A calculation method for load curve of thin-walled angular contact ball bearing based on ellipse truncation

Abstract

The present invention discloses a method for calculating the load curve of a thin-walled angular contact ball bearing based on ellipse truncation. The points on the load curve represent working conditions at the critical state of ellipse truncation. The working condition combination under the load curve makes the bearing not ellipse truncation, and the working condition combination on the load curve makes the bearing ellipse truncation. The method for calculating the load curve of a thin-walled bearing adopted by the invention takes into account the elastic deformation of the bearing ring and its supporting structure and the clearance of the bearing. On this basis, the steel ball-raceway contact load and contact angle solved are closer to the actual situation, and the load curve obtained is more accurate. At the same time, a method for judging whether the thin-walled angular contact ball bearing has ellipse truncation under extreme working conditions is innovatively proposed.

Description

Technical Field

[0001] The invention belongs to the technical field of bearing design, and in particular relates to a calculation method for a load curve of a thin-walled angular contact ball bearing based on ellipse truncation. Background Art

[0002] Bearings are one of the key components of industrial robots, and the most common type is thin-walled angular contact ball bearings. The rib height of thin-walled angular contact ball bearings is smaller than that of ordinary bearings. Under the action of large axial loads, the contact ellipse between the steel ball and the raceway is more likely to exceed the raceway rib and cause ellipse truncation. There are many cases of ellipse truncation in thin-walled angular contact ball bearings used in robots. At present, there is a lack of methods to determine the occurrence of ellipse truncation in thin-walled angular contact ball bearings, that is, there is a lack of methods to determine the load curve of thin-walled angular contact ball bearings based on ellipse truncation.

[0003] On the one hand, it is impossible to determine whether the designed thin-walled angular contact ball bearing will be elliptical truncation, so as to avoid the early failure of the bearing due to insufficient load-bearing capacity of the thin-walled angular contact ball bearing. On the other hand, the traditional bearing analysis theory is based on the rigid ring assumption, which assumes that elastic deformation only occurs in the contact area between the ball and the raceway, and the overall shape of the ring will not change. However, for thin-walled bearings, the ring will undergo a large bending deformation after being loaded, and the deformation of the ring itself must be considered. The deformation of the ring has a great influence on the accurate calculation of the contact load between the ball and the raceway, but there are no relevant reports. Summary of the invention

[0004] The purpose of the present invention is to overcome the deficiencies of the prior art and to provide a method for calculating a load curve of a thin-walled angular contact ball bearing based on ellipse truncation.

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

[0006] A method for calculating the load curve of a thin-walled angular contact ball bearing based on ellipse truncation comprises the following steps:

[0007] 1) Considering the bending deformation of the thin-walled ring, establish the thin-walled ring at any steel ball azimuth angle The calculation formula of the radial deflection caused by the deformation at the point, the flexural deformation of the ring of the thin-walled angular contact ball bearing far exceeds the Hertz contact deformation;

[0008] 2) Establish the contact mechanics model of thin-walled angular contact ball bearings, including geometric equations, physical equations and equilibrium equations. The contact mechanics model is a set of equations that include the axial displacement of the bearing. , radial displacement , angular displacement and ring on either ball Contact azimuth Total radial deflection at ( =1, 2, ..., )of +3 nonlinear equations with unknown variables, which are solved by the Newton-Raphson iterative algorithm, and then the azimuth of any steel ball is obtained by the physical equation Ball-raceway contact load And contact angle , subscript Representative A steel ball, is the number of steel balls;

[0009] 3) Establish the The semi-major axis of the ball-raceway critical contact ellipse and The major semi-axis of the ball-raceway contact ellipse The calculation formula of

[0010] 4) Order Equal to the radial load of the thin-walled angular contact ball bearing under the limit condition , the axial force borne by the bearing and overturning moment In the bearing axial force range ] and overturning moment range ] to obtain the azimuth angle of each steel ball. Contact load at location , contact angle , semi-major axis of critical contact ellipse and the semi-major axis of the contact ellipse ;

[0011] 5) Compare the major semi-axis of the critical contact ellipse and the semi-major axis of the contact ellipse ,if - < ε , ε is the deviation value, and the axial force borne by the bearing is extracted and overturning moment As a point on the load curve;

[0012] 6) Axial force borne by the bearing and overturning moment Take the next value and repeat steps 4) and 5) until you get 10-20 points ( , ),

[0013] All the points obtained ( , ) are connected to obtain the load curve of the thin-walled angular contact ball bearing based on ellipse truncation. The points on the load curve represent the working conditions at the critical state of ellipse truncation. The working condition combination under the load curve makes the bearing not ellipse truncation, and the working condition combination on the load curve makes the bearing ellipse truncation.

[0014] As one of the pre-selected solutions, the equilibrium equation is,

[0015]

[0016] in, , and are the axial force, radial force and overturning moment borne by the bearing respectively. and Respectively represent the angular position The ball-raceway contact load and contact angle at is the pitch diameter of the bearing, The contact angle of the ring at any ball The total radial deflection at . =1, 2, ..., , is the number of steel balls, is the elastic modulus of the bearing material, and are the bending moments of inertia of the outer and inner rings, and are the groove curvature radii of the inner and outer rings of the bearing before loading, is a constant, is the azimuth of the ball-raceway contact force, is the radial force on the inner ring.

[0017] As one of the pre-selected solutions, the geometric equation is,

[0018]

[0019]

[0020] and are the groove curvature radius coefficients of the inner raceway and the outer raceway, is the diameter of the steel ball, is the axial clearance of the bearing, is the distance between the inner and outer raceways before the bearing is loaded. is the azimuth of the steel ball after loading The distance between the inner and outer raceway grooves, is the initial contact angle, , , are the axial, radial and angular displacements of the bearing, respectively. , is the pitch diameter of the bearing.

[0021] As one of the pre-selected solutions, the physical equation is:

[0022]

[0023]

[0024] is the load-displacement coefficient, and is the contact angle of any steel ball after the bearing is loaded Contact load and contact angle.

[0025] As one of the pre-selected solutions, when the contact ellipse does not exceed the raceway rib, The semi-major axis of the ball-raceway critical contact ellipse It is expressed as, ; is the angle corresponding to the inner raceway rib, , Indicates the height of the left rib of the inner raceway. Indicates the azimuth of the steel ball after loading The radius of curvature of the groove of the inner ring raceway, , is the radius of curvature of the inner ring of the bearing before loading,

[0026] For the The semi-major axis of the ball-raceway contact ellipse,

[0027] In the formula is the equivalent elastic modulus of the two contacting objects, is the ellipse eccentricity parameter, is the complete elliptic integral of the first kind, It represents the sum of the inner raceway groove curvatures of all ball-raceway contacts after loading.

[0028] The advantages and beneficial effects of the present invention are:

[0029] The thin-walled bearing load curve calculation method adopted by the invention takes into account the elastic deformation of the bearing ring and its supporting structure and the clearance of the bearing. On this basis, the steel ball-raceway contact load and contact angle solved are closer to the actual situation, and the load curve obtained is more accurate. At the same time, a method for judging whether an elliptical truncation occurs in a thin-walled angular contact ball bearing under extreme working conditions is innovatively proposed. This method not only improves the calculation method of the thin-walled bearing load curve, but also provides a theoretical basis for the optimization design of bearing structural parameters. BRIEF DESCRIPTION OF THE DRAWINGS

[0030] Figure 1 The azimuth angle of the ring on the ball Schematic diagram of radial deflection caused by deformation at .

[0031] Figure 2 This is the position change relationship of the center of curvature of the inner and outer raceway grooves before and after the bearing is loaded.

[0032] Figure 3 Schematic diagram of the contact elliptical rib height.

[0033] Figure 4 Schematic diagram of the truncation phenomenon of the contact ellipse.

[0034] Figure 5 Load curve of thin-walled angular contact ball bearing based on ellipse truncation.

[0035] For ordinary technicians in this field, other relevant drawings can be obtained based on the above drawings without any creative work. DETAILED DESCRIPTION

[0036] 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 specific embodiments.

[0037] The technical solution adopted by the present invention to solve the above technical problems is:

[0038] A method for determining a load curve of a thin-walled angular contact ball bearing comprises the following steps:

[0039] The first step is to consider the bending deformation of the thin-walled ring and establish the thin-walled ring at any steel ball azimuth angle. The calculation formula of radial deflection caused by deformation at ;

[0040] For thin-walled angular contact ball bearings, under the action of large external force (pressure) (greater than 0.8GPa, such as 1GPa), the bending deformation of the thin-walled bearing ring is particularly significant, far exceeding the Hertz contact deformation. In this case, the influence of the radial deflection of the ring on the contact deformation of the ball-raceway must be considered. Figure 1Shows the outer and inner rings at any ball azimuth

[0041] Radial deflection caused by deformation at and Then any azimuth angle of the steel ball Total radial deflection of the bearing ring at for:

[0042] (1)

[0043] in, is the azimuth of the steel ball at any The radial deflection of the outer ring at is the azimuth of the steel ball at any Radial deflection of the inner ring.

[0044] Refer to the bending deformation calculation formula of thin-walled ring (refer to relevant literature), ignore the mass of the steel ball, and combine the attached Figure 1 , then any azimuth Total contact deflection under for:

[0045] (2)

[0046] in, =1,2,... , is the number of steel balls; E is the elastic modulus of the bearing material, and are the bending moments of inertia of the outer and inner rings, and are the groove curvature radii of the inner and outer rings of the bearing before loading, represents the ball-raceway contact force, is a constant, is the azimuth of the steel ball, is the azimuth of the ball-raceway contact force, is the radial force on the inner ring.

[0047] The second step is to establish the contact mechanics model of thin-walled angular contact ball bearings and the azimuth angle of the steel ball. Contact load between ball and raceway and contact angle The calculation formula of

[0048] Generally, the outer ring of the bearing is fixed and the inner ring is rotating. Given the bearing structure parameters, material parameters and load conditions, the position change relationship of the inner and outer raceway groove curvature center before and after the bearing is loaded is as follows: Figure 2Before the bearing is loaded, there is no contact deformation between the steel ball and the raceway, and the distance between the inner and outer raceway grooves is As follows

[0049] (3)

[0050] In formula (3): and are the groove curvature radius coefficients of the inner raceway and the outer raceway, is the diameter of the steel ball, is the axial clearance of the bearing.

[0051] When the bearing is loaded, that is, a large load is applied, the curvature center of the inner raceway groove changes relative to the curvature center of the outer raceway groove. Move to .Depend on Figure 2 From the geometric relationship in the figure, we can see that at the azimuth angle of the steel ball The distance between the inner and outer raceway grooves It is found by the following formula:

[0052] (4)

[0053] In the formula, is the initial contact angle, , , are the axial, radial and angular displacements of the bearing, respectively. , is the pitch diameter of the bearing.

[0054] Arbitrary steel ball azimuth At the total contact deformation of the ball and raceway The expression is:

[0055] (5)

[0056] According to the load-displacement relationship of the ball-raceway contact, any ball azimuth Contact load between ball and raceway It is obtained by the following formula:

[0057] (6)

[0058] In the formula is the load-displacement factor.

[0059] After the bearing is loaded, at any azimuth angle of the steel ball At, the contact angle between the ball and the raceway The expression is:

[0060] (7)

[0061] Assume that the axial force, radial force and overturning moment borne by the bearing are , and , the external load applied to the inner ring should be balanced with the load applied by the steel ball on the inner ring, then the equilibrium equation of the inner ring can be written as:

[0062] (8)

[0063] The contact mechanics model of thin-walled angular contact ball bearings includes geometric equations (3)-(4), physical equations (6)-(7) and equilibrium equations (8). The model is a set of unknown variables (bearing axial displacement , radial displacement , angular displacement and The nonlinear equations (k=1,2,...,z) z is the number of balls) can be solved by the Newton-Raphson iterative algorithm. Then, the ball-raceway contact load can be obtained by equations (6) and (7): And contact angle .

[0064] The traditional bearing contact mechanics model is based on the rigid ring assumption, which assumes that elastic deformation only occurs in the contact area between the ball and the raceway, while the overall shape of the ring does not change. This assumption is only applicable to the analysis of bearings with large wall thickness, but not to thin-walled bearings, mainly because thin-walled rings can undergo large elastic deformation. The model proposed in this invention takes into account the elastic deformation of the ring and the influence of clearance (Formula 3), thereby obtaining more accurate bearing contact characteristics.

[0065] The third step is to establish The semi-major axis of the ball-raceway critical contact ellipse and The major semi-axis of the ball-raceway contact ellipse The calculation formula of

[0066] The elastic deformation of the thin-walled ring can change the contact angle between the ball and the raceway, and thus change the position of the contact ellipse between the two. The geometric relationship of the ball-raceway contact ellipse inside the bearing is as follows: Figure 3 As shown. Figure 3 The geometric relationship in the inner raceway corresponds to the angle of the left rib It can be expressed as:

[0067] (9)

[0068] is the height of the left rib of the inner raceway (e.g. Figure 3 ), Indicates the azimuth of the steel ball after loading The radius of curvature of the groove of the inner ring raceway,

[0069] (10)

[0070] like Figure 4 As shown, when The left end point of the major semi-axis of the ball-raceway contact ellipse just reaches the end point of the left flange of the inner ring of the bearing, that is, the angle corresponding to the critical ellipse truncation state. Written as:

[0071] (11)

[0072] For the The major semi-axis of the ball-raceway critical contact ellipse is:

[0073] (12)

[0074] When the contact ellipse does not exceed the raceway rib,

[0075] (13)

[0076] Therefore, the The semi-major axis of the ball-raceway critical contact ellipse

[0077] (14)

[0078] According to the knowledge of elastic mechanics, The major semi-axis of the ball-raceway contact ellipse The calculation formula is as follows:

[0079] (15)

[0080] In the formula For the The contact load between the ball and the raceway, is the equivalent elastic modulus of the two contacting objects, is the ellipse eccentricity parameter (obtained from the manual), is the complete elliptic integral of the first kind (obtained from the manual), It represents the sum of the inner raceway groove curvatures of all ball-raceway contacts after loading.

[0081] When any steel ball azimuth angle At the point where the long semi-axis of the ball-raceway contact ellipse is When the major axis of the contact ellipse between the ball and raceway is When the contact ellipse is just in the critical state of being truncated, when the long semi-axis of the contact ellipse between the ball and the raceway is When the ellipse is truncated, no ellipse truncation occurs.

[0082] Step 4: Establish the load curve of thin-walled angular contact ball bearing based on ellipse truncation

[0083] 1. Radial force on the bearing Equal to the radial load of the thin-walled angular contact ball bearing under the limit condition , the axial force borne by the bearing and overturning moment In the bearing axial force range ] and overturning moment range ] continuously within the range of , and the value can be selected according to the predetermined step size required. The radial load under the bearing limit condition can be obtained by using formulas (1)-(8) Under given overturning moment and axial load ( , ) under the action of each steel ball azimuth Contact load at location and contact angle ;

[0084] 2. Calculate the angle corresponding to the inner raceway rib according to formulas (9)-(10) ; According to formula (14), find The semi-major axis of the ball-raceway critical contact ellipse ; According to formula (15), find The major semi-axis of the ball-raceway contact ellipse ;

[0085] 3. Comparison and ,if - < ε , ε is the deviation value, and the axial force borne by the bearing is extracted and overturning moment As a point on the load curve;

[0086] 4. Axial force on the bearing and overturning moment Get the next value ， ( and Repeat steps 1-3 until 10-20 points are obtained ( , );

[0087] 5. All the points obtained ( , ) are connected to obtain the load curve of the thin-walled angular contact ball bearing based on ellipse truncation. A curve can be formed by multiple points using broken line connection or curve fitting. The points on the load curve represent the working conditions at the critical state of ellipse truncation. The working condition combination under the load curve can prevent the bearing from ellipse truncation, and the working condition combination on the load curve can cause the bearing to ellipse truncation.

[0088] Based on this load curve, it can be determined whether the thin-walled angular contact ball bearing can meet the extreme working conditions without elliptical truncation, that is, to determine the ultimate load-bearing capacity of the thin-walled angular contact ball bearing without elliptical truncation.

[0089] Beneficial effects: The thin-walled bearing load curve calculation method adopted by the present invention takes into account the elastic deformation of the bearing ring and its supporting structure and the clearance of the bearing. On this basis, the steel ball-raceway contact load and contact angle solved are closer to the actual situation, and the load curve obtained is more accurate. At the same time, a method for judging whether an elliptical truncation occurs in a thin-walled angular contact ball bearing under extreme working conditions is innovatively proposed. This method not only improves the calculation method of the load capacity of thin-walled bearings, but also provides a theoretical basis for the optimization design of bearing structural parameters. Specific embodiments

[0091] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0092] Taking a certain type of thin-walled angular contact ball bearing as an example, the structural parameters and material parameters are as follows:

[0093] Bearing inner diameter =145mm, bearing outer diameter =115mm, steel ball diameter =9.525mm, number of steel balls =37, bearing material =205GPa, bearing axial clearance =-0.03mm, inner raceway groove curvature radius coefficient =0.525mm, outer raceway groove curvature radius coefficient =0.515mm, initial contact angle o =40°, Poisson's ratio =0.3, height of left rib of inner ring of bearing =3.746mm, inner ring groove curvature radius before loading =4.905mm, outer ring groove curvature radius before loading = 5.001mm.

[0094] The calculation method of the load curve of a thin-walled angular contact ball bearing based on ellipse truncation provided by the present invention is used to draw the bearing load curve, and then determine the ultimate load capacity of the thin-walled angular contact ball bearing. The specific process is as follows:

[0095] Step 1: Consider the bending deformation of the thin-walled ring and establish the thin-walled ring at any steel ball azimuth angle Radial deflection caused by deformation at The calculation formula of

[0096] Considering that the thin-walled angular contact ball bearing will produce significant elastic deformation of the ring under the action of large external force, which far exceeds the Hertz contact deformation. In this case, the influence of radial deflection at any ring position on the contact deformation must be considered. According to relevant literature and the force conditions of the ring, the thin-walled ring at any steel ball azimuth angle is obtained. The calculation formula of radial deflection caused by deformation at ;

[0097] Step 2: Establish the contact mechanics model of thin-walled angular contact ball bearings and the azimuth angle of the steel ball Contact load between ball and raceway and contact angle The calculation formula of

[0098] Step 3: Create the The semi-major axis of the ball-raceway critical contact ellipse and The major semi-axis of the ball-raceway contact ellipse The calculation formula of

[0099] Step 4: Establish the load curve of thin-walled angular contact ball bearing based on ellipse truncation;

[0100] (1) Using the Newton-Raphson method to solve equations (3)-(8), the user proposed the limit radial load of thin-walled angular contact ball bearings: =2KN, then =2KN, each working condition ( , ) Axial displacement of lower bearing , radial displacement , angular displacement and , the azimuth angle of any steel ball in the bearing can be calculated by formulas (6) and (7): Contact load between ball and raceway and at any azimuth of the steel ball At, the contact angle between the ball and the raceway ;

[0101] (2) According to formulas (9)-(10), calculate the angle corresponding to the inner raceway rib ; According to formula (14), find The semi-major axis of the ball-raceway critical contact ellipse ; According to formula (15), find The major semi-axis of the ball-raceway contact ellipse ;

[0102] (3) Comparison and ,if - < ε , ε is the deviation value, and the axial force borne by the bearing is extracted and overturning moment As a point on the load curve;

[0103] (4) Axial force borne by the bearing and overturning moment Get the next value ， ( and Continuously take values), repeat steps (1)-(3) until 15 points are obtained ( , );

[0104] (5) All the points obtained ( , ) are connected to obtain the load curve of thin-walled angular contact ball bearing based on ellipse truncation, as shown in Figure 5 As shown, the points on the load curve represent the working conditions at the critical state of ellipse truncation. The working condition combination under the load curve can prevent the bearing from ellipse truncation, and the working condition combination on the load curve can cause the bearing to ellipse truncation.

[0105] Step 5: Based on the above thin-walled angular contact ball bearing load curve, determine whether the selected thin-walled angular contact ball bearing will be elliptical truncation under the operating conditions provided by the user, as follows:

[0106] Users have proposed that the limit radial load of thin-walled angular contact ball bearings =2KN, axial limit load =10KN, ultimate overturning moment =100N·m. Then Figure 5 In the load curve of thin-walled angular contact ball bearing based on ellipse truncation, point A is marked (100 N·m, 10 KN). Point A is below the load curve, which means that ellipse truncation will not occur.

[0107] 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 calculating the load curve of a thin-walled angular contact ball bearing based on ellipse truncation, characterized in that: The following steps are included: 1) Considering the bending deformation of the thin-walled ring, establish the thin-walled ring at any steel ball azimuth angle The calculation formula of the radial deflection caused by the deformation at the point, the flexural deformation of the ring of the thin-walled angular contact ball bearing far exceeds the Hertz contact deformation; 2) Establish the contact mechanics model of thin-walled angular contact ball bearings, including geometric equations, physical equations and equilibrium equations. The contact mechanics model is a set of equations that include the axial displacement of the bearing. , radial displacement , angular displacement and ring on either ball Contact azimuth Total radial deflection at ( =1, 2, ..., )of The nonlinear equations with unknown variables are solved by the Newton-Raphson iterative algorithm. Then, the azimuth angle of any steel ball is obtained by the physical equation. Ball-raceway contact load And contact angle , subscript Representative A steel ball, is the number of steel balls; 3) Establish the The semi-major axis of the ball-raceway critical contact ellipse and The major semi-axis of the ball-raceway contact ellipse k The calculation formula of 4) Order Equal to the radial load of the thin-walled angular contact ball bearing under the limit condition , the axial force borne by the bearing and overturning moment In the bearing axial force range ] and overturning moment range ] to obtain the azimuth angle of each steel ball. Contact load at location , contact angle , semi-major axis of critical contact ellipse and the semi-major axis a of the contact ellipse k ; 5) Compare the major semi-axis of the critical contact ellipse and the semi-major axis of the contact ellipse ,if - < ε , ε is the deviation value, and the axial force borne by the bearing is extracted and overturning moment As a point on the load curve; 6) Axial force borne by the bearing and overturning moment Take the next value and repeat steps 4) and 5) until you get 10-20 points ( , ), All the points obtained ( , ) are connected to obtain the load curve of the thin-walled angular contact ball bearing based on ellipse truncation. The points on the load curve represent the working conditions at the critical state of ellipse truncation. The working condition combination under the load curve makes the bearing not ellipse truncation, and the working condition combination on the load curve makes the bearing ellipse truncation.

2. A method for calculating the load curve of a thin-walled angular contact ball bearing based on ellipse truncation as claimed in claim 1, characterized in that: The equilibrium equation is, in, , and are the axial force, radial force and overturning moment borne by the bearing respectively. and Respectively represent the angular position Ball-raceway contact load and contact angle at is the pitch diameter of the bearing, The contact angle of the ring at any ball The total radial deflection at =1,2,..., , is the number of steel balls, is the elastic modulus of the bearing material, and are the bending moments of inertia of the outer and inner rings, and are the groove curvature radii of the inner and outer rings of the bearing before loading, is a constant, is the azimuth of the ball-raceway contact force, is the radial force on the inner ring.

3. The method for calculating the load curve of a thin-walled angular contact ball bearing based on ellipse truncation as claimed in claim 1, characterized in that: The geometric equation is, and are the groove curvature radius coefficients of the inner raceway and the outer raceway, is the diameter of the steel ball, is the axial clearance of the bearing, The distance between the inner and outer raceway grooves before the bearing is loaded; is the azimuth angle of the steel ball after loading The distance between the inner and outer raceway grooves, is the initial contact angle, , , are the axial, radial and angular displacements of the bearing, respectively. , is the pitch diameter of the bearing.

4. The method for calculating the load curve of a thin-walled angular contact ball bearing based on ellipse truncation as claimed in claim 1, characterized in that: The physical equation is, is the load-displacement coefficient, and is the contact angle of any steel ball after the bearing is loaded Contact load and contact angle.

5. The method for calculating the load curve of a thin-walled angular contact ball bearing based on ellipse truncation as claimed in claim 1, characterized in that: When the contact ellipse does not exceed the raceway rib, The semi-major axis of the ball-raceway critical contact ellipse It is expressed as, ; is the angle corresponding to the inner raceway rib, , Indicates the height of the left rib of the inner raceway. Indicates the azimuth of the steel ball after loading The radius of curvature of the groove of the inner ring raceway, , is the radius of curvature of the inner ring of the bearing before loading, For the The semi-major axis of the ball-raceway contact ellipse, In the formula is the equivalent elastic modulus of the two contacting objects, is the ellipse eccentricity parameter, is the complete elliptic integral of the first kind, It represents the sum of the inner raceway groove curvatures of all ball-raceway contacts after loading.

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