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ORIGINAL ARTICLE An experimental investigation of spindle rotary error on high speed machining center Lan Jin spindle rotary precision is high conversely the lower the rotary precision Thus the identification of the spindlerotationerrorshasbecomeveryimportant Sucherrors cause degradation in surface finish roundness feature size and feature location In an earlier work Tlustry and Bryan et al 1 2 proposed a method for measuring spindle error motions with a master ball while machining and generated a basecircle for better visualizing the motion ofaxisofrotation After this many machine tool testing methods were adopted in many international standards such as American National Standards Institute ANSI and ISO and ANSI especially drew up the standard for the test of the spindle 3 With the development of high speed and high precision machine tools the high speed rotation and the built in motor alsointroducelargeamountsofheatandrotatingmassintothe system requiring precisely regulated cooling lubrication and balancing As a result the thermal and mechanical behaviors of high speed motorized spindles have become very difficult to predict for spindle designers and users 4 The measure ment and evaluation of rotation error of tool spindle is more important for evaluating the performance of high speed CNC computer numerical control machine tools However the analysis of spindle rotation errors can not only predict the quality of the machined part but also be used to evaluate the machine tool precision for purchasing and maintenance pur poses Insomespindlemeasurementsystems 5 10 aprecise sphere or cylinder and multiple probes are used to inspect the spindle axis rotary error for the case of a rotating sensitive direction It is desirable to separate out unnecessary data such as the roundness error and the eccentricity error of a precise sphere or cylinder when the principle of the spindle error measurement and means cannot be fundamentally changed Thus many error separation methods have been developed to L Jin L Xie School of Mechanical thermal growth is normal according to the JB T10801 2 2007 standards 18 2 3 Mathematical models 2 3 1 Calculation of eccentricity and initial phase angle Theartifactmustbeconsideredfirstbecauseithaseccentricity and roundness errors Initially a simulation is carried out to analyzethechangeofmeasurementdatabasedoneccentricity The eccentricity e is expressed as the distance between the origin point of the reference coordinate system and the center of the rotating axis O 19 The eccentricity of the master cylinderandtheinitialphaseangleofthesensorscanbetested by the method described in Fig 2 a As shown in Fig 2 b a dial gauge is used to measure the master cylinder in a static and low speed state Point A shown in Fig 2 c is the initial position of the dial gauge The values of points B C D and E measured by the dial gauge are m emax n and emin 328Int J Adv Manuf Technol 2014 70 327 334 Fig 1 Distribution sensors Fig 2 Calculation ofeccentricity and initial phase angle Int J Adv Manuf Technol 2014 70 327 334329 The eccentricity can be expressed as e emax emin 2 1 where e is the eccentricity of the master cylinder In the triangle OGF shown in Fig 2 b OG e OF m n 2 Therefore the initial phase of the sensor X is arcsinOF OG arcsinm n 2e 2 2 3 2 Construction of the error model The measurement ofthe spindle error isdirectlyinfluencedby the out of roundness of the master cylinder and the Fig 3 Schematic diagram of the measurement Fig 4 Spindle error measurement system 330Int J Adv Manuf Technol 2014 70 327 334 eccentricity of the master cylinder Specifically the eccentric error is present in the measuring signals and it decreases the precision of the measuredspindle error especiallyinthe high precision measurements Therefore several attempts have beenmadetoseparatetheseerrors Generally Fourieranalysis is used to calculate the influence of eccentricity of the master cylinder on the machine spindle for measured data sets The integration scheme is used to calculate appropriate Fourier coefficients for the eccentricity or once around or fundamen tal frequency of the gauge data These Fourier coefficients are then used to reconstruct the once around The once around waveform can then be subtracted from the entire data set so that only the second order and higher harmonics of the error Table 1 The dependency of speed with respect to unbalancing Spindle speed rpm Rotary error mm X axisY axis maxminmaxmin 1 000 26 4 23 224 8 17 9 2 000 28 7 24 422 2 22 1 3 000 21 5 19 322 0 23 9 4 000 20 8 23 322 2 20 1 5 000 23 7 24 829 4 22 7 6 000 27 8 28 830 7 23 3 7 000 33 1 47 940 1 20 2 8 000 14 3 25 238 8 23 1 9 000 22 9 36 341 4 24 5 10 000 22 1 34 444 3 26 1 11 000 26 2 40 149 8 26 5 13 000 27 2 41 147 1 50 4 14 000 32 8 45 630 4 47 3 15 000 42 4 47 141 7 50 1 Fig 6 Rotation sensitive directional error motions at 15 000 rpm ab cd X axis eccentricity error signalsX axis rotation error Y axis eccentricity error signals Y axis rotation error Fig 5 a Eccentricity error signals ex b rotation error x x x t ex c eccentricity error signals ey and d rotation error y y x t ey Int J Adv Manuf Technol 2014 70 327 334331 are included However Fourier analysis can introduce some new errors affecting the results of measurement Therefore the best method is to calculate the eccentricity error first and only then to remove it from the gauge data The two sensors are used to measure the spindle error as shown in Fig 3 As the measuring signal includes both the rotation error and the shape error the mathematical model must be constructed to remove them The measuring signal obtained via the sensors can be deduced from the schematic diagram shown in Fig 4 The location designated by the point O in Fig 3 represents the center of the spindle Point O1 is the center of the master cylinder R is the radius of the master cylinder In a frame of reference in which the spindle is stationary and the sensors move r is the radius of the trace of the sensor motion The initialpositionofsensorXisatpointA asshowninFig 3 the initial phase of which is After rotating through angle the sensor X moves to point B The length of BC is the error causedbyaneccentric mountingofthe mastercylinder Ifx t represents the output of the sensor X x the spindle error of direction x exthe eccentricity error of direction x and the shape error at the angular velocity these variables are related in the following expression as x t x ex 3 With the use ofa high accuracy mastercylinder the round ness error can be considered to be negligible Therefore Eq 3 can be simplified to Eq 4 as follows x t x ex 4 ThedistanceLxfromthespindlecentertothesurfaceofthe master cylinder is Lx ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi R2 esin 2 q ecos 5 The radius r of the trace of the sensor motion can then be obtained as r ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi R2 esin 2 q ecos 6 Therefore the eccentricity error excan be expressed as ex Lx r 7 Since the sensors X and Yare aligned at an angle of 90 as shown in Fig 3 output y t of the sensor Y can be described by the following equations Ly ffiffiffi ffiffiffi ffiffiffi ffiffiffi ffiffiffi ffiffiffiffi ffiffiffiffi ffiffiffiffi ffiffiffiffi ffiffiffiffi ffiffiffiffi ffiffiffiffi ffiffiffiffi R2 esin 90 2 q ecos 90 8 ey Ly r 9 y t y ey 10 2 3 3 Evaluation of the rotation error After removing the eccentricity error the results of the X and Yerror motions can be used to produce an error motion plot for a given rotation sensitive direction The calculation of D t the motion error is performed from Eq 11 below D t r0 xt sin t yt cos t 11 where rois the radius of an arbitrarily chosen base circle x and yare the measurederror motions isthe rotationrateof the sensors and t is the measuring time To evaluate the error motion value a least squares fit of the data is performed to position the error motion on a least squares center The total radial error is determined to be the maximum value of the spindle error less the minimum value with respect to the least squares center The position a b of the least squares center and the radius D of the least squares are calculated in confor mity with the ANSI Standard B89 3 4 M as follows a 2 X k 1 n xk 12 b 2 X k 1 n yk 13 D 2 X k 1 n dk 14 Where a and b correspond to the X and Y locations of the least squares circle center respectively n is the number of discrete data points and xkand ykcorrespond to the values of the ithdata points in the X and Y direction respectively 11 The distance DKfrom the position a b to the position xk yk can be expressed as Dk ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi xk a 2 yk b 2 q 15 Therefore the spindle error is f Dkmax Dkmin 16 332Int J Adv Manuf Technol 2014 70 327 334 3 Experimental measurement 3 1 Measuring system 3 1 1 Total system Figure 4 shows the measuring system for a rotating spindle installed inthe testing platform A precision mastercylinderis mounted on the tool position of the spindle The roundness of the master cylinder has an accuracy of better than 0 001 m The device for mounting the two sensors is shown in Fig 4 The sensor mounting device not only supports the sensors but also keeps the sensors at the same cross sectional circle of the master cylinder With the sensor mounting bracket the mea surement signals can minimize the setup error of the sensors During measurement sensor signals are acquired by an LMS Test lab collector where the signals are pretreated and then transferred into the receiving computer for further analysis 3 2 Test results The dependency of speed with respect to unbalancing is listed in Table 1 The critical speed of this spindle is of 7 000 r min After using the error separation model described in the previous section to remove the setup error of the master cylinder the rotating error and the setup eccentricity of the master cylinder are subtracted from the measured data The variations in the output are as depicted in Fig 5 The least squares method was used for each revolution which has removed the setup error associated with the master cylinder The lateral spindle error is 0 005 mm at 15 000 rpm as shown in Fig 6 To demonstrate the feasibility of the proposed spindle measuring system and the error separation method experi ments on a high speed horizontal machining center with dif ferent spindle speeds A trace of the rotating master cylinder was measured by the method proposed previously in this study The least squares method was used for each revolution and the setup eccentricity of the master cylinder was subtracted from the measured data The overall test results are summarized in Table 2 Table 2 summarizes the measured rotation error at different speeds The maximum error of the spindle rotation as shown in Table 2 is 7 000 rpm which may fall into the resonance region The general trend of rotation errors is indicated in Fig 7 It can be seen that the rotation error increases as the speed increases with the exception of the region around 7 000 rpm The rotation static error of this spindle is 4 m while the test result is 5 m Measured rotation errors are very close to their real values These measurement results confirm the fea sibility of the proposed spindle measuring system 4 Conclusion Thispaperdescribestheprincipleofanewdevicedetermining spindle rotation error using a method of error separation at 15 000 rpm The results of our measuring experiments using the system proposed above along with appropriate signal processing techniques demonstrate that the new device for measuringthe rotationerror ofa high speedspindle isfeasible Table 2 The overall test results of the spindle error measurements Spindle Speed r min 1 0003 0005 0007 0009 00010 00012 00013 00014 00015 000 Rotary error mm 4 624 644 685 184 934 744 784 864 895 10003000500070009000 10000120001300014000 15000 4 5 5 5 5 spindle speed r min rotary error mm Fig 7 The trend of rotary errors Int J Adv Manuf Technol 2014 70 327 334333 and that the mathematical model of error separation can ef fectively isolate the eccentricity error caused by setup error Compared with filtering for error separation use of the meth od of error separation described in this paper can obviate the undesirableremovalofsomecomponentsofthesignalswhich we would otherwise need References 1 Tlustry J 1959 System and methods for testing machine tools Microtechnic 13 162 2 Bryan JB Clouser RW Holland E 1967 Spindle Accuracy Ma chinist 4 149 164 3 Jyweb W Y Chen C J 2004 The development of a high speed spindle measurement system using a laser diode and a quadrants sensor Int Mach Tool 45 1162 1170 4 Lin C W Tu JF Kamman J 2003 An integrated thermo mechanical dynamic model to characterize motorized machine tool spindles during very high speed rotation Int Mach Tool 43 1035 1050 5 Mitsui K 1983 Development of a new measuring method for spindlerotationaccuracybythreepointsmethod ProcIntMachTool 23 115 121 6 Chapman PD 1985 A capacitance based ultra precision spindle error analyser Precis Eng 7 3 129 137 7 Shinno H Mitsui K Tatsue Y Tanaka N Omino T Tabata T 1987 A new method for evaluating error motion of ultra precision spindle Ann CIRP 36 1 381 384 8 Zhang GX Wang RK 1993 Four point method of roundness and spindle error measurements Ann CIRP 42 1 593 596 9 LeeES WiHG 1998 Acomprehensivetechniqueformeasuringthe three dimensionalpositioningaccuracyofarotatingobject IntJAdv Manuf Technol 14 330 335 10 ChoiJ P LeeS J KwonH D 2003 Roundnesserrorpredictionwith a volumetric error model including spindle error motions of a ma chine tool Int J Adv Manuf Technol 21 923 928 11 HeQ X ZhangH R YangJ 1999 Rotaryprecisionanalysiso