Modeling and mechanical characteristics analysis of matched angular contact ball bearings under combined loads

Abstract

Although bearings have been used in many rotating machineries, the literatures relative to matched bearings are very scarce. The distributions of contact load and contact angle between bearing parts are the key indicators to reflect the operating condition of bearings, which can directly influence the wear rate and fatigue life of bearings. Owing to the complex and compact structure of matched bearings, it is hard to judge and evaluate what happens inside the matched bearings by experiments. To solve this problem, this paper presents a novel method to determine the distributions of contact load and contact angle of matched bearings under complex working conditions. Firstly, to establish the nonlinear relationship between internal deformations of balls and applied loads, a comprehensive model is proposed by using the coordinate transformation method and the Hertz contact theory. Then, the accuracy of the proposed model is validated by comparing the stiffness values with those reported in the existing reference. Finally, a pair of 7206B bearings is used as a case to introduce the use of the proposed model. The influences of installation type, preload, combined loads on the distributions of contact load and contact angle of matched bearings are detailed analyzed and studied.

Introduction

Angular contact ball bearings installed in pairs have obvious advantages than single bearings, which can provide higher load carrying capacity and better deformation resisting performance. ¹ Consequently, matched angular contact ball bearings are widely applied in numerous kinds of rotating mechanical systems due to their excellent mechanical properties and high reliability. As the fundamental components in complex machinery systems, their mechanical characteristics play a crucial role in the transmission properties of force and vibration of the whole system. Although matched bearings have a large number of applications in engineering practice, the relevant literature on matched bearings are very scarce. Owing to the different installation types and mechanical behaviors of bearings, the modeling approach of single bearings cannot be applied to the analysis of matched bearings. Accordingly, a method for the modeling of matched bearings is imminent.

Over the past years, many scholars have carried out a great number of studies on single ball bearings from different aspects, such as vibration characteristics, ² fatigue life, ³ fault diagnosis, ⁴ stiffness calculation, ⁵ and heat analysis, ⁶ etc. Aktürk et al. ² investigated the vibration of a shaft-ball bearing system by a nonlinear three-degree-of-freedom model, and the result indicates that the ball number and preload can significantly affect the dynamic characteristics of the rotating system. Pattabhiraman et al. ³ developed a procedure to evaluate the fatigue life of silicon nitride hybrid ball bearings under various uncertainties, and the results indicate that the failure probability can be reduced by decreasing the maximum crack size and increasing the fatigue threshold. Cui et al. ⁴ presented a new method for defective ball bearings to determine their local defects quantitatively and locally. Sheng et al. ⁵ calculated the speed-varying stiffness of ball bearings based on the quasi-static model and the implicit differential method. Yan et al. ⁶ proposed a precise numerical model for ball bearings to investigate the thermal dissipation of bearing cage at ultra high rotating speed. The distributions of contact load and contact angle can directly reflect the operating condition of bearings. Accordingly, the study on the distributions of contact load and contact angle also has attracted attention of domestic and foreign scholars. Liao and Lin ⁷ established a simple, three-dimensional expression to calculate the elastic deformation of a ball bearing, in which the contact angle was considered as a function of the position angle of balls. Amasorrain et al. ⁸ developed a calculation procedure to determine the load distribution of a four contact point slewing bearing. Daidié et al. ⁹ analyzed the variations of contact load and contact angle of a slewing ball bearing under different loading cases by a 3D simplified finite element model. Petersen et al. ¹⁰ studied the influence of a raceway defect on the load distribution of a ball bearing, and the result indicates that the load on the defect-free raceway section increases with the increasing circumferential extent of defect. Li et al. ¹¹ presented a mechanical model for an angular contact ball bearing with a localized defect on its outer race, and the influences of operating condition and defect size on the variations of contact angle and contact load were analyzed. Liu et al. ¹² used an analytical calculation method to investigate the influence of combined loads including radial and axial loads on the distributions of contact load and contact angle of ball bearings. Zhang et al. ¹³ proposed an improved quasi-static model for ball bearings to analyze the distributions of contact load and contact angle, in which the influence of ring misalignment was considered.

The research objects of the above-mentioned works mainly aim at single bearings. So far, the scientific literature about matched bearings is quite sparse. The theory and research achievements about matched bearings are far less abundant than those of single bearings. Therefore, the research on matched bearings has begun to attract increasingly attention from scholars. Bercea et al.^14,15 proposed a set of general formulas to investigate the load characteristics of double-row bearings with different installation types. Li and Shin ¹ presented a dynamic thermo-mechanical model to analyze the influence of bearing installation type on the thermo-dynamic behaviors of a spindle-bearing system, and the results indicate that the bearing orientation has obvious influence on the stiffness of whole structure. Karacay and Akturk ¹⁶ investigated the vibration behaviors of a spindle supported by a pair of angular contact ball bearings using a five-degree-of-freedom model. Gunduz et al.^17,18 proposed an analytical model to study the modal characteristics of double row angular contact ball bearings. Zhang et al. ¹⁹ constructed a universal model to calculate the preload, stiffness and contact angle of matched angular contact ball bearings only subjected to an axial load. Yang et al. ²⁰ employed the experimental method to determine the stiffness values of angular contact ball bearings with different configuration forms. Lin and Jiang ²¹ proposed an improved quasi-static model to study the influences of initial contact angle, installation type, and pretension method on the stiffness values of duplex angular contact ball bearings. Yang et al. ²² presented an analytical model to analyze the influences of combined loads and angular misalignment on the load distribution of double-row tapered bearings. The matched bearings have been widely used in numerous kinds of rotating equipments. The distributions of contact load and contact angle are the key indicators to reflect the operating condition of matched bearings, but the related literature for matched angular contact ball bearings has not been reported to date. To the author’s best knowledge, mainly owing to the geometry complexity of matched angular contact ball bearings. This prompts us to carry out the present work to provide a useful and meaningful reference for the research theory of matched bearings.

In this paper, a comprehensive model is constructed to study the loading characteristics of matched angular contact ball bearings. The article outline is the following. In Section 2, a five-degree-of-freedom model is constructed based on the coordinate transformation method. Then the contact load and contact angle can be determined by the proposed model. In Section 3, the proposed model can be verified by comparing the calculated stiffness values with those reported in the existing literature. In Section 4, the influences of installation type, preload and combined loads on the distributions of contact load and contact angle are investigated minutely. In the end, some main conclusions are given briefly.

Modeling method

Harris and Kotzalas ²³ have given several mechanical equilibrium equations to calculate the load distribution and displacement of single bearings. However, these equations are inapplicable in the calculation of matched bearings due to the different structures between single and matched bearings. All the installed bearings are interacted on each other. The relationship between internal geometrical deformations and complicated working conditions needs to be determined. Based on the coordinate transformation method, a comprehensive model can be constructed by considering the force balance relation of each bearing in the modeling of matched bearings. To carry out the simulation of matched bearings, the analyzed model must be simplified to some extent. Thus, five assumptions are adopted in this study:

• The manufacturing error and assembly tolerance of matched bearings can be neglected.

• The influences of retainer, sealing ring and other elements on the loading behaviors of matched bearings can be neglected.

• The contact deformations between rolling elements are in elastic range.

• The rotating speed of matched bearings is low, so the centrifugal force and gyroscopic torque acting on each ball can be neglected.

• The influences of temperature variation, lubrication condition and other factors on the loading properties of matched bearings can be neglected.

Coordinate transformation

Generally, the matched angular contact ball bearings are often installed by the styles of face-to-face arrangement (DF) and back-to-back arrangement (DB), as depicted in Figure 1. The matched bearings installed in the arrangement of DF or DB can withstand axial loads in different directions. In the arrangement of DF, the load lines of left and right balls point toward the inside of matched bearings. For the DB arrangement, the load lines of left and right balls point toward the outside of matched bearings. In Figure 1, $a_{\mathrm{mj}}$ is the contact angle between the j-th ball and the race, and the subscript m denotes the installed location of single bearing (m is L or R). S represents the distance between the ball centers of the two installed bearings. $d_m$ indicates the pitch diameter.

Figure 1.

Schematic cross-sections of DF and DB arrangements: (a) face to face arrangement and (b) back to back arrangement.

The coordinate transformation method can be applied to establish the load-deformation relationship of different elements of matched bearings. To analyze conveniently, six auxiliary coordinate systems can be adopted in this study depicted in Figures 2 and 3.

• Three initial coordinate systems: a global coordinate system $o_c-x_c{y}_c{z}_c$ , whose origin coincides with the geometrical center of matched bearings. Two local coordinate systems $o_L-x_L{y}_L{z}_L$ and $o_R-x_R{y}_R{z}_R$ , which coincide with the geometrical centers of left and right bearings, respectively.

• Three transformed coordinate systems: $o{\prime }_c-x{\prime }_{c}y{\prime }_{c}z{\prime }_c$ , $o{\prime }_L-x{\prime }_{L}y{\prime }_{L}z{\prime }_L$ , and $o{\prime }_R-x{\prime }_{R}y{\prime }_{R}z{\prime }_R$ , which initially coincide with the fixed coordinate systems $o_c-x_c{y}_c{z}_c$ , $o_L-x_L{y}_L{z}_L$ , and $o_R-x_R{y}_R{z}_R$ , respectively.

Figure 2.

Coordinate system of DF arrangement.

Figure 3.

Coordinate system of DB arrangement.

The above coordinate systems follow the right-hand rule. Generally, it can be assumed that the combined loads are applied on the geometrical center of matched bearings and the force balance can be conveniently studied by keeping the outer races fixed in space. For the usual case, the matched bearings can subjected to loads in five directions: an axial load $F_z$ , two radial loads $F_x,F_y$ , and two tilting moments $M_x,M_y$ . Accordingly, $o_c-x_c{y}_c{z}_c$ moves along three linear displacements ${\delta }_x,{\delta }_y,{\delta }_z$ , and two angular rotations ${\theta }_x,{\theta }_y$ . The coordinate transformation between $o_c-x_c{y}_c{z}_c$ and $o{\prime }_c-x{\prime }_{c}y{\prime }_{c}z{\prime }_c$ can be written as:

$$\left[\begin{array}{c}{}^{o_c}\mathit{p}\\ 1\end{array}\right]=\left[\begin{array}{cc}{}_{{o^{\prime }}_c}{}^{o_c}R& {}^{o_c}{\mathit{p}}_{{o^{\prime }}_c}\\ 000& 1\end{array}\right]\left[\begin{array}{c}{}^{{o^{\prime }}_c}\mathit{p}\\ 1\end{array}\right]$$

(1)

with

$${}^{o_c}\mathit{p}={\left[\begin{array}{ccc}{x}_c\hfill & y_c\hfill & z_c\hfill \end{array}\right]}^T$$

(2)

$${}^{o_c}{\mathit{p}}_{{o^{\prime }}_c}={\left[\begin{array}{ccc}{\delta }_x\hfill & {\delta }_y\hfill & {\delta }_z\hfill \end{array}\right]}^T$$

(3)

$${}^{{o\prime }_c}\mathit{p}={\left[\begin{array}{ccc}{x\prime }_c\hfill & {y\prime }_c\hfill & {z\prime }_c\hfill \end{array}\right]}^T$$

(4)

$${}_{{o\prime }_c}{}^{o_c}R=\left[\begin{array}{ccc}{C}_{{\theta }_y}\hfill & 0\hfill & S_{{\theta }_y}\hfill \\ \multicolumn{1}{c}{S_{{\theta }_x}{S}_{{\theta }_y}}& C_{{\theta }_x}\hfill & -S_{{\theta }_x}{C}_{{\theta }_y}\hfill \\ \multicolumn{1}{c}{-C_{{\theta }_x}{S}_{{\theta }_y}}& S_{{\theta }_x}\hfill & C_{{\theta }_x}{C}_{{\theta }_y}\hfill \end{array}\right]$$

(5)

where the triangle functions $S_{{\theta }_x},C_{{\theta }_x},S_{{\theta }_y}$ and $C_{{\theta }_y}$ represent $\sin {\theta }_x$ , $\cos {\theta }_x$ , $\sin {\theta }_y$ and $\cos {\theta }_y$ respectively. The coordinate of $o{\prime }_m$ in $o{\prime }_c-x{\prime }_{c}y{\prime }_{c}z{\prime }_c$ is ${\left[00c_1\frac{S}{2}\right]}^T$ . The position vector of $o{\prime }_m$ in $o_c-x_c{y}_c{z}_c$ can be derived as $[c_1\frac{S}{2}{S}_{{\theta }_y}+{\delta }_x-c_1\frac{S}{2}{S}_{{\theta }_x}{C}_{{\theta }_y}+{\delta }_y{c}_1\frac{S}{2}{C}_{{\theta }_x}{C}_{{\theta }_y}+{\delta }_z{\theta }_x{\theta }_y{]}^T$ by equation (1). According to Figures 2 and 3, the position vector of $o{\prime }_m$ in $o_m-x_m{y}_m{z}_m$ is derived as ${[c_1\frac{S}{2}{S}_{{\theta }_y}+{\delta }_x{c}_1{c}_2(-c_1\frac{S}{2}{S}_{{\theta }_x}{C}_{{\theta }_y}+{\delta }_y)c_1{c}_2{\delta }_z{\theta }_x{c}_1{c}_2{\theta }_y]}^T$ . Hence, the coordinate transformation between $o_m-x_m{y}_m{z}_m$ and $o{\prime }_m-x{\prime }_{m}y{\prime }_{m}z{\prime }_m$ can be written as:

$$\left[\begin{array}{c}{}^{o_m}\mathit{p}\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]=\left[\begin{array}{cc}{}_{{o\prime }_m}{}^{o_m}R\hfill & {}^{o_m}{\mathit{p}}_{{o\prime }_m}\hfill \\ \multicolumn{1}{c}{000}& 1\hfill \end{array}\right]\left[\begin{array}{c}{}^{{o\prime }_m}\mathit{p}\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]$$

(6)

with

$${}^{o_m}\mathit{p}={\left[\begin{array}{ccc}{x}_m\hfill & y_m\hfill & z_m\hfill \end{array}\right]}^T$$

(7)

$${}^{o_m}{\mathit{p}}_{{o\prime }_m}={\left[\begin{array}{ccc}{c}_1\frac{S}{2}{S}_{{\theta }_y}+{\delta }_x\hfill & c_1{c}_2(-c_1\frac{S}{2}{S}_{{\theta }_x}{C}_{{\theta }_y}+{\delta }_y)\hfill & c_1{c}_2{\delta }_z\hfill \end{array}\right]}^T$$

(8)

$${}^{{o\prime }_m}\mathit{p}={\left[\begin{array}{ccc}{x\prime }_m\hfill & {y\prime }_m\hfill & {z\prime }_m\hfill \end{array}\right]}^T$$

(9)

$${}_{{o\prime }_m}{}^{o_m}R=\left[\begin{array}{ccc}{C}_{{\theta }_y}\hfill & 0\hfill & c_1{c}_2{S}_{{\theta }_y}\hfill \\ \multicolumn{1}{c}{S_{{\theta }_x}{S}_{{\theta }_y}}& C_{{\theta }_x}\hfill & -S_{{\theta }_x}{C}_{{\theta }_y}\hfill \\ \multicolumn{1}{c}{-c_1{c}_2{C}_{{\theta }_x}{S}_{{\theta }_y}}& S_{{\theta }_x}\hfill & C_{{\theta }_x}{C}_{{\theta }_y}\hfill \end{array}\right]$$

(10)

where the coefficient $c_1$ is dependent on the subscript m, $c_1=\{\begin{array}{cc}-1\hfill & m=L\hfill \\ \multicolumn{1}{c}{1}& m=R\hfill \end{array}$ ; the coefficient $c_2$ is determined by the bearing installation types, $c_2=\{\begin{array}{cc}1\hfill & \mathrm{face}-\mathrm{to}-\mathrm{face}\hfill \\ \multicolumn{1}{c}{-1}& \mathrm{back}-\mathrm{to}-\mathrm{back}\hfill \end{array}$ .

In $o{\prime }_m-x{\prime }_{m}y{\prime }_{m}z{\prime }_m$ , the coordinate of inner race groove curvature center at the j-th ball can be expressed as ${\left[R_i\cos {\psi }_{\mathrm{mj}}{R}_i\sin {\psi }_{\mathrm{mj}}0\right]}^T$ , where $R_i$ represents the orbit radius of inner race curvature center; ${\psi }_{\mathrm{mj}}=2\pi (j-1)/Z$ is the angular position of the j-th ball and Z denotes the ball number in a bearing. The angular position of balls is shown in Figure 4 (The left view of matched bearings depicted in Figure 1). Hence, its coordinate in $o_m-x_m{y}_m{z}_m$ can be calculated by equation (6), which can be derived as equation (11).

$$\begin{array}{c}\left[\begin{array}{c}{x}_{\mathrm{imj}}\hfill \\ \multicolumn{1}{c}{y_{\mathrm{imj}}}\\ \multicolumn{1}{c}{z_{\mathrm{imj}}}\end{array}\right]=\hfill \\ \left[\begin{array}{c}{R}_i\cos {\psi }_{\mathrm{mj}}{C}_{{\theta }_y}+c_1\frac{S}{2}{S}_{{\theta }_y}+{\delta }_x\hfill \\ \multicolumn{1}{c}{R_i\cos {\psi }_{\mathrm{mj}}{S}_{{\theta }_x}{S}_{{\theta }_y}+R_i\sin {\psi }_{\mathrm{mj}}{C}_{{\theta }_x}+c_1{c}_2(-c_1\frac{S}{2}{S}_{{\theta }_x}{C}_{{\theta }_y}+{\delta }_y)}\\ \multicolumn{1}{c}{-c_1{c}_2{R}_i\cos {\psi }_{\mathrm{mj}}{C}_{{\theta }_x}{S}_{{\theta }_y}+R_i\sin {\psi }_{\mathrm{mj}}{S}_{{\theta }_x}+c_1{c}_2{\delta }_z}\end{array}\right]\hfill \end{array}$$

(11)

Figure 4.

Angular position of balls.

In this research, the outer races are fixe. Accordingly, in $o_m-x_m{y}_m{z}_m$ , the coordinate of outer race groove curvature center at the j-th ball can be derived as:

$$\left[\begin{array}{c}{x}_{\mathrm{omj}}\hfill \\ \multicolumn{1}{c}{y_{\mathrm{omj}}}\\ \multicolumn{1}{c}{z_{\mathrm{omj}}}\end{array}\right]=\left[\begin{array}{c}{R}_o\cos {\psi }_{\mathrm{mj}}\hfill \\ \multicolumn{1}{c}{R_o\sin {\psi }_{\mathrm{mj}}}\\ \multicolumn{1}{c}{(1-f_i-f_o)D_b\sin a_o-{\delta }_p}\end{array}\right]$$

(12)

in which $R_o$ represents the orbit radius of outer race curvature center; $D_b$ refers to the ball diameter; $a_o$ denotes the initial contact angle; $f_i$ and $f_o$ represent the groove curvature coefficients of inner and outer races, respectively; ${\delta }_p$ indicates the axial deformation of balls under preload $F_p$ that can be calculated by the method proposed in Zhang et al. ¹⁹

Equilibrium equations of matched bearings

The geometric relationship between the centers of the j-th ball and the races is depicted in Figure 5. When the bearing is unloaded, $o_{\mathrm{imj}0}$ and $o_{\mathrm{omj}}$ are the groove curvature centers of inner and outer races, respectively; $o_{\mathrm{bmj}0}$ is the center of the j-th ball; $A_o$ is the distance between points of $o_{\mathrm{imj}0}$ and $o_{\mathrm{omj}}$ . When the bearing is subjected to the preload and external loads, the internal deformation will be produced to balance the combined loads. $o_{\mathrm{bmj}0}$ and $o_{\mathrm{imj}0}$ will move to new points $o_{\mathrm{bmj}}$ and $o_{\mathrm{imj}}$ , respectively; $A_{\mathrm{mj}}$ is the distance between points of $o_{\mathrm{bmj}}$ and $o_{\mathrm{imj}}$ ; $a_{\mathrm{mj}}$ is the new contact angle; $A_{\mathrm{rmj}}$ and $A_{\mathrm{zmj}}$ are the radial and axial distances between points of $o_{\mathrm{bmj}}$ and $o_{\mathrm{imj}}$ , respectively. To study the mechanical characteristics of matched bearings, the relationship between external load vector $\mathit{F}={\left[F_x{F}_y{F}_z{M}_x{M}_y\right]}^T$ and displacement vector $\mathbf{\delta }={\left[{\delta }_x{\delta }_y{\delta }_z{\theta }_x{\theta }_y\right]}^T$ needed to be determined firstly.

Figure 5.

Relative positions between bearing elements.

The distance $A_o$ can be written as:

$$A_o=(f_i+f_o-1)D_b$$

(13)

Based on the geometric relationship demonstrated in Figure 5, the radial and axial distances $A_{\mathrm{rmj}}$ and $A_{\mathrm{zmj}}$ can be calculate by:

$$\{\begin{array}{c}{A}_{\mathrm{rmj}}=\sqrt{{\left(x_{\mathrm{imj}}\right)}^2+{\left(y_{\mathrm{imj}}\right)}^2}-R_o\hfill \\ \multicolumn{1}{c}{A_{\mathrm{zmj}}=z_{\mathrm{imj}}-z_{\mathrm{omj}}}\end{array}$$

(14)

The distance $A_{\mathrm{mj}}$ can be derived through the Pythagorean proposition:

$$A_{\mathrm{mj}}=\sqrt{{\left(A_{\mathrm{rmj}}\right)}^2+{\left(A_{\mathrm{zmj}}\right)}^2}$$

(15)

The normal deformation ${\delta }_{\mathrm{mj}}$ of the j-th ball can be calculated by:

$${\delta }_{\mathrm{mj}}=\{\begin{array}{cc}{A}_{\mathrm{mj}}-A_o\hfill & \mathrm{if}{\delta }_{\mathrm{mj}}>0\hfill \\ \multicolumn{1}{c}{0}& \mathrm{if}{\delta }_{\mathrm{mj}}\le 0\hfill \end{array}$$

(16)

Therefore, the normal contact load $Q_{\mathrm{mj}}$ applied on the inner race can be obtained based on the Hertz contact theory as follows:

$$Q_{\mathrm{mj}}=\{\begin{array}{cc}{K}_H{\left({\delta }_{\mathrm{mj}}\right)}^{\frac{3}{2}}\hfill & \mathrm{if}{\delta }_{\mathrm{mj}}>0\hfill \\ \multicolumn{1}{c}{0}& \mathrm{if}{\delta }_{\mathrm{mj}}\le 0\hfill \end{array}$$

(17)

where $K_H$ named as Hertz contact coefficient related to contact shape and material properties. The sine and cosine functions of contact angel $a_{\mathrm{mj}}$ can be expressed as:

$$\{\begin{array}{c}\sin a_{\mathrm{mj}}=\frac{A_{\mathrm{zmj}}}{A_{\mathrm{mj}}}\hfill \\ \multicolumn{1}{c}{\cos a_{\mathrm{mj}}=\frac{A_{\mathrm{rmj}}}{A_{\mathrm{mj}}}}\end{array}$$

(18)

Based on the analysis above, the load balance equations of inner races can be obtained as:

$$\{\begin{array}{c}{F}_x=\sum _{m=L}^R\sum _{j=1}^Z{Q}_{\mathrm{mj}}\cos a_{\mathrm{mj}}\cos {\psi }_{\mathrm{mj}}\hfill \\ \multicolumn{1}{c}{F_y=\sum _{m=L}^R\sum _{j=1}^Z{c}_1{c}_2{Q}_{\mathrm{mj}}\cos a_{\mathrm{mj}}\sin {\psi }_{\mathrm{mj}}}\\ \multicolumn{1}{c}{F_z=\sum _{m=L}^R\sum _{j=1}^Z{c}_1{c}_2{Q}_{\mathrm{mj}}\sin a_{\mathrm{mj}}}\\ \multicolumn{1}{c}{M_x=\sum _{m=L}^R\sum _{j=1}^Z{Q}_{\mathrm{mj}}\sin {\psi }_{\mathrm{mj}}[\sin a_{\mathrm{mj}}(\frac{d_m}{2}-\frac{D_b}{2}\cos a_{\mathrm{mj}})-c_2\cos a_{\mathrm{mj}}(\frac{S}{2}-c_2\frac{D_b}{2}\cos a_{\mathrm{mj}})]}\\ \multicolumn{1}{c}{M_y=\sum _{m=L}^R\sum _{j=1}^Z{Q}_{\mathrm{mj}}\cos {\psi }_{\mathrm{mj}}[c_1\cos a_{\mathrm{mj}}(\frac{S}{2}-c_2\frac{D_b}{2}\sin a_{\mathrm{mj}})-c_1{c}_2\sin a_{\mathrm{mj}}(\frac{d_m}{2}-\frac{D_b}{2}\cos a_{\mathrm{mj}})]}\end{array}$$

(19)

Figure 6 shows the iterative flowchart of the proposed model. As demonstrated, the error ${\mathrm{\Delta }}_f$ can be derived by the differentials between the two sides of equation (19). The iteration will be terminated when the error ${\mathrm{\Delta }}_f$ is smaller than the predetermined error $e_f$ . Accordingly, the displacement vector $\mathbf{\delta }={\left[{\delta }_x{\delta }_y{\delta }_z{\theta }_x{\theta }_y\right]}^T$ can be calculated by equation (19) based on the Newton-Raphson method. The contact loads and contact angles can be obtained by substituting $\mathbf{\delta }$ into equations (14)−(17).

Figure 6.

Flowchart of the proposed model.

Model validation

To validate the proposed model, the stiffness coefficients of matched bearings obtained by the proposed model are compared with those in the existing literature. Based on the definition of stiffness, the five-DOF stiffness matrix can be formulated by taking the partial derivatives of external load vector $\mathit{F}={\left[F_x{F}_y{F}_z{M}_x{M}_y\right]}^T$ to displacement vector $\mathbf{\delta }={\left[{\delta }_x{\delta }_y{\delta }_z{\theta }_x{\theta }_y\right]}^T$ , the explicit expressions of principal diagonal stiffnesses are listed in Appendix A. In Gunduz and Singh, ¹⁷ for the angular contact ball bearings installed in DB and DF arrangements, Gunduz and Singh have given their stiffness values under the condition that the external loads are applied on the center of left end of shaft. The bearing parameters used by Gunduz and Singh are listed in Table 1. This study applies the external loads to the geometrical center of matched bearings. Accordingly, for the sake of comparison, an additional bending moment should be considered in our proposed model to ensure the working condition is the same with that in Gunduz and Singh, ¹⁷ which can be obtained via multiplying the radial load $F_z$ by the distance $Z_G$ . The external load vector in Gunduz and Singh ¹⁷ is [1000 N, 0, 3000 N, 0, 0], so the loads used in our proposed model can be derived as [1000 N, 0, 3000 N, 0, 74 N·m/rad]. Table 2 shows the comparison of stiffness values obtained through our model with those of Gunduz and Singh. ¹⁷ As shown in Table 2, the stiffness values in this study have good coherence to the results of Gunduz and Singh. ¹⁷ Therefore, the proposed model for matched bearings is proved to be correct and effective. So far, the scientific documentation relative to the matched bearings is quite insufficient. This study provides a practical method for the research of matched bearings. Compared with the method in Gunduz and Singh, ¹⁷ the coordinate transformation method is used to describe the internal load distribution and deformations of matched bearings, which is easy to be understood and can be expanded conveniently for the modeling of assembly bearings installed with multiple single bearings.

Table 1.

Parameters of the example case in Gunduz and Singh. ¹⁷

Parameter	Value
Number of balls Z	14
Hertz contact coefficient KH (N/m1.5)	1.2491 × 1010
Pitch diameter dm (mm)	69.5
Initial contact angle ao (°)	30
Distance between the centers of inner and outer races Ao (mm)	0.52
Distance between the centers of left and right balls S (mm)	10
Radial load Fx (N)	1000
Axial load Fz (N)	3000
Distance between the center of left end of shaft to the geometrical center of left bearing ZL (mm)	64
Distance between the center of left end of shaft to the geometrical center of right bearing ZR (mm)	84
Distance between the center of left end of shaft to the geometrical center of matched bearings ZG (mm)	74

Table 2.

Comparison of the results obtained by the proposed model with those in Gunduz and Singh. ¹⁷

Arrangement form	Stiffness coefficient	Result in Gunduz and Singh 17	Present result	Error (%)
DB arrangement	$k_{\mathrm{xx}}$ (N/m)	4.019 × 108	4.160 × 108	3.5
	$k_{\mathrm{yy}}$ (N/m)	3.276 × 108	3.135 × 108	4.3
	$k_{\mathrm{zz}}$ (N/m)	2.826 × 108	2.888 × 108	2.2
	$k_{{\theta }_x{\theta }_x}$ (N·m/rad)	1.666 × 105	1.591 × 105	4.5
	$k_{{\theta }_y{\theta }_y}$ (N·m/rad)	1.981 × 105	2.054 × 105	3.7
DF arrangement	$k_{\mathrm{xx}}$ (N/m)	4.419 × 108	4.601 × 108	4.1
	$k_{\mathrm{yy}}$ (N/m)	3.413 × 108	3.225 × 108	5.5
	$k_{\mathrm{zz}}$ (N/m)	3.423 × 108	3.515 × 108	2.7
	$k_{{\theta }_x{\theta }_x}$ (N·m/rad)	1.834 × 105	1.741 × 105	5.1
	$k_{{\theta }_y{\theta }_y}$ (N·m/rad)	2.585 × 105	2.686 × 105	3.9

Results and discussion

Based on the proposed model, a pair of angular contact ball bearings (Type: NSK7206B) with DF and DB arrangements can serve as a research case. Table 3 shows the specifications of NSK7206B bearing. Preload, structure parameters, operating condition and installation type play vital roles in the bearing application. The loading characteristics of matched bearings are intimately affected by the above parameters. In Section 4, the influences of these parameters on the distributions of contact load and contact angle of matched bearings under complex working conditions are analyzed in detail.

Table 3.

Specifications of NSK7206B bearing.

Parameters	Values
Number of balls Z	13
Diameter of ball Db (mm)	8.800
Initial contact angle ao (°)	40
Preload Fp (N)	480
Pitch diameter dm (mm)	46.0855
Inner race groove curvature radius ri (mm)	4.6041
Outer race groove curvature radius ro (mm)	4.6041
Density of races and balls ρ (kg/mm3)	7.8 × 10−6

Influence of preload

The angular contact ball bearings installed in pairs must be preloaded to eliminate the clearance between rolling elements and thus to improve their operation performance. To investigate the influence of preload on the distributions of contact load and contact angle of matched bearings, take F  = [500 N, 500 N,1000 N, 5 N·m/rad, 5 N·m/rad] for example and keep up other parameters. Figures 7 and 8 demonstrate the change trends of contact load and contact angle with the increase of preload. The color numbers in all the following figures represent the average values of contact load and contact angle. As depicted in Figures 7 and 8, with the variation of preload, the distribution curves of contact load and contact angle have different shapes and show the complicated nonlinear behaviors. When the ball is out of action, its contact angle and contact load are equal to zero. By increasing the preload, the number of laden balls, the mean values and distribution uniformities of contact load and contact angle increase correspondingly. The distribution uniformities of contact load and contact angle of DB arrangement are better than those of DF arrangement. Hence, it can be concluded that the operation performance of DB arrangement is better than that of DF arrangement due to the more uniformed distributions of contact load and contact angle. For the DF arrangement, the right bearing with axial load and preload in the same direction has the highest contact load and contact angle. Due to the different structure with the DF arrangement, the left bearing in the DB arrangement has the highest contact load and contact angle. In addition, the contact load in DF arrangement is larger than that in DB arrangement.

Figure 7.

Contact load with different preloads: (a) DF arrangement and (b) DB arrangement.

Figure 8.

Contact angle with different preloads. (a) DF arrangement and (b) DB arrangement.

Influence of initial contact angle

Initial contact angle is a key factor to influence the bearing load carrying ability. In actual application, the angular contact ball bearings with angles of 15°, 25°, 40°, and 60° are most widely used in various rotating systems. To study the influence of these angles on the distributions of contact load and contact angle, a combined load vector F  = [500 N, 500 N,1000 N, 5 N·m/rad, 5 N·m/rad ] can be applied on the geometrical center of matched bearings. The variations of contact load and contact angle of matched bearings with different initial contact angles are shown in Figures 9 and 10, respectively. It can be observed that the number of laden balls and the distribution uniformities of contact load and contact angle decrease with the increasing initial contact angle, which indicates that the initial contact angle can significantly affect the loading characteristics of matched bearings. The reason for this phenomenon is that the greater the initial contact angle is, the greater the axial bearing capacity is, but the smaller the preload deformation of balls is. With the increase of initial contact angle, the average contact angle increases correspondingly, but the average contact load decreases first and then increases. According to the curve shapes demonstrated in Figures 9 and 10, it can be concluded that the load and angle distributions of DB arrangement are more stable than those of DF arrangement.

Figure 9.

Contact load with different initial contact angles: (a) DF arrangement and (b) DB arrangement.

Figure 10.

Contact angle with different initial contact angles: (a) DF arrangement and (b) DB arrangement.

Influence of external loads

The external loads have an important influence on the mechanical behaviors of matched bearings. To investigate the influence of radial loads on the distributions of contact load and contact angle of matched bearings, the radial loads $F_x$ and $F_y$ ranging from 1000 to 5000 N are applied. Figures 11 and 12 show the variations of contact load and contact angle of matched bearings under varying radial loads. As demonstrated, with the increase of radial loads, the mean value of contact load increases, but the mean value of contact angle, the distribution uniformities of contact load and contact angle decrease. The number of laden balls decreases firstly with the increasing radial loads and then becomes to a constant when the radial loads reach a certain level. It also can be observed that the greater the radial load is, the greater the mean value of contact load is, but the smaller the mean value of contact angle is. Under pure radial loads, for matched bearings installed in either DF arrangement or DB arrangement, the mean values of contact load and contact angle of left bearing are equal to those of right bearing. In addition, the matched bearings installed in DF and DB arrangements are equivalent in terms of performance.

Figure 11.

Contact load with different radial loads: (a) DF arrangement and (b) DB arrangement.

Figure 12.

Contact angle with different radial loads: (a) DF arrangement and (b) DB arrangement.

To study the influence of axial load on the loading characteristics of matched bearings, the distribution curves of contact load and contact angle with different axial loads are depicted in Figures 13 and 14, respectively. As demonstrated, the contact load and contact angle are uniformly distributed when the matched bearings are under pure axial load. For the bearing with axial load and preload in the same direction, the contact load and contact angle increase accordingly with the increase of axial load. However, for the bearing with axial load and preload in opposite directions, the contact load and contact angle decrease with the increase of axial load. According to equations (16), (17), and (19), it can be concluded that the matched bearings will be ineffective when the ball deformation of a bearing reduces to zero. As depicted in Figures 13 and 14, the contact load and contact angle of the bearing with axial load and preload in the opposite directions are equal to zero when $F_{\mathrm{z}}\ge 2000$  N, resulting in the fails of the left bearing in DF arrangement and the right bearing in DB arrangement. Moreover, it also can be observed that the mechanical properties of matched bearings installed in DF and DB arrangements are equivalent under pure axial load condition.

Figure 13.

Contact load with different axial loads: (a) DF arrangement and (b) DB arrangement.

Figure 14.

Contact angle with different axial loads: (a) DF arrangement and (b) DB arrangement.

To study the influence of tilting moments on the loading characteristics of matched bearings, the tilting moments $M_x$ and $M_y$ ranging from 5 to 20 N·m/rad are applied. The distribution curves of contact load and contact angle with different tilting moments are shown in Figures 15 and 16, respectively. From Figures 15 and 16, it can be seen that the left and right bearings in either DF arrangement or DB arrangement have the same mean values of contact load and contact angle. With the increasing tilting moments $M_x$ and $M_y$ , the mean value of contact load increases correspondingly, but the mean value of contact angle, the number of laden balls, and the distribution uniformities of contact load and contact angle decrease instead. Moreover, it also can be observed that DB arrangement is more suitable to withdraw the tilting moments than DF arrangement as a result of its longer load center distance.

Figure 15.

Contact load with different tilting moments: (a) DF arrangement and (b) DB arrangement.

Figure 16.

Contact angle with different tilting moments: (a) DF arrangement and (b) DB arrangement.

Conclusions

The main purpose of this paper is to study the distribution characteristics of contact load and contact angle of matched bearings under complex working conditions. For this purposes, a novel model of matched bearings is established by using the universal coordinate transformation method. The distributions of contact load and contact angle of matched bearings under different installation types, preloads, operating conditions are studied in detail. The main conclusions can be drawn:

1. With the increasing preload, the number of laden balls, the mean values and distribution uniformities of contact load and contact angle increase correspondingly. Under the preload and external loads, the distribution uniformities of contact load and contact angle of DB arrangement are better than those of DF arrangement.

2. With the increasing initial contact angle, the mean value of contact angle increases, the mean value of contact load decreases first and then increases, the number of laden balls and the distribution uniformities of contact load and contact angle decrease. Moreover, the variations of contact load and contact angle of DF arrangement are more obvious than those of DB arrangement.

3. With the increasing pure radial loads, the mean value of contact load increases, but the mean value of contact angle and the distribution uniformity of contact load and contact angle decrease. The number of laden balls reduces at first with the increase of radial loads and then becomes to a constant when the radial loads reach a certain level. With the increasing pure axial load, the contact load and contact angle of balls are uniformly distributed and the matched bearings will fail when the deformation of each ball in a bearing is equal to zero. Accordingly, it is significant to emphatically study the axial mechanical properties of matched bearings. Under the pure radial loads or axial load, the loading characteristics of DF and DB arrangements are equivalent. With the increase of pure tilting moments, the mean value of contact load increases, whereas the mean value of contact angle, the number of laden balls and the distribution uniformities of contact load and contact angle decrease. DB arrangement is more suitable to withdraw the tilting moments than DF arrangement as a result of its longer load center distance.

This paper reveals the distribution characteristics of contact load and contact angle of matched bearings under different working conditions, which can offer some references to the selection, design and assembling of angular contact ball bearings.