Rapid extraction method for tool-workpiece time-varying geometric meshing state

Abstract

The invention relates to a rapid extraction method for a tool-workpiece time-varying geometric meshing state, which comprises the following steps of: firstly, rapidly extracting a tool-workpiece geometric meshing boundary, and reconstructing and replacing a local cut material of a workpiece by using a past adjacent tool at a current tool location point; a tool-workpiece geometric meshing entity is quickly obtained through Boolean operation among a current tool entity, a workpiece blank entity and a reconstructed workpiece material, and a meshing window state function is efficiently calculated through image processing; then cutter-workpiece relative feeding motion is decomposed into three components of translation along a cutter shaft, translation perpendicular to the cutter shaft and rotation around the shaft, the full-cutting-width instantaneous cutting layer thickness of decomposition motion is solved by constructing a geometric coordination relation, and then an explicit calculation formula of the instantaneous cutting layer thickness in the actual meshing state is obtained; the method has first-order calculation time and space complexity, the calculation precision consistent with that of a traditional method can be obtained, and stable and efficient characterization of the meshing state under the complex working condition is achieved.

Description

Technical Field

[0001] This invention relates to the field of machine tool processing technology, and more specifically to a method for rapidly extracting the time-varying geometric meshing state of a tool and a workpiece. Background Technology

[0002] The tool-workpiece geometric meshing region, as the instantaneous local contact area between the tool's rotating surface and the workpiece during machining, directly defines the range of their interaction. It is a core parameter for milling force calculation, milling stability prediction, and machining quality assessment, playing a crucial role in measuring machining effects and optimizing processes. However, under complex working conditions such as curved machining paths, multi-axis posture changes, and time-varying part allowances, this meshing state exhibits significant time-varying characteristics, posing numerous challenges to rapidly and accurately acquiring its geometric information. While existing commercial CAM software (such as Catia, NX, and Vericut) can achieve virtual simulation of the cutting process, toolpath verification, and interference and collision prevention, shortening machine tool operation and verification time, none of them can output the tool-workpiece geometric meshing entity model corresponding to each tool position point, making it difficult to support subsequent calculations of key parameters such as milling forces. Even though NCSIMUL software supports real-time sensor data-driven simulation, its core functions still require expensive access, and the simulation calculation kernel programs of all commercial software are closed to users, making it infeasible to directly use existing commercial software to solve the problem of characterizing the meshing region under complex working conditions. The academic field has proposed three types of tool-workpiece geometric meshing state characterization schemes: analytical method, discrete method and solid model method. However, they have inherent limitations such as low computational efficiency, high storage requirements and poor stability, and none of them can meet the requirements of efficient and stable characterization under complex working conditions. Summary of the Invention

[0003] To overcome the shortcomings of the prior art, the present invention aims to provide a rapid extraction method for the time-varying geometric meshing state of the tool-workpiece. It achieves efficient calculation of the tool-workpiece geometric meshing boundary through image processing, establishes a calculation model for the instantaneous cutting layer thickness under the actual meshing conditions of the tool in multi-axis milling process, has first-order computation time and space complexity, and can obtain the same computational accuracy as traditional methods, thus achieving stable and efficient characterization of the meshing state under complex working conditions.

[0004] To achieve the above objectives, the technical solution adopted by the present invention is as follows: A rapid method for extracting the time-varying geometric meshing state of a tool and workpiece includes the following steps: Step 1) Quickly extract the tool-workpiece geometric meshing boundary, reconstruct and replace the locally cut material of the workpiece with the past neighboring tools at the current tool position, quickly obtain the tool-workpiece geometric meshing entity through Boolean operations between the current tool entity, the workpiece blank entity and the reconstructed workpiece material, and efficiently calculate the meshing window state function through image processing. Step 2) Decompose the relative feed motion between the tool and the workpiece into three components: translation along the tool axis, translation perpendicular to the tool axis, and rotation around the axis. Solve the instantaneous cutting layer thickness of the full-cut width by constructing a geometric coordination relationship, and then obtain the explicit calculation formula of the instantaneous cutting layer thickness under the actual meshing state.

[0005] The specific process of step 1) is as follows: 1.1) Search for previously adjacent cut sites: 1) G-code tool position densification preprocessing: After obtaining the G-code through CNC programming, the workpiece machining coordinate system X is obtained through preprocessing. w Y w Z w The tool tip position (XYZ), tool axis vector (IJK), machine tool rotation axis angles (ABC), and G code line number are as follows: N The data is used to preprocess the G-code through interpolation densification; based on discrete line segments on the toolpath, linear interpolation is performed on the position points on the line segments between adjacent tool points, and spherical interpolation is performed on the tool axis vector. The specific calculation formulas are as follows: In the formula: ——No. N One incision site; ——No. N Each tool axis vector; — Interpolation point number; —Total number of interpolation points; — Densification of interpolation points; — Tool axis vector and The angle between them; — Tool axis vector and The angle between them; — Tool axis vector and The angle between them; 2) Search for previously adjacent cut points: Two modes are proposed to search for previously adjacent cut points at the current tool position; the first is an offline calculation mode, which uses G-code obtained from CNC programming to directly input into the calculation program for line-by-line calculation; when the calculation reaches the... N Cutting point At that time, the densification knife site This constitutes the set of previously cut sites; The second method is online calculation mode. First, the G-code is input into the machine tool's CNC system to drive the feed of each axis. Then, the interpolation instruction data, including the current execution line number of the G-code, is read in real time through the CNC system's OPC UA function. NMachine tool axis position and speed, spindle speed and magnification, G54 coordinate zero offset; For the current tool position Its previously nearby cut sites are further divided into three categories: ① Absolutely nearest points This point is in the densification knife site sequence. Among them; ② Dynamic nearest neighbors This point exists in the currently executing... N Dynamic interpolation sequence of G-code Middle; ③ Particularly nearby point This point is located before the current tool position. At a distance; the three types of nearest neighbors must satisfy the following relationship during the search calculation: In the formula: — Tool point for static interpolation; —The tool position point for dynamic interpolation; —Nominal radius of the cutting tool; —Feed per tooth of the cutting tool; Furthermore, the previously adjacent cut sites can be obtained. The sum of the above three sets is shown in the following formula; when summing, for the division... The neighbor sequence outside is sparsely sampled to obtain the sparse neighbor sequence. 1.2) Local Fast Boolean Operations for Tool-Workpiece Geometric Interlocking Entity: The tool-workpiece geometric interlocking entity is quickly obtained by replacing the partially cut material of the workpiece with the reconstructed entity of the previously cut neighboring tools at the current tool position point. Local Boolean operations are performed between the current tool entity, the workpiece blank entity, and the reconstructed workpiece solid entity to obtain the tool-workpiece geometric interlocking entity. The Boolean operation kernel program is a secondary development on the commercial software NX10.0, using a development mode combining NXOpen C++ and NXOpen C (UFUN). The calculation steps are as follows: (1) Design the CAD model of the tool and workpiece blank, use Catia software to program and obtain G code, obtain densified G code through preprocessing, and read it into the secondary development program using the file interface; (2) Set the workpiece machining coordinate system in the program. And obtain the current data from the CNC system interpolation instruction data. tool position and tool axis vector Reconstruct the relative spatial orientation of the tool and workpiece; (3) Search for the previous neighboring cut points at the current tool position. The Parasolid PK kernel function is used to reconstruct the set of previously cut tool entities based on the tool position point and tool axis vector. This replaces the previously cut material in parts of the workpiece. (4) Determine the physical state of the tool at this time With blank solid Intersection status: If they intersect, perform a Boolean intersection operation to obtain the initial meshing body. If they do not intersect, proceed to the next time step. (5) The initial meshing body Material that has been previously cut in the workpiece Perform a Boolean subtraction operation to obtain the tool-workpiece geometric meshing entity at the current tool position. ,Right now: 1.3) Image processing and extraction of the tool-workpiece geometric meshing boundary: 1) Two-dimensional image processing of the meshing entity: For the tool-workpiece geometric meshing entity obtained by Boolean operations, the entity in the tool feed coordinate system is obtained through secondary development of API functions. The bounding box parameters below, i.e., the minimum values ​​of the solid in each axial direction. , and and length , and Rotate and project the solids onto the orthogonal plane. and And obtain its image screenshot in a two-dimensional plane. and In the screenshot It also includes meshing information for both the side and end cutting edges of the tool. It includes the meshing information of the side blades; using OpenCV library functions, the projected screenshot is converted into a binary grayscale image, and pixel parameters along the horizontal and vertical directions are extracted. and Convert the grayscale image into a 0-1 binary matrix, and name them respectively. and In this system, gray pixels are represented by 1 and white pixels by 0; 2) Calculation of the window function for the cutting edge meshing state: (1) The meshing state window function corresponding to the planar image: In the matrix, first along The direction is used to find the index of a non-zero pixel, as shown in the following formula; In the formula: —A function for summing terms; — Lookup function; Calculate along The actual distance represented by each pixel in the direction and each pixel in The actual coordinates corresponding to the direction ,as follows: Similarly, for Direction, first along The direction is used to find the index of a non-zero pixel, as shown in the following formula; Calculate along The actual distance represented by each pixel in the direction and each pixel in The actual coordinates corresponding to the direction ,as follows: Establish the coordinates of discrete points along the circumferential direction of the tool's cutting edge line as follows: In the formula: —Spindle speed; — Spindle rotation cycle; —Number of discrete times per cycle; —The sequence of cutting radii of the tool end face, where Number the cutting points of the end blade; —The sequence of cutting radii of the tool side edge, where Number the cutting points on the side edge; The window function corresponding to the meshing entity is obtained through two-dimensional linear interpolation. ,for: In the formula: —Two-dimensional interpolation function; ; ——No. k Discrete moments; After interpolation, it is converted into a 0-1 binary matrix, and the end edge and side edge parts are separated and stored separately. and In the matrix, as shown in the following equation; In the formula: —The floor function; The window function for calculating the meshing state of the tool tip and side cutting edges is as follows; In the formula: —Cut tooth number; —Actual cutting radius of the tool; —Thickness of the cutting unit on the side edge of the tool; —Helix angle of the blade teeth; —Interdental angle; (2) The meshing state window function corresponding to the planar image: In the matrix, first along The direction is used to find the index of a non-zero pixel, as shown in the following formula; Calculate along The actual distance represented by each pixel in the direction And the actual coordinates of each pixel in that direction. ,as follows: Similarly, for Direction, first along The direction is used to find the index of a non-zero pixel, as shown in the following formula; Calculate along The actual distance represented by each pixel in the direction And the actual coordinates of each pixel in that direction. ,as follows: Establish the coordinates of discrete points along the circumferential direction of the tool's cutting edge line as follows: The window function corresponding to the meshing entity is obtained through two-dimensional linear interpolation. ,for: Then, the above window function is extended to the entire circumference of the tool, that is: After interpolation, it is then converted into a 0-1 binary matrix, as shown in the following formula; Considering the hysteresis angle variation of different cutting teeth and different cutting point heights, obtain The meshing state window function corresponding to the plane is shown in the following formula; With After integration, the meshing state window function of the tool side edge is finally obtained; The specific process of step 2) is as follows: 2.1) Tool geometry parameters under the influence of eccentricity: 1) Parametric definition of tool eccentricity: Tool eccentricity is the deviation of the tool's rotation axis, including the deviation amount. , deflection angle Tool geometry axis deviation, offset including offset amount Offset angle Yaw includes yaw amount and yaw angle ; Tool end-edge eccentricity parameters It is characterized by 3 eccentric states and 7 independent parameters; 2) Tool geometry parameters under the influence of eccentricity: using D , R z , R r , R c , , and H c The geometric profile is constructed using 7 independent parameters; any cutting point on the cutting edge The position is determined by the corresponding axial height radial distance and radial hysteresis angle The determination, in which the radial distance Defined as the distance between this point and the tool axis, the radial hysteresis angle. Defined as the angle between the line connecting this point and the tip of the blade and the tangent to the helix at a height of 0 on the axis. Projection values ​​within the plane; The cutting edge of the cutting tooth is discretized into several cutting units of minute thickness at equal intervals along the axial direction; at this time, the first... i On the first blade tooth j Layer cutting unit cutting point The coordinates in the tool geometry coordinate system are represented as follows: In the formula: —The cutting element number obtained after discretizing the cutting edge and side edge along the cutting edge; When the tool only has a geometric axis deviation, the cutting point of the cutting edge The position in the machine tool coordinate system, that is, the tool rotation coordinate system, is: In the formula: —Tool overhang length; When the spindle rotates t Cutting point of the cutting edge at all times In the tool rotation coordinate system The following position is: When the tool has both geometric axis misalignment and rotation axis misalignment, the cutting point of the cutting edge... In the machine tool coordinate system The following position is: In the formula: —Length of axis deviation; At this point, the cutting point of the cutting edge must be determined under the influence of tool eccentricity. The actual cutting radius is: Furthermore, the height difference between adjacent bottom edges of the tool tip in the machine tool coordinate system is: Due to tool eccentricity, any cutting point on the side edge In the tool rotation coordinate system The coordinates are: Any cutting point on the end blade In the tool rotation coordinate system The coordinates below are: 2.2) Equivalent decomposition of tool-workpiece relative feed motion: In the workpiece machining coordinate system Next, for the first The cutter tip is located at the cutter position when the cutter tip is... Exercise During the process, the feed rate at the tool tip is (This speed is different from the nominal speed in G-code) (value), the feed motion time between the two points is ; During this process, the tool axis vector is changed from... Change to Accordingly, the B and C axes of the machine tool will... Rotate to The rotational speeds are respectively and ; After rotation along the BC axis, the tool axis vector is: In the formula: ; ; ; At this point, the local meshing coordinate system is defined as follows: Define the feed coordinate system as follows: Further analysis shows that the tool feed motion is actually along... Directional translational motion and rotation The linear superposition of the two point rotational motions; further, along... Directional translational motion is decomposed into along Translational superposition of directions at the upper edge For translational movements in a direction, a reference point needs to be defined. Its coordinates are: In the formula: —— and The angle between them; At this time, along Directional translational motion and along The velocity vectors for the directional translational motion are as follows: In addition, around The rotation of a point can actually be regarded as the resultant rotational motion around the BC axis, that is, around the BC axis. The axis rotates. Here, The calculation formula is as follows: Finally, around The rotation angle and angular velocity of the axis are respectively: 2.3) Calculation of instantaneous cutting layer thickness at full width of decomposed motion: 1) Cutting edge Linear feed motion in the direction of: (1) Tool side edge: the edge of the tool In linear feed motion, the tool rotates around its axis at a speed of [missing information]. n Rotate, and at feed rate along Directional feed; t= At time 0, set the first cutting edgez The tangent at =0 and Axis coincidence; at any time t , No. i On the first blade tooth Layer cutting unit The rotation angle of the point is: Take any cross-section on the trajectory surface of the cycloidal motion of the cutting edge of the tool, and the cutting point on the side edge is then determined. The cutting layer thickness is equal to The line segment at the intersection of the line connecting the point and the center of rotation of the tool and the geometric solid of the workpiece, i.e. ,in The point is formed by the first few cutting teeth. At some point in the past The residue left on the machined surface after cutting; time, Point in the feed coordinate system The coordinates below are: at the same time, point Constantly feeding coordinate system The following is represented as: Since the tool feeds along a straight line in the feed coordinate system at a speed of... Since it is a uniform rigid body motion, we have: By combining the above three equations (equation (41), (42), and (43)), the lag time at the side cutting point can be obtained. and instantaneous cutting layer thickness for: (2) Tool end edge: cutting point of the end edge Compared with previous cutting points Given the height difference, we have: 2) Cutting edge Linear feed motion in the direction of: (1) Tool side edge: the edge of the tool During the directional movement, the tool rotates around its axis at a speed of [missing information]. n Rotate and feed rate along Directional feed; take any cross section on the cycloidal motion trajectory surface of the cutting edge, at which point the side cutting edge is located. The cutting layer thickness is equal to The line segment at the intersection of the line connecting the point and the center of rotation of the tool and the geometric solid of the workpiece, i.e. ,in The point is formed by the first few cutting teeth. At some point in the past The residue left on the machined surface after cutting; time, Point in the feed coordinate system The coordinates below are: at the same time, point Constantly feeding coordinate system The coordinates below are also represented as: Here, the feed rate is defined as the rate at which the tool moves downward along the tool axis. A positive value indicates a positive value, and a negative value indicates a negative value; because the tool follows... The direction of the feed rate is Since it is a uniform rigid body motion, the following relationship exists: By combining the above three equations (equation (48), equation (47), and equation (46)), the lag time at the side cutting point can be obtained. and instantaneous cutting layer thickness for: In the formula: It is a real number; (2) Tool end edge: Consider the cutting point on the end edge Compared with previous cutting points The height difference between them gives: 3) Tool wrap Rotary feed motion in direction: (1) Tool side edge: the tool wraps around During the directional movement, the tool rotates around its axis at a speed of [missing information]. n Rotate, and at feed rate Around Directional feed; when the cutting edge curve is in three-dimensional cycloidal motion, take any cross-section of the motion trajectory surface, at this time That is, the side cutting point The thickness of the cutting layer; exist time, Point in coordinate system The coordinates below are: At this time, because the tool axis is at the feed rate Around Direction rotation, Represented as: also, Represented as: Furthermore, vector Represented as: exist time, Point in coordinate system The following is also represented as: Due to the rigid body rotational motion of the tool axis, we get: Using a simplified method, the result is first obtained through numerical calculation. and order Finally, the hysteresis time of the side cutting point is obtained. and instantaneous cutting layer thickness for: (2) Tool end edge: Consider the cutting point on the end edge Compared with previous cutting points The height difference between them gives: 2.4) Calculation of instantaneous cutting layer thickness under actual meshing of the composite motion: the tool along... and The starting point of the instantaneous cutting layer thickness generated by directional translation is The axis, while the tool rotates around The starting point of the instantaneous cutting layer thickness generated by directional rotation is Therefore, all three elements (axis, yoke, and tool axis) need to be aligned to the same starting point in time, and the tool axis needs to be rotated around... The instantaneous cutting layer thickness in the direction is time-shifted as follows: As shown in the following formula, the instantaneous cutting layer thicknesses of the three motion components are superimposed; The instantaneous cutting layer thickness is the maximum value obtained by comparing the minimum of all cutting thicknesses with zero, as shown in the following formula: Finally, the above cutting thickness and meshing state window function are discretized to obtain the instantaneous cutting layer thickness under the actual meshing state, as shown in the following formula: Compared with existing methods, the present invention has the following advantages: (1) This invention proposes a fast calculation method for tool-workpiece geometric meshing entity based on local Boolean operation. The previously cut tools at the current tool position point are used to reconstruct the locally cut material of the workpiece. The geometric meshing entity is obtained quickly through local Boolean operation between the current tool entity, the blank entity and the reconstructed workpiece material. The efficient calculation of the tool-workpiece geometric meshing boundary is achieved through image processing.

[0006] (2) This invention takes into account the actual three-dimensional cycloidal motion of the cutting edge of the tool and the eccentric state of the tool. The tool feed motion is decomposed into three motion components: translation along the tool axis, translation perpendicular to the tool axis, and rotation around the axis. By constructing a geometric coordination relationship, the calculation formula for the instantaneous cutting layer thickness at full cutting width is derived. Combined with the tool tooth meshing state window function, a calculation model for the instantaneous cutting layer thickness under the actual meshing conditions of the tool in the multi-axis milling process is established.

[0007] (3) The invention was verified in a five-axis milling case. The results showed that the invention has first-order computation time and space complexity, which is better than the traditional method (at least second-order complexity ≥ O(N2)), and can achieve the same computational accuracy as the traditional method. Attached Figure Description

[0008] Figure 1 This is a schematic diagram of the G-code tool point densification preprocessing in the example.

[0009] Figure 2 This is a schematic diagram illustrating the calculation of the search for previously adjacent cut sites in an embodiment.

[0010] Figure 3 This is a schematic diagram of local fast Boolean operation for the tool-workpiece geometric meshing in an embodiment.

[0011] Figure 4 The result is the extraction result of the tool side edge meshing state corresponding to the planar image of the example.

[0012] Figure 5 This example demonstrates the parameterized definition of the tool profile geometry.

[0013] Figure 6 This example demonstrates the characterization and decomposition of the tool feed motion in the workpiece machining coordinate system.

[0014] Figure 7 As an example, the cutting tool along The instantaneous cutting layer thickness during directional translational motion.

[0015] Figure 8 The instantaneous cutting layer thickness is shown in the example for synthesizing motion.

[0016] Figure 9 This example compares the accuracy of tool-workpiece geometry meshing extraction calculations. Detailed Implementation

[0017] The present invention will now be described in detail with reference to the embodiments and accompanying drawings.

[0018] A rapid method for extracting the time-varying geometric meshing state of a tool and workpiece includes the following steps: Step 1) Perform rapid extraction of the tool-workpiece geometric meshing boundary. Reconstruct and replace the locally cut material on the workpiece using the past neighboring tools at the current tool position. Quickly obtain the tool-workpiece geometric meshing entity through Boolean operations between the current tool entity, the workpiece blank entity, and the reconstructed workpiece material. Then, efficiently calculate the meshing window state function through image processing. The specific process is as follows: 1.1) Search for previously adjacent cut sites: 1) G-code tool point densification preprocessing: such as Figure 1 As shown, after obtaining the G-code through CNC programming, the X coordinate system of the workpiece machining can be obtained through preprocessing. w Y w Z w The following parameters are used to determine the tool tip position (XYZ), tool axis vector (IJK), machine tool rotation axis angles (ABC), and G-code line number. N To improve the accuracy of subsequent searches for adjacent cut points, the G-code is preprocessed with interpolation densification. Based on discrete line segments on the toolpath, linear interpolation is performed on the position points on the line segments between adjacent cut point locations, and spherical interpolation is performed on the tool axis vector. The specific calculation formulas are as follows: In the formula: ——No. N One incision site; ——No. N Each tool axis vector; — Interpolation point number; —Total number of interpolation points; — Densification of interpolation points; — Tool axis vector and The angle between them; — Tool axis vector and The angle between them; — Tool axis vector and The angle between them; 2) Search for previously adjacent cut sites: such as Figure 2 As shown, two modes are proposed to search for the previous neighboring cut points at the current tool position; the first is the offline calculation mode, in which the G code obtained from CNC programming is directly input into the calculation program for line-by-line calculation; when the calculation reaches the line... N Cutting point At that time, the densification knife site This constitutes a set of previously cut tool locations, and this model is suitable for offline calculations before machine tool processing; The second method is the online calculation mode. In this mode, the G-code is first input into the machine tool's CNC system to drive the feed of each axis. Then, the interpolation instruction data (including the current execution line number of the G-code) is read in real time through the CNC system's OPC UA function. N (Machine tool axis positions and speeds, spindle speed and magnification, G54 coordinate zero offset, etc.). The tool position read at the current moment. It is often a position in the middle of the line segment adjacent to the original tool position, and it is very likely that it does not coincide with any tool position in the G code.

[0019] For the current tool position Its previously adjacent cut sites can be further divided into three categories: ① Absolutely nearest points This point is in the densification knife site sequence. Middle; ② Dynamic nearest neighbor points This point is currently executing the [number]th [stage]. N Dynamic interpolation sequence of G-code Middle; ③ Particularly nearby point This point is before the current tool position. At a distance; the three types of nearest neighbors must satisfy the following relationship during the search calculation: In the formula: — Tool point for static interpolation; —The tool position point for dynamic interpolation; —Nominal radius of the cutting tool; —This article suggests that its value should be the feed per tooth of the tool; Furthermore, the previously adjacent cut sites can be obtained. The sum of the above three sets is shown in the following formula; during the summation process, to avoid the excessive number of neighboring points affecting the speed of subsequent Boolean operations, except for special points... The neighbor sequence outside the sequence is sparsely sampled to obtain a sparse neighbor sequence; after this sparsification... The maximum quantity is calculated as N c The value ranges from 60 to 120; 1.2) Local fast Boolean operations for the tool-workpiece geometric meshing entity: such as Figure 3 As shown, the key to the local Boolean operation method proposed in this invention lies in replacing the locally cut material of the workpiece with the reconstructed entity of the previously neighboring cut tools at the current tool position. The tool-workpiece geometric meshing entity is quickly obtained through local Boolean operations among the current tool entity, the workpiece blank entity, and the reconstructed workpiece entity. This calculation strategy does not pursue gradual global updates of the workpiece model, thus avoiding the problem of explosive growth in computational load in traditional methods. The Boolean operation kernel program is developed on the commercial software NX10.0, using a development model combining NXOpen C++ and NXOpen C (UFUN). The main calculation steps are as follows: (1) Design the CAD model of the tool and workpiece blank, use Catia software to program and obtain G code, obtain densified G code through preprocessing, and read it into the secondary development program using the file interface; (2) Set the workpiece machining coordinate system in the program. And obtain the current data from the CNC system interpolation instruction data. tool position and tool axis vector Reconstruct the relative spatial orientation of the tool and workpiece; (3) Search for the previous neighboring cut points at the current tool position. The Parasolid PK kernel function is used to reconstruct the set of previously cut tool entities based on the tool position point and tool axis vector. This replaces the previously cut material in parts of the workpiece. (4) Determine the physical state of the tool at this time With blank solid Intersection status: If they intersect, perform a Boolean intersection operation to obtain the initial meshing body. If they do not intersect, proceed to the next time step. (5) The initial meshing body Material that has been previously cut in the workpiece By performing a Boolean subtraction operation, the tool-workpiece geometric engagement entity at the current tool position can be obtained. ,Right now: 1.3) Image processing and extraction of the tool-workpiece geometric meshing boundary: 1) Two-dimensional image processing of meshing entities: For the tool-workpiece geometric meshing entity obtained by Boolean operations, the entity in the tool feed coordinate system can be obtained through secondary development of API functions. The bounding box parameters below, i.e., the minimum values ​​of the solid in each axial direction. , and and length , and The entity is rotated and projected onto the orthogonal planes respectively. and And obtain its image screenshot in a two-dimensional plane. and In the screenshot, It also includes meshing information for both the side and end cutting edges of the tool. It includes the meshing information of the side blades. Using OpenCV library functions, the projected image is converted into a binary grayscale image, and pixel parameters along the horizontal and vertical directions are extracted. and Convert the grayscale image into a 0-1 binary matrix, and name them as follows: and In this matrix, gray pixels are represented by 1 and white pixels by 0; thus, the tool-workpiece meshing entity can be replaced by a binary matrix for calculating the meshing boundary. 2) Calculation of the window function for the cutting edge meshing state: (1) The meshing state window function corresponding to the planar image: In the matrix, first along The direction is used to find the index of a non-zero pixel, as shown in the following formula; In the formula: —A function for summing terms; — Lookup function; Then calculate along The actual distance represented by each pixel in the direction And the actual coordinates of each pixel in that direction. ,as follows: Similarly, for Direction, first along The direction is used to find the index of a non-zero pixel, as shown in the following formula; Then calculate along The actual distance represented by each pixel in the direction And the actual coordinates of each pixel in that direction. ,as follows: Furthermore, the coordinates of discrete points along the circumferential direction of the tool's cutting edge are established as follows: In the formula: —Spindle speed; — Spindle rotation cycle; —Number of discrete times per cycle; —The sequence of cutting radii of the tool end face, where Number the cutting points of the end blade; —The sequence of cutting radii of the tool side edge, where Number the cutting points on the side edge; The window function corresponding to the meshing entity is obtained through two-dimensional linear interpolation. ,for: In the formula: —Two-dimensional interpolation function; ; ——No. k Discrete moments; After interpolation, it is converted into a 0-1 binary matrix, and the end edge and side edge parts are separated and stored separately. and In the matrix, as shown in the following equation: In the formula: —The floor function; Considering the hysteresis angle changes of different cutting teeth and different cutting points, the meshing state window function of the tool end edge and side edge is further calculated as follows; In the formula: —Cut tooth number; —Actual cutting radius of the tool; —Thickness of the cutting unit on the side edge of the tool; —Helix angle of the blade teeth; —Interdental angle; (2) The meshing state window function corresponding to the planar image: such as Figure 4 As shown in (a), in In the matrix, first along The direction is used to find the index of a non-zero pixel, as shown in the following formula; Then calculate along The actual distance represented by each pixel in the direction And the actual coordinates of each pixel in that direction. ,as follows: Similarly, for Direction, first along The direction is used to find the index of a non-zero pixel, as shown in the following formula; Then calculate along The actual distance represented by each pixel in the direction And the actual coordinates of each pixel in that direction. ,as follows: Furthermore, the coordinates of discrete points along the circumferential direction of the tool's cutting edge are established as follows: The window function corresponding to the meshing entity is obtained through two-dimensional linear interpolation. ,for: Then, the above window function is extended to the entire circumference of the tool, that is: After interpolation, it is then converted into a 0-1 binary matrix, as shown in the following formula; Considering the hysteresis angle variation of different cutting teeth and different cutting point heights, obtain The meshing state window function corresponding to the plane is shown in the following formula, and the corresponding meshing state curve is shown in the figure. Figure 4 As shown in (b); With After integration, the meshing state window function of the tool side edge is finally obtained, and the corresponding meshing state curve is as follows: Figure 4 As shown in (c); Step 2) Decompose the relative feed motion between the tool and the workpiece into three components: translation along the tool axis, translation perpendicular to the tool axis, and rotation around the axis. Solve for the instantaneous cutting layer thickness at full cut width by constructing geometric coordination relationships, and then obtain the explicit calculation formula for the instantaneous cutting layer thickness under actual meshing conditions; the specific process is as follows: 2.1) Tool geometry parameters under the influence of eccentricity: 1) Parametric definition of tool eccentricity: Tool eccentricity is the deviation of the tool's rotation axis, including the deviation amount. , deflection angle Tool geometry axis deviation, offset including offset amount Offset angle Yaw includes yaw amount and yaw angle ; Tool end-edge eccentricity parameters It is characterized by 3 eccentric states and 7 independent parameters; 2) Tool geometry parameters under the influence of eccentricity: such as Figure 5 As shown in (a), based on the definition of the end mill in the APT system, independent parameters are used. D , R z , R r , R c , , and H c Construct a general geometric profile. For example... Figure 5 As shown in (b), any cutting point on the cutting edge The position is determined by the corresponding axial height radial distance and radial hysteresis angle The determination, in which the radial distance The distance between this point and the tool axis, radial hysteresis angle. The angle between the line connecting this point and the tip of the blade and the tangent to the helix at a height of 0 on the axis is... Projection in a plane.

[0020] To facilitate the description of the effect of tool eccentricity on its geometric parameters, the cutting edge of the tool teeth is discretized into several cutting elements of minute thickness at equal intervals along the axial direction. At this point, the... i On the first blade tooth j Layer cutting unit cutting point The coordinates in the tool geometry coordinate system can be expressed as: In the formula: —The cutting element number obtained after discretizing the cutting edge and side edge along the cutting edge; When the tool only has a geometric axis deviation, the cutting point of the cutting edge The position in the machine tool coordinate system, that is, the tool rotation coordinate system, is: In the formula: —Tool overhang length; When the spindle rotates t Cutting point of the cutting edge at all times In the tool rotation coordinate system The following position is: When the tool has both geometric axis misalignment and rotation axis misalignment, the cutting point of the cutting edge... In the machine tool coordinate system The following position is: In the formula: —Length of axis deviation; At this point, the cutting point of the cutting edge under the influence of tool eccentricity can be obtained. The actual cutting radius is: Furthermore, the height difference between adjacent bottom edges of the tool tip in the machine tool coordinate system is: Due to tool eccentricity, any cutting point on the side edge In the tool rotation coordinate system The coordinates are: Any cutting point on the end blade In the tool rotation coordinate system The coordinates below are: 2.2) Equivalent decomposition of tool-workpiece relative feed motion: like Figure 6 As shown, in the workpiece machining coordinate system Next, for the first The cutter tip is located at the cutter position when the cutter tip is... Exercise During the process, the feed rate at the tool tip is (This speed is different from the nominal speed in G-code) (value), the feed motion time between the two points is ; During this process, the tool axis vector is changed from... Change to Accordingly, the B and C axes of the machine tool will... Rotate to The rotational speeds are respectively and ; After rotation along the BC axis, the tool axis vector is: In the formula: ; ; ; At this point, the local meshing coordinate system is defined as follows: Define the feed coordinate system as follows: Further analysis shows that the tool feed motion is actually along... Directional translational motion and rotation The linear superposition of the two point rotational motions; further, along... Directional translational motion is decomposed into along Translational superposition of directions at the upper edge For translational movements in a direction, a reference point needs to be defined. Its coordinates are: In the formula: —— and The angle between them; At this time, along Directional translation and along The velocity vectors for directional translation are as follows: In addition, around The rotation of a point can actually be regarded as the resultant rotational motion around the BC axis, that is, around the BC axis. Rotation of the axis; here, The calculation formula is as follows: Finally, around The rotation angle and angular velocity of the axis are respectively: 2.3) Calculation of instantaneous cutting layer thickness at full width of decomposed motion: 1) Cutting edge Linear feed motion in the direction of: (1) Side edge of the cutting tool: Figure 7 (a) represents the cutting edge In the linear feed motion of the direction, the tool rotates around its axis at a speed of [missing information]. n Rotate, and at feed rate along Directional feed; in t= At time 0, set the first cutting edge z The tangent at =0 and Axis coincidence; at any time t , No. i On the first blade tooth Layer cutting unit The rotation angle of the point is: Figure 7 In section (b), the cycloidal motion trajectory surface of the cutting edge of the tool is taken. Taking any cross-section of it, the motion process of the cutting unit of the tool's side edge is as follows: Figure 7 As shown in (c). At this time, the cutting point on the side edge The cutting layer thickness is equal to The line segment at the intersection of the line connecting the point and the tool rotation center with the workpiece geometry. ,in The point is formed by the first few cutting teeth. At some point in the past The residue left on the machined surface after cutting; time, Point in the feed coordinate system The coordinates below are: at the same time, point Constantly feeding coordinate system The following can also be represented as: Since the tool feeds along a straight line in the feed coordinate system at a speed of... Since it is a uniform rigid body motion, we have: By combining the above three equations (equations (41), (42), and (43)), the lag time at the side cutting point can be obtained. and instantaneous cutting layer thickness for: (2) Cutting edge of the tool: such as Figure 7 As shown in (d), the cutting point of the end edge. Compared with previous cutting points Given the height difference, we can conclude that: In conclusion, Figure 7 In Figure (e), the instantaneous cutting layer thickness distribution curve of a single cutting tooth of the tool during a single cycle of spindle rotation is shown. Figure 7 In the middle (f), the instantaneous cutting layer thickness distribution curve of multiple cutting teeth of the tool on its trajectory surface is shown. To facilitate the superposition of the cutting layer thickness of subsequent motion components, the instantaneous cutting layer thickness is distinguished as positive and negative. 2) Cutting edge Linear feed motion in the direction of: (1) Tool side edge: the edge of the tool During the directional movement, the tool rotates around its axis at a speed of [missing information]. n Rotate and feed rate along Directional feed. Take any cross-section on the cycloidal trajectory surface of the cutting edge; at this point, the side cutting point... The cutting layer thickness is equal to The line segment at the intersection of the line connecting the point and the center of rotation of the tool and the geometric solid of the workpiece, i.e. ,in The point is formed by the first few cutting teeth. At some point in the past The residue left on the machined surface after cutting. time, Point in the feed coordinate system The coordinates below are: at the same time, point Constantly feeding coordinate system The coordinates below can also be represented as: Here, the feed rate is defined as the rate at which the tool moves downward along the tool axis. A positive value indicates a positive value, and a negative value indicates a negative value. Because the tool follows... The direction of the feed rate is Since it is a uniform rigid body motion, the following relationship exists: By combining the above three equations (equation (48), equation (47), and equation (46)), the lag time at the side cutting point can be obtained. and instantaneous cutting layer thickness for: In the formula: It is a real number; (2) Tool end edge: Consider the cutting point on the end edge Compared with previous cutting points The height difference between them gives us: 3) Tool wrap Rotary feed motion in direction: (1) Tool side edge: the tool wraps around During the directional movement, the tool rotates around its axis at a speed of [missing information]. n Rotate, and at feed rate Around Directional feed. When the cutting edge curve undergoes three-dimensional cycloidal motion, an arbitrary cross-section of the motion trajectory surface is taken. That is, the side cutting point The thickness of the cutting layer; exist time, Point in coordinate system The coordinates below are: At this time, because the tool axis is at the feed rate Around Direction rotation, It can be represented as: also, It can be represented as: Furthermore, vector It can be represented as: exist time, Point in coordinate system The following can also be represented as: Due to the rigid body rotational motion of the tool axis, we can obtain: Since the above equations are a system of nonlinear equations with three unknowns, a simplification method is adopted, and the equations are first obtained through numerical calculation. and order Finally, the hysteresis time at the side cutting point can be obtained. and instantaneous cutting layer thickness for: (2) Tool end edge: Consider the cutting point on the end edge Compared with previous cutting points The height difference between them gives us: 2.4) Calculation of instantaneous cutting layer thickness under actual meshing of the composite motion: the tool along... and The starting point of the instantaneous cutting layer thickness generated by directional translation is The axis, while the tool rotates around The starting point of the instantaneous cutting layer thickness generated by directional rotation is The axis. Therefore, all three must be aligned to the same starting point in time, and the tool axis must be rotated around... The instantaneous cutting layer thickness in the direction is time-shifted as follows: As shown in the following formula, the instantaneous cutting layer thicknesses of the three motion components are superimposed. Figure 8Figure (a) shows the instantaneous cutting layer thickness distribution curve of a single cutting tooth during a single spindle rotation cycle. Figure 8 Figure (b) shows the instantaneous cutting layer thickness distribution curves of multiple cutting teeth on their trajectory surface; Therefore, the instantaneous cutting layer thickness is the maximum value obtained by comparing the minimum value among all cutting thicknesses with zero, as shown in the following formula: Finally, the above cutting thickness and meshing state window function are discretized to obtain the instantaneous cutting layer thickness under actual meshing conditions, as shown in the following formula. Based on this calculation formula, Figure 8 Figure (c) shows the instantaneous cutting layer thickness distribution curve of a single cutting tooth during a single spindle rotation cycle. Figure 8 Figure (d) shows the instantaneous cutting layer thickness distribution curves of multiple cutting teeth on their trajectory surface; Step 3) The method of the present invention is verified through a five-axis milling example; 3.1) Milling Condition: In this example, the workpiece is an S-shaped curved groove feature part made of aluminum alloy 7075. The cutting tool is a 16mm diameter two-tooth flat-bottom milling cutter with a helix angle of 30° and a tip radius of 4mm. The toolpath consists of four S-shaped curves, with the B-axis varying from -5° to +15° and the C-axis varying from -15° to +15°. The radial cutting width is calculated for each of the four paths in the CNC programming. a e The nominal values ​​are 16mm, 4mm, 8mm, and 12mm, respectively, and the axial cutting depth is... a p 12mm, feed per tooth f t 0.1mm / tooth, climb milling.

[0021] 3.2) Tool-Workpiece Geometric Meshing State Extraction Results: Based on the method of this invention, the tool-workpiece geometric meshing entity at each tool position point along the entire path is obtained. In this embodiment, there are a total of 680 tool positions along the entire path, and the meshing entity is successfully extracted in 525 steps, with a success rate of 77.2%.

[0022] 3.3) Computational Efficiency: According to statistics, the average calculation time per step across the entire path is 1.458s, the average calculation time during cutting is 1.576s, and the maximum calculation time per step is 5.508s. The average search calculation time is 0.0021s, accounting for 0.14% of the total time; the average Boolean calculation time is 0.7633s, accounting for 52.36% of the total time; and the average image export time is 0.6925s, accounting for 47.50% of the total time. This invention outperforms traditional methods in terms of average and maximum single-step calculation time and growth rate.

[0023] Traditional global methods require updating the entire workpiece model sequentially and processing it layer by layer, which is time-consuming and prone to calculation failure due to surface damage. This invention uses a local Boolean operation strategy to reconstruct the adjacent material to be cut and combine it with image processing to quickly extract the meshing state, thus avoiding the above problems. By controlling the number of adjacent tools through sparsification, the number of single-step surfaces and the amount of Boolean operations are controllable, and the space and time complexity are both O(N), which is better than traditional methods (at least second-order complexity ≥ O(N2)). Figure 9 The volumes of the geometrically meshing entities calculated by the proposed method and the traditional method were further compared. Since the workpiece model in the traditional method is updated at each step through global Boolean operations, the calculation principle reflects the material removal process of the workpiece, and its calculation results are considered accurate values. As can be seen from the figures, the present invention can obtain results that are relatively consistent with the traditional method, with an average relative error of only 3.81%, indicating that the present invention has sufficient computational accuracy.

Claims

1. A rapid method for extracting the time-varying geometric meshing state of a tool-workpiece, characterized in that, Includes the following steps: Step 1) Quickly extract the tool-workpiece geometric meshing boundary, reconstruct and replace the locally cut material of the workpiece with the past neighboring tools at the current tool position, quickly obtain the tool-workpiece geometric meshing entity through Boolean operations between the current tool entity, the workpiece blank entity and the reconstructed workpiece material, and efficiently calculate the meshing window state function through image processing. Step 2) Decompose the relative feed motion between the tool and the workpiece into three components: translation along the tool axis, translation perpendicular to the tool axis, and rotation around the axis. Solve the instantaneous cutting layer thickness of the full-cut width by constructing a geometric coordination relationship, and then obtain the explicit calculation formula of the instantaneous cutting layer thickness under the actual meshing state.

2. The rapid extraction method according to claim 1, characterized in that, The specific process of step 1) is as follows: 1.1) Search for previously adjacent cut sites: 1) G-code tool position densification preprocessing: After obtaining the G-code through CNC programming, the workpiece machining coordinate system X is obtained through preprocessing. w Y w Z w The following parameters are used to determine the tool tip position (XYZ), tool axis vector (IJK), machine tool rotation axis angles (ABC), and G code line number. N The data is used to preprocess the G-code through interpolation densification; based on discrete line segments on the toolpath, linear interpolation is performed on the position points on the line segments between adjacent tool points, and spherical interpolation is performed on the tool axis vector. The specific calculation formulas are as follows: In the formula: ——No. N One incision site; ——No. N Each tool axis vector; — Interpolation point number; —Total number of interpolation points; — Densification of interpolation points; — Tool axis vector and The angle between them; — Tool axis vector and The angle between them; — Tool axis vector and The angle between them; 2) Search for previously nearby cut points: Two modes are proposed to search for previously nearby cut points at the current tool position; The first method is offline calculation mode, which uses G-code obtained from CNC programming, directly inputting it into the calculation program for line-by-line calculation; when the calculation reaches the... N Cutting point At that time, the densification knife site This constitutes the set of previously cut sites; The second method is the online calculation mode. First, the G-code is input into the machine tool's CNC system to drive the feed of each axis. Then, the interpolation instruction data, including the current execution line number of the G-code, is read in real time through the CNC system's OPC UA function. N Machine tool axis position and speed, spindle speed and magnification, G54 coordinate zero offset; For the current tool position Its previously nearby cut sites are further divided into three categories: ① Absolutely nearest points This point is in the densification knife site sequence. Among them; ② Dynamic nearest neighbors This point exists in the currently executing... N Dynamic interpolation sequence of G-code Middle; ③ Particularly nearby point This point is located before the current tool position. At a distance; the three types of nearest neighbors must satisfy the following relationship during the search calculation: In the formula: — Tool point for static interpolation; —The tool position point for dynamic interpolation; —Nominal radius of the cutting tool; —The value is the feed per tooth of the tool; Previous adjacent cutting sites The sum of the above three sets is shown in the following formula; during the summation process, except for special points... The neighbor sequence outside is sparsely sampled to obtain the sparse neighbor sequence. 1.2) Local Fast Boolean Operations for Tool-Workpiece Geometric Interlocking Entity: The tool-workpiece geometric interlocking entity is quickly obtained by replacing the partially cut material of the workpiece with the reconstructed entity of the previously cut neighboring tools at the current tool position point. Local Boolean operations are performed between the current tool entity, the workpiece blank entity, and the reconstructed workpiece solid entity to obtain the tool-workpiece geometric interlocking entity. The Boolean operation kernel program is a secondary development on the commercial software NX10.0, using a development mode combining NXOpen C++ and NXOpen C (UFUN). The calculation steps are as follows: (1) Design the CAD model of the tool and workpiece blank, use Catia software to program and obtain G code, obtain densified G code through preprocessing, and read it into the secondary development program using the file interface; (2) Set the workpiece machining coordinate system in the program. And obtain the current data from the CNC system interpolation instruction data. tool position and tool axis vector Reconstruct the relative spatial orientation of the tool and workpiece; (3) Search for the previous neighboring cut points at the current tool position. The Parasolid PK kernel function is used to reconstruct the set of previously cut tool entities based on the tool position point and tool axis vector. This replaces the previously cut material in parts of the workpiece. (4) Determine the physical state of the tool at this time With blank solid Intersection status: If they intersect, perform a Boolean intersection operation to obtain the initial meshing body. ; If they do not intersect, proceed to the next time step; (5) The initial meshing body Material that has been previously cut in the workpiece Perform a Boolean subtraction operation to obtain the tool-workpiece geometric meshing entity at the current tool position. ,Right now: 1.3) Image processing and extraction of the tool-workpiece geometric meshing boundary: 1) Two-dimensional image processing of the meshing entity: For the tool-workpiece geometric meshing entity obtained by Boolean operations, the entity in the tool feed coordinate system is obtained through secondary development of API functions. The bounding box parameters below, i.e., the minimum values ​​of the solid in each axial direction. , and and length , and ; Rotate and project the solids onto the orthogonal plane. and And obtain its image screenshot in a two-dimensional plane. and In the screenshot It also includes meshing information for both the side and end cutting edges of the tool. It includes the meshing information of the side blades; using OpenCV library functions, the projected screenshot is converted into a binary grayscale image, and pixel parameters along the horizontal and vertical directions are extracted. and ; Convert the grayscale image into a 0-1 binary matrix, and name them respectively. and In this system, gray pixels are represented by 1 and white pixels by 0; 2) Calculation of the window function for the cutting edge meshing state: (1) The meshing state window function corresponding to the planar image: In the matrix, first along The direction is used to find the index of a non-zero pixel, as shown in the following formula; In the formula: —A function for summing terms; — Lookup function; Calculate along The actual distance represented by each pixel in the direction and each pixel in The actual coordinates corresponding to the direction ,as follows: Similarly, for Direction, first along The direction is used to find the index of a non-zero pixel, as shown in the following formula; Calculate along The actual distance represented by each pixel in the direction and each pixel in The actual coordinates corresponding to the direction ,as follows: Establish the coordinates of discrete points along the circumferential direction of the tool's cutting edge line as follows: In the formula: —Spindle speed; — Spindle rotation cycle; —Number of discrete times per cycle; —The sequence of cutting radii of the tool end edge, where Number the cutting points of the end blade; —The sequence of cutting radii of the tool side edge, where Number the cutting points on the side edge; The window function corresponding to the meshing entity is obtained through two-dimensional linear interpolation. ,for: In the formula: —Two-dimensional interpolation function; ; ——No. k Discrete moments; After interpolation, it is converted into a 0-1 binary matrix, and the end edge and side edge parts are separated and stored separately. and In the matrix, as shown in the following equation; In the formula: —The floor function; The window function for calculating the meshing state of the tool tip and side cutting edges is as follows; In the formula: —Cut tooth number; —Actual cutting radius of the tool; —Thickness of the cutting unit on the side edge of the tool; —Helix angle of the blade teeth; —Interdental angle; (2) The meshing state window function corresponding to the planar image: In the matrix, first along The direction is used to find the index of a non-zero pixel, as shown in the following formula; Calculate along The actual distance represented by each pixel in the direction And the actual coordinates of each pixel in that direction. ,as follows: Similarly, for Direction, first along The direction is used to find the index of a non-zero pixel, as shown in the following formula; Calculate along The actual distance represented by each pixel in the direction And the actual coordinates of each pixel in that direction. ,as follows: Establish the coordinates of discrete points along the circumferential direction of the tool's cutting edge line as follows: The window function corresponding to the meshing entity is obtained through two-dimensional linear interpolation. ,for: Then, the above window function is extended to the entire circumference of the tool, that is: After interpolation, it is then converted into a 0-1 binary matrix, as shown in the following formula; Considering the hysteresis angle variation of different cutting teeth and different cutting point heights, obtain The meshing state window function corresponding to the plane is shown in the following formula; With After integration, the meshing state window function of the tool side edge is finally obtained; 。 3. The rapid extraction method according to claim 2, characterized in that, The specific process of step 2) is as follows: 2.1) Tool geometry parameters under the influence of eccentricity: 1) Parametric definition of tool eccentricity: Tool eccentricity is the deviation of the tool's rotation axis, including the deviation amount. , deflection angle Tool geometry axis deviation, offset including offset amount Offset angle Yaw includes yaw amount. and yaw angle ; Tool end-edge eccentricity parameters It is characterized by 3 eccentric states and 7 independent parameters; 2) Tool geometry parameters under the influence of eccentricity: using D , R z , R r , R c , , and H c The geometric profile is constructed using 7 independent parameters; any cutting point on the cutting edge The position is determined by the corresponding axial height radial distance and radial hysteresis angle The determination, in which the radial distance Defined as the distance between this point and the tool axis, the radial hysteresis angle. Defined as the angle between the line connecting this point and the tip of the blade and the tangent to the helix at a height of 0 on the axis. Projection values ​​within the plane; The cutting edge of the cutting tooth is discretized into several cutting units of minute thickness at equal intervals along the axial direction; at this time, the first... i On the first blade tooth j Layer cutting unit cutting point The coordinates in the tool geometry coordinate system are represented as follows: In the formula: —The cutting element number obtained after discretizing the cutting edge and side edge along the cutting edge; When the tool only has a geometric axis deviation, the cutting point of the cutting edge The position in the machine tool coordinate system, that is, the tool rotation coordinate system, is: In the formula: —Tool overhang length; When the spindle rotates t Cutting point of the cutting edge at all times In the tool rotation coordinate system The following position is: When the tool has both geometric axis misalignment and rotation axis misalignment, the cutting point of the cutting edge... In the machine tool coordinate system The following position is: In the formula: —Length of axis deviation; At this point, the cutting point of the cutting edge must be determined under the influence of tool eccentricity. The actual cutting radius is: Furthermore, the height difference between adjacent bottom edges of the tool tip in the machine tool coordinate system is: Due to tool eccentricity, any cutting point on the side edge In the tool rotation coordinate system The lower coordinate is: Any cutting point on the end blade In the tool rotation coordinate system The coordinates below are: 2.2) Equivalent decomposition of tool-workpiece relative feed motion: In the workpiece machining coordinate system Next, for the first The cutter tip is located at the cutting point, when the cutter tip... Exercise During the process, the feed rate at the tool tip is (This speed is different from the nominal speed in G-code) (value), the feed motion time between the two points is ; During this process, the tool axis vector is changed from... Change to Accordingly, the B and C axes of the machine tool will... Rotate to The rotational speeds are respectively and ; After rotation along the BC axis, the tool axis vector is: In the formula: ; ; ; At this point, the local meshing coordinate system is defined as follows: Define the feed coordinate system as follows: The tool feed motion is actually along... Directional translational motion and rotation The linear superposition of the two point rotational motions; along Directional translational motion is decomposed into along Translational superposition of directions at the upper edge For translational movements in a direction, a reference point needs to be defined. Its coordinates are: In the formula: —— and The angle between them; At this time, along Directional translational motion and along The velocity vectors for the directional translational motion are as follows: In addition, around The rotation of a point can actually be regarded as the resultant rotational motion around the BC axis, that is, around the BC axis. The axis rotates. Here, The calculation formula is as follows: Finally, around The rotation angle and angular velocity of the axis are respectively: 2.3) Calculation of instantaneous cutting layer thickness at full width of decomposed motion: 1) Cutting edge Linear feed motion in the direction of: (1) Tool side edge: the edge of the tool In linear feed motion, the tool rotates around its axis at a speed of [missing information]. n Rotate, and at feed rate along Directional feed; in t= At time 0, set the first cutting edge z The tangent at =0 and Axis coincidence; at any time t , No. i On the first blade tooth Layer cutting unit The rotation angle of the point is: Take any cross-section on the trajectory surface of the cycloidal motion of the cutting edge of the tool, and the cutting point on the side edge is then determined. The cutting layer thickness is equal to The line segment at the intersection of the line connecting the point and the tool's rotation center with the workpiece's geometric solid, i.e. ,in The point is formed by the first few cutting teeth. At some point in the past The residue left on the machined surface after cutting; time, Point in the feed coordinate system The coordinates below are: at the same time, point Constantly feeding coordinate system The following is represented as: Since the tool feeds along a straight line in the feed coordinate system at a speed of... Since it is a uniform rigid body motion, we have: By combining the above three equations, namely equations (41), (42), and (43), the lag time at the side cutting point can be obtained. and instantaneous cutting layer thickness for: (2) Tool end edge: cutting point of the end edge Compared with previous cutting points Given the height difference, we have: 2) Cutting edge Linear feed motion in the direction of: (1) Tool side edge: the edge of the tool During the directional movement, the tool rotates around its axis at a speed of [missing information]. n Rotate and feed rate along Directional feed; take any cross section on the cycloidal motion trajectory surface of the cutting edge, at which point the side cutting edge is located. The cutting layer thickness is equal to The line segment at the intersection of the line connecting the point and the tool's rotation center with the workpiece's geometric solid, i.e. ,in The point is formed by the first few cutting teeth. At some point in the past The residue left on the machined surface after cutting; time, Point in the feed coordinate system The coordinates below are: at the same time, point Constantly feeding coordinate system The coordinates below are also represented as: Here, the feed rate is defined as the rate at which the tool moves downward along the tool axis. A positive value indicates a positive value, and a negative value indicates a negative value; because the tool follows... The direction of the feed rate is Since it is a uniform rigid body motion, the following relationship exists: By combining the above three equations, namely equations (48), (47), and (46), the lag time at the side cutting point can be obtained. and instantaneous cutting layer thickness for: In the formula: It is a real number; (2) Tool end edge: Consider the cutting point on the end edge Compared with previous cutting points The height difference between them gives: 3) Tool wrap Rotary feed motion in direction: (1) Tool side edge: the tool wraps around During the directional movement, the tool rotates around its axis at a speed of [missing information]. n Rotate, and at feed rate Around Directional feed; when the cutting edge curve is in three-dimensional cycloidal motion, take any cross-section of the motion trajectory surface, at this time That is, the side cutting point The thickness of the cutting layer; exist time, Point in coordinate system The coordinates below are: At this time, because the tool axis is at the feed rate Around Direction rotation, Represented as: also, Represented as: Furthermore, vectors Represented as: exist time, Point in coordinate system The following is also represented as: Due to the rigid body rotational motion of the tool axis, we get: Using a simplified method, the result is first obtained through numerical calculation. and order Finally, the hysteresis time of the side cutting point is obtained. and instantaneous cutting layer thickness for: (2) Tool end edge: Consider the cutting point on the end edge Compared with previous cutting points The height difference between them gives: 2.4) Calculation of instantaneous cutting layer thickness under actual meshing of the composite motion: the tool along... and The starting point of the instantaneous cutting layer thickness generated by directional translation is The axis, while the tool rotates around The starting point of the instantaneous cutting layer thickness generated by directional rotation is Therefore, all three (axis, yoke, and tool axis) must be aligned to the same starting point in time, and the tool axis must be rotated around... The instantaneous cutting layer thickness in the direction is time-shifted as follows: As shown in the following formula, the instantaneous cutting layer thicknesses of the three motion components are superimposed; The instantaneous cutting layer thickness is the maximum value obtained by comparing the minimum of all cutting thicknesses with zero, as shown in the following formula: Finally, the above cutting thickness is discretized and mapped to the meshing state window function to obtain the instantaneous cutting layer thickness under the actual meshing state, as shown in the following formula: 。

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