dynamics and vibrations_ notes_ multi-dof vibrations.pdf

05/08/2015 DynamicsandVibrations:Notes:MultiDOFvibrations

http://www.brown.edu/Departments/Engineering/Courses/En4/Notes/vibrations_mdof/vibrations_mdof.htm 1/9

5.5IntroductiontovibrationofsystemswithmanydegreesoffreedomThe simple 1DOF systems analyzed in the preceding section are very helpful to develop a feel for the generalcharacteristicsofvibratingsystems. Theyare toosimpletoapproximatemostrealsystems,however. Realsystemshavemore than justonedegreeof freedom. Realsystemsarealsoveryrarely linear. Youmaybefeelingcheatedarethesimpleidealizationsthatyougettoseeinintrocoursesreallyanyuse?Itturnsoutthattheyare,butyoucan

onlyreallybeconvincedofthisifyouknowhowtoanalyzemorerealisticproblems,andseethattheyoftenbehavejustlikethesimpleidealizations.Themotionofsystemswithmanydegreesof freedom,ornonlinearsystems,cannotusuallybedescribedusingsimpleformulas.Evenwhentheycan,theformulasaresolongandcomplicatedthatyouneedacomputertoevaluatethem.Forthisreason,introductorycoursestypicallyavoidthesetopics.However,ifyouarewillingtouseacomputer,analyzingthemotionofthesecomplexsystemsisactuallyquitestraightforward infact,ofteneasierthanusingthenastyformulaswederivedfor1DOFsystems.Thissectionofthenotesisintendedmostlyforadvancedstudents,whomaybeinsultedbysimplifiedmodels.Ifyouarefeelinginsulted,readon5.5.1Equationsofmotionforundampedlinearsystemswithmanydegreesoffreedom.We always express the equations of motion for a systemwithmanydegreesoffreedominastandardform.Thetwodegreeoffreedomsystemshowninthepicturecanbeusedas an example. We wont go through the calculation indetail here (you should be able to derive it for yourselfdrawaFBD,useNewtonslawandallthattediousstuff),

buthereisthefinalanswer:

To solve vibration problems,we alwayswrite the equations ofmotion inmatrix form. For an undamped system, thematrixequationofmotionalwayslookslikethis

wherexisavectorofthevariablesdescribingthemotion,MiscalledthemassmatrixandK iscalledtheStiffnessmatrixforthesystem.Forthetwospringmassexample,theequationofmotioncanbewritteninmatrixformas

Forasystemwithtwomasses(ormoregenerally,twodegreesoffreedom),MandKare2x2matrices. Forasystemwithndegreesoffreedom,theyarenxnmatrices.

Thespringmasssystemis linear. Anonlinearsystemhasmorecomplicatedequationsofmotion, but these can alwaysbe arranged into the standardmatrix formby assuming thatthe displacement of the system is small, and linearizing the equation of motion. Forexample,thefullnonlinearequationsofmotionforthedoublependulumshowninthefigureare



Here,asingledotoveravariablerepresentsatimederivative,andadoubledotrepresentsasecondtimederivative(i.e.acceleration).Theseequationslookhorrible(andindeedtheyare themotionofadoublependulumcanevenbechaotic),butifweassumethatif , ,and their time derivatives are all small, so that terms involving squares, or products, ofthesevariablescanallbeneglected,thatandrecallthat and forsmallx,theequationssimplifyto

Or,inmatrixform

Thisisagaininthestandardform.Throughout therestof thissection,wewillfocusonexploringthebehaviorofsystemsofspringsandmasses. This isnot because spring/mass systems are of any particular interest, but because they are easy to visualize, and, moreimportantly theequationsofmotionforaspringmasssystemareidentical to thoseofanylinearsystem. Thiscouldinclude a realistic mechanical system, an electrical system, or anything that catches your fancy. (Then again, yourfancymaytendmoretowardsnonlinearsystems,butifso,youshouldkeepthattoyourself).5.5.2Naturalfrequenciesandmodeshapesforundampedlinearsystemswithmanydegreesoffreedom.First, lets reviewthedefinitionofnatural frequenciesandmodeshapes.Recall thatwecansetasystemvibratingbydisplacingitslightlyfromitsstaticequilibriumposition,andthenreleasingit.Ingeneral,theresultingmotionwillnotbe harmonic. However, there are certain special initial displacements thatwill cause harmonic vibrations. Thesespecial initial deflections are called mode shapes, and the corresponding frequencies of vibration are called naturalfrequencies.The natural frequencies of a vibrating system are itsmost important property. It is helpful to have a simpleway tocalculatethem.Fortunately,calculatingnaturalfrequenciesturnsouttobequiteeasy(atleastonacomputer).Recallthatthegeneralformoftheequationofmotionforavibratingsystemis

wherexisatimedependentvectorthatdescribesthemotion,andMandKaremassandstiffnessmatrices.Sinceweareinterestedinfindingharmonicsolutionsforx,wecansimplyassumethatthesolutionhastheform ,andsubstituteintotheequationofmotion

Thevectorsuandscalars thatsatisfyamatrixequationoftheform arecalledgeneralizedeigenvectorsand generalized eigenvalues of the equation. It is impossible to find exact formulas for andu for a largematrix(formulasexistforupto5x5matrices,buttheyaresomessytheyareuseless),butMATLABhasbuiltinfunctionsthatwillcomputegeneralizedeigenvectorsandeigenvaluesgivennumericalvaluesforMandK.Thespecialvaluesof satisfying arerelatedtothenaturalfrequenciesbyThespecialvectorsXaretheModeshapesofthesystem.Thesearethespecialinitialdisplacementsthatwillcausethemasstovibrateharmonically.Ifyouonlywanttoknowthenaturalfrequencies(common)youcanusetheMATLABcommand

d=eig(K,M)This returns a vector d, containing all the values of satisfying (for an nxn matrix, there are usually ndifferentvalues).Thenaturalfrequenciesfollowas .Ifyouwanttofindboththeeigenvaluesandeigenvectors,youmustuse



[V,D]=eig(K,M)This returns two matrices, V and D. Each column of thematrix V corresponds to a vector u that satisfies theequation, and the diagonal elements of D contain thecorrespondingvalueof . Toextracttheithfrequencyandmodeshape,useomega=sqrt(D(i,i))X=V(:,i)Forexample,hereisaMATLABfunctionthatusesthisfunctiontoautomaticallycomputethenaturalfrequenciesofthespringmasssystemshowninthefigure.

function[freqs,modes]=compute_frequencies(k1,k2,k3,m1,m2)

M=[m1,00,m2]K=[k1+k2,k2k2,k2+k3][V,D]=eig(K,M)fori=1:2freqs(i)=sqrt(D(i,i))endmodes=V

end

Youcouldtryrunningthiswith>>[freqs,modes]=compute_frequencies(2,1,1,1,1)Thisgivesthenaturalfrequenciesas ,andthemodeshapesas (i.e.bothmassesdisplaceinthesamedirection)and (thetwomassesdisplaceinoppositedirections.Ifyouread textbooksonvibrations,youwill find that theymaygivedifferent formulasfor thenatural frequenciesandvibrationmodes. (If you read a lot of textbooks on vibrations there is probably something seriously wrongwith yoursociallife).Thisispartlybecausesolving for anduisrathercomplicated(especiallyifyouhavetodothecalculationbyhand),andpartlybecausethisformulahidessomesubtlemathematicalfeaturesoftheequationsofmotionforvibratingsystems.Forexample,thesolutionsto aregenerallycomplex( anduhaverealandimaginaryparts),soitisnotobviousthatourguess actuallysatisfiestheequationofmotion.Itturnsout,however,thattheequationsofmotionforavibratingsystemcanalwaysbearrangedsothatMandKaresymmetric.Inthiscase anduarereal,and isalwayspositiveorzero.Theoldfashionedformulasfornaturalfrequenciesandvibrationmodesshowthismoreclearly.Butourapproachgivesthesameanswer,andcanalsobegeneralizedrathereasilytosolvedampedsystems(seeSection5.5.5),whereasthetraditionaltextbookmethodscannot.5.5.3Freevibrationofundampedlinearsystemswithmanydegreesoffreedom.Asanexample,considera systemwithn identicalmasseswith mass m, connected by springs with stiffness k, asshowninthepicture.Supposethatattimet=0 themassesare displaced from their static equilibrium position bydistances ,andhaveinitialspeeds . Wewould like to calculate the motion of each mass

asafunctionoftime.Itisconvenienttorepresenttheinitialdisplacementandvelocityasndimensionalvectorsuandv,as

,and .Inaddition,wemustcalculatethenaturalfrequencies andmodeshapes ,i=1..nforthesystem.Themotioncanthenbecalculatedusingthefollowingformula

where

Here,thedotrepresentsanndimensionaldotproduct(toevaluateitinmatlab,justusethedot()command).Thisexpressiontellsusthatthegeneralvibrationofthesystemconsistsofasumofallthevibrationmodes,(whichallvibrateattheirowndiscretefrequencies). Youcancontrolhowbigthecontributionisfromeachmodebystartingthe



systemwith different initial conditions. Themode shapes have thecurious property that the dot product of two different mode shapes isalways zero ( , etc) so you can see that if theinitial displacements u happen to be the same as a mode shape, thevibrationwillbeharmonic.The figure on the right animates themotion of a systemwith 6masses,which is set inmotion by displacing the leftmostmass and releasing it.The graph shows the displacement of the leftmostmass as a function oftime. Youcandownload theMATLABcode for thiscomputationhere,and see how the formulas listed in this section are used to compute themotion.Theprogramwillpredictthemotionofasystemwithanarbitrarynumberofmasses,andsinceyoucaneasilyeditthecodetotypeinadifferentmassandstiffnessmatrix,iteffectivelysolvesanytransientvibrationproblem.5.5.4Forcedvibrationoflightlydampedlinearsystemswithmanydegreesoffreedom.It is quite simple to find a formula for the motion of anundamped system subjected to time varying forces. Thepredictionsareabitunsatisfactory,however,becausetheirvibration of an undamped system always depends on theinitial conditions. In a real system, damping makes thesteadystateresponseindependentoftheinitialconditions.However, we can get an approximate solution for lightlydamped systems by finding the solution for an undampedsystem,andthenneglectingthepartofthesolutionthatdependsoninitialconditions.Asanexample,wewillconsiderthesystemwithtwospringsandmassesshowninthepicture.Eachmassissubjectedtoaharmonicforce,whichvibrateswithsomefrequency (theforcesactingonthedifferentmassesallvibrateatthesamefrequency).Theequationsofmotionare

Wecanwritetheseinmatrixformas

or,moregenerally,

Tofindthesteadystatesolution,wesimplyassumethatthemasseswillallvibrateharmonicallyatthesamefrequencyastheforces.Thismeansthat , ,where arethe(unknown)amplitudesofvibrationofthetwomasses.Invectorformwecouldwrite ,where .Substitutingthisintotheequationofmotiongives

ThisisasystemoflinearequationsforX.TheycaneasilybesolvedusingMATLAB.Asanexample,hereisasimpleMATLABfunctionthatwillcalculatethevibrationamplitudeforalinearsystemwithmanydegreesoffreedom,giventhestiffnessandmassmatrices,andthevectorofforcesf.

functionX=forced_vibration(K,M,f,omega)%Functiontocalculatesteadystateamplitudeof%aforcedlinearsystem.%Kisnxnthestiffnessmatrix%Misthenxnmassmatrix%fisthendimensionalforcevector%omegaistheforcingfrequency,inradians/sec.%ThefunctioncomputesavectorX,givingtheamplitudeof%eachdegreeoffreedom%



X=(KM*omega^2)\fend

Thefunctionisonlyonelinelong!Asanexample,thegraphbelowshowsthepredictedsteadystatevibrationamplitudeforthespringmasssystem,forthespecialcasewherethemassesareallequal ,and thespringsallhave thesamestiffness .Thefirstmassissubjectedtoaharmonicforce ,andnoforceactsonthesecondmass. Notethat thegraph shows themagnitude of the vibration amplitude the formula predicts that for some frequencies somemasseshavenegativevibrationamplitudes,butthenegativesignhasbeenignored,asthenegativesignjustmeansthatthemassvibratesoutofphasewiththeforce.

Severalfeaturesoftheresultareworthnoting:

If the forcing frequency is close to any one of the natural frequencies of the system, huge vibration amplitudesoccur. Thisphenomenon isknownasresonance. Youcancheck thenatural frequenciesof thesystemusing thelittlematlabcodeinsection5.5.2 theyturnouttobe and . At thesefrequenciesthevibrationamplitudeistheoreticallyinfinite.

Thefigurepredictsanintriguingnewphenomenon atamagicfrequency,theamplitudeofvibrationofmass1(thatsthemassthat the force acts on) drops to zero. This is called Antiresonance, and it has an important engineering application. Suppose that we have designed a system with a seriousvibrationproblem(liketheLondonMilleniumbridge). Usually,this occurs because some kind of unexpected force is excitingoneof thevibrationmodes in the system. Wecan idealize thisbehavior as amassspring systemsubjected to aforce, as shown in the figure. So how dowe stop the system from vibrating? Our solution for a 2DOF systemshows that a systemwith twomasses will have an antiresonance. Sowe simply turn our 1DOF system into a2DOFsystembyaddinganotherspringandamass,andtunethestiffnessandmassofthenewelementssothattheantiresonanceoccursattheappropriatefrequency.Ofcourse,addingamasswillcreateanewvibrationmode,butwecanmakesurethatthenewnaturalfrequencyisnotatabadfrequency.Wecanalsoaddadashpotinparallelwiththespring,ifwewant thishastheeffectofmakingtheantiresonancephenomenonsomewhatlesseffective(thevibrationamplitudewillbe small,but finite, at the magic frequency),but thenewvibrationmodeswill alsohaveloweramplitudesatresonance.Theaddedspring masssystemiscalledatunedvibrationabsorber. ThisapproachwasusedtosolvetheMilleniumBridgevibrationproblem.

5.5.5TheeffectsofdampingIn most design calculations, we dont worry aboutaccountingfortheeffectsofdampingveryaccurately.Thisis partly because its very difficult to find formulas thatmodeldampingrealistically,andevenmoredifficulttofindvaluesforthedampingparameters.Also,themathematicsrequired to solve damped problems is a bit messy. Oldtextbooksdontcoverit,becauseforpracticalpurposesitisonlypossibletodothecalculationsusingacomputer.Itisnothardtoaccountfortheeffectsofdamping,however,anditishelpfultohaveasenseofwhatitseffectwillbeina



realsystem.Wellgothroughthisratherbrieflyinthissection.Equations ofmotion:The figure shows a damped springmass system. The equations ofmotion for the system caneasilybeshowntobe

To solve these equations,wehave to reduce them to a system thatMATLABcanhandle,by rewriting themas firstorderequations.Wefollowthestandardproceduretodothis define and asnewvariables,andthenwritetheequationsinmatrixformas

(Thisresultmightnotbeobvioustoyou ifso,multiplyoutthevectormatrixproductstoseethattheequationsareallcorrect).Thisisamatrixequationoftheform

whereyisavectorcontainingtheunknownvelocitiesandpositionsofthemass.Free vibration response: Suppose that at time t=0 the system has initial positions and velocities

,andwewishtocalculatethesubsequentmotionofthesystem.Todothis,wemustsolvetheequationofmotion.Westartbyguessing that thesolutionhas the form (thenegativesign is introducedbecauseweexpectsolutionstodecaywithtime).Here, isaconstantvector,tobedetermined.Substitutingthisintotheequationofmotiongives

This isanothergeneralizedeigenvalueproblem,andcaneasilybesolvedwithMATLAB. Thesolution ismuchmorecomplicated for a damped system, however, because the possible values of and that satisfy the equation are ingeneralcomplex thatistosay,each canbeexpressedas ,where and arepositiverealnumbers,and

.ThismakesmoresenseifwerecallEulersformula

(ifyouhaventseenEulersformula,trydoingaTaylorexpansionofbothsidesoftheequation youwillfindtheyaremagicallyequal.IfyoudontknowhowtodoaTaylorexpansion,youprobablystoppedreadingthisagesago,butifyouare still hanging in there, just trustme). So, the solution is predicting that the responsemay be oscillatory, aswewould expect. Once all the possible vectors and have been calculated, the response of the system can becalculatedasfollows:

1.ConstructamatrixH,inwhicheachcolumnisoneofthepossiblevaluesof (MATLABconstructsthismatrixautomatically)

2.Constructadiagonalmatrix (t),whichhastheform

whereeach isoneofthesolutionstothegeneralizedeigenvalueequation.3.Calculateavectora(thisrepresentstheamplitudesofthevariousmodesinthevibrationresponse)thatsatisfies

4.Thevibrationresponsethenfollowsas

Allthematricesandvectorsintheseformulasarecomplexvalued butall theimaginarypartsmagicallydisappear inthefinalanswer.HEALTH WARNING: The formulas listed here only work if all the generalized eigenvalues satisfying

aredifferent.Forsomeveryspecialchoicesofdamping,someeigenvaluesmayberepeated.Inthiscasethe formulawontwork. A quick and dirty fix for this is just to change the damping very slightly, and the problemdisappears.Yourappliedmathcourseswillhopefullyshowyouabetterfix,butwewontworryaboutthathere.



This all sounds a bit involved, but it actually only takes afewlinesofMATLABcodetocalculatethemotionofanydamped system. As an example, a MATLAB code thatanimatesthemotionofadampedspringmasssystemshownin the figure (butwith an arbitrarynumber ofmasses) canbedownloadedhere. Youcanusethecodetoexplorethebehavior of the system. In addition, you can modify thecodetosolveanylinearfreevibrationproblembymodifyingthematricesMandD.Herearesomeanimationsthatillustratethebehaviorofthesystem.Theanimationsbelowshowvibrationsofthesystemwith initial displacements corresponding to the three mode shapes of the undamped system (calculated using theprocedureinSection5.5.2).Theresultsareshownfork=m=1 .Ineachcase,thegraphplotsthemotionofthethreemasses ifacolordoesntshowup,itmeansoneoftheothermasseshastheexactsamedisplacement.

Mode1Mode2Mode3

Noticethat

1. Foreachmode, thedisplacementhistoryof anymass looksvery similar to thebehaviorof adamped,1DOFsystem.

2.Theamplitudeofthehighfrequencymodesdieoutmuchfasterthanthelowfrequencymode.Thisexplainswhyitissohelpfultounderstandthebehaviorofa1DOFsystem.Ifamorecomplicatedsystemissetinmotion,itsresponseinitiallyinvolvescontributionsfromall itsvibrationmodes. Soon,however, thehighfrequencymodesdieout,andthedominantbehaviorisjustcausedbythelowestfrequencymode.Theanimationtothe rightdemonstrates thisverynicely here, the systemwas startedbydisplacingonly the firstmass. The initial response isnotharmonic,but after a short time thehigh frequencymodes stop contributing, and the system behaves just like a 1DOFapproximation. For design purposes, idealizing the system as a 1DOF dampedspringmasssystemisusuallysufficient.Notice also that light damping has very little effect on the natural frequencies andmode shapes so the simple undamped approximation is a good way to calculatethese.Ofcourse,ifthesystemisveryheavilydamped,thenitsbehaviorchangescompletely thesystemnolongervibrates,andinsteadjustmovesgraduallytowardsitsequilibriumposition.Youcansimulatethisbehaviorforyourselfusingthematlabcode tryrunningitwith orhigher.Systemsofthiskindarenotofmuchpracticalinterest.Steadystateforcedvibrationresponse.Finally,wetakealookattheeffectsofdampingontheresponseofaspringmasssystemtoharmonicforces.Theequationsofmotionforadamped,forcedsystemare

Thisisanequationoftheform

wherewehaveusedEulersfamousformulaagain.Wecanfindasolutionto

byguessingthat ,andsubstitutingintothematrixequation



Thisequationcanbesolvedfor .Similarly,wecansolve

by guessing that ,which gives an equation for of the form . You actually dontneedtosolvethisequation youcansimplycalculate byjustchangingthesignofalltheimaginarypartsof .Thefullsolutionfollowsas

Thisisthesteadystatevibrationresponse.Justasforthe1DOFsystem,thegeneralsolutionalsohasatransientpart,whichdependsoninitialconditions.Weknowthatthetransientsolutionwilldieaway,soweignoreit.Thesolutionfory(t)lookspeculiar,becauseofthecomplexnumbers.Ifwejustwanttoplotthesolutionasafunctionoftime,wedonthavetoworryaboutthecomplexnumbers,becausetheymagicallydisappearinthefinalanswer.Infact,if we use MATLAB to do the computations, we never even notice that the intermediate formulas involve complexnumbers. Ifwedoplot the solution, it isobvious that eachmassvibratesharmonically, at the same frequencyas theforce (this isobvious from the formula too). Itsnotworthplotting the function weare reallyonly interested in theamplitudeofvibrationofeachmass.Thiscanbecalculatedasfollows

1.Let , denotethecomponentsof and2.Thevibrationofthejthmassthenhastheform

where

aretheamplitudeandphaseoftheharmonicvibrationofthemass.Ifyouknowalotaboutcomplexnumbersyoucouldtrytoderivetheseformulasforyourself.Ifnot,justtrustme yourmathclassesshouldcoverthiskindofthing.MATLABcanhandleallthesecomputationseffortlessly.Asanexample,hereisasimpleMATLABscriptthatwillcalculatethesteadystateamplitudeofvibrationandphaseofeachdegreeoffreedomofaforcedndegreeoffreedomsystem,giventheforcevectorf,andthematricesMandD thatdescribethesystem.

function[amp,phase]=damped_forced_vibration(D,M,f,omega)%Functiontocalculatesteadystateamplitudeof%aforcedlinearsystem.%Dis2nx2nthestiffness/dampingmatrix%Misthe2nx2nmassmatrix%fisthe2ndimensionalforcevector%omegaistheforcingfrequency,inradians/sec.%Thefunctioncomputesavectoramp,givingtheamplitudeof%eachdegreeoffreedom,andasecondvectorphase,%whichgivesthephaseofeachdegreeoffreedom%Y0=(D+M*i*omega)\f%Theihereissqrt(1)%WedontneedtocalculateY0barwecanjustchangethesignof%theimaginarypartofY0usingthe'conj'command

forj=1:length(f)/2amp(j)=sqrt(Y0(j)*conj(Y0(j)))phase(j)=log(conj(Y0(j))/Y0(j))/(2*i)end

endAgain,thescriptisverysimple.Hereisagraphshowingthepredictedvibrationamplitudeofeachmassinthesystemshown.Notethatonlymass1issubjectedtoaforce.



Theimportantconclusionstobedrawnfromtheseresultsare:

1.Weobservetworesonances,atfrequenciesveryclosetotheundampednaturalfrequenciesofthesystem.2.Forlightdamping, theundampedmodelpredicts thevibrationamplitudequiteaccurately,exceptverycloseto

theresonanceitself(wheretheundampedmodelhasaninfinitevibrationamplitude)3. In a damped system, the amplitudeof the lowest frequency resonance is generallymuchgreater thanhigher

frequencymodes. For this reason, it is often sufficient to consider only the lowest frequencymode in designcalculations. Thismeanswecan idealize thesystemas justasingleDOFsystem,and thinkof itasasimplespringmass system as described in the early part of this chapter. The relative vibration amplitudes of thevariousresonancesdodependtosomeextentonthenatureoftheforce itispossibletochooseasetofforcesthat will excite only a high frequency mode, in which case the amplitude of this special excited mode willexceedalltheothers.Butformostforcing,thelowestfrequencyoneistheonethatmatters.

4.Theantiresonancebehaviorshownbytheforcedmassdisappearsifthedampingistoohigh.

dynamics and vibrations_ notes_ multi-dof vibrations.pdf

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