Cylindrical roller modification method based on non-uniform rational B-spline curve

Abstract

The invention discloses a cylindrical roller modification method based on a non-uniform rational B-spline curve, and relates to the technical field of bearing design. According to the method, a bearing contact mechanical model considering the roller modification effect is constructed, a modification contour is constructed based on a non-uniform rational B-spline curve (NURBS curve), an optimization model with the basic reference rated life of a bearing as a target function is established on the basis of meeting geometric constraint conditions, and an optimization algorithm is adopted for solving to obtain a final modification curve. The established contact mechanical model comprises a bearing static balance model and a roller-raceway contact mechanical model and is used for calculating load distribution and contact stress under the unbalance loading working condition. Compared with a traditional symmetrical logarithmic modification method, the problem that the axial stress of the roller is not uniform under unbalance loading can be effectively solved, the uniformity of contact stress distribution is improved, and the fatigue life of the bearing is prolonged.

Description

Technical Field

[0001] This invention belongs to the field of bearing technology, and in particular relates to a method for shaping cylindrical rollers based on non-uniform rational B-spline curves. Background Technology

[0002] Cylindrical roller bearings are widely used in important transmission applications such as railway axle boxes, high-speed machine tools, rolling mills, and large lifting equipment due to their high radial load capacity, high rigidity, and separable structure. In practical applications, due to large loads and installation errors, the rollers are often under eccentric loading, causing angular misalignment of the contact rollers inside the bearing. This deteriorates the bearing's contact condition, reducing its fatigue life and reliability.

[0003] Due to their structural characteristics, straight-generic cylindrical roller bearings are prone to stress concentration at the roller edges. Therefore, shaping measures are typically employed on the roller generatrix of cylindrical roller bearings to improve stress distribution. Common shaping methods include full convex shaping and logarithmic shaping. In 1981, Johns and Gohar proposed a classic logarithmic shaping method, which has become an important reference and foundation for subsequent research and improvements. The classic logarithmic shaping expression is as follows:

[0004]

[0005] In the formula, Q d The load value of the maximum loaded roller within the bearing is usually selected, l we is the effective length of the roller, v is the Poisson's ratio of the roller and raceway, E is the elastic modulus of the roller and raceway, a and b are the contact half-width and contact half-length, respectively, and x is the coordinate value along the roller axis.

[0006] However, most existing reshaping methods are symmetrical designs. Under eccentric loading conditions, the axial stress distribution of the rollers is significantly asymmetrical, leading to excessive load on one end and premature failure. Therefore, traditional symmetrical reshaping methods have limited adaptability to eccentric loading conditions and are unable to effectively improve the contact performance and stress distribution of the rollers, thus limiting the improvement of bearing life and reliability.

[0007] In summary, given the limited adaptability of existing symmetrical profile modification methods under eccentric loading conditions due to the asymmetry of roller axial stress, it is necessary to develop an asymmetric profile modification method that can adapt to eccentric loading conditions in order to improve the overall performance and service life of bearings. Summary of the Invention

[0008] The purpose of this invention is to provide a cylindrical roller shaping method based on non-uniform rational B-spline curves, in order to address the problem that existing symmetrical shaping methods are not adaptable enough under eccentric loading conditions due to the asymmetry of roller axial stress.

[0009] The technical solution of this invention is: a method for shaping cylindrical rollers based on non-uniform rational B-spline curves (NURBS curves), the method comprising the following steps:

[0010] S1: Construct a contact mechanics model for cylindrical roller bearings that considers the roller profile effect, and use it to calculate and output the load distribution, contact stress and deformation of the bearing under off-center loading conditions.

[0011] Specifically, the contact mechanics model of the cylindrical roller bearing includes:

[0012] First, a bearing static equilibrium model is used to calculate the load distribution and corresponding deformation of each roller under eccentric loading conditions. The load distribution includes the total radial force and overturning moment borne by all rollers, and the deformation includes the radial displacement and tilt angle of the inner ring. This model is established and solved through the following steps:

[0013] 1) Establish the coordination relationship between the radial deformation and tilting deformation of the inner ring and the deformation at the contact positions of each roller;

[0014] 2) Cut each roller into slices of equal thickness along the axial direction, and considering the influence of roller shaping, establish a coordination relationship between the overall deformation of the roller and the deformation of the slices;

[0015] 3) Establish the relationship between the load on the roller slice and the corresponding deformation;

[0016] 4) Establish the static balance relationship between the load on each roller slice and the overall load on the bearing;

[0017] 5) The Newton-Raphson iterative algorithm is used to solve the relational equations in the static equilibrium model, and the calculation results of load distribution and deformation are output to achieve the corresponding solution of the two.

[0018] Second, the roller-raceway contact mechanics model is used to solve for the contact stress distribution of each roller by applying the load distribution and deformation calculated from the bearing static balance model. The roller-raceway contact mechanics model is implemented through the following steps:

[0019] 1) Slice each roller along the axial direction. The number of slices and the division method are consistent with those in the bearing static balance model. Assume that the contact stress inside the slice is uniformly distributed along the axial direction and distributed in the transverse direction according to the Hertzian line contact theory.

[0020] 2) Based on the load and deformation information obtained from the bearing static equilibrium model, and in accordance with the half-space contact theory, establish a relational equation considering the maximum contact stress of each slice;

[0021] 3) The nested Newton-Raphson iterative algorithm is used to solve the relational equations in the roller-raceway contact mechanics model.

[0022] S2: A roller profile model is constructed using NURBS curves, and geometric constraints are set on the NURBS curves to limit the range of their parameter values, thereby ensuring that the generated profile curves meet the basic geometric requirements of the roller profile.

[0023] The geometric constraints include:

[0024] 1) The node vector is set to uniform parameterization, and multiple repeating nodes are set at both ends of the curve so that the NURBS curve passes through the first and last control points;

[0025] 2) The curve degree is 3, and the weight of each control point is the same to ensure that the continuity of the NURBS curve in the parameter range meets the requirements of the roller shaping curve.

[0026] 3) The horizontal coordinates of the control points increase monotonically, while the horizontal coordinates of the first and last control points are fixed, corresponding to the two ends of the effective length range of the roller, respectively.

[0027] 4) The ordinates of the three middle control points are fixed at 0;

[0028] 5) The ordinate of the control points gradually decreases from both ends toward the center.

[0029] S3: Based on the aforementioned cylindrical roller bearing contact mechanics model, an optimized model is constructed for obtaining the profile modification curve. The optimized model includes:

[0030] S3.1: Set the objective function, which is based on the contact performance index between the roller and the raceway and is used to guide the optimization process of the profile curve;

[0031] S3.2: Set optimization variables, which are the changes in the horizontal and vertical coordinates of control points other than the fixed coordinates, to adjust the shape of the NURBS curve;

[0032] S3.3: Based on the geometric constraints, set optimization algorithm constraints to ensure that the NURBS curve continuously meets the basic geometric requirements of the roller profile during the optimization iteration process;

[0033] S3.4: Select an optimization algorithm that supports constraint processing to solve the optimization model and obtain the final shaping curve.

[0034] The objective function is the basic reference rated life of the bearing, which takes into account the effects of roller skew and roller profile modification. The NURBS curve has 7 control points. The changes in the horizontal and vertical coordinates of each control point are calculated relative to the original coordinates of the corresponding control point. The original coordinates are the initial settings of the control points in the unmodified state. Specifically, the original horizontal coordinates are uniformly distributed within the effective length of the roller, and the original vertical coordinates are uniformly set to zero to correspond to the unmodified state. The initial values ​​of all optimization variables are set to zero. The optimization algorithm is sequential least squares quadratic programming.

[0035] This invention effectively overcomes the problem of uneven axial contact stress distribution of rollers under eccentric loading conditions caused by traditional symmetrical profile modification methods, optimizing the uniformity of contact stress between the roller and raceway, thereby effectively improving bearing reliability and fatigue life. Furthermore, this method has good versatility and scalability, applicable not only to profile modification designs of cylindrical roller bearings but also to other roller bearings subjected to eccentric loads, meeting various working conditions and design requirements, and possessing high engineering application value. Attached Figure Description

[0036] Figure 1 This is a schematic diagram of the force model of a cylindrical roller bearing under eccentric loading.

[0037] Figure 2 This is a schematic diagram of the roller slicing method. In the diagram: 1 is the inner ring of the roller, 2 is the roller, and 3 is the outer ring of the roller.

[0038] Figure 3 This is a schematic diagram of a roller shaping model based on NURBS curves.

[0039] Figure 4 This is a comparison diagram of the roller profile based on NURBS curves and the classic Johns-Gohar roller profile in the embodiment. Detailed Implementation

[0040] The following detailed description of the cylindrical roller shaping method based on non-uniform rational B-spline curves of the present invention, with reference to the accompanying drawings and specific embodiments, is provided. These embodiments are merely illustrative of the technical solution of the present invention and are not intended to limit the scope of the invention.

[0041] Example of a method for shaping cylindrical rollers based on non-uniform rational B-spline curves:

[0042] S1: Constructing a contact mechanics model for cylindrical roller bearings

[0043] A contact mechanics model for cylindrical roller bearings, considering the roller profile effect, is constructed to calculate and output the load distribution, contact stress, and deformation of the bearing under off-center loading conditions. The model includes:

[0044] First, the bearing static equilibrium model is used to calculate the load distribution and corresponding deformation of each roller under eccentric loading conditions. The load distribution includes the total radial force and overturning moment borne by all rollers, and the deformation includes the radial displacement and tilt angle of the inner ring. A schematic diagram of the force model of a cylindrical roller bearing under eccentric loading is shown below. Figure 1 As shown. The specific implementation steps are as follows:

[0045] (1) Establish the coordination relationship between the radial deformation and tilting deformation of the inner ring and the deformation at each roller contact position:

[0046]

[0047] In the formula, δ rk θ represents the radial displacement of the k-th roller. k δ represents the relative tilt angle between the inner and outer rings at the k-th roller. r ψ represents the radial displacement of the bearing inner ring, θ represents the tilt angle of the bearing inner ring, and ψ represents the radial displacement of the bearing inner ring. k This indicates the angular position of the k-th roller.

[0048] (2) Figure 2 As shown, each roller is sliced ​​into equal-thickness slices along the axial direction to establish a coordination relationship between the overall deformation of the roller and the deformation of the slices:

[0049] δ rk,j =max(δ rk +x k,j ·tan(θ k )-2·P(x k,j ),0)

[0050] In the formula, δ rk,j Let x represent the deformation of the k-th roller and the j-th slice. k,j Let P(x) represent the x-coordinate of the k-th roller and the j-th slice. k,j ) represents the roller trimming amount of the k-th roller and the j-th slice.

[0051] (3) Establish the relationship between the load on the roller slice and the corresponding deformation:

[0052]

[0053] In the formula, q rk,j This indicates the load on the k-th roller and the j-th slice, l we n represents the effective length of the roller. s Indicates the number of slices.

[0054] (4) Establish the static balance relationship between the loads on each roller slice and the overall load on the bearing:

[0055] For roller k, first establish the roller slice under load q.rk,j and the total force Q acting on the roller rk and torque M mk Relationship:

[0056]

[0057] Establish the total radial force {Q} acting on each roller. rk} and overturning moment {M mk} and the radial force F on the bearing r and the overturning moment M m Relationship:

[0058]

[0059] In the formula, Z represents the number of rollers.

[0060] (5) The Newton-Raphson iterative algorithm is used to solve the relational equations in the static equilibrium model: the radial displacement δ of the bearing inner ring is used as the basis for solving the relational equations. r The tilt angle θ is used as the iteration variable.

[0061] Second, the roller-raceway contact mechanics model is used to solve for the contact stress distribution of each roller by applying the load distribution and deformation calculated from the bearing static balance model. The specific implementation steps are as follows:

[0062] (1) Slice each roller along the axial direction. The number of slices and the division method are consistent with the bearing static balance model. It is assumed that the contact stress inside the slice is uniformly distributed along the axial direction and the lateral contact stress distribution satisfies the Hertzian line contact theory.

[0063] (2) Based on the load and deformation information obtained from the bearing static equilibrium model, and according to the half-space contact theory, a relational equation considering the maximum contact stress of each slice is established:

[0064]

[0065] In the formula, a j h represents the contact half-width of the j-th slice. j p represents the half thickness of the j-th slice. 0j Let represent the maximum contact stress on the j-th slice, v represent the Poisson's ratio of the roller and raceway, E represent the elastic modulus of the roller and raceway, δ represent the radial deformation of the roller, and P(x) represent the maximum contact stress on the j-th slice. i ) represents the reshaping amount on the i-th slice, x i Let Q represent the coordinates of the i-th slice, Q represent the load on the roller, ξ represent the roller's tilt angle, and D represent the coordinates of the i-th slice. ij The coupling relationship between the i-th and j-th slices is represented by the following formula:

[0066]

[0067] In the formula, x i and x j Let x' and y' represent the coordinates of the i-th and j-th slices, respectively, where x' and y' are integration variables.

[0068] The static equilibrium model of the bearing was solved to obtain the loads {Q} on each roller. rk The relative tilt angle θ between the inner and outer rings at each roller angle position. k In the roller-raceway contact mechanics model, for roller k, the load Q on the roller is equal to Q_k. rk The inclination angle ξ of the roller is equal to θ k / 2.

[0069] (3) The nested Newton-Raphson iterative algorithm is used to solve the relational equations in the roller-raceway contact mechanics model: In the outer loop, the radial deformation δ of the roller is used as the iteration variable; in the inner loop, the maximum contact stress {p} of the roller is used as the iteration variable. 0j} is treated as an iteration variable.

[0070] S2: A roller profile modification model is constructed using NURBS curves, and geometric constraints are set on the NURBS curves to limit their parameter value range, thereby ensuring that the generated modification curves meet the basic geometric requirements of the roller profile. The roller profile modification model based on NURBS curves is as follows: Figure 3 As shown. The geometric constraints include the following:

[0071] (1) The node vector is set to uniform parameterization, and multiple repeating nodes are set at both ends of the curve so that the NURBS curve passes through the first and last control points;

[0072] (2) The curve degree is 3, and the weight of each control point is the same to ensure that the continuity of the NURBS curve in the parameter range meets the requirements of the roller shaping curve.

[0073] (3) The horizontal coordinates of the control points are monotonically increasing, while the horizontal coordinates of the first and last control points are fixed, corresponding to the two ends of the effective length range of the roller respectively.

[0074] (4) The ordinates of the three middle control points are fixed at 0;

[0075] (5) The ordinate of the control points gradually decreases from both ends toward the center.

[0076] S3: Based on the bearing contact mechanics model, construct an optimized model for the profile modification curve, the optimized model including:

[0077] S3.1: Set the objective function, which is based on the contact performance index between the roller and the raceway and is used to guide the optimization process of the profile modification curve. The objective function is selected as the basic reference rated life L of the bearing. 10r This basic reference rated life takes into account the effects of roller skew and roller profile modification:

[0078] max[f(X)]=max[L 10r ]

[0079] Basic reference rated life L of bearing 10r Perform according to the method described in ISO / TS16281:2008(E).

[0080] S3.2: Set optimization variables, which are the changes in the horizontal and vertical coordinates of control points other than the fixed coordinates, used to adjust the shape of the NURBS curve:

[0081] (1) The number of control points is selected as 7. The changes in the horizontal and vertical coordinates of the control points are calculated relative to the original coordinates of the corresponding control points. The original coordinates are the initial settings of the control points in the uncorrected state. Let the order of the NURBS curve C(u) be p, and the control points be marked as... Each control point C i =(x cp,i ,y cp,i ,w cp,i ), where x cp,i y cp,i and w cp,i These represent the x and y coordinates and weights of the control points, respectively. The optimization variables can be represented as:

[0082] X NURBS ={Δy cp,0 ,Δy cp,1 ,Δy cp,5 ,Δy cp,6 ,Δx cp,1 ,Δx cp,2 ,Δx cp,3 ,Δx cp,4 ,Δx cp,5}

[0083] In the formula, {Δy cp,i} and {Δx cp,i} represent the changes in the vertical and horizontal coordinates of the i-th control point relative to its original position, respectively.

[0084] (2) The initial values ​​of all the optimization variables are set to 0:

[0085]

[0086] (3) The original horizontal coordinate is evenly distributed within the effective length of the roller, and the original vertical coordinate is uniformly set to zero to correspond to the untrimmed state:

[0087] y 0,cp,i =0i=0,1,...,6

[0088]

[0089] In the formula, x 0,cp,i and y 0,cp,i Let x and y represent the original x and y coordinates of the i-th control point, respectively.

[0090] S3.3: Based on the aforementioned geometric constraints, set constraints for the optimization variables to ensure that the NURBS curve continuously satisfies the geometric constraints of the roller profile during the optimization iteration process:

[0091] The boundary conditions for the optimization variables are:

[0092]

[0093] In the formula, δ max This indicates the maximum allowable amount of reshaping.

[0094] The inequality constraints for the optimization problem are:

[0095]

[0096] S3.4: Select an optimization algorithm that supports constraint handling to solve the optimization model and obtain the final shaping curve. The optimization algorithm selected is the sequential least squares quadratic programming method.

[0097] The cylindrical roller bearing NU2217E was used as the implementation object, and the specific parameters are shown in Table 1. Under the working conditions of a bearing radial load of 55kN, a deflection torque of 110N·m, and a speed of 220rpm, the above modeling and analysis process was used for simulation calculation.

[0098] Table 1. Basic Bearing Parameters

[0099]

[0100] Through the constructed optimization model of the modified curve, the curve results under the above example were obtained, and compared with the classic Johns-Gohar logarithmic curve under the corresponding working conditions, such as... Figure 4As shown. Under this working condition, the fatigue life of the NURBS curve-based modification method is 24,694 hours, while that of the classic Johns-Gohar logarithmic curve is 20,809 hours, an improvement of 18.7%. This verifies that the proposed method can improve the roller contact state and alleviate the problem of axial stress asymmetry under eccentric loading conditions, thus having good eccentric loading adaptability.

[0101] The above embodiments describe the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. Several modifications and improvements can be made to the present invention without departing from its technical principles, and all such modifications and improvements fall within the protection scope of the present invention.

Claims

1. A method for shaping cylindrical rollers based on non-uniform rational B-spline curves, characterized in that, The method includes the following steps: S1: Construct a cylindrical roller bearing contact mechanics model that considers the roller profile effect, which is used to calculate and output the load distribution, contact stress and deformation of the bearing under eccentric loading conditions; the cylindrical roller bearing contact mechanics model includes: bearing static balance model and roller-raceway contact mechanics model. S2: A roller profile model is constructed using NURBS curves, and geometric constraints are set on the NURBS curves to limit the range of their parameter values, thereby ensuring that the generated profile curves meet the basic geometric requirements of the roller profile. S3: Based on the cylindrical roller bearing contact mechanics model, construct an optimized model for obtaining the profile modification curve.

2. The cylindrical roller shaping method based on non-uniform rational B-spline curves according to claim 1, characterized in that, In step S1, the bearing static balance model is used to calculate the load distribution and corresponding deformation of each roller under eccentric loading conditions. The load distribution includes the total radial force and overturning moment borne by each roller as a whole, and the deformation includes the radial displacement and tilt angle of the inner ring. This model is established and solved through the following steps: 1) Establish the coordination relationship between the radial deformation and tilting deformation of the inner ring and the deformation at the contact positions of each roller; 2) Cut each roller into slices of equal thickness along the axial direction, and considering the influence of roller shaping, establish a coordination relationship between the overall deformation of the roller and the deformation of the slices; 3) Establish the relationship between the load on the roller slice and the corresponding deformation; 4) Establish the static balance relationship between the load on each roller slice and the overall load on the bearing; 5) The Newton-Raphson iterative algorithm is used to solve the relational equations in the static equilibrium model, and the calculation results of load distribution and deformation are output to achieve the corresponding solution of the two.

3. The cylindrical roller shaping method based on non-uniform rational B-spline curves according to claim 1, characterized in that, In step S1, the roller-raceway contact mechanics model is used to solve for the contact stress distribution of each roller by calculating the load distribution and deformation obtained from the bearing static balance model. The roller-raceway contact mechanics model is implemented through the following steps: 1) Slice each roller along the axial direction. The number of slices and the division method are consistent with those in the bearing static balance model. Assume that the contact stress inside the slice is uniformly distributed along the axial direction and distributed in the transverse direction according to the Hertzian line contact theory. 2) Based on the load and deformation information obtained from the bearing static equilibrium model, and in accordance with the half-space contact theory, establish a relational equation considering the maximum contact stress of each slice; 3) The nested Newton-Raphson iterative algorithm is used to solve the relational equations in the roller-raceway contact mechanics model.

4. The cylindrical roller shaping method based on non-uniform rational B-spline curves according to claim 1, characterized in that, In step S2, the geometric constraints include: 1) The node vector is set to uniform parameterization, and multiple repeating nodes are set at both ends of the curve so that the NURBS curve passes through the first and last control points; 2) The curve degree is 3, and the weight of each control point is the same to ensure that the continuity of the NURBS curve in the parameter range meets the requirements of the roller shaping curve. 3) The horizontal coordinates of the control points increase monotonically, while the horizontal coordinates of the first and last control points are fixed, corresponding to the two ends of the effective length range of the roller, respectively. 4) The ordinates of the three middle control points are fixed at 0; 5) The ordinate of the control points gradually decreases from both ends toward the center.

5. The cylindrical roller shaping method based on non-uniform rational B-spline curves according to claim 1, characterized in that, In step S3, the optimization model includes: S3.1: Set the objective function, which is based on the contact performance index between the roller and the raceway and is used to guide the optimization process of the profile curve; S3.2: Set optimization variables, which are the changes in the horizontal and vertical coordinates of control points other than the fixed coordinates, to adjust the shape of the NURBS curve; S3.3: Based on the geometric constraints, set optimization algorithm constraints to ensure that the NURBS curve continuously meets the basic geometric requirements of the roller profile during the optimization iteration process; S3.4: Select an optimization algorithm that supports constraint processing to solve the optimization model and obtain the final shaping curve.

6. The cylindrical roller shaping method based on non-uniform rational B-spline curves according to claim 5, characterized in that, In step S3.1, the objective function is the basic reference rated life of the bearing, which takes into account the effects of roller skew and roller profile modification.

7. The cylindrical roller shaping method based on non-uniform rational B-spline curves according to claim 5, characterized in that, In step S3.2, the number of control points for the NURBS curve is 7. The changes in the horizontal and vertical coordinates of the control points are calculated relative to the original coordinates of the corresponding control points. The original coordinates are the initial settings of the control points in the unmodified state. Specifically, the original horizontal coordinates are evenly distributed within the effective length of the roller, and the original vertical coordinates are uniformly set to zero to correspond to the unmodified state. The initial values ​​of the optimization variables are all set to zero.

8. The cylindrical roller shaping method based on non-uniform rational B-spline curves according to claim 5, characterized in that, In step S3.4, the optimization algorithm is the sequential least squares quadratic programming method.

Images