A method for calculating mixed viscoelastic flow lubrication performance of a dynamic load bearing under a journal tilt

Abstract

The application discloses a kind of dynamic load bearing mixed viscoelastic flow lubrication performance calculation method under journal neck inclination, which solves the problem that bearing mixed lubrication performance is calculated by traditional method under the working condition of journal neck inclination of dynamic load sliding bearing system, and the bearing viscoelastic deformation effect caused by significant load fast change is ignored;The application first comprehensively considers oil film cavitation and friction interface surface topography effect, establishes mass conservation average Reynolds equation, then uses two journal neck inclination angles in horizontal and vertical directions to represent the inclination state of bearing, calculates the bearing viscoelastic deformation under dynamic load by using standard linear solid model, corrects the oil film thickness distribution under the influence of bearing viscoelastic deformation and journal neck inclination, finally constructs journal motion equation, updates journal eccentricity and eccentricity velocity by using Newmark method, and iteratively calculates dynamic load bearing mixed viscoelastic flow lubrication performance parameters.

Description

Technical Field

[0001] This invention belongs to the technical field of solving the lubrication performance of dynamic load sliding bearings, specifically relating to a method for calculating the lubrication performance of dynamic load bearings with mixed viscoelastic flow under journal tilt. Background Technology

[0002] The shaft-dynamic load sliding bearing coupling system is a key support transmission system ensuring the integrity of mechanical equipment performance and is widely used in various reciprocating and rotating machinery. During the actual operation of the dynamic load sliding bearing system, journal tilting is unavoidable due to factors such as machining errors, assembly errors, and shaft deformation under load. The presence of journal tilting alters the oil film pressure and thickness distribution of the lubricating oil, increasing friction between the bearing and journal surfaces. In severe cases, it can exacerbate bearing surface wear and even cause system oscillation.

[0003] Under heavy loads, dynamically loaded sliding bearings have small eccentricities and are in a mixed lubrication state of oil film, boundary friction, and dry friction for extended periods. Existing theories of mixed elastohydrodynamic lubrication analyze the lubrication performance of dynamically loaded sliding bearings, using the fully elastic deformation of the bearing surface under high-pressure oil film to correct for oil film thickness distribution and calculate the mixed lubrication performance under dynamic loads. However, metallic materials have been shown to possess viscoelasticity, and the deformation of the bearing surface under load exhibits a coupling of viscosity and elasticity, i.e., the viscoelastic deformation effect. Mixed lubrication performance parameters such as oil film pressure, oil film thickness, rough contact pressure, frictional power consumption, and end leakage flow are core parameters in the lubrication problem of dynamically loaded sliding bearings. Using complete elasticity as the bearing deformation criterion under pressure easily overlooks the initial impact and rough rubbing characteristics of bearing instability under journal tilting conditions, making it difficult to accurately calculate the lubrication parameters of dynamically loaded bearings under harsh operating conditions and to understand the bearing lubrication status. This, in turn, affects the structural optimization and design improvement of dynamically loaded sliding bearings and the monitoring of lubrication performance.

[0004] Therefore, given the unclear lubrication characteristics of hybrid viscoelastic flow in dynamically loaded bearings under journal tilt and the lack of theoretical basis for complex coupled physical field numerical analysis, this invention provides a precise and efficient numerical calculation method for solving the lubrication performance of hybrid viscoelastic flow in dynamically loaded bearings under journal tilt. Summary of the Invention

[0005] The purpose of this invention is to provide a method for calculating the mixed viscoelastic flow lubrication performance of dynamically loaded bearings under journal tilting conditions and the viscoelastic deformation effect caused by pressure on the surface of the sliding bearing under dynamic load.

[0006] The objective of this invention is achieved through the following technical solution:

[0007] First, determine the geometric and material parameters of the journal and bearing, the physical parameters of the lubricating oil, set the initial eccentricity of the journal, the initial eccentricity velocity, the journal tilt angle in the horizontal and vertical directions, and determine the oil inlet pressure.

[0008] Secondly, taking into account the effects of oil film cavitation and friction interface surface morphology, a mass-conserved average Reynolds equation is established. The Reynolds equation is discretized using the finite difference method, and the oil film pressure is solved using an over-relaxation iterative algorithm.

[0009] Next, the viscoelastic deformation of the bearing surface under high-pressure oil film is solved using the standard linear solid model, and the viscoelastic deformation equation of the sliding bearing at any time is constructed.

[0010] Next, the journal tilting state is characterized by two journal tilting angles in the horizontal and vertical directions, and the oil film thickness distribution under viscoelastic deformation and journal tilt correction is established.

[0011] Finally, the journal motion equation is constructed by combining the dynamic load of the sliding bearing, the oil film force, and the rough contact force, and the journal eccentricity and eccentricity velocity are updated to iteratively calculate the lubrication performance parameters.

[0012] A method for calculating the lubrication performance of a dynamically loaded bearing under journal tilting using a mixed viscoelastic flow, characterized by the following steps:

[0013] The first step is to set the initial parameters of the dynamic load sliding bearing system.

[0014] (1) Determine the radius, length, elastic modulus, Poisson's ratio, and physical parameters of the lubricating oil of the journal and bearing in the dynamic load sliding bearing system;

[0015] (2) Set the initial eccentricity and eccentric velocity of the journal. For the dynamic load sliding bearing system, the initial eccentricity is selected from 0 to c (where c is the radial clearance), and the initial eccentric velocity of the sliding bearing system is set to 0 at the initial moment.

[0016] (3) The journal tilt is characterized by the tilt angles in the horizontal and vertical directions, and the oil film pressure distribution is initialized by the oil inlet pressure.

[0017] The second step is to establish the lubrication control equations.

[0018] This invention addresses the roughness effect statistically, using pressure-flow factor and shear-flow factor to characterize the influence of surface roughness on lubrication characteristics. Simultaneously, to account for the impact of lubricating oil film cavitation on lubrication characteristics, a mass-conserved average Reynolds equation is established by introducing the fluid fraction θ using the mass conservation equation.

[0019]

[0020] Where p represents oil film pressure, h represents oil film thickness, and h T This represents the average oil film thickness, where x represents the circumferential coordinate and z represents the axial coordinate. φ x and φ zφ represents the pressure-flow factor in the x and z directions, respectively. s This represents the shear flow factor, and U represents the journal tangential velocity. σ represents the standard deviation of the overall surface roughness of the bearing journal. j and σ b denoted by , and t represents time, respectively, for the surface roughness of the journal and bearing.

[0021] μ is the dynamic viscosity of the lubricating oil calculated using the Barus viscosity-pressure equation, as follows:

[0022]

[0023] Where μ0 represents the dynamic viscosity of lubricating oil under normal pressure.

[0024] Equation (1) is discretized using the second-order central difference scheme of the finite difference method, and then the oil film pressure and fluid fraction are solved using the over-relaxation iterative method:

[0025]

[0026] Where i and j represent the circumferential and axial mesh node numbers, respectively, n represents the time loop step, and k represents the iteration convergence step. and Let represent the oil film pressures obtained by node (i, j) in the k-th and (k-1)-th iterations at the n-th time step, respectively. and Let w represent the fluid fractions obtained by node (i, j) in the k-th and (k-1)-th iterations at the n-th time step. p The over-relaxation parameter representing oil film pressure, w θ The over-relaxation parameter represents the fluid fraction.

[0027] To ensure the convergence of oil film pressure and fluid fraction during the iteration process, the convergence criterion is as follows:

[0028]

[0029] Where N1 and N2 represent the number of nodes in the circumferential and axial directions, respectively, ζ p ζ represents the convergence tolerance of the oil film pressure. θ This represents the convergence tolerance of the fluid fraction.

[0030] The third step is to construct the viscoelastic deformation equation for the bearing.

[0031] The viscoelastic deformation mechanism of a dynamically loaded sliding bearing is characterized using a standard linear solid model, and its equations are as follows:

[0032]

[0033] Where δ(x,z,t) represents the actual deformation of the bearing at the time of this simulation, δ t (x,z,t) represents the fully elastic deformation of the bearing at the time of this simulation, δ l (x,z,t-△t) represents the bearing time-delay deformation at the previous simulation time. τ represents the complete time-delay deformation of the bearing under oil film pressure p at the previous simulation moment, τ represents the viscoelastic relaxation time of the bearing material, q represents the total number of simulation time steps, and Δt represents the time step.

[0034] The fully elastic deformation of the sliding bearing is solved using the Winkler method:

[0035]

[0036] Among them, v b With E b represents the Poisson's ratio and elastic modulus of the bearing, respectively, and l represents the thickness of the sliding bearing.

[0037] The fourth step is to solve for the oil film thickness distribution.

[0038] Considering bearing clearance, eccentricity, journal tilt, and viscoelastic deformation of the bearing, the oil film thickness of a dynamically loaded sliding bearing is:

[0039] h = h0 + h mis +h v (7)

[0040] Where h0 is the oil film thickness component considering the radial clearance between the bearing and the journal, as well as the journal eccentricity, h mis h is the oil film thickness component for journal tilt correction. v The oil film thickness component, h, is the correction for the viscoelastic deformation of the bearing. v =δ(x,z,t).

[0041] The equation for h0 is as follows:

[0042]

[0043] Where c is the radial clearance, X is the eccentricity of the journal center in the x-direction, Y is the eccentricity of the journal center in the y-direction, and y is the coordinate of the oil film thickness direction. This is the journal offset angle.

[0044] This invention employs two tilt angles γ, one horizontal and one vertical. x With γ y To characterize the journal tilting effect, the oil film thickness component equation for journal tilt correction is obtained based on the geometric position relationship of the journal in the bearing hole when the journal is tilted.

[0045]

[0046] Among them, R b B is the bearing radius, and B is the bearing width.

[0047] From formula (9), it can be seen that when γ x =γ y When h = 0, mis =0, at this time formula (7) is transformed into the oil film thickness when the journal is not tilted.

[0048] The fifth step is to establish the journal motion equation.

[0049] The lubrication control equation (1) uses the journal motion equation to iteratively update the eccentricity and eccentricity given the initial eccentricity and eccentricity, thereby correcting the oil film thickness and oil film pressure. In this paper, the journal's center of mass is assumed to be an effective mass particle. The journal motion equation in the dynamic load sliding bearing system is expressed as follows:

[0050]

[0051] Where, m j Indicates journal mass. and These represent the eccentric accelerations in the x and y directions, respectively. and These represent the dynamic loads acting on the sliding bearing system in the x and y directions, respectively. and These represent the oil film forces in the x and y directions, respectively. and These represent the rough contact forces in the x and y directions, respectively.

[0052] In mixed lubrication conditions, oil film pressure and rough contact pressure constitute the oil film force and rough contact force of the bearing, respectively. The oil film force and rough contact force can be obtained by using the complex trapezoidal integral of the pressure:

[0053]

[0054]

[0055] Where, p asp This indicates rough contact pressure.

[0056] The Newmark method is used to solve equation (10) iteratively to calculate the eccentricity that minimizes the difference between the external load and the calculated support load. The convergence criterion used is:

[0057]

[0058] Among them, X n With X n-1Represent the eccentricity in the x-direction calculated at time steps n and n-1, respectively, and the eccentricity in the y-direction. n With Y n-1 ζ represents the eccentricity in the y-direction calculated at time steps n and n-1, respectively. X Let ζ be the convergence tolerance of X. Y Let Y be the convergence tolerance.

[0059] Step 6: Calculate the performance parameters of the mixed lubrication system.

[0060] The mixed lubrication performance of a dynamically loaded sliding bearing under journal tilting conditions includes tilting torque, frictional power consumption, and end leakage flow.

[0061] For bearings with tilted journals, the oil film pressure on both sides of the bearing's central cross-section is asymmetrical. To ensure stable bearing operation, a corresponding torque needs to be applied to the bearing. The tilting torque components in the x and y directions are:

[0062]

[0063] The resultant moment of tilt is:

[0064]

[0065] Under mixed lubrication conditions, the frictional force F f Viscous friction caused by liquid shear Rough contact friction caused by contact with rough peaks It consists of two parts, and the expression is:

[0066]

[0067]

[0068]

[0069] Where κ = 0.02 is the boundary friction coefficient, and φ f For shear flow factor, φ fs For shear stress factor and φ fp This represents the frictional pressure flow factor.

[0070] φ f φ fs and φ fp The following formula is used for calculation:

[0071]

[0072]

[0073]

[0074] Where Λ=h / σ represents the film thickness ratio, g=Λ / 3 is the film thickness judgment index, and e is the natural constant.

[0075] Frictional power consumption is equal to the product of frictional force and tangential velocity.

[0076] P f =|F f U| (22)

[0077] Among them, F f The friction force is calculated by equation (16), where U is the journal tangential velocity.

[0078] When the journal tilts, the end leakage flows Q1 and Q2 of the bearing front and rear faces can be calculated as follows:

[0079]

[0080]

[0081] The total discharge flow rate is:

[0082] Q = Q1 + Q2 (25) Attached Figure Description

[0083] Figure 1 Flowchart of this invention

[0084] Figure 2 Flowchart of the calculation method for the mixed viscoelastic flow lubrication performance of dynamically loaded bearings under journal tilt

[0085] Figure 3 Schematic diagram of journal tilt in dynamic load sliding bearing system

[0086] Figure 4 Steady-state distribution of oil film pressure under different journal tilt angles

[0087] Figure 5 Steady-state distribution of rough contact pressure under different journal tilt angles

[0088] Figure 6 Maximum oil film pressure curves at different journal tilt angles

[0089] Figure 7 Maximum rough contact pressure curves under different journal tilt angles

[0090] Figure 8 Minimum oil film thickness curves at different journal tilt angles

[0091] Figure 9 Inclination moment curves at different journal tilt angles

[0092] Figure 10Friction power consumption curves at different journal tilt angles

[0093] Figure 11 Discharge curves at the lower end of journals with different tilt angles Detailed Implementation

[0094] To better understand the technical solution of the present invention, the specific embodiments of the present invention will be further described in detail below with reference to the accompanying drawings.

[0095] Figure 1 This is a flowchart of the present invention. (For example...) Figure 2 The diagram shows a flowchart of a method for calculating the lubrication performance of a sliding bearing. This invention provides a method for calculating the lubrication performance of a dynamically loaded bearing under journal tilt, comprising the following steps:

[0096] The first step is to set the initial parameters of the dynamic load sliding bearing system.

[0097] (1) Determine the radius, length, elastic modulus, Poisson's ratio, and physical parameters of the lubricating oil of the journal and bearing in the dynamic load sliding bearing system;

[0098] (2) Set the initial eccentricity and eccentric velocity of the journal. For the dynamic load sliding bearing system, the initial eccentricity is selected from 0 to c (where c is the radial clearance), and the initial eccentric velocity of the sliding bearing system is set to 0 at the initial moment.

[0099] (3) The journal tilt is characterized by tilt angles in both horizontal and vertical directions. In this invention, the inlet pressure is 347 kPa, and the oil film pressure distribution is initialized based on the inlet pressure. Figure 3 This is a schematic diagram of the journal tilting state of a dynamic load sliding bearing system.

[0100] The second step is to establish the lubrication control equations.

[0101] This invention addresses the roughness effect statistically, using pressure-flow factor and shear-flow factor to characterize the influence of surface roughness on lubrication characteristics. Simultaneously, to account for the impact of lubricating oil film cavitation on lubrication characteristics, a mass-conserved average Reynolds equation is established by introducing the fluid fraction θ using the mass conservation equation.

[0102]

[0103] Where p represents oil film pressure, h represents oil film thickness, and h T This represents the average oil film thickness, where x represents the circumferential coordinate and z represents the axial coordinate. φ x and φ z φ represents the pressure-flow factor in the x and z directions, respectively. s This represents the shear flow factor, and U represents the journal tangential velocity. σ represents the standard deviation of the overall surface roughness of the bearing journal. j and σ b denoted by , and t represents time, respectively, for the surface roughness of the journal and bearing.

[0104] μ is the dynamic viscosity of the lubricating oil calculated using the Barus viscosity-pressure equation, as follows:

[0105]

[0106] Where μ0 represents the dynamic viscosity of lubricating oil under normal pressure.

[0107] Equation (1) is discretized using the second-order central difference scheme of the finite difference method, and then the oil film pressure and fluid fraction are solved using the over-relaxation iterative method:

[0108]

[0109] Where i and j represent the circumferential and axial mesh node numbers, respectively, n represents the time loop step, and k represents the iteration convergence step. and Let represent the oil film pressures obtained by node (i, j) in the k-th and (k-1)-th iterations at the n-th time step, respectively. and Let w represent the fluid fractions obtained by node (i, j) in the k-th and (k-1)-th iterations at the n-th time step. p The over-relaxation parameter representing oil film pressure, w θ The over-relaxation parameter represents the fluid fraction.

[0110] To ensure the convergence of oil film pressure and fluid fraction during the iteration process, the convergence criterion is as follows:

[0111]

[0112] Where N1 and N2 represent the number of nodes in the circumferential and axial directions, respectively, ζ p ζ represents the convergence tolerance of the oil film pressure. θ This represents the convergence tolerance of the fluid fraction. The steady-state distribution of oil film pressure at different journal tilt angles is shown below. Figure 4 As shown.

[0113] The third step is to construct the viscoelastic deformation equation for the bearing.

[0114] The viscoelastic deformation mechanism of a dynamically loaded sliding bearing is characterized using a standard linear solid model, and its equations are as follows:

[0115]

[0116] Where δ(x,z,t) represents the actual deformation of the bearing at the time of this simulation, δt (x,z,t) represents the fully elastic deformation of the bearing at the time of this simulation, δ l (x,z,t-△t) represents the bearing time-delay deformation at the previous simulation time. τ represents the complete time-delay deformation of the bearing under oil film pressure p at the previous simulation moment, τ represents the viscoelastic relaxation time of the bearing material, q represents the total number of simulation time steps, and Δt represents the time step.

[0117] The fully elastic deformation of the sliding bearing is solved using the Winkler method:

[0118]

[0119] Among them, v b With E b represents the Poisson's ratio and elastic modulus of the bearing, respectively, and l represents the thickness of the sliding bearing.

[0120] The fourth step is to solve for the oil film thickness distribution.

[0121] Considering bearing clearance, eccentricity, journal tilt, and viscoelastic deformation of the bearing, the oil film thickness of a dynamically loaded sliding bearing is:

[0122] h = h0 + h mis +h v (7)

[0123] Where h0 is the oil film thickness component considering the radial clearance between the bearing and the journal, as well as the journal eccentricity, h mis h is the oil film thickness component for journal tilt correction. v The oil film thickness component, h, is the correction for the viscoelastic deformation of the bearing. v =δ(x,z,t).

[0124] The equation for h0 is as follows:

[0125]

[0126] Where c is the radial clearance, X is the eccentricity of the journal center in the x-direction, Y is the eccentricity of the journal center in the y-direction, and y is the coordinate of the oil film thickness direction. This is the journal offset angle.

[0127] This invention employs two tilt angles γ, one horizontal and one vertical. x With γ y To characterize the journal tilting effect, the oil film thickness component equation for journal tilt correction is obtained based on the geometric position relationship of the journal in the bearing hole when the journal is tilted.

[0128]

[0129] Among them, Rb B is the bearing radius, and B is the bearing width.

[0130] From formula (9), it can be seen that when γ x =γ y When h = 0, mis =0, at this time formula (7) is transformed into the oil film thickness when the journal is not tilted.

[0131] The fifth step is to establish the journal motion equation.

[0132] The lubrication control equation (1) uses the journal motion equation to iteratively update the eccentricity and eccentricity given the initial eccentricity and eccentricity, thereby correcting the oil film thickness and oil film pressure. In this paper, the journal's center of mass is assumed to be an effective mass particle. The journal motion equation in the dynamic load sliding bearing system is expressed as follows:

[0133]

[0134] Where, m j Indicates journal mass. and These represent the eccentric accelerations in the x and y directions, respectively. and These represent the dynamic loads acting on the sliding bearing system in the x and y directions, respectively. and These represent the oil film forces in the x and y directions, respectively. and These represent the rough contact forces in the x and y directions, respectively. Figure 5 The steady-state distribution of rough contact pressure under different journal tilt angles.

[0135] In mixed lubrication conditions, oil film pressure and rough contact pressure constitute the oil film force and rough contact force of the bearing, respectively. The oil film force and rough contact force can be obtained by using the complex trapezoidal integral of the pressure:

[0136]

[0137]

[0138] Where, p asp This indicates rough contact pressure.

[0139] The Newmark method is used to solve equation (10) iteratively to calculate the eccentricity that minimizes the difference between the external load and the calculated support load. The convergence criterion used is:

[0140]

[0141] Among them, X n With X n-1Represent the eccentricity in the x-direction calculated at time steps n and n-1, respectively, and the eccentricity in the y-direction. n With Y n-1 ζ represents the eccentricity in the y-direction calculated at time steps n and n-1, respectively. X Let ζ be the convergence tolerance of X. Y Let Y be the convergence tolerance. Figure 6 This represents the maximum oil film pressure curves under different journal tilt angles. Figure 7 The curves represent the maximum rough contact pressure under different journal tilt angles. Figure 8 This represents the minimum oil film thickness curve under different journal tilt angles.

[0142] Step 6: Calculate the performance parameters of the mixed lubrication system.

[0143] The mixed lubrication performance of a dynamically loaded sliding bearing under journal tilting conditions includes tilting torque, frictional power consumption, and end leakage flow.

[0144] For bearings with tilted journals, the oil film pressure on both sides of the bearing's central cross-section is asymmetrical. To ensure stable bearing operation, a corresponding torque needs to be applied to the bearing. The tilting torque components in the x and y directions are:

[0145]

[0146] The resultant moment of tilt is given by formula (15). Figure 9 The tilting moment curves are shown for different journal tilt angles.

[0147]

[0148] Under mixed lubrication conditions, the frictional force F f Viscous friction caused by liquid shear Rough contact friction caused by contact with rough peaks It consists of two parts, and the expression is:

[0149]

[0150]

[0151]

[0152] Where κ = 0.02 is the boundary friction coefficient, and φ f For shear flow factor, φ fs For shear stress factor and φ fp This represents the frictional pressure flow factor. Figure 10 The friction power consumption curves are shown for different journal tilt angles.

[0153] φ f φ fsand φ fp The following formula is used for calculation:

[0154]

[0155]

[0156]

[0157] Where Λ=h / σ represents the film thickness ratio, g=Λ / 3 is the film thickness judgment index, and e is the natural constant.

[0158] Frictional power consumption is equal to the product of frictional force and tangential velocity.

[0159] P f =|F f U| (22)

[0160] Among them, F f The friction force is calculated by equation (16), where U is the journal tangential velocity.

[0161] When the journal tilts, the end leakage flows Q1 and Q2 of the bearing front and rear faces can be calculated as follows:

[0162]

[0163]

[0164] The total end discharge flow rate is given by formula (25). Figure 11 The discharge flow curves at the lower end of the journal with different tilt angles are shown.

[0165] Q = Q1 + Q2 (25).

Claims

1. A method for calculating the lubrication performance of a dynamically loaded bearing under journal tilt, characterized in that, Includes the following steps: The first step is to set the initial parameters of the dynamic load sliding bearing system; (1) Determine the radius, length, elastic modulus, Poisson's ratio, and physical parameters of the lubricating oil of the journal and bearing in the dynamic load sliding bearing system; (2) Set the initial eccentricity and eccentricity of the journal; for the dynamic load sliding bearing system, the initial eccentricity is selected from 0 to c, where c is the radial clearance, and the initial eccentricity of the sliding bearing system is set to 0 at the initial moment. (3) The journal tilt is characterized by the tilt angles in the horizontal and vertical directions, and the oil film pressure distribution is initialized using the oil inlet pressure; The second step is to establish the lubrication control equations; The roughness effect is handled statistically, and the influence of surface roughness on lubrication characteristics is characterized by pressure flow factor and shear flow factor. At the same time, in order to consider the influence of lubricating oil film cavitation on lubrication characteristics, the mass conservation equation is used and the fluid fraction θ is introduced to establish the mass conservation average Reynolds equation. Where p represents oil film pressure, h represents oil film thickness, and h T The average oil film thickness is represented by x, which represents the circumferential coordinate, and z, which represents the axial coordinate; φ x and φ z φ represents the pressure-flow factor in the x and z directions, respectively. s This represents the shear flow factor, and U represents the journal tangential velocity. σ represents the standard deviation of the overall surface roughness of the bearing journal. j and σ b ... μ is the dynamic viscosity of the lubricating oil calculated using the Barus viscosity-pressure equation, as follows: Where μ0 represents the dynamic viscosity of lubricating oil under normal pressure; Equation (1) is discretized using the second-order central difference scheme of the finite difference method, and then the oil film pressure and fluid fraction are solved using the over-relaxation iterative method: Where i and j represent the circumferential and axial mesh node numbers, respectively, n represents the time loop step, and k represents the iteration convergence step. and Let represent the oil film pressures obtained by node (i, j) in the k-th and (k-1)-th iterations at the n-th time step, respectively. and Let w represent the fluid fractions obtained by node (i, j) in the k-th and (k-1)-th iterations at the n-th time step. p The over-relaxation parameter representing oil film pressure, w θ The over-relaxation parameter representing the fluid fraction; To ensure the convergence of oil film pressure and fluid fraction during the iteration process, the convergence criterion is as follows: Where N1 and N2 represent the number of nodes in the circumferential and axial directions, respectively, ζ p ζ represents the convergence tolerance of the oil film pressure. θ The convergence tolerance for the fluid fraction; The third step is to construct the viscoelastic deformation equation of the bearing. The viscoelastic deformation mechanism of a dynamically loaded sliding bearing is characterized using a standard linear solid model, and its equations are as follows: Where δ(x,z,t) represents the actual deformation of the bearing at the time of this simulation, δ t (x,z,t) represents the fully elastic deformation of the bearing at the time of this simulation, δ l (x,z,t-△t) represents the bearing time-delay deformation at the previous simulation time. τ represents the fully time-delayed deformation of the bearing under oil film pressure p at the previous simulation moment, τ represents the viscoelastic relaxation time of the bearing material, q represents the total number of simulation time steps, and Δt represents the time step. The fully elastic deformation of the sliding bearing is solved using the Winkler method: Among them, v b With E b These represent the Poisson's ratio and elastic modulus of the bearing, respectively, and l represents the thickness of the sliding bearing. The fourth step is to solve for the oil film thickness distribution; Considering bearing clearance, eccentricity, journal tilt, and viscoelastic deformation of the bearing, the oil film thickness of a dynamically loaded sliding bearing is: h=h0+h mis +h v (7) Where h0 is the oil film thickness component considering the radial clearance between the bearing and the journal, as well as the journal eccentricity, h mis h is the oil film thickness component for journal tilt correction. v The oil film thickness component, h, is the correction for the viscoelastic deformation of the bearing. v =δ(x,z,t); The equation for h0 is as follows: Where c is the radial clearance, X is the eccentricity of the journal center in the x-direction, Y is the eccentricity of the journal center in the y-direction, and y is the coordinate of the oil film thickness direction. This refers to the journal offset angle; Using two tilt angles γ, one horizontal and one vertical. x With γ y To characterize the journal tilting effect, the oil film thickness component equation for journal tilt correction is obtained based on the geometric position relationship of the journal in the bearing hole when the journal is tilted. Among them, R b B is the bearing radius, and B is the bearing width; From formula (9), it can be seen that when γ x =γ y When h = 0, mis =0, at this time formula (7) is transformed into the oil film thickness when the journal is not tilted; Fifth step, establish the journal motion equation; The lubrication control equation (1) is to iteratively update the eccentricity and eccentricity using the journal motion equation, given the initial eccentricity and eccentricity, in order to correct the oil film thickness and oil film pressure; the journal centroid is assumed to be an effective mass particle in this paper, and the motion equation of the journal in the dynamic load sliding bearing system is expressed as follows: Where, m j Indicates journal mass. and These represent the eccentric accelerations in the x and y directions, respectively. and These represent the dynamic loads acting on the sliding bearing system in the x and y directions, respectively. and These represent the oil film forces in the x and y directions, respectively. and These represent the rough contact forces in the x and y directions, respectively. In mixed lubrication conditions, oil film pressure and rough contact pressure constitute the oil film force and rough contact force of the bearing, respectively. The oil film force and rough contact force can be obtained by using the complex trapezoidal integral of the pressure: Where, p asp Indicates rough contact pressure; The Newmark method is used to solve equation (10) iteratively to calculate the eccentricity with the minimum difference between the external load and the calculated support load; the convergence criterion used is: Among them, X n With X n-1 Represent the eccentricity in the x-direction calculated at time steps n and n-1, respectively, and the eccentricity in the y-direction. n With Y n-1 ζ represents the eccentricity in the y-direction calculated at time steps n and n-1, respectively. X Let ζ be the convergence tolerance of X. Y Let Y be the convergence tolerance; Step 6: Calculate the performance parameters of the mixed lubrication system; The mixed lubrication performance of dynamically loaded sliding bearings under journal tilting conditions includes tilting torque, frictional power consumption, and end leakage flow. For bearings with tilted journals, the oil film pressure on both sides of the bearing's central cross-section is asymmetrical. To ensure stable bearing operation, a corresponding torque needs to be applied to the bearing. The tilting torque components in the x and y directions are: The resultant moment of tilt is: Under mixed lubrication conditions, the frictional force F f Viscous friction caused by liquid shear Rough contact friction caused by contact with rough peaks It consists of two parts, and the expression is: Where κ = 0.02 is the boundary friction coefficient, and φ f For shear flow factor, φ fs For shear stress factor and φ fp Friction pressure flow factor; φ f φ fs and φ fp The following formula is used for calculation: Where Λ=h / σ represents the film thickness ratio, g=Λ / 3 is the film thickness judgment index, and e is the natural constant; For frictional power consumption, it is equal to the product of frictional force and tangential velocity; P f =|F f U| (22) Among them, F f The frictional force is calculated by equation (16), where U is the journal tangential velocity; When the journal tilts, the end leakage flows Q1 and Q2 of the bearing front and rear faces can be calculated as follows: The total discharge flow rate is: Q = Q1 + Q2 (25).

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