A bearing thermo-mechanical coupling analysis method based on finite element thermal networks

Abstract

This invention discloses a bearing thermo-mechanical coupling analysis method based on finite element thermal networks, comprising the following steps: calculating the initial radial deformation of the bearing under assembly conditions and centrifugal force; considering fluid viscous friction, improving the bearing pseudo-dynamic analysis, and calculating the precise distributed heat generation of the bearing; using the finite element thermal network method, precisely applying heat sources to calculate the refined temperature field of the bearing, its shaft, and bearing housing; based on the discrete temperature field, calculating the comprehensive radial deformation of the bearing under assembly conditions, centrifugal force, and temperature rise, and updating the geometric relationship; updating the heat generation and temperature field through pseudo-dynamic analysis, and iterating through pseudo-dynamic distributed heat generation-temperature field-comprehensive deformation coupling until the lubricating oil outlet temperature converges; outputting the bearing kinematic and dynamic parameters and the bearing temperature field. This invention achieves precise positioning of the bearing heat source and refined temperature distribution of the bearing, its shaft, and bearing housing, providing theoretical guidance for precise thermal analysis in the design and application of high-speed ball bearings.

Description

Technical Field

[0001] This invention relates to the field of bearing thermo-mechanical coupling analysis technology, which is an interdisciplinary field of mechanical engineering and engineering thermophysics, and particularly to a bearing thermo-mechanical coupling analysis method based on finite element thermal networks and pseudo-dynamic theory and its application. Background Technology

[0002] High-speed ball bearings are core components of transmission systems in high-end equipment such as aero-engines and gas turbines, and their thermodynamic properties have a decisive impact on the overall accuracy, reliability, and service life of the equipment. Although considerable research has been accumulated in this field, and various thermo-coupling analysis frameworks have been constructed, limitations still exist in dimensions such as the realism of model mechanisms, the efficiency of numerical calculations, and the depth of system coupling, which restricts their accuracy and effective application in engineering practice.

[0003] Currently, the overall heat generation calculation based on empirical formulas cannot reflect the independent action mechanism of multiple friction pairs inside the bearing, resulting in coarse quantification of heat sources and poor universality under different structural and operating parameters. Although the local heat generation calculation method has higher theoretical accuracy by analyzing the interaction between components, the quasi-static model often ignores key factors such as cage dynamics and lubrication status. While the full dynamic model can describe the component motion in detail, the computational complexity is extremely high after embedding coupled iterations. In addition, most existing local methods do not systematically introduce the fluid viscous friction effect (dragging force, churning torque) under different lubrication methods (jet lubrication, under-ring lubrication), resulting in deviations in the calculation of bearing motion state and distributed heat generation.

[0004] For temperature field analysis, the mainstream methods are the traditional thermal network method and the three-dimensional finite element method. The traditional thermal network method has a limited node scale, simplifying each component into a homogeneous body, and thus cannot detect local temperature gradients and heat concentration phenomena in critical areas. While the three-dimensional finite element method offers good spatial resolution, its complex modeling and high computational resource consumption make it difficult to achieve rapid iterative solutions in engineering design. The inherent trade-off between accuracy and efficiency between these two methods is a major obstacle to achieving refined thermal state assessment of bearing systems.

[0005] At the level of system coupling mechanisms, existing research has constructed a thermo-mechanical coupling framework combining quasi-static / dynamic models with the thermal network method. However, due to mechanistic biases in the heat generation input and the lack of spatial analytical capability in the temperature field model, the entire system still struggles to accurately reflect the feedback effects of thermal deformation on bearing geometry, kinematics, and dynamics, thus failing to improve prediction accuracy. Essentially, existing coupling frameworks suffer from bottlenecks in the realism and refinement of sub-models.

[0006] In summary, existing heat generation calculation models struggle to balance accuracy and efficiency, and generally fail to systematically consider the fluid viscous friction effect under different lubrication methods, leading to inaccurate heat source positioning and quantification distortion. Existing temperature field solution methods inherently contradict computational efficiency and spatial resolution; traditional thermal network methods are too coarse, while three-dimensional finite element methods are computationally expensive and difficult to achieve efficient solutions for refined temperature fields. Existing coupled frameworks, due to the insufficient accuracy of their dependent heat generation and temperature field sub-models, cannot accurately characterize the closed-loop feedback effect of thermal deformation on the internal geometric relationships, kinematics, and dynamic behavior of bearings, thus limiting overall prediction accuracy.

[0007] The current technology fails to achieve efficient and high-precision fusion of high-fidelity heat generation, high-resolution temperature fields, and thermodynamic coupling iteration. Therefore, inventing a bearing thermodynamic coupling analysis method that significantly improves model mechanism, computational efficiency, and coupling logic has become a key technical problem urgently needing to be solved in this field. Summary of the Invention

[0008] The purpose of this invention is to overcome the shortcomings of existing technologies and provide a bearing thermo-mechanical coupling analysis method based on finite element thermal networks. This method achieves accurate analysis of bearing thermo-mechanical coupling by constructing a closed-loop iterative solution framework that integrates a high-fidelity heat generation model, a high-resolution temperature field model, and thermo-mechanical coupling.

[0009] To achieve the above objectives, the technical solution adopted by the present invention is as follows: a bearing thermo-mechanical coupling analysis method based on a finite element thermal network, characterized by including the following steps: Step 1: Calculate the initial radial deformation of the bearing inner ring, outer ring, rolling elements, shaft, and bearing housing under the action of assembly interference and centrifugal force; Furthermore, the initial radial deformation of the bearing inner ring, outer ring, shaft, and bearing housing is calculated using the axisymmetric equilibrium equation of a thick-walled circular ring in cylindrical coordinates in elasticity mechanics. Based on the assumption of plane strain or plane stress, a comprehensive radial deformation calculation formula considering assembly, centrifugal force, and temperature rise is obtained. For the shaft, the corresponding formula is used depending on whether it is a hollow or solid structure. The bearing interference fit pressure is calculated based on the relationship between interference and fit pressure proposed by Harris, and the initial radial deformation at each key position is solved accordingly. The axisymmetric equilibrium equations are as follows:

[0010] In the formula Radial coordinates, Radial stress, For circumferential stress, For density, It is the angular velocity of rotation; The introduced geometric equations and constitutive equations are as follows:

[0011]

[0012] In the formula For radial displacement, For radial normal strain, For circumferential positive strain, For axial normal strain, For axial stress, , , These are Poisson's ratio, elastic modulus, and coefficient of thermal expansion, respectively. For temperature rise; Considering the axial positioning of the bearing's inner and outer rings, i.e., the axial deformation of the bearing's inner and outer rings is constrained, the plane strain assumption is adopted. , The formula for calculating the comprehensive radial deformation considering assembly, centrifugal force, and temperature rise is obtained as follows:

[0013] In the formula and These are the inner and outer diameters of the ring, respectively. and It is the pressure applied radially to the inner and outer surfaces of the ring. This is the radial temperature rise distribution function; For bearing housings, if they are fixed by sleeves or other structures, i.e., the axial position is constrained, the radial deformation calculation formula based on the plane strain assumption is used. For shafts, only the portion mating with the bearing inner ring is considered. If it is a hollow shaft, the radial deformation calculation formula based on the plane strain assumption is used; if it is a solid shaft, the radial deformation calculation formula based on the plane strain assumption with an inner diameter of 0 is used.

[0014] If the axial position of the bearing housing is not constrained, the comprehensive radial deformation calculation formula derived through the plane stress assumption is used:

[0015] The relationship between interference and fitting pressure proposed by Harris and It can be calculated as:

[0016] If the bearing inner ring is interference-fitted with the shaft, the stress on the outer surface of the inner ring will be... , inner surface stress of the shaft There is a stress distribution between the inner surface of the inner ring and the shaft. :

[0017] Thus, the inner channel was calculated. Comprehensive radial deformation at the location Inner ring inner surface Comprehensive radial deformation at the location outer surface of shaft Comprehensive radial deformation at the location ; If the bearing outer ring and bearing housing have an interference fit, the stress on the inner surface of the outer ring will be... stress on the outer surface of the bearing housing There is stress distribution between the outer surface of the outer ring and the inner surface of the bearing housing. :

[0018] Thus, the outer channel was calculated. Comprehensive radial deformation at the location Outer ring outer surface Comprehensive radial deformation at the location and the inner surface of the bearing housing Comprehensive radial deformation at the location ; Similarly, for a solid sphere, the diameter is... The temperature rise causes any radius The radial deformation at that point is:

[0019] In the initial deformation calculation, the initial temperature field of the bearing, its shaft, and bearing housing is uniform and without temperature difference. The overall deformation is caused only by assembly and centrifugation. The calculation results are as follows: , , , , , , At this time, the bearing interference fit is updated synchronously. , This is the initial interference.

[0020] Step 2: Based on the initial radial deformation obtained in Step 1, an improved pseudo-dynamic model that comprehensively considers fluid viscous friction under different lubrication methods is used to perform steady-state analysis of the bearing and solve for the accurate kinematic and dynamic parameters of the bearing. Furthermore, the improved pseudo-dynamic model includes the displacement-deformation coupling relationship of bearing elements and the bearing pseudo-dynamic equilibrium equations. The displacement-deformation coupling relationship of bearing elements is used to calculate the contact angle and elastic approximation between the ball and the inner and outer rings after dynamic loading. The bearing pseudo-dynamic equilibrium equations include the rolling element equilibrium equations, the cage equilibrium equations, and the inner ring equilibrium equations, and systematically introduce fluid viscous friction under jet lubrication and under-ring lubrication, including drag force and churning torque, and are solved using an iterative method. The precise kinematic and dynamic parameters of the bearing include ball displacement, rotational speed, revolution speed, contact angle, attitude angle, cage displacement, inner ring displacement, deflection angle, and the interaction relationship between the inner ring, outer ring, ball, cage, and lubricating oil. The displacement-deformation coupling relationship of the bearing element includes: The inner ring is subjected to combined external loads. , , , , Afterwards, it will occur relative to the outer circle. , , , , Five degrees of freedom displacement, simultaneously causing elastic convergence of the sphere and its inner and outer rings. , From the displacement-deformation compatibility condition, the distance between the center of curvature of the inner groove after dynamic loading and the center of curvature of the outer groove at rest is:

[0021] In the formula The number of the rolling element. This is the original contact angle for angular contact ball bearings. For double-inner-ring ball bearings gasket corner , For the thickness of the gasket, For the original radial clearance, The diameter of the sphere, and These are the inner and outer groove curvature radius coefficients, respectively. The position angle of the ball, and These are the distances between the center of curvature of the sphere after dynamic loading and the outer groove when at rest, respectively; therefore, the formula for calculating the contact angle between the sphere and the inner and outer rings after dynamic loading is:

[0022] Then calculate the elastic approximation of the ball with respect to its inner and outer rings. , :

[0023] The aforementioned bearing pseudo-dynamic equilibrium equations include: Fluid viscous friction includes drag force and churning torque, drag force The rotation of the rolling elements is affected by the lubricating oil, resulting in an oil churning torque. and The rotation of the rolling elements and the cage is affected by the lubricating oil, respectively. The rolling element equilibrium equations are as follows:

[0024] In the formula The normal contact forces between the ball and the inner and outer rings, The drag force of the oil film on the ball, The rolling friction resistance at the contact area between the ball and the inner and outer rings. The horizontal component of the resultant hydrodynamic pressure acting on the center of the sphere. The force acting between the ball and the cage pocket in the normal direction. and These are the rolling friction resistance and sliding friction resistance in the contact inlet area between the ball and the cage pocket, respectively. and It is an inertial force. and For gyro torque, , and These are the three-dimensional rotational components of the sphere. , and The inertial torque is calculated using the finite difference method; The cage equilibrium equations are:

[0025] In the formula The total number of rolling elements, The diameter of the pitch circle. , and To maintain the forces and frictional torques between the cage and guide ring under hydrodynamic lubrication, To maintain the weight of the rack; The inner circle equilibrium equations are:

[0026] In the formula .

[0027] Step 3: Based on the bearing kinematics and dynamics parameters obtained in Step 2, calculate the precise distributed heat generation of each friction pair inside the bearing; Furthermore, the precise distributed heat generation of the bearing includes heat generation from elastic hysteresis friction between the rolling elements and the inner and outer raceways. Heat generation from differential sliding friction Heat generation from spin sliding friction Friction between the rolling elements and the cage pockets generates heat. The friction between the cage and the guide ring generates heat. Heat is generated by the drag friction between the rolling elements and the lubricating oil. The churning torque and frictional heat generated by the rolling elements and cage ; The heat generated by elastic hysteresis friction between the rolling elements and the inner and outer raceways is as follows:

[0028] In the formula The elastic hysteresis coefficient of the material. This refers to the rotational speed of the inner or outer ring of the bearing. The revolution speed of the rolling element. This refers to the distance between the point of contact between the ball and the inner and outer raceways and the axis of rotation of the bearing. Let the principal curvature of the contact surface be a function of the contact surface; The heat generated by the differential sliding friction between the rolling elements and the inner and outer raceways is:

[0029] In the formula The differential sliding speed between the rolling element and the raceway on the contact surface; The heat generated by the spin friction between the rolling element and the inner and outer raceways is as follows:

[0030] In the formula These are the spin components of the rolling element on the inner and outer raceways. The coefficient of spin friction between the rolling element and the raceway. and These are the major and minor semi-axis lengths of the ellipses representing the contact ellipses between the rolling element and the inner and outer raceways, respectively. It is an elliptic integral of the second kind; The frictional heat generated between the rolling elements and the cage pocket is as follows:

[0031] The frictional heat generated between the cage and the guide ring is:

[0032] In the formula The rotational angular velocity of the inner or outer ring. To maintain the angular velocity of the frame's rotation.

[0033] The heat generated by drag friction between the rolling elements and the lubricating oil is:

[0034] If it is jet lubrication, take the above formula; if it is ring lubrication, take the below formula. The frictional heat generated by the churning torque is:

[0035]

[0036] Step 4: Apply the finite element thermal network method to discretize the bearing system into a structured mesh, and accurately apply a heat source based on the distributed heat generation obtained in Step 3. Obtain a refined temperature field by solving the steady-state thermal balance equations. Furthermore, the finite element thermal network method includes: (1) Method construction: Combine the heat network method to construct the heat balance equation matrix of all nodes; (2) Mesh generation: Discretize the bearing inner ring, outer ring, ball, cage, shaft and bearing housing into a structured mesh, and reasonably divide it into a specific number of elements according to the accuracy requirements and computing resources, and set environmental nodes and lubricating oil nodes; (3) Heat source loading: The distributed heat generation of the bearing calculated in step three is distributed and loaded onto the corresponding contact area unit or node according to the preset rules; (4) Heat transfer analysis: Calculate the thermal resistance of heat conduction and heat transfer between unit cells or nodes; (5) Temperature field solution: Based on the heat transfer relationship between each unit or node, the proposed multidimensional sparse matrix is ​​constructed by combining the methods, and the bearing temperature field is obtained by solving the Gaussian elimination method. The method described combines the thermal network method and the finite element method to provide a finite element thermal network method suitable for ball bearings, their shafts, and bearing housings. A thermal network is formed by dividing the network into nodes and establishing the thermal resistance between the nodes. For a given node in the thermal network... The heat balance equation can be written as follows:

[0037] In the formula Represents nodes Nodes with heat transfer relationships, For nodes The total number of nodes with heat transfer relationships. For temperature The function, For nodes With nodes Thermal resistance between them for Time's up The time interval of time, For the internal heat source per unit volume of the node, For node volume, For node density, Let be the specific heat capacity of the node; the first term on the left-hand side of the equation represents the heat transferred from surrounding nodes to the node. The heat generated by the node is represented by the second term on the left side of the equation, and the term on the right side of the equation is the net increase in node energy. During steady-state heat transfer, the temperature at any node in the system is constant. Neglecting radiative heat transfer, the right-hand side of the equation is 0. Let... The total number of system nodes is If the thermal resistance between nodes with no heat transfer relationship is taken as infinite, its reciprocal If it is 0, then:

[0038] In the formula , For nodes All hot-loaded; make:

[0039] The matrix of heat balance equations for all nodes is then:

[0040] The specific meshing is as follows: the bearing inner ring, outer ring, ball, and cage are discretized into a structured mesh along the axial section of the shaft and bearing housing, divided into several unit cells; the inner ring is divided radially into... Layer, each layer along the axial direction Each unit body includes a left retaining edge, a left channel, a right channel, and a right retaining edge, respectively. , , and Each unit body; the outer ring is divided radially into... Layer, each layer along the axial direction Each unit body includes a left retaining edge, a left channel, a right channel, and a right retaining edge, respectively. , , and Each unit body; the cage is divided into two parts: guide rails and crossbeams. The guide rails are radially divided into... Layer, each layer along the axial direction Each unit body, the lintel section is divided into: Each unit body; the axis is divided radially into... Layer, each layer along the axial direction Each unit body, at the leftmost point, at the point where it mates with the left side flange of the inner ring, at the point where it mates with the left side channel of the inner ring, at the point where it mates with the right side channel of the inner ring, at the point where it mates with the right side flange of the inner ring, and at the rightmost point respectively , , , , , Each unit body; the bearing housing is divided radially into... Layer, each layer along the axial direction Each unit body, at the leftmost point, at the point where it mates with the left side flange of the outer ring, at the point where it mates with the left side channel of the outer ring, at the point where it mates with the right side channel of the outer ring, at the point where it mates with the right side flange of the outer ring, and at the rightmost point respectively , , , , , Each unit body; the sphere is divided radially into... Layer, each layer along the polar angle direction One unit cell; the number of unit cells is reasonably selected according to the accuracy requirements and computing resources; and one environmental node, one lubricating oil inlet node, one lubricating oil heat exchange node, and one lubricating oil outlet node are set. The heat source distribution is specifically as follows: the heat generated in the contact area between the ball and the inner raceway is... The ball and the inner raceway each receive half of the load, which is then applied to the unit cells in the contact area between the ball and the inner raceway; the heat generated in the contact area between the ball and the outer raceway is... The ball and the outer race each receive half of the load, which is then applied to the unit cells in the contact area between the ball and the outer race ball; the friction between the ball and the cage pocket generates heat. The ball and cage are each allocated half, and are respectively loaded onto the unit body in the contact area between the ball and cage pockets; the friction between the cage and the guide ring generates heat. The cage and guide ring are each allocated half; if the outer ring is guided, the load is applied to the unit body in the contact area between the cage and the outer ring; if the inner ring is guided, the load is applied to the unit body in the contact area between the cage and the inner ring. Heat is generated by the drag friction between the ball and the lubricating oil. The frictional heat generated by the churning torque of the ball The ball and lubricating oil are each distributed equally, and are respectively applied to the unit body in the contact area between the ball and the lubricating oil and the lubricating oil heat exchange node; the oil stirring torque of the cage generates frictional heat. The cage and lubricating oil are each allocated half, and are respectively loaded onto the unit body in the contact area between the cage and the lubricating oil and the lubricating oil heat exchange node; The heat transfer analysis specifically involves: each unit cell and node possesses physical properties and temperature, and undergoes heat conduction or convection heat transfer with surrounding continuous unit cells or nodes; heat conduction includes planar contact heat conduction, annular radial contact heat conduction, unit spherical shell radial contact heat conduction, unit spherical shell polar angle contact heat conduction, and oil film contact heat conduction, with the following thermal resistances:

[0041]

[0042]

[0043]

[0044]

[0045] In the formula and The thermal conductivity of the two unit cells is... and The thickness of the two unit cells, The contact area between the two unit cells; y is the axial length of the unit cell. , and The radii of the two unit cells are respectively ; , and The polar angles of the unit spherical shell are respectively , , and The radii of the unit spherical shell are respectively ; The Hertzian contact area between the ball and the inner and outer raceways. For oil film thickness, The thermal conductivity of the lubricating oil; Convective heat transfer includes convective heat transfer between the lubricating oil and the balls, inner ring, outer ring, cage, shaft, and bearing housing, as well as convective heat transfer between the surrounding environment and the hollow shaft and bearing housing. Its convective thermal resistance is:

[0046] In the formula For convective heat transfer area, The average convective heat transfer coefficient between the bearing, its shaft, and bearing housing and the lubricating oil is: "+" indicates that the outer ring of the bearing rotates, and "-" indicates that the inner ring of the bearing rotates. The kinematic viscosity of the lubricating oil. The bearing ring speed, The Prandtl number represents the flow rate of the lubricating oil; the convective heat transfer coefficient between the shaft and bearing housing and the environment is... , The thermal conductivity of air. For the inner diameter of the hollow shaft or the outer diameter of the bearing housing, the Nusselt number is... , and These are the Reynolds number and Prandtl number for airflow, respectively. Lubricating oil inlet node With lubricating oil outlet node The relationship between them is , For quality flow, Total heat generated by the bearing; Lubricating oil inlet node Heat exchange nodes with lubricating oil The relationship between them is , It serves as the heat source for the lubricating oil heat exchange node; The temperature field solution is specifically as follows: all unit cells and nodes are connected to form a thermal network through thermal resistance. For each unit cell or node... In general, it can have heat transfer relationships with a maximum of four surrounding unit cells or nodes, thus obtaining a steady-state unit cell or node. The heat balance equation matrix is:

[0047] In the formula To be connected with unit or node The unit or node codes that have heat transfer relationships are combined according to the above formula to construct the multidimensional sparse matrix proposed in (1), and solved by Gaussian elimination method. The obtained temperature field includes the coordinates, width, height, and temperature of each unit of the bearing, its shaft, and bearing housing.

[0048] Step 5: Based on the bearing temperature field obtained in Step 4, calculate the combined radial deformation of the bearing, its shaft, and bearing housing under the combined effects of assembly interference, centrifugal force, and temperature rise, so as to update the internal geometric relationship of the bearing. Furthermore, the comprehensive radial deformation is calculated using the comprehensive radial deformation formula mentioned in step one, and the additional deformation caused by temperature rise is calculated by numerical integration based on the discrete temperature field distribution obtained by the finite element thermal network model, thereby obtaining the component deformation under the combined effects of assembly, centrifugation, and temperature rise, and the bearing interference is updated simultaneously. The comprehensive radial deformation calculation is based on a finite element thermal network model, where the temperature field inside the annulus is radially discretely distributed, ignoring the axial temperature distribution. Therefore, at any radius... The integral of temperature rise at that point is calculated as follows:

[0049] In the formula Let be the number of annexes in the integration interval. For the first The temperature rise of each ring, For the first The outer diameter of the ring, This is the inner diameter of the first ring; The temperature field inside the sphere is radially discretely distributed; neglecting the circumferential temperature distribution, then at any radius... The integral of temperature rise at that point is calculated as follows:

[0050] In the formula Let be the number of spherical shells in the integration interval. For the first The temperature rise of the spherical shell, For the first The outer diameter of the spherical shell, ; This allows for the calculation and updating of the overall radial deformation of components under the combined effects of assembly, centrifugation, and temperature rise. , , , , , , The bearing interference fit is updated synchronously. .

[0051] Step 6: Feed back the updated radial deformation of the bearing components to Step 2, and repeat Steps 2 to 5 for coupled iteration until the change in lubricating oil outlet temperature converges to the set threshold. Furthermore, the convergence criterion for the coupled iteration is that the temperature difference at the lubricating oil outlet between two adjacent iterations is less than 0.1℃. If the convergence condition is not met, the process jumps to steps two through five to update the component deformation caused by the combined effects of assembly, centrifugation, and temperature rise. , , , , , , , In the formula If the convergence condition is met, proceed to step seven.

[0052] Step 7: Output the converged bearing kinematics and dynamics parameters, distributed heat generation, and temperature field; Furthermore, the bearing kinematics and dynamics parameters output in step seven include rolling element displacement, rotational speed, revolution speed, contact angle, attitude angle, cage displacement, inner ring displacement, deflection angle, and the interaction relationship between the inner ring, outer ring, ball, cage, and lubricating oil; distributed heat generation includes elastic hysteresis friction heat generation, differential sliding friction heat generation, spin sliding friction heat generation between the ball and the inner and outer raceways, friction heat generation between the ball and the cage pocket, friction heat generation between the cage and the guide ring, drag friction heat generation between the ball and the lubricating oil, and oil churning torque friction heat generation between the ball and the cage; the temperature field includes the coordinates, width, height, and temperature of each unit of the bearing, its shaft, and bearing housing.

[0053] The beneficial effects of this invention are as follows: This invention provides a bearing thermo-mechanical coupling analysis method based on finite element thermal networks, which has the following innovative effects: (1) Improved the accuracy of heat generation calculation and heat source location. By using the improved pseudo-dynamic model, fluid viscous friction (dragging force, churning torque) is considered as an explicit analysis term, which can more accurately reflect the frictional heat generation mechanism inside the bearing; by calculating the distributed heat generation of each friction pair, a foundation is laid for obtaining a more accurate temperature field.

[0054] (2) It balances the accuracy and efficiency of temperature field solution. The finite element thermal network method adopted, through structured mesh discretization, retains the computational efficiency advantage of the thermal network method while significantly improving the spatial resolution, and can reveal local temperature gradients that are difficult to reflect by the traditional lumped parameter thermal network method. It avoids the problems of complex modeling and high computational cost of the full three-dimensional finite element method, and is more suitable for engineering design and analysis scenarios that require rapid iteration.

[0055] (3) Enhanced the completeness of thermo-mechanical coupling analysis. By constructing a closed-loop iterative framework of pseudo-dynamic distributed heat generation-finite element thermal network temperature field-thermal coupling deformation, this invention can quantitatively examine the influence of structural deformation caused by assembly, centrifugation and temperature rise on the internal geometric relationship of the bearing (such as clearance and contact angle), overcome the limitations of traditional unidirectional coupling analysis models, and make the prediction of bearing dynamic performance more reliable under the condition of considering thermal effects.

[0056] (4) It has clear engineering application value. This method can output more accurate key parameters such as bearing temperature distribution and contact load, which can provide valuable theoretical reference and data support for the thermal management design, life assessment accuracy improvement, and structural optimization of high-speed ball bearings. It is applicable to the thermal design, performance evaluation, and condition prediction of bearing systems in high-end equipment such as aero-engines and gas turbines. Attached Figure Description Figure 1 This is a flowchart of the method of the present invention; Figure 2 This is a schematic diagram of a bearing, its shaft, and bearing housing structure suitable for use in one embodiment of the present invention; Figure 3 This is a schematic diagram of the displacement-deformation coupling relationship of bearing components; Figure 4 yes Figure 2 Schematic diagram of the cage structure of the bearing; Figure 5 yes Figure 2 Schematic diagram of the grid division of the inner and outer rings and cage of the bearing; Figure 6 yes Figure 2 A schematic diagram of the ball mesh division in a bearing; Figure 7 yes Figure 2 Schematic diagram of the grid division for the central shaft and bearing housing; Figure 8 yes Figure 2 The illustrated embodiment uses a schematic diagram of the output temperature field of the present invention.

[0057] In the diagram: 1. Inner ring; 2. Outer ring; 3. Rolling element; 4. Bearing cage; 41. Bearing cage guide rail; 42. Bearing cage crossbeam; 5. Shaft; 6. Bearing housing. Detailed Implementation

[0058] The principles and features of the present invention are described below with reference to the accompanying drawings. The examples given are only for explaining the present invention and are not intended to limit the scope of the present invention.

[0059] To achieve the above objectives, the present invention provides the following specific embodiments: Figure 1 As shown, the bearing thermo-mechanical coupling analysis method based on finite element thermal networks includes the following steps: Step 1: Calculate the initial radial deformation of the bearing inner ring 1, outer ring 2, rolling element 3, shaft 5 and bearing housing 6 under the action of assembly interference and centrifugal force; Step 2: Based on the initial radial deformation obtained in Step 1, an improved pseudo-dynamic model that comprehensively considers fluid viscous friction under different lubrication methods is used to perform steady-state analysis of the bearing and solve for the accurate kinematic and dynamic parameters of the bearing. Step 3: Based on the bearing kinematics and dynamics parameters obtained in Step 2, calculate the precise distributed heat generation of each friction pair inside the bearing; Step 4: Apply the finite element thermal network method to discretize the bearing system into a structured mesh, and accurately apply a heat source based on the distributed heat generation obtained in Step 3. Obtain a refined temperature field by solving the steady-state thermal balance equations. Step 5: Based on the bearing temperature field obtained in Step 4, calculate the combined radial deformation of the bearing, its shaft, and bearing housing under the combined effects of assembly interference, centrifugal force, and temperature rise, so as to update the internal geometric relationship of the bearing. Step 6: Feed back the updated radial deformation of the bearing components to Step 2, and repeat Steps 2 to 5 for coupled iteration until the change in lubricating oil outlet temperature converges to the set threshold. Step 7: Output the converged bearing kinematics and dynamics parameters, distributed heat generation, and temperature field.

[0060] This invention takes a certain type of high-speed angular contact ball bearing as the analysis object, such as... Figure 2 As shown, the shaft 5, which mates with the inner ring 1 of the bearing, and the bearing housing 6, which mates with the outer ring 2, are hollow annular structures. In this embodiment, only the shaft and bearing housing directly connected to the bearing are considered, and the flanges are ignored. Their structural parameters, material properties, and operating parameters are listed in the table. It should be understood that this invention is not limited to a single structure. The complete structure mating with the bearing includes, but is not limited to, components such as shafts (spindle, bushing, sleeve, gasket) and bearing housings (bearing housing, bushing, sleeve, gasket), as well as components of radially locating ball bearings, and can all be analyzed and solved using this invention.

[0061] Step 1, Initial radial deformation calculation: Obtain the initial inputs for the bearing system. These inputs include the bearing's structural parameters (number of rolling elements, rolling element diameter, inner ring inner diameter, inner ring groove bottom diameter, inner ring flange diameter, outer ring flange diameter, outer ring groove bottom diameter, outer ring outer diameter, inner ring width, outer ring width, inner ring groove radius of curvature coefficient, outer ring groove radius of curvature coefficient, cage inner diameter, cage outer diameter, cage pocket diameter, cage width), material parameters (inner ring material, outer ring material, rolling element material, cage material, shaft material, bearing housing material, lubricant grade), fit relationships (interference fit between inner ring and shaft, interference fit between outer ring and housing), and operating conditions (ring rotational speed). Axial force radial force radial force Bending moment Bending moment Oil supply temperature Oil supply flow rate When the ball bearing is working, its axial direction is limited, meaning the axial deformation of the inner ring 1 and outer ring 2 is constrained. The bearing housing 6 is fixed by a sleeve, meaning its axial position is constrained. Shaft 5 is a hollow spindle, and only the part that mates with the inner ring 1 of the bearing is considered. Therefore, the comprehensive radial deformation calculation of the inner ring 1, outer ring 2, shaft 5, and bearing housing 6 adopts the plane strain assumption and the comprehensive radial deformation calculation formula:

[0062] In the formula Radial position, and These are the inner diameter and the outer diameter, respectively. and It is the pressure applied radially to the inner and outer surfaces. , , , These are Poisson's ratio, elastic modulus, and coefficient of thermal expansion, respectively. It is the angular velocity of rotation. This is the radial temperature rise distribution function; The relationship between interference and fitting pressure proposed by Harris and It can be calculated as:

[0063] If the inner ring 1 of the bearing is interference-fitted with the shaft 5, then the stress on the outer surface of the inner ring 1 will be... Surface stress of shaft 5 There is a stress distribution between the inner surface of inner ring 1 and shaft 5. :

[0064] Thus, the inner ring 1 channel was calculated. Comprehensive radial deformation at the location Inner ring 1 inner surface Comprehensive radial deformation at the location outer surface of shaft 5 Comprehensive radial deformation at the location ; If the outer ring 2 of the bearing is interference-fitted with the bearing housing 6, then the stress on the inner surface of the outer ring 2 will be... stress on the outer surface of bearing housing 6 There is a stress distribution between the outer surface of the outer ring 2 and the inner surface of the bearing housing 6. :

[0065] Thus, the outer ring 2 channels were calculated. Comprehensive radial deformation at the location Outer ring 2 outer surface Comprehensive radial deformation at the location and the inner surface of bearing housing 6 Comprehensive radial deformation at the location ; Similarly, for a solid sphere, the diameter is... The temperature rise causes any radius The radial deformation at that point is:

[0066] In the initial deformation calculation, the initial temperature field of the bearing, its shaft, and bearing housing is uniform and without temperature difference. The overall deformation is caused only by assembly and centrifugation. The calculation results are as follows: , , , , , , At this time, the bearing interference fit is updated synchronously. , This is the initial interference.

[0067] Step 2, Bearing pseudo-dynamic analysis: (1) Displacement-deformation coupling relationship of bearing components Inner ring 1 is subjected to external combined loads , , , , Afterwards, it will occur relative to outer ring 2. , , , , Five-degree-of-freedom displacement, simultaneously causing the rolling element 3 to elastically approach the inner ring 1 and the outer ring 2. , .like Figure 3 As shown, The center of curvature of the groove before deformation of the outer ring 2. and These are the center of curvature of the groove in front of the inner ring 1 and the center of the ball, respectively, under dynamic loading. The center of curvature of the groove after deformation of outer ring 2. and These are the centers of curvature of the grooves in inner ring 1 and rolling element 3 after dynamic loading, and the center of the ball, respectively. Based on the displacement-deformation compatibility condition, the distance between the center of curvature of the groove in inner ring 1 after dynamic loading and the center of curvature of the groove in outer ring 2 when at rest is:

[0068] In the formula For the rolling element number 3, This is the original contact angle for a double-inner-ring ball bearing. gasket corner , For the thickness of the gasket, For the original radial clearance, For the diameter of the rolling element 3, and These are the inner and outer groove curvature radius coefficients, respectively. The rolling element is at position 3. and These are the distances between the center of curvature of the rolling element 3 under dynamic loading and the outer ring 2 when stationary; therefore, the formula for calculating the contact angle between the rolling element 3 and the inner ring 1 and outer ring 2 under dynamic loading is:

[0069] Then, the elastic approximation of rolling element 3 with inner ring 1 and outer ring 2 is calculated. , :

[0070] (2) Bearing pseudo-dynamic equilibrium equations Fluid viscous friction includes drag force and churning torque, drag force The churning torque is affected by the lubricating oil during the 3 revolutions of the rolling element. The rotation of rolling element 3 is affected by the lubricating oil; The equilibrium equations for rolling element 3 are as follows:

[0071] In the formula The normal contact force between rolling element 3 and inner ring 1 and outer ring 2. The oil film drag force of rolling element 3, The rolling friction resistance at the contact inlet area between the rolling element 3 and the inner ring 1 and outer ring 2. The horizontal component of the resultant hydrodynamic force acting on the center of the three rolling elements is... The normal force between the rolling element 3 and the cage 4 pocket is... and These represent the rolling friction resistance and sliding friction resistance at the contact inlet area between the rolling element 3 and the cage 4 pocket, respectively. and It is an inertial force. and For gyro torque, , and These are the three-dimensional rotational components of rolling element 3. , and The inertial torque is calculated using the finite difference method; The equilibrium equations for cage 4 are as follows:

[0072] In the formula The total number of rolling elements, The diameter of the pitch circle. , and To maintain the forces and frictional torques between the cage 4 and the outer ring 2 under hydrodynamic lubrication, To maintain the 4-gravity cage; The equilibrium equations for inner circle 1 are:

[0073] In the formula ; By combining the equilibrium equations for rolling element 3, cage 4, and inner ring 1, we can obtain... A nonlinear equation, corresponding to One variable; combining gradient descent and Newton's method, through programming, the displacement, rotation speed, revolution speed, contact angle, and attitude angle of rolling element 3, the displacement of cage 4, the displacement and deflection angle of inner ring 1, and the interaction relationship between inner ring 1, outer ring 2, rolling element 3, cage 4, and lubricating oil are obtained. Step 3, Calculation of distributed heat generation in bearings: (1) The heat generated by the elastic hysteresis friction between the rolling element 3 and the raceways of the inner ring 1 and the outer ring 2 is:

[0074] In the formula The elastic hysteresis coefficient of the material. The rotational speed of the inner ring of the bearing is 1. The rolling element rotates at a speed of 3 revolutions. This refers to the distance between the contact point between the rolling element 3 and the raceways of the inner ring 1 and outer ring 2, and the axis of rotation of the bearing. Let the principal curvature of the contact surface be a function of the contact surface; (2) The heat generated by the differential sliding friction between the rolling element 3 and the raceways of the inner ring 1 and the outer ring 2 is:

[0075] In the formula The differential sliding speed of the rolling element 3 with the raceways of the inner ring 1 and the outer ring 2 on the contact surface; (3) The heat generated by the spin friction between the rolling element 3 and the raceways of the inner ring 1 and the outer ring 2 is:

[0076] In the formula The spin component of rolling element 3 on the raceways of inner ring 1 and outer ring 2. The coefficient of spin friction between rolling element 3 and the raceway. and These are the major and minor semi-axis lengths of the contact ellipses between rolling element 3 and the raceways of inner ring 1 and outer ring 2, respectively. It is an elliptic integral of the second kind; (4) The frictional heat generated between the rolling element 3 and the cage 4 pocket is:

[0077] (5) The frictional heat generated between the cage 4 and the outer ring 2 is:

[0078] In the formula The rotational angular velocity of inner ring 1, To maintain the rotational angular velocity of frame 4.

[0079] (6) The heat generated by the drag friction between the rolling element 3 and the lubricating oil is:

[0080] (7) The frictional heat generated by the stirring torque is:

[0081] (8) The total heat generated by the bearing is:

[0082] Based on the bearing kinematics and dynamics parameters obtained in step two, the precise distributed heat generation of each friction pair inside the bearing is calculated. Step 4, Solving for the bearing temperature field: (1) Grid division like Figure 2 As shown, the inner ring 1, outer ring 2, shaft 5, and bearing housing 6 of the bearing are all axisymmetric bodies of revolution, and therefore can be discretized into a finite number of bodies of revolution; as Figure 4 As shown, the bearing cage 4 includes two parts: a guide rail 41 and a crossbeam 42. Similarly, the guide rail 41 can be discretized into a finite number of rotating bodies; and the crossbeam 42 between every two cage pockets is discretized into one unit, thus there are a total of Z unit bodies. Figure 5 As shown, the inner ring 1 is divided radially into Layer, each layer along the axial direction Each unit body includes a left retaining edge, a left channel, a right channel, and a right retaining edge, respectively. , , and Each unit body; the outer ring 2 is divided radially into... Layer, each layer along the axial direction Each unit body includes a left retaining edge, a left channel, a right channel, and a right retaining edge, respectively. , , and Each unit body; the cage guide rail 41 is radially divided into Layer, each layer along the axial direction Individual units; such as Figure 6As shown, shaft 5 is divided radially into Layer, each layer along the axial direction Each unit body, at the leftmost point, at the point where it mates with the left side flange of the inner ring, at the point where it mates with the left side channel of the inner ring, at the point where it mates with the right side channel of the inner ring, at the point where it mates with the right side flange of the inner ring, and at the rightmost point respectively , , , , , Each unit body; bearing housing 6 is radially divided into Layer, each layer along the axial direction Each unit body, at the leftmost point, at the point where it mates with the left side flange of the outer ring, at the point where it mates with the left side channel of the outer ring, at the point where it mates with the right side channel of the outer ring, at the point where it mates with the right side flange of the outer ring, and at the rightmost point respectively , , , , , Individual units; such as Figure 7 As shown, due to the presence of inner and outer contact angles, the rotational motion of the rolling element 3 in the bearing has an attitude angle. Therefore, the rolling element 3 is discretized symmetrically along its rotation axis into a finite number of unit spherical shells of revolution, and further divided radially into... Layer, each layer along the polar angle direction The number of unit cells is selected reasonably based on accuracy requirements and computational resources. One lubricating oil inlet node, one lubricating oil heat exchange node, and one lubricating oil outlet node are set. Each unit cell and node is assigned a code. The dimensional parameters of each unit cell, including inner diameter, outer diameter, width, and polar angle, are calculated based on structural parameters. Physical properties, including elastic modulus, Poisson's ratio, thermal conductivity, density, specific heat capacity, coefficient of thermal expansion, and kinematic viscosity, are given based on temperature and material parameters. (2) Heat source loading Based on the distributed heat generation calculated in step three and the bearing kinematic parameters calculated in step two, the heat source can be precisely applied to the corresponding unit body; the heat generation in the contact area between rolling element 3 and inner ring 1 raceway is... The rolling element 3 and the inner ring 1 raceway are each allocated half; through attitude angle and inner contact angle Calculate the unit designation of the contact area between rolling element 3 and the raceway of inner ring 1. Due to the high-speed operation of the bearing, rolling element 3 revolves a certain number of times per unit time, thus... The balls are first divided equally. Then, it is evenly distributed to each unit body that contacts the raceway of the inner ring 1 with the rolling element 3; through the inner contact angle Calculate the unit designation of the contact area between the inner ring 1 raceway and the rolling element 3, and then... The heat generated in the contact area between the inner ring 1 raceway and the rolling element 3 is evenly distributed to each unit body; the heat generated in the contact area between the rolling element 3 and the outer ring 2 raceway is... The rolling element 3 and the outer ring 2 raceway are each allocated half; through attitude angle and outer contact angle Calculate the element designation for the contact between rolling element 3 and the raceway of outer ring 2. The balls are first divided equally. Then, it is evenly distributed to each unit body that contacts the raceway of the outer ring 2 with the rolling element 3; through the outer contact angle Calculate the unit designation of the contact area between the raceway of outer ring 2 and rolling element 3, and... Heat is evenly distributed to each unit body in contact with the outer ring 2 raceway and the rolling element 3; frictional heat is generated between the rolling element 3 and the cage 4 pocket. The rolling elements 3 and the cage 4 are each allocated half; Loading the rolling element 3 into a unit with a contact angle of 0; Evenly distributed to cage 4 Each beam unit; frictional heat generation between cage 4 and outer ring 2. The cage 4 and outer ring 2 are each allocated half, and are respectively loaded onto the unit body in the contact area of ​​the cage 4 and outer ring 2; the drag friction between the rolling element 3 and the lubricating oil generates heat. The frictional heat generated by the churning torque of the rolling element 3 The rolling element 3 and the lubricating oil are each distributed equally, and are respectively loaded onto the outermost unit of the rolling element 3 and the lubricating oil heat exchange node; the oil churning torque of the cage 4 generates frictional heat. The cage 4 and the lubricating oil are each distributed in equal portions, and are respectively loaded onto the outermost unit of the cage 4 and the lubricating oil heat exchange node; (3) Heat transfer analysis Each unit cell and node possesses physical properties and temperature, and undergoes heat conduction or convection heat transfer with surrounding continuous unit cells or nodes. Heat conduction includes planar contact heat conduction, annular radial contact heat conduction, unit spherical shell radial contact heat conduction, unit spherical shell polar angle contact heat conduction, and oil film contact heat conduction, with the following thermal resistances:

[0083]

[0084]

[0085]

[0086]

[0087] In the formula and The thermal conductivity of the two unit cells is... and The thickness of the two unit cells, The contact area between the two unit cells; y is the axial length of the unit cell. , and The radii of the two unit cells are respectively ; , and The polar angles of the unit spherical shell are respectively , , and The radii of the unit spherical shell are respectively ; The Hertzian contact area between rolling element 3 and the raceways of inner ring 1 and outer ring 2. For oil film thickness, The thermal conductivity of the lubricating oil; The thermal resistance of each unit body, including the inner ring 1, outer ring 2, cage 3, shaft 5, and bearing housing 6, is calculated through planar contact heat conduction and annular radial contact heat conduction. The thermal resistance of their mating surfaces, including the contact surfaces between the inner ring 1 and shaft 5, and the outer ring 2 and bearing housing 6, is calculated through planar contact heat conduction and annular radial contact heat conduction. The internal thermal resistance of the rolling element 3 is calculated through radial contact heat conduction and polar angle contact heat conduction of the unit spherical shell. The thermal resistance of the contact areas between the rolling element 3 and the inner ring 1 and outer ring 2 is calculated through oil film contact heat conduction. The Hertzian contact area per unit time; Convective heat transfer includes the convective heat transfer between the lubricating oil and the inner ring 1, outer ring 2, rolling elements 3, cage 4, shaft 5, and bearing housing 6. Its convective thermal resistance is:

[0088] In the formula For convective heat transfer area, The average convective heat transfer coefficient between the bearing, its shaft, and bearing housing and the lubricating oil is: , The kinematic viscosity of the lubricating oil. The rotational speed of the inner ring of the bearing is 1. The Prandtl number for lubricating oil flow; the thermal resistance of each unit body in contact with the lubricating oil, including the inner ring 1, outer ring 2, rolling elements 3, cage 4, shaft 5, and bearing housing 6, is calculated using the average convective heat transfer coefficient of the lubricating oil, which is corrected according to actual conditions. Lubricating oil inlet node With lubricating oil outlet node The relationship between them is , For quality flow, Total heat generated by the bearing; Lubricating oil inlet node Heat exchange nodes with lubricating oil The relationship between them is , It serves as the heat source for the lubricating oil heat exchange node; (4) Solving the temperature field For unit cells or nodes In general, it can have heat transfer relationships with a maximum of four surrounding unit cells or nodes, thus obtaining a steady-state unit cell or node. The heat balance equation matrix is:

[0089] In the formula To be connected with unit or node The unit or node codes that have heat transfer relationships are used to combine all unit or node codes into a multidimensional sparse matrix according to the above formula. The coordinates, width, height and temperature of each unit of the bearing, its shaft and bearing housing are obtained by solving the Gaussian elimination method. Step 5, update the comprehensive radial deformation calculation: Calculate the component deformation under the combined effects of assembly, centrifugal force, and temperature rise using the comprehensive radial deformation formula mentioned in Step 1. , , , , , , And update the bearing interference fit accordingly. ; The temperature fields of inner ring 1, outer ring 2, shaft 5, and bearing housing 6 are radially discretely distributed. Ignoring the axial temperature distribution, then at any radius... The integral of temperature rise at that point is calculated as follows:

[0090] In the formula Let be the number of annexes in the integration interval. For the first The temperature rise of each ring, For the first The outer diameter of the ring, This is the inner diameter of the first ring; The internal temperature field of rolling element 3 is radially discretely distributed. Ignoring the circumferential temperature distribution, then at any radius... The integral of temperature rise at that point is calculated as follows:

[0091] In the formula Let be the number of spherical shells in the integration interval. For the first The temperature rise of the spherical shell, For the first The outer diameter of the spherical shell, .

[0092] Step 6, Thermal Coupling Iteration: The updated radial deformation of the bearing components is fed back to step two. Steps two through five are repeated for coupled iteration until the temperature difference between two adjacent iterations of the lubricating oil outlet is less than 0.1℃. If the convergence condition is not met, steps two through five are skipped to update the component deformation affected by the combined effects of assembly, centrifugation, and temperature rise. , , , , , , , In the formula If the convergence condition is met, proceed to step seven.

[0093] Step 7: Output the convergence result: The output bearing kinematics and dynamics parameters include the displacement, rotational speed, revolution speed, contact angle, and attitude angle of rolling element 3; the displacement of cage 4; the displacement and deflection angle of inner ring 1; and the interaction relationships between inner ring 1, outer ring 2, rolling element 3, cage 4, and lubricating oil. Distributed heat generation includes elastic hysteresis friction heat generation, differential sliding friction heat generation, and spin sliding friction heat generation between rolling element 3 and the raceways of inner ring 1 and outer ring 2; friction heat generation between rolling element 3 and the pocket of cage 4; friction heat generation between cage 4 and outer ring 2; drag friction heat generation between rolling element 3 and lubricating oil; and churning torque friction heat generation between rolling element 3 and cage 4. The temperature field includes the coordinates, width, height, and temperature of each unit of the bearing, its shaft, and bearing housing.

[0094] Use such as Figure 2 In the specific embodiment shown, it should be understood that the cage churning torque is not considered in the pseudo-dynamic analysis, and the finite element thermal network temperature field is thermally insulated from the environment by the apparent axis and seat. The input parameters are shown in Table 1, which is a table of structural and material parameters of the bearing, its shaft, and bearing seat. The partitioning is shown in Table 2, which is a table of element and node parameters. Based on Table 2, a pseudo-dynamic distributed heat generation-finite element thermal network temperature field-thermal coupling analysis is performed. After 4 iterations, the convergence condition is reached. Some kinematic and dynamic parameters of the output are shown in Tables 3a and 3b, and the distributed heat generation output is shown in Table 4, which is a table of distributed heat generation results. The output temperature field is post-processed, and the two-dimensional temperature field of the axial section with a position angle of 0° is visualized, as shown in Table 4. Figure 8The results are shown in Table 3a. Table 3a shows the kinematic and dynamic results. As can be seen from Table 3a, the contact angle and contact stress distribution of each ball are uneven after loading, which proves that the pseudo-dynamic model effectively considers the internal load distribution under combined load. Table 3b shows the kinematic and dynamic results. As can be seen from Table 3b, the inner and outer raceways and the balls all undergo a certain amount of radial deformation. These deformations are not negligible for calculating the elastic deformation of the ball and the inner and outer raceways, which reflects the necessity of thermo-mechanical coupling iteration. The distributed heat generation results in Table 4 show that the drag loss of the ball is the main fluid viscous friction heat source, and the differential sliding friction and spin sliding friction between the ball and the raceways are the main mechanical friction heat sources. The inner and outer raceways contribute differently, which verifies the precision of the distributed heat generation model. Figure 8 The method clearly reveals the local high-temperature points in the inner and outer groove regions, which cannot be captured by traditional thermal network models. Moreover, the computational complexity and computation time are much less than those of the full three-dimensional finite element method. The implementation of this embodiment verifies the effectiveness of the method of the present invention. The output results can provide accurate thermal analysis for high-speed ball bearings and provide data support for bearing design and application.

[0095] Table 1

[0096] Table 2

[0097] Table 3a

[0098] Table 3b

[0099] Table 4

[0100] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A bearing thermal-mechanical coupling analysis method based on a finite element thermal network, characterized in that, Includes the following steps: Step 1: Calculate the initial radial deformation of the bearing inner ring, outer ring, rolling elements, shaft, and bearing housing under the action of assembly interference and centrifugal force; The initial radial deformation of the bearing inner ring, outer ring, shaft, and bearing housing is calculated using the axisymmetric equilibrium equation of a thick-walled circular ring in cylindrical coordinates in elasticity mechanics. Based on the assumption of plane strain or plane stress, a comprehensive radial deformation calculation formula considering assembly, centrifugal force, and temperature rise is obtained. For the shaft, the corresponding formula is used depending on whether it is a hollow or solid structure. The bearing interference fit pressure is calculated based on the relationship between interference and fit pressure proposed by Harris, and the initial radial deformation at each key position is solved accordingly. Step 2: Based on the initial radial deformation obtained in Step 1, an improved pseudo-dynamic model that comprehensively considers fluid viscous friction under different lubrication methods is used to perform steady-state analysis of the bearing and solve for the accurate kinematic and dynamic parameters of the bearing. The improved pseudo-dynamic model includes the displacement-deformation coupling relationship of bearing elements and the pseudo-dynamic equilibrium equations of the bearing. The displacement-deformation coupling relationship of the bearing elements is used to calculate the contact angle and elastic approximation between the ball and the inner and outer rings after dynamic loading. The pseudo-dynamic equilibrium equations of the bearing include the rolling element equilibrium equations, the cage equilibrium equations, and the inner ring equilibrium equations, and systematically introduce fluid viscous friction under jet lubrication and under-ring lubrication, including drag force and churning torque, and solve using an iterative method. The precise kinematic and dynamic parameters of the bearing include ball displacement, rotation speed, revolution speed, contact angle, attitude angle, cage displacement, inner ring displacement, deflection angle, and the interaction relationship between the inner ring, outer ring, ball, cage, and lubricating oil. Step 3: Based on the bearing kinematics and dynamics parameters obtained in Step 2, calculate the precise distributed heat generation of each friction pair inside the bearing; The precise distributed heat generation of the bearing includes heat generation from elastic hysteresis friction between the rolling elements and the inner and outer raceways. Heat generation from differential sliding friction Heat generation from spin sliding friction Friction between the rolling elements and the cage pockets generates heat. The friction between the cage and the guide ring generates heat. Heat is generated by the drag friction between the rolling elements and the lubricating oil. The churning torque and frictional heat generated by the rolling elements and cage ; Step 4: Apply the finite element thermal network method to discretize the bearing system into a structured mesh, and accurately apply a heat source based on the distributed heat generation obtained in Step 3. Then, obtain a refined bearing temperature field by solving the steady-state thermal balance equations. The finite element thermal network method specifically includes the following steps: (1) Method construction: Combine the heat network method to construct the heat balance equation matrix of all nodes; (2) Mesh generation: Discretize the bearing inner ring, outer ring, ball, cage, shaft and bearing housing into a structured mesh, and reasonably divide it into a specific number of elements according to the accuracy requirements and computing resources, and set environmental nodes and lubricating oil nodes; (3) Heat source loading: The distributed heat generation of the bearing calculated in step three is distributed and loaded onto the corresponding contact area unit or node according to the preset rules; (4) Heat transfer analysis: Calculate the thermal resistance of heat conduction and heat transfer between unit cells or nodes; (5) Temperature field solution: Based on the heat transfer relationship between each unit or node, the proposed multidimensional sparse matrix is ​​constructed by combining the methods, and the bearing temperature field is obtained by solving the Gaussian elimination method. Step 5: Based on the bearing temperature field obtained in Step 4, calculate the combined radial deformation of the bearing, its shaft, and bearing housing under the combined effects of assembly interference, centrifugal force, and temperature rise, so as to update the internal geometric relationship of the bearing. Step 6: Feed back the updated radial deformation of the bearing components to Step 2, and repeat Steps 2 to 5 for coupled iteration until the change in lubricating oil outlet temperature converges to the set threshold. The convergence criterion for the coupled iteration is that the temperature difference between the lubricating oil outlets in two adjacent iterations is less than 0.1℃. If the convergence condition is not met, the process jumps to steps two through five to update the component deformation caused by the combined effects of assembly, centrifugation, and temperature rise. , , , , , , , In the formula If the convergence condition is met, proceed to step seven. Step 7: Output the converged bearing kinematics and dynamics parameters, distributed heat generation, and temperature field.

2. The bearing thermo-mechanical coupling analysis method based on finite element thermal networks as described in claim 1, characterized in that: The axisymmetric equilibrium equation in step one is as follows: In the formula Radial coordinates, Radial stress, For circumferential stress, For density, It is the angular velocity of rotation; The introduced geometric equations and constitutive equations are as follows: In the formula For radial displacement, For radial normal strain, For circumferential positive strain, For axial normal strain, For axial stress, , , These are Poisson's ratio, elastic modulus, and coefficient of thermal expansion, respectively. For temperature rise; Considering the axial positioning of the bearing's inner and outer rings, i.e., the axial deformation of the bearing's inner and outer rings is constrained, the plane strain assumption is adopted. , The formula for calculating the comprehensive radial deformation considering assembly, centrifugal force, and temperature rise is obtained as follows: In the formula and These are the inner and outer diameters of the ring, respectively. and It is the pressure applied radially to the inner and outer surfaces of the ring. This is the radial temperature rise distribution function; If the bearing housing is fixed by a sleeve or other structure, i.e., its axial position is constrained, then the radial deformation calculation formula based on the plane strain assumption is used. For the shaft, only the portion mating with the inner ring of the bearing is considered. If it is a hollow shaft, the radial deformation calculation formula based on the plane strain assumption is used; if it is a solid shaft, the radial deformation calculation formula based on the plane strain assumption with an inner diameter of 0 is used. If the axial position of the bearing housing is not constrained, the comprehensive radial deformation calculation formula derived through the plane stress assumption is used: The relationship between interference and fitting pressure proposed by Harris and It can be calculated as: If the bearing inner ring is interference-fitted with the shaft, the stress on the outer surface of the inner ring will be... , inner surface stress of the shaft There is a stress distribution between the inner surface of the inner ring and the shaft. : Thus, the inner channel was calculated. Comprehensive radial deformation at the location Inner ring inner surface Comprehensive radial deformation at the location outer surface of shaft Comprehensive radial deformation at the location ; If the bearing outer ring and bearing housing have an interference fit, the stress on the inner surface of the outer ring will be... stress on the outer surface of the bearing housing There is stress distribution between the outer surface of the outer ring and the inner surface of the bearing housing. : Thus, the outer channel was calculated. Comprehensive radial deformation at the location Outer ring outer surface Comprehensive radial deformation at the location and the inner surface of the bearing housing Comprehensive radial deformation at the location ; Similarly, for a solid sphere, the diameter is... The temperature rise causes any radius The radial deformation at that point is: In the initial deformation calculation, the initial temperature field of the bearing, its shaft, and bearing housing is uniform and without temperature difference. The overall deformation is caused only by assembly and centrifugation. The calculation results are as follows: , , , , , , ; At this time, the bearing interference fit is updated synchronously. , This is the initial interference.

3. The bearing thermo-mechanical coupling analysis method based on finite element thermal networks as described in claim 1, characterized in that: The displacement-deformation coupling relationship of the bearing element in step two includes: The inner ring is subjected to combined external loads. , , , , Afterwards, it will occur relative to the outer circle. , , , , Five degrees of freedom displacement, simultaneously causing elastic convergence of the sphere and its inner and outer rings. , From the displacement-deformation compatibility condition, the distance between the center of curvature of the inner groove after dynamic loading and the center of curvature of the outer groove at rest is: In the formula The number of the rolling element. This is the original contact angle for angular contact ball bearings. For double-inner-ring ball bearings gasket corner , For the thickness of the gasket, For the original radial clearance, The diameter of the sphere, and These are the inner and outer groove curvature radius coefficients, respectively. The position angle of the ball, and These are the distances between the center of curvature of the sphere after dynamic loading and the outer groove when at rest, respectively; therefore, the formula for calculating the contact angle between the sphere and the inner and outer rings after dynamic loading is: Then calculate the elastic approximation of the ball with respect to its inner and outer rings. , : The bearing pseudo-dynamic equilibrium equation set includes: Fluid viscous friction includes drag force and churning torque, drag force The rotation of the rolling elements is affected by the lubricating oil, resulting in an oil churning torque. and The rotation of the rolling elements and the cage is affected by the lubricating oil, respectively. The rolling element equilibrium equations are as follows: In the formula The normal contact forces between the ball and the inner and outer rings, The drag force of the oil film on the ball, The rolling friction resistance at the contact area between the ball and the inner and outer rings. The horizontal component of the resultant hydrodynamic pressure acting on the center of the sphere. The force acting between the ball and the cage pocket in the normal direction. and These are the rolling friction resistance and sliding friction resistance in the contact inlet area between the ball and the cage pocket, respectively. and It is an inertial force. and For gyro torque, , and These are the three-dimensional rotational components of the sphere. , and The inertial torque is calculated using the finite difference method; The cage equilibrium equations are: In the formula The total number of rolling elements, The diameter of the pitch circle. , and To maintain the forces and frictional torques between the cage and guide ring under hydrodynamic lubrication, To maintain the weight of the rack; The inner circle equilibrium equations are: In the formula .

4. The bearing thermo-mechanical coupling analysis method based on finite element thermal networks as described in claim 1, characterized in that: The heat generated by the elastic hysteresis friction between the rolling element and the inner and outer raceways in step three is as follows: In the formula The elastic hysteresis coefficient of the material. This refers to the rotational speed of the inner or outer ring of the bearing. The revolution speed of the rolling element. This refers to the distance between the contact point between the rolling element and the inner and outer raceways and the bearing's axis of rotation. Let the principal curvature of the contact surface be a function of the contact surface; The heat generated by the differential sliding friction between the rolling elements and the inner and outer raceways is: In the formula The differential sliding speed between the rolling element and the raceway on the contact surface; The heat generated by the spin friction between the rolling element and the inner and outer raceways is as follows: In the formula These are the spin components of the rolling element on the inner and outer raceways. The coefficient of spin friction between the rolling element and the raceway. and These are the major and minor semi-axis lengths of the ellipses representing the contact ellipses between the rolling element and the inner and outer raceways, respectively. It is an elliptic integral of the second kind; The frictional heat generated between the rolling elements and the cage pocket is as follows: The frictional heat generated between the cage and the guide ring is: In the formula The rotational angular velocity of the inner or outer ring. To maintain the frame's rotational angular velocity, The heat generated by drag friction between the rolling elements and the lubricating oil is: If it is jet lubrication, take the above formula; if it is ring lubrication, take the below formula. The frictional heat generated by the churning torque is: 。 5. The bearing thermo-mechanical coupling analysis method based on finite element thermal networks as described in claim 1, characterized in that: The method construction in step four of the finite element thermal network method combines the thermal network method and the finite element concept. It is a finite element thermal network method applicable to ball bearings, their shafts, and bearing housings. A thermal network is formed by dividing the network into nodes and establishing the thermal resistance between the nodes. For a node in a heat network Thus, the heat balance equation is derived: In the formula Represents nodes Nodes with heat transfer relationships, For nodes The total number of nodes with heat transfer relationships. For temperature The function, For nodes With nodes Thermal resistance between them for Time's up The time interval of time, For the internal heat source per unit volume of the node, For node volume, For node density, Let be the specific heat capacity of the node; the first term on the left-hand side of the equation represents the heat transferred from surrounding nodes to the node. The heat generated by the node is represented by the second term on the left side of the equation, and the term on the right side of the equation is the net increase in node energy. During steady-state heat transfer, the temperature at any node in the system is constant. Neglecting radiative heat transfer, the right-hand side of the equation is 0. Let... The total number of system nodes is If the thermal resistance between nodes with no heat transfer relationship is taken as infinite, its reciprocal If it is 0, then: In the formula , For nodes All hot-loaded; make: The matrix of heat balance equations for all nodes is then: The specific mesh division is as follows: the bearing inner and outer rings, balls, and cage are discretized into a structured mesh along the shaft and bearing housing along the axial section, divided into several unit cells; the inner ring is divided radially into... Layer, each layer along the axial direction Each unit body includes a left retaining edge, a left channel, a right channel, and a right retaining edge, respectively. , , and Each unit body; the outer ring is divided radially into... Layer, each layer along the axial direction Each unit body includes a left retaining edge, a left channel, a right channel, and a right retaining edge, respectively. , , and Each unit body; the cage is divided into two parts: guide rails and crossbeams. The guide rails are radially divided into... Layer, each layer along the axial direction Each unit body, the lintel section is divided into: Each unit body; the axis is divided radially into... Layer, each layer along the axial direction Each unit body, at the leftmost point, at the point where it mates with the left side flange of the inner ring, at the point where it mates with the left side channel of the inner ring, at the point where it mates with the right side channel of the inner ring, at the point where it mates with the right side flange of the inner ring, and at the rightmost point respectively , , , , , Each unit body; the bearing housing is divided radially into... Layer, each layer along the axial direction Each unit body, at the leftmost point, at the point where it mates with the left side flange of the outer ring, at the point where it mates with the left side channel of the outer ring, at the point where it mates with the right side channel of the outer ring, at the point where it mates with the right side flange of the outer ring, and at the rightmost point respectively , , , , , Each unit body; the sphere is divided radially into... Layer, each layer along the polar angle direction One unit cell; the number of unit cells is reasonably selected according to the accuracy requirements and computing resources; and one environmental node, one lubricating oil inlet node, one lubricating oil heat exchange node, and one lubricating oil outlet node are set. The heat source allocation is specifically as follows: the heat generated in the contact area between the ball and the inner raceway is... The ball and the inner raceway each receive half of the load, which is then applied to the unit cells in the contact area between the ball and the inner raceway; the heat generated in the contact area between the ball and the outer raceway is... The ball and the outer race each receive half of the load, which is then applied to the unit cells in the contact area between the ball and the outer race ball; the friction between the ball and the cage pocket generates heat. The ball and cage are each allocated half, and are respectively loaded onto the unit body in the contact area between the ball and cage pockets; the friction between the cage and the guide ring generates heat. The cage and guide ring are each allocated half; if the outer ring is guided, the load is applied to the unit body in the contact area between the cage and the outer ring; if the inner ring is guided, the load is applied to the unit body in the contact area between the cage and the inner ring. Heat is generated by the drag friction between the ball and the lubricating oil. The frictional heat generated by the churning torque of the ball The ball and lubricating oil are each distributed equally, and are respectively applied to the unit body in the contact area between the ball and the lubricating oil and the lubricating oil heat exchange node; the oil stirring torque of the cage generates frictional heat. The cage and lubricating oil are each allocated half, and are respectively loaded onto the unit body in the contact area between the cage and the lubricating oil and the lubricating oil heat exchange node; The heat transfer analysis specifically involves: each unit cell and node possesses physical properties and temperature, and undergoes heat conduction or convection heat transfer with surrounding continuous unit cells or nodes; heat conduction includes planar contact heat conduction, annular radial contact heat conduction, unit spherical shell radial contact heat conduction, unit spherical shell polar angle contact heat conduction, and oil film contact heat conduction, with the following thermal resistances: In the formula and The thermal conductivity of the two unit cells is... and The thickness of the two unit cells, The contact area between the two unit cells; y is the axial length of the unit cell. , and The radii of the two unit cells are respectively ; , and The polar angles of the unit spherical shell are respectively , , and The radii of the unit spherical shell are respectively ; The Hertzian contact area between the ball and the inner and outer raceways. For oil film thickness, The thermal conductivity of the lubricating oil; Convective heat transfer includes convective heat transfer between the lubricating oil and the balls, inner ring, outer ring, cage, shaft, and bearing housing, as well as convective heat transfer between the surrounding environment and the hollow shaft and bearing housing. Its convective thermal resistance is: In the formula For convective heat transfer area, The average convective heat transfer coefficient between the bearing, its shaft, and bearing housing and the lubricating oil is: "+" indicates that the outer ring of the bearing rotates, and "-" indicates that the inner ring of the bearing rotates. The kinematic viscosity of the lubricating oil. The bearing ring speed, The Prandtl number represents the flow rate of the lubricating oil; the convective heat transfer coefficient between the shaft and bearing housing and the environment is... , The thermal conductivity of air. For the inner diameter of the hollow shaft or the outer diameter of the bearing housing, the Nusselt number is... , and These are the Reynolds number and Prandtl number for airflow, respectively. Lubricating oil inlet node With lubricating oil outlet node The relationship between them is , For quality flow, Total heat generated by the bearing; Lubricating oil inlet node Heat exchange nodes with lubricating oil The relationship between them is , It serves as the heat source for the lubricating oil heat exchange node; The temperature field solution is specifically as follows: all unit cells and nodes are connected by thermal resistance to form a thermal network. For each unit cell or node... In general, it can have heat transfer relationships with a maximum of four surrounding unit cells or nodes, thus obtaining a steady-state unit cell or node. The heat balance equation matrix is: In the formula To be connected with unit or node The unit or node codes that have heat transfer relationships are combined according to the above formula to construct the multidimensional sparse matrix proposed in (1), and solved by Gaussian elimination method. The obtained temperature field includes the coordinates, width, height, and temperature of each unit of the bearing, its shaft, and bearing housing.

6. The bearing thermo-mechanical coupling analysis method based on finite element thermal networks as described in claim 1, characterized in that: In step five, the comprehensive radial deformation is calculated using the comprehensive radial deformation formula mentioned in step one. Based on the discrete temperature field distribution obtained from the finite element thermal network model, the additional deformation caused by temperature rise is calculated through numerical integration, thereby obtaining the component deformation under the combined effects of assembly, centrifugation, and temperature rise, and the bearing interference is updated simultaneously.

7. The bearing thermo-mechanical coupling analysis method based on finite element thermal networks as described in claim 6, characterized in that: The comprehensive radial deformation calculation is based on a finite element thermal network model, where the temperature field inside the annulus is radially discretely distributed, ignoring the axial temperature distribution. Therefore, at any radius... The integral of temperature rise at that point is calculated as follows: In the formula Let be the number of annexes in the integration interval. For the first The temperature rise of each ring, For the first The outer diameter of the ring, This is the inner diameter of the first ring; The temperature field inside the sphere is radially discretely distributed; neglecting the circumferential temperature distribution, then at any radius... The integral of temperature rise at that point is calculated as follows: In the formula Let be the number of spherical shells in the integration interval. For the first The temperature rise of the spherical shell, For the first The outer diameter of the spherical shell, ; This allows for the calculation and updating of the overall radial deformation of components under the combined effects of assembly, centrifugation, and temperature rise. , , , , , , ; The bearing interference fit is updated synchronously. .

8. A bearing thermo-mechanical coupling analysis method based on a finite element thermal network as described in any one of claims 1-7, characterized in that: The bearing kinematic and dynamic parameters output in step seven include rolling element displacement, rotational speed, revolution speed, contact angle, attitude angle, cage displacement, inner ring displacement, deflection angle, and the interaction between the inner ring, outer ring, ball, cage, and lubricating oil; distributed heat generation includes elastic hysteresis friction heat generation, differential sliding friction heat generation, spin sliding friction heat generation between the ball and the inner and outer raceways, friction heat generation between the ball and the cage pocket, friction heat generation between the cage and the guide ring, drag friction heat generation between the ball and the lubricating oil, and oil churning torque friction heat generation between the ball and the cage; the temperature field includes the coordinates, width, height, and temperature of each unit of the bearing, its shaft, and bearing housing.

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