Method for modeling instantaneous three-dimensional undeformed chip geometry in cycloidal milling

Abstract

In order to overcome the deficiency of the two-dimensional undeformed chip geometry model in the cycloid milling established by the existing method, the application provides a kind of instantaneous three-dimensional undeformed chip geometry modeling method in the cycloid milling. Firstly, based on the layer cutting idea, a plurality of vertical tool shafts are used to cut the tool and the workpiece simultaneously to obtain a plurality of cutting layers. Then, the cutting type of each cutting layer is judged, and the instantaneous two-dimensional undeformed chip geometry on each cutting layer is solved. For the cutting layer with only single-tooth cutting, the instantaneous two-dimensional undeformed chip geometry generated by the cutting edge participating in cutting at any instant is solved. For the cutting layer with multi-tooth cutting, the instantaneous two-dimensional undeformed chip geometry generated by the single cutting edge participating in cutting at any instant is solved. Finally, the instantaneous two-dimensional undeformed chip geometry on all cutting layers is stacked along the direction parallel to the tool shaft to obtain the instantaneous three-dimensional undeformed chip geometry. The application provides a new theoretical basis and calculation method for the geometry and physical simulation in the cycloid milling.

Description

Technical Field

[0001] This invention belongs to the field of multi-axis CNC milling geometry simulation and cycloidal milling modeling, and relates to a method for solving the three-dimensional geometric model of the undeformed chips of the material removed by each cutting edge of the tool at any instant in cycloidal milling. Background Technology

[0002] In recent years, with the continuous and rapid development of my country's aerospace industry, difficult-to-machine materials, such as titanium alloys and nickel-based superalloys, have been increasingly widely used in the aerospace industry due to their high strength-to-weight ratio and excellent corrosion resistance. However, in the aerospace field, in order to improve carrying efficiency, the weight requirements for parts are stringent, and the material removal volume of parts is large, reaching more than 90% for some parts. This leads to low cutting efficiency and high processing costs.

[0003] To address the aforementioned issues, the academic community has proposed cycloidal milling as a strategy to improve cutting efficiency and reduce cutting costs. Research indicates that cycloidal milling can reduce cutting forces, avoid full-groove milling and abrupt changes in cutting forces, improve the surface quality of machined parts, lower cutting temperatures, provide excellent chip removal performance, exhibit good cutting stability, reduce tool wear, extend tool life, save energy, and is particularly suitable for milling difficult-to-machine materials and high-speed machining.

[0004] The literature “Pleta A, Niaki FA, Mears L. Investigation of chip thickness and force modelling of trochoidal milling[J]. Procedia Manufacturing, 2017, 10: 612-621” studies the two-dimensional undeformed chip geometry in trochoidal milling based on a subcycloidal model and proposes a numerical calculation method for the two-dimensional undeformed chip thickness. However, the method disclosed in this literature does not establish an instantaneous three-dimensional undeformed chip geometry model in trochoidal milling, making it difficult to intuitively demonstrate the undeformed chip generation process and multi-tooth cutting phenomenon. In addition, it does not consider the detailed geometric parameters of the tool, such as the helix angle of the cutting edge and the number of tool teeth, in the modeling process, making it difficult to integrate with the subsequent physical simulation model. Summary of the Invention

[0005] To overcome the shortcomings of existing methods in establishing two-dimensional undeformed chip geometry models in cycloidal milling, this invention proposes an instantaneous three-dimensional undeformed chip geometry modeling method in cycloidal milling.

[0006] The inventive concept of this invention is:

[0007] First, the basic geometric relationships in cycloidal milling were analyzed, and the basic idea of ​​the layer cutting method was explained.

[0008] Next, the definition of instantaneous undeformed chip geometry was proposed, and based on this, the composition and mathematical expression of the geometric boundary of arbitrary instantaneous two-dimensional undeformed chips on arbitrary cutting layers were studied.

[0009] Secondly, the multi-tooth cutting phenomenon on a certain cutting layer was analyzed and the criteria for determining the occurrence of multi-tooth cutting were given.

[0010] Finally, based on the spatial geometric relationship, the instantaneous two-dimensional undeformed chips obtained above in cycloidal milling are stacked to obtain the instantaneous three-dimensional undeformed chip geometric model in cycloidal milling.

[0011] The technical solution adopted by this invention to solve its technical problem is as follows:

[0012] The instantaneous 3D undeformed chip geometry modeling method in cycloidal milling is characterized by the following steps:

[0013] Step 1: Based on the concept of layer cutting, multiple parallel planes perpendicular to the tool axis are used to cut the tool and workpiece simultaneously to obtain multiple cutting layers;

[0014] Step 2: Determine the cutting type of each cutting layer, and solve for the instantaneous two-dimensional undeformed chip geometry on each cutting layer at any given instant:

[0015] For a cutting layer with only single-tooth cutting, solve for the instantaneous two-dimensional undeformed chip geometry generated by the cutting edge participating in the cutting at any instant;

[0016] For the cutting layer in which multi-tooth cutting occurs, solve the instantaneous two-dimensional undeformed chip geometry generated by each individual cutting edge participating in the cutting at any instant;

[0017] Step 3: Stack all the instantaneous two-dimensional undeformed chip geometry on the cutting layer in a direction parallel to the tool axis to obtain the instantaneous three-dimensional undeformed chip geometry.

[0018] Furthermore, the solution steps for the instantaneous two-dimensional undeformed chip geometry generated by a single cutting edge at any instant in step 2 are as follows:

[0019] Step 2.1: Solve for the tool center point at any instant in cycloidal milling, the material removal boundary surface of the previous cycloidal cutting cycle, and the material removal boundary surface to be generated in the current cycloidal cutting cycle;

[0020] Step 2.1.1: Establish the reference coordinate system CSO for describing the cycloidal toolpath. r and CSO q ;

[0021] The reference coordinate system CSO r Origin r Located at the center of the cycloidal toolpath arc segment in the current cycloidal cutting cycle, xr The axis is parallel to the straight section of the cycloidal toolpath and points in the feed direction of the tool. r The axis is parallel to the tool axis and points vertically upwards, y r Determined by the right-hand screw rule;

[0022] The reference coordinate system CSO q Origin q Located at the center of the cycloidal toolpath arc segment in the previous cycloidal cutting cycle, each coordinate axis is relative to the reference coordinate system CSO. r Consistent in direction;

[0023] Step 2.1.2: Calculate the tool center point O at any instant during the current cycloidal cutting cycle. c (x c ,y c ,z c ), in the coordinate system CSO r The coordinates in the graph, the material boundary S1(x) cut off in the previous cycloidal cycle. s1 ,y s1 ,z s1 ) and the boundary surface S2(x) generated by the material to be cut in the current cycloidal cycle. s2 ,y s2 ,z s2 ) and S3(x s3 ,y s3 ,z s3 In the coordinate system CSO r The expression in;

[0024] Step 2.2: In the coordinate system CSO r Establish a method to describe arbitrary cutting layers Π i The kth blade tip point C i,k Local coordinate system CSO of instantaneous position ci,j ;

[0025] Step 2.2.1: For a certain cutting layer Π i Calculate the kth tool tip point C on it. i,k The trajectory of the circle S ei,j In the coordinate system CSO r Center O ci,j coordinates (x) ci,j ,y ci,j ,z ci,j );

[0026] Step 2.2.2: In the cutting layer Π i Above, with the center O ci,j Establish q local coordinate systems CSO for the point. ci,j The local coordinate system CSO ci,jx ci,j The shaft and the cutter follow the cycloidal tool path at the center O. ci,j The feed directions at z are parallel and in the same direction. ci,j axis and z r The axes are parallel and in the same direction, y ci,j The axis is determined by the right-hand rule; q is the total number of times the tool cuts the workpiece in one cycloidal cycle;

[0027] Step 2.3: Calculate the tool tip point of each cutting edge on each cutting layer in coordinate system CSO. r The instantaneous position in;

[0028] Step 2.4: Solve for the parametric expression of the instantaneous two-dimensional undeformed chip geometry generated at any instant on each cutting layer;

[0029] Step 2.5: Solve for the parameters in the parameter expression obtained in Step 2.4 to obtain the instantaneous two-dimensional undeformed chip geometry.

[0030] Furthermore, in step 2, the method for determining the cutting type of each cutting layer is as follows: the method for determining whether a certain cutting layer undergoes multi-tooth cutting is as follows: at any instant, on a certain cutting layer, if there are more than one tool tip point whose instantaneous position angle is within the interval of the corresponding entry angle and exit angle, then multi-tooth cutting occurs on that cutting layer; otherwise, only single-tooth cutting occurs on that cutting layer.

[0031] Furthermore, in step 2.1.2: any instantaneous tool center point O during the current cycloidal cutting cycle. c (x c ,y c ,z c In coordinate system CSO r The coordinates in the middle are calculated according to the following formula:

[0032]

[0033] Where: c is the cycloidal step size; r is the cycloidal radius; t is the time parameter, in seconds; f is the feed rate, in mm / min; time parameter t l =60c / f,t t =60(c+2πr) / f;

[0034] The material boundary S1(x) removed in the previous cycloidal cycle s1 ,y s1 ,z s1 ) and the boundary surface S2(x) generated by the material to be cut in the current cycloidal cycle. s2 ,y s2 ,z s2 ) and S3(x s3 ,y s3 ,zs3 In coordinate system CSO r The expressions in the text are as follows:

[0035]

[0036]

[0037] S3={(x s3 ,y s3 ,z s3 )|x s3 ∈[-c,0],y s3 =-(R+r),z s3 ∈[0,a p ]}

[0038] Where, θ s1 The radian parameter of surface S1 and θ s1 ∈[0,π-arcsin(c / 2(R+r))], where R is the tool radius; when θ s1 When θ ∈ [π-arcsin(c / 2(R+r)),2π], surface S1 does not exist, meaning the material boundary represented by surface S1 is removed by the previous cycloidal cutting cycle; S2 is the surface generated by cutting along the straight segment of the cycloidal toolpath, and S3 is the surface generated by cutting along the circular arc segment of the cycloidal toolpath; s2 It is the radian parameter of surface S2, which is derived from y r The negative half of the axis is measured counterclockwise, and θ s2 ∈[0,π+arcsin(c / 2(R+r))];a p The axial depth of cut is expressed in mm.

[0039] Furthermore, in step 2.2.1: the certain cutting layer Π i The kth blade tip point C i,k The trajectory of the circle S ei,j In the coordinate system CSO r Center O ci,j coordinates (x) ci,j ,y ci,j ,z ci,j Calculate according to the following formula:

[0040]

[0041]

[0042] Where: t ci,j For time parameters, n is the spindle speed, measured in rpm; q is the total number of cuts made by the tool in one cycloidal cycle. td i The time lag parameter introduced by the tool helix angle β represents the time t at which the tool tip on the same cutting edge begins cutting material on the cutting layer Π1. d i After a certain time, it will be in the cutting layer Π i The material is then cut. Z i For cutting layer Π i In CSO r z in r coordinate.

[0043] Furthermore, in step 2.3, any one of the cutting layers Π i The tool tip points of each cutting edge in coordinate system CSO r The instantaneous position is calculated as follows:

[0044] 2.3.1 First blade tip point C i,1 In the reference coordinate system CSO r instantaneous position O pi,1 (x pi,1 ,y pi,1 ,z pi,1 Calculate according to the following formula:

[0045]

[0046] in:

[0047] j = int(t / t1)·N+1, where int is the floor operator; ω is the angular velocity, ω = 2πn / 60;

[0048] Let θ fi,j =ω·(t-int(t / t1)·t1), θ fi,j For the first blade tip point C i,1 In the local coordinate system CSO ci,j The instantaneous position angle in (x) ci,j ,y ci,j ,z ci,j ) is the center O ci,j In the reference coordinate system CSO r Coordinates in;

[0049] 2.3.2 The kth blade tip point C i,k In the reference coordinate system CSO r The instantaneous position in the middle is calculated according to the following conditions, where k≠1:

[0050] Case 1: If ω·t<(k-1)·ψ p In the cutting layer Π i The kth blade tip point C i,kThe trajectory of movement does not exist;

[0051] Case 2: If ω·t≥(k-1)·ψ p And θ fi,j <(k-1)·ψ p Cutting layer Π i The kth blade tip point C i,k In coordinate system CSO r instantaneous position O pi,k (x pi,k ,y pi,k ,z pi,k Calculate according to the following formula:

[0052]

[0053] Let θ fi,j-N+k-1 =θ fi,j -(k-1)·ψ p +2π, θ fi,j-N+k-1 Let C be the kth tip point at instant t. i,k In coordinate system CSO ci,j-N+k-1 The instantaneous position angle in (x) ci,j+k-1 ,y ci,j+k-1 ,z ci,j+k-1 C is the kth blade tip point. i,k The trajectory of the circle S ei,j-N+k-1 The center O ci,j+k-1 In the reference coordinate system CSO r Coordinates in;

[0054] Case 3: If ω·t≥(k-1)·ψ p And θ fi,j ≥(k-1)·ψ p At instant t, the cutting layer Π i The kth blade tip point C i,k In the reference coordinate system CSO r instantaneous position O pi,k (x pi,k ,y pi,k ,z pi,k ) Calculated by the following formula:

[0055]

[0056] Let θ fi,j+k-1 =θ fi,j -(k-1)·ψ p θ fi,j+k-1 Let C be the kth tip point at instant t. i,k In the local coordinate system CSO ci,j+k-1 The instantaneous position angle in (x) ci,j+k-1 ,y ci,j+k-1 ,zci,j+k-1 C is the kth blade tip point. i,k The trajectory of the circle S ei,j+k-1 The center O ci,j+k-1 In the reference coordinate system CSO r The coordinates in the diagram.

[0057] Furthermore, any cutting layer Π in step 2.4 i The specific method for solving the parametric expression of the instantaneous two-dimensional undeformed chip geometry generated at any instant is as follows:

[0058] 2.4.1 Determine the cutting layer Π i The kth blade tip point C i,k The curves corresponding to the outer and inner boundaries of the two-dimensional undeformed chip cut after one revolution are obtained. The parametric expressions of these curves are solved as follows:

[0059] Let Δ1=(x ei,j +c) 2 +y ei,j 2 , (x ei,j ,y ei,j S is the circle of motion trajectory. ei,j and S ei,j-1 intersection point P ei,j The coordinates;

[0060] Case 1: If Δ1≤(r+R) 2 The boundary of a two-dimensional undeformed chip consists of an outer boundary and an inner boundary:

[0061] The outer boundary in the local coordinate system CSO ci,j The Chinese character is represented as: P si,j P ei,j ′={S ei,j |θ∈[θ si,j ,θ ei,j ′]}, where θ si,j and θ ei,j 'These are the cutting layers Π i The entry and exit angles of the j-th cut; P si,j P ei,j ' represents the outer boundary curve;

[0062] Inner boundary at CSO q The Chinese character is represented as: P si,j P ei,j ′={s i |θ∈[θ qi,j ′,θ qi,j ]}, where θ qi,j ' and θ qi,j The tangent points are P and P respectively. ei,j 'and the entry point Psi,j In the reference coordinate system CSO q Parameter expression in P; si,j P ei,j ' is the inner boundary curve;

[0063] Case 2: If Δ1 > (r + R) 2 The boundary of a two-dimensional undeformed chip consists of one outer boundary and two inner boundaries:

[0064] The outer boundary in the local coordinate system CSO ci,j The Chinese character is represented as: P si,j P ei,j ={S ei,j |θ∈[θ si,j ,θ ei,j ]}, where θ ei,j For cutting layer Π i The cutting angle of the j-th cut; P si,j P ei,j The outer boundary curve;

[0065] Inner boundary 1 in coordinate system CSO q The Chinese character is represented as: P si,j P si,j-1 ={s i |θ∈[θ qi,j-1 ,θ qi,j ]}, where θ qi,j-1 Let P be the intersection point. si,j-1 In the reference coordinate system CSO q Parameter expression in P; si,j P si,j-1 For the inner boundary curve 1;

[0066] Inner boundary two in local coordinate system CSO ci,j-1 The Chinese character is represented as: P si,j-1 P ei,j ={S ei,j-1 |θ∈[θ si,j-1 ,θ bi,j-1 ]}, where θ si,j-1 For cutting layer Π i The angle of entry for the (j-1)th cut; θ bi,j-1 For cutting layer Π i The cutting angle of the j-th cut; P si,j-1 P ei,j The second inner boundary curve;

[0067] 2.4.2 Based on 2.4.1, determine the cutting layer Π i The kth blade tip point C i,k For the curves corresponding to the outer and inner boundaries of the instantaneous two-dimensional undeformed chip generated at any instant t, the parametric expressions of these curves are solved respectively, as follows:

[0068] Case 1: If Δ1 > (r + R) 2 And the trajectory of the circle S ei,j-1 With material removal boundary s i intersection point P si,j-1 Located in vector F i,j K i,j To the right, the instantaneous two-dimensional undeformed chip boundary is formed by the circle S at the tool tip's motion trajectory. ei,j curve P on si, j F i,j Curve P located on the material removal boundary si,j P si,j-1 The trajectory of the blade tip is a circle S. ei,j-1 curve P on si,j-1 K i,j And the curve K on the rake face i,j F i,j It consists of four curves, and their parameter expressions are as follows:

[0069] P si,j F i,j ={S ei,j |θ∈[θ si,j ,θ fi,j ]}

[0070] P si,j P si,j-1 ={s i |θ∈[θ qi,j-1 ,θ qi,j ]}

[0071] P si,j-1 K i,j ={S ei,j-1 |θ∈[θ si,j-1 ,θ ki,j-1 ]}

[0072] F i,j K i,j From vector F i,j K i,j It means that F i,j K i,j =(-R·sin(θ) fi,j -θ ci,j ),-R·cos(θ fi,j -θ ci,j ));

[0073] Where, θ fi,j Let C be the kth blade tip point. i,k instantaneous position point F i,j In the local coordinate system CSO ci,jThe parameter expression in θ; si,j For cutting layer Π i The angle of entry for the j-th cut; θ ki,j-1 For the normal vector F i,j K i,j and material removal boundary s i The intersection point in coordinate system CSO ci,j-1 The parameter expression in θ; ci,j Let C be the kth blade tip point. i,k The trajectory of the circle S ei,j Corresponding center point O ci,j The instantaneous position angle;

[0074] Case 2: If Δ1 > (r + R) 2 And the trajectory of the circle S ei,j-1 With material removal boundary s i intersection point P si,j-1 Located in vector F i,j K i,j To the left, the instantaneous two-dimensional undeformed chip boundary is formed by the circle S at the tool tip's motion trajectory. ei,j curve P on si, j F i,j Curve P located on the material removal boundary si,j K i,j And the curve F on the rake face i,j K i,j It consists of three curves, and their parameter expressions are as follows:

[0075] P si,j F i,j ={S ei,j |θ∈[θ si,j ,θ fi,j ]}

[0076] P si,j K i,j ={s i |θ∈[θ ki,j ′,θ qi,j ]}

[0077] F i,j K i,j From vector F i,j K i,j It means that F i,j K i,j =(-R·sin(θ) fi,j -θ ci,j ),-R·cos(θ fi,j -θ ci,j ));

[0078] Where, θfi,j Let C be the kth blade tip point. i,k instantaneous position point F i,j In the local coordinate system CSO ci,j The parameter expression in θ si,j For cutting layer Π i The angle of entry for the j-th cut; θ ki,j 'For point K i,j In coordinate system CSO q The parameter expression in θ; ci,j Let C be the kth blade tip point. i,k The trajectory of the circle S ei,j Corresponding center point O ci,j The instantaneous position angle;

[0079] Case 3: If Δ1≤(r+R) 2 The instantaneous two-dimensional undeformed chip boundary is formed by the circular S-shaped trajectory of the tool tip. ei,j curve P on si,j F i,j Curve F on the rake face i,j K i,j And the curve K located on the material removal boundary i,j P si,j It consists of three curves, and their parameter expressions are as follows:

[0080] P si,j F i,j ={S ei,j |θ∈[θ si,j ,θ fi,j ]}

[0081] F i,j K i,j From vector F i,j K i,j It means that F i,j K i,j =(-R·sin(θ) fi,j -θ ci,j ),-R·cos(θ fi,j -θ ci,j ))

[0082] P si,j K i,j ={s i |θ∈[θ ki,j ′,θ qi,j ]}

[0083] Where, θ fi,j Let C be the kth blade tip point. i,k instantaneous position point F i,j In the local coordinate system CSO ci,jThe parameter expression in θ; si,j For cutting layer Π i The angle of entry for the j-th cut; θ ki,j-1 For the normal vector F i,j K i,j and material removal boundary s i The intersection point in coordinate system CSO ci,j-1 The parameter expression in θ; ci,j Let C be the kth blade tip point. i,k The trajectory of the circle S ei,j Corresponding center point O ci,j The instantaneous position angle.

[0084] The present invention also provides an electronic device, including a memory, a processor, and a computer program stored in the memory and executable thereon; its special feature is that the computer program implements the instantaneous three-dimensional undeformed chip geometry modeling method described above when it runs.

[0085] The present invention also provides a non-volatile computer-readable storage medium on which a computer program is stored; its special feature is that the computer program implements the instantaneous three-dimensional undeformed chip geometry modeling method described above when it is executed.

[0086] The beneficial effects of this invention are:

[0087] 1. Considering the detailed geometric parameters of the tool, the instantaneous three-dimensional undeformed chip geometric model in cycloidal milling is: (1) an important basis for geometric simulation during the cutting process, such as overcutting or undercutting judgment, toolpath verification, and the establishment of dynamic workpiece models; (2) an important input for physical simulation during the cutting process, which can provide key information such as undeformed chip thickness, instantaneous material removal rate, and tool-chip contact area geometry for the prediction of cutting force, cutting power, and cutting temperature. Therefore, this invention, by analyzing the basic geometric relationships in cycloidal milling, based on the layer cutting method and comprehensively considering the detailed geometric parameters of the tool (such as the cutting edge helix angle, the number of tool teeth, and the tool radius), proposes a method for constructing a local coordinate system describing instantaneous two-dimensional undeformed chips, a method for calculating the position of arbitrary instantaneous tool tip, a method for solving the boundary of instantaneous two-dimensional undeformed chips, and a method for solving instantaneous two-dimensional undeformed chips under multi-tooth cutting conditions, and finally establishes a method for solving the geometric model of instantaneous three-dimensional undeformed chips in cycloidal milling. This method can solve for the three-dimensional undeformed chip geometry model at any instant in cycloidal milling with high precision and efficiency. It overcomes the shortcomings of traditional undeformed chip modeling methods, such as not considering detailed tool geometry parameters, difficulty in intuitively displaying the undeformed chip generation process and multi-tooth cutting phenomena, and difficulty in combining with subsequent physical simulation models. This invention provides a new theoretical basis and practical calculation method for subsequent geometric and physical simulation in cycloidal milling, which will deepen the understanding of cycloidal milling in academia and industry and promote the widespread application of cycloidal milling technology in the cutting of difficult-to-machine materials in aerospace.

[0088] 2. The method of the present invention can solve both instantaneous chips and conventional chips. Attached Figure Description

[0089] Figure 1 It is Π i Typical two-dimensional undeformed chip geometry in layered cycloidal milling.

[0090] Figure 2 It is Δ1>(r+R) 2 Time П i Instantaneous two-dimensional undeformed chip geometry in cycloidal milling.

[0091] Figure 3 It is Δ1≤(r+R) 2 Time П i Instantaneous two-dimensional undeformed chip geometry in cycloidal milling.

[0092] Figure 4 Under multi-tooth cutting conditions, П i Instantaneous two-dimensional undeformed chip geometry on a layer.

[0093] Figure 5 It is a two-dimensional undeformed chip generated by the second cutting on layer Π1.

[0094] Figure 6 It is the cutting layer Π i Two-dimensional undeformed chip geometry generated on (i = 1, 2, ..., 11).

[0095] Figure 7 It is an instantaneous two-dimensional undeformed chip on layer Π1.

[0096] Figure 8 It is Π i Instantaneous two-dimensional undeformed chips on layers (i = 1, 2, ..., 11).

[0097] Figure 9 It is a three-dimensional undeformed chip generated in cycloidal milling.

[0098] Figure 10 It is the instantaneous, three-dimensional, undeformed chip generated during cycloidal milling. Detailed Implementation

[0099] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0100] The instantaneous three-dimensional undeformed chip geometry modeling method in cycloidal milling of the present invention specifically includes the following steps:

[0101] Step 1: Solve for any instantaneous tool center point O in cycloidal milling c (x c ,y c ,z c The material removal boundary surface S1 of the previous cycloidal cutting cycle, and the material removal boundary surfaces S2 and S3 to be generated in the current cycloidal cutting cycle:

[0102] Step 1.1 Establish the reference coordinate system CSO r and CSO q ;

[0103] In cycloidal milling, the tool cuts the workpiece along a cycloidal toolpath, which consists of intermittent and tangent circular arcs and straight line segments. To accurately describe the cycloidal toolpath, a reference coordinate system CSO is established to describe the cycloidal toolpath for the current cycloidal cycle. r (x r -y r -z r -O r () and the reference coordinate system CSO used to describe the cycloidal toolpath of the previous cycloidal cycle. q (x q -y q -z q -O q O r and O q O is the origin of the coordinate system. rLocated at the center of the cycloidal toolpath arc segment in the current cycloidal cutting cycle, O q Located at the center of the cycloidal toolpath arc segment of the previous cycloidal cutting cycle. Coordinate system x r -y r -z r -O r In the middle: coordinate axis x r Parallel to the straight segment of the cycloidal toolpath and pointing in the feed direction of the tool; coordinate axis z r Parallel to and perpendicular to the direction of the tool axis, pointing upwards, y r It is determined by the right-hand screw rule.

[0104] Coordinate System CSO q (x q -y q -z q -O q The directions of each coordinate axis in the diagram are related to x. r -y r -z r -O r Consistent and parallel.

[0105] 1.2 Calculate the tool center point O in the current cycloidal cutting cycle c (x c ,y c ,z c In coordinate system CSO r Coordinates in;

[0106] Define the point O as the center of the blade. c When the cycloidal toolpath passes through the starting point (-c, -r, 0) of the straight segment, t = 0. Therefore, in the current cycloidal cutting cycle, the tool center point O... c (x c ,y c ,z c In coordinate system CSO r The value can be calculated using the following formula:

[0107]

[0108] Where: c is the cycloidal step size; r is the cycloidal radius; t is the time parameter, in seconds; f is the feed rate, in mm / min; time parameter t l =60c / f,t t =60(c+2πr) / f.

[0109] 1.3 Calculate the material boundary S1 removed in the previous cycloidal cycle in coordinate system CSO r The expression in;

[0110] The material boundary S1(x) cut off in the previous cycloidal cycle s1 ,ys1 ,z s1 In coordinate system CSO r The middle can be represented by the following formula:

[0111]

[0112] Where, θ s1 The radian parameter of surface S1 and θ s1 ∈[0,π-arcsin(c / 2(R+r))], where R is the tool radius. When θ s1 When ∈[π-arcsin(c / 2(R+r)),2π], surface S1 does not exist, that is, the material boundary represented by surface S1 is removed by the previous cycloidal cutting cycle.

[0113] 1.4 Calculate the boundary surfaces S2 and S3 generated by the material to be cut during the current cycloidal cycle in coordinate system CSO. r The expression in;

[0114] The boundary surface S2(x) generated by the material to be cut in the current cycloidal cutting cycle s2 ,y s2 ,z s2 ) and S3(x s3 ,y s3 ,z s3 In coordinate system CSO r The terms can be represented by the following formulas:

[0115]

[0116] S3={(x s3 ,y s3 ,z s3 )|x s3 ∈[-c,0],y s3 =-(R+r),z s3 ∈[0,a p (1-4)

[0117] Where S2 is the surface generated by the tool cutting along the straight segment of the cycloidal toolpath, and S3 is the surface generated by the tool cutting along the circular arc segment of the cycloidal toolpath; θ s2 It is the radian parameter of surface S2, which is derived from y r The negative half of the axis is measured counterclockwise, and θ s2 ∈[0,π+arcsin(c / 2(R+r))];a p The axial depth of cut is expressed in mm.

[0118] Step 2: In coordinate system CSO r A local coordinate system CSO is established to describe the instantaneous two-dimensional undeformed chip geometry.ci,j

[0119] 2.1 Introducing the concept of layer cutting, using a series of parallel planes Π perpendicular to the cutter axis. i (i = 1, 2, ..., m) Simultaneously cuts the tool and the workpiece, plane Π i (i = 1, 2, ..., m) correspond to each cutting layer in cycloidal milling. To ensure solution accuracy, it is recommended that m ≥ 5. Among them, the plane Π i Defined in coordinate system CSO r In the middle, Π1 = 0, Π m =a p Π i =Z i Z i The value is determined by the method of dividing the parallel plane.

[0120] 2.2 In any cutting layer Π i Above, when the tool moves along the cycloidal toolpath, the tip point C of the cutting edge... i,k The actual trajectory of its motion is a cycloid. And in the cutting layer Π i In the above, the geometric boundary of the two-dimensional undeformed chip formed by the removal of material by the tool is jointly constituted by the movement trajectory of the tool tip and the material boundary.

[0121] In order to solve the cutting layer Π i To solve for the two-dimensional undeformed chip geometry boundary, it is necessary to find the intersections between cycloids and between cycloids and the material boundary. However, these intersections are often difficult to solve, or even lack theoretical solutions. When numerical methods are used, it is difficult to achieve both accuracy and efficiency.

[0122] Therefore, this invention uses a series of circles, each with a radius equal to the tool radius R, distributed according to a certain pattern to approximate the k-th tool tip point C. i,k In the cutting layer Π i The trajectory of movement on the surface.

[0123] To accurately describe the kth blade tip point C i,k To determine the instantaneous position, the cutting layer Π must first be obtained. i The kth blade tip point C i,k The trajectory of the circle S ei,j In coordinate system CSO r Center O ci,j coordinates (x) ci,j ,y ci,j ,z ci,j ) and the tip of the knife C i,k In the trajectory circle S ei,j instantaneous position angle θ fi,jWhere i represents the i-th cutting layer and the tool in the current cycloidal cutting cycle, j represents the j-th cutting of the workpiece, and k represents the k-th (k = 1, 2, ..., N) cutting edge of the tool. The value of k can be calculated by the following formula.

[0124]

[0125] Where mod represents the modulo operation, and N is the number of teeth on the cutting tool.

[0126] When the tool rotates one revolution, the k-th tool tip point C i,k In the cutting layer Π i The above will generate N circular motion trajectories of the tool tip. ei,j ,S ei,j+1 ,…,S ei,j+N-2 ,S ei,j+N-1 Then, the kth blade tip point C i,k The trajectory of the circle S ei,j Corresponding center point O ci,j In coordinate system CSO r coordinates (x) ci,j ,y ci,j ,z ci,j It can be calculated using the following formula:

[0127]

[0128]

[0129] Among them, t ci,j As a time parameter, its value can be calculated using the following formula:

[0130]

[0131] n is the spindle speed, measured in rpm;

[0132] q represents the total number of cuts made by the tool in one cycloidal cycle, and its value can be calculated by the following formula:

[0133]

[0134] t d i The time lag parameter introduced by the tool helix angle β represents the time t at which the tool tip on the same cutting edge begins cutting material on the cutting layer Π1. d i After a certain time, it will be in the cutting layer Π i The material cutting begins. That is, when i = 1, t d i =0; when i≠0, t d i The value can be calculated by the following formula:

[0135]

[0136] In the formula, Z i For cutting layer Π i In CSO r z in r coordinate.

[0137] 2.3 Based on the above formula, the cutting layer Π has been obtained. i The trajectory of the upper blade tip is a circle S. ei,j The center O ci,j (x ci,j ,y ci,j ,z ci,j At this time, in the cutting layer Π i Above the center O ci,j Establish q local coordinate systems CSO with the origin as the origin. ci,j To describe the cutting layer Π i The tip point C of the upper tool during the j-th cutting of the workpiece i,k The corresponding angle of entry θ si,j , tangent angle θ ei,j and instantaneous position angle θ fi,j .

[0138] Local coordinate system (CSO) ci,j x ci,j The shaft and the tool follow the cycloidal toolpath at O ci,j The feed directions at the location are parallel and in the same direction, and their z-axis is... ci,j axis and z r Parallel and in the same direction, their y ci,j The axis is determined by the right-hand rule.

[0139] Step 3: Calculate the cutting layer Π in cycloidal milling i The kth blade tip point C i,k In coordinate system CSO r instantaneous position O pi,k

[0140] C, the tip of the knife i,k instantaneous position O pi,k Its trajectory circle S can be determined by ei,j The center point O ci,j and its trajectory circle S ei,j instantaneous position angle θ fi,j The center point O is uniquely determined by a common factor. ci,j In coordinate system CSO r The coordinates in the equation have been obtained using formulas (2-2) and (2-3) from step 2.2 above. Instantaneous position angle θ fi,j It can be determined by the following rules:

[0141] Rule 3-1:

[0142] When the tip of the knife is C 1,1 Draw across the local coordinate system CSO c1,1 of y c1,1 When the axis is at its positive half-axis, define t = 0, and the tool tip point C 1,1 The trajectory of the circle S e1,1 The center of the circle is at O c1,1 .

[0143] Rule 3-2:

[0144] In the cutting layer Π i Above, tip C i,k Around its trajectory circle S ei,j The center point O ci,j From the local coordinate system CSO ci,j of y ci,j The positive half-axis rotates clockwise. From the tool tip point C... i,k Slide across y ci,j Within time t1 after the positive half-axis, the tool tip point C i,k The trajectory of the circle S ei,j The center point O ci,j The position remains unchanged, where t1 is the time required for the tool to rotate one revolution, t1 = 60 / n. That is, in the local coordinate system CSO... ci,j In the case of instantaneous position angle θ fi,j When ∈[0,2π), the knife tip point C i,k The trajectory of the circle S ei,j The center O ci,j The position remains unchanged.

[0145] Step 3.1 Using the above analysis, the first cutting edge point C of the tool... i,1 In the reference coordinate system CSO r instantaneous position O pi,1 (x pi,1 ,y pi,1 ,z pi,1 It can be calculated by the following formula.

[0146]

[0147] Where: j = int(t / t1)·N+1, int is the floor operator; ω is the angular velocity, ω = 2πn / 60; let

[0148] θ fi,j =ω·(t-int(t / t1)·t1) (3-2)

[0149] θ fi,j For the first blade tip point C i,1 In the local coordinate system CSO ci,jThe instantaneous position angle in (x) ci,j ,y ci,j ,z ci,j ) is the center O ci,j In the reference coordinate system CSO r The coordinates in the diagram.

[0150] Step 3.2 The kth (k≠1) tip point C of the tool i,k In the reference coordinate system CSO r The instantaneous position can be solved using rule 3-3.

[0151] Rule 3-3:

[0152] If ω·t<(k-1)·ψ p In the cutting layer Π i The kth blade tip point C i,k The motion trajectory does not exist, meaning no cutting occurred. Wherein, ψ p The inter-tooth angle and ψ p =2π / N. If ω·t≥(k-1)·ψ p And θ fi,j <(k-1)·ψ p Then at instant t, the cutting layer Π i The kth blade tip point C i,k In coordinate system CSO r instantaneous position O pi,k (x pi,k ,y pi,k ,z pi,k It can be calculated using the following formula:

[0153]

[0154] make

[0155] θ fi,j-N+k-1 =θ fi,j -(k-1)·ψ p +2π (3-4)

[0156] Where, θ fi,j-N+k-1 Let C be the kth tip point at instant t. i,k In coordinate system CSO ci,j-N+k-1 The instantaneous position angle in (x) ci,j+k-1 ,y ci,j+k-1 ,z ci,j+k-1 C is the kth blade tip point. i,k The trajectory of the circle S ei,j-N+k-1 The center O ci,j+k-1 In the reference coordinate system CSO r The coordinates in the diagram.

[0157] If ω·t≥(k-1)·ψp And θ fi,j ≥(k-1)·ψ p Then at instant t, the cutting layer Π i The kth blade tip point C i,k In the reference coordinate system CSO r instantaneous position O pi,k (x pi,k ,y pi,k ,z pi,k It can be calculated using the following formula:

[0158]

[0159] make

[0160] θ fi,j+k-1 =θ fi,j -(k-1)·ψ p (3-6)

[0161] Where, θ fi,j+k-1 Let C be the kth tip point at instant t. i,k In the local coordinate system CSO ci,j+k-1 The instantaneous position angle in (x) ci,j+k-1 ,y ci,j+k-1 ,z ci,j+k-1 C is the kth blade tip point. i,k The trajectory of the circle S ei,j+k-1 The center O ci,j+k-1 In the reference coordinate system CSO r The coordinates in the diagram.

[0162] Step 4: Solve for the cutting layer Π in cycloidal milling i The kth blade tip point C i,k Parametric expression of the instantaneous two-dimensional undeformed chip geometry generated at any instant t

[0163] Step 4.1 Cutting layer Π i Two-dimensional geometric analysis of the undeformed chip boundary:

[0164] Traditional undeformed chip modeling methods only study the undeformed chips generated by one revolution of a single cutting edge (i.e., a single cutting edge), making it difficult to solve the geometric model of undeformed chips generated at any position of the current cutting edge during the cutting process. Here, this invention first gives the definition of instantaneous undeformed chips:

[0165] Definition 4-1:

[0166] At any instant t, there are multiple cutting edges on the tool that are engaged with the workpiece. The geometry of the undeformed chips that have been cut off within the time it takes for these cutting edges to rotate one revolution is called the instantaneous undeformed chip.

[0167] To solve for the cutting layer Π in cycloidal milling i For instantaneous two-dimensional undeformed chips, the cutting layer Π must first be determined. i The undeformed chip boundary generated by a single cutting edge rotating once can be divided into two categories:

[0168] The first category, such as Figure 1 As shown in Figure (a), the two-dimensional undeformed chip boundary is formed by the material removal boundary s. i (The material removal surface and cutting layer formed in the previous cycloidal cutting cycle Π) i The curve P on the intersection line) si,j P si,j-1 S ei,j-1 curve P on si,j-1 P ei,j and S ei,j curve P on si,j P ei,j Enclosed;

[0169] The second category, such as Figure 1 As shown in Figure (b), the two-dimensional undeformed chip boundary is formed by s i curve P on si,j P ei,j 'and S ei,j curve P on si,j P ei,j 'Enclosed'.

[0170] For ease of description, the cutting layer Π in cycloidal milling is given below. i Definitions of the inner and outer boundaries of the undeformed chip in two dimensions:

[0171] Definition 4-2:

[0172] If the two-dimensional undeformed chip boundary curve (family) is at the k-th tool tip point C i,k The trajectory of the circle S ei,j If the curve (family) is above the boundary, then it is called the outer boundary; otherwise, it is called the inner boundary.

[0173] based on Figure 1 and two As can be seen from the definitions of the inner and outer boundaries of the two-dimensional undeformed chip, the outer boundary of the two-dimensional undeformed chip in cycloidal milling is only determined by the boundary at point S. ei,j It is formed by a single curve on the upper boundary; its inner boundary is formed by one or two curves.

[0174] Step 4.2 Determine the cutting layer Π i The number of curves constituting the inner boundary of the two-dimensional undeformed chip:

[0175] Here, rule 4-1 is given to determine the number of curves that constitute the inner boundary of a two-dimensional undeformed chip.

[0176] Rule 4-1:

[0177] make

[0178] Δ1=(x ei,j +c) 2 +y ei,j 2 (4-1)

[0179] Among them, (x ei,j ,y ei,j S is the circle representing the trajectory of the k-th tool tip during the j-th cut. ei,j The trajectory circle S of the k-th tool tip during the (j-1)-th cut. ei,j-1 intersection point P ei,j The coordinates of P ei,j Also known as the cut-out point of the j-th cut. If Δ1>(r+R) 2 The inner boundary of the two-dimensional undeformed chip is composed of two curves, namely P. si,j P si,j-1 and P si,j-1 P ei,j If Δ1≤(r+R) 2 Then the inner boundary of the two-dimensional undeformed chip is determined by only one curve P. si,j P ei,j 'constitute.

[0180] Step 4.3 Solving for the cutting layer Π in cycloidal milling i Parametric representation of traditional two-dimensional undeformed chip geometry:

[0181] Cutting layer Π i The traditional two-dimensional undeformed chip geometry refers to the undeformed chip produced by a single cutting edge of a tool rotating once.

[0182] Based on the aforementioned Definition 4-1, Definition 4-2 and Rule 4-1, the mathematical parameter expression of the traditional two-dimensional undeformed chip boundary curve can be obtained. The following discussion will focus on two cases based on the value of Δ1.

[0183] (1) If Δ1≤(r+R) 2 The outer boundary of the two-dimensional undeformed chip is at S ei,j curve P on si,j P ei,j In the local coordinate system CSO ci,j The middle can be represented as:

[0184] P si,j P ei,j ′={S ei,j |θ∈[θ si,j ,θ ei,j (4-2)

[0185] Where, θ si,j and θ ei,j 'These are the cutting layers Π i The entry and exit angles of the j-th cut. The inner boundary of the two-dimensional undeformed chip is at s. i curve P on si,j P ei,j 'In CSO q The middle can be represented as:

[0186] P si,j P ei,j ′={s i |θ∈[θ qi,j ′,θ qi,j (4-3)

[0187] Where, θ qi,j ' and θ qi,j The tangent points are P and P respectively. ei,j 'and the entry point P si,j In the reference coordinate system CSO q The parameter expression in the text.

[0188] (2) If Δ1>(r+R) 2 The outer boundary of the two-dimensional undeformed chip is at S ei,j curve P on si,j P ei,j In the local coordinate system CSO ci,j The middle can be represented as:

[0189] P si,j P ei,j ={S ei,j |θ∈[θ si,j ,θ ei,j (4-4)

[0190] Where, θ ei,j For cutting layer Π i The cut angle of the j-th cut. The inner boundary of the two-dimensional undeformed chip is formed by two curves, namely P si,j P si,j-1 and P si,j-1 P ei,j Composition, located in s i curve P on si,j P si,j-1 In CSO q The middle can be represented as:

[0191] P si,j P si,j-1 ={s i |θ∈[θ qi,j-1 ,θ qi,j (4-5)

[0192] Where, θ qi,j-1 Let P be the intersection point. si,j-1 In the reference coordinate system CSO q The parameter expression in the text.

[0193] Located in S ei,j-1 curve P on si,j-1 P ei,j In CSO ci,j-1 The middle can be represented as:

[0194] P si,j-1 P ei,j ={S ei,j-1 |θ∈[θ si,j-1 ,θ bi,j-1 (4-6)

[0195] Where, θ si,j-1 For cutting layer Π i The angle of entry for the (j-1)th cut; θ bi,j-1 For cutting layer Π i The cut angle of the j-th cut at CSO ci,j-1 The parameter expression in the text.

[0196] Step 4.4 Solving for the cutting layer Π in cycloidal milling i The parametric expression for the instantaneous two-dimensional undeformed chip geometry generated at any instant t:

[0197] The above steps have yielded the cutting layer Π in cycloidal milling. i The kth blade tip point C i,k The parametric expression for the geometry of the two-dimensional undeformed chip cut after one revolution. Based on this, the cutting layer Π in cycloidal milling can be solved. i The kth blade tip point C i,k The parametric expression of the instantaneous two-dimensional undeformed chip geometry generated at any instant t is discussed in detail below, depending on the case.

[0198] (1) When Δ1>(r+R) 2 When, if the kth blade tip point C i,k instantaneous position angle θ fi,j ∈[θ si,j ,θ ei,j ], i.e., the tip of the knife C i,k The material is being cut, generating instantaneous chips. Based on the aforementioned analysis, the cutting layer Π in cycloidal milling... i The kth blade tip point C i,k The instantaneous two-dimensional undeformed chip boundaries generated are as follows: Figure 2 Figure (a) and Figure 2 As shown in Figure (b). Figure 2In Figure (a), the instantaneous two-dimensional undeformed chip boundary enclosing the shaded area is composed of three curves, namely curve K on the rake face. i, j F i,j At the tip of the knife, point C i,k The trajectory circle S ei,j curve P on si,j F i,j and at the material removal boundary s i curve P on si, j K i,j Among them, F i,j Let C be the kth blade tip point. i,k The instantaneous position of K i,j C i,k The trajectory of the circle S ei,j In F i,j The normal vector F at the location i,j K i,j With material removal boundary s i The intersection. Figure 2 In Figure (b), the instantaneous two-dimensional undeformed chip boundary enclosing the shaded area is composed of four curves, namely curve K on the rake face. i,j F i,j At the tip of the knife, point C i,k The trajectory circle S ei,j curve P on si,j F i,j Located at the material removal boundary s i curve P on si,j P si,j-1 and the trajectory circle S ei,j-1 curve P on si,j-1 K i,j Rule 4-2 is given here to determine when Δ1>(r+R). 2 At that time, the kth blade tip point C i,k At any instant t, in the cutting layer Π i The instantaneous two-dimensional undeformed chips generated are formed by several curves.

[0199] Rule 4-2:

[0200] If the trajectory is a circle S ei,j-1 With material removal boundary s i intersection point P si,j-1 Located in vector F i,j K i,j Above or at vector F i,j K i,j If the left side is the kth blade tip point C, then at this time... i,k The generated instantaneous two-dimensional undeformed chips are bounded by three curves, namely K. i,j Fi,j P si,j F i,j and P si,j K i,j .

[0201] If the trajectory is a circle S ei,j-1 With material removal boundary s i intersection point P si,j-1 Located in vector F i,j K i,j If the right side is [the location of the kth blade tip C], then the kth blade tip point C is [the location of the i,k The generated instantaneous two-dimensional undeformed chips are bounded by four curves, namely K. i,j F i,j P si,j F i,j P si, j P si,j-1 and P si,j-1 K i,j .

[0202] Based on the foregoing analysis and rule 4-2, at any instant t in cycloidal milling, the cutting layer Π i The kth blade tip point C i,k The instantaneous two-dimensional undeformed chip boundary, such as Figure 2 As shown, it can be obtained from the following formula.

[0203] Curve P si,j F i,j In the local coordinate system CSO ci,j The middle can be represented as:

[0204] P si,j F i,j ={S ei,j |θ∈[θ si,j ,θ fi,j ]} (4-7)

[0205] Where, θ fi,j Let C be the kth blade tip point. i,k instantaneous position point F i,j In the local coordinate system CSO ci,j The parameter expression in θ si,j For cutting layer Π i The angle of entry for the j-th cut.

[0206] Curve P si,j K i,j In coordinate system CSO q The middle can be represented as:

[0207] P si,j K i,j ={s i |θ∈[θki,j ′,θ qi,j (4-8)

[0208] Where, θ ki,j 'For point K i,j In coordinate system CSO q The parameter expression in the text.

[0209] Curve P si,j P si,j-1 In CSO q The mathematical expression for this has been given in the preceding analysis, see formula (4-5) above.

[0210] Curve P si,j-1 K i,j In coordinate system CSO ci,j-1 The middle can be represented as:

[0211] P si,j-1 K i,j ={S ei,j-1 |θ∈[θ si,j-1 ,θ ki,j-1 (4-9)

[0212] Where, θ ki,j-1 For the normal vector F i,j K i,j and material removal boundary s i The intersection point in coordinate system CSO ci,j-1 The parameter expression in the text.

[0213] The straight line F on the rake face i,j K i,j It can be derived from vector F i,j K i,j This indicates that it is in CSO r The middle can be represented as:

[0214] F i,j K i,j =(-R·sin(θ) fi,j -θ ci,j ),-R·cos(θ fi,j -θ ci,j (4-10)

[0215] Where, θ ci,j Let C be the kth blade tip point. i,k The trajectory of the circle S ei,j Corresponding center point O ci,j The instantaneous position angle. When the center point O... ci,j When it is on the straight section of the cycloidal toolpath, θ ci,j =0; when the center point O ci,j When it is in the arc segment of the cycloidal toolpath, θci,j The value is determined by the line segment O. ci,j 'O r Rotate counterclockwise to the y r The angle rotated when the axes coincide. Where O ci,j 'O r For O ci,j O r In the coordinate plane x r -O r -y r The projection on the surface.

[0216] (2) When Δ1≤(r+R) 2 At that time, the cutting layer Π i The kth blade tip point C i,k The instantaneous two-dimensional undeformed chip boundary is as follows: Figure 3 As shown, it is bounded by 3 curves, namely P si,j F i,j F i,j K i,j and K i,j P si,j Its mathematical expression has been given in the preceding analysis, specifically see formulas (4-7), (4-10) and (4-8).

[0217] At this point, the cutting layer Π has been determined. i The kth blade tip point C i,k The parametric expression for the instantaneous two-dimensional undeformed chip geometry generated at any instant t, if the cutting layer Π is solved... i The angle of entry θ for the j-th cut si,j Cutting layer Π i The cutting angle θ of the j-th cut ei,j or θ ei,j '、Cutting layer Π i The angle of entry θ for the (j-1)th cut si,j-1 ,、cutting layer Π i The cutting angle θ of the j-th cut bi,j-1 Entry point P si,j In CSO q The parameter expression θ in qi,j , tangent point P ei,j 'In CSO q The parameter expression θ in qi,j '、The trajectory circle S ei,j-1 With material removal boundary s i intersection point P si,j-1 In CSO q The parameter expression θ in qi,j-1 The kth blade tip C i,k instantaneous position point F i,j In the local coordinate system CSOci,j The parameter expression θ in fi,j Point K i,j In coordinate system CSO q The parameter expression θ in ki,j 'and normal vector F i,j K i,j and material removal boundary s i The intersection point in coordinate system CSO ci,j-1 The parameter expression θ in ki,j-1 Then the cutting layer Π in cycloidal milling can be obtained. i An instantaneous two-dimensional undeformed chip at any instant t.

[0218] The methods for solving the above parameters will be explained in subsequent steps.

[0219] Step 5: Solve for the cutting layer Π in cycloidal milling i Parametric expression for instantaneous two-dimensional undeformed chip geometry under multi-tooth cutting conditions

[0220] To solve for the parametric expression of the instantaneous two-dimensional undeformed chip geometry during multi-tooth cutting, the cutting layer Π is first given. i Judgment rules for when multi-tooth cutting occurs.

[0221] Rule 5-1:

[0222] At any instant t, the cutting layer Π i If more than one tool tip has an instantaneous position angle within the range of the corresponding entry and exit angles, then multi-tooth cutting will occur at the cutting layer Π. i It happened above.

[0223] According to rule 5-1, if multi-tooth cutting occurs, the solution method for expressing the instantaneous two-dimensional undeformed chip geometry parameters generated by a single cutting edge can be applied to each cutting edge that participates in the cutting simultaneously.

[0224] Cutting layer Π in cycloidal milling i Typical instantaneous two-dimensional undeformed chips generated during multi-tooth cutting, such as Figure 4 As shown. Point C of the blade tip. i,1 C i,2 and C i,3 The instantaneous positions are represented by marked pentagrams, and it is clear that at this moment, the tip of the knife, C... i,1 and C i,2 Both chips engage with the workpiece material, resulting in the generation of two instantaneous two-dimensional undeformed chips. These two instantaneous two-dimensional undeformed chips can be solved separately according to the method described in step 4.

[0225] Step 6: Solve for the parameters in the parametric expression of the instantaneous two-dimensional undeformed chip geometry in cycloidal milling, and thus obtain the instantaneous two-dimensional undeformed chip geometry.

[0226] As can be seen from steps 4 and 5, in order to solve the cutting layer Π in cycloidal milling i To obtain the instantaneous two-dimensional undeformed chip geometry, it is necessary to first determine the key parameter in the expression describing the chip boundary, namely the entry point P of the j-th cut. si,j In CSO respectively ci,j and CSO q The parameter expression θ in si,j and θ qi,j ; or the entry point P of the (j-1)th cut. si,j-1 In CSO respectively ci,j-1 and CSO q The parameter expression θ in si,j-1 and θ qi,j-1 ; Tangent point P ei,j In CSO respectively ci,j and CSO ci,j-1 The parameter expression θ in ei,j and θ bi,j-1 ; or the tangent point P ei,j 'In CSO respectively ci,j and CSO q The parameter expression θ in ei,j ' and θ qi,j ';Knife tip point C i,k Instantaneous position at CSO ci,j The parameter expression θ in fi,j ; and intersection point K i,j In CSO respectively q and CSO ci,j-1 The parameter expression θ in ki,j ' and θ ki,j-1 .

[0227] The solution methods are given below.

[0228] ①Entry point P si,j and P si,j-1 Solving for corresponding parameters

[0229] P si,j In coordinate system CSO ci,j and CSO q The parameter expression θ in si,j and θ qi,j It can be obtained by solving the following equation.

[0230]

[0231] θ qi,j It can be represented as:

[0232]

[0233] θ si,j It can be represented as:

[0234]

[0235] Where, θ qi,j =max{θ qi,j 1 ,θ qi,j 2}, θ si,j =min{θ si,j 1 ,θ si,j 2 Note that when t∈[0,t... l When θ ci,j =0.

[0236] ② Similarly, the entry point P si,j-1 The corresponding parameter θ qi,j-1 and θ si,j-1 It can also be solved by the above method, simply by removing (x) from formulas (6-1) to (6-3). ci,j ,y ci,j ) and θ ci,j Using (x) ci,j-1 ,y ci,j-1 ) and θ ci,j-1 Replacement; θ ci,j-1 Let the trajectory circle S be ei,j-1 Corresponding center point O ci,j-1 The angular position of the location.

[0237] ③ Tangent point P ei,j and P ei,j Solving for the corresponding parameters

[0238] (1) When Δ1>(r+R) 2 At that time, P ei,j In coordinate system CSO ci,j and CSO ci,j-1 The parameter expression θ in ei,j and θ bi,j-1 It can be obtained from the following equation.

[0239]

[0240] θ bi,j-1 It can be represented as:

[0241]

[0242] Where A = x ci,j-1 -xci,j B = y ci,j-1 -y ci,j ;θ bi,j-1 =max{θ bi,j-1 1 ,θ bi,j-1 2}

[0243] Based on the above equation θ bi,j-1 The value and equation (6-4), θ ei,j It can be represented as:

[0244]

[0245] Where, θ ei,j =min{θ ei,j 1 ,θ ei,j 2}

[0246] (2) When Δ1≤(r+R) 2 At this time, the inner boundary of the two-dimensional undeformed chip geometry consists of only one curve. At this time, the cut-out point P... ei,j 'Is S ei,j and s i The intersection point, in coordinate system CSO ci,j and CSO q The middle is composed of θ respectively ei,j ' and θ qi,j ' indicates. θ ei,j ' and θ qi,j It can be obtained from formulas (6-3) and (6-2) respectively, only θ ei,j '=min{θ si,j 1 ,θ si,j 2}、θ qi,j '=min{θ qi,j 1 ,θ qi,j 2}

[0247] Thus, the cutting layer Π in cycloidal milling has been determined. i All key parameters of the two-dimensional undeformed chip geometry generated by rotating the upper tool tip once.

[0248] Next, we will continue to solve for the other parameters required for the instantaneous two-dimensional undeformed chip geometry.

[0249] ④ Instantaneous blade tip position F i,j Solving for corresponding parameters

[0250] F i,j It is the cutting layer Π iAt any instant t, the kth blade tip point C i,k The corresponding position, in coordinate system CSO ci,j The corresponding position angle is θ fi,j Its value can be obtained from formula (3-2), formula (3-4), or formula (3-6) according to different situations based on rules 3-1 to 3-3.

[0251] ⑤ Intersection point K i,j Solving for corresponding parameters

[0252] K i,j It is the inner boundary of the chip and the normal vector F i,j K i,j The intersection point, the normal vector F i,j K i,j In CSO r The expression can be represented by the following formula.

[0253] y = cot(θ) fi,j -θ ci,j )[xx ci,j -R·sin(θ fi,j -θ ci,j )]+y ci,j +R·cos(θ fi,j -θ ci,j (6-7)

[0254] (1) If K i,j It is the normal vector F i,j K i,j With material removal boundary s i The intersection of K, then i,j In CSO q The expression of θ in the middle parameter ki,j It can be obtained from the following equation.

[0255]

[0256] θ ki,j 'Can be represented as:

[0257]

[0258] Where, θ u =π / 2-θ fi,j +θ ci,j θ ki,j '=min{θ ki,j ' 1 ,θ ki,j ' 2}

[0259] (2) If K i,j It is the normal vector F i,j Ki,j With S ei,j-1 The intersection of K, then i,j In CSO ci,j-1 The expression of θ in the middle parameter ki,j-1 It can be obtained from the following equation.

[0260]

[0261] θ ki,j-1 It can be represented as:

[0262]

[0263] Where A = x ci,j-1 -x ci,j B = y ci,j-1 -y ci,j If the center point of the trajectory of the knife tip is (O) ci,j Or O ci,j-1 If the point is on the straight segment of the cycloidal toolpath, then the center point of the corresponding motion trajectory is at CSO. r The position angle (θ) in ci,j or θ ci,j-1 ) is 0; θ ki,j-1 ={θ ki,j-1 1 |θ ki,j-1 1 ∈[θ si,j-1 ,θ bi,j-1 ]} or θ ki,j-1 ={θ ki,j-1 2 |θ ki,j-1 2 ∈[θ si,j-1 ,θ bi,j-1 ]}.

[0264] Based on the above steps, the cutting layer Π in cycloidal milling was obtained. i All key parameters in the instantaneous two-dimensional undeformed chip geometry expression are substituted into the cutting layer Π obtained in the previous steps. i From the instantaneous two-dimensional undeformed chip geometry expression, the cutting layer Π can be obtained. i The instantaneous two-dimensional undeformed chip geometry.

[0265] Step 7: Solve the instantaneous three-dimensional undeformed chip geometry in cycloidal milling.

[0266] Based on steps 3-6, the cutting layer Π at any instant t during cycloidal milling can be obtained. i The instantaneous two-dimensional undeformed chip geometry generated by the cutting tool. By repeating the above method, the instantaneous two-dimensional undeformed chip geometry generated by the cutting tool on all cutting layers at any instant t can be obtained.

[0267] Then, the instantaneous two-dimensional undeformed chip geometry on all cutting layers is obtained based on its z-axis. r By stacking the coordinates, the instantaneous three-dimensional undeformed chip geometry generated by the tool cutting at any instant t can be obtained.

[0268] Example:

[0269] The cutting tool used in this embodiment is a solid end mill with a radius R = 5 mm, a helix angle β = 38°, and 3 teeth with equal tooth angles. The cutting parameters used are a feed rate f = 900 mm / min, a spindle speed n = 300 rpm, and a radial depth of cut a. p =5mm. The cycloidal parameters used are cycloidal radius r = 4mm and cycloidal step length c = 8mm.

[0270] Based on the method of the present invention, the instantaneous three-dimensional undeformed chip geometry modeling method in cycloidal milling in this embodiment specifically includes the following steps:

[0271] Step 1. Solve for any instantaneous tool center point O in cycloidal milling. c (x c ,y c ,z c The material removal boundary surface S1 of the previous cycloidal cutting cycle, and the material removal boundary surfaces S2 and S3 to be generated in the current cycloidal cutting cycle;

[0272] Step 1.1 Establish the reference coordinate system CSO r and CSO q CSO q Origin q In CSO r The coordinates in the graph are (-8, 0, 0).

[0273] Step 1.2 Define the point O of the cutting edge c When the line passes through the starting point (-8, -4, 0) of the current cycloidal toolpath segment, t = 0s. Then, at CSO... r In the current cycloidal period, at any instant t, the center point O is... c In CSO r The coordinates in the equation can be calculated using the aforementioned formula (1-1);

[0274] Step 1.3 The boundary S1 of the material removed in the previous cycloidal cutting cycle is CSO r The coordinates in the equation can be calculated using the aforementioned formula (1-2).

[0275] Step 1.4 The current cycloidal cutting cycle generates boundary surfaces S2 and S3 in coordinate system CSO. r The terms can be represented by the aforementioned formulas (1-3) and (1-4), respectively;

[0276] Step 2. Solving for Π in cycloidal milling i The local coordinate system CSO describes the instantaneous two-dimensional undeformed chip geometry on the layer. ci,j In CSO r The representation in:

[0277] Step 2.1 Based on the concept of layer cutting, this embodiment utilizes 11 parallel planes Π with equal spacing and perpendicular to the cutter axis. i (i = 1, 2, ..., 11) Simultaneously cuts the tool and the workpiece, plane Π i (i = 1, 2, ..., 11) correspond to the cutting layers in cycloidal milling. Among them, planes Π1, Π3 and Π 11 In CSO r The values ​​can be represented as Z1 = 0 mm, Z3 = 1 mm, and Z... 11 =5mm.

[0278] Step 2.2 in Π i On the layer, a series of circles S with a radius of 5 mm distributed according to a certain pattern are used. ei,j The center of the circle is O ci,j To approximate the knife tip point C i,k In Π i The motion trajectory on the layer. Where i represents the i-th cutting layer and the tool in the current cycloidal cutting cycle, j represents the j-th cutting of the workpiece, and k represents the k-th cutting edge of the tool. The value of k can be obtained by the aforementioned formula (2-1).

[0279] C, the tip of the knife i,k In Π i The motion trajectory circle S on the layer ei,j The center O ci,j In coordinate system CSO r coordinates (x) ci,j ,y ci,j ,z ci,j It can be obtained from the aforementioned formula (2-6);

[0280] Step 2.3 is in Π i The tool tip point C obtained on the layer i,k The trajectory of the circle S ei,j The center O ci,j (x ci,j ,y ci,j ,z ci,j Establish q local coordinate systems CSO with ) as the origin. ci,j To describe the tip point C i,k The angle of entry θ si,j , tangent angle θ ei,j and instantaneous position angle θ fi,j Coordinate system CSO ci,j xci,j The shaft and the tool are in the cycloidal toolpath at O ci,j The feed directions at each location are parallel and consistent, and their z-axis is... ci,j axis and z r Parallel and in the same direction, their y ci,j The axis is determined by the right-hand rule.

[0281] Step 3. Solve for Π i C on the blade tip i,k In CSO r instantaneous position O pi,k coordinate

[0282] C, the tip of the knife i,k instantaneous position O pi,k The coordinates can be derived from the motion trajectory circle S. ei,j The center O ci,j The coordinates (already obtained in step 2) and the tool tip C i,k instantaneous position angle θ fi,j The only certainty. Point C (the tip of the blade). i,k instantaneous position angle θ fi,j The value can be determined by solving the aforementioned rules 3-1 and 3-2.

[0283] Step 3.1 Based on the aforementioned rules 3-1 and 3-2, the first cutting edge point C of the tool... i,1 In the reference coordinate system CSO r instantaneous position O pi,1 (x pi,1 ,y pi,1 ,z pi,1 ) is calculated using the aforementioned formula (3-1);

[0284] Step 3.2 Based on the aforementioned rule 3-3, the k-th (k≠1) tool tip point C of the tool... i,k In the reference coordinate system CSO r Coordinate O in pi,k Solve using formula (3-3) or (3-5).

[0285] Step 4. Solve for Π i Parametric representations of traditional two-dimensional undeformed chip geometry on layers and parametric representations of instantaneous two-dimensional undeformed chip geometry.

[0286] Step 4.1 Analyze the geometric composition of the chip boundary;

[0287] Step 4.2 Determine the number of curves formed by the inner boundary of the chip according to the aforementioned formula (4-1).

[0288] Step 4.3 Based on Definition 4-2 and Rule 4-1, calculate Π. i The parametric expression of the curve bounded by the geometry of traditional two-dimensional undeformed chips on a layer, namely Πi The parametric expression of the two-dimensional undeformed chip geometry generated by any single cutting edge of the tool rotating once is specifically expressed by the above formulas (4-2)-(4-6);

[0289] Step 4.4 Based on the aforementioned definition 4-1 and the aforementioned parameter expression of traditional two-dimensional undeformed chip geometry, the parameter expression of instantaneous two-dimensional undeformed chip geometry is solved, specifically as shown in the above formulas (4-7)-(4-10).

[0290] Step 5. Solve for Π i Parametric expression of instantaneous two-dimensional undeformed chip geometry under multi-tooth cutting conditions on a multi-layer surface.

[0291] According to rule 5-1 above, if multi-tooth cutting occurs, the instantaneous two-dimensional undeformed chip geometry generated by a single cutting edge can be calculated for each cutting edge that participates in cutting simultaneously.

[0292] Step 6. Solve for the key parameters in the instantaneous two-dimensional undeformed chip geometry expression during cycloidal milling.

[0293] ①Entry point P si,j In coordinate system CSO ci,j and CSO q The parameter expression θ in si,j and θ qi,j It can be obtained from the aforementioned formulas (6-3) and (6-2), respectively;

[0294] ②Entry point P si,j-1 In CSO respectively ci,j-1 and CSO q The parameter expression θ in si,j-1 and θ qi,j-1 It can also be calculated using formulas (6-3) and (6-2), simply by removing θ from formulas (6-3) to (6-2). ci,j and (x) ci,j ,y ci,j ) respectively using θ ci,j-1 and (x) ci,j-1 ,y ci,j-1 )replace.

[0295] ③When Δ1>(r+R) 2 At that time, cut point P ei,j In CSO respectively ci,j-1 and CSO ci,j The parameter expression θ in bi,j-1 and θ ei,j The value of Δ1 can be obtained from the aforementioned formulas (6-5) to (6-6), or formulas (6-3) to (6-2);

[0296] ④ Knife tip point C i,kinstantaneous position F i,j Corresponding to the coordinate system CSO ci,j The parameter θ fi,j The part of formula (3-2) in step 3 has been obtained.

[0297] ⑤ Intersection point K i,j It is the inner boundary of the chip and the normal vector F i,j K i,j The intersection of K. i,j It is F i,j K i,j With s i The intersection of K, then i,j In coordinate system CSO q and CSO ci,j-1 The parameter expression θ in ki,j ' and θ ki,j-1 It can be obtained from the aforementioned formulas (6-8) and (6-9) respectively.

[0298] Based on the above process, the value of Π in cycloidal milling can be obtained. i Solving the geometric parameter expression for the instantaneous undeformed chip on the layer involves all parameters.

[0299] Based on the above calculation method and the given cycloidal parameters, cutting parameters, and tool parameters, the undeformed chips generated by the second cut on layer Π1 can be calculated. Since Δ1 < 0, according to rule 4-1 and definition 4-2, the inner boundary of the chip is determined by curve P on s1. s1,2 P e1,2 The outer boundary of the chip is formed by the location of S. e1,2 curve P on s1,2 P e1,2 Composition. θ is obtained by combining the aforementioned method for solving the key parameters defining the chip boundary curve. s1,2 =147.03°, θ e1,2 =184.89°, θ q1,2 =24.42°, θ q1,2 =3.65°. Substituting the above parameters into formulas (4-2) to (4-3) yields the two-dimensional undeformed chips generated by the tool during the second cutting of the material on layer Π1. Figure 5 As shown.

[0300] All cutting layers Π can be obtained using the same method. i On layer (i = 1, 2, ..., 11), the geometry of all two-dimensional undeformed chips generated by the tool cutting material during the third cutting operation and one cycloidal cycle is as follows: Figure 6 As shown. Figure 6 The cross symbol in the middle represents the cutting edge of the tool at the Π i The center point of the motion trajectory on the (i = 1, 2, ..., 11) layer.

[0301] The following solution calculates the instantaneous two-dimensional undeformed chips generated by the tool cutting the material on layer Π1 at t = 0.6076s and t = 0.6178s.

[0302] When t = 0.6076s, the tip point C is obtained according to formulas (3-2) to (3-6). 1,1 C 1,2 C 1,3 The corresponding instantaneous position angles are θ f1,10 =13.64°, θ f1,8 =253.64°, θ f1,9 = 133.64°. Based on formulas (6-1) and (6-4), the entry and exit angles of the tool for the 8th, 9th, and 10th cuts on layer Π1 are respectively (θ...). s1,8 ,θ e1,8 )=(23.34°,185.74°),(θ s1,9 ,θ e1,9 )=(11.00°,185.74°),(θ s1,10 ,θ e1,10 ) = (13.64°, 192.89°).

[0303] Clearly, at t = 0.6076s, the knife tip point C... 1,1 The cutting tip has just come into contact with the material and has not yet produced any chips; point C. 1,2 The workpiece has been cut out, and the resulting undeformed chips P s1,8 P e1,8 P s1,7 This can be expressed by formulas (4-4) to (4-6), in Figure 7 In figure (a), curve P s1,8 P e1,8 P e1,8 P s1,7 and P s1,7 P s1,8 The enclosed shadow represents the tip of the knife, point C. 1,3 The material is being cut, and the instantaneous, undeformed chip geometry P is being generated. s1,9 F 1,9 E 1,9 P s1,8 This can be expressed by formulas (4-5) to (4-9), in Figure 7 Graph (a) is composed of curve P s1,9 F 1,9 F 1,9 E 1,9 E 1,9 P s1,8 and P s1,8 P s1,9 The enclosed shadow represents...

[0304] When t = 0.6187s, the tip point C is obtained according to formulas (6-1) to (6-4). 1,1 C 1,2 C 1,3 The corresponding instantaneous position angles are θ f1,10 =33.67°, θ f1,8 =273.67°, θ f1,9 = 153.67°. Because θ f1,10 θ f1,8 and θ f1,9 Since all points belong to [0, 2π), according to rule 3-2, the center O of the trajectory of each blade tip is... c1,10 O c1,8 and O c1,9 The positions remain unchanged, meaning the cut-out and cut-in angles at each tool tip are the same as at t = 0.6076s. Clearly, at t = 0.6187s, according to rule 5-1, multi-tooth cutting occurs on layer Π1, i.e., at tool tip C... 1,1 and C 1,3 Simultaneously cutting the material. The tool tip point C... 1,1 The instantaneous undeformed chip geometry P s1,10 F 1,10 E 1,10 P s1,9 This can be expressed by formulas (4-5) to (4-9), in Figure 7 In figure (b), curve P s1,10 F 1,10 F 1,10 E 1,10 E 1,10 P s1,9 and P s1,9 P s1,10 The enclosed shadow represents the tip of the knife, point C. 1,2 The workpiece has been cut out, and the undeformed chips generated are consistent with those at t = 0.6076s; the tool tip point C... 1,3 The instantaneous two-dimensional undeformed chip geometry P s1,9 F 1,9 E 1,9 P s1,8 This can be expressed by formulas (4-5) to (4-9), in Figure 7 In figure (b), curve P s1,9 F 1,9 F 1, 9E 1,9 E 1,9 P s1,8 and P s1,8 P s1,9 The enclosed shadow represents...

[0305] All cutting layers Π can be obtained using the same method.i On layer (i = 1, 2, ..., 11), the instantaneous two-dimensional undeformed chips generated by the tool cutting the material at t = 0.6076s and t = 0.6187s are respectively as follows: Figure 8 Figure (a) and Figure 8 As shown in Figure (b). When t = 0.6076 s, as... Figure 8 As shown in Figure (a), only the cutting edge C i,3 Cutting material on all layers, cutting edge C i,1 and C i,2 No material was cut on any cutting layer. At t = 0.6187 s, as... Figure 8 As shown in Figure (b), the cutting edge C i,3 Material is being cut in all layers, with cutting edge C i,1 Material cutting occurred only on layers P1 to P5, specifically at t = 0.6187 s, where multi-tooth cutting occurred on layers P1 to P5, with two chips generated simultaneously. C i,2 No material was cut on any layer.

[0306] Step 7. Solve for the instantaneous three-dimensional undeformed chip geometry in cycloidal milling.

[0307] Based on the above analysis, the arbitrary layer Π has been obtained. i The undeformed chips generated by one revolution of a certain cutting edge of the upper tool, and at any instant t, Π i The instantaneous two-dimensional undeformed chips generated by tool cutting on each layer. By repeating the above method on different cutting layers, the two-dimensional undeformed chips on all layers can be obtained, and then they can be determined based on z. r By stacking the coordinates, we can obtain the three-dimensional undeformed chip generated by a certain cutting edge of the tool rotating one revolution, as well as the three-dimensional instantaneous undeformed chip generated by the tool at any instant t.

[0308] Based on the given parameters and the above method, the three-dimensional undeformed chips generated by the tool during the third cutting of the material in cycloidal milling are obtained as follows: Figure 9 As shown in Figure (a), all the three-dimensional undeformed chips generated in one cycloidal cutting cycle are as follows: Figure 9 As shown in Figure (b).

[0309] Based on the given parameters and the method described above, the instantaneous three-dimensional undeformed chips generated at t = 0.6076s and t = 0.6187s are respectively as follows: Figure 10 Figure (a) and Figure 10 As shown in Figure (b). When t = 0.6076 s, as... Figure 10 As shown in Figure (a), only one three-dimensional undeformed chip is generated at this point. When t = 0.6187 s, as... Figure 10 As shown in Figure (b), two three-dimensional undeformed chips are generated simultaneously, indicating that multi-tooth cutting has occurred.

Claims

1. A method for instantaneous three-dimensional undeformed chip geometry modeling in cycloidal milling, characterized in that, Includes the following steps: Step 1: Based on the concept of layer cutting, multiple parallel planes perpendicular to the tool axis are used to cut the tool and workpiece simultaneously to obtain multiple cutting layers; Step 2: Determine the cutting type of each cutting layer, and solve for the instantaneous two-dimensional undeformed chip geometry on each cutting layer at any given instant: For a cutting layer with only single-tooth cutting, solve for the instantaneous two-dimensional undeformed chip geometry generated by the cutting edge participating in the cutting at any instant; For the cutting layer in which multi-tooth cutting occurs, solve the instantaneous two-dimensional undeformed chip geometry generated by each individual cutting edge participating in the cutting at any instant; Step 3: Stack all the instantaneous two-dimensional undeformed chip geometry on the cutting layer along the direction parallel to the tool axis to obtain the instantaneous three-dimensional undeformed chip geometry; The steps for solving the instantaneous two-dimensional undeformed chip geometry generated by a single cutting edge at any instant in step 2 are as follows: Step 2.1: Solve for the tool center point at any instant in cycloidal milling, the material removal boundary surface of the previous cycloidal cutting cycle, and the material removal boundary surface to be generated in the current cycloidal cutting cycle; Step 2.1.1: Establish the reference coordinate system CSO for describing the cycloidal toolpath. r and CSO q ; The reference coordinate system CSO r Origin r Located at the center of the cycloidal toolpath arc segment in the current cycloidal cutting cycle, x r The axis is parallel to the straight section of the cycloidal toolpath and points in the feed direction of the tool. r The axis is parallel to the tool axis and points vertically upwards, y r Determined by the right-hand screw rule; The reference coordinate system CSO q Origin q Located at the center of the cycloidal toolpath arc segment in the previous cycloidal cutting cycle, each coordinate axis is relative to the reference coordinate system CSO. r Consistent in direction; Step 2.1.2: Calculate the tool center point O at any instant during the current cycloidal cutting cycle. c (x c , y c , z c ), in the coordinate system CSO r The coordinates in the graph, the material boundary S1(x) cut off in the previous cycloidal cycle. s1 , y s1 , z s1 ) and the boundary surface S2(x) generated by the material to be cut in the current cycloidal cycle. s2 , y s2 , z s2 ) and S3(x s3 , y s3 , z s3 In the coordinate system CSO r The expression in; Step 2.2: In the coordinate system CSO r Establish a local coordinate system to describe the instantaneous two-dimensional undeformed chip geometry. CSO ci,j ; Step 2.2.1: For a certain cutting layer Π i Calculate the kth tool tip point C on it. i,k The trajectory of the circle S ei,j In the coordinate system CSO r Center O ci,j coordinates (x) ci,j , y ci,j , z ci,j ); Step 2.2.2: In the cutting layer Π i Above, with the center O ci,j Establish q local coordinate systems CSO for the point. ci,j The local coordinate system CSO ci,j x ci,j The shaft and the cutter follow the cycloidal tool path at the center O. ci,j The feed directions at z are parallel and in the same direction. ci,j axis and z r The axes are parallel and in the same direction, y ci,j The axis is determined by the right-hand rule; q is the total number of times the tool cuts the workpiece in one cycloidal cycle; Step 2.3: Calculate the tool tip point of each cutting edge on each cutting layer in coordinate system CSO. r The instantaneous position in; Step 2.4: Solve for the parametric expression of the instantaneous two-dimensional undeformed chip geometry generated at any instant on each cutting layer; Step 2.5: Solve for the parameters in the parameter expression obtained in Step 2.4 to obtain the instantaneous two-dimensional undeformed chip geometry.

2. The instantaneous three-dimensional undeformed chip geometry modeling method in cycloidal milling according to claim 1, characterized in that: In step 2, the method for determining the cutting type of each cutting layer is as follows: If a cutting layer is not subject to multi-tooth cutting, the method is as follows: At any instant, if there are more than one tool tip point in a cutting layer whose instantaneous position angle is within the range of the corresponding entry angle and exit angle, then multi-tooth cutting occurs on that cutting layer; otherwise, only single-tooth cutting occurs on that cutting layer.

3. The instantaneous three-dimensional undeformed chip geometry modeling method in cycloidal milling according to claim 1 or 2, characterized in that: In step 2.1.2: any instantaneous tool center point O during the current cycloidal cutting cycle. c (x c , y c , z c In coordinate system CSO r The coordinates in the middle are calculated according to the following formula: Where: c is the cycloidal step size; r is the cycloidal radius; t is the time parameter, in seconds; f is the feed rate, in mm / min; time parameter t l =60c / f, t t =60(c+2πr) / f; The material boundary S1(x) removed in the previous cycloidal cycle s1 , y s1 , z s1 ) and the boundary surface S2(x) generated by the material to be cut in the current cycloidal cycle. s2 , y s2 , z s2 ) and S3(x s3 , y s3 , z s3 In coordinate system CSO r The expressions in the text are as follows: Where, θ s1 The radian parameter of surface S1 and θ s1 ∈[0, π-arcsin(c / 2(R+r))], where R is the tool radius; when θ s1 When θ ∈ [π-arcsin(c / 2(R+r)), 2π], surface S1 does not exist, meaning the material boundary represented by surface S1 is removed by the previous cycloidal cutting cycle; S2 is the surface generated by cutting along the straight segment of the cycloidal toolpath, and S3 is the surface generated by cutting along the circular arc segment of the cycloidal toolpath; s2 It is the radian parameter of surface S2, which is derived from y r The negative half of the axis is measured counterclockwise, and θ s2 ∈[0,π+arcsin(c / 2(R+r))];a p The axial depth of cut is expressed in mm.

4. The instantaneous three-dimensional undeformed chip geometry modeling method in cycloidal milling according to claim 3, characterized in that: In step 2.2.1: the certain cutting layer Π i The kth blade tip point C i,k The trajectory of the circle S ei,j In the coordinate system CSO r Center O ci,j coordinates (x) ci,j , y ci,j , z ci,j Calculate according to the following formula: Where: t ci,j For time parameters, n is the spindle speed in rpm; q is the total number of cuts made by the tool in one cycloidal cycle. ;t d i The time lag parameter introduced by the tool helix angle β represents the time t at which the tool tip on the same cutting edge begins cutting material on the cutting layer Π1. d i After a certain time, it will be in the cutting layer Π i The material is then cut. Z i For cutting layer Π i In CSO r z in r coordinate.

5. The instantaneous three-dimensional undeformed chip geometry modeling method in cycloidal milling according to claim 4, characterized in that: Any cutting layer Π in step 2.3 i The tool tip points of each cutting edge in coordinate system CSO r The instantaneous position is calculated as follows: Step 2.3.1 First blade tip point C i,1 In the reference coordinate system CSO r instantaneous position O pi,1 (x pi,1 , y pi,1 , z pi,1 Calculate according to the following formula: in: j = int(t / t1)·N+1, where int is the floor operator; ω is the angular velocity, ω = 2πn / 60; make θ fi,j For the first blade tip point C i,1 In the local coordinate system CSO ci,j The instantaneous position angle in (x) ci,j , y ci,j , z ci,j ) is the center O ci,j In the reference coordinate system CSO r Coordinates in; Step 2.3.2 The kth tool tip point C i,k In the reference coordinate system CSO r The instantaneous position in the middle is calculated according to the following conditions, where k≠1: Scenario 1, if In the cutting layer Π i The kth blade tip point C i,k The trajectory of ψ does not exist; p The inter-tooth angle and ψ p =2π / N; Scenario 2, if Cutting layer Π i The kth blade tip point C i,k In coordinate system CSO r instantaneous position O pi,k (x pi,k , y pi,k , z pi,k Calculate according to the following formula: make θ fi,j-N+k-1 Let C be the kth tip point at instant t. i,k In coordinate system CSO ci,j-N+k-1 The instantaneous position angle in (x) ci,j+k-1 , y ci,j+k-1 , z ci,j+k-1 ) represents the kth blade tip point C i,k The trajectory of the circle S ei,j-N+k-1 The center O ci,j+k-1 In the reference coordinate system CSO r Coordinates in; Scenario 3, if At instant t, the cutting layer Π i The kth blade tip point C i,k In the reference coordinate system CSO r instantaneous position O pi,k (x pi,k , y pi,k , z pi,k ) Calculated by the following formula: make θ fi,j+k-1 Let C be the kth tip point at instant t. i,k In the local coordinate system CSO ci,j+k-1 The instantaneous position angle in (x) ci,j+k-1 , y ci,j+k-1 , z ci,j+k-1 ) represents the kth blade tip point C i,k The trajectory of the circle S ei,j+k-1 The center O ci,j+k-1 In the reference coordinate system CSO r The coordinates in the diagram.

6. The instantaneous three-dimensional undeformed chip geometry modeling method in cycloidal milling according to claim 5, characterized in that: Any cutting layer Π in step 2.4 i The specific method for solving the parametric expression of the instantaneous two-dimensional undeformed chip geometry generated at any instant is as follows: Step 2.4.1 Determine the cutting layer Π i The kth blade tip point C i,k The curves corresponding to the outer and inner boundaries of the two-dimensional undeformed chip cut after one revolution are obtained. The parametric expressions of these curves are solved as follows: make , (x ei,j , y ei,j S is the circle of motion trajectory. ei,j and S ei,j-1 intersection point P ei,j The coordinates; Case 1: If Δ1≤(r+R) 2 The boundary of a two-dimensional undeformed chip consists of an outer boundary and an inner boundary: The outer boundary in the local coordinate system CSO ci,j The Chinese character is represented as: , where θ si,j and θ ei,j 'These are the cutting layers Π i The entry and exit angles of the j-th cut; P si,j P ei,j ' represents the outer boundary curve; Inner boundary at CSO q The Chinese character is represented as: , where θ qi,j ' and θ qi,j The tangent points are P and P respectively. ei,j 'and the entry point P si,j In the reference coordinate system CSO q Parameter expression in P; si,j P ei,j ' is the inner boundary curve; Case 2: If Δ1 > (r + R) 2 The boundary of a two-dimensional undeformed chip consists of one outer boundary and two inner boundaries: The outer boundary in the local coordinate system CSO ci,j The Chinese character is represented as: , where θ ei,j For cutting layer Π i The cutting angle of the j-th cut; P si,j P ei,j The outer boundary curve; Inner boundary 1 in coordinate system CSO q The Chinese character is represented as: , where θ qi,j-1 Let P be the intersection point. si,j-1 In the reference coordinate system CSO q Parameter expression in P; si,j P si,j-1 For the inner boundary curve 1; Inner boundary two in local coordinate system CSO ci,j-1 The Chinese character is represented as: , where θ si,j-1 For cutting layer Π i The angle of entry for the (j-1)th cut; θ bi,j-1 For cutting layer Π i The cutting angle of the j-th cut; P si,j-1 P ei,j The second inner boundary curve; Step 2.4.2, based on 2.4.1, determines the cutting layer Π. i The kth blade tip point C i,k For the curves corresponding to the outer and inner boundaries of the instantaneous two-dimensional undeformed chip generated at any instant t, the parametric expressions of these curves are solved respectively, as follows: Case 1: If Δ1 > (r + R) 2 And the trajectory of the circle S ei,j-1 With material removal boundary s i intersection point P si,j-1 Located in vector F i, j K i,j To the right, the instantaneous two-dimensional undeformed chip boundary is formed by the circle S at the tool tip's motion trajectory. ei,j curve P on si,j F i,j Curve P located on the material removal boundary si,j P si,j-1 The trajectory of the blade tip is a circle S. ei,j-1 curve P on si,j-1 K i,j And the curve K on the rake face i,j F i,j It consists of four curves, and their parameter expressions are as follows: F i,j K i,j From vector F i,j K i,j express, ; Where, θ fi,j Let C be the kth blade tip point. i,k instantaneous position point F i,j In the local coordinate system CSO ci,j The parameter expression in θ; si,j For cutting layer Π i The angle of entry for the j-th cut; θ ki,j-1 For the normal vector F i,j K i,j and material removal boundary s i The intersection point in coordinate system CSO ci,j-1 The parameter expression in θ; ci,j Let C be the kth blade tip point. i,k The trajectory of the circle S ei,j Corresponding center point O ci,j The instantaneous position angle; Case 2: If Δ1 > (r + R) 2 And the trajectory of the circle S ei,j-1 With material removal boundary s i intersection point P si,j-1 Located in vector F i, j K i,j To the left, the instantaneous two-dimensional undeformed chip boundary is formed by the circle S at the tool tip's motion trajectory. ei,j curve P on si,j F i,j Curve P located on the material removal boundary si,j K i,j And the curve F on the rake face i,j K i,j It consists of three curves, and their parameter expressions are as follows: F i,j K i,j From vector F i,j K i,j express, ; Where, θ fi,j Let C be the kth blade tip point. i,k instantaneous position point F i,j In the local coordinate system CSO ci,j The parameter expression in θ si,j For cutting layer Π i The angle of entry for the j-th cut; θ ki,j 'For point K i,j In coordinate system CSO q The parameter expression in θ; ci,j Let C be the kth blade tip point. i,k The trajectory of the circle S ei,j Corresponding center point O ci,j The instantaneous position angle; Case 3: If Δ1≤(r+R) 2 The instantaneous two-dimensional undeformed chip boundary is formed by the circle S at the tool tip's motion trajectory. ei,j curve P on si,j F i,j Curve F on the rake face i,j K i,j And the curve K located on the material removal boundary i,j P si,j It consists of three curves, and their parameter expressions are as follows: F i,j K i,j From vector F i,j K i,j express, Where, θ fi,j Let C be the kth blade tip point. i,k instantaneous position point F i,j In the local coordinate system CSO ci,j The parameter expression in θ; si,j For cutting layer Π i The angle of entry for the j-th cut; θ ki,j-1 For the normal vector F i,j K i,j and material removal boundary s i The intersection point in coordinate system CSO ci,j-1 The parameter expression in θ; ci,j Let C be the kth blade tip point. i,k The trajectory of the circle S ei,j Corresponding center point O ci,j The instantaneous position angle.

7. An electronic device comprising a memory, a processor, and a computer program stored in and executable thereon; characterized in that: The computer program implements the instantaneous three-dimensional undeformed chip geometry modeling method as described in any one of claims 1-6.

8. A non-volatile computer-readable storage medium having a computer program stored thereon; characterized in that: The computer program implements the instantaneous three-dimensional undeformed chip geometry modeling method as described in any one of claims 1-6.

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