振动学系列Vibration3ViscousDamping.docx
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振动学系列Vibration3ViscousDamping.docx
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振动学系列Vibration3ViscousDamping
3.ViscousDamping
Inthespring-masssystemswehavestudiedsofar,theresponsewillcontinuetoinfinity;thatis,itwillneverdieout.Realsystems,ofcourse,dodieouteventually,anditisimportantthatweaccountforthisinourmathematicalmodel.
Tocreatearesponsethatdiesoutovertime,wetypicallyadddampingtothesystem.
Thedampingforceisusuallytakentobeproportionaltothevelocity;thatis:
Asvelocityincreases,sodoesthedampingforce.
Dampingisusuallytheresultofonetypeoffrictionoranother.
Theequationofmotionforthedampedcaseis
Thisisalsoasecond-order,homogeneousdifferentialequation.Thegeneralsolutiontothistypeofequationis
Substitutethisintotheequationofmotion
Herewehavethesamesituationasearlier.Since
wehave
Usingthequadraticformula,wefind
Thisiscalledthe“characteristicequation”ofthesystem.Observingthisequation,weseethattherearethreepossiblesolutiontypes:
1.→arereal
2.→arecomplex
3.→(real)
Letusdefinethe“criticaldampingcoefficient”
Then
Thus,ccristhedampingnecessarytoproducecase3.
Also,letusdefinethe“dampingratio”
Then,wecanrewritethecharacteristicequationas
Again,wehavethreecases
1.ζ>1λ→real
2.ζ<1λ→complex
3.ζ=1λ→“criticallydamped”
Case1:
ζ>1
Inthecasewhereζ>1,theλarereal
Sincewehavetwodistinctroots,thesolutiontotheequationofmotionisalinearcombination
wherewecansolvefora1anda2usinginitialconditions
Solvingbothoftheseequationssimultaneouslygives
Substitutingtheexpressionsforλ1,2givenaboveresultsin
Thistypeofresponseisnotoscillatory.Theresponsedecaysbacktoitsinitialstateexponentially.
Sinceλ1andλ2arenegative,theresponsedecayswithtime.Thiscaseiscalledthe“Overdamped”case.
Case2:
ζ<1
Recall
Ifζ<1,thenζ2–1<0,andweareleftwithacomplexsolution.Letusrearrange
then
Let=“dampednaturalfrequency”.Then
Thegeneralsolutionisstill
buttheλvaluesarecomplex.Wecanagainsolvefora1anda2usinginitialconditions
Solvingbothoftheseequationssimultaneouslygives
Substitutingthisintothegeneralsolutiongives
WenextmakeuseofEuler’sidentity
Substitutingthisintothegeneralsolutiongives
InHomework1,wefoundthatwecouldexpress
Thus
where
Whatdoesthesolutionlooklike?
Theresponseoscillatesasitdecaysexponentially.
Case3:
ζ=1
Inthecriticallydampedcase,wefindthat
Thesolutiontakestheform
Andfromtheinitialconditions
Asbefore,thistakestheformofadecayingexponentialsolution.Thesolutiondoesnotoscillate.
Whatdoeachoftheseresponsesmeanphysically?
Again,thinkofthecardrivingoffthecurb:
theresponseoscillatesaboutanequilibrium
ζ<1
ζ>1
doesnotoscillate,returnstoequilibriumslowly
doesnotoscillate,returnstoequilibriumquickly
ζ=1
Underdampedresponseisdesirableinguitarstrings.
Criticallydampedresponseisdesirableinautomotivesuspensions.
Overdampedresponseisdesirableinadoorclosingmechanism.
Example
Acarwithmass1000kgistobedesignedsuchthatitssuspensioniscriticallydamped.Thesuspensiondeflects10cmunderthecar’sweight.Findkandcforthissuspension.
Solution
Theproblemstatementgivesthestaticrideheightdeflectionas10cm.FromtheBajacardesignexamplewehave
Rearranginggives
Theformulafordampingratiois
Tobecriticallydamped,ofcourse,wemusthaveζ=1.Thus
Ifweaddmasstothecarbyaddinga100kgpassenger,whathappens?
Thedampingratiowillchangeslightly,sincewehaveaddedtothemass.
Thus,thesolutionisslightlyunderdampedandwilloscillate!
Itisimpossibletodesignareal-worldsystemtobecriticallydampedunderallsituations.
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