Solution Manual for Vibrations, 3rd Edition by Edward , 9781108427319, Covering Chapters 1-9 Includes Rationales

Chapter 5 f f




Vibrations


5.1 Overview of Vibrations
f f




5.1.1 Examplesfoffpracticalfvibrationfproblems

Vibrationfisfafcontinuousfcyclicfmotionfoffafstructureforfafcomponent.

Generally,fengineersftryftofavoidfvibrations,fbecausefvibrationsfhavefafnumberfoffunpleasantfeffects:
• Cyclicfmotionfimpliesfcyclicfforces.f Cyclicfforcesfarefveryfdamagingftofmaterials.
• Evenfmodestflevelsfoffvibrationfcanfcausefextremefdiscomfort;
• Vibrationsfgenerallyfleadftofaflossfoffprecisionfinfcontrollingfmachinery.

Examplesfwherefvibrationfsuppressionfisfanfissuefinclude:

Structuralfvibrations.f Mostfbuildingsfarefmountedfonftopfoffspecialfrubberfp
ads,fwhichfarefintendedftofisolatefthefbuildingffromfgroundfvibrations.f Theffig
urefonfthefrightfshowsfvibrationfisolatorsfbeingfinstalledfunderftheffloorfoffafbu
ildingfduringfconstructionf(fromfwww.wilrep.comf)

NofvibrationsfcoursefisfcompletefwithoutfafmentionfoffthefTacomafNarrows
suspensionfbridge.f Thisfbridge,fconstructedfinfthef1940s,fwasfatftheftimeftheflongestfsuspensionfbridgefinfthef
world.f Becausefitfwasfafnewfdesign,fitfsufferedffromfanfunforseenfsourcefoffvibrations.f Infhighfwind,fthefroad
wayfwouldfexhibitfviolentftorsionalfvibrations,fasfshownfinfthefpicturefbelow.




Youfcanfwatchfnewsreelffootagefoffthefvibrationfandfevenftheffinalfcollapsefatfhttp://www.youtube.com/watch?
v=HxTZ446tbzEf Tofthefcreditfoffthefdesigners,fthefbridgefsurvivedfforfanfamazinglyflongftimefbeforefitffinallyf
failed.f Itfisfthoughtfthatfthefvibrationsfwerefafformfoffself-
excitedf vibrationf knownf asf `flutter,’f orf ‘galloping’f Af similarf formf off vibrationf isf knownf tof occur
f

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infaircraftfwings.f Interestingly,fmodernfcablefstayedfbridgesfthatfalsofsufferffromfafnewfvibrationfproblem:fthefc
ablesfarefveryflightlyfdampedfandfcanfvibratefbadlyfinfhighfwindsf(thisfisfafresonancefproblem,fnotfflutter).fYouf
canffindfafdetailedfarticlefonfthefsubjectfatfwww.fhwa.dot.gov/bridge/pubs/05083/chap3.cfm.fSomefbridgefdesi
gnsfgofasffarfasftofincorporatefactivefvibrationfsuppressionfsystemsfinftheirfcables.




Vehiclefsuspensionfsystemsfareffamiliarftofeveryone,fbutfcontinueftofevolvefasfenginee
rsfworkftofimprovefvehiclefhandlingfandfridef(theffigurefabovefisffromfhttp://www.altair
hyperworks.com.f Afradicalfnewfapproachftofsuspensionfdesignfemergedfinf2003fwhenf
afresearchfgroupfledfbyfMalcolmfSmithfatfCambridgefUniversityfinventedfafnewfmecha
nicalfsuspensionfelementftheyfcalledfanf‘inerter’.fThisfdevicefcanfbefthoughtfoffasfafsortfo
ffgeneralizedfspring,fbutfinsteadfoffexertingfafforcefproportionalftofthefrelativefdisplacem
entfoffitsftwofends,fthefinerterfexertsfafforcefthatfisfproportionalftofthefrelativefaccelerati
onfoffitsftwofends.f Anfactualfrealizationfisfshownfinftheffigure.f Youfcanffindfafdetailedfp
resentationfonftheftheoryfbehindfthefdevicefatfhttp://www-
control.eng.cam.ac.uk/~mcs/lecture_j.pdffThef devicefwasfadoptedfinfsecretfbyfthefMc
LarenfFormulaf1fracingfteamfinf2005f(theyfcalledfitfthef‘Jfdamper’,fandfafscandalferupted
finfFormulaf1fracingfwhenfthefRenaultfteamfmanagedftofstealfdrawingsfforfthefdevice,fbu

tfwerefunableftofworkfoutfwhatfitfdoes.f Thefpatentfforfthefdevicefhasfnowfbeenflicensedf
Penskefandflooksftofbecomef afstandardfelementfinfformulaf1fracing.f Itfisfonlyfafmatterfo
fftimefbeforefitfappearsfonfvehiclesfavailableftofthefrestfoffus.

PrecisionfMachinery:fThefpicturefonfthefrightfshowsfonefexamplefoffafprecisionfinstrum
ent.f Itfisfessentialftofisolatefelectronfmicroscopesffromfvibrations.f Aftypicalftransmissio
nfelectronfmicroscopefisfdesignedftofresolveffeaturesfoffmaterialsfdownftofatomicflengthfs
cales.f Iffthefspecimenfvibratesfbyfmorefthanfaffewfatomicfspacings,fitfwillfbefimpossibleft
ofsee!f Thisfisfonefreasonfthatfelectronfmicroscopesfarefalwaysflocatedfinfthefbasementf–
fthefbasementfoffafbuildingfvibratesfmuchflessfthanfthefupperffloors.f ProfessorfK.-

S.fKimfatfBrownfrecentlyfinventedfandfpatentedfafnewfvibrationfisolationfsystemftofsupp
ortfhisfatomicf forcef microscopef onfthef 7thffloorf offthefBarus-
Holleyfbuildingf–fyoufcanffindfthefpatentfatfUnitedfStatesfPatent,fPatentfNumberf7,543,791.

Herefisfanotherfprecisionfinstrumentfthatfisfveryfsensitiveftofvibrations.

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Thefpicturefshowsffeaturesfoffaftypicalfhardfdiskfdrive.fItf
isfparticularlyfimportantftofpreventfvibrationsfinfthefdiskf
stackfassemblyfandfinfthefdiskfheadfpositioner,fsincefanyf
relativefmotionfbetweenftheseftwofcomponentsfwillfmak
efitfimpossibleftofreadfdata.fThefspinningfdiskfstackfasse
mblyfhasfsomefveryfinterestingfvibrationfcharacteristicsf
(whichffortunatelyfforfyou,fisfbeyondfthefscopefoffthisfco
urse).




Vibrationsfarefnotfalwaysfundesirable,fhowever.f Onfoccasion,ftheyfcanf
befputftofgoodfuse.f Examplesfoffbeneficialfapplicationsfoffvibrationsfin
cludefultrasonicfprobes,fbothfforfmedicalfapplicationfandfforfnondestruc
tiveftesting.fThefpicturefshowsfafmedicalfapplicationf offultrasound:fitfis
fanfimagefoffsomeone’sfcolon.f Thisftypefoffinstrumentfcanfresolveffeatu

resfdownftofaffractionfoffafmillimeter,fandfisfinfinitelyfpreferableftofexpl
oratoryfsurgery.f Ultrasoundfisfalsofusedftofdetectfcracksfinfaircraftfandfs
tructures.

Musicalfinstrumentsfandfloudspeakersfarefafsecondfexamplefof
systemsfwhichfputfvibrationsftofgoodfuse.f Finally,fmostfmechanicalfclocksfusefvibrationsftofmeas
ureftime.


5.1.2 VibrationfMeasurement

Whenffacedfwithfafvibrationfproblem,fengineersfgenerallyfstartfbyfmakingfsomefmeasur
ementsftoftryftofisolatefthefcausefoffthefproblem.f Therefareftwofcommonf waysftofmeasu
refvibrations:

1. Anfaccelerometerfisfafsmallfelectro-
mechanicalfdevicefthatfgivesfanfelectricalfsignalf proportionalf tof itsf acceleratio
n.f f Thef picturef showsf af typicalf 3f axis
MEMSfaccelerometerf(you’llfusefonefinfafprojectfinfthisfcourse).f MEMSfaccelerometersfshouldfbefsele
ctedfveryfcarefullyf–
fyoufcanfbuyfcheapfaccelerometersfforflessfthanf$50,fbutfthesefarefusuallyfmeantfjustfasfsensors,fnotfforf

makingfprecisionfmeasurements.f Forfmeasurementsfyou’llfneedftofselectfonefthatfisfspeciallyfdesigned
fforftheffrequencyfrangefyoufarefinterestedfinfsensing.f Thefbestfaccelerometersfarefexpensivef‘inertialfg

rade’fversionsf(suitablefforfso-
calledf‘inertialfnavigtation’finfwhichfaccelerationsfarefintegratedftofdeterminefposition)fwhichfarefoften
fusefKalmanffilteringftoffusefthefaccelerationsfwithfGPSfmeasurements.




2. Afdisplacementftransducerfisfsimilarftofanfaccelerometer,fbutfgivesfanfelectricalfsignalfproportionalftofit
sfdisplacement.

Displacementf transducersf aref generallyf preferablef iff youf needf tof measuref lowf frequencyf vibrations;f accele
rometersfbehavefbetterfatfhighffrequencies.

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5.1.3 FeaturesfoffafTypicalfVibrationfResponse
Displacementforf
Thefpicturefbelowfshowsfaftypicalfsignalfthatfyoufmightfreco Acceleration
rdfusingfanfaccelerometerforfdisplacementf transducer. Peakftofpeakf
amplitude
PeriodfT
Importantffeaturesfoffthefresponsefare


f Thefsignalfisfoftenf(althoughfnotfalways)fperiodic:fthatfi
Time
sftofsay,fitfrepeatsfitselffatffixedfintervalsfofftime.fVibrati
onsfthatfdofnotfrepeatfthemselvesfinfthisfwayf arefsaidftofb
efrandom.f Allfthefsystemsfwefconsiderfinfthisfcoursefwill
fexhibitfperiodicfvibrations.




f ThefPERIODf offthefsignal,fT,f isftheftimefrequiredfforfone
fcompletefcyclefoffoscillation,fasfshownfinfthefpicture.


ThefFREQUENCYfoffthefsignal,ff,f isfthefnumberfoffcyclesfoffoscillationfperfsecond.fCyclesfperfsecon
dfisfoftenfgivenfthefnamefHertz:fthus,fafsignalfwhichfrepeatsf100ftimesfperfsecondfisfsaidftofoscillatefatf1
00fHertz.

Thef ANGULARf FREQUENCYf off thef signal,ff,f isf definedf asf f=f2f ff .f Wef specifyf angular
frequencyf inf radiansf perf second.f Thus,f af signalf thatf oscillatesf atf 100f Hzf hasf angularf frequency
200f radiansfperfsecond.

ff Period,ffrequencyfandfangularffrequencyfarefrelatedfby

1 2
ff =f f=ffff =f
T T


f ThefPEAK-TO-
PEAKfAMPLITUDEfoffthefsignal,fA,fisfthefdifferencefbetweenfitsfmaximumfvaluefandfitsfminimumf
value,fasfshownfinfthefpicture

ThefAMPLITUDEfoffthefsignalfisfgenerallyftakenftofmeanfhalffitsfpeakftofpeakfamplitude.
Engineersfsometimesfusefamplitudefasfanfabbreviationfforfpeakftofpeakfamplitude,fhowever,fsofbefcare
ful.

ThefROOTfMEANfSQUAREfAMPLITUDEfTorfRMSfamplitudefisfdefinedfas
f1f 1/2

f =f ffy(t) dtf 
2
T
 0 