Modeling Method and System for Cutting Force of Unequal Gradient Helical Milling Cutter

Abstract

The present invention discloses a method and system for modeling the cutting force of an unequal variable pitch helical milling cutter. By differentiating the variable helix angle, the milling cutter is divided into several cutting edge micro-elements. The deformation process of the workpiece during the milling process is equivalent to a variable cross-section stepped cantilever beam model to calculate the instantaneous cutting thickness, and the elastic deformation amount is used to calculate the micro-force considering the workpiece deformation. A friction effect force model is introduced to calculate the micro-force of tool wear. The micro-force considering deformation and the micro-force of tool wear are summed and integrated within the helix angle range to obtain the milling force model for machining frame-like parts with unequal variable pitch helical cutters with different wear degrees. Aiming at the problem that the helix angle of each cutting edge of the unequal variable pitch helical cutter is a variable, resulting in the inapplicability of the cutting force model differentiated according to the axial direction, based on the mechanical type II mechanical model, a method of calculating the micro-cutting force with the helix angle as the differential object is proposed to establish a dynamic milling force model for the cutting process of unequal variable pitch helical cutters.

Description

Technical Field

[0001] The present invention relates to the field of metal cutting, and particularly to a cutting technology of an unequal variable pitch helical milling cutter considering workpiece deformation and tool wear. Background Art

[0002] The structure of aircraft body components has developed towards thin-walled, integrated structural load-bearing and functions. Such parts have complex structures, thin walls, weak stiffness, high surface accuracy, and obvious vibrations during side milling. Variable helix and variable pitch cutters are usually used to reduce vibrations. In addition, during the milling process, the elastic deformation is large and the stiffness of the milling area position is uneven. The unequal variable pitch helical milling cutter is designed for such parts. Unequal variable pitch means that the helix angles of each cutting edge of the cutter are not completely equal, and the helix angle of each cutting edge of the cutter is not a fixed value but gradually changes with the axial dimension. The tooth space angles can be the same or different. During the side milling process of the unequal variable pitch helical milling cutter, as the stiffness of the milling area position decreases, the helix angle of the corresponding unequal variable pitch cutter cutting edge gradually increases, so as to realize the reduction of the milling force and vibration with the weakening of the stiffness, and ensure that the weak stiffness frame beam parts have higher surface quality and dimensional accuracy.

[0003] When the unequal variable pitch helical milling cutter cuts weak stiffness frame beam parts, especially titanium alloy parts, good results can be obtained, but the increase in the complexity of its spatial cutting edge brings challenges to the prediction of cutting force. As one of the most important physical quantities in the cutting process, reliable cutting force prediction has important value for production practice and theoretical research. It has a positive impact on the optimization of cutting parameters, the optimization of tool structure parameters, and the improvement of machining accuracy and quality, thereby improving production efficiency and reducing processing costs, and has great economic value and practical significance.

[0004] The existing milling force models for unequal helices are studied for a single fixed helix angle of the cutting edge. Some of the relevant milling force coefficients are related to the helix angle and are fixed values, which are considered as constants during the integration process. Therefore, the traditional milling force prediction method is not applicable to unequal variable pitch helical cutters. The helix angle of any cutting edge of the unequal variable pitch helical milling cutter is a variable. The angular position of any point on the cutting edge is related not only to the tool rotation angle but also to the helix angle at the cutting point position. Especially for the unequal variable pitch end mill with unequal tooth space and cutting edge parameters, the helix edges participate in the cutting at different time intervals, and there are differences in the system time delay effect and the chip thickness per tooth of the tool. At the same time, each shear force and ploughing force milling force coefficient is a function related to the helix angle, and the existing milling force model is not applicable to the prediction of unequal variable pitch helical milling force. Summary of the Invention

[0005] The technical problem to be solved by the present invention is to provide a cutting force modeling method and system for an unequal variable pitch helical milling cutter considering workpiece deformation and tool wear, which is applicable to the prediction of unequal variable pitch helical milling force.

[0006] To solve the above technical problems, the following technical solutions are provided:

[0007] A method for modeling the cutting force of an unequal variable pitch helical milling cutter considering workpiece deformation and tool wear, characterized in that the developed line of the variable pitch helical edge is a set function, and includes the following steps:

[0008] S1: Based on the milling cutter rectangular coordinate system and the micro-element coordinate system, obtain the parameters of the cylindrical unequal variable pitch helical milling cutter, and establish the spatial edge line equation of the unequal variable pitch helical tool;

[0009] S2: By differentiating the helix angle variable of the spatial edge line equation of the unequal variable pitch helical tool, divide the milling cutter into several cutting edge micro-elements, and determine the cutting depth micro-element or the micro-element axial length, and the edge length micro-element or the micro-element cutting edge length in the tool coordinate system;

[0010] S3: Calculate the instantaneous radial cutting angle;

[0011] S4: By comparing the magnitudes of the instantaneous radial cutting angle with the cutting-in angle and the cutting-out angle, establish a window function to determine whether the current micro-element participates in cutting;

[0012] S5: Equivalent the elastic deformation of the workpiece during the milling process to a variable cross-section stepped cantilever beam model. The geometric shape contour of the cantilever beam is in the form of a fixed bottom end. Adopt the Euler-Bernoulli beam theory to calculate the elastic deformation amount caused by the cutting force of the frame-like parts to calculate the instantaneous cutting thickness;

[0013] S6: Collect the workpiece material parameters, and determine the tangential, radial and axial plowing force coefficients or shear force coefficient functions of the current micro-element; calculate the micro-force considering workpiece deformation using the elastic deformation amount;

[0014] S7: Collect the tool wear parameters, introduce the friction effect force model, and calculate the tool wear micro-force including the tangential friction force and the radial pressure per unit edge length of the worn tool flank;

[0015] S8: Calculate the range of the helix angle participating in cutting, or the helix angle values at the starting helix angle of the tool and the helix angle at the axial cutting depth of the tool;

[0016] S9: Sum the micro-forces considering deformation and the tool wear micro-forces;

[0017] S10: Integrate the micro-forces after summing in S9 within the helix angle range in S8 to obtain the milling force model indicated by the predicted cutting forces in the X, Y, and Z directions of the unequal variable pitch helical tool for cutting frame-like parts with different wear degrees.

[0018] In step S1, based on the milling cutter rectangular coordinate system and the micro-element coordinate system, obtain the parameters of the cylindrical unequal variable pitch helical milling cutter, and establish the spatial edge line equation of the unequal variable pitch helical tool; including:

[0019] S11: Simplify the solid end mill into a cylinder, and establish a rectangular coordinate system fixed on the tool; at the same time, establish a cutting edge micro-element coordinate system that rotates with the tool on the cutting edge.

[0020] S12: Obtain the parameters of the cylindrical unequal variable pitch helical end mill, and establish the spatial edge line equation of the unequal variable pitch helical tool.

[0021] In the above technical solution, step S2 is: Obtain the parameters of the cylindrical unequal variable pitch helical end mill, including the number of teeth N, the diameter D, the cutting edge serial number i, 1 ≤ i ≤ N, the helical start angle β imin of the i-th cutting edge, the helical end angle β imax , the tooth space angle where the helical angle β i of the i-th cutting edge is a variable and satisfies β imin ≤ β i ≤ β imax ; establish the spatial edge line equation of the unequal variable pitch helical tool.

[0022] In the above technical solution, the parameters of the cylindrical unequal variable pitch helical end mill required in step S3 include the tool rake angle α 0 , the tool clearance angle γ 0 , the tip edge radius r e , the cutting parameters: the axial cutting depth a p , the radial cutting depth a e ; by differentiating the helical angle, divide the end mill into M cutting edge micro-elements, and determine the cutting depth micro-element or the micro-element axial length, and the edge length micro-element or the micro-element cutting edge length in the tool coordinate system.

[0023] In the above technical solution, the parameters of the cylindrical unequal variable pitch helical end mill required in step S4 include the cutting edge serial number i, the helical angle β i , the tooth space angle It is also necessary to adopt the workpiece material parameters: the shear yield strength τ s , the shear angle φ 0 , the normal friction angle ζ 0 , the chip flow angle η c .

[0024] In the above technical solution, step S6: After obtaining the elastic modulus E of the workpiece material, the workpiece moment of inertia I 1 in the machining area, the workpiece moment of inertia I 2 in the unprocessed area, the cutting depth, and the remaining depth of the workpiece, equivalent the deformation process of the workpiece during the milling process to a variable cross-section stepped cantilever beam model.

[0025] In the above technical solution, to calculate the tangential frictional force and radial pressure per unit edge length of the worn flank of the tool, the flank wear VB of the tool, the tangential stress τ 0 , the normal stress σ 0 , and the fixed width VB of the elastic contact area * are required.

[0026] In the above technical solution, the general form of the equation of the i-th cutting edge line is as follows:

[0027]

[0028] where: i is a positive integer and satisfies 1 ≤ i ≤ N; N is the number of teeth of the tool; β i is the helix angle of the i-th cutting edge line, satisfying β imin ≤ β i ≤ β imax , rad; β imin is the helix start angle, rad; β imax is the helix end angle, rad; D is the tool diameter, mm; is the start tooth space angle between the (i - 1)-th cutting edge and the i-th cutting edge, rad. u i is the functional relationship between the position angle and β i , and f i is the functional relationship between the Z coordinate and β i .

[0029] In the above technical solution, the developed line of the variable helix cutting edge is one of a parabola, a hyperbola, or a logarithmic curve.

[0030] In the above technical solution, when the developed line of the variable helix cutting edge is a quadratic function, the specific form of the equation of the i-th cutting edge line is as follows:

[0031]

[0032] where: N is the number of teeth of the tool; i is a positive integer and satisfies 1 ≤ i ≤ N; β i is the helix angle of the i-th cutting edge line, satisfying β imin ≤ β i ≤ β imax , rad; β imin is the helix start angle of the i-th cutting edge line, rad; β imax is the helix end angle of the i-th cutting edge line, rad; D is the tool diameter, mm; u i is the functional relationship between the position angle and β i , and f i is the functional relationship between the Z coordinate and β i ; l 1 is the edge length of the tool, mm; is the starting tooth space angle between the (i - 1)-th edge and the i-th edge, in rad.

[0033] In the above technical solution, the cutting depth element or the axial length of the element dz in the tool coordinate system i is as follows:

[0034]

[0035] the edge length element or the cutting edge length of the element ds i is as follows:

[0036]

[0037] the cutting depth element or the axial length of the element dz in the tool coordinate system i The quadratic function form is as follows:

[0038]

[0039] the edge length element or the cutting edge length of the element ds i The quadratic function form is as follows:

[0040]

[0041] In the above technical solution, the elastic deformation amount δ of the workpiece caused by the cutting force of the frame beam type part d (z) is:

[0042]

[0043] In the formula: E is the elastic modulus of the workpiece material, in Pa; H 1 is the depth of the machining area, H 1 = a p , in mm; I 1 is the moment of inertia of the workpiece in the machining area, m 4 ; I 2 is the moment of inertia of the workpiece in the unprocessed area, m 4 ; H 2 is the remaining depth of the workpiece, in mm.

[0044] In the above technical solution, the milling force model is as follows:

[0045]

[0046] Among them, F x , F y , F z are the predicted cutting forces in the X, Y, and Z directions of the thin-walled part cut by the variable pitch helical tool, in N; β i is the helix angle of the i-th edge line, satisfying β imin ≤β i ≤βimax , rad; β imin is the helix start angle of the i-th cutting edge, rad; β imax is the helix end angle of the i-th cutting edge, rad; N is the number of teeth of the tool; i is a positive integer and satisfies 1 ≤ i ≤ N; j is the microelement serial number; D is the tool diameter, mm; l 1 is the cutting edge length of the tool, mm; φ ij is the instantaneous position angle, rad; g(φ(β, t)) is a window function to determine whether the microelement u ij participates in cutting.

[0047] The window function expression of g(φ(β, t)) is as follows:

[0048]

[0049] In the formula: φ st is the milling entry angle, rad; φ ex is the milling exit angle, rad; and the two satisfy the relationship: 0 ≤ φ st <φ ex ≤ π;

[0050] (9) In the formula, the entry angle φ st , the exit angle φ ex can be expressed as follows:

[0051]

[0052] In the formula, a e is the cutting width.

[0053] (8) In the formula, the tangential, radial and axial shear force coefficients K j of the microelement u tci , K rci , K aci satisfy the following formula:

[0054]

[0055] In the formula, τ s is the shear yield strength, φ 0 is the shear angle, α 0 is the tool rake angle, ζ 0 is the normal friction angle, η c is the chip flow angle; (8) In the formula, the tangential, radial and axial plowing force coefficients K ij of the microelement u tei , K rei , K aei satisfy the following formula:

[0056]

[0057] In the formula, r e is the radius of the cutting edge circle of the tool tip, and θ f is the splitting angle. Based on the splitting angle theory proposed by AbdelMoneim, it can be known that θ f = ζ 0 ;

[0058] In formula (8), h D (φ ij ) is the instantaneous cutting thickness, and the expression is as follows:

[0059]

[0060] In the formula, f is the feed per revolution, in mm / r; n is the rotational speed, in r / min; is the starting tooth space angle between the (i - 1)-th edge and the i-th edge, in rad; δ d (z) is the elastic deformation of the workpiece;

[0061] In formula (8), β iap is the helix angle value at the axial cutting depth a p of the i-th edge line of the tool, and it satisfies the following formula:

[0062]

[0063] In the above technical solution, when the developed line of the variable helix edge is a quadratic function, the quadratic function form of the milling force model is as follows:

[0064]

[0065] In the formula, β iap satisfies:

[0066]

[0067] In the above technical solution, the friction effect force model is constructed as follows:

[0068] During the cutting process, the tool will gradually wear. The shear force generated by the shear action on the rake face and the friction force and pressure generated by the flank wear together constitute the milling force of the worn tool. Among them, the friction force and the pressing force generated by the flank wear are collectively called the friction effect force. The frictional force is generated by the friction and extrusion between the tool face and the machined surface. Therefore, the friction effect force is related to the flank wear and has nothing to do with the undeformed chip thickness.

[0069]

[0070] In the formula, dF t is the tangential differential resultant force considering tool wear, and dF r is the radial differential resultant force considering tool wear.

[0071] The tangential differential friction force dF in Equation (17) tw and the radial differential pressure dF rw are calculated as follows:

[0072]

[0073] In the formula, F tw (VB) is the tangential friction force per unit edge length of the flank face, and F rw (VB) is the radial pressure per unit width of the flank face. Both are related to the flank face wear and are functions of the flank face wear value VB.

[0074] The tangential friction force F tw (VB) per unit edge length of the flank face and the radial pressure F rw (VB) in Equation (18) are as follows:

[0075]

[0076] In the formula, x is the edge length distance, and VB * is the fixed width of the elastic contact area, and τ 0 and σ 0 are the tangential stress and the normal stress respectively.

[0077] The present invention is directed to general weak-rigid frame beam type workpieces, and thus is applicable to any workpiece with thin walls that can be side-milled.

[0078] Based on the above method, the present invention can also protect a modeling system for the cutting force of an unequal variable pitch helical milling cutter considering workpiece deformation, which is characterized in that it is used to implement the above modeling method.

[0079] Of course, the above method does not exclude being stored as a program on a storage medium. When the program is executed, the above modeling method for the cutting force of an unequal variable pitch helical milling cutter considering workpiece deformation is implemented.

[0080] The present invention discloses a method and system for modeling the cutting force of an unequal variable pitch helical milling cutter considering workpiece deformation and tool wear. By differentiating the variable pitch angle, the milling cutter is divided into several cutting edge micro-elements. The deformation process of the workpiece during the milling process is equivalent to a variable cross-section stepped cantilever beam model to calculate the instantaneous cutting thickness, and the elastic deformation amount is used to calculate the micro-force considering workpiece deformation. A friction effect force model is introduced to calculate the micro-force of tool wear. The micro-force considering deformation and the micro-force of tool wear are summed and integrated within the range of the pitch angle to obtain the milling force model for machining frame-like parts with an unequal variable pitch helical tool with different wear degrees. Aiming at the problem that the cutting force model differentiated axially is not applicable due to the variable pitch angle of each cutting edge of the unequal variable pitch helical tool, based on the mechanical type II mechanical model, a method of calculating the micro-cutting force with the pitch angle as the differentiation object is proposed, and a dynamic milling force model for the cutting process of the unequal variable pitch helical tool is established.

[0081] Compared with the prior art, the beneficial effects produced by the present invention are as follows:

[0082] The dynamic cutting force model of a conventional cylindrical helical milling cutter is generally differentiated axially. During the calculation process of the milling force model, it is considered that the milling force coefficient is a constant and does not participate in the integration of the micro-cutting force. Therefore, the existing milling force model is not applicable to the prediction of the cutting force during the side milling process of an unequal variable pitch helical edge milling cutter.

[0083] Combined with the variable cross-section stepped cantilever beam model, the elastic deformation amount during the cutting process is characterized, the undeformed instantaneous cutting thickness is updated, and an accurate cutting force model for an unequal variable pitch helical tool considering workpiece deformation is established.

[0084] Aiming at the problem of the change in cutting force caused by the progressive wear of the tool, a friction effect force model is introduced to construct a milling force model for an unequal variable pitch helical tool considering tool wear.

[0085] Each milling force coefficient of the present invention is related to the variable pitch angle. Therefore, each micro-cutting force coefficient is a variable. The workpiece deformation amount is calculated through the variable cross-section stepped cantilever beam model, and at the same time, the milling force solution process is differentiated according to the pitch angle; it is more in line with the actual machining process.

[0086] Finally, taking the simplest uniform unequal variable pitch angle for each cutting edge as an example, the present invention accurately predicts the milling force of a quadratic function unequal variable pitch helical tool. BRIEF DESCRIPTION OF THE DRAWINGS

[0087] The present invention will be further described below in conjunction with the drawings and embodiments. In the drawings:

[0088] Figure 1 is the flow chart of the method for modeling the cutting force of an unequal variable pitch helical milling cutter considering workpiece deformation of the present invention.

[0089] Figure 2Schematic diagram of the coordinate system and cutting edge micro - element geometry of the unequal - variable - pitch helical end - mill of the present invention.

[0090] Figures 3A - 3D Schematic diagram of the force analysis of the micro - element of the unequal - variable - pitch helical milling cutter of the present invention.

[0091] Figure 4 Schematic diagram of the undeformed cutting thickness of the instantaneous cutting micro - element considering workpiece deformation in the present invention.

[0092] Figure 5 Equivalent variable - cross - section stepped cantilever beam model of the present invention.

[0093] Figure 6A and Figure 6B Distribution diagram of the tool friction effect force when the flank wear value of the present invention is VB.

[0094] Figure 6B is Figure 6A C - C cross - sectional view of

[0095] Figure 7 Morphology of the cutting micro - element of the worn tool and distribution diagram of the flank stress of the present invention.

[0096] Figure 8 Flow chart of the milling force simulation of the variable - pitch helical tool considering workpiece deformation of the present invention.

[0097] Figure 9 Prediction result of the milling force model of the unequal - variable - pitch variable - helix - edge tool of the present invention.

[0098] Figure 10 Comparison diagram of the predicted value and experimental value of the side - milling cutting force of the variable - pitch helical tool with different wear degrees. Specific implementation mode

[0099] In order to make the purpose, technical solution and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0100] Embodiment 1:

[0101] As Figure 1 shown, according to the method for modeling the cutting force of the unequal - variable - pitch helical milling cutter considering workpiece deformation implemented by the present invention, by differentiating the helix - angle variable, the milling cutter is divided into cutting - edge micro - elements, the deformation process of the workpiece in the milling process is equivalent to a variable - cross - section stepped cantilever beam model, the micro - element force considering workpiece deformation is calculated and stored to calculate the instantaneous cutting thickness; a friction effect force model is introduced to calculate the tangential friction force and radial pressure per unit edge length of the flank of the worn tool; combined with different micro - element instantaneous cutting thicknesses, a milling force model for the unequal - variable - pitch helical tool for cutting box - beam parts with different wear degrees is obtained.

[0102] Specifically, the present invention is implemented according to the following steps:

[0103] When establishing the milling force model, the following two sets of coordinate systems are usually established:

[0104] (1) Tool rectangular coordinate system

[0105] The solid end mill is simplified to a cylinder, and the intersection point of the tool axis and the tool bottom surface is defined as the origin O of the tool rectangular coordinate system t , and the tool axis is defined as the Z t axis (positive upward); the positive direction of the X t axis is the same as the feed direction, and the Y t axis is perpendicular to the workpiece surface (positive outward), and the tool coordinate system O t -X t Y t Z t is established, as shown in Figure 2 .

[0106] (2) Micro-element coordinate system of the cutting edge

[0107] The tool is discretized along the Z axis into M micro-element disks, as shown in Figure 2 . Assuming the number of teeth is N, there are M cutting-edge micro-elements on each micro-element disk. Taking any micro-element cutting edge on any micro-element disk as an example, the center of the micro-element body is defined as the center o, the direction parallel to the tool axis is defined as the a axis (positive upward), the direction pointing to the center of the disk is defined as the r axis (positive inward), and the tangent direction of the disk is defined as the t axis (the positive direction is determined by the right-hand screw rule). The micro-element coordinate system o-tra is established, and the micro-element coordinate system on the i-th (1 ≤ i ≤ N) cutting edge is o i -t i r i a i .

[0108] Let the diameter of a certain cylindrical non-uniform variable helix end mill be D (radius R), the number of tool teeth / flutes be N, and the helix angle of the tool side edge be β = {β 1 , β 2 , …, β i , …, β N}, that is, the helix angle of the i-th cutting edge is β i and β imin ≤ β i ≤ β imax , where β imin and β imax are respectively the minimum and maximum values of the non-uniform variable helix angle of the i-th cutting edge; the tool tooth space angle is the included angle between the i-th cutting edge and the (i - 1)-th cutting edge (i.e., the i-th tooth space angle), and the axial depth of cut is a p , and the radial depth of cut (cutting width) is a e . The position angle at the tip of the i-th cutting edge is φ i . Due to the existence of the helix angle, there is a lag during the milling process. Let the axial microelement at the j-th (1 ≤ j ≤ M) position on the i-th cutting edge be dz i , and the lag angle of this microelement is ψ ij , and the position angle is φ ij . The relationships between the variables are as shown in Figure 2 .

[0109] The spatial equation M(x i , y i , z i ) of the i-th cutting edge line of the unequal variable helix cutter is expressed as follows:

[0110]

[0111] In the formula, u i is the functional relationship between the lag angle ψ ij of the i-th cutting edge and the helix angle β i ; w i is the axial position z i of the i-th developed cutting edge line and the functional relationship with Du i (β i ) / 2. Different values of w i result in different degrees of unequal variation of the cutter edge line, and the functional relationships of different edge lines may be different.

[0112] Suppose that at time t, the included angle between the line connecting the tip of a certain cutting edge of the milling cutter and the origin of the workpiece coordinate system and the positive direction of the Y t axis of the cutter rectangular coordinate system (the included angle between the reverse direction of the microelement r axis and the positive direction of the cutter Y t axis) is θ. Then θ is called the radial position angle (initial position angle) of the first cutting edge and can be expressed as follows:

[0113]

[0114] In the formula, n is the rotational speed, r / min; t is the time, s;

[0115] From Figure 2 , it can be known that the radial position angle φ i at the tip of the i-th cutting edge can be expressed as follows:

[0116]

[0117] Since the helix angle is variable, there is a one-to-one functional relationship between the lag angle and the helix angle. Assume the cutting microelement uij Lag angle ψ ij The relationship is as follows:

[0118] ψ i (β i ) = u i (β i ) (4)

[0119] Therefore, for the differential element u ij the radial position angle φ ij can be expressed as:

[0120]

[0121] To calculate the magnitude of the differential cutting force, the force analysis of the milling differential element u ij is carried out as shown Figures 3A - 3D below.

[0122] Combined with Figures 3A - 3D , according to the mechanical type II mechanical model proposed by ALTINTAS, the three-direction milling forces on the differential element u ij can be expressed by the following formula:

[0123]

[0124] In the formula: dF ti , dF ri , dF ai are the tangential, radial and axial milling forces of the differential element u ij , respectively, in N; K tci , K rci , K aci are the tangential, radial and axial shear force coefficients of the differential element u ij , respectively; K tei , K rei , K aei are the tangential, radial and axial plowing force coefficients of the differential element u ij , respectively; dz i , ds i are the cutting depth differential element and the cutting edge length differential element of the differential element u ij , respectively; φ i (β i , t) is the radial position angle of the differential element u ij at time t, in rad; h D (φ i (β i , t)) is the cutting thickness of the differential element u ij at time t, in mm; g(φ i (β i , t)) is a window function for judging whether cutting occurs;

[0125] Differential element uij Tangential, radial, and axial shear force coefficients K tci , K rci , K aci Satisfy the following formula:

[0126]

[0127] Where τ s is the shear yield strength, φ 0 is the shear angle, α 0 is the rake angle of the cutting tool, ζ 0 is the normal friction angle, η c is the chip flow angle. It can be seen from the formula that since the helix angles of each cutting edge of the unequal variable pitch helical edge tool are not exactly the same and are variables, each coefficient is not a fixed value and is related to the helix angle, and changes with the change of the helix angle.

[0128] The tangential, radial, and axial plowing force coefficients K ij of the microelement u tei , K rei , K aei Satisfy the following formula:

[0129]

[0130] Where r e is the radius of the cutting edge nose, θ f is the splitting angle. Based on the splitting angle theory proposed by AbdelMoneim, θ f = ζ 0 .

[0131] During the milling process of frame-like parts, as the material is continuously removed, the geometric shape of the workpiece continuously changes. The existence of the milling force causes the deformation of the workpiece. As long as the milling force exists, deformation will occur. Especially for thin-walled parts, the stiffness along the wall thickness direction is very low, and large forces will be generated in the tool-workpiece contact area during the cutting process, resulting in workpiece deformation. Therefore, the influence of workpiece deformation along the wall thickness direction should be considered when modeling the milling force. To simplify the solution process, the cutting edge trajectory of the milling cutter is approximated as a circle. Therefore, the shape of the cutting area during side milling is a sector, as shown in Figure 4 .

[0132] There are generally two reasons for the elastic deformation of workpieces: the elastic deformation δ t of the workpiece caused by the cutting force and the elastic deformation δ r caused by the initial residual stress or the residual stress during the cutting process of the workpiece。The deformation amount of the residual stress is relatively small compared with the former and is thus ignored. Since the depth of cut for difficult-to-machine materials is not easily selected to be too large, the frame parts need to be machined in multiple passes to remove the allowance during side milling. Let the depth of the first pass be H 1 , that is, H 1 = a p , and the depth of the second pass be H 2 . In the present invention, the deformation process of the workpiece during the milling process is equivalent to a stepped cantilever beam model with a variable cross-section as Figure 5 shown. The geometric shape profile of the cantilever beam is fixed at the bottom end. The Euler-Bernoulli beam theory is used to calculate the elastic deformation amount caused by the cutting force of the frame beam parts.

[0133] The deflection deformation amount δ d (z) at the free end can be expressed as follows:

[0134]

[0135] In the formula, δ H1 (z 1 ) is the deflection deformation amount at the free end of the machining area, in mm; δ H2 (z 2 ) is the deflection deformation amount at the end of the beam in the unprocessed area, in mm;

[0136] Among them, the deflection deformation amount δ H1 (z 1 ) at the free end of the machining area is caused by the action of the radial milling force F in the H 1 section of the milling area, and the expression is as follows:

[0137]

[0138] In the formula, E is the elastic modulus of the workpiece material, in Pa; I 1 is the moment of inertia of the workpiece in the machining area, in m 4 ;

[0139] In the unprocessed area, the deformation amount in the H 2 section is mainly caused by the combined action of the milling radial force F and the moment, and the expression is as follows:

[0140]

[0141] In the formula, I 2 is the moment of inertia of the workpiece in the unprocessed area, in m 4 ;

[0142] Match the deformation amounts at both ends at the cross-section with the slope k(z):

[0143]

[0144] Furthermore, the piecewise function of the deflection deformation of the simplified beam in the milling process of the frame beam parts can be obtained:

[0145]

[0146] In the formula: E is the elastic modulus of the workpiece material, Pa; H 1 is the depth of the machining area, H 1 = a p , mm; I 1 is the moment of inertia of the workpiece in the machining area, m 4 ; I 2 is the moment of inertia of the workpiece in the unprocessed area, m 4 ; H 2 is the remaining depth of the workpiece, mm;

[0147] Let the tooth feed of the i-th cutting edge be f Ni , then there is

[0148]

[0149] In the formula, f is the feed per revolution, mm / r;

[0150] When the cutting edge of the tool microelement participates in cutting, the instantaneous feed displacement can be ignored, and it can be considered that the tip point starts to contact the surface of the workpiece to be machined at point A ij until the point B ij leaves the machined surface of the workpiece, and the curve experienced on the workpiece surface is shown in Figure A ij B ij , thus, the length of the undeformed cutting thickness h ij of the j-th cutting microelement u Dij on the i-th cutting edge can be expressed as:

[0151]

[0152]

[0153] Since only when the cutting edge contacts the workpiece will cutting occur during side milling, therefore, a window function g(φ ij (β ij ,t)) is needed to judge whether the microelement u ij participates in cutting, and its expression is as follows:

[0154]

[0155] In the formula: φ st is the cutting-in angle of milling, rad; φ ex is the cutting-out angle of milling, rad; and the two satisfy the relationship: 0 ≤ φ st <φ ex ≤ π;

[0156] It can be seen that the key to solving the cutting force micro-element model is to determine the instantaneous cutting thickness model and the magnitudes of the entry and exit angles. The entry angle φ st , the exit angle φ ex can be expressed as follows:

[0157]

[0158] In the formula: is the angle between the (i - 1)-th cutting edge and the i-th cutting edge, that is, the i-th tooth space angle, in rad;

[0159] The cutting depth micro-element (axial length of the micro-element) dz and the cutting edge length micro-element (length of the micro-element cutting edge) ds in the tool coordinate system. Combining with Equation (1), the axial length dz of the micro-element in the tool coordinate system can be obtained:

[0160]

[0161] Then the length ds of the micro-element cutting edge can be expressed as:

[0162]

[0163] Through coordinate transformation, it is transformed into the tool coordinate system:

[0164]

[0165] After arrangement, the cutting force components acting in the rectangular coordinate system can be obtained as follows:

[0166]

[0167] The milling force distributions along the tangential, radial, and axial directions on the effective cutting edge of the tool at any time t are as Figure 3A shown. By integrating along the axial angle β and summing for each tooth, the instantaneous cutting forces acting on the entire milling cutter in the feed, normal, and axial directions can be obtained as follows:

[0168]

[0169] where β iap is the helix angle value of the i-th cutting edge line at the axial position a p of the tool, and it satisfies the following formula:

[0170]

[0171] Taking the simplest uniformly non-equal variable helix angle for each cutting edge line of the tool as an example, that is, each cutting edge of the tool increases in the form of a quadratic function, the dynamic cutting force model for side milling of a non-equal variable helix milling cutter based on the quadratic function is derived to verify the accuracy of the milling force model. The quadratic function is as follows:

[0172] w i (Dψ i (β i ) / 2) = a i (Dψ i (β i ) / 2) 2 +b i (Dψ i (β i ) / 2)+c i (25)

[0173] In the formula, a i , b i and c i are the coefficients of the quadratic function. Since the developed line of the cutting edge passes through the origin of coordinates, c i = 0.

[0174]

[0175] Combining Equation (19) and Equation (20), the following formula for the lag angle can be obtained:

[0176]

[0177] After arrangement, the position angle at time t:

[0178]

[0179] Combining Equation (13), the axial length dz of the microelement in the tool coordinate system can be obtained:

[0180]

[0181] Combining Equation (14), the length ds of the microelement cutting edge can be expressed as:

[0182]

[0183] Substituting the above results into Equation (17), the milling force model of the unequal variable pitch helical end mill based on the quadratic function can be obtained as follows:

[0184]

[0185] Among them, from Equation (18), it can be known that β iap satisfies:

[0186]

[0187] During the milling process, the cutting force generated by the shearing action on the rake face and the frictional force and pressure generated by the flank wear together constitute the milling force of the worn tool. Among them, the frictional force and the pressing force generated by the flank wear are collectively referred to as the frictional effect force. The shearing force is related to the undeformed chip thickness and does not explicitly consider the tool face wear. The frictional force is generated by the friction and extrusion between the tool face and the machined surface. Therefore, the frictional effect force is related to the flank wear and has nothing to do with the undeformed chip thickness.

[0188] For milling, neglecting the influence of the frictional effect force on the axial force, the force analysis of the milling cutter microelement is as Figure 6A and 6B shown. In the figure, dF rc is the radial microelement force, and dF tc is the tangential microelement force. They are caused by the shearing action and act on the rake face of the tool. dF rw and dF tw are the radial microelement pressure and the tangential microelement frictional force respectively. They are caused by the frictional action during the tool edge wear.

[0189] Therefore, by introducing the cutting force change term caused by the flank wear into the microelement milling force, the source of the cutting force can be regarded as composed of the shearing force in the first deformation zone and the frictional force in the third deformation zone, that is:

[0190]

[0191] In the formula, dF t is the resultant tangential microelement force considering tool wear, and dF r is the resultant radial microelement force considering tool wear.

[0192] According to Teitenberg's theory

[134] , the flank wear has no influence on the axial milling force. The calculation methods of the tangential microelement frictional force dF tw and the radial microelement pressure dF rw are as follows:

[0193]

[0194] In the formula, F tw (VB) is the tangential frictional force per unit edge length of the flank, and F rw (VB) is the radial pressure per unit width of the flank. Both are related to the flank wear and are functions of the flank wear value VB.

[0195] According to the research of Lapsley and Waldorf

[135] , the contact area between the worn flank and the workpiece is divided into a plastic flow zone and an elastic contact zone. The tangential stress and the normal stress are both constant values τ in the plastic flow zone 0and σ 0 is distributed in a quadratic form in the elastic contact area, and the stress distribution is as Figure 7 shown, VB p is the demarcation point between the elastic contact area and the plastic contact area.

[0196] The tangential friction force and radial pressure per unit edge length of the flank face are obtained by integrating the shear stress τ(x) and normal stress σ(x) in the contact area between the flank face and the workpiece, and the calculation method is as follows:

[0197]

[0198] where x is the edge length distance.

[0199] When 0 < x ≤ VB p , the tool wear area is in the elastic contact area, and the stress distribution is as follows:

[0200]

[0201] When VB p < x < VB, the tool wear area is in the plastic flow area, and the stress distribution is as follows:

[0202]

[0203] Smithey's research shows that

[136] , when the tool wear reaches a certain level, the width of the elastic contact area remains unchanged, and the width of the plastic flow area increases with the increase of the tool edge wear.

[0204]

[0205] where VB * is the fixed width of the elastic contact area.

[0206] Substituting Equation (3-36) and Equation (3-37) into Equation (3-35), the tangential friction force F tw (VB) and radial pressure F rw (VB) per unit edge length of the flank face are as follows:

[0207]

[0208] In summary, the instantaneous milling force during the cutting process of the unequal variable pitch helical end mill can be obtained by using the above formula.

[0209] Based on the above method, the present invention can also protect a cutting force modeling system for an unequal variable pitch helical end mill considering workpiece deformation, which is characterized in that it is used to implement the above modeling method.

[0210] Of course, the above method does not exclude being stored as a program on a storage medium. When the program is executed, the above method for modeling the cutting force of a non-uniformly variable spiral milling cutter considering workpiece deformation is implemented.

[0211] Embodiment 2: The correctness of the milling force model is verified through MATLAB simulation, and the simulation process is as Figure 8 shown.

[0212] The relevant set parameters are as follows: The tool diameter D = 12 mm, the number of tool teeth / flutes is N = 4, the tool edge length l 1 = 30 mm, the helix angle of the tool side edge is β = {40° - 45°, 37° - 42°, 40° - 45°, 37° - 42°}, the tool tooth space angle is The axial depth of cut is a p = 20 mm, and the radial depth of cut (cutting width) is a e = 0.3 mm. The predicted results of the instantaneous cutting forces in the Fx and Fy directions when the tool rotates one week (360°) are as Figure 9 shown.

[0213] From Figure 9 it can be seen that the simulation results of the numerical model of the milling force of the variable spiral tool established by the present invention are in good agreement with the experimental results. Through calculation, it can be known that the maximum prediction error of the milling force model established by the present invention is 16.87%. The reason for the error may be the forced vibration caused by the centrifugal force and impact force generated by the machine tool system during the cutting process. The certain fluctuations in the peaks of the experimental milling force signal may be caused by the periodic deformation of the workpiece during the side milling process, resulting in periodic fluctuations in the milling force.

[0214] The present invention also performs side milling on parts with tools of different flank wear degrees, and conducts experimental studies on the milling force by using tools with VB = {62.5 μm, 90.1 μm, 113.8 μm} respectively. The comparison between the measured milling force signals and the simulation results is as Figure 10 shown. From Figure 10 it can be seen that the model established by the present invention can predict the milling force under tools with different wear degrees and obtains good results.

[0215] From Figure 10 it can also be obtained that as the tool wears severely, the prediction accuracy will decrease. This is because the cutting force of the worn variable spiral tool not only increases the pressure and friction force, but also the thermo-mechanical coupling relationship in the cutting process is more complex and the prediction is more difficult. Through calculation, it can be known that the maximum prediction error of the worn tool is 19.26%. Therefore, the milling force model established by the present invention can accurately predict the milling force during the cutting process of the variable spiral tool.

[0216] It should be understood that those of ordinary skill in the art can make improvements or modifications based on the above description, and all such improvements and modifications shall fall within the protection scope of the appended claims of the present invention.

Claims

1. A cutting force modeling method for unequally gradient spiral milling cutter considering workpiece deformation and tool wear, characterized in that The gradual spiral blade development line is a setting function, which includes the following steps: S1: Based on the rectangular coordinate system and micro-element coordinate system of the milling cutter, the parameters of the cylindrical unequal-gradient spiral milling cutter are obtained, and the spatial edge line equation of the unequal-gradient spiral cutter is established; S2: By differentiating the helix angle variable of the spatial edge line equation of the unequally variable spiral tool, the milling cutter is divided into a number of cutting edge microelements, and the cutting depth microelement or microelement axial length, edge length microelement or microelement cutting edge length in the tool coordinate system are determined; S3: Calculate the instantaneous radial cutting angle; S4: By comparing the instantaneous radial cutting angle with the cutting-in angle and the cutting-out angle, a window function is established to determine whether the current microelement is involved in cutting; S5: The elastic deformation of the workpiece during milling is equivalent to a variable-section stepped cantilever beam model. The geometric shape of the cantilever beam is a fixed bottom form. The Euler-Bernoulli beam theory is used to calculate the elastic deformation caused by the cutting force of frame beam parts to calculate the instantaneous cutting thickness. S6: Collect workpiece material parameters, determine the tangential, radial and axial plowing force coefficients or shear force coefficient functions of the current microelement; use the elastic deformation to calculate the microelement force considering the deformation of the workpiece; S7: Collect tool wear parameters, introduce friction effect force model, and calculate tool wear micro-element force including tangential friction force per unit blade length of the flank of the worn tool and radial pressure; S8: Calculate the helix angle range involved in cutting, or the helix angle value of the tool starting helix angle and the tool axial cutting depth; S9: sum the micro-element force considering deformation and the micro-element force of tool wear; S10: Integrate the micro-element force after summing up in S9 within the helix angle range of S8 to obtain the milling force model indicated by the predicted cutting force in the three directions of X, Y, and Z when cutting frame beam parts with unequally gradient helical tools of different wear degrees.

2. The method for modeling cutting force of a unequally gradient spiral milling cutter considering workpiece deformation and tool wear according to claim 1 is characterized in that The parameters of cylindrical unequal-gradient spiral milling cutter to be obtained include: number of teeth N ,diameter D , cutting edge number i , 1≤ i ≤ N , No. i Helix starting angle of the cutting edge β imin , helix termination angle β imax , inter-tooth angle φ i , among which i Helix angle of cutting edge β i is a variable and satisfies β imin ≤ β i ≤ β imax ; The tool rake angle is α 0, tool back angle is γ 0, the radius of the cutting edge is r e , cutting parameters: axial cutting depth a p , radial depth of cut a e ; Cutting edge number i , helix angle β i , inter-tooth angle φ i .

3. The method for modeling cutting force of a unequally gradient spiral milling cutter considering workpiece deformation and tool wear according to claim 1 is characterized in that The cutting depth microelement or microelement axial length in the tool coordinate system obtained in step S2 dz i as follows: satisfy β imin ≤ β i ≤ β imax , rad; β imin is the starting angle of the spiral, rad; β imax is the helix termination angle, rad; D is the tool diameter, mm; l 1 is the blade length of the tool, mm; φ i For the i - 1 blade and i The starting tooth angle between the strip edges, rad; u i is the position angle and β i The functional relationship between f i for Z Coordinates and β i The functional relationship between them.

4. The method for modeling cutting force of a unequally gradient spiral milling cutter considering workpiece deformation and tool wear according to claim 1 is characterized in that Step S6: Obtaining the elastic modulus of the workpiece material E , Workpiece inertia moment in the processing area I 1. Moment of inertia of workpiece in unprocessed area I 2. After the cutting depth and the remaining depth of the workpiece are determined, the deformation process of the workpiece during the milling process is equivalent to a variable-section stepped cantilever beam model.

5. The method for modeling cutting force of a unequally gradient spiral milling cutter considering workpiece deformation and tool wear according to claim 1 is characterized in that Step S7: Obtain tool back surface wear VB , tangential stress τ 0, normal stress σ 0, fixed width of the elastic contact area VB * , in order to calculate the tangential friction force and radial pressure per unit cutting edge length of the flank of the worn tool.

6. The method for modeling cutting force of a unequally gradient spiral milling cutter considering workpiece deformation and tool wear according to claim 1 is characterized in that The friction effect force model used in step S7 is as follows: The shear force generated by the shearing action of the front cutting edge and the friction and pressure generated by the wear of the back cutting edge together constitute the milling force of the worn tool. The friction and clamping force generated by the wear of the back cutting edge are collectively referred to as the friction effect force: Where, d F rc is the radial micro-element force, d F tc is the tangential micro-element force, d F t To consider the tangential micro-force after tool wear, d F r is the radial micro-element force after tool wear is considered; among them, the tangential micro-element friction force d F tw and radial micro-element pressure d F rw The calculation method is as follows: In the formula, F tw ( VB ) is the tangential friction force per unit blade length of the back cutting edge, F rw ( VB ) is the radial pressure per unit width of the flank face, which is the flank face wear value VB Function of Tangential friction force per unit blade length on the back face F tw ( VB ) and radial pressure F rw ( VB )as follows: in, x For blade length, VB * is the fixed width of the elastic contact area, τ0 and σ0 are the tangential stress and normal stress, respectively.

7. The method for modeling cutting force of a unequally gradient spiral milling cutter considering workpiece deformation and tool wear according to claim 1 is characterized in that The workpiece material parameters also obtained in step S6 include: shear yield strength τ s , Shear Angle ϕ 0. Normal friction angle ζ 0. Chip flow angle η c .

8. The method for modeling cutting force of a unequally gradient spiral milling cutter considering workpiece deformation and tool wear according to claim 1 is characterized in that The workpiece is a weak rigid frame beam type workpiece with a thin wall and suitable for side milling.

9. A cutting force modeling system for unequally gradient spiral milling cutters considering workpiece deformation and tool wear, characterized in that A method for modeling cutting forces of unequally gradient spiral milling cutters taking into account workpiece deformation and tool wear as described in any one of claims 1 to 8 above.

10. A storage medium storing a computer program, characterized in that When the computer program is executed, the method for modeling cutting force of unequally gradient spiral milling cutter considering workpiece deformation and tool wear as described in any one of claims 1 to 8 is implemented.

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