ACOUSTICSofWOOD Lecture3

, . , Jan Tippner Dep of Wood Science FFWT MU Brno. .jan tippner@mendelu cz

ACOUSTICSofWOODLecture3

Contentoflecture3:

1.Damping

2.Internalfrictioninthewood

ACOUSTICSofWOODLecture3

Contentoflecture3:

1.Damping

2.Internalfrictioninthewood

ACOUSTICSofWOODLecture3

Damping

sinewaveisawaveformgeneratedbyasystemthatischaracterisedbysimpleharmonicmotion

ideal system which exhibits simple harmonic motion is a system that loses no energy (or has its energyreplenishedfromoutsidethesystem)

suchawaveformcanalsobecalledacontinuouswaveformasitcontinuesforeverwithouteventuallyreducingtozerointensity

ACOUSTICSofWOODLecture3

Damping

realsystemsareneverideal;allnaturallyoccuringsystemslooseenergy(eg.asheatduetofriction)

systemlosesenergyasheat(bothinternallyasaconsequenceofheatlossduringphysicaldeformationandexternallyasaconsequenceoffrictionwithair)

thislossofenergyinanoscillatingsystemisknowasdamping;adampedwaveformisalsoknowasanoncontinuouswaveform

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

ACOUSTICSofWOODLecture3

Damping

dampedharmonicoscillatorssatisfythesecondorderdifferentialequation:

where:0istheundampedangularfrequencyoftheoscillatorandisaconstantcalledthedampingratio

foramassonaspringhavingaspringconstantkandadampingcoefficientc:

0=(k/m)

=c/2m 0.

ACOUSTICSofWOODLecture3

Damping

valueofthedampingratio determinesthebehaviorofthesystem.Adampedharmonicoscillatorcanbe:

1.Overdamped(>1)systemreturns(exponentiallydecays) toequilibriumwithoutoscillating; largervaluesofthedampingratio returntoequilibriumslower

2.Criticallydamped( =1)systemreturnstoequilibriumasquicklyaspossiblewithoutoscillating(oftendesired)

3.Underdamped(

ACOUSTICSofWOODLecture3

ExampleofSpringMassSystem

Amassmattachedtoaspringanddamper.ThedampingcoefficientisrepresentedbyB,Fdenotesanexternalforce.

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 >>>>>>

ACOUSTICSofWOODLecture3

ExampleofSpringMassSystem

rearr.to:

where:(undamped)naturalfrequencyofthesystem: thedampingratio:

thenaturalfrequencyrepresentsanangularfrequency,expressedinradianspersecondthedampingratioisadimensionlessquantity

ACOUSTICSofWOODLecture3

ExampleofSpringMassSystem

thedifferentialequationnowbecomes

wecansolvetheequationbyassumingasolutionxsuchthat:

where:theparameter(gamma)is,ingeneral,acomplexnumber.

substitutingthisassumedsolutionbackintothedifferentialequationgiveswhichisthecharacteristicequation.

solvingthecharacteristicequationwillgivetworoots, + and ; andthesolutiontothedifferentialequationis:

where:AandBaredeterminedbytheinitialconditionsofthesystem:

ACOUSTICSofWOODLecture3

ExampleofSpringMassSystem

rearr.to:

where:(undamped)naturalfrequencyofthesystem: thedampingratio:

thenaturalfrequencyrepresentsanangularfrequency,expressedinradianspersecondthedampingratioisadimensionlessquantity

ACOUSTICSofWOODLecture3

Aharmonicoscillatorcanbe:

1.Overdamped(>1)systemreturns(exponentiallydecays)toequilibriumwithoutoscillating;largervaluesofthedampingratio returntoequilibriumslower2.Criticallydamped ( =1) system returns toequilibriumasquicklyaspossiblewithoutoscillating (oftendesired)3.Underdamped(

ACOUSTICSofWOODLecture3

Dependenceofthesystembehavioronthevalueofthedampingratio ,forunderdamped,criticallydamped,overdamped,andundampedcases,forzerovelocityinitialcondition.

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):

ACOUSTICSofWOODLecture3

2.Overdamping( >1)When >1,thesystemisoverdampedandtherearetwodifferentrealroots.Thesolutiontothemotionequationis:

whereAandBaredeterminedbytheinitialconditionsofthesystem:

ACOUSTICSofWOODLecture3

3.Underdamping(0

ACOUSTICSofWOODLecture3

LogarithmicDecrementofDamping

Logarithmicdecrement,,isusedtofindthedampingratioofanunderdampedsysteminthetimedomain.Thelogarithmicdecrementisthenaturallogoftheamplitudesofanytwosuccessivepeaks:

wherex0 isthegreaterofthetwoamplitudesandxn istheamplitudeofapeaknperiodsaway;thedampingratioisthenfoundfromthelogarithmicdecrement:

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.

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