>> P.534
RAbbe’sprinciplestatesthat“maximumaccuracyisobtainedwhenthescaleandthemeasurementaxesarecommon”.Thisisbecauseanyvariationintherelativeangle(θ)ofthemovingmeasuringjawonaninstrument,suchasacaliperjawmicrometer,causesdisplacementthatisnotmeasuredontheinstrument’sscaleandthisisanAbbeerror(ε=ℓ-Linthediagram).Spindlestraightnesserror,playinthespindleguideorvariationofmeasuringforcecanallcause(θ)tovary,andtheerrorincreaseswithR.Hertz'sFormulaHertz’sformulagivetheapparentreductionindiameterofspheresandcylindersduetoelasticcompressionwhenmeasuredbetweenplanesurfaces.Theformulaisusefulfordeterminingthedeformationofaworkpiececausedbythemeasuringforceinpointandlinecontactsituations.PSøDPLøD(a)(b)SpherebetweentwoplanesCylinderbetweentwoplanesAssumingthatthematerialissteelandunitsareasfollows:Modulusofelasticity:E=205GPaAmountofdeformation:δ(µm)Diameterofsphereorcylinder:D(mm)Lengthofcylinder:L(mm)Measuringforce:P(N)a)Apparentreductionindiameterofsphereδ1=0.823√P~2/Db)Apparentreductionindiameterofcylinderδ2=0.094×(P/L)3√1~/DMajorMeasurementErrorsoftheScrewMicrometerErrorcauseMicrometerfeederrorAnvilangleerrorMaximumpossibleerrorPrecautionsforeliminatingerrorsErrorthatmightnotbeelminatedevenwithprecautions3µm1.Correctthemicrometerbeforeuse.±1µm±5µmassumingtheerrorofahalfangleis15minutes1.Measuretheangleerrorandcorrectthemicrometer.2.Adjustthemicrometerusingthesamethreadgageastheworkpiece.±3µmexpectedmeasurementerrorofhalfangleDuetoanvildifference+10µmInfluenceofmeasuringforce±10µmAngleerrorofthreadgage±10µm+3µm+3µm+3µm1.Useamicrometerwithalowmeasuringforceifpossible.2.Alwaysusetheratchetstop.3.Adjustthemicrometerusingathreadgagewiththesamepitch.1.Performcorrectioncalculation(angle).2,Correctthelengtherror.3,Adjustthemicrometerusingthesamethreadgageastheworkpiece.JIS2gradeerrorofhalfangle±229minutes-91µm+71µm1.Minimizetheangleerrorasmuchaspossible.2.Measuretheangleerrorandperformcorrectioncalculation.3.Usethethree-wiremethodforalargeangleerror.(±117+40)µmAggregatevalueofpossibleerrorsErrorofhalfangle±8µmat±23minutes+26µm-12µm1.Performcorrectioncalculation.2,Adjustthemicrometerusingthesamethreadgageastheworkpiece.±1µmL(3+±25)µmLengtherrorofthreadgageWorkpiecethreadangleerrorCumulativeerrorℓLεθMicrometersDifferenceinThermalExpansionBetweenMicrometerandLengthStandardAbbe’sPrinciple125225325425525Nominallength(mm)*Valuesarenotguaranteedvaluesbutexperimentalvalues.0°C20°C10°C+3+2+1+0-1-2-3Differenceinexpansion(µm)Theabovegraphshowstheresultsforeachofthesizesfrom125through525mmateachtemperatureunderthefollowingtime-seriesconditions.1.Themicrometeranditsstandardwereleftataroomtemperatureof20ºCforabout24hours.2.Thestartpointwasadjustedusingthemicrometerstandardafterthetemperatureofthemicrometerandthestandardstabilized.3.Themicrometerwithitsstandardwereleftatthetemperaturesof0ºCand10ºCforaboutonehour.4.Measurementofeachstartpoint.Thisgraphshowsthatboththemicrometeranditsstandardmustbeleftatthesamelocationforatleastseveralhoursbeforeadjustingthestartpoint.Hooke'sLawHooke’slawstatesthatstraininanelasticmaterialisproportionaltothestresscausingthatstrain,providingthestrainremainswithintheelasticlimitforthatmaterial.EffectofChangingSupportMethodandOrientation(Unit:µm)Thetablesbelowshowhowthezeropointchangesduetodifferencesinsupportorientation—such“Supportedonlyatthecenter,”“Supportedatthecenterinalateralorientation,”and“Supportedbyhanddownward”—afterzerosettinginthe“Supportedatthebottomandcenter”case.Ifthestartpointisnotalignedusingthesamesupportmethodandorientationastheactualmeasurement,thevalueswillchangefromthepositionsshowninthetablebelow.Therefore,aligningthestartpointusingthesamesupportmethodandorientationasthemeasurementisrecommended.SupportingmethodSupportedatthebottomandcenterSupportedonlyatthecenterMaximummeasuringlength(mm)Attitude325425525625725825925102500000000-5.5-2.5-5.5-11.0-9.5-18.0-22.5-26.0SupportingmethodSupportedatthecenterinalateralorientation.Supportedbyhanddownward.AttitudeMaximummeasuringlength(mm)3254255256257258259251025+1.5+2.0-4.50.0-9.5-5.0-14.0-5.0-4.5-10.5-10.0-5.5-19.0-35.0-27.0-40.0*Valuesarenotguaranteedvaluesbutexperimentalvalues.13-7Micrometers
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