机械振动加州大学University of CaliforniaChapter 1 Oscillatory Motion.docx

机械振动加州大学University of CaliforniaChapter 1 Oscillatory Motion.docx

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机械振动加州大学UniversityofCaliforniaChapter1OscillatoryMotion

Chapter1OscillatoryMotion

∙Mostengineeringmachinesandstructures:

–Experiencevibrationtosomedegree

–Allbodieswithmassandelasticitycapableofvibration

∙Oscillatorysystems:

–Linearsystemswiththeprincipleofsuperpositionandthemathematicaltechniquesavailable

–Techniquesfortheanalysisofnonlinearsystemsarelesswellknown,anddifficulttoapply.

∙Freevibrations:

–Systemsoscillateundertheactionofforcesinherentinthesystemitself

–Externalimpressedforcesareabsent

–Vibrateatoneormoreofitsnaturalfrequencies

∙Forcedvibrations:

–Whentheexcitationisoscillatory,thesystemisforcedtovibrateattheexcitationfrequency.

–Resonanceifthefrequencyofexcitationcoincideswithoneofthenaturalfrequencies

–Thefailureofmajorstructuressuchasbridges,buildings,orairplanewingsisawesomepossibilityunderresonance.

–Thecalculationofthenaturalfrequenciesisofmajorimportanceinthestudyofvibration.

∙Damping:

-Energydissipationbyfrictionandotherresistances

-Importanceinlimitingtheamplitudeofoscillationatresonance

∙DegreesofFreedom（DOF）:

-Thenumberofindependentcoordinatesrequiredtodescribethemotionofasystem

-ThreeDOFforaparticleandsixDOFforarigidbody

-InfiniteDOFforacontinuouselasticbody

-RigidbodyassumptionforfiniteDOF

-Infact,asurprisinglylargenumberofvibrationproblemscanbetreatedwithsufficientaccuracybyreducingthesystemtoonehavingafewDOF.

1.1HARMONICMOTION

∙PeriodicMotion:

-Themotionisrepeatedinequalintervaloftime,

-istheperiodoftheoscillation

-Thefrequencyoftheoscillation

-Theperiodicmotioncanbeexpressedas

∙

HarmonicMotion:

-Thesimplestformofperiodicmotion

-Demonstratedbyasuspendedmass（Fig.1.1.1）

-Themotioncanbeexpressedas:

（1.1.1）

whereAistheamplitudeofoscillation,measuredfromtheequilibriumpositionofthemass,andistheperiod.

∙Harmonicmotionisoftenrepresentedastheprojectiononastraightlineofapointthatismovingonacircleatconstantspeed.

-Withtheangularspeed

（circularfrequency）,

（1.1.2）

-Becausethemotionrepeatsitselfin2radians,

（1.1.3）

-Thevelocityandaccelerationofharmonicmotion

（1.1.4）

（1.1.5）

-Thevelocityandaccelerationarealsoharmonicwiththesamefrequencyofoscillation,butleadthedisplacementby

and

radians,respectively.

-FromEqs.（1.1.2）and（1.1.5）,

（1.1.6）

-Inharmonicmotion,theaccelerationisproportionaltothedisplacementandisdirectedtowardtheorigin.

-BecauseNewton’ssecondlawofmotionstatesthattheaccelerationisproportionaltotheforce,harmonicmotioncanbeexpectedforsystemswithlinearspringswithforcevaryingaskx.

Exponentialform

∙Euler’sequation:

（1.1.7）

∙AvectorofamplitudeArotatingatconstantangularspeedcanberepresentedasacomplexquantityzintheArganddiagramasshowninFig.1.1.4:

-

（1.1.8）

-zisreferredtoasthecomplexsinusoid.

∙Therulesofexponentialoperations:

and

Multiplication

Division

Powers

PROBLEMS

1.2Anaccelerometerindicatesthatastructureisvibratingharmonicallyat82cpswithamaximumaccelerationof50g.Determinetheamplitudeofvibration.

1.4Findthesumoftwoharmonicmotionsofequalamplitudebutofslightlydifferentfrequencies.Discussthebeatingphenomenathatresultfromthissum.

1.5Expressthecomplexvector4+3iintheexponentialformAei.

1.7Showthatthemultiplicationofavectorz=Aeitbyirotatesitby90.

1.2PERIODICMOTION

∙Fourierseries:

–Anyperiodicmotioncanberepresentedbyaseriesofsinesandcosinesthatareharmonicallyrelated.

–Ifx（t）isaperiodicfunctionoftheperiod,itisrepresentedas

（1.2.1）

where

.

–Todetermineanandbn,wemultiplybothsidesofEq.（1.2.1）bycosntorsinntandintegrateeachtermovertheperiod.

–Bythefollowingrelations,

–

and

∙Fourierseriesintermsoftheexponentialfunction:

–

–Eq.（1.2.1）canbeexpressedas

where

–

∙WhenthecoefficientsoftheFourierseriesareplottedagainstfrequencyn,theresultisaseriesofdiscretelinescalledtheFourierspectrum

∙FastFouriertransform（FFT）

PROBLEMS

1.9DeterminetheFourierseriesfortherectangularwaveshowninFig.P1.9.

1.3VIBRATIONTERMINOLOGY

∙PeakValue:

-Ingeneral,themaximumstressthatthevibratingpartisundergoing.

∙AverageValue:

-Asteadyorstaticvalue,somewhatlikethedclevelofanelectricalcurrent

（1.3.1）

-Theaveragevalueforahalf-cycleof

is

∙MeanSquareValue:

-Thesquareofthedisplacementgenerallyisassociatedwiththeenergyofthevibration.

（1.3.2）

-If

itsmeansquarevalueis

∙RootMeanSquare（rms）value:

-Thesquarerootofthemeansquarevalue

-ThermsofsinewaveofamplitudeAis

.

∙Decibel（dB）:

-Intermsofapowerratio,

（1.3.3）

-Sincethepowerisproportionaltothesquareoftheamplitudeorvoltage,thedecibelisoftenexpressedintermsofthefirstpowerofamplitudeorvoltageas

（1.3.4）

-Whentheupperlimitofafrequencyrangeistwiceitslowerlimit,thefrequencyspanissaidtobeanoctave.

PROBLEMS

1.17Writetheequationforthedisplacementsofthepistoninthecrank-pistonmechanismshowninFig.P1.17,anddeterminetheharmoniccomponentsandtheirrelativemagnitudes.If

whatistheratioofthesecondharmoniccomparedtothefirst?

1.22ThecalibrationcurveofapiezoelectricaccelerometerisshowninFig.P1.22wheretheordinateisindecibels.Ifthepeakis32dB,whatistheratiooftheresonanceresponsetothatatsomelowfrequency,say,1000cps?