1、11FTM04AGMA Technical PaperFirst InternationalInvolute GearComparisonBy F. Hrtig, W. Adeyemi, andK. Knielm, Physikalisch-Technische BundesanstaltFirst International Involute Gear ComparisonF. Hrtig, W. Adeyemi, K. Kniel, Physikalisch-Technische BundesanstaltThe statements and opinions contained here
2、in are those of the author and should not be construed as anofficial action or opinion of the American Gear Manufacturers Association.AbstractSevennationalmetrologyinstitutescomprisedofEuropeanandnon-Europeanmemberstatesparticipatedinthe implementation of the first international comparison of involu
3、te gear standards. Participants include:Germany, China, Japan, Thailand, United Kingdom, Ukraine and USA. This comparison was organized bythe European Association of National Metrology Institutes (EURAMET). The German metrology institutePhysikalisch-Technische Bundesanstalt (PTB) was chosen as the p
4、ilot laboratory as well as theorganiser. Threetypical involute gear standards donated by the PTB were deployed for this comparison,namely:profile-,helix-andpitch-standard. Inthefinalanalysisoftheresultsobtainedfromallparticipants,theweighted mean was evaluated as reference value from all 28 measured
5、 parameters. Consequently,competency of each participant was estimated by calculating the normalized error En. The approach usedinvolves evaluating the compatibility of the obtained results taking into account their measurementuncertainties. However,largedistributionoftheresultsthatconformwithrangeo
6、ftodaysrequiredtolerancesin industry pose a lot of questions. This presentation would present the measurement setup, the measuredparameters, and most importantly the comparison results from all participants which are anonymouslyrepresented. Furthermore, mishandling of standards as experienced during
7、 the comparison would beillustrated.Copyright 2011American Gear Manufacturers Association1001 N. Fairfax Street, 5thFloorAlexandria, Virginia 22314October 2011ISBN: 978-1-61481-003-23 11FTM04First International Involute Gear ComparisonF. Hrtig, W. Adeyemi, and K. Kniel, Physikalisch-Technische Bunde
8、sanstaltBackgroundInternational comparisons are required to ensure the compatibility and reliability of measurement resultsamongdifferentcountries. Sofarinthefieldofeconomicallyimportantandhighaccurateinvolutegearmeas-urement, no international comparison measurement has been conducted. Therefore, it
9、 was imperative toorganize this comparison among some national metrology institutes (NMIs) or designated institutes (DIs)which usually represents the highest metrological competence inside each country. The rules of thecomparison following internationally agreed documents published by the Bureau int
10、ernational des Poids etMesures (BIPM) 1 which task is to ensure world-wide uniformity of measurements and their traceability totheInternationalSystemofUnits(SI). TheBIPMdoesthiswiththeauthorityoftheConventionoftheMetre,adiplomatic treaty between fifty-five nations. The terminology and symbols used i
11、n this paper follow actualdocuments of the BIPM and the International organization for Standardization (ISO) 2, 3, 4, 5.Thisfirst comparisonwas initiatedby thePTB. Following theregional meetingin 2007,EURAMET undertheTechnicalCommitteeofLength(TC-L)5decidedtoimplement thiscomparison asregional compa
12、risonwiththeinvolvementofothernon-Europeanparticipants. ThePTBwaschosenasa pilotlaboratory aswell astheorganizer. The choice of standards to be used, parameters to be measured, potential participants, scheduleandmethodappliedforthecirculationofthestandardswerealldecidedatthesubsequentmeetings,whilet
13、heprotocol adopted was later communicated to all partners. Each participant is allocated equal amount of spe-cifiedtimetocarryoutthemeasurementsattheirinstituteafterwhichthestandardshavetobesenttothenextparticipant. The measurement comparison was implemented between the monthof July2008 toSeptember2
14、010. In total, seven NMIs participated in the study. Three involute gear standards that are typically used inthe industries were chosen for this comparison.Description of standards and measurement parametersThe standards deployed for this comparison represents the three most typically measured stand
15、ards of theinvolute gears in the industry; namely profile-, helix- and pitch-standards (Figure 1). These standards weredeveloped by thePTB andmanufactured fromhigh-alloy steelin morethan thirtyyears ago. PTBas thepilotlaboratory and one of the main team players donated these standards as one of the
16、major contributions tosupport the implementation of the program. The choice of helix-, profile-, and pitch standards for this meas-urement comparison and their suitability were based on a number of factors. Among them are the measure-ment accuracies and long history of measurement stability which ha
17、ve been observed by the PTB since thetime they were manufactured. Furthermore, geometrical parameters of these standards are other importantattributes for their choice, particularly; their reference bands and flanks possessed significantly small formand roughness errors. Each of the standards was me
18、asured by all participants and the results were collatedand evaluated according 610.Profile standardFigure 1a shows the involute profile standard used for the measurement. In a classical design it consists oftwo base discs each of db49,997 mm and one centered involute shape. Table 1 delineates gear
19、parameterswhich are necessary to measure the involute profile standard on a coordinate measuring machine or othergear inspection devices. The following typical measurement parameters for the profile evaluation werechosen according to 7-11:S Profile slope deviation fHain mmS Profile form deviation ff
20、ain mmS Profile total deviation Fain mm4 11FTM04a) b) c)Figure 1. a) Involute profile standard; b) Helix slope standard; c) Pitch standardTable 1. Parameters of the three standardsGear parameter Valuea) Involute profile:Pressure angle n20Helix angle 0Normal module mn2.9559134 mmFace width b 3,2 mmNu
21、mber of teeth z 18b) Helix:Helix angle 0 15 30 45Face width b 75 mm 75 mm 75 mm 75 mmTransversal module mt4mm 4mm 4mm 4mmNumber of teeth z 50 50 50 50Pressure angle n20 20 20 20c) Pitch:Normal module mn4mmNumber of teeth z 37Tip diameter da156 mmFacewidth b 32 mmPressure angle n20Measurement procedu
22、reThisprofilestandardwasmeasuredalongthesurfaceofleftflankatthecenterofthetooth. Themeasurementpointswereselectedatequidistancefromoneanotheroverthelengthofroll. A sphericalstylus tipof 8 mmindiameter was chosen for the measurement because of the following advantages: it offers guaranteed com-parabi
23、lity as being observed since the profile standard has beenacquired, andsecondly, itenables toreducethe influence of form errors of the flank surface. The evaluated parameters were measured within thefollowing limits.Start of profile evaluation at 1 mm expressed in length of rollEnd of profile evalua
24、tion at 18 mm expressed in length of roll5 11FTM04Measurement referencesThe reference axis of the standard was numerically determined. For this purpose, the reference bands ofapproximately50 mmindiameteroftheprofilestandardwereprobedinthecenterofthediscs. Ineachofthetransverse planes at least 36 poi
25、nts at equally spaced distances were measured over the circumference.Through the points, a circle was fitted in accordance with the least squares method and the center wasdefined. Theaxisofthegearstandardwasdefinedfromthecentersofthetwocircles. Thereferencepointforthe height of the profile measureme
26、nt was determined at the top of the tooth, 2 mm from the tip circle in thedirection of the reference axis.Helix standardFigure 1b presents aclassical helixstandard. It embodiesfour differenthelix angles(0,15,30,45)whichareatthelefthandsideaswellasatthe righthand sideof thegear. The measurementswere
27、conductedonlyon the right flank. The corresponding gear parameters are listed in the Table 1. The following typicalmeasurement parameters for the helix evaluation were chosen according to 7-11:S helix slope deviation fHin mmS helix form deviation ffin mmS helix total deviation Fin mmMeasurement proc
28、edureThehelixslopemeasurementswereperformedonameasurementcylinderat dM=204 mm. Thediameterofthe stylus sphere used is approximately 8.0 mm. The evaluation is conducted at the range of Lb=70mmonthe gearing.Measurement referencesThereferenceaxisofthestandardwasnumericallydetermined. Forthispurpose,the
29、tworeferencecylindersof the gear standard were probed. The measurement points were arranged in two end face planes. Theendface planes were located at a distance of 43 mm from the lateral surface of the cylinders with 30 mm indiameter. In each of these transversal planes at least 36 points, which wer
30、e distributed equally spaced overthecircumference,wererecorded. Throughthepoints,acirclewasfittedinaccordancewiththeleastsquaresmethod. The axis of the gear standard was defined from the centers of the two circles.Pitch standardFigure 1c illustrates the pitch standard. The specified gearing paramete
31、rs embodied in the standard aredelineated in Table 1. The following typical measurement parameters for the pitch evaluation were chosenaccordingto7-11:S Cumulative pitch deviation FPin mm (left- and right flank)S Single pitch deviation fPin mm (left- and right flank)Measurement procedureThepitchstan
32、dardwasmountedonthemeasuringmachinebyfixing itwith aninternal three-jawchuck attheinnersideofthehollow shaft. Thepitch wasmeasured ina single-flankmode. The diameterof stylussphereused was 3.0 mm, while the diameter of the measurement circle was dm= 148 mm.Reference axisThe reference axis of the sta
33、ndard was numerically determined. Forthis purpose,two circlesat twodifferentlocationsintheboreweremeasured,oneat10 mmfromthereferencesurface(upperside)ofthegearstand-ard, the other one at 40 mm. In each case at least 36 points, which were distributed equally spaced over thecircumference,wererecorded
34、. Throughthesespoints,acirclewasfittedinaccordancewiththeleastsquaresmethodandthecenterwasdetermined. Theaxisofthegearstandardwasdefinedfromthecentersofthetwocircles.6 11FTM04Measurement instructions and standards handlingTaking into account the geometrical parameters of each standard, the measureme
35、nt procedures, andguidelinesoftraceabilityanduncertaintyforthecomparisonweredevelopedandcirculatedtoallparticipants.Lengths are required to be measured traceable to the latest realization of the meter as set out in the current“MiseenPratique”12irrespectiveoftheinstrumentused. Themeasurementofthetemp
36、eratureisbasedonthe use of international temperature scale of 1990 (ITS-90). Similarly, the uncertainty of measurement isestimated according to the ISO Guide to the Expression of Uncertainty of Measurement 13.The procedures for the packaging and handling of the standards were stated in the protocol
37、adopted for thecomparison. The standards were circulated to the participants in a customized self-containment casedesigned for safe transportation. In addition, the case prevents surface scratches, contamination to thestandards. The packaging cases are all portable enough to be sent by any courier s
38、ervices.Similarly, recommendations on physical inspections of the standards prior to measurement and after themeasurementweregiventotheparticipants. Thecirculationofthestandardswascarriedoutinaloopamongthepartners. Eachpartnerwasgivenasufficienttimetocarryoutthemeasurementbeforesendingthestand-ard t
39、o the next partner.Evaluation of reference values and comparisonThe NMIs are the highest metrological authority. Therefore, the reference valuesmust bedetermined onthebasisofthereceivedmeasurementresults. However,theguidelinesaslaiddownbytheBIPMallowtheuseofdifferent methods for the evaluation of re
40、ference values. These methods include: simple mean, weightedmean and median. This condition allows the participants to choose any evaluation method of their ownchoice. Consequently, for a correctinterpretation ofthe results,an extensiveknowledge onthe evaluationofcomparison is essential. In general,
41、 the most commonly used evaluation method is weighted means x.Therefore,ithasbeenadaptedinthisinvolutecomparison. Accordingtoequation 1itconsidersthemeasure-mentvalueandthecorrespondingmeasurementuncertaintywhichfinallyreflectsthemeasurementconditionand competence of each of the participating labora
42、tories. Each of the participating NMIs reports itsmeasurement results according the ISO directive GUM 13.(1)x =ni = 11ni = 11u xi21u2xixiAcheckforstatisticalconsistencyoftheresultswiththeirassociateduncertaintiescanbemadebycalculationof the Enfactor for each laboratory and for each measurement param
43、eter.The internationally agreed parameter which show if the individual value xNMItogether with its determinedstandard measurement uncertainty uNMIand the standard uncertainty of the comparison uintis reliable incomparison with the calculated reference value x is express as the Envalue according to e
44、quation 2.(2)En=xNMI xu2NMI+ u2intThe absolute value Enmust be less than 1 to indicate that the laboratory is in theposition toobtain aqualifiedresult. That means the measurement value and its corresponding standard measurement uncertainty arecomparable to other NMIs.Contrary to national comparison
45、14 where the Enfactor is calculated on the basis of the expandedmeasurement uncertainty U95(k =2),Enfactor in international comparisons is calculated on the basis of the7 11FTM04standard measurement uncertainty u. This implies that the conditions for the international test are muchstronger! The resu
46、lts of the Encriteria can be seen in table 2. The cell of the measurement parameters ishighlighted in grey color when Enfactor is greater 1.Measurement results and analysisThe total measured parameters for all the standards (profile, helix and pitch) were 28 in number. They wereanalyzed and evaluate
47、d. Envalues for each participant were evaluated as presented in Table 2.Themostrelevantvaluesthatindicatethegeometricalcompetenceandcorrectevaluationofeachparticipantare the results of the slope error for profile and helix measurements and the total error for pitch measure-ments. Therobustnessofthes
48、lopeevaluationforprofileandhelixmeasurementsisbasedontheregressionalgorithm where a single outlier has only a small effect. On the contrary it does for the form and total errors.For pitch measurement evaluation, the probability thatan outlierinfluences thevalue ofthe totalpitch errorismuch smaller a
49、s it appears in single pitch error.Table 2. Overview of the comparability of the measurement results; highlighted grey colouredcells indicate where comparability factor Enis not fulfilleda b c d e f gProfilefH0.27 0.11 0.30 0.89 0.47 1.20 2.68ff0.27 0.18 0.27 0.00 0.18 0.12 0.38F0.10 0.03 0.35 0.13 0.13 0.55 0.25Helix 0fH0.12 0.78 0.28 0.37 0.58 1.66Noresultsff0.67 0.05 0.55 0.46 0.64 0.45F0.70 0.03 0.45 0.41 0.44 0.29Helix 15left handfH0.49 0.83 0.22 0.76 0.45 0.80ff0.13 0.31 0.42 0.24 0.21 0.07F0.
