Effects of Ball Groupings on Ball Passage Vibrations of a Linear Guideway Type Ball Bearing Pitching and Yawing Ball Passage Vibrations

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1 Hiroyuki Ohta 1 Department of Mechanical Engineering, Nagaoka University of Technology, Kamitomioka, Nagaoka, Niigata, Japan ohta@mech.nagaokaut.ac.jp Yoshiki Kitajima LG Technology Department, NSK Precision Co., Ltd., 1-1 Ohnuma, Hanyu, Saitama, Japan Soichiro Kato LG Technology Department, NSK Precision Co., Ltd., 78 Toriba, Maebashi, Gunma, Japan Yutaka Igarashi Research Department, NSK Precision Co., Ltd., 78 Toriba, Maebashi, Gunma, Japan Effects of Ball Groupings on Ball Passage Vibrations of a Linear Guideway Type Ball Bearing Pitching and Yawing Ball Passage Vibrations The effects of ball groupings on the pitching and yawing ball passage vibrations of linear guideway type ball bearings (linear ball bearings) under low-speed operation were studied. For this study, the test linear ball bearings (which can retain the ball grouping in operation) with three types of ball groupings were manufactured, and the pitching and yawing ball passage vibrations of each test linear ball bearing were measured using a laser autocollimator. Moreover, a calculation method of the ball passage vibrations for a linear ball bearing with an arbitrary ball grouping was presented. According to the presented method the ball passage vibrations for three types of ball groupings were calculated. The experimental and calculated results show that the occurrence of the pitching and yawing ball passage vibrations was affected by the ball groupings. For the occurrence, the wave forms, and the amplitude of the pitching and yawing ball passage vibrations for the ball groupings, the calculated results based on the presented method almost matched the experimental results. DOI: / Keywords: ball bearing, linear bearing, ball passage vibration, ball grouping 1 Introduction Linear guideway type ball bearings linear ball bearings are widely used in machine tools, robots, and precision x-y tables 1. Although linear bearings have many advantages compared with sliding guides, they often lead to vibrations Ball passage vibrations periodical posture changes of the carriage with a ball passage frequency are typical vibrations of linear bearings. Kasai et al. 6 8 observed the ball passage vibrations of recirculating linear ball bearings in operation. They also pointed out that the ball passage vibration was reduced by using a carriage with crowning. Shimizu 9,10 presented the calculation method of ball passage vibrations of the table supported by the recirculating linear ball bearings. He also proposed an optimum circular crowning to reduce the ball passage vibration based on his experimental results. Ohta et al. 11 presented a theoretical design method of the crowning to reduce the ball passage vibrations of linear bearings. Kato and Matsumoto 12 showed a calculation method of the pitching and vertical ball passage vibrations for a ball grouping where the balls in the upper rows and in the lower rows are assumed to be staggered by half of the distance s between adjacent balls. They presumed that the ball passage vibrations are affected by ball groupings. Since the ball grouping in the recirculating linear ball bearings which were used in the previous studies is not retained in operation, it was difficult to evaluate the effects of the ball groupings on the ball passage vibrations. Therefore, the effects of the ball groupings on the ball passage vibrations have not yet been examined thoroughly. 1 Corresponding author. Contributed by the Tribology Division of ASME for publication in the JOURNAL OF TRIBOLOGY. Manuscript received June 14, 2006; final manuscript received September 18, Review conducted by Liming Chang. Paper presented at the STLE/ASME 2006 International Joint Tribology Conference, San Antonio, TX, October 22 25, In this paper, the effects of the ball groupings on the pitching and yawing ball passage vibrations under-low speed operation are examined by using the nonrecirculating linear ball bearings which can retain the ball grouping in operation. 2 Experiments 2.1 Test Bearings and Ball Groupings. In order to examine the effects of the ball groupings on the ball passage vibrations, nonrecirculating linear ball bearings were manufactured and used as test bearings. The photograph of a test linear ball bearing is shown in Fig. 1. One test bearing consists of one rail, one carriage, one retainer, and balls. The number of rows is four. In the test bearing, the ball grouping can be changed by changing the pocket locations in the retainer. All the test bearings are preloaded by oversizing of the balls. The test bearing specifications are shown in Table 1. The ball groupings of the test bearings are shown in Fig. 2. In Fig. 2, n 1 n 4 show the number of the loaded balls in the rows i=1 4, respectively. In this study, three types of ball groupings are considered as shown in Fig. 2. The features of each grouping are follows: 1. Ball grouping I. The ball locations in the two upper rows i=1,2 are shifted half the distance s between the adjacent balls to those in the two lower rows i=3,4 ; 2. Ball grouping II. The ball locations in the two left rows i =2,3 are shifted s/2 to those in the two right rows i =1,4 ; and 3. Ball grouping III. The ball locations in the second row i =2, in the third row i=3, and in the fourth row i=4 are shifted s/2, s/3, and s/5, to those in the first row i=1, respectively. 2.2 Experimental Method. The experimental apparatus is shown in Fig. 3. The rail of the test bearing is fixed on the concrete bed by bolts. The carriage of the test bearing is driven at a 188 / Vol. 129, JANUARY 2007 Copyright 2007 by ASME Transactions of the ASME

2 Fig. 1 Test bearing ball grouping I certain linear velocity. The pitching and yawing motion of the carriage was detected by using a laser autocollimator Chuo Precision Industrial Co., Ltd: LAC-S and a mirror, and was stored in a personal computer. The positive and negative directions of the pitching and yawing motion of the carriage were defined as shown in Fig. 3. Figure 4 shows the mounting position of the mirror. Because the gravitational center G m of the mirror was right above that of the carriage G c, the gravitational center G of the mirror attached the mirror was also right above G c. In the measurement using the laser autocollimator and the mirror, the motion caused by the deformation of the rail as well as the ball passage vibrations were detected 11. To avoid the effect of the motion caused by the deformation of the rail, a high pass filter was used. The cutoff frequency was 0.8f p, where f p is the ball passage frequency. The ball passage frequency f p is given by f p = V 2s 1 In the measurement, the linear velocity V of the carriage was varied in four steps in the range of m/s. Except for the vertical load caused by the weight of the carriage and mirror, external loads were not applied to the test bearing. The test bearings were lubricated with mineral oil ISO VG Experimental Results Table 1 Test bearing specifications Asssembly height m Carriage width m Carriage length L 1 m Carriage height m Rail width m Rail length L m Rail height m Ball diameter m Distance between adjacent balls s m Number of rows 4 Contact angle deg 50 Oversize of balls 0 m Crowning length L C m Crowing radius R m 1.1 Fig. 2 Ball arrangements The typical measured wave forms of the pitching and yawing ball passage vibrations are shown in Fig. 5. In Fig. 5, T p =1/ f p is the ball passage period. It is clear from Fig. 5 that the measured wave forms of the ball passage vibrations are affected by the ball groupings. In ball grouping I, the pitching ball passage vibration occurs, while the yawing ball passage vibration does not occur. In ball grouping II, the pitching ball passage vibration does not occur, while the yawing ball passage vibration occurs. In ball grouping III, both the pitching and yawing ball passage vibrations occur. The effect of the linear velocity V of the carriage on the measured rms amplitude of the ball passage vibrations in each ball grouping is shown in Fig. 6. It is clear from Fig. 6 that the measured amplitude of the pitching and yawing ball passage vibrations does not significantly change with linear velocity V, within the experimental condition linear velocity V= m/s. Journal of Tribology JANUARY 2007, Vol. 129 / 189

3 Fig. 3 Experimental apparatus 4 Discussion 4.1 Calculation Method of Ball Passage Vibrations for Linear Bearings with Arbitrary Ball Groupings. In this section, a calculation method of the ball passage vibrations for a linear bearing with an arbitrary ball grouping is discussed. Figure 7 shows the coordinate system for the test bearing. The coordinate system O-xyz is a moving coordinate which moves with the carriage in the longitudinal direction of the rail at the linear velocity V. The origin O of coordinates coincides with the position of the gravitational center G of the carriage with the mirror, on the condition that the distances between the races of the carriage and rail are equidistance reference condition. In Fig. 7, a is the distance from the origin O to the contact point of the upper rows of the carriage and the balls in the direction parallel to the z axis; b is the distance from the origin O to the contact point of the lower rows of the carriage and the balls in the direction parallel to the z axis; c is the distance from the origin O to the contact point of the rows of the carriage and the balls in the direction parallel to the y axis, is the contact angle; L 1 is the carriage body length; and L c ;is the crowning length. In the experiment in this study, only the vertical load caused by the weight of the carriage and mirror was applied to the test bearing. However, to present a wide use calculation method of Fig. 6 Effect of linear velocity on rms amplitude of ball passage vibrations Fig. 4 Center of gravity ball passage vibrations, we consider a loading condition in which the horizontal load F H, the vertical load F V, the rolling moment M R, the pitching moment M P, and the yawing moment M Y are applied to the carriage. Under the loading condition, the horizontal displacement u, the vertical displacement v, the rolling angle, the pitching angle, and the yawing angle occur as shown in Fig. 7, total contact deformation ij at jth ball in ith row is given by Fig. 5 Measured wave forms of ball passage vibrations V=1.67Ã10 4 m/s 190 / Vol. 129, JANUARY 2007 Transactions of the ASME

4 Table 2 Values of a, b, c, K, and W a m b m c m K N/m W N 9.92 Fig. 7 Coordinates of test bearing in integers which are not over Z Z real number. From the Hertzian theory, the normal force Q ij between jth ball and raceways of ith row is given by 13 Q ij = K 1.5 ij : ij 0 Q ij =0: ij 0 6 where K is the load deflection factor which is determined by the material and the geometry of the ball and the raceways. The static balance of the forces horizontal and vertical forces and moments rolling, pitching, and yawing moments can be written as n 1 Q 1j + n 2 n 3 Q 2j + n 4 Q 3j Q 4j cos = F H 7 1j = e u cos v sin + a cos + c sin x 1j sin + x 1j cos C x 1j 2j = e + u cos v sin a cos c sin x 2j sin x 2j cos C x 2j 3j = e + u cos + v sin b cos + c sin + x 3j sin x 3j cos C x 3j 4j = e u cos + v sin + b cos c sin + x 4j sin + x 4j cos C x 4j 2 where e is the total contact deformation of a ball on the straight area L 1 2L c of the raceways of the carriage under the reference condition. Considering the carriage flexibility under preloading, e is given by 6 e = where 0 is the oversize of the balls. In Eq. 2, C x is the crowning drop at x. For circular arc crownings, C x is given by 11 C x = R R 2 x + L 1 2 c 2 L : L 1 2 x L L c C x =0: L L c x L 1 2 L c C x = R R 2 x L 2 1 c 2 + L : L 1 2 L c x L 1 2 where R is the crowning radius. x ij is x coordinates of the jth ball in the ith row in the carriage under a certain linear velocity V, and int is given by x ij = j 1 s + x i0 + Vt 2 L 1 +2x i0 + Vt s 5 2s where t is time; s is the distance between the adjacent balls; x i0 is x coordinates of the first ball in the ith row at t=0; and L 1 /2 x i0 s L 1 /2. By setting x i0, an arbitrary ball grouping can be expressed. In Eq. 5, int Z represents the maximum 3 4 n 1 n 2 Q 1j n 1 Q 1j n 2 Q 2j + n 3 Q 3j + n 4 Q 4j sin = F V 8 n 3 n 4 Q 2j c sin + a cos + Q 3j Q 4j c sin b cos = M R n 1 n 2 n 3 n 4 Q 1jx 1j Q 2jx 2j + Q 3jx 3j + Q 4jx 4j sin = M P n 1 n 2 n 3 n 4 Q 1jx 1j Q 2jx 2j Q 3jx 3j + Q 4jx 4j cos = M Y By using Eqs. 2 11, the horizontal, vertical, rolling, pitching, and yawing ball passage vibrations for an arbitrary ball grouping can be calculated based on the static balance of the forces and moments. 4.2 Comparison of Experimental Results with Calculated Results. Based on the calculation method described in Sec. 4.1, the ball passage vibrations of the test bearings with ball groupings I III were calculated under the loading condition in the experiment F V = W, F H =M R =M P =M Y =0. In the calculations, the values listed in Tables 1 and 2, and the values of x i0 which correspond to the ball grouping listed in Table 3 were used. The value of K in Table 2 was calculated theoretically from the design values of the ball-race configuration and material properties of the Table 3 Values of x i0 for ball groupings I III Ball grouping x 10 x 20 x 30 x 40 I L 1 2 L 1 2 L 1 2+s 2 L 1 2+s 2 II L 1 2 L 1 2+s 2 L 1 2+s 2 L 1 2 III L 1 2 L 1 2+s 2 L 1 2+s 3 L 1 2+s 5 Journal of Tribology JANUARY 2007, Vol. 129 / 191

5 Fig. 8 Calculated wave forms of ball passage vibrations V=1.67Ã10 4 m/s test bearings, using Hertzian theory 13. The typical calculated wave forms of the ball passage vibrations for ball groupings I III are shown in Fig. 8. It is clear from Fig. 8 that the calculated wave forms of the ball passage vibrations are affected by the ball groupings. In ball grouping I, the vertical and pitching ball passage vibrations occur. In ball grouping II, the horizontal and yawing ball passage vibrations occur. In ball grouping III, the horizontal, vertical, rolling, pitching, and yawing ball passage vibrations occur. By comparing Figs. 8 and 5, it is clear that the calculated results on the occurrence, the time wave forms of the pitching, and yawing ball passage vibrations for ball groupings I III are almost matched with the experimental results. The calculated rms amplitude of the pitching and yawing ball passage vibrations is drawn in Fig. 6. The calculated rms amplitude of the yawing ball passage vibration in ball grouping I and that of the pitching ball passage vibration in ball grouping II are zero. From Fig. 6, it is clear that the calculated rms amplitude of the pitching and yawing ball passage vibrations is almost matched with the measured rms amplitude. The effect of ball groupings on the horizontal, vertical, and the rolling ball passage vibrations has not been verified in the experiment, since the autocollimator cannot measure the horizontal, vertical, and the rolling ball passage vibrations. This will be examined in a future study. 5 Conclusions The following conclusions are drawn from the experimental results and discussion: 1. The occurrence of the pitching and yawing ball passage vibrations were affected by the ball groupings; 2. The measured rms amplitude of the pitching and yawing ball passage vibrations did not significantly change with linear velocity V, within the experimental condition linear velocity V= m/s ; 3. Based on the static balance of forces and moments, a calculation method of the ball passage vibrations for a linear bearing with an arbitrary ball grouping was presented; and 4. For the occurrence, the time wave forms and rms amplitude of the pitching and yawing ball passage vibrations for the ball groupings, the calculated results based on the presented method were almost matched with the experimental results. Acknowledgment This research was supported in part by scientific research funds fundamental research C, No from the Ministry of Education, Science and Culture, Japan. Nomenclature a distance from origin O to the contact point of the upper rows of the carriage and the balls in the direction parallel to the z axis m b distance from the origin O to the contact point of the lower rows of the carriage and the balls in the direction parallel to the z axis m c distance from the origin O to the contact point of the rows of the carriage and the balls in the direction parallel to the y axis m C x crowning drop at x m f p ball passage frequency Hz F H horizontal load N F V vertical load N M P pitching moment Nm M R rolling moment Nm M Y yawing moment Nm i row number j loaded ball number in a row 192 / Vol. 129, JANUARY 2007 Transactions of the ASME

6 K load deflection factor which is determined by the material and the geometry of the ball and the raceways N/m 3/2 L 1 carriage body length m L c crowning length m n 1 n 4 number of the loaded balls in the rows i=1 4, respectively. O-xyz Cartesian coordinates Q ij normal force Q ij between ith ball and raceways of jth row N R crowning radius m s distance between the adjacent balls m t time s T p ball passage period s u horizontal displacement of the ball passage vibration m v vertical displacement of the ball passage vibration m V linear velocity of the carriage m/s W total weight of the carriage and the mirror N x ij x coordinates of jth ball in ith row in the carriage under a certain linear velocity V Z real number contact angle deg pitching angle rad e total contact deformation of a ball on the straight area of the raceways of the carriage under the reference condition m References ij total contact deformation at jth ball in ith row m o oversizing of the balls m 1 The Japan Society for Precision Engineering, 2000, Present and Future Technology of Ultraprecision Positioning, FUJI Technosystem, Tokyo, Japan. 2 Teramachi, A., 2000, Introduction to Linear Systems, The Nikkan Kogyo Shimbun, Tokyo, Japan. 3 Grunau, A., and Giese, P., 1991, Scwingungsverhalten von Linearwalzfuhrungen für Werkzeugmashinen, Konstruktion, 43, pp Schneider, M., 1991, Statisches Und Dynamisches Verhalten Beim Einsatz Linearer Schienenführungen Auf Wälzlagerbasis Im Werkzeugmaschinenbau, Carl Hanser Verlag, Munchen, Wien, Germany. 5 Ohta, H., and Hayashi, E., 2000, Vibration of Linear Guideway Type Recirculating Linear Ball Bearings, J. Sound Vib., 235 5, pp Kasai, S., Tsukada, T., Ozawa, N., and Kato, S., 1985, Precision Linear Guides, NSK Tech. J., 645, pp Kasai, S., Tsukada, T., and Kato, S., 1987, Precision Linear Guides for Machine Tools, NSK Tech. J., 647, pp Kasai, S., Tsukada, T., and Kato, S., 1989, Recent Technical Trends of Linear Guides, NSK Tech. J., 649, pp Shimizu, S., 1990, Load Distribution and Accuracy/ Rigidity of Linear Motion Ball Guides System, J. Jpn. Soc. Precis. Eng., 56 8, pp Shimizu, S., 1992, Study on Accuracy Average Effect of Linear Motion Ball Guides System, J. Jpn. Soc. Precis. Eng., 58 11, pp Ohta, H., Kato, S., Matsumoto, J., and Nakano, K., 2005, A Design of Crowning to Reduce Ball Passage Vibrations of a Linear Guideway Type Recirculating Linear Ball Bearings, ASME J. Tribol., 127, pp Kato, S., and Matsumoto, J., 2000, Recent Developments in Highly Precise NSK Linear Guides, Motion & Control, 9, pp Harris, T. A., 2001, Rolling Bearing Analysis, 4th ed., Wiley, New York. Journal of Tribology JANUARY 2007, Vol. 129 / 193