Full complement CARB bearing design method based on elastohydrodynamic lubrication bearing model

Abstract

The invention belongs to the field of bearing design theories, and discloses a full complement CARB bearing design method based on an elastohydrodynamic lubrication bearing model. The viscosity-pressure coefficient, the basic viscosity and the elastic modulus are integrated to oil film thickness calculation for the first time by establishing a joint objective function of the maximum dynamic load and the elastic flow lubrication minimum film thickness, and the dynamic load coefficient is adopted to correct the actual working condition deviation. A complete system containing 21 constraints is constructed, and five dimensions of geometric compatibility, contour matching, motion compensation, intensity life and manufacturing boundary are covered. The design variables including the curvature radius, the roller diameter and the pitch circle diameter are solved through a multi-objective optimization algorithm, collaborative maximization of the bearing capacity and the lubricating performance is achieved, and the problem that in a traditional design theory, curvature parameters are difficult to accurately determine is solved.

Description

TECHNICAL FIELD

[0001] The present application belongs to the field of bearing design theory, and relates to a full CARB bearing design method based on an elastohydrodynamic lubrication bearing model. BACKGROUND

[0002] As a special self-aligning, single-row toroidal roller bearing, CARB bearing has excellent aligning ability, high load-carrying capacity and compact radial size due to its unique structure. These characteristics make it an important choice in the application of harsh working conditions such as heavy load, impact load, shaft / housing deflection, etc.

[0003] However, although CARB bearings are widely used in industrial practice, a key technical challenge is the lack of specific, detailed and unified design specifications in their design field. Compared with standard cylindrical roller or ball bearings with cages, the design process is significantly different:

[0004] Structural complexity: Traditional simplified design methods based on Hertz contact theory are suitable for point contact and infinite line contact. For CARB bearings, the roller and raceway are not infinite line contact but modified line contact, and the modified line contact needs to be considered in load-carrying capacity calculation.

[0005] Key role of elastohydrodynamic lubrication (EHL): Due to the modified line contact between the roller and the raceway, and the formation of an extremely thin lubrication film under heavy load, low speed or high temperature conditions. At this time, the lubrication state is no longer simple boundary lubrication or hydrodynamic lubrication, but elastohydrodynamic lubrication state needs to be considered, which is the combined action of elastic deformation of the contact surface, viscosity-pressure relationship of lubricating oil and extrusion flow effect.

[0006] Load distribution model is missing: The existing general design methods or standards (such as ISO 281 or similar derivative standards) are mainly for bearings with cages. The basic dynamic load rating formula for life calculation and the internal load distribution calculation model cannot be directly applied to the structural characteristics of full CARB bearings. In the design, the curvature of the roller and raceway is the key factor restricting the design, so the method must be guided to design the optimal curvature.

[0007] To solve this design problem, a CARB bearing design method needs to be developed. Therefore, a full CARB bearing design method based on an elastohydrodynamic lubrication bearing model is proposed to guide the design. SUMMARY

[0008] In view of the deficiencies of the existing design theory and guidance method, the present application provides a design model, taking the dynamic load and the minimum film thickness of elastohydrodynamic lubrication as the objective function, and maximizing the objective function to find the optimal curvature, so as to solve the curvature design problem of CARB bearing.

[0009] The technical solution of the present invention:

[0010] A design method for a full complement CARB bearing based on the elastohydrodynamic lubrication load-bearing model includes the following steps:

[0011] (1) Constructing multi-objective functions;

[0012] According to the operation requirements, different objective functions of rolling bearings are proposed. The objective function is defined as the maximum value of the load-bearing life and the minimum film thickness of the elastohydrodynamic lubrication, which can be expressed as:

[0013]

[0014] The expression of dynamic load capacity C is composed of the reduction factor λν considering edge load and non-uniform stress, roller diameter D r , pitch circle diameter D m , number of roller rows i, roller length l, number of rollers Z, ratio of roller diameter to pitch circle diameter composition;

[0015] The minimum film thickness of the elastohydrodynamic lubrication is determined by the minimum film thickness of the bearing inner ring (h min ) inner and the minimum film thickness of the bearing outer ring (h min ) outer composition.

[0016] Aiming at the lubrication behavior of the fluid dynamic pressure oil film between two elastic bodies in relative rolling or rolling with sliding, the elastic fluid dynamic pressure lubrication theory is proposed. Using the elastic fluid dynamic pressure lubrication theory, the minimum oil film thickness for elastohydrodynamic lubrication is calculated. The minimum oil film thickness for elastohydrodynamic lubrication is:

[0017]

[0018] Where: α p is the viscosity-pressure coefficient; η0 is the dynamic viscosity at normal pressure; u is the average speed between the roller and the inner and outer ring surfaces; R d is the equivalent curvature radius; E0 is the equivalent elastic modulus; q is the load per unit contact length;

[0019] Without considering the roller slip surface, the average speed between the roller and the inner and outer ring surfaces is:

[0020]

[0021] Where: n is the inner ring speed, r / min;

[0022] (2) Building a constraint system;

[0023] Maximizing dynamic load-bearing capacity, maximizing fatigue life, and minimizing friction power consumption are defined as core optimization objectives, forming a multi-objective optimization problem. Flexible adjustment of design preferences is achieved through weight coefficients. Constraints are set, including the material strength limit (maximum contact stress), the minimum oil film thickness (avoiding direct metal contact), the size limit (installation space limitation), and the lower limit of fatigue life to ensure the feasibility of the optimization results. Based on the above description, 21 constraints are proposed as follows:

[0024] Constraints 1 and 2: The bearing's mean diameter should be selected between the bearing's inner and outer diameters to ensure space for the sharp chamfers of the inner and outer rings;

[0025] D m -(d+2r3+D r )≥0

[0026] (D-2r1-D r )-D m ≥0

[0027] Where d is the inner diameter of the bearing, D is the outer diameter of the bearing, r3 is the radial chamfer of the inner ring, and r1 is the radial chamfer of the outer ring;

[0028] Constraints 3 and 4: The range of the average roller diameter is derived from both strength and geometry considerations. The lower limit of the average roller diameter is obtained from the contact stress expression, and the upper limit is obtained from the geometry. The minimum value of the average roller diameter should be able to withstand the contact stress generated by the external load.

[0029]

[0030] Where Q is the roller contact load, l e The effective length of the roller is the difference between the roller length and the roller chamfer, r 1min With r 3min is the minimum value of the radial chamfer between the outer ring and the inner ring during design, and δ is the roller crown;

[0031] Constraints 5 and 6: The minimum number of rollers is determined based on the lower limit of the bearing pitch diameter and the upper limit of the roller average diameter. Similarly, the maximum number of rollers is determined based on the upper limit of the bearing pitch diameter and the lower limit of the roller average diameter.

[0032]

[0033] Constraint 7: The maximum roller length is the bearing width minus the axial space reduction caused by self-alignment;

[0034]

[0035] Where B is the bearing width, d2 is the diameter of the boundary line between the inner raceway and the width direction of the bearing, r gis the roller chamfer radius, θ is the maximum self-aligning angle of the bearing;

[0036] Constraints 8 and 9: The diameter of the rolling element should be selected within a certain range, where K Dmin and K Dmax is the roller diameter constraint constant, which determines the possible minimum and maximum diameters of the rolling element;

[0037]

[0038] 0.4≤K Dmin ≤0.5

[0039] 0.6≤K Dmax ≤0.7

[0040] Constraints 10 and 11: To ensure the bearing's operational flexibility, the difference between the bearing's pitch diameter and its mean diameter must be less than a given value. Therefore, the following two constraints must be met:

[0041] D m -(0.5-e)(D+d)≥0

[0042] (0.5+e)(D+d)-D m ≥0

[0043] 0.03≤e≤0.08

[0044] Constraints 12 and 13: To ensure the working strength requirements, the thickness of the bottom of the bearing outer raceway should not be less than εD r , where ε is an unknown constraint constant, and the thickness of the outer ring should be large enough to allow chamfering;

[0045] 0.5(DD r -D m )-εD r ≥0

[0046] 0.3≤ε≤0.4

[0047] 0.5(D-D0)-2r1≥0

[0048] D o :D m +D r

[0049] Constraint 14: Since the inner ring always bears more stress than the outer ring, the thickness of the outer ring needs to be restricted; the inner ring is designed to be stronger than the outer ring:

[0050] (D i -d)-(DD o )≥0

[0051] Di :D m -D r

[0052] D o :D m +D r

[0053] Constraints 15-18: Since the curvature difference is always between 0 and 1, actual constraints are imposed on the roller mean diameter, pitch diameter, roller curvature radius, and raceway radius.

[0054]

[0055] Among them, R g is the roller profile curvature radius, R is the outer raceway groove curvature radius, and r is the inner raceway groove curvature radius;

[0056] Constraints 19 and 20: Effective length l of the bearing e and width B limit the maximum length of the roller, constrained as follows:

[0057] βB-1 e ≥0, 0.7≤β≤0.85

[0058] B-2r1-l e -2r g ≥0

[0059] Constraint 21: The thickness of the outer ring should be large enough so that, in the worst case, the maximum dynamic shear stress occurs at the center of the outer ring;

[0060]

[0061] Z static =0.626b

[0062] Where b is the length of the minor semi-axis of the contact ellipse area;

[0063] (3) Using optimization algorithms to solve the optimal values ​​of design variables;

[0064] The genetic algorithm is used to realize the multi-objective optimization design of bearings. The steps are as follows:

[0065] 1) First, input the bearing design parameters, including the inner diameter, outer diameter, width, roller diameter, roller length, roller crown, and inner and outer ring raceway curvature. Then, complete the construction of multi-objective functions for roller diameter, roller length, and inner and outer ring curvature, define the optimization variable range, initialize the genetic algorithm tool, initialize individuals, and register the evolutionary operator, and finally complete the population initialization.

[0066] 2) The initialized population undergoes an initial population evaluation before entering an evolutionary cycle: The cycle first determines whether the preset maximum number of generations has been reached. If so, the optimal frontier solution is extracted and all optimal solutions are output, terminating the process. If not, a descendant population is generated, and then the new descendant population is evaluated. The evaluation process integrates the calculation of multiple objective functions and 21 constraints, and constraint violation penalties are imposed based on the constraint violation values.

[0067] 3) The new offspring after evaluation is merged with the parent generation, and the new population is screened through the selection operation. The evolution cycle is entered again and iterated until the maximum generation termination condition is met, and finally the optimized solution for the bearing design is output.

[0068] Beneficial effects of the present invention: The present invention uses a genetic algorithm to perform multi-objective optimization design on key bearing design parameters (such as roller diameter, roller length, roller convexity, inner and outer ring raceway curvature). The present invention directly constructs a multi-objective function of roller diameter, roller length and inner and outer ring curvature, which can weigh multiple key performance indicators in a single optimization process, significantly improve design efficiency, avoid the tediousness and one-sidedness of the traditional serial optimization method, and provide designers with a series of excellent candidate solutions with balanced performance. At the same time, based on the algorithm rules, it automatically completes iterative processes such as population initialization, evolution (crossover, mutation), evaluation and selection, greatly reducing the workload of manual trial and error and repeated calculations, significantly shortening the design cycle and accelerating the product development process. This automated, efficient and global multi-objective optimization design method for bearings can generate a large number of excellent design solutions that meet stringent engineering constraints, greatly improving bearing design efficiency, optimizing key bearing performance indicators, and providing sufficient basis for design decisions, ultimately promoting the development of bearing products with higher performance and longer life. BRIEF DESCRIPTION OF THE DRAWINGS

[0069] Figure 1 It is the flow chart of optimization design algorithm.

[0070] Figure 2 This is a schematic diagram of the CARB bearing dimensions.

[0071] Figure 3 It is a schematic diagram of the optimization design calculation results. DETAILED DESCRIPTION

[0072] The specific implementation of the present invention is further described below in conjunction with the accompanying drawings and technical solutions.

[0073] A design method for a fully complemented CARB bearing based on the "elastohydrodynamic lubrication load model" is proposed. A certain rolling bearing is used as the research object. Its geometric and load parameters are as follows: bearing outer diameter D = 150 mm, bearing inner diameter d = 100 mm, width B = 50 mm; basic dynamic load rating C = 330 kN; inner and outer ring raceway curvature radius is 320 mm; rolling element diameter Dr =12.9mm, length l = 38mm; pitch circle diameter D m =136.6mm, number of rolling elements Z = 30. The above parameters form the basic reference system for the design evaluation of this bearing. The design steps are as follows:

[0074] (1) Construction of multi-objective functions

[0075] According to the operating requirements, different objective functions of rolling bearings can be proposed. The most important ones are the load life and wear life. Therefore, the objective function is defined as the maximum value of the load life and the minimum film thickness of the elastohydrodynamic lubrication, which is expressed as:

[0076]

[0077] The expression of dynamic load capacity C is composed of the reduction factor λν considering edge load and non-uniform stress, roller diameter D r , pitch circle diameter D m , number of roller rows i, roller length l, number of rollers Z, ratio of roller diameter to pitch circle diameter composition;

[0078] The minimum film thickness of the elastohydrodynamic lubrication is determined by the minimum film thickness of the bearing inner ring (h min ) inner and the minimum film thickness of the bearing outer ring (h min ) outer composition.

[0079] The study of the lubrication behavior of the hydrodynamic oil film between two elastic bodies in relative rolling or rolling-sliding motions led to the development of the elastohydrodynamic lubrication (EHL) theory. Using the elastohydrodynamic lubrication theory, the minimum oil film thickness can be calculated. Taking these facts into account, another goal is the minimum EHL oil film thickness. The minimum oil film thickness for the inner and outer rings is:

[0080]

[0081] Where: α p is the viscosity-pressure coefficient; η0 is the dynamic viscosity at normal pressure; u is the average speed between the roller and the inner and outer ring surfaces; R d is the equivalent curvature radius; E0 is the equivalent elastic modulus; q is the load per unit contact length;

[0082] Without considering the roller slip surface, the average surface velocity is:

[0083]

[0084] Where: n is the inner ring speed, r / min;

[0085] (2) Construction of constraint system;

[0086] Maximizing dynamic load-bearing capacity, maximizing fatigue life, and minimizing frictional power consumption are defined as core optimization objectives, forming a multi-objective optimization problem (MOOP). Flexible adjustment of design preferences is achieved through weight coefficients. Physical constraints such as material strength limit (maximum contact stress), minimum oil film thickness (to avoid direct metal contact), dimensional boundaries (installation space restrictions), and fatigue life lower limit are set to ensure the engineering feasibility of the optimization results. Based on the above description, 21 constraints are proposed as follows:

[0087] Constraints 1 and 2: The bearing's mean diameter should be chosen between the bearing's inner and outer diameters to ensure space for the sharp chamfers of the inner and outer rings.

[0088] D m -(d+2r3+D r )≥0

[0089] (D-2r1-D r )-D m ≥0

[0090] Where d is the inner diameter of the bearing, D is the outer diameter of the bearing, r3 is the radial chamfer of the inner ring, and r1 is the radial chamfer of the outer ring.

[0091] Constraints 3 and 4: The range of the average roller diameter is determined by both strength and geometry considerations. The lower limit of the average roller diameter is determined by the contact stress expression, and the upper limit is determined by the geometry. The minimum average roller diameter should be sufficient to withstand the contact stress generated by the external load.

[0092]

[0093] Where Q is the roller contact load, l e is the effective length of the roller (the difference between the roller length and the roller chamfer), r 1min With r 3min It is the minimum value of the radial chamfer of the outer ring and the inner ring during design, and δ is the roller crown.

[0094] Constraints 5 and 6: Based on the lower limit of the bearing pitch diameter and the upper limit of the roller average diameter, the minimum number of rollers is obtained. Similarly, based on the upper limit of the bearing pitch diameter and the lower limit of the roller average diameter, the maximum number of rollers is obtained. Therefore, in the present invention,

[0095]

[0096] Constraint 7: The maximum roller length is the bearing width minus the axial space reduction caused by self-alignment

[0097]

[0098] where B is the bearing width, d2is the diameter of the inner raceway and the width direction intersection line, r g is the roller chamfer radius.

[0099] Constraints 8 and 9: the diameter of the rolling element should be chosen within a certain range, where K Dmin and K Dmax are the roller diameter constraint constants (which determine the possible minimum and maximum diameter of the rolling element), the corresponding constraint conditions are given as:

[0100]

[0101]

[0102] 0.4≤K Dmin ≤0.5

[0103] 0.6≤K Dmax ≤0.7

[0104] Constraints 10 and 11: in order to ensure the flexibility of the bearing operation, the difference between the pitch diameter and the average diameter of the bearing should be less than a certain given value. Therefore, the following two constraints need to be met:

[0105] D m -(0.5-e)(D+d)≥0

[0106] (0.5+e)(D+d)-D m ≥0

[0107] 0.03≤e≤0.08

[0108] Constraints 12 and 13: in order to ensure the working strength requirement, the thickness of the bottom of the outer raceway of the bearing should not be less than εD r , where ε is an unknown constraint constant, and the thickness of the outer ring should be large enough to allow chamfering. Therefore, in the present application:

[0109] 0.5(D-D r -D m )-εD r ≥0

[0110] 0.3≤ε≤0.4

[0111] 0.5(D-D0)-2r1≥0

[0112] D o :D m +D r

[0113] Constraint 14: In practice, the inner ring always experiences more stress than the outer ring; this requires a limitation on the thickness of the outer ring. In the bearing of the present patent, the inner ring is designed to be stronger than the outer ring:

[0114] (D i -d)-(D-D o )≥0

[0115] D i :D m -D r

[0116] D o :D m +D r

[0117] Constraints 15-18: The curvature difference is always a number between 0 and 1, which imposes practical constraints on the average diameter of the rollers, the pitch diameter, the roller curvature radius and the raceway radius.

[0118]

[0119] where R g is the roller profile curvature radius, R is the outer raceway groove curvature radius, and r is the inner raceway groove curvature radius;

[0120] Constraints 19 and 20: The constraint on the length of the rollers is a key issue in the design of roller bearings, the effective length l e and the width B of the bearing limit the maximum length of the rollers, the constraints can be written as

[0121] βB-l e ≥0, 0.7≤β≤0.85

[0122] B-2r1-l e -2r g ≥0

[0123] Constraint 21: The thickness of the outer ring should be large enough and should be such that, in the worst case, the maximum dynamic shear stress occurs at the center of the outer ring.

[0124]

[0125] Z static =0.626b

[0126] where b is the length of the short semi-axis of the contact ellipse area.

[0127] (3) Use optimization algorithm to solve the optimal value of design variable

[0128] In the structural design of rolling bearing systems, core geometric parameters such as the inner diameter, outer diameter, and width are typically considered as fixed constraints. To determine the key design solutions that meet service life and performance requirements, a multi-objective optimization framework is required. This framework uses the bearing's load capacity under operating load, the material's inherent stress limits, the roller profile crown design, and the intrinsic properties of the lubricant as core input parameters and boundary conditions. Numerical optimization algorithms are then applied to iteratively solve the problem and identify one or more Pareto-optimal design solutions.

[0129] This paper uses Genetic Algorithm (GA) to achieve dual-objective optimization design of bearings. The overall process is shown in the figure. The algorithm process revolves around bearing design optimization, such as Figure 1 The specific steps are as follows:

[0130] 1) First, input the bearing design parameters (bearing inner diameter, outer diameter, width, roller diameter, roller length, roller crown, inner and outer ring raceway curvature), and then complete the dual objective function construction (roller diameter, roller length, inner and outer ring curvature), optimize the variable range definition, initialize the genetic algorithm tool, initialize the individual and register the evolutionary operator, and then complete the population initialization.

[0131] 2) The initialized population undergoes an initial population evaluation before entering an evolutionary loop: the loop first determines whether the preset maximum number of generations has been reached. If so, the optimal frontier solution is extracted and all optimal solutions are output, and the process terminates. If not, the offspring population is generated first, and then the new offspring is evaluated (the evaluation process integrates the objective function calculation and the calculation of 21 geometric constraints, and imposes constraint violation penalties based on the constraint violation values).

[0132] 3) The new offspring after evaluation is merged with the parent generation, and the new population is screened through the selection operation. The evolution cycle is entered again and iterated until the maximum generation termination condition is met, and finally the optimized solution for the bearing design is output.

[0133] Through algorithm optimization, the key parameters of the bearing are obtained:

[0134] Rolling element characteristics: number of rollers Z = 32, roller diameter D r =12.203mm, pitch diameter D m =133.80mm, effective contact length l e =36.30mm, roller profile radius R g =290.27mm, inner and outer raceway groove curvature radius R = r = 299.25mm. Based on the optimized parameters, the rated dynamic load C = 363017.15N is calculated, which meets the actual dynamic load requirements.

Claims

1. A design method for a fully complemented CARB bearing based on an "elastohydrodynamic lubrication load-bearing model," characterized in that: Here are the steps: (1) Constructing multi-objective functions; According to the operation requirements, different objective functions of rolling bearings are proposed. The objective function is defined as the maximum value of the load-bearing life and the minimum film thickness of the elastohydrodynamic lubrication, which can be expressed as: The expression of dynamic load capacity C is composed of the reduction factor λν considering edge load and non-uniform stress, roller diameter D r , pitch circle diameter D m , number of roller rows i, roller length l, number of rollers Z, ratio of roller diameter to pitch circle diameter composition; The minimum film thickness of the elastohydrodynamic lubrication is determined by the minimum film thickness of the bearing inner ring (h min ) inner and the minimum film thickness of the bearing outer ring (h min ) outer composition; Aiming at the lubrication behavior of the fluid dynamic pressure oil film between two elastic bodies in relative rolling or rolling with sliding, the elastic fluid dynamic pressure lubrication theory is proposed. Using the elastic fluid dynamic pressure lubrication theory, the minimum oil film thickness for elastohydrodynamic lubrication is calculated. The minimum oil film thickness for elastohydrodynamic lubrication is: Where: α p is the viscosity-pressure coefficient; η0 is the dynamic viscosity at normal pressure; u is the average speed between the roller and the inner and outer ring surfaces; R d is the equivalent curvature radius; E0 is the equivalent elastic modulus; q is the load per unit contact length; Without considering the roller slip surface, the average speed between the roller and the inner and outer ring surfaces is: Where: n is the inner ring speed, r / min; (2) Building a constraint system; Maximizing dynamic load-bearing capacity, maximizing fatigue life, and minimizing friction power consumption are defined as core optimization objectives, forming a multi-objective optimization problem. Flexible adjustment of design preferences is achieved through weight coefficients. Constraints are set, including the material strength limit (maximum contact stress), the minimum oil film thickness (avoiding direct metal contact), the size limit (installation space limitation), and the lower limit of fatigue life to ensure the feasibility of the optimization results. Based on the above description, 21 constraints are proposed as follows: Constraints 1 and 2: The bearing's mean diameter should be selected between the bearing's inner and outer diameters to ensure space for the sharp chamfers of the inner and outer rings; D m -(d+2r3+D r )≥0 (D-2r1-D r )-D m ≥0 Where d is the inner diameter of the bearing, D is the outer diameter of the bearing, r3 is the radial chamfer of the inner ring, and r1 is the radial chamfer of the outer ring; Constraints 3 and 4: The range of the average roller diameter is derived from both strength and geometry considerations. The lower limit of the average roller diameter is obtained from the contact stress expression, and the upper limit is obtained from the geometry. The minimum value of the average roller diameter should be able to withstand the contact stress generated by the external load. Where Q is the roller contact load, l e The effective length of the roller is the difference between the roller length and the roller chamfer, r 1min With r 3min is the minimum value of the radial chamfer between the outer ring and the inner ring during design, and δ is the roller crown; Constraints 5 and 6: The minimum number of rollers is determined based on the lower limit of the bearing pitch diameter and the upper limit of the roller average diameter. Similarly, the maximum number of rollers is determined based on the upper limit of the bearing pitch diameter and the lower limit of the roller average diameter. Constraint 7: The maximum roller length is the bearing width minus the axial space reduction caused by self-alignment; Where B is the bearing width, d2 is the diameter of the boundary line between the inner raceway and the width direction of the bearing, r g is the roller chamfer radius; Constraints 8 and 9: The diameter of the rolling element should be selected within a certain range, where K Dmin and K Dmax is the roller diameter constraint constant, which determines the possible minimum and maximum diameters of the rolling element; 0.4≤K Dmin ≤0.5 0.6≤K Dmax ≤0.7 Constraints 10 and 11: To ensure the bearing's operational flexibility, the difference between the bearing's pitch diameter and its mean diameter must be less than a given value. Therefore, the following two constraints must be met: D m -(0.5-e)(D+d)≥0 (0.5+e)(D+d)-D m ≥0 0.03≤e≤0.08 Constraints 12 and 13: To ensure the working strength requirements, the thickness of the bottom of the bearing outer raceway should not be less than εD r , where ε is an unknown constraint constant, and the thickness of the outer ring should be large enough to allow chamfering; 0.5(D-D r -D m )-εD r ≥0 0.3≤ε≤0.4 0.5(D-D0)-2r1≥0 D o :D m +D r Constraint 14: Since the inner ring always bears more stress than the outer ring, the thickness of the outer ring needs to be restricted; the inner ring is designed to be stronger than the outer ring: (D i -d)-(D-D o )≥0 D i :D m -D r D o :D m +D r Constraints 15-18: Since the curvature difference is always between 0 and 1, actual constraints are imposed on the roller mean diameter, pitch diameter, roller curvature radius, and raceway radius. Among them, R g is the roller profile curvature radius, R is the outer raceway groove curvature radius, and r is the inner raceway groove curvature radius; Constraints 19 and 20: Effective length l of the bearing e and width B limit the maximum length of the roller, constrained as follows: βB-l e ≥0，0.7≤β≤0.85 B-2r1-l e -2r g ≥0 Constraint 21: The thickness of the outer ring should be large enough so that, in the worst case, the maximum dynamic shear stress occurs at the center of the outer ring; WITH static =0.626b Where b is the length of the minor semi-axis of the contact ellipse area; (3) Using optimization algorithms to solve the optimal values ​​of design variables; The genetic algorithm is used to realize the multi-objective optimization design of bearings. The steps are as follows: 1) First, input the bearing design parameters, including the inner diameter, outer diameter, width, roller diameter, roller length, roller crown, and inner and outer ring raceway curvature. Then, complete the construction of multi-objective functions for roller diameter, roller length, and inner and outer ring curvature, define the optimization variable range, initialize the genetic algorithm tool, initialize individuals, and register the evolutionary operator, and finally complete the population initialization. 2) The initialized population undergoes an initial population evaluation before entering an evolutionary cycle: The cycle first determines whether the preset maximum number of generations has been reached. If so, the optimal frontier solution is extracted and all optimal solutions are output, terminating the process. If not, a descendant population is generated and then evaluated. The evaluation process integrates the calculation of multiple objective functions and 21 constraints, adds a penalty function, and implements constraint violation penalties based on the constraint violation value. 3) The new offspring after evaluation is merged with the parent generation, and the new population is screened through the selection operation. The evolution cycle is entered again and iterated until the maximum generation termination condition is met, and finally the optimized solution for the bearing design is output.

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