1. Introduction
Ship manufacturing is the process of transforming designed and drawn ship drawings into real ships while ensuring their operational performances under the control of normal technical indicators, including shaft alignment, the installation of shaft systems, propellers, etc. During the ship construction phase, the installation quality of the propulsion shaft system directly affects the lubrication statuses of the bearings. The quality of the shaft system installation not only affects the ship construction progress, but also the safety, stability, and reliability of the shaft system [
1,
2,
3]. During the trial voyage of a 22,000 m
3 liquefied gas ship in 2004, the installation error of its stern bearing caused serious damage to the inner surface of the fore stern bearings of the stern tube [
4]. Therefore, studying the sensitive relationship between bearing installation error and the bearing film thickness, and clarifying the impacts of relevant parameters on the bearing lubrication performance, is of great significance to accurately analyze the bearing lubrication status, ensuring the safe, stable, and reliable operation of the shaft system.
Currently, many scholars have conducted in-depth research in the fields of manufacturing errors and the lubrication of inclined bearings [
5]. Li et al. [
6] studied the effects of manufacturing errors and micro grooves on the static, dynamic, and stability characteristics of water-lubricated radial bearings (WLJBs). The mathematical expressions for manufacturing errors and surface micro grooves were given, and the steady-state and non-stationary Reynolds equations were calculated using the linear perturbation method and the finite difference technique. Hailong C et al. [
7] conducted numerical and experimental studies on the influences of manufacturing errors on the static performances of aerostatic porous journal bearings and simulated the effects of circumferential waviness, taper, and concavity on the bearing film thickness. Zoupas L et al. [
8] studied the impacts of manufacturing errors on the performances of large tilting pad thrust bearings after conducting a CFD-based thermal fluid dynamics analysis of individual bearing pads. Jianlin C et al. [
9] used a numerical model to investigate the effects of wear and shaft shape error defects on the frictional dynamic response of water-lubricated bearings under nonlinear propeller disturbances. The above scholars analyzed the impacts of manufacturing parameter errors on bearing lubrication characteristics.
Qilin L et al. [
10] established a transient fluid structure coupling dynamic model for bearings while considering bidirectional shaft inclination and lining deformation and explored the effects of the eccentricity ratio, shaft inclination angle, and bearing structural parameters on lubrication and dynamic characteristics. Xingxin L et al. [
11] proposed a mixed lubrication (ML) deterministic model based on the unified Reynolds equation model to investigate the effect of surface roughness on the lubrication performances of elastic-supported water-lubricated tilting pad thrust bearings. Lili W et al. [
12] established a mathematical model while considering shape and position errors, studied the lubrication performances of hybrid ceramic radial bearings with four concave capillary flow limiters with different journal inclination angles and cylindricities, and analyzed the effect of eccentricity on film lubrication performance. Jan P [
13] proposed a two-dimensional calculation model based on the theory of thermal elastic deformation lubrication, which uses the Newton method to calculate the equilibrium tilt of the backing plate. The above scholars analyzed the influences of tilt bearing parameters on bearing lubrication characteristics.
The stern bearing of the ship’s propulsion system may have obvious vibration and noise problems caused by friction in the event of poor lubrication [
14]. Zhen Y et al. [
15] used the fluid solid coupling (FSI) method to analyze the elastohydrodynamic lubrication performances of bearing sleeves and studied the effects of the eccentricity ratio, rotational speed, elastic modulus distribution, and thickness distribution of bearing sleeves on their lubrication performances. Changxin L et al. [
16] proposed the concept of end offset of the outer ring and studied the relationship between the forming accuracy of self-lubricating spherical sliding bearings and bearing life. Based on the influence of offset on the bearing contact stress and strain, the reasonable range of offset was analyzed. Biao L et al. [
17] studied the journal bearing system and established a hydrodynamic lubrication model for radial bearings that considers axial motion and the misalignment of the journal. The axial motion of the journal has a significant impact on the lubrication characteristics of misaligned journal bearings. Liangyan Z [
18] studied the effect of stress couple on the lubrication performances of inclined bearings. The above scholars studied the impacts of the bearing sleeve, outer ring end offset, and misalignment on bearing lubrication performance, but there are few studies on the lubrication characteristics of inclined bearings that consider bearing installation errors.
This article takes sliding bearings in ship shafting as the research object and establishes a hydrodynamic lubrication model of sliding bearings while considering installation errors to address the issue of installation errors of ship stern bearings. The finite difference method and super-relaxation iteration method are used to solve the problem, and the influences of bearing installation errors on bearing lubrication characteristic parameters are explored.
Figure 1 shows a schematic diagram of the installation error of ship shaft bearings, defining the direction of the Cartesian coordinate system. The Z-axis represents the coordinate of the shaft length direction, the X-axis represents the coordinate of the shaft circumference direction, and the Y-axis represents the radial coordinate of the shaft system.
Figure 1 introduces two types of bearing installation errors: vertical installation error and lateral installation error. Vertical installation error refers to the installation error of the bearing in the Y-axis direction, and lateral installation error refers to the installation error of the bearing in the X-axis direction.
3. Numerical Solution and Model Validation
Using the finite difference method to discretize the Reynolds equation, a circumferential and axial grid density of 120 × 60 is selected, and the bearing film is solved force using the Simpson method. The solution process is shown in
Figure 3. First, the initial film pressure is given, and the dimensionless film thickness equation for the inclined bearings is solved using the finite difference method. Then, the bearing film pressure is solved using the Newtonian fluid lubrication model. The specific difference equation is
Equations (18)–(20) are substituted into the dimensionless Reynolds equation, where the discrete coefficients are
,
,
,
,
,
,
If
, the calculation is completed; otherwise, the iteration calculation is repeated.
To verify the correctness of the fluid lubrication model for inclined bearings including the installation errors established in this article, a bearing example is selected for the verification of the inclined bearing lubrication program that includes installation errors. Due to the lack of publicly published articles on the lubrication characteristics of inclined bearings that include installation errors, the calculation results of the inclined bearings in reference [
23] were compared using the bearing data from reference [
23] (radius of 30 mm, bearing width of 66 mm, radius clearance of 0.03 mm, rotational speed of 3000 r/m, and lubricating oil viscosity of 0.009 Pa·s) and compared with the calculation results of references [
10,
18,
23]; the results are shown in
Table 1. From
Table 1, it can be seen that the error between the calculation results of the bearing lubrication model established in this paper and the results of reference [
10] is within 5%, the error between the results of reference [
18] is within 1%, and the error between the results of reference [
23] is within 4%. Therefore, the bearing lubrication model established in this paper has high accuracy.
4. Calculation and Analysis
Taking the stern bearing of a large ship’s shaft system as the research object, the lubrication characteristics of the bearing, including the bearing installation errors, are analyzed. As shown in
Figure 1, the stern bearing of the shaft system consists of one fore stern bearing and one aft stern bearing, and the bearing parameters are shown in
Table 2.
Figure 4 shows the cloud diagram of the film pressure distribution of the stern bearing under different vertical installation errors. As the vertical installation error of the stern bearing gradually increases, the maximum film pressure of the stern bearing gradually increases. From
Figure 4, it can be seen that the maximum film pressure position of the stern bearing shifts from the center position of the axial distance to the position with a smaller axial distance. This indicates that the force bearing point of the stern shaft shifts towards the position with a smaller axial distance, and the load on the stern bearing increases at the position with a smaller axial distance. The lubrication state at this position decreases, making it more prone to dry friction and causing bearing wear and other fault problems.
Figure 5 shows the cloud diagram of the film pressure distribution of the stern bearing under different lateral installation errors. As the lateral installation error of the stern bearing gradually increases, the maximum film pressure of the stern bearing gradually increases. From
Figure 5, it can be seen that the maximum film pressure position of the stern bearing shifts from the center position of the axial distance to the position with a larger axial distance. This indicates that the force bearing point of the stern shaft shifts towards the position with a larger axial distance, and the load on the stern bearing increases at the position with a larger axial distance. The lubrication state at this position decreases, making it more prone to dry friction and causing bearing wear and other fault problems.
In summary, the installation error of the stern bearing can lead to an increase in the film pressure at both ends of the stern bearing in the axial direction, leading to a decrease in the bearing lubrication status. This indicates that problems such as poor lubrication and wear failure of the stern bearing caused by the installation error of the stern bearing occur at both ends of the stern bearing.
Figure 6 shows the impacts of installation errors on the minimum film thickness and maximum film pressure of the stern bearing. The solid line represents the impacts of vertical installation errors on the bearing, and the dashed line represents the impacts of lateral installation errors on the bearing.
Figure 6a shows the effects of installation error of the aft stern bearing on the minimum film thickness and maximum film pressure of the aft stern bearing.
Figure 6b shows the changes in the minimum film thickness and maximum film pressure of the fore stern bearing caused by the installation error of the aft stern bearing.
Figure 6c shows the changes in the minimum film thickness and maximum film pressure of the aft stern bearing caused by the installation error of the fore stern bearing.
Figure 6d shows the effects of the installation error of the fore aft bearings on the minimum film thickness and maximum film pressure of the fore aft bearings. From
Figure 6, it can be seen that as the installation error of the stern bearing increases, the minimum film thickness of the stern bearing decreases and the maximum film pressure increases. Moreover, as the installation error increases, its impact on the minimum film thickness decreases and the maximum film pressure becomes increasingly significant.
From
Figure 6a,b, it can be seen that as the installation error increases, the impact of vertical installation error on the maximum film pressure is greater than that of the lateral installation error. This indicates that the sensitivity of the stern bearing to vertical installation error is greater than that to lateral installation error; from
Figure 6c,d, it can be seen that as the installation error increases, the influence of lateral installation error on the maximum film pressure is greater than that of vertical installation error. This indicates that the sensitivity of the fore stern bearings to lateral installation error is greater than that to vertical installation error; the reason for this phenomenon is the influence of the propeller load on the aft stern shaft.
From
Figure 6a,b, it can be seen that as the installation error increases, the vertical installation error of the aft stern bearing has a greater impact on the maximum oil film pressure than the lateral installation error, indicating that the sensitivity of the aft stern bearing to vertical installation error is greater than the lateral installation error; from
Figure 6c,d, it can be seen that as the installation error increases, the influence of the lateral installation error of the fore stern bearing on the maximum oil film pressure is greater than that of vertical installation error, indicating that the sensitivity of the fore stern bearings to lateral installation error is greater than that to vertical installation error. The reason why the vertical installation error of the aft stern bearing has a significant impact on the large oil film pressure is that the aft stern shaft bears the influence of the propeller load, which puts a large vertical load on the aft stern bearing. When the shaft system is working, some deformation may occur due to weight, and the impact of this deformation on bearing installation error should not be ignored. The reason is that the wear of ship shaft system bearings is caused by changes in the bearing load and axis state, which causes damage to the lubrication states of the bearings, further leading to bearing wear and vibration faults. The factors that affect the statuses of the shaft axis and bearing load include the quality of shaft alignment and the influence of multiple sources of loads such as environment and navigation motion, among which installation error is the key factor for the quality of shaft alignment. At the same time, due to the influence of the propeller load, the load on the stern bearing is relatively high, and installation errors have greater impacts on the stern bearing, causing bearing wear problems to be more common in the stern bearing. Therefore, the impacts of installation errors on bearing lubrication characteristics should not be ignored.
Figure 7 shows the effects of installation error on the lateral and vertical film forces of the aft stern bearing, while
Figure 8 shows the effects of installation error on the lateral and vertical film forces of the fore stern bearing. The lateral axis represents the axial distance of the bearing, where 0 to 1 represents the position of the stern bearing, 0 represents the leftmost position of the bearing, and 1 represents the rightmost position of the bearing. The vertical axis represents the magnitudes of lateral and vertical film forces, with positive and negative indicating the directions of the film forces.
Figure 7a,b show the effects of vertical installation errors on the vertical and lateral film forces of the aft stern bearing, respectively.
Figure 7c,d show the effects of lateral installation errors on the vertical and lateral film forces of the aft stern bearing. From
Figure 7, it can be seen that the film forces at both ends of the stern bearing vary significantly, indicating that the sensitivities of the lateral and vertical film forces at both ends of the stern bearing are greater than that of the middle part. From
Figure 7a,b, it can be seen that as the vertical installation error increases, the vertical film force of the aft stern bearing gradually increases between 0 and 0.3, and decreases between 0.3 and 1.0. The lateral film force of the aft stern bearing gradually increases between 0 and 1. From
Figure 7c,d, it can be seen that as the lateral installation error increases, the vertical film force of the aft stern bearing gradually decreases between 0 and 0.5, and increases between 0.5 and 1.0. The lateral film force of the aft stern bearing gradually increases between 0 and 1.
Figure 8a,b show the effects of vertical installation errors on the vertical and lateral film forces of the fore stern bearings, respectively.
Figure 8c,d show the effects of lateral installation errors on the vertical and lateral film forces of the fore stern bearings. From
Figure 8, it can be seen that the film forces at both ends of the fore stern bearings vary significantly, indicating that the sensitivities of the lateral and vertical film forces at both ends of the fore stern bearings are greater than that of the middle part. From
Figure 8a,b, it can be seen that as the vertical installation error increases, the vertical film force of the fore stern bearings gradually increases between 0 and 0.4, and decreases between 0.4 and 1.0. The lateral film force of the fore stern bearings gradually increases between 0 and 1. From
Figure 8c,d, it can be seen that as the lateral installation error increases, the vertical film force of the fore stern bearings first decreases and then increases between 0 and 0.5, and gradually increases between 0.5 and 1.0.
Figure 8a,b show the effects of vertical installation errors on the vertical and lateral film forces of the fore stern bearings, respectively.
Figure 8c,d show the effects of lateral installation errors on the vertical and lateral film forces of the fore stern bearings. From
Figure 8, it can be seen that the film forces at both ends of the fore and stern bearings vary significantly, indicating that the sensitivities of the lateral and vertical film forces at both ends of the fore stern bearings are greater than that of the middle part. From
Figure 8a,b, it can be seen that as the vertical installation error increases, the vertical film force of the fore stern bearings gradually increases between 0 and 0.4, and decreases between 0.4 and 1.0. The lateral film force of the fore stern bearings gradually increases between 0 and 1. From
Figure 8c,d, it can be seen that as the lateral installation error increases, the vertical film force of the fore stern bearings first decreases and then increases between 0 and 0.5, and gradually increases between 0.5 and 1.0. The lateral film force of the fore stern bearings first decreases and then increases between 0 and 0.6, and gradually increases between 0.6 and 1.0. From
Figure 7 and
Figure 8, it can be seen that the horizontal oil film force of the bearing changes direction with the increase in installation error, and a horizontal installation error is more likely to cause changes in the direction of the bearing’s horizontal oil film force.