Pseudo-statics calculation method for pretightening force of asymmetric back-to-back tapered roller bearing of wind power main shaft

Abstract

The invention belongs to the technical field of design, installation and adjustment of main bearings of wind generating sets, and discloses a quasi-statics calculation method for pre-tightening force of asymmetric back-to-back tapered roller bearings of a wind power main shaft. And solving the relative displacement of the inner ring and the outer ring of the bearing under the multi-load working condition by virtue of bearing analysis software Romax. Based on the relative displacement, the internal geometrical relationship, the material attribute and the contact deformation characteristic of the bearing are deeply combined, and a dominant quantitative correlation model between the axial pretightening force and the actually measured / simulated displacement is established through multi-physical field coupling relationship analysis. By using the model, the optimal pre-tightening force values required by the large tapered roller bearing and the small tapered roller bearing which are used in pairs in the wind turbine generator main shaft system are reversely calculated. According to the method, a systematic solution with theoretical support is provided, the pre-tightening force setting accuracy and calculation efficiency are remarkably improved, and a key calculation basis is provided for reliable design of a wind power main shaft bearing system and pre-tightening force optimization regulation and control.

Description

Technical Field

[0001] This invention belongs to the technical field of wind turbine generator main bearing design and assembly, and relates to a pseudo-static calculation method for the preload of an asymmetric back-to-back tapered roller bearing for wind turbine main shaft. Background Technology

[0002] Single-row tapered roller bearings are key support components in the main drivetrain of large wind turbine generators. They are typically arranged in back-to-back pairs in the main shaft system to withstand the axial thrust and radial loads transmitted from the rotor, as well as additional bending moments caused by tower deformation and nacelle deflection. Applying appropriate axial preload during assembly can eliminate internal clearance, improve system rigidity, suppress squealing vibrations and roller slippage, and is one of the key process parameters for ensuring the operational accuracy and fatigue life of the drivetrain.

[0003] Currently, there are two main approaches to determining the preload of tapered roller bearings for wind turbine main shafts in engineering: one is based on empirical formulas or clearance-preload curves, which indirectly controls the preload by measuring the difference in axial displacement between the inner and outer rings or adjusting the shim thickness. This type of method is mostly derived from machine tool spindles or automotive wheel hub bearings and is poorly adapted to the large size, low speed, and complex load spectrum of megawatt-level wind turbine main shaft bearings. The other approach is based on numerical simulation methods using finite element or multibody dynamics models, relying on commercial software such as Romax or self-developed programs, to repeatedly calculate the preload until the stiffness and temperature rise indicators are met. This modeling and solution process is complex and computationally expensive, which is not conducive to rapid parameter optimization in the engineering design stage.

[0004] In the study of preload mechanism, Bercea et al. analyzed the relationship between the initial axial compression of paired tapered roller bearings and bearing life in their paper "Optimum initial axial compression due to preload in an arrangement of two tapered roller bearings," but this was still based on an ideal symmetrical arrangement and simplified load conditions. Hagiu and Gafitanu studied the correlation between preload and life for machine tool spindle ball bearings, focusing mainly on ball bearings rather than large-size tapered roller bearings. Domestic scholars such as Luo Jiwei systematically summarized the analysis and calculation methods for rolling bearings, pointing out that preload, interference fit, and other assembly parameters have a significant impact on bearing stiffness and life, requiring refined calculation and optimization during the design phase.

[0005] For large asymmetric back-to-back tapered roller bearing assemblies such as those used in wind turbine main shafts, the following problems are also common:

[0006] First, existing empirical methods usually assume that the load is distributed among the rollers according to a simple cosine law, which makes it difficult to reflect the variation of the bearing area and the non-uniform load distribution characteristics under the coupled action of axial pre-displacement and radial displacement.

[0007] Secondly, although numerical simulation methods can provide relatively accurate preload and contact load distribution, they require the establishment of complex three-dimensional models and the calculation of a large number of parameters in a loop, making it difficult to quickly reverse engineer the optimal preload on the engineering site.

[0008] Third, for asymmetrical back-to-back tapered roller bearings such as those used in wind turbine main shafts, there is still a lack of analytical calculation methods that use a small amount of displacement results as input and explicitly solve for the required preload within a quasi-static framework. Summary of the Invention

[0009] This invention addresses the following shortcomings of existing methods for determining the preload of asymmetric back-to-back tapered roller bearings for wind turbine main shafts: the lack of a clear analytical relationship between the preload and the relative displacement of the inner and outer rings of the bearing; the difficulty of using empirical formulas to account for different assembly tolerances and load conditions; the large computational load of full three-dimensional numerical simulation and the difficulty of parameter sensitivity analysis; and the insufficient consideration of the influence of the load-bearing area and roller load distribution under complex loads, leading to deviations in the preload setting from the optimal value, thereby affecting the stiffness and fatigue life of the main shaft system.

[0010] The purpose of this invention is to propose a quasi-static calculation method based on load distribution coefficient and integral coefficient. Under given geometric-operating condition parameters such as bearing radial displacement, axial pre-displacement, and nominal contact angle, the method can quickly solve the axial preload required by an asymmetric back-to-back tapered roller bearing under the target operating condition, and can compare and verify the results with simulation platforms such as Romax.

[0011] The technical solution of the present invention:

[0012] A pseudo-static calculation method for the preload of an asymmetric back-to-back tapered roller bearing for wind turbine main shafts, comprising the following steps:

[0013] (1) Establish a calculation model for a multi-degree-of-freedom tapered rolling element bearing:

[0014] First, establish A computational model for a multi-degree-of-freedom tapered rolling element bearing with an asymmetric back-to-back three-degree-of-freedom single-row tapered rolling element bearing is presented. This multi-degree-of-freedom tapered rolling element bearing computational model treats the bearing inner ring as a rigid body with three degrees of freedom to characterize the overall deformation of the bearing inner ring in three directions under external load. At the same time, three degrees of freedom are established for each rolling element to describe the spatial attitude change of the rolling element relative to the inner and outer rings of the bearing.

[0015] The bearing inner ring is considered as a rigid body with three degrees of freedom, namely radial displacement. axial displacement and bending angles For the first A rolling element is introduced with three degrees of freedom: radial displacement. axial displacement and overturning angle ;in, The asymmetric back-to-back three-degree-of-freedom single-row tapered rolling element bearing forms a multi-degree-of-freedom tapered rolling element bearing with 3Z+3 degrees of freedom. Represented by a generalized coordinate vector, we have:

[0016] q = [ d r , I R , d a , I R , i b , d x j 1 , d y j 1 , ψ j 1 , … , d x j Z , d y j Z , ψ j Z ] T

[0017] The geometric characteristics of asymmetric back-to-back three-degree-of-freedom single-row tapered rolling element bearings are uniformly defined as a geometric parameter interface. Through the geometric parameter interface, the geometric characteristics of different batches of asymmetric back-to-back three-degree-of-freedom single-row tapered rolling element bearings or bearings under different error conditions are read into the multi-degree-of-freedom tapered rolling element bearing calculation model, realizing configurable modeling of geometric parameters. Among them, geometric characteristics include geometric parameters and manufacturing error parameters. Geometric parameters include rolling element diameter, effective contact length of rolling element, raceway cone angle, raceway modification amount, flange position, bearing inner ring radius, and bearing pitch circle diameter.

[0018] (2) Establish the contact deformation compatibility relationship and load balance equation:

[0019] Based on the established calculation model of multi-degree-of-freedom tapered rolling element bearings, the contact deformation coordination relationship between the rolling elements and the inner ring raceway, outer ring raceway and flange is further established, and the mechanical equilibrium equation of a single rolling element is given, thus forming a complete set of nonlinear equations.

[0020] First, there are the coordinates and geometric relationships of the sliced ​​unit. For the first... One rolling element, along the effective contact length Divided into If there are 10 slices of equal thickness, then the thickness of each slice is:

[0021]

[0022] Establish a system with the center of the rolling element as the origin, along the axial direction of the rolling element. axis, then the first The center coordinates of each slice are:

[0023]

[0024] in, , representing half the number of slices; the circumferential position angle of the rolling element is:

[0025]

[0026] The diameter of the rolling element at the slice point is denoted as:

[0027]

[0028] in, The diameter of the middle part of the rolling element. The cone angle of the rolling element;

[0029] First, regarding the contact deformation of the inner and outer ring raceways of the bearing, let the degrees of freedom of the rolling elements in the radial and axial directions be respectively... and The nominal contact angle of the bearing outer ring is ;No. Contact deformation of each rolling element in the outer raceway Represented as:

[0030]

[0031] Second, the contact deformation between the rolling elements and the inner ring raceway should be considered, taking into account the axial displacement of the rigid body of the bearing inner ring. radial displacement and the overturning angle of the rolling element in the circumferential direction The nominal contact angle of the bearing inner ring is Then the first Contact deformation of the rolling elements at the inner ring raceway of the bearing Expressed as:

[0032]

[0033] Third, the contact deformation between the rolling element and the flange is the nominal contact angle characterizing the rollover of the rolling element. The correction defines the rolling element overturning angle. The actual contact angle of the rolling element after rolling for:

[0034]

[0035] Then the first Contact deformation of the rolling element at the flange Written as:

[0036]

[0037] Next are the nonnegative deformation constraints and contact loads. All the contact deformations mentioned above must satisfy the nonnegative constraints:

[0038]

[0039] Therefore, the non-negative part of the contact deformation of the retaining edge is recorded as:

[0040]

[0041] when When there is no contact at the corresponding position, the contact load is taken as zero.

[0042] For bending deformation correction and slice element bending moment arm, consider the applied bending moment. The bending deformation of the bearing inner ring under the action of force is assumed to conform to the bending theory of beams, and the reference bending deformation of the inner ring is denoted as... The radial clearance is In the The first rolling body At each slice, the correction for bending deformation is written as:

[0043]

[0044] in, The rolling body shaping function is defined as follows:

[0045] P ( x k ) = − 4 . 5 × 1 0 − 4 D m l n [ 1 − ( 2 x k L Re ) 2 ]

[0046] in, It is from the center of the rolling element to the first The distance between the centers of each slice;

[0047] For rolling bodies, only in To generate bending contact load during positive time, the following definition applies:

[0048]

[0049] No. The moment arm of the contact load of the slice relative to the center of the right end of the rolling element is denoted as . When an external bending moment is applied At that time, there were:

[0050] l k M = + [ L R e 2 − x t ( k ) ] c o s α 1 2 − D k 2 s i n α 1 2

[0051] when ,have:

[0052] l k M = − [ L R e 2 + x t ( k ) ] c o s α 1 2 − D k 2 s i n α 1 2

[0053] In constructing the vertical moment balance equation, it is also necessary to distinguish the moment arms of the left and right halves of the slice. and Its expression is:

[0054]

[0055] The moment arm of the flange load relative to the center of the rolling element is denoted as . , written as:

[0056]

[0057] in, The inner ring radius of the bearing. The diameter of the rolling element's pitch circle. The diameter of the right-hand center of the rolling element is given by the following formula:

[0058]

[0059] According to the Hertzian line contact stiffness relationship, the first The first rolling body The normal contact load of each slice on the inner ring side is:

[0060]

[0061] No. The slice normal load of each rolling element on the outer ring side is:

[0062]

[0063] in, Here is the line contact stiffness coefficient. This is the Reusner correction factor, used to correct for local load distribution along the length of the rolling element;

[0064] Summing all slices along the rolling element axis yields the first... The total normal load of the contact area between the rolling element and the inner and outer rings:

[0065]

[0066] Edge contact load utilizes corrective deformation The calculation is as follows:

[0067]

[0068] in, The contact stiffness coefficient of the flange; the first The bending internal moment of each rolling element is obtained by superimposing the bending moment contributions of each slice:

[0069]

[0070] Establish the mechanical equilibrium equations for a single rolling element, and for the th rolling element... The force balance equations for each rolling element in the bearing radial direction (y-direction) and axial direction (x-direction), and the moment balance equation in the z-direction, are given, where the z-direction is perpendicular to the plane formed by x and y. Let the equivalent concentrated radial load acting on the rolling element be... Then the equilibrium equation in the y-direction is:

[0071]

[0072] The equilibrium equation in the x-direction is:

[0073]

[0074] With the center of the rolling element as the reference point, the torque balance in the z-direction can be written as:

[0075]

[0076] in, For concentrated loads relative to the center of the rolling element The moment arm; the moment arms of the left and right halves of the slice are defined as follows: and , This represents half the number of rolling element slices;

[0077] (3) Establish the global moment equilibrium equation and use an iterative algorithm to solve for the load distribution and displacement field:

[0078] After obtaining the relationship between contact deformation and contact load, radial force balance equation, axial force balance equation and balance bending moment equation are established based on the overall static equilibrium condition, forming a high-dimensional nonlinear equation set about the bearing inner ring degree of freedom and each rolling element degree of freedom, and solving it using an iterative algorithm.

[0079] First, we need to establish the expression for the line contact stiffness coefficient. With the effective length of the rolling element The relationship is written as:

[0080]

[0081] Secondly, a radial force balance equation is constructed to ensure that the sum of the radial components of the bearing outer ring contact load is equal to the external radial load. Balance, written as:

[0082]

[0083] Next, the axial force balance equation is constructed, taking the axial load... This is related to the axial component of the contact load on the outer ring of the bearing:

[0084]

[0085] The next step is to construct the equilibrium bending moment equation, coupling the bending effect through slice elements and lever arm integration, denoted as... For the first Given the lever arm of a slice relative to the neutral axis of bending, then:

[0086]

[0087] The radial force balance equation, axial force balance equation and bending moment balance equation of the bearing, together with the rolling element x, y and z three-dimensional balance equations constructed in step (2), constitute a 3Z+3-dimensional nonlinear equation system.

[0088] The radial load of the bearing is given according to the actual working conditions. Axial load and additional bending moment Engineering experience values ​​are selected as the degree-of-freedom vector. The Levenberg-Marquardt nonlinear least squares iterative algorithm is used to numerically solve the above nonlinear equations. The iteration accuracy is controlled by a dual convergence criterion of displacement tolerance and function tolerance. Once the iteration converges, the axial displacement of the bearing inner ring in three degrees of freedom is obtained. radial displacement With corner The distribution of rolling elements and the contact load distribution between each rolling element and its slice unit and the outer ring of the bearing. ;

[0089] (4) Mechanical transfer of angular contact ball bearings:

[0090] First, establish the basic relationship between the center thrust load of a tapered roller bearing:

[0091]

[0092] in, This is the working contact angle; to obtain a linear load-displacement relationship, it is necessary to define the assembly pre-displacement. With initial contact angle Geometric constraints:

[0093]

[0094] in, This is the effective length coefficient of the roller. Where the diameter is the roller diameter;

[0095] When a bearing enters its working state, the normal displacement of the bearing originates from the working contact angle. With initial contact angle Differences:

[0096]

[0097] Based on Hertzian contact theory, a nonlinear relationship between load and displacement is further established:

[0098]

[0099] in, This is the contact stiffness coefficient;

[0100] (5) Combined load response of tapered roller bearings:

[0101] radial displacement With axial displacement Under the combined effect, at any angle position The deformation pattern exhibits spatial distribution characteristics:

[0102]

[0103] Spatial nonuniformity of center thrust load through distribution coefficient Quantification:

[0104]

[0105] Based on this, the angular attenuation model of roller load is expressed as follows:

[0106] Q ψ = Q m a x [ 1 − 1 2 e ( 1 − c o s ψ ) ] 1 . 1 1

[0107] In the formula, It is the load on each rolling element. The maximum load received by all rolling elements;

[0108] The angular attenuation model described above will mathematically attenuate to 0, so it is necessary to determine the boundary of the effective load-bearing zone, which is defined by the load angle threshold:

[0109]

[0110] To maintain static equilibrium, the sum of the forces acting on the rolling elements in the radial and axial directions must equal the load applied in those directions:

[0111]

[0112]

[0113] Load critical angle The spatial range of the actual load-bearing rollers was defined, becoming the theoretical boundary for integral calculations; ultimately, the system equilibrium equations constructed the constitutive equations for macroscopic loads and microscopic contact:

[0114]

[0115]

[0116] Where, radial integral and thrust integral It carries the core information of spatial distribution:

[0117] J r ( e ) = 1 2 π ∫ − ψ l + ψ l [ 1 − 1 2 e ( 1 − c o s ψ ) ] n c o s ψ d ψ

[0118] J a ( e ) = 1 2 π ∫ − ψ l + ψ l [ 1 − 1 2 e ( 1 − c o s ψ ) ] n d ψ

[0119] (6) The final expression for the preload:

[0120] Based on previous theoretical findings, preload The closed-form solution ultimately condenses into:

[0121]

[0122] This expression constitutes the final form of the theoretical derivation.

[0123] The beneficial effects of this invention are as follows: This invention proposes a quasi-static calculation method based on a three-degree-of-freedom slice model and Reusner's modified theory, effectively solving the problems of insufficient accuracy of existing empirical formulas and the huge time consumption of three-dimensional finite element simulation calculations. By introducing an asymmetric geometric parameter interface and slice differential technology, this invention can accurately capture the edge effects and non-uniform load distribution caused by roller overturning. The relative error between its calculation results and the professional simulation software Romax is less than 4%. It realizes rapid and high-precision analytical solution of the preload of wind turbine main shaft bearings under complex working conditions, significantly improving the reliability of bearing life prediction and the efficiency of assembly process optimization. Detailed Implementation

[0124] The specific embodiments of the present invention will be further described below in conjunction with the technical solution.

[0125] To verify the accuracy and engineering applicability of the quasi-static calculation method for the preload of asymmetric back-to-back tapered roller bearings on wind turbine main shafts of this invention, an example of an asymmetric back-to-back single-row tapered roller bearing on the main shaft of a MW-class wind turbine is used. Under external load conditions with radial loads, the load calculation results of the method of this invention and the Romax platform are compared and analyzed. The bearing conditions are shown in Table 1:

[0126] surface Bearing operating condition table

[0127]

[0128] Under typical operating conditions, axial load, radial load, and bending moment load are used as input conditions and substituted into the three-degree-of-freedom slice model established in this invention and the standard bearing analysis module of the Romax platform, respectively. The distribution law of the contact load between the inner and outer raceways and the rollers along the roller axis is calculated. The specific calculation results are shown in Table 2.

[0129] Table 2 Comparison and analysis of 3DOF and Romax calculation results

[0130]

[0131] As shown in Table 2, the maximum relative error between the method of the present invention and the calculation by the Romax platform does not exceed 4%, indicating that the prediction results of the method of the present invention on the peak contact load and load distribution characteristics are highly consistent with those of commercial software.

[0132] In summary, by comparing the load calculation results with those of the Romax platform, it can be demonstrated that the pseudo-static calculation method for the preload of asymmetric back-to-back tapered roller bearings for wind turbine main shafts proposed in this invention has good calculation accuracy and engineering applicability, and can be used for the preload assessment of tapered roller bearings for wind turbine main shafts.

Claims

1. A method for calculating the pre-tightening force of a wind turbine main shaft asymmetric back-to-back tapered roller bearing by quasi-statics, characterized in that, The steps are as follows: (1) Establish a multi-degree-of-freedom tapered roller bearing calculation model: First, establish A computational model for a multi-degree-of-freedom tapered rolling element bearing with an asymmetric back-to-back three-degree-of-freedom single-row tapered rolling element bearing is presented. This multi-degree-of-freedom tapered rolling element bearing computational model treats the bearing inner ring as a rigid body with three degrees of freedom to characterize the overall deformation of the bearing inner ring in three directions under external load. At the same time, three degrees of freedom are established for each rolling element to describe the spatial attitude change of the rolling element relative to the inner and outer rings of the bearing. The bearing inner ring is regarded as a rigid body with three degrees of freedom, i.e. radial displacement , axial displacement and bending angle ; for the first rolling body, three degrees of freedom are introduced, i.e. radial displacement , axial displacement and overturning angle ; wherein, ； The asymmetric back-to-back three-degree-of-freedom single-row tapered roller bearing forms a 3Z+3 degree-of-freedom multi-degree-of-freedom tapered roller bearing calculation model, which is expressed by a generalized coordinate vector, and has: q = [ δ r , I R , δ a , I R , θ b , δ x j 1 , δ y j 1 , ψ j 1 , … , δ x j Z , δ y j Z , ψ j Z ] T ； The geometric characteristics of the asymmetric back-to-back three-degree-of-freedom single-row tapered roller bearing are uniformly defined as a geometric parameter interface, and the geometric characteristics of different batches of asymmetric back-to-back three-degree-of-freedom single-row tapered roller bearings or under different error conditions are read into the multi-degree-of-freedom tapered roller bearing calculation model through the geometric parameter interface, realizing the configurable modeling of geometric parameters; wherein, The geometric characteristics include geometric parameters and manufacturing error parameters, and the geometric parameters include roller diameter, effective contact length of roller, raceway taper angle, raceway modification amount, rib position, bearing inner ring radius and bearing pitch circle diameter; (2) Establish the contact deformation coordination relationship and load balance equation: On the basis of the established multi-degree-of-freedom tapered roller bearing calculation model, the contact deformation coordination relationship between the roller and the inner ring raceway, the outer ring raceway and the rib is further established, and the mechanical balance equation of a single roller is given, thereby forming a complete nonlinear equation set; First is the slice unit coordinate and geometric relationship, for the first Rolling body, along the effective contact length Divided into Equal thickness slice, each slice thickness is: ； With the center of the rolling body as the origin, the axial direction of the rolling body is established axis, the first slice center coordinate is: Wherein, representing half the number of slices; the circumferential position angle of the rolling body is: ； The diameter of the roller at the slice is denoted as: ； Wherein, is the diameter of the middle part of the rolling body, is the cone angle of the rolling body; First, regarding the contact deformation of the inner and outer ring raceways of the bearing, let the degrees of freedom of the rolling elements in the radial and axial directions be respectively... and The nominal contact angle of the bearing outer ring is ;No. Contact deformation of each rolling element in the outer raceway Represented as: ； Second, the contact deformation between the rolling elements and the inner ring raceway should be considered, taking into account the axial displacement of the rigid body of the bearing inner ring. radial displacement and the overturning angle of the rolling element in the circumferential direction The nominal contact angle of the bearing inner ring is Then the first Contact deformation of the rolling elements at the inner ring raceway of the bearing Expressed as: ； Third, the contact deformation of the rolling element and the rib, for representing the nominal contact angle of the rolling element overturning The definition of the modified rolling element overturning angle The actual contact angle of the rolling element after the modification of the rolling element overturning angle is: ； then the first contact deformation of the rolling body at the stop edge is written as ； Then the non-negative deformation constraint and the contact load, all the above contact deformations need to satisfy the non-negative constraint: ； Therefore, the non-negative part of the rib contact deformation is denoted as: ； When the corresponding position is considered to be out of contact, the contact load is taken as zero; For bending deformation correction and slice element bending moment arm, consider the applied bending moment. The bending deformation of the bearing inner ring under the action of force is assumed to conform to the bending theory of beams, and the reference bending deformation of the inner ring is denoted as... The radial clearance is In the The first rolling body At each slice, the correction for bending deformation is written as: ； Wherein, For the rolling body shape function, define as: P ( x k ) = − 4 . 5 × 1 0 − 4 D m l n [ 1 − ( 2 x k L Re ) 2 ] ； Wherein, is the distance from the center of the rolling body to the center of the slice; For rolling bodies, only the For positive camber contact load, define: ； The first slice contact load is recorded with respect to the bending moment arm of the right end center of the rolling body ; when the external bending moment is applied, there is: l k M = + [ L R e 2 − x t ( k ) ] c o s α 1 2 − D k 2 s i n α 1 2 ； When There are: l k M = − [ L R e 2 + x t ( k ) ] c o s α 1 2 − D k 2 s i n α 1 2 ； In the construction of the vertical direction moment balance equation, the bending moment arm of the left half and the right half of the slice also need to be distinguished and The expression is: ； The bending moment arm of the rim load relative to the center of the rolling body is denoted by and is written as: ； Wherein, is the radius of the inner ring of the bearing, is the pitch diameter of the rolling elements, is the right end center diameter of the rolling elements, as follows: ； According to the Hertz linear contact stiffness relationship, the normal contact load of the first rolling body on the side of the first slice of the inner ring is: the first slice of the inner ring is:​ ； No. The slice normal load of each rolling element on the outer ring side is: ； Wherein, is the line contact stiffness coefficient, is the Reusner correction factor for the local load distribution along the length of the rolling body; Summing all slices along the rolling element axis yields the first... The total normal load of the contact area between the rolling element and the inner and outer rings: ； Rim contact load utilizes correction deformation The calculation is: ； Wherein, C is the contact stiffness coefficient of the shoulder; and The bending internal moment of the first rolling body is obtained by superimposing the bending moment contributions of each slice: ； The mechanical equilibrium equation is established for the single rolling body, and the force equilibrium equations in the y direction of the bearing radial direction and the x direction of the bearing axial direction and the torque equilibrium equation in the z direction perpendicular to the xy plane direction are established for the first rolling body. The equivalent concentrated radial load acting on the rolling body is The equilibrium equation in the y direction is: ； The balance equation in the x direction is: ； Taking the center of the roller as the reference point, the z direction torque balance is written as: ； Wherein, force arm for a concentrated load relative to the center of the rolling element the bending moment arm for the left and right halves of the slice is defined as and , , represents half of the number of slices of the rolling element (3) Establish the global torque balance equation and solve the load distribution and displacement field by using the iterative algorithm: After obtaining the contact deformation and contact load relationship, the radial force balance equation, the axial force balance equation and the balance bending moment equation are established based on the overall static balance condition, forming a high-dimensional nonlinear equation set about the bearing inner ring freedom and each roller freedom, and the iterative algorithm is used for solving; First, the expression of the linear contact stiffness coefficient is established, and the relationship between the linear contact stiffness coefficient and the effective length of the rolling element is written as: ​​ ； Secondly, the radial force balance equation is constructed to ensure that the sum of the radial components of the contact loads of the bearing outer ring and the external radial load balance, written as: ； Then the axial force balance equation is constructed, which relates the axial load to the axial component of the contact load with the bearing outer ring: ； The next step is to construct the equilibrium bending moment equation, which couples the bending effects by integrating the slice elements with the force arms, denoted as the first slice relative to the bending neutral axis, then: ； The radial force balance equation, the axial force balance equation and the bending moment balance equation of the bearing and the x, y, z three-direction balance equations of the roller constructed in step (2) together constitute a 3Z+3 dimensional nonlinear equation set; According to the actual working condition, the radial load of the bearing outer load is given , the axial load and the applied bending moment , the engineering experience value is selected as the degree of freedom vector ; the Levenberg-Marquardt nonlinear least square iteration algorithm is used to solve the above nonlinear equation group, and the iteration precision is controlled through the double convergence criterion of displacement tolerance and function tolerance; when the iteration converges, the axial displacement , the radial displacement and the rotation angle of the bearing inner ring under three degrees of freedom are obtained, and the contact load distribution of each rolling body and each slice unit and the bearing outer ring is obtained ; (4) Mechanical migration of angular contact ball bearing: First, the basic relationship of the center thrust load of the tapered roller bearing is established: ； Wherein, Working contact angle; to get a linear load-displacement relationship, a pre-assembly displacement needs to be defined Geometric constraint relationship with initial contact angle of the assembly ； Wherein, is the effective length coefficient of the roller, is the diameter of the roller; When the bearing enters the working state, the normal displacement of the bearing is generated from the working contact angle and the difference between the initial contact angle ​ ； Based on the Hertz contact theory, the nonlinear relationship between load and displacement is further established: Wherein, Kt is the contact stiffness coefficient; (5) Compound load response of tapered roller bearing: Under the combined action of the radial displacement and the axial displacement , the deformation law at any angular position presents spatial distribution characteristics: ； Spatial non-uniformity of central thrust loads by distribution coefficients Quantification: ； Based on this, the angular attenuation model of the roller load is expressed as: Q The above angular attenuation model will attenuate to 0 in mathematical results, so it is necessary to determine the boundary of the effective load bearing area, which is defined by the load angle threshold: = Q m a x [ 1 − 1 2 In order to maintain static balance, the sum of the forces of the roller in the radial direction and the axial direction must be equal to the acting load in that direction: ( 1 − c o s Wherein, ) ] 1 . 1 1 ； wherein is the load received by each rolling element, is the maximum load received by all rolling elements; (6) Final expression of pre-tightening force: ； This expression constitutes the ultimate form of theoretical derivation. ； ； Load critical angle The space range of the actual load roller is delimited, and becomes the theoretical boundary of the integral calculation. Finally, the system equilibrium equation constructs the macro load and micro contact constitutive equation: ； ； ​ Radial integration and thrust integration carrying the core information of the spatial distribution: J r ( ​ ) = 1 2 π ∫ − ​ l + ​ l [ 1 − 1 2 ​ ( 1 − c o s ​ ) ] n c o s ​ d ​ ； J a ( ​ ) = 1 2 π ∫ − ​ l + ​ l [ 1 − 1 2 ​ ( 1 − c o s ​ ) ] n d ​ ； ​ Based on the previous theoretical results, the closed-form solution of the pre-tightening force is finally condensed as ； ​

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