Angular contact ball bearing dynamic stiffness calculation method

Abstract

This invention discloses a method for calculating the dynamic stiffness of an angular contact ball bearing, comprising the following steps: S1, calculating the original contact angle α0; S2, applying a preload F. a Calculate the increased contact angle α; S3. Calculate the normal contact load Q between the inner ring and the ball using the contact angle α. i Using Q i The total load-deformation constant K between the ball and the inner and outer rings n The preload F applied under static conditions is calculated. a The subsequent axial displacement δ1; S4, and the contact angle α between the steel ball and the inner ring is calculated based on the change in contact angle Δα. i And the contact angle α between the steel ball and the outer ring e S5. Utilizing the normal contact load Q between the inner ring and the sphere. i and the normal contact load Q between the outer ring and the ball e δ was calculated i and δ e S6, using α i α e δ i and δ e The displacement δ2 generated after the inner ring rotates is calculated; S7, the total displacement δ is calculated using δ1 and δ2, and the dynamic stiffness of the bearing is calculated using the total displacement δ. This invention can calculate the dynamic stiffness of angular contact ball bearings, providing a basis for the design of high-speed ACBBs.

Description

Technical Field

[0001] This invention relates to the field of bearing analysis and design technology, specifically to a method for calculating the dynamic stiffness of angular contact ball bearings. Background Technology

[0002] Angular contact ball bearings (ACBBs) are widely used in various mechanical equipment and can withstand combined loads of axial force, radial force, and torque. In existing design methods, the determination of ACBB main parameters is primarily based on the roll ratio, static stiffness, and rated dynamic load. However, the calculation formula for dynamic stiffness is complex and difficult to apply in practice. For high-speed rotor shaft systems, dynamic stiffness is a key parameter for the operating characteristics of the bearing, and is crucial for spindle design, characteristic analysis, machining capability assessment, and use. To provide a basis for the design of high-speed ACBBs, this invention proposes a method for calculating dynamic stiffness. Summary of the Invention

[0003] The purpose of this invention is to solve the problems existing in the prior art and provide a method for calculating the dynamic stiffness of angular contact ball bearings.

[0004] To address the shortcomings of the aforementioned technical problems, the present invention provides a method for calculating the dynamic stiffness of angular contact ball bearings, comprising the following steps:

[0005] S1. Measure the accuracy parameters of the angular contact bearing rings and calculate the original contact angle α0;

[0006] S2, Apply preload F a Calculate the increased contact angle α;

[0007] S3. Calculate the normal contact load Q between the inner ring and the ball using the contact angle α. i Utilizing the normal contact load Q between the inner ring and the ball i The total load-deformation constant K between the ball and the inner and outer rings. n The preload F applied under static conditions is calculated. a The subsequent axial displacement δ1;

[0008] S4. Solve for the change in contact angle Δα, and calculate the contact angle α between the steel ball and the inner ring based on the change in contact angle Δα. i And the contact angle α between the steel ball and the outer ring e ;

[0009] S5. Utilizing the normal contact load Q between the inner ring and the ball. i and the normal contact load Q between the outer ring and the ball e δ was calculated i and δ e ;

[0010] S6, using α i αe δ i and δ e The displacement δ2 is calculated to be generated after the inner ring rotates.

[0011] S7. Calculate the total displacement δ using δ1 and δ2, and then calculate the dynamic stiffness of the bearing using the total displacement δ.

[0012] As an optimization of the dynamic stiffness calculation method for angular contact ball bearings of the present invention: the accuracy parameters in step S1 include the outer ring groove bottom diameter d. e Inner groove bottom diameter d i , steel ball diameter D w Inner groove curvature coefficient f i and the outer groove curvature coefficient f e .

[0013] As an optimization of the dynamic stiffness calculation method for angular contact ball bearings of the present invention, the calculation formula for the original contact angle α0 in step S1 is as follows:

[0014]

[0015] in, The radial clearance is calculated using the following formula:

[0016]

[0017] Where, d e The outer groove bottom diameter, d i D is the diameter of the inner groove bottom. w The diameter of the steel ball;

[0018] Where, r i and r e These are the inner and outer channel radii, respectively, and their calculation formulas are as follows:

[0019]

[0020] Among them, f i f is the curvature coefficient of the inner groove. e is the curvature coefficient of the outer groove.

[0021] As an optimization of the dynamic stiffness calculation method for angular contact ball bearings of the present invention: the increased contact angle α in step S2 is calculated using the following formula:

[0022]

[0023] Where Z is the number of balls;

[0024] Among them, K nLet be the total load-deformation constant between the sphere and the inner and outer rings, which satisfies the following condition:

[0025]

[0026] Among them, K i K is the load-deformation constant between the rolling element and the inner ring. e K is the load-deformation constant between the rolling element and the outer ring. i and K e The following conditions must be met:

[0027]

[0028] Where, ∑ρ=ρ 1I +ρ 1Π +ρ 2I +ρ 2Π

[0029]

[0030] in, D wp The diameter of the center circle of the sphere is represented as: D wp =0.5(d) e +d i );

[0031] F(ρ) can be calculated using the following formula:

[0032]

[0033] n is determined by F(ρ) δ The value;

[0034] η is the elastic constant of the inner / outer ring in contact with the steel ball, expressed as:

[0035]

[0036] E1 and E2 are the elastic moduli of the ring and the ball, respectively, and μ1 and μ2 are the Poisson's ratios of the ring and the ball, respectively.

[0037] As an optimization of the dynamic stiffness calculation method for angular contact ball bearings of the present invention: the normal contact load Q between the inner ring and the ball i The calculation formula is as follows:

[0038] Q i =Q e =Q ia sinα

[0039] in,

[0040] The preload F is applied in the static statea The formula for calculating the subsequent axial displacement δ1 is as follows:

[0041]

[0042] As an optimization of the dynamic stiffness calculation method for angular contact ball bearings in this invention: the change in contact angle Δα is directly obtained by solving the fminbnd function in MATLAB.

[0043] As an optimization of the dynamic stiffness calculation method for angular contact ball bearings of this invention: the contact angle α between the steel ball and the inner ring i And the contact angle α between the steel ball and the outer ring e The calculation formula is as follows:

[0044]

[0045] As an optimization of the dynamic stiffness calculation method for angular contact ball bearings of this invention: δ i and δ e The calculation formula is as follows:

[0046]

[0047] in,

[0048] The formula for calculating δ2 is as follows:

[0049] δ2=δ i sinα i +δ e sinα e .

[0050] As an optimization of the dynamic stiffness calculation method for angular contact ball bearings of this invention: total displacement δ and dynamic stiffness K d The calculation formula is as follows:

[0051] δ=δ1+δ2

[0052]

[0053] This invention can calculate the dynamic stiffness of angular contact ball bearings, providing a basis for the design of high-speed ACBBs. Attached Figure Description

[0054] Figure 1 A schematic diagram of the initial state after ACBB assembly;

[0055] Figure 2 A schematic diagram showing the state of ACBB in the shaft system after applying preload Fa;

[0056] Figure 3 A schematic diagram of the forces acting on the inner ring of the ACBB after applying a preload Fa;

[0057] Figure 4 This is a schematic diagram of the forces acting on the outer ring of the ACBB after applying the preload Fa.

[0058] Figure 5 This is a schematic diagram of the force state of ACBB after the bearing rotates;

[0059] Figure 6 This is a flowchart of the dynamic stiffness calculation using differential calculation;

[0060] Figure 7 This is the curve showing the variation of axial stiffness with preload. Detailed Implementation

[0061] To better understand the present invention, the following embodiments further illustrate the content of the present invention, but the content of the present invention is not limited to the following embodiments.

[0062] <Technical Parameter Relationship>

[0063] like Figure 1 As shown, this is the assembled state of the ACBB before the preload is applied.

[0064] The formula for calculating the initial contact angle α0 is as follows:

[0065]

[0066] The radial clearance is obtained through equation (2).

[0067]

[0068] in

[0069] d e —Diameter of the outer groove bottom;

[0070] d i —Diameter of the inner groove bottom;

[0071] D w —Diameter of the steel ball.

[0072] Inner and outer channel radii r i and r e They are respectively

[0073] r i =f i D w (3)

[0074] r e =f e D w (4)

[0075] in

[0076] f i —Inner groove curvature coefficient;

[0077] f e —Outer groove curvature coefficient.

[0078] <Contact Deformation Calculation>

[0079] like Figure 2 As shown: The state of ACBB in the shaft system after applying preload Fa.

[0080] The contact angle increases to α.

[0081]

[0082] in

[0083] Z—Number of balls;

[0084] K n —The total load-deformation constant between the sphere and the inner and outer rings satisfies

[0085]

[0086] K i —Load-deformation constant between the rolling element and the inner ring;

[0087] K e —Load-deformation constant between the rolling element and the outer ring.

[0088] K i and K e for

[0089]

[0090] in

[0091] ∑ρ=ρ 1I +ρ 1∏ +ρ 2I +ρ 2Π (8)

[0092] ρ 1I ρ 1∏ ρ 2I and ρ 2∏ for

[0093]

[0094]

[0095]

[0096]

[0097] in

[0098]

[0099] D wp —The diameter of the center circle of the sphere is denoted as

[0100] D wp =0.5(d) e +d i (14)

[0101] Based on the calculation results of equation (15), refer to Table 6-1 in "Rolling Bearing Design Principles" (edited by Deng Si'er, Jia Qunyi, and Xue Jinxue, second edition) to determine n. δ The value of .

[0102]

[0103] η is the elastic constant of the inner / outer ring in contact with the steel ball, expressed as:

[0104]

[0105] E1, E2 — Elastic moduli of the two materials;

[0106] μ1, μ2 — Poisson's ratio of the two materials.

[0107] like Figure 3 The figure shows the stress situation of the inner ring.

[0108] like Figure 4 The figure shows the stress situation on the outer ring.

[0109]

[0110] Q i =Q e =Q ia sinα (18)

[0111] Q i Q e —Normal load at the contact point between the inner and outer rings;

[0112] Q ne Q ni — Tangential load at the contact point between the inner and outer rings.

[0113] In a static state, the axial displacement after applying preload is given by the formula.

[0114]

[0115] Axial displacement under high-speed rotation conditions

[0116] After the bearing rotates, the ACBB operates under the following forces: Figure 5 As shown.

[0117] n i Let the inner ring rotate at a constant speed, and the outer ring be fixed. Then the orbital speed n of the sphere is... m for

[0118]

[0119] β is the rotation attitude angle. Under high-speed conditions, the ball can be considered as being controlled by the outer raceway, therefore β satisfies...

[0120]

[0121] in

[0122]

[0123]

[0124] β can be obtained from equation (21). The centrifugal force of the ball is...

[0125]

[0126] in

[0127] m g —Ball mass;

[0128] ω g —The angular velocity of the ball's revolution, expressed as

[0129]

[0130] Q e It can be decomposed into axial load Q ea and radial load Q er Q i It can be decomposed into axial load Q ia and radial load Q ir According to the force balance equation, equations (23) and (24) are satisfied.

[0131] Q ia =Q ea (26)

[0132] Q ir +F c =Q er (27)

[0133] in

[0134]

[0135] Based on geometric relationships, we can obtain

[0136] Q ia cotα i +F c =Q er (29)

[0137] Furthermore, it can be deduced that

[0138]

[0139] in

[0140] α e —Contact angle between the steel ball and the outer ring;

[0141] α i —Contact angle between the steel ball and the inner ring.

[0142] It can be approximated that the change in contact angle Δα satisfies

[0143] α e =α-Δα (31)

[0144] α i =α+Δα (32)

[0145] The inner and outer rings are respectively at Q i and Q e Under the action, the elastic deformation of the ball and the inner and outer rings is

[0146]

[0147]

[0148] in

[0149]

[0150]

[0151] δ i and δ e These represent the deformation along the normal direction of the contact point, and the axial displacement generated during the rotation of the inner ring.

[0152] δ2=δ i sinα i +δ e sinα e (37)

[0153] <Dynamic Stiffness Calculation>

[0154] The outer ring is fixed, at F aUnder the action of force, the displacement of the bearing inner ring at high speed is divided into two parts. The first part is under static conditions, when F is applied. a The resulting displacement is δ1. After the inner ring rotates, the resulting displacement is δ2, and the total displacement δ of the inner ring is...

[0155] δ=δ1+δ2 (38)

[0156] Therefore, the dynamic stiffness K of the bearing d for

[0157]

[0158] static stiffness K s for

[0159]

[0160] Differential calculation can be used; the flowchart for dynamic stiffness calculation is as follows: Figure 6 As shown.

[0161] <Case Study Analysis>

[0162] Taking 7006-ACBB as an example, the evolution law between the main design parameters and dynamic stiffness is analyzed. The bearing design parameter range is shown in the table below.

[0163] At a rotational speed of 20000 r / min, the stiffness changes with the preload as follows: Figure 7 As shown.

[0164] The specific embodiments of the present invention have been described above. It should be understood that the present invention is not limited to the specific embodiments described above, and those skilled in the art can make various modifications or variations within the scope of the claims, which do not affect the essence of the present invention.

Claims

1. A method for calculating the dynamic stiffness of an angular contact ball bearing, characterized in that, Includes the following steps: S1. Measure the accuracy parameters of the angular contact bearing rings and calculate the original contact angle α0; S2, applying a pre-tightening force F a , calculating the increased contact angle a; S3, the normal contact load Q of the inner ring and the ball is calculated by using the contact angle a i , the axial displacement δ1 after the pre-tightening force F is applied in the static state is calculated by using the normal contact load Q of the inner ring and the ball i and the total load-deformation constant K between the ball and the inner and outer rings n a ​​ Normal contact load Q of inner ring with ball i The calculation formula is as follows: in, Z represents the number of balls; The preload F is applied in the static state a The formula for calculating the subsequent axial displacement δ1 is as follows: ； S4. Solve for the change in contact angle Δα, and calculate the contact angle α between the steel ball and the inner ring based on the change in contact angle Δα. i And the contact angle α between the steel ball and the outer ring e ; Contact angle α between the steel ball and the inner ring i And the contact angle α between the steel ball and the outer ring e The calculation formula is as follows: ； S5. Utilizing the normal contact load Q between the inner ring and the ball. i and the normal contact load Q between the outer ring and the ball e The deformation δ in the normal direction of the contact point was calculated. i and δ e ; δ i and δ e The calculation formula is as follows: in, K i K is the load-deformation constant between the rolling element and the inner ring. e This is the load-deformation constant between the rolling element and the outer ring; Q ia For Q i Decomposed axial load, Q ea For Q e Decomposed axial load; The formula for calculating δ2 is as follows: ； S6, using α i α e δ i and δ e The displacement δ2 is calculated to be generated after the inner ring rotates. S7. Calculate the total displacement δ using δ1 and δ2, and then calculate the dynamic stiffness of the bearing using the total displacement δ. Total displacement δ and dynamic stiffness K d The calculation formula is as follows: 。 2. The method for calculating the dynamic stiffness of an angular contact ball bearing as described in claim 1, characterized in that, The accuracy parameters in step S1 include the outer groove bottom diameter. d e 、 Inner groove bottom diameter d i steel ball diameter D w Inner groove curvature coefficient f i and the outer groove curvature coefficient f e .

3. The method for calculating the dynamic stiffness of an angular contact ball bearing as described in claim 1, characterized in that, The formula for calculating the initial contact angle α0 in step S1 is as follows: in, The radial clearance is calculated using the following formula: Where, d e The outer groove bottom diameter, d i D is the diameter of the inner groove bottom. w The diameter of the steel ball; Where, r i and r e These are the inner and outer channel radii, respectively, and their calculation formulas are as follows: in, f i Inner groove curvature coefficient 、f e for Outer groove curvature coefficient.

4. The method for calculating the dynamic stiffness of an angular contact ball bearing as described in claim 1, characterized in that, The increased contact angle α in step S2 is calculated using the following formula: Where Z is the number of balls; Among them, K n Let be the total load-deformation constant between the sphere and the inner and outer rings, which satisfies the following condition: Among them, K i K is the load-deformation constant between the rolling element and the inner ring. e K is the load-deformation constant between the rolling element and the outer ring. i and K e The following conditions must be met: in, in, D w Let D be the diameter of the steel ball. wp The diameter of the center circle of the sphere is represented as: , d e The diameter of the outer groove bottom ，d i The diameter of the inner groove bottom; Calculate using the following formula : pass Sure n δ The value; η The elastic constants of the inner / outer ring in contact with the steel ball are expressed as follows: E 1. E 2 represents the elastic modulus of the ring and the ball. μ 1. μ 2 represents the Poisson's ratio of the ring and the ball.

5. The method for calculating the dynamic stiffness of an angular contact ball bearing as described in claim 1, characterized in that: The change in contact angle Δα is obtained directly using the fminbnd function in MATLAB.

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