Linear and arc combined convex optimization method for cylindrical roller

Abstract

The invention relates to a straight line and circular arc combined convex shape optimization method for a cylindrical roller, and aims to solve the problems that in existing logarithmic shape modification fitting, circular arc segmentation connection points depend on experience setting, and adaptability is poor. According to the method, a corresponding line-arc combination curve model is constructed based on a logarithmic modification curve, and the line-arc combination curve is iteratively optimized through an optimization method to obtain an optimal line-arc combination curve, so that the optimal line-arc combination curve approaches the logarithmic modification curve. The modeling process comprises segmentation point setting, connection point generation, arc parameter solving and curve construction, the optimization target is to minimize fitting error quadratic sum, and the geometric rationality and convexity of the curve are constrained and guaranteed. The adopted optimization algorithm is a sequence least square quadratic programming (SLSQP) algorithm. According to the method, fitting precision and engineering applicability are remarkably improved, and the method has good universality and expansibility, is suitable for modification design of various roller bearings and has engineering application value.

Description

Technical Field

[0001] This invention belongs to the field of bearing technology, and particularly relates to an optimization method for the straight-line and circular-arc combination convex shape of cylindrical rollers. Background Technology

[0002] In the field of rolling bearings, when cylindrical rollers are in the form of unmodified straight lines, edge stress concentration is prone to occur at their ends, leading to localized fatigue failure and reducing the overall service life of the bearing. To alleviate this problem, researchers have proposed various roller modification methods, the most common of which include full-circular arc modification, partial-circular arc modification, and logarithmic modification. Among these, logarithmic modification, due to its ability to effectively achieve uniform axial load distribution, exhibits superior stress distribution performance compared to other modification methods in most applications and has become an important solution in roller modification research and application.

[0003] As early as 1939, Lundberg first proposed the concept of logarithmic modification of roller generatrices, laying the foundation for subsequent research. However, his model gave a modification amount that tended to be infinite at the roller ends, which did not conform to actual machining and structural requirements. To address this, in 1981, Johns and Gohar improved the method and proposed a more engineering-applicable classical logarithmic modification expression, which is still widely used today and serves as the basis for improved models.

[0004] The shaping expression is as follows:

[0005]

[0006] 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.

[0007] While logarithmic shaping offers significant performance advantages, its actual machining is quite difficult, especially under CNC conditions where it is hard to achieve directly. Therefore, in engineering practice, multi-segment circular arcs are typically used to approximate the logarithmic shaping curve to reduce machining difficulty. However, existing methods usually rely on experience to determine the connection points between the arc segments, lacking a systematic modeling and optimization strategy. This empirical segmentation method has poor adaptability in different application scenarios, specifically: firstly, it requires multiple trial-and-error adjustments during the design phase, increasing the workload and time cost of modeling and verification; secondly, the fitting accuracy of the logarithmic shaping curve is limited, leading to deviations between the fitted shaping profile and the target curve, thus limiting the bearing's performance.

[0008] In summary, to address the problem that the connection point positions of the arc segments in existing logarithmic shaping fitting methods rely on empirical settings and lack adaptability in different application scenarios, it is necessary to develop a method that can adaptively determine the connection point positions based on the shape of the shaping curve in order to improve the fitting effect. Summary of the Invention

[0009] This invention aims to provide a method for optimizing the straight-line-circular arc combined convex profile of cylindrical rollers, overcoming the problems of existing logarithmic profile fitting methods where the position of the arc segment connection point depends on empirical settings and has poor adaptability in different application scenarios, thereby improving fitting accuracy and engineering applicability.

[0010] This invention proposes a method for optimizing the straight-line-circular-arc combination convex profile of cylindrical rollers, the steps of which are as follows:

[0011] Step 1: Construct the corresponding line-arc combination curve model based on the logarithmic shaping curve;

[0012] Step 2: Iteratively optimize the line-arc combination curve using optimization methods to obtain the optimal line-arc combination curve, making the optimal line-arc combination curve approach the logarithmic shaping curve.

[0013] Step one specifically includes the following steps:

[0014] S1.1: Based on the logarithmic shaping curve and its coordinate system, insert the horizontal coordinate values ​​of the dividing points on the horizontal axis. The number of dividing points is even and not less than 6, dividing the domain into an odd number of intervals, where the middle interval is used to construct a straight line segment and the remaining intervals are used to construct a circular arc segment.

[0015] S1.2: Generate a set of connection points, wherein the connection points are the endpoints of line segments or arc segments, specifically including:

[0016] (1) The connection point corresponding to the two middle dividing points has the same horizontal coordinate as the corresponding dividing point and the vertical coordinate is 0;

[0017] (2) The connection points corresponding to the remaining dividing points have the same horizontal coordinate as the corresponding dividing points, and the vertical coordinate is the value of the point on the logarithmic shaping curve.

[0018] S1.3: Establish a set of relational equations based on the shape characteristics of the line-arc combination curve to solve for the center position and radius of the arc segment. The relational equations require:

[0019] (1) All arc segments pass through the connection points at the ends of the corresponding intervals;

[0020] (2) All adjacent segments must be tangent;

[0021] S1.4: Based on the solution results of the aforementioned relational equation, construct the line-arc combination curve model, specifically including:

[0022] (1) Construct a line segment that passes through the endpoints of the middle interval;

[0023] (2) In the remaining intervals, construct the corresponding circular arc curves based on the center position and radius obtained from the solution;

[0024] (3) Connect the line segment with the circular arc curve to form a line-arc composite curve model.

[0025] Step two specifically includes the following steps:

[0026] S2.1: Define an objective function, which is used to evaluate the similarity between the logarithmic shaping curve and the obtained line-arc combination curve model;

[0027] S2.2: The optimization variable is set as the offset of the x-coordinate of each dividing point (excluding the first and last dividing points) relative to the initial distribution, used to adjust the shape of the line-arc combination curve. The initial x-coordinate position of the dividing point is set to ensure that the line-arc combination curve generated by it meets the geometric requirements of the roller profile;

[0028] S2.3: Set optimization algorithm constraints to ensure that the line-arc combination curve continuously meets the geometric requirements of the roller profile during the optimization iteration process;

[0029] S2.4: Using an optimization algorithm that supports constraint processing, solve the optimization model to obtain the optimal line-arc combination curve.

[0030] Furthermore, in step one, the logarithmic shaping curve is defined in a two-dimensional Cartesian coordinate system, with the horizontal axis coinciding with a generatrix of the unshaped roller, the origin located at the midpoint of that generatrix, and the vertical axis passing through the axis and perpendicular to the horizontal axis. The domain of the logarithmic shaping curve is within the effective length range of the roller.

[0031] Furthermore, the input for the iterative optimization in step two is the number of arc segments and the preset logarithmic shaping curve. By continuously adjusting the positions of the remaining dividing points except for the first and last dividing points, the shape of the line-arc combination curve is changed, and the output is the optimal line-arc combination curve. During the process, it is necessary to call the line-arc combination curve model obtained in step one, or the line-arc combination curve obtained in the previous iteration. The iterative optimization method is the Sequential Least Squares Quadratic Programming (SLSQP) algorithm.

[0032] Furthermore, the objective function is used to evaluate the sum of squared errors between the input logarithmic shaping curve and the line-arc combination curve. Specifically, it involves uniformly generating no less than 100 sampling points within the domain, obtaining the ordinate values ​​of the two curves at each sampling point, calculating the squared errors, and summing them.

[0033] Furthermore, the initial x-coordinate positions of the dividing points are uniformly distributed within the defined domain.

[0034] Furthermore, the constraints of the iterative optimization algorithm include:

[0035] (1) The x-coordinate value of the dividing point does not exceed the valid domain range;

[0036] (2) The abscissa value of the dividing point is kept to increase in a strictly monotonically, and the straight line segment in the line-arc combination curve is not less than the preset minimum length, which is set by the operator according to the actual application requirements.

[0037] (3) The line-arc combination curve satisfies the convexity condition that the second derivative is non-negative.

[0038] The beneficial effects of this invention are as follows: The method of this invention can effectively overcome the problems of existing logarithmic shaping and fitting methods, such as the dependence of the arc segment connection point position on empirical settings and poor adaptability in different application scenarios. Furthermore, this method has good versatility and scalability, and is not only applicable to the logarithmic shaping design of cylindrical roller bearings, but also to the logarithmic shaping simplification of other roller bearings, meeting various working conditions and design requirements, and possessing high engineering application value. Attached Figure Description

[0039] Figure 1 This is a schematic diagram of the optimization method for a combination of straight lines and circular arcs to form a convex shape.

[0040] Figure 2 This is the optimal line-arc combination curve with 6 dividing points in the example.

[0041] Figure 3 This is the optimal line-arc combination curve with 8 dividing points in the example. Detailed Implementation

[0042] The following detailed description of the linear-arc combination convex optimization method for cylindrical rollers according to the present invention, with reference to the accompanying drawings and specific embodiments, is provided. This embodiment is merely for illustrating the technical solution of the present invention and is not intended to limit the scope of the invention.

[0043] An embodiment of a method for optimizing the convex shape of a cylindrical roller by combining straight lines and circular arcs:

[0044] Step 1: Construct the corresponding line-arc combination curve model based on the logarithmic shaping curve; including the following steps:

[0045] S1.1: Based on the logarithmic shaping curve and its coordinate system, the logarithmic shaping curve is defined in a two-dimensional Cartesian coordinate system. The horizontal axis coincides with a generatrix of the unshaped roller, and the origin is located at the midpoint of that generatrix. The vertical axis passes through the axis and is perpendicular to the horizontal axis. The domain of the logarithmic shaping curve lies within the effective length of the roller. The horizontal coordinate values ​​of dividing points are inserted on the horizontal axis. The number of dividing points is even and not less than 6, dividing the domain into an odd number of intervals. The middle interval is used to construct straight line segments, and the remaining intervals are used to construct circular arc segments.

[0046] like Figure 1 As shown, assuming a pre-defined logarithmic shaping curve, P Log (x), x∈[-l] we / 2,l we / 2], where l we This is the effective length value of the roller. Let the x-coordinate of the input separator point be... Where n d The number of dividing points satisfies:

[0047]

[0048] Thus, n is obtained. d -1 interval:

[0049] I i =[x i ,x i+1 ], i = 0, 1, ..., n d -2

[0050] Where, index m is defined d =n d / 2-1, the corresponding interval This is the middle interval, used to construct line segments.

[0051] S1.2: Generate a set of connection points, wherein the connection points are the endpoints of line segments or arc segments, specifically including: First, the connection points corresponding to the two middle dividing points, whose x-coordinates are the same as the corresponding dividing points, and whose y-coordinates are 0; Second, the connection points corresponding to the remaining dividing points, whose x-coordinates are the same as the corresponding dividing points, and whose y-coordinates are the values ​​of the points on the logarithmic shaping curve:

[0052] Let the set of connection points be . The generation method is as follows:

[0053]

[0054] In the formula, y j =P Log (x j ) indicates connection point J j The ordinate.

[0055] S1.3: Establish a set of relational equations based on the shape characteristics of the line-arc combination curve to solve for the center position and radius of the arc segment. The relational equations include:

[0056] (1) The connection point at the end of the corresponding interval for each arc segment:

[0057]

[0058] In the formula, x o,i y o,i and r i Represent the i-th interval I respectively i The corresponding arc curve has the x-coordinate of its center, y-coordinate of its center, and radius.

[0059] (2) All adjacent segments must be tangent. The tangency between adjacent arc segments is ensured by the consistent slope of the tangent lines at the connection points:

[0060]

[0061] The middle straight line segment is tangent to the adjacent circular arc segment, requiring that the x-coordinate of the arc's center is consistent with the x-coordinate of the corresponding endpoint of the straight line segment, i.e.:

[0062]

[0063] S1.4: Based on the solution results of the aforementioned relational equation, construct the line-arc combination curve model, specifically including:

[0064] (1) Construct a line segment passing through the endpoints of the intermediate interval;

[0065] (2) In the remaining intervals, construct the corresponding circular arc curves based on the center position and radius obtained from the solution;

[0066] (3) Connect the line segment with the arc curve to form the line-arc composite curve P. SA (x) and output:

[0067]

[0068] Step 2: Iteratively optimize the line-arc combination curve using an optimization method. The input for iterative optimization is the number of arc segments and a preset logarithmic shaping curve. By continuously adjusting the positions of the remaining dividing points (excluding the first and last dividing points), the shape of the line-arc combination curve is changed to obtain the optimal line-arc combination curve, making it approach the logarithmic shaping curve. The initial abscissa positions of the dividing points are uniformly distributed within the defined domain. This includes the following steps:

[0069] S2.1: Define an objective function, which is to minimize the sum of squared errors between the logarithmic shaping curve and the line-arc combination curve. Specifically, generate no less than 100 sampling points uniformly within the domain, obtain the ordinate values ​​of the two curves at each sampling point, calculate the squared errors, and sum them.

[0070] min[f(X)] = min[SSE]

[0071] In the formula, SSE represents the sum of squared errors, which is calculated as follows:

[0072]

[0073] In the formula, n sa This indicates the number of sampling points.

[0074] S2.2: The optimization variable is set as the offset of the x-coordinate of each dividing point (excluding the first and last dividing points) relative to the initial distribution. This is used to adjust the shape of the line-arc combination curve. The initial x-coordinate position of the dividing point is set to ensure that the line-arc combination curve generated by it meets the geometric requirements of the roller profile.

[0075]

[0076] In the formula, Δx j This represents the offset of the x-coordinate of the j-th separator point relative to the x-coordinate position of the initial separator point, with all initial values ​​set to 0. The x-coordinate positions of the initial separator points are uniformly distributed within the defined domain.

[0077] S2.3: Set optimization algorithm constraints to ensure that the line-arc combination curve continuously meets the geometric requirements of the roller profile during the optimization iteration process. The optimization algorithm constraints include:

[0078] (1) The x-coordinate value of the dividing point does not exceed the valid domain range:

[0079]

[0080] In the formula, x 0,j This represents the initial x-coordinate position of the j-th dividing point.

[0081] (2) The x-coordinate value of the dividing point remains strictly monotonically increasing, and the straight line segment in the line-arc combination curve is not less than a preset minimum length, which is set by the operator according to actual application requirements:

[0082]

[0083] In the formula, l st,min This indicates the preset minimum length, which is set to the effective length l in this case. we 30%.

[0084] (3) The line-arc combination curve satisfies the convexity condition that the second derivative is non-negative:

[0085]

[0086] S2.4: Using an optimization algorithm that supports constraint handling, solve the optimization model and output the adaptive straight-line-circular-arc combination shaping curve. The optimization algorithm is Sequential Least Squares Quadratic Programming (SLSQP).

[0087] Using a Johns-Gohar logarithmic profile modification curve as the implementation object, the effective length of the roller is 24.6 mm, and the design load of the modification curve is taken as 20 kN. Through the constructed optimization model of the modification curve, the curve results under the above example are obtained and compared with the logarithmic curve under the corresponding working condition. The number of dividing points is taken as 6 and 8, respectively, as shown below. Figure 2 and Figure 3 As shown, the adaptive straight-line-circular arc combined shaping curve fits the logarithmic curve well under this working condition, and the fitting effect improves with the increase of the number of circular arc segments. This verifies the good fitting performance of the proposed method for the logarithmic shaping curve of cylindrical rollers.

[0088] 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 optimizing the convex shape of a cylindrical roller by combining straight lines and circular arcs, characterized in that, The steps are as follows: Step 1: Construct the corresponding line-arc combination curve model based on the logarithmic shaping curve; Step 2: Iteratively optimize the line-arc combination curve using optimization methods to obtain the optimal line-arc combination curve, making the optimal line-arc combination curve approach the logarithmic shaping curve. Step one specifically includes the following steps: S1.1: Based on the logarithmic shaping curve and its coordinate system, insert the horizontal coordinate values ​​of the dividing points on the horizontal axis. The number of dividing points is even and not less than 6, dividing the domain into an odd number of intervals, where the middle interval is used to construct a straight line segment and the remaining intervals are used to construct a circular arc segment. S1.2: Generate a set of connection points, wherein the connection points are the endpoints of line segments or arc segments, specifically including: (1) The connection point corresponding to the two middle dividing points has the same horizontal coordinate as the corresponding dividing point and the vertical coordinate is 0; (2) The connection points corresponding to the remaining dividing points have the same horizontal coordinate as the corresponding dividing points, and the vertical coordinate is the value of the point on the logarithmic shaping curve. S1.3: Establish a set of relational equations based on the shape characteristics of the line-arc combination curve to solve for the center position and radius of the arc segment; the relational equations require: (1) All arc segments pass through the connection points at the ends of the corresponding intervals; (2) All adjacent segments must be tangent; S1.4: Based on the solution results of the aforementioned relational equation, construct the line-arc combination curve model, specifically including: (1) Construct a line segment that passes through the endpoints of the middle interval; (2) In the remaining intervals, construct the corresponding circular arc curves based on the center position and radius obtained from the solution; (3) Connect the line segment with the circular arc curve to form a line-arc composite curve model; Step two specifically includes the following steps: S2.1: Define an objective function, which is used to evaluate the similarity between the logarithmic shaping curve and the obtained line-arc combination curve model; S2.2: The optimization variable is set as the offset of the horizontal coordinate of each dividing point other than the first and last dividing points relative to the initial distribution, which is used to adjust the shape of the line-arc combination curve; the initial horizontal coordinate position of the dividing point is set to ensure that the line-arc combination curve generated by it meets the geometric requirements of the roller profile. S2.3: Set optimization algorithm constraints to ensure that the line-arc combination curve continuously meets the geometric requirements of the roller profile during the optimization iteration process; S2.4: Using an optimization algorithm that supports constraint processing, solve the optimization model to obtain the optimal line-arc combination curve.

2. The method for optimizing the straight-line and circular-arc combination convex shape of a cylindrical roller according to claim 1, characterized in that, In step one, the logarithmic shaping curve is defined in a two-dimensional Cartesian coordinate system. The direction of the horizontal axis coincides with a generatrix of the unshaped roller, the origin is located at the midpoint of the generatrix, the vertical axis passes through the axis and is perpendicular to the horizontal axis, and the domain of the logarithmic shaping curve is within the effective length of the roller.

3. The method for optimizing the straight-line and circular-arc combination convex shape of a cylindrical roller according to claim 1, characterized in that, The input for the iterative optimization in step two is the number of arc segments and the preset logarithmic shaping curve. By continuously adjusting the positions of the remaining dividing points except for the first and last dividing points, the shape of the line-arc combination curve is changed, and the output is the optimal line-arc combination curve. During the process, it is necessary to call the line-arc combination curve model obtained in step one, or the line-arc combination curve obtained in the previous iteration. The iterative optimization method is the Sequential Least Squares Quadratic Programming (SLSQP) algorithm.

4. The method for optimizing the straight-line and circular-arc combination convex shape of a cylindrical roller according to claim 1, characterized in that, The objective function is used to evaluate the sum of squared errors between the input logarithmic shaping curve and the line-arc combination curve. Specifically, it involves uniformly generating no less than 100 sampling points within the domain, obtaining the ordinate values ​​of the two curves at each sampling point, calculating the squared errors, and summing them.

5. The method for optimizing the straight-line and circular-arc combination convex shape of a cylindrical roller according to claim 1, characterized in that, The initial x-coordinates of the dividing points are uniformly distributed within the defined domain.

6. The method for optimizing the straight-line and circular-arc combination convex shape of a cylindrical roller according to claim 1, characterized in that, The constraints of the iterative optimization algorithm include: (1) The x-coordinate value of the dividing point does not exceed the valid domain range; (2) The abscissa value of the dividing point is kept to increase in a strictly monotonically, and the straight line segment in the line-arc combination curve is not less than the preset minimum length, which is set by the operator according to the actual application requirements. (3) The line-arc combination curve satisfies the convexity condition that the second derivative is non-negative.

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