formulae sheet final

7/28/2019 Formulae Sheet Final

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HacettepeUniversity,MechanicalEngineeringDepartment

AutomotiveEngineeringProgram

OMU418:MECHANICALVIBRATION

1

FREEVIBRATIONSOFUNDAMPEDSDOFSYSTEMS

EquationofMotion 0SubjecttoInitialConditions0 0 GeneralSolution

where

Period: Frequency: FREEVIBRATIONSOFDAMPEDSDOFSYSTEMS

EquationofMotion 0 2 0SubjecttoInitialConditions0 0 CriticalDampingCoefficient

2

Nondimensionaldampingratio

2For 1.Thegeneralsolutionis: e eCase1: 1UnderdampedFreeVibrationsGeneralSolution sin Where Dampednaturalfrequencyandperiod

1

Logarithmic

Decrement

for

one

cycle

21Forsmalldampingratio,Logarithmicdecrementcanbeapproximatedas 2Dampingratiointermsof 4 LogarithmicDecrement forncycle 1

TotalEnergyoftheSystem

12 12TheRatiooftheenergydissipatedbetweensuccessive

cycles

TheRatiooftheenergydissipatedbetweenmsuccessive

cycles

intermsoflogarithmicdecrement

Case2: 1CriticallyDampedFreeVibrations

Case3: 1OverdampedFreeVibrations 2 1

1

1

HARMONICEXCITATIONOFSDOFSYSTEMS

Equationof

Motion

ForcedResponseofanUndampedSystemdueto

SingleFrequencyExcitation

EquationofMotion sin CASE

A:

If

ParticularSolution sin TotalResponse Applyingtheinitialconditionstototalresponse sincos 1 cos sin

sin

CASEB:If

2 cos TotalResponse Applyingtheinitialconditionstototalresponse cos 2sin 2 cos ForcedResponseofaDampedSystemduetoSingle

FrequencyExcitation

EquationofMotion

2 ParticularSolution Where 2 2 Definefrequencyratio

Definemagnificationfactor


2/3




2

11 2The

following

are

noted

about

maginification

factor

1. 1 when 0. In this case the excitation force isconstant and the forcedeveloped in the springofmassspringdashpotsystem isequaltothevalueoftheeciting

force.

2. 0as Theamplitudeoftheforcedresponse isverysmallforhighfrequencyexcitations

3. Foragivenvalueof,decreaseswithincreasing4. Themagnification factor growswithout bound only for 0. For 0 , the magnification factor has a

maximumvalueforsomevalueof5. For 0 , the maximum value of magnification

factoroccursforfrequencyratioof 12 6. Thecorrespondingmaximumvalueof

is

1217. For ,maximum value of i.e. 0 occurs for 0.For ,monotonicallydecreasesThenondimensionalformofis Eqn(A)Thefollowingarenotedaboutphaseangle1. The forcedresponseand theecitation forceare inphase

for 0.For 0,therepsonseand theexcitationareinphaseonlyfor

0

2. If 0 and 1, then . If 0, then theexcitation is a pure sine wave while the steadystateresponseisapurecosinewave.Theexcitationisinphase

with thevelocity.Thedirectionofexcitation isalwaysas

thedirectionofmotion

3. If 0and 1, then .The response leadstheexcitation

4. If 0and 1, then .The signof the steadystateresponseisoppositethatoftheexciatation

ForcedResponseduetoExcitationWhoseAmplitudeis

ProportionaltotheSquareoftheExcitationFrequency

EquationofMotion

2 FormofExcitationForce sin sin ParticularSolution Define Eqn(B)

isstillthesame.UseEqn(A)anditsproperties

Thefollowingarenotedaboutmaginificationfactor 1. 0,ifandonlyif 0,forallvaluesof2. 1forlargeandforallvaluesof3. growswithoutboundnear 1for 04. For0 , hasamaximumforafrequencyratio

112

5. For0 ,themaximumvalueofcorrespondstothefrequencyratioandisgivenby 1216. For ,doesnot reachamaximum.grows from

zerofor 0andapproachesoneforlargeApplication:RotatingUnbalance

EquationofMotion 2 FormofExcitationForce

sin

Define ThedefinitionisthesameasinEqn(B).ResponseduetoHarmonicExcitationofSupport

EquationofMotion

Relativemotionofstructurew.r.t.base Ifthebasedisplacementisdefinedas Then Where isdefinedinEqn(B)and isdefinedinEqn(A). And Eqn(C) Eqn(C)Thefollowingarenotedabout1. Theamplituderatioisnearunityforsmall2. For all , 0 1, the amplitude ratio grows until it

reachesamaximumforfrequencyratioof

12 18 1


3/3




3

3. The maximum amplitude ratio corresponding to thefrequencyratio, 4

4. The amplitude ratio has a value of one for 2,independentofthevalueof

5. For 2

,the amplituderatioislargerforsmallervalue

of.However,for 2,theamplituderatio issmallerforsmallervalueof.6. For all values of , the amplitude ratio is less than one

whenandonlywhen 2.

Fig.1.Thegraphofmagnificationfactorv.s.

Fig.2.Thegraphofv.s.

Fig.3.Thegraphof

v.s.

Fig.4.ThegraphofTv.s.r

VIBRATIONISOLATION

MotionIsolation

Ifthebasedisplacementisdefinedas Theresponseofthesytstemis Forisolationwewant

1

1ifandonlyif 2SeeFig.4foragraphofTanduseEqn(C)tocomputeTandForceIsolation

Iftheharmonicexcitationforceis

sin

Theforce

transmitted

to

ground

is

where Forforceisolation 1when 2If is constantor is constant is constant in forcingdue tounbalance i.e. theexpression forTcanbeused

SeeFig.4foragraphofTanduseEqn(C)tocomputeTand

formulae sheet final

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