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Chapter10Vibrations
MainContents
§10.1
UndampedFreeVibration§10.2
UndampedForcedVibration§10.3ViscousDampedFreeVibration§10.4
ViscousDampedForcedVibrationVibration
istheperiodicmotionofabodyorsystemofconnectedbodiesdisplacedfromapositionofequilibrium.
Vibrationbelongstothesecondcategoryofkineticproblems
-solvemotionproblemsthroughknownactiveforces
Researchmethodofvibrationproblems
Motionanalysis;
Forceanalysis;
Selecttheappropriatekinetictheorem;
Establishandsovledifferentialequationsofmotion.
Selecttheappropriategeneralizedcoordinates;IntroductionDifferentialequationsoffreevibration
10.1UndampedfreevibrationEstablish
Oxaxis,theequilibriumpositionofparticleMas
coordinateoriginO,whenthepositionofMisx,elasticforceFactingontheMcanbedescribedasBecauseSo(naturalfrequency)Let
Wehave
thisisahomogeneous,second-order,linear,differentialequationwithconstantcoefficients.Usingthemethodsofdifferentialequations,itsgeneralsolutionis
definingnewconstantsC
andϕC
amplitudeofvibration
ϕphaseangleτ
periodf
frequecy10.1Undampedfreevibration
Example10-1mvldetermien(1)vibrationregulationofblock;(2)maxiumtensioninthewirerope.10.1UndampedfreevibrationAblockm=6000kgissuspendfromwireropethatpassesoverawheel.Theelasticmodulusandcross-sectionalareaareE=200GPaandA=2.89×10-4m2,respectively.Theblockisbeingloweredwithcostantvelocityv=0.25m/s,whentheblockisloweredtobel=25m,thewireropesuddenlyjams,
Solution:thesystemcanbeconsideredasspring-blocksystem,thestiffnessofspringis
N/m
Wecanassumethatwhenthewireropeisjammed,thetimeist=0,attheinstant,thepositionofblockisinitialequilibriumposition.Choosethedownwarddistancexofblockasgeneralizdcoordinate,thenvibrationequationofthesystemismkequilibriumpositionOxthesolutionisUsinginitialconditionHence,
10.1UndampedfreevibrationExample10-1S-1
m
(2)maxiumtensioninthewireropeChoosetheblockasobjectofstudymkOxFT10.1UndampedfreevibrationkN
equilibriumposition
10.2UndampedforcedvibrationForcedvibrationsis
casuedbyanexternalperiodicorintermittentforceappliedtothesystem.
DifferentialequationofvibrationorThesolutionis
xciscomplementarysolution;
xpisparticularsolution
10.2UndampedforcedvibrationAmplitudeofforcedvibration
themagnificationfactorMFisdefinedas(1)(2)(3)(4)
10.2Undampedforcedvibrationɷ
<<
ɷn
;MF≈1thevibrationoftheblockwillbeinphasewiththeappliedforceF;resonanceɷ>ɷn
;thevibrationoftheblockisoutofphasewiththeforceɷ>>ɷn
;theblockremainsalmoststationary
10.3ViscousdampedfreevibrationDamping
isattributedtotheresistancecreatedbythesubstance,suchaswater,oil,orair,inwhichthesystemvibrates.Thetypeofforcedevelopedundertheseconditionsiscalledaviscousdampingforce.Themagnitudeofthisforceisexpressedbyanequationofformc
—coefficientofviscousdampingor
(1)SolutionisSubstitutingthissolutionintotheequation(1)Sinceisneverzero,itssolutionis
10.3ViscousdampedfreevibrationDifferentialequationofvibrationThegeneralsolutionofequation(1)isalinearcombinationofexponentialswhichinvolvesλ1
andλ2
.Definingthecriticaldampingcoefficientcc
todiscussthreepossiblecombinations
ofλ1
andλ2
(1)Overdampedsystemwhenc>cc
,therootsλ1and
λ2arebothreal,thegeneralsolutionofequation(1)canbewrittenas(2)Criticallydampedsystem
ifc=cc
,theλ1=λ2=-cc/2m=-ɷn,thegeneralsolutionofequation(1)is
10.3Viscousdampedfreevibration(3)Underdampedsystem
Whenc<cc,therootsλ1andλ2arecomplexnumbers,thegeneralsolutionis
Dandϕ
areconstantsgenerallydeterminedfromtheinitialconditionsoftheproblem.Theconstantɷdisthedampednaturalfrequencyofthesystem.Ithasavalueoftheratioc/ccisthedampingfactor.
10.3ViscousdampedfreevibrationIfadashpotisattachedtotheblockandspringshowninthisfigure,thedifferentialequationis
(2)
Becauseallsystemscontainfriction,however,thissolutionwilldampenoutwithtime.Onlythexpwillremain,itsgeneralsolutionisthesumofacomplementarysolutionxcandaparticularsolutionxp。Substitutingitintoequation(2),thenSincethisequationholdsforalltime,theconstantcoefficie
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