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NonlinearDyn
/s11071-021-06538-5
ORIGINALPAPER
Slippingrollingtransitionsofabodywithtwocontactpoints
MateAntali·GaborStepan
Received:17October2020/Accepted:12May2021
©TheAuthor(s)2021
AbstractInthispaper,thegeneralkinematicsand1Introduction
dynamicsofarigidbodyisanalysed,whichisincon-
tactwithtworigidsurfacesinthepresenceofdryfric-Inmodellingandanalysisofmechanicalsystems,one
-ofthemostchallengingissuesisthecontactprob-
tactpoint,-lembetweenthebodies,whichleadstononsmooth
pointrollingcase,thecontactforcesareundetermined;-
consequently,theconditionofthestaticfrictionforcesicalsources,fromtheunilateralpropertyofthenormal
cannotbecheckedfromtheCoulombmodeltodecidecontactforces,andfromthedryfrictioncharacteristics
whethertwo-,thisofthetangentialcontactforces.
issuecanberesolvedwithinthescopeofrigidbodyAtthetangentialcontactofrigidbodies,wedis-
dynamicsbyanalysingthenonsmoothvectorfieldoftinguishaslippingcontactstateandastaticcontact
,thesestatesaredefined
codimension-2discontinuities,amethodispresentedlocally,andthecumulatedfrictioneffectcanbecom-
todeterminetheconditionswhenthetwo-pointrollingputednumericallyorcanbeinterpretedbydifferent
[13,14].However,whenthestiffnesses
ofthecontactingbodiesarelargeenough,therigidbody
KeywordsNonsmoothdynamics·Filippovsystems·modelisanacceptableassumption,andatthecontact
Coulombfriction·Slippingstateofthediscretecontactpoint,wecansharplydis-
tinguishtheslippingandrolling–,
theCoulombmodelandsimilardryfrictionmodels
includebothaconstraintofrolling–stickingandaforce
lawofslipping(see[16],[17],or[21]foranoverview).
(B)·
DepartmentofAppliedMechanics,BudapestUniversityofbetweenthetwostates,aswell;thesetransitionschange
TechnologyandEconomics,Budapest1521,Hungarynotjusttheequationsbutalsothedimensionofthe
e-mail:******@.
-
MTA-BMEResearchGrouponDynamicsofMachinesandels,thepresenceofmultiplecontactpointsbringsout
Vehicles,HungarianAcademyofSciences,Budapest1521,
Hungarymanyissuesrelatedtomultiplecontacts,includingthe
e-mail:******@
123:.
,
alreadyatasinglethree-dimensionalrigidbodywith2Mechanicalmodelofarigidbodywithtwo
twocontactpoints,
Whenarigidbodyisincontactwithtworigidsur-
faces,fourdifferentcontactstatesarepossiblefromConsiderthemotionofarigidbodywhichisinnormal
thecombinationofrollingandslippingateachcon-contactwithtwofixed,rigidsurfacesatthepointsP+
-pointrolling,andP−,
thecontactforcesareundetermined,whichmakesitandthegeometryofthesurfacesensurethatthecontact
impossibletocheckthelimitationsoffrictionforces,pointspersistcontinuouslyduringthemotion,andno
andthus,-
seemsthattheproblemisbeyondtheavailabletoolsofmalcontactpersistsinaregularstateandcompilations
rigidbodymechanics,andoneshouldincludemodelsofthePainleveparadox[6]donotoccur.
withelasticdeformationtoresolvetheindeterminacy
-
ever,
bodymodel:bycarefulanalysisofthevectorfieldof+−
theresultingdynamicalsystem,itispossibletofindtheAtacertainconfigurationofthebody,letn,ndenote
thenormalunitvectorsofthesurfacesatP+andP−.
trajectoriesofslipping–rollingtransitionswithoutcal-
,LetCdenotethecentreofgravityofthemovingbody,
andletthelocationsofthecontactpointsarer+=
weusethetoolsofanalysisofcodimension-2disconti-−−→−−→
CP+andr−=CP−().
nuitiesofvectorfields,whichwaspresentedin[3]and
[4].Fortheconvenientkinematicdescriptionoffriction
Thescenarioofarigidbodywithtwocontactpointseffects,weintroduceorthonormalbasesatthecontact
canbefoundinseveralmechanicalapplications,,letusconsiderthevectorr+−r−point-
astherailwaywheelsets[2,11,19],therollingelementsingfromP−toP+,anddefinethecorrespondingunit
ofbearings[10,20,29],thecompressedelementsofthe
tensegritystructures[24],orintherotatingballflowme-
ter[7,23],whichwasanalysedbytheauthorsin[1].
Theexampleofaballwithtwocontactpointshasbeen
recentlyusedfordemonstratinganewapproachoffric-
tionmodels[26].
Thepaperisorganizedasfollows:,the
mechanicalmodelispresentedcontainingtherigid
bodywithtwocontactpoints,andthenecessaryformu-
,
thenonsmoothbehaviourispresentedinthestatespace,
andwedemonstratethatthedirectdescriptionofthe
transitionsisnotpossibleduetotheindeterminacyof
,theconceptsofpossibleand
realizablerollingstatesarepresentedinthecaseofa
,andthecondi-
,theresultsaredemonstratedonamechanical
examplewithclosedformcalculations.
.
Thefigureshowsasphereincontactwithtwoplanes,butthe
analysisisvalidforgeneralgeometriesoftherigidbodyandthe
rigidsurfaces
123:.
Slipping–rollingtransitionsofabodywithtwo...
vectora×(v+−v−)=r+−r−·−r+−r−·a,·a.
r+−r−
a=.(1)(9)
r+−r−
Then,byusingthenotation
Then,twotangentialunitvectorscanbedefinedateach
contactpoint:a:=a,(10)
a×n+a×n−fortheangularvelocitycomponentparalleltoa,the
t+=,t−=,(2)angularvelocityvectorcanbeexpressedintheform
1a×n+1a×n−
v+−v−
=a·a+a×+−.(11)
r−r
t+=n+×t+,t−=n+×t−.(3)
2121Consequently,(8)and(11)showthatinagiven
configuration,thevelocitystateofthebodycanbe
Weexcludethedegeneratecaseswhenthetwocon-describeduniquelybythefourvariablesu+,u−,u
11a
tactpointscoincide,or,r+−r−isparallelton+and.
a
orn−.Then,(1)–(3)providetheorthonormalbasesTherollingofthebodyatP+ischaracterizedby
(n+,t+,t+)and(n−,t−,t−).++
1212v=0,whichisequivalenttou1=ua=-
ilarly,therollingatP−ischaracterizedbyv−=0,
whichisequivalenttou−=u=0.
1a
Atthecontactpoints,–Eulerequations
aredenotedby
+++++AtthecontactpointsP+andP−,thecontactforces
v=vP+=u1t1+u2t2,
−−−−−(4)aredenotedbyF+andF−,respectively().
v=vP−=u1t1+u2t2,
Theyareexpressedintheform
,but
theyarerelatedbythereductionformulaofrigidbodyF+=N+n++T+,F−=N−n−+T−,(12)
kinematics,
v+−v−=×(r+−r−),(5)whereN+>0andN−>0arethenormalforce
components,andthetangentialforcesaregivenby
whereistheangularvelocityvectorofthebodyand
×denotescrossproduct.+++++−−−−−
Thescalarproductoftheright-handsideof(5)byT=T1t1+T2t2,T=T1t1+T2t2.(13)
,wecanintroducethevariable
Allotherexternalloadsarereducedintothecentreof
ua=v+,a=v−,a,(6)
gravityCofthebody,leadingtotheresultantexternal
whichisthecommonvelocitycomponentofP+,
P−(4)and(6),wegettheNewton–Eulerequationsoftherigidbodybecome
u+=c+uandu−=c−u,where+−
2a2amv˙C=F+F+Fe,
++−−(14)
J˙+×(J)=r×F+r×F+Me,
c+=1/t+,a,c−=1/t−,a.(7)
22
whereJisthemassmomentofinertiaofthebody,
Then,thevelocities(4)become
v=v+−×r+.(15)
v+=u+t++c+ut+,C
11a2
−−−−−(8)
v=u1t1+,and
Thecrossproductof(5)byagivesthedotdenotesdifferentiationwithrespecttothetime.
123:.
,
Tosimplifythenotations,weimplicitlyassumed,
thatthefrictioncoefficientμisthesameatthetwo
contactpoints,whichrestrictioncanbereleasedifnec-
-
tioncoefficientsarethesame,whichcanbegeneralized
.
Notethatinsteadofthepresentedbasicdescription,
thecontactlawscouldbealternativelyinterpretedas
set-valuedforcelaws(see[8]and[28]).
Theslippingorrollingstatesofthetwocontactpoints
leadtofourdifferentkinematiccasesofthebody.
Throughoutthepaper,thesecasesarereferredtoby
:
gravityC–CaseSS:slipping–slippingcase,thebodyisslip-
pingatbothP+andP−,
–CaseSR:slipping–rollingcase,thebodyisslipping
+−
atPandrollingatP,
–CaseRS:rolling–slippingcase,thebodyisrolling
Tocompleteourmodel,weassumethesimpleCoulomb+−
,thepointP+atPandslippingatP,
–CaseRR:rolling–rollingcase,thebodyisrolling
canbeeitherinslippingstatecharacterizedby+−
atbothPandP.
+
v>0,(16)Notethatweassumeapermanentnormalcontact
v+wherethereisnoseparationorimpactsbetweenthe
T+=−μN+,(17)
v+
withdynamiceffectsfromnormalcontact,severalother
or,inrollingstatecharacterizedbykinematiccasesappearattwocontactpointsalreadyin
problemsintwodimensions[15,18,27].Withtheloss
v+=0,(18)ofcontactinspatialproblemswithtwocontactpoints,
++weobtainatleastninekinematiccases[1].Inthispaper,
T≤μN.(19)
weassumethatthenormalcontactisensured,leading
−tothefourkinematiccaseslistedabove.
Similarly,theslippingandrollingstatesofPare
givenby
3Nonsmoothdynamics
v−>0,(20)
v−
T−=−μN−,(21)
v−
Thesixdegreesoffreedomofafreerigidbodyare
andreducedbytwobecauseofthetwonormalcontactcon-
,atleastinthevicinityofaninitialcon-
v−=0,(22)figuration,theconfigurationspaceofcontactingbody
−−canbeparametrizedbygeneralizedcoordinatesinthe
T≤μN,(23)
form
=(q1,q2,q3,q4).(24)
123:.
Slipping–rollingtransitionsofabodywithtwo...
Weparametrizethevelocitystateofthebodybythe–thefrictionlaw(21)hasadiscontinuity.
vector
Thesesets(30)and(31)arecodimension-2sub-
s=(s,s,s,s)=u+,u−,u,,(25)
whichquantitiesarecalledthequasi-velocities(see[9],codimension-3subspace
#+++−
)ofthesystem.=∩=x∈X:u1=u1=ua=0.
Thevectorsr+,r−,n+andn−dependonthegener-
(32)
alizedcoordinatesq,andthemomentofinertiatensor
Jdependsonq,,weassumethatthePurelyfromkinematicpointofview,thelocationof
:
Inaddition,allthesedependenciesareassumedtobe#
–CaseRR(rolling–rolling)islocatedat.
smooth.+#
–CaseRS(rolling–slipping)islocatedat\.
Then,dynamicsofthemovingbodyisdetermined−#
–CaseSR(slipping–rolling)islocatedat\.
inastatespaceXcomposedfromthegeneralizedcoor-+
–CaseSS(slipping–slipping)islocatedatX(∪
dinates(24)andthequasi-velocities(25),−
).
x=(q,s)∈X⊂R8.(26)
Thepropertiesofthefourkinematiccasescanbefound
ThedynamicsinXisgovernedbyasetoffirst-orderinTable1.
–Eulerequa-
dependslinearlyonthefirstderivativesofthegeneral-tions(14)andtheappropriateconditionsfrom(16)–
izedcoordinates,thesederivativescanbeexpressedin(23),wecanderivethedifferentialequationintheform
theform(29).(AlternativelytotheNewton–Eulerequations,the
q˙=K(q)s,(27)kinematicconstraintscouldbedirectlyincludedtothe
rollingcasesbytheGibbs–Appellequationfrom[9],
whereK(q)isafour-by-fourmatrixdependi