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机械类外文文献翻译
英文原文
APracticalApproachtoVibrationDetectionandMeasurement
——PhysicalPrinciplesandDetectionTechniques
By:
JohnWilson,theDynamicConsultant,LLC
Thistutorialaddressesthephysicsofvibration;dynamicsofaspringmasssystem;damping;displacement,velocity,andacceleration;andtheoperatingprinciplesofthesensorsthatdetectandmeasuretheseproperties.
Vibrationisoscillatorymotionresultingfromtheapplicationofoscillatoryorvaryingforcestoastructure.Oscillatorymotionreversesdirection.Asweshallsee,theoscillationmaybecontinuousduringsometimeperiodofinterestoritmaybeintermittent.Itmaybeperiodicornonperiodic,i.e.,itmayormaynotexhibitaregularperiodofrepetition.Thenatureoftheoscillationdependsonthenatureoftheforcedrivingitandonthestructurebeingdriven.
Motionisavectorquantity,exhibitingadirectionaswellasamagnitude.Thedirectionofvibrationisusuallydescribedintermsofsomearbitrarycoordinatesystem(typicallyCartesianororthogonal)whosedirectionsarecalledaxes.Theoriginfortheorthogonalcoordinatesystemofaxesisarbitrarilydefinedatsomeconvenientlocation.
Mostvibratoryresponsesofstructurescanbemodeledassingle-degree-of-freedomspringmasssystems,andmanyvibrationsensorsuseaspringmasssystemasthemechanicalpartoftheirtransductionmechanism.Inadditiontophysicaldimensions,aspringmasssystemcanbecharacterizedbythestiffnessofthespring,K,andthemass,M,orweight,W,ofthemass.Thesecharacteristicsdeterminenotonlythestaticbehavior(staticdeflection,d)ofthestructure,butalsoitsdynamiccharacteristics.Ifgistheaccelerationofgravity:
F=MA
W=Mg
K=F/d=W/d
d=F/K=W/K=Mg/K
DynamicsofaSpringMassSystem
Thedynamicsofaspringmasssystemcanbeexpressedbythesystem'sbehaviorinfreevibrationand/orinforcedvibration.
FreeVibration.Freevibrationisthecasewherethespringisdeflectedandthenreleasedandallowedtovibratefreely.Examplesincludeadivingboard,abungeejumper,andapendulumorswingdeflectedandlefttofreelyoscillate.
Twocharacteristicbehaviorsshouldbenoted.First,dampinginthesystemcausestheamplitudeoftheoscillationstodecreaseovertime.Thegreaterthedamping,thefastertheamplitudedecreases.Second,thefrequencyorperiodoftheoscillationisindependentofthemagnitudeoftheoriginaldeflection(aslongaselasticlimitsarenotexceeded).Thenaturallyoccurringfrequencyofthefreeoscillationsiscalledthenaturalfrequency,fn:
(1)
ForcedVibration.Forcedvibrationisthecasewhenenergyiscontinuouslyaddedtothespringmasssystembyapplyingoscillatoryforceatsomeforcingfrequency,ff.Twoexamplesarecontinuouslypushingachildonaswingandanunbalancedrotatingmachineelement.Ifenoughenergytoovercomethedampingisapplid,themotionwillcontinueaslongastheexcitationcontinues.Forcedvibrationmaytaketheformofself-excitedorexternallyexcitedvibration.Self-excitedvibrationoccurswhentheexcitationforceisgeneratedinoronthesuspendedmass;externallyexcitedvibrationoccurswhentheexcitationforceisappliedtothespring.Thisisthecase,forexample,whenthefoundationtowhichthespringisattachedismoving.
Transmissibility.Whenthefoundationisoscillating,andforceistransmittedthroughthespringtothesuspendedmass,themotionofthemasswillbedifferentfromthemotionofthefoundation.Wewillcallthemotionofthefoundationtheinput,I,andthemotionofthemasstheresponse,R.TheratioR/Iisdefinedasthetransmissibility,Tr:
Tr=R/I
Resonance.Atforcingfrequencieswellbelowthesystem'snaturalfrequency,R
I,andTr
1.Astheforcingfrequencyapproachesthenaturalfrequency,transmissibilityincreasesduetoresonance.Resonanceisthestorageofenergyinthemechanicalsystem.Atforcingfrequenciesnearthenaturalfrequency,energyisstoredandbuildsup,resultinginincreasingresponseamplitude.Dampingalsoincreaseswithincreasingresponseamplitude,however,andeventuallytheenergyabsorbedbydamping,percycle,equalstheenergyaddedbytheexcitingforce,andequilibriumisreached.Wefindthepeaktransmissibilityoccurringwhenff
fn.Thisconditioniscalledresonance.
Isolation.Iftheforcingfrequencyisincreasedabovefn,Rdecreases.Whenff=1.414fn,R=IandTr=1;athigherfrequenciesR EffectsofMassandStiffnessVariations.FromEquation
(1)itcanbeseenthatnaturalfrequencyisproportionaltothesquarerootofstiffness,K,andinverselyproportionaltothesquarerootofweight,W,ormass,M.Therefore,increasingthestiffnessofthespringordecreasingtheweightofthemassincreasesnaturalfrequency.
Damping
Dampingisanyeffectthatremoveskineticand/orpotentialenergyfromthespringmasssystem.Itisusuallytheresultofviscous(fluid)orfrictionaleffects.Allmaterialsandstructureshavesomedegreeofinternaldamping.Inaddition,movementthroughair,water,orotherfluidsabsorbsenergyandconvertsittoheat.Internalintermolecularorintercrystallinefrictionalsoconvertsmaterialstraintoheat.And,ofcourse,externalfrictionprovidesdamping.
Dampingcausestheamplitudeoffreevibrationtodecreaseovertime,andalsolimitsthepeaktransmissibilityinforcedvibration.ItisnormallycharacterizedbytheGreekletterzeta(
),orbytheratioC/Cc,wherecistheamountofdampinginthestructureormaterialandCcis"criticaldamping."Mathematically,criticaldampingisexpressedasCc=2(KM)1/2.Conceptually,criticaldampingisthatamountofdampingwhichallowsthedeflectedspringmasssystemtojustreturntoitsequilibriumpositionwithnoovershootandnooscillation.Anunderdampedsystemwillovershootandoscillatewhendeflectedandreleased.Anoverdampedsystemwillneverreturntoitsequilibriumposition;itapproachesequilibriumasymptotically.
Displacement,Velocity,andAcceleration
Sincevibrationisdefinedasoscillatorymotion,itinvolvesachangeofposition,ordisplacement(seeFigure1).
Figure1.Phaserelationshipsamongdisplacement,velocity,andaccelerationareshownonthesetimehistoryplots.
Velocityisdefinedasthetimerateofchangeofdisplacement;accelerationisthetimerateofchangeofvelocity.Sometechnicaldisciplinesusethetermjerktodenotethetimerateofchangeofacceleration.
SinusoidalMotionEquation.Thesingle-degree-of-freedomspringmasssystem,inforcedvibration,maintainedataconstantdisplacementamplitude,exhibitssimpleharmonicmotion,orsinusoidalmotion.Thatis,itsdisplacementamplitudevs.timetracesoutasinusoidalcurve.GivenapeakdisplacementofX,frequencyf,andinstantaneousdisplacementx:
(2)
atanytime,t.
VelocityEquation.Velocityisthetimerateofchangeofdisplacement,whichisthederivativeofthetimefunctionofdisplacement.Forinstantaneousvelocity,v:
(3)
Sincevibratorydisplacementismostoftenmeasuredintermsofpeak-to-peak,doubleamplitude,displacementD=2X:
(4)
Ifwelimitourinteresttothepeakamplitudesandignorethetimevariationandphaserelationships:
(5)
where:
V=peakvelocity
AccelerationEquation.Similarly,accelerationisthetimerateofchangeofvelocity,thederivativeofthevelocityexpression:
(6)
and
(7)
where:
A=peakacceleration
Itthuscanbeshownthat:
V=
fD
A=2
2f2D
D=V/
f
D=A/2
2f2
Fromtheseequations,itcanbeseenthatlow-frequencymotionislikelytoexhibitlow-amplitudeaccelerationseventhoughdisplacementmaybelarge.Itcanalsobeseenthathigh-frequencymotionislikelytoexhibitlow-amplitudedisplacements,eventhoughaccelerationislarge.Considertwoexamples:
•At1Hz,1in.pk-pkdisplacementisonly~0.05gacceleration;10in.is~0.5g•At1000Hz,1gaccelerationisonly~0.00002in.displacement;100gis~0.002in.
MeasuringVibratoryDisplacement
OpticalTechniques.Ifdisplacementislargeenough,asatlowfrequencies,itcanbemeasuredwithascale,calipers,orameasuringmicroscope.Athigherfrequencies,displacementmeasurementrequiresmoresophisticatedopticaltechniques.
High-speedmoviesandvideocanoftenbeusedtomeasuredisplacementsandareespeciallyvaluableforvisualizingthemotionofcomplexstructuresandmechanisms.Thetwomethodsarelimitedbyresolutiontofairlylargedisplacementsandlowfrequencies.Strobelightsandstroboscopicphotographyarealsousefulwhendisplacementsarelargeenough,usually>0.1in.,tomakethempractical.
Thechangeinintensityorangleofalightbeamdirectedontoareflectivesurfacecanbeusedasanindicationofitsdistancefromthesource.Ifthedetectionapparatusisfastenough,changesofdistancecanbedetectedaswell.
Themostsensitive,accurate,andpreciseopticaldeviceformeasuringdistanceordisplacementisthelaserinterferometer.Withthisapparatus,areflectedlaserbeamismixedwiththeoriginalincidentbeam.Theinterferencepatternsformedbythephasedifferencescanmeasuredisplacementdownto<100nm.NISTandothernationalprimarycalibrationagenciesuselaserinterferometersforprimarycalibrationofvibrationmeasurementinstrumentsatfrequenciesupto25kHz.
ElectromagneticandCapacitiveSensors.Anotherimportantclassofnoncontact,special-purposedisplacementsenso