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."
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.