PhysicsLettersA 170 (1992) 384—392 PHYSICSLETTERS ANorth-Holland

Limit cyclesin an elasto-plasticoscillator

RudraPratap,SubrataMukheijeeandFrancisC. MoonCornell University, Ithaca,NY14853, USA

Received20April 1992; revisedmanuscriptreceived3 September1992; acceptedforpublication3 September1992Communicatedby A.R. Bishop

A simplysupportedelasto-plasticbeamis modeledasa singledegreeoffreedomoscillatorwith bilinearhysteresis.Kinematichardeningis usedas a problemparameterandanexplicit mapis obtainedfor plastic cyclesunderfree vibrations. This mapisanalyzedand a bifurcation of thehardeningparameteris discussed.An elasto-plasticlimit cycleis shownto exist in a certainrangeof theparameter.Limit cyclesarealsoshownto existunderperiodic impulseforcingof theabovesystemandtheirstabilityis discussedwith thehelpof anexplicit Poincarémap.

1. Introduction plastic oscillatorunderfree vibration.Two distinctsubintervalsarerecognizedin this parameterspace

Thereare numerousphysicalsystemswhich are anda theoremis formulatedtodescribethedynamicmodeledwith piecewiselinearequations.Onemay behaviorin theseregions.In thesecondsubintervalclassify thesesystemsinto two categories:onewith- an elasto-plasticlimit cycle is shown to exist. Nu-out hysteresisand the other with hysteresis.Me- mericalsimulationsare presentedto verify the re-chanicalsystemswith oscillatingpartsandintermit- sults. We also briefly discussthe asymptoticap-tent contactsor impactsbelongto the first category proach to this limit cycle. Next we considerthewhile friction oscillatorsor elasto-plasticoscillators responseof the systemunderperiodicimpulseforc-areexamplesof the secondcategory.Although there ing. We seeka steadystatesolution in the form of ahavebeenmany studiesof dynamicalbehaviorof symmetric periodic motion and obtain an explicitsuchsystems,the use of moderntechniquesof dy- Poincarémap.From thelinearizationof the map itnamicalsystemsis relativelyrecentin thesestudies. isshownthatthe oscillatorhasanasymptoticallysta-Most of the modernanalyseshavebeencarriedout ble limit cycle.for systemsin the first category(seeref. [11 for a The model is shown in fig. I. Massm is attachedlist ofrelevantpapers)andfor friction oscillators[2— to two rigid links eachof length 1 with negligible in-4]. Earlier researchon dynamicsof systemswith bi- ertia. An elasto-plastictorsionalspringwith abilin-linear hysteresisinclude mostly the calculation of earhysteresisis attachedatthemasspoint.The springsteadystatesolutionsundersinusoidalloading [5] characteristicis shown in fig. 2 where spring mo-orstability analysesofbeamcolumnsundertimede-pendenttransverseloading [6]. More recentpaperson elastic-perfectly-plasticbeamsby Symonds[7,81 Elasto-plasucSpring~ 7 - -- --

andPoddaret al. [91havenumericallyinvestigated - - -

chaotic motion of such beamsusing the Shanley a

model. In this studywe usea simplemodel which qhas only materialnonlinearity (no geometricnon- I’linearity) andconsiderthe kinematichardeningas ____________________ 2

a parameterof the problem.We show a qualitativebifurcation in the dynamicbehaviorof the elasto- Fig. 1. 1-DOFelasto-plasticbeammodel.

384 0375-9601/92/$ 05.00© 1992ElseyierSciencePublishersB.V. All rightsreserved.

Volume170, numberS PHYSICSLETTERSA 16 November1992

ment is plottedagainstnormalizedrotation vasde-fined by eq. (3). K is the spring constantin elastic M

regionsAB and CD, and?12K is the spring constant ________

in plastic regionsBC andDE where~~2e[0, 1] is the —

kinematichardeningparameter.On BC the rateof M ~2~

rotationof the springis positive,hencewecall it theprogressiveplasticstateandon ED the rateof rota- K

tion is negative,hencewe call it the regressiveplasticstate. Thesestatesare denotedby P~and P re- /spectively.Let q be the verticaldisplacementof the / 2M

massand a be the correspondingrotation in the n’~..__:~‘ /spring.Thenthe non-dimensionalizedequationsof / I ~motion underfree vibrationare

(a) Elasticstate: /~+co2~=a2c~*, ~*=const. (1) D —

E

(b) Plasticstate:2 2 Fig. 2. Characteristicdiagramof theelasto-plasticspring.

~+(w,1)2c~=~w sgn(~)(l—~),

= (t5— ~sgn(~)(1 — ~2) , (2) anda zeroin the subscriptdenotesthe initial value

of the variable.where Determinationofonsetofplasticstates.According

~= q~iit~ to the rulesof kinematichardening,the yield linesBC andED are fixed in the M—v space.The differ-

non-dimensionalizeddisplacement, ence between the moments at elastic unloading

4K/mi2 (pointC) or reloading(point E), andthenextplas-tic yield, has to be a constant,2M~Therefore,for a

v=a/ã=2~’, givenvalueof~*,thevaluesof i~atanyield point innon-dimensionalizedrotation, (3) eitherthe P+ or the P statecan be easily deter-

mined by solvingthe equationsof linesCD andBCandc~isthelimiting valueofrotationfor first plastic or ED (fig. 2) simultaneously.Thus wegetyield and~ and v” (seefig. 2) are the plasticdis- 1 ~placementand plastic rotation, respectively,corre- ~ ±= ±— + ~ (5)spondingto an elasticstate. 2 1 —

Substitutingthis valuein (4), the time requiredto

2. Analytical solution reachan yield point canbe explicitly obtained.

Equations(1) and(2) canbe solvedexplicitlyand (b) Plasticvibrations:transition points betweenthe three statesdeter- 1 _~2

mined asfollows: t)=R~cos[w,i(t—tr)—O~]——~—~--, (6)(a) Elastic vibration:

where~(t) =Recos[w(t_tg) _Oe]+~:~ (4)

R _,./[~P+(,12l)/2,,2]2+(~P+/w,7)2where

~ _~( ~r/w,iRe=~/(~_~*)2+(~e/~)2 —tan (~2_1 )/2~2

= 1 ( ~ The equationsfor the P - stateareobtainedby sub-stituting — (I —,f) in placeof 1 —~ in the above.

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Volume 170, numberS PHYSICSLETTERSA 16 November1992

Transition to elastic state. The condition for this 2

transition is ~=0, sincefor unloading~0 andforreloading~ 0. By substitutingthis conditionin eq. 1 ~12 0;Q”~

(6) transitiontimesanddisplacementsareobtained ~ .1explicitly. 0 ~ 0.2

2.1. Determinationofplasticcyclesunderfree 1 ~ ‘..~ 0.3

vibration~ 2- S’.. 0.5

Let the systemstartoscillationswith anygivensetof initial conditions.Sincewe know the solution of -

theequationfor eachbranchof the spring state,we ‘S., .8

may be ableto predictthe numberof plastic cyclesthesystemwill go throughbeforesettlingdownto itsfinal elasticoscillationswith a net plastic displace-ment.Since thetransitionstatesin eachcaseareex-plicitly known,we mustsolve for subsequentphase -2 -1 0 1 2 3 4 5 6

changesiteratively. Yp+Let Y~+and Y~,_denotethe displacementat the .

+ . . Fig. 3. Iterativemapfor plasticcyclesfor differentvaluesof 112.onsetof unloadingfrom theP state(point C in fig.2) andthe pointof reloading(point E) fromthe P-

state respectively.Substitutionof subsequenttran- with either of the branchesincluding the startingsition conditionsandsomealgebraicmanipulations point Y~+or Y~_.Figures4aand4b show two suchgive iterativepaths for different valuesof i~

2.The totalnumberof plasticcycles,for example is threein fig.

= r~[1 — ~ (7) 4a.Fromthis figureit isclearthat oncea pathenters

Y~+=r~[—l+./~Yp_+l)2+2r] , (8) regionOBA orOCD (fig. 4a),it cannotundergofur-ther plastic displacement.The last value of Y~±or

where Y~,_determinesthe net plastic displacement~

througheq. (2). The oscillator settlesdownto ape-r= ~ r> OV,

12< 1 . riodic elasticorbit centeredat ~*• Thus we may call

regionsOBA and OCD elastic trapping regions. InTheseequationssuggesta returnmap for the deter- fig. 4b,however,every pathmusteventuallyleadtominationof plasticcycles.Thismaphastwo branches the samepoint, I Y~± = 0.5, through, theoretically,givenby eqs.(7) and(8). Themap is shown in fig. infinitely many plastic cycles. This point corre-3 for variousvaluesof the hardeningparameter~ spondsto net zero plastic displacementandhence

I Y~,±=0.5 representthe thresholdvalues of dis- the final periodic elastic oscillationsmust be ofplacementsfor reachingthe next plastic state.This unique amplitude and centeredat the origin irre-map has a reflection symmetry which is expected spectiveof the initial conditions! Thus, we haveafrom the symmetry of Y~±and Y~_in fig. 2. This uniqueelasticlimit cycleaslongastheoscillatorfirstmap is slightly different from conventionalone-di- goes through at leastone plastic cycle. Inside thismensionalmapsin the sensethat iterationsarenot cycle, of course, we have infinitely many periodicdonewith respectto the identity line. To find the elasticorbits. To distinguishthis cycle from theusualnumberofplasticcyclesonestartsfrom thefirst value definition of a limit cycle we call our limit cycle anof Y~±or Y~andfollows a stair-like pathbetween elasto-plasticlimit cycleto signify the fact that thisthe two branchesuntil thereis no further intersec- cycleisobtainedonlyif theoscillatorhasgonethroughtion with the next branch.The numberof plastic at leastoneplasticcycle.Thislimit cycle is shownincyclesis simply thenumberof encountersof this path fig. 5 as obtainedby direct numericalintegrationof

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Volume 170, number5 PHYSICSLETTERSA 16 November1992

f3~~IIINNN\ .0i5 l.050051I5

0 0.5 1 1.5 2 2.5 3 3.5 4 Displacement

Fig. 5. Elasto-plasticlimit cycleobtainedby numericalintegra-tion for 112=0.7.

are shown in fig. 3 by dotted lines. For the elastic

perfectly plasticcase,i.e. ~2 = 0, therecannotbemorethan two excursionsto the plastic branches.From

-1 fig. 3 it is clearthat the elastictrappingregionsde-

1 5 creasewith increasingvaluesof ~2 andvanishalto-- getherfor 112=0.5. Thus,qualitatively,this valueof

-2 thehardeningparameterrepresentsa bifurcationforD free vibrationsof the oscillator. All the featuresof

-23 themapdiscussedabovecannow besummarizedinthe following theorem.

-3Theorem.Forunforcedoscillationsof a singlede-

-3.5 gree of freedompiecewiselinear elastoplasticoscil-lator of the type representedby eqs. (1) and (2),

~ ~ 2:5 ~ 3.5 ~ there exists a bifurcation value of the kinematichardeningparameterbelowwhich thereisa compact

Yp+ subspaceof equilibrium points for the final elastic

Fig. 4. Stair-pathiteration, (a) 112=0.3, (b) 112=0.5. periodic oscillations and abovewhich there is a

uniqueelasto-plasticlimit cycle.

eqs. (1) and (2) for threedifferent trajectoriesfor

112 = 0.7. Now lookingbackat fig. 3 onecanobserve Proofof this theoremis basedon constructionofthat the featuresdiscussedabovefor figs. 4a and4b the elastictrappingregionandis givenin ref. [101.arerepresentativeof a set of valuesof parameter112 An upperboundon netplasticdisplacementsis alsoFigure 4a sharesfeatureswith iterative maps for obtainedin ref. [101.0<112<0.5 andfig. 4b sharesits featureswith those It is worth notingthatin fig. 4a,if thestartingvalueof 0.5~ 1,2< 1. Mapsfor thetwo limiting valuesof 112 of Y~+is to the left of pointD thenthe systemgets

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trappedin region OCD with a net negativeplastic reacha valueof zero.Basedon energy calculationsdisplacementwhile anotherY~+to theright of point [10], the oscillationsof the plastic rotation canbeD takesthesystemin regionOBA with a netpositive expressedasplastic displacement.Thus, for two initial kicks of 4 2 2 * 2 r’ *

differentmagnitudes,but in the samedirection, the i v~ = (21, D+ C ~ ) I v,, I + I Vfl J (9)n+l D’l+ 2C *

final elasticvibrationsmayhavenetplasticdisplace- 1, V~

mentsin the sameor in the oppositedirectionof the whereinitial kick. Suchcounterintuitivebehaviorhasbeennoticed before [7] in an elasto-plasticbeamwith c= 2772_1 , D= 1geometricnonlinearitiespresent.Here,however,only 1 11 1 11material non-linearities have been considered.It Since we are interestedin the asymptoticapproachshouldbenotedagainherethatthiscounterintuitive to thelimit cycle, we restrictourselvesto very smallbehavioris only possiblefor 772e(0, 0.5). Another v~.UsingTaylorexpansionof (9) we getimportantdynamicfeatureof thesystemin this sub-interval of 1,2 is the existenceof an Infinite number I v~÷iI = (2,12_1)1 I +277~Iv~12 . (10)of limit cyclesin the elastictrappingregionsOCD Specialcases.(a) ,~2= 0.5: Substitutingthis valueandOBA. Initial conditionsdeterminewhich limit in (10) the linear term dropsout andwe getcycle on OBA or OCD is obtained.Point 0 (or A

* ~,4 *2 11and D) correspondsto I Y~±I =0.5 and represents Vn+i ~?1 Vn

the limit cycle centeredat the origin (i.e. the net In fig. 7a, resultsobtainedfrom directnumericalin-plastic displacement,~ ‘= 0). In size,this isthe larg- tegrationof threedifferent trajectories(shownbyest possiblelimit cycle and it correspondsto the ~ threedifferentsymbols)arecomparedwith eq.(11)cillations betweenthe elasticlimit on branchAB of exhibiting excellentagreementfor very small v~

fig. 2. On the otherhand,point B andC in fig. 4a (b) 0.5<772<1: Forverysmallvaluesofv~we maycorrespondto the maximumvalueof the net plastic neglectthe secondordertermin eq. (10) andgetdisplacementandhencethesmallestlimit cycle. Any * 2 * 12otherlimit cycle on OCD or OBA hasa sizebetween I Vn+i I = (277 — 1)1Vn I .

thesetwo extremes.Thus, from a purely sizepoint In fig. 7b thenumericalvaluesobtainedfor threedif-of view, thelimit cyclesform anannulusfor 772<0.5. ferent trajectoriesare comparedwith analyticalval-Theseare,of course,centeredat different valuesof uesasobtainedby botheqs.(12) and (10). It isclear~* In figs. 6a—6d,theselimit cyclesareshownasob- that the approachis linear very close to the limittamedby direct numericalintegrationof different cycle.trajectoriesfor differentvaluesof 772~As the valueof From eq.(11) we seethat the rateof approachto772 increases,theseparationof limit cyclesdecreases, the limit cycle is quadraticin plastic displacementsthe differencein sizedecreasesandfinally all limit andhencenumericallythe cycle is obtainedin justcyclescollapseinto one unique limit cycle for the a few iterations.We shall call this approachsuper-critical valueof 772 = 0.5. Thisis commensuratewith critical to distinguishfrom the linear rategivenbythe shrinkingsize of the elastictrappingregionsfor eq. (12). Onceagain, ,~2= 0.5 distinguishesitself ashighervaluesof 772 andits completedisappearance the bifurcationvalue of the parameter.We cannot,at 772=0.5. however,characterizethisbifurcationin termsof any

For 772e [0.5, 1), the systemhasa uniqueelasto- of the standardbifurcationsbecausethe standardplastic limit cycle. Theoretically the oscillatorwill methodsof characterizationfor continuoussystemstakean infinite numberof plastic cyclesandhence do not applyhere.The collapseof aninfinite num-infinite time to reachthe limit cycle. For practical berof limit cyclesinto oneuniquelimit cycle is notpurposesit is important to know the rate of ap- typical of any standardbifurcationphenomenoninproachto this limit cycle. From the map in fig. 2 it finite dimensions.is clearthat theplasticrotationsof thespring,v~,al-ternatebetweenthe P±andthe P— statesandfinally

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Volume170, number5 PHYSICSLETTERSA 16 November1992

a b0.5 .- ... 0.5 ,

IM5005ii5 ~5 105005 1i5

Displacement Displacement

~ ~. .--- ~ d

1 05115 5 1 1 15

Displacement Displacement

Fig. 6. Limit cyclesfor ,~ 0.5: (a)a~=0.2,(b) ~ (c) i~=0.4,(d) 112=0.5.

3. Limit cyclesunderperiodic impulseforcing elasticstateat point (1) with knownvaluesof (~, ~

s~)andreachthe plasticstate,P~, at point (2). TheIn thissectionwe studythe responseofthe system valuesof (~, ~*) areknownat this point from elastic

subjectedto periodicimpulsesas shown in fig. 8a. statesolutions.The systemcontinuesto point (3)Sincethesystemisunderfreevibrationbetweentwo whereit receivesan impulseand goesto point (4).consecutiveimpulses,it is possibleto look for a pe- Thedynamicsfrompoint (4) to (1) is thesamedueriod onemotion analytically.Underthis loadingwe to the symmetry in the equationsand the loading.requirethatthe displacementbecontinuouswhereas Thedisplacementandvelocity at point (4) cannowthe velocity is changedperiodicallyby the magni- be written in terms of thoseat point (1):tudeof the impulse. In fig. 8b a possiblesymmetricperiodoneorbit is shown.Let the systemstart in an

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Volume 170, number5 PHYSICSLETTERSA 16 November1992

a

0.45

0.4

0.35

0.3.~. 0.25+

0.2

0.15

0.1

0.05

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Iv*(n)l

0.6 b

0.5

0.4

+ ./~ 0.3* /

Iv*(n)I

Fig. 7. Convergenceof plasticrotationsto thelimit cyclefor (a) 112 = 0.5, numericalresults: (0), (a), (+); analyticalresultsfromeq.(11): (—~-~-);(b) 172=0.7,numericalresults: (0), (a), (+); analyticalresultsfromeq.(12): (——);eq. (10): ( ).

~4=(s~1+l—r)cosØ where

+~f~/w77)2(2~i +l)sinø+r, (13) Øz~w77T

~ (w77)2(2~1+ 1) cosØ _77[cost (~‘~)—cos~(~‘+i_~)]

—W77(~1+ 1 —r) sinØ—y, (14)Re=~/(~1~*)2+ (~~/w)2.

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Volume170, number5 PHYSICSLETFERSA 16 November1992

____________________ a

1 0.6 ,/“~24 T 0.8 ~=io

/ :14 00.4, ~ 0(a) 116 ‘.., ~

0.2 .

~ c :~: : -: — ~ , a

0••

2) -0.2 ** * ***** ~000(i)~ ~ -0.4 ~

-0.6

-0.8

-1-1 -0.5 0 0.5

ReaI()~)(4)

2,~

Co 1/3

1.81.6

(b)1.4

Fig. 8. (a) Periodicimpulse;(b) A symmetricperiod-oneorbit. 1.2Regionof StableLimit Cycles

Thus, half of the Poincarémap,~h, is analytically ~— 1

determined.To studythe stability of the periodone0.8

motion,eigenvaluesof thelinearizationof thePoin-caremapare required.Thiscanbe obtainedby first 0.6linearizing ~h about a periodic orbit and thencal-

0.4culatingthe eigenvaluesof [DPh 2~Theperiodicor-bits for any given Tandy maybe obtainedby solv- 0.2ing the nonlinearalgebraicequations

C0 5 10 15 20

(15)T

(16)Following thesesteps,periodic orbitsfor anygiven Fig. 9. (a) Eigenvaluesof linearizedPoincarémap. (b) Region

set of parameterscanbeobtainedandtheir stability of stablelimit cyclesin T—y spacefor w=) and112=0.3.determined.Here, we presentthe results thus ob-tained,for representativevaluesof w and j2~In fig. whenpoints (2) and (3) in fig. 8b coincideand the9a eigenvalues~.areshownin the complexplanefor limiting elasticperiodicorbit is obtained.The limita set of amplitudesandperiod of the impulseforc- cyclesthusobtainedhavealsobeenverifiedby directing. All the eigenvalueslie within the unit circle numericalsimulations[10].showingasymptoticstability of the periodicorbits.Theboundaryof existenceofthesestablelimit cyclesin fig. 9b is determinedby taking the limiting case

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Volume 170, number5 PHYSICSLETTERSA 16 November1992

4. Conclusions impulseforcing,stablelimit cyclesareshownto exist

for all pairsof (T, y) which arecapableof forcingA singledegreeof freedomelasto-plasticoscillator theoscillatorintoa plasticstatebetweenconsecutive

isgovernedby piecewiselinearODEs.Thedynamics impulses.of thesystemjumpsfrom onebranchto anotherde-pendingonthe flow rulesof plasticity.Forkinematichardening,the plastic yield lines are fixed in mo- Acknowledgementment—rotationspace.Underunforcedvibration, thedynamicsof the oscillatorcanbe analyticallydeter- We wish to thank ProfessorPhilip Holmes formined by piecingtogetherthe solutionsof eachun- many helpful suggestionsanddiscussions.This re-earbranch.Theoscillatorexhibitsdifferentbehavior searchhas beensupportedby NSF grant numberdependingon the valueof thekinetic hardeningpa- MSS-9016626 to CornellUniversity.rameter.In fact, there is a bifurction valueof thisparameterwhich dividesthe rangeof the parameterin two subintervals.In onesubintervalthe oscillator Referencesgetsinto anelastictrappingregionaftera finite num-berof plastic cycleswhich aredeterminedfrom the [1] SW.ShawandP. Holmes,J.SoundVibr. 90 (1983) 129.

iterativemapobtainedanalytically.Thefinal elastic [2] SW.Shaw,J. SoundVibr. 108 (1986) 305.

stateof thesystemdependson theinitial conditions. [3] I. Grabec,Phys.Lett. A 117 (1986) 384.

In the other subinterval,however, there exists a [4] B.F.FeenyandF.C.Moon, Phys.Lett. A 141 (1989)397.[5] T.K. Coughy,Trans. ASME J. Appl. Mech. (1960)640.

unique elasto-plasticlimit cycle which is obtained [6] G. Ballio, Meccanica(1970)85.

irrespectiveof the initial conditionsof the oscillator [7] P.S.SymondsandT.X. Yu, ASME J.Appl. Mech.52 (1985)

as the numberof plastic cyclesgoesto infinity. The 189.

final elasticstatein thiscasehaszeronetplastic dis- [8]J.Y. Lee, P.S.Symondsand G. Borino, private commu-

placement.Very closeto the limit cycle the ampli- nication.[9] B. Poddar,F.C.Moon andS. Mukherjee,J. Appl. Mech. 55

tudeof plastic displacementsdecreaseslinearly with (1988) 185.respectto the previousamplitude for 772>0.5 and [10] R.Pratap,S. MukherjeeandF.C.Moon, in preparationfor

quadraticallyfor 772 = 0.5. Undersymmetricperiodic J.SoundVibr.

392