Analyzing the Dynamic Characteristics of Angular Contact Ball Bearings Considering Three-Dimensional Localized Raceway Defects

Abstract

Current research on localized raceway defects of angular contact ball bearings (ACBB) mainly focuses on assuming that localized raceway defects are cube-shaped defects characterized using a half-sine displacement excitation function. However, the assumption of a cube-shaped defect cannot accurately reflect the morphological characteristics of a localized raceway defect, and the half-sine displacement excitation function cannot be used to accurately describe the relationship between the geometric positions of rolling element and raceway in the region of localized raceway defects. In this study, a comprehensive dynamic model of an ACBB considering a three-dimensional localized raceway defect is established based on the nonlinear Hertz contact theory in conjunction with the outer raceway control theory using the improved Newton–Raphson iteration method. Three localized raceway defect distribution types, namely symmetric, offset, and deflection distributions, are considered. The established model is verified by comparing the results of the proposed model with those of existing literature. The dynamic characteristics of the ACBB were analyzed by investigating the effects of the geometrical size and distribution types on the time-varying contact angles, contact forces, and diagonal stiffness of the ACBB. The investigation results show that the appearance of localized raceway defect leads to the time-varying curves of contact angles, contact forces and diagonal stiffness having Λ- and V-shaped mutations in some time intervals; The variation tendencies of the Λ- and V-shaped mutations are significant with the increase in defect radial depth H, defect axial width a and angular distance θ_b. The increase in defect eccentric distance L is beneficial to the rolling elements disengaging from the defect area and it can weaken the influence of localized raceway defect on the time-varying contact and stiffness characteristics of ACBB. The time-varying contact and stiffness characteristics appear to change significantly when the defect deflection angle α_β increase to α_γ. The results of this study provide a theoretical basis for the fault diagnosis of localized raceway defects in ACBB.

Introduction

As essential supporting components, angular contact ball bearings are widely used in numerous rotating mechanical equipment, including machine tools, aeroengines, and gearbox casings. The dynamic performance of angular contact ball bearings significantly affects the stability, reliability, and accuracy of rotating mechanical equipment. Defects in angular contact ball bearings are the most common cause of failure in rotating mechanical equipment. Defects in angular contact ball bearings are caused by operating conditions and manufacturing errors. Angular contact ball bearing defects can be divided into two types: distributed and localized. Distributed defects include the roughness, waviness, and misalignment of raceways and the off-size and waviness of rolling ball elements, which are caused by manufacturing errors. Localized defects include pits, spalls, and cracks, and are caused by operating conditions. The distributed defect is explicit and has little effect on the dynamic characteristics of the angular contact ball bearing when the defect is limited to a certain range according to a series of criteria. Localized defects are implicit, and their appearance has an important effect on the dynamic response of angular contact ball bearings and endangers the overall security performance of the rotating mechanical equipment in operation. To enhance the diagnostic efficiency for localized defects, an analytical numerical modeling must be established to accurately predict the dynamic response features of angular contact ball bearings with localized defects.

Singh et al. and Liu et al. [1, 2] developed an explicit dynamic finite element model to predict the dynamic response of a rolling bearing with a raceway defect using LS-DYNA. Qin et al. [3, 4] proposed a novel fault dynamic model for analyzing the time- and frequency-domain characteristics of angular contact ball bearings using the Newton-Raphson method in conjunction with the Runge–Kutta method, and the fault excitation is simulated by employing the B-spline fitting displacement excitation method. Jafari et al. [5] established a dynamic model of angular contact ball bearings with and without an outer raceway defect to predict the influence of spall defects on bearing failure using the Runge–Kutta method with a variable time step. Kong et al. [6] presented a vibration model of a ball bearing to study its vibration mechanism of ball bearings based on the Hertzian contact stress distribution. Niu et al. [7] proposed a dynamic model of an angular contact ball bearing with ball defects to investigate the vibration mechanism and diagnose ball defects more effectively. Liu et al. [8] proposed a new dynamic model for investigating the vibration response of a ball bearing owing to a localized surface defect on its races, according to the Hertzian elastic contact theory. Xu et al. [9] established a mechanical model of a defective angular contact ball bearings (ACBB) considering the centrifugal force and gyroscopic moment of the rolling element to analyze the influence of the defect size on the bearing performance. He et al. [10] proposed an accurate quantitative estimation method based on instantaneous vibration energy analysis for estimating the size of the localized defect in the ball bearing, which is a new idea. Gomez et al. [11] analyzed the instantaneous angular speed of a deep-groove ball bearing with localized defects subjected to nonstationary conditions by establishing a Hertzian contact angular contact ball bearing model. Liu et al. [12] proposed quasi-static modeling of a roller bearing with a natural defect to analyze the influences of natural defects on the contact and stiffness characteristics of roller bearings. Patil et al. [13] presented a numerical analytical model for forecasting the effects of localized defects on the vibration response of a ball bearing based on Hertzian contact deformation theory. Yang et al. [14] established a rotor-bearing casing system to analyze the vibration signatures of rolling element bearings with localized faults using numerical and experimental methods. Shah et al. [15, 16–17] developed a dynamic model for predicting the vibration behavior of deep-groove ball bearings considering localized race defects by employing the nonlinear Hertzian contact theory in conjunction with the fourth order Runge-Kutta method. Liu et al. [18, 19–20] proposed a new modeling method for defect extension and morphology to analyze the effects of defect-edge features on the vibration behaviors of cylindrical roller bearings with localized defects. Jiang et al. [21] developed a complete bearing dynamics model for solving the vibration response of a bearing with a three-dimensional defect. Wen et al. [22] proposed a complete multi-DOF dynamic modeling to investigate the dynamic behavior of angular contact ball bearings by considering localized surface defects. Niu et al. [23] experimentally developed a dynamic model to analyze the vibration characteristics of a cylindrical roller bearing considering roller defects. Refs. [24, 25] estimated the defect size in rolling element bearings by establishing a new dynamic analysis modeling based on the Hertzian contact theory. Li et al. [26] established a mechanical model of an angular contact ball bearing with a localized raceway defect to analyze the effects of such a defect on the contact angle and load distribution. Zhao et al. [27] established a nonlinear dynamic model to explore the influence of race defects on the nonlinear dynamic characteristics of rolling element bearings based on Hertz theory. Bastami et al. [28] developed a physical model to study the variation tendencies of the statistical features of the vibration patterns of defective rolling element bearings. Refs. [29, 30] established a five-degree-of-freedom nonlinear dynamic model to analyze the changes in the contact angle and contact force of an angular contact ball bearing under external load conditions.

According to the above results, the localized raceway defect of an angular contact ball bearing is assumed to be a cube-shaped defect and is generally characterized by a half-sine displacement excitation function. However, the cube-shaped defect cannot accurately reflect the morphological characteristics of the localized raceway defect, and the half-sine displacement excitation function cannot be used to accurately describe the relationship between geometric positions of the rolling element and the raceway in the region of localized raceway defects. Therefore, this study proposes a comprehensive dynamic modeling of an angular contact ball bearing considering a three-dimensional localized raceway defect to precisely predict the dynamic response characteristics, including the time-varying contact angle, contact force, and stiffness. An angular contact ball bearing was selected as the objective, and the localized raceway defect was assumed to be distributed in the outer raceway of the ACBB, because the outer raceway can withstand a greater contact load. Localized raceway defects are regarded as curve-shaped defects, and three kinds of three-dimensional localized raceway defect distributions, namely symmetric, offset, and deflection distribution are considered, as shown in Figure 1. The geometrical features of a localized raceway defect are expressed by the relative changes in the radial distance between the centers of the rolling element and inner ring in the defect area. The above dynamic modeling is established based on the nonlinear elastic Hertz contact theory in conjunction with the outer raceway control theory and is solved by employing the improved Newton–Raphson iteration method. Based on the established dynamic model, the effects of the localized raceway defect size and distribution type on the time-varying contact and stiffness characteristics of an ACBB with three-dimensional localized raceway defect are investigated systematically. The results of this study can provide a theoretical basis for fault diagnosis of localized raceway defects in ACBBs.

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Figure 1

Types of distribution in a three-dimensional localized raceway defect in an ACBB: (a) Symmetric distribution, (b) Offset distribution, (c) Deflection distribution

Theoretical Modeling

Modeling of Three-Dimensional Localized Raceway Defect

Modeling of Symmetrical Distribution

The system presented in this study is illustrated in Figure 1. The systematic distribution of three-dimensional localized raceway defects is shown in Figure 2(a). The symmetric distribution can be interpreted as the centerline of the localized defect coinciding with the centerline of the outer raceway in the axial direction. In Figure 2(a), a and b represent the axial width and circumferential length of the upper surface of the localized raceway defect, respectively; θ_b denotes the angular distance of the localized defect between the starting and ending edges of the defect; H expresses the radial depth of the localized raceway defect; R_o denotes the radius of the outer raceway. To analyze the relative change in the radial distance between the centers of the rolling element and the inner ring in the defect area, a two-dimensional cross-sectional view is shown in Figure 2(b).

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Figure 2

Symmetric distribution of localized raceway defect of ACBB: (a) Three-dimensional view, (b) Two-dimensional cross-section view

As shown in Figure 2(b), O_b and O_b1 denote the ideal and actual positions of jth rolling element center in the defect area, respectively, O represents the center of the outer raceway, θ_i represents the angle between the azimuth angle of the starting edge of the defect and the position angle of jth rolling element in the defect area, the value range of θ_i is 0 ≤ θ_i ≤ θ_b, R_b denotes the radius of the rolling element. h denotes the relative change in the radial distance between the centers of the jth rolling element and inner ring in the defect area and it can also represent the radial distance between O_b and O_b1, the expression for h using geometric analysis is determined as follows:

1

$$\begin{array}{c}h=\left\{\begin{array}{cc}{R}_b-R_o(1-cos{\theta }_i)-\sqrt{R_b^2-{\left(R_{o}sin{\theta }_i\right)}^2},& 0\le {\theta }_i<\frac{{\theta }_b}{2},\\ R_b-R_o\left(1-cos{\theta }_{\mathit{bi}}\right)-\sqrt{R_b^2-{\left(R_{o}sin{\theta }_{\mathit{bi}}\right)}^2},& \frac{{\theta }_b}{2}<{\theta }_i\le {\theta }_b,\end{array}\right)\\ {\theta }_{\mathit{bi}}={\theta }_b-{\theta }_i.\\ \end{array}$$

It is necessary to point out that the above formulation of h only reflects the geometrical features of the specific size of the localized defect, and it cannot comprehensively describe the geometrical features of the symmetric distribution of the three-dimensional localized raceway defect. To compensate for the above deficiencies, the symmetric distribution of the three-dimensional localized raceway defect is divided into four types based on the geometrical size of the three-dimensional localized defect. The above four types of distributions are described intuitively in Figure 3 as follows.

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Figure 3

Two-dimensional cross-section analysis of localized raceway defect: (a) h<H, (b) h=H, (c) h>H

As shown in Figure 3, ${\phi }_s$ and ${\phi }_e$ denote the azimuth angles of the starting and ending edges of the defect, respectively, R_ib represents the radial distance between the inner ring center O_i and jth rolling element center O_b, and the change of R_ib with azimuth angle ${\phi }_j$ is shown in Figure 3.

For the first type of symmetric distribution of three-dimensional localized raceway defects, h can be expressed as follows:

2

$$\begin{array}{c}\text{When}a\ge 2\sqrt{R_b^2-{\left[R_o\left(1-cos\frac{{\theta }_b}{2}\right)+R_{\mathit{bo}}\right]}^2},\\ \text{When}{h}_{\mathit{max}}=R_b-R_o\left(1-cos\frac{{\theta }_b}{2}\right)-R_{\mathit{bo}}\le H,\\ h=\left\{\begin{array}{cc}\begin{array}{c}{R}_b-R_o\left(1-cos{\phi }_{\mathit{js}}\right)\\ -\sqrt{R_b^2-{\left(R_{o}sin{\phi }_{\mathit{js}}\right)}^2,}\\ \end{array}& {\phi }_s\le mod({\phi }_j,2\pi )<{\phi }_s+\frac{{\theta }_b}{2},\\ \begin{array}{c}{R}_b-R_o\left(1-cos{\phi }_{\mathit{ej}}\right)\\ -\sqrt{R_b^2-{\left(R_{o}sin{\phi }_{\mathit{ej}}\right)}^2},\\ \end{array}& {\phi }_s+\frac{{\theta }_b}{2}\le mod({\phi }_j,2\pi )\le {\phi }_e,\\ 0,& \text{other},\end{array}\right)\\ {\phi }_{\mathit{js}}={\phi }_j-{\phi }_s,{\phi }_{\mathit{ej}}={\phi }_e-{\phi }_j,R_{\mathit{bo}}=\sqrt{R_b^2-{\left(R_{o}sin\frac{{\theta }_b}{2}\right)}^2},\\ \end{array}$$where ${\phi }_j$ denotes the azimuth angle of jth rolling element and mode represents the remainder function.

For the second type of symmetric distribution of three-dimensional localized raceway defects, h can be determined as follows:

3

$$\begin{array}{c}\text{When}a\ge 2\sqrt{R_b^2-{\left[R_o(1-cos\frac{{\theta }_b}{2})+R_{\mathit{bo}}\right]}^2},\\ \text{When}{h}_{\mathit{max}}=R_b-R_o(1-cos\frac{{\theta }_b}{2})-R_{\mathit{bo}}>H,\\ {\theta }_c=arccos\left[\frac{H^2+\left(2R_o+2H\right)\left(R_o-R_b\right)}{2R_o^2-2R_o\left(R_b-H\right)}\right],\\ h=\left\{\begin{array}{cc}\begin{array}{c}{R}_b-R_o\left(1-cos{\phi }_{\mathit{js}}\right)\\ -\sqrt{R_b^2-{\left(R_{o}sin{\phi }_{\mathit{js}}\right)}^2,}\\ \end{array}& {\phi }_s\le mod({\phi }_j,2\mathrm{\pi })<{\phi }_s+{\theta }_c,\\ H,& {\phi }_s+{\theta }_c\le mod({\phi }_j,2\pi )<{\phi }_e-{\theta }_c,\\ \begin{array}{c}{R}_b-R_o\left(1-cos{\phi }_{\mathit{ej}}\right)\\ -\sqrt{R_b^2-{\left(R_{o}sin{\phi }_{\mathit{ej}}\right)}^2}\\ \end{array}& {\phi }_e-{\theta }_c\le mod({\phi }_j,2\pi )\le {\phi }_e\\ 0,& \text{other,}\end{array}\right)\end{array}$$where θ_c denotes the angular distance between the starting edge of the defect and initial angular position corresponding to h_max, which equals H.

For the third type of symmetric distribution of three-dimensional localized raceway defects, the expression for h can be written as follows:

4

$$\begin{array}{c}\text{When}a<2\sqrt{R_b^2-{\left[R_o\left(1-cos\frac{{\theta }_b}{2}\right)+R_{\mathit{bo}}\right]}^2},\\ h_b=R_b-\sqrt{R_b^2-{\left(0.5a\right)}^2}\le H,\\ {\theta }_a=arccos\left[1-\frac{{\left(0.5a\right)}^2}{2R_o^2-2R_o\sqrt{R_b^2-{\left(0.5a\right)}^2}}\right],\\ h=\left\{\begin{array}{cc}\begin{array}{c}{R}_b-R_o\left(1-cos{\phi }_{\mathit{js}}\right)\\ -\sqrt{R_b^2-{\left(R_{o}sin{\phi }_{\mathit{js}}\right)}^2,}\\ \end{array}& {\phi }_s\le mod({\phi }_j,2\pi )<{\phi }_s+{\theta }_a,\\ R_b-\sqrt{R_b^2-{\left(0.5a\right)}^2,}& {\phi }_s+{\theta }_a\le mod({\phi }_j,2\pi )<{\phi }_e-{\theta }_a,\\ \begin{array}{c}{R}_b-R_o\left(1-cos{\phi }_{\mathit{ej}}\right)\\ -\sqrt{R_b^2-{\left(R_{o}sin{\phi }_{\mathit{ej}}\right)}^2},\\ \end{array}& {\phi }_e-{\theta }_a\le mod({\phi }_j,2\pi )\le {\phi }_e,\\ 0,& \text{other,}\end{array}\right)\end{array}$$

5

$$\begin{array}{c}\text{When}a<2\sqrt{R_b^2-{\left[R_o(1-cos\frac{{\theta }_b}{2})+R_{\mathit{bo}}\right]}^2},\\ h_b=R_b-\sqrt{R_b^2-{\left(0.5a\right)}^2}>H,\\ h=\left\{\begin{array}{cc}\begin{array}{c}{R}_b-R_o\left(1-cos{\phi }_{\mathit{js}}\right)\\ -\sqrt{R_b^2-{\left(R_{o}sin{\phi }_{\mathit{js}}\right)}^2},\\ \end{array}& {\phi }_s\le mod({\phi }_j,2\pi )<{\phi }_s+{\theta }_c,\\ H,& {\phi }_s+{\theta }_c\le mod({\phi }_j,2\pi )<{\phi }_e-{\theta }_c,\\ \begin{array}{c}{R}_b-R_o\left(1-cos{\phi }_{\mathit{ej}}\right)\\ -\sqrt{R_b^2-{\left(R_{o}sin{\phi }_{\mathit{ej}}\right)}^2},\\ \end{array}& {\phi }_e-{\theta }_c\le mod({\phi }_j,2\pi )\le {\phi }_e,\\ 0,& \text{other},\end{array}\right)\end{array}$$where θ_a denotes the angle distance between the starting defect edge, and the initial angular position corresponding to h_b is H.

For the fourth type of symmetric distribution of three-dimensional localized raceway defects, the expression for h can be found in Eq. (5). In sum, the universal expression for h considering the symmetric distribution of the localized raceway defects, can be written as follows:

6

$$\begin{array}{c}{h}_1=R_b-R_o\left(1-cos{\phi }_{\mathit{js}}\right)-\sqrt{R_b^2-{\left(R_{o}sin{\phi }_{\mathit{js}}\right)}^2},\\ h_2=R_b-R_o\left(1-cos{\phi }_{\mathit{ej}}\right)-\sqrt{R_b^2-{\left(R_{o}sin{\phi }_{\mathit{ej}}\right)}^2},\\ h_3=R_b-\sqrt{R_b^2-{\left(0.5a\right)}^2},\\ h=\left\{\begin{array}{cc}min\left(h_1,h_3,H\right),& {\phi }_s\le mod({\phi }_j,2\pi )<{\phi }_s+\frac{{\theta }_d}{2},\\ min\left(h_2,h_3,H\right),& {\phi }_s+\frac{{\theta }_d}{2}\le mod({\phi }_j,2\pi )<{\phi }_e,\\ 0,& \text{other}.\end{array}\right)\\ \end{array}$$

Modeling of Offset Distribution

The offset distribution of three-dimensional localized raceway defects is shown in Figure 4(a). The offset distribution can be interpreted as the deviation between the centerline of the local defect and the centerline of the outer raceway in the axial direction. The symbols in Figure 4(a) are consistent with those in Figure 2(a), except L, the symbol of L denotes the amount of axial deviation between the centerline of the localized defect and the centerline of the outer raceway. To analyze the geometrical features of the offset distribution conveniently, a two-dimensional sectional view is given in Figure 4(b). As shown in Figure 4(b), the jth rolling element completely disengages from the localized raceway defect area when the deviation distance reaches 0.5a.

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Figure 4

Offset distribution of the localized raceway defect of ACBB: (a) Three-dimensional view, (b) Two-dimensional sectional view

According to the derivation process of the geometrical features of the symmetric distribution of three-dimensional localized raceway defects, the universal expression of h considering the offset distribution of the localized raceway defect can be determined as follows:

7

$$\begin{array}{c}\text{When}0\le L<0.5a,\\ h_1=R_b-R_o\left(1-cos{\phi }_{\mathit{js}}\right)-\sqrt{R_b^2-{\left(R_{o}sin{\phi }_{\mathit{js}}\right)}^2},\\ h_2=R_b-R_o\left(1-cos{\phi }_{\mathit{ej}}\right)-\sqrt{R_b^2-{\left(R_{o}sin{\phi }_{\mathit{ej}}\right)}^2},\\ h_3=R_b-\sqrt{R_b^2-{\left(0.5a-L\right)}^2},\\ h=\left\{\begin{array}{cc}min\left(h_1,h_3,H\right),& {\phi }_s\le mod({\phi }_j,2\pi )<{\phi }_s+\frac{{\theta }_d}{2},\\ min\left(h_2,h_3,H\right),& {\phi }_s+\frac{{\theta }_d}{2}\le mod({\phi }_j,2\pi )<{\phi }_e,\\ 0,& \text{other},\end{array}\right)\\ \text{when}L\ge 0.5a,h=0.\\ \end{array}$$

According to the above formulation, it is obvious that the symmetric distribution can be viewed as a special case of the offset distribution.

Modeling of Deflection Distribution

The deflection distribution of three-dimensional localized raceway defect is shown in Figure 5(a). The deflection distribution can be interpreted as the center line of the localized defect, and the center line of the outer raceway exhibits angle deflection. The symbols in Figure 5(a) denote the same parameters as those in Figure 2(a) except α_β; α_β denotes the deflection angle. As the deflection angle increases, the effective angular distance of localized defect between the starting defect edge and ending defect edge changes, as shown in Figure 2(b). The symbol ${\theta }_b^{\prime }$ represents the effective angular distance of the localized defect under the deflection angle α_β, and the specific expression for ${\theta }_b^{\prime }$ can be obtained in the following form according to the geometrical size of the localized raceway defect.

8

$${\theta }_b^{\prime }=\left\{\begin{array}{c}{\theta }_b\left./\phantom{{\theta }_{b}cos{\alpha }_{\beta }}\right)cos{\alpha }_{\beta },{\alpha }_{\beta }<{\alpha }_{\gamma },\\ a\left./\phantom{asin{\alpha }_{\beta }}\right)sin{\alpha }_{\beta },{\alpha }_{\beta }\ge {\alpha }_{\gamma },\\ \end{array}\right){\alpha }_{\gamma }=arctan\left(\frac{a}{R_o{\theta }_b}\right),$$where α_γ denotes the critical deflection angle.

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Figure 5

Deflection distribution of localized raceway defect of ACBB: (a) Three-dimensional view, (b) Contact surface view

Owing to the appearance of the deflection angle, the geometrical feature analysis of the deflection distribution of three-dimensional localized raceway defect is very complex. For convenience, the relative change in the radial distance between the center of rolling element and that of the inner ring of the ACBB, considering the deflection distribution of the localized raceway defect, can be determined according to the trajectory of the jth rolling element in the defect area. The trajectory of the jth rolling element can be divided into two types based on the deflection angle.

For the first type, the deflection angle is assumed as 0<α_β<α_γ, the trajectory of the jth rolling element in the defect area is shown in Figure 6.

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Figure 6

Trajectory of the jth rolling element under deflection distribution (0 < α_β < α_γ): (a) I–IV, (b) II–III

As shown in Figure 6, I, II, III, and IV denote the localized raceway defect edges. The symbol “•” expresses the contact point between the jth rolling element and the starting and ending edges of the defect; The symbol “•” represents the contact point of the jth rolling element with the defect edge at a certain moment and this contact point is caused by deflection angle. The symbol “−” denotes the axial distance between “•” and the center line of axial direction of outer raceway. Before continuing with the discussion, a geometrical assumption must be made, which is that the right triangles with curved sides approximately satisfy the Pythagorean theorem. I–IV and II––III represent the two typical contact conditions. In the defect area, the relative change in the radial distance between center of the jth rolling element and that of the inner ring of the ACBB under contact conditions I-IV can be determined as follows:

9

$$h_{I-IV}=\left\{\begin{array}{c}{R}_b-\sqrt{R_b^2-{\left(R_o{\theta }_i\left./\phantom{R_o{\theta }_{i}tan{\alpha }_{\beta }}\right)tan{\alpha }_{\beta }\right)}^2},0\le {\theta }_i\le \frac{{\theta }_b^{\prime }}{2},\\ R_b-\sqrt{R_b^2-{\left(R_o\left({\theta }_b^{\prime }-{\theta }_i\right)\left./\phantom{R_o\left({\theta }_b^{\prime }-{\theta }_i\right)tan{\alpha }_{\beta }}\right)tan{\alpha }_{\beta }\right)}^2},\frac{{\theta }_b^{\prime }}{2}\le {\theta }_i\le {\theta }_b^{\prime }.\\ \end{array}\right)$$

The expression for h under contact conditions II-III in the defect area can be written as follows:

10

$$\begin{array}{c}{h}_{II-III}=\left\{\begin{array}{cc}{R}_b-\sqrt{R_b^2-{\left(\frac{a}{2cos{\alpha }_{\beta }}-R_{\mathit{obi}}tan{\alpha }_{\beta }\right)}^2},& 0\le {\theta }_i\le \frac{{\theta }_b^{\prime }}{2},\\ R_b-\sqrt{R_b^2-{\left(\frac{a}{2cos{\alpha }_{\beta }}-{\overline{R}}_{\mathit{obi}}tan{\alpha }_{\beta }\right)}^2},& \frac{{\theta }_b^{\prime }}{2}\le {\theta }_i\le {\theta }_b^{\prime },\end{array}\right)\\ R_{\mathit{obi}}=\left(\frac{R_o{\theta }_b^{\prime }}{2}-R_o{\theta }_i\right),{\overline{R}}_{\mathit{obi}}=\left[\frac{R_o{\theta }_b^{\prime }}{2}-R_o\left({\theta }_b^{\prime }-{\theta }_i\right)\right].\\ \end{array}$$

According to the above discussion and the geometrical size of the localized raceway defect, the universal expression for h considering the deflection distribution of the localized raceway defect can be ascertained as follows:

11

$$\begin{array}{c}\text{When}0<{\alpha }_{\beta }<{\alpha }_{\gamma },\\ h_1^{\prime }=R_b-R_o\left(1-cos{\phi }_{\mathit{js}}^{\prime }\right)-\sqrt{R_b^2-{\left(R_{o}sin{\phi }_{\mathit{js}}^{\prime }\right)}^2},\\ h_2^{\prime }=R_b-R_o\left(1-cos{\phi }_{\mathit{ej}}^{\prime }\right)-\sqrt{R_b^2-{\left(R_{o}sin{\phi }_{\mathit{ej}}^{\prime }\right)}^2},\\ h_{I-IV-1}=R_b-\sqrt{R_b^2-{\left(R_o{\phi }_{\mathit{js}}^{\prime }\left./\phantom{R_o{\phi }_{\mathit{js}}^{\prime }tan{\alpha }_{\beta }}\right)tan{\alpha }_{\beta }\right)}^2},\\ h_{I-IV-2}=R_b-\sqrt{R_b^2-{\left(R_o{\phi }_{\mathit{ej}}^{\prime }\left./\phantom{R_o{\phi }_{\mathit{ej}}^{\prime }tan{\alpha }_{\beta }}\right)tan{\alpha }_{\beta }\right)}^2},\\ h_{II-III-1}=R_b-\sqrt{R_b^2-{\left[\frac{a}{2cos{\alpha }_{\beta }}-\left(\frac{R_o{\theta }_b^{\prime }}{2}-R_o{\phi }_{\mathit{js}}^{\prime }\right)tan{\alpha }_{\beta }\right]}^2},\\ h_{II-III-2}=R_b-\sqrt{R_b^2-{\left[\frac{a}{2cos{\alpha }_{\beta }}-\left(\frac{R_o{\theta }_b^{\prime }}{2}-R_o{\phi }_{\mathit{ej}}^{\prime }\right)tan{\alpha }_{\beta }\right]}^2},\\ h=\left\{\begin{array}{cc}min\left(h_1^{\prime },h_{I-IV-1},h_{II-III-1},H\right),& {\phi }_s^{\prime }\le mod({\phi }_j,2\pi )<{\phi }_s^{\prime }+\frac{{\theta }_b^{\prime }}{2},\\ min\left(h_2^{\prime },h_{I-IV-2},h_{II-III-2},H\right),& {\phi }_s^{\prime }+\frac{{\theta }_b^{\prime }}{2}\le mod({\phi }_j,2\pi )<{\phi }_e^{\prime },\\ 0,& \text{other},\end{array}\right)\\ {\phi }_{\mathit{js}}^{\prime }={\phi }_j-{\phi }_s^{\prime },{\phi }_{\mathit{ej}}^{\prime }={\phi }_e^{\prime }-{\phi }_j,\\ \end{array}$$where ${\phi }_s^{\prime }$ and ${\phi }_e^{\prime }$ are the effective azimuth angles of the starting and ending defect edges, respectively. The specific expressions of $h_1^{\prime }$ and $h_2^{\prime }$ can be determined by h₁ and h₂ of the symmetric distribution of the localized raceway defects.

For the second type, the deflection angle is selected as α_β≥ α_γ, the trajectories of the jth rolling element in defect area are shown in Figure 7.

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Figure 7

Trajectory of the ith rolling element under deflection distribution (α_β ≥ α_γ): (a) α_β = α_γ, (b) α_β > α_γ.

As shown in Figure 7, the contact conditions between the jth rolling element and defect edge are consistent at certain moments when the deflection angle are select as α_β= α_γ and α_β> α_γ respectively. According to the definition of the contact condition in Figure 6, the contact condition in Figure 7 is defined as II–III. In the defect area, the expression for the relative change in the radial distance between center of the jth rolling element and that of the inner ring of the ACBB can be written as follows:

12

$$h_{II-III}^{\prime }=\left\{\begin{array}{cc}{R}_b-\sqrt{R_b^2-{\left(R_o{\theta }_{i}tan{\alpha }_{\beta }\right)}^2},& 0\le {\theta }_i\le \frac{{\theta }_b^{\prime }}{2},\\ R_b-\sqrt{R_b^2-{\left(R_o\left({\theta }_b^{\prime }-{\theta }_i\right)tan{\alpha }_{\beta }\right)}^2},& \frac{{\theta }_b^{\prime }}{2}\le {\theta }_i\le {\theta }_b^{\prime }.\end{array}\right)$$

Based on the above discussion and the geometric size of the localized raceway defect, the universal expression of h considering the deflection distribution of the localized raceway defect can be ascertained as follows:

13

$$\begin{array}{c}\text{When}{\alpha }_{\beta }\ge {\alpha }_{\gamma },\\ h_{II-III-1}^{\prime }=R_b-\sqrt{R_b^2-{\left(R_o{\phi }_{\mathit{js}}^{\prime }tan{\alpha }_{\beta }\right)}^2},\\ h_{II-III-2}^{\prime }=R_b-\sqrt{R_b^2-{\left(R_o{\phi }_{\mathit{ej}}^{\prime }tan{\alpha }_{\beta }\right)}^2},\\ h=\left\{\begin{array}{cc}min\left(h_1^{\prime },h_{I-II-1}^{\prime },H\right),& {\phi }_s^{\prime }\le mod({\phi }_j,2\pi )<{\phi }_s^{\prime }+\frac{{\theta }_b^{\prime }}{2},\\ min\left(h_2^{\prime },h_{I-III-2}^{\prime },H\right),& {\phi }_s^{\prime }+\frac{{\theta }_b^{\prime }}{2}\le mod({\phi }_j,2\pi )<{\phi }_e^{\prime },\\ 0,& \text{other},\end{array}\right)\\ \end{array}$$where the specific expressions for $h_1^{\prime }$ and $h_2^{\prime }$ can be found in Eq. (11).

Modeling of the ACBB Considering Localized Raceway Defect

As discussed, the three-dimensional localized raceway defect was modelled by considering three types of distributions, and the universal expressions for h for the different types of distributions of the localized raceway defect were determined. Because the localized raceway defect is located at the outer raceway of the ACBB in this study, h can be reflected in the outer raceway groove curvature center of the ACBB. To establish the dynamic modeling of the ACBB considering localized raceway defects, a geometrical relationship analysis of the internal structure of the ACBB must first be performed. The global structure of the ACBB, considering the localized raceway defect is shown in Figure 8. Figure 8 shows that the ACBB is composed of an inner raceway, outer raceway, rolling element, and cage. Five degrees of freedom, x, y, z, θ_x, and θ_y are taken into account and the Cartesian coordinate system O-xyz is selected as the reference coordinate system in this study. The symbol φ_j denotes the azimuth angle, and the corresponding expression is given in the following discussion.

[See PDF for image]

Figure 8

Global view of the internal structure of an ACBB considering localized raceway defect: (a) Global three-dimensional view, (b) Global two-dimensional sectional view

To derive the geometrical relationship in the following discussion, the jth rolling element is selected as the objective, and a local view of the internal structure of the ACBB is shown in Figure 9.

[See PDF for image]

Figure 9

Local view of the internal structure of an ACBB considering a localized raceway defect: (a) Local three-dimensional view, (b) Local two-dimensional sectional view

As shown in Figure 9, O_i, O_o and O_b denote the initial position of inner raceway and outer raceway groove curvature centers and rolling element center, respectively. The above three centers are collinear when the ACBB is at rest. $O_i^{\prime }$ and $O_b^{\prime }$ represent the final position of inner raceway groove curvature center and rolling element. r_i and r_o denote the inner and outer raceway groove curvature radius, respectively;$O_o^{\prime }$ denotes the position of outer raceway groove curvature center in the defect area. Meanwhile, it is necessary to note that the position of the outer raceway groove curvature center remains constant in the non-defect area because the outer raceway is assumed to be stationary. To intuitively analyze the relative location of the internal structure of the ACBB considering the localized raceway defect, the change of relative location of the jth rolling element, inner raceway, and outer raceway are shown in Figure 10. In Figure 10, B_rj and B_zj represent the radial and axial distances of the initial and final positions of the groove curvature center of the inner raceway; the derivations for B_rj and B_zj can be found in Ref. [31], where A_rj and A_zj denote the radial and axial distances of the groove curvature centers of the inner and outer raceway, respectively, and the corresponding expressions can be written as follows:

14

$$\begin{array}{c}\left\{\begin{array}{c}{A}_{\mathit{rj}}=\left(r_i+r_o-D_b-c_i-c_o\right)sin{\alpha }_o+B_{\mathit{rj}}-h\left({\phi }_j\right)sin{\alpha }_o,\\ A_{\mathit{zj}}=\left(r_i+r_o-D_b-c_i-c_o\right)cos{\alpha }_o+B_{\mathit{zj}}-h\left({\phi }_j\right)cos{\alpha }_o,\\ \end{array}\right)\\ \left\{\begin{array}{c}{B}_{\mathit{rj}}={\delta }_z+r_p\left({\theta }_{x}cos{\phi }_j-{\theta }_{y}sin{\phi }_j\right),\\ B_{\mathit{zj}}=z_p\left({\theta }_{y}sin{\phi }_j-{\theta }_{x}cos{\phi }_j\right)+xsin{\phi }_j+ycos{\phi }_j,\\ \end{array}\right)\\ \end{array}$$where the α_o denotes the initial contact angle; c_i and c_o indicate the radial clearance between rolling element and the inner raceway and the outer raceway, respectively; δ_x, δ_y, δ_z, θ_x, and θ_y denote the displacement variables corresponding to the five degrees of freedom x, y, z, θ_x, and θ_y, respectively; D_b denotes the diameter of rolling element; r_p and z_p denote the radial and axial distances, respectively, between inner raceway groove curvature center and inner ring center, the expressions for r_p and z_p can be ascertained as follows:

15

$$\left\{\begin{array}{c}{r}_p=D_m\left./\phantom{D_{m}2}\right)2+\left(r_i-0.5D_b-c_i\right)cos{\alpha }_o,\\ z_p=\left(r_i-0.5D_b-c_i\right)sin{\alpha }_o,\\ \end{array}\right)$$where D_m denotes the pitch diameter of the ACBB. The azimuth angle φ_j is expressed as follows:

16

$${\phi }_j={\omega }_{c}t+\frac{2\pi }{N}\left(j-1\right),j=1,2,\cdots ,N,$$where ω_c and t represent the angular speed of the cage and time variable, respectively, and N denotes the number of rolling elements. As shown in Figure 10, α_ij and α_oj denote the inner and outer raceway contact angles, respectively, and the sine and cosine forms of α_ij and α_oj are defined as follows according to the description of the changes in the relative positions in the internal structure of the ACBB in Figure 10.

17

$$\begin{array}{c}sin{a}_{\mathit{ij}}=\frac{A_{\mathit{zj}}-X_{\mathit{zj}}+h\left({\psi }_j\right)sin{\alpha }_o}{r_i-0.5D_b-c_i+{\delta }_{\mathit{ij}}},\\ sin{a}_{\mathit{oj}}=\frac{X_{\mathit{zj}}-h\left({\psi }_j\right)sin{\alpha }_o}{r_o-0.5D_b-c_o+{\delta }_{\mathit{oj}}},\\ cos{a}_{\mathit{ij}}=\frac{A_{\mathit{rj}}-X_{\mathit{rj}}+h\left({\psi }_j\right)cos{\alpha }_o}{r_i-0.5D_b-c_i+{\delta }_{\mathit{ij}}},\\ cos{a}_{\mathit{oj}}=\frac{X_{\mathit{rj}}-h\left({\psi }_j\right)cos{\alpha }_o}{r_o-0.5D_b-c_o+{\delta }_{\mathit{oj}}},\\ \end{array}$$where X_rj and X_zj denote the radius and axial distances between jth rolling element and groove curvature center of the outer raceway; δ_ij and δ_oj represent the contact deformation caused by the contact between the jth rolling element and the inner and outer raceways. Meanwhile, it is worth noting that the X_rj, X_zj, δ_ij, δ_oj are unknown variables with regard to jth rolling element. A set of equilibrium equations based on the Pythagorean theorem is determined as follows:

18

$$\left\{\begin{array}{c}E{q}_1={\left[A_{\mathit{rj}}-X_{\mathit{rj}}+h\left({\psi }_j\right)cos{\alpha }_o\right]}^2\\ +{\left[A_{\mathit{zj}}-X_{\mathit{zj}}+h\left({\psi }_j\right)sin{\alpha }_o\right]}^2-{\left(r_i-0.5D_b-c_i+{\delta }_{\mathit{ij}}\right)}^2=0,\\ E{q}_2={\left[X_{\mathit{rj}}-h\left({\psi }_j\right)cos{\alpha }_o\right]}^2+{\left[X_{\mathit{zj}}-h\left({\psi }_j\right)sin{\alpha }_o\right]}^2\\ -{\left(r_o-0.5D_b-c_o+{\delta }_{\mathit{oj}}\right)}^2=0.\\ \end{array}\right)$$

[See PDF for image]

Figure 10

Changes in the relative positions in the internal structure of ACBB considering a localized raceway defect

The force analysis of the jth rolling element of the ACBB for obtaining a set of force equilibrium equations with regard to the jth rolling element is shown in Figure 11.

[See PDF for image]

Figure 11

Force analysis of the jth rolling element of the ACBB considering localized raceway defect

According to the force balance principle, the set of equilibrium equations can be expressed as follows:

19

$$\left\{\begin{array}{c}E{q}_3=Q_{\mathit{ij}}cos{\alpha }_{\mathit{ij}}-Q_{\mathit{oj}}cos{\alpha }_{\mathit{oj}}\\ +\frac{{\lambda }_{\mathit{ij}}{M}_{\mathit{gj}}}{D}sin{\alpha }_{\mathit{ij}}-\frac{{\lambda }_{\mathit{oj}}{M}_{\mathit{gj}}}{D}sin{\alpha }_{\mathit{oj}}+F_{\mathit{cj}}=0,\\ E{q}_4=Q_{\mathit{ij}}sin{\alpha }_{\mathit{ij}}-Q_{\mathit{oj}}sin{\alpha }_{\mathit{oj}}\\ -\frac{{\lambda }_{\mathit{ij}}{M}_{\mathit{gj}}}{D}cos{\alpha }_{\mathit{ij}}+\frac{{\lambda }_{\mathit{oj}}{M}_{\mathit{gj}}}{D}cos{\alpha }_{\mathit{oj}}=0,\\ \end{array}\right)$$where Q_ij and Q_oj denote the contact forces of the inner and outer raceways, respectively, which can be determined based on the nonlinear Hertz contact theory, and the corresponding expressions can be found in Eq. (20); λ_ij and λ_oj represent the raceway control parameters, and the values of λ_ij and λ_oj are selected as λ_ij=0 and λ_oj=2, respectively, because the outer raceway control theory is chosen in this study. F_cj and M_gj denote the centrifugal force and gyroscopic moment with regard to jth rolling element, respectively, and the corresponding expression can be seen in Eq. (21):

20

$$Q_{\mathit{ij}}=\left\{\begin{array}{c}{K}_{\mathit{ij}}{\delta }_{\mathit{ij}}^{3/2},{\delta }_{\mathit{ij}}\ge 0,\\ 0,{\delta }_{\mathit{ij}}<0,\\ \end{array}\right)Q_{\mathit{oj}}=\left\{\begin{array}{c}{K}_{\mathit{oj}}{\delta }_{\mathit{oj}}^{3/2},{\delta }_{\mathit{oj}}\ge 0,\\ 0,{\delta }_{\mathit{oj}}<0,\\ \end{array}\right)$$where K_ij and K_oj denote the contact coefficients, and the corresponding expressions can be found in Ref. [32].

21

$$\begin{array}{c}{F}_{\mathit{cj}}=\frac{1}{2}m{D}_m{\omega }_i^2{\left(\frac{{\omega }_m}{{\omega }_i}\right)}^2,M_{\mathit{gj}}=J\left(\frac{{\omega }_b}{{\omega }_i}\right)\left(\frac{{\omega }_m}{{\omega }_i}\right){\omega }_i^{2}sin{\beta }_j,\\ {\left(\frac{{\omega }_m}{{\omega }_i}\right)}_j=\frac{1-\gamma cos{\alpha }_{\mathit{ij}}}{1+cos\left({\alpha }_{\mathit{ij}}-{\alpha }_{\mathit{oj}}\right)},tan{\beta }_j=\frac{sin{\alpha }_{\mathit{oj}}}{cos{\alpha }_{\mathit{oj}}+\gamma },\\ {\left(\frac{{\omega }_b}{{\omega }_i}\right)}_j=\frac{-1}{\left(\frac{cos{\alpha }_{\mathit{oj}}+tan{\beta }_{j}sin{\alpha }_{\mathit{oj}}}{1+\gamma cos{\alpha }_{\mathit{oj}}}+\frac{cos{\alpha }_{\mathit{ij}}+tan{\beta }_{j}sin{\alpha }_{\mathit{ij}}}{1-\gamma cos{\alpha }_{\mathit{ij}}}\right)\gamma cos{\beta }_j},\\ \end{array}$$where m and J denote the mass and moment of inertia of jth rolling element, respectively, and the corresponding expressions can be written as $m=\rho \pi D_b^3\left./\phantom{\rho \pi D_b^{3}6}\right)6;J=\rho \pi D_b^5\left./\phantom{\rho \pi D_b^{5}60}\right)60$, respectively; ω_i represents the angular speed of the inner ring, ω_mj and ω_bj denote the angular speeds of revolution and spin of jth rolling element, respectively, and β_j denotes the helical angle. Based on the above discussion, a set of nonlinear equilibrium equations including Eq₁, Eq₂, Eq₃, and Eq₄ were determined. The unknown variables X_rj, X_zj, δ_ij, and δ_oj can be obtained by solving the above nonlinear equilibrium equations. In this study, the above equilibrium equations were solved by employing the improved Newton-Raphson iterative method because of its strong nonlinearity.

To solve the displacement variables δ_x, δ_y, δ_z, θ_x, θ_y, the inner ring of ACBB is chosen as the objective and the force analysis of inner ring is performed. According to the force balance principle, the set of equilibrium equations of the inner ring can be determined by considering the action results of N rolling elements.

22

$$\begin{array}{c}{f}_1=F_x-\sum _{j=1}^N\left(Q_{\mathit{ij}}cos{\alpha }_{\mathit{ij}}+\frac{{\lambda }_{\mathit{ij}}{M}_{\mathit{gj}}}{D}sin{\alpha }_{\mathit{ij}}\right)sin{\phi }_j=0,\\ f_2=F_y-\sum _{j=1}^N\left(Q_{\mathit{ij}}cos{\alpha }_{\mathit{ij}}+\frac{{\lambda }_{\mathit{ij}}{M}_{\mathit{gj}}}{D}sin{\alpha }_{\mathit{ij}}\right)cos{\phi }_j=0,\\ f_3=F_z-\sum _{j=1}^N\left(Q_{\mathit{ij}}sin{\alpha }_{\mathit{ij}}-\frac{{\lambda }_{\mathit{ij}}{M}_{\mathit{gj}}}{D}cos{\alpha }_{\mathit{ij}}\right)=0,\\ f_4=M_x-\sum _{j=1}^N\left[\left(Q_{\mathit{ij}}sin{\alpha }_{\mathit{ij}}-\frac{{\lambda }_{\mathit{ij}}{M}_{\mathit{gj}}}{D}cos{\alpha }_{\mathit{ij}}\right)R_i-{\lambda }_{\mathit{ij}}{f}_i{M}_{\mathit{gj}}\right]cos{\phi }_j=0,\\ f_5=M_y+\sum _{j=1}^N\left[\left(Q_{\mathit{ij}}sin{\alpha }_{\mathit{ij}}-\frac{{\lambda }_{\mathit{ij}}{M}_{\mathit{gj}}}{D}cos{\alpha }_{\mathit{ij}}\right)R_i-{\lambda }_{\mathit{ij}}{f}_i{M}_{\mathit{gj}}\right]sin{\phi }_j=0,\\ \end{array}$$where F_x, F_y, F_z, M_x, M_y represent the external load components in the different directions of the ACBB. Based on the above established nonlinear equilibrium equations including f₁, f₂, f₃, f₄, and f₅, the displacement variables δ_x, δ_y, δ_z, θ_x, and θ_y can be solved by using the improved Newton Raphson iteration method.

Modeling Solution of the ACBB Considering a Localized Raceway Defect

Following the above derivations, modeling of the ACBB considering the raceway defect was established. The solution process for the development of a model of the ACBB considering localized raceway defects is introduced systematically in this section. The modeling solution can be divided into three steps. First, the relative change of radial distance between rolling element center and inner ring center is determined by analyzing the geometrical features of the three-dimensional localized raceway defect. Then, the unknown variables X_rj, X_zj, δ_ij, and δ_oj are determined using the improved Newton Raphson iteration method to solve a set of nonlinear equilibrium equation including Eq₁, Eq₂, Eq₃, and Eq₄. Finally, the displacement variables δ_x, δ_y, δ_z, θ_x, and θ_y are ascertained by employing the improved Newton Raphson iteration method to solve a set of nonlinear equilibrium equation which include f₁, f₂, f₃, f₄, and f₅. Meanwhile, it is worth noting that the unknown variables X_rj, X_zj, δ_ij, and δ_oj have an implicit relationship with the displacement variables δ_x, δ_y, δ_z, θ_x, and θ_y. This implicit relationship is derived as follows:

23

$$\begin{array}{c}\left[\begin{array}{c}\frac{\partial X_{\mathit{rj}}}{\partial \mathit{u}}\\ \frac{\partial X_{\mathit{zj}}}{\partial \mathit{u}}\\ \frac{\partial {\delta }_{\mathit{ij}}}{\partial \mathit{u}}\\ \frac{\partial {\delta }_{\mathit{oj}}}{\partial \mathit{u}}\end{array}\right]=-\frac{1}{\frac{\partial \mathit{Eq}}{\partial \mathit{\delta }}}\left[\begin{array}{c}\frac{\partial \mathit{Eq}}{\partial \left(\mathit{u},X_{\mathit{zj}},{\delta }_{\mathit{ij}},{\delta }_{\mathit{oj}}\right)}\\ \frac{\partial \mathit{Eq}}{\partial \left(X_{\mathit{rj}},\mathit{u},{\delta }_{\mathit{ij}},{\delta }_{\mathit{oj}}\right)}\\ \frac{\partial \mathit{Eq}}{\partial \left(X_{\mathit{rj}},X_{\mathit{zj}},\mathit{u},{\delta }_{\mathit{oj}}\right)}\\ \frac{\partial \mathit{Eq}}{\partial \left(X_{\mathit{rj}},X_{\mathit{zj}},{\delta }_{\mathit{ij}},\mathit{u}\right)}\end{array}\right],\\ \frac{\partial \mathit{Eq}}{\partial \left(\mathit{u},X_{\mathit{zj}},{\delta }_{\mathit{ij}},{\delta }_{\mathit{oj}}\right)}=\left[\begin{array}{cccc}\frac{\partial E{q}_1}{\partial \mathit{u}}& \frac{\partial E{q}_1}{\partial X_{\mathit{zj}}}& \frac{\partial E{q}_1}{\partial {\delta }_{\mathit{ij}}}& \frac{\partial E{q}_1}{\partial {\delta }_{\mathit{oj}}}\\ \frac{\partial E{q}_2}{\partial \mathit{u}}& \frac{\partial E{q}_2}{\partial X_{\mathit{zj}}}& \frac{\partial E{q}_2}{\partial {\delta }_{\mathit{ij}}}& \frac{\partial E{q}_2}{\partial {\delta }_{\mathit{oj}}}\\ \frac{\partial E{q}_3}{\partial \mathit{u}}& \frac{\partial E{q}_3}{\partial X_{\mathit{zj}}}& \frac{\partial E{q}_3}{\partial {\delta }_{\mathit{ij}}}& \frac{\partial E{q}_3}{\partial {\delta }_{\mathit{oj}}}\\ \frac{\partial E{q}_4}{\partial \mathit{u}}& \frac{\partial E{q}_4}{\partial X_{\mathit{zj}}}& \frac{\partial E{q}_4}{\partial {\delta }_{\mathit{ij}}}& \frac{\partial E{q}_4}{\partial {\delta }_{\mathit{oj}}}\end{array}\right],\\ \end{array}$$

24

$$\begin{array}{c}\left\{\begin{array}{c}\frac{\partial E{q}_1}{\partial \mathit{u}}=\frac{\partial E{q}_1}{\partial A_{\mathit{rj}}}\frac{\partial A_{\mathit{rj}}}{\partial \mathit{u}}+\frac{\partial E{q}_1}{\partial A_{\mathit{zj}}}\frac{\partial A_{\mathit{zj}}}{\partial \mathit{u}},\\ \frac{\partial E{q}_2}{\partial \mathit{u}}=\frac{\partial E{q}_2}{\partial A_{\mathit{rj}}}\frac{\partial A_{\mathit{rj}}}{\partial \mathit{u}}+\frac{\partial E{q}_2}{\partial A_{\mathit{zj}}}\frac{\partial A_{\mathit{zj}}}{\partial \mathit{u}},\\ \frac{\partial E{q}_3}{\partial \mathit{u}}=\frac{\partial E{q}_3}{\partial A_{\mathit{rj}}}\frac{\partial A_{\mathit{rj}}}{\partial \mathit{u}}+\frac{\partial E{q}_3}{\partial A_{\mathit{zj}}}\frac{\partial A_{\mathit{zj}}}{\partial \mathit{u}},\\ \frac{\partial E{q}_4}{\partial \mathit{u}}=\frac{\partial E{q}_4}{\partial A_{\mathit{rj}}}\frac{\partial A_{\mathit{rj}}}{\partial \mathit{u}}+\frac{\partial E{q}_4}{\partial A_{\mathit{zj}}}\frac{\partial A_{\mathit{zj}}}{\partial \mathit{u}},\\ \end{array}\right)\\ \frac{\partial \mathit{Eq}}{\partial \mathit{\delta }}=\left[\begin{array}{cccc}\frac{\partial E{q}_1}{\partial X_{\mathit{rj}}}& \frac{\partial E{q}_1}{\partial X_{\mathit{zj}}}& \frac{\partial E{q}_1}{\partial {\delta }_{\mathit{ij}}}& \frac{\partial E{q}_1}{\partial {\delta }_{\mathit{oj}}}\\ \frac{\partial E{q}_2}{\partial X_{\mathit{rj}}}& \frac{\partial E{q}_2}{\partial X_{\mathit{zj}}}& \frac{\partial E{q}_2}{\partial {\delta }_{\mathit{ij}}}& \frac{\partial E{q}_2}{\partial {\delta }_{\mathit{oj}}}\\ \frac{\partial E{q}_3}{\partial X_{\mathit{rj}}}& \frac{\partial E{q}_3}{\partial X_{\mathit{zj}}}& \frac{\partial E{q}_3}{\partial {\delta }_{\mathit{ij}}}& \frac{\partial E{q}_3}{\partial {\delta }_{\mathit{oj}}}\\ \frac{\partial E{q}_4}{\partial X_{\mathit{rj}}}& \frac{\partial E{q}_4}{\partial X_{\mathit{zj}}}& \frac{\partial E{q}_4}{\partial {\delta }_{\mathit{ij}}}& \frac{\partial E{q}_4}{\partial {\delta }_{\mathit{oj}}}\end{array}\right],\\ \mathit{E}\mathit{q}\mathbf{=}\left[\begin{array}{cccc}E{q}_1& E{q}_2& E{q}_3& E{q}_4\end{array}\right],\\ \mathit{\delta }={\left[\begin{array}{cccc}{X}_{\mathit{rj}}& X_{\mathit{zj}}& {\delta }_{\mathit{ij}}& {\delta }_{\mathit{oj}}\end{array}\right]}^{\text{T}},\\ \mathit{u}={\left[\begin{array}{ccccc}{\delta }_x& {\delta }_y& {\delta }_z& {\theta }_x& {\theta }_y\end{array}\right]}^{\text{T}}.\\ \end{array}$$

In this study, an improved Newton–Raphson iteration method is proposed based on the traditional Newton–Raphson iteration method. The advantage of the improved Newton–Raphson iteration method is that it can adjust the iteration step size by introducing an iteration step size adjustment factor thereby improving the convergence of the modeling solution. The numerical expression for the improved Newton–Raphson iteration method can be written as follows:

25

$${\mathit{\delta }}^{i+1}={\mathit{\delta }}^i-{\left[h_1\frac{\partial \mathit{Eq}}{\partial \mathit{\delta }}\right]}^{-1}\mathit{Eq},{\mathit{u}}^{i+1}={\mathit{u}}^i-{\left[h_2\frac{\partial \mathit{f}}{\partial \mathit{u}}\right]}^{-1}\mathit{f},$$

26

$$\begin{array}{c}\mathit{f}={\left[\begin{array}{ccccc}{f}_1& f_2& f_3& f_4& f_5\end{array}\right]}^{\text{T}},\\ {\mathit{K}}_{\mathit{ACBB}}=\frac{\partial \mathit{f}}{\partial \mathit{u}}=\left[\begin{array}{ccccc}\frac{\partial f_1}{\partial {\delta }_x}& \frac{\partial f_1}{\partial {\delta }_y}& \frac{\partial f_1}{\partial {\delta }_z}& \frac{\partial f_1}{\partial {\theta }_x}& \frac{\partial f_1}{\partial {\theta }_y}\\ \frac{\partial f_2}{\partial {\delta }_x}& \frac{\partial f_2}{\partial {\delta }_y}& \frac{\partial f_2}{\partial {\delta }_z}& \frac{\partial f_2}{\partial {\theta }_x}& \frac{\partial f_2}{\partial {\theta }_y}\\ \frac{\partial f_3}{\partial {\delta }_x}& \frac{\partial f_3}{\partial {\delta }_y}& \frac{\partial f_3}{\partial {\delta }_z}& \frac{\partial f_3}{\partial {\theta }_x}& \frac{\partial f_3}{\partial {\theta }_y}\\ \frac{\partial f_4}{\partial {\delta }_x}& \frac{\partial f_4}{\partial {\delta }_y}& \frac{\partial f_4}{\partial {\delta }_z}& \frac{\partial f_4}{\partial {\theta }_x}& \frac{\partial f_4}{\partial {\theta }_y}\\ \frac{\partial f_5}{\partial {\delta }_x}& \frac{\partial f_5}{\partial {\delta }_y}& \frac{\partial f_5}{\partial {\delta }_z}& \frac{\partial f_5}{\partial {\theta }_x}& \frac{\partial f_5}{\partial {\theta }_y}\end{array}\right],\\ \end{array}$$where h₁ and h₂ denote the iteration step adjustment factors for the solution of δ and u; i represents the number of the iteration and the value of i varies from 1 to 10,000; K_ACBB denotes the stiffness matrix of ACBB considering a localized raceway defect. To describe the above solution process intuitively, a flowchart of the modeling solution is shown in Figure 12. ε₁ and ε₂ denote the precision of the iteration solution, and their values are selected as 10⁻⁸ mm.

[See PDF for image]

Figure 12

The flow chart for modeling solution

Numerical Discussion on ACBB Considering the Localized Raceway Defect

Based on the above theoretical modeling, the numerical modeling of the ACBB considering localized raceway defects was established, and the solution process of the established numerical modeling is introduced in detail. A numerical discussion of the ACBB considering localized raceway defects is presented in this section. The numerical analysis is divided into two parts. First, the established numerical modeling of the ACBB considering localized raceway defects is validated to demonstrate that it can be used to analyze the time-varying contact and stiffness characteristics of the ACBB considering localized raceway defect. Then, the influences of the localized raceway defect geometrical size and distribution types on the time-varying contact and stiffness characteristics of the ACBB are investigated systematically. Unless otherwise indicated, the radical clearances c_i and c_o of ACBB are set to 0.

Model Verification

The model verification was performed by comparing the results calculated by the established numerical model with the corresponding results from the existing literature. The model verification was divided into two parts after including the contact angle and diagonal stiffness of the ACBB. First, the contact angle calculated by the proposed model is compared with the corresponding result from the existing literature. FAG B7014AC was selected as the objective, and the structural parameters of FAG B7014AC are listed in Table 1. Only axial force F_z is considered, and the comparison results of the contact angles are shown in Figure 13.

Table 1. The structure parameters of angular contact ball bearing FAG B7014AC

Structure parameter	Value
Ball diameter D (mm)	12.5
Number of ball N	19
Initial contact angle αo (°)	25
Elastic modulus E (MPa)	2.06 × 105
Poisson’s ratio ν	0.3
Density ρ (kg/m3)	7850
Inner raceway curvature radius ri (mm)	6.54
outer raceway curvature radius ro (mm)	6.54
Inner raceway radius Ri (mm)	38.5
Outer raceway radius Ro (mm)	51
Pitch radius Dm (mm)	89.5

[See PDF for image]

Figure 13

Comparison results for the contact angle of angular contact ball bearings

As shown in Figure 13, the comparison results of the contact angles, including the inner raceway and outer raceway contact angles, indicate that the calculation results of the proposed modeling and the corresponding results from Ref. [33] are in good agreement.

Subsequently, the diagonal stiffness calculated using the proposed model was compared with the experimental and numerical results in the existing literature. Meanwhile, it is necessary to note that only the diagonal stiffness is discussed in this paper because it truly reflects the stiffness characteristics of the ACBB. The RPF7039 is chosen as the objective and the structural parameters of the RPF7039 can be seen in Table 2. The external load working conditions of the RPF7039 are selected as: F_x= F_y=500 N; F_z= 2000 N; M_y= M_z= 1 N·m; n = 0 kr/min. The comparison results for the diagonal stiffness are given in Table 3.

Table 2. The structure parameters of RPF7039

Structure parameter	Value
Ball diameter D (mm)	17.7
Total number of ball N	12
Free contact angle αo (°)	40
Elastic modulus E (MPa)	2.06 × 105
Poisson’s ratio ν	0.3
Density ρ (kg/m3)	7850
Inner raceway curvature radius ri (mm)	9.16
outer raceway curvature radius ro (mm)	9.38
Inner raceway radius Ri (mm)	22.5
Outer raceway radius Ro (mm)	50
Pitch radius Dm (mm)	72.5

Table 3. Comparison results of diagonal stiffness of ACBB

Diagonal stiffness	Kxx (N/mm)	Kyy (N/mm)	Kzz (N/mm)	Kθxθx (N·mm/rad)	Kθyθy (N·mm/rad)
Present	172215.4	155670.0	226159.8	140917683.6	158307690.5
Ref. [34]	166573.8	166573.8	225376.5	144386234.1	144386234.1
Experiment [35]	172131.6	155520.7	226011.1	139304365.4	157160958.5
EA (%)	0.049	0.096	0.066	1.16	0.73
EB (%)	3.23	7.11	0.28	3.65	8.13

As shown in Table 3, the symbol EA represents the relative error between the results of the presented model and experiment, and EB denotes the relative error between the values in literature and those obtained from the experiment EA and EB are defined as in Eq. (27). The comparison results of the diagonal stiffness, including the radial stiffness, axial stiffness, and angular stiffness, indicate that the results calculated by the proposed model and the existing experimental results are in good agreement, with a maximum of EA being less than 1.2%.

27

$$\begin{array}{c}EA=\left|\text{Present}-\text{Experiment}\right|\left./\phantom{\left|\text{Present}-\text{Experiment}\right|\text{Experiment}}\right)\text{Experiment},\\ EB=\left|\text{Literature}-\text{Experiment}\right|\left./\phantom{\left|\text{Literature}-\text{Experiment}\right|\text{Experiment}}\right)\text{Experiment}.\\ \end{array}$$

Based on the comparison results above, it can be concluded that the established numerical model can be used to investigate the time-varying contact and stiffness characteristics of the ACBB by considering localized raceway defects.

Analysis of the Time-Varying Characteristics of the ACBB Considering the Geometrical Size of Defect

The influences of the geometrical size, including the radial depth H of the defect, axial width a of the defect, angular distanceθ_b, eccentric distance L of the defect, and defect deflection angle α_β of three-dimensional localized raceway defect, on the contact and stiffness characteristics of the ACBB are investigated in this section. ACBB b218 is selected as objective and the structural parameters of the ACBB are listed in Table 4. The external load working conditions of the ACBB are set as F_x= F_y= 200 N; F_z= 1000 N; M_y= M_z= 0 N·mm; n = 2 kr/min, in the following discussion. The azimuth angle of the localized raceway defect center is selected as 270°, which can be interpreted as the mean value between the azimuth angles of the starting and ending defect edges.

Table 4. The structure parameters of ACBB b218

Structure parameter	Value
Ball diameter D (mm)	22.225
Total number of ball N	16
Free contact angle αo (°)	40
Elastic modulus E (MPa)	2.075 × 105
Poisson’s ratio ν	0.3
Density ρ (kg/m3)	7800
Inner raceway curvature radius ri (mm)	11.62812
outer raceway curvature radius ro (mm)	11.62812
Inner raceway radius Ri (mm)	51.3969
Outer raceway radius Ro (mm)	73.8632
Pitch radius Dm (mm)	125.2601

First, the influence of the radial depth H of the defect on the contact and diagonal stiffness characteristics of the ACBB was investigated, and the corresponding results are shown in Figures 14 and 15. The geometrical size of three-dimensional localized raceway defect is selected as L = 0, a = 3 mm, θ_b= 4, and α_β= 0°. The third rolling element of the ACBB was selected as the objective for the following discussion.

[See PDF for image]

Figure 14

Influence of the defect radial depth on the time-varying contact characteristics of the ACBB: (a) Inner raceway contact angle, (b) Outer raceway contact angle, (c) Inner raceway contact force, (d) Outer raceway contact force

[See PDF for image]

Figure 15

The influence of defect radial depth on time-varying diagonal stiffness characteristics of ACBB: (a), (b) Radial stiffness, (c) Axial stiffness, (d), (e) Angle stiffness

As shown in Figure 14, time-varying contact angles of the inner and outer raceway exhibit opposite variation tendencies, and the time-varying contact forces of the inner and outer raceway exhibit the same variation tendency as time t increases. The mean value time-varying contact angle of the inner raceway is larger than that of the time-varying contact angle of the outer raceway.

The mean value of the inner raceway time-varying contact force was larger than that of the time-varying contact force of the outer raceway. With the appearance of localized raceway defects, the time-varying curves of the inner and outer raceway contact angles have Λ- and V-shaped mutations, respectively, in some time domains. However, the time-varying curves of the inner and outer raceway contact forces have both Λ- and V-shaped mutations in some time domains. The peak values of Λ- and V-shaped mutations increased with the defect radial depth H.

As shown in Figure 15, the time-varying curves of the diagonal stiffness, including the radial stiffness, axial stiffness, and angle stiffness, have Λ- and V-shaped mutations in some time domains with the emergence of localized raceway defects. Compared with other diagonal stiffnesses, the time-varying curve of the radial stiffness K_xx exhibits only a V-shaped mutation in some time intervals, and the peak value of the V-shaped mutation increases with the defect radial depth H. The time-varying curves of the other diagonal stiffnesses exhibit both the Λ- and V-shaped mutations in some time intervals, and the peak values of the Λ- and V-shaped mutations increase with the defect radial depth H.

Second, the influence of the defect axial width a on the contact and diagonal stiffness characteristics of the ACBB was analyzed, and the corresponding results are shown in Figure 16 and Figure 17. The geometrical size of three-dimensional localized raceway defect is selected as L = 0, H = 0.1 mm, θ_b= 4, and α_β= 0°.

[See PDF for image]

Figure 16

The influence of axial width a of the defect on time-varying contact characteristics of the ACBB

[See PDF for image]

Figure 17

The influence of defect axial width a on time-varying diagonal stiffness characteristics of ACBB

As shown in Figure 16, with an increase in the defect axial width a, the time-varying curves of the contact angle and contact force have both Λ- and V-shaped mutations in some time intervals, and the variation tendencies of the Λ- and V-shaped mutations are consistent with those in Figure 14.

Third, the influence of angular distance θ_b on the contact and diagonal stiffness characteristics of the ACBB was investigated, and the corresponding results are shown in Figures 18 and 19. The geometrical size of three-dimensional localized raceway defect is selected as L = 0, a = 1 mm, H = 0.1 mm, and α_β= 0°.

[See PDF for image]

Figure 18

The influence of angular distance θ_b on time-varying contact characteristics of ACBB

[See PDF for image]

Figure 19

The influence of angular distance θ_b on time-varying diagonal stiffness characteristics of ACBB

As shown in Figure 18, with an increase in angular distanceθ_b, the time-varying curves of the contact angle and contact force have both Λ- and V-shaped mutations in some time intervals. The peak values of Λ- and V-shaped mutations first increase and then remain constant with an increase in angular distanceθ_b. Meanwhile, it is worth mentioning that the number of Λ-shaped mutations increases when the peak value of Λ-shaped mutation remains constant.

As can be seen from Figure 19, the time-varying curves of the diagonal stiffness have both Λ- and V-shaped mutations with an increase in the angular distanceθ_b. Compared with the other diagonal stiffnesses, the time-varying curve of the radial stiffness K_xx has a V-shaped mutation in some time intervals; the peak value of the V-shaped mutation increases first and then remains constant with an increase in the angular distance θ_b. The time-varying curves of the other diagonal stiffnesses have both Λ- and V-shaped mutations in some time intervals, and the peak values of Λ- and V-shaped mutations first remain constant as the angular distanceθ_b increases. The number of Λ-shaped mutations increases significantly when the peak value of Λ-shaped mutation remains constant. The Λ-shaped mutation of the time-varying curves appears as a platform in some time intervals for the diagonal stiffness K_yy and Κ_θxθx.

Fourth, the influence of the eccentric distance L of the defect on the contact and diagonal stiffness characteristics of the ACBB is discussed, and the corresponding results are shown in Figures 20 and 21. The geometrical size of three-dimensional localized raceway defect is selected as a = 1 mm, H = 0.05 mm, θ_b= 6°, and α_β= 0°.

[See PDF for image]

Figure 20

Influence of defect eccentric distance L on time-varying contact characteristics of ACBB

[See PDF for image]

Figure 21

Influence of the defect eccentric distance L on time-varying diagonal stiffness characteristics of ACBB

As shown in Figure 21, the peak values of the Λ- and V-shaped mutations in the time-varying curves of the diagonal stiffness in some time intervals decrease with an increase in the defect eccentric distance L. The variation tendencies of the time-varying curves of the diagonal stiffness are opposite to those in Figure 15.

Finally, the influence of the defect deflection angle α_β on the contact and diagonal stiffness characteristics of the ACBB is discussed, and the corresponding results are shown in Figure 22 and Figure 23. The geometrical size of the three-dimensional localized raceway defect is selected as L = 0, a = 0.02 mm, H = 0.05 mm, θ_b= 2°.

[See PDF for image]

Figure 22

Influence of the defect eccentric distance L on time-varying contact characteristics of ACBB

[See PDF for image]

Figure 23

Influence of the defect deflection angle α_β on time-varying diagonal characteristics of ACBB

As shown in Figure 22, the time-varying curves of the contact angle and contact force exhibit a sudden change with an increase in the defect deflection angle α_β, and the above mutation occurs between α_β= 0.3° and α_β= 0.5°. Based on the geometrical size of three-dimensional localized raceway defect, the value of critical deflection angle α_γ is between 0.3° and 0.5°. The appearance of critical deflection angle α_γ can change the contact condition between rolling element and localized raceway defect edges. It can be concluded that the above mutation phenomenon is caused by the critical deflection angle α_γ. When the defect deflection angle α_β< α_γ or α_β≥ α_γ, the time-varying curves of contact angle and contact force have slight change with the increase in the defect deflection angle.

As shown in Figure 23, the time-varying curves of diagonal stiffness change slightly with an increase in the defect deflection angle α_β when the defect deflection angle α_β< α_γ. However, the time-varying curves of diagonal stiffness change significantly with increase in the defect deflection angle α_β when the defect deflection angle α_β≥ α_γ. The time-varying curves of radial stiffness K_xx and other diagonal stiffness decrease and increase, respectively, with an increase in the defect deflection angle α_β. Meanwhile, the time-varying curves of diagonal stiffness have V-shaped mutations in some time intervals when the defect deflection angle α_β≥ α_γ.

Conclusions

1. The effects of the defect radial depth H and axial width a on the time-varying contact and stiffness characteristics of the ACBB are consistent. With an increase in the defect radial depth H and axial width a, the time-varying curves of the contact angles and diagonal stiffness exhibited Λ- and V-shaped mutations in some time intervals, and the peak values of the Λ- and V-shaped mutations evidently increased.

2. The time-varying curves of the contact angles and diagonal stiffness exhibit Λ- and V-shaped mutations in some time intervals when the angular distance θ_b increased. The peak values of the Λ- and V-shaped mutations initially increased and then remained constant as the angular distance θ_b increased. The number of Λ-shaped mutations evidently increased when the peak values of the Λ- and V-shaped mutations remained constant.

3. The peak values of the Λ- and V-shaped mutations caused by localized raceway defects decreased with increasing defect eccentric distance L. When the defect eccentric distance L was close to 0.5a, the Λ- and V-shaped mutations disappeared. This phenomenon can be interpreted as the rolling elements of the ACBB gradually disengaging the defect area as the defect eccentric distance L increases.

4. The increase of defect deflection angle α_β has a slight effect on the time-varying contact and stiffness characteristics of the ACBB when the defect deflection angle α_β< α_γ; the increase in the defect deflection angle α_β has an important effect on the time-varying stiffness characteristics of ACBB when the defect deflection angle α_β≥ α_γ. Meanwhile, the contact condition between the rolling element and the localized raceway defect edge changed as the defect deflection angle increased.

5. This research reveals the influence mechanism of three-dimensional local raceway defects on the dynamic characteristics of ACBB systematically, which can provide theoretical basis and technique guidance for the designation and manufacture of ACBB. The formation and propagation of three-dimensional local raceway defects of ACBB need to be further studied deeply in future work.

Acknowledgements

Not applicable.

Authors' Contributions

Qingshan Wang was in charge of the entire trial, Zhen Li wrote the manuscript, and Ruihua Wang assisted with the sampling and laboratory analyses. All the authors have read and approved the final version of the manuscript.

Funding

Supported by National Natural Science Foundation of China (Grant No. 52075554), Hunan Provincial Natural Science Foundation of China (Grant No. 2022JJ20070), Innovation-Driven Research Program of Central South University of China (Grant No. 2023CXQD049), and State Key Laboratory of High Performance Complex Manufacturing of China (Grant No. ZZYJKT2021-07).

Data availability

The data that supports the findings of this study are available within the article.

Declarations

Competing Interests

The authors declare no competing financial interests.

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