D.P. & Soom A. (1991) ASME J.Tribology, 113, 87-92. Ibrahim R.A. & Rivin E. (1994) ASME Appl.Mech Rev. 47, 209-253. Jarvis R.P. & Mills B. (1963) Proc.Inst.Mech.Engrs. 178 Pt.I, 847-866. Johnson K.L. Contact Mechanics, C.U.P. (1985). Ko P.L. & Brockley (1970) ASME J.Lub.Tech. 92, 543-549. Kragelski I.V. see Friction & Wear, 1965, Butterworths, London. p. 197. Martins J.A.C., Oden J.T. & Simoes F.M.F. (1990) Int.J.Engng.Sci. 28, 29-92. Martins J.A.C., Faria L.O. & Guimaraes J. (1995) ASME J.Vib.& Acoustics 117, 445-451. Mottershead J.E. & Chan S.N. (1995) ASME J.Vib.& Acoustics, 117, 161-163. Mottershead J.E., Outang H. & Cartmell M.P. (1996) in Advances in Automotive Braking Ed. D.C.Barton, Mech.Eng.Publ. Ltd. pp.49-65. Nishiwaki,M. (1993) Proc.Inst.Mech.Engrs. 207 Pt.D, J.Auto.Eng. 195-202. Popp K.& SteUter P. (1990) Phil.Trans.R.Soc.Lond. A332, 89-105. Rice J.R. & Ruina A.L. (1983) ASME J.Appl.Mech. 50, 343-349. Schumacher R.T. & Woodhouse J. (1995) Contemporary Physics, 36, 79-92. Smith J.H. & Woodhouse J. (2000) J.Mech.Phys.Solids, 48, 1633-81. Spurr R.T. (1961-62) Proc.Inst.Mech.Engrs. (Auto.Div.) 1, 33-52. Tolstoi D.M. (1967) Wear, 10 199-213.
APPENDIX Here we follow the presentation of the Greenwood & Williamson theory of rough surfaces contact in Johnson (1985) pp.411-414. The asperity heights will be assumed to be exponentially distributed, i.e. r
(C / cr)exp(z / cr)
If the force Pi to compress a single asperity through a displacement 6/is given by 1% = g( 6i ), the total load per unit nominal area is: P = CNI p e x p ( - d / o)
where I p ~,f o g(6 / o) exp( --6 / cr).d( 6 / or), We now express the asperity friction force Fi by Fi = NVi = fl6L), where # is not necessarily constant, but can vary with asperity contact pressure and hence dii. Thus the total friction force per unit nominal area is: F - CNI F exp(-d / or)
where I F ,, f o r ( 6 / o r ) exp(-6/cr).d(6 ~or). Note that: F/N = IF [ IF, = constant
even though the local friction at the asperity contacts is an arbitrary function of asperity pressure.