vibrazione

Basicconcepts

Definition

Vibration(alsoknownasoscillation)referstoaprocessofstatechange.Thatis,thereciprocatingmotionoftheobject.

Inhighschoolphysics,thereareonlyfoursimplestmotionsthatcanbestudiedquantitatively(theformulamethod,drawingmethod,andlistmethodcanbeusedtogivedefinitevalues):uniformlyvariablelinearmotion,uniformlycircularmotion,Projectilemotionandsimpleharmonicmotion.

Complicatedsportscanbequalitativelystudiedbyrelyingonthesefoursports.

Ifyouinsistonquantitativelystudyingcomplexmotions,youwillalsorelyonthesefourtypesofmotionsforapproximateresearch.

Horum quatuor simplicium motuum, motus uniformiter variabiles et projectilem motus praecedentes, et relatio inter motum motus (positionis, velocitatis) et temporis istopologici (unum ad unum-correspondentiam) et non iterabilem.

Conceptus

Vibrationisoneofthemostcommonphenomenainnature.Fromtheuniversetothesmallsubatomicparticles,thereisnovibration.Variousformsofphysicalphenomena,includingsound,light,heat,etc.,containvibration.Peoplecannotlivewithoutvibration:thebeatingoftheheart,thevibrationoftheeardrumandthevocalcordsareallindispensablefunctionsofthehumanbody;humanvisionisstimulatedbylight,andlightisessentiallyanelectromagneticvibration;youcannotlivewithoutsoundAndmusic,andsoundproduction,transmissionandreceptionareinseparablefromvibration.Inthefieldofengineeringtechnology,vibrationphenomenaarealsoeverywhere.Forexample,thevibrationofbridgesandbuildingsundergustsorearthquakes,thevibrationofaircraftandshipsinnavigation,thevibrationofmachinetoolsandtoolsduringprocessing,thevibrationofvariouspowermachinery,theself-excitedvibrationinthecontrolsystem,etc..

Inmanycases,vibrationisconsideredanegativefactor.Forexample,vibrationcanaffectthefunctionsofprecisioninstrumentsandequipment,reduceprocessingaccuracyandsmoothness,andaggravatefatigueandwearofcomponents,therebyshorteningtheservicelifeofmachinesandstructures.Vibrationmayalsocauselargedeformationanddamagetothestructure.Somebridgeshavebeencausedbyvibration.Andcollapse;theflutterofaircraftwingsandthebuffetingofwheelsoftencauseaccidents;thevibrationofvehicles,shipsandcabinswilldegradetheloadingconditions;strongvibrationandnoisewillcauseseriouspublichazards.

However,vibrationalsohasitspositiveside.Forexample,vibrationisthebasisofcommunications,radio,television,radarandothertasks.Sincethe1950s,manyproductionequipmentandprocessesusingvibrationhaveappearedoneafteranother.Forexample,vibrationtransmission,vibrationscreening,vibrationgrinding,vibrationpolishing,vibrationpiledriving,vibrationeliminationofinternalstressandsoon.Theyhavegreatlyimprovedworkingconditionsandincreasedlaborproductivitybytenorahundredtimes.Itcanbeexpectedthatwiththecontinuousprogressofproductionpracticeandscientificresearch,theuseofvibrationwillincreasedaybyday.

Althoughvibrationphenomenaindifferentfieldshavetheirowncharacteristics,theyoftenhavesimilarmathematicaldescriptions.Itisonthebasisofthiscommonalitythatitispossibletoestablishaunifiedtheorytodealwithvariousvibrationproblems.Vibrationissuchabasicsubject.Itusesmathematics,physics,experimentsandcalculationtechniquestoexplorethemechanismofvariousvibrationphenomenaandclarifythebasiclawsofvibrationinordertoovercomethenegativefactorsofvibrationandmakeuseofitspositivefactorstobereasonable.Providetheoreticalbasisforsolvingvariousvibrationproblemsencounteredinpractice.

Classificationofvibration

Accordingtowhetheritcanbedescribedbyadefinitetimefunctionrelationship,vibrationisdividedintotwocategories,namelydeterministicvibrationandrandomvibration(non-deterministicvibration).Deterministicvibrationcanbedescribedbyacertainmathematicalrelationship,andforaspecifiedmoment,acorrespondingfunctionvaluecanbedetermined.Randomvibrationhasrandomcharacteristics,andtheresultsofeachobservationaredifferent.Itcannotbedescribedbyprecisemathematicalrelationsandcannotpredicttheprecisevalueatanymomentinthefuture.Instead,itcanonlybedescribedbyprobabilityandstatistics.Forexample:Anearthquakeisakindofrandomvibration.

Deterministicvibrationisdividedintoperiodicvibrationandnon-periodicvibration.Periodicvibrationincludessimpleharmonicperiodicvibrationandcomplexperiodicvibration.Simpleharmonicperiodicvibrationcontainsonlyonevibrationfrequency.Thecomplexperiodicvibrationcontainsmultiplevibrationfrequencies,andtheratioofanytwovibrationfrequenciesisarationalnumber.Non-periodicvibrationincludesquasi-periodicvibrationandtransientvibration.Quasi-periodicvibrationhasnoperiodicity,andtheratioofatleastonevibrationfrequencytotheothervibrationfrequencyamongthemultiplevibrationfrequenciescontainedisirrational.Transientvibrationsarevibrationsthatcanbedescribedbyvariousimpulsefunctionsordecayfunctions.

Constantcircularmotionandsimpleharmonicmotion

Experspectiva (seu "macroview"), itisperiodica, saepe. Standinginaperiodoftime (vel "microscopice"), itistopologica et non iteranda. Haec igitur duo exercitia sunt multo complicata quam prima duo exercitia.

Simpleharmonicvibration

Definition

Simpleharmonicvibrationcanberegardedasauniformcircularmotionthatdecomposesintwodirectionsorthogonal(thatis,perpendiculartoeachother)(thatis,Projection),themovementinanyonedirectionissimpleharmonicvibration.Itcanbeseenthatsimpleharmonicmotionismuchmorecomplicatedthanuniformcircularmotion.

Theprojectilemotioncanbedecomposedinto:anorthogonallinearmotionwithauniformspeedandanotherlinearmotionwithauniformspeed.Therefore,theprojectilemotionismuchmorecomplicatedthanalinearmotionwithauniformspeed.

Intheprocessoforthogonaldecompositionofuniformcircularmotion,theoriginalcentripetalforcewithconstantmagnitudebecomesarestoringforcewhosemagnitudeanddirectionchangeperiodically.Simpleharmonicvibrationiscomplexenough.Therefore,thevibrationisquantitativelystudieduptothesimpleharmonicvibration.

However,themicroscopicconditionsofvibrationthatweusuallyencounteraremuchmorecomplicatedthansimpleharmonicvibrations.Therefore,thetransitionfromstudyingsimpleharmonicvibrationtostudyingvibration,thermalvibration,etc.requiresinsight,imagination,abstractthinking,andlogicalreasoning.

Features

Thecharacteristicsofsimpleharmonicvibrationare:1.Thereisanequilibriumposition(theonlypositionwherethevibratorshouldbestationaryafterthemechanicalenergyisexhausted).2.Thereisarestoringforcewhosesizeanddirectionarechangedperiodically.3.Singlefrequencyandconstantamplitude.

Thevibratorisanabstractionofthevibratingobject:Ignoretheshapeandsizeoftheobject,andusethemasspointtoreplacetheobjectforresearch.Thismasspointthatreplacesthevibratingobjectiscalledavibrator.

Thepositionofthevibratoratacertainmomentisrepresentedbydisplacementx.Thedisplacementxisthedistanceanddirectionofthe"positionofthevibratoratacertainmoment"obtainedbytakingtheequilibriumpositionasthereference(basepoint-datumpoint).

Whenwestudyuniformlyvariablelinearmotionandprojectilemotion,thereferencepointisselectedatthestartingpointofthemotion.Whenwestudyuniformcircularmotionandsimpleharmonicvibration,thereferencepointisselectedatthecenterofthecircleorattheequilibriumposition(fixedpoint).

Thereferenceobjectshouldoriginallybeapointthatremainsstationary(orassumedtobestationary)duringtheresearchprocess.Ourphysicalthinkingisto"studyfromadefinitequantityandaconstantquantity."

vibration

Thereisanessentialdifferencebetweenthedeterminedquantityandtheconstantquantity.Whenstudyingthelinearmotionwithuniformspeedandtheprojectilemotion,thereferencepointisselectedatthestartingpointofthemotion.Thisisacertainamount,butnotnecessarilyaconstantamount.Especiallywhenweconductasegmentedstudy,theendofeachstageisthestartingpointofthenextstage.Wechoosethestartingpointofthemovementasthereferencepoint,whichcansimplifytheresearchprocess.Thisissubjecttotheprincipleof"simplifyingcomplexity"inphysicsresearch.Therefore,wedonothesitatetochoosedifferentreferencepointsatdifferentresearchstages.

Inthestudyofuniformcircularmotionandsimpleharmonicmotion,duetotheperiodicityonthemacroscopicandthetopologyonthemicroscopic,theproblemisverycomplicated,soitisimpossibletochoosethestartingpointofthemotionanduseitasthereferencepointforresearch.Tochooseadefiniteandunchangingcenterorbalancepositionanduseitasareferencepointforresearchisalsosubjecttotheprincipleof"simplifyingcomplexity"inphysicsresearch.

Inasimpleharmonicvibration,theamplitudeAisthemaximumvalueofthedisplacementx,whichisaconstantquantity.

Theshortesttimerequiredforthevibratortoreturnfromacertainstate(positionandvelocity)tothatstateiscalledaperiodT.Thevibrationofthevibratorinaperiodiscalledatotalvibration.The"number"oftotalvibrationofthevibratorinonesecondiscalledfrequencyf.

TheperiodTisthetimeofonefullvibration,andthefrequencyfisthenumberoffullvibrationsinonesecond,soTf=1(equivalenceformula1ofthefourequations)

Thefrequencyω(pronounced[oumiga])isthecentralanglecorrespondingtoonesecond.Thecentralanglecorrespondingtoafullvibrationis2π(thatis,360degrees).Thisborrowedtheconceptofuniformcircularmotion.Inauniformcircularmotion,ωiscalledtheangularvelocity.Whenauniformcircularmotionisorthogonallydecomposedintoasimpleharmonicmotion,theangularvelocityisconvertedintoacircularfrequency.(Somepeoplecallthecircularfrequencytheangularfrequency)

Patet, ω=2πf.

ωT=2π(equivalenceformula2ofthefourquations)

Demum definias numerum offlulvibrationsperminutas "rotationes". Plane, n=60f.

Quattuor quantitatum fT, f, ω, n, nota est, et aliae tres notae, quae quattuor formulis formulis mutuo permutatis "quattuor aequivalentes" dicuntur.

Aslongastheobjectreciprocatesperiodically,itisvibration.Forexample,whenplayingaball,itsv-tdiagramcorrespondstothesawtoothwaveinelectricalengineering,soitisalsovibration.Somepeoplesay:"Theballhasnoequilibriumposition,ortheequilibriumpositionisnotatthecenterofsymmetryofmotion,soitcannotbeconsideredasvibration."Thosewhosaythat,certainlydidnotlearnelectricalengineeringwell.

ThereisabranchofmathematicscalledFourierintegral,whichcandecomposeanyvibrationintoseveralsimpleharmonicvibrations.Thefrequencyofthesesimpleharmonicvibrationsisanintegermultipleoftheoriginalvibration,andthemainfrequency(fundamentaltone)oftheoriginalvibrationistheminimumfrequencyofthesesimpleharmonicvibrations.

Amplitudoism.

TheprocessbywhichthehumaneardistinguishesthesoundingbodyistheprocessofusingFourierintegralspontaneously,automatically,andinstinctively,whichisveryclever.

Becausethefrequencyofthesoundisdeterminedbythesoundsource,nomatterhowthesoundwavepropagatestoourears,westillaccuratelyidentifythecharacteristicsofthestartingsoundbody.

Vibrationinabroadsense

Inabroadsense,vibrationreferstotheprocessinwhichtheparametersdescribingthestateofthesystem(suchasdisplacementandvoltage)alternateupanddownitsreferencevalue.Inanarrowsense,itreferstomechanicalvibration,thatis,vibrationinamechanicalsystem.Electromagneticvibrationistraditionallycalledoscillation.Themechanicalsystemcanmaintainvibrationandmusthaveelasticityandinertia.Duetoelasticity,whenthesystemdeviatesfromitsequilibriumposition,itwillgeneraterestoringforce,promptingthesystemtoreturntoitsoriginalposition;duetoinertia,thesystemaccumulateskineticenergyintheprocessofreturningtotheequilibriumposition,sothatthesystemmovesacrosstheequilibriumpositiontotheotherside.Itispreciselybecauseofthemutualinfluenceofelasticityandinertiathatthesystemvibrates.Accordingtothedegreeoffreedomofsystemmovement,therearesingle-degree-of-freedomsystemvibration(suchasthevibrationofapendulum)andmulti-degree-of-freedomsystemvibration.Thefinitemulti-degree-of-freedomsystemcorrespondstothediscretesystem,anditsvibrationisdescribedbyordinarydifferentialequations;theinfinitemulti-degree-of-freedomsystemcorrespondstothecontinuoussystem(suchasrods,beams,plates,shells,etc.),anditsvibrationisdescribedbypartialdifferentialequations.Asystemthatdoesnotexplicitlycontaintimeintheequationiscalledanautonomoussystem;asystemthatexplicitlycontainstimeiscalledanon-autonomoussystem.Accordingtotheforceofthesystem,therearefreevibration,attenuatedvibrationandforcedvibration.Accordingtothepropertiesofelasticforceanddampingforce,therearelinearvibrationandnonlinearvibration.Vibrationcanbedividedintodeterministicvibrationandrandomvibration.Thelatterhasnodeterministiclaw,suchasbumpsinthecourseofavehicle.Vibrationisacommonphenomenoninnatureandengineering.Thenegativeaspectsofvibrationare:itaffectsthefunctionsofinstrumentsandequipment,reducestheworkingaccuracyofmechanicalequipment,increasescomponentwear,andevencausesstructuralfatiguedamage;thepositiveaspectsofvibrationare:therearemanyequipmentandprocessesthatrequirevibration(suchasvibrationtransmission,vibrationgrinding),Vibratingpiles,etc.).Thebasictaskofvibrationanalysisistodiscussthesystem'sexcitation(i.e.input,referstotheexternaldisturbanceofthesystem,alsoknownasinterference),response(i.e.output,referstothereactionofthesystemafterbeingexcited)andthedynamiccharacteristicsofthesystem(orphysicalparameters).Relationshipbetween.Afterthe1960s,majoradvancesincomputersandvibrationtestingtechnologyopenedupbroadprospectsforthecomprehensiveutilizationofanalysis,experiment,andcalculationmethodstosolvevibrationproblems.

Mechanicalvibration

Definition

Mechanicalvibrationisthereciprocatingmotionofanobject(orpartofanobject)neartheequilibriumposition(thepositionwhentheobjectisstationary).Therearedifferentclassificationmethodsformechanicalvibration.Accordingtothecauseofthevibration,itcanbedividedintofreevibration,forcedvibrationandself-excitedvibration;accordingtothelawofvibration,itcanbedividedintosimpleharmonicvibration,non-harmonicperiodicvibrationandrandomvibration;accordingtothecharacteristicsofthestructuralparametersofthevibrationsystem,itcanbedividedintolinearvibrationAndnonlinearvibration;accordingtothecharacteristicsofvibrationdisplacement,itcanbedividedintotorsionalvibrationandlinearvibration.

Freevibration:Thevibrationofthemechanicalsystemaftertheexcitationorconstraintisremoved.Thevibrationismaintainedonlybyitselasticrestoringforce,andwhenthereisdamping,thevibrationwillgraduallyattenuate.Thefrequencyoffreevibrationisonlydeterminedbythephysicalpropertiesofthesystemitself,whichiscalledthenaturalfrequencyofthesystem.

Forcedvibration:Thevibrationofthemechanicalsystemcausedbycontinuousexternalexcitation.Simpleharmonicexcitationisthesimplestcontinuousexcitation.Forcedvibrationincludestransientvibrationandsteady-statevibration.Thetime-varyingvibrationthatoccurswithinaperiodoftimeatthebeginningofvibrationiscalledtransientvibration.Afterashortperiodoftime,thetransientvibrationdisappears.Thesystemcontinuouslyobtainsenergyfromtheoutsidetocompensatefortheenergydissipatedbythedamping,soitcanmakecontinuousconstantamplitudevibration.Thefrequencyofthisvibrationisthesameastheexcitationfrequency,whichiscalledsteady-statevibration.Forexample,avibrationexciterisinstalledinthemiddleofabeamfixedatbothends,andthecontinuousconstant-amplitudevibrationofthebeamaftertheexciterisactivatedforashorttimeisasteady-statevibration,andthefrequencyofthevibrationisthesameasthefrequencyoftheexciter.Whenthesystemissubjectedtoexternalforceorotherinput,itscorrespondingoutputiscalledresponse.Whenthefrequencyoftheexternalexcitationisclosetothenaturalfrequencyofthesystem,theamplitudeofthesystemwillincreasesharply.Resonanceoccurswhentheexcitationfrequencyisequaltotheresonancefrequencyofthesystem.Resonancemustbepreventedwhendesigningandusingmachinery.Forexample,inordertoensurethesafeoperationofrotatingmachinery,theworkingspeedoftheshaftshouldbeoutsideacertainrangeofitscriticalspeed.

Self-excitedvibration:Innonlinearvibration,thesystemisonlymaintainedbytheexcitationgeneratedbyitself.Inadditiontothevibrationcomponents,theself-excitedvibrationsystemalsohasnon-oscillatingenergysources,adjustmentlinksandfeedbacklinks.Therefore,itcanalsoproduceastableperiodicvibrationwhenthereisnoexternalexcitation,andthealternatingforcethatmaintainstheself-excitedvibrationisgeneratedbythemotionitselfandcontrolledbythefeedbackandadjustmentlinks.Assoonasthevibrationstops,thisalternatingforcealsodisappears.Theself-excitedvibrationhasnothingtodowiththeinitialconditions,anditsfrequencyisequaltoorclosetothenaturalfrequencyofthesystem.Forexample,theflutterofthewingsduringtheflightoftheaircraft,thecrawlingofthemachinetooltablewhenmovingatlowspeedontheslidingguiderail,theswingoftheclockpendulumandthevibrationofthestringsareallself-excitedvibrations.

Applicationofvibrationinthemachineryindustry

Theapplicationofvibrationinmachineryisverycommon.Forexample,inthevibratingscreeningindustry,thebasicprincipleisbasedontheheavyhammerinstalledattheupperandlowerendsofthemotorshaft.(Unevenheavyhammer),whichconvertstherotatingmotionofthemotorintoahorizontal,vertical,andinclinedthree-dimensionalmotion,andthentransmitsthismotiontothescreensurface.Ifyouchangethephaseangleoftheupperandlowerweights,thedirectionoftraveloftherawmaterialscanbechanged.

Thehazardandprotectionofvibrationtothehumanbody

1.Vibrationsourcesencounteredinproduction

(1) Rivetingmachinam, Pneumatica instrumenta, ut cautes et scopas;

(2) Powertoolssuchaselectricdrills,electricsaws,forestrychainsaws, molentes, politores, molentes, viam manutenendiampingmachinas, etc.;

(3) Diesellocomotiva, naves, motorcycles et alia vecturae vecturae;

(4) Tractores, messores, trituratores et agricultural culturae.

Secundo, impactofvibration de variis hominum systematibus

(1) Changesinelectroencephalogramma;mutationes inconditionedreflexlatency, sympathetichyperfunction; instabilitas sanguinis pressus, arrhythmia, etc.; Theskinsensoryfunctionis reducitur, sicut calor, calor, dolor, praecipue vibrationes, quae in primis videntur.

(2)Vibrationof40~300Hzcancausechangesinthemorphologyandtensionofperipheralcapillaries,manifestedasperipheralvasospasmandabnormalcerebralbloodflowdiagram;bradycardiaandsinusrhythmcanoccurintheheart.Qiandintra-room,indoor,atrioventricularconductionblock,etc.

(3) Decresedgripstrength, abnormalelectromyogrammum, musculum fibrillationem, muscleatrophy et dolorem, etc.

(4)Large-amplitudevibrationbelow40Hzcaneasilycausechangesinbonesandjoints.Bonetradeformation,osteoporosis,boneandjointdeformationandnecrosiscanbeseenonboneX-rayfilms.

(5)Thehearingchangescausedbyvibrationarecharacterizedbyhearinglossinthe125-250Hzfrequencyband,butintheearlystage,thehearinglossisstilldominatedbythehighfrequencyband,andthenthelowfrequencyhearinglossoccurs.Vibrationandnoisehaveacombinedeffect.

(6)Long-termuseofvibrationtoolscancauselocalvibrationdiseases.Localvibrationdiseaseisadiseasemainlycausedbyperipheralcirculatorydisorders,anditcanalsoaffectthenervesandmotorfunctionsofthelimbs.Thesiteofthediseaseisusuallyattheendoftheupperextremity,andthetypicalmanifestationisparoxysmalwhiteningofthefingers(referredtoaswhitefingers).Inmycountry,localvibrationdiseasewasdesignatedasanoccupationaldiseasein1957.

(7)Thefactorsthataffectvibrationarevibrationfrequency,accelerationandamplitude.Thehumanbodyonlyproducesvibrationsensationto1~1000Hzvibration.Frequencyplaysanimportantroleinthepathogenesis.30~300Hzmainlycausesperipheralvasospasmandwhitefingers.Whenthefrequencyisthesame,thegreatertheacceleration,thegreaterthedamage.Vibrationswithlargeamplitudeandlowfrequencymainlyactonthevestibularorgansandcancauseinternalorganstoshift.Whenthefrequencyisconstant,thegreatertheamplitude,thegreatertheimpactonthebody.Coldisoneoftheimportantexternalconditionsfortheonsetofvibrationdisease.Coldcanreducebloodflowandchangebloodcirculation,leadingtoinsufficientlocalbloodsupplyandpromotingtheoccurrenceofvibrationdisease.Thelongertheexposuretovibration,thehighertheincidenceofvibrationdisease.Restatworkhasapositiveeffectonthepreventionofvibrationdiseases.Thesensitivityofapersontovibrationisrelatedtothepositionofthebody.Thehumanbodyissensitivetoverticalvibrationwhenstandingup;itissensitivetohorizontalvibrationwhenlyingdown.Somejobsrequireaforcedposture,evenifthechest,abdomenorlowerlimbsareclosetothevibratingobject,theharmofvibrationisevengreater.Whenthehardnessoftheprocessedpartsishigh,theworkerswillsuffermoredamage,andthevibrationwithhighimpactforcewilleasilycausethebonesandjointstodisease.

3.Controlofvibrationhazards

Reformtheprocess,fundamentallyeliminateandreducetheoperationofhand-heldpneumatictools,usinghydraulicpressure,welding,andbondingInsteadofriveting;improvepneumatictools,adopteffectivevibrationreductionmeasures,andreformthepositionoftoolexhaust;useautomaticandsemi-automaticcontroldevicestoreducedirectcontactwiththevibratingbody;thosewhoholdvibrationtoolsshouldweardouble-layerpadsandfingerlessGlovesorpaddedfoamplasticfingerlessgloves,andpayattentiontokeepwarmandcold;fornewworkers,apre-employmentphysicalexamination,patientswithvasospasm,acralvasculardisorders,andneuritis,areprohibitedfromperformingvibrationoperations;workersexposedtovibrationshouldberegularlyTheintervalbetweenphysicalexaminationsshouldbe2to3years;necessarytreatmentsshouldbegiventopatientswithvibrationdiseases,andthosewithrepeatedcomplaintsshouldberemovedfromthevibrationworkposition.

Theactualvibrationproblemisoftenintricateandcomplicated,anditmayincludeseveralaspectssuchasidentification,analysis,andsynthesisatthesametime.Usuallytheactualproblemisabstractedasamechanicalmodel,whichisessentiallyasystemidentificationproblem.Theprocessofsolvingthesystemmodelformulationisessentiallytheprocessofvibrationanalysis.Analysisisnottheendoftheproblem.Theresultsoftheanalysismustalsobeusedtoimprovethedesignortroubleshoot(actualorpotential).Thisistheproblemofvibrationsynthesisordesign.

Themethodtosolvethevibrationproblemisnothingmorethantheoreticalanalysisandexperimentalresearch,whicharecomplementarytoeachother.Inthetheoreticalanalysisofvibration,alargenumberofmathematicaltools,especiallytheincreasingdevelopmentofdigitalcomputers,providepowerfulmeanstosolvecomplexvibrationproblems.Sincethemid-1960s,therehavebeenmajorbreakthroughsandadvancesinvibrationtestingtechnology,whichinturnopenedupbroadprospectsforexperiments,analysisandresearchonvibrationproblems.Seelinearvibration,nonlinearvibration,randomvibration.

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