Surface topography prediction method based on ball-end cutter model and variable-step iterative algorithm

Abstract

The invention relates to a surface topography prediction method based on a ball-end cutter model and a variable step size iterative algorithm, which is based on the ball-end cutter model, combines and applies a dynamic variable time step size strategy, accurately divides cutting and non-cutting tracks, and remarkably improves the simulation efficiency and precision. Through multi-coordinate system dynamic modeling and a specific function, the eccentricity of the ball-end milling cutter is compensated, the abrasion is accurately modeled, dynamic physical coupling is combined, the actual machining state of the ball-end milling cutter is accurately simulated, the simulation precision and authenticity are improved, and the high-precision requirement of aerospace parts can be met. Meanwhile, through modular design, a complex curved surface tool path is accurately simulated, and through adjustment of evaluation parameter threshold values, the method can be expanded to surface topography prediction of various difficult-to-machine materials, process parameters are rapidly iterated, and the process optimization period is greatly shortened.

Description

Technical Field

[0001] The present invention relates to the technical field of mechanical machining surface quality prediction, and particularly to a method for predicting the surface topography of ball-end milling. Background Art

[0002] In the field of aerospace manufacturing, the high-efficiency and precision machining of complex parts such as engine blades and turbine disks is one of the core technical challenges. The surface topography quality of such parts directly affects key issues such as the fatigue life, aerodynamic performance, and assembly accuracy of the parts. To meet the stringent surface quality requirements, traditional processes rely on a large number of trial cutting experiments to adjust parameters, but there are problems such as high cost, long cycle, and small application range.

[0003] With the progress of numerical control technology and simulation methods, surface topography prediction based on numerical simulation has become a key means to optimize the process. However, the particularity of aerospace complex surface machining poses higher challenges to surface topography simulation technology.

[0004] When a ball-end mill is machining a curved surface, the contact point of the cutting edge changes continuously with the attitude of the ball-end mill, resulting in significant dynamic effects such as cutting force and vibration, which increases the complexity of the simulation. At the same time, the surface roughness of aerospace parts usually requires R a ≤0.8 μm, and the simulation model is required to have sub-micron resolution. The traditional Z-MAP method relies on dense grid division, resulting in an exponential increase in the amount of calculation and making it difficult to meet the real-time requirements of engineering.

[0005] Currently, although the mainstream surface topography simulation methods have their own advantages, they all have significant limitations. The solid modeling method simulates material removal through Boolean operations, but ignores the complex structural characteristics of the dynamic interaction between the ball-end mill and the workpiece; the numerical calculation method can efficiently solve the trajectory of the ball-end mill, but it is difficult to expand the consideration of dynamic physical factors, resulting in limited prediction accuracy; the traditional Z-MAP method updates the surface height through discrete grid projection, although it can take into account both accuracy and dynamic effects, but in high-resolution simulation, due to the fixed time step, there is redundant calculation in the non-cutting stage, resulting in low efficiency.

[0006] In recent years, related research has tried to optimize the Z-MAP algorithm through linear interpolation, Newton iteration method or point cloud envelope method, but these methods still rely on a constant time step and fail to fundamentally solve the contradiction between efficiency and accuracy. Summary of the Invention

[0007] The purpose of the present invention is to avoid the deficiencies of the prior art and provide a surface topography prediction method based on a ball-end mill model and a variable-step iteration algorithm. By using a dynamic simulation strategy, a surface topography prediction model is established, which significantly reduces the computational complexity while ensuring accuracy, and realizes efficient surface topography prediction under any cutting conditions to meet the urgent needs of aerospace high-efficiency and precision manufacturing.

[0008] To achieve the above object, the technical solution adopted by the present invention is a surface topography prediction method based on a ball-end mill model and a variable-step iterative algorithm, including the following steps: Step 1: In the cutting edge geometric model of the structure of the part of the ball-end mill involved in cutting, establish a spatial coordinate system for describing the position of the cutting edge of the ball-end mill, use the spatial coordinate system to determine the positions of the cutting edge discrete points of all the cutting edges of the ball-end mill, that is, obtain the coordinate expressions of the cutting edge discrete points. At the same time, use the normal distribution random quantity function in the expressions of the cutting edge discrete points to characterize the passivation form of the real cutting edge of the ball-end mill during the cutting process; Among them, the spatial coordinate system includes the ball-end mill coordinate system, the machine tool spindle coordinate system, the workpiece reference coordinate system, and the workpiece coordinate system; Step 2: Based on the expressions of the cutting edge discrete points, describe the movement trajectories of the cutting edge discrete points in the spatial coordinate system through multiple calculations of the three-dimensional homogeneous coordinate transformation matrix. Thus, in the workpiece reference coordinate system, establish a cutting edge movement trajectory calculation equation for describing the cutting edge movement trajectory; Step 3: Divide the cutting edge movement trajectories corresponding to the cutting edge discrete points into non-cutting trajectories in the non-cutting stage and cutting trajectories in the cutting stage according to the cutting state, and calculate the time step lengths of the cutting edge discrete points in the non-cutting stage and the cutting stage; Step 4: Combine the cutting edge discrete point time step length with the Z-MAP algorithm, and based on the milling process data for describing the movement state of the ball-end mill, iteratively update to obtain the height information of the workpiece machining surface after the end of each cutting edge movement trajectory, that is, obtain the three-dimensional surface topography of the workpiece machining surface; Among them, the machining program data includes the tool path file for determining the movement path of the cutting edge discrete points and the cutting process parameters of the movement speed of the cutting edge discrete points themselves.

[0009] Further, the spatial coordinate system described in Step 1 specifically includes: Taking the center of the ball of the ball-end mill as the origin O, taking the axis of the ball-end mill as the Z T axis, taking the direction of the connection line from the tip of any cutting edge to the Z T axis as the X T axis, thus, determining the Y T axis according to the right-hand coordinate system rule, and establishing the ball-end mill coordinate system; the ball-end mill coordinate system can rotate around the center of rotation of the machine tool spindle; Based on the X T axis and Y T axis directions of the ball-end mill coordinate system, taking the machine tool spindle as the Z S axis, and establishing the machine tool spindle coordinate system; Based on the origin O of the coordinate axes in the ball-end mill coordinate system, with the cutting feed pitch direction as the X G axis, the feed direction of the ball-end mill as the Y G axis, and determining the Z G axis according to the right-hand coordinate system rule, a workpiece reference coordinate system is established; Based on the coordinate axis directions of the workpiece reference coordinate system, with any vertex on the bottom surface of the workpiece as the origin, a workpiece coordinate system is established; Furthermore, using the space coordinate system to determine the positions of the discrete points of the cutting edges of all the cutting edges of the ball-end mill, for any cutting edge j among the N cutting edges in the cutting edge geometric model, the expression for the discrete point L of any cutting edge j is: , In the formula, R is the radius of the ball-end mill, γ is the helix angle, N is the total number of cutting edge teeth, E(σ) is the normal distribution random variable function, θ is the line connecting the discrete point of the cutting edge and the center of the ball of the ball-end mill, and the projection of θ on the X T OY T plane and the angle with the X T axis.

[0010] Further, the established cutting edge motion trajectory calculation equation in the second step is, for any cutting edge j among the N cutting edges in the cutting edge geometric model, the motion trajectory calculation equation for any discrete point L of the cutting edge j, specifically: , where θ is the line connecting the discrete point of the cutting edge and the center of the ball of the ball-end mill, t is the cutting edge motion time, and (x WL , y WL , z WL ) is the position coordinate of the discrete point L in the workpiece coordinate system; M1 to M5 are respectively the rotation, eccentric translation, spindle rotation, pose adjustment, and feed transformation matrices of the ball-end mill, and the specific expressions are: The rotation matrix M1 is: , The eccentric translation matrix M2 is: , where δ is the radial eccentric angle of the ball-end mill; the spindle rotation matrix M3 is: , where φ0 is the initial phase angle of the cutting edge and ω is the angular velocity of the machine tool spindle; The pose adjustment matrix M4 is: , where α is the rake angle of the ball-end mill and β is the side rake angle of the ball-end mill; The feed transformation matrix M5 is as follows: , where f p is the machining row spacing of the ball-end mill, f v is the feed rate of the ball-end mill, i is any cutting row among the N cutting edges, and (x0, y0, z0) is the initial position coordinates of the ball center of the ball-end mill in the workpiece coordinate system.

[0011] Further, the specific content of Step 3 is as follows: During any rotation period of the ball-end mill in the cutting process, for any cutting edge j among the N cutting edges in the cutting edge geometric model, there is: Since the discrete points of the cutting edge at the initial position are all above the workpiece grid height W Z , that is, there is: Z L >W Z , Z L is the height of any discrete point L of any cutting edge j at time t. At this time, the discrete points of the cutting edge are in the first non-cutting stage. According to the maximum discrete time step Δt, calculate the cutting edge movement trajectory, and record the cutting edge discrete point counting variable in the first non-cutting stage as A = A + 1 until the discrete points of the cutting edge leave the first non-cutting stage, that is, Z L ≤W Z ; As the discrete points of the cutting edge leave the first non-cutting stage, that is, Z L ≤W Z , at this time, the discrete points of the cutting edge are in the cutting stage. Record the cutting edge discrete point counting variable in the cutting stage as B = B + 1 until the discrete points of the cutting edge leave the cutting stage, that is, Z L >W Z ; Continuing, when the discrete points of the cutting edge leave the cutting stage, that is, Z L >W Z , at this time, the discrete points of the cutting edge are in the second non-cutting stage. Record the cutting edge discrete point counting variable in the second non-cutting stage as C = C + 1 until the rotation period of the ball-end mill ends; Among them, the initial counting variable values of A in the cutting edge discrete point counting variable in the first non-cutting stage, B in the cutting edge discrete point counting variable in the cutting stage, and C in the cutting edge discrete point counting variable in the second non-cutting stage are all 0; That is, the division of the cutting trajectories and non-cutting trajectories of the discrete points of all N cutting edges is completed; Furthermore, let the time step Δt1 in the first non-cutting stage = AΔt; the time step Δt2 in the cutting stage = Δt; the time step Δt3 in the second non-cutting stage = CΔt; During the rotation cycle of the ball-end milling cutter with continuous cutting, the time step is dynamically selected according to the positional relationship between the discrete points of the cutting edge and the workpiece grid. Specifically: If the discrete point of the cutting edge is in the first non-cutting stage, the time step Δt1 is adopted. If the discrete point of the cutting edge is in the cutting stage, the time step Δt2 is adopted; otherwise, the time step Δt of the second non-cutting stage is adopted. 3； Among them, Δt is the maximum discrete time step of the cutting edge movement trajectory determined according to the target accuracy of the predicted output result of the surface topography, and the specific expression is: , In the formula, A c is the workpiece grid accuracy, and the value of A c is used to determine the target accuracy achieved by the predicted result of the ball-end milling cutter's milling surface topography; θ max is the maximum axial angle of the part of the cutting edge participating in cutting; R is the radius of the ball-end milling cutter.

[0012] Furthermore, the specific content of step four is as follows: First, initialize the discrete accuracy A c of the machined surface of the workpiece, the discrete accuracy Δθ of the cutting edge, the maximum discrete time step Δt of the cutting edge movement trajectory, and construct the discrete point cloud of the machined surface. Then, according to the actual size or model of the workpiece, determine the position coordinates of the discrete point cloud in the workpiece coordinate system, and then convert the discrete point cloud into the point cloud height matrix W Z ; Finally, traverse all discrete points of the cutting edge. Based on the tool path file and cutting process parameters, calculate the movement trajectory of the discrete points of the cutting edge according to the determined time step of the discrete points of the cutting edge, and update the point cloud height matrix W Z of the workpiece machining surface for the coordinate points in the workpiece reference coordinate system corresponding to the discrete points of the cutting edge in the cutting stage, that is, the predicted workpiece machining surface topography is obtained.

[0013] Furthermore, it also includes the step of evaluating the accuracy of the surface topography prediction by using the profile arithmetic mean deviation R a and the surface arithmetic mean height S a .

[0014] Furthermore, the profile arithmetic mean deviation R a and the surface arithmetic mean height S a are respectively: , , In the formula, Z iis the height value of the sampling grid point, Z mid is the height value of the calculation reference plane; m and n are the number of grid divisions along the X and Y axes in the workpiece coordinate system, respectively.

[0015] The beneficial effects of the present invention are as follows: The present invention uses a dynamic variable time step strategy to accurately divide the cutting and non-cutting trajectories, skipping the non-cutting areas, greatly reducing the redundant calculation amount, and significantly improving the simulation efficiency and accuracy; it also compensates for the eccentricity of the ball-end mill through multi-coordinate system dynamic modeling and specific functions, accurately models the wear, and combines dynamic physical coupling to accurately simulate the actual machining state of the ball-end mill, comprehensively improving the accuracy and authenticity of the simulation, and meeting the high-precision requirements of aerospace parts.

[0016] At the same time, through modular design, this method can flexibly adapt to a variety of machining scenarios. It can not only support the dynamic analysis of the ball-end mill attitude of five-axis machine tools based on a specific coordinate transformation matrix, accurately simulate complex curved surface tool paths, but also expand to the surface topography prediction of a variety of difficult-to-machine materials by adjusting the evaluation parameter threshold; and it can quickly iterate process parameters, helping to significantly shorten the process optimization cycle. Brief Description of the Drawings

[0017] Figure 1 is a schematic diagram of the position for establishing the space coordinate system of the present invention; Figure 2 is a schematic diagram of the movement trajectory of the discrete points on the cutting edge of the present invention; Figure 3 is a schematic diagram of the Z-MAP algorithm flow based on the variable time step strategy of the present invention; Figure 4 is a schematic diagram of the comparison of the simulation time lengths between the improved Z-MAP method and the traditional Z-MAP method of the present invention; Figure 5 is the first comparison schematic diagram of the simulated predicted surface topography and the actual measured surface topography under variable tilt angles of the present invention; Figure 6 is the second comparison schematic diagram of the simulated predicted surface topography and the actual measured surface topography under variable tilt angles of the present invention; Figure 7 is the third comparison schematic diagram of the simulated predicted surface topography and the actual measured surface topography under variable tilt angles of the present invention; Figure 8 is the fourth comparison schematic diagram of the simulated predicted surface topography and the actual measured surface topography under variable tilt angles of the present invention; Figure 9 is the comparison schematic diagram of the simulated predicted roughness and the actual measured roughness under variable tilt angles of the present invention.

[0018] In the figure: 1. Ball-end milling cutter coordinate system; 2. Workpiece reference coordinate system; 3. Machine tool spindle coordinate system; 4. Workpiece coordinate system; 5. Period start point; 6. Cutting depth; 7. First non-cutting stage; 8. Cutting stage; 9. Second non-cutting stage; 10. Period end point; Specific implementation manner The principles and features of the present invention will be described below in conjunction with the accompanying drawings. The examples given are only used to explain the present invention and are not intended to limit the scope of the present invention.

[0019] To achieve the above object, the present invention provides the following specific implementation manner: Example 1: As Figures 1-3 shown, a surface topography prediction method based on a ball-end cutter model and a variable step size iteration algorithm includes the following steps: S01. In the cutting edge geometric model of the structure of the part of the ball-end milling cutter participating in cutting, establish a space coordinate system for describing the position of the cutting edge of the ball-end milling cutter. As Figure 1 shown, the space coordinate system specifically includes: Taking the center of the ball of the ball-end milling cutter as the origin O, taking the axis of the ball-end milling cutter as the Z T axis, taking the direction of the line connecting the tip of any cutting edge to the Z T axis as the X T axis, and thus, determining the Y T axis according to the right-hand coordinate system rule, establishing the ball-end milling cutter coordinate system 1; the ball-end milling cutter coordinate system 1 can rotate around the machine tool spindle as the center of rotation; Based on the X T axis and the Y T axis directions of the ball-end milling cutter coordinate system 1, taking the machine tool spindle as the Z S axis, establishing the machine tool spindle coordinate system 3; Based on the origin O of the coordinate axes of the ball-end milling cutter coordinate system, taking the cutting feed direction as the X G axis, the ball-end milling cutter feed direction as the Y G axis, and determining the Z G axis according to the right-hand coordinate system rule, establishing the workpiece reference coordinate system 2; Based on the coordinate axis directions of the workpiece reference coordinate system 2, taking any vertex on the bottom surface of the workpiece as the origin, establishing the workpiece coordinate system 4.

[0020] S02. Use the space coordinate system to determine the positions of the cutting edge discrete points of all the cutting edges of the ball-end milling cutter, that is, obtain the coordinate expression of the cutting edge discrete points. At the same time, use the normal distribution random quantity function in the expression of the cutting edge discrete points to characterize the passivation form of the real cutting edge of the ball-end milling cutter during the cutting process. Specifically: For any cutting edge j among the N cutting edges in the cutting edge geometric model, the expression of the discrete point L of any cutting edge j is: , wherein, R is the radius of the ball-end mill, γ is the helix angle, N is the total number of cutting edges of the teeth, E(σ) is the normal distribution random variable function, θ is the line connecting the discrete point of the cutting edge and the center of the ball of the ball-end mill, and in the ball-end mill coordinate system X T OY T plane projection and X T axis included angle.

[0021] S03. Based on the expression of the discrete points of the cutting edge, through the calculation of the three-dimensional homogeneous coordinate transformation matrix for many times, the motion trajectory of the discrete points of the cutting edge in the space coordinate system is described. Thus, in the workpiece reference coordinate system 2, a cutting edge motion trajectory calculation equation for describing the motion trajectory of the cutting edge is established. That is, for any cutting edge j among the N cutting edges in the cutting edge geometric model, there is a motion trajectory calculation equation for any discrete point L of the cutting edge j, specifically: , wherein, θ is the line connecting the discrete point of the cutting edge and the center of the ball of the ball-end mill, t is the cutting edge motion time, (x WL , y WL , z WL ) are the position coordinates of the discrete point L in the workpiece coordinate system; M1 to M5 are respectively the rotation, eccentric translation, spindle rotation, pose adjustment and feed transformation matrices of the ball-end mill, and the specific expressions are: The rotation matrix M1 is: , The eccentric translation matrix M2 is: , wherein, δ is the radial eccentric angle of the ball-end mill; The spindle rotation matrix M3 is: , wherein, φ0 is the initial phase angle of the cutting edge, and ω is the angular velocity of the machine tool spindle; The attitude adjustment matrix M4 is: , wherein, α is the rake angle of the ball-end mill, and β is the side tilt angle of the ball-end mill; The feed transformation matrix M5 is: , wherein, f p is the machining stepover of the ball-end mill, f v is the feed rate of the ball-end mill, i is any cutting row among the N cutting edges, and (x0, y0, z0) are the initial position coordinates of the center of the ball of the ball-end mill in the workpiece coordinate system.

[0022] S04. During any ball-end milling cutter rotation period in the cutting process, for any cutting edge j among the N cutting edges in the cutting edge geometric model, there is: Since the discrete points of the cutting edge at the initial position are all above the workpiece grid height W Z That is, there is: Z L >W Z , Z L is the height of any discrete point L of any cutting edge j at time t. At this time, the discrete points of the cutting edge are in the first non-cutting stage. According to the maximum discrete time step Δt, calculate the cutting edge movement trajectory, and record the cutting edge discrete point counting variable in the first non-cutting stage as A = A + 1 until the discrete points of the cutting edge leave the first non-cutting stage, that is, Z L ≤W Z ; As the discrete points of the cutting edge leave the first non-cutting stage, that is, Z L ≤W Z , at this time, the discrete points of the cutting edge are in the cutting stage. Record the cutting edge discrete point counting variable in the cutting stage as B = B + 1 until the discrete points of the cutting edge leave the cutting stage, that is, Z L >W Z ; Continuing, the discrete points of the cutting edge leave the cutting stage, that is, Z L >W Z , at this time, the discrete points of the cutting edge are in the second non-cutting stage. Record the cutting edge discrete point counting variable in the second non-cutting stage as C = C + 1 until the ball-end milling cutter rotation period ends; Among them, the initial counting variable values of A in the cutting edge discrete point counting variable in the first non-cutting stage, B in the cutting edge discrete point counting variable in the cutting stage, and C in the cutting edge discrete point counting variable in the second non-cutting stage are all 0; That is, the division of the cutting trajectories and non-cutting trajectories of the discrete points of all N cutting edges is completed.

[0023] S05. Determine the maximum discrete time step Δt of the cutting edge movement trajectory according to the target accuracy of the surface topography prediction output result. The specific expression is: , In the formula, A c is the workpiece grid accuracy, and the value of A c is used to determine the target accuracy reached by the ball-end milling cutter milling surface topography prediction result; θ max is the maximum axial angle of the cutting part of the cutting edge; R is the radius of the ball-end milling cutter.

[0024] S06. Let the time step Δt1 in the first non-cutting stage be AΔt; the time step Δt2 in the cutting stage be Δt; the time step Δt3 in the second non-cutting stage be CΔt; Then, in the rotation period of the continuously cutting ball-end milling cutter, the time step is dynamically selected according to the position relationship between the discrete points of the cutting edge and the workpiece grid, specifically: If the discrete point of the cutting edge is in the first non-cutting stage, the time step Δt1 is adopted; If the discrete point of the cutting edge is in the cutting stage, the time step Δt2 is adopted, otherwise the time step Δt3 in the second non-cutting stage is adopted.

[0025] S07. Initialize the discrete accuracy A of the workpiece's machined surface c , the discrete accuracy Δθ of the cutting edge, the maximum discrete time step Δt of the cutting edge movement trajectory, and construct the discrete point cloud of the machined surface.

[0026] S08. According to the actual size or model of the workpiece, determine the position coordinates of the discrete point cloud in the workpiece coordinate system, and then convert the discrete point cloud into a point cloud height matrix W Z .

[0027] S09. Traverse all discrete points of the cutting edge. Based on the machining program data, calculate the movement trajectory of the discrete points of the cutting edge according to the determined time step of the discrete points of the cutting edge. For the coordinate points in the workpiece reference coordinate system corresponding to the discrete points of the cutting edge in the cutting stage, update the point cloud height matrix W Z of the workpiece's machined surface, that is, obtain the predicted topography of the workpiece's machined surface; Among them, the machining program data includes the tool path file for determining the movement path of the discrete points of the cutting edge and the cutting process parameters of the movement speed of the discrete points of the cutting edge itself.

[0028] S10. Use the profile arithmetic mean deviation R a and the surface arithmetic mean height S a to evaluate the accuracy of the surface topography prediction. The profile arithmetic mean deviation R a and the surface arithmetic mean height S a are respectively: , , In the formula, Z i is the height value of the sampling grid point, Z mid is the height value of the calculation reference plane; m and n are the number of grid parts divided along the X and Y axes in the workpiece coordinate system respectively.

[0029] As Figures 1-9 shown, in order to further illustrate the solution and technical effect of the present invention, the following specific examples are provided: A surface topography prediction method based on a ball-end mill model and a variable step-size iterative algorithm, comprising the following steps: Step 1, based on the cutting edge geometric model, construct a spatial coordinate system required to describe the position and movement of the cutting edge, as Figure 1 shown, including the ball-end mill coordinate system 1, the spindle coordinate system 3, the workpiece reference coordinate system 2, and the workpiece coordinate system 4; wherein, the cutting edge geometric model includes the ball-end mill radius R, the helix angle γ, the number of teeth N, and the true cutting edge during the cutting process is characterized by introducing a normal distribution random quantity function E(σ) for the passivation form of the ball-end mill; Accurately describing the position and movement of the cutting edge is the basis for accurately simulating and predicting the milling process. By establishing the ball-end mill coordinate system 1, the spindle coordinate system 3, the workpiece reference coordinate system 2, and the workpiece coordinate system 4, the relative position relationship between the ball-end mill, the spindle, and the workpiece can be clearly defined, providing a unified and accurate coordinate framework for subsequent kinematic and dynamic analyses; Step 2, describe the movement trajectory of the discrete points of the cutting edge in the spatial coordinate system through multiple three-dimensional homogeneous coordinate transformation matrices, and combine the radial eccentricity distance e, the radial eccentricity angle δ, the rake angle α, the side tilt angle β of the ball-end mill, and the angular velocity ω of the machine tool spindle to construct an equation to achieve the calculation of the movement trajectory of the cutting edge in the workpiece coordinate system; Accurately calculating the movement trajectory of the cutting edge is crucial for predicting the machining effect and optimizing the machining process. Describing the movement trajectory of the discrete points of the cutting edge through multiple three-dimensional homogeneous coordinate transformation matrices takes into account various motion states and attitude parameters of the ball-end mill, and can more realistically reflect the cutting process; Step 3, determine the maximum discrete time step of the cutting edge movement trajectory according to the target accuracy of the surface topography prediction output result; then calculate the division position and duration of the cutting trajectory and the non-cutting trajectory of each discrete point of the cutting edge within any one ball-end mill rotation period during the cutting process, and then traverse each discrete point of the cutting edge to calculate the division position and duration of its cutting trajectory and non-cutting trajectory; While meeting the output accuracy requirements, reasonably control the calculation amount, avoiding the problems of insufficient accuracy or waste of computing resources that may be brought by a fixed step size. By dynamically adjusting the step size, fine calculation is carried out in the cutting stage to ensure accuracy, and a larger step size is adopted in the non-cutting stage to improve efficiency, balancing accuracy and computational efficiency; Step 4, combine the cutting edge discrete point time step and the Z-MAP algorithm to obtain an improved Z-MAP algorithm; generate a three-dimensional surface topography by real-time updating the coordinate information of the workpiece machining surface, and use the profile arithmetic mean deviation R a and the surface arithmetic mean height S a to evaluate the accuracy of the surface topography prediction; Using the profile arithmetic mean deviation R aArithmetical mean height S of dough kneading a Evaluating the simulation accuracy can quantify the degree of fit between the simulation results and the actual measurement results, which helps to optimize the processing technology, predict the surface quality of the workpiece in advance, and improve the accuracy and reliability of the workpiece processing.

[0030] Connecting the above steps 1 to 4 in series, the time step of the discrete points of the cutting edge and the Z-MAP algorithm process are obtained as Figure 3 shown; The following is a verification and explanation with a specific numerical example 1: Specific numerical example 1: It is a plane milling experiment of a ball-end mill. The workpiece material is TC4; the number of teeth of the ball-end mill is 4, the diameter is 8 mm, and the helix angle γ is 20°; the spindle speed n is 4000 r / min, and the feed speed f is 1000 mm / min, as Figure 2 shown, the cutting depth 6, that is, a p is 0.3 mm, the rake angle α is 60°, and the side rake angle β is 0°; after the workpiece is processed, a Bruker NPFLEX three-dimensional optical profiler is used to measure its three-dimensional surface topography. The test lens is 2.5 times, and the reflectivity is 0.1% - 100%; Based on the above selected process parameters, taking the principle of sufficient output of three-dimensional surface topography characteristics, the output size is determined to be 1.2 mm × 1.2 mm × 2.3 mm; based on this target output size, the discrete accuracy A c of the workpiece processing surface is adopted from 50 to A c = 200, a total of 16 accuracy levels, as Figure 2 shown. By determining the cycle start point 5 and the cycle end point 10, the division of the first non-cutting stage 7, the cutting stage 8, and the second non-cutting stage 9 is completed, and the time steps of each stage are calculated respectively, and finally the three-dimensional surface topography calculation output is carried out; To compare the improvement effect of the method provided by the present invention compared with the traditional Z-MAP method in the total simulation time; the comparison of the total simulation time of the two methods is as Figure 4 shown. Under the condition of the same grid accuracy, the Z-MAP method improved based on the variable time step strategy generally improves the simulation efficiency by more than 3 times compared with the traditional Z-MAP method; Specific numerical example 2: According to the process parameters and experimental settings selected in specific numerical example 1, a single-factor experiment with only the side rake angle β changed from 0° to 330° is carried out, and the three-dimensional surface topography calculation output is carried out respectively; The comparison of some output results with the measured results is as Figures 5-8 shown; in Figure 5 are the simulated predicted surface topography and the actual measured surface topography with the side rake angle β = 0° respectively; in Figure 6 are the simulated predicted surface topography and the actual measured surface topography with the side rake angle β = 30° respectively;Figure 7 respectively show the simulated predicted surface topography and the actually measured surface topography with the roll angle β = 60°; in Figure 8 respectively show the simulated predicted surface topography and the actually measured surface topography with the roll angle β = 90°.

[0031] To quantitatively evaluate the accuracy of the method provided by the present invention, the profile arithmetic mean deviation R a and the surface arithmetic mean height S a are calculated for each group; the comparison of the calculation results with the measured values is as Figure 9 shown, further indicating that the improved Z-MAP method based on the variable time step strategy not only achieves a significant improvement in simulation efficiency but also ensures the prediction accuracy.

[0032] In summary, the present invention has significant improvements in both simulation efficiency and accuracy. The algorithm has strong universality and supports the expansion of complex working conditions. From specific example 1, it can be seen that under the condition of the same grid accuracy, the simulation efficiency of this method is generally more than 3 times higher than that of the traditional Z-MAP method; specific example 2 shows that this method not only achieves a significant improvement in simulation efficiency but also ensures the prediction accuracy by calculating the profile arithmetic mean deviation R a and the surface arithmetic mean height S a and comparing with the measured values, providing strong support for the surface topography prediction and process parameter optimization of complex parts machining in aerospace.

[0033] The above are only the preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. A surface topography prediction method based on a ball-end cutter model and a variable step-size iterative algorithm, characterized in that It includes the following steps: Step 1: In the cutting edge geometric model of the structure of the part of the ball-end mill involved in cutting, establish a spatial coordinate system for describing the position of the cutting edge of the ball-end mill. Use the spatial coordinate system to determine the positions of the discrete points of the cutting edge of all the cutting edges of the ball-end mill, that is, obtain the coordinate expressions of the discrete points of the cutting edge. At the same time, use the normal distribution random variable function in the expressions of the discrete points of the cutting edge to characterize the passivation form of the ball-end mill during the cutting process of the actual cutting edge; Among them, the spatial coordinate system includes the ball-end mill coordinate system, the machine tool spindle coordinate system, the workpiece reference coordinate system, and the workpiece coordinate system; Step 2: Based on the expressions of the discrete points of the cutting edge, through multiple calculations of the three-dimensional homogeneous coordinate transformation matrix, describe the movement trajectory of the discrete points of the cutting edge in the spatial coordinate system. Thus, in the workpiece reference coordinate system, establish a cutting edge movement trajectory calculation equation for describing the cutting edge movement trajectory; Step 3: Divide the cutting edge movement trajectory corresponding to the discrete points of the cutting edge into a non-cutting trajectory in the non-cutting stage and a cutting trajectory in the cutting stage according to the cutting state, and calculate the time step of the discrete points of the cutting edge in the non-cutting stage and the cutting stage; Step 4: Combine the time step of the discrete points of the cutting edge with the Z-MAP algorithm, and based on the milling process data used to describe the movement state of the ball-end mill, iteratively update to obtain the height information of the workpiece machining surface after the end of each cutting edge movement trajectory, that is, obtain the three-dimensional surface topography of the workpiece machining surface; Among them, the machining process data includes the tool path file for determining the movement path of the discrete points of the cutting edge and the cutting process parameters of the movement speed of the discrete points of the cutting edge themselves.

2. The surface topography prediction method based on the ball-end mill model and the variable step size iteration algorithm according to claim 1, wherein The spatial coordinate system described in Step 1 specifically includes: Taking the center of the ball nose cutter as the origin O, and the axis of the ball nose cutter as the Z T axis, taking the direction of the connecting line from the tip of any cutting edge to the Z T axis as the X T axis, thus, determining the Y T axis according to the right-hand coordinate system rule, and establishing the coordinate system of the ball nose cutter; the coordinate system of the ball nose cutter can rotate around the center of rotation of the machine tool spindle; X-axis and Y-axis directions based on the coordinate system of the ball-end milling cutter, with the machine tool spindle as the Z-axis, a machine tool spindle coordinate system is established; T axis and Y T axis direction, taking the machine tool spindle as the Z S axis, the established machine tool spindle coordinate system; Based on the origin O of the coordinate axis of the ball-end milling cutter, with the cutting feed direction as the X G axis, the feed direction of the ball-end milling cutter as the Y G axis, and the Z G axis is determined according to the right-hand coordinate system rule, the workpiece reference coordinate system is established; A workpiece coordinate system established with any vertex on the bottom surface of the workpiece as the origin based on the axis direction of the workpiece reference coordinate system; Furthermore, use the spatial coordinate system to determine the positions of the discrete points of the cutting edge of all the cutting edges of the ball-end mill. For any cutting edge j among the N cutting edges in the cutting edge geometric model, the expression of the discrete point L of any cutting edge j is: ， Wherein, R is the radius of the ball-end milling cutter, γ is the helix angle, N is the total number of cutting edges of the teeth, E(σ) is the function of the normal distribution random variable, θ is the line connecting the discrete point of the cutting edge and the center of the ball of the ball-end milling cutter, and is the included angle between the projection of the line on the XOY plane and the X axis in the coordinate system of the ball-end milling cutter. T OY T plane and the projection on the X T axis.

3. The surface topography prediction method based on the ball-end mill model and the variable step size iteration algorithm according to claim 1, characterized in that The cutting edge movement trajectory calculation equation established in Step 2 is, for any cutting edge j among the N cutting edges in the cutting edge geometric model, the movement trajectory calculation equation of any discrete point L of the cutting edge j, specifically: ， Among them, θ is the line connecting the discrete point of the cutting edge and the center of the ball end mill, t is the cutting edge movement time, (x WL , y WL , z WL ) is the position coordinate of the discrete point L in the workpiece coordinate system; M1 to M5 are respectively the rotation, eccentric translation, spindle rotation, pose adjustment, and feed transformation matrices of the ball-end mill, and the specific expressions are: The rotation matrix M1 is: ， The eccentric translation matrix M2 is: ， Among them, δ is the radial eccentric angle of the ball-end mill; The spindle rotation matrix M3 is: ， Among them, φ0 is the initial phase angle of the cutting edge, and ω is the angular velocity of the machine tool spindle; The pose adjustment matrix M4 is: ， Among them, α is the rake angle of the ball-end mill, and β is the side inclination angle of the ball-end mill; The feed transformation matrix M5 is: ， Among them, f p is the machining row spacing of the ball end mill, f v is the feed rate of the ball end mill, i is any cutting row among N cutting edges, and (x0, y0, z0) is the initial position coordinate of the ball center of the ball end mill in the workpiece coordinate system.

4. The surface topography prediction method based on the ball-end mill model and variable-step iterative algorithm according to claim 1, characterized in that The specific content of Step 3 is: During any rotation period of the ball-end mill during the cutting process, for any cutting edge j among the N cutting edges in the cutting edge geometric model, there is: Since the discrete points of the cutting edge at the initial position are all above the workpiece grid height W Z That is: Z L >W Z , Z L is the height of any discrete point L of any cutting edge j at time t. At this time, the discrete points of the cutting edge are in the first non-cutting stage. According to the maximum discrete time step Δt, the cutting edge movement trajectory is calculated, and the cutting edge discrete point counting variable in the first non-cutting stage is recorded as A = A + 1 until the discrete points of the cutting edge leave the first non-cutting stage, that is, Z L ≤W Z ; As the discrete points of the cutting edge leave the first non-cutting stage, i.e., Z L ≤W Z , at this time, the discrete points of the cutting edge are in the cutting stage, and the cutting edge discrete point counting variable in the cutting stage is recorded as B = B + 1 until the discrete points of the cutting edge leave the cutting stage, i.e., Z L >W Z ; Continuing, the discrete points of the cutting edge leave the cutting stage, i.e., Z appears again L > W Z , at this time, the discrete points of the cutting edge are in the second non-cutting stage, and record the cutting edge discrete point counting variable of the second non-cutting stage as C = C + 1 until the end of the ball-end mill rotation period; Among them, the initial count variable values of A in the cutting edge discrete point count variable of the first non-cutting stage, B in the cutting edge discrete point count variable of the cutting stage, and C in the cutting edge discrete point count variable of the second non-cutting stage are all 0; That is, the division of the cutting edge discrete point cutting trajectory and the non-cutting trajectory of all N cutting edges is completed; Furthermore, let the time step Δt1 of the first non-cutting stage = AΔt; the time step Δt2 of the cutting stage = Δt; the time step Δt3 of the second non-cutting stage = CΔt; Then, in the rotation period of the continuously cutting ball-end milling cutter, the time step is dynamically selected according to the position relationship between the cutting edge discrete point and the workpiece grid, specifically: If the cutting edge discrete point is in the first non-cutting stage, the time step Δt1 is adopted; If the discrete points of the cutting edge are in the cutting stage, the time step Δt2 is adopted; otherwise, the time step Δt of the second non-cutting stage is adopted. 3； Among them, Δt is the maximum discrete time step of the cutting edge movement trajectory determined according to the target accuracy of the surface topography prediction output result, and the specific expression is: ， Where, A c is the workpiece grid accuracy, and the value of A c is used to determine the target accuracy achieved by the prediction result of the milling surface topography of the ball-end mill; θ max is the maximum axial angle of the cutting edge part participating in cutting; R is the radius of the ball-end mill.

5. The surface topography prediction method based on the ball-end mill model and the variable step size iteration algorithm according to claim 1, wherein The specific content of step four is: First, initialize the discrete precision A of the workpiece's machining surface c , the discrete precision Δθ of the cutting edge, the maximum discrete time step Δt of the cutting edge's motion trajectory, and construct the discrete point cloud of the machining surface; Then, according to the actual size or model of the workpiece, determine the position coordinates of the discrete point cloud in the workpiece coordinate system, and then convert the discrete point cloud into a point cloud height matrix W Z ; Finally, traverse all the discrete points of the cutting edge. Based on the tool path file and the cutting process parameters, calculate the motion trajectory of the discrete points of the cutting edge according to the determined time step of the discrete points of the cutting edge, and for the coordinate points in the workpiece reference coordinate system corresponding to the discrete points of the cutting edge in the cutting stage, update the point cloud height matrix W Z of the workpiece machining surface, that is, obtain the predicted workpiece machining surface topography.

6. The surface topography prediction method based on the ball-end mill model and the variable step size iteration algorithm according to any one of claims 1-5, characterized in that It also includes steps of using the arithmetic mean deviation of the profile R a and the arithmetic mean height of the surface S a to evaluate the accuracy of the surface topography prediction.

7. The surface topography prediction method based on the ball-end mill model and variable-step iterative algorithm according to claim 6, characterized in that, The profile arithmetic mean deviation R a and the surface arithmetic mean height S a are respectively as follows: ， ， where Z i is the height value of the sampling grid point, and Z mid is the height value of the calculation reference plane; m and n are the numbers of grid divisions along the X and Y axes in the workpiece coordinate system, respectively.

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