Yugoslav J ournal of Operat ion!! Research
6 (1996), Number 1. 85- 100
,
PREVENTIVE MAINTENANCE O~' UN ITS SUB.JECT
TO TWO KINDS OF WEA R FAILURE
C. DUBOKA, Z. ARSEN IC,.J. TODOROVIC
University of Belgrade. Fnculty of.\Iechllnical Enj;ineen'np.
'rV-l1000 Belgrade. 27. Mart 80
Abstra ct . There a re mechanical systems in which the most important failures
originate from excessive wear. Wear fail ures gradually develop ar-d can tx> rather I"'RI'i1y
monitored or inspected. Design of such syst e ms usually enables one or more
adjustme nts to he performed before a failu re requiring re placement of frict ion elements
takes place. It means that such a un it has two kinds of failures : ant' requu-ing
adjustment and a not he r requiring replaceme nt. when preventive maintenance of a
frict ion mechanism comprisi ng several elements deteriorating by weur, each
cha rac t er ized by a specific wear int e ns ity. l.e. by different failure rates, is cons idered it
is a problem how to determine periods of prevent ive maintenance Sf} as to provide the
best satisfact io n of a defined cri te rion, i.e. the bes t out puts. T he p ape r presents an
approac h to the problem solut ion whe n t he avai lability or the operat ional readiness are
used as optimizat ion criterion . T he model is developed for braking systems of motor
vehicles used in large fleets and ha ving differe nt brakes in th e Iront an d t he rear
wheels .
Kl'ywordK: Preventive maintena nce, maintena nce optimization, availabi lity, operational
readiness, maintenance modelling, motor vehicle brake, .....oar failu re
I. INT RODUCTlO:-l
A procedure for opt imization of prevent ive maintena nce of u ni ts s ubject t o two
kinds of wear fail ure is presented in the paper. Friction brakes used in motor vehicles
are obse rved. Wear in such mechanis ms resu lts d irec tly from brake operation . i.e. from
bra ke a pplication . Wear failures take place wh en the brake lining' wear exceeds cr itical
value with respect to brake performance . Two differen t kinds of wear failure may
occu r : (a) a brake is worn -out partly or tb l a brake is worn-out totally . A partly wornout brake may be re newed by wear compensation i.e. by brlike adjustment while 8 to ta-
86
C. Duboka, Z. Arsenic, J . Todorovic I Preventive Maintenance ofUmts Subject
Ily worn-out brake may be renewed only by component replacement.
When preventive maintenance of a friction mechanism comprising a number of
ele ments deteriorating by wear is considered, where each element is characterized by
specific wear rate, i.e. by a diffe rent reliability fun ction, the problem is to determine the
periods of maintenance actions so as to provide the best satisfaction of a defined
criterion, i.e. to minimize total mai ntenance costs or t o maxi mize availab ility . To
op ti mize the preventive mai nt en ance of braking systems used in road vehicl es, a
mai ntenance engineer in a motor vehicle fleet has t o find the "best" solutions for the
following:
(i) t he number of brake adjustments between t wo consecutive brake component
replacements which means t he relationship between the pe riods of brake adj ust ments
and brake component replacements, and
the relationship between the pe riods of preventive maintenance for front and rear
brakes.
( ii)
2. ENGINEERING BACKGROUND
Braking of road vehicles and other moving mac hinery is an actio n by which
driver may control vehicle speed and stop it if necessary.
Braki ng of vehicles is provided by a system called braki ng system. which is
required to fulfill road safety, dependability and ot her requ irements. From the energy
point of view, braking is a process in which kinet ic energy of a vehicle is transformed.
into other kinds of energy. It is normally realized by friction forces developed between
moving and stationary parts of a brake. Wear develops duri ng each brake application. It
results directly from continuous and repeated action of friction forces developed at
friction surfaces. i.e. the surfaces between moving and stationary brake parts.
A braking system consists of seve ral subsystems and components, including a
number of brakes. The number of brakes in a vehicle is related. to the number of
wheels. In a two-axle vehicle there are four brakes, two in the front and t wo in the rear
axle.
Road safety requirements impose t hat all brakes of a vehicle must act
simu ltaneous ly and into accordance with requirements . Otherwise the vehicle stability
while braking woul d be endangered . In t he engineering sense it means that a good
"brake balance" between fro nt and rear brakes has to be provided . T he problem is t hat
front and rear brakes in a vehicle usually differ by type, kind and dimensions (in one
axl e brakes have to be seme). That is why "brake tuning" makes a rat her difficult
engineering task, not only with regard to the braking system design but to its
maintenance as well.
A good "brake tuning" requires a good wear balance between all b rakes in a
vehicle. particularly between front and rear. Since the clearance between moving and
stationary parts of a brake (when it is off t he action) has a direct influence on vehicle
stability during braking, as it is explained above, a good "brake t uning" is dependent on
the wear properties of th e frict ion pair elements, and in particul ar on the wear
properties of elements mad e of friction meterlals . brake linings or pads. Clearance
C, Duboka . Z, ArSt'n ic,,1 T odorovic / I'rl"\'t'ntl\'l' Main t enance of Unu s Subject
87
must he maintained wit hin d etermi n ed tnlerance lim its - othe rwise a brak e wou ld not
be a ble to fu lfill Its m ission . Therefore, a la rge clea ra nce result ing fro m a worn-out
brake, i.e. worn-out frict ion element ("lining") a nd ('xcpl'ding- t he prcdotcrmi r n-d
t hres hold represe nt s an importa nt and typi cal hrakr- failure req uir int::" muintcna nco.
After seve ra l adjustments . tilt' brake lining becom e worn -Hut com ph-tely a nd furt her
adjustments arc not possible. In this case a brake m ay he repa ired by re placing the
worn-out lining by a new o ne . It is i rnpor tnn t to ha ve in mind thai fro m thl'
teclmoln,...i cal point of view brake liOlng replace me nt a lways co mpri ses hrnke
adjustment , meani ng that bra ke linin g replacement ill; more com plic an-d a nd mon- ti me
co nsu m ing maintenance act ivity .
3. nn: APPHOACII TO TH~; I'HOIlLEM SOLU TION
Pr even t ive ma int enance ha Nhl'('n described in man y pape r s lind in a number
of cases it wa.... rel ated to wear fail ures la. 7, 12. 21. 221_It is so mct tmoa considered a....
the most difficult act ivity in the field of maintenance 16) . particularly because nf tilt'
d ifficulties in esta blishing: the re lations between t he co mponents reliability a nd t he
frequen cy of perform in g preventive ma inte na nce. However. tilt' he m-fits wh ich
preventive maintenance is provid in g- for pra ct icall y all technical syst ems is widely and
ge nerally accepted as a n obvious fact 17. 12, 201 . T hat 's why preventive maintenance
attract s so much a ttent io n ,
The t radit io nal method for solv ing different pre ventive main tenan ce tasks is
by modelling the most important fact ors affect ing- preventive muintcnancc 12, 4, 1:J, 181.
These models usually a rc the relati on ship bet w een periods or prevent ive maintenance
and the most important o ut put charac te rist ics : mainte nance cos t JX'r u nit ti me or
co mpo ne nts uvnilnbility. So me other op t im ization cri teria are proposed as well 11 3, 18,
191·
Because of co mplexity of the problem . modell ing is a lways based on so me
assumpt ions o r s implifi ca t ions . The q uality of a model depends wry much on t ill'
reality of the assumptions , The sa me applies to the wo rt hin ess of t hi' est imat es made by
a mode l.
To model prevent ive mai nte na nce for the problem ex plained in the
introductory sec t io n. i.e. for a brakin g syst em hllvin J,: two kinds of fa ilure : pa rt ly worn o ut a nd com ple tely warn-ou r brake linin gs. t he following questions have to he
di scussed :
•
•
•
•
the nature of fa ilures in braking syst e ms a nd tilt' s ignifica nce of wear-failu res,
the rela t ionship be t ween two ki nds or wear fa ilu re in a brake (a re they dependent
or not" ),
t he cha ract e r of wour failure occurrence or prohahility de nsity fun ct ions for hot h
kinds or we a r fa ilure, which means the charact er of the brake reliability fu net ion.
t he e ffective ness o r pre ventive mai nte nance with regard to two kinds of wear
failure, i.o. whether adjust men ts and lining re placeme nt renew the braking system.
restorin g it to the "as new" co nd it ion,
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C. Duboka, Z. Arseruc• .J Todorovic I l' reVt'ntlvc Mai ntena nce of Unu s Subjl'ct
~9
3.2. T he Relationship Betw('e n Two Kinds of Wellr Fai lure
The preve ntive maintenance of sys tems having tw o or mo re types of failu res
was discussed in several pa pe rs 13. I i . 221. T he failu res a re considered as independent
a nd not in ter related. However. the wear fai lures analyzed in this pa per tpart fy wornout a nd co mpletely worn-out linings failure ) are dependent . T hi s re sul ts fro m t he
followi ng:
•
•
•
the mech an is ms by which failures uf hoth kinds are origin ated is the same, i.e. they
bot h o rigi nat e from the same de te riorat ion process.
t he completely worn-out lining failu re occurs a fter a number of adjust ments have
bee n done, i.e . after severa l occ ur rences of pa rtly worn-out lin ing fa ilu res,
the two kinds of wear failure a rt' mut uall y exclusiv e: at 11 moment a brake ca n fail
on ly because it is ei t her partly worn-out or because it is co mpletely worn-out ; a
completely worn-out brake ca n not be pa rtly worn out at the same t ime.
3 .3 , T he C h ar-acter- of Wear Failurt·s
The expected average number of failu res ,v(T) in a cycle of le ngt h T is a very!
im portant funct ion fur olanning preventive maintenance. Fu nction N { T) depends on
the effectiveness of mainte nance act ivities . If hot h of preventive and co rrective
maintenance activities r estore the compo ne nt to the "as new" condition N (T) represents
an ord inary re newal function . If on ly prcvenuve mamtenance rest ores the component
t o the "as new" co nd ition . N\T) is a delayed re newa l process [I] .
Funct ion .V(T) depends on the character of the fail u re r at e , i.e. on the
re lia bility function . T h is dependence is very co mpl ex and t herefore there a re various
appr oaches t o its modelling. A ve ry conve nient estimat e fo r fu nct ion N<T) is gi ve n in
the for m ( 13):
N( T ) < F(t) I fW )
(1)
whe re F {t) is unreliability function . R(t) is rcliabihtv function . and t represents t ime.
Acce pt ing t hat expression (1) provides a good estima te for the avera ge number
of fa ilure s in a cycle of length T . it is necessary! t o assume t hat reliability fu nc t ions ....-ub
rega r d t o fa ilures a r c analyzed in t his paper. The comprehensive bwes ngatio n of
braking system r eliahility r eali zed in la bora t ory a nd und er se rvice co nd itions gave a
good basis for that. T hese investigations ha ve shown that wear fa ilures in vehicle
br akes represent a Wei hu ll process (8 , 10, 16, 2:J}. which can be interpreted by the
Wetbu ll di strihuti on 114.', i.e.:
(2 )
where:
RWftl - the re lia bility funct ion r ela ted to brake wear fa ilu res ,
11, Jl - the Wc ibu ll parameters .
90
C. Duboka, z. Arsenic, J. Todorovic I Preventive Mainte nan ce of Units Subject
It is worthwhile to outli ne t hat al most in al l tests t he values for the Weibull
shape parameter P lied in the interval from 2,5 t o 4, conform ing to the previous ly
explained increasing rate of wear fail ure occurrence a nd its nonstationary character.
On this basis. and having in mind that bot h kinds of wea r failure discussed
here originate from the same random process, reliability functions for t he two kinds of
wear failure ca n be expressed by means of t he Weibull probability distribution having
the same value of the shape parameter . It has been proved in a number of expe riments
and surveys. This can be expressed in the following form :
(3)
whe re ;
R (t) . the reliability funct ion re lated to partly worn-ou t li ning failures,
/1;(0 . t he reliab ility fu nction related to completely worn -oul lin ing failures,
II p' llc - the scale pa rameters for partly and completely worn-out lining failures,
respecti vely,
f\ - common shape parameter {II = Pp = Pc).
3 .4 . T he Ef fect iveness of the Preventive Maintenance of Brakes
A very important feature of maintenance is related to t he degree of a
com pone nt rene.....al. The general attitude is that both preventive and corrective
maintenance restore a component to the "as new" condition, meaning t hat t he ren ewal
process is an ordinary one [I] . From engineering practice, ho.....ever, it is well known
that maintenance in many cases does not provide a real "re newal". A component's or
sys tem's performance are poorer after maintenance then before. T he decrease in
reliability after maintenance is even more probab le, particularly wit h regard to the
residual lives (17). It means that in general it is not correct to consider mai ntenance as
action restoring a technical system t o the "as new" con dition.
However. it is not easy to check or test this st at eme nt in practical situat ions.
Most probably, it is reason that in spite of all sho rtage this assu mpt ion is usually
accepted for modelling t he maintena nce policy for solving practical maint ena nce t asks
(I I. Therefore . t he assum pt ion t hat bot h bra ke adjustment and lin ing replacements
restore brake to the "as new " condit ion is accepted for t he analyses in t his paper also.
T he target of the analysis presented he re, i.e. the development of a proced ure
which will provide improved and mo re efficient maintenance planning for front and
rear brakes in motor vehicles, s upports this assumption . Namely, for the problem
observed in this paper it is primarily important to achieve the "best" relat ionship
between necessary maintenance actions, so as to satisfy the relevant criterion . The
provisions for providing a "highe r reliability or longer component life are not in the
focu s of our attent ion. ThE' differences in residual life and reliability after subsequent
adjustments may be therefor!' ign ored.
M
C. lluboka , Z. Ar~o ic.' , J Todoro vic I l'r l"\'('o l lv,-" Ma iott>nanCl" of Un us Subject
91
3 ,5. The !\1;tintl"nanCl' Poli cy For Braking System
It is not a quest ion whether the maintenance of motor vehicles and their
com ponents has to be prevent ive. T his policy is t h e only acceptable and obvious , The
question is , however, which type of preventive mnintcnanco is the most co nvenient for
determined vehicle types and components and for specific us.a ge condit ions .
The Ro n-co ndi t io n- preventive maintenance has proven very effec tive in man)'
eases, including- mruntenunce of motor vehicles. pa rt icul a rly if used in large fleet s [B] .
It means t hat ma int e na nce ac t ion s depend Oil t he st a te of the unit , estuhlishod hy
conti nuo us monitoring or pe r iodical checking- hy mean s (If appropriate instruments .
sensors o r s im ila r.
This type of prcvcnuve maintenance is probably the best nne for ma intenance
of vehicle braking syst e ms and brakes al so , However , since brakes have an e xcept ional
impact on road sa fety, the preventive maintenance a ccording to the "ha rd-t ime limit"
concept has t o be applied as well. It means t hat adjustments and replacements of
frict io n lin in gs have to be done no t only o n the hasis of the "con dition" estuhlislu-d by
pe ri od ica l check ing, hut a lso a fter certai n intervals of t ime, The "hurd-t ime limit"
m a inte na nce means that adjustments and re placements of non failod dements have to
be done before failure occurs, i.e. before a poor condition of an clement is established .
This maintenance concept is very conveni ent from the technological vievvpoi nt a lso,
offering possibilities for higher o pera ti ona l readiness and hi gher availability, It IS
obvious t ha t for an effective maintenance it would be co nvenie nt if time or mileage
period fo r "ha rd- t ime" replaceme nt represent s a multiple of the time or mileage period
for "ha rd-ti me" brake adju stment. Thi s will be e xpla ined later.
3 .6. The Criterion of Optimization
The time or mileage periods of preventive maintenance have to he determined
so us to satisfy the requ irements com in g: out from the defined target or goal fun ction ,
Ma inte na nce model s are the best too] for so lving: th is problem , Mai nt e na nce models
give "t he best" sol ut ion for t he accepted opt im iza t ion cr iterion ,
This typical opt imization problem usually boil s down to the minimization of
total maintenance costs or to the maximization of availability (1 , 2, 1:1, 18, 20), It may
be expressed by the following :
cm = ICI'M
and
A (7'J =
(4 1
+ N(7) CCMJ/T
11' · (I'M
- N (TJiC,uJlT
(5)
whe re :
C(Tl . the total maintenance cost of component per unit t ime over ti me interval 1',
C p \ / . the average oost of preventive maintenance,
NCr, . the expected average number of co mpo ne nt failures ove r time interval T,
92
C. Duboka, Z. Arscmc, ,J Todorovic I Preve ntive Maintenance of Units Subject
ceM - t he ave rage cost of cor rec ti ve maintenance,
T · t ime in te rval,
A (Tl - the com ponent availability per unit ti me over time interval T ,
l p.,, ' the average down t ime rela ted to preventive maintenance over t ime inte rval T.
lc,,, - the ave rage down ti me re lated to correct ive ma intenance ove r t ime inte rval T ,
The cost optimi zation cr iteri on is used almost general ly but the a vailability
optimization criterion is often more conve nient for mai ntenance of various mech a nical
compon ents and sys te ms. In particular, it is the case wit h utility and public transport
systems which have to provide t he highest possible quality of se rvice meaning a very
high ava ilability . That's why t his cri teri on is accepted for t he task wh ich is el abora ted
in this paper.
4. THE PREVENTIVE l\!AINT ENAl'lCE MODEL
Although t he availability of a technical syst em over a period of time T can be
exp ressed in the following gene ral form :
(6'
where t u is lip t WI C a nd td is down time ove r a t ime per-iod T in further a nalyses a more
conven ient for m as give n by expression (5) will be applied.
Dealin g wit h two kinds of failu re, namely partly a nd completely worn-ou t
linings (using indexes p a nd c, respect ively, as marked before ) availability can be
expressed in t wo forms , each one related to one kind of fa ilure. He nce:
(7')
(7 " ,
a nd
whe re:
Ap {Tp ) a nd A c(Tc ) • availa bility related t o partly Ipl a nd complet ely (c) worn-out lin in gs
over t he time inte rvals Tp and 7'e, respect ively,
IpAt and tpM • the average down t im e related t o tw o ki nd s of prevent ive
(PKW acti olls ,e namely : adj ustment caused hy part ly (p ) worn-out
replace me nt ca used by completely (e) worn-out lini ngs . respect ively,
ICA! and tCAf • t he average down time re lated to two kinds of correcti ve
e
(C1W acti on s. namely: adjust ment caused by partly (p) worn-out
re placeme nt caused by com pletely (c) worn-out linin gs, res pectively .
Using e xpression
(1 ) ,
maintena nce
lin ings a nd
maintenance
linings and
formulae (7') a nd (7") become:
(8'1
a nd
(8",
It means that the optimal periods for preve ntive adjust me nt and replacement
a re determined by the minimi zing of the fu nct ions (8'l and (B") , i.e.:
C. Duboka. Z. Arsenic, J , Todorovic I l' n'ventlv(' Maintenance of Unus SubJt'Ct
93
(S ' ,
and
(9")
Hen ce , for a se t of brake linin gs (in nne bra ke) t here are two optimums for
preventive maintenance, one for preventive adjustme nt a nd anot her for preventive
replacement. The same applies for each bra ke. It means th at if fou r brakes in a vehicle
an' considered separately, t he expressions (9') and 19" , will give t'i~ht optimums, i.e .
eigh t preventive mainte na nce pe riods providing maximum availability with regard to
both kinds of wenr failure. Such a mainten a nce schedule wou ld obv iously he very
in efficient a nd expensive.
In reality , this s it uation is simplified by the fact th a t t wo bra kes fitted at on e
axle may be considered to be s im ilar. It means th at for a t wo-axle vehicle there are onlv
four optimums: two for prevent ive adjustment of hot h fr ont a nd rear bra kes and two f~r
preventive replacements of bot h front and rear brakes. These optimums ¢ving the
highest a vai lability wit h regard to preventive adjustment and replacement of brakes
fitted to front (f) a nd rear (r) wheel s in such a vehicle may he expressed by t hc following
notat ions :
•
Af/,(T(p ) = t he maxim um availability of front brakes (f) wit h respec t t o their
preventive adjust me nt ove r the optim um pertod of preventive adjust ment T(p
associated with partly (p) worn-out brakes,
I
•
A(c(Tfc> = the maximum a vail ability of fro nt brakes If) with respect to their
preve ntive replaceme nt over the opt im um pe r iod of preventive replacement T(c
associated with completely (cJ worn-out brakes,
• Arp(Trp J : the maximum availability of rear brakes (r) wit h res pect to t heir
preventive adjustment over the optimu m pe riod of preventive adjustment T rp
associated wi t h partly (pI worn -out brakes.
• A rc(Trcl = the ma ximu m availability of rear brakes (r) wit h respect to their
prevent ive replace me nt ove r t he optimum period of preventive replacement T rc
associated with completely (e) worn-out brakes.
Such a ma inte na nce schedule will be still complicated. inefficie nt and
expensive. It is logi cal therefore t o com bi ne some of the maintenance activities so fLS to
pe rfo r m t hem s im ulta neously in t he cou rse of the sa me do w n time. In general, it would
cont r ibute to more effective maintenance cont rol and management and w ill decrease
operational costs.
Howe ver, s ince t he rates of fai lure occurrence for front and rea r brakes are not
the same , simul taneous maintena nce wou ld mean t hat at brakes having more favo rable
reliability t he mai nt e na nce action would be performed ea rlier then needed . This may
increase the maintenance costs. Howe ve r, in the case ofhrake adjustment acti on s, this
in crease in costs will he neglunbl e . Therefore, si mul taneous adjustment of all brakes III
a vehicle is a gene ral practi ce in all mainten ance schedules .
If re placement of brakes is considered. th e premature replace me nt may cause
important adduionel e xpe nses. Simultaneous replace ment of all brakes in a vehicle may
he ra t ional on ly if relia bility of the front brake is simila r to t ha t of the rear brake.
giving ra ther small di fference between thei r mean lives . Since it usually is not the case,
maintenance sched ules generally prescr ibe the replaceme nt of brakes at each vehicle
axle to be realized separately. If such an approach is applied in the cas e of a two-axle
vehicle there will be three optim um ma intena nce periods, one for simultaneous adjust-
!H
(. Dubokn, Z AJllol·mc. J TodoroVic P rew-ntr ve ~f aml<'n Bn Cf' rof Umtll Rub,''C1
men t of all brakes lind two fur separate replncement of front and rear bru kea ,
respec t IVI'ly_
This problem may be si mplified if ce rtam relatmna between three optimal
maintenance p" rmd H would Ill' es mbbebed It may be achieved hy pllttm~ the
n-lationship hf'lWI'''1l nme intervals for rcpl.ux-mcnte lind for ndju etmenta to 1)('
eXJlfl'l'l fl..d hy I nu-gr-r. IlS explai lwd III Fil,,'tJre I lind by means 01" the follow i n~ ex presainn :
"rel'l" '" M lind 'I'rel'lp "" N
(10 /
·
1111' optimum ))NIUd of
,\rand N mtl'gN s ,
T
s rrnulta neous adjustme-nt of all front and rPM brnkce.
k
2k
WEEK
nk
+
t
ADJUSTMENT
REPlECE MENT
FRONT
3k
+
+
•
•
k N
•
k M
REPL ECE MENT
REAR
.1"
FiJ{url' I
An adrhtinnnl convenience may be achieved if the ratio between figures M a nd
N may al so be mad.. a w hole number. i.e. If the wear balance bet .....een front lind rear
brakes may he oxpressod as an i n t p~er C} , or :
MIN =
(I
(II)
5 . AN EXAMPLE
To illustrate the modr-la an example of braking sys t e ms of two-axil' publ ic
transport vehicles uM'(1 within a lurge ON't .....ill he demonst mtcd The vehicles have to
provide the hi ~h('st possible quality of service or nvuilahifity. It means that
mnintenunce pol icy explained nbcve, reprl 'st' n ti n ~ a combination of "on-co ndi ti on" and
"hard-ti nu- lim it" conrr-pta, has 10 hI' o pt im ized wit h regard t o fleet availuhihty . From
tilt' previou s invt·"tit.::at iolls under lI sa ~e co ndit io ns , charnc tenzcd by a n nverugc ve hicle
Il SIl I-:(' inu-ns uy of about 15 kill JlN hour o r uhout 2nO km per day (8,10. 2:l] it may he
nseu mod that n-liability of t hI' fro nt (J) and n-n r (r ) hrukca co n he ex pre ssed by the
following fu nctions:
C. Duboka, z , Arsenic, J . Todorovic J P reven uve Maintl'nance of Unils Subject
•
•
•
•
95
Front brake adjustment : Rfj (t) = exp (·W 3()() )3)
Fro nt brake re placement:
(I ) = exp [ -(1/1000)3 1
Rear brake adjustment: Rrp (t) = exp (-(t/600 )41
Rear brake replace ment : R rc ( t ) :: exp 1-(/12500 )4)
If,c
Besides, it will be assu med that the average time for preve ntive adjustment is
tpm :: 5h (for al l brakes in the vehicle) and the average rime for preventive
repiRcement is tp m = 50h (for o ne brake ), while corresponding average times for
correc t ive maintena~ce of brakes are tern =: ISh and tern = 150h .
With these ligu res, expressio ns ~' ) a nd (8") willcgive t he availability functions
with regard t o all four ma intenance actions . These function s are plotted in Figure 2. It
can be seen that the maximums of all four curv es are achieved fOT ve ry different ti mes
T • taking values from 156 to 1402 hou rs. It proves tha t it is difficult to design s uch
maintenance program me which would link al l fou r maintenance actions so a.s to obtain
a co m mon opt imum .
1
REAR BRAKE - ADJUSTMENT
-
J.= 337 h I A = .9808
,
,..
I
.. . ...
FRONT BRAKE ·- ADj USTMENT
T = 15tH ; A = .9555
-c .88
:i
.:
-'
en
>
-c
. .... . ..
.94
•
:5~
-
/
-
- ,
'" ,
r
I
.82
REAR BRAKE - REPLACEMENT
T = 1402 h I A = .9553
\,
\
,
FRONT BRAKE - REPLACEMENT
I
T = 519h ; A = .8774
t
,I
.76
I
f
I
.7 -'-----'--
o
-
r -500
- - , --
-
--,----
1000
1500
-
-,---2000
TIME T I h J - ..
•' i gure 2
- - - r2500
C. Duboka, Z, Arsenic, J . Todorovic I Preventive Maintenance of Units Subject
96
Th e picture is a little bit better if only adjustments of brakes on the front and
the rear whee ls are considered : t he optimum s correspond to 156 and 337 hours . In t his
CMe it see ms reasonable to perform simu ltaneous adjustments of both front and rear
brakes, after a period of time which will be th e maximum of the mult iple of two
availability functions. The plot of this new function is presented. in Figure 3., pointing
out that the common maximum for both adjustment functions is obtained for 186
hou rs, or about 30 hours above the optimu m obtained for only the front brake
adjust ment, but even 150 hours below the opti mum for the rear brake adjustment. It is
important to notice, however, t hat the differences in availability by pe rforming separate
adju stment of each brake are not very high: availability of the front brake adjustment
for T = 186 hours decrease to 0.9537 or for only 0.2%, while corresponding decrease for
rear brake adjust ment availability reaches about 10.5%.
1
,
.8
•I
--,
-c
,
,
-----------T=156 h;A= .928
,,
,
•
•,
,,
.6
,•
,,
,
,,
,,
•
,,,
1: .4 ,
,
co
,
:5
,,
~
-c
2 I'
~
t
- - -----,----, - --
0
120
240
- , - - - --,-360
480
-
-
-,600
TIME T( h l - .
Figure 3
A good reason for accepting such a "joint optimization" of brake adj ustme nt
cernes out also from tech nological reasons. It is an us ual practice in motor vehicle fleets
similar to the one under observation to organize preven tive mai ntenance in segments
wh ich corres pond to certai n time intervals, like one day, one week, a mont h or so.
Havin g in mind the usage intensity mentioned above (250 km per day in average ) the
C. Duboka, Z. Arse nic. J . Todorovic I Preventtve Main te nan ce or Un its Subject
97
optimum period for front brake adjustment corresponds to app. 2400 km and for the
rear to app. 5000 km . In other words, front brakes have to be adjusted every 10 days
and rear brakes every 20 days, or about every two and every three weeks. For a joint
optimum of 186 hours or 2800 km the adjustment has to be done every 11 days. Since in
a public tra nsport a week means in ave rage 6 working days or 1500 km , brake
adjustments have to be performed approxi mately every 2 weeks or after 3000 km or
every 200 hours of operation, It is evidently not too far from the optimu ms obtained for
the fro nt a nd t he rear brake sepa rately.
T he sit uation would be quite different if "joint optimization" of preve ntive
replaceme nt of t he front a nd rear brakes is conside red . The optimums obtained for
these acti on s a re rather far-off 5 19 a nd 1402 hou rs , as given in Figu re 2. It means that
t he replaceme nt of the fro nt brakes has to be perfor med roughly every 8.000 km or
after one month of service. whil e t hese figu res for the rear brakes are 20 .000 km or
t h ree mont hs . It results from signi ficant wear u nbal a nce bet ween fron t and rear
brakes, which originates from design reasons and different brake loading. This can be
seen even more clearly if the joi nt maximum of two derived availability fun ct ions are
considered . as it is done above. The diagram plotted in Figu re 4 shows that the joint
maximum is obtained for T = 629 hours. what is rather far away from both optimums
obtained for each functi on separately. If this joint optimum wou ld be applied . it w;11
mean that both front and rear brakes have to be re placed after roughly 40 days. For t he
front brakes the d ifference is not too large. but t he rear brakes would be replaced more
the n two ti mes earlier t hen necessary. T his is obviously quite uneconomical. T he refore
there are no real possibilit ies to combine these two preventive action s tog-et her, so as to
obt ai n a joint optimum.
,
•,
--c
.B
,
,
,
6 ~
-----
----
--, ,
,
,, ,
,
~
"
- - -- - -- --
T= 629 n : A= .82 7
~
"'
:3
,, ,
4,
,
,
,
-c
,
,
2- ,
,, ,
,
,
0
300
600
1200
900
TI M E
Figure 4
T
Ihi
-.
1500
C. Duboka, Z. Arse nic, J . Todorovic I Preventive Main tenance of Units Subject
98
However, a very posit ive effect can be achieved if the time periods for
preventive replacements are put in the relation to correspondi ng intervals for brake
adj ust ments, as it is explai ned by expressions (l0). Namely , the comparison of the
optimum obtained gives:
Tfc lTp = 5191200 = 2,60 = M
Trc lTp = 14021200 = 7,01 = N
The figures 2,60 and 7,01 have to be rounded up so as to become integers and
to give the ratio between t hem as the integer Q as well (express ion 11).
It is easy t o conclude that there are two solut ions: M can be rounded up to 2 or
3 and N to 8 or 9, giving Q = 3 or 4. In such a way the periods of preventive
replacem ent are Tfc = 400 or 600 hours and Trc = 1600 or 1800 hours.
Using the diagrams shown in Figures 2 and 3 ., these two cases are compared in
Table 1.
Tab le 1
T
T
A IT
1.600
1.800
400
600
I
0.95386
0.94924
It may be seen that t he seco nd combinat ion with M = 3 and N = 9 is better,
giving: higher availability . It means that the repl acem ent of fr ont brakes has to be done
every 6 weeks and for the rear brakes eve ry 18 weeks. Such a maintenance sched ule is
illustrated in Figu re 5. Preliminary t rial of this maintenance plan has proven this
programme .
W E EK
2468 ' 2 16 ~ ~ 28 32 ~ 40 44 48 52 ~
I 1111111 1111 111111111 II I II11 111 1I11I1 11111 1I 11II111 II II I
ADJUSTMENT
FRONT
REPL ECEMENT
FRONT
REPLECEMENT
REAR
0
0
0
0
0
0
0
0
ADJUSTM ENT
REAR
F igure 5
0
0
0
0
C. Duboka, Z. Arllt'nit , J Todorovic I Preventive Maintenance of U mts Subject
99
7. CONC LUSIONS
1. T he mechanical sys te ms in which the most important failures origi nate from
excessive wear are s pecific in several as pects. Firstly, failures originating fr om wear
may be of two kinds, one requiring adjust me nt and another requiri ng replacemen t .
Secondly, different kinds of wear failure have a direct im pact on the mainte na nce
policy, creating specific problema with regard to preventive maintenance schedules.
Therefore, the optimi zation of main tena nce of this tYJX' of engineering systems
requires a thorou gh understandin g of all releva nt factors.
2.Since t wo kind s of failure in fr iction mecha nisms used in automotive brakes origina te
from the same wear mechanism, it is possibl e to combine cor responding preventive
maintenance actions. One of the solut ions is to des ign such a braking system so as to
obtain the wear balance between front a nd rear brakes providing that intervals for
preventive adjustment a nd preventive replacement of each brakes can be expressed
also as integers. Su ch a maintenance schedu le has proved quite acceptable a nd
satisfactory .
3. lf the opti mization cr iter ion of optimal ity is availability, as it is usually acce pted for
public services and ut ility syet e me, t he optimizution has t o be achieved by successive
approximation, com paring different possible combinations .
Acknowledgment. This paper was prepared under a research project financially
supported by the Ministry for Scie nce and Technology of Serbia .
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