Construction method of bearing mechanical model considering flexibility and thermally induced preload

Abstract

The present invention discloses a method for constructing a bearing mechanical model considering flexibility and thermally induced preload, comprising the following steps: 1) calculating the thermally induced preload and the change in bearing contact angle caused thereby; 2) calculating the bending radial deformation of the inner and outer rings of the bearing; 3) obtaining the calculation formula for the normal load of the rolling elements on the inner ring of the bearing at any position after loading; 4) determining the axial displacement, radial displacement, angular displacement, the amount of bending deformation at the positions of each rolling element, and the load distribution of the bearing, thereby obtaining the mechanical model of the thin-walled double-row angular contact ball bearing considering the influence of flexibility and thermally induced preload. The present invention establishes a rigid-flexible coupling mechanical model of a thin-walled double-row angular contact ball bearing considering thermally induced preload based on the thin-walled ring theory and Hertz contact theory, improves the calculation accuracy of the contact load of the thin-walled double-row angular contact ball bearing in robots and aerospace systems, and improves the accuracy of bearing design.

Description

Technical Field

[0001] The present invention belongs to the technical field of bearing design, and particularly relates to a method for constructing a bearing mechanical model considering flexibility and thermally induced preload. Background Art

[0002] As the most commonly used type of machine tool spindle bearing, the angular contact ball bearing, the preload is one of the key factors affecting its operating state. The magnitude of the preload directly determines the key performance indicators of the bearing, such as rolling element slip, temperature rise, and fatigue life. The preloading methods of angular contact ball bearings can be divided into three types: positioning preloading, constant pressure preloading, and pressure regulating preloading. Currently, the popular bearing preloading method is the combination of bolts and positioning preloading, and the initial preload is applied by adjusting the screwed length of the bolts. However, under the action of factors such as high speed, cutting force, and initial preload, there is severe power loss inside the bearing, resulting in a large temperature rise of the spindle unit components, leading to a large thermally induced preload generated in the bearing. Since the initial preload applied by the combination of bolts and positioning preloading is unknown and it is difficult to compensate for the generated thermally induced preload, it will affect the normal service state of the bearing.

[0003] The thin-walled double-row angular contact ball bearing is increasingly widely used in the fields of robots and aerospace. 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, and the influence of the thermally induced preload on it is even greater. How to take into account the influence of the thermally induced preload during bearing design has become an urgent technical problem to be solved. Summary of the Invention

[0004] The purpose of the present invention is to overcome the deficiencies of the prior art and provide a method for constructing a bearing mechanical model considering flexibility and thermally induced preload, constructing a rigid-flexible combined mechanical model of a thin-walled double-row angular contact ball bearing considering the thermally induced preload, and thus effectively obtaining the design calculation of the contact load.

[0005] The present invention is realized through the following technical solutions:

[0006] A method for constructing a bearing mechanical model considering flexibility and thermally induced preload includes the following steps,

[0007] 1) According to the Hertz contact theory, calculate the thermally induced preload and the change in the bearing contact angle caused by it;

[0008] 2) According to the plane bending theory of thin-walled circular rings, calculate the bending radial deformation of the inner and outer rings of the bearing;

[0009] 3) Take the change in the distance between the curvature centers of the inner and outer grooves of the two rows of bearings at any position before and after loading as the total contact deformation of the rolling element and the inner and outer rings at that arbitrary position after loading, and obtain the calculation formula for the normal load of the rolling element on the inner ring of the bearing at any position after loading;

[0010] 4) Establish the equilibrium equations by considering the equilibrium of the inner ring of the bearing under external loads, thermally induced preload forces, and the forces of all rolling elements, as well as the deformation equilibrium of the bearing rings. When the external loads and torques are given, use the Newton-Raphson method for iterative solution to determine the axial displacement, radial displacement, angular displacement, the 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 considering the influence of flexibility and thermally induced preload forces.

[0011] As one of the implementation methods, the calculation formula for the contact angle between the rolling element and the inner raceway at any position angle of the rolling element after loading is

[0012] ,

[0013] ,

[0014] where 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 the 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 change angle of the contact angle caused by the thermally induced preload force, which is obtained by the following formula

[0015] ,

[0016] where is the axial preload force of the bearing, is the thermally induced preload force of the whole bearing, is the number of rolling elements, is the contact angle at the initial preload force. The calculation formulas for the groove curvature center distances of the left and right raceways after loading are

[0017] ,

[0018] ,

[0019] where , is the contact angle between the rolling element and the inner raceway at any position angle , is the axial displacement of the inner ring of the bearing after loading, is the trajectory radius of the curvature center of the inner raceway, The bending radial deformation of the inner and outer rings at any angular position of the bearing.

[0020] As one of the implementation manners, the radial deformation amount of the inner ring of the bearing is

[0021] ,

[0022] The radial deformation amount of the outer ring of the bearing is

[0023] ,

[0024] The bending radial deformation of the inner and outer rings at any angular position of the bearing is , ,

[0025] Wherein, is the bending moment of inertia of the outer ring cross-section, is the bending moment of inertia of the inner ring cross-section, and are the median circle 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, is the th rolling element-ring contact force, , is the radial force load, is the contact force at the position angle, is the series number.

[0026] As one of the implementation manners, the calculation formula for the normal load of the rolling element on the inner ring of the bearing at any position after loading is

[0027] ,

[0028] When , , , and are the normal loads of the rolling elements on the inner ring of the bearing at any position of the left and right rows after loading, is the load-deformation constant between the rolling element and the inner and outer rings, and are the total elastic deformation amounts at the contact points of the rolling elements in the left and right rows with the inner and outer rings after loading respectively.

[0029] As one of the implementation manners, the balance equation is:

[0030] ,

[0031] In the formula, is the radial force load, is the axial force load, is the moment load, is the bending radial deformation of the inner and outer rings at the position angle of the j-th rolling element. The subscripts 1 and 2 represent the left and right rows of rolling elements respectively. and are the contact angles between the rolling elements and the inner raceway after loading for the left and right rows at the rolling element position angle respectively, is the pitch diameter of the bearing, is the radial deformation of the outer ring of the bearing, is the radial deformation of the inner ring of the bearing.

[0032] As one of the implementation manners, the bearing is a thin-walled double-row angular contact ball bearing.

[0033] The advantages and beneficial effects of the present invention are as follows:

[0034] A thin-walled bearing refers to a bearing with an ultra-thin outer ring and inner ring, and the ratio of its outer diameter to inner diameter is usually less than 1.25. This structure makes the difference between the outer and inner diameters of the bearing very small, realizing the extreme thinness of the bearing wall. In this way, while maintaining high rigidity and load-carrying capacity, the thin-walled bearing has the advantages of light weight and space saving. Therefore, currently, thin-walled double-row angular contact ball bearings are more and more widely used in the fields of robots and aerospace. 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 relatively large after bearing the load. At present, the mechanical models of double-row angular contact ball bearings are all based on Hertz contact theory, with the premise of a rigid ring assumption. The ring deformation is not considered during calculation. And because thin-walled double-row angular contact ball bearings are often used in compact structures, it leads to the accumulation of heat inside the bearing, causing the generation of thermally induced preload, which in turn leads to the change of the contact angle. Ignoring these two factors will cause a large error in analyzing thin-walled bearings. Therefore, the present invention establishes a rigid-flexible coupling mechanical model of a thin-walled double-row angular contact ball bearing considering thermally induced preload based on thin-walled circular ring theory and Hertz contact theory. Through the mechanical model proposed by the present invention, the actual deformation and load-bearing situation of the bearing can be described more accurately, greatly improving the calculation accuracy of the contact load of thin-walled double-row angular contact ball bearings in robot and aerospace systems, improving the accuracy of bearing design, and providing a basis for the continuous development of bearings and the equipment to which they are applied in the future. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] Figure 1 is a schematic diagram of the contact angle change;

[0036] Figure 2 is a schematic diagram of the structure of a double-row angular contact ball bearing;

[0037] Figure 3 Schematic diagram of the bearing under combined loads

[0038] Figure 4 Schematic diagram of the bearing displacement under combined loads

[0039] Figure 5 Is the inner raceway contact line without preload

[0040] Figure 6 Is the outer raceway contact line without preload

[0041] Figure 7 Is the inner raceway contact line with preload

[0042] Figure 8 Is the outer raceway contact line with preload

[0043] In the figure, 1, outer ring; 2, rolling element; 3, inner ring

[0044] For those of ordinary skill in the art, without creative efforts, other relevant drawings can be obtained based on the above drawings Detailed implementation manners

[0045] In order to enable those skilled in the art to better understand the solution of the present invention, the technical solution of the present invention will be further described below in conjunction with the drawings and specific embodiments

[0046] Bearings, especially thin-walled double-row angular contact ball bearings, due to their thin and light design, high rigidity and load-bearing capacity, have been widely used in fields such as aerospace, robotics, automation equipment, and medical devices. It is precisely their application in high-precision and cutting-edge fields that requires high precision, stability, and reliability of the bearings. However, at present, most of the mechanical models of double-row angular contact ball bearings are based on the rigid body assumption, ignoring the flexural deformation of the inner and outer rings of the bearing and the change of the contact angle caused by the change of the preload force after operation. This has little impact on the analysis of ordinary bearings, but when used to analyze thin-walled bearings, due to their thin and light design, they are more likely to be affected by external loads and generate deformation, so the error is relatively large. This greatly limits the accurate description of the model for actual working conditions

[0047] First step, considering the change of the bearing contact angle caused by the change of the preload force, the thermal-induced preload force of the bearing is caused by the thermal expansion of the contact parts due to the temperature rise of the system. According to Hertz contact theory and referring to relevant literature, the calculation formula of the thermal-induced preload force can be expressed as

[0048] (1)

[0049] In the formula Is the thermal-induced preload force of the whole bearing 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 raceways of the inner ring 3 and the outer ring 1.

[0050] Among them, the total thermal deformation of the bearing in the contact direction is calculated by the following formula,

[0051] (2)

[0052] 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 elements, is the initial contact angle, is the thermal expansion coefficient of the inner and outer rings, is the thermal expansion coefficient of the rolling elements, is the outer diameter of the bearing, is the inner diameter of the bearing, is the pitch 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 steady state, is the initial temperature of the inner ring, is the initial temperature of the outer ring, is the temperature of the rolling elements after thermal steady state, is the initial temperature of the rolling elements.

[0053] The thermally induced preload usually increases rapidly in a short time during the bearing startup phase. However, due to the slow temperature difference change of the bearing housing, the thermally induced preload is easily ignored in the initial startup phase. If the thermally induced preload of the bearing increases to approach the dangerous threshold, it is extremely likely to cause bearing wear and premature failure, resulting in large contact stresses. Therefore, the contact stiffness of the inner and outer ring raceways is

[0054] (3)

[0055] In the formula is the contact stiffness between the inner raceway and the rolling elements, is the contact stiffness between the outer raceway and the rolling elements, is the Young's modulus of the rolling elements, is the Poisson's ratio of the rolling elements, 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.

[0056] The curvature sum of the contact points between the inner and outer rings of the bearing is calculated by the following formula:

[0057] (4)

[0058] In the formula: is the curvature of the inner ring contact point, is the curvature of the outer ring contact point, is the diameter of the rolling element, 、 are the curvature radii of the inner and outer raceways, 、 are the curvature radius coefficients of the inner and outer raceways.

[0059] The dimensionless deflection factor is calculated by the following formula:

[0060] (5)

[0061] 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, is the principal curvature difference function between the outer raceway and the rolling element.

[0062] When the rolling element contacts the raceway, the cross-sectional shape of the raceway is concave. Therefore, the principal curvature difference function between the rolling element and the raceway is calculated by the following formula:

[0063] (6)

[0064] When the bearing is running at high speed, the axial preload of the bearing is expressed by the following formula:

[0065] (7)

[0066] In the formula: is the axial preload of the bearing, is the contact angle at the initial preload, is any position of the bearing after loading where the contact angle between the rolling element and the inner raceway is located.

[0067] After the thermal-induced preload is generated, it will change the initial preload contact angle As Figure 1 shown, combined with the Hertz contact theory formula, the changed angle is:

[0068] (8)

[0069] In the formula: is the change angle of the contact angle caused by the thermally induced preload force.

[0070] In the second step, considering the bending radial deformation of the inner and outer rings of the bearing, the structure of the bearing is as Figure 2 shown. The bending deformation of the thin-walled bearing ring belongs to a plane problem. According to the plane bending theory of thin-walled circular rings and referring to relevant literature, the radial deformation of the inner ring of the bearing can be expressed as

[0071] (9)

[0072] The radial deformation of the outer ring of the bearing can be expressed as

[0073] (10)

[0074] In the formula, is the bending moment of inertia of the outer ring cross-section, is the bending moment of inertia of the inner ring cross-section, and are the trajectory radii of the curvature centers 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, is the th rolling element-ring contact force, ; is the contact force at the position angle, is the series, is the position angle of the rolling element.

[0075] The bending radial deformation of the inner and outer rings of the bearing at any position angle is

[0076] (11)

[0077] In the third step, establish the calculation formula for the normal load and of the rolling element on the inner ring of the bearing at any position after loading.

[0078] Assume that the outer ring of the bearing is fixed and the inner ring rotates. The curvature radius coefficients of the inner and outer grooves of the two rows of raceways are the same respectively. Take the left row of bearings as the first row and the right bearings as the second row. Take as the initial contact angle of the bearing in the unloaded state. Then the distance between the curvature centers of the inner and outer grooves at any position before the bearing is loaded is

[0079] (12)

[0080] 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 curvature radius coefficients of the inner and outer raceways respectively.

[0081] When the bearing is under the combined action of axial force , radial force and moment loads, relative axial displacement , relative radial displacement and relative angular displacement will occur between the inner and outer raceways. 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.

[0082] (13)

[0083] In the formula, is the pitch diameter of the bearing, is the diameter of the rolling element.

[0084] As Figure 3 and 4 shown, represents the angle between any other rolling element and the rolling element with the maximum load in any bearing of this row. Because the load distribution is symmetric, so .

[0085] After being loaded, at any position of the two rows of bearings, the distance between the curvature centers of the inner and outer grooves , are respectively,

[0086] (14)

[0087] (15)

[0088] In the formula, is the center distance between the two rows of rolling elements. As Figure 3 and 4 shown, is the groove center distance of the left raceway after being loaded, is the groove center distance of the right raceway after being loaded, and are respectively the axial and radial displacements of the inner ring of the bearing after being loaded, is the angular displacement, is the initial contact angle of the bearing.

[0089] According to Hertz contact theory, for a given contact between a rolling element and a raceway, the load-displacement relationship of the rolling element can be expressed as,

[0090] (16)

[0091] Wherein, is the rolling element load, is the total elastic deformation amount between the inner and outer raceways of the bearing under the action of the load, is the load-displacement constant.

[0092] Under the action of the load, the normal elastic deformation amount between the two raceways separated by the rolling elements is equal to the sum of the elastic deformation amounts of the rolling elements and each raceway, that is,

[0093] (17)

[0094] Wherein, is the normal elastic deformation amount between the two raceways separated by the rolling elements, is the elastic deformation amount at the contact between the rolling element and the inner ring of the bearing, is the elastic deformation amount at the contact between the rolling element and the outer ring of the bearing.

[0095] Therefore, the total load-deformation constant between the rolling elements and the inner and outer rings, and The relationship is,

[0096] (18)

[0097] Wherein, is the total load-deformation constant between the rolling elements 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.

[0098] Thus, the rolling element load in formula (16) and the contact elastic deformation can be expressed as,

[0099] (19)

[0100] Using the change amount of the groove center distance of the raceway before and after loading to represent the sum of the approach amounts of the rolling elements and each raceway, then formula (19) can be expressed as,

[0101] (20)

[0102] Wherein, is the distance between the curvature centers of the inner and outer grooves at any position after the bearing is loaded,

[0103] According to formulas (18) and (19), the load on the rolling elements at any position is expressed as

[0104] (21)

[0105] where, for ball bearings, .

[0106] Substituting the distance between the curvature centers of the inner and outer grooves at any position of the two rows of bearings after loading into the above formula, the normal loads of the rolling elements on the inner ring of the bearing at any position after loading are and , and ,

[0107] (22)

[0108] When , , .

[0109] Step 4: Establish the mechanical equilibrium equation of the bearing. According to the change in the contact angle caused by the thermally induced preload shown in Figure 1 and the geometric relationships shown in Figure 3 and 4 , the contact angles between the rolling elements and the inner raceway at any position and after loading are respectively

[0110] (23)

[0111] (24)

[0112] When , , . The inner ring of the bearing is in equilibrium under the external loads (axial force , radial force and torque ), the thermally induced preload and the forces of all the rolling elements.

[0113] In the radial direction,

[0114] (25)

[0115] where , are the radial forces of the two rows of rolling elements on the inner ring respectively.

[0116] In the axial direction,

[0117] (26)

[0118] In the formula, 、 are respectively the axial forces of the two rows of rolling elements on the inner ring.

[0119] The torque on the inner ring,

[0120] (27)

[0121] In the formula, 、 are respectively the torques of the two rows of rolling elements on the inner ring.

[0122] If the calculation method of Hertz contact theory is adopted to solve the three unknowns 、 and , three mechanical equilibrium equations are required. However, when considering the deformation of the bearing rings, attention should also be paid to the bending deformation at each rolling element position of the rings. Therefore, in addition to equations (25)-(27), the ring deformation equilibrium equation should be

[0123] (28)

[0124] The equilibrium formulas all contain the unknowns 、 、 and the bending deformations of the inner and outer bearing rings. When the external loads 、 and are given, by combining equations (25), (26), (27) and (28) and using the Newton-Raphson method to perform iterative solution on them, the axial displacement , the radial displacement , the angular displacement and the bending deformation at each rolling element position can be determined. Then, the load distribution of the bearing can be determined from 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.

[0125] The following is further illustrated through specific embodiments.

[0126] Take a double-row angular contact ball bearing, which is subjected to a radial load , an axial load , and a moment load , the bearing structure parameters are shown in Table 1. The bearing material parameters are shown in Table 2.

[0127] Table 1 Structural Parameters of Double-Row Angular Contact Ball Bearings

[0128]

[0129] Table 2 Material Parameters of Double-Row Angular Contact Ball Bearings

[0130]

[0131] Without considering the influence of preload, according to the rigid ring assumption (such as Zhu Kai, Zhang Sijia, Luo Zhengbo, etc. 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 non-linear equations to calculate the raceway load. The results show that the maximum raceway load is 1926.3 N. Considering the influence of preload and calculating according to the flexible ring, the maximum raceway load is 1963.1 N. This load difference reflects the real force change of the bearing under different calculation assumptions, indicating that the flexural deformation of the ring has a non-negligible influence on the raceway load.

[0132] Under the rigid ring assumption, it is calculated that the ring will not undergo obvious deformation, so the distribution of the raceway load is relatively idealized. However, in actual working conditions, especially when the bearing is subjected to preload, the ring will undergo small but important elastic deformation, which affects the contact angle, contact stiffness and load distribution of the rolling elements. This deformation results in a larger maximum load on the raceway than under the rigid ring assumption, meaning that under the same external load, the local load borne by the actual rolling elements is higher. By introducing the calculation of the flexible ring, the model can more accurately reflect the real force situation 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.

[0133] Take a double-row angular contact ball bearing, without preload ( Figure 5 , Figure 6 ), and with preload ( Figure 7 , Figure 8Calculate the raceway contact lines in the two cases respectively. It can be seen from the figure that with the change of the preload, the raceway contact lines of the bearing change, indicating that the contact angle will change under the influence of the preload. In bearing design, the change of the contact angle caused by the preload directly affects the stiffness, load-carrying capacity and friction characteristics of the bearing. If this change is not considered, it may lead to deviation in the calculation of the bearing stiffness, affecting the structural matching of the whole machine, and at the same time, it may also accelerate local fatigue damage due to uneven load distribution. In addition, a reasonable preload can compensate for the change of the bearing clearance caused by machining errors or thermal expansion, improving the positioning accuracy and service life of the bearing. Therefore, in the design process, it is necessary to accurately calculate the influence of the preload on the contact angle to optimize the bearing performance. In Figure 5 and Figure 6 In the case of no preload shown, the distribution of the raceway contact lines of the bearing is relatively symmetric, while in Figure 7 and Figure 8 under the action of preload, the raceway contact lines change significantly. The maximum contact angle increases by about 4%, the maximum contact point shifts about 1.2 mm at the outer raceway, and about 0.8 mm at the inner raceway. This shows that the change of the preload will cause the displacement of the raceway contact area, affecting the load distribution of the bearing.

[0134] In actual operation, the change of the contact angle has a direct impact on the running accuracy, life and thermal stability of the bearing. After correctly considering this factor, the stiffness of the bearing under load is more stable, which can improve the overall running accuracy of the machine, especially more obvious in high-precision equipment. In addition, reasonable preload control can optimize the change of the contact angle, reduce the local overload of the rolling elements, extend the service life of the bearing, while reducing the friction loss and improving the energy efficiency. Under the working conditions of high-speed operation or large temperature change, optimizing the change of the contact angle can also improve the thermal stability of the bearing, ensuring the long-term stable operation of the equipment. Therefore, in the thin-walled double-row angular contact ball bearing model, considering the change of the contact angle caused by the change of the preload is essential for the accuracy of the model solution.

[0135] The above has made an exemplary description of the present invention. It should be noted that without departing from the core of the present invention, any simple deformation, modification or equivalent replacement that can be made by those skilled in the art without creative labor 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, thermal induced preload and all rolling element forces, as well as the deformation balance of the bearing ring, the equilibrium equation is constructed. When the external load and torque 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. The mechanical model of the thin-walled double-row angular contact ball bearing under the influence of flexibility and thermal induced preload is obtained. The contact angle between the rolling element and the inner raceway at any position angle of the rolling element after loading 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.

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: 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.

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: 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.

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 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.

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 bearing is a thin-wall double-row angular contact ball bearing.

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