A particle swarm optimization method based on optimal loading contact area of planetary modified gear

Abstract

The present application provides a kind of particle swarm optimization method based on optimal loading contact area of planetary modified gear, mainly including the steps of establishing planetary gear comprehensive modified gear surface equation, calculating modified gear surface contact area and establishing optimization model and solving etc.The present application is based on particle swarm algorithm, and the tooth profile design of planetary gear mechanism is optimized to improve its gear surface contact area, and then improve carrying capacity.Particle swarm algorithm can provide efficient, accurate and easy-to-implement design method for the tooth profile modification of planetary mechanism, so as to improve the quality and performance of gear products, meet the demand of modern mechanical transmission system for high-performance gears.

Description

Technical Field

[0001] This invention belongs to the field of gear transmission design technology, specifically involving a particle swarm optimization method based on the optimal loading contact area of ​​planetary modified gears. Background Technology

[0002] Planetary gear train gear modification is a technique that improves gear meshing performance by modifying the gear tooth profile to increase the loaded contact area on the tooth surface. Traditional tooth profile modification methods rely on empirical rules or trial-and-error processes and simplified mathematical models, which cannot accurately capture the complex dynamic behavior of gear systems. Robust design is a systematic design methodology designed to ensure that a product or system maintains its performance and quality in the face of various uncertainties and changes. Robust design is particularly important in the field of gear transmissions.

[0003] The application of target optimization models in planetary gear modification is a highly specialized field, involving the intersection of multiple disciplines such as mechanical engineering, numerical analysis, materials science, and computer science. With the improvement of computing power and the continuous development of optimization algorithms, modern optimization techniques, represented by particle swarm optimization (PSO), have been rapidly adopted. PSO is an optimization tool based on swarm intelligence, simulating the social behavior of flocks of birds or schools of fish to find optimal solutions. This type of method can find better combinations of tooth profile parameters more quickly, improving design efficiency and reducing reliance on empirical design. PSO provides an efficient, accurate, and easy-to-implement design method for modifying the tooth profile of planetary mechanisms, thereby improving the quality and performance of gear products and meeting the demands of modern mechanical transmission systems for high-performance gears. Simultaneously, the combination of modern finite element methods, computer-aided design, and intelligent algorithms can further promote the improvement of gear system transmission performance. Summary of the Invention

[0004] (a) Technical problems to be solved This invention proposes a particle swarm optimization method based on the optimal loading contact area of ​​planetary modified gears. Based on the comprehensive modified tooth surface of planetary gears, the method takes maximizing the loading meshing imprint on the tooth surface as the optimization objective and determines the optimal parameter combination for the comprehensive modification of planetary gears. This method aims to solve the technical problem of how to continuously iterate the tooth profile and tooth direction modification parameters to optimize the transmission performance of gears, improve gear performance, and reduce design time.

[0005] (II) Technical Solution To address the aforementioned technical problems, this invention proposes a particle swarm optimization method based on the optimal loading contact area of ​​a planetary modified gear. This particle swarm optimization method based on the optimal loading contact area of ​​a planetary modified gear includes the following steps: S1. Establish the comprehensive modified tooth surface equation of the planetary gear: modify the tooth profile and tooth direction of the planetary gear through spline curves, define the tooth profile offset and tooth direction offset respectively, and establish the comprehensive modified tooth surface equation of the planetary gear based on the coordinates of the standard tooth surface point and the offset of the normal vector. S2. Calculate the contact area of ​​the modified tooth surface: Based on the tooth surface contact theory and Hertz contact model, establish the tooth surface contact coordinate system of the modified planetary gear. By solving the tooth surface contact equation, obtain the contact trajectory and contact elliptical region, and then calculate the contact area of ​​the modified tooth surface. S3. Establish and solve the optimization model: With the maximum contact area of ​​the tooth surface as the optimization objective and the offset of the tooth profile and tooth direction as the design variables, construct the optimization function and use the particle swarm optimization algorithm to solve for the optimal shaping parameters.

[0006] Further, in step S1, tooth profile modification is obtained by interpolating or fitting the normal offset of 7 tooth profile sampling points P1~P7 on the standard tooth profile line to obtain a tooth profile spline curve; tooth direction modification is obtained by interpolating or fitting the normal offset of 7 tooth direction sampling points P1~P7 on the tooth direction line to obtain a tooth direction spline curve.

[0007] Furthermore, the tooth profile sampling points P1~P4 are in r b and r s The distribution is equidistant between P4 and P7. r s and r a They are equidistantly distributed; among them, r b The radius of the base circle, r s This is the distance from point P4 to the gear axis, which is also the radius of the tooth profile at point P4. r a The radius of the tooth tip circle is defined; the tooth direction sampling points are equidistantly distributed along the tooth width direction, and the offset is defined along the gear axis direction.

[0008] Furthermore, in step S1, the spline curve is a Bézier curve or a B-spline curve. The offset is interpolated using a cubic or quartic Bézier curve to construct the profile and tooth direction modification curve equations, respectively.

[0009] Furthermore, in step S2, the tooth surface contact equation is established based on the condition that the position vectors and normal vectors of the two tooth surfaces are equal in a fixed coordinate system, and the contact point parameters are solved by discretizing the rotation angle of the large gear to obtain the contact trajectory.

[0010] Furthermore, in step S2, the major and minor semi-axes of the contact ellipse are calculated based on Hertzian contact theory, as shown in the following formula: In the formula, a , b These are the major and minor axes of the contact area ellipse, respectively; e For eccentricity, K (e) E (e) represent the elliptic integral of the first kind and the elliptic integral of the second kind, respectively; A It is a positive constant; E * It is the equivalent elastic modulus; P This represents the maximum pressure in the contact area.

[0011] Furthermore, in step S3, the objective function is optimized as follows: In the formula, min f ( x Let ) be the objective function. s The contact area of ​​the planetary gear tooth surface is used for optimization. The optimization variables include the tooth profile offset and tooth direction offset corresponding to each sampling point.

[0012] Furthermore, in step S3, the optimization variables must satisfy the constraint that the contact area does not exceed the tooth surface boundary. The constraint expression is: In the formula, a i , b i For the short and long semi-axis of the instantaneous contact imprint, a c , b c This represents the boundary of the tooth surface of the planetary gear.

[0013] Furthermore, in step S3, the particle swarm optimization algorithm initializes the particle swarm, iteratively updates the particle velocity and position, tracks the individual optimal solution and the global optimal solution, and finally obtains the optimal combination of shaping parameters.

[0014] Furthermore, in step S3, the update formulas for particle velocity and position are as follows: In the formula, w Inertial weight; c 1. c 2 is a positive learning factor; r 1. r 2 is a random number uniformly distributed between 0 and 1.

[0015] (III) Beneficial Effects This invention proposes a particle swarm optimization method based on optimizing the loading contact area of ​​planetary gear profiles. The method mainly includes establishing the comprehensive profile tooth surface equation of the planetary gear, calculating the contact area of ​​the profiled tooth surface, and establishing and solving the optimization model. Based on the particle swarm algorithm, this invention optimizes the tooth profile design of planetary gear mechanisms to increase their tooth surface contact area, thereby improving load-bearing capacity. The particle swarm algorithm provides an efficient, accurate, and easily implemented design method for modifying the tooth profile of planetary mechanisms, thus improving the quality and performance of gear products and meeting the demands of modern mechanical transmission systems for high-performance gears.

[0016] The beneficial effects of this invention specifically include: 1. Improve gear performance: This invention aims to increase the loading contact area of ​​gears, which can maximize the meshing strength of gears by maximizing the tooth surface contact area; by optimizing the tooth profile design, the contact spot area can be significantly increased, reducing gear meshing vibration and noise, and improving transmission efficiency and smoothness.

[0017] 2. Enhanced reliability: This invention incorporates robust design to improve the overall reliability and durability of the gear system under various uncertain conditions.

[0018] 3. Improved design efficiency: This invention uses a particle swarm search algorithm, which can quickly search for the optimal or near-optimal solution, significantly shortening the design cycle and reducing the iteration and trial-and-error time in traditional design methods. Attached Figure Description

[0019] Figure 1 This is a schematic diagram of gear tooth profile modification according to the present invention. Figure 2 This is the gear modification model in this invention; Figure 3 This is the gear pair contact coordinate system in this invention; Figure 4 This is a schematic diagram of the elastic deformation at the contact point in this invention; Figure 5 This is a diagram of the gear contact trajectory in this invention; Figure 6 This is an iterative diagram of the particle swarm optimization algorithm in this invention. Detailed Implementation

[0020] To make the objectives, contents, and advantages of the present invention clearer, the specific embodiments of the present invention will be described in further detail below with reference to the accompanying drawings and examples.

[0021] This embodiment proposes a particle swarm optimization method based on the optimal loading contact area of ​​a planetary modified gear. This method specifically includes the following steps: S1. Establish the equation for the comprehensive modification of the planetary gear tooth surface. Spline curves are used to modify the tooth surface of planetary gears. The spline curves are obtained by interpolating (or fitting) the corresponding control vertices. For the spline curve of the planetary gear tooth profile, such as... Figure 1 As shown, the position of its control vertex is determined by two factors: the tooth profile offset point and the offset amount. Considering the characteristics of planetary gears, the following method is used to define the planetary gear tooth profile spline curve: Based on the tooth profile of the standard planetary gear tooth surface, seven sampling points P1~P7 are discretized, where P1~P4 are located at... r b and r s The distribution is equidistant between P4 and P7. r s and r a They are equidistant from each other. r b The radius of the base circle, r s This is the distance from point P4 to the gear axis, which is also the radius of the tooth profile at point P4. r a Let be the radius of the tooth tip circle. The offset Δ along the normal vector for the 7 sampling points. p Interpolation (or fitting) is performed to obtain the offset of all points on the tooth profile. All points are then offset along the normal vector according to their corresponding offsets and connected to form a line, resulting in the planetary gear tooth profile spline curve. Let this be... .

[0022] The offset Δ corresponding to the entire grinding wheel profile p Given a function that varies along the tooth height, interpolation is performed using Bézier curves. To achieve higher-order continuity across the entire tooth surface, B-spline curves can also be used for interpolation.

[0023] For a given tooth profile offset point and offset amount, the definitions are shown in Table 1.

[0024] Table 1. Definition of offset of tooth profile sample points P1 P2 P3 P4 P5 P6 P7 Corresponding radius rh rh1 rh2 rh3 rh4 rh5 rh6 rh7 Corresponding offset Δp Δp1 Δp2 Δp3 Δp4 Δp5 Δp6 Δp7 in, coordinate values , A point gear coordinate system on the standard gear tooth profile x g y g The coordinates of the plane.

[0025] The equation for the Bezier curve is expressed as: (1) It can be written in matrix form: (2) Wherein, basis functions (3) These are Bernstein basis functions. b j To control the vertices.

[0026] Bézier curves possess global control properties. This invention employs cubic or quartic curves; taking cubic curves as an example, the Bézier curve function with four control vertices is a cubic polynomial, i.e. j =3, so the basis functions are: (4) in (5) Combining formulas (2) and (4), we obtain the equation of the curve as follows: (6) Take the radius values ​​corresponding to P1~P4 and P4~P7 respectively. r h and tooth profile offset Δ p Using two sets of control points, the corresponding radius values ​​and offsets are interpolated using Bezier curves to obtain the curve equations for the new tooth profile's radius values ​​and offsets. The segments P1 to P4 are: (7) get for (8) Sections P4 to P7 are: (9) get for: (10) Assuming the planetary gear tooth profile is in Figure 1 The gear coordinate system shown x g y g Plane representation Accordingly, the newly defined equation for the planetary tooth profile spline curve is expressed as: (11) In the formula, the upper and lower symbols correspond to the left and right tooth profiles of the planetary gear, respectively.

[0027] Similarly, the offset corresponding to the grinding wheel's motion trajectory is defined by the spline curve offset function. Based on the tooth direction line of the original standard tooth surface, seven sampling points P1~P7 are discretized, as follows: Figure 1 As shown. The tooth offset Δ is the offset of the 7 sampling points along their normal vector. l Interpolation (or fitting) is performed to obtain the offset of all points on the tooth axis. Seven points on the tooth axis are sampled, and these seven points are offset along their normal vectors according to their corresponding offsets and then connected to form a line, resulting in a tooth axis spline curve. In... Figure 1 In the coordinate system shown, find the coordinates of any point on the tooth-direction spline curve, let it be... .

[0028] Seven sample points P1~P7 on the tooth line are equidistantly distributed along the tooth width direction. Furthermore, for the offset Δ... l The definition is along y g Offset along the axis, such as Figure 1 As shown, the definition For the front end, For the back end, F This refers to the tooth width of the planetary gear.

[0029] Similar to the calculation method for the definition of planetary gear tooth profile spline curves, the offset points and offset amounts for the grinding wheel motion trajectory are shown in Table 2.

[0030] Table 2. Definition of offset of sample points of grinding wheel motion trajectory P1 P2 P3 P4 P5 P6 P7 Corresponding to f 0 F / 6 2F / 6 3F / 6 4F / 6 5F / 6 F Corresponding offset Δl Δl1 Δl2 Δl3 Δl4 Δl5 Δl6 Δl7 in, f Points representing radial lines z g Coordinate values.

[0031] Similarly, we can conclude that: The segments P1 to P4 are: (12) get for: (13) Sections P4 to P7 are: (14) get for: (15) The defined parametric equations of the grinding wheel's motion trajectory are in y g z g The parametric equations on the plane are: (16) In the formula, the upper and lower symbols correspond to respectively: y gPositive and negative directions of the axis.

[0032] Combining the tooth profile modification and tooth contour modification of planetary gears, the comprehensive modification tooth surface equation of planetary gears is established: (17) In the formula, coordinate values x g , y g For the point on the standard gear tooth profile Figure 1 The gear coordinate system shown x g y g The expression for a plane; n gx n gy This is the normal vector corresponding to the tooth surface; l 3 represents the parameter in the tooth profile direction; f For the direction of the teeth; p , l These represent the offsets of the tooth profile and tooth direction, respectively. The planetary gear composite profile tooth surface model is as follows: Figure 2 As shown.

[0033] S2. Calculate the contact area of ​​the modified tooth surface. Establish the tooth surface contact coordinate system of the modified planetary gear, such as... Figure 3 As shown, four coordinate systems are established, including two fixed coordinate systems. and ; and These are two fixed coordinate systems, one for the planetary gear and the other for the small gear. The rotation angles of the small gear and the large gear during their motion are represented by... and This indicates that the two angles change over time.

[0034] First, based on formula (17), the tooth surface equation of the planetary gear pair is established, and the tooth surface of the planetary gear is transformed to a fixed coordinate system through coordinate transformation. Down: (18) (19) Among them, the pinion moving coordinate system To a fixed coordinate system rotation transformation matrix for: (20) Large gear moving coordinate system To a fixed coordinate system rotation transformation matrix for: (twenty one) Fixed coordinate system To a fixed coordinate system Transformation matrix for: (twenty two) In the formula, E and These represent the center distance and center distance error values ​​for the large and small gears, respectively.

[0035] In a fixed coordinate system Below, the unit normal vector of the two tooth surfaces is expressed as: (twenty three) (twenty four) Among them, matrix , and Each is a matrix , and The third-order matrix after removing the fourth row and fourth column, n p ( l 3p , f p ) is the normal vector of the tooth surface of the pinion, n g ( l 3g , f g ) is the normal vector of the tooth surface of the large wheel.

[0036] According to tooth surface contact theory, the point vector and unit normal vector of the two contacting tooth surfaces at the contact point are equal, therefore they are in the same coordinate system. The contact equations for the two tooth surfaces of the modified gear are shown below: (25) The contact equation consists of five independent equations with six unknowns. Therefore, the rotation angle of the large gear can be determined based on the actual contact conditions. Discretize, given With certain values, the contact equation can be solved to obtain the other 5 parameter values. Substituting the 6 parameters into the planetary gear tooth surface equations (18) and (19), a series of discrete points on the tooth surface of the planetary gear contact trajectory can be obtained.

[0037] When the friction between the tooth surfaces is not considered, there is only normal contact force between the two tooth surfaces. In the case of point contact, when the two tooth surfaces are lightly loaded, the contact elliptical region is a very small part of the tooth surface (excluding edge contact), and the relative radius of curvature of the two tooth surfaces at the contact point is very small. Therefore, the Hertzian theory of elastic contact is used to analyze the loaded contact of the planetary gear pair.

[0038] The contact point p between the small gear tooth surface ∑1 and the large gear tooth surface ∑2 under normal load F n Under the action of the tooth surface, elastic deformation occurs, and the deformation amounts are respectively δ 1 and δ 2. The deformed tooth surfaces are ∑1' and ∑2', respectively. Since the gear pair is in point contact, the contact point p expands into an elliptical contact region within its tangential plane, with point p as the center of the ellipse.

[0039] The major and minor semi-axes of the planetary gear contact ellipse and contact trajectory are calculated as follows: (26) In the formula, a , b These are the major and minor axes of the contact area ellipse, respectively; e For eccentricity, K (e) E (e) represent the elliptic integral of the first kind and the elliptic integral of the second kind, respectively; A It is a positive constant; E * It is the equivalent elastic modulus; P This represents the maximum pressure in the contact area.

[0040] The contact area of ​​the tooth surface is obtained from the contact length and short semi-axis. s ,like Figure 4 As shown.

[0041] S3. Establishing and solving the optimization model The optimization objective is to maximize the area of ​​the tooth surface contact region, such as... Figure 5 As shown, the objective function is expressed as: (27) In the formula, min f ( x Let ) be the objective function. s This represents the contact area of ​​the planetary gear tooth surface.

[0042] The design optimization variable is represented as: Δ p 1, Δ p 2, Δ p 3, Δ p 4, Δ p 5, Δ p 6, Δ p 7, Δl 1, Δ l 2, Δ l 3, Δ l 4, Δ l 5, Δ l 6, Δ l 7.

[0043] Edge release must not occur in the contact area of ​​the planetary tooth surface during loading, and the range of optimization variables needs to be limited: (28) In the formula, a i , b i For the short and long semi-axis of the instantaneous contact imprint, a c , b c This represents the boundary of the tooth surface of the planetary gear.

[0044] The particle swarm optimization algorithm is used to solve the optimization function. The system is initialized with a set of random solutions, and the optimal solution is searched iteratively. However, without the crossover and mutation operations of the genetic algorithm, the global optimum is found by following the currently found optimal solution.

[0045] The particle swarm optimization algorithm first initializes a swarm of random particles, in d Dimensional space i The current position and velocity vector of each particle are respectively X i =( x i,1 x i,2 … x i,d )and V i =( v i,1 v i,2 … v i,d In each iteration, the particle updates itself by tracking two optimal solutions: one is the optimal solution that the particle itself has found. p i =( p i,1 p i,2 … p i,d The other is the optimal solution found by the entire population. p g =( p g,1 p g,2… p g,d After finding two optimal solutions, the particle updates its velocity and position using the following formula: Figure 6 As shown.

[0046] (29) (30) In the formula, w Inertial weight; c 1. c 2 is a positive learning factor; r 1. r 2 is a random number uniformly distributed between 0 and 1.

[0047] This yields the solution for the tooth surface modification parameters of the planetary gear, thus obtaining the optimal contact performance of the planetary gear.

[0048] The above description is only a preferred embodiment of the present invention. It should be noted that for those skilled in the art, several improvements and modifications can be made without departing from the technical principles of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention.

Claims

1. A particle swarm optimization method based on the optimal loading contact area of ​​a planetary gear, characterized in that, The particle swarm optimization method based on the optimal loading contact area of ​​the planetary gear includes the following steps: S1. Establish the comprehensive modified tooth surface equation of the planetary gear: modify the tooth profile and tooth direction of the planetary gear through spline curves, define the tooth profile offset and tooth direction offset respectively, and establish the comprehensive modified tooth surface equation of the planetary gear based on the coordinates of the standard tooth surface point and the offset of the normal vector. S2. Calculate the contact area of ​​the modified tooth surface: Based on the tooth surface contact theory and Hertz contact model, establish the tooth surface contact coordinate system of the modified planetary gear. By solving the tooth surface contact equation, obtain the contact trajectory and contact elliptical region, and then calculate the contact area of ​​the modified tooth surface. S3. Establish and solve the optimization model: With the maximum contact area of ​​the tooth surface as the optimization objective and the offset of the tooth profile and tooth direction as the design variables, construct the optimization function and use the particle swarm optimization algorithm to solve for the optimal shaping parameters.

2. The particle swarm optimization method based on the optimal loading contact area of ​​a planetary modified gear as described in claim 1, characterized in that, In step S1, the tooth profile modification is obtained by interpolating or fitting the normal offset of 7 tooth profile sampling points P1~P7 on the standard tooth profile line to obtain the tooth profile spline curve; the tooth direction modification is obtained by interpolating or fitting the normal offset of 7 tooth direction sampling points P1~P7 on the tooth direction line to obtain the tooth direction spline curve.

3. The particle swarm optimization method based on the optimal loading contact area of ​​planetary modified gears as described in claim 2, characterized in that, The tooth profile sampling points P1~P4 are in r b and r s The distribution is equidistant between P4 and P7. r s and r a They are equidistantly distributed; among them, r b The radius of the base circle, r s This is the distance from point P4 to the gear axis, which is also the radius of the tooth profile at point P4. r a The radius of the tooth tip circle is defined as follows: the tooth direction sampling points are equidistantly distributed along the tooth width direction, and the offset is defined along the gear axis direction.

4. The particle swarm optimization method based on the optimal loading contact area of ​​a planetary modified gear as described in claim 1, characterized in that, In step S1, the spline curve is a Bézier curve or a B-spline curve. The offset is interpolated using a cubic or quartic Bézier curve to construct the profile and tooth direction modification curve equations, respectively.

5. The particle swarm optimization method based on the optimal loading contact area of ​​a planetary modified gear as described in claim 1, characterized in that, In step S2, the tooth surface contact equation is established based on the condition that the position vectors and normal vectors of the two tooth surfaces are equal in a fixed coordinate system, and the contact point parameters are solved by discretizing the rotation angle of the large gear to obtain the contact trajectory.

6. The particle swarm optimization method based on the optimal loading contact area of ​​a planetary modified gear as described in claim 1, characterized in that, In step S2, the major and minor semi-axes of the contact ellipse are calculated based on Hertzian contact theory, as shown in the following formula: In the formula, a , b These are the major and minor axes of the contact area ellipse, respectively; e For eccentricity, K (e) E (e) represent the elliptic integral of the first kind and the elliptic integral of the second kind, respectively; A It is a positive constant; E * It is the equivalent elastic modulus; P This represents the maximum pressure in the contact area.

7. The particle swarm optimization method based on the optimal loading contact area of ​​a planetary modified gear as described in claim 1, characterized in that, In step S3, the optimization objective function is: In the formula, min f ( x Let ) be the objective function. s The contact area of ​​the planetary gear tooth surface is used for optimization. The optimization variables include the tooth profile offset and tooth direction offset corresponding to each sampling point.

8. The particle swarm optimization method based on the optimal loading contact area of ​​a planetary modified gear as described in claim 1, characterized in that, In step S3, the optimization variables must satisfy the constraint that the contact area does not exceed the tooth surface boundary. The constraint expression is: In the formula, a i , b i For the short and long semi-axis of the instantaneous contact imprint, a c , b c This represents the boundary of the tooth surface of the planetary gear.

9. The particle swarm optimization method based on the optimal loading contact area of ​​a planetary modified gear as described in claim 1, characterized in that, In step S3, the particle swarm optimization algorithm initializes the particle swarm, iteratively updates the particle velocity and position, tracks the individual optimal solution and the global optimal solution, and finally obtains the optimal combination of shaping parameters.

10. The particle swarm optimization method based on the optimal loading contact area of ​​a planetary modified gear as described in claim 9, characterized in that, In step S3, the update formulas for the particle velocity and position are as follows: In the formula, w Inertial weight; c 1. c 2 is a positive learning factor; r 1. r 2 is a random number uniformly distributed between 0 and 1.

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