Precision Instrumentation for Rolling Element Bearing Characterization. Eric R. Marsh, Vincent C. Vigliano, Jeffrey R. Weiss, Alex W.

Precision Instrumentation for Rolling Element Bearing Characterization Eric R. Marsh, Vincent C. Vigliano, Jeffrey R. Weiss, Alex W. Moerlein Machine Dynamics Research Laboratory The Pennsylvania State University 331 Reber Building University Park, PA 16802, USA R. Ryan Vallance Precision Systems Laboratory The George Washington University 738 Phillips Hall 801 22nd St., N.W. Washington, DC, 20052, USA Abstract This article describes an instrument to measure the error motion of rolling element bearings. This challenge is met by simultaneously satisfying four requirements. First, an axial preload must be applied to seat the rolling elements in the bearing races. Second, one of the races must spin under the influence of an applied torque. Third, rotation of the remaining race must be prevented in a way that leaves the radial, axial/face, and tilt displacements free to move. Finally, the bearing must be fixtured and measured without introducing off-axis loading or other distorting influences. In the design presented here, an air bearing reference spindle with error motion of less than 10 nm rotates the inner race of the bearing under test. Non-influencing couplings are used to prevent rotation of the bearing outer race and apply an axial preload without distorting the bearing or influencing the measurement. Capacitive displacement sensors with 2 nm resolution target the nonrotating outer race. The error motion measurement repeatability is shown to be less than 25 nm. The paper closes with a discussion of how the instrument may be used to gather data with sufficient resolution to accurately estimate the contact angle of deep groove ball bearings. 1

1 Introduction Rolling element bearing performance improves over time as a result of ongoing metallurgical, tribological and manufacturing research and development. However, one of the remaining challenges in rolling element bearing applications is measurement of rotational accuracy. The instrument described in this article measures and characterizes the error motion of bearings intended for precision applications. Error motion is movement in the five rigid body degrees of freedom other than pure rotation. These five components of error motion are described with radial, tilt (angular), and axial measurements. However, most applications use at least two bearings to provide tilt stiffness. Accordingly, the radial and axial error motion components shown in Figure 1 are most relevant. Measurements made on the bearing under test reflect the combined contributions of errors in the bearing and the errors of the reference spindle. Figure 2 shows the decomposition of the measurement [4] separating these errors. The left-hand column of polar plots shows the synchronous and asynchronous components of a single radial measurement [15, 23]. The synchronous component is the repeatable path calculated by averaging several consecutive revolutions. Asynchronous error motion is the remaining error representing the revolution-byrevolution deviation from the synchronous component. The synchronous and asynchronous components have an intuitive interpretation in the frequency domain. Given some integer number of revolutions of data (greater than one), the synchronous component is contained in the frequency bins associated with integer multiples of the number of revolutions (or integer values of cycles per revolution or cpr). The asynchronous component appears in the remaining frequency bins, as illustrated in Figure 2. Rolling element bearings generally have proportionally greater asynchronous error motion because of the non-integer relationship between input shaft speed and rolling element rotation. For example, the cage rotation of a 6204 ball bearing is 38.5% of the inner race rotation. An error separation method may be applied to distinguish the synchronous error motion of the ball bearing under test from the reference air bearing spindle error. However, the synchronous error of the reference spindle is small compared to most rolling element bearings and is ignored in the results presented here. This is also true of the asynchronous component, 2

which is about 1 nm for the reference spindle while typically 100 nm for rolling element bearings. Asynchronous error motion presents an additional challenge to the metrologist in that not all of the apparent motion between the displacement sensor and target should be assigned to the bearing under test because of test stand vibration, environmental influences, and other instrumentation or sensor noise. One of the biggest challenges in accurately measuring ball bearings is minimizing the contribution of external effects so that a reliable reading of asynchronous error is achieved. The instrument presented here benefits from previous work in the techniques used to accurately quantify the performance of axes of rotation [10, 14, 16]. Precision engineering pioneers such as Tlusty and Donaldson inspired four decades of work to reduce axis of rotation measurement uncertainty through clever hardware and analytical developments [5, 6, 7, 22]. Sensors and data acquisition systems also advanced to the point where hardware design is often the largest remaining contributor to measurement uncertainty [8, 20, 18]. 2 Background Rolling element bearing measurement techniques may be broadly categorized by whether data are collected in situ or off-line on a dedicated instrument. In situ approaches have received the most interest for condition-based monitoring of mission-critical components. However, the interpretation of measurement results is complicated by the dynamic interaction of the bearing with its surrounding structure and environment [9, 11, 13, 21]. Off-line testing is the best way to reliably isolate a bearing from environmental and structural influences [10]. Properly designed instrumentation also offers a practical means of providing data for bearing model development. The last 30 years have seen several significant improvements in off-line bearing testing equipment and practice. One of the earliest commercially available instruments is the PDI Anderometer R, which characterizes bearing performance with three frequency ranges in units dubbed Anderons (1 Anderon is 1 micro-inch per radian of rotation). The low measurement range is from 1.67 to 10 times the shaft speed, medium is 10 to 60 times the shaft speed, and the high range is 60 to 300 times the shaft speed. A bearing is quantified in each of the three frequency ranges with a single number. 3

In unpublished work, Professional Instruments Co. began designing custom instrumentation for measuring bearing error motion in the early 1980 s in an effort led by Gene Dahl. The significant contribution of their work was the integration of high resolution capacitive displacement sensors with an ultra-precision air bearing spindle to enable accurate measurements with several khz bandwidth. This approach represented a significant departure from the Anderometer in both design and implementation; the PICo instruments allowed data collection over many revolutions and at thousands of data points per revolution, as triggered by optical encoders. Bouchard, Lau, and Talke measured asynchronous radial errors for both ball and fluid bearing spindles in the time and frequency domains using spindles mounted on a vibration isolation table [5]. They also used a capacitive displacement sensor to measure error motion. The asynchronous component was calculated by eliminating the repeatable signal from the timevarying displacement signal between the stationary probe and the rotating spindle. Statistical methods were used to calculate the asynchronous component in both the time and frequency domains. McFadden and Smith developed an experimental test rig to measure the vibration produced by a defect on the inner race of a ball bearing under a constant radial load [17]. Their instrument used an accelerometer to measure vibrations that were later correlated to the shaft rotation frequency. Noguchi et al. returned to the reference spindle-type instrument layout to measure the radial motion of the outer race of a bearing with the inner race rotated by an aerostatic spindle [20]. Load cells measured the axial load and torque while two orthogonal displacement sensors measured the radial error; a rotary encoder triggered data sampling. With the benefit of modern computing and data acquisition hardware, it is now practical to implement the PICo/Noguchi-style instrument with higher usable bandwidth and accuracy. Such a design improves resolution in both the time and frequency domains along with precisely controllable operating conditions to explore the effects of geometric and manufacturing problems such as misalignment and out-of-round components. Furthermore, issues such as bearing load, wear, and lubrication can be exhaustively studied in a controlled manner. 4

3 Bearing Analyzer The bearing analyzer instrument is shown in Figure 3 and Figure 4. Components chosen for the instrument allow testing at axial loads up to 100 N and speeds up to 10 krpm. An ultraprecision air bearing spindle (Professional Instruments BLOCK-HEAD spindle with Heidenhain ERO 1221 rotary encoder) spins the inner race of the bearing under test. The radial error motion of this spindle is less than 10 nm and the axial error motion is less than 5 nm. In most cases, it is reasonable to neglect the small contribution of this error to the bearing measurement. However, it is possible to accurately remove the reference spindle s contribution using an error separation technique such as Donaldson reversal [7]. Capacitive displacement sensors with 10 µm range and 2 nm resolution (Lion Precision C1- C capacitance probe with a DMT10 driver) measure the relative motion between the reference spindle s stator and the outer race. Capacitive sensors require that the target electrode be grounded. In this instrument configuration, the target electrode is the steel bearing retaining cup that does not rotate and is readily grounded. The inner race of the bearing under test is secured to the reference spindle on a lapped spherical carbide pilot sized with a light drive interference (FN) fit. The outer bearing race has a locational interference (LN) fit within the steel bearing retaining cup. The retaining cup also has a torque arm that prevents rotation of the outer race while enabling measurement of the torque. The torque arm prevents rotation but the remaining five degrees of freedom are unconstrained and free to move. This decoupling is achieved with a steel pin with lapped, carbide hemispherical ends that fit into lapped conical features. The axial preload is applied with a similar pin to minimize the transmission of off-axis loads to the bearing. The result is axial load and rotation constraint with the five remaining degrees of freedom almost completely unconstrained and uninfluenced by the test apparatus. A load cell is placed in line with the torsional restraint to allow torque measurement with repeatability of 1 mn-m using a piezoelectric sensor (Kistler 9303). Data acquisition and analysis is carried out using a National Instruments DAQ board (PCI- 6110E) and software written in LabWindows/CVI. Analog anti-alias filtering is done prior to digitization with a Krohn-Hite 3360 tunable filter. All error motion computations are made 5

in accordance with the procedures set by ANSI/AFBMA Standard 13-1987, Rolling Bearing Vibration and Noise (Methods of Measuring) and ANSI/ASME B89.3.4M Axes of Rotation, Methods for Specifying and Testing [3, 4]. The software provides integrated motor control, thermal drift compensation, and data acquisition of the displacement and force sensors synchronized and triggered by the optical rotary encoder. 4 Results Sample measurement results are shown in Figure 5 from single-row, deep groove ABEC 3 6204 bearings. Table 1 lists the standardized 6204 bearing dimensions. Rolling element bearings exhibit significant asynchronous error motion because of the planetary-type kinematics of the inner race, outer race, and rotating balls that lead to noninteger relationships between shaft speed and the key rotational frequencies. Consecutive rotations will not yield identical error motion results even in the absence of measurement errors. However, a series of tests show similar overall characteristics, especially when averaged over multiple revolutions for the purpose of computing the error motion values specified in the ASME/ANSI B89.3.4M standard. The number of revolutions included in a test is chosen by the metrologist [4]. Experience shows that the shape of the synchronous error motion takes quite a few revolutions to emerge, especially for frequency components near, but not precisely equal to, integer multiples of the input (e.g., inner bearing race) speed. In general, averaging additional revolutions of data reduces the synchronous error motion and increases the asynchronous. In the results that follow, 512 revolutions were used in all computations. This number of revolutions was found to give stable results while providing sufficient frequency domain resolution within a reasonable amount of test time. 4.1 Measurement repeatability Table 2 shows data from consecutive measurements taken on a 6204 ball bearing at 1000 RPM for 512 revolutions under 100 N of axial preload. The tabulated motion and torque values have a standard deviation of approximately 15 to 20% of the mean. In the case of the 6204 bearing, 6

the repeatability is better than 100 nm for the overall magnitude of the radial error motion, and torque repeatability is better than 1 mn-m. The most significant (i.e., largest) frequency component in the data tabulated in Table 2 occurs at the cage rotation frequency at 38.5% of the shaft rotation frequency. This component is below the range included in the three frequency bands of the Anderometer. For this reason, the low-medium-high band results, which start at 1.67 times the shaft rotation speed, do not share the same general magnitude as the asynchronous component, which includes all noninteger multiples of the shaft rotation frequency down to zero Hz. The instrument repeatability is also apparent in the frequency domain. Figure 6 shows four FFT plots of the radial error motion calculated from 512 revolutions of data sampled 1024 times per revolution. As before, the results show little variation in consecutive tests. Furthermore, the frequency components associated with particular geometric defects of the ball bearing occur at identical frequencies and similar amplitudes for all tests, as summarized in Table 3. The high consistency in the frequency domain data suggests that the bearing frequency components combine in somewhat different patterns in the time domain because of the non-integer frequencies at which they occur. As a result, the FFTs are very similar while the polar plots show more variation. 4.2 Axial preload testing Testing was carried out to investigate the instrument performance with various axial loads applied to the bearing. Figure 7 shows the synchronous and asynchronous error motion values of a 6204 bearing at axial preload increasing in steps of 10 N. As before, these error motion values are computed from 512 revolutions of data taken at 1000 rpm. Axial loads above 30 N result in consistent values of synchronous error motion in the bearing. The asynchronous component also reaches a roughly constant value above 30 N until it jumps to a second plateau above 80 N. Table 4 shows additional information from the same testing. It is important that the axial load be sufficient to hold the bearing races in proper contact with the rotating rolling elements. The table shows an abrupt transition in error motion and bearing torque between 30 and 40 N. The bearing balls and races are loaded sufficiently to achieve proper contact above 7

30 N for the 6204 deep groove bearing. 5 Discussion The previous section demonstrates the repeatability and resolution of the proposed bearing error motion instrument. This section introduces an application of the instrument to a problem of interest to end users of rolling element bearings as well as the bearing modeling community. An interesting facet of ball bearing analysis is the precise determination of the contact angle between the balls and the inner and outer races. The contact angle tends to vary somewhat with axial load because of the Hertzian contact deformation between the races and the rolling elements. In practice it is difficult to infer the contact angle from error motion data because of finite resolution in the frequency domain. The use of a single measured bearing defect frequency, such as the cage rotation frequency, can be less accurate than simply using the nominal contact angle calculated from the standardized bearing geometry. This is because bearing defect frequencies are not particularly sensitive to contact angle. Despite this challenge, the computation of the effective contact angle is critical in bearing modeling and troubleshooting applications. First, consider the measured cage rotation frequency as a means of estimating the effective contact angle of a ball bearing. The cage rotation frequency f cage is typically slightly less than half the inner race (shaft) rotational frequency f s and is a function of ball diameter b, pitch diameter p, and the contact angle α. f cage = f s 2 (1 bp cos α ) (1) Linearizing the cage rotation frequency with respect to the nominal value of the contact angle α 0 leads to an equation for the sensitivity of contact angle to small changes in cage frequency. or f cage = ( fs 2 ) b p sinα 0 α (2) ( ) 2p fcage α = (3) b sinα 0 f s 8

The sensitivity of the contact angle to the measured cage rotation frequency is limited by the number of revolutions captured in the data sample. For example, 512 revolutions of data yield a frequency resolution of 1/512 of the shaft speed f s. For the 6204 bearing, with a nominal contact angle of 15 degrees, the computed correction in contact angle is nearly two degrees over half the width of one frequency bin. This is insufficient for accurate estimation of the actual bearing contact angle. Therefore, we use several spectral peaks to improve the accuracy of the calculated contact angle. The bearing instrument is well suited for this because the measured spectral peaks may be relied upon to represent bearing harmonics and not other environmental influences. Furthermore, the peaks are sharp and easily identified because the data sampling is triggered by an optical encoder measuring shaft orientation angle, rather than relying on constant-speed rotation during testing. A number of spectral peaks appear in the FFT of the error motion and may be matched to integer combinations of the basic bearing defect frequencies. In addition to the cage rotation frequency, there is the cage rotation relative to the inner race f ci that is typically a little more than half the shaft rotation frequency. f ci = f s 2 (1 + bp cos α ) (4) The ball pass frequencies of the inner and outer races are computed from f c age and f ci using the number of rolling elements n. f bpo = nf cage (5) f bpi = nf ci (6) The rotational frequency of the rolling elements is given by ( f r = f ( ) ) 2 s p b 1 2 b p cos α (7) 9

Finally, a ball defect will appear at twice its rotational frequency f r because the defect will strike the inner and outer race once per revolution. f bp = 2f r (8) These frequencies may be calculated using standardized bearing geometry data with typical error of 0.2% or less, as demonstrated here for a 6204 bearing. To use a set of experimentallymeasured defect frequencies to improve our estimates of the bearing geometry, the defect frequency equations are linearized in terms of small deviations of ball diameter b, pitch diameter p and contact angle α. This yields a set of equations relating the predicted defect frequencies f (using the nominal geometry values b 0, p 0, and α 0, plus a small correction ˆf, to the measured defect frequencies f. f f + ˆf (9) where f = f cage f ci f r f bpo f bpi = f s 2 1 1 + b0 cos α0 p 0 b0 cos α0 p 0 p 0 b 0 b0 cos2 α 0 p 0 n n + b0n cos α0 p 0 b0 cos α0n p 0 (10) ˆf = ˆf cage ˆf ci ˆf r ˆf bpo ˆf bpi =. f i b.. f i p.. f i α. b p α = A x 10

= f s b 0 cos α 0 2p 0 1 b 0 1 p 0 tan α 0 1 b 0 1 p 0 tan α 0 cos α0 b 0 p2 0 cos α 0 b 3 0 cos α0 p 0 + p0 b 2 0 cos 2sin α 0 α0 n b 0 n p 0 ntan α 0 n b 0 n p 0 ntan α 0 b p α (11) These linearized equations may be used to compute the small deviation of the ball and pitch diameters along with the contact angle by matching the predicted and measured defect frequencies appearing in experimentally-gathered FFT spectra. Table 5 shows the measured and predicted error frequencies for a sample 6204 ball bearing measured under a 100 N axial load. The frequencies are listed as multiples of the shaft rotation speed f s. ˆf = f f = TA x (12) The matrix A is 5 3 and relates the frequency corrections to the geometry corrections. The matrix T is of dimension l 5 and contains the integer values relating the 5 bearing defect frequencies to the l spectral peaks identified in the experimental data and listed in the second column of Table 5. Sixteen spectral peaks were used to improve the estimate of bearing geometry under load. Matching 16 peaks allows better correction of the contact angle as well as the effective ball and pitch diameters, both of which are affected by the Hertzian-type contact of the bearing. T T = 1 2 3 4 0 0 1 1 1 0 1 1 0 1 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 2 2 3 0 3 4 4 4 5 5 0 0 0 0 0 1 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (13) The resulting system of equations is overconstrained and ill-conditioned, necessitating a singular value decomposition (SVD) solution. There are 16 equations and just three unknowns (the pitch diameter correction p, ball diameter correction d, and the contact angle correction 11

α). In this example, the singular values are 3.264, 0.491, and 0. The SVD solution eliminates the singularity and solves the problem with a numerically stable algorithm. The correction values are b = 4 µm, p = 1 µm, and α = 0.8 degrees for an effective contact angle of 14.2 degrees. 6 Conclusion This paper presents an instrument designed to measure error motions of precision rolling element bearings at various loads and speeds. The results are repeatable to the 100 nm level and may be used to identify bearing characteristics without the influence of the dynamics of a larger, compliant environment. Sample tests under varying axial load show that the error motion results are stable once sufficient load is applied to properly seat the rolling elements in the raceways. Additional testing of the effective contact angle shows that a significantly improved estimate of bearing geometry is obtained by comparing the predicted bearing fault frequencies with the measured values. This is important in contact angle calculations because without multiple spectral peaks to compare, the contact angle will be inaccurate. 12

References [1] Aini, R., Rahnejat, H., and Gohar, R. International Journal of Machine Tools & Manufacture 30, 1 (1990), 1 18. [2] Aini, R., Rahnejat, H., and Gohar, R. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 209, 2 (1995), 107 114. [3] ANSI/AFBMA. Standard 13-1987, rolling bearing vibration and noise, methods of measuring. Anti-Friction Bearing Manufacturers Association (1987). [4] ANSI/ASME. Standard B89.3.4M, axes of rotation; methods for specifying and testing. American Society of Mechanical Engineers (1985). [5] Bouchard, G., Lau, L., and Talke, F. E. IEEE Transactions on Magnetics MAG-23, 5 (1987), 3687 3689. [6] Deeyiengyang, S., and Ono, K. Journal of Information Storage and Processing Systems 3, 1-2 (2001), 89 99. [7] Donaldson, R. R. Annals of the CIRP 21, 1 (1972), 125 126. [8] Grejda, R. D., Marsh, E. R., and Vallance, R. R. Precision Engineering 29, 1 (2005), 113 123. [9] Gupta, P. Advanced dynamics of rolling elements. Springer-Verlag, New York, 1984. [10] Harris, T. A. Rolling bearing analysis, 4 ed. John Wiley and Sons, NY, 2001. [11] Houpert, L. Journal of Tribology 119 (1997), 851 858. [12] Jones, A. American Society of Mechanical Engineers Transactions Journal of Basic Engineering Series D 82, 2 (1960), 309 320. [13] Kahn, H. Evaluation Engineering 14, 5 (1975), 29 31. [14] Lim, T., and Singh, R. Journal of Sound and Vibration 139, 2 (1990), 201 225. 13

[15] Marsh, E., Couey, J., and Vallance, R. Journal of Manufacturing Science and Engineering, Transactions of the ASME 128, 1 (2006), 180 187. [16] Martin, D. L., Tabenkin, A., and Parsons, F. International Journal of Machine Tool Manufacturers 35, 2 (1995), 187 193. [17] McFadden, P. D., and Smith, J. D. Journal of Sound and Vibration 96, 1 (1984), 69 82. [18] Noguchi, S., Hiruma, K., Kawa, H., and Kanada, T. Precision Engineering 29, 1 (2005), 11 18. [19] Noguchi, S., and Ono, K. Precision Engineering 28, 4 (2004), 409 418. [20] Noguchi, S., Tanaka, K., and Ono, K. IEEE Transactions on Magnetics 35, 2 (1999), 845 50. [21] Tandon, N., and Choudhury, A. Journal of Sound and Vibration 205, 3 (1997), 275 292. [22] Tlusty, J. Microtecnic 13, 4 (1959), 162 178. [23] Vigliano, V. C. Computer control for precision bearing analysis. Master s thesis, The Pennsylvania State University, 2001. [24] Walford, T. L. H., and Stone, B. J. Proceedings of the Institution of Mechanical Engineers, Part C: Mechanical Engineering Science 197 (1983), 225 232. 14

Bore diameter 20 mm Outer diameter 47 mm Ball diameter, b 7.9375 mm Pitch diameter, p 33.5 mm Ball count, n 8 Contact angle, α 15 Table 1: 6204 bearing parameters 15

Error motion (nm) RMS error motion (nm) RMS torque (mn-m) Synch Asynch Asynch Low Med High Low Med High Test P-V P-V 4σ 1.7-10 10-60 60-300 1.7-10 10-60 60-300 1 31 556 292 36 15 2 1.2 1.9 0.2 2 35 514 283 46 16 3 1.5 2.4 0.2 3 26 718 440 43 15 3 2.1 2.2 0.2 4 31 630 382 50 19 3 1.7 3.0 0.2 5 35 588 316 47 17 3 1.7 2.9 0.2 mean 32 601 343 45 16 3 1.7 2.5 0.2 σ 4 78 67 5 2 0 0.3 0.5 0.0 Table 2: Repeatability results for radial error motion and torque. The low, medium, and high ranges are multiples of the input shaft speed, as reported in Anderometer-style measurements. 16

Error motion amplitude at bearing defect frequencies (nm) f c 2f c f bp f bpo f bpi Test (0.385f s) (0.771f s) (2.000f s) (3.084f s) (4.916f s) 1 58 27 3 10 1 2 56 28 4 9 1 3 56 28 3 9 1 4 59 29 3 9 1 mean 57 28 3 9 1 σ 1 1 0 0 0 Table 3: Frequency and amplitude of bearing defect frequencies in consecutive measurements. 17

Total error motion (nm) RMS error motion (nm) RMS torque (mn-m) Synch Asynch Asynch Low Med High Low Med High Test P-V P-V 4σ 1.7-10 10-60 60-300 1.7-10 10-60 60-300 10 31 532 283 46 17 3 1.61 2.96 0.23 20 35 536 311 43 15 2 1.50 2.08 0.19 30 30 527 283 48 17 3 1.67 2.96 0.23 40 16 149 106 18 4 3 0.73 0.12 0.22 50 16 145 93 17 4 3 0.75 0.12 0.23 60 16 152 99 18 5 3 0.71 0.15 0.24 70 15 155 97 18 5 3 0.72 0.17 0.22 80 15 220 124 17 5 3 0.70 0.18 0.38 90 15 244 126 16 6 3 0.73 0.21 0.25 100 15 224 121 16 6 3 0.75 0.23 0.21 Table 4: Radial error motion and bearing torque with varying axial preload. 18

Error Frequency Measured Predicted Corrected Predicted Corrected amplitude (nm) component frequency frequency frequency error (%) error (%) 58.5 f cage 0.385 0.386 0.385-0.2-0.1 26.8 2f cage 0.770 0.771 0.770-0.2-0.1 13.7 5f r 3f cage 8.838 8.842 8.843 0.0-0.1 9.6 f bpo 3.082 3.085 3.082-0.1 0.0 9.3 2f r f cage 3.615 3.614 3.614 0.0 0.0 9.2 2f r + f cage 4.387 4.385 4.385 0.0 0.0 6.2 4f r + f cage 8.387 8.384 8.384 0.0 0.0 4.0 4f r 8.000 7.999 7.999 0.0 0.0 3.8 3f r + f cage 6.385 6.385 6.385 0.0 0.0 3.0 4f cage 1.541 1.542 1.541-0.1 0.0 2.8 f r 2.000 2.000 2.000 0.0 0.0 2.7 2f bpo 6.162 6.169 6.163-0.1 0.0 2.4 3f r f cage 5.615 5.614 5.614 0.0 0.0 2.4 5f r f cage 9.617 9.613 9.614 0.0 0.0 2.2 4f r f cage 7.617 7.613 7.614 0.1 0.0 2.1 3f cage 1.156 1.157 1.156 0.0 0.1 Table 5: 6204 bearing fault frequencies normalized by inner race speed. 19

Captions Figure 1. Axial (a) and radial (b) error motion associated with a rolling element bearing. The proposed instrument can also be used to measure the remaining rigid body motion (tilt) when appropriate. Figure 2. Separation of measurement data into synchronous and asynchronous components of rolling element bearing error motion and artifact form error (cpr is cycles per revolution; a two-lobe error will appear in the cpr = 2 frequency bin). Note that the 1 cpr bin (eccentricity) is always removed from radial measurements. Figure 3. Bearing analyzer with motorized air bearing reference spindle. Figure 4. Cross sectional view of the bearing error motion instrument. Figure 5. Sample polar plot and FFT results from a 6204 bearing. The synchronous error is quite small (33 nm) but the asynchronous error motion has a total excursion of 289 nm during the 512 revolutions of the test. However, 95% of the error motion values fall within a 160 nm band as suggested by the nearly normal distribution. The FFT allows identification of individual frequency components. Figure 6. Repeatability results from four consecutive tests on a single 6204 bearing. Figure 7. Radial error motion of a 6204 bearing as a function of axial preload. 20

532 600 Synchronous error (nm) 229 40 150 32 16 15 0 0 10 20 30 40 50 60 70 80 90 100 Axial Load (N) Asynchronous error (nm)

Measurement locations Preload force Axial preload pin Bearing retainer cup Bearing Pilot Chuck adapter Spacer Spindle rotor A Torque arm Spindle stator Mounting flange Frameless motor Rotary encoder Section A-A Non-rotating components are hatched A

Axial load piston Bearing retaining cup with torque arm Radial error motion probe locations Bearing under test (inside cup) rot. Motorized air bearing master spindle with rotary encoder

Axial error motion Displacement indicator Radial error motion (a) (b) Non-rotating components are cross-hatched

0 1 2 cpr 4 5 6 0 1 2 cpr 4 5 6 75 f c 75 f c Radial error (nm) 2f c f bpo f bp 4-f c 4+f c Radial error (nm) 2f c f bpo f bp 4-f c 4+f c 0 1 2 cpr 4 5 6 0 1 2 cpr 4 5 6 75 f c 75 f c Radial error (nm) 2f c f bp f bpo 4-f c 4+f c Radial error (nm) 2f c f bpo f bp 4-f c 4+f c

nm 80 40 nm 80 40 0 2 4 6 8 cpr 0 2 4 6 8 cpr FFT FFT Error motion of ball bearing plus spindle: integer Fourier components Synchronous ball bearing error motion Raw probe data Initial processing Error separation (requires at least one additional measurement) Ball bearing Asynchronous: non-integer Fourier components Synchronous spindle error motion Reference spindle nm 80 FFT nm 80 FFT 40 40 0 2 4 6 8 cpr 0 2 4 6 8 cpr

125 nm f cage Synch 33 nm Asynch P-V 289 nm 4σ 160 nm -250 nm 250 2f cage f ballpass fouter 0 0 2 4 cpr 6 8 10