# Prediction Method for the Radial Runout of Inner Ring in Cylindrical Roller Bearings

Authors: Yongjian Yu; Guoding Chen; Jishun Li; Yujun Xue; Bitao Pang
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

The motion error of assembled bearing depends on the geometric profile of bearing components. Therefore, it is crucial to establish the relationship between geometric error of bearing components and motion error of assembled bearing, which contributes to improving the rotational accuracy of assembled bearing in the design and machining of the bearing. The main purpose of this research is to propose an accurate method for predicting the radial runout of inner ring based on the geometrical constraint model of cylindrical roller bearings. In the geometrical constraint model, dimension and form errors in the inner raceway, the outer raceway, and rollers are considered, and the change of contact positions between the raceways and rollers caused by geometric errors of bearing components is taken into account. This method could predict the radial runout of inner ring after bearing components with geometric error are assembled. In order to testify the validity of the proposed prediction method, two particular cases in which the profiles of the inner raceway are circle and ellipse are selected, and the analysis algorithms for the radial runout of inner ring are derived. Two analytical results obtained from the analysis algorithms validate accuracy and effectiveness of the proposed prediction method.


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## Body

## 1. Introduction

Rolling bearing is mainly used to support shafting, keeping motion accuracy of shafts. Rotational accuracy of assembled bearing, as an important parameter to measure dynamic performance of the roller bearing, often directly determines the accuracy of mechanical systems [1, 2]. Geometric error is always present in manufactured bearing components, which is one of the important factors that cause the motion error of assembled bearing. Therefore, it is significant to establish the relationship between geometric errors of bearing components and rotational accuracy of assembled bearing.At present, for the rotational accuracy of ball bearings, most researchers focus on shaft centerline orbit and nonrepetitive runout of ball bearings. Okamoto et al. [3] developed a model to predict shaft centerline orbit of ball bearings based on form error of the outer raceway. Noguchi et al. [4–7] presented a computational model about nonrepetitive runout of ball bearings considering form error of the inner and outer raceways. Tada [8] established a three-dimensional model to calculate nonrepetitive runout of angular contact ball bearings based on the waviness of bearing components. Jang et al. [9] characterized the source of nonrepetitive runout (NRRO) of ball bearings and the transmission of NRRO from the ball bearing to the disk. Yang et al. [10–12] established a mathematical model and a five-degree-of-freedom model to research the influences of geometric error on the NRRO of an angular contact ball bearing. Chen et al. [13, 14] developed a one-dimensional mathematical model with algebraic equation considering roundness error of the raceways and diameter error of the rollers. The influence of diameter differences among the rollers on the load performance in a cylindrical roller bearing was investigated. Liu et al. [15] developed a computational model about nonrepetitive runout of high-speed angular contact ball bearings based on five-freedom quasi-statics method. Li and Mao [16] developed a five-freedom static model for nonrepetitive runout of deep groove ball bearings. Bhateja and Pine [17] established a prediction model for rotational accuracy of hollow roller bearings considering dimension difference of the rollers. Wang [18] presented a model to predict geometric accuracy of a cylindrical roller bearing considering form error of the inner or outer raceway. In previous research, the effect of roundness error of the inner and outer raceways on rotational accuracy of cylindrical roller bearings was researched, respectively. A computational model for the radial runout of inner ring based on form error of the inner raceway was established [19], and a computational model for the radial runout of outer ring considering form error of the outer raceway was developed [20, 21].In these current researches, the methods for rotational accuracy of cylindrical roller bearings were proposed considering form error of the inner raceway or the outer raceway or dimension error of the rollers, in which it was assumed that the contact positions between the rollers and the raceways do not change. However, the rotational accuracy of assembled bearing should result from the interaction among geometric errors of the inner raceway, the outer raceway, and the rollers. Meanwhile, the contact positions between the rollers and the raceways change with geometric errors of bearing components; thus the influence of the whole profiles of bearing components on the rotational accuracy of the bearing could be reflected perfectly and accurately. Therefore, it is very necessary to carry out some further researches to develop a method to accurately predict the rotational accuracy of cylindrical roller bearings. For this purpose, this research proposes a prediction method for the radial runout of inner ring in cylindrical roller bearings based on the geometrical constraint model of cylindrical roller bearings. The geometrical constraint model considers the change of contact positions between the raceways and rollers and the dimension and form errors in the raceways and rollers. The proposed method could accurately predict the radial runout of inner ring under given or known dimension and form errors of the inner raceway, the outer raceway, and rollers. This prediction method could be used to research the influences of geometric errors of bearing components and their coupling effect on the rotational accuracy of assembled bearing, thus establishing the relationship between geometric errors of bearing components and the radial runout of inner ring in cylindrical roller bearings.


## 2. Basic Assumptions

According to the international standards for rotational accuracy measurement of a rolling bearing, when the radial runout of a rolling bearing is measured, a ring is fixed, and another ring is rotating at very slow speed and its vertical displacement is measured as its radial runout value every rotational angle of the ring. To make the raceways contact with the rollers, a small measuring load is applied to the bearing, which can not cause obvious elastic deformation of bearing components. According to the above measurement operating conditions, a prediction method for the radial runout of inner ring in cylindrical roller bearings is presented, which is based on the following assumptions:(1)
There are no axial geometric errors in the raceways and rollers.(2)
The rollers are distributed in the circumferential direction uniformly, without considering the influence of cage.(3)
There is no slip at the rolling element-raceway contact.(4)
Lubrication in the bearing is not taken into account.


## 3. Characterization Method for Geometric Error of Bearing Components

As for the above assumptions, only dimension and form errors in the raceways and rollers are taken into account. The geometric error of bearing components can be expressed with addition of the dimension and roundness errors in the raceways and rollers. Hereinto, the roundness error item is described by the Fourier series in a polar coordinates system.(1)ΔSθ=Δr+ΔR=Δr+∑m=1∞Cmcos⁡mθ+φm,where ΔS is the geometric error of bearing components. Δr is the dimension error of bearing components. ΔR is the roundness error of bearing components. m is the order of a harmonic component. θ is the polar angle. C is the amplitude of a harmonic component. φ is the phase angle of a harmonic component.The geometric error of bearing components can be given arbitrarily. However, the geometric error of bearing components is unknown for manufactured bearing components, which needs to be obtained through experiments. The dimension errors of different circumferential positions are measured by the bearing comparator. The dimension error of bearing components is equal to the average value of dimension errors of different circumferential positions. The roundness error of manufactured bearing components is expressed by Fourier series. In order to get harmonic orders, amplitudes, and phase angles of roundness error, the roundness instrument is used to collect the data of running surface profile of manufactured bearing components. These parameters are obtained with Fourier transformation of the data of running surface profile, and thus the roundness error of bearing components is restructured.


## 4. Geometrical Constraint Model of Cylindrical Roller Bearings

Due to geometric error of bearing components, the raceways and rollers shapes are noncircular profile, as shown in Figure1. A fixed global Cartesian coordinate system (X,Y) is set up at the center of the bearing with the origin O coinciding with the center of the outer raceway. Also, a moving Cartesian coordinate system (xi,yi) is fixed to the center of the inner raceway, which translates with the inner ring and is always parallel to the global coordinate (X,Y), respectively. Another moving Cartesian coordinate system (xrj,yrj) is fixed to the center of the jth roller, which translates with the jth roller and is always parallel to the global coordinate (X,Y), respectively. The inner ring rotates, and the outer ring is fixed. When a small measuring load is applied vertically downward to the inner ring, the inner ring moves down and eventually reaches a stable state. In that case, the horizontal displacement of the centeroi of the inner raceway is Δx, and the vertical displacement of the center oi of the inner raceway is Δy; meanwhile it is the radial runout value of inner ring, too.Figure 1
Geometrical constraint model of cylindrical roller bearings.The inner ring rotates anticlockwise, and rollers revolve anticlockwise; in the meantime rollers spin clockwise. The rotation angle of the inner ring isα. The autorotation and orbital angle of rollers are calculated by (2) and (3), respectively.(2)γ1=di+de2-4Dw2di+deDwα,(3)γ2=0.5α1-2Dwdi+de,where γ1 and γ2 is the autorotation and orbital angle of rollers, respectively. Dw is the roller diameter. di and de are the diameter of inner raceway and outer raceway, respectively.The inner raceway, outer raceway, and thejth roller radii Ri, Re, and Rrj at the different azimuth angles corresponding with coordinate systems xioiyi, XOY, and xrjorjyrj are expressed by (4)Riθi=di2+Δdi2+ΔRi,Reθe=de2+Δde2+ΔRe,Rrjθrj=Dw2+ΔDj2+ΔRrj,where Δdi, Δde, andΔDare the diameter error of the inner raceway, the outer raceway, and rollers, respectively, and the subscript j denotes the roller number (j=1,2,…,Z). ΔRi, ΔRe, and ΔRrare the roundness error of the inner raceway, the outer raceway, and rollers, which are expressed by (5)~(7), respectively.(5)ΔRiθi=∑m=1∞Cimcos⁡mθi-α+φim,(6)ΔReθe=∑m=1∞Cemcos⁡mθe+φem,(7)ΔRrjθrj=∑m=1∞Crjmcos⁡mθrj+γ1+φrjm,where Ci and Ce are the amplitude of harmonic components. φi, φe, and φr are the phase angles of harmonic components.


## 5. Prediction Method for the Radial Runout of Inner Ring

It is assumed that the inner ring rotates and the outer ring is fixed. The position of the inner ring varies with the inner ring rotating. When the inner ring rotates an angle, due to the gravity and the geometrical constraint of bearing components, the inner ring moves in the vertical and horizontal directions until it contacts with the partial rollers at the bottom of the bearing and reaches a stable position, which is unique. In order to obtain the stable position of the inner ring, a prediction method for the radial runout of inner ring is presented. Its calculation process is as follows: The first step is to calculate coordinates of the center of each roller at the bottom of the bearing. The rollers at the bottom of the bearing purely roll to a new position with the inner ring rotating. Each roller is moved along the radial direction with a given step, until it contacts with the outer raceway. The coordinates of the center of each roller are calculated. The second step is to obtain the contact statuses of the inner raceway in different positions of the inner ring. The inner ring moves both vertically and horizontally at a given step in a certain range, and the position relation (contact, separation, interference) between the inner raceway and each roller at the bottom of the bearing is obtained in different positions, and the number and position angle of rollers which contact with the inner raceway are obtained. Thus, the contact statuses of the inner raceway are obtained in different positions. The third step is to obtain the coordinates of the center of the inner raceway. According to the contact statuses of the inner raceway in the different positions of the inner ring, the stable position of the inner ring where the inner ring is stable is distinguished by the criterion for stable contact between the inner raceway and rollers from some positions of the inner ring; thus the coordinates of the center of the inner ring are obtained. Therefore, the radial runout value of inner ring is obtained every rotation angle of the inner ring. Hereinto, the difference between the maximal radial runout value and the minimal radial runout value is the radial runout of inner ring.


### 5.1. Calculation of Coordinates of the Center of Rollers

To obtain the coordinates of the center of rollers at the bottom of the bearing when the rollers contact with the outer raceway, it is assumed that a roller at the bottom of the bearing is moved along the radial direction at a given step until it contacts with the outer raceway; thus the distance between the center of the roller and the center of the outer raceway is obtained. According to the geometric relationship among bearing components, the coordinates of the center of the roller are calculated through the distance. Hereinto, the key problem is to calculate the shortest distance between the outer raceway and the surface of the roller when the roller is moved to a position, thus determining whether the roller contacts with the outer raceway. For this purpose, a calculation method of the shortest distance is proposed.The inner ring rotates an angleα, and the jth roller is moved to a position with the distance Dej between the center of the jth roller and the center of the outer raceway. The geometric relationship between a roller and the outer raceway is shown in Figure 2. The distance between any point A on the outer raceway and any point B on the surface of the jth roller is expressed by (8)AB=Aorj-Borj,where Aorj is the distance between the center of the jth roller and any point A on the outer raceway, which is calculated by (9). Borj is the radius of any point B on the surface of the jth roller when the jth roller spins around its axis at an angle of γ1, which is calculated by (10). (9)Aorj=OA2+Oorj2-2OA×Oorjcos⁡Δθe,where OA is the radius of any point A on the outer raceway, which is calculated by (11). Oorj is the distance between the center of the outer raceway and the center of the jth roller, which is known, namely, Dej. Δθe is the angle between any point A on the outer raceway and the center of the jth roller in the circumferential direction, -arcsin⁡Dw/de≤Δθe≤arcsin⁡Dw/de.(10)Borj=RrjθB=Dw2+ΔDj2+∑m=1∞Crjmcos⁡mθB+mγ1+φrjm,(11)OA=ReθA=de2+Δde2+∑m=1∞Cemcos⁡mθA+φem,whereθA is the polar angle of any point A on the outer raceway.θB is the polar angle of any point B on the surface of the jth roller, which is related to the location of the jth roller and B point, as described in next paragraph.Figure 2
Geometric relationship between a roller and the outer raceway.
(a)
(b)
(c)
(d)When thejth roller is located in the third quadrant and B point is located above the line Oorj, the geometric relationship between the jth roller and the outer raceway is shown in Figure 2(a), θB=ϕej+βj-π; thereinto, ϕej=arccos⁡Aorj2+Oorj2-OA2/2Aorj×Oorj, βj is the position angle of rollers, and βj=2πj-1/Z+γ2, Z is roller number. When the jth roller is located in the third quadrant and B point is located below the line Oorj, the geometric relationship between the jth roller and the outer raceway is shown in Figure 2(b),θB=π-ϕej+βj. When the jth roller is located in the fourth quadrant and B point is located above the line Oorj, the geometric relationship between the jth roller and the outer raceway is shown in Figure 2(c), θB=βj-π-ϕej. When the jth roller is located in the fourth quadrant and B point is located below the line Oorj, the geometric relationship between the jth roller and the outer raceway is shown in Figure 2(d),θB=βj-π+ϕej.In (8), when the angleΔθevaries in a certain range, there is a point A on the outer raceway and a point B on surface of the jth roller, which make the distance between point A and point B reach a minimum value, namely, the shortest distance between the outer raceway and the surface of the jth roller.When the shortest distance between the outer raceway and surface of thejth roller is more than given tolerance, the jth roller does not contact with the outer raceway, continuing to move the jth roller along the radial direction. When the absolute value of the shortest distance is less than given tolerance, the jth roller contacts with the outer raceway. Thus, the distanceDejbetween the center of the jth roller and the center of the outer raceway is obtained. According to the geometric relationship among bearing components, the coordinates of the center of the jth roller are calculated by (12). In the same way, the coordinates of the center of each roller at the bottom of the bearing are calculated. (12)Xrj=Dejcos⁡βj,Yrj=Dejsin⁡βj.


### 5.2. Contact Statuses of the Inner Raceway in Different Positions

When the inner ring is located in a position, the contact status of the inner raceway depends on the position relationship (contact, separation, and interference) between the inner raceway and each roller at the bottom of the bearing. In order to obtain the position relationship between the inner raceway and each roller, a calculation method for the shortest distance between the surface of a roller and the inner raceway is proposed.When the inner ring rotates an angleα, and it is shifted to a position, the geometric relationship between the jth roller and the inner raceway is shown in Figure 3. The distance between any point C on the inner raceway and any point D on the surface of the jth roller is expressed by (13)CD=Corj-Dorj,where Corj is the distance between any point C on the inner raceway and the center of the jth roller, which is calculated by (14). Dorj is the radius of any point D on the surface of the jth roller when the jth roller spins around its axis at an angle of γ1, which is given by (15).(14)Corj=Coi2+oiorj2-2Coi×oiorjcos⁡Δθi,where Coi is the radius of C point on the inner raceway, which is expressed by (16). oiorj is the distance between the center of the inner raceway and the center of the jth roller.Δθi is the angle between any point C on the inner raceway and the center of the jth roller in the circumferential direction, -arcsin⁡Dw/di≤Δθi≤arcsin⁡Dw/di. (15)Dorj=RrjθD=Dw2+ΔDj2+∑m=1∞Arjmsin⁡mθD+mγ1+φrjm,(16)Coi=RiθC=di2+Δdi2+∑m=1∞Cimsin⁡mθC-mα+φim,where θC is the polar angle of any point C on the inner raceway. θD is the polar angle of any point D on the surface of the jth roller, which is related to the location of the jth roller and D point, as described in next paragraph.Figure 3
Geometric relationship between a roller and the inner raceway.
(a)
(b)
(c)
(d)When thejth roller is located in the third quadrant and D point is located above the line oiorj, the geometric relationship between the jth roller and the inner raceway is shown in Figure 3(a), θD=ψj+ϕij; thereinto, ϕij=arccos⁡Corj2+oiorj2-Coi2/2Corj×oiorj, ψj=arctan⁡Yi-Yrj/Xi-Xrj, and Xi and Yi are the coordinates of the center of the inner raceway. When the jth roller is located in the third quadrant and D point is located below the line oiorj, the geometric relationship between the inner raceway and the jth roller is shown in Figure 3(b), θD=ψj+2π-ϕij. When the jth roller is located in the fourth quadrant and D point is located above the line oiorj, the geometric relationship between the inner raceway and the jth roller is shown in Figure 3(c), θD=π-ψj-ϕij. When the jth roller is located in the fourth quadrant and D point is located below the line oiorj, the geometric relationship between the inner raceway and the jth roller is shown in Figure 3(d), θD=π-ψj+ϕij.In (13), when the angle Δθi varies in a certain range, there is a point C on the inner raceway and a point D on the surface of the jth roller, which make the distance between point C and point D reach a minimum value, namely, the shortest distance between the inner raceway and the surface of the jth roller. When the shortest distance is more than given tolerance, the jth roller is separate from the inner raceway. When the absolute value of the shortest distance is less than given tolerance, the jth roller contacts with the inner raceway. When the shortest distance is less than negative given tolerance, there is interference between the jth roller and the inner raceway. Therefore, the position relationship between the jth roller and the inner raceway is obtained.In the same way, the position relationship between the inner raceway and each roller at the bottom of the bearing is obtained, and the number and position angle of rollers which contact with the inner raceway are obtained. Therefore, the contact status of the inner raceway is obtained when the inner ring is located in a position. The inner ring is moved to a new position, and so on; the contact statuses of the inner raceway are obtained in different positions of the inner ring.


### 5.3. Coordinates of the Center of the Inner Raceway

In order to obtain the stable position of the inner ring from different positions when the inner ring rotates an angleα, according to the conditions that the stable position of the inner ring satisfies, the criterion for stable contact between the inner raceway and rollers is presented as follows:(
1
) Noninterference between the inner raceway and rollers: To obtain the stable position of the inner ring, different positions of the inner ring are given. When the inner ring is located in some positions, there is interference between the inner raceway and rollers. Obviously, there is no interference among bearing components when the inner ring is located in the stable position.(
2
) On any side of y-axis, there is at least one roller contacting with the inner raceway. Due to the gravity and the measuring load, when one roller contacts with the inner raceway, the inner ring is not stable. When two rollers are located on one side of y-axis, the inner ring is not stable, too. When there is at least one roller contacting with the inner raceway on each side of y-axis, the inner ring is stable.(
3
) The angle α1 between the negative half-axis of y-axis and the angular bisector of the angle between the leftmost and rightmost roller which contact with the inner raceway is the smallest. When the angle α1 is getting smaller, the center of the inner ring is getting closer to the y-axis and the potential energy of the inner ring is getting smaller. There are many positions of the inner ring which meet the above two conditions. In these positions of the inner ring, the inner ring is stable in this position where the angle α1 is the smallest. Two positions of the inner ring are shown in Figure 4. The angle α1 is less in the first position than in the second position. Therefore, the first position is the stable position of the inner ring.Figure 4
Contact status of the inner raceway.
(a)
The first contact status of the inner raceway
(b)
The second contact status of the inner racewayAccording to the contact statuses of the inner raceway in different positions, the stable position of the inner ring is discriminated from some positions by the criterion for stable contact between the inner raceway and rollers. Thus, the coordinates of the center of the inner raceway are obtained when the inner ring rotates an angleα. And so on, the coordinates of the center of the inner raceway are obtained every rotation angle of the inner ring.


## 5.1. Calculation of Coordinates of the Center of Rollers

To obtain the coordinates of the center of rollers at the bottom of the bearing when the rollers contact with the outer raceway, it is assumed that a roller at the bottom of the bearing is moved along the radial direction at a given step until it contacts with the outer raceway; thus the distance between the center of the roller and the center of the outer raceway is obtained. According to the geometric relationship among bearing components, the coordinates of the center of the roller are calculated through the distance. Hereinto, the key problem is to calculate the shortest distance between the outer raceway and the surface of the roller when the roller is moved to a position, thus determining whether the roller contacts with the outer raceway. For this purpose, a calculation method of the shortest distance is proposed.The inner ring rotates an angleα, and the jth roller is moved to a position with the distance Dej between the center of the jth roller and the center of the outer raceway. The geometric relationship between a roller and the outer raceway is shown in Figure 2. The distance between any point A on the outer raceway and any point B on the surface of the jth roller is expressed by (8)AB=Aorj-Borj,where Aorj is the distance between the center of the jth roller and any point A on the outer raceway, which is calculated by (9). Borj is the radius of any point B on the surface of the jth roller when the jth roller spins around its axis at an angle of γ1, which is calculated by (10). (9)Aorj=OA2+Oorj2-2OA×Oorjcos⁡Δθe,where OA is the radius of any point A on the outer raceway, which is calculated by (11). Oorj is the distance between the center of the outer raceway and the center of the jth roller, which is known, namely, Dej. Δθe is the angle between any point A on the outer raceway and the center of the jth roller in the circumferential direction, -arcsin⁡Dw/de≤Δθe≤arcsin⁡Dw/de.(10)Borj=RrjθB=Dw2+ΔDj2+∑m=1∞Crjmcos⁡mθB+mγ1+φrjm,(11)OA=ReθA=de2+Δde2+∑m=1∞Cemcos⁡mθA+φem,whereθA is the polar angle of any point A on the outer raceway.θB is the polar angle of any point B on the surface of the jth roller, which is related to the location of the jth roller and B point, as described in next paragraph.Figure 2
Geometric relationship between a roller and the outer raceway.
(a)
(b)
(c)
(d)When thejth roller is located in the third quadrant and B point is located above the line Oorj, the geometric relationship between the jth roller and the outer raceway is shown in Figure 2(a), θB=ϕej+βj-π; thereinto, ϕej=arccos⁡Aorj2+Oorj2-OA2/2Aorj×Oorj, βj is the position angle of rollers, and βj=2πj-1/Z+γ2, Z is roller number. When the jth roller is located in the third quadrant and B point is located below the line Oorj, the geometric relationship between the jth roller and the outer raceway is shown in Figure 2(b),θB=π-ϕej+βj. When the jth roller is located in the fourth quadrant and B point is located above the line Oorj, the geometric relationship between the jth roller and the outer raceway is shown in Figure 2(c), θB=βj-π-ϕej. When the jth roller is located in the fourth quadrant and B point is located below the line Oorj, the geometric relationship between the jth roller and the outer raceway is shown in Figure 2(d),θB=βj-π+ϕej.In (8), when the angleΔθevaries in a certain range, there is a point A on the outer raceway and a point B on surface of the jth roller, which make the distance between point A and point B reach a minimum value, namely, the shortest distance between the outer raceway and the surface of the jth roller.When the shortest distance between the outer raceway and surface of thejth roller is more than given tolerance, the jth roller does not contact with the outer raceway, continuing to move the jth roller along the radial direction. When the absolute value of the shortest distance is less than given tolerance, the jth roller contacts with the outer raceway. Thus, the distanceDejbetween the center of the jth roller and the center of the outer raceway is obtained. According to the geometric relationship among bearing components, the coordinates of the center of the jth roller are calculated by (12). In the same way, the coordinates of the center of each roller at the bottom of the bearing are calculated. (12)Xrj=Dejcos⁡βj,Yrj=Dejsin⁡βj.


## 5.2. Contact Statuses of the Inner Raceway in Different Positions

When the inner ring is located in a position, the contact status of the inner raceway depends on the position relationship (contact, separation, and interference) between the inner raceway and each roller at the bottom of the bearing. In order to obtain the position relationship between the inner raceway and each roller, a calculation method for the shortest distance between the surface of a roller and the inner raceway is proposed.When the inner ring rotates an angleα, and it is shifted to a position, the geometric relationship between the jth roller and the inner raceway is shown in Figure 3. The distance between any point C on the inner raceway and any point D on the surface of the jth roller is expressed by (13)CD=Corj-Dorj,where Corj is the distance between any point C on the inner raceway and the center of the jth roller, which is calculated by (14). Dorj is the radius of any point D on the surface of the jth roller when the jth roller spins around its axis at an angle of γ1, which is given by (15).(14)Corj=Coi2+oiorj2-2Coi×oiorjcos⁡Δθi,where Coi is the radius of C point on the inner raceway, which is expressed by (16). oiorj is the distance between the center of the inner raceway and the center of the jth roller.Δθi is the angle between any point C on the inner raceway and the center of the jth roller in the circumferential direction, -arcsin⁡Dw/di≤Δθi≤arcsin⁡Dw/di. (15)Dorj=RrjθD=Dw2+ΔDj2+∑m=1∞Arjmsin⁡mθD+mγ1+φrjm,(16)Coi=RiθC=di2+Δdi2+∑m=1∞Cimsin⁡mθC-mα+φim,where θC is the polar angle of any point C on the inner raceway. θD is the polar angle of any point D on the surface of the jth roller, which is related to the location of the jth roller and D point, as described in next paragraph.Figure 3
Geometric relationship between a roller and the inner raceway.
(a)
(b)
(c)
(d)When thejth roller is located in the third quadrant and D point is located above the line oiorj, the geometric relationship between the jth roller and the inner raceway is shown in Figure 3(a), θD=ψj+ϕij; thereinto, ϕij=arccos⁡Corj2+oiorj2-Coi2/2Corj×oiorj, ψj=arctan⁡Yi-Yrj/Xi-Xrj, and Xi and Yi are the coordinates of the center of the inner raceway. When the jth roller is located in the third quadrant and D point is located below the line oiorj, the geometric relationship between the inner raceway and the jth roller is shown in Figure 3(b), θD=ψj+2π-ϕij. When the jth roller is located in the fourth quadrant and D point is located above the line oiorj, the geometric relationship between the inner raceway and the jth roller is shown in Figure 3(c), θD=π-ψj-ϕij. When the jth roller is located in the fourth quadrant and D point is located below the line oiorj, the geometric relationship between the inner raceway and the jth roller is shown in Figure 3(d), θD=π-ψj+ϕij.In (13), when the angle Δθi varies in a certain range, there is a point C on the inner raceway and a point D on the surface of the jth roller, which make the distance between point C and point D reach a minimum value, namely, the shortest distance between the inner raceway and the surface of the jth roller. When the shortest distance is more than given tolerance, the jth roller is separate from the inner raceway. When the absolute value of the shortest distance is less than given tolerance, the jth roller contacts with the inner raceway. When the shortest distance is less than negative given tolerance, there is interference between the jth roller and the inner raceway. Therefore, the position relationship between the jth roller and the inner raceway is obtained.In the same way, the position relationship between the inner raceway and each roller at the bottom of the bearing is obtained, and the number and position angle of rollers which contact with the inner raceway are obtained. Therefore, the contact status of the inner raceway is obtained when the inner ring is located in a position. The inner ring is moved to a new position, and so on; the contact statuses of the inner raceway are obtained in different positions of the inner ring.


## 5.3. Coordinates of the Center of the Inner Raceway

In order to obtain the stable position of the inner ring from different positions when the inner ring rotates an angleα, according to the conditions that the stable position of the inner ring satisfies, the criterion for stable contact between the inner raceway and rollers is presented as follows:(
1
) Noninterference between the inner raceway and rollers: To obtain the stable position of the inner ring, different positions of the inner ring are given. When the inner ring is located in some positions, there is interference between the inner raceway and rollers. Obviously, there is no interference among bearing components when the inner ring is located in the stable position.(
2
) On any side of y-axis, there is at least one roller contacting with the inner raceway. Due to the gravity and the measuring load, when one roller contacts with the inner raceway, the inner ring is not stable. When two rollers are located on one side of y-axis, the inner ring is not stable, too. When there is at least one roller contacting with the inner raceway on each side of y-axis, the inner ring is stable.(
3
) The angle α1 between the negative half-axis of y-axis and the angular bisector of the angle between the leftmost and rightmost roller which contact with the inner raceway is the smallest. When the angle α1 is getting smaller, the center of the inner ring is getting closer to the y-axis and the potential energy of the inner ring is getting smaller. There are many positions of the inner ring which meet the above two conditions. In these positions of the inner ring, the inner ring is stable in this position where the angle α1 is the smallest. Two positions of the inner ring are shown in Figure 4. The angle α1 is less in the first position than in the second position. Therefore, the first position is the stable position of the inner ring.Figure 4
Contact status of the inner raceway.
(a)
The first contact status of the inner raceway
(b)
The second contact status of the inner racewayAccording to the contact statuses of the inner raceway in different positions, the stable position of the inner ring is discriminated from some positions by the criterion for stable contact between the inner raceway and rollers. Thus, the coordinates of the center of the inner raceway are obtained when the inner ring rotates an angleα. And so on, the coordinates of the center of the inner raceway are obtained every rotation angle of the inner ring.


## 6. Verification of the Proposed Prediction Method

The proposed prediction method is verified through two particular cases in which the profiles of the inner raceway are circle and ellipse. For two particular cases, the motion error of the bearing can be derived theoretically. The prediction results obtained from the proposed prediction method are compared with the analytical results calculated by the derived analysis algorithms for motion error of the bearing. Main parameters of a cylindrical roller bearing with code NU209 is as shown in Table1.Table 1
Main parameters of cylindrical roller bearing.
Parameters
Value
Inner raceway diameter (mm)
54.991
Outer raceway diameter (mm)
75.032
Roller diameter (mm)
10
Number of roller
14


### 6.1. Verification of the Prediction Method with Inner Raceway Profile for Circle

#### 6.1.1. Analysis Algorithm for the Radial Runout of Inner Ring with Inner Raceway Profile for Circle

There is no geometric error in the inner raceway, the outer raceway, and rollers. Due to the existence of radial internal clearance, the inner raceway only contacts with two nethermost rollers while the bearing is not loaded, as indicated in Figure5. When the inner raceway, the outer raceway, and rollers are without geometric error, the analysis algorithm for the radial runout of inner ring is derived as follows.Figure 5
Geometric relationship among bearing components without geometric error.The inner ring rotates an angleα; the geometric relationship among bearing components is shown in Figure 5. According to the geometric relationship, (17) are obtained.(17)or1oi=Xi-Xr12+Yi-Yr12=di+Dw2,or2oi=Xi-Xr22+Yi-Yr22=di+Dw2,where Xr1, Yr1, Xr2, and Yr2 are the coordinates of the center of two nethermost rollers, Xr1=0.5de-Dwcos⁡β1, Yr1=0.5de-Dwsin⁡β1, Xr2=0.5de-Dwcos⁡β2, and Yr2=0.5de-Dwsin⁡β2; hereinto, β1, β2 are the position angle of rollers.The analysis algorithm for the radial runout of inner ring is composed of (17). Equations (17) are simultaneous nonlinear equations with unknown Xi and Yi. They may be solved by Newton-Raphson method. Having obtained Xi and Yi, the radial runout value of inner ring is obtained when the inner ring rotates an angel α.


#### 6.1.2. Verification of the Proposed Prediction Method

The correctness of the proposed prediction method for the radial runout of inner ring is validated via the analytical results obtained from the above analysis algorithm. According to the standard of rolling bearings-radial internal clearance, the radial internal clearance of cylindrical roller bearings with code NU209 varies in the range of 0.03~0.05 mm. Therefore, the radial internal clearances of 0.031 mm, 0.041 mm, and 0.05 mm are used to calculate the radial runout value of inner ring by the proposed prediction method and analytical algorithm, respectively. The inner ring rotates one revolution; the prediction results and the analytical results are shown in Figure6. Comparing prediction results with analytical results, the biggest relative error is 0.097%.Figure 6
Comparison between prediction results and analytical results.
(a)
Radial internal clearance for 0.031 mm
(b)
Radial internal clearance for 0.041 mm
(c)
Radial internal clearance for 0.05 mm


### 6.2. Verification of the Prediction Method with Inner Raceway Profile for Ellipse

#### 6.2.1. Analysis Algorithm for the Radial Runout of Inner Ring with Inner Raceway Profile for Ellipse

The profile of the inner raceway is ellipse, and profiles of other bearing components are circle. When the ellipticity in the inner raceway is less, the inner raceway only contacts with two nethermost rollers, as shown in Figure7. When the profile of the inner raceway is ellipse, the analysis algorithm for the radial runout of inner ring is derived as follows.Figure 7
Geometric relationship among bearing components with inner raceway profile for ellipse.The inner ring rotates an angleα; the geometric relationship among bearing components is shown in Figure 7. According to the geometric relationship, (18) and (19) are obtained.(18)or1oi=Xi-Xr12+Yi-Yr12=Eoi+Dw2,(19)or2oi=Xi-Xr22+Yi-Yr22=Foi+Dw2,where Eoi is the radius of E point on the inner raceway, which is calculated by (20). Foi is the radius of F point on the inner raceway, which is calculated by (21).(20)Eoi=abacos⁡θE-α2+bsin⁡θE-α2,(21)Foi=abacos⁡θF-α2+bsin⁡θF-α2,where a and b are semimajor and semiminor axis of ellipse, respectively. θE is the polar angle of point E on the inner raceway, θE=π+arctan⁡Yi-Yr1/Xi-Xr1. θF is the polar angle of point F on the inner raceway,θF=2π+arctan⁡Yi-Yr2/Xi-Xr2.The analysis algorithm for the radial runout of inner ring is composed of (18) and (21). Equations (18) through (21) are simultaneous nonlinear equations with unknown Xi and Yi. They may be solved by Newton-Raphson method. Having obtained Xi and Yi, the radial runout value of inner ring is obtained when the inner ring rotates an angel α.


#### 6.2.2. Verification of the Proposed Prediction Method

To verify the correctness of the proposed prediction method for the radial runout of inner ring, different ellipticity in the inner raceway is selected. The semimajor axis of ellipse is 27.4955 mm, and semiminor axis of ellipse is 27.4905 mm, 27.492 mm, and 27.4935 mm which are used to calculate the radial runout value of inner ring by the proposed prediction method and analysis algorithm, respectively. The inner ring rotates one revolution; the prediction results and the analytical results for the radial runout value of inner ring are shown in Figure8. Comparing prediction results with analytical results, the biggest relative error is 0.15%. The relative error with inner raceway profile for ellipse is larger than that with inner raceway profile for circle. This is due to the assumption that the curvature center of ellipse is approximate to geometric center of ellipse, which results in the error of the analysis algorithm with inner raceway profile for ellipse.Figure 8
Comparison between prediction results and analytical results.
(a)
Semiminor axis for 27.4905 mm
(b)
Semiminor axis for 27.492 mm
(c)
Semiminor axis for 27.4935 mmRegarding Figures7 and 8 results from this study, the validity of the proposed prediction method is proven by analytical results. The proposed prediction method can accurately forecast the radial runout of inner ring.


## 6.1. Verification of the Prediction Method with Inner Raceway Profile for Circle

### 6.1.1. Analysis Algorithm for the Radial Runout of Inner Ring with Inner Raceway Profile for Circle

There is no geometric error in the inner raceway, the outer raceway, and rollers. Due to the existence of radial internal clearance, the inner raceway only contacts with two nethermost rollers while the bearing is not loaded, as indicated in Figure5. When the inner raceway, the outer raceway, and rollers are without geometric error, the analysis algorithm for the radial runout of inner ring is derived as follows.Figure 5
Geometric relationship among bearing components without geometric error.The inner ring rotates an angleα; the geometric relationship among bearing components is shown in Figure 5. According to the geometric relationship, (17) are obtained.(17)or1oi=Xi-Xr12+Yi-Yr12=di+Dw2,or2oi=Xi-Xr22+Yi-Yr22=di+Dw2,where Xr1, Yr1, Xr2, and Yr2 are the coordinates of the center of two nethermost rollers, Xr1=0.5de-Dwcos⁡β1, Yr1=0.5de-Dwsin⁡β1, Xr2=0.5de-Dwcos⁡β2, and Yr2=0.5de-Dwsin⁡β2; hereinto, β1, β2 are the position angle of rollers.The analysis algorithm for the radial runout of inner ring is composed of (17). Equations (17) are simultaneous nonlinear equations with unknown Xi and Yi. They may be solved by Newton-Raphson method. Having obtained Xi and Yi, the radial runout value of inner ring is obtained when the inner ring rotates an angel α.


### 6.1.2. Verification of the Proposed Prediction Method

The correctness of the proposed prediction method for the radial runout of inner ring is validated via the analytical results obtained from the above analysis algorithm. According to the standard of rolling bearings-radial internal clearance, the radial internal clearance of cylindrical roller bearings with code NU209 varies in the range of 0.03~0.05 mm. Therefore, the radial internal clearances of 0.031 mm, 0.041 mm, and 0.05 mm are used to calculate the radial runout value of inner ring by the proposed prediction method and analytical algorithm, respectively. The inner ring rotates one revolution; the prediction results and the analytical results are shown in Figure6. Comparing prediction results with analytical results, the biggest relative error is 0.097%.Figure 6
Comparison between prediction results and analytical results.
(a)
Radial internal clearance for 0.031 mm
(b)
Radial internal clearance for 0.041 mm
(c)
Radial internal clearance for 0.05 mm


## 6.1.1. Analysis Algorithm for the Radial Runout of Inner Ring with Inner Raceway Profile for Circle

There is no geometric error in the inner raceway, the outer raceway, and rollers. Due to the existence of radial internal clearance, the inner raceway only contacts with two nethermost rollers while the bearing is not loaded, as indicated in Figure5. When the inner raceway, the outer raceway, and rollers are without geometric error, the analysis algorithm for the radial runout of inner ring is derived as follows.Figure 5
Geometric relationship among bearing components without geometric error.The inner ring rotates an angleα; the geometric relationship among bearing components is shown in Figure 5. According to the geometric relationship, (17) are obtained.(17)or1oi=Xi-Xr12+Yi-Yr12=di+Dw2,or2oi=Xi-Xr22+Yi-Yr22=di+Dw2,where Xr1, Yr1, Xr2, and Yr2 are the coordinates of the center of two nethermost rollers, Xr1=0.5de-Dwcos⁡β1, Yr1=0.5de-Dwsin⁡β1, Xr2=0.5de-Dwcos⁡β2, and Yr2=0.5de-Dwsin⁡β2; hereinto, β1, β2 are the position angle of rollers.The analysis algorithm for the radial runout of inner ring is composed of (17). Equations (17) are simultaneous nonlinear equations with unknown Xi and Yi. They may be solved by Newton-Raphson method. Having obtained Xi and Yi, the radial runout value of inner ring is obtained when the inner ring rotates an angel α.


## 6.1.2. Verification of the Proposed Prediction Method

The correctness of the proposed prediction method for the radial runout of inner ring is validated via the analytical results obtained from the above analysis algorithm. According to the standard of rolling bearings-radial internal clearance, the radial internal clearance of cylindrical roller bearings with code NU209 varies in the range of 0.03~0.05 mm. Therefore, the radial internal clearances of 0.031 mm, 0.041 mm, and 0.05 mm are used to calculate the radial runout value of inner ring by the proposed prediction method and analytical algorithm, respectively. The inner ring rotates one revolution; the prediction results and the analytical results are shown in Figure6. Comparing prediction results with analytical results, the biggest relative error is 0.097%.Figure 6
Comparison between prediction results and analytical results.
(a)
Radial internal clearance for 0.031 mm
(b)
Radial internal clearance for 0.041 mm
(c)
Radial internal clearance for 0.05 mm


## 6.2. Verification of the Prediction Method with Inner Raceway Profile for Ellipse

### 6.2.1. Analysis Algorithm for the Radial Runout of Inner Ring with Inner Raceway Profile for Ellipse

The profile of the inner raceway is ellipse, and profiles of other bearing components are circle. When the ellipticity in the inner raceway is less, the inner raceway only contacts with two nethermost rollers, as shown in Figure7. When the profile of the inner raceway is ellipse, the analysis algorithm for the radial runout of inner ring is derived as follows.Figure 7
Geometric relationship among bearing components with inner raceway profile for ellipse.The inner ring rotates an angleα; the geometric relationship among bearing components is shown in Figure 7. According to the geometric relationship, (18) and (19) are obtained.(18)or1oi=Xi-Xr12+Yi-Yr12=Eoi+Dw2,(19)or2oi=Xi-Xr22+Yi-Yr22=Foi+Dw2,where Eoi is the radius of E point on the inner raceway, which is calculated by (20). Foi is the radius of F point on the inner raceway, which is calculated by (21).(20)Eoi=abacos⁡θE-α2+bsin⁡θE-α2,(21)Foi=abacos⁡θF-α2+bsin⁡θF-α2,where a and b are semimajor and semiminor axis of ellipse, respectively. θE is the polar angle of point E on the inner raceway, θE=π+arctan⁡Yi-Yr1/Xi-Xr1. θF is the polar angle of point F on the inner raceway,θF=2π+arctan⁡Yi-Yr2/Xi-Xr2.The analysis algorithm for the radial runout of inner ring is composed of (18) and (21). Equations (18) through (21) are simultaneous nonlinear equations with unknown Xi and Yi. They may be solved by Newton-Raphson method. Having obtained Xi and Yi, the radial runout value of inner ring is obtained when the inner ring rotates an angel α.


### 6.2.2. Verification of the Proposed Prediction Method

To verify the correctness of the proposed prediction method for the radial runout of inner ring, different ellipticity in the inner raceway is selected. The semimajor axis of ellipse is 27.4955 mm, and semiminor axis of ellipse is 27.4905 mm, 27.492 mm, and 27.4935 mm which are used to calculate the radial runout value of inner ring by the proposed prediction method and analysis algorithm, respectively. The inner ring rotates one revolution; the prediction results and the analytical results for the radial runout value of inner ring are shown in Figure8. Comparing prediction results with analytical results, the biggest relative error is 0.15%. The relative error with inner raceway profile for ellipse is larger than that with inner raceway profile for circle. This is due to the assumption that the curvature center of ellipse is approximate to geometric center of ellipse, which results in the error of the analysis algorithm with inner raceway profile for ellipse.Figure 8
Comparison between prediction results and analytical results.
(a)
Semiminor axis for 27.4905 mm
(b)
Semiminor axis for 27.492 mm
(c)
Semiminor axis for 27.4935 mmRegarding Figures7 and 8 results from this study, the validity of the proposed prediction method is proven by analytical results. The proposed prediction method can accurately forecast the radial runout of inner ring.


## 6.2.1. Analysis Algorithm for the Radial Runout of Inner Ring with Inner Raceway Profile for Ellipse

The profile of the inner raceway is ellipse, and profiles of other bearing components are circle. When the ellipticity in the inner raceway is less, the inner raceway only contacts with two nethermost rollers, as shown in Figure7. When the profile of the inner raceway is ellipse, the analysis algorithm for the radial runout of inner ring is derived as follows.Figure 7
Geometric relationship among bearing components with inner raceway profile for ellipse.The inner ring rotates an angleα; the geometric relationship among bearing components is shown in Figure 7. According to the geometric relationship, (18) and (19) are obtained.(18)or1oi=Xi-Xr12+Yi-Yr12=Eoi+Dw2,(19)or2oi=Xi-Xr22+Yi-Yr22=Foi+Dw2,where Eoi is the radius of E point on the inner raceway, which is calculated by (20). Foi is the radius of F point on the inner raceway, which is calculated by (21).(20)Eoi=abacos⁡θE-α2+bsin⁡θE-α2,(21)Foi=abacos⁡θF-α2+bsin⁡θF-α2,where a and b are semimajor and semiminor axis of ellipse, respectively. θE is the polar angle of point E on the inner raceway, θE=π+arctan⁡Yi-Yr1/Xi-Xr1. θF is the polar angle of point F on the inner raceway,θF=2π+arctan⁡Yi-Yr2/Xi-Xr2.The analysis algorithm for the radial runout of inner ring is composed of (18) and (21). Equations (18) through (21) are simultaneous nonlinear equations with unknown Xi and Yi. They may be solved by Newton-Raphson method. Having obtained Xi and Yi, the radial runout value of inner ring is obtained when the inner ring rotates an angel α.


## 6.2.2. Verification of the Proposed Prediction Method

To verify the correctness of the proposed prediction method for the radial runout of inner ring, different ellipticity in the inner raceway is selected. The semimajor axis of ellipse is 27.4955 mm, and semiminor axis of ellipse is 27.4905 mm, 27.492 mm, and 27.4935 mm which are used to calculate the radial runout value of inner ring by the proposed prediction method and analysis algorithm, respectively. The inner ring rotates one revolution; the prediction results and the analytical results for the radial runout value of inner ring are shown in Figure8. Comparing prediction results with analytical results, the biggest relative error is 0.15%. The relative error with inner raceway profile for ellipse is larger than that with inner raceway profile for circle. This is due to the assumption that the curvature center of ellipse is approximate to geometric center of ellipse, which results in the error of the analysis algorithm with inner raceway profile for ellipse.Figure 8
Comparison between prediction results and analytical results.
(a)
Semiminor axis for 27.4905 mm
(b)
Semiminor axis for 27.492 mm
(c)
Semiminor axis for 27.4935 mmRegarding Figures7 and 8 results from this study, the validity of the proposed prediction method is proven by analytical results. The proposed prediction method can accurately forecast the radial runout of inner ring.


## 7. Simulation Examples

Taking NU209 type cylindrical roller bearing as an example, as shown in Table1, numerical examples are provided to verify the universality of the proposed prediction method.Figure9 shows the radial runout value of inner ring when the inner ring rotates 540 degrees; hereinto, the amplitude of roundness error in the inner raceway is 1 μm, and the harmonic order of roundness error is 10. The maximal radial runout value of inner ring is 0.0217 mm, and the minimal radial runout value of outer ring is 0.01956 mm. The radial runout of inner ring is 2.14 μm.Figure 9
Radial runout value of inner ring.Figure10 shows radial runout value of inner ring when the inner ring rotates 540 degrees; hereinto, the amplitude of roundness error in the outer raceway is 1 μm, and the harmonic order of roundness error is 12. The maximal radial runout value of inner ring is 0.02013 mm, and the minimal radial runout value of inner ring is 0.02157 mm. The radial runout of inner ring is 1.44 μm.Figure 10
Radial runout value of inner ring.Figure11 shows radial runout value of inner ring when the inner ring rotates 540 degrees; hereinto, the amplitude of roundness error in all rollers is 0.5 μm, and the harmonic order of roundness error is 8. The maximal radial runout value of inner ring is 0.02153 mm, and the minimal radial runout value of inner ring is 0.01956 mm. The radial runout of inner ring is 1.98 μm.Figure 11
Radial runout value of inner ring.Figure12 shows radial runout value of inner ring when the inner ring rotates 540 degrees; hereinto, the amplitudes of roundness error in all rollers, the inner raceway, and the outer raceway are 0.5 μm, 1 μm, and 1 μm, respectively, and the harmonic orders of roundness error in all rollers, the inner raceway, and the outer raceway are 15, 8, and 18, respectively. The dimension errors in all rollers, the inner raceway, and the outer raceway are 1 μm. The maximal radial runout value of inner ring is 0.02186 mm, and the minimal radial runout value of inner ring is 0.01668 mm. The radial runout of inner ring is 5.186 μm.Figure 12
Radial runout value of inner ring.The radial runout values of inner ring are calculated by the proposed prediction method when there are roundness errors in the inner raceway, the outer raceway, and the rollers. The universality of the proposed prediction method is verified by numerical examples. Therefore, the proposed prediction method can predict the radial runout of inner ring with random dimension and roundness errors in the inner raceway, outer raceway, and rollers.


## 8. Conclusions

A prediction method for the radial runout of inner ring in cylindrical roller bearings is presented in this paper. The correctness of the proposed prediction method is validated via two particular cases that the inner raceway shapes are circle and ellipse. Two analytical results of particular cases for the radial runout value of inner ring are derived, and the results show great coincidence with the prediction results obtained from the proposed prediction method. Numerical examples are provided to verify the universality of the proposed prediction method. This method can predict the radial runout of inner ring after bearing components with dimension and form errors are assembled. Based on the proposed prediction method, the coupling effect of the roller number and roundness errors in the inner raceway, the outer raceway, and rollers on rotational accuracy of assembled bearing can be investigated efficiently. Furthermore, the relationship between harmonic order of roundness error and the roller number which has a great effect on the radial runout of inner ring is established, which may provide the theoretical basis for the precision design and the control of machining process parameters of bearing components.



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*Source: 6584561-2017-05-31.xml*