Research on Rotational Accuracy of Cylindrical Roller Bearings

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ScienceDirect Procedia CIRP 62 (2017) 380 – 385

10th CIRP Conference on Intelligent Computation in Manufacturing Engineering - CIRP ICME '16

Research on Rotational Accuracy of Cylindrical Roller Bearings YU Yongjiana*, CHEN Guodinga, LI Jishunb,c, XUE Yujunc a

School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an 710072, China Henan Key Laboratory for Machinery Design and Transmission System, Luoyang 471003, China c School of Mechanical Engineering, Henan University of Science and Technology, Luoyang 471003, China b

* Corresponding author. Tel.: +86-379-6427-8961; fax: +86-379-6427-8961. E-mail address: [email protected]

Abstract To investigate transferring mechanism of form error of bearing elements and rotational accuracy in cylindrical roller bearings, according to the bearing components motion and geometric relationships, a prediction model for rotational accuracy of cylindrical roller bearings based on roundness error of the inner raceway is presented. The effects of roundness error (i.e., amplitude, order) in the inner raceway, the number of rollers and radial clearance on the radial runout of inner ring are analyzed. Analytic solutions of the radial runout of inner ring are deduced when shapes of bearing elements are an ideal circle, which verifies the correctness of the presented model. The results show that the proposed model has better precision accuracy. © Authors. Published by Elsevier B.V. This ©2017 2016The The Authors. Published by Elsevier B.V.is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and peer-review under responsibility of the International Scientific Committee of “10th CIRP ICME Conference". Peer-review under responsibility of the scientific committee of the 10th CIRP Conference on Intelligent Computation in Manufacturing Engineering Keywords: Cylindrical roller bearing; Rotational accuracy; Radial runout; Roundness error; Form error

1. Introduction The main function of rolling bearing is used to support shafting and maintain its rotational accuracy. The rotary accuracy of the bearing often directly affects the precision of mechanical system[1,2]. During the processing of bearing elements, geometrical error is inevitable. Geometrical error of bearing elements is one of the most important factors which cause the motion error of a bearing. Therefore, it is extremely important to research the influence of geometrical error of bearing elements on rotational accuracy of the bearing. At the present, the research on rotational accuracy of rolling bearing is mainly concerned on non-repetitive runout, orbit of shaft center and radial runout. Okamoto et al. [3] developed a model to predict the orbit of shaft center of a ball bearing based on form errors of outer raceway. The influences of number of balls, dimension error of balls and form error of the raceways on the size and shape of the orbit of shaft center were analyzed. Noguchi et al. [4-6] established a computational model for non-repetitive runout of a ball bearing considering form error of the outer raceway and diameter error of balls, and analyzed the influences of geometrical error of the raceways, ball numbers, diameter

difference and geometrical error of the balls on non-repetitive runout of a ball bearing. Jang al.[7] analyzed the mechanism of resulting in non-repetitive runout of a ball bearing which is used in disks, and characterized the transmission path of nonrepetitive runout of a ball bearing from the bearing to the disk. Yang et al.[8-10] developed a mathematical model and a fivefreedom model to study the effects of geometrical errors on the non-repetitive runout of an angular contact ball bearing, respectively. Liu et al.[11] proposed a computational model about non-repetitive runout of a high-speed angular contact ball bearing based on five-freedom quasi-static model, and studied the influences of the waviness of bearing elements on non-repetitive runout of an angular contact ball bearing. Li et al.[12] established 5-DOF static model of non-repetitive runout of deep groove ball bearings, and investigated the influences of the distribution of diameter errors among rollers and roundness error of the inner or outer raceway on nonrepetitive runout of a deep groove ball bearing. Bhateja et al.[13] established a prediction model for rotational accuracy of the cageless hollow roller radial bearing considering dimension difference of rollers, and analyzed the influences of the change of internal and external diameter of rollers, diameter and the position of rollers on rotational accuracy of

2212-8271 © 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the scientific committee of the 10th CIRP Conference on Intelligent Computation in Manufacturing Engineering doi:10.1016/j.procir.2017.01.002

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bearing. Wang et al.[14] presented a model to predict geometric accuracy of a roller bearing considering form error in the inner or outer raceway, and studied the effects of the diameter error of roller, the amplitude and order of roundness error in the raceways on geometrical precision of the bearing. Previous research work mainly analyzed the influences of the profile of inner and outer raceways on rotational accuracy of bearing, and established a computational model for rotational accuracy of bearing based on roundness error of inner raceway[15]. A computational model for rotational accuracy of bearing based on form error of outer raceway was developed, and studied the effects of roundness error in the outer raceway on the radial runout of outer ring. Roundness error in the outer raceway is experimentally investigated and the function relation between the radial runout of outer ring and harmonic distribution parameters in the outer raceway was obtained [16,17,18]. In this paper, according to the geometrical model of cylindrical roller bearings, considering roundness error in the inner raceway, a prediction method for the radial runout of inner ring is presented. The influences of the amplitude and order of roundness error in the inner raceway, number of rollers and radial clearance on the radial runout of inner ring are analyzed.

inner raceway. m is harmonic number. Cim is the amplitude of the mth harmonic component in inner raceway. Mim is the phase angle of the mth harmonic component in inner raceway.

2. Mathemetical model of rotational accuracy

2.2. Contact state of the inner raceway

In this paper, some assumptions are made as follows: (1) The elastic deformation of bearing components is taken into account. (2) The rollers are uniformly distributed in circumferential direction. (3) There is no relative slip between the rollers and raceways. (4) There are no axial geometric and form errors in raceways and rollers.

It is assumed that every roller below the bearing contacts with the outer raceway. According to geometric model of a cylindrical roller bearing, the center coordinates of the rollers is obtained. The inner ring is moved with a certain step in the plane XY. The position relationship (contact, separate, interference) between the rollers and the inner raceway is obtained by the minimum distance between the surface of rollers and the inner raceway. The contact state of the inner raceway is obtained in each location of inner ring.

not the the the

Y

yi

Zi Ri

'x

'y

Ti

xi

oi

X

Fig. 1. Geometric model of a cylindrical roller bearing

Y

2.1. Geometric model of the bearing yi

Only considering roundness error in the inner raceway, the inner ring is a noncircular contour. The geometric model of a cylindrical roller bearing is shown in Fig. 1. A global Cartesian coordinate system XOY is set up at the center of the bearing. The plane XY coincides with the center plane of outer raceway. Local coordinate system xioiyi is set up at the center of inner raceway, which only translates, and the plane xiyi coincides with the center plane of inner raceway. Outer ring is fixed, and inner ring is rotated in this bearing. 'x is the displacement of inner ring in the horizontal direction, and 'y is the displacement in the vertical direction, which is the radial runout value of inner ring. In coordinate system xioiyi, when the inner ring turned an angle D , the contour radius Ri(ș) of inner raceway is expressed by equation (1).

O

X A B

oi Ti

xi

orj

f

Ri (T i )

d i / 2  ¦ Cim cos( m(T i  D )  M im )

(1)

m 1

Where T i is position angle of any point on the inner raceway in coordinate system xi oi yi . d i is the diameter of

Fig. 2. Geometric relationship between roller and inner raceway

When the inner ring is translated to a certain position, the geometrical relationship between a roller and the inner raceway is shown in Fig. 2. The distance between point E on

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the inner raceway and point F on the surface of the jth roller is calculated by equation (2).

Eorj  Dw / 2  

EF

Where Eorj is the distance between the center of the jth roller

The stable location of inner ring which satifies all conditions of the criterion is got, thus the center coordinates of inner ring is obtained. The displacement of inner ring in the vertical direction is the radial runout value of inner ring when inner ring is rotated an angle D .

and point E , which is calculated by equation(3).

 Eorj

Y

Eo  oiorj  2 Eoi u oiorj cos(T1 )   2 i

2

Where Eoi is the contour radius of point E on the inner raceway, which is

calculated by equation(4). oi orj is the

distance between the center of the inner ring and the center of the jth roller. T1 is the angle between point E on the inner raceway and the center of the jth roller in the circumferential direction,  arcsin( Dw / d i ) d T1 d arcsin( Dw / d i ) .

O

X

f

Eoi

Ri (T E )

d i / 2  ¦ Cim cos( mT E  mD  M im )

(4)

m 1

Where

TE

is the position angle of E point on the inner

raceway. In equation (2), when the angle T1 changes in a certain range, there is a minimum distance between point E on the inner raceway and point F on the surface of the jth roller, which is the minimum distance between the surface of the jth roller and the inner raceway. When the absolute value of the minimum distance is less than the given tolerance, the jth roller contacts with the inner raceway. When the minimum distance is larger than the given tolerance, the jth roller is separate from the inner raceway. When the minimum distance is less than negative tolerance, there is interference between the jth roller and the inner raceway. The position relationship between the jth roller and the inner raceway is gained. The position relationship between every roller below the bearing and the inner raceway is obtained by the same way, and the numbers and position angle of rollers which contact with the inner raceway is gained, thus getting the contact state of the inner raceway when the inner ring is moved to a certain position. Similarly, the contact states of the inner raceway are obtained in all the positions of the inner ring.

 D 

Y

k1

X

k2

o1

kj

o2

oj

(b)

2.3. Determination of the center coordinates of the inner ring In order to obtain the stable position from several positions of the inner ring, the criterion is presented as follows: (1) no interference between the rollers and the inner raceway. When different positions of the inner ring are given, some position of the inner ring is not reasonable, where there is interference between the inner raceway and rollers. Obviously, there is no interference between rollers and the inner raceway in cylindrical roller bearings. (2) At least one roller contact with the inner raceway on both sides of Y axis. When there is only one roller which contacts with the inner raceway, the inner ring is not stable, as shown in Fig. 3a. When there is at least one roller contacting with the inner raceway on any side of Yaxis, the inner ring is stable. (3) the absolute value of the sum of slopes between the center of the rollers which contact with the inner raceway and the center of the outer ring is the minimum, as shown in Fig. 3b.

Fig. 3. Contact status of inner raceway

3. Verification of the presented prediction model

To verify the correctness of the prediction method for the radial runout of inner ring, the analytical algorithm for the radial runout of inner ring is deduced when the shapes of bearing components are an idea circle. The main parameters of a cylindrical roller bearing are shown in Table.1. Table.1 Parameter of cylindrical roller bearing parameter value Diameter of inner raceway/mm 54.991 Diameter of outer raceway/mm 75.032 Diameter of roller/mm 11 Number of roller 14

When shapes of the inner raceway, the outer raceway and the rollers are an idea circle, the inner raceway only contacts

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with two rollers below the bearing because of the radial clearance of bearing, as shown in Fig. 4. Y

4. Results and analysis

4.1. Effect of the amplitude of roundness error in inner raceway on the radial runout of inner ring Fig. 6 shows the effect of the amplitude of roundness error in the inner raceway on the radial runout of inner ring. The radial runout of inner ring increases linearly with the amplitude of roundness error in the inner raceway. The results show that the influence of the amplitude of roundness error on the radial runout of inner ring is significant.

yi

O

xi

oi

6

X

Radial runout of inner ring/ȝm

5

O2

O1

Fig. 4. Calculation of radial runout of inner ring

The inner ring is turned an angle D . According to the geometric relationship, it can be seen that (5) O1oi ( X i  X 1 )2  (Yi  Y1 )2 (d i  Dw ) / 2 ( X i  X 2 )2  (Yi  Y2 )2

O2oi

Where X1, Y1, X2, Y2 are the center coordinates of two rollers at the bottom of bearing, X 1 0.5( d e  Dw ) cos E1 ˈ

0.5( d e  Dw ) sin E1 ˈ X 2

0.5( d e  Dw ) cos E 2 ˈ 0.5( d e  Dw ) sin E 2 , E is the position angle of rollers. By solving simultaneous equation (5) and equation (6), the center coordinates of the inner ring are calculated. The radial runout value of inner ring is obtained when it is turned an angle D . Similarly, the radial runout value of inner ring is obtained with the inner ring turning any angle. The inner ring is rotated a whole circle, and the radial runout values of inner ring are calculated. The simulation results compare with theoretical analytical solutions is shown in Fig. 5. The relative error is less than 0.09%. Therefore, the presented prediction method sufficiently forecasts the rotational accuracy of cylindrical roller bearings. Y1

Y2

Prediction result Analytical result

1.6

2.4

3.2

4.0

4.8

5.6

Amplitude of roundness error/ȝm

Fig. 6. Effects of the amplitude of roundness error on the radial runout of inner ring

4.2. Effect of the order of roundness error in inner raceway on the radial runout of inner ring Fig. 7 shows the effect of the order of roundness error in the inner raceway on the radial runout of inner ring. The radial runout of inner ring fluctuates with the increase of the order of roundness error in the inner raceway. When the order of the roundness error and number of rollers satisfies a certain relationship, the radial runout of inner ring significantly reduces. 2.60

2.55

-0.0205

Radial runout value of inner ring /mm

2

0 0.8

Radial runout of inner ring/ȝm

-0.0204

3

1

(6)

(d i  Dw ) / 2

4

-0.0206

-0.0207

-0.0208

2.50

2.45

2.40

2.35

2.30

-0.0209

2

4

6

8

10

12

Order of roundness error -0.0210

Fig. 7. Effects of the order of roundness error on the radial runout of inner ring

-0.0211 0

50

100

150

200

250

300

350

400

Rotation angle of inner ring /°

Fig. 5. Comparision between prediction results and analytical results

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4.3. Effect of the number of rollers on the radial runout of inner ring Fig. 8 shows the number of rollers on the radial runout of inner ring. The radial runout of inner ring has a decreasing fluctuation trend with the increase of number of rollers. The radial runout of inner ring does not always decrease with the increase of number of rollers. 4.5

Radial runout of inner ring/ȝm

4.0 3.5

Acknowledgements 3.0 2.5 2.0 1.5 1.0 0.5 8

10

12

14 16 number of rollers

18

20

22

Fig. 8. Effects of number of rollers on the radial runout of inner ring

4.4. Effect of radial clearance on the radial runout of inner ring Fig. 9 shows the effect of radial clearance on the radial runout of inner ring. The radial runout of inner ring shows a increasing trend with the radial clearance. When the radial clearance is less, the effect of radial clearance on the radial runout of inner ring is significant. 2.75

2.50

Radial runout of inner ring/ȝm

the radial clearance on the radial runout of inner ring are analyzed, the following conclusions are obtained. (1) The effects of the amplitude of roundness error in the inner raceway on the radial runout of inner ring are significant. (2) The radial runout of inner ring does not always decrease with the increase of number of rollers. The radial runout of inner ring significantly reduces when the order of the roundness error and number of rollers satisfies a certain relationship. (3) The increase in the radial runout of inner ring is proportional with the increase of the radial clearance.

2.25

2.00

1.75

1.50

1.25

1.00 5

10

15

20

25

30

35

40

45

Radial clearance/ȝm

Fig. 9. Effects of radial clearance on the radial runout of inner ring

5. Conclusions

In this paper, the prediction method for the radial runout of inner ring based on the profile of the inner raceway is presented. The effects of the amplitude and order of roundness error in the inner raceway, number of rollers and

The authors gratefully thank the National Natural Science Foundation of China (No. 51375148) and the Project of Basic and Advanced Technology Research of Henan Province of China (No. 142300413217) for financial support. This work is also supported by the Project of Basic and Advanced Technology Research of Henan Province of China (No. 152300410081), the Key Research Program of the Higher Education Institutions of Henan Province (No. 15A460022), and the Program for Innovative Research Team (in Science and Technology) in University of Henan Province (No. 15IRTSTHN008).

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