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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.)
•LinearsystemMechanicalSystems
Linearsystem
y=Ma+Mb=M(a+b)
•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
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
AccelerationForce
Accelerance=ApparentMass=
ForceAcceleration
DisplacementForce
Receptance=DynamicStiffness=
ForceDisplacement
Invibrations,FRFsdependonwhatweareinterestedin.