Least square curve fitting technique for processing time sampled high speed spindle data. S. Denis Ashok and G.L. Samuel*

Transcription

1 256 Int. J. Manufacturing Research, Vol. 6, No. 3, 2011 Least square curve fitting technique for processing time sampled high speed spindle data S. Denis Asho and G.L. Samuel* Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai , India *Corresponding author Abstract: Radial error measurement of a high speed spindle is commonly affected by speed variations, centring error, form error of master cylinder and thermal drift of the spindle. This paper presents a least square curve fitting technique for processing the high speed spindle data using a mathematical model consisting of a second-order polynomial and sum of sinusoidal functions. Re-sampling method is proposed for accounting the spindle speed variations. Performance of the proposed method is compared with Fourier transform based frequency domain filtering method. Simulation and experimental results confirm the accuracy of the proposed method for analysing the time sampled spindle data. [Received 18 March 2010; Revised 4 October 2010; Accepted 20 October 2010] Keywords: high speed spindle; radial error motion; modelling; least square method; modelling and simulation. Reference to this paper should be made as follows: Denis Asho, S. and Samuel, G.L. (2011) Least square curve fitting technique for processing time sampled high speed spindle data, Int. J. Manufacturing Research, Vol. 6, No. 3, pp Biographical notes: S. Denis Asho received his Bachelors Degree in Mechanical Engineering and Masters Degree in Production Engineering from Madurai Kamaraj University, Madurai, India. He has carried out his research wor towards his PhD at Indian Institute of Technology Madras, Chennai. He is currently teaching at Vellore Institute of Technology, Vellore, Tamil Nadu, India. His area of interests includes machine tool metrology, spindle metrology and computer aided inspection. G.L. Samuel is currently woring as Assistant Professor at the Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai. He obtained his BE Degree in Mechanical Engineering from Mysore University in 1991, MTech Degree in Production Engineering and Systems Technology from Kuvempu University in 1994 and PhD Degree in Mechanical Engineering from Indian Institute of Technology Madras in He has been a Post Doctoral Fellow at School of Mechanical Engineering, Kyungpoo National University, South Korea. His active areas of research are: measurements and inspection of freeform surfaces, geometric error compensation in machine tools, and evaluation of form errors, micromachining and laser vision systems. Copyright 2011 Inderscience Enterprises Ltd.

2 Least square curve fitting technique for processing time Introduction With the global trend towards miniaturisation, demand for micro scale components is tremendously increasing in the different application fields such as aerospace and automotive industries, electronics, optics and bio technology. In this context, miniaturised machine tool provides a promising technology for manufacturing miniature components (Masuzawa, 2000). Spindle is one of the important components of the miniaturised machine tool for providing high rotational speed to the micro cutting tools. Out of roundness of bearing races, defects in the ball bearings and structural motion of machine tool affect the perfect rotation of the spindle. Consequently, measurement and analysis of spindle rotation accuracy are very important for verifying the machining accuracy of the miniaturised machine tool. An optical measurement system consisting of a laser source and quadrant photo diode is proposed for measuring the radial runout of the miniature high speed spindle (FujiMai and Mitsui, 2007). A measurement system consisting of a laser diode and quadrant sensor is developed for measuring the spindle errors, spindle speed and spindle indexing (Jywe and Chen, 2005). However, these methods require more setup time to align the path of the laser for spindle error measurement. Capacitive sensor-based measurement techniques are commonly used for measuring the spindle error motions due to the high resolution, adequate sensing range, band width and low cost (Marsh, 2008). Evaluation of spindle error motion requires preprocessing of the spindle data to separate the thermal drift, synchronous and asynchronous components of the spindle data (ANSI/ASME B89.3.4M Standard, 2004). Fourier transform based frequency domain filtering and second-order polynomial fitting methods have been applied for processing the spindle data (Gredja et al., 2005). Averaging method is used for separating synchronous and asynchronous components of the spindle data (Lee et al., 2005). However, averaging method and Discrete Fourier Transform (DFT) are not suitable for time sampled spindle data containing an incomplete spindle revolution (Gredja et al., 2005). Best fit sine wave method is proposed for synchronising the spindle data (Murthy, 2000). This method is not adequate for separating the synchronous components of the spindle data. Least square fitting method is proposed for identifying the synchronous and asynchronous components of the time sampled spindle data (Asho and Samuel, 2009). However, this method does not separate the thermal drift components of the spindle data. This paper presents a least square curve fitting technique for simultaneous separation of thermal drift, synchronous and asynchronous components of spindle data using a mathematical model consisting of a second-order polynomial and a sum of sinusoidal functions. In the present method, a re-sampling method is proposed for overcoming the effect of speed variations. Performance of the proposed analysis method is compared with the DFT method for the samples of the simulated data and measured spindle data. 2 Measurement of radial error motion of high speed spindle In accordance with ANSI/ASME B89.3.4M standard, fixed sensitive radial error motion test is followed for assessing the rotation accuracy of a high speed spindle in a

3 258 S. Denis Asho and G.L. Samuel miniaturised machine tool. A master cylinder is used as a target for measuring the radial errors of the high speed spindle. Table 1 shows the specifications of the capacitive sensor used for the spindle error measurement. Table 1 Technical details and specifications of the capacitive sensor S. No. Specifications Calibration values 1 Range 250 µm 2 Standoff 250 µm 3 Output voltage 10 to 10 VDC 4 Output sensitivity 0.08 V/µm 5 Linearity error 0.05% 6 Pea to pea resolution 47.5nm Preliminary experiments were conducted to verify the performance of the capacitive sensor for the given target displacement. A chec fixture is used for testing the accuracy and linearity of the capacitive sensor for the travel range of 250 µm. Figure 1 shows the output of the capacitive sensor for the given target displacement. Figure 1 Output of the capacitive sensor for the given target displacement It is found that the capacitive sensor has a good linearity within the travel range of ±50 µm. In the present wor, the radial error measurement is carried out within the measuring range of ±15 µm in a temperature controlled environment. A master cylinder of diameter 6.34 mm is used as the target for measuring the radial error motion of the high speed spindle. Figure 2 shows the experimental arrangement for spindle radial error measurement in the miniaturised machine tool. The effect of target shape on the spindle data needs to be corrected for nanometer level evaluation of spindle error motion (Smith et al., 2005). A multiplier value of is applied as a correction factor for the target shape as suggested by the manufacturers of the sensor.

4 Least square curve fitting technique for processing time 259 Figure 2 Experimental arrangement of fixed sensitive radial error motion test for the miniature, high speed spindle 2.1 Discrete time sampling of high speed spindle data A computer-aided data acquisition system is used for sampling the spindle data at discrete time intervals. To avoid the aliasing effects, the sampling frequency (f s ) is fixed using Nyquist criteria such that it is at least twice the product of the desired harmonic cutoff (H) and the spindle rotational frequency (f r ), as given by the following equation (1). f f H 2 (1) s r where H represents the desired harmonic cutoff in Cycle Per Revolution (CPR) and f r corresponds to the spindle rotational frequency for the specified spindle speed (N). In the present wor, a harmonic cutoff value is selected to be 30 CPR. 2.2 Interpretation of time sampled spindle data Discrete time samples of the spindle data obtained at the spindle speed of 25,000 rpm are shown in Figure 3(a). Periodic sinusoidal component in the measured spindle data represents the contribution of centring error of master cylinder and the individual revolution of the spindle. It is difficult to gain more insight into the contributions of spindle thermal drift, form error of master cylinder, synchronous and asynchronous radial error motion in time domain. Hence, the spindle data is analysed in frequency domain for interpreting synchronous and asynchronous components. Harmonic frequency spectrum for discrete time samples of the spindle data is obtained using DFT and it is shown in Figure 3(b). It is noticed that the magnitude of the signal is maximum at 1 CPR and it represents the magnitude of centring error of the master cylinder. There is a spread in the magnitude of fundamental component to the adjacent frequency bins and it is nown as spectral leaage. It is caused due to the spindle speed variations during measurement and the finite length of the spindle data containing incomplete spindle revolution. Spectral leaage affects the accurate interpretation of synchronous and asynchronous components of the spindle data. This effect can be eliminated by synchronising the spindle data to spindle rotation.

5 260 S. Denis Asho and G.L. Samuel Figure 3 Measured spindle data obtained at the speed of 25,000 rpm: (a) time domain and (b) frequency domain (a) (b) 3 Proposed method for synchronising the spindle data Processing algorithms based on fundamental frequency component of the spindle data are commonly used to overcome the errors due to the spindle speed variations (Marsh, 2008). In the present wor, best fit sine wave method (Murthy, 2000) is used to determine the fundamental frequency of the time sampled spindle data. A re-sampling method is presented to synchronise the spindle data using the estimated fundamental frequency. 3.1 Estimation of fundamental frequency of spindle data Best fit sine wave method is applied for estimating the fundamental frequency of the spindle data obtained at the spindle speed of 25,000 rpm. As the rotational frequency of the spindle is 25,000/60 = Hz, a frequency interval of 410 Hz to 418 Hz is selected for accounting the speed variations during spindle error measurement.

6 Least square curve fitting technique for processing time 261 A set of discrete frequency values such as 410, , and 418 are obtained from the specified frequency interval with a resolution of 0.01 Hz. Sine wave with the given discrete frequency value is fitted to the samples of spindle data and the sum of squared residuals is estimated as shown in Figure 4(a). It is found that the minimum of squared residual error corresponds to µm 2 at the frequency value of Hz. Figure 4 Best fit sine wave for the measured spindle data obtained at the spindle speed of 25,000 rpm: (a) determination of fundamental frequency and (b) estimated best fit sine wave (a) (b) Figure 4(b) shows the best fit sine wave for the given samples of spindle data. Frequency of the best fit sine wave provides the time taen to complete one spindle revolution and it is useful in synchronising the spindle data to spindle rotation. 3.2 Re-sampling method for synchronising the spindle data In this wor, a re-sampling method is proposed for synchronising the discrete time samples of the spindle data. Mathematical formulation of the re-sampling method is explained below. A set of m samples of spindle data (m i ) and the corresponding sampling time (t i ) are considered for analysis. It is represented in matrix form using equations (2) and (3).

7 262 S. Denis Asho and G.L. Samuel m = [ m, m,, m ] T (2) i m T t = [ t, t,, t ] i= 1,2,3,, m. (3) i m Re-sampling frequency (f rs ) is determined based on the fundamental frequency of the spindle data (f 0 ) using the following equation (4). frs = 2 Hfˆ. (4) 0 Modified sampling time interval ( t r ) is determined using the re-sampling frequency and it is given by the following equation t = 1/ f. (5) r rs A new sequence of sample times is calculated using the modified sampling time interval ( t r ). t = t where = 0,1, 2, n. (6) r Length of the re-sampled data (n) is determined based on the maximum value of the sampling time (t m ) and it is given by n= t / t. (7) m r Magnitude of the re-sampled spindle data are estimated at the new sequence of sample times (t ) using linear interpolation of the adjacent sample data. It is represented by equation (8). mi+ 1 m i m = m i + ( t ti) such that ( ti < t < ti+1). (8) t t i+ 1 i Re-sampled spindle data (m ) and the corresponding sampling time (t ) are synchronised to spindle rotation. Table 2 shows the magnitude of the re-sampled spindle data and the new sequence of sampling time for the samples of spindle data obtained at the spindle speed of 25,000 rpm. Table 2 Re-sampling spindle data obtained at the spindle speed of 25,000 rpm S. No. Sampling time (10 3 Sec) Magnitude of Spindle data (µm) Re-sampling time (10 3 Sec) Re-sampled Spindle data (µm)

8 Least square curve fitting technique for processing time 263 Re-sampled spindle data requires further processing for separating the contributions of thermal drift, synchronous and asynchronous components. 4 Least square curve fitting method for processing spindle data In the present wor, a mathematical model is proposed for simultaneous separation of thermal drift, synchronous and asynchronous components of the spindle data. Least square curve fitting method is used for accurately estimating the parameters of the model. Mathematical formulation of the proposed method is explained in the following subsections. 4.1 Mathematical representation of spindle data A mathematical model consisting of a constant term, second-order polynomial and sum of sinusoidal functions is proposed for characterising the contribution of mean position of master cylinder, thermal drift and synchronous components of the spindle data, respectively. H 2 = h π 0 + h π 0 h= 1 m C pt pt a cos(2 hft) bsin(2 hft) (9) where a h, b h represent the amplitudes of cosine and sine components of sinusoidal function. C represents the mean value of the spindle data. p 1, p 2 represents the coefficients of second-order polynomial for representing the trend due to thermal drift of spindle. f 0 is the fundamental frequency of the spindle data. H represents the cutoff value for the synchronous components of spindle data in cycles per revolution. Asynchronous components are not included in equation (9) and it is assumed as the residuals of the estimated value using the model. It is given by the following equation e = m m. (10) Equations (9) and (10) is useful in interpreting and separating the various contributions of spindle data such as mean position of master cylinder, thermal drift, synchronous and asynchronous components in the spindle data. However, it is necessary to estimate the parameters of the mathematical model accurately. 4.2 Least square fitting method for model parameter estimation In equation (9), C, p 1, p 2, a h, b h are the linear model parameters and f 0 is the non linear parameter. Here, the fundamental frequency of the spindle data (f 0 ) is already estimated using best fit sine wave method as explained in Section 3.1. Assuming the residuals follow the normal probability distribution, a linear least square method is used for estimating the linear model parameters such as C, p 1, p 2, a h, b h. Ordinary least square method provides equal weights to the spindle data in the parameter estimation. However, asynchronous components of the spindle data vary in each spindle revolution. Hence, a weighted least square method is used for the robust estimation of model parameters. Steps involved in the proposed least square algorithm are explained below.

9 264 S. Denis Asho and G.L. Samuel Step 1: Input data Proposed method requires the n samples of synchronised spindle data m' and the corresponding time values t. A harmonic cutoff value (H) is also required for selecting the number of harmonics to be included in the model Step 2: Ordinary least square method For the given value of fundamental frequency (f 0 ), the linear model for the spindle data can be represented in matrix form as shown in equation (11). m = DX (11) where D is the basis matrix containing n rows and 2H + 3 columns as given by equation (12). 2 1 t1 t1 cos(2 π ft 01) sin(2 π ft 01)... cos(2 πhft 01) sin(2 πhft 01) 2 1 t2 t2 cos(2 π ft 0 2) sin(2 π ft 0 2)...cos(2 πhft 0 2) sin(2 πhft 0 2) D = t t cos(2 π ft 0 ) sin(2 π ft 0 )...cos(2 πhft 0 ) sin(2 πhft 0 ) (12) X is the set of linear model parameter values for as given by equation (13) T X = [ C, p, p, a, b, a, b a, b ]. (13) H H Equation (11) leads to a system of over determined equations. The number of equations is more than the unnown model parameters (i.e., n > 2H+3). In this case, the solution for the model parameter X is obtained by minimising an objective function ρ of the residual error e defined by equation (14). e = ( m DX). (14) Least square method minimises residual between the estimated spindle data at the parameter vector X and the measured spindle data m i, so that the objective function to be minimised for maximum lielihood of the parameters can be obtained as n n 2 2 ρ( e) = ρ( m DX). = 1 = 1 (15) The minimum value of the objective function can be obtained by differentiating with respect to the model parameters and setting the partial derivatives to zero. Let ψ = ρ be the derivative of ρ. It produces a set of estimating equations (16) for the model parameters X. n ψ ( m DX) X = 0. (16) = 1 Model parameter estimation for the ordinary least square method can be obtained as. ˆ T 1 T ( ). X = D D D m (17)

10 Least square curve fitting technique for processing time 265 It is obtained by minimising the sum of squared residual given by n 2 eˆ. = 1 E = (18) Residuals can be estimated using the following equation eˆ = ( m DXˆ ). (19) Step 3: Weighted least square method Weight function ω is defined for the estimated residuals and it is given by ψ ( eˆ ) ω ( eˆ ) =. (20) eˆ In this wor, an iterative method (Leffler and Jay, 2009) is applied for determining the final weights for the residuals using bi-square weighting function. Substituting the values of weighting function to the estimating equations given in equation (21) n ω( m DX) X = 0. (21) = 1 Weighted linear least square estimation of the model parameters can be obtained for the given frequency value (f j ). ˆ T 1 T w ( ω ) ω. X = D D D m (22) Solving the estimating equations is a weighted least square problem that minimises the weighted sum of squared residuals as given below. E w n = ωε. (23) = Residuals for the weighted least square estimation is given by ˆ ε = ω ( m DXˆ ). (24) wi Step 4: Termination of proposed least square algorithm Weights are determined recursively until the change in magnitude of model parameter estimates ( Xˆ w ) reaches less than between any two successive iterations. The final weights are applied to the spindle data and model parameters are estimated using the following equation. X = ( D D) D m = [ Cˆ, p, p, a, bˆ, a, bˆ... a, bˆ ]. (25) ˆ T 1 T ' ˆ ˆ ˆ ˆ ˆ w ωf ωf H H Estimated model parameters are used for separating the thermal drift, synchronous, and asynchronous components of spindle data.

11 266 S. Denis Asho and G.L. Samuel Step 5: Separation of various components of the spindle data Contribution of thermal drift can be estimated using equation (26) mˆ = pˆ t + pt ˆ. (26) 2 t 2 1 Synchronous components of the spindle data is obtained as given by mˆ = DXˆ pˆ t pt ˆ Cˆ. (27) 2 s w 2 1 Synchronous components are further analysed to remove the contribution of centring error of master cylinder as given by equation (28). mˆ = mˆ aˆ cos(2 π fˆt ) bˆ sin(2 π fˆt ). (28) r s In this wor, residuals are considered as the asynchronous components of spindle data. Although the algorithm uses weights for estimating the residuals in equation (24), weights are not considered for estimating the asynchronous components of the spindle data as given by equation (29). ˆ τ = m DXˆ. (29) w The amplitude and phase of harmonic components of the spindle data can be obtained from equations (28) and (29), respectively. Cˆ = aˆ + bˆ (30) 2 2 h h h bˆ 1 h ˆh = tan. aˆ h γ (31) 5 Simulation results Performance of the proposed least square fitting method is verified using the simulated spindle data. Proposed mathematical model is used for generating the simulated spindle data as given by. H 2 = hsin(2 π r + ϕh) + ε h= 1 m C pt p t C hf t. (32) Three simulation case studies are presented by varying the magnitudes of model parameters. Table 3 shows the input values used for the first 10 harmonic components of simulated data for the three cases. Table 3 Magnitudes of harmonic components used in the simulation case studies Input values for magnitude of harmonic components (C h ) (µm) Harmonic number (h) Case-1 Case-2 and Case

12 Least square curve fitting technique for processing time 267 Table 3 Magnitudes of harmonic components used in the simulation case studies (continued) Input values for magnitude of harmonic components (C h ) (µm) Harmonic number (h) Case-1 Case-2 and Case In each simulation case, discrete sample time values are obtained for the sampling frequency of 27 KHz and the spindle data is generated for the spindle speed of 25,000 rpm. For simplicity, mean value of the spindle data (C) is assumed to be zero and typical values for the polynomial coefficients are assumed as p 2 = 15 µm/sec 2 ; and p 1 = 5 µm/sec; The effect of variations in the magnitude of spindle rotational frequency (f r ) and asynchronous components (є ) are simulated using normal distribution random numbers. Asynchronous components are generated using normal distribution random number with a constant variance of 0.01 µm 2. Estimated magnitude of harmonic components using the proposed method is compared with DFT-based filtering method. 5.1 Case 1: Higher magnitude of synchronous components In this case, the magnitude of the centring error and other synchronous components are assumed to be higher than the asynchronous components in the simulated spindle data. DFT-based filtering method is applied to 300 samples of simulated spindle data. Figure 5 shows the fitted curve using proposed method to the simulated data and the filtered synchronous components using DFT. Figure 5 Filtering and fitting results for the simulation case-1 It is found that curve fitted using least square method more closely follows the periodic pattern of the simulated data with a correlation coefficient (R 2 ) of However, synchronous components obtained using the DFT shows deviations in magnitude and

13 268 S. Denis Asho and G.L. Samuel phase for the simulated data and the correlation coefficient is found to be It is due to the spectral leaage effect and the length of the spindle data containing incomplete number of spindle revolutions. 5.2 Case 2: Smaller magnitude of synchronous and asynchronous components To verify the robustness of the proposed method, it is applied to the samples of the spindle data containing the smaller magnitude of synchronous and asynchronous components. Figure 6 shows the fitted curve using the proposed method along with the estimated synchronous components using DFT-based filtering method. It can be seen that despite magnitude variations in the synchronous and asynchronous components, curve fitted using the proposed method follows the periodic trend accurately than the DFT-based filtering method. Figure 6 Filtering and fitting results for simulation case-2 Table 4 shows the estimated values for the harmonic components of the simulated data using the DFT and proposed method. It can be noticed that the fundamental component is accurately identified by the proposed method. As the variance of asynchronous components is comparable with the magnitude of synchronous components of simulated data, estimated magnitudes of the other harmonic components provide more deviations from the input values. However, the values estimated by the proposed method are much closer to the input values than the DFT-based filtering method. This simulation case study proves the robustness of the proposed method in identifying harmonic components in presence asynchronous components with comparable magnitude. Table 4 Comparative estimation results for DFT and the proposed method for the simulation-case-2 Estimated values (µm) Harmonic number (h) Input values DFT Least square method

14 Least square curve fitting technique for processing time 269 Table 4 Comparative estimation results for DFT and the proposed method for the simulation-case-2 (continued) Estimated values (µm) Harmonic number (h) Input values DFT Least square method Case 3: Effect of change in the magnitude of asynchronous components In this case, the effect of change in magnitude of the asynchronous components in each spindle revolution is analysed. The variance values for the normal distribution random numbers are modified in the range of µm 2. However, the model parameters for the harmonic components and thermal drift are ept same values as used in the previous case study. Comparison of estimated values of harmonic components of the simulated data using DFT and the proposed method is given in Table 5. It is inferred that the proposed method provides comparatively closer estimations for the magnitude of harmonic components of simulated data with respect to the corresponding input values model parameters. Simulation case studies proved the robustness and accuracy of the proposed algorithm for interpreting the spindle data, despite variations in the magnitude of synchronous and asynchronous components. Table 5 Comparative estimation results for DFT and the proposed method for the simulation-case-3 Estimated values (µm) Harmonic number (h) Input values DFT Least square method Experimental results Proposed least square method is applied to the measured spindle data obtained at the spindle speed of 25,000 rpm. In the present wor, spindle data containing 20 spindle

15 270 S. Denis Asho and G.L. Samuel revolutions are considered for analysis and synchronised to spindle speed using re-sampling method. A harmonic cutoff value of 30 cycles per revolution is used for estimating the synchronous components of spindle data. Harmonic estimation results of the proposed method are compared with DFT-based filtering method. 6.1 Frequency domain analysis of measured spindle data DFT method is applied for frequency domain analysis of the measured spindle data. Figure 7 shows the single sided frequency spectrum for the measured spindle data. Fundamental harmonic component shows a significant spectral leaage to the adjacent frequency bins. Figure 7 Frequency domain analysis of measured spindle data obtained at speed of 25,000 rpm Harmonic components at the integer multiples of fundamental frequency were filtered in the frequency domain as shown in Figure 8. It shows the frequency spectrum after filtering the synchronous components at the integer multiples of fundamental frequency of measured data. The effect of spectral leaage can be inferred on the magnitude of asynchronous components. This is will lead to inaccurate estimation for the asynchronous components and the evaluation of spindle radial error motion. Figure 8 Filtering of synchronous components in frequency domain

16 Least square curve fitting technique for processing time Comparison of filtering and fitting method Figure 9 shows the fitted curve using the proposed least square fitting method along with the filtered synchronous components obtained using DFT for the samples of spindle data. Figure 9 Least square fitting results for the measured spindle data obtained at speed of 25,000 rpm It can be observed that the fitted curve using the least square method more closely follows the magnitude and phase of the periodic pattern of the spindle data. Correlation coefficient of the fitted curve to the spindle data is found to be However, filtered synchronous components using DFT method show the deviations in the magnitude and phase of the spindle data. Correlation coefficient of the filtered synchronous components to the spindle data is found to be and it is much lesser than the proposed least square method. These results prove the improved performance and suitability of proposed method for analysing time sampled spindle data. Table 6 provides the estimated values for the magnitude of first 10 harmonic components of spindle data using the proposed method and DFT method. Table 6 Comparison of harmonic amplitude estimation for the measured spindle data obtained at 25,000 rpm Estimated values of harmonic amplitudes (µm) Number of harmonics DFT Least square method

17 272 S. Denis Asho and G.L. Samuel It is observed that the fundamental harmonic component estimated by DFT is lesser than the proposed least square method. It is due to the spectral leaage effect to the adjacent frequency bins as shown in Figure 7(a). Magnitudes of other harmonic components are also found to be lesser as compared to the estimated values obtained by the proposed least square method. It is also found mean value of the spindle data (C) as µm and the model coefficients for the thermal drift of the spindle are estimated to be p 2 = µm/sec 2 and p 1 = µm/sec. 6.3 Separated components of the measured spindle data As explained in section 4.2.5, estimated values of the model parameters are used for separating the average position of master cylinder, thermal drift, synchronous and asynchronous components of the spindle data. Sample values of the spindle data and the separated components are summarised in Table 7. Table 7 Samples of synchronised spindle data and the estimated components using the proposed method Separated components of spindle data using the proposed method (µm) S. No. Synchronous Asynchronous Thermal drift Centring error components components It is noticed that the magnitude of centring error is dominant than the synchronous and asynchronous components of the spindle data. Separated components such as centring error, synchronous and asynchronous components of the spindle data are represented in polar plot as shown in Figure 10. Spindle data in the base circle radius of 25 µm is illustrated in Figure 10(a). It can be seen that after removal of centring error components, the polar profile of synchronous components are centred to polar chart centre as shown in Figure 10(b). However, synchronous components of the spindle data requires further analysis for the form error separation of master cylinder. Figure 10(c) shows the asynchronous components of the spindle data and it shows the non repeatable pattern due to the structural motion of the miniaturised machine tool.

18 Least square curve fitting technique for processing time 273 Figure 10 Results of proposed fitting method for the measured spindle data obtained at speed of 25,000 rpm: (a) measured data; (b) synchronous components and (c) asynchronous components (a) (b) (c) 6.4 Validation of proposed analysis method Asynchronous components are graphically analysed using histogram for verifying the normality assumption and it is shown in Figure 11. The shape of the histogram plot resembles the bell curve and this confirms the normality assumption of the asynchronous components. It is also found that count values are less for the tails of histogram. These results prove the suitability of proposed model and least square fitting method for analysing the high speed spindle data. Figure 11 Histogram plot for asynchronous components of the measured spindle data

19 274 S. Denis Asho and G.L. Samuel 7 Conclusions In this paper, a new processing method is proposed for synchronising and analysing the high speed spindle data. A re-sampling method is followed for overcoming the effect of speed variations using the best fit sine wave frequency of the spindle data. To reduce the number of preprocessing steps, a least square curve fitting method is proposed for simultaneous separation of synchronous, asynchronous and thermal drift components of time sampled spindle data using a mathematical model consisting of second-order polynomial and a sum of sinusoidal functions. Performance of the proposed analysis method is verified using the simulated data containing different magnitudes of synchronous and asynchronous components of the spindle data. Simulations results confirm the robustness and accuracy of the proposed method in identifying the components of the spindle data than the DFT-based filtering method. Fitted curve using the proposed method provides a better correlation coefficient of and this confirms the suitability of the proposed model for analysing the spindle data. Normal probability assumption of asynchronous components is verified using histogram plot. Proposed method overcomes the spectral leaage problem of DFT-based filtering method and it is suitable for analysing the time sampled spindle data containing an incomplete number of spindle revolutions. This method can be implemented for online identification and processing of the spindle data at high speed conditions. Acnowledgement The authors than Prof. M.S. Shunmugam, Head of the Department of Mechanical Engineering for providing mechanical micro machining setup to conduct the experiments. References ANSI/ASME B89.3.4M Standard (2004) Axes of Rotation, Methods for Specifying and Testing. Asho, D.S. and Samuel, G.L. (2009) Least square estimation of harmonic components in radial error measurement of a miniaturized machine tool spindle, Proceeding of 24th Annual Meeting of the American Society for Precision Engineering (ASPE), 4 9 October, Montery, CA, USA, pp Fujimai, K. and Mitsui, K. (2007) Radial error measuring device based on auto collimation for miniature ultra high speed spindles, International Journal of Machine Tools & Manufacture, Vol. 47, No. 11, pp Gredja, R.D., Marsh, E.R. and Vallance, R.R. (2005) Techniques of calibrating spindles with nanometer error motion, Precision Engineering, Vol. 29, No. 1, pp Jywe, W.Y. and Chen, C.J. (2005) The development of a high speed spindle measurement system using laser diode and quadrant sensor, International Journal of Machine Tools & Manufacture, Vol. 45, No. 11, pp Lee, J.H., Lee, E.S. and Yang, S.H. (2005) Assessment of radial errors of high speed spindle in a miniaturized machine tool, International Conference of Leading Edge Manufacturing in 21st Century, October, Nagoya, Japan, pp Leffler, K.E. and Jay, D.A. (2009) Enhancing tidal harmonic analysis: Robust (hybrid L 1 /L 2 ) solutions, Continental Shelf Research, Vol. 29, No. 1, pp Marsh, E.R. (2008) Precision Spindle Metrology, DEStech Publications, Inc, Lancaster, PA 17602, USA.

20 Least square curve fitting technique for processing time 275 Masuzawa, T. (2000) State of the art of micromachining, Annals of the CIRP, Vol. 49, No. 2, pp Murthy, T.S.R. (2000) A method for evaluation of spindle running accuracy of high precision spindles by best fit sine wave. Manufacturing Technology Proceeding of 19th AIMTDR, IIT Madras, Chennai, pp Smith, P.T., Vallance, R.R. and Marsh, E.R. (2005) Correcting capacitive displacement measurements in metrology applications with cylindrical artifacts, Precision Engineering, Vol. 29, No. 3, pp List of abbreviations Description Magnitude of harmonic components of the spindle data Mean value of the spindle data Basis matrix of the model Sum of squares of residuals in ordinary least square method Sum of squares of weighted residuals Desired harmonic cutoff in cycle per revolution Set of linear parameters of the proposed model Estimated values of linear parameters of the proposed model Amplitude of cosine and sine components of proposed model (µm) Residual of the spindle data Sampling frequency Abbreviations C h C D E E w H X X w a h, b h e f s Fundamental frequency f 0 Spindle rotational frequency Re-sampling frequency Spindle rotational frequency at the given sampling time t Length of the sampled data Estimated values of radial error data using the model Samples of radial error data m i Re-sampled spindle data m Estimated values for the thermal drift of spindle Estimated value for the synchronous components of the spindle data Estimated value for residual synchronous components of the spindle data Length of the re-sampled data Coefficients for representing the thermal drift p 1, p 2 Sampling time Re-sampling time Maximum value of sampling time Phase of harmonic components of the spindle data Constant increment for obtaining discrete frequency values Re-sampling time interval f r f rs f r m m i m t m h m ri n t i t t m γ h f t r

21 276 S. Denis Asho and G.L. Samuel Weighted residual value Sum of squares of the residual Set of weights in the iteration Set of final weights Derivative of sum squared residual error function Estimated asynchronous components of the spindle data Discrete Fourier Transform Cycles per revolution є ρ ω ω f ψ τ ˆ DFT CPR