| 1.IntroductionIt's well-known that the relation between the stability limited cutting width blim ofmachine cutting system and the spindle speed n is presents a periodically dispersal.Characteristic, the cutting width blim is extremely sensitive to spindle speed n, as shown inFig.1. If the minimum value of limited cutting width and the instable spindle speed rangenBwrelativetothelesserlimitedcuttingwidthcanbepredictedbasedonrecognitionofthedynamicparameters for machine cutting system before machining, the machine operator may select theoptimum spindle speed nopt away from the instable spindle speed range nBw and the largercutting width as cutting parameters, the cutting process can be all along performed in stablestate. On the condition of ensuring machine quality, the potential of machine can be exertedmaximumly,the machiningefficiencywillincreaseobservably, theproductcostwillreduce.2.PredictiononStabilityLimitsforRegenerativeChatterSystemThedifferentialequationofregenerativechattersystemcanbeexpressedasWhere m,k and c are respectively the equivalent mass , stiffness coefficient and dampingcoefficient of the main vibration system ;u is the oriental coefficient concerned with the mainvibration direction and the direction of cutting force ; kc is the cutting stiffness coefficient perunit cutting width; b is the cutting width ; ? is the overlap factor; T is the time for onerotationofworkpiece.By deducing, we can obtain the following formulas for predicting stability limited widthblimanditscorrespondentspindlespeed n? ?1 12 2222 sin 2 tan2(2 ) 11nj? ?? ? ? ?? ??? ?? ? ???? ??????????j ?0 ,1,2????????? (2)? ?lim1 12 2222sin sin 2 tan2c(2 ) 11bkuk??? ? ? ?? ?? ??????? ????? ? ?? ??? ????????? (3)Where ?is the frequency ratio, ?? ? ?n, ?is the chatter frequency , ?n is the naturalfrequencyofthemainchattersystem.Fig.2 shows the prediction results which were obtained by computer simulation based ononebyone identifying thedynamicparametersofthedesignedtestinturning.Fig.2Thepredictionresultsoflimitedcuttingwidth blim versus spindlespeednSimulatedconditions:k ?3 164N/mm,? ?0 .0356,? n ?2 ? ?5 02.5rad/s,k c?2 512.81N/mm2,? ?0 .98,u ?0.493. Experiment System Design and its Dynamic Parameters Identification InTurningInordertoincreasethedistinctnessoftestingeffectinexaminetest,aflexibletoolholderwasartificiallyaddedtotheexperimentalsystem,theflexibletoolholderwasdesignedbyfiniteelementmethod.Based on characteristic of Sinumerik 840D type numerical control system of selected testmachine tool, the spindle speed real-time control system and simultaneous measure system forthesignalsofspindlespeedandvibrationresponsehadbeendeveloped.The identification of the machine system dynamic parameters is the foundation of thestability limits prediction , its accuracy consists in the precision of built model of machiningsystem and identification precision of dynamic parameters.The dynamic parameters of testingmachinesystem( ?n ,?,k)andthedynamicparametersofthecuttingprocesses(kc,?)wererecognized by experimental method, among them , the cutting stiffness coefficient kc is a mostimportant parameter and its accurate recognition is the keyfactor for predicting turningsystemstability limits. In this paper, a vibration exciter was utilized for exciting a vibrationdisplacement in longitudinal feed direction in order to make the cutting process from staticstate to dynamic state, the identifying accuracy of cutting stiffness coefficient kc increasesobviously.4. The Testing Examine of the Prediction Results for Stability Limits in00.511.52800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3400 3600Spindlespeed /rpmLimitedcuttingwidth/mmj=4 j=3 j=2 j=1(blim)min=0.2mm4.1Time-varyingCuttingDepthCuttingTestThe tests were performed on SB-CNC lathe. The straight turning are carried on taperturned surfacesunder different spindle speeds,the cutting depth becomes deeper as the cuttingtool feeds forward towards the left, viz., the cutting tool was working on the condition oftime-varying depth cutting. Oncethe chatterwavesemerge on theworkpiece surface,make themachine stop at once.And then measure the non-wave lengthLlimat the workpiece surfaceandcalculatethevalueoflimitedcuttingwidthblim.Fig.3showsthetestresultsoflimitedcuttingwidth blimversusspindlespeed n.00.511.522.51000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500SpindleSpeed n/rpmLimitedcuttingwidth/mmFig.3Thetestingresultsoflimitedcuttingwidth blim versusspindlespeed nInFig.3,theminimumvaluesoflimitedcuttingwidth?bl i m ?minweremeasuredunderthecondionof ??0 .799,onthesameconditionthepredictionvalueof ?bl im ?minisequalto0.30mm.Ascomparedwiththemeanvalue0.34mmofmeasured ?bl im ?min,itcanbeknownthattheyrelative percentageerrorisequalto11.76%.In order to compare with the prediction results objectively, make three unstable spindlespeed points relative to ? ?bl im minshown in Fig.3 respectively expand 5% toward the right andleft. Thus, we may obtain three spindle speed ranges, they are 1092～1207rpm,1567～1733rpm and1995～2205rpm. The predicted instable spindle speed ranges are 1140～1205rpm,1480～1580rpm and 2120～2305rpm which are relative to blim=0.2～0.22mm inFig.2.To compare the measured instable spindle speed ranges with the predicted spindle speedranges, it may be known that the prediction results are in accord with the testing result on thewhole.4.2Time-varyingSpindleSpeedCuttingTestThe tests were performed on the SB-CNC lathe. In turning, the cutting depth and thefeedrate are constant, the spindle speed is time-variying, the cutting tool was working on thecondition of time-varying speed cutting. The vibration acceleration response signal and thespindlespeedsignalarepickedupsynchronouslyinturningprocess.Thetestingdataofthevariance ??y?2 ofvibrationaccelerationresponsesignalversusspi-ndlespeedarelistedinFig.4, thestraightlineof ??? ?y?2 ???0 .371V2isthecontrolthresholdthatisderivedfromthesurfacewaveness profile ofmachinedsurface.Thespindlespeedrangesre-lativeto ??y2? ? ??? ?y2??? areunstableranges,thespindlespeedrangesrelativeto ??y2? ? ??? y?2??? arestableranges.Tab.1liststheunstablespindlespeedrangesrelativetothelimitedcuttingwidthblim byexperimentalmethodwhilethevalueofcuttingdepthvaryfrom0.2mmto0.8mm.Fig.5liststhelimitedcuttingwidthblimversusthespindlespeed,whichwasdrawnaccordingtotheunstablespindlespeedpointslistedinTab.1.Tab.1 Thetesting resultsofthethespindlespeedversus limitedcuttingwidthCuttingDepthap/mm 0.2 0.21 0.3 0.4 0.5 0.6 0.8LimitedCuttingwidthblim/mm 0.283 0.297 0.42 0.57 0.71 0.85 1.13na1/r·min-1 / 1184 1145 1127 1092 1163 1114na2/r·min-1 / 1192 1183 1234 1197 1345 1332nb1/r·min-1 / 1568 1514 1547 1454 1465 1398nb2/r·min-1 / 1582 1623 1686 1619 1721 1715nc1/r·min-1 / 2148 2121 2081 2127 1983 2052nc2/r·min-1 / 2154 2234 2200 2251 2298 228700.20.40.60.811.21.41.61000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500Spindlespeedn/rpmLimitedcuttingwidth/mmFig.5 Diagramofthelimitedcuttingwidthsblimversusthespindlespeed fromtestingresultsIt can be known from the prediction results shown in Fig.2,at the range of 1000～2500rpm,there are three stability lobes, the instable spindle speed ranges relative to bl im=0.2～0.22mm are respectively 1140～1205rpm,1480～1580rpm and 2120～2305rpm. Thetesting results shown in Fig.5 demonstrates that there indeed three stability lobes and theinstable spindle speed relative to ? ?bl im minare respectively 1188rpm,1575rpm and 2151rpm.00.511.51000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500SpindleSpeedn/rpmTheVarianceofVibretionVccelerationResponse/V2Fig.4ThetestingdataofvibrationresponsesignalversusspindlespeedTestingcondiction: cuttingdepthap=0.3mm??? ?2y?s teel???0.371V2na1 na2 nb1 nb2 nc1 nc2In order to compare with the prediction results objectively, make above three instable spindlespeed points repectively expand 5% towards the right and left. Thus, we may obtain threespindle speed ranges, they are 1128～1247rpm,1496～1654rpm and 2043～2259rpm. Tocompare the measured instable spindle speed ranges with the predicted instable spindle speedranges, itisobviousthattheprediction resultsareinaccordwiththetesting resultsessentially.The minimum value of limited cutting width( ?bl i m ?min= 0.297mm) as shown in Fig.5was obtained bysearching cutting depth method, which was measured under the condition of??0 .905.Onthesametheoverlapfactorcondition,thepredictivevalueof ?bl im ?minisequalto 0.24mm. To compare with the measured value of ?bl im ?min, it can be known that theirrelativepercentageerrorisequalto 19.2%.5.ConclusionEngineers can select cutting parameters properly with the help of the stability limitedprediction results, not only the machining quality may be insured, but also the machiningefficiencycan be increasedobservably. Moreover, the cutting propertyof machine andtoolcanbeenough exerted.The prediction accuracy of stability limited cutting width blimand instable spindle speedranges are essential dependent on the accuracy of the built model and recognized accuracy ofthedynamicparametersofmachinetoolcuttingsystem.Thetesting dataofthestabilitylimitedcuttingwidthblimversusthespindlespeednintur-ningwasobtainedfirstlybytime-varyingcuttingdepthtestsandtime-varyingspindlespeedcuttingtests.Thetestingresultsshowthattheyareagreementwiththeprodictive resultsessen-tially,thepresentedstability predictionmethodforturningsystemiscorrectandfeasible.Forcomparisonwiththetime-varyingcuttingdepthcuttingtest,thetime-varyingspindlespeedcuttingtesthastheadvantagesofsimpleandconvenienceinrealization,ithasgoodpro-spectforapplyingtoidentifystablespindlespeedranges on-lineinfuture.00.20.40.60.811.21.41.6900 1100 1300 1500 1700 1900 2100 2300 2500 2700 2900 3100 3300Spindlespeedn/rpmLimitedcuttingwidth/mm0.2mm 0.24mm??0 .905??0.98Fig.6Thepredictedvalueof(blim)minwhilethe ? varied...
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