A method for determining the limiting speed of high-speed bearing cage

Abstract

The present invention discloses a method for determining the limiting speed of a high-speed bearing cage, comprising: establishing a finite element model of the cage; defining the material of the finite element model; defining the initial temperature of the finite element model; applying boundary conditions to the finite element model; sequentially applying loads, namely, temperature load, cage gravity, impact force of rolling elements on the cage, friction force of each rolling element on the cage, and cage centrifugal force; defining calculation conditions; performing finite element analysis; and performing cage limiting speed analysis. The present invention faithfully reproduces the actual operating conditions and material mechanical properties of the cage, achieving high-precision prediction of the cage limiting speed, and has strong adaptability and a wide range of applications.

Description

Technical Field

[0001] The present invention belongs to the technical field of electric vehicle reducers, and in particular relates to a method for determining a limiting rotational speed of a high-speed bearing retainer. Background Art

[0002] High speed is one of the development trends of electric vehicle drive motors. Existing motors generally run at around 16,000 r / min, with the highest speed reaching over 20,000 r / min. As a matching reducer assembly, the reducer input shaft is directly connected to the drive motor, and its maximum speed also increases linearly and synchronously. At this time, the bearings supporting the input shaft will face severe challenges, especially its important component, the cage. Made of engineering plastic, it has an open structure with low strength and weak rigidity. In the process of separating the rolling elements at equal intervals to ensure that the rolling elements do not fall out of the raceway, it is easy to fail due to disengagement, fracture, and significant deformation due to high-speed operation. To this end, the cage structure design must take into account the high temperature, high friction, and large impact forces during high-speed operation, as well as the centrifugal force that is proportional to the square of the speed. This ensures that the cage's maximum speed is higher than the reducer's maximum design speed, ensures that the cage has good mechanical properties during high-speed operation, and avoids bearing ablation and reducer functional failure caused by cage failure.

[0003] In order to guide the structural design of the cage and effectively avoid failures such as cage disengagement, fracture, and large deformation, the paper "Calculation of the Limiting Speed ​​of Straight Hole Solid Cages under Centrifugal Force" converts the cage into a circular ring with uniform mass, keeps the external dimensions and mass unchanged, and derives the limiting speed formula of the cage based on stress. The advantage of this method is fast calculation speed. The disadvantages are: first, only the influence of centrifugal force is considered, and other loads such as temperature are not considered, resulting in low accuracy in the prediction of the limiting speed of the cage; second, cages have different shapes. When the cage is simplified into a circular ring, its external dimensions are difficult to determine, which causes difficulties in specific application. As a result, this method is only applicable to cages with simple shapes and is not very universal.

[0004] Liu Zejiu et al. mentioned a method for determining the limiting speed of rolling bearings in the book "Application of Rolling Bearings". This method takes into account the bearing structural parameters and the load on the bearing, but does not consider the influence of temperature, and has low accuracy.

[0005] The patent "Method for calculating the life of roller bearing cages by combining bearing dynamics and FEM" (CN202110413593.4) ​​calculates the cage life by considering the fluid dynamic pressure, elastic collision force, centrifugal force, the force of the ring on the cage, and the blocking torque of the non-guide surface of the cage, which makes up for the shortcomings of the bearing life calculation. However, it does not consider the effects of temperature changes and centrifugal force changes. The temperature can change the matching relationship between the cage and the roller and the cage radius, affecting the magnitude of loads such as elastic collision force and centrifugal force, and thus affecting the magnitude of the force on the cage; the change in centrifugal force is related to the rotational speed, which is a prerequisite for predicting the cage's limit speed. The document does not further deduce a method for calculating the cage's limit speed.

[0006] The paper "Dynamic Analysis of Cages Based on ANSYS / LS-DYNA" establishes a set of equilibrium equations based on the dynamic method to obtain the rolling element speed, and then applies the rolling element speed curve to the dynamic finite element model composed of rolling elements and cages to obtain the cage stress distribution and speed fluctuation. The shortcoming is that the method established in this document only considers the mechanical load and does not consider the influence of temperature load. The calculated cage centrifugal force has low accuracy and the cage's limiting speed cannot be obtained.

[0007] In summary, the existing technology cannot effectively guide the structural design of cages of any structural form. Summary of the Invention

[0008] In order to solve the above-mentioned problems existing in the prior art, the present invention provides a method for determining the limiting speed of a high-speed bearing cage, which truly reproduces the actual working state and material mechanical properties of the cage, realizes high-precision prediction of the limiting speed of the cage, has strong adaptability and a wide range of applications.

[0009] The purpose of the present invention is achieved through the following technical solutions:

[0010] A method for determining the limiting speed of a high-speed bearing cage comprises the following steps:

[0011] S1. Establish a finite element model of the cage: The high-speed bearing consists of an outer ring, a cage, rolling elements, and an inner ring. The cage is meshed using second-order tetrahedron elements.

[0012] S2. Define the finite element model material: define the material elastic modulus E, Poisson's ratio μ, thermal expansion coefficient α, and density ρ of the cage finite element model;

[0013] S3. Define the initial temperature of the finite element model: the initial temperature is room temperature, which is defined as 25°C;

[0014] S4. Apply finite element model boundary conditions: fix the cage by applying symmetry constraints;

[0015] S5. Apply load 1: Load 1 is a temperature load, corresponding to the operating temperature of the cage;

[0016] S6. Apply load 2: Load 2 is the weight of the cage, which is applied by defining the gravitational acceleration;

[0017] S7, apply load 3: Load 3 is the impact force F1 of the rolling element on the cage;

[0018] S8, apply load 4: Load 4 is the friction force F of each rolling element on the cage 12 ;

[0019] S9, applying load 5: Load 5 is the cage centrifugal force F2, which is applied by defining the angular velocity w;

[0020] S10, defining calculation conditions: the calculation conditions are composed of the finite element model boundary conditions, load 1, load 2, load 3, load 4 and load 5;

[0021] S11, performing finite element analysis: according to the calculation conditions defined in step S10, the stress and deformation of the cage are calculated taking into account geometric nonlinearity;

[0022] S12. Analyze the cage's limiting speed: The cage's limiting speed is evaluated using two parameters: stress σ and deformation x. Stress σ is used to determine whether the cage will fracture and fail, while deformation x is used to determine whether the cage will fall out and fail due to excessive structural deformation.

[0023] Furthermore, in step S1, the cage is meshed using a symmetric modeling method. First, a symmetry plane is used to cut and remove the cage portion containing half a pocket for finite element meshing, and then the finite element mesh of the entire cage is obtained through an over-symmetric method.

[0024] Furthermore, in step S2, the elastic modulus E changes with the temperature T. When the elastic modulus E at the corresponding temperature T is not directly given and the temperature T is between the given minimum temperature and the given maximum temperature, the two temperatures closest to the temperature T and the corresponding elastic moduli are interpolated to obtain the value; when the elastic modulus E at the corresponding temperature T is not directly given and the temperature T is less than the given minimum temperature or higher than the given maximum temperature, the two temperatures closest to the temperature T and the corresponding elastic moduli are interpolated to obtain the value.

[0025] Furthermore, in step S4, a rectangular coordinate system is established on the geometric center line of the retainer, the center line of the retainer coincides with a coordinate axis of the rectangular coordinate system, the center line and the other two coordinate axes of the rectangular coordinate system form two planes, the two planes intersect with the retainer, constraining the freedom of two rows of nodes at the intersection of the first plane and the inner diameter side of the retainer along the geometric center line of the retainer and the degree of freedom perpendicular to the first plane, and constraining the freedom of two rows of nodes at the intersection of the second plane and the inner diameter side of the retainer along the geometric center line of the retainer and the degree of freedom perpendicular to the second plane.

[0026] Furthermore, in step S5, the operating temperature of the retainer is applied to be no less than 140°C.

[0027] Furthermore, in step S5, the operating temperature of the retainer is applied to be 140°C.

[0028] Furthermore, in step S6, the direction of the gravitational acceleration is perpendicular to the first plane or the second plane.

[0029] Furthermore, in step S7, the impact force F1 is calculated by formula (1):

[0030] F1=m×a (1)

[0031] Where: F1 is the impact force; m is the mass; a is the acceleration;

[0032] The direction of the impact force F1 is opposite to the direction of the revolution of the rolling element. The point of action of the impact force F1 is located at the center of the area between the cage pocket and the rolling element. The impact force F1 on each pocket is 11 Calculated by formula (2):

[0033] F 11 =F1÷n (2)

[0034] Where: F 11 is the impact force received by each pocket; n is the number of pockets.

[0035] Furthermore, in step S7, the impact acceleration a=25g, where g=9.8N / s 2 .

[0036] Furthermore, in step S8, the friction force F 12 Calculated by formula (3):

[0037] F 12 =F `11 ×μ (3)

[0038] Where: F 12 F is the friction force on each pocket; 11 is the impact force on each pocket; μ is the friction coefficient;

[0039] Friction force F 12 The direction of the rolling element's rotation is consistent with the direction of the rolling element's rotation. The friction force F 12 The point of action is the center of the cage pocket and the rolling element action area.

[0040] Furthermore, in step S9, the centrifugal force F2 of the cage is calculated by formula (4):

[0041] F2=m×r×w 2 (4)

[0042] Where: F is the centrifugal force; m is the mass; r is the radius of rotation; w is the angular velocity, w = 2πR, R is the rotation speed of the cage;

[0043] The relationship between the cage speed R and the reducer input shaft speed R' is as follows:

[0044]

[0045] Where: R' is the speed of the reducer input shaft; D w is the rolling element diameter; D is the bearing pitch diameter;

[0046] When the centrifugal force F2 of the cage is applied, the influence of the dynamic imbalance of the cage and the clearance between the cage pocket and the rolling element must be considered.

[0047] Furthermore, in step S9, the influence of the dynamic imbalance of the cage is taken into account by adjusting the position of the cage's rotating axis, and the rotating axis passes through the cage's center of mass position; theoretically, the cage's center of mass position is located on the cage's geometric center line, but due to the influence of manufacturing errors, the actual cage's center of mass position usually deviates from the cage's geometric center line. This deviation causes dynamic imbalance of the cage, affects the cage's strength and stiffness, and thereby reduces the cage's maximum speed.

[0048] Furthermore, in step S9, the influence of the clearance between the cage pocket and the rolling element is taken into account by adjusting the intersection angle α between the cage rotation axis and the cage geometric centerline. The intersection angle α is calculated by equations (6) and (7):

[0049] α=L÷D (6)

[0050] D=D1+D2 (7)

[0051] Where: α is the angle between the cage's rotation axis and the cage's geometric centerline; L is the maximum distance the rolling elements can move from one end to the other along the cage's centerline when the cage is fixed; D is the bearing pitch diameter; D1 is the bearing's outer diameter; and D2 is the bearing's inner diameter.

[0052] Furthermore, in step S12, the deformation x refers to the maximum opening amount of the pocket opening position;

[0053] When one of the following two conditions is met, the calculation stops and the cage limit speed is R. Otherwise, the cage speed R in step S9 is adjusted and steps S10 to S12 are repeated.

[0054] Condition 1: The stress σ satisfies formula (8) and the deformation x is not greater than the allowable opening x0;

[0055] Condition 2: The deformation x satisfies formula (9) and the stress σ is not higher than the allowable stress σ0;

[0056] 0.95σ0≤σ≤σ0 (8)

[0057] 0.9x0≤x≤x0 (9)

[0058] Where: σ is the maximum stress on the cage; σ0 is the allowable stress; x is the maximum opening of the cage pocket opening; x0 is the allowable opening.

[0059] Furthermore, in step S12, the allowable stress σ0 is the material strength limit, which decreases with increasing temperature; and the allowable opening x0 is the rolling element diameter.

[0060] The present invention has the following beneficial effects:

[0061] ① The present invention not only takes into account the effect of the centrifugal force of the cage, but also takes into account the influence of the cage's operating temperature, dynamic imbalance, gravity, impact force, friction force, the influence of the gap between the pocket and the rolling element, and the influence of the change in the material's elastic modulus with temperature. It more realistically reproduces the actual working state of the cage and the mechanical properties of the material, and achieves high-precision prediction of the cage's limiting speed.

[0062] ② The present invention is applicable to the prediction of the limiting speed of various types of cages and is not affected by the structural shape of the cage. Therefore, the present invention has the characteristics of strong adaptability and a wide range of applications.

[0063] ③ The present invention uses stress and deformation as two parameters to evaluate the mechanical properties of the cage, and the stress evaluation index decreases with increasing temperature, which effectively ensures that the cage will not break, avoids the risk of the cage falling out due to excessive deformation, and improves structural reliability. BRIEF DESCRIPTION OF THE DRAWINGS

[0064] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0065] Figure 1 It is a schematic diagram of the bearing structure;

[0066] Figure 2 It is a schematic diagram of the cage structure;

[0067] Figure 3 This is a schematic diagram of a partial cut of the cage pocket;

[0068] Figure 4 It is a partial schematic diagram of the cage pocket;

[0069] Figure 5 It is a schematic diagram of the force on the cage;

[0070] Figure 6 It is a schematic diagram of the position of the cage rotation axis;

[0071] Figure 7 It is a schematic diagram of bearing structure dimensions;

[0072] Figure 8 yes Figure 6 A partial enlarged schematic diagram of point A in the middle. DETAILED DESCRIPTION

[0073] The technical solution of the present invention is further described below with reference to the accompanying drawings and embodiments:

[0074] This embodiment provides a method for determining a limiting speed of a high-speed bearing cage, comprising the following steps:

[0075] S1. Establish the finite element model of the cage: Figures 1-2 As shown in the figure, the high-speed bearing consists of an outer ring 1, a cage 2, rolling elements 3 and an inner ring 4. The cage is meshed using second-order tetrahedron elements, and other structures are not meshed.

[0076] The cage is meshed using a symmetric modeling method, such as Figures 3-4 As shown, first, the symmetry plane 51 and the plane 52 are used to cut and remove the cage part 21 including half of the pocket 211 from the cage for finite element mesh division, and then the finite element mesh of the entire cage is obtained by a transsymmetric method, thereby improving the symmetry and simulation accuracy of the cage finite element simulation results.

[0077] S2. Define the finite element model material: The cage material is engineering plastic, and its elastic modulus E changes with temperature, as shown in Table 1. When the elastic modulus E at the corresponding temperature T is not directly given and the temperature T is between the given minimum and maximum temperatures, the two temperatures closest to the temperature T and the corresponding elastic moduli can be interpolated. For example, the elastic modulus E at 90°C is obtained by linear interpolation using the elastic moduli corresponding to 23°C and 120°C. When the elastic modulus E at the corresponding temperature T is not directly given and the temperature T is less than the given minimum temperature or higher than the maximum temperature, the two temperatures closest to the temperature T and the corresponding elastic moduli can be extrapolated. For example, the elastic modulus E at 150°C is obtained by linear interpolation using the elastic moduli corresponding to 120°C and 140°C.

[0078] Material Poisson's ratio μ = 0.3, thermal expansion coefficient α = 0.000045, density ρ = 1410 kg / m 3 .

[0079] Table 1 Elastic modulus E

[0080] Serial number Temperature T / ℃ Elastic modulus E / MPa 1 23 10000 2 120 5500 3 140 5300

[0081] S3. Define the initial temperature of the finite element model: the initial temperature is room temperature, equal to 25°C.

[0082] S4. Apply finite element model boundary conditions: fix the cage by applying symmetry constraints.

[0083] like Figure 2 、 Figure 5 As shown, a rectangular coordinate system 6 is established on the geometric center line 5 of the cage. The center line 5 of the cage coincides with the coordinate axis X of the rectangular coordinate system 6. The center line 5 forms two planes XOY and XOZ with the coordinate axis Y and the coordinate axis Z of the rectangular coordinate system 6 respectively. The plane XOY intersects with the cage at the nodes 22 and 23 on the inner diameter side of the cage, constraining the X-direction degree of freedom and the Z-direction degree of freedom of the nodes 22 and 23; the plane XOZ intersects with the cage at the nodes 24 and 25 on the inner diameter side of the cage, constraining the X-direction degree of freedom and the Y-direction degree of freedom of the nodes 24 and 25.

[0084] S5. Apply load 1: Load 1 is a temperature load, corresponding to the working temperature of the cage, and in the embodiment, 140° C. is applied.

[0085] S6. Apply load 2: Load 2 is the gravity of the cage, which is applied by defining the gravity acceleration, and the direction of the gravity acceleration is along the negative direction of the coordinate axis Z.

[0086] S7, apply load 3: Load 3 is the impact force F1 of the rolling element on the cage, which is calculated as shown in formula (1); the direction of the impact force F1 is opposite to the direction of the revolution of the rolling element 3, and the point of action of the impact force F1 is located at the center of the cage pocket and the rolling element action area. The impact force F1 on each pocket is 11 See formula (2) for size; Figure 5 As shown, the rolling element revolution direction is consistent with the cage rotation direction 26, and the impact force F on any pocket is 11 They are equal in size and directed along the tangent direction of the cage rotation direction 26.

[0087] F1=m×a (1)

[0088] Where: F1 is the impact force; m is the mass; a is the acceleration, a = 25g, g = 9.8N / s 2 .

[0089] F 11 =F1÷n (2)

[0090] Where: F 11 is the impact force received by each pocket; n is the number of pockets.

[0091] S8, apply load 4: Load 4 is the friction force F of each rolling element on the cage 12 , which is calculated as shown in formula (3), friction force F 12 The direction of the rolling element's rotation is consistent with the direction of the rolling element's rotation. The friction force F 12 The action point is the center of the cage pocket and the rolling element action area; Figure 5 As shown, the friction force F on any pocket is 12 Equal in magnitude and perpendicular in direction to the impact force F 11 .

[0092] F 12 =F `11 ×μ (3)

[0093] Where: F 12 F is the friction force on each pocket; 11 is the impact force on each pocket; μ is the friction coefficient, μ=0.1.

[0094] S9. Apply load 5: Load 5 is the centrifugal force F2 of the cage, which is applied by defining the angular velocity w. For details, see formula (4). When applying, the influence of the dynamic unbalance of the cage and the influence of the clearance between the cage pocket and the rolling element must be considered.

[0095] F2=m×r×w 2 (4)

[0096] Where: F is the centrifugal force; m is the mass; r is the radius of rotation; w is the angular velocity, w = 2πR, R is the rotation speed of the cage.

[0097] The cage speed R is related to the speed of the reducer input shaft R'. The relationship between the two is shown in formula (5). The initial value of R' is the maximum design speed of the reducer R'. max , that is, R'=R' max =18000r / min;

[0098]

[0099] Where: R' is the speed of the reducer input shaft; D w is the rolling element diameter; D is the bearing pitch circle diameter.

[0100] like Figures 6-7 As shown, the influence of the dynamic unbalance of the cage is taken into account by adjusting the position of the cage rotation axis 7, and the rotation axis 7 passes through the center of mass 29 of the cage. In theory, the center of mass of the cage is located on the cage geometric center line 5, but due to the influence of manufacturing errors, the actual center of mass of the cage usually deviates from the cage geometric center line. This deviation causes dynamic unbalance of the cage, affects the strength and stiffness of the cage, and thus reduces the maximum speed of the cage. In the embodiment, the cage rotation axis determined according to the dynamic unbalance of the cage is located 0.05 mm above the cage geometric center line 5.

[0101] like Figure 7 As shown in the figure, the influence of the clearance between the cage pocket and the rolling element is taken into account by adjusting the intersection angle α between the cage rotation axis and the cage geometric center line. The intersection angle α is calculated by formulas (6) to (7). According to the bearing parameters, α = 0.291°;

[0102] α=L÷D (6)

[0103] D=(D1+D2) / 2 (7)

[0104] Where: α is the angle between the cage's rotation axis and the cage's geometric centerline; L is the maximum distance the rolling elements can move along the cage's centerline from one end to the other when the cage is fixed; D is the bearing pitch diameter; D1 is the bearing's outer diameter; and D2 is the bearing's inner diameter.

[0105] like Figure 6 As shown in the figure, the centrifugal force calculation process of a unit n1 on the cage is explained as an example. The calculation process of the centrifugal force load of other units is the same as that of unit n1. The only difference is the value of the rotation radius r in formula (4). The mass m of unit n1 is 2.58×10 -11 kg, and its distance from the rotation axis 7 is r = 27.7 × 10 -3m, angular velocity around the rotation axis 7 w = R / 60 × 2 × π = 1002.1 / s, the centrifugal force can be obtained from formula (4)

[0106] F2=m×r×w 2 =7.18×10 -7 N.

[0107] S10. Define calculation conditions:

[0108] The calculation condition consists of the boundary conditions in step S4, load 1 in step S5, load 2 in step S6, load 3 in step S7, load 4 in step S8, and load 5 in step S9.

[0109] S11. Perform finite element analysis: Calculate the stress and deformation of the cage by considering geometric nonlinearity according to the calculation conditions defined in step S10.

[0110] S12. Analyze the cage's limiting speed: The cage's limiting speed is evaluated using two parameters: stress σ and deformation x. Stress σ is used to determine whether the cage will fracture, while deformation x is used to determine whether the cage will fall out due to excessive structural deformation. Deformation x refers to the maximum opening of the pocket opening.

[0111] When one of the following two conditions is met, the calculation stops and the cage limit speed is R. Otherwise, the cage speed R in step S9 is adjusted and steps S10 to S12 are repeated.

[0112] Condition 1: The stress σ satisfies formula (8) and the deformation x is not higher than the allowable opening x0

[0113] Condition 2: The deformation x satisfies formula (9) and the stress σ is not higher than the allowable stress σ0

[0114] 0.95σ0≤σ≤σ0 (8)

[0115] 0.9x0≤x≤x0 (9)

[0116] Where: σ is the maximum principal stress on the cage; σ0 is the allowable stress; x is the maximum opening of the cage pocket; x0 is the allowable opening.

[0117] The allowable stress σ0 is the material strength limit, which decreases with increasing temperature. See Table 2 for details. The allowable stress σ0 corresponding to the cage operating temperature in step S5 is equal to 105 MPa. At this time, formula (8) is specifically 99.75 MPa≤σ≤105 MPa; the allowable opening x0 is the rolling element diameter, which is equal to 11.9 mm. At this time, formula (9) is specifically 10.71 mm≤x≤11.9 mm.

[0118] Table 2 Allowable stress σ0

[0119] Serial number Temperature T / ℃ <![CDATA[Allowable stress σ0 / MPa]]> 1 23 210 2 120 110 3 140 105

[0120] like Figure 8 As shown in the figure, when R'=18000r / min, the maximum principal stress of the cage is located at the middle position 212 on the inner diameter side of the cage pocket, σ=95MPa, which does not satisfy formula (8) and is smaller than the target range; the maximum opening of the cage pocket is located at the pocket at position 212, specifically at the opening 213 on the outer diameter side of the pocket, x=10.634mm, which does not satisfy formula (9) and is smaller than the target range; therefore, the initial value of the speed R' in step S9 is too low and should be adjusted.

[0121] Adjust the speed R'=19000r / min, repeat steps S10~S12, the maximum principal stress of the cage is still located at the middle position 212 on the inner diameter side of the cage pocket, σ=104MPa, and its value satisfies formula (8); the maximum opening of the cage pocket is still located at the pocket where position 212 is located, specifically at the opening 213 on the outer diameter side of the pocket, x=10.696mm, and its value does not satisfy formula (9) and is smaller than the target range; at this time, the cage meets condition 1, "the stress σ satisfies formula (8) and the deformation x is not higher than the allowable opening x0", and the calculation is stopped. According to formula (5), the cage limit speed R=10101r / min. At this speed, the cage can work normally without breaking or falling out failure.

Claims

1. A method for determining the limiting speed of a high-speed bearing cage, characterized in that: The steps include: S1. Establish a finite element model of the cage: The high-speed bearing consists of an outer ring, a cage, rolling elements, and an inner ring. The cage is meshed using second-order tetrahedron elements. S2. Define the finite element model material: define the material elastic modulus E, Poisson's ratio μ, thermal expansion coefficient α, and density ρ of the cage finite element model; S3. Define the initial temperature of the finite element model: the initial temperature is room temperature; S4. Apply finite element model boundary conditions: fix the cage by applying symmetry constraints; S5. Apply load 1: Load 1 is a temperature load, corresponding to the operating temperature of the cage; S6. Apply load 2: Load 2 is the weight of the cage, which is applied by defining the gravitational acceleration; S7, apply load 3: Load 3 is the impact force F1 of the rolling element on the cage; S8, apply load 4: Load 4 is the friction force F of each rolling element on the cage 12 ; S9, applying load 5: Load 5 is the cage centrifugal force F2, which is applied by defining the angular velocity w; The centrifugal force F2 of the cage is calculated by formula (4): <h2 style=";text-align:left;direction:ltr">F2 = m×r×w<h2 style=";text-align:left;direction:ltr"> 2 <h2 style=";text-align:left;direction:ltr"> (4) Where: F is the centrifugal force; m is the mass; r is the radius of rotation; w is the angular velocity, w = 2πR, R is the rotation speed of the cage; The relationship between the cage speed R and the reducer input shaft speed R' is as follows: Where: R' is the speed of the reducer input shaft; D w is the rolling element diameter; D is the bearing pitch diameter; When applying the centrifugal force F2 of the cage, the influence of the dynamic imbalance of the cage and the clearance between the cage pocket and the rolling element must be considered; The influence of the clearance between the cage pocket and the rolling element is taken into account by adjusting the intersection angle α between the cage rotation axis and the cage geometric center line. The intersection angle α is calculated by formula (6) and formula (7): α=L÷D (6) D=D1+D2 (7) Where: α is the angle between the cage's rotation axis and the cage's geometric centerline; L is the maximum distance the rolling element can move from one end to the other along the cage's centerline when the cage is fixed; D is the bearing pitch diameter; D1 is the bearing's outer diameter; D2 is the bearing's inner diameter; S10, defining calculation conditions: the calculation conditions are composed of the finite element model boundary conditions, load 1, load 2, load 3, load 4 and load 5; S11, performing finite element analysis: according to the calculation conditions defined in step S10, the stress and deformation of the cage are calculated taking into account geometric nonlinearity; S12. Analyze the cage's limiting speed: The cage's limiting speed is evaluated using two parameters: stress σ and deformation x. Stress σ is used to determine whether the cage will fracture and fail, while deformation x is used to determine whether the cage will fall out and fail due to excessive structural deformation.

2. The method for determining the limiting speed of a high-speed bearing cage according to claim 1, wherein: In step S1, the cage is meshed using a symmetric modeling method. First, a portion of the cage including half a pocket is cut from the cage using a symmetry plane and removed for finite element meshing. Then, a finite element mesh of the entire cage is obtained through a transsymmetric method.

3. The method for determining the limiting speed of a high-speed bearing cage according to claim 1, wherein: In step S2, the elastic modulus E changes with the temperature T. When the elastic modulus E at the corresponding temperature T is not directly given and the temperature T is between the given minimum temperature and the given maximum temperature, the two temperatures closest to the temperature T and the corresponding elastic moduli are interpolated to obtain the value; when the elastic modulus E at the corresponding temperature T is not directly given and the temperature T is less than the given minimum temperature or higher than the given maximum temperature, the two temperatures closest to the temperature T and the corresponding elastic moduli are interpolated to obtain the value.

4. The method for determining the limiting speed of a high-speed bearing cage according to claim 1, wherein: In step S4, a rectangular coordinate system is established on the geometric center line of the retainer, the center line of the retainer coincides with a coordinate axis of the rectangular coordinate system, the center line and the other two coordinate axes of the rectangular coordinate system form two planes, the two planes intersect with the retainer, constraining the freedom of two rows of nodes at the intersection of the first plane and the inner diameter side of the retainer along the geometric center line of the retainer and the degree of freedom perpendicular to the first plane, and constraining the freedom of two rows of nodes at the intersection of the second plane and the inner diameter side of the retainer along the geometric center line of the retainer and the degree of freedom perpendicular to the second plane.

5. The method for determining the limiting speed of a high-speed bearing cage according to claim 1, wherein: In step S7, the impact force F1 is calculated by formula (1): F1=m×a (1) Where: F1 is the impact force; m is the mass; a is the acceleration; The direction of the impact force F1 is opposite to the direction of the revolution of the rolling element. The point of action of the impact force F1 is located at the center of the area between the cage pocket and the rolling element. The impact force F1 on each pocket is 11 Calculated by formula (2): F 11 =F1÷n (2) Where: F 11 is the impact force received by each pocket; n is the number of pockets.

6. The method for determining the limiting speed of a high-speed bearing cage according to claim 1, wherein: In step S8, the friction force F 12 Calculated by formula (3): F 12 =F `11 ×μ (3) Where: F 12 F is the friction force on each pocket; 11 is the impact force on each pocket; μ is the friction coefficient; Friction force F 12 The direction of the rolling element's rotation is consistent with the direction of the rolling element's rotation. The friction force F 12 The point of action is the center of the cage pocket and the rolling element action area.

7. The method for determining the limiting speed of a high-speed bearing cage according to claim 1, wherein: In step S12, the deformation x refers to the maximum opening amount of the pocket opening position; When one of the following two conditions is met, the calculation stops and the cage limit speed is R. Otherwise, the cage speed R in step S9 is adjusted and steps S10 to S12 are repeated. Condition 1: The stress σ satisfies formula (8) and the deformation x is not greater than the allowable opening x0; Condition 2: The deformation x satisfies formula (9) and the stress σ is not higher than the allowable stress σ0; 0.95σ0≤σ≤σ0 (8) 0.9x0≤x≤x0 (9) Where: σ is the maximum stress on the cage; σ0 is the allowable stress; x is the maximum opening of the cage pocket opening; x0 is the allowable opening.

8. The method for determining the limiting speed of a high-speed bearing cage according to claim 1, wherein: In step S12, the allowable stress σ0 is the material strength limit, which decreases as the temperature increases; the allowable opening x0 is the rolling element diameter.

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