A Dynamic Analysis of Angular Contact Ball Bearing 7205C Used for a Scraper Conveyor

Abstract

As core pieces of transport equipment in longwall mining systems, scraper conveyors operate under extremely harsh and dynamic loading conditions. Their operational reliability and service life primarily depend on the performance of critical components within their drive systems, particularly the support bearings. However, complex and often unpredictable load spectra (such as severe impacts, vibrations, and contaminant ingress) pose significant challenges to the dynamic behavior and longevity of these bearings. Traditional static analysis fails to capture their true operating state, as it neglects transient effects, varying contact angles, and internal vibration excitation. This study conducts a comprehensive dynamic analysis of angular contact ball bearing 7205C to elucidate its dynamic response under actual operating conditions of scraper conveyors. Based on Hertzian elastic contact theory and bearing dynamics theory, the comprehensive stiffness of the angular contact ball bearing is derived, and the effects of axial force, rotational speed, and mass eccentricity on bearing performance are analyzed. The findings are expected to provide a theoretical foundation for optimizing bearing selection, predicting service life, and enhancing the overall reliability of mining machinery.

1. Introduction

Scraper conveyors are indispensable backbone equipment in longwall mining systems, and unplanned downtime caused by scraper conveyor failures can paralyze the entire production chain, resulting in significant economic losses. Their operational reliability and stability primarily depend on the performance of critical components within their drive system. Bearings, as core components of the drive system, directly determine the mechanical efficiency and service life of the scraper conveyor through their operational status [1,2,3]. However, dynamic challenges arise from load excitations such as coal impact and chain sprocket meshing forces, posing significant demands on bearing performance and operational status. Consequently, there is an urgent need to investigate the dynamic characteristics of scraper conveyor bearings, providing theoretical support for bearing selection, condition monitoring, and reliability enhancement in heavy machinery [4,5,6].

In terms of dynamic analysis, Xu et al. [7] systematically analyzed the influence of key parameters, such as axial force, on the interactions between rolling elements, retainers, and guide rings within bearings by establishing a comprehensive dynamic model. The study revealed that increased axial force helps suppress retainer motion instability caused by radial forces, thereby enhancing the periodicity of ball–raceway contact. Mo et al. [8] developed a real-time coupled model integrating thermal and dynamic aspects of the gear–rotor–bearing system. Thermal effects induce periodic clearance fluctuations, leading to frequent transitions between meshed and unmeshed states and thereby increasing system instability. A 6-degree-of-freedom dynamics model with multiple faults in the outer raceway was developed by Hu et al. [9]. The effects of the number of faults, spacing angle, and distribution mode on the dynamical characteristics were investigated using phase trajectories, envelope analysis, and Lempel–Ziv complexity (LZC) index. Tian et al. [10] developed a numerical model based on the fluid lubrication theory to determine the dynamic characteristics of the oil film stiffness and damping of a thrust bearing. The results show that at high rotational speeds, the increase in vibration and acoustic radiation is mainly due to the increase in excitation force. At a constant excitation force, higher rotational speed enhances the lubrication performance of the thrust bearing and reduces the vibration energy transfer between the shaft shell. Gao et al. [11] developed a dynamic model for a ceramic electric spindle system that accounts for the influence of internal clearance variations on bearing stiffness characteristics. Thermal deformation leads to nonlinear changes in outer ring fit clearance and internal clearance. Compared with the outer ring mating clearance, the change in internal clearance has a more obvious effect on the motion stability of the ceramic motorized spindle system. Sun et al. [12] investigated the influence of journal deformation-induced misalignment on the nonlinear friction dynamics of rotor–bearing systems. The results indicate that synchronous vibration and oil film oscillation generate significant amplitude, leading to reduced film thickness. Furthermore, increased external loading amplifies both the oil film misalignment angle and amplitude.

In terms of scraper conveyor, Zhang et al. [13] analyzed the meshing process and force conditions between the drive sprocket and ring chain. Considering the impact of rolling bearing installation clearance on nonlinear elastic restoring forces, a nonlinear rigid–flexible coupled dynamic model for the meshing transmission system was constructed. Transmission experiments were conducted to validate the effectiveness of the dynamic model. Zhao et al. [14] developed a rigid–flexible coupled mathematical model for the cutting section of a coal miner. Through a comprehensive analysis of radial bearing loads and rotational speeds, they derived the dynamic response between rolling bearings and shafts under low-speed, heavy-load conditions. Wang et al. [15] constructed a nonlinear dynamic model of the rigid–flexible coupling between chain links, sprockets, and bearings, validating the model’s effectiveness through transmission experiments. The research findings indicate that during transmission, the system exhibits periodic vibration characteristics with step phenomena. Increased sprocket mounting distance, load, and preload exacerbate vibrations in the meshing transmission system. For strong noise vibration signals with impact noise collected from scraper conveyor bearings, Chang et al. [16] proposed an improved ICEEMDAN decomposition combined with wavelet packet denoising method. Verification demonstrated that the proposed denoising algorithm outperforms traditional methods, achieving noise reduction without losing original signal features and effectively improving fault diagnosis classification accuracy.

Compared to previous research focusing on high-speed spindles or general rotating machinery, this work specifically investigates the dynamic performance of angular contact ball bearings under the demanding operating conditions of scraper conveyors, characterized by significant axial loads and mass eccentricity. A comprehensive stiffness model integrating Hertzian contact stiffness and oil film stiffness is derived. The independent effects of axial force, mass eccentricity, and rotational speed on bearing deformation, stiffness, and contact angle are systematically quantified, providing targeted theoretical support for bearing selection and dynamic modeling in heavy-duty conveying systems.

The structure of this paper is as follows: Section 1 provides an overview of the current research status in bearing dynamics; Section 2 establishes a dynamic model for angular contact ball bearing 7205C based on Hertz contact theory and bearing dynamics and derives its comprehensive stiffness; Section 3 thoroughly analyzes the effects of axial force, mass eccentricity, and drive shaft speed on bearing performance (including deformation, stiffness, and contact angle); and Section 4 summarizes the main conclusions drawn from this study.

2. A Dynamic Model of the Angular Contact Ball Bearing

In the dynamic model, the following aspects are discussed (1) the stiffness of the lubricating oil film is considered; (2) since the outer ring of the ball bearing is fixed in the bearing housing, the deformation of the outer ring can be considered negligible; (3) the rotational displacement of the bearing caused by the torsional deformation of the spindle is considered; (4) due to the presence of the cage, the relative positions of the balls within the ball bearing are considered fixed; and (5) the frictional force between the steel ball and the oil film is considered. Figure 1 shows the structure of the scraper conveyor.

Figure 1. The structure of the scraper conveyor.

Figure 2 shows the geometric model of the ball bearing. The coordinate system is established at the center of the inner ring. Z represents the axial coordinate; X and Y represent the radial coordinates; R_p denotes the radius of the pitch circle; O_d denotes the center of the rolling element; O_i and O_e denote the curvature centers of the inner and outer rings; r_i and r_e denote the curvature radii of the inner and outer rings; R_i and R_e denote the radii of the inner and outer rings; α₀ denotes the initial contact angle of the ball bearing; and w denotes the speed of the ball bearing.

Figure 2. A geometric model of the angular contact ball bearing.

During the uniform rotation of the bearing-motorized drive shaft system, the inner ring is fixed on the motorized drive shaft, and the external load [F_x F_y F_z M_x M_y M_z] is transferred to rolling elements and the outer ring. When the ball bearing reaches a state of balanced force, the inner ring produces the deformation [δ_x δ_y δ_z θ_x θ_y θ_z] relative to the outer ring. Figure 3 shows the deformation of the j-th rolling element, and φ_j denotes the position angle of the j-th rolling element. Because the steel balls are distributed circumferentially within the bearing, their positions differ, resulting in varying stress states and deformation amounts. According to the conformability of deformation, the contact angle and total deformation of different rolling elements can be calculated [17].

Figure 3. Deformation of the j-th rolling element.

Figure 4 shows the curvature variation in the inner ring of a ball bearing. When the ball bearing is free from an external force, curvature center O_i of the inner ring and curvature center O_e of the outer ring are in a straight line with rolling element O_d, and α_j denotes the contact angle after the ball bearing is loaded. The distance between curvature centers of the inner and outer rings is given by

$${\mathrm{d}}_0={\mathrm{O}}_{\mathrm{i}}{\mathrm{O}}_{\mathrm{e}}={\mathrm{r}}_{\mathrm{i}}+{\mathrm{r}}_{\mathrm{e}}-{\mathrm{d}}_{\mathrm{g}}$$

(1)

Figure 4. A schematic diagram of the curvature center change.

In Equation (1), d_g denotes the diameter of rolling elements. After the ball bearing is loaded, the force analysis of the j-th rolling element is written by [18]

$${\mathsf{\theta }}_{\mathrm{zj}}={\displaystyle \frac{{\mathrm{M}}_{\mathrm{zj}}\mathrm{L}}{\mathrm{G}{\mathrm{I}}_{\mathrm{p}}}}$$

(2)

$${\mathsf{\delta }}_{\mathrm{zj}}={\mathsf{\delta }}_{\mathrm{z}}+{\mathrm{r}}_{\mathrm{p}}\left({\mathsf{\theta }}_{\mathrm{x}}\sin {\mathsf{\phi }}_{\mathrm{j}}-{\mathsf{\theta }}_{\mathrm{y}}\cos {\mathsf{\phi }}_{\mathrm{j}}\right)$$

(3)

$${\mathsf{\delta }}_{\mathrm{rj}}={\mathsf{\delta }}_{\mathrm{x}}\cos {\mathsf{\phi }}_{\mathrm{j}}+{\mathsf{\delta }}_{\mathrm{y}}\sin {\mathsf{\phi }}_{\mathrm{j}}$$

(4)

$${\mathrm{r}}_{\mathrm{p}}={\mathrm{R}}_{\mathrm{p}}+({\mathrm{r}}_{\mathrm{i}}-0.5{\mathrm{d}}_{\mathrm{g}})\cos {\mathsf{\alpha }}_0$$

(5)

θ_zj denotes the rotation displacement at the position of the j-th rolling element. δ_zj denotes the axial displacement and δ_rj denotes the radial displacement at the position of the j-th rolling element. Due to the high rotational speed of the drive shaft, friction between the inner and outer rings of the bearing must be considered. When the drive shaft is affected by frictional torque M_zj, rotation displacement θ_zj occurs at the middle section of the drive shaft. L denotes the length of the middle segment. G denotes the shear modulus of the drive shaft. I_p denotes the polar moment of inertia, and r_p denotes the distance from the geometric center of the ball bearing to the curvature center of the inner ring.

At different position angles, total deformation δ_j between the inner and outer rings is given by

$${\mathsf{\delta }}_{\mathrm{j}}={\mathrm{d}}_{\mathrm{j}}-{\mathrm{d}}_0$$

(6)

In Equation (6), d_j denotes the distance between curvature centers of the inner and outer rings after deformation:

$${\mathrm{d}}_{\mathrm{j}}=\sqrt{{\left({\mathrm{d}}_0{\sin \mathsf{\alpha }}_0+{\mathsf{\delta }}_{\mathrm{zj}}\right)}^2+{\left({\mathrm{d}}_0{\cos \mathsf{\alpha }}_0+{\mathsf{\delta }}_{\mathrm{rj}}\right)}^2}$$

(7)

In the previous literature, the calculation of ball bearing stiffness has mostly relied on Hertz contact stress theory to compute the elastic deformation between the balls and the inner/outer rings, thereby determining the Hertz contact stiffness of the ball bearing. However, during ball bearing operation, an oil film exists between the balls and the rings, typically placing the bearing in an electrohydrodynamic lubrication state. Therefore, the stiffness calculation of ball bearings comprises both the Hertz contact stiffness and the oil film stiffness. The oil film primarily influences the radial stiffness of the ball bearing, so its effects on other aspects of stiffness are neglected in the calculation.

Figure 5 is the mechanical model of the j-th rolling element. Q_ij and Q_ej denote the external forces of the inner and outer rings on the j-th rolling element; F_ij and F_ej denote the friction forces of the j-th rolling element; α_ij and α_ej denote the contact angles of the rolling element; and F_cj denotes the centrifugal force of the j-th rolling element [19]. According to the force balance of the j-th rolling element,

$${\mathrm{Q}}_{\mathrm{ij}}{\sin \mathsf{\alpha }}_{\mathrm{ij}}-{\mathrm{Q}}_{\mathrm{ej}}{\sin \mathsf{\alpha }}_{\mathrm{ej}}=0$$

(8)

$${\mathrm{Q}}_{\mathrm{ij}}{\cos \mathsf{\alpha }}_{\mathrm{ij}}-{\mathrm{Q}}_{\mathrm{ej}}{\cos \mathsf{\alpha }}_{\mathrm{ej}}+{\mathrm{F}}_{\mathrm{cj}}=0$$

(9)

Figure 5. A mechanical model of the j-th rolling element.

According to Hertz contact stress theory, the relationship between the load and deformation of the j-th rolling element is [20]

$${\mathsf{\delta }}_{\mathrm{b}\mathrm{j}}={\left({\displaystyle \frac{{\mathrm{Q}}_{\mathrm{i}\mathrm{j}}}{{\mathrm{K}}_{\mathrm{h}}}}\right)}^{2/3}$$

(10)

In Equation (10), K_h denotes the coefficient of total elastic deformation between the inner and outer rings. The coefficient of total elastic deformation is given by

$${\mathrm{K}}_{\mathrm{h}}={\displaystyle \frac{1}{{\left({{\mathrm{K}}_{\mathrm{i}}}^{-\frac{2}{3}}+{{\mathrm{K}}_{\mathrm{e}}}^{-\frac{2}{3}}\right)}^{1.5}}}$$

(11)

In Equation (11), K_i denotes the coefficient of the elastic deformation between the rolling element and inner ring. K_e denotes the coefficient of the elastic deformation between the rolling element and outer ring [21]:

$${\mathrm{K}}_{\mathrm{i}}={\left[{\displaystyle \frac{32}{9{\left({{\mathsf{\delta }}^{*}}_{\mathrm{i}}\right)}^3{\mathsf{\lambda }}^2{\mathsf{\rho }}_{\mathrm{i}}}}\right]}^{0.5}$$

(12)

$${\mathrm{K}}_{\mathrm{e}}={\left[{\displaystyle \frac{32}{9{\left({{\mathsf{\delta }}^{*}}_{\mathrm{e}}\right)}^3{\mathsf{\lambda }}^2{\mathsf{\rho }}_{\mathrm{e}}}}\right]}^{0.5}$$

(13)

In Equations (12) and (13), ${{\mathsf{\delta }}^{*}}_{\mathrm{i}}$ and ${{\mathsf{\delta }}^{*}}_{\mathrm{e}}$ denote the constant of relative approach between the inner and outer rings; $\mathsf{\lambda }$ denotes the comprehensive elastic constant: $\mathsf{\lambda }={\displaystyle \frac{\mathrm{E}}{1-{\mathsf{\upsilon }}^2}}$; E denotes the elastic modulus of the ball bearing material; $\mathsf{\nu }$ denotes Poisson’s ratio; and ${\mathsf{\rho }}_{\mathrm{i}}$ and ${\mathsf{\rho }}_{\mathrm{e}}$ are the curvature sum of the contact pair between the inner ring, outer ring and rolling elements:

$${\mathsf{\rho }}_{\mathrm{i}}={\displaystyle \frac{2}{{\mathrm{d}}_{\mathrm{g}}}}+{\displaystyle \frac{1}{{\mathrm{r}}_{\mathrm{i}}}}-{\displaystyle \frac{2}{{\mathrm{d}}_{\mathrm{g}}}}{\displaystyle \frac{\mathsf{\gamma }}{1-\mathsf{\gamma }}}$$

(14)

$${\mathsf{\rho }}_{\mathrm{e}}={\displaystyle \frac{2}{{\mathrm{d}}_{\mathrm{g}}}}+{\displaystyle \frac{2}{{\mathrm{d}}_{\mathrm{g}}}}{\displaystyle \frac{\mathsf{\gamma }}{1+\mathsf{\gamma }}}-{\displaystyle \frac{1}{{\mathrm{r}}_{\mathrm{e}}}}$$

(15)

$$\mathsf{\gamma }={\displaystyle \frac{{\mathrm{d}}_{\mathrm{g}}/2}{{\mathrm{R}}_{\mathrm{i}}+{\mathrm{d}}_{\mathrm{g}}/2}}$$

(16)

For simplified calculations, the drive shaft is treated as a concentrated mass model. Figure 6 shows the simplified model of the ball bearing–drive shaft system. When mass eccentricity d_e exists in the drive shaft, the radial force, F_r, of the ball bearing is given by

$${\mathrm{F}}_{\mathrm{r}\mathrm{x}}=\mathrm{m}{\mathrm{d}}_{\mathrm{e}}{\mathrm{w}}^2\cos \mathrm{wt}$$

(17)

$${\mathrm{F}}_{\mathrm{r}\mathrm{y}}=\mathrm{m}{\mathrm{d}}_{\mathrm{e}}{\mathrm{w}}^2\sin \mathrm{wt}$$

(18)

Figure 6. Simplified model of the ball bearing–drive shaft system.

In Equations (17) and (18), m denotes the mass of the drive shaft and w denotes the speed.

According to the oil film thickness formula of Hamrock–Dowson, oil film thickness h between the inner and outer rings is given by [20]

$$\mathrm{h}=\mathsf{\Lambda }{{\mathrm{F}}_{\mathrm{r}}}^{0.07}$$

(19)

$$\begin{array}{l}\mathsf{\Lambda }=0.44{\mathsf{\mu }}^{0.49}{\left({\mathsf{\eta }}_0{\displaystyle \frac{\mathrm{w}}{2\mathsf{\pi }}}\right)}^{0.7}{\left({\displaystyle \frac{{\mathrm{d}}_{\mathrm{g}}}{2}}\right)}^{0.47}{\left({\mathrm{R}}_{\mathrm{i}}+{0.5\mathrm{d}}_{\mathrm{g}}\right)}^{0.7}\\ \cdot \left[{\left(1-\mathsf{\gamma }\right)}^{1.15}{\left(1+\mathsf{\gamma }\right)}^{0.7}\left(1-{\mathrm{e}}^{0.7{\mathrm{k}}_{\mathrm{i}}}\right)+{\left(1+\mathsf{\gamma }\right)}^{1.15}{\left(1-\mathsf{\gamma }\right)}^{0.7}\left(1-{\mathrm{e}}^{0.7{\mathrm{k}}_{\mathrm{e}}}\right)\right]{\mathsf{\lambda }}^{-0.12}{\mathrm{n}}^{0.07}\end{array}$$

(20)

In Equation (20), μ denotes the viscosity pressure coefficient of lubricating oil; ${\mathsf{\eta }}_0$ denotes the dynamic viscosity of lubricating oil at atmospheric pressure; n denotes the number of rolling elements; and k_i and k_e denote the ellipticity of the inner and outer rings. According to the definition, the stiffness of oil film is given by

$${\mathrm{K}}_{\mathrm{m}}=\underset{\begin{array}{c}∆{\mathrm{F}}_{\mathrm{r}}\to 0\\ ∆\mathrm{h}\to 0\end{array}}{\lim }{\displaystyle \frac{∆{\mathrm{F}}_{\mathrm{r}}}{∆\mathrm{h}}}=0.07\mathsf{\Lambda }{\mathrm{F}}_{\mathrm{r}}^{-1.07}$$

(21)

The comprehensive stiffness of the ball bearing, K_u, is given by

$${\mathrm{K}}_{\mathrm{u}}={\mathrm{K}}_{\mathrm{h}}+{\mathrm{K}}_{\mathrm{m}}$$

(22)

When the inner ring of the ball bearing is in a balanced state, the equilibrium equation is established at the position of φ_j:

$$\left[\begin{array}{c}{\mathrm{F}}_{\mathrm{xj}}\\ {\mathrm{F}}_{\mathrm{yj}}\\ {\mathrm{F}}_{\mathrm{zj}}\\ {\mathrm{M}}_{\mathrm{xj}}\\ {\mathrm{M}}_{\mathrm{yj}}\\ {\mathrm{M}}_{\mathrm{zj}}\end{array}\right]=\left[\begin{array}{c}{\mathrm{Q}}_{\mathrm{ij}}\cos {\mathsf{\alpha }}_{\mathrm{j}}\cos {\mathsf{\phi }}_{\mathrm{j}}\\ {\mathrm{Q}}_{\mathrm{ij}}\cos {\mathsf{\alpha }}_{\mathrm{j}}\sin {\mathsf{\phi }}_{\mathrm{j}}\\ {\mathrm{Q}}_{\mathrm{ij}}\sin {\mathsf{\alpha }}_{\mathrm{j}}\\ {\mathrm{R}}_{\mathrm{p}}{\mathrm{Q}}_{\mathrm{ij}}\sin {\mathsf{\alpha }}_{\mathrm{j}}\sin {\mathsf{\phi }}_{\mathrm{j}}\\ {\mathrm{R}}_{\mathrm{p}}{\mathrm{Q}}_{\mathrm{ij}}\sin {\mathsf{\alpha }}_{\mathrm{j}}\cos {\mathsf{\phi }}_{\mathrm{j}}\\ {\mathrm{R}}_{\mathrm{p}}{\mathrm{F}}_{\mathrm{ij}}\end{array}\right]$$

(23)

By summing the external load on the inner ring, the whole equilibrium equation of the ball bearing can be obtained as follows:

$$\left[\begin{array}{c}{\mathrm{F}}_{\mathrm{x}}\\ {\mathrm{F}}_{\mathrm{y}}\\ {\mathrm{F}}_{\mathrm{z}}\\ {\mathrm{M}}_{\mathrm{x}}\\ {\mathrm{M}}_{\mathrm{y}}\\ {\mathrm{M}}_{\mathrm{z}}\end{array}\right]=\sum _{\mathrm{j}=1}^{\mathrm{n}}\left[\begin{array}{c}{\mathrm{Q}}_{\mathrm{ij}}\cos {\mathsf{\alpha }}_{\mathrm{j}}\cos {\mathsf{\phi }}_{\mathrm{j}}\\ {\mathrm{Q}}_{\mathrm{ij}}\cos {\mathsf{\alpha }}_{\mathrm{j}}\sin {\mathsf{\phi }}_{\mathrm{j}}\\ {\mathrm{Q}}_{\mathrm{ij}}\sin {\mathsf{\alpha }}_{\mathrm{j}}\\ {\mathrm{R}}_{\mathrm{p}}{\mathrm{Q}}_{\mathrm{ij}}\sin {\mathsf{\alpha }}_{\mathrm{j}}\sin {\mathsf{\phi }}_{\mathrm{j}}\\ {\mathrm{R}}_{\mathrm{p}}{\mathrm{Q}}_{\mathrm{ij}}\sin {\mathsf{\alpha }}_{\mathrm{j}}\cos {\mathsf{\phi }}_{\mathrm{j}}\\ {\mathrm{R}}_{\mathrm{p}}{\mathrm{F}}_{\mathrm{ij}}\end{array}\right]$$

(24)

According to the principles of materials mechanics, the relationship between the deformation, q, of ball bearings and the applied load F can be expressed as follows:

$$\mathrm{F}=\mathrm{Kq}$$

(25)

The stiffness matrix, K, of the ball bearing is given by

$$\mathrm{K}=\left[\begin{array}{cccccc}{\displaystyle \frac{\partial {\mathrm{F}}_{\mathrm{x}}}{\partial {\mathsf{\delta }}_{\mathrm{x}}}}& {\displaystyle \frac{\partial {\mathrm{F}}_{\mathrm{x}}}{\partial {\mathsf{\delta }}_{\mathrm{y}}}}& {\displaystyle \frac{\partial {\mathrm{F}}_{\mathrm{x}}}{\partial {\mathsf{\delta }}_{\mathrm{z}}}}& {\displaystyle \frac{\partial {\mathrm{F}}_{\mathrm{x}}}{\partial {\mathsf{\theta }}_{\mathrm{x}}}}& {\displaystyle \frac{\partial {\mathrm{F}}_{\mathrm{x}}}{\partial {\mathsf{\theta }}_{\mathrm{y}}}}& {\displaystyle \frac{\partial {\mathrm{F}}_{\mathrm{x}}}{\partial {\mathsf{\theta }}_{\mathrm{z}}}}\\ {\displaystyle \frac{\partial {\mathrm{F}}_{\mathrm{y}}}{\partial {\mathsf{\delta }}_{\mathrm{x}}}}& {\displaystyle \frac{\partial {\mathrm{F}}_{\mathrm{y}}}{\partial {\mathsf{\delta }}_{\mathrm{y}}}}& {\displaystyle \frac{\partial {\mathrm{F}}_{\mathrm{y}}}{\partial {\mathsf{\delta }}_{\mathrm{z}}}}& {\displaystyle \frac{\partial {\mathrm{F}}_{\mathrm{y}}}{\partial {\mathsf{\theta }}_{\mathrm{x}}}}& {\displaystyle \frac{\partial {\mathrm{F}}_{\mathrm{y}}}{\partial {\mathsf{\theta }}_{\mathrm{y}}}}& {\displaystyle \frac{\partial {\mathrm{F}}_{\mathrm{y}}}{\partial {\mathsf{\theta }}_{\mathrm{z}}}}\\ {\displaystyle \frac{\partial {\mathrm{F}}_{\mathrm{z}}}{\partial {\mathsf{\delta }}_{\mathrm{x}}}}& {\displaystyle \frac{\partial {\mathrm{F}}_{\mathrm{z}}}{\partial {\mathsf{\delta }}_{\mathrm{y}}}}& {\displaystyle \frac{\partial {\mathrm{F}}_{\mathrm{z}}}{\partial {\mathsf{\delta }}_{\mathrm{z}}}}& {\displaystyle \frac{\partial {\mathrm{F}}_{\mathrm{z}}}{\partial {\mathsf{\theta }}_{\mathrm{x}}}}& {\displaystyle \frac{\partial {\mathrm{F}}_{\mathrm{z}}}{\partial {\mathsf{\theta }}_{\mathrm{y}}}}& {\displaystyle \frac{\partial {\mathrm{F}}_{\mathrm{z}}}{\partial {\mathsf{\theta }}_{\mathrm{z}}}}\\ {\displaystyle \frac{\partial {\mathrm{M}}_{\mathrm{x}}}{\partial {\mathsf{\delta }}_{\mathrm{x}}}}& {\displaystyle \frac{\partial {\mathrm{M}}_{\mathrm{x}}}{\partial {\mathsf{\delta }}_{\mathrm{y}}}}& {\displaystyle \frac{\partial {\mathrm{M}}_{\mathrm{x}}}{\partial {\mathsf{\delta }}_{\mathrm{z}}}}& {\displaystyle \frac{\partial {\mathrm{M}}_{\mathrm{x}}}{\partial {\mathsf{\theta }}_{\mathrm{x}}}}& {\displaystyle \frac{\partial {\mathrm{M}}_{\mathrm{x}}}{\partial {\mathsf{\theta }}_{\mathrm{y}}}}& {\displaystyle \frac{\partial {\mathrm{M}}_{\mathrm{x}}}{\partial {\mathsf{\theta }}_{\mathrm{z}}}}\\ {\displaystyle \frac{\partial {\mathrm{M}}_{\mathrm{y}}}{\partial {\mathsf{\delta }}_{\mathrm{x}}}}& {\displaystyle \frac{\partial {\mathrm{M}}_{\mathrm{y}}}{\partial {\mathsf{\delta }}_{\mathrm{y}}}}& {\displaystyle \frac{\partial {\mathrm{M}}_{\mathrm{y}}}{\partial {\mathsf{\delta }}_{\mathrm{z}}}}& {\displaystyle \frac{\partial {\mathrm{M}}_{\mathrm{y}}}{\partial {\mathsf{\theta }}_{\mathrm{x}}}}& {\displaystyle \frac{\partial {\mathrm{M}}_{\mathrm{y}}}{\partial {\mathsf{\theta }}_{\mathrm{y}}}}& {\displaystyle \frac{\partial {\mathrm{M}}_{\mathrm{y}}}{\partial {\mathsf{\theta }}_{\mathrm{z}}}}\\ {\displaystyle \frac{\partial {\mathrm{M}}_{\mathrm{z}}}{\partial {\mathsf{\delta }}_{\mathrm{x}}}}& {\displaystyle \frac{\partial {\mathrm{M}}_{\mathrm{z}}}{\partial {\mathsf{\delta }}_{\mathrm{y}}}}& {\displaystyle \frac{\partial {\mathrm{M}}_{\mathrm{z}}}{\partial {\mathsf{\delta }}_{\mathrm{z}}}}& {\displaystyle \frac{\partial {\mathrm{M}}_{\mathrm{z}}}{\partial {\mathsf{\theta }}_{\mathrm{x}}}}& {\displaystyle \frac{\partial {\mathrm{M}}_{\mathrm{z}}}{\partial {\mathsf{\theta }}_{\mathrm{y}}}}& {\displaystyle \frac{\partial {\mathrm{M}}_{\mathrm{z}}}{\partial {\mathsf{\theta }}_{\mathrm{z}}}}\end{array}\right]$$

(26)

For the nonlinear equations, the Newton–Raphson method was used to solve the following equations:

$${\mathrm{f}}_{\mathrm{m}}\left({\mathsf{\delta }}_{\mathrm{k}}\right)=0$$

(27)

$${\mathrm{f}}_{\mathrm{m}}+{\sum }_{\mathrm{k}=1}^6{\displaystyle \frac{\partial {\mathrm{f}}_{\mathrm{m}}}{\partial {\mathsf{\delta }}_{\mathrm{k}}}}{\mathsf{\epsilon }}_{\mathrm{k}}=0$$

(28)

where m = 1, 2, 3, 4, 5, 6; k = 1, 2, 3, 4, 5, 6; and ε_k is the iteration error, with ε_k ≤ 10⁻⁶. The step size of the contact angle step is π/8.

During the model analysis process, MATLAB (https://www.mathworks.com/products/matlab.html, accessed on 1 June 2025) integrates a highly efficient numerical computation engine with a rich suite of pre-built toolboxes. Its highly optimized matrix operations enable the rapid solution and visualization of complex models, significantly enhancing computational efficiency and reliability. Consequently, MATLAB software was employed to program the computation and visualization of the derived dynamic model.

3. A Dynamic Analysis of the Angular Contact Ball Bearing

In this paper, a 7205C angular contact ball bearing is selected. Table 1 shows the parameters of the 7205C angular contact ball bearing. Model parameters were determined through a combination of engineering estimates and the literature data based on typical operating conditions of scraper conveyor drive systems [13,14]. The selected parameter range is designed to cover typical load and excitation conditions that bearings may experience during low-to-medium-speed, heavy-load operation [20,21].

Table 1. The parameters of the 7205C angular contact ball bearing (NACHI, Toyama, Japan).

Parameters	Values
Radius of pitch circle Rp	27 mm
Radius of inner ring Ri	20 mm
Radius of outer ring Re	48 mm
Number of rolling elements n	16
Radius of curvature of inner ring ri	4.108 mm
Radius of curvature of outer ring re	4.108 mm
Initial contact angle ${\mathsf{\alpha }}_0$	15°
Modulus of elasticity E	206 Gpa
Poisson’s ratio $\mathsf{\nu }$	0.3
Grease model	Changcheng 7018 (Sinopec Lubricant Company Limited, Beijing, China)

3.1. The Effect of Axial Force on the Performance of the Bearing

Figure 7 shows the effect of axial force F_z on total deformation δ (w = 500 rad/s, d_e = 0.01 mm). As shown in Figure 7, total deformation δ exhibits periodic variation with the increase in position angle φ. When the position angle is 0°, the deformation reaches its minimum value. Meanwhile, the maximum deformation occurs at 180°. When the position angle is 90° and 270°, the total deformation is close to 0 mm. This indicates that deformation at these position angles is independent of the axial force. At the same position angle, the total deformation increases as the axial force increases.

Figure 7. Effect of axial force on bearing deformation.

Figure 8 shows the effect of axial force F_z on radial stiffness K and angular stiffness K_θ (w = 500 rad/s, d_e = 0.01 mm). As shown in Figure 8a,b, the radial stiffness of the ball bearing increases with the increase in axial force. Notably, the variation trends of the two-radial stiffness differ. The increase in radial stiffness K_x gradually slows down after axial force F_z reaches 600 N. When axial force F_z is 1000 N, radial stiffness K_x is increased by 27.9%. Meanwhile, radial stiffness K_y is increased by 47.5%. According to Figure 8c,d, the increase in angular stiffness follows a nearly consistent trend with rising axial force. When axial force F_z reaches 1000 N, angle stiffness K_θx is increased by 78.6%, and angle stiffness K_θy is increased by 84.6%. Compared to radial stiffness, the axial force has a greater impact on the angular stiffness.

Figure 8. The effect of axial force on the stiffness of the bearing: (a) radial stiffness K_x; (b) radial stiffness K_y; (c) angular stiffness k_θx; (d) angular stiffness k_θy.

Figure 9 shows the variation trend of contact angle α with the increase in axial force F_z (w = 500 rad/s, d_e = 0.01 mm). It shows that the contact angle exhibits sinusoidal periodic variation at different positions. The contact angle is smallest at φ = 270° and largest at φ = 90°. At φ = 0°, the contact angle remains unchanged, indicating that the contact angle near this position is less affected by axial force. With an increase in axial force F_z, the amplitude variation in contact angle α decreases accordingly. When F_z = 200 N, the amplitude of contact angle α is 0.02°. When F_z = 800 N, the amplitude of contact angle α is approximately 0.0055°. It can be observed that appropriately increasing the axial force of the ball bearing reduces the variation in contact angle α, thereby enhancing the stability of the ball bearing system.

Figure 9. Effect of axial force on contact angle: (a) F_z = 200 N; (b) F_z = 400 N; (c) F_z = 600 N; (d) F_z = 800 N.

3.2. Effect of Mass Eccentricity on the Performance of the Bearing

Figure 10 shows the variation in total deformation δ with mass eccentricity d_e (w = 1000 rad/s, axial force F_z = 1000 N). To distinguish results across different groups, the vertical axis data have been logarithmically scaled. As seen in Figure 10, total deformation δ exhibits periodic variation as mass eccentricity d_e increases. At position angle φ = 180°, the deformation reaches its maximum value. At φ = 0°, the deformation reaches its minimum value. At φ = 90° and φ = 270°, the total deformation tends to be similar, indicating that at these position angles, the deformation is less affected by mass eccentricity d_e. As d_e increases, the total deformation also increases at the same position angle.

Figure 10. Effect of mass eccentricity on bearing deformation.

Figure 11 shows the variation trend of radial stiffness K and angular stiffness K_θ (w = 1000 rad/s, F_z = 1000 N). As shown in Figure 11a,b, when mass eccentricity d_e reaches 0.04 mm, K_x reaches a minimum value of $3.315\times {10}^8\mathrm{N}/\mathrm{m}$. At d_e = 0.1 mm, K_x is increased by 0.13% and K_y is increased by 0.21%. According to Figure 11c,d, a proportional relationship also exists between angular stiffness K_θ and mass eccentricity d_e. Compared with axial force F_z, mass eccentricity d_e has a negligible effect on the stiffness of ball bearings.

Figure 11. The effect of mass eccentricity on the stiffness of the bearing: (a) radial stiffness K_x; (b) radial stiffness K_y; (c) angular stiffness k_θx; (d) angular stiffness k_θy.

Figure 12 shows the variation trend of mass eccentricity d_e on contact angle α (w = 1000 rad/s, axial force F_z = 1000 N). For each set of mass eccentricity d_e, contact angle α exhibits sinusoidal variation with consistent trends. When position angle φ = 22.5°, contact angle α reaches its maximum value. Contact angle α exhibits a minimum value at φ = 202.5°. When φ = 112.5° and 292.5°, the values of contact angle α are both concentrated around 18.5°. This indicates that contact angle α at this position is less affected by d_e. In terms of amplitude, when mass eccentricity d_e is 0.1 mm, the amplitude of the contact angle reaches 2.3°. With the increase in mass eccentricity d_e, the amplitude of change in contact angle α also increases. According to Figure 9, the effect of mass eccentricity on the contact angle is significantly greater than that of the axial force.

Figure 12. The effect of mass eccentricity on the contact angle of the bearing.

3.3. The Effect of Drive Shaft Speed on the Performance of the Bearing

Figure 13 shows the variation trend of radial stiffness K and angular stiffness K_θ with the increase in drive shaft speed w (mass eccentricity d_e = 0.01 mm, axial force F_z = 1000 N). In Figure 13a,b, radial stiffnesses K_x and K_y exhibit opposite trends. With the increase in drive shaft speed, the minimum value of K_x reaches $6.782\times {10}^7$ N/m at w = 1500 rad/s, and the maximum value of K_y is $1.1588\times {10}^8\mathrm{N}/\mathrm{m}$ at w = 1000 rad/s. It can be seen from Figure 13c,d that both angular stiffnesses K_θx and K_θy simultaneously attain their minimum values at 1500 rad/s. When drive shaft speed w is larger than 1500 rad/s, the angular stiffness exhibits a gradual increase.

Figure 13. The effect of drive shaft speed on the stiffness of the bearing: (a) radial stiffness K_x; (b) radial stiffness K_y; (c) angular stiffness k_θx; (d) angular stiffness k_θy.

Figure 14 shows the effect of drive shaft speed on the contact angle of the ball bearing. As seen from Figure 14, the influence of drive shaft speed on the contact angle exhibits a complex trend, which is related to the moving state of the ball bearing. As rotational speed increases, the radial force grows due to the mass eccentricity of the spindle, leading to an increase in the amplitude of the contact angle. When drive shaft speed w reaches 500 rad/s, the amplitude of contact angle α is approximately 0.005°. When w = 2500 rad/s, the amplitude of contact angle α is approximately 0.09°, representing a 3.4% increase in contact angle α. According to Figure 12, the influence of drive shaft speed on the contact angle is smaller than that of mass eccentricity.

Figure 14. The effect of drive shaft speed on the contact angle of the ball bearing.

4. Conclusions

Based on Hertz elastic contact theory and the principles of rolling bearing geometry and kinematics, this paper derives an analytical expression for the dynamic model of angular contact ball bearings. It considers the elastic fluid dynamic lubrication state between the rolling elements and the inner/outer rings. The effects of varying axial forces, drive shaft rotational speeds, and drive shaft mass eccentricity on ball bearing deformation, stiffness, and contact angle are analyzed. The main conclusions are as follows:

(1)

At the same position angle, both the deformation and stiffness of ball bearings are directly proportional to the axial force. When axial force F_z reaches 1000 N, radial stiffness K_x is increased by 27.9% and K_y is increased by 47.5%. Angular stiffnesses K_θx and K_θy are increased by 78.6% and 84.6%, respectively. Compared to radial stiffness, axial force exerts a more pronounced effect on angular stiffness. Simultaneously, increasing the axial force reduces contact angle amplitude, thereby enhancing the operational stability of ball bearing systems.

(2)

As mass eccentricity d_e of the drive shaft increases, deformation δ exhibits periodic variation. When azimuth φ reaches 90° and 270°, the total deformation tends toward the same value, indicating that deformation δ is less affected by mass eccentricity d_e. When mass eccentricity d_e is 0.1 mm, K_x is increased by 0.13% and K_y is increased by 0.21%. Compared to the axial force, the mass eccentricity of the drive shaft has a relatively minor effect on the stiffness of the ball bearing.

(3)

The effect of drive shaft rotational speed on ball bearing performance is relatively complex. As the drive shaft speed increases, the mass eccentricity of the drive shaft causes the amplitude of the contact angle to grow. When drive shaft rotational speed w reaches 2500 rad/s, the amplitude of contact angle α is approximately 0.09°, representing a 3.4% increase in contact angle α.