stle bushing 2004

8/10/2019 Stle Bushing 2004

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Tribology Transactions, 47:257-262, 2004

Copyright C Society of Tribologists and Lubrication Engineers

ISSN: 0569-8197 print / 1547-397X online

DOI: 10.1080/05698190490439175

Evolution of Wear in a Two-Dimensional Bushing

DANIEL J. DICKRELL, III and W. GREGORY SAWYER

Department of Mechanical and Aerospace Engineering

University of Florida

Gainesville, Florida 32611

A model for the evolution of wear for the shaft and bushing

for a simple two-dimensional bushing system was developed

underthe assumptions of uniform contact pressureand constant

applied load. A simplelaboratory apparatus was constructed to

test the model. Two experiments were run; one showed wear on

the shaft only and the other showed wear on the bushing only.

The results showed the predicted linear progression of wear.

KEY WORDS

Bushings; Wear

INTRODUCTION

Improvements in modeling the cycle- or time-dependent pro-

gression of wear in simple mechanisms can aid designers and en-

gineers in predicting the useful lifetimes for machines andsystems

made up of these simple components. This evolution of wear for

individual components can be approached with many different

numerical and analytical techniques, although, to date, physicaltesting remains the gold standard.

Modeling a change in part geometry for a mechanism at any

particular cycle requires knowledge of the contact conditions, tri-

bological data for the materials in contact, and accurate descrip-

tions of the current geometry. Thus, any numerical or analytical

treatment will follow the progression of the mechanisms geom-

etry from the initial conditions forward to any particular cycle of

interest. Pastexperience has shown that estimating a components

geometryatanyparticularcyclebyassumingalinearextrapolation

of the initial contact and wear data can result in gross underesti-

mates (Blanchet(1)) or overestimates (Sawyer(2)).

Successful techniques for predicting worn shapes follow the

progression of wear forward from the initial cycle. However, thisapproach is extremely numerically intensive, and only recently

have the errors associated with making periodic extrapolations

along the way been evaluated (Dickrell, et al. (3)).

Finite element methods are popular computer-aided

engineering techniques that are well utilized in many fields

of life prediction. Unfortunately, due to many difficulties sur-

Presented at the STLE 58th Annual Meeting

in New York City

April 28-May 1, 2003

Final manuscript approved January 8, 2004

Review led by Thierry Blanchet

rounding contact and updating component geometry it is not

widely used to model wear. In cases where finite element models

were coupled with wear models to tackle specific components,

the cycle-by-cycle approach was found to be successful; notably,

Hugnell, et al.(4)modeled a cam-follower contact, Maxian, et al.

(5)modeled a prosthetic hip joint, and Podra and Andersson (6)

modeled a conical spinning contact.In order to reduce the computational intensity surrounding fi-

nite element techniques, other researchers have used various nu-

merical contact models such as elastic foundations (Podra and

Andersson(7)) or a beam-on-elastic foundation (Sawyer (8)) to

both calculate contact pressure and update the surface geometry.

In other cases, the assumption of a concentrated line load and

an associated wear depth per unit line load is utilized to update

the geometry (Dickrell, et al. (3)). Finally, a distribution of point

contacts over a half-space was used to model the evolution of the

wear track shape on the disk surface during a spherically tipped

pin-on-disk test (Jiang and Arnell(9)). Modified discrete element

techniqueshavealsobeenusedbyBellandColgan(10) and Oqvist

(11), who simulated the wear of valve trains and cylinder-on-flatcontacts, respectively.

Thesecomputational approaches have to be compared andval-

idated against physical testing, which, due to the costs and time

required to do such testing may represent a significant obstacle to

further methodology development. There are surprisingly few an-

alytical solutions available. Blanchet(1)developed an expression

for the evolution in wear for a scotch-yoke mechanism, which is

a harmonic oscillator. Sawyer, et al. (12)constructed and instru-

mented this mechanism, validating the model, and demonstrated

the need for cycle-dependent wear rates. An analytical model of

the evolution in wear for a simple offset circular cam was de-

rived by Sawyer(2), and later refined and validated by Dickrell,

et al.(3). These two analytical and closed-form solutions are theonly such solutions known to the authors. It is interesting that

these two mechanisms both have sinusoidal motions, yet in the

scotch-yoke mechanism the dynamics cause increasing amounts

of wear with each cycle, while in the offset circular cam the decay-

ing spring deflections cause decreasingamounts of wearwith each

cycle.

In this article a third analytical model will be developed and

experimentally validated. This modelwill solvefor the progression

in wear for a simple two-dimensional shaft-bushing pair, in which

both the shaft and the bushing are assumed to be rigid, although

neither with infinite wear resistance.

257


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258 D. J. DICKRELL, III AND W. GREGORYSAWYER

NOMENCLATURE

Fv = external force

hn = bushing recession at cyclen

hn = change in bushing recession at cycle n

Kb = wear rate of bushing

Ks = wear rate of shaft

n = cycle number

Pn = interface pressure at cyclen

Rn = shaft radius at cyclen

Rn = change in shaft radius at cyclen

w = depth of contact into the page

= contact subtend angle

MODELING

The bushing system, which is comprised of a rotating shaft and

stationary bushing, is shown in Fig. 1. For this analysis, the shaft

rotates at a steady speed under a constant applied vertical force

Fv, and all effects of friction in the interface are neglected. The

shaft and bushing are assumed to be in contact over an angle of

2; the initial length of contact is then given by 2R, where Ris

the initial radius of the shaft.

The pressure distribution that acts through thecontactlengthis

assumed to be uniform. There are numerous contact solutions for

this two-dimensional contact, such as Johnson (13) and Persson

(14). Such solutions do not support this simple assumption, but

the reasonableness of the uniform pressure assumption may be

argued. First, regardless of the pressure distribution acting on the

shaft,each location on theshaft sweeps through thesame distribu-

tion and thus each point recedes the same amount; so an initially

circular shaftremains circular throughout. Second, the shaftleaves

an arcuate impression in the bushing through the contacting re-

gion. Since wear depth occurs normal to the sliding interface, the

assumption of uniform pressure will preserve an arcuate shape, at

the expense of not preserving a recession of points along the di-

rection of global displacement. This uniform pressure assumption

is illustrated in Fig. 2.

Other assumptions used in the wear modeling of the shaft-

bushing pairare that thesubtendangle remains constant,and the

Fig. 1Schematic drawing of a simple shaft-bushingsystem as modeled

in this article. The angle describes the arc over which contact

occurs,Fvis the applied load, and is the rotating speed.

wear rates ofthe shaft and bushing are constantand not a function

of time or cycle number. The assumption of a constant subtend

angle is particularly dubious as almost certainly any experiment

or application begins with some prescribed contact that may be

changing continuously during the life of the components.

The mathematics begin by modeling the recession of a differ-

ential element on the shaft Rn (the subscript n denotes cycle

number) that moves through the contact length 2Rn under the

uniform pressure distribution Pn. The wear rate of the shaft ma-terial is given by Ks , which is not a function of cycle number and

has units of volume lost per normal load per distance of sliding

(mm3/[Nm]). The recession of points normal to the surface along

the shaft after one cycle are most easily found by integrating along

the length of contact as shown in Eq. [1].

Rn =

2Rn0

Ks Pnds = 2Rn Pn Ks [1]

The pressure at any cycle Pnis found by balancing the applied

vertical force Fv with the integrated vertical projections of the

contact pressure, Pncos. This is shown in Eq. [2] and is solved

Fig. 2Schematic drawing of a simple shaft-bushing system where the

contact pressure, which is assumed uniform, is shown acting on

the half-bushing and shaft. As modeled, wear proceeds on each

surface where the dashed line denotes an earlier shape and the

solid lines denote a current shape.


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Evolution of Wear in a Two-Dimensional Bushing 259

TABLE 1PROGRESSION OF THESHAFTRADIUS FOR THEFIRST

THREECYCLES

Cycle Rn

1 R1 = R0 2R0 Fv Ks2wR0sin()

= R0 KsFv

w

sin()

2 R2 = R1 R1 = (R0 KsFv

w

sin())

2R1 Fv Ks2wR1sin()

= R0 2KsFv

w

sin()

3 R3 = R2 R2 = (R0 2KsFv

w

sin()) 2R2 Fv Ks2wR2sin()

= R0 3KsFv

w

sin()

n Rn = R0 nKsFv

w

sin()

in Eq. [3] following the assumption that the subtend angle is

constant during operation.

Fv = 2

0

w Rn Pncos()d [2]

Pn =Fv

2w Rnsin() [3]

The radius of the shaft at any cycle is simply a function of the

previous cycles shaft radius minus the change in shaft radius as a

result of the previous cycle; this is shown in Eq.[4]. The sequential

application and substitution of Eqs. [1], [3], and [4] (shown in

Table 1) yields a simple closed-form expression of shaft radius for

any cycle number, which is given by Eq. [5]. In this expression R0

is the initial shaft radius, and everything but the subtend angle

is a model input (the subtend angle will be addressed shortly).

Rn+1 = Rn Rn [4]

Rn = R0 nKs Fv

w

sin() [5]

A similar procedure is used to model the evolution of bushing

wear. First, the depth of wear per cycle is given by Eq. [6]. Thesliding distance for each element in contact is the circumference

of the shaft, 2 Rn.

hn =

2 Rn0

Kb Pnds = 2 Rn Pn Kb [6]

Following the techniques used previously for the shaft evolu-

tion, the recession of bushing material as a function of cycle num-

ber is given by Eq. [7]. Table 2 shows the successive substitutions

of Eqs. [3], [6], and [7] that yields Eq. [8].

hn+1 = hn +hn [7]

hn =nKb Fv

w

sin() [8]

Figure 3 shows the progression of wear of both shaft and bush-

ingas developedin Eqs. [5]and [8]. The primaryunknownquantity

TABLE 2PROGRESSION OF THE BUSHING RECESSION FOR THE FIRST

THREECYCLES

Cycle hn

1 h1 = h0 +h0 = 2R0 P0 Kb = 2 R0 Fv Kb2wR0sin()

= Kb Fv

w

sin()

2 h2 = h1 +h1 = (Kb Fv

w

sin())+

2 R1 Fv Kb2wR1sin()

= 2Kb Fv

w

sin()

3 h3 = h2 +h2 = (2Kb Fv

w

sin())+ 2 R2 Fv Kb

2wR2sin()=

3Kb Fvw

sin()

n hn = nKb Fv

w

sin()

Fig. 3Schematic drawing of thepredicted shape fora mutuallywearing

simple shaft-bushing system. A vector loop, which sets up the

scalar equationsthat will eventually define theangle, is shown.

in themodelis thesubtend angle, . Thisanglecan bederived geo-

metrically using a series of closure equations, which is illustrated

in Fig. 3 and given by Eqs. [9]-[11].

L1 = R0 + hn Rn [9]

R0sin() L2cos() Rnsin() = 0 [10]

R0cos() + L2sin() Rncos() = L1 [11]

These closure Eqs. [9]-[11] can be solved with Eqs. [5] and [8]

for. The result, Eq.[12], is a transcendental equation that cannot

be explicitly solved for . Introducing a dimensionless ratio of

wear rates (given in Eq. [13]), Eq. [12] can be rearranged into the

explicit forms shown by Eqs. [14] and [15].

1

cos () = 1 +

Kb

Ks

[12]

Kb =Kb

Ksand Ks =

Ks

Kb[13]

Kb =

1

cos() 1

[14]

Ks =

1cos()

1

[15]

These expressions are plotted using the subtend angle as the

independent variable and then inverting the axes, which creates a

graph or look-up table of subtend angle as a function of the ratio

of the wear rates. This is illustrated in Fig. 4, where the ratio of

wear rates goes from 0 to 1 and back to 0. This corresponds to the

physical situation of an infinitely wear-resistant bushing (Kb = 0),

transitioning to a mutually wearing-shaftand bushing (K

b = K

s =1), and terminating at an infinitely wear-resistant shaft (Ks = 0).

Using solutions to the subtend angle in terms of the ratio

of the wear rates, with the other geometric and system variables,

predictions in the bushing system can be made as a function of

number of cycles. Of some importance is the narrowness of the

range in which non-extreme subtend angles are predicted (i.e.,

wear rates need to be well within one order of magnitude to have

noticeable wear on each component).

EXPERIMENTAL APPARATUS

A dedicated apparatus was constructed to study the wear of a

bushing system that mimics the assumptions in modeling; namely,


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Fig. 4A graphical solution to the transcendental equation for the angle

and the corresponding worn shapes of the bushing system.

two-dimensional, steady rotational shaft speed and constant verti-

cal load. A schematic ofthe apparatus is givenin Fig.5. A series of

linear bearings with four posts was used to prescribe the smooth

linear motionof a housing assembly that contained a 13-mm-thick

bushing specimen. A 3/8 kW DC motor with a speed-reducing

Fig. 5Schematic of the experimental apparatus.

transmission and a timing belt and pulley drive system was con-

nected directly to the 19-mm-diameter shaft. A DC motor con-

troller was used to set and maintain the desired speed, which was

120 rpm for these experiments. A dead weight load of 222 N was

placed on top of the bushing housing, which was constrained to

move vertically as previously described.The bushing was carefully

prepared to have a 19-mm-diameter recess and carefully alignedwith the shaft to make a conformal 180 arc of contact (note:there

is no bottom to the bushingit is, in effect, a half-bushing).

Both the shaft and the bushing were measured using a co-

ordinate measuring machine prior to testing. This machine out-

puts a point cloud of data with manufacturer-reported uncertain-

ties on the order of 60 m. At the conclusion of the test, the

bushing system was disassembled and the surfaces were again

scanned.

In situ measurements of theslider displacementwere recorded

manually using a dial indicator that was positioned over a refer-

ence location on the housing. This measurement includes wear of

both the shaft and bushing, and, in order to decouple this, tests

were periodically interrupted and measurements of the shaft di-

ameter were made using a digital handheld caliper that has re-

ported uncertainties on the order of 30 m.

RESULTS

The results from two experiments that were run will be re-

ported and discussed in this section. The first experiment was a

self-mated brass bushing system. Both the brass shaft and brass

plate were purchasedfrom thesame provider andwere reportedly

the same 360-alloy material (61.5% Cu, 35.5% Zn, and 3.0% Pb).

The surface scans from the coordinate measuring machine for the

shaft areshownin Fig.6(a). The experimentran forapproximately

84,000 cycles. In situ measurements using thedial indicatorand di-

rect shaft measurements using the calipers are shown are shown

in Fig. 6(b).

A second experiment was run using a 360-alloy brass shaft

with a polytetra-fluoroethylene (PTFE)bushing material. Fig. 7(a)

gives thecoordinate measuringmachine scan of thePTFE bushing

after 276,000 cycles. Changes in the displacement of the housing

assembly are attributed to changes in the bushing material. This is

confirmedby three observations:(1) no measurablechanges in the

shaft, (2) agreement between the measurement from the coordi-

nate measuring machines, and (3) agreement between gravimetric

analysis and calculated volumes of material removed from the co-

ordinate measuring machine data.

DISCUSSION

The results from the brass versus PTFE experiments were ex-

actly as hypothesized. The wear rates calculated by fitting the

model to the experimental data are Kb = 8.4 104 mm3/(Nm),

which is within a range of published wear rates for PTFE. The cur-

vature in the graph of bushing wear depth versus number of cycles

(Fig. 7) suggests that there may be a cycle-dependent change in

wearrate of thePTFE.Considering thenumber of cycles run, over

250,000, such a long transient behavior for PTFE is questionable.

A second hypothesis for this curvature is that the normal load is

being reduced as thecarriage holding thePTFE bushing translates

down the four supporting shafts. Care was taken to minimize this


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Evolution of Wear in a Two-Dimensional Bushing 261

Fig. 6Results from experiments with a self-mated brass bushing sys-

tem. On the top, point cloud data collected from the shaft speci-

menusingthe coordinate measuringmachineare plotted. On the

bottom, a plot of the raw data collected during the experiment

and a curvefit with the model are shown.

effect in the design of the apparatus. The coordinate measuring

machine measurements agree extremely well with the measure-

ments made using the dial indicator, so there is likely not a large

accumulation of debris on the shaft. Although it is not clear what

causes the curvature, a large amount of PTFE debris was gen-

erated during the test, and by the end of the test a continuous

thin transfer film was present on the shaft; so perhaps an ever-

improving transferfilm protective ability was reducing the wear

rate of the PTFE. Unfortunately, no interrupted examinations of

the shaft were made and no torque measurements were made.

The results of this self-mated brass-on-brass experiment,

shown in Fig. 6, are curious. It was originally hypothesized that

the wear rates of the two brass samples should be very close inthis self-mated contact, and it was expected that wear should be

observed on both the shaft andthe bushing. However, this was not

seen in the experiment, as only the shaft showed measurable wear

forthe pair. Further, thewear rate of thebrass appearsexcessively

high (even greater than the PTFE). A series of tests were run on a

pin-on-flat reciprocating tribometer; this tribometer is described

in detail in Sawyer, et al. (15). Pin specimens of 360-alloy brass

(61.5% Cu, 35.5% Zn, 3.0% Pb) were machined out of the round

and plate stock that the shaft and bushing were created from.

Flat counterfaces were also machined out of the same stock. The

cuboidal pins measured 6.35 mm 6.35 mm 12.7 mm and the

counterface samples were made such that they could accommo-

Fig. 7Results from experiments with a brass shaft and a polytetrafluo-

roethylene bearing. On the top, point cloud data collected fromthe bushing specimen using the coordinate measuring machine

areplotted.On thebottom, a plot of theraw data collectedduring

the experiment and a curve fit with the model are shown.

date up to 40 mm of reciprocating contact. Severalspecimens were

machined to determine if grain orientation in thestock played any

part in the wear behavior.

A pneumatically applied normal load of 650 N, corresponding

to a contact pressure of approximately 16 MPa, was applied for

the duration of the test (this is a reasonable mimic of the average

contact pressure experienced during the bushing tests). A slidingspeed of 50.8 mm/s and a per-cycle sliding distance of 50.8 mm

were kept constant during testing. The total distance of sliding

was 152 m. A final mean wear rate on the counterface samples

was computed as K= 1.05 103 mm3/(Nm) for all combina-

tions of shaft and bushing material; the standard deviation was

(K) = 8.0 105 mm3/(Nm). In all combinations the counter-

face experienced significant wear while the pin wear was inconse-

quential. This is consistent with what was observed experimentally

as elements on the bushing experience continuous contact (pin)

and the elements on the shaft move though the contact zone each

cycle (counterface). The friction coefficients for all tests remained

steady at around = 0.25.


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Previous investigations on the wear properties of/ brass,

such as 360-alloy brass, have reproduciblygeneratedwear rates on

the order ofK= 104 mm3/(Nm) when sliding against a smooth

hard surface. Archard and Hirst (16) reported a wear rate of

K= 6 104 mm3/(Nm) for an / brass on an unlubricated

tool steel counterface in 1956. This wear rate is characteristically

severein terms of the wear behavior of free-machining brasses.A mild wearmode is alsoseen under certainconditions; namely,

low contact pressures and sliding speeds.

Brass wear has many influencing parameters that affect the

severity of wear: contact pressure,sliding speed, temperature, ma-

terial combination, and environment. The transition from mild to

severe wear can be seen over a wide range of these experimen-

tal conditions, but there is little to indicate that any appreciable

mild wear occurred in the self-mated brass from the experimen-

tal conditions used in pin-on-flat or bushing testing. Historically

it has been found that if contact pressures exceed some threshold

value, which may be as low as 0.2 MPa (Lancaster(17)), the real

area of contact jumps suddenly and large brass particles detach

and accumulate as an extremely hard transfer film. This trans-

ferfilm undergoes a cyclic buildup and detachment process that

generates extremely high wear rates. Friction coefficient values of

0.2-0.3 have also been seen in self-mated brass tests undergoing

severe wear (Hutchings(18)).

A hypothetical explanation of why the brass member that sees

intermittent contact is the member that undergoes severe wear is

that thestationary member (either thepin or the bushing) is likely

the site of the accumulation. The accumulation probably protects

this member at the expense of the other; these layers have been

seen previously by Lancaster(17)and have measured a hardness

of over 200 HV compared to 150 HVfor the bulk material. Debris

was observed on the bushing of these tests, as is evident in Fig. 7,

where the differences between the indicator measurements and

the shaft radius measurements are attributed to such buildup.

It is intriguing that neither experiment showed wear on both

surfaces. Oneoutcomefrom themodelingis that thesubtendangle

is extremely sensitive to the ratios in wearrate,and variations that

are orders of magnitude in wear rate are at the extremes. Thus,

in order to realize wear on both the components, the wear rates

may need to be unrealistically similar. There is some anecdotal ev-

idence that abrasive wear resulting from introduced third bodies

such as sand may cause wear on both surfaces; this was not ex-

perimentally investigated in this study, although the mathematics

developed should be applicable to such a process given sufficient

understanding of the wear mechanism.

CONCLUSIONS

1. A model for a mutually wearing shaft and bushing pair was

derived as a function of experimental parameters and cycle

number.

2. An experimental apparatus was constructed and instrumented

for dry-sliding contacts of a shaft and bushing pair. A criti-

cal parameter that determines the behavior of the system was

found to be the ratio of the wear rates of the bushing and shaft

material.

REFERENCES

(1) Blanchet, T. A. (1997), The Interactionof Wearand Dynamicsof a Simple

Mechanism,J. Tribol., 119, pp 597-599.

(2) Sawyer, W. G. (2001),Wear Predictions for a Simple-Cam Including the

Coupled Evolution of Wear and Load,Lubr. Eng., September, pp 31-36.

(3) Dickrell, D. J., Dooner, D. B. and Sawyer, W. G. (2003), The Evolution of

Geometry for a Wearing Circular Cam: Analytical and Computer Simula-

tion with Comparison to Experiment,J. Tribol., 125, pp 187-192.

(4) Hugnell, A. B.-J., Bjorklund, S. and Andersson, S. (1996), Simulation of

the Mild Wear in a Cam-Follower Contact with Follower Rotation,Wear,

199, pp 202-210.

(5) Maxian, T. A., Brown, T. D., Pedersen, D. R. and Callaghan, J. J. (1996),A Sliding-Distance-CoupledFiniteElementFormulationfor Polyethylene

Wear in Total Hip Arthroplasty,J. Biomech.,29, pp 687-692.

(6) Podra, P. and Andersson, S. (1999), Finite Element Analysis Wear Sim-

ulation of a Conical Spinning Contact Considering Surface Topography,

Wear, 224, pp 13-21.

(7) Podra, P. and Andersson, S. (1997), Wear Simulation with the Winkler

Surface Model,Wear, 207, 79-85.

(8) Sawyer, W. G. (Accepted)Surface Shape and Contact Pressure Evolution

inTwoComponentSurfaces:Applicationto Copper Chemical-Mechanical-

Polishing.Tribol. Lett.

(9) Jiang, J. and Arnell R. D. (1998) On the Running-in Behaviour of

Diamond-Like Carbon Coatings under the Ball-on-Disk Contact Geome-

try,Wear,217, pp 190-199.

(10) Bell, J. C. and Colgan, T. (1991), Pivoted-Follower Valve Train Wear:

Criteria and Modeling,Lub. Eng.,47, February, pp 114-121.

(11) Oqvist, M. (2001), Numerical Simulations of Mild Wear Using Updated

Geometry with Different Step Size Approaches,Wear

,249

, pp 6-11.(12) Sawyer, W. G., Diaz, K. I., Hamilton, M. A. and Micklos, B. (2003),Evalu-

ation of a Model for the Evolution of Wear in a Scotch-Yoke Mechanism,

J. Tribol., 125, pp 678-681.

(13) Johnson, K. L. (1985), Contact Mechanics, Cambridge University Press,

Cambridge.

(14) Persson, A., (1964), On the Stress Distributionof Cylindrical Elastic Bod-

ies in Contact,diss., Chalmers Tekniska Hogskola, Goteborg, Sweden.

(15) Sawyer, W. G., Freudenberg, K. D., Bhimaraj, P. and Schadler, L. S. (2003),

A Study on the Friction and Wear Behavior of PTFE Filled with Alumina

Nanoparticles,Wear, 254, pp 573-580.

(16) Archard,J.F.and Hirst,W.(1956), TheWearof Metals under Unlubricated

Conditions,Proc. R. Soc. Lond., Ser. A,236, pp 397-410.

(17) Lancaster, J. K. (1963),The Formation of Surface Films at the Transition

between Mild and Severe Metallic Wear,Proc. R. Soc. Lond., Ser. A,273,

pp 466-483.

(18) Hutchings, I. M. (1992), Tribology: Friction and Wear of Engineering Ma-

terials, CRC Press, Boca Raton, FL.

stle bushing 2004

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