Research on Lubrication Characteristics of Ship Stern Bearings Considering Bearing Installation Errors

: The installation quality of a propulsion shaft system directly affects the lubrication statuses of the bearings. The quality of the shaft system installation not only affects the progress of ship construction, but also the safety, stability, and reliability of the shaft system. 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. An 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 lubrication status of the bearing. Poor lubrication and wear faults of the stern bearing are prone to occur at both ends of the stern bearing. As the installation error of the stern bearing increases, the minimum ﬁlm thickness of the stern bearing decreases and the maximum ﬁlm pressure increases, and as the installation error increases, the sensitivity of the aft stern bearing to the vertical installation error is greater than that of the lateral installation error, and the sensitivity of the fore stern bearing to the lateral installation error is greater than that of the vertical installation error. The sensitivity of the lateral and vertical ﬁlm forces at both ends of the aft stern bearing and the fore stern bearing is greater than that of the middle part; the installation error of bearings has a signiﬁcant impact on the lubrication characteristics of bearings.


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 CFDbased 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.
Lubricants 2023, 11, x FOR PEER REVIEW 3 of 15 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.

Bearing Film Thickness Equation Including Installation Errors
The film thickness expression of the bearing lubrication medium is the fundamental equation for solving lubrication characteristics.To facilitate the calculation of parameters such as the bearing journal film thickness and eccentricity, a Cartesian coordinate system is established [19].Using the center of the bearing section in the plane where the axial section of the journal is located as the coordinate origin, Z represents the axial coordinate (bearing length direction), X represents the circumferential coordinate of the bearing, and Y represents the radial coordinate of the bearing (film thickness direction).As shown in Figure 2a,c

Mathematical Model 2.1. Bearing Film Thickness Equation Including Installation Errors
The film thickness expression of the bearing lubrication medium is the fundamental equation for solving lubrication characteristics.To facilitate the calculation of parameters such as the bearing journal film thickness and eccentricity, a Cartesian coordinate system is established [19].Using the center of the bearing section in the plane where the axial section of the journal is located as the coordinate origin, Z represents the axial coordinate (bearing length direction), X represents the circumferential coordinate of the bearing, and Y represents the radial coordinate of the bearing (film thickness direction).As shown in Figure 2a,c, the schematic diagram of the bearing structure without considering installation errors is shown, and γ is the radial inclination angle.The journal rotates clockwise at an angular velocity of ω, and θ is the circumferential angle.O L , O 0 and O z are the centers of the axial end-plane, axial mid-plane, and each axial section, respectively; ϕ 0 and ϕ z are the attitude angle and eccentricity at the axial mid-plane of bearing; and e 0 and e z are the eccentricities of the middle section and each section.By projecting the centers of each axial section along the z-axis direction to the middle section, the eccentricity and offset angle at any section can be obtained based on the structural schematic diagram.As shown in Figure 2b,d From Figure 2b, based on the geometric relationship, the expression for the radial inclination angle γ M and the center distance ' z e of any section along the axial direction, including the vertical installation error, can be obtained as follows: From Figure 2d, based on the geometric relationship, the expressions for the center distance '' z e and offset angle z ϕ ' of any section along the axial direction, including the vertical and lateral installation errors, can be obtained as follows: ( ) As shown in Figure 2a,c, the center distance e z and the offset angle ϕ z of any section along the axial direction can be obtained based on geometric relationships.The expression is as follows: (1) From Figure 2b, based on the geometric relationship, the expression for the radial inclination angle γ M and the center distance e z of any section along the axial direction, including the vertical installation error, can be obtained as follows: From Figure 2d, based on the geometric relationship, the expressions for the center distance e z and offset angle ϕ z of any section along the axial direction, including the vertical and lateral installation errors, can be obtained as follows: Generally, γ M is smaller, and it can be considered that tan γ M ≈ γ M .Therefore, considering the inclination of the journal, the film thickness h of any cross section can be expressed as follows: c is the bearing clearance.

Reynolds Equation and Boundary Conditions for Bearings
The Reynolds equation can be derives using the N-S equation and continuity equation [20]: In the formula, U 1 and U 2 are the velocities of the upper and lower surfaces of the journal, ρ is the density of the lubricating medium, h is the thickness of the lubricating medium, and p is the pressure of the lubricating medium.
It can be simplified to the Reynolds equation of the equal viscosity density and isothermal only for the pressure flow and shear flow, as shown below [21].
The mathematical expression for the rupture point and regeneration position of the film is shown in Equation ( 10) [22]: In the formula, n is the normal vector of the boundary; V is the movement speed of the lubricating oil; and Z is the volume ratio of the film in the rupture zone.
The dimensionless definition of the main variable is shown in Equation ( 11): In the formula, P is the dimensionless film pressure; H is the dimensionless film thickness; λ is the dimensionless axial coordinate; L is the length of the bearing; and θ is the dimensionless circumferential coordinate.
By substituting the above variables into Equation ( 8), the dimensionless form of the Reynolds equation can be obtained as follows:

Bearing Friction Coefficient
Using the Reynolds boundary conditions, and assuming that the film ruptures at angle θ c , the pressure boundary condition is as follows [18]: The bearing capacity components of the film in the lateral and vertical directions are In the formula, F x is the film force in the vertical direction, and F y is the film force in the lateral direction.
The bearing capacity and its deviation angle are The expression for the shear stress acting on the surface of the journal is The dimensionless friction force F f and friction coefficient f on the surface of the journal are

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 A i,j , B i,j , C i,j , D i,j , E i,j , F i,j , A i,j P i+1,j + B i,j P i−1,j + C i,j P i,j+1 Order: If J/T < 1 × 10 5 , the calculation is completed; otherwise, the iteration calculation is repeated.
, 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 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.

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.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 max- 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.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 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 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   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.
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 Figures 7 and 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.

Conclusions
The installation errors of bearings have significant impacts on the lubrication characteristics of bearings.Based on the results obtained, the main conclusions are as follows: 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 Figures 7 and 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.

Conclusions
The installation errors of bearings have significant impacts on the lubrication characteristics of bearings.Based on the results obtained, the main conclusions are as follows: (1) 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 lubrication status of the bearing.Poor lubrication and wear faults of the stern bearing are prone to occur at both ends of the stern bearing.(2) 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 impacts on the minimum film thickness reduction and maximum film pressure become increasingly significant.(3) The sensitivity of the aft stern bearing to vertical installation error is greater than that to lateral installation error, while the sensitivity of the fore stern bearing to lateral installation error is greater than that to vertical installation error.These phenomena are caused by the influence of the propeller load on the aft stern shaft.(4) The sensitivities of the lateral and vertical film forces at both ends of the aft stern bearing and the fore stern bearing are greater than that of the middle part.

Figure 1 .
Figure 1.Schematic diagram of installation error of ship shafting bearings.
, the schematic diagram of the bearing structure without considering installation errors is shown, and γ is the radial inclination angle.The journal rotates clockwise at an angular velocity of ω , and θ is the circumferential angle.L O , 0 O and z O are the centers of the axial end-plane, axial mid-plane, and each axial section, respectively; 0 ϕ and z ϕ are the attitude angle and eccentricity at the axial mid-plane of bearing; and 0 e and z e are the eccentricities of the middle section and each section.By projecting the centers of each axial section along the z-axis direction to the middle section, the eccentricity and offset angle at any section can be obtained based on the structural schematic diagram.As shown in Figure 2b,d, the schematic diagram of the bearing structure with installation errors included is shown.The red coordinate axis represents the Cartesian coordinate system with installation errors included.Figure 2b shows a schematic diagram of the bearing that includes the vertical installation error, v IE is the vertical installation error of the bearing, Figure 2d is a schematic diagram of the bearing that includes the lateral installation errors, and h IE is the lateral installation error of the bearing.

Figure 1 .
Figure 1.Schematic diagram of installation error of ship shafting bearings.

Figure 2 .
Figure 2. Schematic diagram of bearing structure with installation errors included.(a,c) No installation error; (b) vertical installation error; (d) lateral installation error.

Figure 2 .
Figure 2. Schematic diagram of bearing structure with installation errors included.(a,c) No installation error; (b) vertical installation error; (d) lateral installation error.

Figure 3 .
Figure 3. Flow chart for numerical solution of equation.

Figure 3 .
Figure 3. Flow chart for numerical solution of equation.
Lubricants 2023, 11, x FOR PEER REVIEW 9 of 15 wear failure of the stern bearing caused by the installation error of the stern bearing occur at both ends of the stern bearing.

Figure 4 .
Figure 4. Cloud chart of film pressure distribution of stern bearing under different vertical installation errors: (a)

Lubricants 2023 , 15 Figure 5 .
Figure 5. Cloud chart of film pressure distribution of stern bearing under different lateral installation errors: (a)

Figure 6 .
Figure 6.Effects of installation error on minimum film thickness and maximum film pressure of stern bearing: (a,b): Aft stern bearing installation error; (c,d): Fore stern bearing installation error.

Figure 6 .
Figure 6.Effects of installation error on minimum film thickness and maximum film pressure of stern bearing: (a,b): Aft stern bearing installation error; (c,d): Fore stern bearing installation error.
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.Lubricants 2023, 11, x FOR PEER REVIEW 12 of 15 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

Figure 7 .
Figure 7. Effects of installation error on lateral and vertical film forces of aft stern bearings: (a,b): v IE ; (c,d): h IE .

Figure 7 .
Figure 7. Effects of installation error on lateral and vertical film forces of aft stern bearings: (a,b): IE v ; (c,d): IE h .

Figure
Figure7a,b show the effects of vertical installation errors on the vertical and lateral film forces of the aft stern bearing, respectively.Figure7c,dshow the effects of lateral installation errors on the vertical and lateral film forces of the aft stern bearing.From Figure7, 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 Figure7a,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 Figure7c,d, it can be seen that as the

Figure 8 .
Figure 8. Effects of installation error on lateral and vertical film forces of fore stern bearings: (a,b): v IE ; (c,d): h IE .

Figure 8 .
Figure 8. Effects of installation error on lateral and vertical film forces of fore stern bearings: (a,b): IE v ; (c,d): IE h .

Table 1 .
Comparison of calculation results for maximum film pressure of bearings at different tilt angles.