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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.Insomespindlemeasurementsystems510,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) 70:327334 Fig. 1 Distribution sensors Fig. 2 Calculation ofeccentricity and initial phase angle Int J Adv Manuf Technol (2014) 70:327334329 The eccentricity can be expressed as: e emax emin=21 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) 70:327334 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 26.423.224.817.9 2,000 28.724.422.222.1 3,000 21.519.322.023.9 4,000 20.823.322.220.1 5,000 23.724.829.422.7 6,000 27.828.830.723.3 7,000 33.147.940.120.2 8,000 14.325.238.823.1 9,000 22.936.341.424.5 10,000 22.134.444.326.1 11,000 26.5 13,000 50.4 14,000 32.845.630.447.3 15,000 42.447.141.750.1 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) 70:327334331 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 ex4 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 esin2 q ecos6 Therefore, the eccentricity error excan be expressed as: ex Lxr7 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 Lyr9 y t y ey10 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 errormotion 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 t11 where rois the radius of an arbitrarily chosen base circle, x andyare 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 k1 n xk12 b 2 X k1 n yk13 D 2 X k1 n dk14 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 xka2 ykb2 q 15 Therefore, the spindle error is: f DkmaxDkmin16 332Int J Adv Manuf Technol (2014) 70:327334 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) 70:327334333 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. 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