This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Technical Paper Wear Predictions for a SimpleCam Including the Coupled Evolution of
Wear and Load© Predictions of wear based on theforces and slip
t‘eiocities caicuiated using the unworn ﬁrst cycie
geometry will not accurateiy predict a tnechanisrn. ’s
useful! life. This is because there is a conpiing be—
tween the contact conditions and the geometry of
the components, it"i’tic'i’t are changing as they wear.
The abiiity to accurateiv predict how a mechanism
tL‘iii perform over an extended number of cycies
requires knowiedge of how the components are
changing during operation. Using an Archard’s
wear constant, a CiOSﬂiﬁJt'ftt expression describ
ing the coupled evolution of the contact toads and
wearfor a t'rircuiar—cam with aﬂatfacedfoiiower
is developed. Further, these closed ﬁrrm expres—
siottsfor food and wear, which are given in terms
of the number of com—cycles n, are nort—
dimensionaiized by the cam eccentricity, width, and
the Archarri’s wear constant. The non—dintensionai
wear and ioad are functions ofthe number ofcycies
and the nondimensional group termed wear—com—
pliance, which is the product oftheArchard 's wear
constant andr the spring constant over the cam
width. Nondimensional ciosedform equations are
oiso developed for an uncoupled evolution oj‘ge
otnetry and wear; and the predictions in aseﬁti i i f'e
are comparedfor the coupieti and uncoupied equa—
tions. The unconpied ivearpredictions always over
predict wear, and greatly over predict wear under
conditions of high spring sttﬂitess, iow wear re—
sistance, narrow cam widths and iarge nutnhers
ofcycies. It is suggested that a coupling ofthe evo
iution of geometry and contact conditions must be
accounted for when making iije predictions of
wearing mechanisms based soieiy on wear: Final manuscript approved February ‘t, 2001
Review led by Barry Riddle W. GREGORY SAWYER (Member, STLE)
University of Florida Department of Mechanical Engineering
Gainesville, Florida 32611 KEY WOFIDS Life Prediction Methods; Cams; Wear and Failure INTRODUCTION In many mcchanical systems, it is desirable to design components for
inl‘initc lil‘c. Unfortunately, such designs are practically impossible to real—
i7.c, and subsequently engineers are constrained to design for finite life.
While there are many established life prediction theories for material fail—
ure in mechanical designs (cg. fatigue. fracture, and creep) there are no
available theories that include the effects of wear. Thus, designers must
rely on experience and available rankings of material wear resistance to
guide the designs and material sclcctions. Predictions of wear based on the
forces and slip velocities calculated for an unworn mechanism may greatly
under predict or. worse yet. greatly ovcr prcdict the useful life, This is a
result of the basic coupling between thc contact conditions and the geom—
etry of the components, which are changing as they wear. if the effects of
wear could be accounted for in a simple and formulaic method. material
sclcctions could be based on function and expected life, and the mechanism’s
components could be designed to enhance function. expected life, or. po—
tentially, both function and life. The ability to accurately predict how a mechanism will perform over an
extended number ofcyclcs requires knowledge of how the components are
changing during operation. The geometric evolution of these componcnts
as a result of wear requires, at a minimum, knowledge of the complete
history of contact locations. pressures, and component wear—rates. As wear
occurs, the changes in the geometry change the kinematics, which influ—
ence the contact loads, contact locations, slip velocities, and sliding dis—
tanccs. For many simple mechanisms. closed form solutions can be devel—
oped to predict displacements, sliding velocities, and contact loads (texts
on the subject have been written by Shigley and Uicker (1995) it), Sncck
(1991) {2), and many others); for more complex mechanisms, many com
nicrcially available software packages numerically evaluate the kinematics
and dynamics (cg. ProMechanica, Adams, Working Model, and Autolcv). Over the past decade, efforts have been made to include wear within
numerical and finite element methods. These models use an Archat‘d’s wear (Continued on next page) Journal of the Society of Tribologists and Lubrication Engineers 31 (Continued from previous page) Fig. 1—A schematic drawing of a circularcam with a flat
faced follower, the angular coordinate 9, and the vectors for the radius R, eccentricity E, and po sition ,5 are shown. As the circular cam rotates, the inital centroid described by the eccentricity
vector sweeps out a circular path as shown by
the dashed line. model for the incremental wear predictions. Podra and
Andersson {3) used a finite element analysis to predict the
wear ofa hetnispherically tipped pin sliding against a plane,
and found good agreement with pin—on—disk experiments.
Podra and Andersson (4) performed a similar finite element
analysis for a conical spinning contact and compared these
results to an analytical model with favorable agreement. Podra
and Andersson (5} also simulated the wear for a sliding
sphereon—flat and cylinder—on—flat configuration, compar
ing the results of a finite element simulation to a numerical
simulation, which used an elastic foundation or Winkler
model to calculate the contact pressures. The wear predic—
tions and contact pressures calculated using the Winkler
model were in good agreement with the finite element analy—
ses. with the greatest deviation occurring during the early
sliding distances where the contacts were under their most
concentrated conditions. Flodin and Andersson (6) used the
Winkler model and numerically evaluated the mild wear oc—
curring in spur gears. allowing the pressures to evolve as the
surfaces wear. Hugnell andAndersson {7) and later Hugnell,
et al. {8) simulated the mild wear in a cam—follower contact.
allowing the shape and pressures to evolve as a function of
the number of cam rotations; these models had a non—rotat~
ing and rotating cam follower, respectively. Maxian. et al.
(till—(H) used ﬁnite element analysis with coupled evolutions
in shape and pressures for the polyethylene components in
total hip replacements. The calculated worn geometries and
the rates of material loss were in good agreement with clini—
cal observations for a variety of designs. Kurtz. et a]. H2)
extended the work of Maxian, et al. (9)41!) to include wear
on both the articulating and backside of the polyethylene
component. Sui, et al. {)3} studied the wear of
Po]ytetrafluoroelhylene lip seals using an iterative rezoning 32 procedure and finite elements to predict the worn geometry
of such seals. Barecki and Scieszka {14) numerically simu
lated the wear process and changes in the pressure distribu
tion of the friction linings in a posttype brake system. Closed form analytical solutions that allow geometries
and load to evolve for simple mechanisms are rare. The only
such solution that the author is aware of is by Blanchet U5),
who modeled the coupled evolution in geometry and load
for a simple Scotch—Yoke mechanism. This mechanism has
a crank with a rigidly attached pin, which is located some
distance away from the line of rotation, and slides within a
vertical slot connected a mass. The resulting mass position.
mass velocity. and pintslot reaction forces are sinusoidal funes
tions. As wear occurs, the slot evolves into an ellipse result
ing in larger displacement, higher velocities, and greater pint
slot reaction forces. Thus, the cyclic rate of change of the
depth of wear into the slot is a continuously increasing func—
tion with number of cycles. Through recursive application
of the equations for depth of wear and contact force, Blanchet
()5) developed a closed—form solution capable of predicting
the evolution in the pin and slot geometries as well as the
kinematics and dynamics of the Scotch Yoke as a function of
the number of crank revolutions. ANALYSIS A closed form expression describing the coupled evolu—
tion ofthe contact loads and the wear for a circular cam with
a ﬂatfaced follower (Fig. l) is developed. In this two—di—
mensional mechanism of width w, the cam is a circle of ini—
tial radius R that rotates about an origin point located a dis—
tance e away from the cam’s center. Wear of the cam due to
sliding against the flatfaced follower is considered; wear on
the follower is not included. An angular coordinate (9) rela—
tive to the initial center of the circular cam is used to deﬁne
the point of contact; 8 is also a cam—attached angular coordi
nate. The depth of wear along the cam surface is denoted by
Ami. where the subscript (8) denotes that it is a function of
the angular coordinate. A closure equation for the circular cam with a ﬂatfaced
follower is given by Hq, [1]. In this equation ,6 is the posi—
tion vector from the origin of cam rotation to the contact—
point on the flat—faced follower. e is the eccentricity vector
from the origin of cam rotation to the initial center of the circular cam. and fr is the radial vector going from the ini—
tial cam center to the contact—point. The ttnit vectors :3 and j are the horizontal and vertical unit vectors respectively, as
shown in Fig. l. p=E5+RzecosQi+esin6j+Rj [I] in this simple cam mechanism. the contact location on the follower face is described by the 9' component and the compression in the spring is described by the } component.
The angular coordinate 8 describes both the rotation of the
cam (see Fig. l) and the corresponding angular location of September 2001 LUBRICATION ENGINEERING Fig. 2—A schematic drawing of a circularcam with a flat
faced follower, after a certain number of cycles
the evolution in shape of the cam is shown. The
corresponding angular coordinate 8, which de
scribes the location of contact on the cam, and vectors for the radial distance R, eccentricity £3, and position ,6 are shown. The dashed circle corresponds to the nth cycle while the solid circle
is the profile for the (n+1) cycle. contact on the follower Fig. 2). As shown in Fig. l, 9 =
0, 21c when the eccentricity vector is horizontal and the cor—
responding contact point on the cam is at 8 + in“? (see Fig.
2). Using an Archard’s wear constant. Sawyer {16} developed
an expression for the incremental change in wear depth (dh)
for a differential element moving through a two—dimensional
linc contact with a pressure distribution given by PM. where
s is the coordinate axis through the contact. This expression
is given in Eq. [2]. and the total accumulation of depth of
wear from the entrance to the exit of the contact is given by
Eq. [3], where K is the Archard’s wear constant= L is the
length of contact, w is the width, and FD is the normal load. dh=xega or L
h. = K Pmds = —” [3]
{J ' For the circular—cam mechanism, an initial contact force
F is deﬁned for the 6 = 0 location. As the cam rotates= the
eccentricity causes this force to vary according to Eq. [4],
where k (Ntrn) is the spring constant. Using Eq. [3], the re—
sulting depth of wear along the cam surface after the ﬁrst
cycle of loading is given by Eq. [5]. F Egg:r+amw m TABLE luvANALYTICAL EXPRESSIONS FOR WEAR DEPTH AS A FUNCTION or
ANGULAR COORDINATE 8, SPRING CONSTANT k. ARCHARD'S WEAR
CONSTANT K, WIDTH w, INI'HAL CONTACT Ponce F . AND ECCENTRLCITY e
FOR THF. FIRST EIGHT CYCLES. n . iq) ......._____.__ . . .. . 0 .
l K! +eksmq!
i W
Klf.'l'ek5i"qlt—tx+2w]
2 W!
Mtge —3tht+.1s"l
3 w .
F  ' . . . .
Wt—t—‘x‘Hrssummit» p4..'
_4_.__ W “Lem...” . ' .__..___
AGE .t. , . , . . .
wow —5t K'\.'+]Jr K'n" —]U.ltKn.'+5u':
5 a. . .
WFFK‘warnuser“; +2nt=x3u"—15ixu‘ + F:ny
6 W .
I”: k _ ..___... .. I \ I
wrac —tt’K‘w+ Ell‘K'w'  iii"KM" a 351mm — Mix.3 + 7311'":
7 W
mpk‘K‘aar"x‘.u—2sx‘x‘w‘+sct‘x‘w'_itu'x‘.n I stirs1t" llama is”
8 “ A =§tf+wsnm m 9'. —
[ ’1” ‘H; As shown. greatly exaggerated, in Fig. 2, the compres
sion in the spring and the resulting contact forces are re—
duced in the following cycle (n+1) because of the wear in—
curred during the current cycle (It). The cumulative depth of
wear Am” and contact force Fm” for an arbitrary number of cycles are given in Eqslﬁ] and [7]. respectively: K AND.” : A[_3}.n— +;(F[‘_H}Ji—) [email protected]—Amﬂ) in fin.” These equations can be evaluated numerically cycle—by—
cycle or analytically starting from the expression for anzl
and the initial condition Atom] and marching forward in cycles
applying Eqs. [6] and [7] recursively. The resulting depths
of wear as a function of q for the first eight cam revolutions
are shown in Table 1. The resulting depth of wear Aim and
the contact forces Fm” can be described exactly by the sum— mations given in Eqs. [8] and [9], respectively. K F+eksint9i "" t _,tK ""‘HJ
am— [ )2 H ( l m w M, Ma —i]l w (Continued on next page) Journal of the Society of Tribologists and Lubrication Engineers 33 (Continued from previous page) [r i n 
wT—wwvrsm — anh__ M. n M _
M '__\II ‘. _ M "._ \ ‘ s. y M
I w 'x . .‘ ,l ._
" H E I": ‘3‘ a. si.
'3 ' '
I I_ R. I‘
 ~ 5‘
xi, ': .
 \I‘ 'I ll III
.gl ~3
,l E .
' 't_ \._ , i
3 ': 'i
'. 's I'.
. i
 .' 1':  r
..r._~m ;.  I Fig. 3—A plot of Eq. [16], which is the ratio of the non
dimensional wear depth predictions of the
coupled Eq. [13] and the uncoupled Eq. [15], for
values of ,8" [ram 1011 0'8 over a range in cycles
from 110la is shown. (a) linear
(b) logarithmic = (r + saksin9)[l _ " '1_’..“ _I[ 4"" 19] w Emilia—i). w The group kK/w is a non—dimensional product. of the lin—
ear spring constant and the linear wear rate; for the purpose
ofthis manuscript, it is termed “wearcompliance“ and given
the symbol The other non—dimensional groupings are
wear—depth. Eli10m: Awmle and load, F*: FKr‘ew. Thus, Eqs.
[8] and I9] can be written in a non—dimensional form, as
shown in Eqs. [I0] and [I I ]. n—I (Sr“HM : (Fe: + Sin ﬂﬁjii (_ﬁs]l’—]—EJ] [IUJ i:l_l as = {F + small — ﬂat—H)” ' I I u The series g_ll_lf"_ly:__.i;""' is recognized as the function 1'5"“ '["’r"3"I_ This greatly simpliﬁes Eqs. [IO] and I I ll, and pro—
vides closed form expressions for the depth of wear and load
as shown in Eqs. ll2l and ['13]. 34 65;.)_”=(F*+ﬁistn9) 1* l—(l—ﬁ"=)” l2j
[3 2U?“ +ﬁ*sin6)(l —l3*]" 113] DISCUSSION As visible in Eqs. [12] and Hill= the sinusoidal depen—
dence on contact force and wear predicts a maximum depth
of wear at 8 = 7:32 and a minimum at 8 : 3nf2. Sinusoidal
dependencies are also predicted if no coupling between the
wear and the subsequent loading are assumed. and, instead,
the initial loads are used to make predictions ofthe changes
in geometry and performance. The resulting equation pre—
dicts a linearly increasing depth of wear with number of
cycles: this is shown in Eq. [14] and in a non—dimensional
form in liq, [15]. Flip = (I? +t3'isin8)(I 43*)” Ml tim'uripirri _ HKLIE 8k sin 6) IV I'15l Atoll.” The fractional difference between the non—dimensional
wear predictions of the coupled and uncoupled equations,
respectively, is shown in Ed [16] and plotted vs. number of
cycles in Fig. 3 for various values of non—dimensional wear
compliance 55%» :(FO—ﬁir) l16
maaiwmi "‘8 3 m; [6M The meaningful limits on are from zero to one. The
wear compliance [3* must be greater than zero, which corre
sponds to the condition of inﬁnite wear resistance K : 0, no
spring stiffness k = 0, or inﬁnite width. The upper limit on comes from the condition where the flatfaced follower
loses contact. with the circular cam, a condition described by
F* < 0. This condition would occur first at the location 8 :
3nl2. and as shown in Eq. [ 1?]. if the cam does not lose con—
tact on the first cycle (Ff > b*)._ then the available ranges of
[W that ensure a continuous contact between the follower are
for [3* less than one. 1:" =(1~"“" —B'“)(1—ﬁ"‘]" 20 In] Ill] 3 El and [4* > then qus.  l2] and [I3] predict
positive values ofcontact force 17* and wear 6* for all values
of 8 at any number of cycles n. Both the contact force and
the cyclic rate of change of depth of wear (db‘i‘ldn) asymp—
totically approach zero as the number ofcycles approaches September 2001' LUBRICATION ENGINEERING L'L‘ Fig. 4—Plots. (a) a plot of Eq. [21], which gives the life at which
the cam wear proceeds through the center
of rotation in number of cycles nmx, over a
range of [5* from 110"3 and various values of
the ratio {so1)l{s(o1)) from 10410“ lb) an overlayplot oi the lite prediction on the
ratio of the nondimensional wear depth pre
dictions of the coupled and uncoupled equa
tions. no. There is a physical limitation on the amount of wear than
can be accommodated by the circular cam mechanism. The
wear depth can’t penetrate through the center of rotation for
the cam: the earliest occurrence would be at the 9 = 3m“?
location. This condition is described dimensionally and non—
dimensionally by Eq. [l8] where c is the ratio of the eccen—
tricity over the radius (8 : er‘R] A ,. <R—e; <——1 [181
'Jr [Q—'—],n 8 Substituting 8 2 Ball into Eq. [ 12], simplifying and then substituting this result into Eq. [18] for at“ ,_,_ gives Eq. [19']. which if satisfied ensures that. the depth of wear has not pro
ceeded through the center of rotation. (Fi—ﬁ’i);(t—(l—ﬁi)“)<%—l [19] As previously described. F ’‘ must. be greater than and
[3* must be less than one in order for the cam to remain in continuous contact with the follower. Describing in terms of lit“ by introducing a new parameter ot, where : 0t B’k
and 0t > 1, simpliﬁes [it]. [19] as shown in Eq. [20]. Addi—
tionally, this equation can be easily solved, yielding expres—
sions in terms of 0'. and E. for the critical maxi mum wear com—
pliance [if for a cam design that will survive a given num—
ber of cycles n, or the maximum number of cycles nm“ that
can be survived for a given value of wear compliance Because [if must be greater than zero and less than one, and
rim< must be greater than one, it is evident from Eq. [2]] that the ratio (E.0tl}t(e(Ctl)) must be greater than zero and less
than one. [a—l)(t—[l—ﬁi]")<l—l 120} 1 SUI—l
505—1 " 8(05—1] = [ — ; plum l ix)
2 Figure 4(a) shows the maximum life nmax over a range
of values of and varying values of the ratio (smudge.
l)); the maximum life is an extremely strong function of this
ratio and a much weaker function of As shown in Fig. 3
the uncoupled wear predictions can greatly over—predict the
amounts of wear realized under conditions of high l3* and
large numbers ofcyclcs. Equation [l6]. which is the ratio of
the fractional difference in the non—dimensional wear pre—
dictions of the coupled and uncoupled equations, is a func—
tion of wear compliance and number ofeycles, and the maxi—
mum number of cycles before failure is a function of the
ratio (ecu Uttefa— l )) and wear compliance. Thus, ratio of the
nondimensional wear prediction of the coupled and un—
coupled equations can be plotted for various values of (80:—
llttefot— l )) and cycles to failure nm. Figure 4(b) is an over—
lay plot of the fractional difference in the nondimensional
wear predictions ofthe coupled and uncoupled equations us.
both 1) the number of cycles at various values of B41, and 2)
the cycles to failure rim“x at various ratio’s of (see 1 )t(e(0t 1)).
As the ratio (err—l )tfetot— l )) decreases greater life and greater
differences between the coupled and uncoupled predictions
are realized. While the variable n‘m represents a nurnberofcycles when
wear proceeds all the way through the rotation point of the
cam. it is almost certain that the limit of acceptable perfor—
mance of the cam system will occur before this. lfthc maxi—
mum allowable amount of wear for a specific system 5*W is
deﬁned. Eq. I 2l c...
View
Full Document
 Spring '08
 MEI

Click to edit the document details