机械振动.docx

机械振动.docx

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机械振动

FundamentalsofVibration

Lecture1

JiLinSectionofMechanicalDesignandTheorySchoolofMechanicalEngineeringShandongUniversity

VibrationProblems

1.Vibrationsareusuallysmall,oscillatorymotionaboutastaticequilibriumposition.

2.Mostengineeringstructuresvibrate.（←Rotatingmachinery）PracticalExamplesEffects

3.Becominglighter,faster,quieter,andmoreflexibleareoftenmorepronetovibrations.

4.Engineersneedtobeequippedwiththeknowledgerequiredtotacklevibrationproblemsencounteredinindustry

→tounderstand,model,analysis,designandtreat

GeneralAim

Tointroducestudentswithlittleornopreviousexperienceofmechanicalvibrations,andwithquitedifferentbackgrounds,tothebasicconceptsofvibrationalbehavior,toprovideageneralintroductiontovibrationmodeling,analysisandcontrol.

CourseOverview

1.CourseContent→9Sections

2.Resources

1）Lecturenotes（includingexamplesandproblems,mainlywrittenbyProf.Mace,ISVR,Univ.ofSoton）

2）MechanicalVibrationsbyS.S.Rao,AddisonWesleyPublishing.（Coretext）

3）机械振动基础,胡海岩,航空工业出版社（Secondarytext）

3.CreditValue:

2points

4.FormalContactHours:

32

5.AssessmentAssignments（80%）+Attendance（10%）+Others（10%）

Introduction

1.Terminology

Free/forcedvibration;Damped/undampedsystem;Linear/non-linearsystem;Deterministic/randomVibration;Discrete/continuoussystem.

2.BasicPrinciples（→tofindsystemequations）NewtonLaws;Work-energy;Impulsemomentum;Lagrange’sequation

3.BasicConcepts

Degreeoffreedom;Simpleharmonicmotion;Complexexponentialnotation（C.E.N）;Frequencyresponsefunction（FRF）

Fundamentals

•Forvibrationtooccurweneed

?

mass

?

stiffnessk

?

Theothervibrationquantityisdamping

c

Systemvibratesaboutitsequilibriumposition

IngredientsofVibration

Mass

→storeofkineticenergy

Stiffness

→storeofpotential（strain）energyDamping:

→dissipatesenergy

Force

→provideenergy

Vibro-acousticProblems

InteriorNoise

EffectsofVibration

1.Largedisplacementsandstresses（esp.resonance）

2.Fatigue

3.Noise,sound

4.Breakage,wear,improperoperation

5.Physicaldiscomfort,physiologicaleffects

6.Instabilities（flutter,galloping）

FreeVibration←noexternalforcesact

•Systemvibratesatitsnaturalfrequency

Fundamentals-damping

MechanicalSystems

?

Systemsmaybelinearornonlinear

?

LinearSystems←（idealization）

1Outputfrequency=Inputfrequency

2Ifthemagnitudeoftheexcitationischanged,theresponsewillchangebythesameamount

3Superpositionapplies

（Non-linearsystemsarenotconsideredinthiscourse.）

MechanicalSystems

•LinearsystemMechanicalSystems

?

Linearsystem

y=Ma+Mb=M（a+b）

MechanicalSystems

•Nonlinearsystem

Containnonlinearspringsanddampers;Donotfollowtheprincipleofsuperposition

•outputcomprisesfrequenciesotherthantheinputfrequency•outputnotproportionaltoinput

NewtonLaws

Force=mass×accelerationMoment=rotationinertia×angularacceleration

Work-energy

=kineticenergy+potentialstrain）energy

Energy（Workofexternalforces=changeinenergy

Impulse-momentumtheorem

Impulse=changeinmomentum

Lagrange’sequation

Systematicmethod（seethelastsection）

DegreesofFreedom（DOFs）－Modelling

NumberofDOFs=numberofindependentcoordinatesweusetodescribethemotion

Coordinatesmaybedisplacementsofsomepoints,rotation,relativedisplacement,other（modalamplitudes）.

Numberdependson1）howcomplexthesystemis;2）howwechoosetomodelit;3）modellingsimplificationsandassumptions;4）whatwewantfromthemodel.（←FEA?

SEA?

）

Harmonicmotion

2πradians

Solutioncanbewrittenasanyof

xt（）=Asin（ωt）+Bcos（ωt）t+）

xt（）=Csin（ωφ（←sinusoidalort+）timeharmonic）

xt（）=Dcos（ωθ

ω

frequency:

（rad/）f=（cycle/sec

ωsond）

2πperiod:

T（,）

stimepercycle（）:

==

A22

amplitudemagnitudeCD+Bmeanvalue:

x=0

212C

meansquarevalue:

x=C→..rms

rmsvalue:

x=

2

2

dx

velocityx

dt

2

dx

accelerationx

dt2

ComplexExponentialNotation

b

x=Acosφ+iAsinφx=A（cosφ+isinφ）

+real

+imaginary

i

phase

Sox=Aeφ

Euler’sEquation

±iφ

e=cosφ±isinφmagnitude

22−

magnitude

x

=A=

a+bphaseφ=tan1（ba

）

ComplexExponentialNotation（C.E.N）

Timeharmonicquantitywrittenasxt（）=

Inthe“real”worldweseeRe{X（t）}

xiX=ωeitω

Timederivatives

x2eωiωXit

differentialequation→algebraicequationMakelifeeasybutintroducecomplexnumbers.

it+

（ωφ）

xt（）=

X

e

magnitudephase

Deterministicvibration

Forceandresponseknown+predictable

（e.g.rotatingmachinery,impulse,ect.）

RandomvibrationForceandresponseunknown/unpredictable

e.g.unevenroad,wind,turbulenceboundarylayers（TBL）

DiscreteSystems

finitenumberofrigidmasses

+masslessstiffnesselements

Multi-degree-of-freedom（lumpedparametersystems）

（Nmodes,Nnaturalfrequencies）

x3

x1

x2

x4

Continuoussystems

Systemshavingdistributedmassandstiffness（Infinitenumberofdegrees-of-freedom）

e.g.beams,platesetc.

Example-beam

FrequencyResponseFunctions（FRFs）

Definethesystemintermsofresponsetosinusoidalinputs（e.g.harmonicforceexcitations）.

FRF:

Theratiooutput/inputofasysteminsteady-statewhentimeharmonic.

it

e.g.force（input）

f=Feω

it

displacement（output）

x=Xeω

itratioof（complex）

XeωX

FRF→

itω=

amplitude,doesnot

FeF

dependontime

Complex,（usually）frequencydependent;

magnitudephase∠H（）ω

H（ω）

HarmonicForcesWeoftendealwithtimeharmonicbehaviour.

MainReasons

1.oftenhaveharmonicforces,e.g.rotatingmachine;

2.oftenhaveperiodicforcescomprisingharmoniccomponents,e.g.Fourierseries;

3.generalforcestransformedasasumofharmonicsbyFourierTransform.

HarmonicResponse

FrequencyResponseFunction（FRF）Theratiooutput/inputofasysteminsteady-statewhentimeharmonic.NotethatV=iX;A=ω

ωiV

FrequencyResponseFunctions（FRFs）

AccelerationForce

Accelerance=ApparentMass=

ForceAcceleration

DisplacementForce

Receptance=DynamicStiffness=

ForceDisplacement

Invibrations,FRFsdependonwhatweareinterestedin.