Construction method of bearing mechanical model considering flexibility and thermally induced pretightening force

Abstract

The invention discloses a method for constructing a bearing mechanical model considering flexibility and thermally induced pretightening force, which comprises the following steps of: 1) calculating the thermally induced pretightening force and the change of a bearing contact angle caused by the thermally induced pretightening force; 2) calculating bending radial deformation of inner and outer rings of the bearing; 3) obtaining a calculation formula of a normal load of a rolling body at any position of the loaded bearing to a bearing inner ring; and 4) determining axial displacement, radial displacement, angular displacement, bending deformation of each rolling body position and load distribution of the bearing, thereby obtaining the thin-wall double-row angular contact ball bearing mechanical model considering the influence of the flexibility and the thermally induced pre-tightening force. According to the method, the thin-wall double-row angular contact ball bearing rigid-flexible combined mechanical model considering the thermally induced pretightening force based on the thin-wall circular ring theory and the Hertz contact theory is established, the calculation precision of the contact load of the thin-wall double-row angular contact ball bearing in a robot and an aerospace system is improved, and the accuracy of bearing design is improved.

Description

Technical Field

[0001] The invention belongs to the technical field of bearing design, and in particular relates to a method for constructing a bearing mechanics model taking flexibility and thermally induced preload into consideration. Background Art

[0002] Angular contact ball bearings are the most commonly used type of machine tool spindle bearings. Preload is one of the key factors affecting their operating status. The size of the preload directly determines the key performance indicators of the bearings, such as rolling element slippage, temperature rise and fatigue life. The preload methods of angular contact ball bearings can be divided into three types: positioning preload, fixed pressure preload and pressure-adjusting preload. The current popular bearing preload method is to use a combination of bolts and positioning preload, and apply the initial preload by adjusting the bolt screw length. However, the high speed, cutting force and initial preload cause severe power loss inside the bearing, causing a significant temperature rise in the spindle unit assembly, resulting in a large thermally induced preload in the bearing. The preload method using a combination of bolts and positioning preload has an unknown initial preload and is difficult to compensate for the generated thermally induced preload, which in turn affects the normal service state of the bearing.

[0003] Thin-walled double-row angular contact ball bearings are increasingly used in robotics and aerospace fields. Due to the thin-wall structural characteristics, the ratio of the diameter of the bearing ring to its thickness is very large. After bearing the load, the ring is greatly bent and deformed, and the thermally induced preload has a greater impact on it. How to take into account the influence of thermally induced preload when designing bearings has become a technical problem that needs to be solved urgently. Summary of the invention

[0004] The purpose of the present invention is to overcome the shortcomings of the prior art and provide a method for constructing a bearing mechanical model taking into account flexibility and thermally induced preload, a rigid-flexible mechanical model of a thin-walled double-row angular contact ball bearing taking into account thermally induced preload is constructed, and the design calculation of the contact load is effectively obtained.

[0005] The present invention is achieved through the following technical solutions: A method for constructing a bearing mechanical model considering flexibility and thermally induced preload comprises the following steps: 1) According to Hertz contact theory, calculate the thermally induced preload and the resulting change in bearing contact angle; 2) Calculate the bending radial deformation of the inner and outer rings of the bearing based on the thin-walled ring plane bending theory; 3) The change in the distance between the inner and outer groove curvature centers at any position of the two rows of bearings before and after loading is taken as the total contact deformation between the rolling element and the inner and outer rings at that position after loading, and the calculation formula for the normal load of the rolling element on the inner ring of the bearing at any position of the bearing after loading is obtained; 4) Considering the balance of the inner ring of the bearing under the external load, thermally induced preload and the forces of all rolling elements, as well as the deformation balance of the bearing ring, the equilibrium equation is constructed. When the external load and moment are given, the Newton-Raphson method is used for iterative solution to determine the axial displacement, radial displacement, angular displacement, bending deformation of each rolling element position and the load distribution of the bearing, thus obtaining the mechanical model of the thin-walled double-row angular contact ball bearing under the influence of flexibility and thermally induced preload.

[0006] As one implementation method, the contact angle between the rolling element and the inner raceway at any position angle of the rolling element after being loaded is calculated as follows: , , In the formula, is the initial contact angle, is the distance between the curvature centers of the inner and outer grooves at any position before the bearing is loaded, is the center distance between two rows of rolling elements, is the angular displacement of the inner ring of the bearing after loading, is the position angle of the rolling element, is the radial displacement of the inner ring of the bearing after loading, is the distance between the curvature centers of the inner and outer grooves at any position after the bearing is loaded, is the contact angle change caused by the thermally induced preload, which can be obtained by the following formula: , In the formula, is the axial preload of the bearing, is the thermally induced preload of the bearing as a whole, is the number of rolling elements, is the contact angle at the initial preload, and the calculation formula for the center distance of the groove curvature of the left and right raceways after loading is: , , In the formula, , Any position angle The contact angle between the rolling element and the inner raceway at is the axial displacement of the inner ring of the bearing after loading, is the trajectory radius of the inner raceway curvature center, It refers to the radial deformation of the inner and outer rings at any angle of the bearing.

[0007] As one embodiment, the radial deformation of the inner ring of the bearing for, , Radial deformation of the bearing outer ring for, , The radial deformation of the inner and outer rings at any angle of the bearing is , , in, is the bending moment of inertia of the outer ring section, is the bending moment of inertia of the inner ring section, and are the centerline radii of the inner and outer rings respectively, is the elastic modulus of the inner ring material, is the elastic modulus of the outer ring material, For the The contact force between the rolling element and the ring, , is the radial force load, Contact force Position angle, For the series.

[0008] As one implementation method, the normal load calculation formula of the rolling element at any position of the bearing on the inner ring of the bearing after loading is: , when hour, , , and is the normal load of the rolling element on the inner ring of the bearing at any position of the left and right rows of the bearing after loading, is the load-deformation constant between the rolling element and the inner and outer rings, and They are the total elastic deformation at the contact between the left and right rows of rolling elements and the inner and outer rings after loading.

[0009] As one implementation method, the equilibrium equation is: , In the formula, is the radial force load, is the axial force load, is the moment load, is the radial bending deformation of the inner and outer rings at the position angle of the jth rolling element. The subscripts 1 and 2 represent the left and right rows of rolling elements, respectively. , They are rolling element position angles The contact angle between the rolling elements and the inner raceway after the left and right rows are loaded. is the bearing pitch diameter, is the radial deformation of the bearing outer ring, is the radial deformation of the bearing inner ring.

[0010] As one implementation manner, the bearing is a thin-walled double-row angular contact ball bearing.

[0011] The advantages and beneficial effects of the present invention are: Thin-walled bearings refer to bearings with ultra-thin outer and inner rings, and the ratio of their outer diameter to inner diameter is usually less than 1.25. This structure makes the difference between the inner and outer diameters of the bearing very small, and realizes the extremely thin bearing wall. In this way, thin-walled bearings have the advantages of lightweight and space saving while maintaining high rigidity and load-bearing capacity. Therefore, thin-walled double-row angular contact ball bearings are increasingly used in robotics and aerospace fields. Due to the structural characteristics of the thin wall, the ratio of the diameter of the bearing ring to its thickness is very large, and the bending deformation of the ring is large after bearing the load. The current mechanical models of double-row angular contact ball bearings are based on Hertz contact theory, with the rigid ring assumption as the premise, and the ring deformation is not considered during calculation. In addition, since thin-walled double-row angular contact ball bearings are often used in compact scenarios, heat accumulates inside the bearing, causing the generation of heat-induced preload, which leads to changes in the contact angle. Ignoring these two factors will result in large errors in the analysis of thin-walled bearings. Therefore, the present invention establishes a rigid-flexible mechanical model of a thin-walled double-row angular contact ball bearing taking into account thermally induced preload based on thin-walled ring theory and Hertz contact theory. The mechanical model proposed in the present invention can more accurately describe the actual deformation and load-bearing conditions of the bearing, greatly improving the calculation accuracy of the contact load of thin-walled double-row angular contact ball bearings in robots and aerospace systems, improving the accuracy of bearing design, and providing a basis for the future sustainable development of bearings and the equipment used in them. BRIEF DESCRIPTION OF THE DRAWINGS

[0012] Figure 1 Schematic diagram of contact angle change; Figure 2 It is a schematic diagram of the structure of a double-row angular contact ball bearing; Figure 3 It is a schematic diagram of bearing load under combined load; Figure 4 Schematic diagram of bearing displacement under combined load; Figure 5 It is the inner raceway contact line without preload; Figure 6 It is the outer raceway contact line without preload; Figure 7 It is the inner raceway contact line under preload; Figure 8 It is the outer raceway contact line under preload condition.

[0013] In the figure, 1, outer ring; 2, rolling element; 3, inner ring.

[0014] For ordinary technicians in this field, other relevant drawings can be obtained based on the above drawings without any creative work. DETAILED DESCRIPTION

[0015] In order to enable those skilled in the art to better understand the technical solution of the present invention, the technical solution of the present invention is further described below in conjunction with the accompanying drawings and specific embodiments.

[0016] Bearings, especially thin-walled double-row angular contact ball bearings, have been widely used in aerospace, robotics, automation equipment, medical equipment and other fields due to their light and thin design, high rigidity and load-bearing capacity. It is precisely because of their application in high-precision fields that the accuracy, stability and reliability of bearings are very high. However, the current mechanical models of double-row angular contact ball bearings are mostly based on the rigid body assumption, ignoring the flexural deformation of the inner and outer rings of the bearings and the change in contact angle caused by the change in preload after operation. It has little effect on the analysis of ordinary bearings, but when used to analyze thin-walled bearings, due to their light and thin design, they are more susceptible to deformation caused by external loads, so the error is large. This greatly limits the model's accurate description of actual working conditions.

[0017] The first step is to consider the change in bearing contact angle caused by the change in preload. The thermally induced preload of the bearing is caused by the thermal expansion of the contact parts due to the temperature rise of the system. According to the Hertz contact theory and referring to relevant literature, the calculation formula of the thermally induced preload can be expressed as: (1) In the formula, is the thermally induced preload of the bearing as a whole, is the total contact stiffness coefficient of each rolling element 2, is the total thermal deformation of the rolling elements under bearing preload, , is the contact stiffness between the inner ring 3 and the outer ring 1 raceway.

[0018] Among them, the total thermal deformation of the bearing in the contact direction is Calculated by the following formula, (2) In the formula, is the axial thermal deformation of the inner and outer rings, is the radial thermal deformation of the inner and outer rings, is the thermal deformation of the rolling element, is the initial contact angle, is the thermal expansion coefficient of the inner and outer rings, is the thermal expansion coefficient of the rolling element, is the bearing outer diameter, is the bearing inner diameter, is the spacing of the bearing rolling elements, is the temperature of the inner ring after thermal steady state, is the temperature of the outer ring after thermal stabilization, is the initial temperature of the inner ring, is the initial temperature of the outer ring, is the temperature of the rolling element after thermal stabilization, is the initial temperature of the rolling element.

[0019] Thermally induced preload usually increases rapidly in a short period of time during the start-up phase of the bearing. However, due to the slow change in the temperature difference of the bearing seat, the thermally induced preload is easily ignored during the initial start-up phase. If the thermally induced preload of the bearing increases to a level close to the danger threshold, it is very likely to cause wear and premature failure of the bearing, resulting in greater contact stress. Therefore, the contact stiffness of the inner and outer raceways is for, (3) In the formula is the contact stiffness between the inner raceway and the rolling element, is the contact stiffness between the outer raceway and the rolling element, is the Young's modulus of the rolling element, is the Poisson's ratio of the rolling element, is the curvature of the bearing inner ring, is the curvature of the bearing outer ring, is the dimensionless deflection factor of the inner ring, is the dimensionless deflection factor of the outer ring.

[0020] The curvature of the contact point of the inner and outer rings of the bearing is calculated by the following formula: (4) In the formula, is the curvature of the inner ring contact point, is the curvature of the outer ring contact point, is the rolling element diameter, , are the inner and outer channel curvature radii, , is the curvature radius coefficient of the inner and outer channels.

[0021] Dimensionless deflection factor Calculated by the following formula, (5) In the formula, is the dimensionless deflection factor of the inner raceway, is the dimensionless deflection factor of the outer raceway, is the principal curvature difference function between the inner raceway and the rolling element, It is a function of the difference in principal curvature between the outer raceway and the rolling element.

[0022] When the rolling element contacts the raceway, the cross-sectional shape of the raceway is concave, so the principal curvature difference function between the rolling element and the raceway is calculated by the following formula: (6) When the bearing is running at high speed, the axial preload of the bearing is expressed by the following formula: (7) In the formula, is the axial preload of the bearing, is the contact angle at the initial preload, Any position of the bearing after loading The contact angle between the rolling element and the inner raceway.

[0023] The thermally induced preload will change the initial preload contact angle like Figure 1 As shown, combined with the Hertz contact theory formula, the angle of change for, (8) In the formula, is the contact angle change caused by thermally induced preload.

[0024] The second step is to consider the bending radial deformation of the inner and outer rings of the bearing. The structure of the bearing is as follows: Figure 2 As shown in the figure, the bending deformation of the thin-walled bearing ring belongs to a plane problem. According to the plane bending theory of thin-walled annulus and referring to relevant literature, the radial deformation of the inner ring of the bearing is It can be expressed as, (9) Radial deformation of the bearing outer ring It can be expressed as, (10) In the formula, is the bending moment of inertia of the outer ring section, is the bending moment of inertia of the inner ring section, and are the trajectory radii of the curvature centers of the inner and outer rings, is the elastic modulus of the inner ring material, is the elastic modulus of the outer ring material, For the The contact force between the rolling element and the ring, ; Contact force Position angle, is the series, is the position angle of the rolling element.

[0025] The inner and outer rings are bent and deformed radially at any angle of the bearing for, (11) The third step is to establish the normal load of the rolling element on the inner ring of the bearing at any position of the bearing after loading. and The calculation formula for .

[0026] Assume that the outer ring of the bearing is fixed, the inner ring rotates, and the inner and outer groove curvature radius coefficients of the two rows of raceways are consistent. The bearings in the left row are taken as the first row, and the bearings on the right row are taken as the second row. is the initial contact angle of the bearing under no-load condition, then the distance between the center of curvature of the inner and outer grooves at any position before the bearing is loaded is, (12) In the formula, is the distance between the curvature centers of the inner and outer grooves at any position before the bearing is loaded, is the total curvature, , , are the inner and outer channel curvature radius coefficients respectively.

[0027] Axial force on bearing , radial force and torque Under the combined action of load, the inner and outer ring grooves produce relative axial displacement , relative radial displacement and relative angular displacement Before the external load acts, the trajectory radii of the curvature centers of the two rows of inner raceways on the inner ring of the bearing are equal.

[0028] (13) In the formula, is the bearing pitch diameter, is the rolling element diameter.

[0029] like Figure 3 and 4 As shown, It represents the angle between any other rolling element in any bearing in this column and the rolling element with the maximum load. Because the load distribution is symmetrical, .

[0030] Any position of two rows of bearings after loading Distance between the center of curvature of inner and outer grooves , They are respectively, (14) (15) In the formula, is the center distance between two rows of rolling elements, such as Figure 3 and 4 As shown, is the left raceway groove center distance after loading, is the right raceway groove center distance after loading, and are the axial and radial displacements of the inner ring of the bearing after loading, is the angular displacement, is the initial contact angle of the bearing.

[0031] According to Hertz contact theory, for a given rolling element and raceway contact, the load-displacement relationship of the rolling element can be expressed as, (16) In the formula, is the rolling element load, is the total elastic deformation between the inner and outer raceways of the bearing under load, is the load-displacement constant.

[0032] Under load, the normal elastic deformation between two raceways separated by rolling elements is equal to the sum of the elastic deformation of the rolling element and each raceway, that is, (17) In the formula, is the normal elastic deformation between the two raceways separated by the rolling elements, is the elastic deformation at the contact point between the rolling element and the inner ring of the bearing, It is the elastic deformation at the contact point between the rolling element and the outer ring of the bearing.

[0033] Therefore, the total load-deformation constant between the rolling element and the inner and outer rings is The relationship is, (18) In the formula, is the total load-deformation constant between the rolling element and the inner and outer rings, is the load-deformation constant between the rolling element and the inner ring, is the load-deformation constant between the rolling element and the outer ring.

[0034] So formula (16) rolling element load Elastic deformation with contact It can be expressed as, (19) The sum of the approach distances between the rolling element and each raceway is expressed by the change in the raceway center distance before and after loading, then formula (19) It can be expressed as, (20) In the formula, is the distance between the curvature centers of the inner and outer grooves at any position after the bearing is loaded, According to formulas (18) and (19), the load on the rolling element at any position is The expression of is, (twenty one) In the formula, for ball bearings, .

[0035] The two rows of bearings under load will be placed at any position Distance between the center of curvature of inner and outer grooves , Substituting the above formula, we can get the normal load of the rolling element on the inner ring of the bearing at any position after loading: and , (twenty two) when hour, , .

[0036] The fourth step is to establish the mechanical balance equation of the bearing. Figure 1 The change in contact angle due to the thermally induced preload is shown as well as Figure 3 and 4 The geometric relationship shown in the figure shows that the bearing is in any position after loading. The contact angle between the rolling element and the inner raceway and They are respectively, (twenty three) (twenty four) when hour, , The inner ring of the bearing is subjected to external load (axial force , radial force and torque ), thermally induced preload and all rolling element forces are in equilibrium.

[0037] In radial direction, (25) In the formula, , They are the radial forces exerted by the two rows of rolling elements on the inner ring.

[0038] In the axial direction, (26) In the formula, , They are the axial forces exerted by the two rows of rolling elements on the inner ring.

[0039] The torque on the inner ring is (27) In the formula, , They are the moments acting on the inner ring by the two rows of rolling elements.

[0040] If the Hertz contact theory is used to calculate the three unknown quantities, , and , three mechanical equilibrium equations are required. However, when considering the deformation of the bearing ring, it is also necessary to pay attention to the bending deformation of the ring at each rolling element position. , so, in addition to equations (25)-(27), the ring deformation equilibrium equation should be, (28) The equilibrium formula contains unknown quantities. , , And the inner and outer rings of the bearing are bent and deformed A series of nonlinear equations, when the external load is given , and When , the axial displacement can be determined by combining equations (25), (26), (27) and (28) and iteratively solving them using the Newton-Raphson method. , radial displacement , angular displacement , Bending deformation at each rolling element position The load distribution of the bearing can be determined by equation (22). In summary, the mechanical model of the thin-walled double-row angular contact ball bearing considering the influence of flexibility and thermally induced preload can be obtained.

[0041] The following is further explained by means of specific embodiments. Take a double row angular contact ball bearing, which is subjected to radial load , axial load , moment load , bearing structure parameters are shown in Table 1. Bearing material parameters are shown in Table 2.

[0042] Table 1 Structural parameters of double row angular contact ball bearings Table 2 Double row angular contact ball bearing material parameters Without considering the influence of preload, according to the rigid ring assumption (such as Zhu Kai, Zhang Sijia, Luo Zhengbo, et al. Design and life and reliability research of double-row angular contact ball bearings [J]. Mechanical Engineer, 2024(6): 136-141, 145.), the Newton-Raphson method is used to iteratively solve the nonlinear equations to calculate the raceway load. The results show that the maximum raceway load is 1926.3N. Considering the influence of preload, according to the flexible ring calculation, the maximum raceway load is 1963.1N. This load difference reflects the actual force change of the bearing under different calculation assumptions, indicating that the influence of the flexural deformation of the ring on the raceway load cannot be ignored.

[0043] Under the assumption of rigid rings, the calculation assumes that the rings will not deform significantly, so the distribution of the raceway load is idealized. However, in actual working conditions, especially when the bearing is subjected to preload, the rings will undergo small but significant elastic deformation, affecting the contact angle, contact stiffness and load distribution of the rolling element. This deformation causes the maximum load on the raceway to be greater than under the assumption of rigid rings, which means that under the same external load, the actual rolling element is subjected to a higher local load. By introducing the calculation of flexible rings, the model can more accurately reflect the actual force conditions of the bearing, improve the reliability of the calculation results, and provide a more accurate basis for bearing selection and optimization for high-precision mechanical equipment.

[0044] Take a double row angular contact ball bearing and place it without preload ( Figure 5 , Figure 6 ) and with preload ( Figure 7 , Figure 8 ) The raceway contact line is calculated in two cases. It can be seen from the figure that the raceway contact line of the bearing changes under the change of preload, which means that the contact angle will change due to the influence of preload. In bearing design, the change of contact angle caused by preload directly affects the stiffness, load-bearing capacity and friction characteristics of the bearing. If this change is not taken into account, it may lead to deviations in the calculation of bearing stiffness and affect the structural matching of the whole machine. It may also accelerate local fatigue damage due to uneven load distribution. In addition, a reasonable preload can compensate for changes in bearing clearance caused by processing errors or thermal expansion, and improve the positioning accuracy and service life of the bearing. Therefore, during the design process, the effect of preload on the contact angle must be accurately calculated to optimize bearing performance. Figure 5 and Figure 6 In the case of no preload as shown, the bearing raceway contact line distribution is relatively symmetrical, while in Figure 7 and Figure 8 Under the action of preload, the raceway contact line changes significantly. The maximum contact angle increases by about 4%, the maximum contact point is offset by about 1.2mm on the outer raceway, and about 0.8mm on the inner raceway. This shows that the change in preload will cause the raceway contact area to shift, affecting the load distribution of the bearing.

[0045] In actual operation, the change of contact angle has a direct impact on the running accuracy, life and thermal stability of the bearing. After properly considering this factor, the stiffness of the bearing under load is more stable, which can improve the overall running accuracy of the machine, especially in high-precision equipment. In addition, reasonable preload control can optimize the change of contact angle, reduce local overload of rolling elements, extend the service life of bearings, reduce friction loss and improve energy efficiency. Under high-speed operation or large temperature changes, optimizing the change of contact angle can also improve the thermal stability of bearings and ensure long-term stable operation of equipment. Therefore, in the thin-walled double-row angular contact ball bearing model, considering the change of contact angle caused by the change of preload is essential for the accuracy of model solution.

[0046] The present invention is described above by way of example. It should be noted that, without departing from the core of the present invention, any simple deformation, modification or other equivalent replacement that can be made by those skilled in the art without inventive effort falls within the protection scope of the present invention.

Claims

1. A method for constructing a bearing mechanical model considering flexibility and thermally induced preload, characterized in that: The following steps are included: 1) According to Hertz contact theory, calculate the thermally induced preload and the resulting change in bearing contact angle; 2) Calculate the bending radial deformation of the inner and outer rings of the bearing based on the thin-walled ring plane bending theory; 3) The change in the distance between the inner and outer groove curvature centers at any position of the two rows of bearings before and after loading is taken as the total contact deformation between the rolling element and the inner and outer rings at that position after loading, and the calculation formula for the normal load of the rolling element on the inner ring of the bearing at any position of the bearing after loading is obtained; 4) Considering the balance of the inner ring of the bearing under the external load, thermally induced preload and the forces of all rolling elements, as well as the deformation balance of the bearing ring, the equilibrium equation is constructed. When the external load and moment are given, the Newton-Raphson method is used for iterative solution to determine the axial displacement, radial displacement, angular displacement, bending deformation of each rolling element position and the load distribution of the bearing, thus obtaining the mechanical model of the thin-walled double-row angular contact ball bearing under the influence of flexibility and thermally induced preload.

2. The method for constructing a bearing mechanical model taking into account flexibility and thermally induced preload as claimed in claim 1, characterized in that: The formula for calculating the contact angle between the rolling element and the inner raceway at any position angle of the rolling element after loading is: ， ， In the formula, is the initial contact angle, is the distance between the curvature centers of the inner and outer grooves at any position before the bearing is loaded, is the center distance between two rows of rolling elements, is the angular displacement of the inner ring of the bearing after loading, is the position angle of the rolling element, is the radial displacement of the inner ring of the bearing after loading, is the distance between the curvature centers of the inner and outer grooves at any position after the bearing is loaded, is the contact angle change caused by the thermally induced preload, which can be obtained by the following formula: ， In the formula, is the axial preload of the bearing, is the thermally induced preload of the bearing as a whole, is the number of rolling elements, is the contact angle at the initial preload, and the calculation formula for the center distance of the groove curvature of the left and right raceways after loading is: ， ， In the formula, , Any position angle The contact angle between the rolling element and the inner raceway at is the axial displacement of the inner ring of the bearing after loading, is the trajectory radius of the inner raceway curvature center, It refers to the radial deformation of the inner and outer rings at any angle of the bearing.

3. The method for constructing a bearing mechanical model taking into account flexibility and thermally induced preload as claimed in claim 1, characterized in that: Radial deformation of the bearing inner ring for, ， Radial deformation of the bearing outer ring for, ， The radial deformation of the inner and outer rings at any angle of the bearing is , , in, is the bending moment of inertia of the outer ring section, is the bending moment of inertia of the inner ring section, and are the centerline radii of the inner and outer rings respectively, is the elastic modulus of the inner ring material, is the elastic modulus of the outer ring material, For the The contact force between the rolling element and the ring, , is the radial force load, Contact force Position angle, For the series.

4. The method for constructing a bearing mechanical model taking into account flexibility and thermally induced preload as claimed in claim 1, characterized in that: The calculation formula for the normal load of the rolling element on the inner ring of the bearing at any position after loading is: ， when hour, , , and is the normal load of the rolling element on the inner ring of the bearing at any position of the left and right rows of the bearing after loading, is the load-deformation constant between the rolling element and the inner and outer rings, and They are the total elastic deformation at the contact between the left and right rows of rolling elements and the inner and outer rings after loading.

5. The method for constructing a bearing mechanical model taking into account flexibility and thermally induced preload as claimed in claim 1, characterized in that: The equilibrium equation is: ， In the formula, is the radial force load, is the axial force load, is the moment load, is the radial deformation of the inner and outer rings at the position angle of the jth rolling element. The subscripts 1 and 2 represent the left and right rows of rolling elements, respectively. , They are rolling element position angles The contact angle between the rolling elements and the inner raceway after the left and right rows are loaded. is the bearing pitch diameter, is the radial deformation of the bearing outer ring, is the radial deformation of the bearing inner ring.

6. The method for constructing a bearing mechanical model taking into account flexibility and thermally induced preload as claimed in claim 1, characterized in that: The bearing is a thin-wall double-row angular contact ball bearing.

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