Formed grinding tooth axial modification method based on four-axis motion simplified model and numerical control gear forming grinding system

Abstract

The invention relates to the technical field of gear precision machining, in particular to a four-axis movement simplified model-based axial modification method for formed grinding teeth and a numerical control gear forming grinding system. A gear forming grinding tooth direction modification method based on a four-axis linkage simplified model comprises the following steps that (1) a corresponding grinding wheel revolution curved surface is calculated based on theoretical spiral tooth surface parameters, and a space meshing coordinate system is established; according to the method, three-axis compensation and polynomial interpolation are adopted, the five-axis modeling difficulty is remarkably reduced, and the method is adaptive to a common gear grinding machine; and the production and processing efficiency is high, and industrial deployment and popularization are facilitated.

Description

Technical Field

[0001] This invention relates to the field of precision gear machining technology, specifically a method for profile grinding and tooth profile modification based on a simplified four-axis motion model, and a CNC gear profile grinding system. Technical Background

[0002] CNC form grinding machines are crucial equipment for machining large-size, high-precision gears, widely used in the manufacture of transmission gears for heavy equipment such as wind power, tunnel boring machines, ships, and construction machinery. Typically, form grinding is the final step in gear machining, used to eliminate deformation errors caused by heat treatment and improve the precision of hardened tooth surfaces. To improve gear contact conditions, especially avoiding edge contact, and to enhance transmission accuracy and smoothness, large-size gears usually undergo tooth profile modification. Form grinding machines feature multi-axis linkage, can machine both internal and external gears, and offer high degrees of freedom. Therefore, it is considered to add multi-axis motion during the grinding process for minor adjustments, achieving tooth profile topology modification. Existing tooth profile modification methods include adding radial motion, adding angular motion, and five-axis linkage trajectory modification.

[0003] Traditional methods, while simple in their radial and angular additional motion forms, are prone to causing tooth surface distortion errors during profile grinding. While the five-axis linkage method offers high profile correction accuracy, it faces challenges such as complex modeling, numerous optimization parameters, heavy computational burden, and poor stability, making it difficult to widely apply in ordinary gear grinding equipment. Furthermore, some low-end gear grinding machines have low A-axis positioning accuracy, objectively hindering their ability to support high-precision five-axis additional motion machining processes. Summary of the Invention

[0004] The purpose of this invention is to provide a tooth profile modification method for forming grinding based on a simplified four-axis motion model and a CNC gear forming grinding system.

[0005] The principle of this invention is as follows: by supplementing with additional tangential motion (Y-axis), a four-axis linkage control of X, Y, Z, and C is formed to control the torsion error in the grinding of the double tooth surface. Simultaneously, 3rd-5th order polynomials are used to describe the motion of each axis. This simplified model can significantly reduce the computational cost of matrix solving and improve iterative stability. Furthermore, during the iterative optimization process, an improved Levenberg-Marquardt (LM) algorithm based on a trust region strategy is introduced to overcome the unsolvable problem of matrix singularity and stabilize the computational cost of the additional motion.

[0006] To solve the above-mentioned technical problems, the technical solution adopted in this patent is as follows: A gear forming grinding tooth profile modification method based on a simplified four-axis linkage model includes the following steps: (1) Based on the theoretical helical tooth surface parameters, calculate the corresponding grinding wheel rotation surface and establish a spatial meshing coordinate system; (2) Establish a four-axis linkage model in the gear processing machine tool, where: the X-axis is the radial feed axis, the Y-axis is the tangential feed axis, the Z-axis is the gear stroke axis, and the C-axis is the gear rotation axis; (3) The motion of the X, Y and C axes is modeled using third- to fifth-order polynomials respectively. The linkage of the A-axis (grinding wheel swing axis) is not considered, and only the fixed installation angle is set. (4) Derive the contact line constraint conditions based on the meshing principle and construct the residual objective function; (5) The improved Levenberg-Marquardt (LM) algorithm based on the trust region strategy is used to iteratively optimize the additional motion parameters of the multi-axis motion and solve the additional motion trajectory. (6) The optimized motion trajectories of each axis are output to the CNC system to guide the tooth profile modification process during the forming grinding process.

[0007] The sand profile is a rotating surface obtained by rotating around the grinding wheel axis, calculated based on a mathematical model, and its surface normal vector is obtained by differentiating the profile parameters.

[0008] The polynomial trajectory model is of third order and is expressed as follows: ; Where t is the normalized time parameter, DX i DY i ψ i The coefficients are polynomials, and the motion trajectories on the Y and C axes are consistent with these.

[0009] The objective function is the sum of squared residuals of the tooth surface error at the contact point, and the optimization objective is to minimize the residuals. Its sensitivity to parameters is represented by the Jacobian matrix. The Jacobian matrix only contains the partial derivative terms of the X, Y, and C axes, and does not contain the A axis.

[0010] The improved Levenberg-Marquardt (LM) algorithm based on the trust region strategy includes the following steps: (1) Initialize the residual function and damping coefficient λ; (2) Construct a trust region descent model and dynamically adjust λ based on the descent ratio ρ; (3) If the descent effect is significant, accept the current step size and decrease λ; otherwise, increase λ and iterate again. (4) The convergence criterion is that the change in the residual objective function is less than the set threshold or the number of iterations reaches the upper limit.

[0011] The iterative update formula for the improved Levenberg-Marquardt (LM) algorithm based on the trust region strategy is as follows: ; Where J is the Jacobian matrix, r is the current residual vector, and λ is the non-negative damping factor, which is dynamically adjusted to balance the descent rate and convergence stability. A trust region factor ρ is introduced to determine the descent trend of the objective function and adjust the value of λ accordingly.

[0012] The trust region descent ratio ρ satisfies the following formula: ; ; ; The decision to accept the current step size and adjust λ is based on the magnitude of ρ. A larger ρ indicates a greater decrease in the objective function, in which case the current step size is accepted and λ is decreased; conversely, a smaller ρ is rejected and λ is increased.

[0013] During the optimization iteration process, key parameters are selected with the help of strategies to improve the convergence speed. Generally, residual convergence can be completed in only 2-3 iterations, which has the ability to quickly optimize local areas.

[0014] A CNC gear forming grinding system includes: a forming grinding machine bed with a four-axis linkage mechanism of X, Y, Z, and C; a CNC unit configured to read the optimized multi-axis motion trajectory and drive each axis to perform linkage shaping; a machining module for performing double-sided grinding of the gear according to the trajectory; and a three-coordinate measuring system for in-machine detection of gear shaping accuracy and feedback of shaping results.

[0015] The grinding module, in conjunction with a third- to fifth-order polynomial interpolation control program, achieves coordinated four-axis motion, ensuring continuous tooth surface forming and smooth trajectory. The CNC gear forming grinding system is suitable for forming grinding processes such as typical tooth direction drum-shaped shaping and symmetrical tooth direction shaping, and is especially suitable for low- to medium-configuration gear grinding machines without A-axis dynamic control capability, achieving efficient and high-precision tooth direction shaping.

[0016] The beneficial effects of this invention are: (1) The use of three-axis compensation and polynomial interpolation significantly simplifies the difficulty of five-axis modeling and is compatible with ordinary gear grinding machines; (2) The improved Levenberg-Marquardt (LM) algorithm based on the trust region has good stability and fast convergence speed, and avoids solution failure caused by matrix singularity; (3) The optimization process involves fewer variables and less computational burden, making it suitable for online or fast offline computation; (4) The generated tooth surface topology has high accuracy, which can effectively eliminate tooth surface distortion error and improve the consistency of double-sided grinding; (5) It has high production and processing efficiency, which is conducive to industrial deployment and promotion. Attached Figure Description

[0017] Figure 1 This is a schematic diagram of a spatial meshing coordinate system.

[0018] Figure 2 This is a schematic diagram of the structure of the CNC five-axis forming gear grinding machine used in this invention.

[0019] Figure 3 This is a schematic diagram of the axial cross-section of the grinding wheel and the distribution of the normal vector.

[0020] Figure 4 This is a schematic diagram of the additional motion trajectories of each axis obtained based on four-axis motion shaping optimization.

[0021] Figure 5 This is a schematic diagram of the topological deviation distribution in tooth profile modification.

[0022] Figure 6 This is a schematic diagram illustrating the convergence speed of the improved Levenberg-Marquardt (LM) algorithm based on the trust region strategy. Detailed Implementation

[0023] The implementation steps of this invention are as follows: Step 1: Based on the theoretical helical tooth surface parameters, calculate the corresponding sand profile and establish a spatial meshing coordinate system; Step 2: Set up a four-axis linkage model in the gear processing machine tool, with the X-axis for radial feed, the Y-axis for tangential feed, the Z-axis for tooth stroke, and the C-axis for gear rotation; Step 3: Model the motion of the X, Y, and C axes using third- to fifth-order polynomials respectively, without considering the linkage of the A-axis.

[0024] Step 4: Based on the contact line constraints, establish the residual objective function and generate the Jacobian matrix; Step 5: Iteratively solve for the additional motion trajectory using the improved Levenberg-Marquardt (LM) algorithm based on the trust region; Step Six: Transmit the optimization results to the CNC system to achieve four-axis linkage machining; The improved Levenberg-Marquardt (LM) algorithm based on trust regions includes the following steps: Step 1: Initialize the weight matrix λ ≥ 0 and calculate the residuals; Step 2: Construct a trust region descent model and dynamically adjust λ according to the changes in the objective function; Step 3: If the decrease is significant, accept the step size and decrease λ; if the decrease is not significant, increase λ and iterate again. Step 4: The convergence criterion is that the change in the objective function is lower than the set threshold or the number of iterations exceeds the upper limit.

[0025] The machine tool configuration includes: The bed of the forming gear grinding machine is equipped with a four-axis linkage mechanism (X, Y, Z, C), a CNC unit, a machining module, and a three-coordinate measurement system.

[0026] Based on the known parameters of the gear, the axial profile of the grinding wheel is established. The axial profile of the grinding wheel can be expressed as: (1) This profile rotates around the grinding wheel axis to obtain the grinding wheel's surface of rotation in space.

[0027] (2) Where u is the axis profile parameter, M is the phase parameter. wq A matrix describing the shape of the sand as it rotates around the axis of the grinding wheel.

[0028] The unit normal vector of the grinding wheel surface is: (3) in, , , .

[0029] By transforming matrix M gw The surface r of the grinding wheel during machine tool processing can be obtained. w In gear coordinate system S g The family of surfaces formed in the process: (4) In the formula, t is the independent variable of the motion function of each CNC axis of the machine tool, which is dimensionless.

[0030] (5) In the formula , , i It can be either C or A. The A-axis is set to a fixed value, equal to the grinding wheel mounting angle Σ.

[0031] The X, Y, and C-axis motions are described using 3rd to 5th order polynomials. The additional motion tooth profile control for each machine tool axis can be achieved by adjusting the polynomial coefficients. The polynomial interpolation description for each machine tool axis motion is as follows: (6) Where B is the tooth width, DX0, DX1, ..., , These are the polynomial coefficients.

[0032] Based on the meshing principle, the contact line of grinding wheel is solved using formulas.

[0033] (7) The speed of the grinding wheel relative to the gear.

[0034] Substituting the variables, we get: (8) Let a, b, and c be the algebraic terms of equation (8), then equation (8) can be simplified as follows: (9) Where a, b, and c are functions of the grinding wheel axial profile parameter u, and are related to the gear rotation angle. Irrelevant. If ,but (10) By combining it with equations (2) and (4), the coordinates of the contact line in the grinding wheel coordinate system and the gear coordinate system can be obtained.

[0035] The following is the solution for the additional motion along multiple axes: The additional four-axis motion for tooth profile modification on a gear grinding machine can be achieved by adjusting the coefficients of various polynomials, as defined. (11) In the formula, n = 12.

[0036] Topological tooth surface This can be rewritten as relating to the polynomial coefficients. Topological functions of variables The influence of the polynomial coefficients of each motion axis on the tooth surface topology is expressed as follows: (12) In the formula, This represents the correction amount for the polynomial coefficients of each motion axis. The deviation between the discrete points on the tooth surface and the theoretical topological modification value can be written as... (13) In the formula, m is the number of discrete points on the tooth surface. (Abbreviated as...) (14) J is the Jacobian matrix.

[0037] Solving This yields the additional motion trajectories along the X, Y, and C axes for tooth profile modification.

[0038] The steps for applying the improved Levenberg-Marquardt (LM) algorithm based on the trust region to solve the above equations are as follows: To avoid the Jacobian matrix during iteration (k is the iteration number) If a singular or near-singular condition occurs, resulting in no solution for the entire iteration, the improved LM algorithm used in this invention, combined with a trust region strategy, introduces non-negative parameters. Its iteration steps are: (15) parameter It is based on the trust region strategy selection, assuming (16) in , Based on the following trust domain policy update (17) In the formula, , .

[0039] Define the actual decrease and the estimated decrease in the k-th iteration as follows: (18) (19) Using the ratio of the two Deciding whether to accept an iteration step and adjusting the iteration parameters Factor values. Generally, The larger the value, the better the objective function. The more it falls, the more it is accepted. Expected next iteration Longer, therefore smaller Conversely, consider refusing to accept. Increase In this article, μ, λ, ρ 1 、 ρ 2 、l All selections are made within a certain range to ensure the stability and robustness of the algorithm, and to facilitate optimized configuration for different gear machining tasks.

[0040] The optimized motion trajectories of the X, Y, and C axes are stored in the CNC unit of the machine tool for machining steps such as centering, flushing, rough grinding, and fine grinding.

[0041] The gear tooth direction accuracy is detected by using an in-machine coordinate measuring system.

[0042] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an improper limitation of the invention.

[0043] The above are merely preferred embodiments of the present invention. The scope of protection of the present invention is not limited to the above embodiments. All technical solutions that fall within the scope of the present invention are within the scope of protection of the present invention. It should be noted that for those skilled in the art, any improvements and modifications made without departing from the principle of the present invention should be considered within the scope of protection of the present invention.

Claims

1. A method for tooth direction modification in gear form grinding based on a simplified model of four-axis linkage, characterized in that, The method comprises the following steps: (1) Based on the theoretical helical tooth surface parameters, the corresponding grinding wheel rotary surface is calculated, and a space meshing coordinate system is established; (2) A four-axis linkage model is established on the gear forming grinding machine, wherein: the X-axis is the radial feed axis, the Y-axis is the tangential feed axis, the Z-axis is the tooth direction stroke axis, and the C-axis is the gear rotation axis; (3) The three-five order polynomials are used for trajectory modeling of the X, Y and C axes respectively, the A-axis (the grinding wheel swing axis) linkage is not considered, and only the fixed installation angle is set; (4) The contact line constraint condition is derived based on the meshing principle, and a residual objective function is constructed; (5) The improved Levenberg-Marquardt (L-M) algorithm based on the trust region strategy is used to iteratively optimize the additional motion parameters of the multi-axis, and the additional motion trajectory is solved; (6) The optimized motion trajectory of each axis is output to the numerical control system, and is used for guiding the tooth direction modification processing in the gear forming grinding process.

2. The gear forming grinding tooth direction modification method based on the four-axis linkage simplified model according to claim 1, characterized in that: the grinding wheel profile is a rotary surface obtained by rotating around the grinding wheel axis, and is calculated based on a mathematical model, and a surface normal vector is obtained by derivation of profile parameters. The polynomial trajectory model is in the form of three-five orders, and is expressed as:

3. The gear form grinding tooth direction modification method based on a four-axis linkage simplified model according to claim 1 or 2, characterized in that: The objective function is the residual sum of squares of tooth surface errors at the contact points, and the optimization target is to minimize the residual, and the sensitivity of the parameters is represented by a Jacobian matrix; the Jacobian matrix only includes partial derivative terms of the X, Y and C three axes, and does not include the A axis. ； where t is a normalized time parameter, DX i The Y-axis and C-axis motion trajectories are consistent with this.

4. The gear form grinding tooth direction modification method based on a four-axis linkage simplified model according to claim 1, characterized in that: The improved Levenberg-Marquardt (L-M) algorithm based on the trust region strategy comprises the following steps:

5. The gear form grinding tooth direction modification method based on a four-axis linkage simplified model according to claim 1 or 2, characterized in that: (1) The residual function and the damping coefficient λ are initialized; (2) A trust region descent model is constructed, and λ is dynamically adjusted based on the descent ratio ρ; (3) If the descent effect is significant, the current step is accepted and λ is reduced, otherwise λ is increased for reiteration; (4) The convergence criterion is that the variation of the residual objective function is less than a set threshold value or the iteration number reaches an upper limit.

6. The gear forming grinding tooth direction modification method based on the four-axis linkage simplified model according to claim 1, characterized in that: The iterative update formula of the improved Levenberg-Marquardt (L-M) algorithm based on the trust region strategy is: Wherein J is the Jacobian matrix, r is the current residual vector, λ is the non-negative damping factor, and is dynamically adjusted to balance the descent speed and convergence stability; ； And a trust region factor ρ is introduced, which is used to judge the descending trend of the objective function, and the λ value is adjusted accordingly. The trust region descent ratio ρ satisfies the following formula:

7. The gear form grinding tooth direction modification method based on a four-axis linkage simplified model according to claim 5, characterized in that: According to the size of ρ, it is determined whether to accept the current step and the adjustment strategy of λ. The larger ρ is, the more the objective function decreases, and the current step is accepted and λ is reduced, otherwise λ is rejected and increased. ； ； ； In the optimization iteration process, the key parameters are selected by means of the strategy, and the convergence speed is improved, and generally only 2-3 iterations are needed to complete the residual convergence, and the method has the ability of fast local optimization.

8. The gear form grinding tooth direction modification method based on a four-axis linkage simplified model according to claim 1, characterized in that: It comprises:

9. A CNC gear form grinding system suitable for use in the method of any one of claims 1-8, characterized in that, A gear forming grinding machine bed body with X, Y, Z and C four-axis linkage mechanisms; ​ A numerical control unit is configured to read the optimized multi-axis motion trajectory and drive each axis to perform linkage modification; A processing module is configured to perform double-face grinding processing on the gear according to the trajectory; A three-coordinate measurement system is configured to detect the modification accuracy of the gear in machine and feed back the modification result.

10. The CNC gear-forming grinding system of claim 9, wherein: The grinding processing module cooperates with a three-to-five order polynomial interpolation control program to realize four-axis coordinated motion, guarantee the continuity of tooth surface forming and the smoothness of the trajectory; and the numerical control gear forming grinding system is suitable for typical tooth direction drum modification, symmetric tooth direction modification and other forming grinding processes, and is particularly suitable for medium and low configuration gear grinding machines without A-axis dynamic control capability, to realize efficient and high-precision tooth direction modification.

Images