A method and system for designing half-tooth and full-tooth profiles based on LM iteration and gears.

Abstract

This invention relates to the field of circular arc gear design methods, and particularly to a method and system for designing half-tooth and full-tooth profiles based on LM iteration, and related to gears. The half-tooth profile comprises three sequentially connected curves: the addendum arc, the involute, and the root arc. This invention provides the equations for these three curves, extracts points B and C as the desired endpoints of the involute, and uses the LM iteration algorithm to solve for the starting and ending angles of γ. Several values ​​are sequentially taken from each of the three curves and substituted into the corresponding addendum arc model, involute model, and root arc model to obtain the coordinates of several points. New curves are then established using these coordinates and concatenated as spline curves to form a smooth, high-precision half-tooth profile. This solves the technical problem of low accuracy in existing half-tooth profile design methods during the transition between the circular arc and the involute, resulting in unsmooth curves.

Description

Technical Field

[0001] This invention relates to the field of circular arc gear design methods, and in particular to a semi-tooth profile design method based on LM iteration, a full tooth profile design method using the semi-tooth profile design method, a full tooth profile design system using the full tooth profile design method, and an involute circular arc gear using the semi-tooth profile design method. Background Technology

[0002] Existing design methods for involute circular arc gears employ standard involute tooth profiles and rely on parametric formulas such as module and pressure angle to calculate the tooth profile. The circular arcs (tooth tip and tooth root portions) of involute circular arc gears are typically achieved through radial truncation or simple circular arc transitions.

[0003] Disadvantages of existing technologies: Existing involute circular arc gear design methods use standard involute tooth profiles, which form a semi-tooth profile by connecting three curves: the addendum arc, the standard involute curve, and the root arc. These semi-tooth profiles are then mirrored and spliced ​​to form the single tooth profile of the involute circular arc gear, and the tooth profile is formed based on this single tooth profile. However, the traditional method of forming a semi-tooth profile by connecting the addendum arc, the standard involute curve, and the root arc has inherent flaws. For gears with a small number of teeth (z < 10), there are inconsistencies in the root transition curve, leading to stress concentration and shortened lifespan. Furthermore, the use of standard involute tooth profiles results in abrupt curvature changes at the arcs (addendum and root), easily causing interference during meshing, significantly increasing noise and vibration. This makes it impossible to accurately control the addendum diameter using empirical formulas, requiring repeated trial and error adjustments. This leads to excessively long optimization algorithm processing times (>30 minutes / tooth profile) and poor machining data compatibility, ultimately resulting in an uneven and rough tooth profile. Summary of the Invention

[0004] To address the technical problem of low accuracy in existing semi-tooth profile design methods, which results in uneven and non-smooth curves during the transition between arcs and involute curves, this invention provides a semi-tooth profile design method based on LM iteration, a full-tooth profile design method using the semi-tooth profile design method, a full-tooth profile design system using the full-tooth profile design method, and an involute arc gear using the semi-tooth profile design method.

[0005] The first aspect of this invention provides a semi-tooth profile design method based on LM iteration, wherein the semi-tooth profile is mirrored and spliced ​​to form a single tooth profile of an involute circular arc gear. The semi-tooth profile includes three sequentially connected curves: a tooth tip arc, an involute curve, and a tooth root arc; the endpoint of the tooth tip arc... B The starting point of the tooth root arc C These are the two endpoints of the involute; the semi-tooth profile design method includes:

[0006] Provide three corresponding models of the tooth tip arc, the involute, and the tooth root arc in the same coordinate system;

[0007] According to the endpoint B Known angle parameters u B ,starting point C Known angle parameters u C The starting angles of the two endpoints are unknown. c B and termination angle c C The endpoint was obtained based on the three models. B Coordinates, starting point C coordinates, including c B and c C The coordinates of the two endpoints;

[0008] The problem of coinciding the coordinates of points B and C with the coordinates of the two endpoints is transformed into a nonlinear least squares problem. An objective function f(γ) is established to minimize the combined angle γ. The LM iterative algorithm is used to solve f(γ) by minimizing the sum of squared residuals, in order to find the optimal starting angle of γ. c B and termination angle c C :

[0009] ;

[0010] In the formula, R(γ) is the residual vector, representing c B The error between the coordinates of the endpoint and the coordinates of point B, or c C The error between the coordinates of the endpoint and the coordinates of point C; γ = i + , i Let be the rotation angle of a point in the coordinate system about the origin. The angle of counterclockwise rotation of the involute around the origin;

[0011] between 0 and u B Between, starting angle c B and termination angle c C Between, 0 and u CBetween these points, several values ​​are sequentially taken and substituted into the corresponding tooth tip arc model, involute model, and tooth root arc model to obtain the coordinates of several points. The difference between two adjacent values ​​is n times the step size used by the LM iterative algorithm.

[0012] In the coordinate system, all point coordinates are drawn sequentially and connected sequentially to form a semi-tooth-shaped profile.

[0013] Furthermore, the coordinate system is a two-dimensional coordinate system XY; the three models of the tooth tip arc, involute, and tooth root arc are as follows:

[0014] ;

[0015] ;

[0016] ;

[0017] In the formula, ( x AB , y AB ), ( x BC , y BC ), ( x CD , y CD The coordinates of any point on the addendum arc, involute arc, and root arc are given below. r Let be the radius of the tooth tip arc and the tooth root arc. R Let be the pitch circle radius of the gear. u α , u n These are the angle parameters for the tooth tip arc and tooth root arc, respectively, 0≤ u α ≤ u B , 0≤ u n ≤ u C ;R b is the base circle radius; z is the number of teeth.

[0018] Furthermore, based on the base circle radius R of the gear b The number of teeth z and the normal phase pressure angle α are designed respectively. r , u B 、u C , :

[0019] .

[0020] Furthermore, n is 1.

[0021] Furthermore, the number of point coordinates in any one of the tooth tip arc model, involute model, and tooth root arc model is not less than 100.

[0022] Furthermore, the number of point coordinates in any one of the tooth tip arc model, involute model, and tooth root arc model is 200-300.

[0023] The second aspect of the present invention also provides a design method for a full tooth profile, which uses the above-mentioned half tooth profile design method based on LM iteration to generate a half tooth profile, mirrors the half tooth profile and splices them together to form a single tooth profile, and generates a full tooth profile by rotating and arraying several single tooth profiles according to the center of the gear corresponding to the half tooth profile.

[0024] A third aspect of the present invention also provides a full tooth profile design system, including a storage module, a processing module, and a computer program stored on the storage module, wherein the processing module executes the computer program to implement the steps of the above-described full tooth profile design method.

[0025] Furthermore, the full tooth profile design system also includes a visualization module; the visualization module is used to model the full tooth profile to obtain a three-dimensional model of the gear corresponding to the full tooth profile.

[0026] The fourth aspect of the present invention also provides an involute circular arc gear, which is manufactured by processing a half-tooth profile of a gear, wherein the half-tooth profile of the gear is the aforementioned half-tooth profile.

[0027] Compared with the prior art, the present invention has the following beneficial effects:

[0028] 1. This invention provides three curve equations: the addendum arc, the root arc, and the involute. Points B and C are extracted as the desired endpoints of the involute. The LM iterative algorithm is used to solve for the starting and ending angles of γ, ensuring that the two endpoints of the involute are equivalent to points B and C on the addendum and root arcs within a certain accuracy range. This yields an involute equation that meets the connection accuracy requirements. To further improve the smoothness of the semi-tooth profile, several values ​​from each of the three curve segments are sequentially substituted into the corresponding addendum arc model, involute model, and root arc model to obtain coordinates of several points. These coordinates are used to establish new curves, which are then concatenated as spline curves to form a smooth, high-precision semi-tooth profile. This overcomes the problem of curve inconsistency at the B and C connection points in traditional methods, thus solving the technical problem of low accuracy leading to unsmooth curves in existing semi-tooth profile design methods during the transition between arcs and involute curves.

[0029] 2. Based on the LM iterative algorithm, this invention improves the connection accuracy at the connection points B and C of the involute BC, increasing the accuracy by two orders of magnitude compared to traditional approximation methods. It solves the problem of stress concentration at the tooth root of gears with few teeth, eliminates meshing interference and transmission instability, and breaks the barrier of experience dependence on traditional design parameters. Attached Figure Description

[0030] Figure 1 This is a flowchart of a semi-tooth profile design method based on LM iteration.

[0031] Figure 2 This is a schematic diagram of a circular arc tooth-shaped coordinate system.

[0032] Figure 3 This is a schematic diagram of a semi-tooth-shaped profile.

[0033] Figure 4 This is a schematic diagram of the full tooth profile. Detailed Implementation

[0034] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0035] An embodiment of the present invention provides an involute circular arc gear. The involute circular arc gear is designed using a full tooth profile design method. After the full tooth profile is visualized (3D modeling), the involute circular arc gear is produced based on this visualized graphic (it can also be considered as indirectly designing an involute circular arc gear using a half tooth profile). The full tooth profile design method includes: mirroring and splicing the half tooth profiles to form a single tooth profile of the involute circular arc gear; and rotating and arraying several single tooth profiles according to the center of the involute circular arc gear to generate the full tooth profile of the involute circular arc gear (the involute circular arc gear is the gear corresponding to the half tooth profile). Furthermore, the present invention also provides a full tooth profile design system, including a storage module, a processing module, and a computer program stored on the storage module. It may also include a visualization module. The processing module executes the computer program to implement the steps of the above-described full tooth profile design method. The visualization module is used to model the full tooth profile to obtain a 3D model of the gear corresponding to the full tooth profile.

[0036] The design method for the semi-tooth profile is as follows: Figure 1 As shown, Figure 1 The semi-tooth profile design method based on LM iteration is as follows:

[0037] 1. The semi-tooth profile is designed based on three curves in a two-dimensional coordinate system: the addendum arc, the involute curve, and the root arc. The semi-tooth profile includes three sequentially connected curves: the addendum arc, the involute curve, and the root arc; the endpoint of the addendum arc... B The starting point of the tooth root arc C These are the two endpoints of the involute. Therefore, based on the design requirements, we first provide three corresponding curve models: the addendum circle, the involute, and the root circle.

[0038] 1.1 Tooth tip radius:

[0039] ;

[0040] 1.2 Involute:

[0041] ;

[0042] 1.3 Tooth root arc:

[0043] ;

[0044] Of the three curve equations above, ( x AB , y AB ), ( x BC , y BC ), ( x CD , y CD The coordinates of any point on the addendum arc, involute arc, and root arc are given below. r Let be the radius of the tooth tip arc and the tooth root arc. R Let be the pitch circle radius of the gear. u α , u n These are the angle parameters for the tooth tip arc and tooth root arc, respectively, 0≤ u α ≤ u B , 0≤ u n ≤ u C ;R b is the base circle radius; z is the number of teeth.

[0045] 1.4 Key Parameter Definition Formula:

[0046] To improve the accuracy of the connection points of the three curve segments—the addendum arc, the involute arc, and the root arc—based on the base circle radius R of the gear... b The number of teeth z and the normal phase pressure angle α are designed respectively.r , u B 、u C , :

[0047] .

[0048] As shown in the parameter definition formula, when the parameter r、u B 、u C 、 When the functional relationship shown in the above formula is true, it indicates that the curve is smooth and continuous (base circle radius R). b The number of teeth z and the normal pressure angle α are input parameters and are known.

[0049] 2. Based on the destination B Known angle parameters u B (After defining the key parameters, u B It is considered to be known; but in the absence of a definition, u B This indicates the angle parameter of the tooth tip arc at the endpoint. B The value at that location, u C Similarly), starting point C Known angle parameters u C The starting angles of the two endpoints are unknown. c B and termination angle c C The endpoint is obtained based on the above three models. B Coordinates, starting point C coordinates, including c B and c C The coordinates of the two endpoints:

[0050] In this step, based on the known angle parameters u B、 Known angle parameters u C This allows us to solve for the parameters of the specific equations for the tooth tip arc and tooth root arc, thereby obtaining the endpoint. B Coordinates, starting point C The coordinates; and obtain the coordinates containing c B and c C The coordinates of the two endpoints. Note this. cB and c C It's still unknown, so we can't smoothly connect the addendum arc, root arc, and involute line yet. However, since the curves of the addendum arc and root arc are known, we just need to gradually make points B and C on the involute line approach the known angle parameters of the addendum arc and root arc. u B、 Known angle parameters u C The two corresponding endpoints will yield an approximately smooth curve (i.e., a target involute curve that allows the semi-tooth profile to achieve the required precision). Note that the endpoint... B ,starting point C It needs to be distinguished from points B and C on the involute.

[0051] 3. The problem of coinciding the coordinates of points B and C with the coordinates of the two endpoints is transformed into a nonlinear least squares problem. An objective function f(γ) is established to minimize the combined angle γ. The LM iterative algorithm is used to solve f(γ) by minimizing the sum of squared residuals to find the optimal starting angle of γ. c B and termination angle c C :

[0052] ;

[0053] In the formula, R(γ) is the residual vector, representing c B The error between the coordinates of the endpoint and the coordinates of point B, or c C The error between the coordinates of the endpoint and the coordinates of point C; γ = i + , i Let be the rotation angle of a point in the coordinate system about the origin. The angle of counterclockwise rotation of the involute around the origin is given.

[0054] In this step, the starting angle is solved using the LM iterative algorithm. c B and termination angle c CThen, the specific parameters of the involute model are obtained, allowing the involute curve to be drawn. In the same two-dimensional Cartesian coordinate system, after drawing the addendum arc, involute, and root arc in sequence, the addendum arc and root arc that define the semi-tooth profile can be obtained, as well as the involute that infinitely approximates the endpoint B of the addendum arc and the starting point C of the root arc. Thus, the ideal involute has been obtained, and based on this involute, the ideal smooth semi-tooth profile can be obtained. However, after obtaining the involute, it is not possible to directly connect the two ends of the involute to the end point B of the tooth tip arc and the starting point C of the tooth root arc, respectively. Since the tooth tip arc and tooth root arc are part of a circle, and the involute of this invention is part of an infinitely extending involute, the curvature is inconsistent. If they are simply connected, it will cause abrupt curvature changes at the connection points of the tooth tip arc, tooth root arc and involute, making the connection points very uncoordinated. Such a tooth profile will also have unreasonable stress during use after manufacturing. Therefore, it is necessary to further optimize the connection method between the three curves to truly obtain the required smooth and high-precision semi-tooth profile curve.

[0055] Furthermore, regarding the solution of the optimal starting angle c B and termination angle c C Explanation of the necessity: Solving for the optimal starting angle c B and termination angle c C This is essential. For the three curve segments—the addendum arc, the involute, and the root arc—if the starting and ending angles of the involute are not determined before connecting the three segments, the connection points between the two endpoints of the involute and the addendum and root arcs cannot be determined, thus failing to improve the accuracy of the connection. However, once points B and C are determined, the optimal connection points are found, facilitating subsequent LM iterative approximation of the involute and thus contributing to the establishment of a smooth curve.

[0056] 4. Between 0 and u B Between, starting angle c B and termination angle c C Between, 0 and u C Between these points, several values ​​are sequentially taken and substituted into the corresponding tooth tip arc model, involute model, and tooth root arc model to obtain the coordinates of several points. The difference between two adjacent values ​​is n times the step size used by the LM iterative algorithm. In the coordinate system, all point coordinates are sequentially drawn and connected to form a semi-tooth profile.

[0057] In this step, the tooth tip arc, involute, and tooth root arc are connected to obtain the optimal starting angle. c B and termination angle c C Then, the coordinates of the involute curve and several points on the curve are obtained. At this point, by sequentially substituting several values ​​into the corresponding tooth tip arc model, involute model, and tooth root arc model, several point coordinates are obtained (the difference in values ​​is n times the step size used in the LM iterative algorithm, where n=1; during the value selection process, there may be cases where the values ​​cannot be integer multiples, in which case the remaining margin insufficient for the step size can be ignored. Based on industry experience, the number of point coordinates for any one of the tooth tip arc model, involute model, and tooth root arc model should not be less than 100, with a better range being 200-300), thus obtaining a series of point coordinates. These point coordinates are drawn sequentially and connected sequentially to form a spline curve. Since the spline curve in this step is obtained by connecting the point coordinates extracted from the tooth tip arc, involute, and tooth root arc, it must be a smooth connection based on the spline curve. There is no situation where B and C are directly connected to the end point B and the starting point C at the breakpoint, so there will be no abrupt change in curvature at the breakpoint. Therefore, the final spline curve obtained is the smooth semi-tooth profile that is desired.

[0058] like Figure 2 As shown, Figure 2 This is a schematic diagram of a circular arc tooth-shaped coordinate system.

[0059] Based on the above parameters, establish three independent curve equations: tooth tip arc AB, involute BC, and tooth root arc CD.

[0060] Before proceeding, let's first introduce the three-section tooth profile of the circular arc gear:

[0061] ① Establish a coordinate system with the pitch circle center as the origin for the tooth tip arc AB. The tooth tip arc AB is an arc with radius r, and its center is located on the y-axis at a distance R from the origin. Use angular parameters. u α Describe an approximate circular arc.

[0062] ② The involute BC: The standard involute unfolds from the x-axis, but in this design, it needs to be rotated. The angle is used to connect the two arcs. Through coordinate transformation, the equation of the rotated involute is obtained, with the parameter γ.

[0063] ③ The root arc CD is located at the bottom of the tooth groove, with its center on the center line of the tooth groove and a radius of r. It is represented by the angular parameter. u n describe.

[0064] The conditions for a smooth connection are: the curves are continuous in position and tangent at points B and C, meaning that the coordinates of two adjacent curve segments at the connection point are the same, and the tangent directions of two adjacent curve segments at the connection point are the same.

[0065] This design uses coordinate transformation in MATLAB to obtain the mathematical model of the circular arc gear tooth profile and the constraints for smooth curve connection. Based on the gear meshing principle, the feasibility of circular arc gear meshing and the meshing process are analyzed.

[0066] In the same Cartesian coordinate system, the point (x, y) is rotated around the origin by an angle θ (assuming counterclockwise rotation is positive) to obtain the point (x1, y1). According to the geometric relationship in the figure, we can obtain:

[0067] ;

[0068] Converted to matrix form, it becomes:

[0069] ;

[0070] The above formula is a template for a general transformation.

[0071] Using a "circular arc-involute-circular arc" tooth profile, as shown in Figure 2, a coordinate system is established, and the parametric equations of the three curve segments are established as follows:

[0072] The methods for establishing the equation for the tooth tip arc AB include:

[0073] with u α Create an expression for the parameter variable:

[0074] ;

[0075] In the formula, the coordinates of any point on the tooth tip arc AB are ( x AB , y AB ), R The pitch circle radius (in mm) of the gear with the semi-tooth profile. u α The angle parameter variable is the tooth tip arc AB, and its value range is 0 ≤ u α ≤ u B ; u B The angle parameter of the tooth tip arc AB u α The angle value at point B is taken.

[0076] The methods for establishing the equation of the involute BC include:

[0077] For a typical involute, we establish an equation with the involute angle θ as a parameter. When its starting point is on the x-axis, its rectangular coordinate expression is:

[0078] ;

[0079] For the tooth profile in this paper, the involute segment in the tooth profile is as follows: Figure 2 As shown, this is achieved by rotating the involute represented by the above equation counterclockwise around the origin. The mathematical model derived from the angle is obtained through coordinate transformation, as shown in the following process:

[0080] The equation of the involute starting on the axis is:

[0081] ;

[0082] After coordinate transformation, it becomes:

[0083] ;

[0084] After simplification, we get:

[0085] ;

[0086] Let γ = i + and with γ- Substituting the parameter into the above equation, we obtain the expression for the involute BC as follows:

[0087] ;

[0088] The methods for establishing the equation of the tooth root circular arc CD include:

[0089] with u n Create an expression for the parameter variable:

[0090] ;

[0091] Wherein, the coordinates of any point on the root arc are (x CD y CD ), u C The angle parameter of the tooth root arc CD u n The angle at point C is taken as 0 ≤ u n ≤ u C .

[0092] Draw the semi-tooth profile based on the constraint model (constraint conditions) of the spline curve with three independent curve equations and preliminary constraints:

[0093] The two segments of the circular arc curve are smoothly connected to the involute segment. Points A and B are the connection points between the two segments and the involute segment, respectively. Therefore, points A and B must satisfy the following conditions: their coordinate values ​​on the two curves are equal, and the derivatives of the two curves at the connection point are the same. Thus, the continuity condition for the circular arc curve and the involute segment can be obtained as follows:

[0094] At point B:

[0095] ;

[0096] At point C:

[0097] ;

[0098] Solving equations I and II simultaneously, we get:

[0099] ;

[0100] For an involute, there is a relationship R between its base circle radius and pitch circle radius. b =Rcosα, where α is the pressure angle. Based on geometric relationships, the above equation can be transformed into the following equation:

[0101] ;

[0102] This formula represents a constraint model, where r is the radius of the tooth tip arc AB; R b γ is the base circle radius; u is the angular parameter of the involute BC; γ = θ + θ is the angle of the involute's starting point on the X-axis. The angle by which the involute rotates counterclockwise around the origin; u B The angle parameter of the tooth tip arc AB u α The angle at point B is taken as 0 ≤ u α ≤ u B ; u C The angle parameter of the tooth root arc CD u n The angle at point C is taken as 0 ≤ u n ≤ u C .

[0103] The above describes the preliminary mathematical solution process for establishing the constraint model (circular arc-involute-circular arc tooth profile). The actual solution process involves compiling the formulas in MATLAB and solving the equations based on the Levenberg–Marquardt iterative algorithm, which facilitates a smooth and even connection between the circular arc and the involute.

[0104] 5. After establishing the three curve segments, the parameters of the involute BC are still undetermined. The LM algorithm is then called to ensure that the parameters of the involute curve (x...) are determined. BC (γ),y BC The coordinates of (γ) and the target point (end point B, start point C) almost completely coincide (with an error within 0.0002 mm), thus determining the starting and ending angles of parameter γ. The LM algorithm is an existing LM iterative algorithm.

[0105] The derivation and application of the LM algorithm are as follows:

[0106] Its essence is to find the parameter γ such that the (x) on the involute curve BC (γ),y BC The coordinates of (γ) completely coincide with those of the target point (end point B, start point C) (with an error within 0.0002 mm). This can be transformed into a nonlinear least squares problem: finding γ to minimize the objective function f(γ) = ||Error(γ)|| 2 .

[0107] The following is the derivation of the algorithm:

[0108] 5.1 Problem Definition and Objective Function.

[0109] We need to solve a system of linear equations. For example, for point B, we hope to:

[0110] ;

[0111] Where R(γ) is the residual vector, which represents the error between the current involute point and the target point.

[0112] The LM algorithm solves this problem by minimizing the sum of squared residuals, i.e., minimizing the objective function f(γ):

[0113] ;

[0114] Where m = 2 (containing both x and y components). Adding 1 / 2 is to simplify the form when differentiating.

[0115] 5.2 Core idea of ​​the algorithm: Taylor expansion and linear approximation.

[0116] The LM algorithm is an iterative algorithm that, in the k-th iteration, aims to find a step size d such that γ(k+1) =γ (k) +d can make the value of the objective function f(γ) smaller.

[0117] To find the optimal step size d, the residual vector R(γ) is... (k) +d) in γ (k) Perform a first-order Taylor expansion at this point:

[0118] ;

[0119] Wherein, J(γ) (k) ) is R(γ) in γ (k) The Jacobian matrix at γ. In this example, since γ is a scalar, the Jacobian matrix is ​​a 2×1 vector:

[0120] ;

[0121] 5.3 From Gauss-Newton method to LM algorithm.

[0122] To minimize the objective function that approximates the above approximation, let its derivative with respect to d be zero:

[0123] ;

[0124] After simplification, the core equations of the Gauss-Newton method are obtained:

[0125] ;

[0126] Gauss-Newton method in J T When the J matrix is ​​benign (non-singular, non-ill-conditioned), it converges quickly. If the matrix is ​​ill-conditioned, the solution will be unstable.

[0127] Levenberg-Marquardt introduced a damping factor μ≥0 to solve this problem, thus obtaining the core equation of the LM method:

[0128] ;

[0129] Where I is the identity matrix.

[0130] 5.4 The role and iterative process of the damping factor μ.

[0131] The damping factor μ is used to dynamically adjust the behavior of the algorithm.

[0132] When μ is very small (approaching 0), the algorithm is close to the Gauss-Newton method, which converges quickly by utilizing second-order information.

[0133] When μ is large, the matrix Dominant, the equation is approximated as μd≈-J T R, or the step size equivalent to the steepest descent method (gradient descent), ensures that the algorithm can stably move in the correct direction even when it is far from the minimum.

[0134] Finally, the algorithm iteratively adjusts the value of μ and solves for the step size d until the residual ||R(γ) is reached. (k) If the tolerance is less than the preset tolerance, find the optimal parameter γ that can accurately connect the involute with the front and rear arcs.

[0135] 6. For example Figure 3 , Figure 4 As shown, after solving for the starting and ending angles (domains) of γ using the LM method, the domains of the proposed points B and C, and the other two circular arcs AB and CD, can be obtained, thus generating three independent and complete curves. Then, by substituting the domains of the three independent curves into the constraint model, at 0 and... u B Between, starting angle c B and termination angle c C Between, 0 and u C Between these points, several values ​​are sequentially taken and substituted into the corresponding tooth tip arc model, involute model, and tooth root arc model to obtain the coordinates of several points. The difference between two adjacent values ​​is n times the step size used by the LM iterative algorithm (n can be 1). In the coordinate system, all point coordinates are sequentially drawn and connected to form a semi-tooth profile, thereby generating a smooth and complete semi-tooth profile ABCD.

[0136] 7. Based on the already generated semi-tooth profile ABCD, mirror the semi-tooth profile ABCD and splice them together to form a single tooth profile; generate a full tooth profile by rotating and arraying several single tooth profiles according to the center of the gear corresponding to the semi-tooth profile ABCD.

[0137] 8. Perform deduplication and data export steps: generate CSV and TXT files.

[0138] 9. Perform calculations and verification: Calculate the addendum circle diameter, dedendum circle diameter, and whether the error meets the preset values. If they do, generate a visual graphic; otherwise, redesign.

[0139] This design utilizes MATLAB's fsolve nonlinear equation solver, specifically invoking the LM iterative algorithm, to achieve a smooth connection between the three curve segments: the addendum arc AB, the involute BC, and the root arc CD. Although fsolve offers various algorithm options, including the default Trust-region-dogleg and Trust-region, the LM iterative algorithm demonstrates a more significant advantage over other algorithms in the specific application scenario of this design.

[0140] The LM iterative algorithm was chosen as the primary choice for this design mainly due to its excellent robustness and computational efficiency. First, the algorithm is less sensitive to initial points; its unique damping mechanism makes it behave like gradient descent when far from the optimal solution, thus providing excellent stability while ensuring global convergence. It is the "out-of-the-box" choice for dealing with poor initial points. Second, the LM iterative algorithm directly improves the condition number of the matrix through an explicit damping mechanism, making it most capable of handling ill-conditioned systems. Furthermore, for the small-to-medium scale problems involved in this design, the LM iterative algorithm has a near-quadratic local convergence speed (degenerating to the Gauss-Newton method when the residual is small), and often requires fewer iterations to reach convergence, resulting in leading overall computational efficiency.

[0141] From the perspective of principle mechanism and final accuracy, the Levenberg-Marquardt algorithm transforms the nonlinear least squares problem into a linear problem by using a "trust region" strategy through linear approximation of the model function, resulting in faster iteration speed and more accurate calculation. Based on these advantages, this design employs the LM iterative algorithm, successfully obtaining a smoother and more even connection curve. The final designed gear achieves micron-level positioning accuracy, with the error at the critical connection point (B / C points) strictly controlled within 0.0002mm, meeting the design requirements of a high-precision circular arc gear.

[0142] This invention improves the connection accuracy at connection points B and C of the involute BC, solving the problem of stress concentration at the tooth root of gears with few teeth, and eliminating meshing interference and transmission instability. It breaks through the experience-dependent barrier of traditional design parameters, proposing a closed-loop design of parameters through addendum circle constraints. It achieves efficient and high-precision manufacturing, solving the problems of excessively long optimization algorithm processing time (>30 minutes / tooth profile) and poor machining data compatibility. An automatic de-emphasis cloud generation algorithm (tolerance 1e-5mm) is developed, outputting standard 3D machining files within 5 seconds.

[0143] In summary, this invention employs the Levenberg-Marquardt optimization algorithm to solve for the involute-circular arc fusion point, and achieves micron-level positioning accuracy (B / C point error <0.0002mm) through the LM iterative algorithm. This invention uses the LM iterative algorithm to solve for the spatial matching relationship between the involute parametric equations and the arc endpoints, accurately solving for the involute-circular arc connection point, improving accuracy by two orders of magnitude compared to traditional approximation methods, and achieving high-precision tooth profile generation. This invention only requires a base circle radius R. b The three parameters, number of teeth z and normal pressure angle α, can generate a semi-tooth profile. Based on the constraint model, this invention optimizes the tooth number and root undercut topology, breaks through the traditional tooth number limit of involute gears, and can achieve high-precision tooth profile design for small-tooth gears.

[0144] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0145] The embodiments described above are merely illustrative of several implementations of the present invention, and while the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the invention patent. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these all fall within the protection scope of the present invention. Therefore, the protection scope of this invention patent should be determined by the appended claims.

Claims

1. A method for designing a half-tooth profile based on LM iteration, the half-tooth profile comprising three curve segments of a tip circle arc, an involute, and a root circle arc connected in sequence; the end point of the tip circle arc B , the start point of the root circle arc C are two end points of the involute, respectively. characterized in that The half-tooth profile design method comprises: The coincidence of the end point B coordinate, the start point C coordinate and the two end point coordinates of the involute is converted into a nonlinear least square problem, a minimization objective function f(γ) about the combined angle γ of the involute is set up, the LM iterative algorithm is adopted to solve f(γ) to find the optimal start angle γ Gamma B and the termination angle Gamma C : ； where R(γ) is the residual vector, representing Gamma B the error between the end point coordinates of the line and the end point B coordinates, or Gamma C the error between the end point coordinates of the line and the start point C coordinates; γ = Theta + , Theta the angle of rotation of the point on the coordinate system around the origin, the angle of rotation of the involute around the origin counterclockwise. The three models of the addendum circle arc, the involute, and the dedendum circle arc are provided under the same coordinate system; in [0, u B ], [ Gamma B , Gamma C ], [0, u C ], a plurality of values are sequentially taken respectively and substituted into the addendum circle arc model, the involute model, and the dedendum circle arc model to obtain a plurality of point coordinates respectively, and a difference between two adjacent values is a step length adopted by the LM iteration algorithm; u B is a value of an angle parameter of the addendum circle arc at the end point B , u C is a value of an angle parameter of the dedendum circle arc at the start point C . All point coordinates are sequentially drawn and connected to form a half-tooth profile.

2. The LM iteration based half tooth profile design method of claim 1, wherein, The half-tooth profile design method further comprises: According to the end point B angle parameter u B , the start point C angle parameter u C , the starting angle Gamma B and the ending angle Gamma C , the end point B coordinates, the start point C coordinates, the two end point coordinates Gamma B and Gamma C are obtained according to the three models.

3. The LM iteration based half tooth profile design method of claim 2, wherein, The coordinate system is a two-dimensional coordinate system X-Y; the three models of the addendum arc, the involute, and the dedendum arc are respectively: ； ； ； In the formula, ( x AB , y AB ), ( x BC , y BC ), ( x CD , y CD The coordinates of any point on the addendum arc, involute arc, and root arc are given below. r Let be the radius of the tooth tip arc and the tooth root arc. R Let be the pitch circle radius of the gear. u α , u n These are the angle parameters for the tooth tip arc and tooth root arc, respectively, 0≤ u α ≤ u B , 0≤ u n ≤ u C ;R b is the base circle radius; z is the number of teeth.

4. The LM iteration based half-tooth profile design method of claim 3, wherein, According to the base circle radius R of the gear b , the number of teeth z, and the normal pressure angle a are designed respectively r , u B 、u C , : 。 5. The LM iteration based half tooth profile design method of claim 1, wherein, The number of point coordinates of any one of the addendum arc model, the involute model, and the dedendum arc model is not less than 100.

6. The LM iteration based half tooth profile design method of claim 1, wherein, The number of point coordinates of any one of the addendum arc model, the involute model, and the dedendum arc model is 200-300.

7. A full-tooth profile design method characterized by, The half-tooth profile is generated by using the LM iteration-based half-tooth profile design method according to any one of claims 1 to 6, the half-tooth profile is mirrored and spliced to form a single-tooth profile, and a full-tooth profile is generated according to a plurality of single-tooth profiles of a gear corresponding to the half-tooth profile.

8. A full-tooth profile design system comprising a storage module, a processing module and a computer program stored on the storage module, characterized in that, The processing module executes the computer program to realize the steps of the full-tooth profile design method according to claim 7.

9. The full-tooth profile design system of claim 8, wherein, The full-tooth profile design system further comprises a visualization module; the visualization module is used to model the full-tooth profile to obtain a three-dimensional model of a gear corresponding to the full-tooth profile.

10. A involute circular arc gear, which is produced by using a half-tooth profile of the gear, characterized in that, The half-tooth profile of the gear is a half-tooth profile designed according to the LM iteration-based half-tooth profile design method according to claim 1.

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