Engineering Basic Mechanics Ⅱ Dynamics 工程基础力学 Ⅱ 动力学 课件 Chapter 10 Vibrations

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