A method for optimizing the tooth surface accuracy of drum-shaped helical gears based on multi-axis additional motion

Abstract

This invention discloses a method for optimizing the tooth surface accuracy of helical gears based on multi-axis additional motion. The method includes: establishing a spatial coordinate system between the grinding wheel and the gear based on a five-axis linkage CNC forming grinding machine structure; calculating the influence of the three-axis additional motion on the left and right tooth surface distortion and the influence on the tooth profile slope; and using a differential evolution algorithm to jointly optimize the three-axis additional motion quantities, so that the actual modified tooth surface contact line matches the theoretical modified tooth surface contact line. This invention reduces tooth surface errors by optimizing multi-axis additional motion. During forming grinding, helical gears experience tooth surface distortion due to errors. By using a double-sided grinding process and optimizing the influence of the three-axis additional motion on tooth surface distortion, the tooth surface accuracy is improved. Compared with single-sided grinding, double-sided grinding has the advantages of high efficiency, high precision, and strong applicability.

Description

Technical Field

[0001] This invention relates to the fields of gear tooth profile precision machining and error analysis between theoretical and actual tooth profiles, and particularly to a method for optimizing the tooth surface precision of drum-shaped helical gears based on multi-axis additional motion. Background Technology

[0002] Drum gears are characterized by high dimensional accuracy, complex structure, and high degree of integration. Compared with spur gears, they have advantages such as strong load-bearing capacity, large angular displacement compensation, good tooth surface meshing, and low noise. They are widely used in civilian products such as couplings. However, during profile grinding, the kinematic geometry of the grinding wheel causes the grinding trajectory to deviate from the standard helix. This affects the alignment of the contact line with the gear during tooth profile modification. Therefore, the modified tooth surface cannot fully meet the design requirements during the modification process, resulting in tooth surface errors. As the amount of tooth profile modification increases, the tooth surface error will further increase.

[0003] The fundamental error of the tooth surface is an inherent problem in the grinding process of helical gears with tooth profile modification. Unlike other errors, it is caused by the non-uniform cutting of the tooth surface during the tooth profile modification process. How to effectively reduce the tooth surface distortion error and thus improve the grinding accuracy of the modified tooth surface is the problem that this invention aims to solve.

[0004] CN116079157A discloses a method for tooth surface modification of asymmetric gears using a worm wheel grinding based on multi-axis additional motion. This method derives the surface equation of the asymmetric worm wheel grinding based on asymmetric tooth profile and meshing principle, establishes a model for asymmetric gear machining using a worm wheel generating grinding, selects grinding point clouds of a specific pair of left and right tooth surfaces of the asymmetric gear, meshes the asymmetric gear tooth surface and selects mesh points, calculates the normal deviation between the actual grinding surface mesh points and the theoretical asymmetric tooth surface mesh points using vector dot product, defines the CNC axis as a fourth-order polynomial, uses the polynomial coefficients as optimization parameters, and sets the optimization objective as minimizing the difference between the actual tooth surface deviation and the target tooth surface deviation. An optimization model is established and solved using a sensitivity matrix and the least squares method, thereby shortening the manufacturing cycle and reducing processing costs. However, this method does not eliminate gear tooth surface distortion error, and the wear accuracy of the modified tooth surface still has an impact. Summary of the Invention

[0005] The purpose of this section is to outline some aspects of embodiments of the present invention and to briefly describe some preferred embodiments. Simplifications or omissions may be made in this section, as well as in the abstract and title of this application, to avoid obscuring the purpose of these documents; however, such simplifications or omissions should not be construed as limiting the scope of the invention.

[0006] In view of the aforementioned existing problems, the present invention is proposed.

[0007] Therefore, the technical problem solved by this invention is: how to effectively eliminate gear tooth surface distortion error and improve the grinding accuracy of modified tooth surface.

[0008] To solve the above-mentioned technical problems, the present invention provides the following technical solution: based on the structure of a five-axis linkage CNC forming gear grinding machine, a spatial coordinate system is established between the grinding wheel and the gear;

[0009] Calculate the effects of the additional triaxial motion on the torsion of the left and right tooth surfaces and the influence of the tooth profile slope, respectively.

[0010] The differential evolution algorithm is used to jointly optimize the additional motion of the three axes, so that the actual modified tooth surface contact line matches the theoretical modified tooth surface contact line.

[0011] As a preferred embodiment of the method for optimizing the tooth surface accuracy of drum-shaped helical gears based on multi-axis additional motion described in this invention, the spatial coordinate system includes a tool coordinate system, a workpiece coordinate system, and a fixed coordinate system;

[0012] The standard helical tooth surface equation is obtained based on the spatial coordinate system. By calculating the profile and tooth direction modification of the drum-shaped tooth, coordinate transformation is performed to obtain the tooth surface modification equation of the gear.

[0013] The tooth surface modification equation is combined with the tooth surface contact condition equation to calculate the theoretical contact line. The theoretical contact line is then transformed to the grinding wheel coordinate system through the coordinate transformation equation and projected onto the grinding wheel shaft section.

[0014] The profile is fitted using the least squares method to obtain the profile of the grinding wheel shaft section;

[0015] By combining the known equations of the grinding wheel's rotating surface with the contact condition equations of the grinding wheel's rotating surface, the actual contact line is calculated, and then transformed to the gear coordinate system.

[0016] The modification amount is allocated to the additional motion of the three axes and substituted into the gear inertial coordinate system to obtain the modified tooth surface equation.

[0017] As a preferred embodiment of the method for optimizing the tooth surface accuracy of drum-shaped helical gears based on multi-axis additional motion described in this invention, a standard helical tooth surface equation is established according to the spatial coordinate system, and its general mathematical expression is as follows:

[0018]

[0019] Where, r b Let θ be the base circle radius, σ0 be the tooth groove half angle, μ be the involute parameter, p be the helical parameter, and θ be the helical motion parameter.

[0020] As a preferred embodiment of the method for optimizing the tooth surface accuracy of a drum-shaped helical gear based on multi-axis additional motion described in this invention, the modification amounts of the tooth profile and tooth direction of the drum-shaped gear are as follows:

[0021]

[0022]

[0023] Where, α ce To measure the pressure angle of a circle, r ce To measure the radius of a circle, since the displacement coefficient used in the experiment is 0, the measuring circle is directly calculated based on the pitch circle, ε. 左 ε 右 This refers to the tooth profile modification amount on both the left and right sides.

[0024] As a preferred embodiment of the method for optimizing the tooth surface accuracy of drum-shaped helical gears based on multi-axis additional motion described in this invention, the general mathematical expression for the theoretical contact line is as follows:

[0025]

[0026] Where a is the center distance between the gear and the grinding wheel, and ∑ is the angle between the z-axis of the forming grinding wheel and the z-axis of the gear.

[0027] As a preferred embodiment of the method for optimizing the tooth surface accuracy of drum-shaped helical gears based on multi-axis additional motion described in this invention, the general mathematical expression for the profile of the grinding wheel shaft section is as follows:

[0028]

[0029] Where R is the profile equation of the grinding wheel shaft section, f(R) is the profile function, and f'(R) is the derivative of f(R) with respect to R.

[0030] As a preferred embodiment of the method for optimizing the tooth surface accuracy of drum-shaped helical gears based on multi-axis additional motion described in this invention, the general mathematical expression of the tooth profile modification equation is as follows:

[0031]

[0032] Where △x and △y are the additional motion quantities along the x-axis and y-axis, respectively, and △c is the additional rotation quantity along the c-axis.

[0033] As a preferred embodiment of the method for optimizing the tooth surface accuracy of drum-shaped helical gears based on multi-axis additional motion described in this invention, the joint optimization includes:

[0034] Use the additional motion along the x-axis, y-axis, and c-axis as the initial population;

[0035] Starting with the candidate solutions randomly generated from the initial population, denoted as:

[0036] x i,G =(x 1,G ,x 2,G ,...,x D,G i = 1, 2, ..., NP;

[0037] A mutation is generated by the difference between two random individuals, a scaling factor, and a target vector;

[0038] Based on the current individuals in the initial population, several vectors are randomly selected for difference operations to generate a mutation difference vector;

[0039] A new candidate solution, i.e., a test vector, is generated based on the combination of the mutation vector and the parent vector.

[0040] According to the greed principle, from the test individual u iG and the original individual x iG The individual with the best function value is selected to enter the next generation. After obtaining the optimal optimization amount, the additional motion of the three axes is calculated in reverse until the optimal solution is obtained.

[0041] The beneficial effects of this invention are as follows: This invention aims to improve the tooth surface accuracy of drum-shaped helical gears by optimizing the additional motion of the three axes. Based on the forming grinding mechanism, it comprehensively considers the influence of additional motion on the left and right tooth surface twisting and the relative tooth surface twisting. By optimizing the additional motion of the multi-axis, the tooth surface error is reduced. Drum-shaped gears will have tooth surface twisting due to errors during forming grinding. By using double-sided grinding process and optimizing the influence of the additional motion of the three axes on the tooth surface twisting, the tooth surface accuracy is improved. Compared with single-sided grinding, double-sided grinding has the advantages of high efficiency, high precision and strong applicability. Attached Figure Description

[0042] To more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings used in the description of the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort. Wherein:

[0043] Figure 1 This is a schematic diagram of the tooth profile modification based on the normal mapping of the method for optimizing the tooth surface accuracy of a drum-shaped helical gear according to the multi-axis additional motion shown in this invention.

[0044] Figure 2(a) is a schematic diagram of the influence of the y-axis additional motion on the deviation of the slope of the tooth profile on the left and right sides of the drum-shaped helical gear tooth surface accuracy optimization method based on multi-axis additional motion shown in this invention.

[0045] Figure 2(b) is a schematic diagram of the influence of the x-axis additional motion on the deviation of the slope of the tooth profile on the left and right sides of the drum-shaped helical gear tooth surface accuracy optimization method based on multi-axis additional motion shown in this invention.

[0046] Figure 2(c) is a schematic diagram of the influence of the C-axis additional motion on the tooth profile slope deviation of the drum-shaped helical gear tooth surface accuracy optimization method based on multi-axis additional motion shown in this invention.

[0047] Figure 3 This is a schematic diagram of the differential evolution algorithm for the method of optimizing the tooth surface accuracy of a drum-shaped helical gear based on multi-axis additional motion, as shown in this invention.

[0048] Figure 4 This is a simulation diagram of the bulging amount of the tooth profile on both sides of the drum-shaped helical gear tooth surface accuracy optimization method based on multi-axis additional motion as shown in this invention.

[0049] Figure 5 This is a schematic diagram of the simulation results of the left and right tooth profile optimization of the drum-shaped helical gear tooth surface accuracy optimization method based on multi-axis additional motion as shown in this invention. Detailed Implementation

[0050] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the specific embodiments of the present invention will be described in detail below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments.

[0051] Based on the embodiments of this invention, all other embodiments obtained by those skilled in the art without inventive effort should fall within the scope of protection of this invention.

[0052] Many specific details are set forth in the following description in order to provide a full understanding of the invention. However, the invention may also be practiced in other ways different from those described herein, and those skilled in the art can make similar extensions without departing from the spirit of the invention. Therefore, the invention is not limited to the specific embodiments disclosed below.

[0053] Example 1

[0054] The fundamental error of the tooth surface is an inherent problem in the grinding process of helical gears with tooth profile modification. Unlike other errors, it is caused by the non-uniform cutting of the tooth surface during the tooth profile modification process. The embodiments of the present invention, through algorithm optimization and through simulation analysis and grinding experiments, have proven that the tooth surface distortion error can be effectively reduced, thereby improving the grinding accuracy of the modified tooth surface.

[0055] According to an embodiment of the present invention, in combination Figures 1-3 The schematic diagram shown illustrates a method for optimizing the tooth surface accuracy of a drum-shaped helical gear based on multi-axis additional motion, which specifically includes the following steps:

[0056] S1: Based on the structure of a five-axis linkage CNC forming gear grinding machine, establish a spatial coordinate system between the grinding wheel and the gear. Note that the following points need to be explained in this step:

[0057] The standard helical tooth surface equation is obtained based on the spatial coordinate system. By calculating the profile and tooth direction modification of the drum-shaped tooth, coordinate transformation is performed to obtain the tooth surface modification equation of the gear.

[0058] The tooth surface modification equation is combined with the tooth surface contact condition equation to calculate the theoretical contact line. The theoretical contact line is then transformed to the grinding wheel coordinate system through the coordinate transformation equation, and then projected onto the grinding wheel shaft section.

[0059] The profile is fitted using the least squares method to obtain the profile of the grinding wheel shaft section;

[0060] By combining the known equations of the grinding wheel's rotating surface with the contact condition equations of the grinding wheel's rotating surface, the actual contact line is calculated, and then transformed to the gear coordinate system.

[0061] The modification amount is allocated to the additional motion of the three axes and substituted into the gear inertial coordinate system to obtain the modified tooth surface equation.

[0062] As an example, spatial coordinate systems include tool coordinate system, workpiece coordinate system, and fixed coordinate system.

[0063] The standard helical tooth surface equation is established based on a spatial coordinate system, and its general mathematical expression is as follows:

[0064] (1)

[0065] Where, r b Let θ be the base circle radius, σ0 be the tooth groove half angle, μ be the involute parameter, p be the helical parameter, and θ be the helical motion parameter.

[0066] The modification amounts for the tooth profile and tooth direction of the drum-shaped tooth are as follows:

[0067]

[0068]

[0069] Where, α ce To measure the pressure angle of a circle, r ce To measure the radius of a circle, since the displacement coefficient used in the experiment is 0, the measuring circle is directly calculated based on the pitch circle, ε. 左 ε 右 This refers to the tooth profile modification amount on both the left and right sides.

[0070] The general mathematical expression for the theoretical contact line is as follows:

[0071] (2)

[0072] Where a is the center distance between the gear and the grinding wheel, and ∑ is the angle between the z-axis of the forming grinding wheel and the z-axis of the gear.

[0073] Based on the selected grinding wheel rotation surface, establish the general contact condition formula for inverse calculation, namely:

[0074] (3)

[0075] The general mathematical expression for the profile of the grinding wheel shaft section is as follows:

[0076]

[0077] Where R is the profile equation of the grinding wheel shaft section, f(R) is the profile function, and f'(R) is the derivative of f(R) with respect to R.

[0078] The general mathematical expression for the equation of the tooth profile modification surface is as follows:

[0079] (4)

[0080] Where △x and △y are the additional motion quantities along the x-axis and y-axis, respectively, and △c is the additional rotation quantity along the c-axis.

[0081] As an example, the above content can be illustrated by the following table:

[0082] Table 1: Meaning of Formula Characters.

[0083]

[0084] S2: Calculate the effects of the additional triaxial motion on the torsion of the left and right tooth surfaces and the influence of the tooth profile slope, respectively.

[0085] The following points need to be explained in this step:

[0086] The additional motion of the three axes affects the deviation of the right tooth profile slope due to the radial motion of the Y and X axes and the rotation of the C axis. The Y axis is radially moved. The offset of the Y axis and X axis changes due to the different pressure angles at the tooth root and tooth tip, which causes the deviation of the tooth profile slope.

[0087] Rotation of the C-axis changes the phase of the tooth profile, but does not affect the profile shape itself or the pressure angle at each point. It is related to the radii of the addendum circle and the dedendum circle.

[0088] The general mathematical expression for the deviation is as follows:

[0089]

[0090]

[0091] fH c =Δc a -Δc f =r a Δc-r f Δc

[0092] Preferably, the additional motion of the three axes has a similar effect on the slope deviation of the left tooth profile as on the right tooth profile, but in the opposite direction. The principle is as follows: during grinding, the right side of tooth 1 and the left side of tooth 2 are machined first, then the right side of tooth 2 and the left side of tooth 3 are machined, and so on until one revolution is completed.

[0093] S3: Utilize the differential evolution algorithm to jointly optimize the additional motion quantities of the three axes, ensuring that the actual contact line of the modified tooth surface matches the theoretical contact line. This step also requires clarification regarding the joint optimization, which includes:

[0094] Use the additional motion along the x-axis, y-axis, and c-axis as the initial population;

[0095] Starting with candidate solutions randomly generated from the initial population, denoted as:

[0096] x i,G =(x 1,G ,x 2,G ,...,x D,G i = 1, 2, ..., NP;

[0097] A mutation is generated by the difference between two random individuals, a scaling factor, and a target vector;

[0098] Based on the current individual in the initial population, several vectors are randomly selected for difference operations to generate a mutation difference vector;

[0099] A new candidate solution, i.e., a test vector, is generated based on the combination of the mutation vector and the parent vector.

[0100] According to the greed principle, from the test individual u iG and the original individual x iG The individual with the best function value is selected to enter the next generation. After obtaining the optimal optimization amount, the additional motion of the three axes is calculated in reverse until the optimal solution is obtained.

[0101] It should be noted that the range of values ​​for the optimization object is set according to the range of deviation values ​​in S2. The deviations caused by the additional motion along the x, y, and c axes are substituted into the algorithm and optimized. The optimization flowchart is as follows. Figure 3 As shown, when the optimal solution is obtained, the additional motion of the three axes can be calculated by back-calculating the aforementioned formula for S2, thus obtaining the optimal solution. The simulation diagram of the final optimization result is shown below. Figure 4 , Figure 5 As shown.

[0102] Reference Figure 4 and Figure 5 As can be seen intuitively, this invention reduces tooth surface error by optimizing multi-axis additional motion. During the forming grinding of drum gears, tooth surface distortion will occur due to errors. By using double-sided grinding process and optimizing the influence of three-axis additional motion on tooth surface distortion, the tooth surface accuracy is improved. Compared with single-sided grinding, double-sided grinding has the advantages of high efficiency, high precision and strong applicability.

[0103] The aforementioned methods for coordinate vector transformation and error calculation can be performed using existing technologies and methods, and will not be elaborated upon in this example.

[0104] It should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the spirit and scope of the technical solutions of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.

Claims

1. A method for optimizing the tooth surface accuracy of a barrel helical gear based on multi-axis additional motion, characterized in that, include: Based on the structure of a five-axis linkage CNC forming gear grinding machine, a spatial coordinate system between the grinding wheel and the gear is established. The spatial coordinate system includes a tool coordinate system, a workpiece coordinate system, and a fixed coordinate system; The standard helical tooth surface equation is obtained based on the spatial coordinate system. By calculating the profile and tooth direction modification of the drum-shaped tooth, coordinate transformation is performed to obtain the tooth surface modification equation of the gear. The general mathematical expression of the standard helical tooth surface equation is as follows: Where, r b The radius of the base circle, Where is the tooth groove half-angle, μ is the involute parameter, and p is the helical parameter. These are the parameters for helical motion; The modification amounts for the tooth profile and tooth direction of the drum-shaped tooth are as follows: in, α ce To measure the pressure angle of a circle, r ce To measure the radius of a circle, since the displacement coefficient used in the experiment is 0, the measuring circle is directly calculated based on the pitch circle. ε 左 , ε 右 This refers to the tooth profile modification amount on both the left and right sides; The tooth surface modification equation is combined with the tooth surface contact condition equation to calculate the theoretical contact line. The theoretical contact line is then transformed to the grinding wheel coordinate system through the coordinate transformation equation and projected onto the grinding wheel shaft section. The general mathematical expression for the theoretical contact line is as follows: Where 'a' is the center distance between the gear and the grinding wheel. The angle between the z-axis of the forming grinding wheel and the z-axis of the gear; The profile is fitted using the least squares method to obtain the profile of the grinding wheel shaft section; The general mathematical expression for the profile of the grinding wheel shaft section is as follows: Where R is the equation of the profile of the grinding wheel shaft section. It is a profile function; By combining the known equations of the grinding wheel's rotating surface with the contact condition equations of the grinding wheel's rotating surface, the actual contact line is calculated, and then transformed to the gear coordinate system. The modification amount is allocated to the additional motion of the three axes and substituted into the gear inertial coordinate system to obtain the modified tooth surface equation; Calculate the effects of the additional triaxial motion on the torsion of the left and right tooth surfaces and the influence of the tooth profile slope, respectively. The differential evolution algorithm is used to jointly optimize the additional motion of the three axes so that the actual modified tooth surface contact line matches the theoretical modified tooth surface contact line. The joint optimization includes: Use the additional motion along the x-axis, y-axis, and c-axis as the initial population; Starting with the candidate solutions randomly generated from the initial population, denoted as: A mutation is generated by the difference between two random individuals, a scaling factor, and a target vector; Based on the current individuals in the initial population, several vectors are randomly selected for difference operations to generate a mutation difference vector; A new candidate solution, i.e., a test vector, is generated based on the combination of the mutation vector and the parent vector. According to the greedy criterion, the individual u iG and the original individual x iG with the optimal function value is selected into the next generation, and after obtaining the best optimization quantity, the additional motion of the three axes is back calculated until the optimal solution is obtained.

2. The method for optimizing the tooth surface accuracy of a drum-shaped helical gear based on multi-axis additional motion according to claim 1, characterized in that, The general mathematical expression for the tooth profile modification equation is as follows: Where △x and △y are the additional motion quantities along the x-axis and y-axis, respectively, and △c is the additional rotation quantity along the c-axis.

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