Dynamic Modeling and Vibration Response

Linkai Niu Graduate Research Assistant State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710049, PR China e-mail: [email protected]

Hongrui Cao1 Associate Professor State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710049, PR China e-mail: [email protected]

Zhengjia He Professor State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710049, PR China e-mail: [email protected]

Yamin Li Graduate Research Assistant State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710049, PR China e-mail: [email protected]

1

Dynamic Modeling and Vibration Response Simulation for High Speed Rolling Ball Bearings With Localized Surface Defects in Raceways A dynamic model is developed to investigate vibrations of high speed rolling ball bearings with localized surface defects on raceways. In this model, each bearing component (i.e., inner raceway, outer raceway and rolling ball) has six degrees of freedom (DOFs) to completely describe its dynamic characteristics in three-dimensional space. Gyroscopic moment, centrifugal force, lubrication traction/slip between bearing component are included owing to high speed effects. Moreover, local defects are modeled accurately and completely with consideration of additional deflection due to material absence, changes of Hertzian contact coefficient and changes of contact force directions due to raceway curvature variations. The obtained equations of motion are solved numerically using the fourth order Runge–Kutta–Fehlberg scheme with step-changing criterion. Vibration responses of a defective bearing with localized surface defects are simulated and analyzed in both time domain and frequency domain, and the effectiveness of fault feature extraction techniques is also discussed. An experiment is carried out on an aerospace bearing test rig. By comparing the simulation results with experiments, it is confirmed that the proposed model is capable of predicting vibration responses of defective high speed rolling ball bearings effectively. [DOI: 10.1115/1.4027334] Keywords: dynamic modeling, high speed rolling bearing, localized defects, vibration response, fault diagnosis

Introduction

High speed rolling bearings are widely used in aero-engines, high speed spindles, and other rotational machinery. Localized defects, such as pits, cracks, and spalls, may be generated in bearings during operations. When rolling elements roll over defects, shock pulses with short duration are generated and excessive vibration may be excited, which may lead the whole system to failure. Therefore, condition monitoring and fault diagnosis of rolling bearings are crucial for the prevention of system failure. In practice, vibration signals are used widely for the fault diagnosis of rolling bearings [1–11]. Fault features are extracted from measured data with the aid of advanced signal processing techniques. However, when bearing defects occur, the generated impacts can cause system vibration at many frequencies from different structures. Therefore, sometimes it is difficult to explain the results of data processing. For the defect detection and fault diagnosis, fault mechanism analysis is essential, which is similar to the anatomy and the pathology that give reasons and characteristics of pathological changes in the medical science. In order to study the fault mechanism, mathematical models are usually employed to reveal the generation mechanism and characteristics of defects, which provide theoretical proofs for running state identification of rolling bearings. By now, many models of rolling bearings have been proposed, and they can be classified into quasi-static and dynamic models [12], both of which were adopted by researchers to study 1 Corresponding author. Contributed by the Manufacturing Engineering of ASME for publication in the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received December 28, 2013; final manuscript received March 27, 2014; published online May 21, 2014. Assoc. Editor: Tony Schmitz.

dynamics of rolling bearings with localized defects. The quasistatic model was firstly proposed by Jones [13]. In this model, force and moment equilibrium equations were given for raceways and rolling elements. These equations include centrifugal forces and gyroscopic moments, together with the externally applied forces and moments which were then solved by Newton–Raphson method. Based on the quasi-static model, the effects of bearing raceway’s distributed faults on system vibration responses were studied by Jang [14] and Bai [15]. Recently, Cao [16] simulated vibration responses of a machine tool spindle system with localized bearing surface defects by using Jones’ model, and sensor placement optimization was also discussed. However, the quasistatic model is based on the ‘raceway control’ hypothesis, which restricts the ball to roll either on outer raceway or inner raceway. This kinematic constraint limits the model in dealing with problems involving lubricant behaviors and transient motions for high speed conditions. In a dynamic model, equilibrium equations used in the quasistatic model are replaced by differential equations of motion for each bearing component, which solves most problems in the quasi-static model. All transient behaviors and lubrication effects can be simulated with the dynamic model. The simplest dynamic model of rolling bearing was firstly proposed by Sunnersj€o [17], which has two DOFs. In this model, only the two translational DOF of inner raceway in radial directions were included, and contacts between balls and raceways were treated as nonlinear contact springs by using Hertzian theory. Many researchers investigated dynamic modeling of rolling bearings with localized defects based on the 2-DOF model [18–27]. Shao [18] investigated both the time-varying deflection excitation and the time-varying contact stiffness excitation produced by localized surface defects for cylindrical roller bearings. Rafsanjani [19] studied nonlinear

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dynamic behaviors of rolling bearing systems with surface defects. Liu [20] expressed the amplitude and time duration of the impulse generated when a ball passes over a local defect by using piecewise response functions. Patil [21] studied the effects of defect size and position on vibration amplitudes and spectral components. Kankar and Pandya analyzed dynamic responses of high speed rotor-ball bearing systems due to raceway waviness [22] and combined localized defects [23], respectively. In order to investigate the interaction between bearing ring and housing, Feng [24] added another two translational DOFs to the 2-DOF model to describe the ring/housing contact. Sawalhi [25,26] added an extra DOF to Feng’s model to represent a typical high frequency for bearings with localized surface defects, and this model was further integrated into a bearing-gear-rotor system to simulate dynamic behaviors of a gearbox with localized bearing faults. More recently, Babu [27] added a rotational DOF to the inner raceway based on the 2-DOF model to study vibration responses of a rotor supported by two angular contact ball bearings. Since only two DOFs of the inner raceway were considered in the above dynamic models and transient motions of each bearing component cannot be studied accurately, more complicated models were developed in the past [28–38]. Harsha proposed a dynamic bearing model based on Lagrange’s equations [28]. On the basis of this model, Harsha studied the stability of a rotor bearing system due to surface waviness and number of balls [29], and the nonlinear dynamics of balanced rotor supported by bearings with surface waviness and internal clearance [30]. Arslan [31] proposed a bearing-rotor system dynamic model, which treated the shaft and balls as masses and raceways as contact springs. Vibrations of the bearing with and without defects were studied in both time domain and frequency domain. Patel [32] studied dynamic characteristics of deep groove ball bearings with single and multiple surface defects. In this model, each bearing component has two translation DOFs in a plane. Nakhaeinejad [33] developed a dynamic model of rolling element bearing using bond graph method, in which each element has one rotation DOF and one translation DOF. Cao [34] investigated the effects of localized surface defects on vibration responses of a double-row spherical roller bearing. Tadina [35] modeled the outer raceway using finite element method to take its flexible deformations into account. In order to consider lubrication effects, Sassi [36] and Choudhury [37] proposed a lumped mass model of a bearing-rotor system to take damping effects due to oil films into account, but lubrication tractive forces were not considered. Bogdevicˇius [38] investigated five cases of various defects, including single and multiple defects on different components based on a dynamic model established by Lagrange scheme. In all of these models, rolling elements just have planar motions, and relative slippage between rolling elements and raceways were not considered. For high speed bearings which have intensive transient motions during operations, a full three-dimensional dynamic model is necessary. The earliest three-dimensional dynamic model for rolling ball bearings was proposed by Gupta [12,39,40]. He completely considered three-dimensional and time-varying transient motions of each bearing component. The lubrication traction/slip effect was also included. Later, Stacke [41,42] from SKF developed a fully dynamic bearing model which can deal with not only Hertzian contact, but also general contact as well. In Stacke’s model, the parallel computation technique was introduced to increase the numerical integration efficiency. Ghaisas [43] proposed a dynamic model of rolling ball bearing by using discrete element method to investigate three-dimensional motions of each bearing component. Based on Ghaisas’ model, Ashtekar [44] studied the contact pressure distributions between rolling elements and raceways due to localized surface defects. In Ashtekar’s another paper [45], a defect was modeled by changing Hertizan contact relationships. Despite the fruitful achievements of dynamic modeling of rolling bearings with localized defects, their effectiveness is usually weakened under high speed conditions by some limitations. For high speed rolling bearings, their dynamic behaviors are 041015-2 / Vol. 136, AUGUST 2014

significantly influenced by centrifugal forces, gyroscopic and spin effects, relative slippage and lubrication traction between contact bodies, which need detailed modeling of three-dimensional motions of bearing components and behaviors of lubrications. Moreover, as a ball rolls over defect areas, additional displacements are induced due to material absence. Additionally, Hertzian contact coefficients and contact force directions are changed in defect zones due to the geometric characteristic variations of raceways. Therefore, in order to describe transient motions of high speed rolling bearings with defects, a complete model considering both high speed effects and influences of defects is needed. In this paper, a dynamic model for high speed rolling ball bearings with localized surface defects in raceways is proposed based on Gupta’s model. The three-dimensional motions of bearing components, relative slippage, and tractive forces are taken into account. Furthermore, the variations when a ball rolls over defect zones, i.e., additional displacements due to material absence, changes of Hertzian contact coefficients and contact force directions are also considered. The simulation results are validated with experiments on an aerospace bearing test rig. The comparisons show the ability and effectiveness of the present model in predicting vibration responses for high speed rolling ball bearings with localized surface defects.

2

System Modeling

In the present paper, some realistic assumptions and considerations are listed below: • • • •

Mass and geometric centers are all coincident with each other for each bearing component. Bearing components are treated as rigid except for contact zones. Thermal effects are negligible. Interactions between cage/balls and cage/raceways are ignored.

2.1 Modeling of Bearing Dynamics. Geometric interactions between balls and raceways are described to determine normal forces firstly (Sec. 2.1.1). Then, relative slip velocities are obtained by subdividing the contact ellipse into elementary strips (Sec. 2.1.2). Moreover, lubricant tractive forces are determined by using obtained normal forces, relative slip speeds and the traction model of lubricant (Sec. 2.1.3). Finally, the net forces and moments can be obtained (Sec. 2.1.4). 2.1.1 Ball/Raceway Interactions. The geometrical interaction between a ball and a raceway is shown in Fig. 1. Position vector r r

Fig. 1 Ball/raceway interaction

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locates the center of the raceway Or to inertial frame i ðxi ; yi ; zi Þ. Vector r cr locates the groove center of the raceway relative to the raceway center, and this vector is described in raceway fixed frame r ðxr ; yr ; zr Þ. Axis xr is the rotation axis of the raceway. The center of the ball Oa is located relative to the inertial frame by vector r b . In addition, it is convenient to define a ball azimuth frame a ðxa ; ya ; za Þ such that za axis is parallel to the radial component of r b , axis xa is parallel to the inertial xi axis, and axis ya is determined by the right-hand screw rule. Now, the interaction between the ball and the raceway will be determined by locating the center of the ball relative to the raceway center. This vector is denoted by r b r in Fig. 1 and is given as rb r ¼ rb rr

(1)

Using the superscript i for the inertial frame and r for the raceway fixed frame, the transformation of vector r b r from the inertial frame to the raceway fixed frame is written as r rb r ¼ Tir r ib r

where subscripts 2 and 3 denote the second and the third components of vector r rb r . Moreover, another frame called azimuth-in-raceway frame ar ðxar ; yar ; zar Þ can be defined by rotating the raceway fixed frame along its x axis by angle hrbr . This frame is not shown in Fig. 1. The relative position between the raceway curvature center and the ball center is given by

Then, contact angle a will be given as ar r a ¼ arctan bc1 r ar bc3

(8)

j2 ¼ j2 ðvcrb1 ; vcrb2 Þ

(9)

Then, multiplication of the normal force by the traction coefficient will give the tractive force f cT . 2.1.4 Net Forces and Moments. Once the normal contact force Qc and the tractive force f cT are known, the net force can be given as Fc ¼ Qc þ f cT

(10)

The force acting on the raceway will be equal and opposite to that acting on the ball. Then, the moments about mass centers for balls and raceways are expressed respectively as Mcb ¼ r ccp ðFc Þ

(11)

Mcr ¼ r cpr Fc

(12)

(5)

(6)

where r ccp and r cpr are position vectors locate a point in contact zone relative to the ball center and the raceway center, respectively. 2.2 Modeling of Localized Surface Defects. When a ball rolls over the defect, normal motions of all bearing components will be disturbed. Three main variations arise, i.e., • • •

where f is the raceway curvature factor and D is the ball diameter. The contact force between the ball and the raceway is given by Hertzian point contact theory [46] ( Kd1:5 d > 0 (7) Q¼ 0 d0 where K is the Hertzian contact stiffness coefficient. In the contact frame, contact force vector Qc is described as ð 0 0 Q Þ. 2.1.2 Relative Slip Velocities. Tractive forces between balls and raceways mainly rely on their relative slip velocities. Since the contact ellipse is quite narrow for most bearings, variations of slip along yc axis are neglected in Gupta’s model [12]. Journal of Manufacturing Science and Engineering

j1 ¼ j1 ðvcrb1 ; vcrb2 Þ

(4)

where superscript ar denotes the azimuth-in-raceway frame, and subscripts 1 and 3 denote the first and the second components of vector r ar b c. It is convenient to establish a new frame called contact frame c ðxc ; yc ; zc Þ (Fig. 1) whose origin is located at the center of contact ellipse. Once the vector r b c is transformed from the azimuth-inraceway frame to the contact frame by contact angle a, the elastic deformation between the ball and the raceway is calculated with d¼ r cbc3 ðf 0:5ÞD

2.1.3 Lubrication Tractive Forces. In a contact ellipse, tractive forces are determined according to the first and the second components of relative velocity vcrb . Its second component vcrb2 is located along the rolling direction and its first component vcrb1 is normal to vcrb2 . Given traction model of certain lubricant, traction coefficients corresponding to the slip velocities can be determined.

(2)

where Tir is the relevant transformation matrix. Then, the azimuth angle of the ball described in the raceway fixed frame is given as r r br2 (3) hbrr ¼ arctan r rbr3

r bc ¼ r br r cr

Furthermore, the contact ellipse is divided into several elementary strips, and the slip velocity over an elementary strip is determined by certain point in this strip [12,47]. Conveniently, ball velocities can be defined in the inertial cylindrical frame in terms of components x_ ball , r_ball , and h_ball . Then, local slip velocity vrb at any point in a contact ellipse can be given in terms of raceway and ball velocities. Detail expressions for these velocities can be found in [12]. Once the relative velocity is calculated, the lubrication tractive force can be obtained from the traction model of certain lubricant.

Additional deflection due to material absence is introduced (Sec. 2.2.1). The change of curvature radius of a raceway in defect zone alters the Hertzian contact coefficients (Sec. 2.2.2). An additional tangential component of the contact force in defect zone is generated compared with the original normal condition (Sec. 2.2.3).

2.2.1 Additional Deflection Introduced by Localized Defect. As a ball rolls over the defect, the total deflection d0 between them is the difference between defect characteristic dd (see Fig. 2, the largest value of dd is equal the depth of the defect) and elastic deformation d ( d dd jhbd j < he 0 (13) d ¼ d other where hbd is the difference between angular positions of the ball and the defect, and he is half the angle of the defect in the AUGUST 2014, Vol. 136 / 041015-3


The angle between plane O00 ABC and plane O0 AC is b. Angle he is determined from the geometric relationship (see Fig. 3) rd1 wd =2 (15) arcsin he ¼ rd1 rd2 where rd1 ðdm =2=cos bÞ6ðD=2Þ and rd2 6ðD=2Þ cos b þðdm =2Þ. The signs “6” refers to outer and inner raceways, respectively. Now, it is convenient to establish a new coordinate frame called defect frame d xd ; yd ; zd whose origin is located at the center of the defect zone, and the yd zd plane is located in plane O00 ABC (see Fig. 3). The profile of the defect in yd zd plane can be described as deflection induced by the a function as zd ¼ zd ðyd Þ. The additional defect can be determined by zd ( zd jhbd j < he (16) dd ¼ 0 other

Fig. 2 Angular position of the defect in bearing circumference

circumference of a raceway (see Fig. 2). The angle hbd at time t can be determined as ( for outer raceway modðhball ; 2pÞ hdinitial hbd ¼ modðhball ; 2pÞ modðhdinitial þ X i t; 2pÞ for inner raceway (14) where hdinitial is the initial position of the defect in raceway circumference, X i is the shaft rotation speed, and mod() is the function calculating the remainder after division. For the defect located in stationary outer raceway, its angular position hd always equals hdinitial . But for the inner raceway, angle hd changes with the shaft rotation at each time step. The position relationship between hball and hd will determine whether the ball rolls into the defect or not. As shown in Fig. 3, a defect is localized in a raceway. The center of the defect is Od, and its middle cross section in rolling direction is plane ABC. Angle hd is defined in plane O0 AC which is parallel to the rotation plane of the bearing. The width and the depth of the defect (wd and hd) are determined in plane O00 ABC.

2.2.2 Changes of Directions of Contact Forces. In defect zone, the direction of contact force Qq0 between the ball and the raceway is not coincident with OqOb which connects the ball center Ob and the original curvature center Oq, and an additional tractive force Qq0 1 along the tangential direction of the original normal raceway can be decomposed from Qq0 . In other words, the contact force Qq0 can be decomposed into two components, i.e., the tractive and the normal components (Qq0 1 and Qq0 2 in Fig. 4) in the original contact frame c. In order to introduce these two forces into dynamic equations, the contact force Qq0 should be described in the original contact frame. Once the function of defect the position of the curve is given, point Oq0 (its coordinate is xdoq ydoq zdoq in the defect frame) in yd zd plane can be obtained by using differential geometry theory [48]. Position vector r d which locates the point Oq0 relative to the contact point q0 (its coordinate is xdq0 ydq0 zdq0 in the defect frame) in yd zd plane can be expressed as r dd ¼ 0 ydoq zdoq 0 ydq0 zdq0

(17)

Then, the angle between vector r d and axis zd can be expressed as hddd ¼ arctan

d r d2 r dd3

(18)

Fig. 3 Geometric description of a defect in a raceway

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Fig. 5 Geometry of contacting bodies Fig. 4 Variations of contact force direction and raceway curvature

coordinate frame called contact-in-defect frame q0 qA0 new 0 0 x ; yq ; zq whose origin is located at the contact point q0 can be defined by rotating the defect frame around its xd axis by hddd . Then, the contact force Qq0 can be transformed from the contact-in-defect frame to the original contact frame by the following expression: Qcq0

¼ Tq0 c Qq0

(19)

where Tq0 c is the transformation matrix from the contact-in-defect frame to the original contact frame. 2.2.3 Changes of Hertzian Contact Stiffness Coefficient. As mentioned above, another important characteristic variation when a ball rolls over the defect is the Hertzian contact relationship. It is obvious that when a ball strikes the defect, the curvature properties differ from those on the normal raceway. For the contact between the ball and the defect depicted in Fig. 5, the radii of curvature are rI 1 ¼

D ; 2

rI2 ¼

D ; 2

rII 1 1;

rII 2 ¼ jr d j

(20)

where vector r d is defined in Sec. 2.2.2. Then, the Hertzian contact stiffness coefficient can be obtained by Hertzian point contact theory [46]. 2.3 Dynamic Equations. The dynamic equations for different bearing components will be described in different coordinates according to their kinematic characteristics. The equations of motion for a ball are conveniently written in the inertial cylindrical coordinate frame 8 mball x€ball ¼ Fballx > < mball r€ball mball rball h_2ball ¼ Fballr (21) > : _ _ mball rball hball þ 2mball r_hball ¼ Fballh where mball is the mass of a ball, and ðFballx ; Fballr ; Fballh Þ are the three components of the applied force vector in the cylindrical coordinate. Journal of Manufacturing Science and Engineering

The translational motions of any raceway are described in the inertial Cartesian coordinate 8 m x€ ¼ Fx > < r mr y€ ¼ Fy > : mr €z ¼ Fz

(22)

where mr is the mass of a raceway, and Fx ; Fy ; Fz are the components of the applied force vector in inertial Cartesian coordinate. The rotational motion of any bearing component can be described using Euler equations of motion 8 I x_ ðI2 I3 Þx2 x3 ¼ M1 > < 1 1 I2 x_ 2 ðI3 I1 Þx3 x1 ¼ M2 > : I3 x_ 3 ðI1 I2 Þx1 x2 ¼ M3

(23)

where ðI1 ; I2 ; I3 Þ are the principal moments of inertia, ðx1 ; x2 ; x3 Þ are the components of angular velocity vector, and ðM1 ; M2 ; M3 Þ are the components of applied moment vector. In order to simulate the vibration responses of measured points on the pedestal of rolling bearing, another two translational DOFs are introduced in the present model for the pedestal, as shown in Fig. 6. The corresponding dynamic equations are expressed as ( mp y€p þ cpy y_p þ kpy yp ¼ Fpy (24) mp €zp þ cpz €zp þ kpz zp ¼ Fpz where mp is the mass, kpy and kpz denote the stiffness, cpy and cpz denote the damping coefficients, and Fpy and Fpz are the loads acting on the pedestal.

3

Simulations and Discussions

The general solutions of the dynamic equations (Eqs. (21)–(24)) mainly involve position vectors and velocities of bearing components, and these equations are solved numerically by using fourth order Runge–Kutta–Fehlberg scheme with stepchanging criterion. At each time step t, the relationship between hball and hd is checked to determine whether a ball is in the defect zone. Then, forces and moments acting on each bearing AUGUST 2014, Vol. 136 / 041015-5


Table 1 Parameters for the simulated bearing- pedestal system Number of balls (z) Ball diameter (D), m Pitch diameter (dm), m Initial contact angle (a0), degree Curvature factor of inner raceway (fi) Curvature factor of outer raceway (fo) Clearance, m Damping due to lubricant (cbr), Ns/m Stiffness of pedestal (kpy, kpy), N/m Damping of pedestal (cpy, cpy), Ns/m

Fig. 6

14 12.7 103 70 103 30 0.515 0.52 0 20 15 106 1800

Pedestal model

Fig. 8 Traction model of the lubricant in simulation

speed) X i is constant. The numbering of the balls is anticlockwise in the raceway circumference as shown in Fig. 2. Parameters for the simulated bearings are listed in Table 1. The defect is modeled as quadratic curves. The lubricant model used in the current simulation is shown in Fig. 8. Since the present analysis mainly focuses on the modeling of localized surface defects, a simplified traction model of lubricant is adopted (see Fig. 8). This traction model is applicable for solid lubricants [12]. Elastohydrodynamic (EHD) effects, such as oil film damping and stiffness, have a certain effect on the dynamics of high speed rolling bearings. Oil film stiffness and damping coefficients depend considerably on shaft speed and external loads, which will result in different vibration characteristics for a bearing under different operating conditions. The EHD effect is beyond the scope of the current analysis, which will be investigated in the future work. In Secs. 3.1, 3.2, and 3.3, a localized surface defect is located in the outer raceway to investigate the motion characteristics when a ball rolls over the defect. In Sec. 3.4, the effect of radial loads on bearing dynamics is discussed. In Sec. 3.5, vibration characteristics of bearings with outer raceway and inner raceway defects are studied in both time domain and frequency domain.

Fig. 7 Flow chart for numerical computation

component are obtained based on their interactions. The flow chart of the numerical computation is provided in Fig. 7. In the simulation procedure, the outer raceway is considered to be stationary, and the inner raceway rotation speed (shaft rotation 041015-6 / Vol. 136, AUGUST 2014

3.1 Motion Characteristics of a Ball When It Rolls Over the Defect. In ball bearings, every ball exhibits similar motion characteristics when a pure axial thrust load is applied. Therefore, in order to investigate the general motion of defective ball bearings, a pure axial load of 3000 N is applied on the inner raceway in this section. Moreover, the shaft rotation speed is 10000 r/min, and the width and depth of the defect are 2.0 mm and 0.5 mm, respectively. As mentioned above, when jhbd j (jmodðhball ; 2pÞ hd j for outer raceway) is less than he, the ball is in the defect zone. As shown in Fig. 9, points A and E can be related to entry and exit points, respectively. Furthermore, the stage between A and C corresponds to the defect angle he. Transactions of the ASME


It can be found from Fig. 10(a) that, after the ball impacts the outer raceway at point D and then exits the defect at point E, contact forces at ball/outer raceway and ball/inner raceway reduce to 0 successively, and they reach in phase motion after some revolutions. This shows that, after the ball impacts and then exits the defect, the ball jumps between inner raceway and outer raceway successively. The origination of this phenomenon can be related to the zi component of impact force (see Fig. 10(d)), which results in corresponding accelerations for the ball to jump between these two raceways.

Fig. 9 Relationship between jmodðhball ; 2pÞ hd j and he

The contact forces between the ball and raceways when it rolls over the defect are shown in Fig. 10. In Fig. 10, points A, C, and E have the same meanings in Fig. 9. The contact force between the ball and the outer raceway drops to 0 suddenly at the entry point due to material absence (Figs. 10(a)–10(c)). Since the elastic deformation between the ball and the inner raceway cannot disappear immediately, the ball removes from the inner raceway gradually corresponding to the region from point A to point B. Then, the ball neither contacts with inner nor outer raceways until it impacts the trailing edge of the defect at point D (impact point). This shows that a contact-separation stage exists when a ball rolls over a defect zone. When the ball impacts the defect, the contact force between them achieves a large value at this point in a very short time, and this maximum contact force can be related to impact force. Finally, the ball exits the defect zone at point E.

3.2 Effect of Rotation Speed. In this section, the defect width is still set as 2.0 mm, and a pure axial load of 3000 N is applied on the inner raceway. Figure 11 shows the relationship between the contact force at ball/outer raceway and the shaft speed. It can be seen that the impact effect becomes more severe as the shaft speed increases until it reaches certain values (9000–10,000 r/min in this case, as shown in Fig. 11(a)). From Fig. 11(b), it can also be observed that the higher the shaft speed is, the earlier the ball drops into the defect zone. This phenomenon can be explained in this way, as the shaft speed increases, the ratio of ball orbit speed h_ball to shaft speed also increases in lubricated angular ball bearings acted on by pure axial loads [47]. Accelerations of the pedestal in zi direction under relatively low speed (3000 r/min) and high speed (10,000 r/min) are shown in Figs. 12 and 13, respectively. It can be found that at low speed, acceleration exhibit abundant oscillation in unit revolution than that at high speed. It can be seen from Figs. 12(b) and 13(b) that the impulse in acceleration results from the impulse of contact force at impact point. This can be recognized as the generation mechanism of the impact pulse of faulty bearings. Since the

Fig. 10 Motion characteristics of a ball when it rolls over the defect: (a) contact forces, (b) the detail view of contact forces, (c) the schema when a ball rolls over the defect, and (d) components of impact force

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Fig. 11 Relationships between contact forces and shaft speeds: (a) the maximum contact force and (b) entry points at different shaft speeds

Fig. 12 Vibration responses of pedestal at 3000 r/min (a) radial acceleration and (b) detail views of acceleration and contact force

Fig. 13 Vibration responses of pedestal at 10,000 r/min (a) radial acceleration and (b) detail views of acceleration and contact force

contact force at impact point is smaller in low speed case, the amplitude of acceleration is smaller consequently. When shaft speed increases, the second impulse (i.e., the impulse at impact point) becomes significant. When a ball rolls over a defect, two impulses are generated. The first one results from the entry of a ball into the defect, and the second one originates from the impact between the ball and the trailing edge of the defect, and these two impulses are 180 deg shift in phase (see Figs. 12(b) and 13(b)). This phenomenon was also reported by Sawalhi [49] and Dowling [50] based on experiments. In Sawalhi’s paper [49], the first impulse (i.e., the impulse 041015-8 / Vol. 136, AUGUST 2014

at entry point) was explained as a step response generated from the change of radial radius (i.e., rball described in Sec. 1) of the rolling element. In the current analysis, the first impulse is mainly caused by the sudden loss of contact between the ball and the defect, and this will result in a corresponding acceleration for rball to change. In Ref. [49], the origination of the impulse at impact point was treated as step changes in the rolling element’s travel path and velocity. In the current analysis, it can be found that the impact force will cause the ball to accelerate or decelerate instantaneously in certain directions when the ball impacts the trailing edge of the Transactions of the ASME


Fig. 14 Relationship between the maximum contact force and defect width

defect. This will change the rolling directions and velocities of the ball. Sawalhi also reported that another pulse existed behind the impulse at the impact point. The space between them was reported independent of the shaft speed. These two impulses were treated as a “beating” effect related to a small difference in resonance frequencies. However, the “beating” effect cannot be observed in the present analysis. One possible reason is the vibration sources which cause the “beating” effect are not modeled in the present model. Furthermore, when a ball exits a defect, a motion of high frequency can be found in Figs. 12 and 13. This motion is mainly caused by the jump of the ball between inner and outer raceways after it exits the defect.

Fig. 16 Contact forces of a radial loaded bearing

3.3 Effect of Defect Size. In this section, the shaft speed is 9000 r/min, and a pure axial load of 3000 N is applied on the inner raceway. Figure 14 shows the relationship between the impact force and the defect width. When the defect width is small, the impact force increases as the defect width increases. However, when the defect width reaches a certain value, the impact force becomes lighter as the defect width increases. It can be seen that the impact force and the defect width have a nonlinear relationship with each other. Accelerations resulted from different defect widths are shown in Fig. 15. The maximum of acceleration and the defect width also exhibit similar nonlinear relationship. Moreover, by comparing Figs. 14 and 15, it can be found that, despite a small defect and a large defect may generate equal impact force (such as wd ¼ 1.0 mm and 3.0 mm in this simulation case), the oscillation of acceleration after the ball exits the defect becomes more severe as the defect width increases.

3.4 Effect of Radial Load. It is well known that, the vast majority of ball bearings are operated under dominant radial loads. The main type of such bearings is the deep groove ball bearing whose contact angle is 0 deg. Thus, the vibration characteristics of a deep groove ball bearing applied by a pure radial load are discussed in this section. The simulated bearing has the same parameters as listed in Table 1 and Fig. 8 except for the contact angle. The radial load is 1000 N, and the shaft speed is 10,000 r/min. The width and depth of the defect are 2.0 mm and 0.2 mm, respectively. For a radial loaded bearing, contact forces are different with respect to the ball’s position. Figure 16 shows the contact forces between ball 1 and raceways. When the ball rotates around the bearing’s center axis, contact forces change periodically. As a result, the bearing can be divided into the loaded and unloaded zones according to the contact force distribution, as shown in Fig. 16. Moreover, it can be seen from Fig. 16 that the contact force of outer raceway is always larger than that of inner raceway due to centrifugal forces. From the distribution of contact force, it can be expected that the impact force when a ball rolls over a defect (the maximum contact force at impact point in Sec. 3.1) will differ with different defect positions (i.e., hd). Figure 17 shows the changes of the impact force with respect to different hd. It indicates that the impact force distribution is nearly the same as the contact force distribution in Fig. 16. When the defect is located in the loaded zone, the impact force is larger. Moreover, Fig. 17 shows that the impact force at hd ¼ 2.094 rad is larger than that at hd ¼ 4.189 rad although these two positions should result in the same contact force. One possible reason is that the impact directions between the ball and the trailing edge of the defect are different, which will result in different coefficients of Hertzian contact stiffness.

Fig. 15 Relationship between radial acceleration and defect width

Fig. 17 Relationship between the maximum contact force and hd


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Fig. 18 Radial acceleration and contact force

Figure 18 illustrates the contact force and the acceleration of pedestal in zi direction when ball 1 rolls over a defect located in the outer raceway. In Fig. 18, the defect position hd ¼ 2.966 rad. It can be found that the bearing exhibits the same dynamic characteristics as discussed in Secs. 3.1 and 3.2. 3.5 Feature Extraction for Localized Defect in Raceways. In this section, vibration responses simulation and analysis of rolling bearing with localized defects in raceways are carried out to provide proofs for defect detection and fault diagnosis. An axial load of 4000 N and a radial load of 1000 N are applied on the inner raceway. The width and depth of the defect are 1.0 mm and 0.2 mm, respectively. Parameters of the simulated bearing are listed in Table 1. The shaft speed is 9900 r/min, and the corresponding shaft rotation frequency fs is 165 Hz. Furthermore, characteristic fault frequencies, i.e., ball pass frequency on outer raceway (BPFO) and ball pass frequency on inner raceway (BPFI), are about 973.5 Hz and 1336.5 Hz, respectively. These frequencies are calculated on the assumption of pure rolling. 3.5.1 Defects on Outer Raceway. The acceleration of pedestal in zi direction is shown in Fig. 19. As shown in Fig. 19(a), an impulse is generated once a ball rolls over the defect. These impulses exhibit regular patterns due to stationary defect. Figure 19(b) is the detail view of the impulse. The time between two

impulses is about 1.02 103 sec. This indicates the BPFO here is about 980.4 Hz. Figure 19(c) is the frequency spectrum. From 0 Hz to 10,000 Hz, the maximum component is 5885 Hz, and the lower and larger ones surround it are 4904 Hz and 6866 Hz, respectively. Moreover, it can be found that the differences between 5885 Hz/ 4904 Hz and 6866 Hz/5885 Hz are all both BPFO. These indicate that the 5885 Hz is one of the natural frequencies and this frequency is modulated by BPFO. In the envelope spectrum (Fig. 19(d)), the dominant one is 980.7 Hz, and others are its superharmonics. Frequency 980.7 Hz can be related to BPFO. In order to understand the origination of this natural frequency (5885 Hz in Fig. 19(c)), an impact force of 1000 N (the force is modeled as a triangular form and its duration is about 104 s, as shown in Fig. 20(a)) is added on the radial direction of the outer raceway of a healthy bearing. Figure 20(b) is the vibration responses, and the corresponding frequency spectrum is shown in Fig. 20(c). Two distinct components can be found in the frequency spectrum, one is 593.7 Hz (fel) in low frequency domain, and the other one is 5488 Hz (feh) in high frequency domain. In order to demonstrate that the 5488 Hz is one of the natural frequencies, three different axial loads with the same radial impact force (Fig. 20(a)) are applied to study the vibration responses. The two distinct components, fel and feh, under different loading conditions are listed in Table 2. It can be seen that, feh is proportional to axial loads. However, fel still remains constant. In Gupta’s works [12,51], the frequency feh which is proportional to external loads was called ‘Bearing Kinematic Frequency’, and was recognized as one of natural frequencies of ball motion. This frequency can be defined on the assumption that response at this frequency is a sinusoidal motion with a fixed frequency [12,51] fe ¼

1 2p

rffiffiffiffiffiffiffiffiffiffiffi Q mball l

(25)

where l is the effective length of the oscillating pendulum. Thus, the origination of frequency 5885 Hz in Fig. 19(c) can be explained by Eq. (25). The difference between 5885 Hz in Fig. 19(c) and 5488 Hz in Fig. 20(c) may be caused by the reason that the localized defect changes the contact force distribution and effective length of ball motion in Eq. (25). Furthermore, in order to understand the origination of fel, bearings with different

Fig. 19 Vibration responses of the bearing with defected outer raceway: (a) radial acceleration, (b) detail view of radial acceleration, (c) frequency spectrum, and (d) envelope spectrum

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Fig. 20 Investigation of fel and feh: (a) impact force, (b) the corresponding vibrations, and (c) frequency spectrum

Table 2

fel and feh under different external loads

External loads (N) Fa ¼ 3000, Fr ¼ 1000 Fa ¼ 4000, Fr ¼ 1000 Fa ¼ 5000, Fr ¼ 1000

fel (Hz)

feh (Hz)

594.5 593.7 593.1

5328 5488 5636

Table 3 fel and feh under different kpy and kpz (Fa 5 5000 N, Fr 5 1000 N) Pedestal stiffness (N/m)

fel (Hz)

feh (Hz)

kpy ¼ 5 107, kpz ¼ 5 107 kpy ¼ 15 106, kpz ¼ 15 106 kpy ¼ 5 106, kpz ¼ 5 106

1038 593.1 380.4

5698 5636 5619

stiffness of pedestal under the action of the impact force are simulated. As shown in Table 3, fel changes severely under the influence of pedestal stiffness. Thus, the pedestal stiffness can be regarded as the main origination of fel. Moreover, it can be found that the characteristic fault frequencies calculated here are different with those calculated on pure rolling assumption. In a dynamic model, none of kinematic constraints and assumptions is used. Therefore, fault frequencies obtained by dynamic analysis are more accurate and reasonable. 3.5.2 Defects on Inner Raceway. A localized defect on inner raceway will rotate with the shaft’s rotation and go through the loaded zone of bearing every rotation. This mechanism shows that the vibration of inner raceway with defect will be modulated by the shaft rotation. Figure 21(a) shows the acceleration of the pedestal in zi direction, and its detail view can be found in Fig. 21(b).

Fig. 21 Vibration responses of the bearing with defected inner raceway: (a) radial acceleration, (b) detail view of radial acceleration, (c) frequency spectrum, and (d) envelope spectrum


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Table 4

Parameters of the tested bearing

Number of balls (z) Ball diameter (D), m Pitch diameter (dm), m Initial contact angle (a0), deg Clearance, m

27 11.12 103 117 103 15 0

Fig. 22 Orbit speed of ball 1 under both axial and radial loads

The maximum peaks of acceleration responses are related to the positions where the maximum contact force occurs, and the interval between two maximum peaks is related to one cycle of the shaft. It can also be found that time intervals between two neighboring impulses have some fluctuations (see Fig. 21(b)). This is mainly because that the orbit speed (i.e., h_ball ) of a ball has oscillations under the action of both axial and radial loads (see Fig. 22). The frequency spectrum is shown in Fig. 21(c). In this spectrum, the dominant component is the natural frequency which is excited when a ball rolls over the defect. Figure 21(d) shows the envelope spectrum. In the spectrum, shaft revolution frequency (164.8 Hz) and its superharmonic (329.2 Hz) can be found. Moreover, frequency 1329 Hz is related to BPFI. Furthermore, other components are superharmonics of BPFI and their sidebands. Moreover, it can be seen from Fig. 21(a) that when the inner raceway rotates, the sign of impulse of acceleration changes periodically. This phenomenon can be explained that, as shown in Fig. 23, the zi component of impact force due to inner raceway defect changes its sign periodically when the inner raceway rotates.

Fig. 25 A spall of the failure bearing

Fig. 23 Components of impact forces result from inner raceway defect

Fig. 24 Setup of the aerospace bearing test rig

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Fig. 26 Experimental results: (a) radial acceleration, (b) frequency spectrum, and (c) envelope spectrum

4

Experimental Verification

In order to show the capability of our proposed model, an experiment is presented here. This experiment is a part of the bearing life prediction program tested on an aerospace bearing test rig held in Xi’an Jiaotong University [52,53]. The test rig is shown in Fig. 24. The rig is driven by a motorized spindle connected with the shaft through a coupling. The tested bearing is a H7018C angular contact ball bearing. Parameters of the tested bearing are listed in Table 4. The shaft rotation speed is 6000 r/ min. The characteristic frequency of inner raceway fault BPFI of the tested bearing on the pure rolling assumption is about 1473.9 Hz. A radial load of 11 kN and an axial load of 2 kN are applied to the test bearing by the hydraulic loading system. Two accelerometers are mounted on the sleeve that is connected with the outer ring of the bearing. After about 146 h, the width of the spall is measured about 1.0 mm as shown in Fig. 25.

The experiment results are shown in Fig. 26. In time domain responses, we can find some distinct impulses in the loaded zone of the bearing, and they are marked with red circles in Fig. 26(a). In frequency spectrum (Fig. 26(b)), the dominant component is 4624 Hz, and none of distinct fault features can be found in this spectrum. In envelope spectrum (Fig. 26(c)), fault feature BPFI (1475 Hz) is evident, and this frequency has sidebands of shaft rotation frequency. In simulation procedure, the assumed traction model of lubricant is shown in Fig. 8. The adopted loading condition is the same as that in the experiment. Parameters of the simulated bearing are listed in Table 4. Moreover, stiffness and damping coefficients of the pedestal in Sec. 3 are adopted for this simulation procedure. The detailed calculation of parameters of the pedestal is beyond the scope of the current analysis. Simulation results are shown in Fig. 27. Figure 27(a) is the contact force distribution of the bearing which can be divided into loaded and unloaded zones. It can be seen in Fig. 27(b) that

Fig. 27 Simulated results: (a) contact force distribution, (b) radial acceleration, (c) frequency spectrum, and (d) envelope spectrum


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acceleration responses in loaded zone are much more severe than the ones in unloaded zone. However, the amplitude of acceleration is much larger than experiments. This is mainly because the transfer path, which will result in attenuation of the signal, is not considered in the present paper. In frequency spectrum (Fig. 27(c)), the dominant component is the natural frequency (6161 Hz) which is excited by the fault when a ball rolls over it. In the frequency spectrum, none of distinct fault features can be detected. Figure 27(d) is the envelope spectrum. The shaft rotation frequency fs (99.8 Hz) and its superharmonic (199.6 Hz) are evident in this spectrum. Furthermore, a component 1476 Hz which has sidebands spaced at fs exists. This frequency can be related to BPFI. This indicates that the vibration signals is modulated by fs. These frequency components in simulation results can also be found in experiments. It should be noted that the differences between simulated and experimental results are mainly related to unknown oil lubricant parameters and curvature factors of raceways, and also the transfer path and thermal effect neglected in this paper.

5

Discussions on Limitations of the Present Model

As demonstrated above, the proposed model is effective in modeling and predicting transient motion characteristics of high speed rolling ball bearings with localized raceway defects. But some drawbacks and limitations still exist in the present model which will be improved and investigated in our future works. •

• •

•

6

The width of the defect is assumed larger than the semi major axis of contact ellipse, which is suitable for large scale of spalls. The vibration response due to surface damages with small size such as micropitting is not covered in our model. Furthermore, the effect of the defect depth is not considered. EHD effects, which could influence the dynamic characteristics of bearings, are not considered. Transfer path, which will modify the vibration signal from the point where it is generated to the point where it is measured, is not considered in the present model. The effect of localized defects on cage instabilities and the result in bearing vibrations are ignored in the present model. However, cage instability plays an important role in fault evolutions of high speed bearings.

Conclusion

Dynamic modeling and vibration responses simulation are essential for fault mechanism studies to provide proofs for defect detection and fault diagnosis. In this paper, dynamic characteristics of high speed ball bearings with localized surface defects are systematically investigated on the basis of Gupta’s model. Variations of three important factors, i.e., additional deflections, Hertzian contact stiffness coefficient, and directions of contact forces due to localized defects are all integrated into Gupta’s model to completely describe transient motions of defective high speed ball bearings. Based on the present analysis, the following conclusions are drawn. •

•

•

The variations of contact forces between balls and raceways when a ball rolls over the defect zone demonstrate that there exists a contact-separation stage that the ball neither contacts with inner raceway nor outer raceway. Parametric studies are carried out for the effects of different defect widths and inner raceway rotation speeds on bearing vibration responses. These studies demonstrate that the severity of vibration responses exhibit nonlinear relationships with both shaft speed and defect width. In a dynamic model, none of kinematic constraints and assumptions is used. Thus, fault frequencies obtained by dynamic analysis are more accurate and reasonable.

Acknowledgment This work is supported by National Natural Science Foundation of China (No. 51105294 and 51035007), National Basic Research 041015-14 / Vol. 136, AUGUST 2014

Program of China (No. 2011CB706606), and the Fundamental Research Funds for the Central University.

Nomenclature a ¼ ball azimuth frame ar ¼ azimuth-in-raceway frame BPFI ¼ ball passing frequency on inner raceway (Hz) BPFO ¼ ball passing frequency on outer raceway (Hz) c ¼ contact frame cbr ¼ damping coefficient at ball/raceway contact (Ns/m) cpy, cpz ¼ damping coefficients of pedestal (Ns/m) d ¼ defect frame D ¼ ball diameter (m) f ¼ raceway curvature factor fe ¼ natural frequency of ball motion (Hz) fel ¼ distinct component in low frequency domain (Hz) feh ¼ distinct component in high frequency domain (Hz) fi, fo ¼ curvature factor of inner and outer raceways, respectively fs ¼ shaft rotation frequency (Hz) f T ¼ tractive force vector (N) F ¼ net force vector (N) Fa ¼ axial load, N Fr ¼ radial load (N) ðFballx ; Fballr ; Fballh Þ ¼ components of the applied force vector in cylindrical frame (N) Fx ; Fy ; Fz ¼ components of the applied force vector in inertial Cartesian coordinate (N) hd ¼ depth of a defect (m) i ¼ inertial frame ðI1 ; I2 ; I3 Þ ¼ three principal moments of inertia (Nm2) K ¼ Hertzian contact stiffness coefficient (N/m1.5) kpy, kpz ¼ stiffness of pedestal (N/m) l ¼ equivalent length of the oscillating pendulum (m) mball ¼ mass of a rolling ball (Kg) mp ¼ mass of the pedestal (Kg) mr ¼ mass of a raceway (Kg) Mb ¼ moment vector acting on ball centers (Nm) Mr ¼ moment vector acting on raceway centers (Nm) ðM1 ; M2 ; M3 Þ ¼ three components of applied moment vector (Nm) q0 ¼ contact-in-defect frame Q ¼ contact force (N) Qq0 ¼ contact force in defect zone (N) r ¼ radius of curvature of contacting body (m) r ¼ raceway fixed frame r b ¼ position vector locating the center of a ball to the inertial frame (m) r br ¼ position vector locating the center of a ball to the center of a raceway (m) r bc ¼ position vector locating the center of a ball to the curvature center of a raceway (m) r cr ¼ position vector locating the groove center of a raceway relative to the raceway center (m) r d ¼ position vector locating the curvature center of a defect relative to the contact point (m) rcp, rpr ¼ position vectors locating a point in contact zone relative to the ball center and raceway center, respectively (m) Transactions of the ASME


rr ¼ position vector locating the center of a raceway to the inertial frame (m) t ¼ time (sec) Tir ¼ transformation matrix between inertial frame and raceway fixed frame Tq0 c ¼ transformation matrix between defect frame and contact-in-defect frame vrb ¼ local slip velocity at a contact point (m/s) wd ¼ width of a defect (m) (x, y, z) ¼ coordinate axes ðx_ ball ; r_ball ; h_ball Þ ¼ components of ball velocities described in inertial cylindrical frame (m/s m/s, rad/s) a ¼ contact angle (degree) d ¼ elastic deformation (m) dd ¼ geometric characteristic of defect (m) d0 ¼ total deflection between ball and raceway in defect zone (m) Dt ¼ time increment (sec) hbr ¼ azimuth angle of a ball described in raceway fixed frame (rad) hbd ¼ the difference between angular positions of the ball and defect (rad) hb ¼ ball azimuth position (rad) hd ¼ angular position of a defect (rad) hdd ¼ angle between vector r d and axis zd (rad) hdinitial ¼ initial angular position of a defect in raceway circumference (rad) he ¼ half the angle of the defect in the circumference of a raceway (rad) ðx1 ; x2 ; x3 Þ ¼ components of a angular velocity vector (rad) Xi¼ shaft rotation speed (rad/s)

Subscripts 1, 2, 3 ¼ corresponding components of vectors I1;I2; II1; II2 ¼ two bodies of revolution (I, II) in contact, each has two principal planes (1,2).

Superscripts a¼ ar ¼ c¼ d¼ q0 ¼ r¼

ball azimuth frame azimuth-in-raceway frame contact frame defect frame contact-in-defect frame raceway fixed frame

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