Reduction of NRRO in ball bearings for HDD spindle motors

Precision Engineering 28 (2004) 409–418

Reduction of NRRO in ball bearings for HDD spindle motors Shoji Noguchi a,∗ , Kyosuke Ono b,1 a b

Department of Mechanical Engineering, Faculty of Science and Engineering, Tokyo University of Science, 2641 Yamazaki, Noda, Chiba 278-8510, Japan Tokyo Institute of Technology, Graduate School of Science and Engineering, 2-12-1 O-okayama, Meguro-ku, Tokyo 152-8552, Japan Received 19 June 2003; received in revised form 24 December 2003; accepted 9 January 2004

Abstract In this paper, it is theoretically and experimentally analyzed that the non-repetitive run-out (NRRO) of a ball bearing is caused by geometrical errors of the inner and outer races and the balls, and the number of balls. The results are summarized as follows: (1) As for the geometrical errors of the inner and outer races, it is possible to reduce the NRRO to be less than 1 nm by choosing 12 or 18 number of balls, even if the inner and outer races have a harmonic undulation of less than the 11th order. (2) The NRRO of a ball bearing with 12 balls is very small even if the geometrical errors of ball bearing parts are large. It was confirmed that 12 was an optimal number of balls for ball bearings used for HDD spindle motors. © 2004 Elsevier Inc. All rights reserved. Keywords: NRRO; Ball bearing; Ball number; Spindle motor; HDD

1. Introduction An increase in the speed and precision of ball bearings for information technology equipment has been very strongly desired. Especially for ball bearings for HDD spindle motors, requirements for reducing non-repetitive run-out (NRRO) are high as recording density continues to increase. In the year 2000, requirements for NRRO was less than 0.05 ␮m [1].While reducing geometrical and dimensional errors of the inner and outer rings and the balls can reduce NRRO, the degree to which such improvements can be effective is limited by the costs associated with improving processing accuracy. We, therefore, found it necessary to devise design enhancements to further reduce NRRO. In this report, we present the results of our research into design changes for reducing the NRRO of ball bearings. Our research followed these basic steps: (1) We used a program to calculate the NRRO of ball bearings by themselves by taking error factors and the num∗ Corresponding

author. Tel.: +81-4-7122-9596; fax: +81-4-7123-9814. E-mail addresses: [email protected] (S. Noguchi), [email protected] (K. Ono). 1 Tel.: +81-3-5734-2171; fax: +81-3-5734-2892. 0141-6359/$ – see front matter © 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.precisioneng.2004.01.005

ber of balls as parameters. We clarified the influence of the error factors on the NRRO and determined design specifications that would reduce NRRO. (2) We made samples of the ball bearings with these specifications and then through testing, confirmed the validity of our theoretical analysis in reducing NRRO. 2. NRRO calculation program for single bearing First, we summarize the program we used to calculate the NRRO for single ball bearing. As the NRRO in the radial direction is important for HDDs, we considered the ball bearing in a two-dimensional plane model and calculated the shaft center behavior in the radial direction in response to the balance between the Hertzian contact forces of the inner and outer rings with the balls [2]. A model of the ball bearing for the NRRO analysis is shown in Fig. 1. The center of the outer race is set as O and O–XY is taken, and the coordinates of the center of the inner race O is set as x, y. As the geometrical error of the raceway surface and balls is minimum, it is assumed that all balls are in elastic contact with the inner and outer rings and that the balls are rolling without sliding. The position angle of each ball and autorotation angle is given by the following equation, where

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Fig. 1. Model of ball bearing for NRRO calculation.

Fig. 2. Evaluation method of NRRO (Lissajou’s figure).

the inner ring rotation angle is ωt, ball diameter is d, ball pitch circle diameter (PCD) is D, and the number of balls is Z. First ball position kth ball position Ball’s autorotation angle

Ωt = ωt 21 (1 − d/D) 2π(k − 1) θk = Ωt + Z ωb t = ωt 21 (D/d − d/D)

(1)

At first, we assumed that the deformation is the same for the inner and outer rings. When setting the ball diameter at the contact point of the kth ball as dk (ωb t) and the radius of the inner and outer rings as r(θ k ) and R(θ k ), respectively, if the shaft center runs out to x, y, the kth ball’s elastic displacement amount, δk , is given by the following equation. δk = 21 {r(θk ) + dk (ωb t) − R(θk ) + x cos θk + y sin θk }

(2)

r(θ k ), R(θ k ), and dk (ωb t) are given by the following equation when they are expressed in Fourier series, after the addition of the reference radius and respective geometrical error in the circumferential direction:
r(θk ) = r0 + N n=1 arn cos(nθk + φrn )
N R(θk ) = R0 + n=1 aRn cos(nθk + φRn ) (3)
dk (ωb t) = 2rbk0 + N n=1 abk0 {cos(nωb t + φbkn ) +cos(nωb t + π + φbkn )} The force acting on the balls and the inner and outer rings under elastic displacement amount, δk , is given by Eq. (4), with the constant C from the Hertz’s elastic contact theory: Pk = Cδ1.5 k

(4)

As the force acting on the balls, and inner and outer rings varies depending on the position of each ball, the shaft center moves to the position where the resultant force becomes zero. From the balance of internal force, the coordinates of the shaft center (x, y) is obtained by solving the following simultaneous equations: X direction

Z  k=1

Cδ1.5 k cos θk = 0

(5)

Y direction

Z  k=1

Cδ1.5 k sin θk = 0

(6)

As Eqs. (5) and (6) are non-linear equations, we seek the numerical solution of the unknown figures x and y using the Newton–Raphson method. By repeating the calculations while changing the inner ring rotation angle, the shaft center behavior corresponding to the rotation angle can be calculated. To evaluate NRRO, we adopted Lissajou’s figure to keep the common base with the NRRO evaluation of a ball bearing by itself described in Section 4 [3]. As shown in Fig. 2, considering as NRRO the line fluctuation width in the radial direction of the behavior of shaft center per turn is overwritten and we sought its maximum value and mean value.

3. Calculation results and considerations 3.1. Calculation conditions The bearing specifications necessary to calculate NRRO are the ball pitch circle diameter and the ball diameter. The program determines the radii of the outer and inner ring raceways by adding or subtracting the ball radius to ball PCD. The conditions for the calculations in this report are shown in Table 1. In the calculation results presented throughout this report, the conditions listed in Table 1 were applied, except for the error component given to the number of balls and parts of the bearing. Also, in the calculation results, values less than 1 nm are listed as 0 because the detection resolution power of the sensor in the actual NRRO measurement is 1 nm. Table 1 Calculation conditions Ball PCD (mm) Ball diameter (mm) Initial elastic displacement (␮m) Number of data partitions (per revolution) Number of calculation laps Convergence condition of Newton–Raphson method

9.7 1.0 3.0 256 30 10−4

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Table 2 Maximum NRRO value calculated by changing ball number and integral harmonic undulation of outer race (NRRO is expressed in ␮m, 0 means less than 1 nm) Ball number (Z)

Undulation (n) 2

3

4

5

6

7

8

9

10

7 8 9 10 11 12 13 14 15 16 17 18 19 20

0.007 0 0.001 0 0 0 0 0 0 0 0 0 0 0

0.129 0.006 0 0.006 0.001 0 0.001 0 0 0 0 0 0 0

0.130 0 0.130 0 0.006 0 0.006 0 0.001 0 0.001 0 0 0

0.007 0.006 0.130 0 0.129 0 0 0.006 0 0.006 0 0 0.001 0

2.001 0 0 0 0.129 0 0.130 0 0 0 0.006 0 0.006 0

0 1.999 0.001 0.006 0.006 0 0.130 0 0.130 0 0 0 0 0.006

2.003 0 2.005 0 0.001 0 0 0 0.129 0 0.131 0 0 0

0.007 1.999

0 0 2.006 0 2.003 0 0.001 0 0 0 0 0 0.131 0

3.2. Influence of the geometrical errors of the outer ring raceway Table 2 lists maximum NRRO values in the radial direction when a sine wave is given as the geometrical error of the outer raceway and assuming that the inner raceway is a perfect circle, the balls are perfectly round, and there is no size variation. We set the error amplitude an to 1 ␮m (2 ␮m when converted to out-of-roundness). This value is larger than the actual one in order to make a clearer correlation between the number of integral harmonic undulations (undulation number)

2.008 0 0 0.006 0.006 0 0 0.131 0 0.131 0

and the number of balls with NRRO. We calculated the number of balls, Z, from 7 to 20, and the undulation number, n, from 2 to 10 based on the actual raceway measurement results [4]. The analysis of the calculation results in Table 2 indicates that there is a regular correlation between the number of balls, Z, and integral harmonic undulation number, n, for the NRRO values. The NRRO values in Table 2 can be classified into the following five groups: I. For n = Z ± 1, the values are close to 2 ␮m (bold-italic figures).

Fig. 3. Relation between ball and undulation of the outer ring according to the rotation of the inner ring.

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II. For n = (Z ± 1)/2, the values are close to 0.13 ␮m (boldfaced figures). III. For n = (mZ ± 1)/3 (m is a natural number), the values are close to 0.006 ␮m (italic figures). IV. For n = (mZ ± 1)/4 (m is a natural number), the values are 0.001 ␮m (underlined figures). V. For Z = mn (m is natural number), the values are less than 1 ␮m. Among the relational expressions above, the well-known ones are I and IV, which reflect the relation between the number of balls and the integral harmonic undulation number with vibration in the radial and axial directions, respectively [5]. The other relations listed have not been known until now. Shaft center behavior with the number of balls and integral harmonic undulation of the outer race, which validates the relational expressions of I, II, and V is shown in Fig. 3. This model assumes that the outer ring is fixed and the inner ring is rotating. With relational expression I, each time a ball moves to the peak or valley of the undulation, the gap in the up/down or right/left direction varies and the force acting on the ball becomes asymmetric. Therefore, the inner ring moves to the place where the acting forces are balanced. When this movement cycle is synchronized with the rotation of the inner ring, NRRO is zero. However, inner ring movement cycle caused by the ball passage is the product of the revolution cycle of the balls times the number of balls. If the inner ring rotation is considered as the reference cycle, it becomes NRRO. In relational expression II also, as the number of undulations of the outer ring becomes half of the model of relational expression I, the qualitative explanation becomes the same with that for relational expression I. The cases of relational expressions III and IV are similar. However, using relational expression I as a reference, the maximum NRRO value is about 1/15 for II and about 1/315 for III. To reduce NRRO, it is important to reduce the error amplitude of the undulations number, which validates relational expression I but when the required accuracy to NRRO becomes several tens of nanometers, influence of relational expressions II or III becomes non-negligible. On the other hand, for relational expression V, as the forces generated by the ball passage through the peaks and valleys of the undulation in the raceway are internally balanced, the inner ring does not move in the radial direction. In actual ball bearings, as there is a contact angle, the balance of forces is adjusted by the displacement in the axial direction. Up to this point, we have discussed the situations where NRRO occurs. However, with 12 and 18 balls, NRRO is less than 1 nm when the number of integral harmonic undulations is up to 10 undulations as a single sine wave. With 12 and 18 balls, it is thought that NRRO was less than 1 nm because the relation between the number of balls and undulations does not meet the criteria of relations I–IV within the range up to 10 undulations; and in many cases, relation V became valid. Up to this point, we have considered the geometrical error of a single sine wave but in an actual raceway, there are

Fig. 4. Waviness of outer ring for NRRO simulation.

multiple geometrical error components of up to 10 undulations. Therefore, we have calculated NRRO for the profile composed of sine waves having diverse amplitudes and numbers of undulations. The raceway surface profile obtained by composition of a given amplitude and number of undulations of a sine wave is shown in Fig. 4. The amplitude is a half-amplitude value and except for seven undulations, the undulations are determined considering the harmonic analysis results of the actual out-of-roundness measurements of the outer ring. As seven undulations is the number of undulations that affects NRRO for a large number of balls, it is set to a value larger than the actual one. In the composed profile, the phase difference of each sine wave was set to 0. The relation between the number of balls and the maximum NRRO value is shown in Fig. 5. For the calculations similar to the case of Table 2, error components of the inner ring and balls were set to zero. As the seven undulations were given a large value, in the case of eight balls, the maximum NRRO value became prominent. For other cases, it appears generally that the greater the number of balls, the smaller the NRRO with one exception being that the NRRO of 11 balls is larger than that of 10. The reason for this is believed to be that there are many undulations being influenced by NRRO up to 10 undulations. Also, for composed geometrical error, with 12 and 18 balls, the maximum NRRO value is very small: 0.001 ␮m with 12 balls and less than 1 nm with 18 balls. Considering the above, it became clear that 12 and 18 balls were effective in reducing NRRO. For 7, 8, and 9 balls, the maximum NRRO value when the composed sine wave of Fig. 4 existed alone is

Fig. 5. Relation between the number of balls and the maximum NRRO value in Fig. 4.

S. Noguchi, K. Ono / Precision Engineering 28 (2004) 409–418 Table 3 Components of maximum NRRO value in Fig. 4 Undulation, n (␮m)

Ball number (Z) 7

8

9

2 (0.050) 3 (0.015) 4 (0.007) 5 (0.005) 6 (0.005) 7 (0.015) 8 (0.004) 9 (0.003) 10 (0.002)

0 0.0001 0 0 0.0100 0 0.0080 0 0

0 0 0 0 0 0.0301 0 0.0060 0

0 0 0 0.0001 0 0 0.0080 0 0.0040

Sum of components Calculated value

0.0181 0.0178

0.0361 0.0359

0.0121 0.0118

shown in Table 3. Here, we listed the values up to 0.1 nm to clarify the relation between NRRO in single and composed profiles. The maximum NRRO value in the composed profile is nearly equal to the sum of the maximum NRRO value of the sine waves making it clear that the influence of geometrical errors in NRRO is linear. In this section, we gave a geometrical error to the outer race but it was confirmed that when a geometrical error is given to the inner ring, the results are similar to those in Table 2 and Fig. 5.

413

the cage revolution component decreases uniformly. The reason for this is thought to be the imbalance of contact force in the circumferential direction generated by the size variations of balls being relieved by increasing the contact points (load sustaining points) with the inner and outer rings as the number of balls increases. However, the NRRO reduction curve inclination is sharp when the number of balls is low and more moderate when balls are numerous; convergence starts from around 15 balls. When the size of one ball is varied and as the number of balls increases, NRRO can be suppressed finally to about one-fifth of the given size variation. It was also confirmed that even if a negative size variation was applied to a ball, the calculation results would be the same as in Fig. 6. Next, we examine NRRO when more than one ball is made larger or smaller. It is thought that NRRO varies depending on ball disposition when there is a size variation between numerous balls. Taking into account ball disposition, we calculated the NRRO of a ball bearing with eight balls, which is a standard number for ball bearings for spindle motors. The calculation result on the maximum NRRO value when a positive size variation from 0.01 to 0.1 ␮m is applied to the balls is shown in Fig. 7. This disposition number indicates the position

3.3. Influence of differences in diameter of balls While it is understood that rotation accuracy is reduced by NRRO in the revolution cycle of the cage when there is size variation among the balls [5,6], quantitative examination has not been sufficient. We calculated NRRO when size variation is applied to the balls installed in one ball bearing taking into account ball disposition while supposing the inner and outer races were perfect circles and the balls were perfect spheres. The relation between the number of balls and NRRO when a ball is made larger than the others by 0.05 ␮m is shown in Fig. 6. The conditions of the calculations are the same as those for geometrical error of the outer race. It was observed that as the number of balls increases, the maximum NRRO value of

Fig. 6. Relation between number of balls and NRRO when one ball is larger by 0.05 ␮m.

Fig. 7. Relation between ball location and NRRO when some balls are larger than normal size.

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of the larger balls. Results indicated that NRRO is the largest when larger balls are placed next to each other. If larger balls are placed far from each other, NRRO decreases, and for even numbers of balls, when larger balls are positioned 180◦ opposite from each other, NRRO is less than 1 nm. It is believed that as the number of larger balls positioned consecutively increases, eccentricity of the inner ring under a balanced condition becomes larger, and it rotates in synchronization with the cage causing a larger NRRO. When balls of the same size variation are positioned 180◦ opposite to each other, distribution of the contact force becomes uneven. But as eccentricity of the inner ring does not occur, the situation becomes the same as in Fig. 3(c) and the center of the inner ring does not move. 3.4. Influence of the geometrical errors of the balls When ball size varies, the generation cycle of NRRO matches with ball revolution cycle but when there are geometrical errors, the NRRO generation cycle becomes much shorter and it is known that such cycle is a product of the ball rotation cycle and the number of undulations contained in the geometrical error [5]. Supposing that the inner and outer races are perfectly circular and the diametric size variation of the balls is zero, the maximum NRRO value in the radial direction when a single sine wave is applied to one ball is shown in Table 4. The geometrical error of the balls is very small and the maximum contained undulation components is 6. The conditions of the calculation are the same as in Table 1. From these calculation results, the following is deduced: (1) When the undulation number of the geometrical error is even, the maximum NRRO value decreases uniformly as the number of balls increases.

(2) When the undulation number of the geometrical error is odd, the geometrical error of the ball does not affect NRRO. If there is a geometrical error on the balls, a perfect sphere of standard size (diameter) becomes irregular, thus, the variation of the size of a ball is considered when it comes in contact with the inner and outer raceway. When a sine wave geometrical error with odd-numbered undulations is applied to a ball, the error at the position where the ball makes contact with the inner and outer raceway is of the same magnitude but the symbol is different. Therefore, as the diameter does not change, a change in distance of inner and outer raceway does not occur, even when there is geometrical error and NRRO is not observed. When sine wave geometrical errors with even-numbered undulations are applied to balls, change of ball diameter between the inner and outer raceway becomes double of the geometrical error amplitude and NRRO is observed. As a phenomenon, this is the same as the case when a geometrical error was applied to the balls in the preceding section, in which NRRO became small because contact force was dispersed as the number of balls increased.

4. Verification by experiments We learned through the theoretical analysis in the preceding section that when the number of balls is 12 or 18, the geometrical error component of up to 10 undulations contained in the inner and outer races does not affect NRRO, and the influence of the ball size and geometrical error can be better relieved with more than the currently common numbers of balls of 8 or 10. We, therefore, measured NRRO under different conditions using bearing SR1810 with 12 balls to verify our theoretical conclusions on NRRO reduction. 4.1. Experimental conditions

Table 4 Maximum NRRO value calculated by changing ball number and integral harmonic undulation of one ball (NRRO is expressed in ␮m, 0 means less than 1 nm) Ball number (Z)

7 8 9 10 11 12 13 14 15 16 17 18 19 20

Undulation (n) 2

3

4

5

6

0.059 0.051 0.046 0.041 0.037 0.034 0.032 0.029 0.027 0.026 0.024 0.023 0.022 0.021

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0.059 0.052 0.046 0.041 0.038 0.035 0.032 0.030 0.028 0.026 0.024 0.023 0.022 0.021

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0.060 0.052 0.046 0.042 0.038 0.035 0.032 0.030 0.028 0.027 0.026 0.023 0.022 0.021

The principal experimental conditions are listed in Table 5. Bearing SR1810 is often used in the swing arm support unit of magnetic disk devices. As its race rings are extremely thin, it is thought that the geometrical error of its raceways is larger than that of other ball bearings. However, its size is similar to spindle motor ball bearings and it has a plas-

Table 5 Experimental conditions Test ball bearing Ball used in SRI810 (Da , mm; grade 5) Number of data (per revolution) Number of circles for NRRO evaluation Axial pre-load (N) Rotating speed (rpm) Lubricant

SR1810 (Ø7.938 × Ø12.7 × 3.967) 1.2 256 50 9.8 300 Spindle oil (VG 12)

S. Noguchi, K. Ono / Precision Engineering 28 (2004) 409–418

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Table 6 Diametrical error of balls in one bearing from normal size (normal size = 1202 mm) Ball number

Diametrical error (␮m)

l 2 3 4 5 6 7 8 9 10 11 12

+0.003 −0.013 +0.009 −0.036 −0.009 +0.016 −0.016 +0.019 +0.010 +0.023 −0.012 +0.012

tic cage. Mounting and dismounting are also easy. For these reasons, we used this bearing in our experiments. The conditions of our experiment were set as close as possible to those of the theoretical calculations but the revolution number of measurement was increased considering the stability of the NRRO evaluation value. To measure as accurately as possible the NRRO caused by the number of balls and structural components such as geometrical errors, it is necessary to minimize the influence of lubrication and inertia. We, therefore, adopted minimum oil lubrication with a low-viscosity oil and measured it at 300 rpm. Also, to eliminate high-frequency noise components from the measurements, we used a phase compensation-type digital filter [8] to prevent the influence of phase delay [7]. The cut-off value of the low pass filter was set at 100 undulations per rotation so that the run-out component of the geometrical error of all parts of the bearing passed without attenuation. In the NRRO measurement, frequency components caused by the errors of the bearing parts led to problems. We, therefore, provided an FET analyzer so that the analysis of the frequency and amplitude of each component of NRRO could be done simultaneously. However, because NRRO of ball bearing is often specified by an overall value, the evaluation of NRRO in this report also uses overall values.

Fig. 8. Geometrical errors of ball bearing parts (SR1810): (a) outer raceway; (b) inner raceway; and (c) ball.

that the theoretic analysis described in Sections 2 and 3 is valid and can be used to simulate NRRO. 4.3. Measurement results from 50 normal products

4.2. Comparison of calculated and measured values The measured geometrical error of bearing parts and harmonic analysis results are shown in Fig. 8. Size variation of the balls is described in Table 6. As to the ball shape, only representative one is shown. Table 7 compares the NRRO calculated with the input error component of the outer race and the balls with the actual measured NRRO. The calculated values agreed well with the measured values. We confirmed Table 7 Maximum NRRO comparison between theory and experiment Theoretical maximum NRRO value (␮m) Experimental maximum NRRO value (␮m)

0.043 0.046

The distribution of the maximum NRRO value of 50 SR1810 bearings is shown in Fig. 9. When compared to the already published results of the NRRO of the spindle motor ball bearing 695 (with eight balls) [9], it is hard to say that the number of balls had resulted in a better NRRO because other factors such as the shape of the inner and outer ring and the ball accuracy are different. However, there is a general tendency for NRRO to become smaller. Comparing the distributions, SR1810 was 0.045 ␮m, more than 40% smaller than the 0.08 ␮m of 695. Geometrical error of the inner and outer races is rather large compared to the spindle motor raceways and the accuracy class of the balls is high. Considering these factors, we believe that as suggested by Fig. 5 employing 12 balls was effective in reducing NRRO.

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Fig. 9. NRRO distribution of SR1810 (n = 50). Fig. 10. Form error of the same inner raceway used.

4.4. Measurement results when the number of balls was changed SR1810 has 12 balls, a number with numerous divisors. Even if the number of balls is three, four or six, balls can be positioned at equal distance from each other. Therefore, we measured NRRO using 12 balls whose dimension had been measured and one plastic cage with one set of inner and outer rings, changing at random the number of balls. When changing the inner and outer rings to eliminate the influence of ball error components, particularly with 3, 4, 6, and 12 balls, balls were placed in the same disposition to be measured. As the axial load corresponding to the preload, we adopted the value of the following expression so that the load becomes the same for 12 balls and 1 ball: Axial load = number of balls ×

9.8 12

(N)

clear from Fig. 8, NRRO became extremely large as a result of the influence from 2 and 4 undulation components, which are the biggest geometrical errors contained in the raceway. It is thought that NRRO became small because the number of influential undulations was reduced, and their amplitude also became small as the number of balls increased and the number of undulations of the inner and outer race geometrical errors affecting NRRO increased. The absolute value of

(7)

The change in the maximum NRRO value when the number of balls was changed is shown in Table 8. We analyzed five bearings; it was observed that in all cases, as the number of balls decreased, the maximum NRRO value became larger. It is believed that the change in NRRO is attributable to changes in the number of balls as the inner and outer rings remained unchanged. When the number of balls is set to Z, the number of undulations of the inner and outer races’ geometrical errors, which has the largest influence on NRRO is given by Y = nZ ± 1 (n is a natural number). It is thought that NRRO increased with the number of undulations of the inner and outer races being affected by the decreasing number of balls. Especially when the number of balls was three, as it is Table 8 The maximum NRRO value in the case of changing ball number (unit of NRRO: ␮m) Bearing number

l 2 3 4 5

Ball number 12

6

4

3

0.042 0.052 0.051 0.048 0.054

0.128 0.122 0.098 0.106 0.117

0.272 0.306 0.205 0.310 0.296

0.695 0.968 0.639 0.002 0.606

Fig. 11. Form error of outer raceway given five undulations: (a) before and (b) after.

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Table 9 The result of harmonic analysis (unit: ␮m) Undulation

Ring Inner raceway used in common

Outer raceway given five summits

Outer raceway given seven summits

Before

After

Before

After

2 3 4 5 6 7 8 9 10 11 12 13 23 25

0.027 0.021 0.002 0.003 – 0.007 0.005 – – – – 0.001 – –

0.060 0.025 0.027 0.007 0.008 0.001 0.002 0.005 0.002 0.003 0.006 0.002 0.002 –

0.023 0.013 0.015 0.058 0.010 0.014 0.012 0.009 0.011 0.017 0.006 0.006 0.003 –

0.085 0.016 0.007 0.005 0.002 0.002 0.003 0.003 0.001 0.006 0.001 0.002 – –

0.033 0.048 0.012 0.004 0.009 0.025 0.014 0.007 0.005 0.001 0.001 0.006 – –

Total of nZ ± l

0.001

0.007

0.026 (+0.019)

0.008

0.007 (−0.001)

(–) means less than 0.001 ␮m.

the diametric size variation of 12 balls used for this measurement was about 0.06 ␮m (Table 6). 4.5. Measurement results when a specific number of undulations is applied to outer race as geometrical error Usually two or four undulations are dominant for the inner and outer races and as the number of undulations increases, the amplitude decreases. When the number of balls is 12, as the preceding section has clarified, geometrical error up to 10 undulations contained in the raceway does not theoretically affect NRRO. Thus, to verify the theory, we measured NRRO using a bearing of which the outer race has a specific number of undulations. The inner ring, cage, and 12 balls used the same and we caused a geometrical error of a specific number of undulations on the outer race by scratching the raceway at equidistant locations. The profile of the inner race and outer race before and after scratching a specific number of undulations are shown in Figs. 10–12. Harmonic analysis results are shown in Table 9. Measurements were performed not at the bottom of the raceway but near the place where the balls actually made contact. Five and seven undulations were aimed and it cannot be said in uniform but without making out-of-roundness too large, we could increase the amplitude of specific numbers of undulations. The maximum NRRO value of bearings with a common inner ring, cage, balls, Table 10 The maximum NRRO value in the case of giving specific undulations in outer raceway

Given five undulations Given seven undulations

Before deformation (␮m)

After deformation (␮m)

0.048 0.058

0.083 (+0.035) 0.056 (−0.002)

Fig. 12. Form error of outer raceway given seven undulations: (a) before and (b) after.

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and outer ring before and after the scratching are shown in Table 10. Comparing NRRO before and after the scratching of a specific number of undulations, with five undulations, NRRO increased but with seven, it decreased slightly. Examining Table 9 carefully, it is clear that NRRO is affected more by the nZ ± 1 undulations component (11, 13, 23, and 25 undulations), which changed when being scratched than by increasing or decreasing the specific undulation amplitude. With the outer ring with seven scratched undulations, the total of nZ ± 1 undulation component was reduced from 0.008 to 0.007 ␮m. With the outer ring with five scratched undulations, the total of nZ ± 1 undulation component was increased by 0.019 ␮m from 0.007 to 0.026 ␮m. In the theoretic calculations in the preceding section, the variation of the maximum NRRO value in regard to the change of nZ ± 1 undulation component (half-amplitude value) of the outer race appeared to be about two times larger. The variation of the maximum NRRO value in Table 6 became about double the variation amount of the total of the nZ ± 1 undulation component, approximately matching the theory. The experiment clarified also that if the number of balls is 12, NRRO is not affected, even if specific undulation components of less than 10 undulations become large.

(3) We clarified that NRRO decreases uniformly as the number of balls increases when dimensional difference is applied to one ball. (4) Through tests using the same inner and outer rings and changing the number of balls, we clarified that NRRO becomes smaller as the number of balls increases. (5) We determined that when more than one ball has dimensional differences, NRRO is affected by the position of the balls. When balls that are different from the majority of the balls are positioned side-by-side, NRRO becomes larger but if the same balls are placed 180◦ opposite to each other, NRRO is not affected. (6) It became clear that NRRO appears when the undulation number of the geometrical error of the balls is even and that as the number of balls increases, NRRO decreases uniformly. However, when the undulation number of the geometrical error is odd, the geometrical error of the balls does not affect NRRO. (7) For design specifications that can reduce NRRO, we determined that the number of balls should be even and as large as possible. However, considering other factors such as ease of assembly and cage strength, 12 balls is the optimum.

5. Conclusion

References

Taking the number of balls and the geometrical error components of the outer race and balls as parameters, using the two-dimensional plane model, we calculated the theoretical NRRO of the ball bearings and determined the design specifications to reduce NRRO. We then verified the effectiveness of the specifications through experiments. We obtained the following conclusions:

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(1) The calculated ball bearing NRRO after having input geometrical and dimensional errors coincided well with the measured NRRO, thus, validating the theoretical analysis method. (2) The theory and experiment demonstrated that low frequency undulation components of up to 10 undulations, which are major components of the outer and inner ring geometrical error do not affect NRRO when 12 or 18 balls are employed.