Impact mechanics

Generallywhenthestrengthofmachineelementsareconsidereditisassumedthattheloadingis staticorappliedgradually.Thisloadingconditionisoftennotthecase,theloadingmaybecyclic requiringassessmentforfatigue. Fatigue oritmayinvolveimpactorsuddenlyapplied loads.Whenloadsareappliedsuddenlyandwhentheloadsareappliedasimpactloadsthe resultingtransientstresses(anddeformations)inducedinthemachineelementsaremuchhigher thaniftheloadsareappliedgradually.Thiseffectisshowninthediagrambelow

Itisnormalpracticetodesignmachinessuchthatimpactloadsareeliminatedorreducedby inclusionofshockabsorbers.Inclusionoflowcost,massproduced,shockabsorberscanvirtually eliminatetheincreasedstressesanddeformationsresultingfromimpactloads. Mostductilematerialshavestrengthpropertieswhichareafunctionoftheloadingspeed.Themore rapidtheloadingthehigherthetensileandultimatestrengthsofthematerials..Twostandard tests,theCharpyandIzod,measuretheimpactenergy(theenergyrequiredtofractureatestpiece underanimpactload),alsocalledthenotchtoughness. Thedetailedassessmentofthestrengthofmachineelementsunderimpactloadingregimesinvolves useofadvancedtechniquesincludingFiniteElementAnalysis.Impactloadsresultinshockwaves propogatingthroughtheelementswithpossibleseriousconsequences.Itispossibletocompletea relativelysimplystressevaluationforsuddenlyappliedandimpactloadsbyusingtheprincipleof conservationofenergyandconditionalthatthematerialsconsideredareoperatingwithintheirelastic regions. Theequationsaremostaccurateforrelativelyheavyimpactingmassesmovingatlowvelocitieson impact.

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 A=Area(m ) 2

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E=ModulusofElasticity(N/m )

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h=Dropdistance(m)

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k=Stiffness(N/m)

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M=Mass-movingbody(kg)

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M1=Massofstaticbarorbeam(kg)

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K=Factortoallowforenergylossat impact

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l=lengthofbar(l)

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tp=timeperiod(s)

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v=velocity(m/s)

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W=Weight-movingbody(N)

       

V = Velocity (m/s) 3 w = Specific Weight (kg/m ) 2 σ = stress (N/m ) 2 σ stat = stress resulting from static load(N/m ) δ = deflection (m) δ stat = deflection resulting from static load(m) μ = Ratio Moving Mass/Stationary bar  β = Constant = A Sqrt(wEg/W) - see text

Importantnote:Thenotesbelowrepresentaverysimpleviewoftheloadingconditionanddonot considermorerealcaseinvolvingshockwavesbeingpropagatedthroughtheloadedmemberorthe movingmass ConsideraloadingregimeasshownbelowwitharingofMassM(kg)withweightW=M.g(N)being droppedthroughadistancehontoacollarsupportedbyaverticalbarwhichbehavesasaspring withastiffnessofk(N/m). 2

2

Thesupportbarhasalengthl(m),anAreaA(m )WithamodulusofelasticityE(N/m )

Inpracticetheweightwouldimpactontothesupportwhichwouldelasticallydeformuntilallofthe potentialenergyhasbeenabsorbed.Thesupportwouldthencontractinitiatingdampedoscillations untilthesystemassumesastablestaticposition.Theequationsbelowdeterminetheinitial maximumdeformationwhichprovidesthemosthighlystressedcondition.

Inaccordancewithconservationofenergythepotentialenergyoftheweightisconvertedtoelastic strainenergy.

Thismaybeexpressedasaquadraticequation

Thisissolvedforthemaximumdeflectionδmaxasfollows

Theweightappliedgraduallywouldresultinadeflectionδstthus

Note: Thestiffnessk=Force/Deflection=F/δ:E=stress/strain=σ/e=(F/A)/(δ/l)Thereforek=EA/l

Substitutingthisintotheequationforδmaxresultsin

Thiscanbeexpressedas

Theresultantmaximumforceissimply δ

andtheresultantmaximumstress= σ

Thismaybeexpressedas

Forthecalculationofthestressduetoasuddenlyappliedloadwithh=0 σ

σ

Importantnote:Thenotesbelowrepresentaverysimpleviewoftheloadingconditionanddonot considermorerealcaseinvolvingshockwavesbeingpropagatedthroughtheloadedmemberorthe movingmass Impactloadsbasedpimarilyonkineticenergye.ghorizontalimpactsaretreatedslightly differently.Fortheseapplicationsthekineticenergyisconvertedintostoredenergyduetoelasticity oftheresistingelement. ConsideraMassM(kg)withavelocityofvimpactingonacollarwhichissupportedbyabarwitha stiffnessofK(N/m)-Ignoringgravitationalforces. 2

ThekineticenergyofthemassMv /2istransformedintostoredenergyinthesupport.

Theresultingequationis

Theresultantmaximumdeflectionequals..

Notingthatthestaticdeflection=Wl/AEthisequationcanbewritten

Theequivalentmaximumstress=

Notingthatthestaticdeflection=Wl/AEthisequationcanbewritten

Usingsimilarprinciplesasexpressedaboveitcanbeeasilyprovedusingprinciplesofconservation ofenergythat.

and

Iftheimpactishorizontalinsteadofvertical.Theresultingdeformationandstressresultingfromthe impactare

Note:Theaboveapproximaterelationshipscanbeenappliedgenerallytomoststructuralsystems subjecttodistortionwiththeelasticrangewhensubjecttoimpactloading

Theaboveequationsareveryapproximateandincludemanyassumptions.Averyimportant 2 assumptionisthatalloftheenergy(basedonhorv )isusedupinproducingthesamedistortionas wouldresultfromstaticloading.Inreality,somekineticenergyislostininternalfriction.Accountcan 2 betakenfortheselossesbymultiplyinghorv byanappropriatefactorK.Thisfactorisderivedfrom theMassofthemovingbody(M)andthemassofthebeamorbar(M1).Thefactorisdifferentfor differentloadingsystemsassfollows. 2

2

Intheequationsabovehorv wouldbereplacedbyK.horKv  TheKfactorisnearestunitywhenMislargecomparedtoM1.Asanexamplefortheaxialimpact ifM1/Missay0,1thenK=0,95.. ifM1/Missay10thenK=0,15.. Thisisillustratedinthefigurebelow

1)AmovingmassMstrikingaxiallyoneendofabarofmassM1.Theotherendofthebarbeing fixed...

2)AmovingmassMstrikingtraverselytheendofabeamofmassM1...

3)AmovingmassMstrikingtraverselythecenterofabeamwithsimplesupportedendsofmass M1....

4)AmovingmassMstrikingtraverselythecenterofabeamwithfixedendsofmassM1...

Whenaimpactforceissuddenlyappliedtoandelasticbody,awaveofstressispropogated travelingthroughthebodywithavelocity..

2

w=weight/unitvolume(kg/m ),v=velocity(m/s) Theunsupportedbarsubjecttothelongitudonalimpactfromarigidbodywithvelocityvexperiences awaveofcompressivestressofintensityσ.

Ifthemassofthemovingbodyisverylargecomparedtothemassofthebarthewaveof compressionbouncesbackfromthefarendofthebarasawaveoftensionandreturnstothestruck endafteratimeperiod.

Ifthemassofthemovingbodyisverylargecomparedtothebarsothatitcanbeconsideredinfinite thenafterbreakingcontactthemovingbarwillmoveawayfromtheimpactingmasswithavelocity of .Themovingbarwillbestressfree. Ifthemassoftheimpingingbodyisμtimethemassofthebarthenthebarwillmoveawaywithan averagevelocityof

Themovingbarisleftvibratingwithastressintensityof

Forthecaseofabarwithoneendfixed,thewaveofcompressivestressresultingfromtheimpact ontheunsupportedendisreflectedbackunchangedfromthesupported(fixedend)andcombines withtheadvancingwavestoproduceamaximumstressapproximatelyequalto..

μ=MassofMovingBody/MassofBar