5
Friction
5.1
5.2
Introduction
Qualitative Ranges of Friction
Lubricated Friction • Rolling Friction • The Elastic Case •
The Plastic Case • The Viscoelastic Case • Effect of Fracture
Properties of Materials
5.3
5.4
5.5
5.6
5.7
5.8
Early Concepts on the Causes of Friction
Adhesion, Welding, and Bonding of the Three Major
Classes of Solids
The Formation and Persistence of Friction Controlling
Surface Films
Experiments that Demonstrate the Influence of
Films on Surfaces
Mechanisms of Friction
Measuring Friction
Unsteady Data and Vibration of Test Machines • The Sources
of Unsteady Data • Interaction of Inherent Unsteady Data
with the Dynamics of a Test Machine
Kenneth C Ludema
University of Michigan
5.1
5.9 Test Machine Design and Machine Dynamics
5.10 Tapping and Jiggling to Reduce Friction Effects
5.11 Equations and Models of Friction
Static Friction • The Stribeck Curve
Introduction
Friction is the resistance to the sliding of one solid body over or along another, as solid bodies are
ordinarily understood in the macroscopic world. High friction is desirable between tires and roads
(coefficients of which range from about 0.5 to 1.2), between mechanical parts that are bolted together
and in countless other examples. Low friction is desirable between sliding parts in computer hard disk
systems, in engines, in door latches and in many other mechanical devices. Whether friction is low or
high is of little consequence in many applications, provided it is predictable (i.e., reasonably constant)
and does not cause noise or vibrations. The most notable exception to the latter is the stringed instrument
that uses a bow to create music.
Friction often seems constant, regular, and well behaved, but that is observed mostly in mechanical
devices that operate under relatively constant sliding conditions averaged over time. Actually, virtually
every sliding condition produces a different level of friction. One reason is that surface materials continually change with time. Even in a vacuum, the surfaces of materials A and B are likely not really A
and B, but rather have a higher composition of an alloying element that has migrated to the free surface
from an alloyed substrate. Then, surfaces in air are likely to oxidize: even some oxide ceramics will oxidize
and some also hydrolyze on their surfaces. These coatings or films will alter frictional behavior. Further,
in “clean” air, surfaces attract gases to adsorb upon them, producing a layer of partially oriented liquid
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that has very different properties from the amorphous condensed phase of the gas. In addition, in the
usual atmosphere in which parts slide, there is water vapor, and there are hydrocarbons (e.g., turpenes
from trees and oil vapor from machinery) and other “contaminants” (e.g., human fingerprints) of every
description that find their way into sliding contacts. The substances that settle on A and B may be specific
to A and B, but A and B rarely “touch” each other. In this sense, tires do not actually “meet the road”
either, but rather are separated by thin films of water and other substances. Very thin films of water can
produce the high friction of tires, even at low shear rates: at 10 mph, on a water film 10–6 inch thick, the
ratio of viscous retarding force to normal force (µ?) is about 0.66. If high friction of wet surfaces seems
incongruous, consider another common human experience: everyone who plays softball or baseball
knows that the bat is less likely to slip from the hands if the hands (and bat handle) are moistened a bit
(Murakami et al., 1995). (Viscous friction is a much better explanation for this phenomenon than is the
influence of “meniscus forces” discussed in Section 5.5 below.)
The magnitude or “level” of friction is often expressed in terms of the coefficient of friction, µ, which
is the force, F, to slide divided by the force or load, W, pressing the two solid bodies together, µ = F/W.
Where an object is set upon a flat and horizontal plane, this load is the weight of the object: it is often
referred to as the normal load, or the applied load. Expressing friction in terms of a “coefficient” is a
misleading practice, suggesting possibly that the “coefficient” is an intrinsic property of materials, and that
friction force is always proportional to the applied load. The latter may be true over narrow ranges of
applied load but is not generally true. For example, sliding resistance has been measured when no external
load is applied: clean metal surfaces can be drawn together by atomic bonding forces only, and two wetted
surfaces can be drawn together by meniscus forces, both of which are described in following sections.
Friction had long been assumed to be invariant under all conditions (i.e., independent of load, sliding
speed, contact area, and all other operational variables) and graced with the term “Coulomb friction.”
It is safe to say that Coulomb friction is an unusual phenomenon, safe because a few tests will prove it so.
(The Baldwin Testing Company, the manufacturer of tensile test machines, at one time distributed
bronze plaques on which were the words, “One Test is Worth a Thousand Opinions.”)
One goal in this introduction is to point out the futility of trying to find useful values of coefficient
of friction in books or published papers. Most published values were measured in research laboratories,
usually with simple devices (though perhaps controlled by a computer), and rarely at practical sliding
speeds and contact conditions. The data in themselves are usually reliable and highly reproducible, but
they apply only to the conditions in that test. The tests may have been designated as standard tests by
ASTM or other standards-making bodies, but such standards apply only to the methods of doing the
test. Tabular values of friction are also frequently found in computer-based design packages, which seems
to add a measure of legitimacy to the values. Most often only a single value of coefficient of friction is
admissible or allowed to be entered into the algorithms and equations, some have room for two values,
a “static” (or starting) coefficient of friction and a “dynamic” (or kinetic, or sliding) coefficient of friction,
and a few can accommodate more complete frictional behavior as discussed in Section 5.11. The unresolved issue remains, however, of finding valid values.
Though fixed (single) values of friction are attractive and widely available, ranges of values taken from
measurements over wide ranges of sliding conditions are more realistic. A chart of friction ranges is given
in Figure 5.1. Even these data have little meaning since specific values of friction are not connected with
specific sliding conditions. For example, though handbooks often publish values for dry steel at 0.2, one
can measure values ranging from 0.1 to 1.1, and perhaps even higher, depending on sliding conditions.
The value 0.2 is probably correct for several applications, but no one knows a priori whether the application at hand is one of the well-characterized applications. In order to infer this to be the case one must
make sure that the application at hand is similar in every detail to previous, well-behaved applications.
Many published papers state, or strongly infer, that similitude is sufficiently shown by matching some
combination of contact pressure and sliding speed only: rarely do these papers show supporting data or
mention the importance of the ever-present oxides, adsorbed gases, and contaminants. No papers have
yet provided methods to predict the coefficient of friction for any dry sliding pair within an order of 10
of measured values without introducing some questionable assumptions.
© 2001 by CRC Press LLC
FIGURE 5.1
A table of values of the coefficient of friction of various materials and conditions of sliding.
A topic closely related to the values of the coefficient of friction is the tendency or likelihood of sliding
surfaces to vibrate or make noise. Sometimes this is called stick-slip, which will be discussed in Sections 5.8
and 5.9. Vibration is often seen as a system problem and, indeed, noisy vibrations can often be suppressed
by applying damping somewhere in the system surrounding the sliding members. However, it seems
more sensible to avoid initiating the vibration by choosing appropriate sliding materials. This is not
readily done because there is almost never enough information in tables of friction coefficient that indicate
the tendency for materials to cause vibration during sliding. This problem, too, requires close attention
to proper simulation between old and new designs, and may also require some testing to resolve.
Though we usually separate discussions of lubrication from discussions of friction, this is unreasonable
to do. Friction is the resistance to sliding, and fluid films also offer resistance to sliding. Furthermore,
all surfaces in air are lubricated, but by invisible layers of “lubricant.” The term “lubricated” appears to
have two meanings, either “not dry,” or “having deliberately exposed a sliding pair to a lubricant.” Thick
film lubrication is a very well developed field, whereas “thinner film” lubrication, or “boundary film”
lubrication often involves some chemistry and is somewhat more of an empirical art. Lubrication-bycontaminant film or by “residual” films is scarcely mentioned, probably because they are invisible and
few people appear to know what to do with these films even if their presence is acknowledged. Actually
© 2001 by CRC Press LLC
films of deliberately applied lubricants are sometimes thinner and less effective than films of “contaminants,” but one advantage of deliberate lubrication is that the chemistry of the interface region can be
influenced by choice of lubricant. Furthermore, a continuous stream of lubricant can be supplied if
desired, whereas a “contaminant” film is less reliably re-formed. Friction will be influenced by the
thickness and mechanical properties of all these films.
Furthermore, the entire chemistry in the contact region between two solids changes with time, that
is, time of both sliding and of standing still. During sliding time the sliding speed, ambient temperature,
contact pressure, duty cycles, changes in the roughness of the solids, and several more variables influence
the chemistry. Chemistry can also change while surfaces are not sliding, whether contacting or not.
The following paragraphs will expand on the points summarized above. The major purpose of this
chapter is to discuss the many variables that influence friction so that designers may have confidence
that friction values cannot be reliably predicted. This is an amazingly liberating concept and it encourages
designers to focus their attention on the important aspects of frictional similitude between old products
and “upgraded” products. Where similitude is questionable, testing should be done, and testing must be
“simulative” testing, i.e., testing components that are very nearly like production parts in the way they
will eventually operate. Testing can be a lengthy, expensive and risky procedure, if it is not guided by
good conceptual or physical models of friction, and this tends to deter good testing. A frequent consequence of confused thinking on friction (and wear) is that these issues are postponed to very late in the
design process, or perhaps ignored until warranty claims on production parts become very high.
5.2
Qualitative Ranges of Friction
Though absolute values of friction cannot be predicted, general ranges of friction are fairly easy to define.
Very low friction can be achieved either by rolling, or by “elastic contact” between solid surfaces that have
low bonding force between them. Low friction can be achieved by fluid film lubrication. Intermediate
friction levels are found in materials that are moderately hard and not carefully cleaned. The highest
friction is usually found with soft metal parts that are well cleaned, though “clean” is a relative term: high
friction is usually accompanied by much plastic flow in surfaces. Ceramic material pairs often have higher
friction as a class than do metals, though they are generally much harder than most metals, but this is
usually because many metals have soft oxide coatings on them, whereas few ceramics do. Solid lubricants
often produce friction at intermediate-to-high levels (and are used primarily for the prevention of galling
and other severe forms of wear rather than for low friction). Rubber has high friction when sliding against
almost any other solid, and the friction is inversely related to the hardness of the rubber. The friction of
plastics, particularly “filled” plastics, against almost any other material can be low but usually is not very
high either. The friction of rubber and plastics often varies in a manner that reflects their viscoelastic
properties, which properties are not prominent in polymer-based composites (filled plastics).
5.2.1 Lubricated Friction
The most completely studied mode of sliding is the liquid lubricated state. Longest life is achieved by
maintaining complete separation of the sliding members with a “thick fluid film,” which is accomplished
by selecting combinations of lubricant viscosity, bearing length, and shaft/bearing clearance to suit the
geometry, the speed of shaft rotation, and the load on the system. (Cameron [1966] and Barwell [1979]
are two good books on the practical aspects of hydrodynamic lubrication.) Generally the coefficient of
friction is to be minimized for system efficiency, but there is little return on achieving values of coefficient
of friction much less than about 0.1.
Relatively few systems are expected to operate continuously at optimum friction conditions: most
systems start and stop often or are overloaded on occasion. At low speeds a thick fluid film cannot be
sustained, leading to wear of the shaft and/or bearing, and the friction usually increases. A curve widely
attributed to Stribeck (Stachowiak et al., 1993) portrays the variations in friction over a range of the
ratio ZN/p, where Z is the viscosity of the lubricant, N is the rotational speed of the shaft, and p is the
© 2001 by CRC Press LLC
boundary
lubrication
log of friction
coefficient
hydro-dynamic
"friction"
mixed
lubrication
(undefined)
axis
log of ZN/p
FIGURE 5.2 A Stribeck curve showing the variation in sliding resistance of a shaft in a bearing over a range of
viscosity, Z, shaft speed, N, and contact pressure, p, between the shaft and bearing.
nominal contact pressure between the shaft and bearing. A typical Stribeck curve is shown in Figure 5.2,
which is often used to show zones or regimes of operation of a shaft in a bearing, in terms of the
effectiveness of lubrication.
The thick fluid film, or hydrodynamic regime is of the form of calculations by Petrov showing increased
viscous losses as ZN/p increases. At some point, when ZN/p diminishes, it is thought that some asperities
on each surface collide and “adhere” together, to cause higher friction: in the extreme, sufficient adhesion
causes seizure of the sliding members. The region in which asperity collision is thought to occur is
referred to as the “mixed-film” region because contacting bodies are thought to be supported on a mixture
or combination of asperity–asperity contact points and fluid regions between asperities. In the mixedfilm region the friction is high and totally unpredictable: published Stribeck curves often suggest values
of 0.2 or more but this is largely speculation. The value of 0.2 is the most common assumption for dry
sliding of steel and brass, which is probably taken as typical of very poor lubrication.
The left-hand section of the mixed lubrication region is called the “boundary lubrication” regime
when there are active chemical compounds or “additives” in the lubricant. This region is identified by
the lack of serious wear problems even though the surfaces are not totally separated. The additives are
referred to as “boundary additives” because they form substances that coat the boundaries of the sliding
pairs with a “protecting” substance. The existence of such films is verified, but their thickness and
mechanical properties have not been well characterized.
The usual Stribeck curve is a composite of the analytical Petrov type of curve on the right, and the
empirical work by McKee and McKee (1929) for minimally lubricated surfaces on the left. The major
value of the Stribeck representation is that when tests or experiments are run in the lower ranges of
ZN/p, variations in friction can be used to estimate the adequacy of lubrication and likelihood of
progression of surface damage.
5.2.2 Rolling Friction (Tabor, 1955)
A hard ball or cylinder that rolls along the surface of a softer material (or a soft roller on a hard surface)
deforms the softer material. This requires energy, which is particularly evident when the softer material
deforms plastically. Energy is also required to deform materials in the elastic and viscoelastic ranges of
strain, but in the case of these latter materials some energy is restored when the roller moves on and the
material returns to its near original state. All materials consume energy when cyclically strained in the
elastic range, in amounts ranging from 0.5 to ≈4% of the input strain energy for ceramic materials and
metals, and about 20% in the case of the rubber in auto tires.
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Rolling balls and tires slide a small amount against a contacting flat surface because the softer body
is deforming laterally in the contact region while rolling, sliding in one direction in the “front half ” of
contact and in the opposite direction in the “back half ” of contact. (Sliding between regions of a tire
against a flat surface may be demonstrated by rolling a tire over a sheet of paper.) Lubrication reduces
this sliding friction and results in lower overall rolling loss, as in the case of automobile tires on wet
roads. The effect is small because the sliding losses are much smaller than deformation losses in the
materials substrate.
5.2.3 The Elastic Case (McClelland and Glosli, 1992)
When two clean asperities of very low slope (<2°) are in atomic contact, and where the bonding forces
of the atoms are weak (as with most covalently and ionically bonded solids where lattice matching of
exposed lattice planes is rare), there is not likely to be very high sliding resistance. When an atom in one
asperity moves from one location on the opposing surface to an adjacent location, significantly less force
is required than to pull the asperities apart in a direction normal to the surface. The reason is that as a
laterally moving atom is being pulled away from one stationary surface atom, it moves into the force
field of an adjacent atom, increasing the energy of the atom left behind while decreasing the energy of
the one ahead. The transition occurs with little net energy rise on the overall surface, particularly if
laterally moving atoms move between equivalent atomic positions.
The transition is likely not smooth. Rather, the separating atoms “snap apart” from each other and the
approaching atoms will “snap together.” The result is a ripple of vibrations in atomic lattices, which
vibrations are not totally conservative: some energy is lost in each cycle of the vibration, and eventually
vibration ceases. As a historical note, the idea that frictional energy might be dissipated as lattice vibrations
was introduced in 1929. G.A. Tomlinson (1929) published a paper on friction in which he proposed an
irreversibility of atomic bonding due to energy lost in lattice vibrations. His model was rather mathematical
in nature and he predicted the friction of several materials based on their lattice properties. Beare and
Bowden (1935) measured the friction of these same materials and found no connection: they concluded
that it was not possible to predict friction in the mathematical format of Tomlinson. It appears today as
if Tomlinson’s intuition was correct, but he picked the wrong properties or the wrong materials in his paper.
Research by Israelachvili (1995) is under way on finding material pairs that have much lower friction
than any material pair has today. One important condition for low friction is that the asperities not be
stressed beyond the elastic limit. (The presumed elastic behavior of asperities at the beginning of sliding
is discussed in Section 5.11.) Contact stresses can be minimized by limiting the slope of the sides of
asperities (as with smooth surfaces), or in the ultimate by contacting on flat crystallographic planes.
5.2.4 The Plastic Case
If the combined normal and friction force is great enough, it will cause plastic flow in the asperities.
Asperities of high slope (>2°), as in the case of rough surfaces, will plastically deform readily upon
application of a normal load alone (Tabor, 1952). Then, from solid mechanics we learn that when the
ratio of shear stress to normal stress on asperities is less than about 0.3 (Childs, 1992), passing asperities
will “iron” each other down by plastic deformation so that the surface becomes smoother on average,
and the surfaces harden as sliding continues. When the ratio of shear to normal stress is greater than 0.3,
asperities are likely to be stretched (plastically deformed) in the direction of sliding, sometimes producing
a rougher surface than before sliding. These surfaces harden more quickly than do the previously
mentioned surfaces, and plastic deformation energy will appear as friction.
It would seem that the friction of very ductile (soft) materials, on average should be greater than that
of hard materials because of the great amount of plastic flow in the surfaces during sliding, except that
very ductile materials are also usually weaker materials. (In air and other lubricants, the thickness and
structure of surface substances will likely be influenced by the substrate material on which the films form,
and that will also influence friction.)
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FIGURE 5.3
of load.
The viscoelastic response of two different arrays of springs and dashpots to imposed loads and release
FIGURE 5.4 Sketch of two properties, tangent modulus, and damping loss of a viscoelastic material during cyclic
tensile or torsion tests.
5.2.5 The Viscoelastic Case
When sliding a very smooth plastic (linear polymer) material against any other smooth solid, the friction
shows no patterns that can be identified with substrate properties: the properties of the physically
adsorbed films are more prominent. When the asperities of a hard surface, or when a spherical slider
slides along a surface of the plastic materials, clear viscoelastic effects are seen in the friction measurements. Viscoelastic effects are often modeled as various arrays of springs and dashpots (Ferry, 1980), as
shown in Figure 5.3, and there are two prominent results, as sketched in Figure 5.4. One is that in cyclic
straining the effective elastic (tangent) modulus is seen to increase very considerably (2 to 3 orders of
10) when increasing the strain rate (or frequency of oscillation at constant strain). The other is that the
damping loss (loss modulus) is low at the low and at high strain rates, but much higher in an intermediate
range of strain rate. The full range of these variations covers about 8 orders of 10 of strain rate, and the
highest damping loss can be about 8 times higher than the low values.
Pin-on-disk friction of plastics varies also by 5 to 1, or even 10 to 1 over a wide range of sliding speeds.
Thus, whereas the coefficient of friction of Teflon® is widely reported as 0.17, it has been measured to
© 2001 by CRC Press LLC
FIGURE 5.5 Sketch of the movement of waves of detachment (Schallamach waves) during sliding of soft rubber
along a flat surface.
cover a range from 0.10 to 0.80! A plotted curve of friction vs. sliding speed has the same appearance as
a curve of damping loss vs. strain rate, which can be taken as strong evidence that the viscoelastic
properties of the plastic influences the friction. Furthermore, temperature has the same influence on
measured values of friction (Grosch, 1963) as on the measured values of damping loss. This is shown in
Figure 5.4. It is likely, however, that damping loss is not the relevant viscoelastic property to connect with
friction, but rather the connection can be made through the tangent modulus property (Ludema and
Tabor, 1966).
Soft rubber exhibits interesting behavior when sliding on a smooth counterface, which was described
by A. Schallamach (1963) and is sketched in Figure 5.5. Upon the start of sliding with uniform contact
applied pressure, there is shear throughout the body of the rubber, but because the contact pressure is
reduced at the rear of contact the rubber slips first at the rear. This slipping alters the stress state in the
rubber, such that the contact pressure just ahead of the slipped region to the rear is reduced. Slip then
occurs in this region, which then alters the contact pressure further ahead. A wave of slip occurs and
moves forward at about 10 times the sliding speed. With continued sliding, standing waves are established,
which often have frequencies in the audible range. In the case of very soft rubber the waves of slip become
waves of detachment in that the rubber actually separates from the flat plate. Good lubrication reduces
the Schallamach waves such that sliding is quiet at moderate sliding speeds, but the waves are reestablished
when the sliding speed becomes low and the lubricant film is thin. The viscoelastic component of rubber
friction is also diminished when there is good lubrication, as shown in Figure 5.6 (Wassink, 1996).
The surface of the rubber does not slide the distance of translation of the entire body, so that
Schallamach waves diminish friction: the reduction is not exactly proportional to the “slip distance
reduction” due to the waves because some energy is expended in the passage of the waves along the
rubber, which is measured as friction.
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3
2.5
dry sliding
1.5
2.5 orders
of ten
coefficient of friction
2
1
0.5
lubricated
sliding
0
10
-5
0
10
sliding speed, mm/s
10
5
10
10
FIGURE 5.6 The viscoelastic nature of rubber friction, in dry sliding and as progressively attenuated by incursion
of lubricant into the contact region as sliding speed increases. The curves are constructed by viscoelastic transformation, using data over ranges of temperature and sliding speed.
5.2.6 Effect of Fracture Properties of Materials
Some friction is due to the loosened particles in the sliding interfaces. Metal surfaces (except gold) have
oxides on them which come loose during sliding, and over time some substrate metal is removed as well,
either by adhesive tearing or by fatigue failure. Ceramic materials are more likely to have grains loosened
because of poor bonding, which will fragment into smaller bits and cover a surface: grains can be loosened
after only a few cycles of shear strain. Brittle polymers fragment somewhat also but bits of the softer
polymers will be removed by adhesion and circulate within the contact region.
The nature of the loosened particles is difficult to express. Some oxides scratch the metal surfaces
beneath, which suggests that the oxide is abrasive: others are not abrasive. The general definition of the
hardness and brittleness of materials in the sliding interface becomes unclear when a high contact pressure
is applied. These terms are usually mentioned in the context of tensile properties of materials where no
hydrostatic pressure is applied. For example, the true strain at fracture of 1018 annealed steel is somewhat
larger than 1, but from hardness tests of surfaces that have been sliding together for a long time it would
appear that the metal had strained to a true strain as high as 10. Some work has been done on the strain
accumulations and fracturing of materials under the conditions that exist in a sliding contact (Kapoor,
1994).
5.3
Early Concepts on the Causes of Friction
In the earlier days, friction was thought to be due to the “interlocking of asperities” (Ludema, 1979).
Asperities are the very small (usually microscopic) bumps or protuberances on most surfaces, which
constitute the “roughness” of surfaces. Actually, asperities “interfere” with each other’s passage rather than
interlock, but interestingly one modern author explained the “interlocking” mechanism as equivalent to the
action of the fabric Velcro® (Victor, 1961), found in many garments, shoes, floor mats, and other products.
There were some objections to the “interlocking” theory over 100 years ago (Bowden and Tabor, 1954):
some pointed out that once a group of asperities on one surface had ascended the slopes of the asperities
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FIGURE 5.7 Sketch of real contact area between two bodies of surface dimension A and B: the summation of the
small regions in this sketch is about 0.07AB and is referred to as the “real” area.
on the opposite surface, which required energy to do, no further energy would be required to continue
sliding. Rebuttals developed but none were persuasive, probably because the necessary tools to observe
the discussed phenomena (asperities, etc.) were not available. Some “interlocking” likely does occur where
one surface has hard grains, constituents, or compounds protruding from the surface, which plow grooves
in the opposite surface. Such plowing produces plastic deformation in the opposite surface, which
consumes energy.
The most serious doubts about the interlocking concept came with the experiments of W.B. Hardy
(1923). He transferred a single molecular layer of a fatty acid from the surface of water to a glass plate.
From his work on the sizes of these molecules he was sure that a monolayer was thinner by at least an
order of 10 than the average height of the roughness (asperities) on the glass surface. When sliding clean
glass on clean glass he measured a coefficient of friction of about 0.6, whereas when sliding clean glass
on the glass plate coated with a thin film of fatty acid the coefficient of friction was about 0.06. Interlocking
or interference could clearly not explain such great difference in friction. Adhesion, cohesion, atomic
attraction, molecular attraction, and similar terms were used then to explain friction. F.P. Bowden and
D. Tabor (1964) and several others vigorously pursued the idea of adhesion as the source of friction.
Bowden and Tabor and their many Ph.D. students showed that friction was proportional to the real area
of contact between two surfaces, which was taken as strong support of the adhesion concept. This was
a formidable task because the real area of contact is ultimately the summation of thousands of microscopic
points of contact within a nominal area of contact, as sketched in Figure 5.7: these details cannot be seen
while sliding occurs. Further, the actual shear strength within those microscopic areas of contact was not
known either, and is not really known to this day. Bowden, Tabor, and students used very many methods
to estimate and infer what was occurring on surfaces, and the preponderance of their evidence surely
showed that friction is not primarily due to mechanical interlocking of asperities. They and several others
developed physical models of friction, usually around the assumption that friction is the product of real
contact area and the shear strength of the material in the asperities. One model is expressed in the
following terms: µ = AsS/Aw3Y (Greenwood et al., 1955), where µ is the coefficient of friction, S is the
shear strength of the bond between asperities, 3Y is the pressure hardness of the weakest of the sliding
materials, Y is the yield strength, and A is the cross-sectional area of shear (As) and the area carrying the
normal load (Aw).
Assuming that Aw ≈ As, µ ≈ S/3Y, which is about 0.16 or 0.2 depending on which flow rules of metal
one favors. The major problem with this conclusion is that µ of all materials would be about the same.
Tabor recognized this as erroneous, and over 40 years ago (Tabor, 1959) developed a model in which the
sliding resistance is offered by shearing of a substance between sliding bodies, a solid substance with a
flow (shear) strength. This model could be adapted for the case of viscous substances in the interface,
but whether solid or viscous, the properties of the interfacial substances could only be assumed. This
model further assumes that the real contact area will grow as shear stresses develop in the asperities.
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Though adhesion appears to be a better explanation for friction than is interlocking of asperities, it
is probably the major mechanism only when surfaces are clean and sliding in a vacuum. Analysis of many
published papers that invoke the “adhesion theory” of friction suggests that most authors believe adhesion
to be complete and full-strength welding or bonding at every asperity contact site. This cannot be a valid
general assumption for two reasons:
1. Bonding of atomically clean ceramics and polymers is seldom “full strength,” and,
2. Most systems function in atmospheres that provide substances (contaminants) that adsorb to solid
surfaces, which diminishes bonding strength considerably.
Bond strength and persistence of surface contaminants will be discussed in the following two sections.
5.4
Adhesion, Welding, and Bonding of the Three Major
Classes of Solids
Adhesion, welding, etc., usually refer to resistance to separating bodies from each other in the direction
normal to these surfaces. When two atomically clean, smooth, and conforming pieces of the same metal
are brought into contact in a perfect vacuum, against matching atomic or lattice arrays, they will become
one piece leaving no evidence of the former boundaries or surfaces. There is no need to apply pressure,
heat or rubbing energy: atomic bonding occurs spontaneously. This is called “adhesion,” though some
argue that the term “cohesion” should be used where identical materials bond together. The original two
pieces of metal will not readily “slide” over or along each other, and in fact no definable interface survives
along which sliding can occur in preference to shearing of substrate material. For most common metals
(there are a few exceptions among the metals), if the lattice arrays are even significantly misaligned they
will still bond with high strength. By contrast, ceramic materials will bond together at high strength only
when lattice arrays are perfectly matched and aligned. Mismatch occurs where ceramics of different atom
spacing are in contact or where the lattices in contacting asperities are rotated or tilted (ever so) slightly
relative to each other. The strength (Van Vlack, 1959) of bond ranges is as low as 2.5% of the maximum
if lattices are mismatched, so that when multicrystalline ceramics slide over each other there will be
varying degrees of lattice matching, and therefore varying friction with distance of sliding. For the third
class of materials, linear polymers, bond strength will be fairly high whatever the relative alignment.
Bonding between atoms may be described in terms of their electron structures (VanVlack, 1959). In
the current “shell theory” of electrons the number of electrons with negative charge should balance the
positive charge on the nucleus, and there should be no net electrostatic attraction between atoms.
However, within clusters of atoms the valence electrons (those in the outer shell) take on two different
duties. In the case of some elements, particularly those that do not conduct heat or electricity very well
and are referred to as insulators, a pair of electrons orbit around two adjacent nuclei and constitute the
“s” bond. These bonds are very specific as to allowed angle between pairs of nuclei, and any attempt to
displace an atom from its “assigned” position results in much reduced bonding strength. This explains
the brittle nature of most monolithic insulators (including ceramic materials), and the difficulty in
bonding one group of ceramic atoms to another.
The few additional electrons in insulators and all of the electrons of conducting elements (the metals
mostly) become “delocalized,” that is, they do not retain allegiance to any single or pair of nuclei. These
delocalized electrons set up standing waves among a wide group of nuclei: the average energy state of
the delocalized electrons is lower than the energy state of valence electrons of single atoms, and this is
the energy of bonding between atoms within a solid. This group bonding is known as “π bonding,” and
not referred to as “a bond.” The angular relationship between pairs of atoms in π bonding is influenced
more by atomic packing than by variations in bond strength, which explains both the ductile nature of
most metals and their ability to bond well under most conditions of misalignment.
© 2001 by CRC Press LLC
Some solid materials such as polymers are made up of long chains as a “backbone” (usually carbon
or silicon) of a molecule. Bond angles along this chain are very specific. For example, the bond angle of
carbon is 109.5°, and considerable effort is required to bend this bond to any other static angle. These
bonds do vibrate, however, over a range of angle around the central value of 109.5°, and they also vibrate
in rotational modes. But, a long chain of carbon atoms, as in a polymeric material, does not satisfy the
potential bond energy of carbon atoms. Carbon can connect with up to four other atoms usually; two
of which are satisfied by connection to others in the chain, leaving two bonds to connect with side atoms
or chains of atoms. In this array some electrons are not involved in the “s” bonds and these develop the
π bonding. Where segments of two or more adjacent molecules lie moderately parallel to each other, π
bonding is particularly strong, packing these molecular segments into a crystalline order and imparting
more elastic than viscous behavior to these regions.
5.5
The Formation and Persistence of Friction Controlling
Surface Films
After a “new” solid surface has been created it exists in a high state of energy, of the order of 1 J/m2
(VanVlack, 1959). The energy state of electrons at the surface of a solid is higher (less delocalized) than
that in the substrate, which is seen in the form of surface atoms (ions) being ≈ 95% the “size” of substrate
atoms. This energy constitutes the potential for bonding of one solid body to another solid body or to
a liquid or a gas.
Surface energy can be reduced by bringing any atom or molecule to that surface. The first layer of
adsorbed atoms reduces the surface energy the most, but if that layer is nearly the same material as the
new surface, then that layer becomes the new high energy surface. Molecules of liquid or gas approaching
an energetic surface will surely not have the same atomic arrangement as the solid surface and will
therefore not reduce the surface energy of that surface very much: of the order of 10% of the surface
energy. The first molecular layer is aligned as best it can, which is to say that some order is imposed
upon this first group of otherwise randomly oriented molecules. The first layer of molecules does not
totally satisfy the surface free energy of the solid, so energy is available to attract a second layer and
impose some order within it as well, and so it transpires in 4 or 5 more layers. Order in these layers has
actually been observed and experiments have shown that the viscosity of these layers is directly related
to the degree of order (vanAlsten et al., 1990; Pashley and Israelachvili, 1984; Gee et al., 1990). The
viscosity of the first layers of the liquid is very high, perhaps 1000 times its “bulk” viscosity. The viscosity
of succeeding layers decreases as films become thicker, apparently getting to bulk viscosity at about 5 to
10 molecular-layer thickness. The affected films might be 10 to 20 nm thick (e.g., 0.3 nm for computer
hard disks), which is small compared with the height of asperities on very smooth technological surfaces,
which would be about 25 nm. It is apparent that these “fluid” films could be very tenacious, that is, hard
to displace even by repeated sliding. Consider the persistence of grease or other high-viscosity substances
on a glass plate: continued rubbing with a finger over such substances does decrease their thickness but
requires much effort to remove entirely. Of course, the layer may be regularly renewed if sliding occurs
in an atmosphere containing the proper molecules.
The liquid nature of adsorbed water is seen in the behavior of heads and the hard disks of computers.
While the disk is rotating the read/write head “flies” over the disk and adsorbed gas with very small
spacing. When the disk stops for a short time, the slider settles closely enough to the disk so that the
adsorbed films on each surface “wets” the other (Li, 1989). This is shown in Figure 5.8.
The force pulling the head to the disk is sometimes referred to as a “meniscus force” and is equivalent
to the action of a meniscus in contact with the wall of a container or the surface of tubes inserted vertically
into a liquid. This is shown in Figure 5.9. If the wetting angle of the liquid to the material of a tube is
less than 90°, the meniscus will lift a column of liquid in a tube, whereas if the wetting angle is greater
than 90°, the meniscus will suppress the liquid in the tube below the general level of liquid outside of
© 2001 by CRC Press LLC
SLIDER
meniscus
FLAT DISK OR PLATE
liquid that wets
the beaker and
tube
FIGURE 5.9
tube
Sketch of the “liquid bridge” that forms when two films of adsorbed liquid touch each other.
tube
FIGURE 5.8
liquid that does
not wet the
beaker or tube
Sketch of the effects of surface tension or meniscus forces acting on a liquid surface at a solid boundary.
the tube. The wetting angle is equivalent to the “contact angle” at the edge of contact between a drop of
liquid and a horizontal flat plane.
Meniscus forces can influence friction by drawing two bodies together, sometimes with a greater force
than an applied external force. The result will be a simultaneous enlarging of the “wetted” area, and a
thinning of the fluid film, two conditions that increase the force required to shear the fluid. The drive
motor of a typical hard disk drive (particularly those in laptop computers) does not have sufficient torque
to start rotation when there is a high sliding resistance: this phenomenon is known as “stiction” (sticktion?). The problem can be alleviated by deliberately roughening a “track” on which the slider is programmed to “park.”
Meniscus forces also have a significant effect on the sliding friction of very soft lubricated rubber. This
occurs by the meniscus force attracting the rubber to the other surface and thereby increasing the area
of close proximity between the two. Meniscus forces also “hold” a thin polymeric film to a glass surface
and allow many insects to walk on vertical surfaces.
Adsorbed films of liquid and gas build up at very great rates in Earth’s atmosphere. About 99% of a
solid surface will be covered by a monolayer of adsorbed gas in about 1.2 × 10–8 sec (Ludema, 1996).
After about 10 seconds the film is about 1.5 nm thick. After a day these films grow to thicknesses in the
range of 4 to 12 nm. Adsorbed gas films can usually be driven off surfaces by heating, in which case the
surface energy of the solid surface beneath returns to a higher energy state. This reversible process is
referred to as physical adsorption. Films of ordinary liquids can also be “driven off ” by heating, as is the
case with solvents. Many other liquids such as oil, are more likely to dissociate or oxidize rather than to
simply boil off.
When a surface is covered with adsorbed O2, CO2, or H2O, the oxygen in these adsorbates is available
to oxidize the substrate surface. About 10 times the energy is released in oxidation as in physical adsorption.
Oxidation begins on metals (except on gold) and some ceramic material in approximately 10–7 seconds:
in about 10–4 seconds a monolayer of oxide (or other compounds) has formed (Evans, 1960) and the films
grow at about the same rate as physically adsorbed films grow. The oxidation process cannot be reversed
by heating alone. When chemical reactions occur during adsorption the process is known as chemical
adsorption, or more simply, chemisorption. Physically adsorbed films form on top of the oxide.
Most of the above-described films are invisible to the unaided eye or by microscopy (light or electron).
Neither are they detectable with many instruments: for example, even several layers of water on a
© 2001 by CRC Press LLC
crystalline solid cannot be detected by infrared spectroscopy, but ellipsometry can detect fractions of a
monolayer of water.
The properties of oxides cover a wide range, but the ratio of the mechanical properties of the oxides
to those of the substrate is the relevant quantity. For coatings (including oxides) that are softer (lower
shear strength) than the substrate, the friction is lowered. The opposite could probably be said of hard
coatings on soft substrates, but the picture in this case is less clear. Hard coatings debond or fragment
at some load, causing higher friction than for sliding on a solid body of high hardness. At very high loads
the soft substrate will plastically deform, which also contributes to high friction.
5.6
Experiments that Demonstrate the Influence
of Films on Surfaces
One of the mysterious aspects of research reports and published papers is that sliding surfaces are
discussed as if they have no contaminant or other substances on them, whereas virtually everyone else,
educated and uneducated, child and adult, knows that such substances are ubiquitous. Films of “contaminants” accumulate on windows and mirrors to the point of being visible as a haze, and they form
without people touching them. (These may be films of water from vapor or films of condensed hydrocarbon gases emitted from nylon, vinyl polymers, etc.) Fingerprints appear on all objects handled by
people, and since they are often visible their thicknesses must be in the range of the wavelength of light
(≈1 µm). It is common knowledge that fingerprint films and other films influence friction. When these
films are removed, the surfaces are “squeaky clean,” a term taken from the practice of rubbing a (clean)
finger along the wetted edge of a glass container: if the friction is low, the glass is “dirty,” but if squeaking
(stick-slip) is heard and friction is high, the glass is considered “clean.” Very thin films do indeed influence
friction, but the published evidence is sparse.
Measures of cleanliness are not often discussed in research papers, but cleaning methods are. A frequent
litany of how surfaces have been cleaned, might include the following, “sprayed with acetone, held in
ultrasonically agitated isopropanol for 10 minutes, dried with an electric hair dryer, and held in a
desiccator for 24 hours.” Of course, one problem in measuring cleanliness is that there are no convenient
instruments that can detect very thin films of contaminants, and if such measurements were to be made,
the measurements would need to be made in an atmosphere where the contaminants in the laboratory
would not form a new contaminant film!
A few experiments were run to measure the influence of contaminant films on the friction of steel on
steel at low contact pressures (100 MPa) and very low sliding speed (23 µm/s). The experiments were
done with a 6.2-mm-diameter cylinder (roller from a roller bearing) sliding on a disk of 50 mm diameter.
The cylinder was steel of 58Rc, the disks were steel of 51Rc, with a roughness in the range of 0.06 µm
Ra or better. The disks were “contaminated” by several means, and cleaned by several common laboratory
methods as indicated in Table 5.1. The thickness of the films was measured with an ellipsometer that
displayed its results in the form of the Mueller Matrix transformation between four incident Stokes
parameters and four reflected Stokes parameters. The internal consistency of the numbers in the transformation assured that valid film thickness measurements were made, and at the same time the index of
refraction of each film was displayed.
The coefficient of friction vs. measured film thickness is plotted in Figure 5.10. The overwhelming
trend is toward lower friction with thicker films. The reason for some inconsistency is probably that each
type of contaminant, before and after being diluted or converted by cleaning fluids, has a different viscosity
or molecular ordering. However, the data from each type of contaminant and cleaning methods do show
some order.
The clear point in Figure 5.10 is that friction is not as much controlled by the properties of the substrate
material as by the varied thickness and (probably) composition of invisible contaminant films. Handbook
values of friction, which were taken from surfaces covered with one type and thickness of contaminant
cannot be applied to surfaces covered with another type and thickness of contaminant.
© 2001 by CRC Press LLC
TABLE 5.1 Sample Data on the Cleaning Method, Thickness of Film, Index of Refraction of the Films,
and Coefficient of Friction of Steel Covered with the Films
Contaminant
Vacuum pump oil
Light mineral oil
Normal finger print
Greasy finger print
Cleaning Method
Film Thickness, Å
Refractive Index
Friction Coefficient
None
Acetone spray
Isopropanol-soaked paper
None
Isopropanol spray
Sprayed with acetone
None
Toluene spray
Acetone spray
Ultrasonic acetone bath
Isopropanol-soaked paper
None
Acetone spray
2461.4
5.374
<1
2524.6
13.95
<1
156.4
59.44
24.19
7.053
<1
1.085
2.476
120.2
1.79
0.082
0.296
0.467
0.262
0.522
0.552
0.0337
0.191
0.314
0.482
0.528
0.351
0.07
1.077
1.071
1.835
2.167
2.51
2.48
Note: Thickness and friction data are plotted in Figure 5.10.
FIGURE 5.10
5.7
The effect of four contaminants on the friction of steel.
Mechanisms of Friction
The simple model in which friction is the product of real contact area times the strength of bond in that
area was appropriate in the early days of friction knowledge but is clearly limited in light of modern
understanding of surface geometry, surface chemistry, and strains in surface materials. But papers continue to be published in which the limited concepts continue to be advanced, probably arising from
limited perspective of friction mechanisms.
Adhesion concepts are often used in attempts to understand “mixed film” or “boundary film” lubrication. The common assumption was that lubricant films could be thick enough to prevent most asperities
on one surface contacting asperities on the other surface, but a few stand high enough to “contact” each
other. Friction is then explained in terms of a combination of adhesion of “contacting asperities” plus
© 2001 by CRC Press LLC
FIGURE 5.11
Three sources of sliding resistance (friction) between lubricated surfaces.
the viscous drag in regions that are not in “contact.” Even this concept needs to be updated to account
for the influence of the very thin films of lubricant, which reduce the strength of bond between asperities
and have their own distinctive shear-resisting properties (either viscous or solid). A measurement of
electrical resistance across the interface of nominal contact regions is sometimes made to estimate the
relative amount of asperity (adhesive) contact, but the results are strongly influenced by the high resistivity
of very thin layers of contaminant (lubricants) on most asperities. Likely the greater part of the applied
load is carried on these thin layers. Sliding with sufficient speed or applied load will, of course, change
the nature of the insulating layer and will expose more asperities to atomic contact, but the extent of the
change is usually not even approximately indicated by electrical contact resistance measurements.
In summary, friction of surfaces that are not rigorously clean is seen to consist of the following three
components, which may be visualized with the aid of Figure 5.11:
1. Shear of those real contact (load carrying) regions that are in (solid) atom–atom (adhesive) contact
2. Shear of those real contact (load carrying) regions where the solid asperities are not in atom–atom
(adhesive) contact but are separated by interposing films of contaminant/boundary/chemisorbed/etc., substances
3. Viscous shear of fluid between the noncontacting regions of the apparent area of contact. The nature
of surface roughness influences some of these components and so will the amount, shape, and
properties of loosened particles of substrate material and oxides (e.g.) in the nominal contact area.
5.8
Measuring Friction
The measurement of the coefficient of friction involves measuring two quantities, namely F, the force
required to initiate and/or sustain sliding, and N, the normal force holding two surfaces together. (Recall
that atomic bonding forces and/or meniscus forces may also be present as an unmeasured normal force.)
Some of the earliest measurements of the coefficient of friction were done by an arrangement of pulleys
and weights shown in Figure 5.12. Increasing load P is applied until sliding begins and one obtains the
FIGURE 5.12
A device for measuring static or starting friction.
© 2001 by CRC Press LLC
FIGURE 5.13
The inclined plane method of measuring the static coefficient of friction.
F
static, or starting coefficient of friction, with µs = Ps /N = -----s . If the kinetic coefficient of friction, µk is
W
desired, a weight is applied to the string and the slider is moved manually and released. If sliding ceases,
more weight is applied to the string for a new trial until sustained sliding of uniform velocity is observed.
F
In this case, the final load Pk is used to obtain µ k = Pk /N = ----k- .
W
A second convenient system for measuring friction is the inclined plane shown in Figure 5.13. The
measurement of the static coefficient of friction consists simply in increasing the angle of tilt, α, of the
plane until the object begins to slide down the inclined plane: the tangent of the angle of tilt, tan α, is
the coefficient of static friction. If the kinetic coefficient of friction is required, the plane is tilted at some
chosen angle and the slider is advanced manually: if the object stops sliding, the proper angle of tilt has
not been set. When an angle of tilt is found at which sustained sliding of uniform velocity occurs, the
tangent of that angle is the kinetic coefficient of friction. Data will vary according to one’s judgment of
the uniformity of sliding speed.
Other friction force measuring devices range from the simple spring scale to transducers that produce
an electrical signal in proportion to an applied force. The deflection of the holder of one of the sliding
members can be measured by capacitance sensors, inductance sensors, piezoelectric materials, optical
interference, moire fringes, light beam deflection, and several other methods. The strain in the specimen
holder can be measured by strain gages, acoustic emission, etc. The most widely used, because of its
simplicity and reliability is the strain gage system.
Just as there are many sensing systems available, there are also many designs of friction measuring
machines: these can be classified in terms of range of load, range of speed, atmosphere in which they
function, reciprocating vs. continuous motion, rotating vs. linear motion, spherical or cylindrical or flat
(e.g.) shapes of sliding members, etc. Many machine configurations are available for purchase but all are
variants of the basic motion-producing and force-measuring components.
Only the pin-on-disk (pin-on-flat plate) geometry will be discussed here, where a round-tip pin is
held by a cantilever-shaped force transducer and it slides against a disk. The disk is usually rotated by a
drive system, either directly from a motor as sketched in Figure 5.14, or through gears or belts. The disk
is held on a pedestal, which is held on a shaft that is supported by bearings. This geometry is among the
least useful for purposes of simulating practical hardware but it is easy to explain: and the results are
quickly available for publication!
5.8.1 Unsteady Data and Vibration of Test Machines
The focus in the following paragraphs is on the dual problem of unsteady data from test machines, and
vibration of test machines. Virtually all test machines produce unsteady and time-varying data, which
behavior is disconcerting where smooth and constant values of friction are expected (to accuracies of
±0.01!). The variations range from about ±5% for most practical sliding pairs to more than ±35% for
dry sliding of soft metals. Unsteady friction force appears in the test machine as a vibration, which may
© 2001 by CRC Press LLC
FIGURE 5.14
The components of a pin-on-disk friction test machine.
range in amplitude from very small to very large: the latter may be annoying to the operator, or even
damaging to the machine.
Smooth data can sometimes be obtained by artificial means. One way is by changing the test conditions,
such as by changing the applied load or the sliding speed. Machine vibrations can be reduced by increasing
its stiffness, or by applying damping in appropriate places. Smooth data can also be artificially obtained
by filtering the data stream or by using a data acquisition system with low frequency response to eliminate
the higher frequencies so that the data appear more stable.
Where data are displayed in rough form, as with a chart recorder, it is tempting to obtain a “best
estimate” line through the traces to obtain a single value of friction. This is reasonable to do where
variations are small, but when variations are larger and approach the stick-slip state, reasonable values
of friction cannot be obtained. True stick-slip is uncommon, though frictional vibrations are commonly
referred to as stick-slip. In the actual stick-slip condition the sliding bodies momentarily stop sliding and
then slide some short distance before becoming stationary again. In such cases, the value of µs may
possibly be estimated from the maximum force measured when slip starts, as indicated by the arrow in
Figure 5.15a. The shape of the curve in Figure 5.15b, prior to the maximum, reflects only the system
stiffness and intended speed of sliding. When slip begins, the slip portion is usually not recorded in
sufficient detail to determine µk, however. In general, it is incorrect to assume that µk is the average of
© 2001 by CRC Press LLC
(c)
(d)
applied force velocity
(b)
applied force velocity
(a)
time
in actual
stick slip
there is
virtually
zero sliding
speed at the
"stick" point
in the cycle
time
time
time
frictional
oscillations
occur without
"stick" but is a
quasi-harmonic
motion imposed
upon an average
velocity
FIGURE 5.15 The momentary sliding speed and force applied to a slider in the stick-slip condition (a and b) and
in the more common sliding condition (c and d).
peaks and minima in the excursions though such estimates would not introduce much error if the data
were in the form of Figures 5.15c and d.
Neither of the above solutions is wise without consideration of the effects of the solution. If the lab
tester is to be used to simulate production hardware in service, filtering the data from the lab tester will
not produce stable sliding in the production hardware, and the same logic applies to the other solutions
to unsteady data. More important, the average friction values in a vibrating machine are usually different
from the friction values in a quiet machine for nominally identical test conditions: instantaneous contact
pressures and sliding speeds are different in the two.
5.8.2 The Sources of Unsteady Data
Unsteady data may be classified as two major types, those that are independent of machine dynamics
and those that result from interaction between inherent frictional behavior and machine dynamics.
There are at least four sources of unsteady data that are independent of machine dynamics. These are:
1. Surface roughness, with wavelength of the order of, or larger than, the size of contact regions,
which usually produces high frequency variations in friction.
2. Inherent and repeatable variation of friction along a sliding path, under all steady-state conditions
with ideal specimens and machines, usually producing high frequency variations in friction. These
variations are greatest in the case of dry metals, ceramics, and hard polymers: they are the least
in the case of well-lubricated surfaces or when a metal sphere slides on a soft plastic. A sketch of
such data is shown in Figure 5.16a.
3. Inherent and repeatable variations of low frequency that are cyclical in nature due to misaligned
test machine parts, bad bearings, poorly made specimens, or vibrations transmitted from outside
the machine, etc. Figure 5.16b shows such cyclical variations superposed upon noncyclical data.
4. A long-term change due to changing conditions in the contact region of specimens. This is shown
in Figure 5.16c.
© 2001 by CRC Press LLC
FIGURE 5.16
Three forms of surface traces and read-out.
5.8.3 Interaction of Inherent Unsteady Data with the Dynamics
of a Test Machine
Unsteady data can interact with the dynamics of test machines to produce two affects: one is to alter the
dynamic content of inherent friction data, and the other is to “feed back” sliding conditions upon the
sliding pair that are different from the intended conditions.
The data from transducers in friction test machines usually contain a range of frequencies (which can
be separated by Fourier analysis). Some of these frequencies will be below and some above the several
natural frequencies of a transducer and test machine system. The frequencies that are lower than the
natural frequency of the test machine components will induce vibrations in those components in phase
with the driving frequency; those near but still below the natural frequency of the components will induce
very large amplitudes. The frequencies that are higher than the natural frequency of the machine components will induce vibrations that are 180° out of phase from the driving amplitude, which amplitudes
will increase near the natural frequencies again. This phenomenon is shown in Figure 5.17.
Several causes of machine coupled vibration are described:
1. Vibrations resulting from the coupling between the elasticity of a machine and a negative slope
of friction vs. sliding speed. When a mechanical driver within an elastic frame or machine exerts
force upon some other part of the system to slide it, the system will distort in proportion to the
resistance to slide. A “prime mover” is represented in Figure 5.18 as pulling a sliding body with
an elastic coupler or spring. Beginning with zero force in the spring, the prime mover moves,
© 2001 by CRC Press LLC
amplitude
amplitude of the
driving force
natural frequency of
vibration of the system
driving
force
frequency of the
imposed force
M
the mass vibrates in the mass vibrates 180° out
phase with the driver of phase from the driver
FIGURE 5.17 Amplitude of motion of a mass, M, subjected to a force of constant amplitude and variable frequency,
through a spring, without damping.
prime
mover
spring
FIGURE 5.18
sliding
object
Schematic sketch of the important elements of a typical elastic machine in which sliding occurs.
stretching the spring. The slider moves only when sufficient force has developed in the spring to
meet the static friction force to slide the body. After sliding begins, and assuming a negative slope
of dynamic friction vs. sliding speed, the stretched spring exerts more force upon the slider than
is required to keep the object moving, so the excess spring force accelerates the object. The object
now moves faster than the prime mover, shortening the spring beyond the point required to keep
the object moving at the speed of the prime mover: it “overshoots” and slows down. At the new
lower speed its friction increases which hastens further slowing, eventually to a speed less than
that of the prime mover. This oscillation continues as long as there is sliding in the general speed
range in which there is a negative slope of friction, as sketched in Figure 5.19. From the general
principles of mechanical dynamics, the frequency will depend on the mass of the slider, the stiffness
of the spring and on the nature of damping in the sliding interface and in the overall system.
These events take place in all noisy and vibrating brakes and clutches, for example.
Where the spring is very flexible and the slope of friction curve is steep, the slider speed may
range from zero to some finite value, and this behavior is commonly referred to as stick-slip, as
speed of the
prime mover
µ
range of speed
variation of the slider
sliding speed
FIGURE 5.19
One frictional behavior that encourages frictional oscillations.
© 2001 by CRC Press LLC
mica
ordered molecular layers
disorder due to shearing
ordered molecular layers
mica
FIGURE 5.20 A sketch of the ordered array of molecules near a solid surface, with some disorder occurring in the
center of the film due to shearing.
noted before. Most practical sliding systems that vibrate do so without the sliding velocity actually
diminishing to zero: actual stick-slip is not common in machinery. Zero velocity is, of course, a
relative term. For most people, it is a velocity that is too slow to “see.” Section 2 below discusses
motion on the molecular scale, and Section 5.11.1 discusses “breakaway” friction in µm dimensions, both of which are difficult to “see.”
The description above applies to the case where all points on an interface oscillate in phase with
each other. More likely every contact point on a surface acts quite independently, due to the
random physical interaction between asperities and due to local variations in friction, such that
there are many interacting elastic waves in the sliding surfaces, which produces the noise discernible
from all sliding and rolling solids.
The above paragraphs describe the case of dry sliding. If lubricant is present in adequate
quantity, the lubricant film is likely to be thicker at high sliding speed than at rest. This will have
the effect of increasing the (negative) slope of friction vs. sliding speed, and generate frictional
vibrations more readily or more severely.
2. Frictional oscillations have also been seen when shearing films of a solvent of a few molecules
thick between two molecularly smooth flat surfaces. When there is no sliding the atoms of the
solid impose structure or lattice order upon the molecules of the liquid, extending to three or
more molecular dimensions from each solid surface: such a system is sketched in Figure 5.20.
Shear displacement of that ordered structure requires a progressively larger force with an increase
in displacement. When the “shear strength” of the ordered solvent is reached, the lattice order
changes progressively toward the unordered liquid structure in a manner equivalent to the generation of dislocations in plastically shearing metal. The shear resistance diminishes as shear
progresses. This behavior is equivalent to a negative slope of the friction vs. velocity curve.
Whether or not frictional oscillations (stick-slip) on the scale related to order–disorder phenomenon in nanodimension-thick solvents could couple with the larger–scale phenomenon
remains to be seen. Few machines are lubricated with solvents, of course, but there is likely to be
a similar order–disorder phenomenon in hydrocarbon lubricants, particularly in the lubricants
containing molecules with branched side-chains (boundary additives). These molecules form
strong bonds with metal ions (in the oxides), which then assume the lattice order in the oxides.
The order–disorder phenomenon in these molecules will be of much larger dimension than that
of solvents, likely increasing the possibility of some synchronized vibrational modes with the
vibration frequencies in machinery.
3. A general increase in coefficient of friction with increase in normal load (or pressure) produces
the same type of behavior as that produced by a decrease in friction with an increase in sliding speed.
4. Sometimes “hot spots” may be generated on surfaces, which produce variations in friction. Hot
spots are local regions on a nominal contact region that heat and expand quickly, which propels
the regions of the sliding bodies apart momentarily. Hot spots occur because in most machines,
sliding components do not contact each other with uniform contact pressure, either because they
do not conform to each other geometrically, or because they are not held rigidly. A hot spot
© 2001 by CRC Press LLC
exacerbates the inherent nonuniformity of contact which develops increasing amplitude of vibration and variation of friction as time progresses. Clutches and brakes are subject to this type of
vibration because their surface temperatures rise quickly to rather high values.
5. Second-order effects arise from the effects of specimen waviness, for example, where the upper
specimen bounces along the plate. These impulses may have the effect of providing impulses in
both contact pressure and sliding speed, which often have their separate effects on µ.
5.9
Test Machine Design and Machine Dynamics
Likely, someday, the fundamental frictional behavior for some sliding pairs will be found, but that appears
to be a distant prospect. Until then, testing will be necessary and product design would be greatly
simplified if any convenient test machine could be used and operated in any convenient manner to
measure that friction. Sometimes useful data can be derived from a simple test machine, but not often,
and rarely for a novice. A skilled operator can often interpret laboratory data in a way that applies to
production hardware, but not always. The shape of specimens, the manner of preparing surfaces, the
manner of “break-in” at the beginning of sliding, the sliding cycles and stopping times, and several other
issues can be important and must be considered in simulative testing. Furthermore, the dynamics of a
test machine influences test results, particularly the severity and nature of the vibrations and variations
in data. These variations in turn influence the data obtained from a sliding pair to varying degrees. It is
instructive to operate two machines with identical specimens and observe the differences in the results
between the two: even two identical machines can produce different results, depending on the condition
of the bearings, the fit of the components, etc.
As to test machines, adequate simulation requires careful attention to the dynamics of data acquisition
systems as well as to machine dynamics. These are rather specialized topics beyond the scope of this
chapter, but a simple discussion of the behavior of two cantilevered force transducers will introduce the
subject of dynamics adequately for present purposes.
One common cantilevered force transducer is sketched in Figure 5.21. The cantilever is firmly fixed
to, and projects from a vertically adjustable base. The adjustable base lowers the pin into contact with
FIGURE 5.21
A simple cantilever force transducer for measuring friction.
© 2001 by CRC Press LLC
FIGURE 5.22
The hinged friction force transducer.
the disk, and is adjusted downward still more to produce a normal load between the two. The disk rotates
and applies a force, F, to the pin. The force, F, bends the cantilever, which produces a maximum strain
near the root of the cantilever. Strain gages are applied on the vertical surfaces of the cantilever. The
vertical force can also be measured by strain gages applied near the root of the cantilever, on the horizontal
(upper and lower) surfaces (not shown in Figure 5.21). One advantage of this configuration is that
specimen (vertical) loading can be varied by moving the base up or down with a servomotor. One
disadvantage is that the specimen loading will vary where the disk surface is not flat or not perpendicular
to its support shaft. (Servo systems could be devised to compensate for such machine errors.)
Another common type of cantilevered force is the hinged cantilever transducer system sketched in
Figure 5.22. The load is applied directly in some manner upon the cantilever or pin holder. Either a mass
or a force can be applied anywhere along the cantilever, or upon an extension of the cantilever beyond
the pin holder. A mass will alter the vibration characteristics of the cantilever, whereas a low-mass force,
as applied by a mass hanging on a weak spring or a force applied by an air cylinder, will have less effect.
An important feature of both designs sketched in Figures 5.21 and 5.22 is that the contact area between
the pin and specimen plate is usually located on neither the vertical nor the horizontal center line of the
transducer bar. This is seen in detail in Figure 5.23, and it is apparent that a friction force applies a
FIGURE 5.23
View of the end of a friction force transducer showing three positions of the pin.
© 2001 by CRC Press LLC
FIGURE 5.24
Sketch of a tilted transducer bar.
moment to the cantilever. When the pin is in the leading position relative to the vertical axis of the
transducer, a frictional impulse rotates the transducer counterclockwise, imparting a lifting impulse upon
the transducer, which increases the vertical load on the sliding contact region (by bending the cantilever
upward in Figure 5.21 and by accelerating a mass upward in Figure 5.22). This action constitutes a
coupling between the vertical and horizontal mode of deflection of the transducer, which exacerbates
the effect of inherent variations in friction force (depending to some extent on the match between the
spacing of inherent frictional variations and natural frequency of bar vibration). By contrast, a frictional
impulse upon a slider in the lagging position will also couple the vertical and horizontal deflection modes
but in a manner that will diminish the effect of inherent variations in friction force. When the slider is
in the middle position a small impulse would produce very little coupling. Where the contact region
between the pin and specimen plate is above the centerline of the transducer bar, the effect is the opposite
of that described above.
A second type of coupling may occur when the root of a transducer in Figure 5.21 is tilted in the
manner shown in Figure 5.24, and the stiffness in the horizontal direction is lower than that in the vertical
direction. When the disk surface moves in the direction shown, a friction force, F, will be exerted that
will bend the transducer in a direction having an upward component, by an amount dependent on the
angle ε. In the case shown, a friction force will have the effect of reducing the vertical load on the sliding
contact. When ε is in the opposite sense, a friction force will have the opposite effect.
The above discussions focus on the vibration of a flexible transducer bar alone. The influence of the
surrounding support structure and the influence of damping is a much more complicated matter,
particularly where there is damping in the contacting interface between the sliding specimens.
5.10
Tapping and Jiggling to Reduce Friction Effects
One of the practices in the use of instruments is to tap and/or jiggle to obtain accurate readings. Tapping
the face of a meter or gage probably causes the sliding surfaces in the gage to separate momentarily,
reducing friction resistance to zero. The sliding surfaces (shafts in bearings, or racks on gears) will advance
some distance before contact between the surfaces is reestablished. Continued tapping will allow the
surfaces to progress until the force to move the gage parts is reduced to zero.
© 2001 by CRC Press LLC
Jiggling is best described by using the example of a shaft advanced axially through an O-ring. Such
motion requires the application of a force to overcome friction. Rotation of the shaft also requires
overcoming friction, but rotation reduces the force required to effect axial motion. In lubricated systems
the mechanism may involve the formation of a thick fluid film between the shaft and the O-ring. In a
dry system an explanation may be given in terms of components of forces. Frictional force usually acts
in the exact opposite direction from the direction of relative motion between sliding surfaces. If the shaft
is rotated at a moderate rate, there will be very little frictional resistance to slow axial motion.
Jiggling, fiddling, and coaxing would appear to be an anachronistic practice in this age of computerbased data acquisition systems. To some extent, instruments are better and more precise than they were
only 20 years ago, but friction remains unchanged. It is instructive to tap transducer heads and other
sensors now and then, even today.
5.11
Equations and Models of Friction
There are many equations in the technical literature relating the coefficient of friction, µ, to various
material properties and sliding conditions (Meng and Ludema, 1995). Virtually all of them are empirical
in nature, having been derived from experimental data, with a constant or two to correspond with
measured values of friction. Most friction tests are done over a range of sliding speed, V, a range of
applied load, W, two or three harnesses, H, and often temperature, T. The resulting equations would be
of the form, µ = kWaVbHcTd, and the exponent on each variable is likely to have some value other than one.
Most often an equation is constructed by varying the load, for example, over some range while holding
H, V, and T constant at some convenient value, and then treating each other variable in turn. This is
done in the unrealistic expectation that the effect of each variable is independent of the others. The most
curious variable to include is temperature, as if temperature by itself has some effect. It probably influences
several other quantities in the contact region which, in turn, influences friction. Temperature, both from
ambient heating and from frictional energy, affects lubricant viscosity and oxidation rate, for example,
and the latter is likely not to be reversible.
The major value of a complete friction equation for a particular system is to characterize the functioning of that system under dynamic conditions, that is, when operating under nonsteady-state conditions. Examples of such systems are artificial skeletal joints, robot joints, automotive steering assist devices
(power steering), and computer hard disks as mentioned before. Modelers of such systems search in vain
for equations with “friction as an output.” They generally prefer linear models or apply effort to linearize
existing nonlinear models; apparently, stability of the mathematical expressions becomes an issue with
nonlinear models. A complete equation for a “new” system requires extensive testing to derive. It is highly
unlikely that an equation developed for one mechanical system can, at all, be applied to any other system.
An example of friction modeling is seen in the designing of industrial robots and automotive power
steering systems. Robot joints operate at fairly constant temperature but cover a wide range of sliding
speed, stopping, starting, reversing, etc. Where a robot is used to place an object precisely at some location
in a short time, the motor drive must ramp down at the proper rate, and this rate depends on the rate
of change of friction as sliding speed changes. In essence, the motor must be programmed to meet the
frictional resistance to be encountered in the joint as well as to accommodate the momentum (and
inertia) of all moving masses. In automotive power steering systems, control is employed to multiply the
effort of a human operator to turn the direction of a car in a predictable manner; control is made more
difficult in this case by large temperature changes and, in hydraulic systems, by the wide range of friction
in the shaft seals due to the wide ranges of hydraulic pressures required to meet highly variable turning
resistances.
Convenient representations of lubricated friction used by control designers are of the form shown in
Figure 5.25 (Armstrong-Hélouvry, 1991). The first curve shows the force required to initiate sliding. This
is a curve dominated by the elastic “wind-up” of the system between the power source and the sliding
surfaces. The second applies to the sliding condition and is divided into four regimes: static friction, a
transition from standing to the sliding condition, a level of friction at low sliding speed known as the
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friction force
friction force
static friction
time (distance)
kinetic
friction
viscous friction
sliding speed
Representation of friction in lubricated joints used by control engineers.
static friction
FIGURE 5.25
transition region between
static and kinetic friction
standing time
FIGURE 5.26
Representation of the increase in static friction with time of standing.
kinetic friction, and a regime with positive slope labeled as viscous friction. The curve in Figure 5.25 has
the appearance of a Stribeck curve with a spike at zero velocity representing static friction.
5.11.1
Static Friction
The magnitude of the static or starting friction of lubricated systems varies considerably according to
standing time, or time at rest, as shown in Figure 5.26. This is most easily, but not totally satisfactorily
explained as resulting from the squeezing out of lubricant during the standing time so that asperities
continue to come into contact over the start-up time; it is surely not connected with the change in solidto-solid adhesion forces with shear rate.
The transition from the static condition, before sufficient force is applied to slide is not expressed in
terms of sliding speed but rather in terms of displacement by control engineers. The events at start-up are
described in terms of breakaway or frictional lag. Some displacement, perhaps as large as 5 µm in the case
of lubricated metals, may be required to effect complete breakaway. The breakaway phenomenon is
described by Dahl (1987) in terms similar to the fracture of a brittle tensile specimen: two bonded asperities
shear elastically until the bond fractures between the two, and a lower friction follows. Apparently, in the
Dahl model, new points of adhesion are less firmly bonded during sliding than after standing for some
time. The Dahl model is difficult to validate but some expression of the type of the Dahl model provides
a link between force-displacement (static) behavior and force-velocity (dynamic) behavior after breakaway,
through information on the acceleration of the body. The “stopping” phenomenon is the inverse of the
breakaway phenomenon, but the magnitude of the displacement involved is usually too small to affect
robot accuracy. The magnitude is readily measurable, however, in the case of a rubber seal on a shaft.
Stick-slip is ordinarily described as involving alternate stopping and sliding of one body along another.
Close examination shows that supposed zero velocity of a sliding interface during continuous movement of
a prime mover is actually very slow sliding and just not perceptible moving. It is then interesting to speculate
what is the minimum velocity of sliding in other noisy systems such as in the case of squeaking floor boards
in an old house, or squeaking plastic panels above the instrument panel in a car, or squealing tires of cars.
© 2001 by CRC Press LLC
friction force
sliding speed
FIGURE 5.27
5.11.2
Hysteresis effect in lubricated sliding when the sliding speed varies.
The Stribeck Curve
The mixed or boundary lubrication region of the Stribeck curve (see Section 5.2) is a negative slope
which tends to induce frictional vibrations (see Sections 5.8 and 5.9). This slope can be minimized, by
a properly formulated lubricant. The lowest value of friction is taken as the kinetic friction, and to the
right the Petrov form of curve is approximated as a straight line. This reflects the behavior of a lubricated
shaft in a bearing with fixed spacing or clearance between the two, rather than the elastohydrodynamic
case where film thickness varies with sliding speed.
Most lubricated systems exhibit a hysteresis effect, that is, the friction follows a different path when
the velocity is decreasing compared with increasing velocity. A simple hysteresis curve is shown also in
Figure 5.27, but in some practical systems, the two curves are quite different from each other, producing
different vibration characteristics. Whether or not this effect is included in control algorithms depends
on the magnitude of allowed control error.
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