acoust lect damping
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ACOUSTICSofWOOD Lecture3
, . , Jan Tippner Dep of Wood Science FFWT MU Brno. .jan tippner@mendelu cz
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ACOUSTICSofWOODLecture3
Contentoflecture3:
1.Damping
2.Internalfrictioninthewood
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ACOUSTICSofWOODLecture3
Contentoflecture3:
1.Damping
2.Internalfrictioninthewood
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ACOUSTICSofWOODLecture3
Damping
sinewaveisawaveformgeneratedbyasystemthatischaracterisedbysimpleharmonicmotion
ideal system which exhibits simple harmonic motion is a system that loses no energy (or has its energyreplenishedfromoutsidethesystem)
suchawaveformcanalsobecalledacontinuouswaveformasitcontinuesforeverwithouteventuallyreducingtozerointensity
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ACOUSTICSofWOODLecture3
Damping
realsystemsareneverideal;allnaturallyoccuringsystemslooseenergy(eg.asheatduetofriction)
systemlosesenergyasheat(bothinternallyasaconsequenceofheatlossduringphysicaldeformationandexternallyasaconsequenceoffrictionwithair)
thislossofenergyinanoscillatingsystemisknowasdamping;adampedwaveformisalsoknowasanoncontinuouswaveform
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ACOUSTICSofWOODLecture3
Damping
dampedwaveformcandieoutquicklyorslowly;waveformthatdiesoutquicklyissaidtobestronglydampedasitlosesenergyquickly;waveformthatdiesoutslowlyissaidtobeweaklydampedasitlosesenergyslowly
dampingisnotjustacharacteristicofsystemsthatgeneratenoncontinuoussinewavelikepatterns,dampingisacharacteristicofsystemsthatproducesoundswithverycomplexspectralpatterns
inphysics,damping isanyeffect that tends to reduce theamplitudeofoscillations inanoscillatorysystem,particularlytheharmonicoscillator
inmechanics,frictionisonesuchdampingeffect.FormanypurposesthefrictionalforceFfcanbemodeledas
beingproportionaltothevelocityvoftheobject:
Ff=cv
where:cistheviscousdampingcoefficient,giveninunitsofnewtonsecondspermeter
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ACOUSTICSofWOODLecture3
Damping
dampedharmonicoscillatorssatisfythesecondorderdifferentialequation:
where:0istheundampedangularfrequencyoftheoscillatorandisaconstantcalledthedampingratio
foramassonaspringhavingaspringconstantkandadampingcoefficientc:
0=(k/m)
=c/2m 0.
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ACOUSTICSofWOODLecture3
Damping
valueofthedampingratio determinesthebehaviorofthesystem.Adampedharmonicoscillatorcanbe:
1.Overdamped(>1)systemreturns(exponentiallydecays) toequilibriumwithoutoscillating; largervaluesofthedampingratio returntoequilibriumslower
2.Criticallydamped( =1)systemreturnstoequilibriumasquicklyaspossiblewithoutoscillating(oftendesired)
3.Underdamped(
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ACOUSTICSofWOODLecture3
ExampleofSpringMassSystem
Amassmattachedtoaspringanddamper.ThedampingcoefficientisrepresentedbyB,Fdenotesanexternalforce.
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ACOUSTICSofWOODLecture3
ExampleofSpringMassSystem
idealmassspringdampersystemwithmassm(inkilograms),springconstantk(innewtonspermeter)andviscousdamperofdampingcoefficientc(innewtonsecondspermeterorkilogramspersecond)issubjectto
anoscillatoryforce............................................andadampingforce...................................................................
treatingthemassasafreebodyandapplyingNewton'ssecondlaw(F=ma),thetotalforceFtotonthebodyis
where:aistheacceleration(inmeterspersecondsquared)ofthemassandxisthedisplacement(inmeters)ofthemassrelativetoafixedpointofreference
Ftot=Fs+Fd >>>>>>
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ACOUSTICSofWOODLecture3
ExampleofSpringMassSystem
rearr.to:
where:(undamped)naturalfrequencyofthesystem: thedampingratio:
thenaturalfrequencyrepresentsanangularfrequency,expressedinradianspersecondthedampingratioisadimensionlessquantity
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ACOUSTICSofWOODLecture3
ExampleofSpringMassSystem
thedifferentialequationnowbecomes
wecansolvetheequationbyassumingasolutionxsuchthat:
where:theparameter(gamma)is,ingeneral,acomplexnumber.
substitutingthisassumedsolutionbackintothedifferentialequationgiveswhichisthecharacteristicequation.
solvingthecharacteristicequationwillgivetworoots, + and ; andthesolutiontothedifferentialequationis:
where:AandBaredeterminedbytheinitialconditionsofthesystem:
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ACOUSTICSofWOODLecture3
ExampleofSpringMassSystem
rearr.to:
where:(undamped)naturalfrequencyofthesystem: thedampingratio:
thenaturalfrequencyrepresentsanangularfrequency,expressedinradianspersecondthedampingratioisadimensionlessquantity
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ACOUSTICSofWOODLecture3
Aharmonicoscillatorcanbe:
1.Overdamped(>1)systemreturns(exponentiallydecays)toequilibriumwithoutoscillating;largervaluesofthedampingratio returntoequilibriumslower2.Criticallydamped ( =1) system returns toequilibriumasquicklyaspossiblewithoutoscillating (oftendesired)3.Underdamped(
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ACOUSTICSofWOODLecture3
Dependenceofthesystembehavioronthevalueofthedampingratio ,forunderdamped,criticallydamped,overdamped,andundampedcases,forzerovelocityinitialcondition.
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ACOUSTICSofWOODLecture3
the behavior of the system depends on the relative values of the two fundamentalparameters, thenaturalfrequency0andthedampingratio
inparticular,thequalitativebehaviorofthesystemdependscruciallyonwhetherthequadraticequationfor hasonerealsolution,tworealsolutions,ortwocomplexconjugatesolutions.
1.Criticaldamping( =1) When =1,thereisadoubleroot (definedabove),whichisreal.Thesystemissaidtobecriticallydamped. Acriticallydampedsystemconvergestozerofasterthananyother,andwithoutoscillating.
Inthiscase,withonlyoneroot ,thereisinadditiontothesolutionx(t)=e t asolutionx(t)=te t :
whereAandBaredeterminedbytheinitialconditionsofthesystem(usuallytheinitialpositionandvelocityofthemass):
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ACOUSTICSofWOODLecture3
2.Overdamping( >1)When >1,thesystemisoverdampedandtherearetwodifferentrealroots.Thesolutiontothemotionequationis:
whereAandBaredeterminedbytheinitialconditionsofthesystem:
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ACOUSTICSofWOODLecture3
3.Underdamping(0
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ACOUSTICSofWOODLecture3
LogarithmicDecrementofDamping
Logarithmicdecrement,,isusedtofindthedampingratioofanunderdampedsysteminthetimedomain.Thelogarithmicdecrementisthenaturallogoftheamplitudesofanytwosuccessivepeaks:
wherex0 isthegreaterofthetwoamplitudesandxn istheamplitudeofapeaknperiodsaway;thedampingratioisthenfoundfromthelogarithmicdecrement:
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ACOUSTICSofWOODLecture3
Qfactor
thequalityfactororQfactorisadimensionlessparameterthatdescribeshowunderdampedanoscillatororresonatoris,orequivalently,characterizesaresonator'sbandwidthrelativetoitscenterfrequencyhigherQindicatesalowerrateofenergylossrelativetothestoredenergyoftheoscillator;theoscillationsdieoutmoreslowly (apendulumsuspended fromahighqualitybearing,oscillating inair,hasahighQ,whileapendulumimmersedinoilhasalowone)forasingledampedmassspringsystem,theQfactorrepresentstheeffectofsimplifiedviscousdampingordrag,wherethedampingforceordragforceisproportionaltovelocity.TheformulafortheQfactoris:
where M is the mass, k is the spring constant, and D is the damping coefficient, defined by the equationFdamping=Dv,wherevisthevelocity
bandwidth,f, of a damped oscillator is shown on a graph of energy versus frequency.The Q factor of thedampedoscillator,orfilter,isf0/f.ThehighertheQ,thenarrowerand'sharper'thepeakis.
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ACOUSTICSofWOODLecture3
Simulationbyfiniteelementmethod
Ansyssoftware
unsteady:fulltransientsolution
1D(beam),2D(shell),ev.3D(solid)
1Dsolution: http://homepages.strath.ac.uk/~clas16/~fyfe/ansys/dynamic/transient/transient.html
http://www.mece.ualberta.ca/tutorials/ansys/IT/Transient/Transient.html
preprocessing(geometry,physics),solution,postprocessing(timehistory)
modalanalysis>>>>>fulltransientanalysis
adaptationforwood(changesofmaterialmodel)
http://homepages.strath.ac.uk/~clas16/~fyfe/ansys/dynamic/transient/transient.htmlhttp://www.mece.ualberta.ca/tutorials/ansys/IT/Transient/Transient.html -
ACOUSTICSofWOODLecture3
Contentoflecture3:
1.Damping
2.Internalfrictioninthewood
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