A critical analysis on friction force models in dynamical systems

Uni e sidade do Minho
Escola de Engenha ia
Uni e sidade do Minho
Escola de Engenha ia
F ancisco Miguel Gonçal es No ais
A c i ical analysis on ic ion o ce
models in dynamical sys ems
dezemb o de 2024
A c i ical analysis on ic ion o ce models in dynamical sys ems
F ancisco Miguel Gonçal es No ais
UMinho | 2024
Uni e sidade do Minho
Escola de Engenha ia
F ancisco Miguel Gonçal es No ais
A c i ical analysis on ic ion o ce models
in dynamical sys ems
Disse ação de Mes ado
Mes ado em Engenha ia Mecânica
Á ea de especialização em Conceção e Cons ução Mecânica
T abalho e e uado sob a o ien ação do
P o esso Dou o João Paulo Flo es Fe nandes
P o esso Dou o Ped o Filipe Lima Ma ques
dezemb o de 2024
ii
DIREITOS DE AUTOR E CONDIÇÕES DE UTILIZAÇÃO DO TRABALHO POR TERCEIROS
Es e é um abalho académico que pode se u ilizado po e cei os desde que espei adas as eg as e
boas p á icas in e nacionalmen e acei es, no que conce ne aos di ei os de au o e di ei os conexos.
Assim, o p esen e abalho pode se u ilizado nos e mos p e is os na licença abaixo indicada.
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p e is as no licenciamen o indicado, de e á con ac a o au o , a a és do Reposi ó iUM da Uni e sidade
do Minho.
Licença concedida aos u ilizado es des e abalho
A ibuição-NãoCome cial
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h ps://c ea i ecommons.o g/licenses/by-nc/4.0/
iii
ACKNOWLEDGEMENTS
The execu ion o his hesis in ol ed a signi ican amoun o wo k, dedica ion and esilience om
mysel o ca y ou all he asks ha I se o accomplish. Howe e , all hese e o s would ha e been in
ain wi hou he help o se e al indi iduals.
In he i s place, I would like o exp ess my mos since e and p o ound g a i ude o my supe iso s,
P o esso Paulo Flo es and P o esso Filipe Ma ques. To P o esso Paulo Flo es o he in i a ion o wo k
wi h him and o sha ing his expe ise and knowledge wi h me. To P o esso Filipe Ma ques o his
expe ise and pa ience in he coun less long mee ings and labo a o y wo k. Thank you o you iendship
and ca e ulness h oughou his wo k and o belie ing in my po en ial, e en when I el ha I was no
good enough. I was an hono o wo k wi h he bes om he bes and I hope o one day each you le el.
A special acknowledgmen o Enginee Filipe Ma ques om he Mechanical Enginee ing
Depa men wo kshops o he mul iple sugges ions o imp o e he expe imen al appa a us design and
o all he lessons augh abou manu ac u ing echniques. I would like also o hank all he indi iduals
om Labo a ó io de Ensaio de Ma e iais o le ing me use he equi ed equipmen o he expe imen al
appa a us and o all he ips gi en.
I wan o gi e a special hanks o all my iends om high school (whe e some o hem we e me
way be o e ha ) and om he “Roubocopo” g oup o all he laughs and good momen s h oughou he
yea s. I am g a e ul o you e o s o upli my spi i s du ing di icul imes.
To my gi l iend, Ma a, a huge acknowledgemen o he unwa e ing lo e, suppo and belie in
my capaci ies. Also, hank you o all you pa ience and unde s anding.
To my b o he , Fe nando, o his ue iendship and suppo , e en du ing he momen s when his
ac ions es my pa ience.
The inal and mos p o ound acknowledgmen is ese ed o my pa en s, o whom I dedica e his
wo k. I am e e nally g a e ul o hei unwa e ing suppo and lo e h oughou my li e. Fo he coun less
oppo uni ies hey ha e p o ided me, he sac i ices hey ha e made, and o hei cons an
encou agemen o become a be e pe son, bo h p o essionally and pe sonally. Thank you o he cons an
desi e o see me succeed and achie e he bes .

i
STATEMENT OF INTEGRITY
Decla o e a uado com in eg idade na elabo ação do p esen e abalho académico e con i mo que não
eco i à p á ica de plágio, nem a qualque o ma de u ilização inde ida ou alsi icação de in o mações
ou esul ados em nenhuma das e apas conducen e à sua elabo ação.
Mais decla o que conheço e que espei ei o Código de Condu a É ica da Uni e sidade do Minho.
RESUMO
A c i ical analysis on ic ion o ce models in dynamical sys ems
As sociedades a uais êm en en ado bas an es desa ios no que diz espei o à economia de
ecu sos na u ais, eciclagem e eu ilização des es mesmos ecu sos, assim como na p ocu a po no as
on es de ene gia mais sus en á eis. O con ac o e mo imen o ela i o en e as supe ícies de
componen es mecânicos são esponsá eis pela dissipação de pe cen agens conside á eis de ene gia
sob a o ma de uído, calo , en e ou as, con ibuindo pa a um desgas e acele ado e alha p ema u a
des es mesmos componen es.
A co e a modelação e p e isão do compo amen o dinâmico de componen es e mecanismos dos
mais a iados ipos de máquinas e equipamen os eco endo a e amen as compu acionais é c ucial
pa a o co e o con olo do seu uncionamen o e p olongamen o da sua ida ú il, possibili ando a
economia de ecu sos na u ais e ene gé icos que de ou a o ma se iam gas os na manu a u a de
componen es de subs i uição.
Des e modo, es e abalho cen a-se, p incipalmen e, na análise c í ica de modelos de o ça de
a i o eó icos, compa ando o desempenho compu acional des es na p e isão do compo amen o
dinâmico de sis emas mecânicos de complexidade c escen e na p esença de enómenos dissipa i os
elacionados com as o ças de a i o que se desen ol em en e os di e en es ipos de con ac o.
Depois da análise dos concei os undamen ais e eó icos associados aos enómenos de a i o são
analisadas as o mulações dos modelos de a i o es á icos e dinâmicos pa a con ac os secos,
comummen e aplicados pa a a simulação des es enómenos dissipa i os em p oblemas que se inse em
no quad o da dinâmica de sis emas mul ico po.
O p oje o de uma bancada simples e e sá il pa a a de e minação de dados ela i os a o ças de
a i o é ap esen ado com o obje i o de calib a e p e e os pa âme os numé icos associados aos
di e en es modelos de a i o exis en es na li e a u a e que melho pe mi em ep esen a as p op iedades
es u u ais, ísicas e cinemá icas de um de e minado sis ema mecânico.
O es udo da in luência dos modelos de o ça de a i o ap esen ado nes e abalho culmina na
análise de um sis ema mul ico po, onde é ei a uma análise quali a i a e quan i a i a dos e ei os des es
enómenos dissipa i os no compo amen o dinâmico da suspensão de um ca o do ipo Fo mula S uden .
Pala as-cha e: Bancada Expe imen al, Dinâmica de Sis emas Mul ico po, Modelos de A i o,
Simulação Compu acional
i
ABSTRACT
A c i ical analysis on ic ion o ce models in dynamical sys ems
Cu en socie ies ha e been acing conside able challenges ega ding he managemen o na u al
esou ces, ecycling, and he sea ch o new and mo e sus ainable ene gy sou ces. The con ac and
ela i e mo emen be ween he su aces o mechanical componen s a e esponsible o he dissipa ion
o signi ican amoun s o ene gy, which is was ed in he o m o noise, hea , among o he s, con ibu ing
o accele a ed wea and p ema u e ailu e o hese componen s.
P ope modeling and p edic ion o he dynamic beha io o componen s and mechanisms in
a ious ypes o machines and equipmen u ilizing compu a ional esou ces is c ucial o con olling hei
ope a ion and ex ending hei li espan, enabling he conse a ion o na u al and ene gy esou ces ha
would o he wise be spen on manu ac u ing eplacemen componen s.
Thus, his wo k ocuses p ima ily on he c i ical analysis o heo e ical ic ion o ce models,
compa ing hei compu a ional pe o mance in p edic ing he dynamic beha io o mechanical sys ems
o inc easing complexi y in he p esence o dissipa i e phenomena ela ed o he ic ion o ces de eloped
be ween di e en con ac ypes.
A e analyzing he undamen al and heo e ical concep s associa ed wi h ic ion phenomena,
s a ic and dynamic ic ion models o d y con ac s which a e commonly applied o simula ing hese
dissipa i e phenomena in p oblems wi hin he amewo k o mul ibody sys ems dynamics a e examined
in de ail.
The design o a simple and e sa ile expe imen al appa a us o de e mining da a ela ed o
ic ion o ces is p esen ed wi h he aim o calib a ing and p edic ing he nume ical pa ame e s associa ed
wi h di e en ic ion models ound in he li e a u e, which bes ep esen he s uc u al, physical, and
kinema ic p ope ies o a speci ic mechanical sys em.
The s udy o he in luence o ic ion o ce models p esen ed in his wo k culmina es in he
analysis o a mul ibody sys em, whe e a quali a i e and quan i a i e analysis o he e ec s o hese
dissipa i e phenomena on he dynamic beha io o he suspension sys em o a Fo mula S uden ca is
ca ied ou .
Keywo ds: Compu a ional Analysis, Expe imen al Appa a us, F ic ion Models, Mul ibody Dynamics
ii
TABLE OF CONTENTS
Acknowledgemen s ........................................................................................................... iii
Resumo ..............................................................................................................................
Abs ac ............................................................................................................................. i
Table o Con en s .............................................................................................................. ii
Lis o Abb e ia ions ........................................................................................................ xiii
Lis o Figu es .................................................................................................................. xi
Lis o Symbols ................................................................................................................. xxi
Lis o Tables ................................................................................................................ xxxii
1. In oduc ion ................................................................................................................ 1
1.1 Mo i a ion ............................................................................................................................... 2
1.2 Scope and Objec i es .............................................................................................................. 4
1.3 S a e o he A o F ic ion Fo ce Models ................................................................................. 5
1.4 O ganiza ion o he Disse a ion .............................................................................................. 9
1.5 Con ibu ions o his Wo k ..................................................................................................... 11
2. Li e a u e Re iew ...................................................................................................... 12
2.1 Mul ibody Dynamics .............................................................................................................. 12
2.2 Expe imen al Appa a us o S udy F ic ion Phenomena .......................................................... 17
2.3 In luence o F ic ion on Vehicle Suspension Sys ems ............................................................ 19
3. A B ie His o y abou F ic ion .................................................................................... 23
3.1 F ic ion in he Ea ly Human Ci iliza ions ............................................................................... 23
3.2 F ic ion S udies by Leona do da Vinci ................................................................................... 24
3.3 F ic ion S udies by Guillaume Amon ons ............................................................................... 26
3.4 F ic ion S udies by Leonha d Eule ....................................................................................... 30
3.5 F ic ion S udies by Cha les Augus Coulomb ......................................................................... 31
xi
LIST OF FIGURES
Figu e 1.1 – Final ene gy consump ion by sec o in he Eu opean Union in 2022 (Adap ed om Eu os a ,
2024) ..................................................................................................................................................... 2
Figu e 2.1 – Abs ac ep esen a ion o a gene alized mul ibody sys em. ............................................. 12
Figu e 3.1 – Rep esen a ion o an Egyp ian indi idual using a bow d ill (Dowson, 1998). .................... 23
Figu e 3.2 – T anspo a ion o an Egyp ian colossus om he omb o Tehu i-He ep, El-Be sheh (Dowson,
1998). .................................................................................................................................................. 24
Figu e 3.3 – Leona do Da Vinci’s ic ion expe imen al appa a us: (a) Sliding blocks in a ho izon al su ace
disposed in a ious con igu a ions (MacCu dy, 1955); (b) Expe imen al appa a us used o measu e
ic ion using a weigh connec ed o a sliding block (Pi enis e al., 2014). ............................................ 25
Figu e 3.4 – Expe imen al appa a us de eloped by Guillaume Amon ons o measu e ic ion. A specimen
BB is sliding along he es su ace AA while being p essed downwa ds by a CCC lexible body (
lèche
).
The sp ing connec ed o he scale is used o measu e he ic ion o ce (Hu chings, 2021). ................ 26
Figu e 3.5 – a) Se en een h cen u y wo kshop wi h wo ke s polishing glass; b) Componen s o he
wo kbench used o polish glass (Hu chings, 2021). ............................................................................. 27
Figu e 3.6 – Guillaume Amon on’s expe imen using a se o specimen blocks A ubbing agains a se o
specimen blocks B while being p essed down by he weigh body C (Se ano e al., 2015). ................ 29
Figu e 3.7 – Amon ons’ heo y on ic ion being desc ibed by he sp ing beha iou o aspe i ies.
(Desplanques, 2014). .......................................................................................................................... 29
Figu e 3.8 – Schema ic ep esen a ion o su ace oughness as a se o iangula -shaped aspe i ies in a
saw oo h pa e n by Eule . Each side o he iangula shape exhibi s an inclina ion, α, in ela ion o he
ho izon al plane (Besson, 2013). ......................................................................................................... 30
Figu e 3.9 – Adap ed Eule ’s inclined plane schema ic (Dowson, 1998).............................................. 31
Figu e 3.10 – Coulomb’s expe imen al appa a us o he s udy o sliding ic ion. Adap ed om Coulomb
(1821).................................................................................................................................................. 32
Figu e 3.11 – In e locking phenomena be ween aspe i ies su aces. Adap ed om (Coulomb, 1821). 34
Figu e 3.12 – Coulomb’s expe imen al appa a us o he s udy o olling ic ion (Coulomb, 1821). .... 34
Figu e 4.1 - Con ac be ween aspe i ies in he in e ace o wo su aces. Con ac only occu s in he do s,
which explains he di e ence be ween he eal a ea and he appa en a ea o con ac . ....................... 39
Figu e 4.2 – Schema ic ep esen a ion o sec ioned aspe i ies, co esponding o he eal a ea o con ac ,
and he appa en a ea o con ac . ........................................................................................................ 39

x
Figu e 4.3 – Simple ep esen a ion o he con ac and plas ic de o ma ion o aspe i ies (a) ini ial s a ic
posi ion (b) de o ma ion o aspe i ies due o he exis ence o a angen ial eloci y,
. Adap ed om
G een (1955). ...................................................................................................................................... 40
Figu e 4.4 – Aspe i ies o a ough su ace in con ac wi h a smoo h su ace o he applied load W. An
inc ease in he alue o W leads o he de o ma ion o he con ac ing aspe i ies and mo e aspe i ies come
in o con ac wi h he smoo h su ace, inc easing he o al eal con ac a ea, A. ................................... 41
Figu e 4.5 – Fo ces ac ing on a block placed in an inclined plane. ...................................................... 42
Figu e 4.6 – Inclined plane posi ioned a di e en angles o inclina ion. In he bo om igh scheme, he
block s a s sliding along he plane. ..................................................................................................... 43
Figu e 4.7 – G aph o he a ia ion o ic ion o ce be ween s a ic and kine ic s a es as a unc ion o he
applied ex e nal o ce
. ..................................................................................................................... 44
Figu e 4.8 – G aphical ep esen a ion o he Coulomb d y ic ion o ce model as a unc ion o angen ial
eloci y. ................................................................................................................................................ 45
Figu e 4.9 – Mac oscopic sliding mo ion o a block and zoom exhibi ing he mic oscopic phenomenon
ha explains sliding ic ion as elas ic and plas ic de o ma ion o aspe i ies. ........................................ 46
Figu e 4.10 – Rolling ic ion o a disc in a plane a) s a ic si ua ion and p essu e dis ibu ion b) kine ic
si ua ion due o he applica ion o he o ce . ...................................................................................... 47
Figu e 5.1 – G aphical ep esen a ion o he Coulomb d y ic ion model exhibi ing he s a ic ic ion
(s ic ion) phenomena and he Coulomb ic ion o ce le el o non-null alues o angen ial ela i e eloci y,
. ...................................................................................................................................................... 52
Figu e 5.2 – G aphical ep esen a ion o he Coulomb ic ion model conside ing iscous e ec . ........ 53
Figu e 5.3 – G aphical ep esen a ion o he Coulomb ic ion model conside ing he S ibeck e ec . . 54
Figu e 5.4 – Coulomb ic ion o ce model wi h iscous and S ibeck e ec s. ...................................... 56
Figu e 5.5 – F ic ion o ce,
, a ia ion as a unc ion o ime ,
, associa ed wi h he s ick-slip
phenomena o a) low eloci ies, b) high ela i e eloci ies. .................................................................. 57
Figu e 5.6 – P e-sliding displacemen and associa ion wi h he mic o-slip and mac o-slip phenomena.58
Figu e 5.7 – G aphical ep esen a ion o he ic ional lag phenomena. ............................................... 59
Figu e 5.8 – Theo e ical schema iza ion o he non- e e sibili y o ic ion o ce. In he di ec ion om 1 o
2, he sys em is in an accele a ion phase, while om 3 o 4 is in a decele a ion phase. ...................... 61
Figu e 6.1 – S a ic ic ion o ce models di ided by models wi h s ic ion and wi hou s ic ion ca ego ies.
............................................................................................................................................................ 64
x i
Figu e 6.2 – Linea egula iza ion o he Coulomb ic ion model p oposed by Be na d (1974). .......... 65
Figu e 6.3 – G aphical ep esen a ion o he Th el all egula iza ion s a egy o he Coulomb ic ion
model. .................................................................................................................................................. 66
Figu e 6.4 – Rooney and De a i’s s a ic ic ion model g aphical ep esen a ion. ................................. 68
Figu e 6.5 – G aphical ep esen a ion o Amb ósio’s egula iza ion app oach o Coulomb ic ion model.
............................................................................................................................................................ 69
Figu e 6.6 – G aphical ep esen a ion o he Ka nopp s a ic ic ion o ce model. ................................ 70
Figu e 6.7 – G aphical ep esen a ion o he Bengisu and Akay s a ic ic ion o ce model. ................. 71
Figu e 6.8 – Ande sson e al. s a ic ic ion o ce model g aphical ep esen a ion. ............................... 75
Figu e 6.9 – G aphical ep esen a ion o Aw ejcewicz e al. s a ic ic ion o ce model. ........................ 78
Figu e 6.10 – G aphical ep esen a ion o he linea s a ic ic ion o ce model wi h s ic ion. ............... 81
Figu e 6.11 – Schema ic ep esen a ion o he one deg ee o eedom benchma k p oblem. .............. 83
Figu e 6.12 – Main window o he compu a ional code c ea ed in MATLAB o s udy he benchma k
p oblem. .............................................................................................................................................. 84
Figu e 6.13 – F ic ion o ce models selec ion window o he MATLAB code c ea ed. ........................... 84
Figu e 6.14 – S a ic ic ion o ce models wi hou s ic ion window om he MATLAB compu a ional code
o de ine he speci ic pa ame e s o each model. ................................................................................. 85
Figu e 6.15 – G aphical esul s o he benchma k p oblem o he s a ic ic ion o ce models wi hou
s ic ion a) Posi ion [m] s Time [s] g aph b) Rela i e eloci y [m/s] s Time [s] g aph c) Accele a ion
[m/s2] s Time [s] g aph d) F ic ion o ce [N] s Time [s] g aph using he ode45 sol e . ..................... 87
Figu e 6.16 – S a ic ic ion o ce models wi h s ic ion window om he MATLAB compu a ional code o
de ine he speci ic pa ame e s o each model. ..................................................................................... 90
Figu e 6.17 – G aphical esul s o he benchma k p oblem o he s a ic ic ion o ce models wi h s ic ion
a) Posi ion [m] s Time [s] g aph b) Rela i e eloci y [m/s] s Time [s] g aph c) Accele a ion [m/s2] s
Time [s] g aph d) F ic ion o ce [N] s Time [s] g aph using he ode45 sol e . .................................... 92
Figu e 7.1 – Analyzed dynamic ic ion o ce models. .......................................................................... 97
Figu e 7.2 – Analogy o b is le elas ic and plas ic de o ma ion wi h ma e ial’s beha io a) b i le ma e ial
b) duc ile ma e ial. Adap ed om Dahl (1968). .................................................................................... 99
Figu e 7.3 – Schema ic ep esen a ion o b is le de lec ion and loca ion o he bond be ween aspe i ies.
.......................................................................................................................................................... 101
x ii
Figu e 7.4 – Physical in e p e a ion o b is le in e ac ion be ween su aces and b is le de lec ion, z, which
is in luenced by he b is le s i ness and damping, ep esen ed by he sp ing and he dampe elemen s,
espec i ely. ....................................................................................................................................... 104
Figu e 7.5 – a) Schema ic ep esen a ion o an aspe i y wi h d being he dis ance be ween e e ence
planes o con ac ing su aces and h he ela i e heigh o a di e en e e ence plane b) Aspe i y’s
de o ma ion, δ, due o angen ial ela i e mo ion. .............................................................................. 106
Figu e 7.6 – Physical ep esen a ion o he elas o-plas ic dynamic ic ion model whe e he displacemen
x o he sliding body is di ided in o an elas ic displacemen po ion z and a plas ic displacemen w. 109
Figu e 7.7 – Physical in e p e a ion o he Gene alized Maxwell Slip ic ion model as he pa allel
in e ac ion o N elemen a y single s a e ic ion models. .................................................................... 111
Figu e 7.8 – Con ac be ween o di e en bodies (Body 1 and Body 2) in wo di e en ins an s,
and
+
. .............................................................................................................................................. 116
Figu e 7.9 – a) Rep esen a ion o maximum s a ic
(𝑧s max), and kine ic (𝑧k max) de lec ions o a gi en
ac ing o ce 𝐹 in a ce ain ime ins an and b) o ano he ime ins an gi en by 𝑡 + ∆𝑡, whe e he b is le
de lec ion shows an expansion and o a ion a ound he common angen ial plane. ........................... 116
Figu e 7.10 – Dynamic ic ion o ce models window om he MATLAB compu a ional code o de ine he
speci ic pa ame e s o each model..................................................................................................... 119
Figu e 7.11 – G aphical esul s o he benchma k p oblem o he dynamic ic ion o ce models a)
Posi ion [m] s Time [s] g aph b) Rela i e eloci y [m/s] s Time [s] g aph c) Accele a ion [m/s2] s Time
[s] g aph d) F ic ion o ce [N] s Time [s] g aph using he ode45 sol e . ........................................... 121
Figu e 7.12 – Zoomed egion o he ic ion o ce s ime g aph. ....................................................... 122
Figu e 7.13 – G aph o ic ion o ce [N] as a unc ion o ela i e eloci y [m/s] o dynamic ic ion models.
.......................................................................................................................................................... 123
Figu e 7.14 – G aph o ic ion o ce [N] as a unc ion o displacemen [m] o dynamic ic ion models.
.......................................................................................................................................................... 124
Figu e 8.1 – CAD model o he expe imen al wo kbench designed in SolidWo ks. ............................. 128
Figu e 8.2 – Real-wo ld manu ac u ed wo kbench o ic ion o ce da a acquisi ion. ......................... 129
Figu e 8.3 – F ee body diag am o he slide du ing a o wa d mo ion cycle. ..................................... 129
Figu e 8.4 – Elec onic equipmen u ilized o acqui e analog o ce da a om he load cell. ............... 130
Figu e 8.5 – Illumina ion LED and came a u ilized o he slide ’s acking p ocedu e. ...................... 131
Figu e 8.6 – Diag am o he di e en da a sou ces, equipmen s and used so wa es o ob ain ic ion
o ce da a. .......................................................................................................................................... 132
x iii
Figu e 8.7 – Displacemen p o ile o he slide o he conduc ed dynamic es s. ............................... 133
Figu e 8.8 – A e age ic ion o ce- ime expe imen al da a ob ained om he ecip oca ing mo ion es .
.......................................................................................................................................................... 134
Figu e 8.9 – A e age ic ion o ce-displacemen expe imen al da a ob ained om he ecip oca ing
mo ion es . ........................................................................................................................................ 134
Figu e 8.10 – A e age ic ion o ce- ela i e eloci y esul s ob ained om he ecip oca ing mo ion es .
.......................................................................................................................................................... 135
Figu e 8.11 – Compa ison o he li e a u e and op imized pa ame e s o he LuG e ic ion o ce model
ela i ely o he expe imen al da a as a unc ion o a) displacemen b) ela i e eloci y. ..................... 141
Figu e 9.1 – Comple e CAD model o he FSUM p o o ype ca o 2023. ............................................ 145
Figu e 9.2 – Subsys ems o he FSUM ca mul ibody model a) chassis b) s ee ing sys em................ 145
Figu e 9.3 – Suspension subsys ems o he FSUM ca mul ibody model a) on le b) ea le . ....... 146
Figu e 9.4 – Di e en s ages o o a ion o he B yan angles con en ion. a) Re e ence global sys em; b)
Fi s o a ion c) Second o a ion d) Thi d o a ion. .............................................................................. 148
Figu e 9.5 – Flowcha o he de e mina ion o Eule pa ame e s om B yan angles con en ion. ..... 150
Figu e 9.6 – Real-wo ld kinema ic join s used o connec he di e en bodies o he on le suspension
subsys em.......................................................................................................................................... 151
Figu e 9.7 – Rep esen a ion o he mul ibody dynamics compu a ional app oach o sol e he o e
cons ained p oblem in a double wishbone A-a m ype suspension a) Sphe ical join s in bo h A-a m
moun ing poin s o he chassis ha gene a e an o e cons ained model b) Re olu e join ha sol es he
o e cons ained p oblem while being kinema ically equi alen . ......................................................... 152
Figu e 9.8 – Replacemen o eal-wo ld join s wi h kinema ically equi alen join s in he shock abso be
a) Real wo ld e olu e join s on bo h ends o he SA b) Uni e sal join s eplacemen on bo h ends o he
SA. ..................................................................................................................................................... 153
Figu e 9.9 – Real-wo ld kinema ic join s used o connec he di e en bodies o he le ea suspension.
.......................................................................................................................................................... 153
Figu e 9.10 – Real-wo ld kinema ic join s connec ing he di e en bodies ha compose he s ee ing
subsys em.......................................................................................................................................... 154
Figu e 9.11 – Deg ees o eedom associa ed o he ca 's chassis – 3 ansla ions and 3 o a ions. .. 155
Figu e 9.12 – Ti e es ing machine (Adap ed om (Calspan, 2019). ................................................. 156
Figu e 9.13 – SAE i e o ces and momen s axis sys em. .................................................................. 158
Figu e 9.14 – Inpu s and ou pu s o he nonlinea empi ical Magic Fo mula Ti e Model. ................... 159
xix
Figu e 9.15 – DNM RCP-2S shock abso be u ilized o he suspension subsys em o he FSUM ca . 162
Figu e 9.16 – Sec ion iew o a downhill bicycle shock abso be . The a ows indica e he c i ical con ac
zones whe e he ela i e mo ion be ween su aces gene a es ic ion................................................. 162
Figu e 9.17 – Gene ic schema ic o a kinema ic cylind ical join and allowed deg ees o eedom. .... 163
Figu e 9.18 – Cylind ical join wi h eac ion o ces and momen s de e mined om he Lag ange
mul iplie s ec o ................................................................................................................................ 165
Figu e 9.19 – Schema ic sec ion iew o a shock abso be wi h he possible con ac poin s conside ed.
.......................................................................................................................................................... 167
Figu e 9.20 – Schema ic o he SA p ojec ed in he ajnj plane and he esul an eac ion o ces and
momen s............................................................................................................................................ 168
Figu e 9.21 – Schema ic o he SA p ojec ed in he mjaj plane and he co esponding eac ion o ces and
momen s............................................................................................................................................ 169
Figu e 9.22 – F on and ea sp ings cu es conside ing bumps ops. ................................................ 175
Figu e 9.23 – T ack layou and dimensions o se e e lane-change maneu e acco ding o ISO 3888-2.
.......................................................................................................................................................... 177
Figu e 9.24 – a) S ee ing wheel inpu angles de ined o he se e e lane-change maneu e a he ini ial
speed o 72 km/h b) T ajec o y o he cen e o g a i y o he ca du ing he maneu e . ................... 178
Figu e 9.25 – P elimina y analysis o he FSUM ca beha io du ing he obs acle a oidance maneu e a)
Ti e e ical o ces on all ou wheels b) Sp ing de o ma ion on all ou shock abso be s o he ca . .. 178
Figu e 9.26 – Compa ison o he shock abso be o ce o he on le wheel o he FSUM ca conside ing
an ideal no ic ion scena io and conside ing ic ion o ces calcula ed by di e en models................ 180
Figu e 9.27 – De ailed iews o he o al o ce ac ing on he shock abso be du ing he e asi e maneu e
o a a) ebounding phase b) comp ession phase. ............................................................................. 180
Figu e 9.28 – To al ic ion o ce ac ing on he on le shock abso be du ing he e asi e maneu e .
.......................................................................................................................................................... 181
Figu e 9.29 – Ve ical displacemen o 40 mm applied o bo h on and ea wheels on he ou -pos e
ig wi h a

ad phase delay. ............................................................................................................. 183
Figu e 9.30 – Ti e modeling as a se o h ee sp ing-dampe elemen s o he ou -pos e ig simula ion.
.......................................................................................................................................................... 184
Figu e 9.31 – To al shock abso be o ce o he on le wheel du ing he ou -pos e ig es . ........ 185
Figu e 9.32 – De ailed iews o he shock abso be s mo ion du ing he ou -pos e ig simula ion a)
Rebounding phase b) Comp ession phase. ........................................................................................ 186

xx
Figu e 9.33 – To al ic ion o ce e i ied in he on le shock abso be du ing he ou -pos e ig
simula ions. ....................................................................................................................................... 187
Figu e 9.34 – To al ic ion o ce as unc ion o he ela i e angen ial eloci y be ween he shock abso be
bodies du ing he ou -pos e ig es . ................................................................................................. 188
xxi
LIST OF SYMBOLS
Chap e 2
Symbol
SI Uni
Desc ip ion
DoF
n
-
Numbe o Deg ees o F eedom
b
n
-
Numbe o Bodies
i
q
-
Gene alized coo dina e ec o o i-index body
i
-
T ansla ional coo dina es o i-index body in global coo dina es
i
p
-
Eule pa ame e s ec o o i-index body

ad
Angle o o a ion
e
u
-
Uni ec o
i
-
Gene alized eloci ies ec o o i-index body
i
-
Vec o o ansla ional eloci ies
i
-
Vec o o ansla ional accele a ions
i

-
Vec o o angula eloci ies
i

-
Vec o o angula accele a ions
( )
, q
-
Se o holonomic cons ain s
s
Time a iable

-
Veloci y cons ain equa ions in ma ix o m
D
-
Jacobian ma ix

-
Righ -hand side o he eloci ies cons ain equa ions

-
Accele a ion cons ain equa ions
i
-
Gene alized accele a ions ec o

-
Righ -hand side o he accele a ion cons ain equa ions

-
Mass ma ix
g
-
Gene alized o ces and gy oscopic momen s

-
Vec o o Lag ange mul iplie s

-
Veloci y eedback con ol pa ame e o Baumga e s abiliza ion me hod
xxii

-
Posi ion eedback con ol pa ame e o Baumga e s abiliza ion me hod
Chap e 3
Symbol
SI Uni
Desc ip ion

ad
Inclina ion angle
N
Ex e nal o ce
g
N
G a i y o ce
Chap e 4
Symbol
SI Uni
Desc ip ion
i
N
In insic ic ional o ce
a
N
Fo ce equi ed o shea adhe ed junc ions
d
N
Fo ce associa ed wi h de o ma ion ene gy
m/s
Rela i e angen ial eloci y
A
m2
To al eal a ea o con ac
W
N
No mal load
N
Ex e nal o ce
g
N
G a i y o ce
n
N
No mal o ce
N
F ic ion o ce
m
kg
Mass o he block
g
m/s2
G a i y accele a ion

ad
Inclina ion angle

-
F ic ion coe icien
k

-
Kine ic ic ion coe icien
s

-
S a ic ic ion coe ic
e
N
Resul an o ex e nal o ces applied
C
N
Coulomb ic ion o ce magni ude
M
N·m
Rolling esis ance momen
xxiii
Chap e 5
Symbol
SI Uni
Desc ip ion
k

-
Kine ic ic ion coe icien
s

-
S a ic ic ion coe icien
s
N
S a ic ic ion o ce magni ude
C
N
Coulomb ic ion o ce magni ude
m/s
Rela i e angen ial eloci y
N
Viscous ic ion o ce

-
Non-linea eloci y dependen a iable
( )
-
A bi a y eloci y dependen ic ion unc ion
S
m/s
S ibeck eloci y

-
Cons an ac o de ined o con ol he shape o he S ibeck cu e

-
Fac o dependen on he geome y o con ac ing su aces

-
Speci ic cu e i ing pa ame e

-
Speci ic cu e i ing pa ame e
s
Time
k
N/m
Cons an slope o he elas ic ange o ic ion o ce-displacemen
ela ionship
s
m
Displacemen

s
Small ime in e al
u
N
F ic ion o ce beha io o he uppe cu e accele a ion phase
l
N
F ic ion o ce beha io o he lowe cu e decele a ion phase
a
-
Pa ame e o adjus he accele a ion phase slope
b
-
Pa ame e o adjus he decele a ion phase slope
xxx
c i
c
N·s/m
C i ical damping
m
kg
Ti e mass
i
a
-
Longi udinal uni ec o
j
m
-
Uni ec o
j
n
-
Uni ec o
d
-
Vec o de ined be ween
i
P
and
j
P
poin s

-
Se o cons ain equa ions
g
-
Vec o o ex e nal gene alized o ces
M
-
Mass ma ix
-
Accele a ion ec o
T
D
-
T anspose o he Jacobian ma ix

-
Vec o o Lag ance mul iplie s
ai
,
mj
,
sP
i
,
sP
j
,
d
-
Skew-symme ic ma ices o he espec i e ec o s
1 2 3 4
, , ,    
-
Reac ion o ces
( )
i

-
Gene alized ic ion o ces ec o ob ained om eac ion o ces
1C
P
-
Poin con ac 1
2C
P
-
Poin con ac 2
d
-
No m o ec o
d
ii
P
-
Auxilia poin in i-index body
jj
P
-
Auxilia poin in j-index body
ii
d
m
Immu able dimension on i-index body
jj
d
m
Immu able dimension on j-index body
m/s
Tangen ial ela i e eloci y
N CP
N
No mal o ce in con ac poin
CP
N
F ic ion o ce in con ac poin
CP
n
N·m
F ic ion o que in con ac poin

xxxi
s
k
N/m
Sp ing s i ness
pl
N
P eload
bs
k
N/m
Bums op s i ness

m
Sp ing de o ma ion
d
c
N·s/m
Damping coe icien

m/s
Time a e o sp ing de o ma ion


-
Baumga e s abiliza ion me hod pa ame e s
s
k
-
Vec o o sp ing s i nesses
 
  
  
-
Vec o o sp ings de o ma ions
d
c
-
Vec o o damping cons an s
 
  
  
-
Vec o o ime- a e o sp ing de o ma ion
 
T
i i i
  
-
Di ec ion o ac ua ion o sp ing-dampe o ces in local
coo dina es
xxxii
LIST OF TABLES
Table 1.1 – Ene gy expended o o e come ic ional o ces (Adap ed om (Holmbe g & E demi , 2017).
.............................................................................................................................................................. 3
Table 2.1 – Quan i ica ion o ic ion o ce in suspension sys ems. ...................................................... 22
Table 4.1 – Summa y o he mos ele an ic ion laws. Adap ed om Mo e al. (2009). .................... 38
Table 4.2 – Fac o s in luencing ic ional beha iou (Adap ed om Blau (2001)) ................................. 48
Table 6.1 – Benchma k p oblem simula ion pa ame e s. .................................................................... 83
Table 6.2 – S a ic ic ion o ce models wi hou s ic ion speci ic pa ame e s. ....................................... 85
Table 6.3 – Sol e and a e age simula ion ime o he s a ic ic ion o ce models wi hou s ic ion. .... 89
Table 6.4 - S a ic ic ion o ce models wi h s ic ion speci ic pa ame e s. ............................................. 90
Table 6.5 – Sol e and a e age simula ion ime o he s a ic ic ion o ce models wi h s ic ion. ......... 94
Table 7.1 – Dynamic ic ion o ce models speci ic pa ame e s. ........................................................ 119
Table 7.2 – Sol e and a e age simula ion ime o he dynamic ic ion o ce models. ..................... 125
Table 8.1 – Op imized pa ame e s i ed o he LuG e model by he MATLAB gene ic algo i hm. ....... 139
Table 8.2 – Compa ison o he pe cen di e ence be ween li e a u e alues and op imiza ion i ed alues
o he LuG e ic ion o ce model....................................................................................................... 139
Table 9.1 – Calcula ion o he o al DoF o he FSUM ca mul ibody model........................................ 155
Table 9.2 – Ti e s uc u al and geome ic inpu da a ......................................................................... 160
Table 9.3 – S a ic and kine ic ic ion coe icien s o con ac poin s 1 and 2 o he shock abso be (Axsom,
2023; Giesbe s, 2012). ..................................................................................................................... 171
Table 9.4 – F on and ea shock abso be s and bump s ops pa ame e s. ........................................ 175
Table 9.5 – Sol e and s abiliza ion me hod op ions o he equa ions o mo ion. .............................. 176
Table 9.6 – Pa ame e s o he ic ion o ce models u ilized in he FSUM ca shock abso be s. ....... 179
Table 9.7 – Compa ison o he simula ion ime o he e asi e maneu e o he ideal non- ic ion scena io
and he di e en simula ions conside ing s a ic and dynamic ic ion o ce models. Values inside b acke s
co espond o quan i ies no malized o he lowes alue. ................................................................... 182
Table 9.8 – Compa ison o he simula ion ime and numbe o unc ion e alua ions o he ou -pos e ig
es . Values inside b acke s co espond o quan i ies no malized o he lowes alue. ........................ 188
1
1. INTRODUCTION
The apid ad ancemen in compu a ional powe o e ecen decades has ans o med he design
and alida ion p ocesses o mechanical sys ems. The g owing ma ke compe i i eness demands
op imized p oduc de elopmen while minimizing cos s, esou ces, and ime. These cons ain s ende
adi ional ial-and-e o p ocedu es imp ac ical due o hei ine iciencies. Addi ionally, he inc easing
complexi y o mechanical sys ems has exace ba ed he challenges o pe o ming pu ely analy ical
analyses. Consequen ly, Compu e -Aided Enginee ing (CAE) has eme ged as an essen ial ool in mode n
enginee ing p ac ices, acili a ing enhanced eliabili y and e iciency ac oss he p oduc de elopmen
cycle.
CAE ools ha e become indispensable o hei abili y o pe o m simula ions and analyses ha
encompass mul idisciplina y pe spec i es, enabling enginee s o achie e highe pe o mance le els in
sys em design. This shi om expe imen al o ial-based me hods o compu a ional app oaches e lec s
he demand o in eg a ed and ealis ic simula ions ha conside all ac o s signi ican ly impac ing he
pe o mance o a p oduc . These me hodologies ha e become c ucial ac oss all enginee ing domains,
pa icula ly in ields like mul ibody sys em dynamics and i s applica ion o a eas such as au omo i e,
obo ics, ae onau ical and ae ospace, which a e cha ac e ized by hei subs an ial economic and
echnological ele ance.
As he ield ad ances, i inc easingly necessi a es he inco po a ion o complex phenomena such
as ic ion, which can signi ican ly in luence dynamic beha io . F ic ional o ces play a pi o al ole in
de e mining sys em pe o mance, ye accu a ely modeling hese o ces emains a challenge due o hei
nonlinea and complex na u e. Unde s anding he in luence o ic ional phenomena is he e o e c ucial
o achie ing ealis ic and eliable simula ions in dynamical sys ems.
This wo k aims o explo e and e alua e a ious ic ion o ce models employed in he simula ion o
mechanical sys ems. By c i ically analyzing hei applicabili y, limi a ions, and impac on sys em
dynamics, i is possible o enhance he ideli y o dynamic simula ions and add ess he g owing demands
o mode n enginee ing challenges.
2
1.1 Mo i a ion
The p ima y aim o his s udy is o in es iga e he impac o ic ion modeling on he dynamic
beha io o mechanical sys ems. F ic ion is commonly desc ibed as a dissipa i e o ce ha ac s as a
esis ance be ween wo con ac ing su aces ha exhibi a ela i e angen ial mo ion be ween hem.
The inc easing e olu ion and de elopmen o mode n socie ies highly elies in indus ial ac i i ies
ha can only ope a e when anspo a ion and powe gene a ion sec o s exis alongside. These sec o s
in ol e he mo emen o people and a wide a ie y o ma e ials, u ilizing di e se machines and
mechanical sys ems composed o nume ous mo ing componen s wi h in e ac ing su aces. The smoo h,
eliable, and du able ope a ion o such machines is c i ically dependen on he e ec i e con ol o ic ion
a hese in e ac ing su aces (Holmbe g & E demi , 2017).
The dissipa i e na u e o ic ion o ces indica es ha hese o ces lead o a loss o ene gy ha in
he majo i y o cases is no eco e able. Figu e 1.1 shows he inal ene gy consump ion da a by sec o in
he Eu opean Union in 2022 om Eu os a (2024).
Figu e 1.1 – Final ene gy consump ion by sec o in he Eu opean Union in 2022 (Adap ed om Eu os a , 2024)
Holmbe g & E demi (2017) es ima ed om he ene gy consumed in a pape mill and in he mining
indus y ha he o e all ene gy consump ion o o e come ic ion in indus y is oughly 20% o he o al
supplied ene gy. In he anspo a ion sec o , he e ec s o ic ion in ene gy consump ion we e calcula ed
wi h g ea de ail o he passenge ehicles, ucks and buses, eaching a alue o a ound 32% o o al
ene gy usage jus o o e come ic ion. Al hough he e was no enough a ailable da a abou he a ia ion
and ma ine sec o s, i was es ima ed ha he sha e o ene gy was ed o o e come ic ion is abou 10%
4%
25%
31%
13%
27%
Final Ene gy Consump ion by Sec o (EU, 2022)
O he
Indus y
T anspo
Se ices
Households
3
and 20%, espec i ely, which leads o an es ima e 30% o o al supplied ene gy consump ion o o e come
ic ion in he anspo a ion sec o (Holmbe g & E demi , 2017).
In wha conce ns he se ices and households ca ego y, he e is a wide ange o ene gy con e sion
and u iliza ion echnologies, such as HVAC sys ems, ligh ing, household appliances and business
equipmen . The mechanical sys ems which need ene gy and hus ic ion can impac hei pe o mance
a e mainly en ila ion sys ems, ans, pumps, e c., whe e an es ima e o 10% o he o al ene gy is used
o o e come ic ion (Holmbe g & E demi , 2017). In a e age, conside ing he economic sec o s explo ed,
a o al o 20% o he global ene gy use is u ilized o o e come ic ion. Table 1.1 summa izes he
pe cen age o o al ene gy ha is used o o e come ic ion in he economic sec o s conside ed.
Table 1.1 – Ene gy expended o o e come ic ional o ces (Adap ed om (Holmbe g & E demi , 2017).
Economic sec o
Pe cen age o o al ene gy use o
o e come ic ion [%]
Indus y
20
T anspo a ion
30
Se ices and Households
10
A e age
20
Cu en ly, signi ican e o s a e ocused on de eloping mo e ene gy-e icien ehicles and
machines, d i en no only by economic conside a ions, bu also by he need o mee CO2 emission
educ ion a ge s se by he Kyo o P o ocol. In ce ain coun ies, hese e o s a e u he mo i a ed by he
desi e o a oid subs an ial go e nmen -imposed inancial penal ies (Holmbe g & E demi , 2017).
Al hough ic ion is commonly seen as an undesi able phenomenon, he e a e se e al examples o
i s u ili y ole in e e yday li e. Some examples a e ela ed o he ic ion o ces gene a ed in he con ac
pa ch o au omobile i es wi h he g ound, he high ic ion o ces needed in ca s disk b akes, he high
ic ion o ces associa ed wi h clu ch- ype ansmissions, he ic ion be ween he soles o shoes and he
g ound, o he ic ion o ce ha allows a obo g ippe o pick and place objec s. On he o he hand, he e
a e se e al applica ions whe e ic ion is undesi able and should be minimized such as bea ings and
o he machine elemen s, since i s dissipa i e e ec s cause empe a u e ise and o he phenomena ha
can lead o excessi e wea and ul ima ely o he p ema u e and/o unexpec ed ailu e o mechanical
componen s and sys ems.

4
The la es echnological ad ancemen s ha can educe ic ion a e associa ed wi h he use o new
syn he ic lub ican s, which con ain an i-wea and an i- ic ion addi i es, ha pe o m be e han adi ional
mine al oil lub ican s and new ma e ials and new su ace ea men s ha educe ic ion be ween
con ac ing su aces, as well as he in oduc ion o supe -ha d and sel -lub ica ing pa icles wi hin he
ma e ial’s s uc u e using addi i e manu ac u ing echniques (Holmbe g & E demi , 2017).
A key echnology ha allows o be e comp ehend, model and p edic ic ional beha io be ween
con ac ing bodies is he use o compu a ional me hods. The u iliza ion o mul iscale in eg a ed ma e ial
modelling and simula ion based on sophis ica ed compu e codes, which can in eg a e a ious me hods
such as ini e elemen analysis and mul ibody dynamics, pe mi s he p edic ion and co ec ion o he
dynamical beha io o mechanical sys ems conside ing ic ion phenomena. By simula ing he beha io
o a sys em in a i ual en i onmen , i is possible o educe mone a y expenses bo h in designing, building
and es ing p o o ypes, as well as du ing he sys em’s use ul li e, since he inal design should be
op imized in e ms o ene gy consump ion o ake in o accoun ic ion o ces, and by knowing how many
mo e cycles he componen /sys em has le un il ailu e, conside ing ic ion phenomena. Compu a ional
modeling o ic ion phenomena has pa amoun impo ance in he design o con ol sys ems, because
using a sui able ic ion o ce model, he sys em’s con olle can compensa e o he e ec s o ic ion
wi hou equi ing high gain con ol loops (Canudas De Wi and Lischinsky, 1995).
To summa ize, he p esen ed wo k is mo i a ed by iden i ying ic ion phenomena and he in luence
ha ic ion o ces ha e in he dynamic beha io o mechanical sys ems h ough compu a ional and
expe imen al app oaches in d y con ac s.
1.2 Scope and Objec i es
The main scope o his wo k cen e s on he c i ical analysis and e alua ion o ic ion o ce models
in dynamic mechanical sys ems. F ic ion, a nonlinea and mul i ace ed phenomenon, plays a pi o al ole
in in luencing he beha io o mechanical sys ems.
Accu a ely modeling and unde s anding ic ion is essen ial o de eloping simula ions ha e lec
eal-wo ld dynamics, enabling enginee s o op imize sys em pe o mance, du abili y, and e iciency. This
wo k del es in o he heo e ical ounda ions, p ac ical implemen a ions, and expe imen al alida ions o
a ious ic ion models, aiming o b idge he gap be ween heo e ical insigh s and hei p ac ical
applica ions.
5
In o de o explo e he di e en dimensions in which he c i ical analysis o ic ion o ce models is
di ided, i is o pa amoun impo ance o es ablish a di ision be ween he asks equi ed o analyze he
impac o he dissipa i e e ec s o ic ion in he dynamic beha io o mechanical sys ems in a sys ema ic
manne . The subsequen opics indica e he objec i es ha mus be ul illed:
• p esen a b ie e iew o he mos impo an con ibu ions h oughou he cen u ies o he
scien i ic knowledge o ic ion;
• p esen and explo e in de ail he di e en phenomena associa ed wi h ic ion;
• e iew he mos u ilized d y ic ion o ce models in li e a u e in he con ex o mul ibody
dynamics simula ions;
• de elop an in ui i e and simple in e ace o es di e en s a ic and dynamic ic ion o ce
models wi hin he con ex o a simple one deg ee o eedom benchma k p oblem;
• design and manu ac u e a simple and e sa ile expe imen al appa a us o ob ain ic ion
o ce da a;
• de elop an op imiza ion ou ine in MATLAB o p edic and calib a e ic ion o ce model
pa ame e s;
• de elop a igid mul ibody model o he 2023 Fo mula S uden o Uni e si y o Minho
(FSUM) ca p o o ype, comp ising he modeling o i e-g ound con ac u ilizing he Magic
Fo mula (MF) i e model calib a ed wi h expe imen al da a and subjec he ob ained model
in a a ie y o dynamic es o assess he in luence o inco po a ing ic ion phenomena in
he beha io o he suspension subsys em.
Th oughou he di e en chap e s and sec ions o his wo k is possible o e i y he
accomplishmen s o he a o emen ioned opics.
1.3 S a e o he A o F ic ion Fo ce Models
F ic ion is a o ce ha opposes he ela i e mo ion be ween wo su aces in con ac . I encompasses
a ious phenomena, including d y ic ion, which occu s be ween unlub ica ed su aces, and lub ica ed
ic ion, which in ol es su aces wi h a lub ica ing laye . Each o hese ic ional in e ac ions is
cha ac e ized by dis inc mechanisms and beha io s ha in luence he esis ance o mo ion.
The impo ance o s udying ic ion phenomena was ecognized se e al cen u ies ago, being he
i s wo ks de eloped by Da Vinci, ollowed by Amon ons and Coulomb (Dowson, 1998). The Coulomb
ic ion model is a ounda ional model o desc ibing ic ion and se es as he basis o many subsequen
6
ic ion models. I is a ela i ely simple app oach, equi ing only a single pa ame e , he coe icien o
ic ion, o cha ac e ize he ic ional o ce. Howe e , he Coulomb model has signi ican limi a ions,
pa icula ly i s inabili y o cap u e many complex ic ional phenomena. One o i s main p oblems is he
discon inui y in he ic ion o ce a null ela i e eloci y be ween he con ac ing bodies. This discon inui y
poses a majo challenge in dynamic simula ions, as i in oduces nume ical ins abili ies.
Fu he esea ch in o ic ional beha io has sugges ed ha ic ion o ces a e g ea e when an
objec is a es (Mo in, 1833; Rabinowicz, 1951). This led o he di e en ia ion o wo dis inc ic ion
coe icien s: one o s a ic ic ion and ano he o kine ic ic ion. These a ia ions in ic ional o ce
con ibu e o he occu ence o he s ick-slip phenomenon (Rabinowicz, 1957, 1961).
The con inuous p og ess made o be e unde s and and cha ac e ize ic ion led o he
iden i ica ion o se e al complex phenomena such as iscous and S ibeck e ec s, p e-sliding
displacemen , mic o-slip and ic ional lag (Ande sson e al., 2007; Olsson e al., 1998). Se e al ic ion
models ha e been p oposed in li e a u e. They can be di ided in o mul iple ca ego ies based on he
capabili y o modeling ce ain phenomena, complexi y o simply by ch onological o de o appea ance.
The mos common c i e ion used o di e en ia e ic ion models consis s o dynamic and s a ic models
(Ande sson e al., 2007a; Ma ques e al., 2016; Olsson e al., 1998; Pennes ì e al., 2016). Usually,
s a ic ic ion models a e simple in e ms o ma hema ical o mula ions and do no ha e he abili y o
cap u e complex phenomena, being su icien ly accu a e in desc ibing he beha io o ic ion o ces a
s eady s a e. Dynamic ic ion models a e capable o modeling complex phenomena due o he use o
ex a s a e a iables, which al hough inc eases he accu acy o he ic ion o ce esul s, no mally
p oduces mo e compu a ional e o and less e icien simula ion imes.
Di e en wo ks a e ound in li e a u e ega ding he di e en phenomena associa ed wi h ic ion
and e iew wo ks conca ena ing he s a e o he a s a ic and dynamic ic ion o ce models. Olsson e
al. (1998) p esen ed a e iew o he ecognized complex ic ion phenomena and he ic ion models o
in e es o au oma ic con ol. Rega ding ic ion phenomena, s a ic ic ion (s ic ion) is iden i ied as being
he ic ion o ce le el eached be o e a sys em ini ia es mac oscopic mo ion, being he b eakaway o ce
he le el o o ce equi ed o lea e he s a ic s a e. The mic oscopic mo ion e i ied be ween he ully s a ic
s a e and he b eakaway o ce le el is desc ibed as p e-sliding displacemen (Olsson e al., 1998). The
ic ional lag phenomenon is associa ed wi h he hys e esis ela ionship e i ied be ween ic ion o ce and
eloci y whe e o dec easing eloci ies he ic ion o ce is lowe han o inc easing eloci ies. This highly
non-linea phenomenon is one o he main easons ha dynamic ic ion o ces a e equi ed o desc ibe
he physical beha io o con ac ing su aces (Olsson e al., 1998). The s a ic ic ion o ce models
7
p esen ed a e mainly based on he Coulomb ic ion model conside ing iscous and S ibeck e ec s as
well as s ic ion. The only s a ic egula ized app oaches p esen ed o ic ion modeling a e he Ka nopp
and he A ms ong ic ion models. Fo he dynamic ca ego y, Dahl, B is le Model, Rese In eg a o ,
Bliman and So ine and LuG e models a e p esen ed in a comp ehensi e way. The LuG e ic ion o ce
model is u he analyzed by compa ing he nume ic esul s wi h expe imen al da a ob ained om a
labo a o y es ig.
Ande sson e al. (2007) p esen ed a ious ic ion models ailo ed o di e en sliding con ac
condi ions, including d y, bounda y, and mixed lub ica ion scena ios. The models analyzed include he
Coulomb ic ion model, which is widely used bu o en inadequa e o accu a ely ep esen ing ic ion
beha io ; he iscous ic ion model, which simpli ies simula ions bu may no e lec eal condi ions; and
he S ibeck ic ion model, which accoun s o lub ica ion e ec s and a ying ic ion wi h sliding speed.
Addi ionally, he pape in oduces combined models, such as he Coulomb and iscous ic ion
model, which aim o mi iga e he limi a ions o indi idual models. The Dankowicz and Dahl models a e
also discussed, ocusing on hei applicabili y in con ol enginee ing and hei ep esen a ion o ic ion as
a unc ion o displacemen . The au ho s emphasize he impo ance o conside ing mic o-slip phenomena
and he s ochas ic na u e o ic ion, sugges ing ha ic ion can be modeled as a s ochas ic p ocess
in luenced by he in e ac ions o su ace aspe i ies. Rega ding he expe imen al appa a us, he au ho s
u ilized a one-deg ee-o - eedom (1 DoF) dynamic sys em o s udy he beha io o di e en ic ion models
h ough nume ical simula ions conduc ed in MATLAB.
Ma ques e al. (2016) p o ides a e y comple e e iew and compa ison o a ious ic ion o ce
models o mul ibody mechanical sys ems in a d y sliding egime. The au ho s ca ego ize ic ion models
in o wo p ima y g oups: s a ic models, which desc ibe s eady-s a e ic ion beha io s, and dynamic
models, which inco po a e addi ional s a e a iables o cap u e ime-dependen ic ion e ec s. Complex
phenomena such as s ick-slip, iscous ic ion, and he S ibeck e ec a e also desc ibed in a
comp ehensi e manne wi hin he ic ion o ce models ha conside hem. A compa a i e analysis o he
compu a ional e iciency o he di e en ic ion o ce models was conduc ed o he ode45 and ode15s
sol e s o MATLAB.
The basic Coulomb model se es as a ounda ion, desc ibing ic ion as a o ce p opo ional o
he no mal o ce ha ac s in he opposi e di ec ion o he ela i e mo ion be ween con ac ing bodies. Due
o i s limi a ions, a ian s such as models wi h iscous and S ibeck componen s we e de eloped o
add ess speed-dependen and ansi ional ic ion phenomena. Dynamic models, like he LuG e model,
14
and
i
p
ep esen s he ec o o Eule pa ame e s o body i, which can be w i en in he ollowing o m
 
T
T
0 1 2 3 cos cos
22
ie
e e e e


==


pu
(2.4)
whe e

ep esen s he angle h ough which he body mus o a e abou a unique axis, gi en by he uni
ec o
e
u
o align i s e e ence ame wi h he global e e ence ame. Al hough using Eule pa ame e s
equi es ou coo dina es o desc ibe he o ien a ion o he body, he sys em e ains only h ee
independen coo dina es due o he cons ain ha
i
p
emains a uni ec o .
Rega ding gene alized eloci ies and accele a ions, he p oblem o singula posi ions does no
exis . Thus, he gene alized eloci ies o a igid body can be w i en as
 
T
i i i
= 
(2.5)
in which
i
is he ec o o he ansla ional eloci ies and
i

is he ec o o angula eloci ies o body
i. By di e en ia ing Equa ion (2.5), he gene alized accele a ions ec o is ob ained being de ined as
 
T
i i i
= 
(2.6)
whe e
i
is he ec o o he ansla ional accele a ions and
i

is he ec o o angula accele a ions.
The deg ee o accu acy wan ed o a ce ain dynamic analysis dic a es he ype o join s modeled
o simula e he connec ions be ween he di e en bodies o a mul ibody sys em ha cons ain hei
ela i e mo ion. Two main ca ego ies o join s can be dis inguished in he amewo k o mul ibody
dynamics: ideal join s and impe ec join s. The ideal join s a e simply kinema ic ela ions de ined be ween
he connec ed bodies ha eso on algeb aic equa ions ha cons ain he mo ion h ough ec o s,
coo dina es o poin s, e c. The impe ec join s a e usually based on con ac -impac e en s and can model
mo e complex phenomena such as clea ance, ic ion and wea . Despi e he enhanced ealis ic beha io
o impe ec join s, hei compu a ional e iciency is no mally lowe compa ed o he simple kinema ic
join s.
Ideal join s a e ma hema ically de ined by kinema ic cons ain equa ions, whe e he numbe o
independen equa ions ep esen s he numbe o cons ained deg ees o eedom. Kinema ic cons ain s
a e classi ied as holonomic o nonholonomic, depending on whe he hey can be in eg a ed o o m a
posi ion cons ain o no , espec i ely (Flo es, 2015). Addi ionally, cons ain s a e ca ego ized as

15
heonomic i hey explici ly depend on ime o scle onomic i no (Flo es, 2015). The e o e, in a
cons ained mul ibody sys em, he se o holonomic cons ain equa ions can be exp essed as ollows
( )
, =q0
(2.7)
whe e
q
deno es he ec o o he gene alized coo dina es o he sys em and
is he ime a iable. The
eloci ies cons ain equa ions a e ob ained om di e en ia ing Equa ion (2.7) in o de o ime and a e
ep esen ed in ma ix o m as
−=D 0
(2.8)
in which
D
is he Jacobian ma ix o he cons ain equa ions,
ep esen s he ec o o he gene alized
eloci ies o he sys em and

ep esen s he igh -hand side o he eloci ies cons ain equa ions. The
second de i a i e o Equa ion (2.7) wi h espec o ime yields he cons ain equa ions a accele a ion
le el de ined as
=−D 0 
(2.9)
whe e
is he ec o o gene alized accele a ions and

is he igh -hand side ec o o he accele a ion
cons ain equa ions.
In his wo k, he equa ions o mo ion o he mul ibody sys em a e de i ed acco ding o New on-
Eule ’s me hodology. The New on-Eule equa ions es ablish a ela ion be ween he mo ion o he cen e
o mass o he igid bodies and he ex e nal and eac ion o ces and momen s o he kinema ic cons ain s,
which a e o mula ed as
T
M +D = g
(2.10)
whe e
M
ep esen s he mass ma ix o he sys em which encompasses bo h he mass and he ine ia
enso o he sys em’s bodies and he ec o
g
deno es he ec o o he ex e nal gene alized o ces and
gy oscopic momen s. The e m
T
D
includes he eac ion o ces and momen s due o he kinema ic
join s in he global coo dina e sys em, in which

deno es he ec o o he Lag ange mul iplie s, whe e
i s size is equal o he numbe o holonomic cons ain s, wi h each elemen ep esen ing he eac ion
o ce o momen associa ed wi h a speci ic kinema ic cons ain .
In dynamic analysis, a unique solu ion is ob ained when he algeb aic cons ain equa ions a he
accele a ion le el a e conside ed simul aneously wi h he di e en ial equa ions o mo ion (Flo es, 2015).
The ma ix o m o he dynamic equa ions o mo ion o a mul ibody sys em can be ob ained by in oducing
16
Equa ion (2.9) in Equa ion (2.10) which esul s in he ollowing sys em o di e en ial algeb aic equa ions
(DAE)
T

   
=
   

   

g
MD
D0
(2.11)
ha is hen sol ed o accele a ions ec o ,
, and Lag ange mul iplie s ec o ,

. A each in eg a ion
ime s ep, he accele a ion ec o ,
, along wi h he eloci y ec o ,
, is in eg a ed o compu e he
sys em’s eloci ies and posi ions o he subsequen ime s ep. This p ocess is i e a i ely epea ed un il
he inal analysis ime is eached (Flo es, 2015).
Sol ing Equa ion (2.11) de e mines he accele a ions o he sys em a each ime s ep, bu does
no ensu e ha he posi ion and eloci y cons ain s a e sa is ied, since he cons ain s a bo h he posi ion
and eloci y le els a e no explici ly included in he equa ions o mo ion. Consequen ly, e o s in oduced
h ough nume ical in eg a ion can lead o iola ions o bo h posi ion and eloci y cons ain equa ions. To
add ess cons ain iola ions, se e al me hods a e a ailable, ca ego ized in o h ee main app oaches:
cons ain s abiliza ion echniques, coo dina e pa i ioning me hods, and di ec co ec ion o mula ions,
which a e well de ailed and analyzed in li e a u e (Ma ques e al., 2017).
In his wo k, he Baumga e s abiliza ion echnique (Baumga e, 1972) was in oduced in he
dynamic equa ions o mo ion in o de o keep he iola ion o he cons ain s unde accep able alues. I
should be no ed ha he cons ain s abiliza ion app oaches do no ully elimina e he iola ions bu allows
o keep hei alues unde con ol. In his cons ain s abiliza ion app oach, he accele a ion cons ain
equa ions a e modi ied o include wo addi ional e ms ha p o ide eedback con ol o co ec iola ions
o he posi ion and eloci y cons ain s. The sys em o equa ions o Equa ion (2.11) can hen be ew i en
as
T
2


   
=
   

   

g
MD
D0  −   − 
(2.12)
in which

and

a e posi i e alues ha ep esen he eedback con ol pa ame e s o he eloci y
and posi ion cons ain iola ions, espec i ely.
The nume ical esolu ion o Equa ion (2.12) should be ca e ully add essed. The mul ibody
dynamics equa ions o mo ion a e di e en ial and algeb aic equa ions (DAE), bu i is use ul o con e
hem o o dina y di e en ial equa ions (ODE), since in he la e o m i is possible o sol e he equa ions
h ough nume ical in eg a ion algo i hms ha a e a he simple o implemen (Flo es, 2015). The mos
17
widely used nume ical in eg a ion me hods in he li e a u e include he Eule me hod, Runge-Ku a
me hods, and Adams p edic o -co ec o me hods (Flo es, 2015), which a e no he mos accu a e no
e icien due o hei ixed in eg a ion ime-s ep na u e. Howe e , MATLAB and o he p og amming
languages ha e ODE sol e s al eady in eg a ed ha can u ilize a iable ime-s ep in eg a o s, which
esul s in imp o ed accu acy and e iciency o he solu ion o highly non-linea p oblems.
2.2 Expe imen al Appa a us o S udy F ic ion Phenomena
In he middle cen u ies o he p e ious millennium, many scien is s we e in e es ed in
unde s anding he dissipa i e phenomena associa ed wi h pa s ha exhibi ed ela i e mo ion be ween
hem, which led o wea and p ema u e ailu e o mechanical componen s.
Leona do da Vinci cons uc ed simple wooden de ices wi h di e en weigh s o in es iga e ic ion
o ces, d i en by his in e es in op imizing mechanical sys ems like gea s and pulleys. Guillaume
Amon ons de eloped a ious sliding appa a uses using blocks, pla es and opes o measu e ic ional
esis ance, aiming o add ess mechanical ine iciencies in machines. Cha les-Augus in Coulomb designed
mo e sophis ica ed appa a uses o analyze ic ion, mo i a ed by he need o imp o e na al and s uc u al
enginee ing. These ools we e buil o sys ema ically measu e and quan i y ic ion o sol ing eal-wo ld
mechanical challenges.
Expe imen al appa a uses ha e been pi o al o unde s anding ic ion phenomena, p o iding
con olled en i onmen s o obse e and quan i y ic ional e ec s. F om hys e e ic beha io s o s ick-slip
dynamics and non- e e sible ic ion cha ac e is ics, hese se ups ha e allowed esea che s o alida e
heo e ical models and e ine hem.
The e a e appa a uses buil speci ically o s udy ce ain isola ed ic ion phenomena. The wo k by
Aw ejcewicz & Olejnik (2007), o example, consis s o a wo-deg ee-o - eedom mechanical sys em wi h
a eedback- ein o ced labo a o y ig, whe e a block mo ing on a bel coupled wi h sp ings and senso s
allows o measu e bo h ic ion o ce and displacemen in o de o explo e s ick-slip dynamics and de elop
a ic ion model accoun ing o he beha io a bo h posi i e and nega i e angen ial eloci ies. Wojewoda
e al. (2008) buil a se up designed o s udy hys e e ic beha io s in d y ic ion, ocusing on p e-sliding
displacemen and non- e e sible ic ion cha ac e is ics. The appa a us cap u es phenomena such as he
S ibeck e ec and a ying b eakaway o ces in o de o in es iga e hys e esis du ing ansi ions om
s a ic o dynamic ic ion and alida e ic ion models h ough expe imen al esul s compa ed wi h
nume ical simula ions.
18
The phenomenon o non- e e sibili y is ano he complex phenomenon associa ed wi h ic ion, ha
due o i s complexi y and s ochas ic na u e, has led o he cons uc ion o expe imen al appa a us
dedica ed o cap u e his speci ic phenomenon. Wie cig och e al. (1999) buil a Coulomb oscilla o
equipped wi h a pneuma ic ac ua o o con ol he no mal o ce, allowing p ecise measu emen o ic ion
dynamics and non- e e sibili y o ic ion o ce du ing accele a ion and decele a ion phases. This
appa a us allowed o in es iga e non- e e sible ic ion cha ac e is ics and alida e heo e ical ic ion
models h ough dynamic expe imen a ion. Guo e al. (2008) conduc ed simila s udies u ilizing a
ecip oca ing mo ion es ig wi h a ib a ion able and lase displacemen senso s o measu e ic ional
o ces du ing sinusoidal and ec angula wa e mo ions wi h he goal o modeling and iden i ying non-
e e sible ic ion cha ac e is ics, including hys e esis in p e-sliding and sliding egimes, and o analyze
he e ec s o ela i e eloci y and accele a ion in he o e all dynamic esponse o he sys em.
F ic ion phenomena iden i ica ion and modeling a e impo an o de elop ene gy e icien and high
p ecision mechanical and elec omechanical sys ems. Thus, i is o pa amoun impo ance o also build
ailo ed made expe imen al igs o es eal-wo ld sys ems in o de o co ec ly p edic and compensa e
o he dissipa i e e ec s o ic ion o ces. The wo k o Kebai i e al. (2015) desc ibes an appa a us o
accu a ely measu ing he in luence o ic ion in modeling ac ua o dynamics, inco po a ing s a ic and
LuG e ic ion models, and e alua ing hei pe o mance unde a ious ope a ing condi ions. The
expe imen al ig includes a DC mo o , a wo-s age spu gea sys em, a helical sp ing, and posi ion senso s
o measu e ic ion o que and ac ua o displacemen , simula ing eal-wo ld engine condi ions ha a ec
he ope a ion o a Bosch GPA-S elec omechanical ac ua o used in Diesel engines o con ol ai in ake.
The inco po a ion o ic ion phenomena in au oma ion and mecha onic sys ems, such as
pneuma ic and hyd aulic cylinde s o obo s, is c ucial o ensu ing p ecision and eliabili y in dynamic
con ol sys ems. F ic ion, as a dissipa i e phenomenon, signi ican ly in luences sys em beha io ,
especially a low speeds o du ing ansi ions be ween s a ic and dynamic s a es. Accu a e iden i ica ion
and modeling o ic ion allow con ol sys ems o mi iga e i s nega i e e ec s, enhancing pe o mance in
e ms o s abili y, accu acy, and ene gy e iciency.
Felix & Sil ei a (2018) buil a low-cos ig using a Raspbe y Pi con olle and pneuma ic
ins umen a ion o measu e ic ion o ces in eal ime. The se up calcula es ic ion based on p essu es
in ac ua o chambe s and pis on dynamics o analyze he s a ic and dynamic ic ion beha io s o
pneuma ic ac ua o s, enabling p ecise pa ame e iden i ica ion o ic ion compensa ion in indus ial
posi ioning sys ems. Wakasawa e al. (2018) pe o med es s in a pneuma ic cylinde d i en by a s eppe
mo o , equipped wi h load cells, and p essu e and eloci y senso s o isola e and measu e ic ion
19
con ibu ions om od and pis on seals unde a ying lub ica ion condi ions. Feng e al. (2019) pe o med
an iden i ica ion p ocedu e o nonlinea ic ion pa ame e s using an imp o ed S ibeck model and
implemen ed ic ion compensa ion, imp o ing ajec o y accu acy and mi iga ing low- eloci y issues in
elec o-hyd aulic sys ems. The eal-wo ld applica ion consis ed o a obo ic exca a o a m equipped wi h
hyd aulic cylinde s, p essu e senso s, and displacemen senso s o measu e ic ion o ces du ing
con olled mo emen s a cons an eloci ies. Qian e al. (2022) s udied he impac o pis on speed,
p essu e, and p essu e di e ences on ic ion in pneuma ic cylinde s u ilizing a ic ion es ig ea u ing
wo pneuma ic cylinde s moun ed ace- o- ace wi h hei chambe p essu es balanced o elimina e
p essu e e ec s on o ce measu emen s, enabling p ecise and isola ed e alua ion o ic ion o ces.
The expe imen al es ig buil and es ed in he con ex o his wo k ocuses speci ically on he
pa ame e es ima ion o nume ical ic ion o ce models, allowing a be e unde s anding o he in luence
o pa ame e selec ion in he dynamical beha io o mechanical sys ems.
2.3 In luence o F ic ion on Vehicle Suspension Sys ems
F ic ion is modeled in suspension sys ems h ough a ious o mula ions such as ini e elemen
analysis, ib a ions and mul ibody dynamics. In highly complex and ealis ic simula ions, mul iple
o mula ions can be employed in a single model, by combining ini e elemen analysis and mul ibody
dynamics, o example.
The in luence o ic ion in he dynamical beha io o suspension sys ems can be explo ed by wo
di e en pe spec i es: com o and NVH (Noise Vib a ion and Ha shness) equi emen s, usually o capi al
impo ance o egula passenge ehicles o dynamic pe o mance, o spo s and acing ca s. P ima ily,
ic ion o ces a ec he e ical dynamics o a ehicle. The inc eased sales o elec ic ehicles in ecen
yea s b ough pa icula in e es in s udying he in luence o ic ion in suspension sys ems, because he
subs i u ion o he in e nal combus ion engine, ha p oduces highe noise and ib a ion le els compa ed
o an elec ic o hyb id powe ain, leads o an inc eased pe cep ion o he noise gene a ed by suspension
componen s, educing he o e all NVH quali y o a ce ain ehicle.
The opic o ic ion in luence analysis in shock abso be 's pe o mance is no a no el y in li e a u e.
Se e al wo ks explo e he c i ical ole o ic ion in shock abso be s and suspension sys ems, ocusing on
i s e ec on ehicle dynamics and ide com o .
Liza aga e al. (2008) p o ide insigh s in o he dual ole o ic ion in suspension sys ems and i s
in luence on bo h com o and ene gy dissipa ion. Thei esea ch add esses limi a ions in exis ing ic ion

20
models by p oposing a new o mula ion ha combines he S ibeck e ec wi h iscous ic ion, imp o ing
he unde s anding o shock abso be beha io . A specialized hyd aulic es ing de ice was de eloped o
isola e and measu e ic ion, minimizing hyd aulic o ces' in luence. Expe imen al da a con i med he
p esence o he S ibeck e ec , highligh ing s ick-slip phenomena and asymme ical ic ion
cha ac e is ics du ing ebound and comp ession phases.
Benini e al. (2017) explo ed he impac o ic ion in suspension sys ems o acing ca s, inding
ha ic ion signi ican ly a ec s he ehicle’s e ical dynamics, especially a low equencies nea he
ca 's na u al pi ch equency. Using a Fo mula 4 ca es ed on a Fou Pos e Rig, he s udy highligh s ha
ic ion, pa icula ly in he on suspension, ac s as an addi ional damping elemen , in luencing ehicle
beha io du ing dynamic es ing. F ic ion’s ole in e ical esponse and ansien maneu e s indica es
ha igno ing i in simula ions can esul in an inaccu a e assessmen o he o e all ehicle dynamic
beha io .
Kö ne & Maye (2020) ocus on he ic ion cha ac e is ics o dampe s in ela ion o d i ing com o
and ehicle dynamics. The s udy e eals ha ic ional o ces a dampe con ac su aces, be ween seals
made om ubbe and o he ma e ials, signi ican ly in luence he dampe ’s esponse o exci a ion and
he ini ial b eakaway o ce a he s a o mo emen . The esea ch demons a ed ha in e nal p essu e
and ela i e eloci y impac ic ion. Unlike pu e Coulomb ic ion, dampe ic ion exhibi s elas o-
hyd odynamic cha ac e is ics, in luenced by mo emen di ec ion and su ace condi ioning. Al hough
ic ional o ces a e gene ally lowe han hyd aulic damping o ces, he au ho s s a ed ha unde s anding
hem is c i ical o dampe op imiza ion and com o enhancemen .
He zog & Augsbu g (2021) p esen ed a wo-pa s udy on modeling ic ion in shock abso be s.
The i s pa de elops a dynamic ic ion model wi hin a ini e elemen amewo k, ocusing on unwan ed
s a ic ic ion gene a ed by seals and guides. Using a hyb id model, he esea che s in eg a ed physical
and dynamic ic ion models, applying he LuG e model o simula e con ac ic ion. This model, alida ed
h ough high accu acy simula ions in Ansys Mechanical, accu a ely ep esen s shock abso be ic ion
dynamics, enhancing shock abso be design e iciency by educing physical es ing needs. The second
pa (He zog & Augsbu g, 2021) alida es he ic ion model using an expe imen al se up wi h wo
specialized es igs, which measu e ic ion a key con ac poin s in shock abso be s, such as he od
guide assembly and pis on/ ube in e aces. The da a ob ained con i med he simula ion model's accu acy
in p edic ing ic ion ac oss a ious componen s, hough some de ia ions we e no ed in he loa ing
pis on/ ube in e ace due o he complexi y o seal de o ma ion modeling. The alida ed model e ec i ely
aids shock abso be design and op imiza ion.
21
L e al. (2021) examined he kinema ics and compliance (K&C) cha ac e is ics o double wishbone
ai suspensions (DWAS), add essing limi a ions in adi ional models ha o en o e look ic ion and join
clea ances. This esea ch inco po a es a nonlinea dynamic model using ADAMS/VIEW, in eg a ing
lexible componen s and ic ion o ces o enhance suspension simula ion accu acy. Expe imen al
alida ion using a K&C es ig demons a ed ha including ic ion and join clea ances in he model
signi ican ly imp o ed p edic ions o suspension pe o mance, accu a ely eplica ing o ce- ela ed and
displacemen - ela ed beha io s obse ed in eal-wo ld condi ions. The indings unde sco e he
impo ance o hese ac o s in suspension design, since hey ha e subs an ial impac s on kinema ics,
hough hey a ec compliance less.
Deubel e al. (2024) in es iga ed he impo ance o accu a ely modeling ic ion in shock abso be s,
no ing ha ic ion cha ac e is ics a e o en ea ed as quasi-s a ic, despi e being in luenced by dynamic
condi ions. Using modi ied al e- ee shock abso be s, he esea che s obse ed dynamic ic ion
p ope ies, which e ealed ha dynamic ic ion o ces signi ican ly exceed quasi-s a ic ic ion o ces,
especially a lowe eloci ies (up o 60 mm/s) and unde subs an ial side o ces. The s udy ound ha
side o ce and eloci y should be conside ed in ic ion analysis, as con en ional models unde es ima e
o al shock abso be o ces by neglec ing dynamic ic ion as a key componen . This unde s anding can
lead o imp o emen s in ac i e suspension con ol and ehicle simula ions by allowing be e
pa ame iza ion o shock abso be models.
A common p oblem ac oss he a ious s udies on he ic ion ole in he dynamic beha io o
suspension sys ems is he gene al way ic ion is s udied wi hin hese sys ems, whe e eal wo ld oad
e en s a e no conside ed no a clea quan i ica ion o he ic ion e ec is made (Deubel & P okop, 2024).
Ano he no able issue is he limi ed a ailabili y o s udies ha p o ide accu a e, eliable da a on ic ion
le els wi hin ehicle suspension sys ems. This sca ci y applies bo h o o e all suspension ic ion and o
ic ion in speci ic componen s. Fu he mo e, he e is a lack o comp ehensi e esea ch on how a ious
pa ame e s, such as empe a u e and ampli ude, may in luence suspension ic ion (Deubel & P okop,
2024).
The limi ed numbe o s udies in his a ea may be a ibu ed o se e al challenges: (i) he need
o ex ensi e measu emen s o achie e accu a e and ep oducible ic ion da a, (ii) he signi ican e o
equi ed o model ic ion wi h p ecision, bo h in e ms o cap u ing all he phenomena associa ed wi h
ic ion, as well as he equi ed compu a ional powe , and (iii) he inhe en di icul ies in p ac ically
alida ing how a ia ions in ic ion a ec ehicle dynamics (Deubel & P okop, 2024).
Table 2.1 shows a summa y o he quan i ica ion o ic ion o ce in luence on suspension sys ems.
22
Table 2.1 – Quan i ica ion o ic ion o ce in suspension sys ems.
Au ho s
Da a Sou ce
F ic ion Fo ce Values
Kölsch (1994)
In o ma ion ob ained om Deubel &
P okop (2024) pape .
25 N
Mange (1995)
In o ma ion ob ained om Deubel &
P okop (2024) pape .
30 – 45 N
Liza aga e al. (2008)
Expe imen al es o a shock
abso be in a dedica ed es ig and
nume ical simula ion using he
Coulomb ic ion model modi ied o
accoun o he S ibeck e ec .
Nume ical Simula ion
Comp ession: 32 N
Rebounding: 23 N
Expe imen al Da a
Comp ession: 22 N
Rebounding: 15 N
Fujimo o e al. (2018)
MATLAB/Simulink nume ical da a
compa ison wi h expe imen al da a
20 – 150 N, depending on he ype o
shock abso be and damping le el
Wegene (2020)
Lis o on and ea axles da a o
a ious ehicles. No in o ma ion
abou es condi ions.
75 – 350 N
Kö ne & Maye (2020)
Expe imen al da a ob ained om
dedica ed es bench.
Asymme ical beha io o he shock
abso be , anging om 20 o 70 N o
comp ession and 10 o 40 N o
ebounding, depending on exci a ion
equency and p essu e.
He zog & Augsbu g (2021)
FEA analysis in Ansys and
expe imen al da a om dedica ed
es ig.
Values anging om 35 N o
comp ession o 30 N o ebounding.
Deubel & P okop (2024)
Analy ical o mula ion based on
kinema ic cons ain o ces and
expe imen al da a om ull-scale
ou -pos e ig
62 – 134 N o di e en combina ions o
displacemen and wheel la e al o ces.
In his wo k, a model o he Fo mula S uden o Uni e si y o Minho will be used o assess he
in luence o ic ion o ces in he dynamic beha io o suspension sys ems and i s in luence on he o e all
ehicle dynamics.
23
3. A BRIEF HISTORY ABOUT FRICTION
F ic ion ep esen s a challenging phenomenon wi hin he dynamic beha io o mechanical sys ems
since ancien imes. Ea ly ci iliza ions, such as he Egyp ians, de eloped ingenious me hods o manage
ic ion while mo ing massi e s ones and s a ues.
The Renaissance ma ked a u ning poin wi h Leona do da Vinci’s pionee ing expe imen s, laying
he ounda ion o he s udy o ic ion. Following he i s scien i ic me hodology o s udy ic ion p esen ed
by Da Vinci, Guillaume Amon ons o malized he i s laws o ic ion, ollowed by he c i ical e inemen s
o Leonha d Eule and Cha les-Augus in de Coulomb.
This chap e aces he e olu ion o ic ion’s unde s anding, om he p ac ical applica ions o each
his o ic e a o he ou come scien i ic knowledge o he las cen u y.
3.1 F ic ion in he Ea ly Human Ci iliza ions
I is known ha h oughou he ages he impo ance o ic ion and esis ance o mo ion has no
doub been ecognized. E idence om ea ly ci iliza ions, such as ock a and eng a ings, indica e an
awa eness o his phenomena’s in luence on e e yday asks. The Ea ly Human Ci iliza ions, mo e
speci ically he Sume ian and Egyp ian ci iliza ions, ha e de eloped ools in which he ic ion phenomena
we e bene icial and o g ea use such as he so-called d ills ha employed al e na ing o a y mo ion and
we e de eloped o p oduce i e and holes (Dowson, 1998). Figu e 3.1. shows he speci ic example o an
Egyp ian indi idual using a bow d ill.
Figu e 3.1 – Rep esen a ion o an Egyp ian indi idual using a bow d ill (Dowson, 1998).
30
3.4 F ic ion S udies by Leonha d Eule
The i s ma hema ical s udies o ic ion we e a ibu ed o Leonha d Eule in he 18 h cen u y.
Eule (1750b) ea ed he aspec s conce ning he equilib ium s a e and uni o m mo ion o bodies,
de eloping ma hema ical o mula ions o desc ibe hese phenomena.
The i s ma hema ical app oach o ic ion p oposed by Eule was he in e locking aspe i y heo y,
whe e he su aces o con ac ing bodies we e ep esen ed by a se o iangula -shaped aspe i ies in a
saw oo h pa e n, which in e lock pe ec ly and a e inclined a an angle,

, in ela ion o he ho izon al
plane (Besson, 2013). Figu e 3.8 shows a schema ic ep esen a ion o he saw oo h pa e n cons i u ed
by small iangula -shaped aspe i ies ha o igina e he oughness o a su ace.
Figu e 3.8 – Schema ic ep esen a ion o su ace oughness as a se o iangula -shaped aspe i ies in a saw oo h
pa e n by Eule . Each side o he iangula shape exhibi s an inclina ion, α, in ela ion o he ho izon al plane
(Besson, 2013).
The block placed in an inclined plane, ep esen ed in Figu e 3.9, is in equilib ium s a e when he
ex e nal applied o ce, , equi ed o mo e he block up he plane and he ic ional o ce ac ing on he
block, , exhibi he ollowing equali y ela ion
g g
sin cos
  
=  =
(3.1)
U ilizing he angen igonome ic unc ion, i is possible o ew i e Equa ion (3.1) in a mo e
simple and con enien o m
sin an
cos

  

=  =
(3.2)
which leads o he conclusion ha Eule demons a ed he dependency ela ion o he ic ion coe icien ,

, wi h he angen o he inclined plane angle, 𝛼. Mo eo e , i can also be a ibu ed o Eule he
in oduc ion o he g eek le e , 𝜇, in o he ic ion s udy a ea o ep esen he coe icien o ic ion.

31
Ano he impo an con ibu ion o Eule o he scien i ic knowledge o ic ion is he dis inc ion
made be ween s a ic ic ion and kine ic ic ion. The dis inc ion be ween s a ic and kine ic ic ion was
also made using he inclined plane appa a us ep esen ed in Figu e 3.9.
Figu e 3.9 – Adap ed Eule ’s inclined plane schema ic (Dowson, 1998).
Al hough Eule did no p esen a jus i ica ion ha allowed o explain he loss o mechanical ene gy
due o ic ion, since ha p oblem did no belong o he scien i ic pa adigm o he 18 h cen u y,
demons a ions we e made and led o he conclusion ha i he plane was inclined un il eaching a limi
slope whe e he s a ic equilib ium was e i ied, hen a e y small angle inc ease would esul in a
dis u bance o he equilib ium s a e o he block and i would exhibi a as linea mo ion down he plane.
This obse a ion con adic ed Eule ’s i s hypo hesis ha a small angle inc ease would esul in he
block’s mo ion wi h a e y small eloci y, concluding ha he kine ic ic ion, associa ed wi h he mo ion
o a body, mus assume a smalle alue in compa ison o he s a ic ic ion (Dowson, 1998).
3.5 F ic ion S udies by Cha les Augus Coulomb
Cha les Augus Coulomb (1736 – 1806) is undoub edly a pionee esea che in he scien i ic a ea
o ic ion. As desc ibed by Dowson (1998), Coulomb’s s udies on ic ion a e a good example o high-
quali y scien i ic wo k p omp ed by p ac ical p oblems.
The base o Coulomb’s ic ion wo ks elies on he de elopmen s by Amon ons, al hough Coulomb
demons a ed h ough se e al expe imen s ha some o his conclusions con adic ed he ones o mula ed
by Amon ons. Coulomb s udied he sliding and olling ypes o ic ion, p oposing expe imen al alida ion
o hei beha io in p ac ical la ge-scale applica ions, mainly ela ed o pulleys, caps ans, and he
launching o ships on slipways (Dowson, 1998).
32
Figu e 3.10 shows he appa a us de eloped by Coulomb o s udy he sliding ic ion be ween plane
su aces, which is composed o a solid wooden able, se ing as he lowe su ace o he con ac ing pai ,
ep esen ed by (a), and by a sledge, ep esen ed by (b), ha can ha e ails o a ious wid hs and ma e ials
a ached.
Figu e 3.10 – Coulomb’s expe imen al appa a us o he s udy o sliding ic ion. Adap ed om Coulomb (1821).
The in luence o he no mal o ce in he ic ion o ce ha ac s in he e e se di ec ion o he
sledge sliding mo ion was s udied by placing di e en weigh s on he sledge o Figu e 3.10 (d) schema ic.
The sliding o ce was exe ed by placing a weigh along he leng h o a ba which is connec ed o a pulley
ha pushed he sledge o e he able h ough a ope. The da a ob ained wi h his expe imen was o
a ious combina ions o ibological pai s o ma e ials such as oak, g een oak, guaiac wood, i and elm
o he woods, and i on and coppe o wha conce ns he me allic ma e ials. D y and lub ica ed condi ions
we e analyzed using lub ican s such as axle g ease, allow, soo , wa e and oli e oil, bo h in smoo h and
ough su aces, o p essu es anging up o app oxima ely 30.4 MPa and maximum sliding eloci ies o
abou 18 km/h. The s a ic s a e o mo ion ime was also an impo an a iable in Coulomb’s heo ies o
ic ion and his expe imen al da a showed s a iona y imes anging om 0.5 s up o 4 days (Besson,
2013; Dowson, 1998).
33
Coulomb in es iga ed he in luence o ou di e en ypes o ac o s ha can ha e a signi ican
impac on ic ion: (i) he na u e o he ma e ials in con ac as well as he e ec o d y and lub ica ed
condi ions, (ii) he a ea o con ac , (iii) he no mal load suppo ed by he su aces, which Coulomb
desc ibed as p essu e, and (i ) he du a ion o he con ac be ween su aces (Besson, 2013; Dowson,
1998). The main con ibu ions o ic ion knowledge esul ing om he expe imen al indings o Coulomb
can hen be exposed: (i) he ic ion exhibi s an ini ial ise bu soon eaches a maximum o ibological
con ac s o wood sliding on wood unde d y condi ions, being he o ce o ic ion essen ially p opo ional
o he load; (ii) he ic ion o ce o wood sliding on wood is essen ially p opo ional o he load a any
speed, bu he kine ic ic ion o ce, no mally associa ed o he s eady s a e sliding mo ion, has a lowe
magni ude compa ed o he s a ic ic ion o ce, especially hose seen a e long pe iods o es ; (iii) he
ic ion o ce o me als sliding o e me als wi hou lub ican is p opo ional o he load and he e is no
di e ence be ween he magni ude o bo h s a ic and kine ic ic ion; (i ) o he me al-wood ibological
pai unde d y condi ions, he s a ic ic ion ises a a small a e, aking some hou s o e en days o each
i s limi , while o me al-me al pai s his limi is eached almos immedia ely, and o wood-wood pai s
his s a ic ic ion ise akes up a sho minu e ime. I was also e i ied ha o wood-wood and me al-
me al pai s, he kine ic ic ion o ce magni ude unde d y condi ions is almos independen o he
angen ial eloci y, while o wood-me al con ac s, he ic ion o ce magni ude inc eases wi h he inc ease
o angen ial eloci y be ween con ac ing su aces (Dowson, 1998).
The con ibu ions by Coulomb o he de elopmen o ic ion did no only encompass he p ac ical
e ec s o ic ion, bu also del ed in o he heo e ical explana ion o ic ion phenomena. Acco ding o
Coulomb, ic ion could ha e wo possible causes, one o hem being he in e locking o aspe i ies on
su aces, as ep esen ed in Figu e 3.11, ha can only be sepa a ed by bending, b eaking o ising abo e
o he , o by adhesion/cohesion phenomena e i ied be ween he su aces in con ac (Besson, 2013).
Coulomb e i ied ha only he in e locking o su ace aspe i ies was esponsible o he ic ion
phenomena, while cohesion ep esen s a small in luence, since ic ion is p opo ional o he no mal load
and independen o he con ac a ea. The cohesion ep esen s an ine i able inc ease o ic ion because
he e a e mul iple con ac poin s be ween su aces, bu he magni ude o he e e ed con ac s, al hough
no being null, can be negligible o a mac oscopic analysis (Besson, 2013).
The aspe i ies o con ac ing su aces, a e he applica ion o a angen ial o ce, de o m un il a
c i ical poin whe e he opposing aspe i ies o he adjacen su ace slip ou o he in e locking mesh and
sliding mo ion ini ia es, as i is ep esen ed in he Figu e 3.11 (c) scheme. Since he de o ma ion o
aspe i ies is assumed o be cons an in he sliding phase, he ic ion o ce is also cons an , ac ha led
34
a hi d law o ic ion o be a ibu ed o Coulomb whe e is s a ed ha “kine ic ic ion is independen o
he sliding eloci y” (Dowson, 1998).
Figu e 3.11 – In e locking phenomena be ween aspe i ies su aces. Adap ed om (Coulomb, 1821).
Ano he impo an aspec o Coulomb’s ic ion disco e ies is co ela ed o expe imen al
appa a uses buil o s udy he go e ning phenomena o olling ic ion. Figu e 3.12 shows a wo kbench
p ojec ed o s udy he olling ic ion phenomenon, whe e Coulomb s a ed ha olling ic ion was
dependen on he no mal eac ion o ce, al hough p esen ing a magni ude lowe ha sliding ic ion and
ha he esis ance o olling was in e sely p opo ional o he adius o he olle (Dowson, 1998).
Figu e 3.12 – Coulomb’s expe imen al appa a us o he s udy o olling ic ion (Coulomb, 1821).
I is clea ha Coulomb was one, i no he mos in luen ial esea che in he s udy o ic ion,
pa ing he way o he subsequen esea ch indings in he 20 h cen u y by in oducing and de eloping he
concep s o su ace oughness, aspe i y de o ma ion, adhesion and b eaking o aspe i ies, which a e
a guably he mos impo an concep s in he scien i ic domain o ic ion.
35
3.6 O he S udies on F ic ion
Following he essen ial con ibu ions made o he knowledge o ic ion igh a e he Indus ial
Re olu ion, ha emain unchanged o su e ed small imp o emen s un il p esen , in his sec ion a e
b ie ly p esen ed new in e p e a ions o he ic ion phenomenon, p ima ily associa ed o wea e ec s.
The ini ial signi ican heo e ical inqui ies in o he mechanisms o wea we e conduc ed by Holm,
Bu well and S ang, A cha d, and A cha d and Hi s (Besson, 2013). These esea che s p esen ed a
cohe en heo e ical analysis, in oducing models o desc ibe and explain wea and p o iding o mulas
o calcula ion o he wea a e as he olume o ma e ial emo ed pe uni leng h o wo k. Mo e
speci ically, Holm p oposed an adhesi e mechanism a he a omic le el, sugges ing ha bodies in con ac
only made con ac a he peaks o he su ace i egula i ies (aspe i ies). This concep , la e expanded
upon by Bowden and Tabo , in ol ed Holm assuming ha wi h each app oach o wo a oms be ween he
con ac ing aspe i ies, exis ed a cons an p obabili y o an a om being de ached om he su ace o ei he
o he wo bodies (Besson, 2013).
The comp ehensi e syn hesis o esea ch on ic ion, lub ica ion, and wea can be ound in he
collec i e wo ks o Bowden and Tabo , encompassed in hei wo olumes published in 1950 and 1964
(Bowden & Tabo , 1950, 1964). Despi e some knowledge gaps o incomple e aspec s, hei con ibu ions
es ablished he ounda ion o he mode n s udy o ic ion and se ed as a p ima y e e ence in he ield.
Bowden and Tabo p o ided ex ensi e expe imen al esul s, pa icula ly ocused on me als and wood,
co e ing undamen al aspec s o he subjec such as su ace s uc u e, opog aphy, sliding and olling
ic ion, lub ica ion, and wea (Dowson, 1998).
One o he mos signi ican con ibu ions o Bowden and Tabo was he in oduc ion o he
concep o he eal a ea o con ac , comp ising nume ous small egions known as aspe i ies o junc ions
o con ac . These au ho s demons a ed he s ong dependence o ic ion o ce be ween sliding su aces
on he eal con ac a ea and de eloped an in e p e a ion o Amon ons' laws based on an adhesion model
called he adhesi e (plas ic) junc ion model, which emains a undamen al e e ence in he ic ion
domain. To suppo hei model, Bowden and Tabo del ed in o he opog aphy o su aces and aspe i ies,
especially in me als (Besson, 2013).
Addi ionally, Bowden and Tabo explo ed ic ion om he pe spec i e o pu ely elas ic sliding
p ocesses. In such cases, hey applied he heo y de eloped by He z in 1881 conce ning he
de o ma ions o con ac be ween wo elas ic solids unde load. The He z con ac heo y, s ill conside ed
alid oday, o ms he basis o calcula ing aspe i ies' de o ma ions in ic ion models.

36
3.7 Summa y and Discussion
This chap e p esen ed an o e iew o he i s scien i ic con ibu ions o he knowledge o ic ion
phenomena. Al hough he e we e no scien i ic me hodologies a he ime, ea ly ci iliza ions such as he
Egyp ians ecognized he exis ence o dissipa i e e ec s be ween con ac ing su aces when sliding la ge
objec s in he g ound, leading o he use o ancien lub ican s o smoo h hese p ocesses.
Mo i a ed by imp o ing his designs and comp ehend he physical phenomena ha go e ned his
designs, Leona do Da Vinci can be conside ed he i s scien is o p opose a esea ch me hodology o
s udy ic ion. The expe imen al appa a uses o Da Vinci, despi e i s simplici y, allowed him o being he
i s o s udy he in luence o appa en con ac a ea upon ic ional esis ance, c ea ing he emb yogenesis
o he well-known ic ion law ha speci ies ha he appa en a ea o con ac does no in luence he ic ion
esponse.
Guillaume Amon ons p oposed and buil mo e sophis ica ed expe imen al wo kbenches, u ilizing
mechanisms inspi ed by indus ial ools o ha e a, such as polishing glass ables. By es ing di e en
pai s o con ac ing ma e ials and he addi ion o coa s o po k a ha ac ed as lub ican s, Amon ons was
able o o mula e wo o he mos widely ecognized laws o ic ion: (i) he o ce o ic ion is di ec ly
p opo ional o he applied load and (ii) he o ce o ic ion is independen o he appa en a ea o con ac .
Mo eo e , Amon ons was also esponsible o associa ing ic ion wi h he de o ma ion o aspe i ies, which
cons i u es nowadays he undamen al basis o many ic ion o ce models.
The heo e ical wo ks o Leonha d Eule we e mainly associa ed wi h he p oposal o he
in e locking aspe i y heo y o explain he ic ional beha io be ween con ac ing su aces. The main
con ibu ion o Eule o ic ion’s scien i ic knowledge is associa ed wi h he dis inc ion o s a ic and kine ic
ic ion coe icien s, h ough s udies based on inclined planes. The in oduc ion o he G eek le e

o
e e o he ic ion coe icien is also a ibu ed o his au ho .
Cha les Augus Coulomb is p obably he mos widely ecognized and associa ed name wi h ic ion,
c ea ing many expe imen al appa a uses based on la ge scale eal-wo ld applica ions. Coulomb’s wo ks
elies on he de elopmen s by Amon ons, al hough con adic ing some heo ies o mula ed by he la e .
Al hough he hi d undamen al law o ic ion is a ibu ed o Coulomb, which s a es ha he
kine ic ic ion is independen o he sliding eloci y, he i s wo undamen al laws a e associa ed wi h
Amon ons. The undamen al laws o ic ion should be e e ed o as Amon ons-Coulomb ic ion laws
ins ead o only a ibu ing c edi o he la e , as i is done by many au ho s. The na u e o con ac ing
ma e ials, d y o lub ica ed condi ions, con ac a ea, no mal load, and con ac du a ion signi ican ly
in luence ic ion we e also s udied by Coulomb, as well as o he key a iables include su ace oughness,
37
aspe i y de o ma ion, adhesion, and he b eaking o aspe i ies, o ming he undamen al concep s in he
scien i ic s udy o ic ion.
To conclude, he s udies pe o med in he pas millennium pa ed he way o a solid scien i ic
esea ch me hodology associa ed wi h ic ion, leading o nume ous ma hema ical and phenomenological
models ha can accu a ely p edic he dynamic beha io o mechanical sys ems conside ing he
dissipa i e e ec s o ic ion.
38
4. FRICTION – CONCEPT, DISTINCTION AND INFLUENCING FACTORS
F ic ion is a phenomenon which is obse ed in a g ea a ie y o si ua ions. This subjec is s udied
in he con ex o ibology, a wo d ha de i es om he G eek oo
ibos
, which means “ ubbing”, and
he su ix
-logy
ha ep esen s he “s udy o science o ”, and encompasses he s udy o lub ican s,
lub ica ion, ic ion, wea and bea ings (Dowson, 1998). Since his science co e s a wide ange o
scien i ic a eas, i can be desc ibed as an in e disciplina y subjec ha equi es he expe ise s udy o
physicis s, chemis s, and mechanical enginee s, as well as ma e ials scien is s and me allu gis s
(Hu chings & Shipway, 2017).
In a mo e common app oach, ic ion can be uni e sally desc ibed as he esis ance o ela i e
angen ial mo ion which occu s be ween wo in e ac ing su aces. The e a e se e al laws ha a emp o
desc ibe he ic ion phenomenon, being he mos ele an ones exposed in Table 4.1.
Table 4.1 – Summa y o he mos ele an ic ion laws. Adap ed om Mo e al. (2009).
Na u e o he ic ion heo y
F ic ion law
Mac oscale
Amon on’s law
Bowden and Tabo
Single aspe i y
Non-adhesi e – based on He z model
Adhesi e
Mul i-aspe i y pic u e o nanoscale con ac
Non-adhesi e
Adhesi e
Since he main objec i e o his wo k is o make a c i ical analysis o he ic ion o ce models in
dynamical sys ems, he desc ip ion and analysis will be ocused on he mac oscale ic ion concep laws.
The mos u ilized and accep ed mac oscale heo y o explain ic ion is ela ed o he exis ence o
aspe i ies ep esen ing he opog aphy o a ce ain su ace, known as he oughness o a su ace. Bowden
and Tabo (1950, 1964) made aluable con ibu ions o ibology science ha a e s ill ele an and used
oday, e en i mo e ad anced heo ies exis nowadays. These au ho s we e esponsible o he
in oduc ion o he concep o eal a ea o con ac , composed o nume ous small egions, known as
aspe i ies o con ac junc ions, and demons a ed ha he ic ional o ce be ween sliding su aces hea ily
elied on his eal con ac a ea. Figu e 4.1 exhibi s he eal a ea o con ac be ween wo su aces
cha ac e ized by he black do s.
39
Figu e 4.1 - Con ac be ween aspe i ies in he in e ace o wo su aces. Con ac only occu s in he do s, which
explains he di e ence be ween he eal a ea and he appa en a ea o con ac .
A sec ion iew o he g aphical ep esen a ion o he con ac be ween aspe i ies is p esen ed in
Figu e 4.2 exhibi ing he physical di e ence be ween he eal and appa en a ea o con ac .
Figu e 4.2 – Schema ic ep esen a ion o sec ioned aspe i ies, co esponding o he eal a ea o con ac , and he
appa en a ea o con ac .
I is hen in ui i e o unde s and ha he eal a ea o con ac is a ela i ely small ac ion o he
appa en a ea o con ac . Bowden & Tabo (1950) de e mined o a ious loads he a io be ween he
appa en a ea o con ac and he eal a ea o con ac in s eel-on-s eel ibological pai s h ough elec ical
conduc i i y me hods, concluding ha he eal con ac o a ea is 10-5 o 10-2 imes smalle han he
appa en a ea.
These au ho s de eloped an in e p e a ion o Amon ons' laws based on an adhesion model called
he adhesi e (o plas ic) junc ion model, which emains a undamen al e e ence ill p esen days. In his
Real a ea o con ac
Appa en a ea o con ac
46
lead o nume ical ins abili ies in dynamic simula ions o mechanical sys ems. This unce ain y in he
ic ion o ce alue in he nea by egion o he null alue o angen ial eloci y equi es egula iza ion
echniques ha p e en nume ical ins abili ies om occu ing du ing simula ions.
4.1.3 Sliding F ic ion
The sliding ic ion is ela ed o he esis ance ha an objec o e s when i slips o e ano he .
This ype o dynamic ic ion is di ec ly ela ed o he heo y ha s a es ha he su aces a e composed
o aspe i ies wi h di e en leng hs and heigh s. The mo emen is hen cha ac e ized by he ela i e mo ion
be ween he aspe i ies o bo h su aces in a ela i ely la ge a ea o con ac and he esis ance o mo ion,
i.e., he ic ion o ce, esul s om he shea ing o junc ions o med as a esul o adhesion phenomena
a he in e ace (Bowden & Tabo , 1950; G eenwood e al., 1961; Pe sson, 1998).
In Figu e 4.9 is possible o obse e he classical example o sliding ic ion ma e ialized by a
sliding block and i is also ep esen ed a zoom-in o he shea ing de o ma ion o aspe i ies in he in e ace
o bo h su aces in con ac .
Figu e 4.9 – Mac oscopic sliding mo ion o a block and zoom exhibi ing he mic oscopic phenomenon ha explains
sliding ic ion as elas ic and plas ic de o ma ion o aspe i ies.
4.1.4 Rolling F ic ion
Rolling ic ion exis s when wo con ac ing bodies exhibi olling mo ion be ween hem. This ype
o ic ion can be desc ibed h ough wo di e en si ua ions ha can be e i ied in Figu e 4.10: a) a olling
disc in a s a iona y s a e on a plane su ace whe e he con ac p essu e ansla es in o a no mal esul an
o ce,
n, which is no mal o he con ac ing su ace and aligned wi h he weigh ac ing on he olling body;
b) when he disc s a s o oll due o an ex e nal o ce,
, he con ac p essu e dis ibu ion ollows he
de o ma ion disc-plane, which esul s in a ansla ion o he con ac poin by a dis ance, d, in ela ion o
g
n

47
he cen e o g a i y o he disc. This displacemen o he con ac poin o igina es a momen ,
M
, named
olling esis ance momen , ha opposes o he olling mo emen o he disc and is di ec ly ela ed o he
de o ma ion o he con ac ing bodies.
a)
b)
Figu e 4.10 – Rolling ic ion o a disc in a plane a) s a ic si ua ion and p essu e dis ibu ion b) kine ic si ua ion
due o he applica ion o he o ce .
Mo eo e , his su ace con ac ha is gene a ed due o he elas ic de o ma ion o he con ac
su aces o bo h bodies can be explained by hys e esis losses. Tabo (1952, 1955) and A ack (1955,
1958) demons a ed ha he on po ion o he con ac o a sphe e wi h he plane ansla es in o elas ic
wo k, due o comp ession s esses, ha would be equally compensa ed by he ea po ion o he ci cle
o con ac when i eco e s elas ically and mo es he sphe e o wa d, due o ension s esses on he ea
o he sphe e. Tha is no e i ied in eali y due o he hys e esis phenomenon, so he o ce equi ed o
mo e he sphe e along he plane is non-null, which hen allows o conclude ha exis s a esis ance o ce
ac ing on he con ac su ace (G eenwood e al. 1961). Acco ding o Dahl (1968), ension and
comp ession p ope ies p incipally go e n he ic ion p ocess.
The olling o m o mo ion is mo e complex han sliding mo ion and mos o he knowledge ha
exis s nowadays is ela ed o Reynolds (1876), He z (1886) and Hea co e (1921) (Doménech e al.
1987). One ac ha con i ms his s a emen was made by Reynolds whe e wha e e ma e ials ha
compose he olling and plane bodies, he e will always be slipping mo ion in he con ac su ace, al hough
he magni ude o i migh be oo mino o be measu ed, so olling mo ion hen combines olling and
sliding mo ions which makes i complex o cha ac e ize (Reynolds, 1875).
p
n
g
d
g
p
n
M
48
4.2 Fac o s in luencing ic ional beha io
The esis ance o sliding p ima ily mani es s in he egions nea and be ween solid su aces.
Iden i ying he speci ic a ibu es o con ac condi ions and ma e ials ha con ibu e signi ican ly o ic ion
o ce is a signi ican challenge in de eloping ic ion es s and analy ical models.
F ic ion models inco po a e a ious app oaches, including geome ic conside a ions such as
su ace oughness and aspe i y in e locking, mechanical p ope ies-based a iables in ol ing shea
p ope ies o solids and subs ances be ween su aces, being he mos common si ua ion he exis ence o
lub ican s, luid dynamics p inciples, elec os a ic o ces be ween su ace a oms, chemical compa ibili y,
su ace empe a u e, and o he a iables ega ding he dynamics o he sys em in analysis, such as
ela i e sliding eloci y, load o ime o con ac , o example. Gi en he di e si y o hese app oaches, he
numbe o po en ial a iables o use in p edic i e ic ion models can be subs an ial. Table 4.2 exhibi s
a summa y o he mos common ac o s in luencing ic ional beha io acco dingly o Blau (2001).
Table 4.2 – Fac o s in luencing ic ional beha iou (Adap ed om Blau (2001))
Ca ego y
Fac o
Con ac geome y
Con o mi y o he componen s: mac oscale ma ing o shapes
Su ace oughness: mic oscale ea u es – aspe i y shape, size
dis ibu ion
Fluid p ope ies and low
Lub ica ion egime; bounda y, mixed, elas ohyd odynamic,
hyd odynamic
Lub ican ilm hickness, p essu e, empe a u e, shea s ess
and iscosi y
Oxida ion and acidi ica ion o lub ican s
Rela i e mo ion
Unidi ec ional o ecip oca ing mo ion
Mo ion a ia ion – accele a ions, pauses, s ick-slip
Applied o ces
Magni ude and a ia ion o he no mal o ce
Thi d-bodies
Pa icles en ained in he lub ican and wea pa icles
Ex e nal con aminan s
Tempe a u e
The mal e ec s on ma e ial p ope ies: he moelas ic
ins abili ies, duc ili y inc ease
The mal e ec s on lub ican p ope ies
S i ness and ib a ions
Con ac compliance
Damping o ic ional o ex e nal ib a ions
49
Due o he nume ous po en ial ac o s in luencing ic ion, i is essen ial o pinpoin he key a iables
ele an o each speci ic case. This allows o he selec ion o app op ia e es me hods o simula ions
ailo ed o each pa icula case s udy (Blau, 2001).
In he con ex o his wo k, he e ec o each a iable p esen ed in Table 4.2 will no be analyzed
in de ail, since i is no he main ocus o he c i ical and p ac ical analysis o ic ion o ce models in
dynamical sys ems conduc ed in his wo k.
4.3 Summa y and Discussion
In his chap e , a summa ized p esen a ion o he main aspec s o ic ion was elabo a ed, ocusing
p ima ily on he concep , dis inc ion and in luencing ac o s ha a ec he ela i e mo ion be ween
con ac ing bodies.
The dis inc ion be ween s a ic and kine ic ic ion coe icien s was made conside ing he p ac ical
and in ui i e example o he inclined plane, simila ly o wha has been done in he wo ks o Eule . The
conduc ed analysis allowed o ecognize he exis ence o highe ic ion o ces a s a ic s a es o mo ion,
compa ed o he lowe and cons an alue o kine ic ic ion o ce associa ed wi h sliding mo ion. Al hough
he exposi ion o hese concep s was made using he inclined plane as a p ac ical example, o he no so
in ui i e examples, such as he p essu e cone heo y (Flo es e al., 2023), could be u ilized o explain he
di e ences be ween s a ic and ic ion coe icien s.
The Coulomb ic ion o ce, conside ed he p edecesso o all mode n ic ion o ce models, was
analyzed bo h in e ms o i s gene al g aphical ep esen a ion and ma hema ical ep esen a ion. The
simplici y o his model is associa ed wi h he ac ha only needs one inpu pa ame e o be ully de ined
– he kine ic ic ion coe icien ,
k

. Despi e i s udimen a y na u e, implemen ing his ic ion model in
compu a ional models is complex due o he non-di e en iable beha io nea he null alue o ela i e
eloci y, which can cause nume ical ins abili ies. Regula iza ion echniques, which will be u he
discussed in he nex chap e s, a e equi ed o ensu e s able dynamic simula ions.
Bo h sliding and olling ic ion phenomena we e also analyzed in a simple and e ec i e manne .
In one hand, sliding ic ion is mainly associa ed wi h he heo y ha su aces consis o aspe i ies o
a ying sizes, wi h mo emen a ising om hei ela i e mo ion ac oss a la ge con ac a ea. F ic ion o ce
o igina es om he shea ing o junc ions o med by adhesion a he in e ace. On he o he hand, olling
ic ion is mo e complex han sliding ic ion due o hys e esis, whe e elas ic eco e y a he ea egion
o he con ac a ea does no ully compensa e o comp ession a he on . This esul s in a non-ze o
50
o ce associa ed o he mo ion o olling objec s. Addi ionally, olling mo ion inhe en ly in ol es mino
slipping a he con ac su ace, combining olling and sliding mo ions, which complica es i s
cha ac e iza ion.
A summa y o ac o s in luencing ic ional beha io was also ga he ed in his chap e , anging
om con ac geome y o luid p ope ies and low associa ed wi h lub ican s, ela i e mo ion be ween
bodies, applied o ces, exis ence o hi d-bodies, empe a u e and s i ness and ib a ions, al hough he
e ec s o each a iable a e no p esen ed in he con ex o his wo k.
51
5. FRICTION PHENOMENA
F ic ion is an inhe en ly non-linea phenomenon, shaped by a ange o complex sub-phenomena
ha in luence he in e ac ions be ween con ac ing su aces. While many o hese sub-phenomena a e
well unde s ood and suppo ed by ma hema ical o mula ions, inco po a ing hem in o comp ehensi e
ic ion o ce models emains a challenge. This complexi y a ises om he di e si y o ma e ials,
lub ica ion egimes, kinema ic a iables like ela i e eloci y, and s ochas ic ac o s ha impac su ace
in e ac ions. Mo eo e , cha ac e izing each phenomenon in e ms o a nume ic alue a ibu ed o a
model pa ame e is also a challenging ask.
Classical models, such as Coulomb, o e simpli ied app oxima ions o de e mine he magni ude
o he dissipa i e e ec s o ic ion, bu ail o cap u e c i ical aspec s like s a ic ic ion exceeding he
kine ic ic ion o ce le els o he in ica e ansi ions be ween mo ion s a es. O e ime, ad ancemen s in
ibology ha e unco e ed new ic ional beha io s, allowing o mo e de ailed and ealis ic ep esen a ions
o su ace con ac phenomena. The e olu ion o compu a ional powe has u he enabled he accu a e
simula ion o hese phenomena, imp o ing p edic ions o he beha io o dynamic mechanical sys ems.
This chap e p o ides a s uc u ed explo a ion o key ic ion phenomena, p og essing in o de o
inc easing complexi y. Topics include s ic ion, b eakaway o ce and dwell- ime, iscous e ec s, he
S ibeck e ec , s ick-slip mo ion, p e-sliding displacemen and mic o-slip, ic ional lag, and non-
e e sibili y. Toge he , hese phenomena o m he ounda ion o unde s anding he a iables and physical
assump ions ma e ialized in he ma hema ical o mula ions o ic ion o ce models.
5.1 S a ic ic ion (s ic ion), b eakaway o ce and dwell- ime
I is in ui i e o unde s and ha he o ce needed o main ain he mo emen o a ce ain objec
is lowe han he o ce equi ed o ini ia e he mo ion. Wi h his ac in mind and analyzing he g aphical
ep esen a ion o he Coulomb ic ion o ce model, i can be con adic i e in he i s place, since an
inc easing alue o ela i e angen ial eloci y be ween su aces does no ansla e in o a educ ion in
ic ion o ce, due o he ac ha his ic ion model does no conside any di e en ia ion be ween he
s a ic ic ion o ce le el and he Coulomb (kine ic) ic ion o ce le el.
Conside ing he p e iously men ioned phenomenon, se e al s udies ha e been de eloped o
explain i . Mo in (1833) in oduced he idea o a ic ion o ce associa ed wi h a s a ic s a e ha is highe
han he Coulomb ic ion and Rabinowicz (1951) was one o he mos in luen ial scien is s in he s udy

52
o he na u e o s a ic and kine ic coe icien s o ic ion (Mo in, 1833; Rabinowicz, 1951). Th ough se e al
expe imen s, he la e au ho was able o add ess he ansi ion be ween s icking and sliding mo ion and,
consequen ly, be ween s a ic and kine ic ic ion as a unc ion o he displacemen o a block sliding along
an inclined plane.
Due o adhesi e bonds ha a e gene a ed be ween he aspe i ies o bo h su aces in con ac and
he shea o ces de eloped in he ini ial phase o mo ion o o e come he esis ance o e ed by hese
same aspe i ies, i is possible o explain why he s a ic coe icien o ic ion, deno ed as
s

, is highe
han he kine ic coe icien o ic ion,
k

. This means ha in he diag am o ic ion o ce in unc ion o
he angen ial ela i e eloci y, he e is a s a ic ic ion o ce
s
, highe han he Coulomb ic ion o ce,
C
, ha now ep esen s unequi ocally he kine ic ic ion o ce acco ding o Figu e 5.1.
Figu e 5.1 – G aphical ep esen a ion o he Coulomb d y ic ion model exhibi ing he s a ic ic ion (s ic ion)
phenomena and he Coulomb ic ion o ce le el o non-null alues o angen ial ela i e eloci y,
.
The in oduc ion o he s ic ion phenomenon in he Coulomb ic ion o ce model leads o he
ollowing o mula ion
( )
C
e s e
sgn( ) i 0
min , sgn( ) i 0



==


(5.1)
whe e
s
ep esen s he s a ic ic ion o ce le el (s ic ion).
Associa ed wi h his s ic ion phenomenon is commonly in oduced he b eakaway o ce concep .
The b eakaway o ce is associa ed wi h he maximum ic ion o ce alue, hus co esponding o he s a ic
s
– s
C
– C
53
ic ion o ce le el. Due o he high nonlinea i ies associa ed wi h his phenomenon, he measu emen o
i is qui e challenging and complex, which can explain he inaccu acy some imes e i ied in con ol
applica ions du ing accele a ion and decele a ion phases o mo ion. These inaccu acies a ise om he
dependency o he s a ic ic ion on he ime o con ac , which is e e ed o as dwell- ime dependency.
The ansi ion om null ela i e eloci ies un il eaching he b eakaway o ce le el du ing he s ick phase
occu s in a ini e bu con inuous ime in e al, ha is o en di icul o cap u e and model. Only highly
complex dynamic models such as he one p oposed by Gon hie e al. (2004) a e capable o modeling
and p edic ing his phenomenon.
5.2 Viscous e ec
In he ea ly 19 h cen u y, wi h he inc easing in e es and de elopmen o hyd odynamics heo ies,
he iscous e ec was in oduced in he Coulomb ic ion model due o he in luence o lub ican s in he
su ace’s con ac . This beha io is ma e ialized by he luid lub ican laye ha exis s be ween he wo
ubbing su aces. In his case, he iscous beha io is conside ed o be a linea unc ion o he angen ial
ela i e eloci y as ep esen ed in Figu e 5.2 (Olsson e al., 1998).
Figu e 5.2 – G aphical ep esen a ion o he Coulomb ic ion model conside ing iscous e ec .
The linea inc easing beha io o he ic ion o ce due o iscous e ec s indica es ha he
lub ica ion egime is he ull- ilm lub ica ion ype. The Coulomb ic ion o ce model o mula ion
conside ing he iscous e ec componen can hen be w i en as
C
– C
54
( )
( )
C
e C e
sgn( ) i 0
min , sgn i 0
+


==


(5.2)
in which
is a iscous ic ion coe icien ela ed o he iscosi y o he luid.
Al hough he ela ion be ween ic ion o ce and eloci y is conside ed o be linea , o ha e a
be e i o expe imen al da a, his e ec can be desc ibed by a non-linea dependence on eloci y using
an ex a a iable,

, which depends on he geome y o he applica ion (Olsson e al., 1998), making
he second pa cel o he i s b anch o Equa ion (5.2) equal o
sgn( )

(5.3)
5.3 S ibeck e ec
The iscous e ec men ioned in he p e ious sec ion desc ibed he ela ionship be ween ic ion
o ce and ela i e angen ial eloci y in he p esence o a lub ica ed con ac be ween he su aces as a
linea ela ion, bu his phenomenon only cap u es he beha io o a ull- ilm lub ica ion si ua ion (Olsson
e al., 1998).
To be able o cap u e mul iple s ages o he lub ica ion egime ela ed o di e en angen ial
ela i e eloci ies, he S ibeck e ec was added o he Coulomb d y ic ion model (S ibeck, 1902). This
phenomenon is esponsible o ensu ing a con inuous dec ease om s a ic o kine ic ic ion as can be
obse ed in Figu e 5.3.
Figu e 5.3 – G aphical ep esen a ion o he Coulomb ic ion model conside ing he S ibeck e ec .
s
– s
C
– C
55
This con inuous d op is cha ac e ized by a dec easing ic ion o ce wi h inc easing ela i e
angen ial eloci y,
, a low eloci y, and is esponsible o cap u ing he ansi ion om a bounda y
lub ica ion egime o a mixed- ilm lub ica ion, being desc ibed nume ically by he ollowing o mula ion
( )
e S e
( ) i 0
min , sgn( ) i 0



==


(5.4)
whe e
()
is an a bi a y eloci y dependen ic ion unc ion. Many au ho s in li e a u e p oposed
ma hema ical exp essions o ep esen he S ibeck e ec as a o m o a eloci y dependen ic ion
unc ion.
Tus in (1947) u ilized an exponen ial exp ession o es ablish a ela ionship be ween he
angen ial eloci y,
, and he S ibeck eloci y,
S
( )
S
C s C 1
e−

= + − −



(5.5)
Bo & Pa elescu (1982) also p oposed an exponen ial unc ion o ep esen he S ibeck e ec
whe e he o al ic ion o ce is de e mined by he ollowing unc ion
( ) ( )
C s C
e 
−
= + −
(5.6)
whe e

is a cons an ac o de ined o con ol he shape o he S ibeck cu e and

is a ac o
dependen on he geome y o he con ac ing su aces.
Hess and Soom (1990) de ined a ma hema ical exp ession o nume ically desc ibe he S ibeck
cu e, ypically known as Lo en zian model, and ha is o mula ed as
( )
sC
C
2
S
1
−
= + +

+

(5.7)
whe e
is a e m used o desc ibe he iscous e ec phenomena.
Popp & S el e (1990) desc ibed he dec easing cha ac e is ic o he ic ion o ce associa ed wi h
he S ibeck e ec using he ollowing o mula ion
2
sC
s
() 1
−
= + + 
+
(5.8)
62
5.8 Summa y and Discussion
In his chap e , a comple e and de ailed p esen a ion and analysis o he p incipal phenomena
associa ed wi h ic ion ha a e conside ed in he ma hema ical o mula ions o a ious ic ion o ce
models was conduc ed. Each phenomenon, including s ic ion, b eakaway o ce, dwell- ime, iscous
e ec s, he S ibeck e ec , s ick-slip mo ion, p e-sliding displacemen and mic o-slip, ic ional lag, and
non- e e sibili y, was explo ed o highligh i s unique con ibu ions o he o e all ic ional beha io
be ween con ac ing su aces.
Unde s anding hese phenomena is essen ial o ad ancing he accu acy o ic ion o ce models.
Each con ibu es o he complex dynamics obse ed in eal-wo ld mechanical sys ems, ye hei nonlinea
and in e dependen na u e poses signi ican challenges o in eg a ion in o mul ibody simula ions.
Accu a e modeling equi es no only a ho ough comp ehension o hese mechanisms bu also obus
ma hema ical o mula ions ha can eplica e hei e ec s unde a ying condi ions, such as changes in
ela i e eloci y, ma e ial p ope ies, and lub ica ion egimes.
The necessi y o inco po a ing hese phenomena in o ic ion models becomes e iden when
conside ing he p ecision equi ed in enginee ing applica ions. Failu e o adequa ely ep esen hese
beha io s can lead o disc epancies be ween simula ed and obse ed pe o mance, po en ially
comp omising he eliabili y o a ious sys ems.
Wi h ad ancemen s in compu a ional capabili ies, i is now possible o inco po a e hese
phenomena in o inc easingly sophis ica ed models. Consequen ly, such enhanced models a e
indispensable o op imizing sys em pe o mance, imp o ing du abili y, and educing ene gy losses
ac oss a b oad ange o enginee ing applica ions.
In conclusion, he phenomena discussed in his chap e unde sco e he complexi y o ic ion
phenomena and he necessi y o accu a ely p edic ing hei e ec s on he dynamic beha io o
mechanical sys ems.

63
6. STATIC FRICTION MODELS
The Coulomb ic ion model is cha ac e ized by a simple g aphical shape whe e he ic ion o ce,
gi en by he p oduc be ween he kine ic ic ion coe icien ,
k

, and he no mal eac ion,
n
, assumes
a posi i e alue o a ce ain di ec ion o ela i e angen ial eloci y and a symme ic alue o ic ion o ce
when he sys em in analysis changes i s di ec ion. This model can be de ined as an empi ical model since
i s g aphical ep esen a ion de i ed simply om expe imen s and obse a ions made by Coulomb.
Despi e i s simplici y a i s sigh , gi en by he ac ha i s implemen a ion only equi es one
pa ame e , he kine ic ic ion coe icien ,
k

, he Coulomb ic ion model becomes a e y complex
p oblem o sol e when ying o implemen i in compu a ional applica ions. The obse ed challenge s ems
om he discon inui y associa ed wi h he ic ional o ce alue co esponding o a ela i e eloci y o null
magni ude. When he alue o ela i e angen ial eloci y is null, his ic ion o ce model canno be
desc ibed by any ype o ma hema ical unc ion since i iola es one o he basic p inciples o a gi en se
o da a being desc ibed by a unc ion – he e is a alue o he domain, in his case he null alue o
angen ial eloci y, ha has mul iple and unde e mined co esponden ange alues, in his case ic ion
o ce alues, which means ha he model is non-di e en iable in he null ela i e eloci y egion.
Conside ing his compu a ional implemen a ion p oblem, se e al s a egies we e de eloped
h oughou he yea s o su pass his nume ical ins abili y and “smoo h” he ansi ion nea he null alue
o angen ial eloci y so ha ic ion can be desc ibed by a ma hema ical con inuous unc ion.
The models analyzed in his sec ion, which can be i ed in he s a ic ca ego y, ha e a common
a iable in common – he angen ial eloci y, . This ela i e eloci y de eloped in he in e ace be ween
con ac ing bodies is e y impo an in he con ex o his ype o models since mos o hei o mula ions
a e made as a unc ion o he a o emen ioned eloci y. The exis ence o a ela i ely small numbe o
pa ame e s needed o de ine hese models, limi s hei capaci y o modelling complex phenomena ela ed
o physical in e ac ion be ween he su aces
The s a ic ic ion o ce models s udied a e di ided in o wo main ca ego ies as he Figu e 6.1
diag am indica es: s a ic wi hou s ic ion, whe e he maximum alue o ic ion o ce is he kine ic one,
also ecognized as he Coulomb ic ion o ce le el,
C
, and he s a ic wi h s ic ion models, ha as he
name sugges s, a e capable o modelling he maximum alue o ic ion o ce as he s a ic ic ion o ce
le el,
s
.
64
Figu e 6.1 – S a ic ic ion o ce models di ided by models wi h s ic ion and wi hou s ic ion ca ego ies.
I is impo an o no e ha he models in each ca ego y a e p esen ed in an ascending o de o
appea ance, wi h he oldes models being p esen ed i s un il eaching he mos ecen o mula ions.
Fu he mo e, one o he main objec i es o his wo k is o in es iga e he compu a ional pe o mance o
hese ic ion models, so only egula ized app oaches o s a ic ic ion models a e s udied and simula ed
h ough a benchma k p oblem.
6.1 S a ic ic ion o ce models wi hou s ic ion
6.1.1 Be na d model (1974)
Be na d (1974) p oposed one o he simples egula iza ion app oaches o he Coulomb ic ion
o ce model by conside ing a linea ansi ion om a nega i e alue o Coulomb ic ion o ce,
C
−
, o a
posi i e alue,
C
+
. Figu e 6.2 shows he smoo hened egula iza ion modeled by Be na d.
S a ic ic ion o ce models
wi hou S ic ion wi h S ic ion
Be na d (1974)
Rooney and De a i (1982)
Amb ósio (2003)
Ka nopp (1985)
Bengisu and Akay (1994)
A ms ong-Hélou y e al. (1994)
Holla s (1994)
Ande sson e al. (2007)
Wojewoda e al. (2007)
Aw ejcewicz e al. (2008)
Specke e al. (2014)
Linea wi h s ic ion
(Ma ques e al.) (2016)
B own and McPhee (2016)
Th el all (1978)
65
Figu e 6.2 – Linea egula iza ion o he Coulomb ic ion model p oposed by Be na d (1974).
In ma hema ical e ms, his model can be desc ibed by he ollowing sys em o equa ions
C l
C l
min( ,1) i 0
max( , 1) i 0
k
k


=−

(6.1)
whe e C is he Coulomb ic ion o ce magni ude,
l
k
ep esen s a coe icien de ined o de e mine he
speed o ansi ion om nega i e alues o posi i e alues o ic ion o ce, which in p ac ical e ms
ep esen s he linea slope o his ic ion o ce model and is he ela i e angen ial eloci y be ween
he con ac ing su aces.
As a clea ad an age o his model can be e e ed he con inuous ansi ion be ween posi i e
and nega i e alues o ic ion o ce using a simple (linea ) ma hema ical unc ion ha has small
compu a ional cos in compa ison wi h o he egula iza ion s a egies, bu he disad an ages ela ed o i
a e he sha p slope change a he poin s whe e he ic ion o ce eaches i s maximum alue, namely in
he ansi ion be ween he linea slope and he Coulomb ic ion o ce le el, and he ac o no being
capable o cap u e he s ic ion phenomenon. Despi e some o he g aphical ep esen a ions o his model
exhibi ing a eloci y ole ance, which is he alue o angen ial eloci y ha ma ks he ansi ion be ween
s icking and sliding phases, he ma hema ical o mula ion does no conside any ype o equa ion o
es ablish his ansi ion, being ully de ined by he ansi ion cons an slope.
C
– C
66
6.1.2 Th el all model (1978)
As i was p e iously s a ed in he Be na d (1974) model sec ion, one o i s main disad an ages
is he sha p ansi ion be ween he linea slope and he cons an slope associa ed wi h he s eady-s a e
ic ion o ce alue, which in his speci ic case, is ep esen ed by he Coulomb ic ion o ce le el,
C
. To
o e come his p oblem, a cu e shape de ined by an exponen ial unc ion was de ined by Th el all (1978),
which led he ic ion o ce o be ma hema ically exp essed as
( )
0
3
C
1 sgn
e
−


=−


(6.2)
whe e he cons an 3 is esponsible o ensu ing ha a
0
=
he ic ion o ce is app oxima ely equal
o
C
0.95 
(6.3)
being
0
he eloci y a which ic ion is eloci y independen (Th el all, 1978). When he dynamic sys em
in analysis eaches he
0
alue, he ic ion o ce assumes he s eady s a e magni ude, which in he
case o his model ep esen s he Coulomb ic ion o ce alue. Figu e 6.3 exhibi s he g aphical
ep esen a ion o he Th el all model.
Figu e 6.3 – G aphical ep esen a ion o he Th el all egula iza ion s a egy o he Coulomb ic ion model.
C
– C
0
- 0
67
The alue o he pa ame e
0
is selec ed by he model use , howe e , i should no be e y
small, i.e., close o ze o, because his would esul in a beha io simila o he Coulomb ic ion model,
hus in oducing a discon inui y nea he alue o null angen ial eloci y.
Rega ding his model, Ma ques e al. (2016) p oposed an al e na i e e sion o i ep esen ed
o mula ed as
( )
0
3
C
1
1sgn
1
e
e
−
−


−
=
−


(6.4)
Besides eaching he Coulomb ic ion o ce, he modi ica ion o he model ensu es ha he
ansi ion be ween he aising s icking phase and he sliding phase is made in a smoo he and con inuous
manne wi hou yielding nume ical ins abili ies.
6.1.3 Rooney and De a i model (1982)
Ano he ela i ely simple app oach, in ma hema ical e ms, o a smoo hened Coulomb ic ion
model was p oposed by Rooney and De a i (1982). In his case, a igonome ic unc ion, mo e
speci ically a hype bolical angen unc ion,
anh
, is used, being he ic ion o ce de ined as
C 0
0
C 0
anh i
i
 

=


(6.5)
whe e
0
ep esen s he eloci y ole ance. This pa ame e de ines by i sel he wid h o he egion in
which he ic ion o ce is dependen on he angen ial eloci y. Figu e 6.4 exhibi s he g aphical
ep esen a ion o he Rooney and De a i egula iza ion app oach o he Coulomb ic ion o ce model.

68
Figu e 6.4 – Rooney and De a i’s s a ic ic ion model g aphical ep esen a ion.
I is possible o no ice ha he Rooney and De a i’s model conside s ha o angen ial eloci ies
supe io o he eloci y ole ance (
0

), he ic ion o ce is equal o he Coulomb ic ion o ce, as i
is demons a ed by he second b anch o Equa ion (6.5), no ha ing a s a ic h eshold ic ion o ce o
ep esen he s ic ion phenomenon. The posi i e aspec s o conside ing his ic ion o ce model a e
ela ed o he simple compu a ional implemen a ion, he need o only de ining wo pa ame e s, he kine ic
ic ion coe icien ,
k

, and eloci y ole ance,
0
, and i s e y sui able o high-speed cyclic mo ions
which makes i e y applicable o au omo i e (engine, ansmission and d i e ain’s mo ing pa s) and
indus y applica ions (Rooney and De a i, 1982). Downsides o he use o his ic ion model can be
ela ed o he inexpe ience o lack o sensi i i y o he use in de ining an adequa e alue o he eloci y
ole ance,
0
, ha p o ides bo h p ecise esul s and minimizes he compu a ional ime.
6.1.4 Amb ósio model (2003)
The ic ion o ce model p oposed by Amb ósio (2003) o applica ion in mul ibody con ac
p oblems cha ac e izes i sel as a o m o egula iza ion o he Coulomb ic ion model based in a linea
ela ion be ween he symme ic maximum alues o ic ion o ce,
C
−
and
C
+
, espec i ely. The
o mula ion o his model is made using he ollowing piecewise unc ion
C
– C
0
- 0
69
0
0
0 1
10
C 1
0 i
i
sgn( ) i

−

=  
−



(6.6)
whe e
0
and
1
a e eloci y ole ances ha es ablish he beginning o he s ic ion phase whe e he
ic ion o ce assumes a non-null alue and he ansi ion om he s icking phase o he sliding s a e,
espec i ely. This model p e en s he ic ion o ce om changing di ec ion nea he null alue o ela i e
angen ial eloci y, bu as a disad an age, i also conside s a null alue o ic ion o ce o small sliding
eloci ies. Also, i is no capable o modelling he s ic ion phenomenon, eason why is inse ed in he
wi hou s ic ion s a ic ic ion o ce models ca ego y. Figu e 6.5 ep esen s he g aphical aspec o his
ic ion model.
Figu e 6.5 – G aphical ep esen a ion o Amb ósio’s egula iza ion app oach o Coulomb ic ion model.
Ano he disad an age ha a ise om he use o his model is ela ed o he sha p ansi ions
e i ied be ween he h esholds
0
and
1
.
C
- C
- 1- 0
0 1
70
6.2 S a ic ic ion o ce models wi h s ic ion
6.2.1 Ka nopp model (1985)
In o de o educe he p obabili y o exis ence o nume ical ins abili ies nea he alue o null
angen ial ela i e eloci y, Ka nopp (1985) p oposed a model whe e he ic ion o ce,
, is a unc ion
o he angen ial eloci y,
, o wha is conside ed he s eady s a e o he dynamical sys em in analysis.
In he nea by egion o null angen ial eloci y, a small eloci y ange is de ined by he conjunc ion o
condi ions
0 0
−  
, whe e
0
ep esen s a eloci y h eshold.
Since his model is a egula iza ion o he Coulomb ic ion model, hen i can be exp essed as
( )
( )
( )
0
e s e 0
i
min , sgn i



=


(6.7)
whe e
( )
is a unc ion o angen ial eloci y, ha in p ac ical e ms, ep esen s he alue o Coulomb
ic ion o ce,
( )
C
=
, o alues o angen ial eloci y ou side he ange de ined by he eloci y
h eshold
0
. Figu e 6.6 p esen s a g aphical ep esen a ion o his ic ion model.
Figu e 6.6 – G aphical ep esen a ion o he Ka nopp s a ic ic ion o ce model.
Inside he ha ched egion o he Ka nopp g aphical ep esen a ion, which is delimi ed by he
eloci y h eshold alue and i s symme ical alue,
0
and
0
−
, espec i ely, he angen ial eloci y is
s ( ) = C
- ( ) = - C- s
0
- 0
71
conside ed o be null, le ing he sys em change i s s a e and i s co esponding esponse om he kine ic
ic ion o ce alue o a s a ic (s ic ion) ic ion o ce alue,
s
. Thus, he ic ion o ce is de e mined as
(i) he alue needed o keep he sys em a ze o eloci y, ha is, he s a ic ic ion o ce le el,
s
, o (ii)
he b eakaway o ce h eshold ha ma ks he ansi ion o he kine ic ic ion o ce alue,
C
, which is
associa ed wi h he sliding phase.
6.2.2 Bengisu and Akay (1994)
As he Ka nopp model allowed he modelling o he s ic ion phenomenon, a simila s a egy was
adop ed by Bengisu and Akay (1994) in o de o model he S ibeck e ec , wi h he pu pose o being able
o co ec ly and mo e ealis ically ep esen he ic ion phenomena in p ac ical applica ions.
Ma hema ically, he p oposed model can be desc ibed by he ollowing unc ion
C
sgn( )(1 )[1 ( 1) ]
e e

−−
= − + −
(6.8)
whe e

and

a e pa ame e s ha can be ei he ela ed o he su ace oughness o he bonding abili y
o he su aces and
ep esen s he a io o he asymp o ic alue o he cu e o i s maximum, ha is,
he a io be ween he s a ic and kine ic ic ion o ce alues
sn
kn


=
(6.9)
Figu e 6.7 exhibi s he g aphical ep esen a ion o his d y ic ion model.
Figu e 6.7 – G aphical ep esen a ion o he Bengisu and Akay s a ic ic ion o ce model.
s
C
- C
- s
0
- 0
78
( )
( )
s e 0 e s
s 0 e s e
e
s e 0 e s e
0
( sgn( )) i
(2 1) sgn( ) i 0
,sgn( ) i 0
sgn( ) i
A
A
− + +   

−     

=    



(6.22)
whe e
2
2
00
32
A

=−


(6.23)
and
s
ep esen s he maximum alue o s a ic ic ion o ce and
0
being a eloci y ole ance simila ly
o he ones de ined in p e iously analyzed models o ma k he ansi ion be ween s icking and sliding
phases.
I is e i iable ha o alues o angen ial eloci y below he h eshold de ined by he eloci y
ole ance,
0
, he ic ion o ce is no only dependen o he angen ial eloci y, bu also dependen o
he ex e nal esul an o ce,
e
, as i was s a ed in he beginning o his sec ion. Figu e 6.9 shows he
g aphical ep esen a ion o his model in which he black lines ep esen he possible beha io s o his
ic ion model o he minimum and maximum alues o ex e nal esul an o ce.
Figu e 6.9 – G aphical ep esen a ion o Aw ejcewicz e al. s a ic ic ion o ce model.
s
e
- s
0
- 0
( )
( )
Maximum cu e o ( )
Minimum cu e o ( )

79
6.2.8 Specke e al. model (2014)
The s a ic ic ion model p oposed by Specke e al. (2014), alongside he Coulomb ic ion, is
capable o modeling he iscous and S ibeck e ec s h ough con inuous unc ions, which makes i ideal
o compu a ional applica ions due o i s capabili ies o a oiding nume ical ins abili ies.
The Coulomb ic ion in he analyzed model is desc ibed by he ollowing o mula ion
C k n
0
anh


=

(6.24)
whe e
k

is he kine ic ic ion coe icien ,
n
is he no mal o ce,
is he angen ial eloci y and
0
is a ansi ion eloci y ha ma ks he shi om s icking o sliding phase. This eloci y is one o he
a gumen s o he hype bolic angen unc ion used o egula ize he Coulomb ic ion o ce model, hus
being esponsible o he shape o his unc ion.
In wha conce ns he S ibeck e ec , his model conside s ha i can be modeled as
2
Sp
11
2
Sp
s C Sp
0 Sp
anh
c e



−







= − −




(6.25)
whe e
s
is he s a ic ic ion o ce le el,
C
is he Coulomb ic ion o ce le el,
Sp
is he S ibeck peak
eloci y ha de ines he ic ion o ce decay due o he S ibeck e ec , and
c
is a damping ic ion
coe icien .
The inal e m o his ic ion o ce model is ela ed o he iscous e ec and is exp essed as
c =
(6.26)
whe e he meaning o each a iable has al eady been desc ibed be o e.
The comple e o mula ion o his ic ion o ce model can be ob ained by summing he Coulomb,
S ibeck and iscous e ec e ms as ollows
2
Sp
11
2
Sp
k n s C Sp
0 0 Sp Viscous e ec
Coulomb S ibecke ec
anh anh
c e c




−





   
= + − − +

   
   

(6.27)
80
Like o he ic ion models, he disad an age o he Specke e al. model esides in he de ini ion
o he pa ame e alues, since i can only be made in a p ecise way ia expe imen al se ups. None heless,
he ansi ion eloci y and he S ibeck peak eloci y can be app oxima ed as
S 0 Sp
42 
(6.28)
being
S
he S ibeck eloci y.
An in e es ing ac o his model is ha he au ho s pe o med an adap a ion o i o a dynamic
ic ion model by in oducing a linea pa ame e - a ying (LPV) sys em in a o m o a i s -o de lowpass
il e , which allowed o cap u e hys e esis and memo y e ec s ha a e complex phenomena no mally
only modelled by dynamic ic ion o ce models (Specke e al. 2014).
6.2.9 Linea s a ic wi h s ic ion model (2016)
Ano he egula iza ion app oach o he Coulomb ic ion o ce model discon inui y a null ela i e
angen ial eloci y is he linea ic ion model wi h s ic ion (Ma ques e al., 2016). In p ac ical e ms, his
model conside s a peak o ic ion o ce co esponding o he alue o s a ic ic ion which occu s a he
eloci y h eshold,
0
, which ma ks he ansi ion whe e he ic ion o ce is independen o he angen ial
eloci y,
.
The ansi ion om he s a ic ic ion o ce le el,
s
, o kine ic ic ion o ce,
C
, e i iable in
he Coulomb ic ion o ce wi h s ic ion, is also cha ac e ized by ano he nume ical discon inui y, since
his ansi ion occu s o he null alue o angen ial eloci y. To o e come his p oblem, he s a egy
ollowed was o conside a linea unc ion be ween
0
and a second eloci y h eshold named
1
, ha
allows a con inuous ansi ion om s a ic o kine ic ic ion o ce le el.
The model can be ma hema ically exp essed by he ollowing sys em o equa ions (Ma ques e
al., 2016)
( )
s 0
0
0
s s C 0 1
10
C 1
sgn( ) i
sgn( ) i
sgn( ) i
 



−


= − −  

−






(6.29)
81
Figu e 6.10 shows he g aphical ep esen a ion o he linea ic ion model wi h s ic ion.
Figu e 6.10 – G aphical ep esen a ion o he linea s a ic ic ion o ce model wi h s ic ion.
An o e iew o he p esen ed model allows o conclude ha i s ma hema ical o mula ion is e y
simple o implemen in a compu a ional code and he ype o equa ions used (linea ) lead o e y low
compu a ional cos s. The disad an ages shown by his model a e he lack o capabili y o modelling mo e
complex phenomena such as S ibeck and iscous e ec s, and he sha p ansi ions be ween he di e en
b anches o he model’s ma hema ical o mula ion, ha could lead o nume ical ins abili ies.
6.2.10 B own and McPhee model (2016)
B own and McPhee (2016) p oposed a eloci y-based ic ion o ce model capable o cap u ing
s a ic, dynamic and iscous beha io s. These h ee di e en beha io s a e ep esen ed by h ee non-
coupled e ms which can be exp essed as
( )
n
0
C s C
2
2
0 n
Coulomb Viscous e ec
s eady s a e dynamic ic ion 0
S a ic ic ion (s ic ion)
anh 4 anh 4
13
44

   
= + − +
   

   


+




(6.30)
whe e he i s e m ep esen s he s eady s a e dynamic ic ion, which co esponds o he Coulomb
ic ion o ce le el, he second e m he s a ic ic ion (s ic ion) and he hi d one he iscous ic ion. This
non-coupled app oach is e y use ul and simple o implemen when ce ain ic ion phenomena, o
s
C
1
0
- 1- 0
- C
- s
82
example he iscous e ec , is no a a iable o in e es o he analysis o a ce ain sys em, which is done
by emo ing he co esponding e m om Equa ion (6.30).
In wha ega ds he meaning o he a iables,
C
ep esen s he Coulomb ic ion o ce le el,
is he angen ial eloci y,
0
is a h eshold o ole ance eloci y ha ma ks he ansi ion om s ick
o slip phase, which occu s o he maximum alue o s a ic ic ion o ce,
s
,

is he iscous
coe icien o ic ion,
n
is he no mal con ac o ce and
n
is a ansi ion no mal o ce which is used o
de ec when he iscous ic ion e ec s should be accoun ed. When
n n

, he iscous ic ion e m
has a signi ican impac on he ic ion o ce alue. Wi hou conside ing iscous e ec s, he hi d e m o
Equa ion (6.30) can be neglec ed and he ic ion o ce can be simply calcula ed as
( )
0
C s C 2
2
0
0
anh 4
13
44

= + −
 
 

+




(6.31)
This ic ion o ce model e eals a simple app oach o egula ize he Coulomb ic ion model,
h ough he use o a hype bolic angen unc ion and is able o cap u e s ic ion and iscous e ec in an
in ui i e o implemen model.
6.3 Benchma k p oblem – s a ic ic ion o ce models
To e alua e he beha io and compu a ional e iciency o he s a ic ic ion o ce models p esen ed
and analyzed in he p e ious sec ions, i was necessa y o selec a dynamic p oblem wi h signi ican
ele ance and well documen ed in li e a u e, ha is, a benchma k p oblem. This simple sys em is widely
used in li e a u e o alida e ic ion o ce models, and i was i s p oposed by Rabinowicz (1956).
Figu e 6.11 exhibi s he one deg ee o eedom sys em composed by a block wi h mass
m
which
is connec ed o a ixed e e ence by a sp ing wi h s i ness
k
and is placed on a con eyo bel ha mo es
wi h a linea speed
b
.
83
Figu e 6.11 – Schema ic ep esen a ion o he one deg ee o eedom benchma k p oblem.
Using he equa ions o dynamics i is possible o desc ibe he mass mo emen o his mechanical
sys em h ough he ollowing di e en ial equa ion
2
2
dx
m kx
d = − −
(6.32)
whe e
ep esen s he angen ial ic ion o ce.
The simula ion pa ame e s chosen o s udy he pe o mance, beha io and e iciency o he s a ic
ic ion o ce models analyzed in his sec ion a e p esen ed in Table 6.1, being pa icula ly based on he
ones used by Gon hie e al. (2004) and Ma ques e al. (2016).
Table 6.1 – Benchma k p oblem simula ion pa ame e s.
Pa ame e
Symbol
Value
Pa ame e
Symbol
Value
Mass o he block
m
1 kg
Ini ial eloci y
0
0.1 m/s
G a i y accele a ion
g
9.81 m/s2
Time-s ep
-
0.0001 s
Con eyo bel speed
b
0.1 m/s
Simula ion ime
-
40 s
Sp ing s i ness cons an
k
2 N/m
ODE Sol e
-
ode45, ode15s,
ode23s, ode78
Kine ic ic ion coe icien
μk
0.1
Sol e Rela i e
Tole ance, RelTol
-
1e-8
S a ic ic ion coe icien
μs
0.15
Sol e Absolu e
Tole ance, AbsTol
-
1e-8
Ini ial posi ion
x0
0 m
These pa ame e s we e hen in oduced in he in e ace shown in Figu e 6.12 which is an
in eg an pa o a compu a ional code dedica edly p og ammed in MATLAB o s udy he beha io o a
a ie y o ic ion o ce models in an in ui i e way.
k
m
b
x

84
Figu e 6.12 – Main window o he compu a ional code c ea ed in MATLAB o s udy he benchma k p oblem.
A e in oducing all he pa ame e s in he main window and hi ing he
Nex
bu on, he use is
aken o ano he window, p esen ed in Figu e 6.13, whe e i is possible o choose he models ha wan
o be s udied.
Figu e 6.13 – F ic ion o ce models selec ion window o he MATLAB code c ea ed.
85
In he nex sec ions a mo e de ailed s udy o he s a ic ic ion o ce models wi h and wi hou
s ic ion will be made using he p og ammed MATLAB code.
6.3.1 Pe o mance analysis o s a ic ic ion o ce models wi hou s ic ion
A e selec ing all he s a ic ic ion o ce models wi hou s ic ion inse ed in he MATLAB code, i
is necessa y o de ine he alues o ce ain speci ic pa ame e s o each model. Table 6.2 displays he
a ious pa ame e s conside ed o his ca ego y o models.
Table 6.2 – S a ic ic ion o ce models wi hou s ic ion speci ic pa ame e s.
Pa ame e
Symbol
Value
T ansi ion coe icien (Be na d)
kl
2000
Veloci y h eshold (Th el all, Rooney and
De a i, Th el all modi ied)
0
0.001 m/s
Veloci y h eshold (Amb ósio)
0
0.0001 m/s
Veloci y h eshold (Amb ósio)
1
0.001 m/s
The p e ious pa ame e s we e hen de ined in he s a ic ic ion o ce wi hou s ic ion window in
each espec i e box as isible in Figu e 6.14.
Figu e 6.14 – S a ic ic ion o ce models wi hou s ic ion window om he MATLAB compu a ional code o de ine
he speci ic pa ame e s o each model.
86
A e clicking he
Sol e
bu on, i is possible o ob ain he g aphs ha quan i y he beha io o he
sys em in analysis in e ms o posi ion, ela i e eloci y, accele a ion, and ic ion as unc ions o he
simula ion ime and which a e ep esen ed in Figu e 6.15 om a) o d).
a)
b)
87
c)
d)
Figu e 6.15 – G aphical esul s o he benchma k p oblem o he s a ic ic ion o ce models wi hou s ic ion a)
Posi ion [m] s Time [s] g aph b) Rela i e eloci y [m/s] s Time [s] g aph c) Accele a ion [m/s2] s Time [s] g aph
d) F ic ion o ce [N] s Time [s] g aph using he ode45 sol e .
A gene al o e iew o all he g aphs allows o conclude ha all he s a ic models wi hou s ic ion
ha e ema kably simila beha io s. Figu e 6.15 a) exhibi s an ini ial linea slope ha co esponds o he
94
Al hough no ep esen ed in Table 6.5, he a e age alue o he simula ion imes o all he sol e s
yields he Aw ejcewicz ic ion model as he mos e icien model in e ms o compu a ional elapsed ime,
while he Ande sson e al. model is he leas e icien . These esul s show once again ha he ode23s is
no a e y eliable sol e , since ha o he Ka nopp ic ion o ce model, he p oblem did no con e ge
o a solu ion.
Table 6.5 – Sol e and a e age simula ion ime o he s a ic ic ion o ce models wi h s ic ion.
Model
Sol e
A e age
Simula ion
Time [s]
Model
Sol e
A e age
Simula ion
Time [s]
Ka nopp
ode45
1.4239
Specke
ode45
1.5924
ode15s
1.9092
ode15s
2.2854
ode23s
DNC*
ode23s
1.4331
ode78
2.2380
ode78
2.4739
Bengisu and Akay
ode45
1.7400
Linea wi h
s ic ion
ode45
1.7217
ode15s
2.4848
ode15s
2.3644
ode23s
1.3448
ode23s
1.1405
ode78
2.6984
ode78
2.6100
Ande sson
ode45
2.6665
Holla s
ode45
1.6364
ode15s
2.3516
ode15s
2.2919
ode23s
1.2756
ode23s
1.2626
ode78
4.2416
ode78
2.5369
Aw ejcewicz
ode45
1.4188
B own and
McPhee
ode45
1.8278
ode15s
2.2201
ode15s
2.5609
ode23s
1.1813
ode23s
1.4607
ode78
2.1865
ode78
2.8658
*DNC – Did No Con e ge

95
6.4 Summa y and Discussion
This chap e has p esen ed a comp ehensi e analysis o 14 s a ic ic ion o ce models, comp ising
4 s a ic models wi hou s ic ion and 10 s a ic models wi h s ic ion. Each model ep esen s a egula ized
al e na i e o he Coulomb ic ion model, ha is, hey do no exhibi a nume ical discon inui y nea he
alue o null angen ial eloci y. The analyses ocused p ima ily on he compu a ional pe o mance,
physical ealism, and abili y o hese models o inco po a e phenomena such as iscous e ec s and he
S ibeck cu e.
The s a ic ic ion models wi hou s ic ion demons a ed ema kable compu a ional e iciency, wi h
beha io s ha a e la gely consis en ac oss all models. Howe e , hei inabili y o simula e s ic ion
phenomena limi s hei applicabili y in scena ios whe e a ealis ic ep esen a ion o s a ic ic ion is
equi ed. These models each only he kine ic ic ion o ce le el du ing simula ions, as e iden om hei
ic ion o ce s. ime esponses. While he absence o s ic ion simpli ies hei implemen a ion and
imp o es compu a ional pe o mance, i also cons ains hei use in simula ions demanding highe
ideli y.
In con as , he s a ic ic ion models wi h s ic ion o e enhanced capabili ies o ep esen ic ion
phenomena mo e ealis ically. These models success ully simula e he s ic ion phase, achie ing he
expec ed maximum s a ic ic ion o ce le el in line wi h heo e ical expec a ions. The dis inc ion be ween
s icking and sliding phases was e iden , wi h he sliding phase exhibi ing as e ansi ions compa ed o
he oscilla o y s ick-slip beha io obse ed in s a ic models wi hou s ic ion. This imp o ed ealism,
howe e , comes a he cos o inc eased compu a ional complexi y, which is s ill ela i ely low.
The compu a ional e iciency o he models was e alua ed using a benchma k dynamic p oblem
based on he 1 DoF p oblem p oposed by Rabinowicz, o which an in ui i e g aphical in e ace was
c ea ed in MATLAB. Fo s a ic models wi hou s ic ion, he Th el all modi ied model eme ged as he mos
compu a ionally e icien , achie ing an a e age simula ion ime o 1.2801 s using he ode23s sol e .
Howe e , he Coulomb ic ion model, while simple in concep , exhibi ed he leas compu a ional
e iciency and ailed o con e ge unde ce ain condi ions, unde sco ing i s limi a ions in nume ical
applica ions due o ins abili ies in he non-di e en iable egion o null ela i e eloci y.
Among s a ic models wi h s ic ion, he Linea wi h s ic ion model demons a ed he sho es
a e age simula ion ime o 1.1405 s wi h he ode23s sol e . Con e sely, he Ande sson e al. model was
he leas compu a ionally e icien , wi h an a e age ime o 4.2416 s using he ode78 sol e . The esul s
also indica ed ha he eliabili y o sol e s like ode23s can a y signi ican ly, as e idenced by non-
con e gence in some scena ios.
96
The pe o mance disc epancies among sol e s, as obse ed in his s udy, u he unde sco e he
impo ance o selec ing app op ia e nume ical sol e s o ic ion simula ions. The esul s sugges ha
while ode23s o en yields as e compu a ions, i s eliabili y can be p oblema ic, necessi a ing ca e ul
conside a ion o sol e choice based on he speci ic ic ion model and dynamical sys em in analysis.
The indings o his chap e ein o ce he impo ance o ailo ing s a ic ic ion o ce models o he
speci ic needs o a gi en applica ion. S a ic ic ion models wi hou s ic ion a e well-sui ed o scena ios
equi ing high compu a ional e iciency and mode a e ideli y, while s a ic models wi h s ic ion p o ide a
be e ep esen a ion o eal-wo ld ic ion phenomena, by being able o ep esen ing he s a ic ic ion
o ce phenomenon, ha ing simila compu a ional demands.
97
7. DYNAMIC FRICTION MODELS
S a ic ic ion models a e no capable o cap u ing ce ain complex phenomena, such as
hys e esis o ic ional lag. Wi h his p oblem in mind, a mo e complex and p ecise ca ego y o ic ion
models is in oduced – he dynamic ic ion o ce models.
Dynamic ic ion models no only depend on he angen ial ela i e eloci y be ween con ac ing
su aces bu also inco po a e a s a e a iable, o en ep esen ed by b is le de lec ion, as mos dynamic
models a e o mula ed a he aspe i y le el. A s a e a iable can be de ined as a ma hema ical a iable
ha cap u es he condi ion o con igu a ion o a ce ain dynamic sys em in a ce ain ins an o ime. This
a iable is hen explici ly upda ed using an upda e exp ession a he beginning o end o each ime s ep
o he nume ical simula ion.
The use o a s a e a iable enables he ma hema ical o mula ion o dynamic ic ion models o
be desc ibed by con inuous unc ions ha no only co ec ly desc ibe he ic ion phenomenon a he
aspe i y le el, bu also elimina e he nume ical ins abili ies no mally e i ied in he p oximi y egion o he
null alue o angen ial eloci y.
Despi e i s enhanced p ecision in p edic ing he ic ion o ce ac ing on he con ac in e ace
be ween bodies ha exhibi ela i e mo ion, hei inc eased compu a ional cos and complexi y make
some o hese models ha dly usable o p ac ical enginee ing asks. Ano he majo p oblema ic associa ed
wi h hese aspe i y le el models is he di icul y o choose pa ame e alues o model he physical beha io
o su ace b is les, which o en equi es calib a ion p ocedu es u ilizing expe imen al da a.
In his sec ion, simila ly o wha has been done o he s a ic models, he dynamic ic ion o ce
models analyzed appea in ch onological o de o appea ance, being he s udied ones p esen ed in he
schema ic o Figu e 7.1.
Figu e 7.1 – Analyzed dynamic ic ion o ce models.
Dahl
(1968)
Liang e al.
(2012)
Dynamic ic ion o ce models
B is le
(1991)
Rese In eg a o
(1991)
LuG e
(1995)
Dankowicz
(1999)
Leu en
(2000)
Elas o-Plas ic
(2000)
GMS
(2004)
Gon hie e al.
(2004)
Bliman and So ine
(1993)
98
7.1 Dynamic ic ion o ce models analyzed – o mula ions
7.1.1 Dahl model (1968)
A new ma hema ical model o solid ic ion was o mula ed and p esen ed by Dahl (1968) wi h he
main objec i e o desc ibing ic ion o bo h sliding and olling ic ion ha can be used in simula ions o
dynamic sys ems in ol ing mechanical elemen s ha a e subjec ed o ic ion.
The Coulomb ic ion concep was in oduced based on expe imen al da a and i is widely
accep ed as a physical mac o-phenomenon. In his model, a new heo y eme ged by es ablishing a
compa ison be ween ic ional beha io and he quasi-s a ic p ope ies o ma e ials. When an ex e nal
o ce is applied o a ma e ial, due o he ine ial p ope ies o i , a esis ance o he de o ma ion is o e ed
by he ma e ial. In ic ional e ms, his esis ance o de o ma ion can be hough o as he ic ion o ce
ha de elops a he in e ace be ween he bodies and p e en s he angen ial ela i e mo ion be ween
hem.
Wi h ha being said, s ic ion, he maximum alue o s a ic ic ion, can be di ec ly co ela ed wi h
he ul ima e ensile s ess,
UTS

, o a s ess-s ain cu e, since om his poin onwa ds, he applied o ce
will lead o a con inuum plas ic de o ma ion o he b is les be ween bo h con ac ing su aces un il
Coulomb ic ion o ce is eached. This kine ic ic ion o ce le el can be di ec ly co ela ed wi h he up u e
poin o a ma e ial in a s ess-s ain cu e whe e he con ac ing su aces will exhibi ela i e angen ial
mo ion, ha is, will s a o slide (Dahl 1968).
I was ound ha his analogy o de o ma ion o ma e ials can be dis inc when b i le and duc ile
ma e ials a e in con ac due o di e ences be ween he s ess-s ain cu es o hese ypes o ma e ials.
Thus, o ma e ials ha ha e a b i le beha io , he s ic ion (s a ic ic ion) and Coulomb ic ion a e
indis inguishable, as i can be obse ed in
Figu e 7.2 a) and o duc ile ma e ials, he ul ima e s ess poin will co espond o he maximum
s a ic ic ion alue (s ic ion) and Coulomb ic ion o ce will be co ela ed wi h up u e poin as i is
ep esen ed in
Figu e 7.2 b) and as i was s a ed in he p e ious pa ag aph.
99
a)
b)
Figu e 7.2 – Analogy o b is le elas ic and plas ic de o ma ion wi h ma e ial’s beha io a) b i le ma e ial b) duc ile
ma e ial. Adap ed om Dahl (1968).
In wha conce ns he ma hema ical o mula ion o his model, Dahl (1968) desc ibed his
de o ma ion and hys e e ic beha io o ma e ials as dependen o bo h he ela i e displacemen and
eloci y o he con ac ing su aces, being he s ess s ain cu e de ined by a di e en ial equa ion as
0
CC
1 sgn 1 sgn( )
d
dx



= − −


(7.1)
whe e
co esponds o he ic ion o ce,
x
is he displacemen ,
0

deno es he s i ness o he b is les
and

is a pa ame e ha de ines he shape o he ma e ial cu e, being dependen o he ma e ial and
usually a ying be ween 0 and 1.
I is possible o no ice ha Equa ion (7.1) is a unc ion o displacemen , bu acco ding o Dahl
(1968) i can be di e en ia ed in o de o ime, leading o
( )
0
CC
1 sgn 1 sgn
d d dx d
d dx d dx



= = = − −


(7.2)
In he case o conside ing he pa ame e

being equal o 1, which is he mos common case,
hen Equa ion (7.2) can be w i en as
( )
0
C
1 sgn
d
d


=−


(7.3)
The Dahl model is conside ed o be a dynamic ic ion model, so i mus ha e a s a e a iable
which desc ibes he dynamic beha io o he sys em in analysis. As his o mula ion conside s he
Rup u e
S ic ion and Coulomb
F ic ion Le els


Rup u e
Coulomb
F ic ion
Le el
Ul ima e
S ess
S ic ion
Le el



100
exis ence o b is les, hen he s a e a iable is he b is le de lec ion, ep esen ed by he pa ame e z, and
assuming ha hese en i ies exhibi a linea sp ing beha io ha can be desc ibed as
0
z

=
(7.4)
hen he a e o b is le de lec ion is de ined as
0
C
1 sgn( )
dz z
d


=−


(7.5)
which when he sys em eaches he s eady s a e yields
C
sgn( ) =
(7.6)
being Equa ion (7.6) simply he Coulomb ic ion o ce model.
The analyzed app oach allows o model bo h p e-sliding displacemen as well as Coulomb ic ion,
h ough he in oduc ion o a s a e a iable ela ed o he b is le’s de lec ion, bu he main d awbacks o
his model a e i s incapabili y o modelling he S ibeck e ec and o cap u e s ic ion, which led o he
o mula ion o new ic ion models based on he Dahl model, such as he Bliman and So ine (1993,
1991) and he LuG e model (Canudas De Wi e al., 1995).
7.1.2 B is le model (1991)
The con inuous sea ch o a ic ion o ce model able o ealis ically model he physical
phenomena and simul aneously exhibi good compu a ional pe o mance was he main goal o Haessig
and F iedland (1991). These au ho s s a ed ha ic ion o ce should be de ined as a unc ion o
displacemen a he han angen ial eloci y, so he model p oposed mus be able o desc ibe he s icking
phase o ic ion as a unc ion o displacemen and du ing he slipping phase he ic ion o ce could be
desc ibed as a unc ion o angen ial eloci y.
The ic ion be ween su aces is hen gi en by he in e ac ion be ween a N numbe o b is les in
which each b is le con ibu es wi h an in ini esimal ac ion o he o al ic ion o ce. Wi h ha being said,
he o al ic ion o ce can be exp essed as
( )
i i i
1
N
i
x b

=
=−

(7.7)
whe e
i

is he s i ness o each b is le,
i
b
is he e e ence posi ion ela ed o he ini ial bond o he b is les
and
i
x
is he is he ela i e posi ion o he b is les.
101
Figu e 7.3 illus a es he physical in e ac ion be ween wo b is les and he co esponding
de lec ion o he lowe one.
Figu e 7.3 – Schema ic ep esen a ion o b is le de lec ion and loca ion o he bond be ween aspe i ies.
The di e ence be ween he las wo a iables indica ed be o e (
i
b
and
i
x
) ep esen s he ela i e
displacemen ha should be used as a s a e a iable o desc ibe he s icking phase o he ela i e mo ion
be ween su aces. When he s ain a a ce ain b is le exceeds i s limi ,
( )
i i i
xb = −  
, he up u e
o ha bond occu s, and a new bond is o med in a new loca ion, which is calcula ed om he p e ious
one.
The bond be ween a igid b is le and an elas ic b is le is des oyed when a c i ical de lec ion poin
is eached and hen a new bond is andomly c ea ed be ween he igid b is le and ano he di e en elas ic
b is le. The new b is le loca ion can be de e mined as
' i i
uni o m( )sgn( )
i
bb

= + 
(7.8)
whe e
i
b
is he e e ence bond loca ion and

he new b is le ange.
Al hough he b is le model was designed o accu acy, his comes a a cos , ha in his case is
he compu a ional e iciency. Despi e conside ing a ini e ela i ely small numbe o b is les, ha should
be less han 50 as s a ed by he au ho s (Haessig & F iedland, 1991), because o he spacing be ween
b is les and he discon inui ies ela ed o he snapping o b is le bonds, his model becomes
x
xibi
102
compu a ionally ine icien , equi ing e y small in eg a ion s eps o p o ide p ecise esul s. This is he
main eason why his model is no commonly used.
7.1.3 Rese In eg a o model (1991)
The same au ho s o he b is le model exposed in he p e ious sec ion p oposed a new app oach
o educe he compu a ional ime equi emen s, while s ill being capable o accu a ely ep esen ing and
modelling he s ick-slip ic ion phenomena. Haessig and F iedland (1991) o mula ed a model ha does
no conside he use o mul iple b is les o accu a ely ep esen he bonding e ec o s ick-slip phenomena
in he s icking phase, bu ins ead conside s his s ick phase o he ic ion o ce as a unc ion o posi ion
o a single b is le.
The inpu o he in eg a o used in his model is he s ain a e o he b is le de o ma ion, ha can
assume he ollowing alues
0
0
0 i 0
i 0
z z z
dz
z z z
d
   

=   

(7.9)
whe e
is he angen ial ela i e eloci y, z is he b is le de lec ion and
0
z
he p e-de e mined maximum
alue o he b is le de lec ion.
To de e mine he co esponden ic ion o ce o his model i is necessa y o use wo di e en
equa ions: one o ep esen he s icking phase un il eaching he b eakaway o ce and ano he o desc ibe
he sys em’s beha io du ing he sliding phase as i is exp essed below
0 1 0
0 0
( )(1 ) i
( ) sgn( ) i
dz
a z z
d
z z z


+ + 

=


(7.10)
whe e
0

is he b is le s i ness,
a
is a coe icien ela ed o he s ic ion phenomena, no mally
ep esen ing he a io be ween he s ic ion load,
s
, and he sliding s eady-s a e load,
C
, and
1

is a
damping coe icien used o elimina e unwan ed oscilla ions in he s icking phase and can be de ined as
10
0.707 m

=
(7.11)
whe e
m
ep esen s he mass o he sliding body.
103
In conclusion, his model equi es signi ican ly less compu a ional ime han he p e iously
p oposed model (B is le model, 1991), since i exhibi s a smoo h and con inuous beha io , so he small
ime s eps associa ed wi h b eaking he b is les is no a manda o y equi emen .
7.1.4 Bliman and So ine model (1993)
The ic ion models de eloped by Bliman and So ine (1993; 1991) a e dynamical models based
on he wo ks o Rabinowicz (1951; 1957; 1961). The models p esen ed in his sec ion ul il bo h
dependence on he sign unc ion o he angen ial eloci y,
sgn( )
, and he space a iable s, which
esul s in he in eg al o m
0()s d=  

(7.12)
o eplace he ime a iable,
, wi h he space a iable, s, h ough a a iable ans o ma ion p ocess.
Conside ing ha
s
x
is he absolu e ela i e displacemen o he con ac ing bodies, hen he model
can be exp essed as a linea sys em o equa ions
s
ss
s s
dx Ax B
ds
Cx D
=+


=+

(7.13)
This model depends on he sign unc ion o he angen ial eloci y,
sgn( )
, which in his case is
ma hema ically ep esen ed by he a iable
s
, being
( )
s
sgn =
The i s model analyzed is a i s -o de model, cha ac e ized by i s educed complexi y, which
uses he ollowing a iables
1
110A B C D


= − = = =
which esul s in he model o m
1
1
d d ds d
d ds d ds


= = = −


(7.14)
This o mula ion is ema kably simila o he Dahl model, whe e
1
C1
1


= = =
110
in which
ss
()F
is he ic ion o ce alue in s eady s a e, i is an in ege coe icien and
( , )z

is a
piecewise con inuous unc ion used o cap u e s ic ion, since i allows he elas ic displacemen o be
de eloped un il eaching he b eakaway o ce. This unc ion depends on he egime (elas ic o plas ic) o
he displacemen as i is ep esen ed
ba
ba ss
ss
0 i sgn( ) sgn( )
0 1 i sgn( ) sgn( ) ( )
( , ) 1 i sgn( ) sgn( ) ( )
0 i sgn( ) sgn( )
z z z
z z z z
z z z z
z


=  

  =   

==  



(7.33)
whe e
ba
z
is he b eakaway b is le de lec ion and
ss
()z
is he s eady-s a e b is le de lec ion, which
co esponds o he maximum alue o de lec ion,
ss max
()z z=
. The ela ion
( )
ba
ss
0.7
z
z 
can be used
o de e mine he alue o he b eakaway de lec ion,
ba
z
.
In he speci ic case o small ib a ions, he
( , )z

unc ion o he
ba ss
()z z z 
in e al
can be equal o
ss ba
ss ba
()
11
2
( , ) sin
2 ( ) 2
z z
z
z z z

+


−


=   +
−



(7.34)
The use o he sine unc ion is explained by he ac o being e y use ul o s udy he ansi ion
beha io om s a ic o kine ic ic ion o ce le els in ic ion o ce models. Despi e i s accu acy, his model
s ill p esen s some p oblems ela ed o he de e mina ion o he ma hema ical desc ip ion o 𝛼(𝑧, 𝑣𝑡)
unc ion, since i s shape canno be in ui i ely chosen due o he inexis ence o a physical ela ion wi h he
ic ion phenomena.
7.1.9 Gene alized Maxwell Slip model (2003)
The Gene alized Maxwell Slip Model (GMS), p oposed by Lampae e al. (2003), is a ic ion
o ce model based in physical p ope ies o in e ac ing su aces which simula es he con ac physics a
he aspe i y le els. This model can p edic p e-sliding egime, S ibeck e ec , ic ional lag, b eakaway
o ce and s ic ion beha io s.

111
The e a e wo main ca ego ies o phenomena in oduced in his model: (i) ic ion mechanics and
(ii) he aspe i y con ac scena io. In wha conce ns he ic ion mechanics, i is possible o e alua e and
analyze phenomenological mechanisms such as no mal c eep o con ac ing aspe i ies, adhesion aising
and hys e esis losses due o aspe i y de o ma ion. Rega ding aspe i y con ac , his ca ego y can be
di ided in o h ee di e en phases: (i) he aspe i y o one con ac su ace is ee, hus no con ac ing o
su e ing any kind o bending solici a ion om he o he su ace; (ii) he aspe i y o one su ace is in
con ac wi h he opposed su ace which leads o a de o ma ion ha can be exp essed as a unc ion o
he ela i e displacemen be ween he in e ac ing su aces; (iii) he su aces a e no longe in con ac and
all he de o ma ion ene gy is dissipa ed.
This analysis leads o he ep esen a ion o ic ion as he pa allel in e ac ion o 𝑁 elemen a y
single s a e ic ion models as i is ep esen ed in Figu e 7.7.
Figu e 7.7 – Physical in e p e a ion o he Gene alized Maxwell Slip ic ion model as he pa allel in e ac ion o N
elemen a y single s a e ic ion models.
Each one o he elemen s o his model is desc ibed by he same dynamical model, which is
ma e ialized by he bending beha io o b is les as sp ings, whe e he displacemen and eloci y applied
o each one o he elemen a y models is he same. The s icking and sliding phases o each one o he
elemen a y ic ion models is e alua ed o desc ibe he gene al esponse o he sys em in analysis.
(…)
(…)
x
z1
zi
k1
ki
kN
zN
112
The GMS model is based on h ee ic ion p ope ies: (i) he S ibeck cu e o cons an eloci ies,
(ii) he hys e esis unc ion wi h non-local memo y in he p e-sliding egime, and (iii) he ic ional memo y
in he sliding egime (Lampae e al. (2003)).
Al-Bende e al. (2005) s a ed ha i an indi idual ic ion elemen i is s icking, hen he b is le
de lec ion a e can be exp essed as
i
dz
d =
(7.35)
whe e
is he angen ial eloci y. This egime main ains un il he de lec ion o he b is le equals he alue
o he S ibeck eloci y in sliding,
( )
i i
z s =
.
Fo he slipping phase o an indi idual ic ion elemen i, he b is le de lec ion a e can be
desc ibed as
( )
ii
i
i
sgn( ) 1
dz z
C
d s

=−


(7.36)
whe e
i
C
is an a ac ion pa ame e which ep esen s a gain, simila o he p opo ional e m o a PID
con olle , ha de e mines how as
i
z
con e ges o
i
s
. Equa ion (7.36) is alid un il he ela i e eloci y
o he elemen a y model eaches a null alue.
Since he s icking and slipping equa ions o he s a e a iable z a e w i en o each indi idual
ic ion elemen i, hen o de e mine he o al ic ion o ce ac ing on a sys em is necessa y o sum all he
ou pu s o he N elemen a y ic ion models as
( ) ( )
i
i i i
i1
Ndz
k z
d

=

= + +



(7.37)
whe e
i
k
is he s i ness o each elemen a y model sp ing,
i

is a damping coe icien and
( )
is a
iscous unc ion which is de ined o model he iscous componen . This unc ion is e e ed o as eloci y
s eng hening, since i is p opo ional o he angen ial eloci y.
I is possible o no ice ha each elemen is cha ac e ized by ou di e en pa ame e s: a s i ness
i
k
, a damping coe icien
i

, an a ac ion pa ame e
i
C
and a S ibeck eloci y unc ion
i
s
. The
de ini ion o
i
s
in oduces one mo e unknown pa ame e
i

, which is he ecip ocal o he S ibeck
eloci y o elemen i. To sol e his p oblem wi hou in oducing addi ional unknown pa ame e s no
113
sac i icing he essence o he p oblem, a
i
a iable mus be de ined which ac s as a scale pa ame e
o all he elemen al ic ion models exis ing on he sys em.
Al hough being e y p ecise and ai h ul o expe imen al da a, his dynamic ic ion o ce model
has he disad an age o equi ing a lo o compu a ional esou ces, needing a ound 1000 equi alen
aspe i ies o be modeled o ob ain smoo h esponses. Thus, i s main applica ions can be associa ed o
he alida ion o simple ic ion o ce models which a e mo e sui able o p ac ical con ol and mul ibody
applica ions.
7.1.10 Gon hie e al. model (2004)
The Gon hie model, p oposed by Gon hie e al. (2004), is a egula ized h ee-dimensional
con ac o ce model wi h asymme ic damping and dwell- ime dependen ic ion. Since he ocus o his
wo k is o desc ibe he ic ion o ce models in a comp ehensi e way, only he angen ial componen o
his model will be s udied in de ail.
In his model, he ic ion o ce is also de ined as a unc ion o b is le de lec ion, simila ly o he
app oach p oposed in he LuG e model by Canudas De Wi e al. (1995), being exp essed as
b 0 1
dz
z
d

=+
(7.38)
whe e
0

is he s i ness o he b is les, z he b is le de lec ion,
1

a damping coe icien and
dz
d
he a e
o b is le de lec ion.
The no el y in oduced wi h his model is di ec ly ela ed o he b is le s a e o de lec ion. This
de lec ion was hen di ided in o wo componen s: a s a ic a e o de lec ion de ined o he s icking phase
s
dz
d
, and a dynamic a e o de lec ion de ined o he sliding phase
sl
dz
d
. The gene al o mula ion o he
de lec ion a e is p esen ed below
s sl
(1 )
dz dz dz
ss
d d d
= + −
(7.39)
whe e
s
is a s icking s a e unc ion de ined as
2
S
se

−

=
(7.40)
in which
ep esen s he angen ial ela i e eloci y and
S
he S ibeck eloci y.
114
I is hen in ui i e o unde s and ha when
ends o ze o, he de lec ion a e becomes equal
o he s a ic de lec ion a e,
s
dz dz
d d
=
, and when
s
ends o ze o, he de lec ion a e is p ac ically equal
o he dynamic de lec ion a e,
sl
dz dz
d d

. Mo eo e , in he s icking egime, he b is le de lec ion will
co espond o he angen ial eloci y be ween he in e ac ing su aces,
s
dz
d =
, and in he sliding
egime he b is le de lec ion will be de ined in e ms o he Coulomb ic ion o ce
C
, since i ep esen s
he s eady s a e o he sys em in analysis.
The Coulomb ic ion o ce opposes he ex e nal o ce applied o a sys em and can be de ined as
( )
C C 0 0
di , =
(7.41)
whe e C ep esen s he Coulomb ic ion o ce le el and
( )
0 0
di ,
is a piecewise unc ion o bo h
angen ial eloci y
and a small eloci y ole ance
0
which e u ns a uni ec o along he
di ec ion
exp essed as
( )
0
3
0 0
0
0 0 0
i
di , 31 i
22



=


−





(7.42)
The eloci y ole ance,
0
, is used o educe he discon inui ies in eloci y di ec ion and should
be a leas one en h o
S
, being commonly conside ed o be equal o one hund ed h o
S
.
Combining Equa ion (7.39) wi h he assump ion ha
sl
dz dz
d d

esul s in a dynamic b is le
de lec ion equal o
0
C
11
1
sl
dz z
d


=−
(7.43)
which leads o a gene al b is le de lec ion equa ion o mula ed as
22
SS 0
C
11
1
1
dz e e z
d


   
−−
   
   



= + − −




(7.44)
115
To co ec ly model he ansi ion exis en in he s ick-slip mo ion due o he ic ional lag
associa ed wi h he dwell- ime, i is necessa y o include a new s a e a iable
dw
s
. This a iable is a
unc ion de ined by wo ime cons a s ha allow he co ec modelling o dwell- ime e ec on he maximum
s a ic ic ion o ce and can be exp essed as
( )
( )
dw dw
dw
dw
dw dw
b
1i 0
1i 0
s s s s
s
s s s s
− − 


=
− − 


(7.45)
whe e
dw

is he dwell- ime dynamics ime cons an and
b

is he b is le dynamics ime cons an which
can be de ined as
1
b
0


=
.
The maximum s a ic ic ion (s ic ion) o ce can be de ined as
( )
max C s C dw
s= + −
(7.46)
Conside ing ha he ic ion o ce applied by he b is le
b
should be equal o he Coulomb
ic ion o ce
C
when he con ac ing bodies a e sliding,
b C
=
, hen he ic ion o ce can be
modelled as
b 2 b max
b
max 2 b max
b
i
i


+


=+


(7.47)
whe e
2

is a iscous damping coe icien .
This ic ion model is hen cha ac e ized by eigh (o se en, i i is conside ed ha he eloci y
ole ance, 0, is de ined as a unc ion o he S ibeck eloci y, S) di e en pa ame e s:
k

,
s

,
0

,
1

,
2

,
0
,
S
and
dw

.
7.1.11 Liang e al. (2012)
The Liang e al. (2012) b is le model is an ex ension o he b is le model p oposed by Haessig
and F iedland (1991) o h ee dimensional mul ibody sys ems ha can be used o desc ibe he angen ial
con ac be ween gene al igid o nea - igid body con ac dynamics applica ions.

116
This model consis s in ea ing he con ac be ween in e ac ing su aces as mul iple indi idual
con ac s, ma e ialized by b is les. In Figu e 7.8 is ep esen ed he con ac be ween wo di e en bodies,
Body 1 and Body 2 in wo di e en ime ins an s,
and
+
.
Figu e 7.8 – Con ac be ween o di e en bodies (Body 1 and Body 2) in wo di e en ins an s,
and
+
.
Fo a ime ins an ,
, he con ac poin be ween bo h bodies is he poin
1
P
. I he sys em is in
a s icking phase, hen o he ins an
+
he wo bodies emain in he same ela i e posi ion, so he
poin o con ac be ween bodies 1 and 2 is s ill
1
P
. On he o he hand, i one body mo es ela i ely o he
o he , hen he con ac poin will change om
1
P
o
2
P
. Then, he mo ion o hese b is les can be
desc ibed as a sp ing like beha io ha exhibi s s e ch and o a ion a ound he common angen ial plane,
being he ic ion o ce compu ed as he a e age displacemen o hese elemen s, as depic ed in Figu e
7.9 a) and b).
Figu e 7.9 – a) Rep esen a ion o maximum s a ic
(𝑧s max), and kine ic (𝑧k max) de lec ions o a gi en ac ing
o ce 𝐹 in a ce ain ime ins an and b) o ano he ime ins an gi en by 𝑡 + ∆𝑡, whe e he b is le de lec ion
shows an expansion and o a ion a ound he common angen ial plane.
Body 2
Body 1 a ins an Body 1 a
ins an
Tangen ial plane
a ins an
Tangen ial plane
a ins an
a) b)
117
The main ad an ages o his model a e ha i is a s andalone model, so i can be easily
implemen ed and in eg a ed in o an exis ing gene al dynamics sol e o mul ibody dynamical sys ems,
and i is able o desc ibe bo h s a ic and dynamic ic ion in he same model and simula e usual ic ional
phenomena (Liang e al. 2012).
In wha conce ns each con ac poin , he h ee-dimensional ic ion o ce can be desc ibed by
Hooke’s law in he o m o
0
z

=−
(7.48)
whe e
0

is he b is le s i ness and
z
is a h ee-dimensional ec o ha ep esen s he a e age b is le
de lec ion o displacemen , which can be exp essed as
( )
0
0 max
max max
( ) i ( )
()
( ) i ( )
()
z d z z
z
z z z
+


=




(7.49)
whe e
0
is he ini ial ime ins an co esponding wi h he beginning o con ac ,
is he cu en ime o
con ac ,
()
ep esen s he h ee-dimensional ec o o angen ial eloci y and
()
is simply he
magni ude o angen ial eloci y.
The maximum b is le de lec ion in a gi en ime ins an ,
max ()z
, can be de ined as
C
max 0
0
max
s
max 0
0
()
( ) i
() ()
( ) i
k
s
z
z
z


=


=
=


(7.50)
in which
C
is he Coulomb ic ion o ce ac ing in each ins an ,
0
is a eloci y h eshold ha ma ks he
ansi ion be ween s icking and sliding phases and
s
is he s a ic ic ion o ce ac ing in each ins an . I
is also impo an o no ice ha his de ini ion o he maximum b is le de lec ion makes a dis inc ion
be ween alues o he maximum s a ic de lec ion
smax ( )z
, and maximum kine ic de lec ion
k max ()z
,
as i is isible in Figu e 7.9.
The model o mula ion and he co esponding illus a ion p esen ed in Figu e 7.9 a) and b) show
ha his model app oxima es he b is le beha io o he one exhibi ed by a linea sp ing. Thus, his can
lead o nume ical ins abili ies in he ansi ion be ween s ick and slip s a e o mo ion due o excessi e
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s i ness and/o oscilla o y beha io o he sys em. To p e en he noise gene a ed by high equency
ic ion o ce a ia ion, ha can lead o nume ical discon inui ies, i is in oduced a b is le damping
coe icien
1

which leads he ic ion o ce ma hema ical exp ession o be equal o
0 1
dz
z
d

= − −
(7.51)
This ic ion o ce model educes o he Coulomb ic ion o ce model when he ic ion is in sliding
egime and while he sys em in analysis is in he s icking phase, he ic ion o ce magni ude is below he
one co esponding o he s ic ion h eshold. To ully de ine his ic ion o ce model i is necessa y o
speci y he alues o i e pa ame e s: he kine ic coe icien o ic ion
k

, he s a ic coe icien o ic ion
s

, he eloci y h eshold
0
, he b is le s i ness
0

, which should be de ined, by ule o humb, wi h
a alue o one o de o magni ude (101) highe han he la ges dominan s i ness o he sys em in
analysis, and he b is le damping coe icien
1

.
7.2 Benchma k p oblem – dynamic ic ion models
A e p esen ing he mos commonly u ilized dynamic ic ion o ce models in li e a u e, i becomes
impe a i e o assess he compu a ional beha io and e iciency o his ca ego y o models. The MATLAB
compu a ional p og am b ie ly desc ibed in he p e ious chap e abou s a ic ic ion o ce models, was
also adap ed o s udy he one deg ee o eedom dynamic sys em o he sliding block using dynamic
ic ion o ce models.
The selec ion o his ype o model implies he de ini ion o he speci ic pa ame e s o each model.
Table 7.1 displays he a ious pa ame e s conside ed o he dynamic ca ego y o ic ion o ce models.
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Table 7.1 – Dynamic ic ion o ce models speci ic pa ame e s.
Pa ame e
Symbol
Value
Pa ame e
Symbol
Value
Ini ial b is le de lec ion
(common o all models)
0
z
0 m
S ic ion- s eady
s a e a io
(Rese In eg a o )
a
0.52
S i ness coe icien
(common o all models)
0

105 N/m
In ege powe
(Elas o-Plas ic)
i
1
Damping coe icien (LuG e,
Rese In eg a o , Elas o-
Plas ic, Gon hie , Liang)
1

√105 N⋅s/m
Ini ial dwell- ime
(Gon hie )
dw
s
0
Viscous coe icien (LuG e,
Elas o-Plas ic, Gon hie )
2

0.1 N⋅s/m
Dwell- ime
dynamics cons an
(Gon hie )
dw

2
S ibeck eloci y (LuG e,
Elas o-Plas ic, Gon hie )
S
0.001 m/s
Veloci y h eshold
(Liang e al.)
0
0.001 m/s
The p e iously speci ied pa ame e s we e hen in oduced in he espec i e in e ace window o
he MATLAB p og am shown in Figu e 7.10.
Figu e 7.10 – Dynamic ic ion o ce models window om he MATLAB compu a ional code o de ine he speci ic
pa ame e s o each model.