Dynamic modeling and vibration response simulations of angular contact ball bearings with ball defects considering the three-dimensional motion of balls

Author’s Accepted Manuscript Dynamic modeling and vibration response simulations of angular contact ball bearings with ball defects considering the three-dimensional motion of balls Linkai Niu, Hongrui Cao, Xiaoyan Xiong www.elsevier.com/locate/jtri

PII: DOI: Reference:

S0301-679X(16)30484-4 http://dx.doi.org/10.1016/j.triboint.2016.12.011 JTRI4495

To appear in: Tribiology International Received date: 22 September 2016 Revised date: 28 November 2016 Accepted date: 8 December 2016 Cite this article as: Linkai Niu, Hongrui Cao and Xiaoyan Xiong, Dynamic modeling and vibration response simulations of angular contact ball bearings with ball defects considering the three-dimensional motion of balls, Tribiology International, http://dx.doi.org/10.1016/j.triboint.2016.12.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Dynamic modeling and vibration response simulations of angular contact ball bearings with ball defects considering the three-dimensional motion of balls Linkai Niua,*, Hongrui Caob, Xiaoyan Xionga a.

Key Laboratory of Advanced Transducers and Intelligent Control System, Ministry of Education, Taiyuan University of Technology, Taiyuan 030024, China

b.

Stake Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710049, China

*Corresponding author. Tel/Fax: +86 0351 6014551. ([email protected]) ABSTRACT A dynamic model to investigate the dynamics and vibration responses of angular contact ball bearings with ball defects is proposed. In the proposed model, each ball has 6 degrees of freedom (DOFs), meaning that the three-dimensional motion of balls is considered. The results show that whether a ball hit a race and the frequency property of the envelope spectrum largely depend on the geometrical characteristics and operating conditions. This increases the difficulty in diagnosing the ball defect when using vibration-based monitoring techniques. The current investigation suggests that the vibration response simulations should be carried out to investigate the vibration mechanism and to diagnose the ball defect more effectively. Keywords: Angular contact ball bearings; ball defect; bearing dynamics; fault diagnosis Nomenclature a , b : Lengths of the semi-major axis and the semi-minor axis of the contact ellipse. A , B , C , D : Four orbital positons of a ball. A1 , A2 : The left and the right ends of the contact ellipse. c y , cz :The damping coefficients of the bearing housing along the yi and zi directions. d m : Pitch diameter of the bearing. D : Ball diameter. f bdf : Ball defect frequency. f i : Groove curvature factor of inner race. f o : Groove curvature factor of outer race. f s : Rotation frequency of the bearing. Fhy , Fhz : The forces acting on the bearing housing along the yi and zi directions. F : Force vector. J : Principal moment of inertial. k y , k z : The stiffness coefficients of the bearing housing along the yi and zi directions. K : Hertzian contact stiffness coefficient. m : Mass of a bearing component. mh : Mass of the bearing housing. M : Moment vector. Pmax : The maximum Hertzian contact stress at the contact ellipse. 1

Q : Contact force. rb : Position vector locating the ball center relative to the origin of the inertial frame. rbc : Position vector locating the ball center relative to the race groove curvature center.

rbcb : The vector rbc described in the ball-fixed frame. rbcd : The vector rbc described in the defect frame. rbdd : Position vector locating the ball center relative to the defect center, described in the defect frame. rbr : Position vector locating the ball center relative to the race center. rcp : Position vector locating the race groove curvature center relative to a point p on the edge of

the defect. rcpd : The vector rcp described in the defect frame. rcr : Position vector locating the race groove curvature center relative to the race center.

rpd : Position vector locating a point p on the edge of the defect relative to the defect center. d rpd : The vector rpd described in the defect frame. rr : Position vector locating the race center relative to the origin of the inertial frame.

r : Acceleration vector. Tbd : Transformation matrix from the ball-fixed frame to the defect frame. Tib : Transformation matrix from the inertial frame to the ball-fixed frame. u : Relative sliding velocity. vx , v y : Relative sliding velocities along the xk and yk directions. wd : Defect width. yh , zh : The displacements of the bearing housing along the yi and zi directions. yh , zh : the velocities of the bearing housing along the yi and zi directions. yh , zh :The accelerations of the bearing housing along the yi and zi directions. z : Number of balls.  : Contact angle.  i : Contact angle of inner race. ini : Initial contact angle.  o : Contact angle of outer race.  b : Angle between the zd axis and the zb axis.  b : Angle between the vector rbc and the zd axis.  d : Circumference angle of the defect in the plane xb zb .  b : Geometrical interaction between a race and a ball.  bp : Geometrical interaction between a race and a ball defect. b1 : Orbital position of ball 1.  : Angle between the yd axis and the vector rpd .  : Friction coefficient.  : Angle between the speed ωb and the rotation axis of the bearing. i : Rotation speed of the inner ring. ωb : Rotation speed of a ball about its own axis. ω : Angular acceleration. Ob xb yb zb : Ball-fixed frame. Od xd yd zd : Defect frame. Oi xi yi zi : Inertial frame. Ok xk yk zk : The frame established at the contact ellipse. Or xr yr zr : Race-fixed frame. ,  ,  : Three attitude angles of a frame relative to another frame. 2

b , b , b : Three attitude angles of the ball-fixed frame relative to the inertial frame. r , r , r : Three attitude angles of the race-fixed frame relative to the inertial frame.

1 Introduction Rolling element bearings as one of the most important components of the system, have been widely used in modern industrial machineries. Bearing defects largely threaten the safety and the reliability of the whole machine. As the vibration response of the bearing could be significantly changed when the bearing has surface defects on the races and the rolling elements, vibration-based monitoring has been widely investigated by many researchers to diagnose bearing defects. Extracting the fault features indicating the occurrence of bearing defects from the vibration signals using advanced signal processing techniques is one of the key procedures of vibration-based fault diagnosis. Moreover, the vibration response simulations of bearings with certain defects under certain operating conditions are useful to understand the generation mechanism of the fault features in the vibration signals and to diagnose the bearing defect more effectively [1-3]. Dynamic analysis is capable of investigating the vibration response mechanism of a bearing with certain defects and is crucial to vibration-based fault diagnosis. Because of the importance of dynamic analysis, many researchers have proposed certain dynamic models to investigate the vibration responses of defective bearings. In the published models, the dynamics and vibration responses of ball bearings with race defects have been widely investigated [4-24]. Some of the important issues that have been studied in the available investigations include the effects of defect edge [4], multi-defect race [7], high-speed condition [11, 12] and race inhomogeneity [23] on the vibrations of a ball bearing. Moreover, the dynamics as a ball rolls over a race defect [5, 11, 12, 15, 24], the effects of race defect on bearing stiffness [14] and contact stiffness [6, 10], the wear evolution of race [16, 17] and the impact force generated due to race defects [9, 18, 19] have also been studied. However, compared with race defects, there are relatively few investigations on the dynamics and vibration responses for ball bearings with ball defects. Choudhury [25] studied the vibration responses of a radial loaded bearing using a discrete spring-mass-dashpot system that has 3 degrees of freedom (DOFs). In Choudhury’s model, the defect is modeled as a series of pulses of excitation forces. Sassi [26] also proposed a lumped spring-mass system with 3 DOFs to study the effect of ball defects on the vibrations of a ball bearing. In Sassi’s model, the defect is modeled as a series of impact forces that are determined using the kinematic energy method. Sawalhi [27, 28] proposed a dynamic model to investigate the effects of bearing defects on the vibrations of a bearing-gear-rotor system. In Sawalhi’s model, the defect is modeled by changing the geometrical properties of the bearing components (i.e., races and rolling elements). Arslan [29] proposed a spring-mass model to investigate the effect of the defects on the vibrations of a shaft supported by 3

angular contact ball bearings. The defect is modeled by considering additional clearance that is introduced by the defect. Rafsanjani [30] investigated the nonlinear dynamics and stabilities of a bearing-shaft system considering the effect of race and ball defects. Mishra [31] developed a dynamic model to take account of the effects of the race and the ball defects on the dynamics of a deep groove ball bearing based on the bond graph method. Additionally, Yuan [32] investigated the vibration responses of a ball bearing-rotor system when the bearing has single and compound multi-defects (including ball defects). In the published works on ball defects [25-32], it is assumed that the ball only has planar motions, that the defect rotates around the ball axis at the ball defect frequency (BDF) that is given by Eq. (1) (Eq. (1) is given for a bearing with stationary outer ring) and that the ball is capable of striking the outer and inner races as it rotates a cycle around the ball axis. f bdf 

2   fsdm   D 1   cos    2D   dm   

(1)

where f s is the rotation frequency of the bearing, d m is the pitch diameter, D is the ball diameter, and  is the contact angle. However, Eq. (1) is determined based on simple kinematics and pure rolling assumptions [33], and the motion of a ball is constricted on a plane when deriving Eq. (1). Moreover, the contact angles of the inner race (  i ) and the outer race (  o ) are assumed to be equal when deriving Eq. (1). The angle between the rotation speed of a ball about its own axis ( ωb ) and the rotation axis of the bearing (i.e., the angle  shown in Fig. 1) equals the contact angles of the inner and outer races, as shown in Fig. 1(a). This ensures that the defect on a ball is capable of striking both the outer and inner races as it rotates a cycle about the ball axis, which is the basic assumption for published works on ball defects [25-32]. However, the actual motion of a ball in angular contact ball bearings is rather complex, and the motion of a ball is a combination of rolling and sliding with the total effect of the centrifugal force, gyroscopic moment, and spinning motion (refer to Fig. 1(b)). These complex dynamics make the angles  i ,  o and  unequal [34] (as shown in Fig. 1(b)). The difference in these angles makes the ball defect such that it may not strike one of the races when the ball defect rotates around the ball axis at the speed of ωb . This property makes the vibration responses of an angular contact ball bearing with ball defects rather complex and increases the difficulty in diagnosing the ball defects using vibration-based monitoring techniques. Indeed, for angular contact ball bearings,  i is generally larger than  o under the action of the centrifugal force. Moreover, the angle  significantly depends on the moments acting on the ball that are mainly determined based on the friction force and also the position vector that locates the contacting point relative to the ball center [35]. These factors are also functions of the geometrical parameters and the operating conditions of the bearing [35]. In order to reasonably investigate these complex mechanical properties, the motion of a ball should be described in the 4

three-dimensional space (thus, each ball has 6 DOFs) To date, many researchers have developed complex dynamic models to take account of the effects of the relative slip, the centrifugal force, and the gyroscopic and spinning motions of balls. One such model was proposed by Gupta [36-38]. In the Gupta model, the motion of each bearing component is described in three-dimensional space (each bearing component has 6 DOFs), and the centrifugal force and the gyroscopic and the spinning motions of a ball are all considered. Moreover, the lubrication and the cage effects can also be studied using the Gupta model. However, the localized defects are not considered in the Gupta model. In recent years, Niu [11, 12] and Wang [39] extended the Gupta model to investigate the effect of race defects on the dynamics of ball bearings [11, 12] and roller bearings [39]. However, in Niu’s model [11, 12], the effect of the ball defect was not considered. The aim of the current paper is to develop a dynamic model to investigate the dynamics and vibration responses of an angular contact ball bearing with ball defects based on the Gupta model and to discuss the effect of the three-dimensional motion of a ball on the vibration responses of bearings with ball defects for fault diagnosis purpose. The defect modeled in this paper is the material absence due to pitting and spall. In the proposed model, the ball defect is modeled in the three-dimensional space. As the model is proposed based on Gupta model, the motion of a ball is described in the three-dimensional space (i.e., each ball has 6 DOFs), and the effects of the relative slip, the centrifugal force and the gyroscopic and the spinning motions can be considered. The effects of the three-dimensional motion of a ball on the vibration responses of the bearing with ball defects are discussed both in time and frequency domains. Because the envelope spectrum is widely used to diagnose the bearing faults in vibration-based monitoring techniques, vibrations in the frequency domain are mainly discussed by the envelope spectrum in the current paper.

2 Model development In this paper, the vibration of an angular contact ball bearing with ball defects is investigated based on the Gupta model. The basic concept and modeling procedure of the Gupta model are discussed in Section 2.1. The Gupta model has been well described in Refs. [36-38] and had been verified experimentally [38] however, in order to make the current paper more consistent, only the parts that are essential to the current investigation (especially for the modeling of the ball defect discussed in Section 2.2) are discussed. Moreover, because the interaction between a race and a ball (also a ball defect) is the main focus of the current paper, only the ball/race interaction is discussed in Section 2.1. Discussions on the ball/cage pocket and cage/guiding ring interactions can be found in Refs. [36, 38]. Additionally, in practice, the vibration signal is always collected by accelerometers mounted on the bearing housing that connects the outer race. The modeling of the bearing housing in the 5

current investigation is discussed in Section 2.3. 2.1 Gupta model The ball/race interaction is shown in Fig. 2. The ball-fixed frame Ob xb yb zb and the race-fixed frame Or xr yr zr (these body-fixed frames are fixed in the ball and the race and move together with the movements of these components) are located at the center of the ball and the race. In the inertial frame Oi xi yi zi , the positions of the ball center Ob and the race center Or can be described by vectors rb and rr , respectively. Vectors rb and rr are known at the beginning of each time step of the numerical integration of dynamic equations. Moreover, the attitude angles of the ball-fixed frame and the race-fixed frame relative to the inertial frame, i.e., b , b , b  and

r , r , r  are also known at the beginning of each time step. Thus, the relative position between the ball center and the race center is rbr  rb  rr

(2)

Additionally, the relative position between the race groove curvature center (point c shown in Fig. 2) and the ball center can be written as rbc  rbr  rcr

(3)

where the vector rcr locates the position of the race groove curvature center relative to the race center that is also known at the beginning of each time step. Next, the geometrical interaction between the race and the ball can be given as   rbc  f o D for outer race   rbc  f i D for inner race

b  

(4)

where f o and f i are the groove curvature factors of the outer and inner races, respectively. Based on the geometrical interaction, the contact force can be determined by using the Hertzian point contact theory [33]: Q  K b1.5

(5)

where K is the Hertzian contact stiffness coefficient. The maximum contact stress can be determined as [33] Pmax 

3Q 2πab

(6)

where a and b are the lengths of the semi-major and semi-minor axes of the contact ellipse, respectively. In Eqs. (5) and (6), K , a and b largely depend on the material and geometrical properties of the ball and the race groove, and can be determined based on the Hertzian point contact theory [33]. When the contact force is determined, the friction force can be determined by multiplying the contact force by the friction coefficient. When calculating the friction force, the distribution of relative sliding velocities in the contact ellipse, which affects the friction coefficient and the 6

direction of the friction force, should be determined. Fig. 3 schematically shows the distribution of relative sliding velocities in the contact ellipse. In Fig. 3, Ok is the center of the contact ellipse. The xk and the yk axes are along the major and the minor axes of the contact ellipse, respectively. The relative sliding velocities between a ball and a race in the contact ellipse are the functions of the abscissa xk :  vx  vx ( xk )  v y  v y ( xk )

(7)

where vx and v y are the relative sliding velocities along the xk and the yk directions, respectively (Fig. 3). Generally, there are 0, 1, or 2 solutions for the equation vy ( xk )  0 based on the operational and the geometrical parameters of the bearing. The point which satisfies the equation vy ( xk )  0 can be recognized as the rolling center [35] (there is one rolling center in Fig. 3(a), and two rolling centers exist in Fig. 3(b)). The rolling center divides the contact ellipse into several sub-regions, and the directions of v y in two adjacent sub-regions are different. Additionally, the relative sliding velocity v y is resulted from spinning motions, and the relative sliding velocity vx is caused by the gyroscopic motion of the ball. When the relative sliding velocities are calculated, the friction coefficients can be determined based on certain lubricant models. When the forces (including friction force and contact force) are determined, the moment acting on the bearing component can be determined by the cross product of the force vector and the position vector that locates the contacting point to the center of the bearing component. The translation and rotation of a bearing component are described by Newton’s law and Euler’s equation, respectively (see Ref. [38] for more details). mr  F

(8)

Jω  M

(9)

where m is the mass of the component, r is the translational acceleration vector, F is the force vector, J is the principal moment of inertial, ω is the angular acceleration vector and

M is the moment vector. 2.2 Ball defect model In order to investigate the effect of material absence (the material absence can be caused by pitting and spall) on the vibrations of an angular contact ball bearing, a simple defect is modeled as shown in Fig. 4(a). In order to calculate the interaction between the defect and the race and to define the position of the defect in the ball, a coordinate frame, called the defect frame, Od xd yd zd is established on the defect. Origin Od is located at the center of the defect, and the zd axis coincides with the line connecting the ball center Ob and the defect center Od . Moreover, plane xd zd is coplanar with plane xb zb . The position of the defect in the ball can be determined by 7

angle  b , as shown in Fig. 4(b). In the defect frame, the vector locating the defect center relative to the ball center ( rbd ) can be written as   r  0 0  

2  2  D   wd         2  2   

d bd

T

(10)

where D is the ball diameter, wd is the width (i.e., the diameter) of the defect (Fig. 4(b)) and superscript d means that the vector rbd is described in the defect frame. The circumference angle of the defect in plane xb zb is  wd    D

 d  arcsin 

(11)

In order to determine whether the race has potential to strike the defect, angle  b (i.e., the angle between the vector rbc and the zd axis) should be calculated. When  b   d , the race has potential to strike the defect. The determination of angle  b is discussed as follows. When vector rbc is determined based on the Gupta model in the inertial frame, vector rbc can be transformed and described in the defect frame as rbcd  Tbd Tibrbc

(12)

where Tbd   0,  d ,0 is the transformation matrix from the ball-fixed frame to the defect frame, Tib  b , b , b  is the transformation matrix from the inertial frame to the ball-fixed frame, and

superscript d means that vector rbc is described in the defect frame. The theory of the transformation matrix can be found in Appendix A. Based on vector rbcd , angle  b , can be determined as  

rbcd 3

 

r   r 

 b  arctan 

d 2 bc1

2 d bc 2

    

(13)

where subscripts 1, 2 and 3 indicate the first, second and third components of vector rbcd , respectively. As discussed above, when  b   d , the race has potential to strike the defect. Under this condition, the interaction between the defect and the race can be determined as follows. For any point p on the edge of the defect, the vector locating this point to the defect center described in the defect frame can be written as  w sin  r   d 2  d pd

8

wd cos  2

 0 

T

(14)

where  is the angle between the yd axis and vector rpd (Fig. 4(a)). Next, vector rcp , locates point p relative to race groove curvature center c , can be described in the defect frame as d rcpd  rbcd  rbdd  rpd

(15)

Then, the geometrical interaction between the race and the defect at point p can be given as  rcpd  f o D for outer race   bp   d   rcp  f i D for inner race

(16)

Because vector rpdd is a function of the angle  , geometrical interaction  bp is also a function of angle  . When determining the geometrical interaction between the race and the defect, Eq. (16) is calculated for every point on the edge of the defect based on the defined domain of angle  (i.e., [π, π] ), and the maximum  bp will be adopted as the geometrical interaction between the race and the defect. When the geometrical interaction is obtained, the contact force can be determined based on the Hertzian contact theory. Indeed, when the race contacts the defect, non-Hertzian contact occurs. However, the non-Hertzian contact is beyond the scope of the current investigation. Moreover, when the contact force is determined, the friction force and moment acting on the race and the ball can be determined based on the Gupta model. 2.3 Bearing housing model In the current paper, two additional DOFs are added to take into account the vibrations of the bearing housing, as shown in Fig. 5. The corresponding dynamic equations are mh yh  c y yh  k y yh  Fhy   mh zh  cz zh  k z zh  Fhz

(17)

where mh is the mass of the bearing housing, yh and zh are the displacements of the bearing housing along the

yi

and zi directions, respectively,

yh ,

zh

and

yh ,

zh

are the

corresponding velocities and accelerations, c y and cz are the damping coefficients, respectively, and k y and k z are the stiffness coefficients. Moreover, Fhy and Fhz are the forces acting on the bearing housing along the yi and zi directions, respectively.

3 Results and discussions The defect model discussed in Section 2.2 and the bearing housing model discussed in Section 2.3 are integrated into the Gupta model. The dynamic equations given in Eqs. (8) and (9) are a set 9

of second-order differential equations with stiff properties [40]. As a result, an explicit method, i.e., fourth-order Runge-Kutta scheme with step-changing criterion [41] is used to integrate the equations numerically. All the computer programs are coded in FORTRAN 95 language. At every time step, the relationship between the angles  b and  d is calculated. When  b   d , the defect model discussed in Section 2.2 is used to determine the geometrical interaction between the race and the defect. The parameters of the investigated bearing system are listed in Table 1. Some of the parameters listed in Table 1 are the same as those used by Gupta to investigate the general dynamic motions of ball bearings [38, 42]. The defect diameter listed in Table 1 is the typical size of a large spall [17, 43]. Moreover, the damping and stiffness coefficients listed in Table 1 are selected from Ref. [28]. Additionally, because the main focus of the current investigation is the effect of the defect, not the lubrication, a simple lubricant model is used to determine the friction coefficient,  , based on the relative slip velocity, u [44, 45]: 0.08 u   0.08



u  1 m  s1 u  1 m  s1

(18)

The friction coefficients at the ball/cage pocket and the cage/guiding ring are set to as a constant value (0.1) because of the pure sliding properties [44] at these contacting surfaces. Because the angles  i ,  o and  largely depend on the bearing geometrical and operating conditions, four different sets of geometrical and operating conditions are adopted in the simulation studies to investigate the vibration responses of ball bearings with ball defects. These four conditions are given in Table 2. Additionally, regarding geometrical parameters, the initial contact angle, ini , largely affects the angles  i ,  o and  . As a result, in the current simulation, two different initial contact angles, i.e., 0 (condition 3 in Table 2, corresponding to deep groove ball bearing) and 20 (conditions 1, 2, and 4 in Table 2, corresponding to angular contact ball bearing) are investigated. The deep groove ball bearing (condition 3) is used for comparison purpose. 3.1 Conditions 1 and 2 3.1.1 Condition 1 In the first condition, a pure axial load of 1000 N is applied on the bearing, and the rotation speed of the inner ring i (the outer ring is fixed in space) is 5000 r min-1 ( f s is approximately 83.33 Hz). Based on Eq. (1), the BDF is approximately 223 Hz (note that the result of Eq. (1) is used for reference only). Additionally, in this simulation, we assume that the defect is generated as a result of the ball/inner race interaction that indicates that the defect is capable of striking the inner race under condition 1 (the initial position of the defect relative to the races under condition 1 is schematically shown in Fig. 6(a)). In order to show the periodicity of the bearing motion, all of the vibration responses are plotted 10

in 5 inner ring revolutions in the paper. Angles  b and  d are shown in Fig. 7(a). It can be seen that,  b   d periodically for the outer race, while  b is always larger than  d for the inner race, indicating that the defect strikes the inner race periodically while it never strikes the outer race. This phenomenon can also be seen in the contact force and the acceleration of the bearing housing, as shown in Figs. 7(b) and 7(c), respectively. It can be seen that the time interval between two adjacent impulses in the contact forces and the accelerations corresponds to BDF, indicating that the ball defect only strikes one race as it rotates. The corresponding envelope spectrum is given in Fig. 7(d); the most evident frequency component is the BDF (224.1 Hz). Angles  i ,  o and  are shown in Fig. 7(e). It can be seen that angles  i ,  o and  are not equal under this condition. When the rotation of a ball about its own axis is ωb , the defect also rotates around the ball axis at ωb . As shown in Fig. 6(a), if the ball defect hit the inner race under condition 1, the ball defect passes through the outer race at point L when the defect rotates around the ball axis at ωb , and it largely deviates from the contacting point of the outer race (point Oko shown in Fig. 6(a)). From this case study, it can be found that for angular contact ball bearings, the rotation speed of the ball about its own axis is not parallel to the rotation speed of the bearing, and that the contact angles of the inner and the outer races are not equal. As a result, the ball defect may strike only one race as it rotates along the ball axis at the speed ωb . To investigate the motion of the ball under condition 1 more clearly, the distribution of relative sliding velocities in the contact ellipse is discussed. As the motions of a ball at different orbital positions are nearly the same when an angular contact ball bearing is loaded by pure axial loads, the motion of ball 1 (the numbering of balls is shown in Fig. 5) when its orbital position is 360 degree is adopted to investigate this issue. The relative velocities vx and v y at the ball/inner race and the ball/outer race contacts are given in Table 3 and Table 5, respectively (in Table 3 and Table 5, the orbital position of ball 1 is denoted by b1 ). Moreover, the size of the contact ellipse and the maximum contact stress are given in Table 4 and Table 6. It should be noted that non-Hertzian contact occurs when a ball defect hits a race. However, the current investigation is not capable of dealing with the non-Hertzian contact. Therefore, the contact stress in the paper are calculated for a bearing with defect free. In Table 3 and Table 5, A1 , Ok , and A2 represent the left end, the center and the right end of the contact ellipse. The number of rolling center can be determined by analyzing the signs (positive and negative) of v y at points A1 , Ok , and A2 . Take the ball/inner race contact for instance. The velocity v y at points A1 , Ok , and A2 are 6.41102 m·s , 2.56 102 m·s and 2.80 101 m·s (refer to Table 3). It can be found that -1

-1

-1

the sign of v y changes once as it varies along the major axis of the contact ellipse, which indicates that there is one solution for the equation vy ( xk )  0 . Thus, there is one rolling center in 11

the contact ellipse. By analyzing the relative sliding velocities listed in Table 3 and Table 5, it can be found that there is one rolling center along the major axis of the contact ellipse, both for the ball/inner race and the ball/outer race contacts. Moreover, the relative velocity vx is caused by the gyroscopic motion, and the nonzero v y are resulted from the spinning motion. Based on the above discussion, it can be found that the motion of the ball under condition 1 it a combination of rolling, spinning and gyroscopic motions. The complex motions make the relationship between angles  i ,  o and  , and consequently the vibration responses of the angular contact ball bearing with ball defects rather complicated. Despite angles  i ,  o and  , it can also be found that whether the defect strikes a race when it rotates around the ball axis also depends on the initial position of the defect relative to the races. As a comparison, Fig. 6(b) shows another initial position of the defect different from that of condition 1 as shown in Fig. 6(a). It can be expected that the defect will pass through the inner and the outer races at points Li and Lo that largely deviate from contacting points Oki and Oko . Thus, although angles  i ,  o and  shown in Fig. 6(b) are the same as those in condition 1 as shown in Fig. 6(a), the defect will not strike any of the races because of the initial position of the defect. Fig. 8 shows the contact force and the acceleration of the bearing housing under the condition shown in Fig. 6(b). No obvious impulses can be found in Fig. 8 to indicate the ball defect. The effect of the initial position of the defect will be discussed further in the following sections. 3.1.2 Condition 2 As discussed above, whether the ball defect strikes a race depends on angles  i ,  o and  , and the initial position of the defect. Angles  i ,  o and  largely depend on the operating conditions. Additionally, the initial position of the defect relative to the races changes with the change of the operating conditions. This indicates that if a ball defect hits a race under a certain operating condition, the defect may not hit the race when the operating condition is changed. In order to show this, the axial load in condition 1 is changed to 2000 N, and the corresponding vibration responses are shown in Fig. 9. The angles  b and  d are shown in Fig. 9(a). It can be seen in Fig. 9(a) that when the operating condition is changed from condition 1 to condition 2, the angles  b of the inner and the outer races are both larger than  d , indicating that the ball defect does not hit the outer and inner races any longer. Figs. 9(b) and 9(c) show the contact force and the acceleration of the bearing housing. It can be seen that when the geometrical and operating conditions are changed from condition 1 to condition 2, no obvious impulses indicating the ball defect can be found in the contact force (Fig. 9(b)) and the acceleration of the bearing housing (Fig. 9(c)). The motion of a ball under this condition is also investigated by the distribution of relative sliding velocities in the contact ellipse. Take the motion of ball 1 when its orbital position is 360 12

degree for instance, as shown in Table 3 and Table 5. It can be found from Table 3 and Table 5 that there are two rolling centers at the ball/inner race contact and there is one rolling center at the ball/outer race contact. Moreover, velocities vx are smaller compared with those under condition 1. This shows that the ball motion under condition 2 is also a combination of rolling, spinning and gyroscopic motions. However, the gyroscopic motion is not as prominent as that under condition 1. This is mainly because that large axial load results in large contact force, and consequently large friction force in the contact region, which is capable of balancing the gyroscopic moment [35]. By comparing the vibration responses of conditions 1 and 2, it can be found that whether the ball defect hits a race or not largely depends on the operating conditions. If the ball defect is generated under certain operating conditions, the ball defect could not be detected by vibrations when the operating condition is changed. This shows the difficulty in diagnosing the ball defect. Under this condition, other detection techniques, such as ferrographic analysis, can be introduced to effectively diagnose the ball defect. 3.2 Condition 3 In this section, the dynamics and vibrations when the initial contact angle is zero and a pure radial load of 1000 N is loaded (corresponding to deep groove ball bearing) are investigated. The initial position of the defect relative to the races is shown in Fig. 10(a), and corresponding vibration responses are shown in Fig. 11. Angles  b and  d are shown in Fig. 11(a). It can be found that the  b of the inner and the outer races will both be smaller than  d , indicating that the defect is capable of passing through the contacting regions of the inner and outer races, and that both have potentials to strike the races. The contact angles of the inner and outer races are nearly the same (both changes around 0 as shown in Fig. 11(b)). Moreover, angle  nearly equals zero (Fig. 11(b)), indicating that speed ωb is parallel to the bearing rotation axis. Because speed ωb is parallel to the bearing rotation axis and  i nearly equals  o , the ball defect is capable of striking both the outer and inner races as it rotates around the ball axis at ωb . Additionally, the acceleration responses can be classified into loaded zone and unloaded zone, as shown in Fig. 11(d). The impulses in the loaded zone are severer than those in the unloaded zone. Moreover, the time interval between two adjacent impulses in the loaded zone corresponds to 2BDF, while the time interval between two adjacent impulses in the unloaded zone corresponds to BDF. The reason is discussed as follows. As shown in Fig. 12, four orbital positions, i.e., A, B, C, and D are investigated. When the ball moves from position A to position B, and from position C to position D, the defect rotates about half a cycle around the ball axis. When the ball is in the unloaded zone (positions A and B), the ball contacts closer to the outer race under the action of the centrifugal force. Moreover, the angles  b of the inner and outer races are smaller than  d periodically. Therefore, when the ball defect passes through the outer race, the defect is capable of striking the outer race, and generates an impulse as shown in Fig. 11(d) (in Fig. 11(d), the 13

impulses generated at the outer race is denoted by “o”). Additionally, because the ball contacts closer to the outer race in the unloaded zone, the defect cannot impact the inner race as the defect passes through the inner race (position B shown in Fig. 12). Therefore, the impulses generated in the unloaded zone are all resulted from the impacts between the ball defect and the outer race, and thus the time interval between two adjacent impulses in the unloaded zone corresponds to BDF. However, in the loaded zone, the ball is compressed between the inner race and the outer race. Thus, the ball defect can impact both the inner race (position C shown in Fig. 12) and the outer race (position D shown in Fig. 12) as the ball defect rotates one cycle around the ball axis. Therefore, the time interval between the two impulses in the loaded zone correspond to 2BDF as shown in Fig. 11(d) (In Fig. 11(d), the impulses generated at the inner race are denoted by “I”). The corresponding envelope spectrum is shown in Fig. 11(e) in which the 2BDF (445.3 Hz) is more evident than BDF (222.9 Hz). Moreover, the cage rotation frequency f c (33.6 Hz) and its super-harmonic 2 f c , and sidebands spaced at f c around BDF and 2BDF can also be found in the envelope spectrum (such as BDF  fc , 2BDF  fc , 2BDF  2 fc shown in Fig. 11(e)). This is mainly because the ball passes through the loaded zone periodically at a speed of f c when the bearing rotates. This property of the envelope spectrum of a bearing with a ball defect is the most fundamental vibration mechanism of rolling element bearings with a defect on a rolling element [3], and has been verified in experiments for deep groove ball bearings [46]. By comparing Figs. 7(d) and 11(e), it can be seen that the frequency property of the envelope spectrum of a ball bearing with a ball defect largely depends on the geometrical and operating conditions of the bearing. The ball motion under this condition is investigated as follows. As the contact force of a ball changes with the rotation of the bearing when a radial load is loaded, the motions of ball 1 at two different orbital positions are investigated. The first orbital position is 360 degree (the contact force is minimum at this position). The other orbital position is 180 degree (the contact force is maximum at this position). The relative velocities vx and v y at the ball/inner race and the ball/outer race contacts are given in Table 3 and Table 5, respectively. Moreover, the size of the contact ellipse and the maximum contact stress are given in Table 4 and Table 6. It should be noted that the ball does not contact the inner race when its orbital position is 360 degree as discussed above. It can be found that there are two rolling centers both at the ball/inner race and the ball/outer race contacts. Moreover, velocities vx are relatively very small compared with those under conditions 1 and 2. This shows that the ball motion under this condition is also a combination of rolling, spinning and gyroscopic motions. However, the rolling motion is more dominant under this condition, and the gyroscopic motions are not as dominant as those under conditions 1 and 2. As a comparison, another initial position of the ball defect is shown in Fig. 10(b). 14

Corresponding contact forces and accelerations of the bearing housing are shown in Fig. 13. It can be expected that the defect will pass through the inner and the outer races at points Li and Lo that largely deviate from contacting points Oki and Oko ; consequently, no obvious impulses will be generated in the vibration signals indicating the ball defect (Fig. 13). A real example is the famous experimental project for the fault diagnosis of deep groove ball bearings that was conducted at Case Western Reserve University [46 47]. There are 161 sets of data in total [46]. In the experimental data, the ball defect has been proven to be the most difficult fault to diagnose [47]. Ref. [46] reported that most of the data sets for ball defects have no obvious impulses indicating the ball defect. Based on the current investigation, a possible reason can be a result of the following. In the experiments [47], after a defect was artificially seeded on a ball, the ball was reinstalled in the bearing to carry out the following experiments. However, when reinstalling the ball, the initial position of the defect cannot be ensured to strike the races, and the defect cannot strike one of the races when the defect rotates about the ball axis. 3.3 Condition 4 A more complex condition is an angular contact ball bearing that is loaded by both axial and radial loads. Under this condition, the contact forces (Fig. 14(a)), the angles  i ,  o and  (Fig. 14(b)) and the initial position of the defect relative to the races all change periodically with the rotation of the bearing. As the bearing is loaded by both axial and radial loads, the contact force of a ball varies as it rotates around the bearing axis. Therefore, the motions of ball 1 at two different orbital positions, i.e., 180 degree and 360 degree are investigated. The relative sliding velocities at the ball/inner race and the ball/outer race contacts are given in Table 3 and Table 5, respectively. Moreover, the size of the contact ellipse and the maximum contact stress are given in Table 4 and Table 6. When the orbital position is 180 degree, there is one rolling center at the ball/inner race contact, and there are two rolling centers at the ball/outer race contact. However, when the orbital position is 360 degree, there is only one rolling center at the ball/outer race contact, and there is no rolling center at the ball/inner race contact. This shows that the motion of a ball changes largely as it rotates around the bearing axis, and thus make the angles  i ,  o and  change. Angles  b and  d are shown in Fig. 14(c). It can be seen that, as angles  i ,  o and  change with time (Fig. 14(c)),  b is smaller than  d at some time instant both for the inner and outer races. However, unlike condition 3, the relationship between  b and  d changes irregularly with time, indicating that the defect contacts the races irregularly over time. As a result, the contact properties between the defect and the races and also the acceleration of the bearing housing (Fig. 14(d)) become rather complicated. The envelope spectrum is shown in Fig. 14(e). It can be seen that, although the rotation speed of the bearing and the initial contact angle under condition 4 are the same as those under condition 1, the BDF shown in Fig. 14(e) is different from that shown in Fig. 7(d). Moreover, as the ball 15

periodically passes through the loaded zone, cage rotation frequency f c and its super-harmonic 2 f c , and also the sidebands spaced at f c around BDF can be found in Fig. 14(e). However,

except for the sidebands spaced at f c , much more complex sidebands around the BDF can be found in Fig. 14(e) because the defect contacts the races irregularly over time. From the above investigations it can be found that the envelope spectrum of a ball bearing with ball defects largely depends on the geometrical and operating conditions of the bearing. Because the frequency property of the envelope spectrum is important in diagnosing the bearing fault [1], the current investigation suggests that dynamic simulation should be carried out to thoroughly understand the vibration responses under different geometrical and operating conditions and to diagnose the ball fault more effectively. Conclusions In the current paper, a dynamic model is proposed to investigate the dynamics and vibration responses of angular contact ball bearings with ball defects. Based on the current investigation, the following conclusions can be drawn. For angular contact ball bearings, the motion of a ball is a combination of rolling, spinning and gyroscopic motions. The rolling motion of a ball in an angular contact ball bearing is not as dominant as that in a deep groove ball bearing. The complex motions make the relationship between the angles  i ,  o and  , and consequently the vibration of the bearing with a ball defect rather complicated compared with deep groove ball bearings. Moreover, whether a ball defect hits a race depends both on the initial position of the defect relative to the race and the complex motion of the ball. Additionally, when the bearing is loaded by both axial and radial loads, the frequency characteristics of the envelope spectrum are much more complex than that of a deep groove ball bearing which is loaded by a pure axial load. Additionally, the BDF shown in the envelope spectrum may be changed with operating condition changes.

Acknowledgements Special thanks to Prof. Zhengjia He for his great contributions to start this investigation. This work was supported by the National Natural Science Foundation of China (No. U1510206). Appendix A. Transformation matrix between coordinate frames. The transformation between two coordinate frames can be achieved by three successive rotations based on the right-hand rule. When the rotation angles of three successive rotations are η, ξ, and λ, the transformation matrix can be written as [38]  cos  cos  cos cos   sinsincos  sinsin  cos sincos     T    cos  sin  cos cos   sinsinsin sincos   cos sinsin    sin  sincos  cos cos    16

(A.1)

When the transformation matrix from frame s to frame t is Tst and a vector r described in the frame s is r s (the superscript s means that the vector r is described in the frame s), then the vector r s can be transformed and described in frame t as r t  Tst r s

(A.2)

where the superscript t means that the vector r is described in the frame t.

References [1] Rai A, Upadhyay SH. A review on signal processing techniques utilized in the fault diagnosis of rolling element bearings. Tribol Int 2016;96:289-306. [2] El-Thalji I, Jantunen E. A summary of fault modelling and predictive health monitoring of rolling element bearings. Mech Sys Signal Process 2015;60-61:252-272. [3] Randall RB, Antoni J. Rolling element bearing diagnostics-A tutorial. Mech Sys Signal Process 2011;25(2):485-520. [4] Liu J, Shao Y. A new dynamic model for vibration analysis of a ball bearing due to a localized surface defect considering edge topographies. Nonlinear Dynamics 2014;79(2):1329-1351. [5] Liu J, Shao Y, Lim TC, Vibration analysis of ball bearings with a localized defect applying piecewise response function. Mech Mach Theory 2012;56:156-169. [6] Liu J, Shao Y, Zhu WD, A new model for the relationship between vibration characteristics caused by the time-varying contact stiffness of a deep groove ball bearing and defect sizes. J Tribol 2015;137(3):031101. [7] Patel VN, Tandon N, Pandey RK, A dynamic model for vibration studies of deep groove ball bearings considering single and multiple defects in races. J Tribol 2010;132(4):041101. [8] Patil MS, Mathew Jose, Rajendrakumar PK, Desai Sandeep. A theoretical model to predict the effect of localized defect on vibrations associated with ball bearing. Int J Mech Sci 2010;52(9):1193-1201. [9] Ashtekar A, Sadeghi F, Stacke L-E, A new approach to modeling surface defects in bearing dynamics simulations. J Tribol 2008;130(4):041103. [10] Ashtekar A., Sadeghi F, Stacke L-E, Surface defects effects on bearing dynamics. Proc Inst Mech Eng Part J: J Eng Tribol 2010;224(1):25-35. [11] Niu L, Cao HR, He ZJ, Li YM. A systematic study of ball passing frequencies based on dynamic modeling of rolling ball bearings with localized surface defects. J Sound Vib 2015;357:207-232. [12] Niu L, Cao H, He Z, Li Y. Dynamic modeling and vibration response simulation for high speed rolling ball bearings with localized surface defects in raceways. J Manuf Sci Eng 2014;136(4):041015. [13] Cao H, Niu L, He Z. Method for vibration response simulation and sensor placement optimization of a machine tool spindle system with a bearing defect. Sensors 2012;12:8732-8754. [14] Petersen D, Howard C, Sawalhi, Ahmadi AM, Singh S. Analysis of bearing stiffness variations, contact forces and vibrations in radially loaded double row rolling element bearings with raceway defects. Mech Sys Signal Process 2015;50-51:139-160. 17

[15] Ahmadi AM, Petersen D, Howard C. A nonlinear dynamic vibration model of defective bearings – The importance of modelling the finite size of rolling elements. Mech Sys Signal Process 2015;52-53:309-326. [16] El-Thalji I, Jantunen E. Dynamic modelling of wear evolution in rolling bearings. Tribol Int 2015;84:90-99. [17] El-Thalji I, Jantunen E. Fault analysis of the wear fault development in rolling bearings. Eng Fail Anal 2015;57:470-482. [18] Khanam S, Dutt JK, Tandon N. Impact force based model for bearing local fault identification. J Vib Acoust 2015;137:051002. [19] Khanam S, Tandon N, Dutt JK, Multi-event excitation force model for inner race defect in a rolling element bearing. J Tribol 2015;138:011106. [20] Sopanen J, Mikkola A. Dynamic model of a deep-groove ball bearing including localized and distributed defects. Part 1: Theory. Proc Inst Mech Eng Part K: J Multi-body Dyna 2003;217(3)201-211. [21] Sopanen J, Mikkola A. Dynamic model of a deep-groove ball bearing including localized and distributed defects. Part 2: Implementation and results. Proc Inst Mech Eng Part K: J Multi-body Dyna 2003;217(3):213-223. [22] Kiral Z, Karagülle H. Simulation and analysis of vibration signals generated by rolling element bearing with defects. Tribol Int 2003;36(9):667-678. [23] Wang W-Z, Zhang S-G, Zhao Z-Q, Ai S-Y. Effect of the Inhomogeneity in Races on the Dynamic Behavior of Rolling Bearing. J Vib Acoust 2015;137(6):061015. [24] Cui L, Zhang Y, Zhang F, Zhang J, Lee S. Vibration response mechanism of faulty outer race rolling element bearings for quantitative analysis. J Sound Vib 2016;364:67-76. [25] Choudhury A, Tandon N. Vibration response of rolling element bearings in a rotor bearing system to a local defect under radial load. J Tribol 2006;128:252-261. [26] Sassi S, Badri B, Thomas M. A numerical model to predict damaged bearing vibrations. J Vib Control 2007;13(11):1603-1628. [27] Sawalhi N, Randall RB. Simulating gear and bearing interactions in the presence of faults. Part II: Simulation of the vibrations produced by extended bearing faults. Mech Syst Signal Process 2008;22:1952-1966. [28] Sawalhi N, Randall RB. Simulating gear and bearing interactions in the presence of faults. Part I. The combined gear bearing dynamic model and the simulation of localized bearing faults. Mech Syst Signal Process 2008;22:1924-1951. [29] Arslan H, Aktürk N. An investigation of rolling element vibrations caused by local defects. J Tribol 2008;130:041101. [30] Rafsanjani A, Abbasion S, Farshidianfar A, Moeenfard H. Nonlinear dynamic modeling of surface defects in rolling element bearing systems. J Sound Vib 2009;319(3-5):1150-1174. [31] Mishra C, Samantaray AK, Chakraborty G. Bond graph modeling and experimental verification of a novel scheme for fault diagnosis of rolling element bearings in special operating conditions. J Sound Vib 2016;377:302-330. [32] Yuan X, Zhu YS, Zhang YY. Multi-body vibration modelling of ball bearing-rotor system considering single and compound multi-defects. Proc Inst Mech Eng Part K: J Multi-body Dyn 2014;228:199-212. [33] Harris TA, Kotzalas MN. Rolling bearing analysis: Essential concepts of bearing technology. 18

[34] [35] [36] [37] [38] [39]

[40] [41] [42] [43]

[44] [45] [46] [47]

CRC Press, Boca Raton, 2007. Bozet JL, Servais C. Influence of the balls kinematics of axially loaded ball bearings on Coulombic frictional dissipations. J Tribol 2017;139:011502. Harris TA, Kotzalas MN. Rolling bearing analysis: Advanced concepts of bearing technology. CRC Press, Boca Raton, 2007. Gupta PK. Dynamics of rolling-element bearings. 3. Ball bearing analysis. J Tribol 1979;101(3):312-318. Gupta PK. Dynamics of rolling-element bearings. 4. Ball bearing results. J Tribol 1979;101(3):319-326. Gupta PK. Advanced dynamics of rolling elements. New York: Springer-Verlag; 1984. Wang F, Jing M, Yi J, Dong G, Liu H, Ji B. Dynamic modelling for vibration analysis of a cylindrical roller bearing due to localized defects on raceways. Proc Inst Mech Eng Part K: J Multi-body Dyn 2015;229:39-64. Stacke L-E, Fritzson D. Dynamic behavior of rolling bearings: simulations and experiments. Proc Inst Mech Eng Part J: J Eng Tribol 2001;215: 499-508. Sauer T. Numerical analysis. Boston, USA: Person Education; 2011. Gupta PK, Winn LW, Wllcock DF. Vibrational characteristic of ball bearings. J Tribol 1977;99:284-289. Al-Ghamd AM, Mba D. A comparative experimental study on the use of acoustic emission and vibration analysis for bearing defect identification and estimation of defect size. Mech Sys Signal Process 2006;20:1537-1571. Ghaisas N, Wassgren CR, Sadeghi F. Cage instabilities in cylindrical roller bearings. J Tribol 2004;126(4):681-689. Niu L, Cao H, He Z, Li Y. An investigation on the occurrence of stable cage whirl motions in ball bearings based on dynamic simulations. Tribol Int 2016;103:12-24. Simth WA, Randall RB. Rolling element bearing diagnostics using the Case Western Reserve University data: A benchmark study. Mech Syst Signal Process 2015; 64-65: 100-131. Case Western Reserve University Bearing Data Center Website .

19

Table 1 Parameters of the investigated bearing system. Parameter

Value

Number of balls, z

14 [42]

Ball diameter, D (m)

12.7E-3 [42]

Pitch diameter, d m (m)

70E-3 [42]

Radial clearance (m)

0

Cage pocket-ball clearance (m)

1.0E-4

Cage-guiding ring clearance (m)

2.5E-4

Guide ring

Outer ring

Defect diameter, wd (m)

3.0E-3

Groove curvature factor of outer race, f o

0.52 [42]

Groove curvature factor of inner race, f i

0.515 [42]

Mass of bearing housing, mh (kg)

5 -1

Damping coefficients, c y , c z (N s m )

1.8E3 [28] 1.5E7 [28]

-1

Stiffness coefficients, k y , k z (N m ) Elastic modulus of ball (N m-1) Elastic modulus of outer race (N m-1) Elastic modulus of inner race (N m-1) Elastic modulus of cage Poisson ratio of ball Poisson ratio of outer race Poisson ratio of inner race Poisson ratio of cage Density of ball (Kg m-3) Density of inner race (Kg m-3) Density of outer race (Kg m-3) Density of cage (Kg m-3)

2E11 2E11 2E11 1.1E10 [38] 0.25 0.25 0.25 0.45 [38] 7.75E3 7.75E3 7.75E3 2.03E2 [38]

Table 2 Geometrical and operating conditions used in Section 3. Conditions

Description

Condition 1

Fa  1000 N , Fr  0 N , i  5000 r  min 1 , ini  20

Condition 2

Fa  2000 N , Fr  0 N , i  5000 r  min 1 , ini  20

Condition 3

Fa  0 N , Fr  1000 N , i  5000 r  min 1 , ini  0

Condition 4

Fa  1000 N , Fr  2000 N , i  5000 r  min 1 , ini  20

20

Table 3 Distribution of relative sliding velocities in the contact ellipse (ball/inner race contact).

θb1 Condition

vx (m·s-1)

vy (m·s-1)

Number of

(°)

A1

Ok

A2

A1

Ok

A2

rolling center

1

360

-7.73E-2

-7.83E-2

-7.73E-2

6.41E-2

2.56E-2

-2.80E-1

1

2

360

-4.57E-2

-4.66E-2

-4.57E-2

-2.64E-2

4.14E-2

-3.11E-1

2

3

180

-8.85E-5

-8.72E-5

-8.85E-5

-1.55E-1

3.78E-2

-1.51E-1

2

360

-

-

-

-

-

-

-

180

-1.85E-2

-1.91E-2

-1.85E-2

4.08E-2

6.07E-2

-5.35E-1

1

360

-7.66E-1

-7.68E-1

-7.66E-1

2.36E-2

4.88E-2

7.72E-1

0

4

Table 4 The size of the contact ellipse and the maximum Hertzian contact stress (ball/inner race contact). Condition

θb1/°

a/m

b/m

Pmax / Gpa

1

360

1.01E-3

9.14E-5

9.32E-1

2

360

1.26E-3

1.14E-4

1.17E0

3

180

1.20E-3

1.07E-4

1.11E0

360

-

-

-

180

1.52E-3

1.38E-4

1.41E0

360

3.58E-4

3.25E-5

3.30E-1

4

Table 5 Distribution of relative sliding velocities in the contact ellipse (ball/outer race contact). vx (m·s-1)

θb1 Condition

vy (m·s-1)

Number of

(°)

A1

Ok

A2

A1

Ok

A2

rolling center

1

360

-7.75E-2

-7.83E-2

-7.75E-2

1.21E-1

-1.40E-2

-9.74E-4

1

2

360

-4.59E-2

-4.66E-2

-4.59E-2

2.19E-1

-2.23E-2

-3.57E-4

1

3

180

-1.31E-4

-1.30E-4

-1.31E-4

8.14E-2

-2.00E-2

7.90E-2

2

360

-5.03E-4

-5.03E-4

-5.03E-4

9.74E-3

-2.47E-3

9.66E-3

2

180

-1.87E-2

-1.91E-2

-1.87E-2

1.33E-1

-3.23E-2

1.31E-2

2

360

-7.65E-1

-7.67E-1

-7.65E-1

7.16E-2

-8.88E-3

-5.47E-4

1

4

21

Table 6 The size of the contact ellipse and the maximum Hertzian contact stress (ball/outer race contact). Condition

θb1/°

a/m

b/m

Pmax / Gpa

1

360

8.87E-4

1.19E-4

8.69E-1

2

360

1.10E-3

1.48E-4

1.08E0

3

180

1.04E-3

1.41E-4

1.02E0

360

3.64E-4

4.93E-5

3.56E-1

180

1.32E-3

1.77E-4

1.29E0

360

4.26E-4

5.73E-5

4.14E-1

4

(a)

(b)

Fig. 1 Contact angles and rotation speed of a ball around its own axis. (a) Angles  i ,  o and  are determined based on pure rolling assumptions and simple kinematic relations and (b) the angles  i ,  o and  when considering the three-dimensional motion of a ball.

22

Fig. 2 The ball/race interaction. This figure is a simplified version of that originally shown in Ref. [38].

23

Fig. 3 Distribution of relative sliding velocities in the contact ellipse. (a) The contact ellipse has one roller center, and (b) the contact ellipse has two rolling centers. This figure is a simplified version of that originally shown in Ref. [35].

24

(a)

(b)

Fig. 4 The interaction between the ball defect and a race. (a) Basic geometric property of the defect and (b) position of the defect on the ball.

25

Fig. 5 Model of the bearing housing.

26

(a)

(b)

Fig. 6 Position of the ball defect relative to the races of an angular contact ball bearing. (a) Position in condition 1, and (b) another defect position.

27

(a) b and d [degree]

20 15 10 5 b (inner race)

0 15

(b)

(c) -2

1/BDF

Inner race Outer race

180 175 15

16 17 18 19 Number of inner ring revolutions

(e)

0.01

0.005

200

400 600 Frequency (Hz)

800

0 -0.5

1000

1/BDF

16 17 18 19 Number of inner ring revolutions

20

25

i , o and  [degree]

-2

Amplitude [ms ]

BDF(224.1 Hz)

0 0

0.5

-1 15

20

0.015

20

1 zi acceleration [ms ]

Contact force [N]

190 185

d

16 17 18 19 Number of inner ring revolutions

195

(d)

b (outer race)

24 23

i

22 21 15

16

17

18

o



19

20

Number of inner ring revolutions

Fig. 7 Dynamics and vibration responses when the bearing is operated under condition 1. (a) Angles  b and  d , (b) the contact forces, (c) the acceleration of the bearing housing along the zi direction, (d) the envelope spectrum of the acceleration of the bearing housing, and (e) angles  i ,  o and  .

28

(a)

Inner race Outer race

-2

190 185 180 175 170 15

16 17 18 19 Number of inner ring revolutions

3 zi acceleration [ms ]

Contact force [N]

195

-9

(b)

200

20

x 10

2 1 0 -1 -2 -3 15

16 17 18 19 Number of inner ring revolutions

20

Fig. 8 Vibration response of the bearing (the positon of the defect relative to the race is shown in Fig. 6(b)). (a) Contact forces, and (b) acceleration of the bearing housing.

29

(a) b and d [degree]

20 15 10 5 b (inner race)

0 15

(b)

-2

350

16 17 18 19 Number of inner ring revolutions

2 zi acceleration [ms ]

Contact force [N]

Inner race Outer race

20

-9

(c)

360

340 15

d

16 17 18 19 Number of inner ring revolutions

380 370

b (outer race)

20

x 10

1 0 -1 -2 15

16 17 18 19 Number of inner ring revolutions

20

Fig. 9 Dynamic and vibration responses when the bearing is operated under condition 2. (a) Angles  b and  d , (b) the contact forces, (c) the acceleration of the bearing housing along the zi direction, and (d) angles  i ,  o and  .

30

(a)

(b)

Fig. 10 Position of the ball defect relative to the races of a deep groove ball bearing. (a) Position used in condition 3, and (b) another defect position.

31

(a)

20

b and d [degree]

b (inner race)

5

16 17 18 19 Number of inner ring revolutions

(c)

2

i

o

400

1 0 -1

20

Loaded zone Unloaded zone

 Contact force [N]

i , o,  [degree]

d

10

0 15

(b)

b (outer race)

15

Inner race Outer race

300 200 1/BDF

100 1/(2BDF)

-2

zi acceleration [ms ]

15

O

(e)

O

1/(2BDF)

I

10 5

I

I

O

1/BDF

O

0 15

20

O

O O

O O

20

0.25 fc (33.6 Hz)

I

O

0 -5

16 17 18 19 Number of inner ring revolutions

0.2

-2

(d)

16 17 18 19 Number of inner ring revolutions

Amplitude [ms ]

-2 15

O

-10

0.15

2fc (67.2 Hz) 2BDF (445.3 Hz)

0.1 BDF(222.9 Hz) BDF+fc BDF-f

0.05

c

2BDF-fc 2BDF-2fc

2BDF+fc 2BDF+2fc

Loaded zone Unloaded zone

-15 15

16 17 18 19 Number of inner ring revolutions

20

0 0

100

200 300 400 Frequency [Hz]

500

600

Fig. 11 Dynamic and vibration responses when the bearing is operated under condition 3. (a) Angles  b and  d , (b) angles  i ,  o and  , (c) the contact forces, (d) the acceleration of the bearing housing along the zi direction, and (e) the envelope spectrum of the acceleration of the bearing housing.

32

Fig. 12 Four orbital positions (A, B, C, and D) of a ball (the centrifugal force is not shown for positions C and D).

33

(a)

(b)

400

Contact force [N]

Inner race Outer race 300 200 100 0 15

16 17 18 19 Number of inner ring revolutions

20

16 17 18 19 Number of inner ring revolutions

20

-2

zi acceleration [ms ]

0.2 0.1 0 -0.1 -0.2 15

Fig. 13 Vibration response of the bearing (the positon of the defect relative to the race is shown in Fig. 10(b)). (a) Contact forces, and (b) acceleration of the bearing housing.

34

(a) Contact force [N]

800 Inner race Outer race

600 400 200 0 15

(b)

16 17 18 19 Number of inner ring revolutions

(c)

30 20 10

i 0 15

(d)

o

16 17 18 19 Number of inner ring revolutions

5 b (inner race)

20

5 0 -5

20

b (outer race)

d

16 17 18 19 Number of inner ring revolutions

20

0.1

Amplitude [m s-2]

-2

zi acceleration [ms ]

10

0 15

(e)

16 17 18 19 Number of inner ring revolutions

15



10

-10 15

20

b and d [degree]

i , o,  [degree]

40

20

0.08 0.06 0.04

fc (36 Hz)

BDF-fc BDF(229.5 Hz)

2fc

BDF+fc

0.02 0 0

100

200

300

400

500

Frequency [Hz]

Fig. 14 Dynamic and vibration responses when the bearing is operated under condition 4. (a) the contact forces, (b) angles  i ,  o and  , (c) angles  b and  d , (d) the acceleration of the bearing housing along the zi direction, and (e) the envelope spectrum of the acceleration of the bearing housing.

Highlights  A dynamic model to investigate the vibration responses of angular contact ball bearings with ball defects is proposed.  The effects of the three-dimensional motion of balls are considered.  Whether a ball defect hit a race and the frequency property of the envelope spectrum depend on geometrical characteristics and operating conditions.

35