Method and system for predicting cutting force and judging stability in cavity helical milling process

Abstract

The application provides a cavity spiral milling process cutting force prediction and stability discrimination method and system, comprising: selecting a machining path parameter according to the size of a cavity to be machined, and generating a spiral machining tool path; based on the tooth track arc assumption, geometric modeling is performed on the machining engagement area at each tool point, and the tooth cutting in and out angle is calculated; a cavity spiral milling cutting force model is established based on the geometric modeling results, and accurate prediction of transient cutting force in the cavity spiral milling process is realized; considering the dynamic cutting thickness regeneration effect, a spiral milling dynamics equation dependent on the feed direction is established, and stability analysis is performed on the representative feed direction along the spiral tool path, so as to determine the stability of the cavity spiral milling process; and a time domain method based on the discontinuous Galerkin method is proposed to improve the calculation accuracy and efficiency of the stability analysis. The application realizes accurate prediction of cutting force and stability determination in the cavity spiral milling process.

Description

TECHNICAL FIELD

[0001] The present application relates to cavity machining technology, in particular, to a method for predicting cutting force and judging stability in cavity spiral milling process. BACKGROUND

[0002] Cavity machining is widely used in aerospace manufacturing, mold and automobile industry, which has the characteristics of large material removal and long processing cycle. The traditional cavity machining adopts the ring cutting method or the ring cutting method to generate tool path, which will appear frequent acceleration and deceleration during processing, which restricts the efficiency of cavity machining. Therefore, a high-efficiency cavity machining method-spiral milling appears. When spiral milling is carried out, the tool removes material along the spiral tool path in the form of spline curve, the tool movement path is smooth and the processing condition changes gently, which can effectively realize high-speed milling and improve the efficiency of cavity machining.

[0003] However, the tool feed direction and machining engagement area change constantly in the process of cavity spiral milling, and the machining mechanism is relatively complex. In order to maximize the material removal rate under the premise of ensuring the processing quality, it is necessary to reasonably select the processing parameters. Cutting force is a very important physical quantity in the process of milling, and its size directly affects the processing state and tool life. The stability analysis of cutting process helps to select reasonable processing parameters to effectively avoid the occurrence of regenerative chatter and realize the smooth operation of the machining process. Therefore, it is of great significance to carry out the modeling of cavity spiral milling cutting mechanics and dynamics.

[0004] The document "Kardes, N, Altintas, et al. Mechanics and Dynamics of the Circular Milling Process. [J]. Journal of Manufacturing Science&Engineering, 2007." considers the change of tool feed direction and machining engagement area in circular milling, and establishes a cutting force prediction model and a machining chatter prediction model for circular milling. For circular milling, the tool path and machining profile can be expressed by implicit equations, and the analytical solution of the intersection point can be obtained by solving the equation set, and then the machining engagement area of the tool and the workpiece can be obtained. However, for the spiral tool path of cavity machining, the tool path and machining profile are usually expressed by parametric equations, and the intersection point can only be solved numerically, which brings great challenges to the accurate prediction of cutting force and stability determination in the process of cavity spiral milling. SUMMARY

[0005] In view of the defects in the prior art, the purpose of the present application is to provide a method and system for predicting cutting force and judging stability in cavity spiral milling process.

[0006] The application provides a cutting force prediction and stability discrimination method in a cavity spiral milling process, comprising the following steps:

[0007] A tool path generation step: generating a smooth spiral milling tool path in the form of a B-spline curve according to a known shape and size of a cavity to be machined and a set maximum span;

[0008] A modeling step: based on a tool tooth track circular arc assumption, geometric modeling is performed on a machining engagement area at each discrete tool position point of the smooth spiral milling tool path;

[0009] A cutting force prediction step: based on the modeling result of the modeling step, a cavity spiral milling cutting force model is established to realize prediction of transient cutting force in the cavity spiral milling process;

[0010] A stability judgment step: based on the modeling result of the modeling step, a spiral milling dynamics equation dependent on a feeding direction is established by considering dynamic cutting thickness regeneration effects, stability analysis is performed on a representative feeding direction along the spiral tool path, and thus the stability of the cavity spiral milling process is judged;

[0011] An optimization step: based on the spiral milling dynamics equation, a time domain method based on an intermittent Galerkin method is adopted for optimization.

[0012] Preferably, the tool path generation step comprises the following steps:

[0013] An elliptic partial differential equation with Dirichlet boundary conditions is solved by using a finite element method, an equivalent closed field curve is generated according to the set maximum span, a transition curve is generated after the field curve is divided, a group of spiral point columns are obtained, and a smooth spiral milling tool path in the form of a B-spline curve is obtained by fitting the spiral point columns by using a B-spline curve.

[0014] Preferably, the modeling step comprises the following steps:

[0015] Geometric modeling is performed on the machining engagement area by solving the intersection of the tool and the workpiece surface, the intersection relationship between the tool circumference with the tool position point C(x C ,y C ) as the center and the machined surface is used to divide the cavity spiral milling process into two sub-stages: in the cutting-in stage, the tool circumference intersects with the initial circumference surface of the inner boundary; in the steady state stage, the tool circumference intersects with the machined surface formed by the spiral tool path;

[0016] For the cutting-in stage, the intersection P2(x2,y2) is solved by solving the equation group of the tool circumference and the initial circumference surface:

[0017]

[0018] Where r is the tool radius, R i is the inner boundary radius;

[0019] For steady stage, by discretizing the curve parameter u, the discrete tool position C(x C ,y C ) and corresponding feed direction angle θ are obtained, then the discrete machined surface profile point P1(x1,y1) is calculated:

[0020]

[0021] For given tool position C(u i ), the discrete machined surface profile point is calculated by equation (2), the distance between the discrete machined surface profile point and the tool position is obtained, to determine the intersection points between the tool circumference and the discrete machined surface profile, the point corresponding to the maximum curve parameter among these intersection points is the intersection point P2(x2,y2);

[0022] After the intersection point P2(x2,y2) is determined by geometric modeling, the cutting in angle φ st and the cutting out angle φ ex are calculated, for finish milling, the cutting out angle is fixed as π, the cutting in angle is determined by the intersection point P2(x2,y2);

[0023]

[0024] Preferably, the cutting force forecasting step comprises:

[0025] By densely discretizing the smooth helical milling tool path in the form of B-spline curve, the spline curve length between adjacent discrete tool positions is approximated by the straight line distance between them, thereby obtaining the relationship between the motion distance l along the tool path and the curve parameter u, when the tool feed speed V and the spindle speed Ω are determined, the time t(u) corresponding to each tool position and the feed per tooth f t are obtained:

[0026]

[0027]

[0028] where N t is the number of teeth;

[0029] The milling force is obtained by using the discrete summation strategy, the milling cutter containing the helix angle effect is divided into N A disks along the tool axis, for each tool position, the position angle of the cutting microelement of the jth tooth of the kth disk in the tool coordinate system X c -Y c is

[0030] φ j,k= (2pQ / 60) t(u) - Q(u) + (j - 1) - 2p / N - (i - 1) - (2-dz-tanp / D) (6)

[0031] where dz is the axial depth of cut of the disc. p is the helix angle of the tool, and D is the diameter of the tool;

[0032] According to the circular arc approximation assumption of the tool tooth locus, the instantaneous undeformed chip thickness of the cutting microelement is determined as follows:

[0033] h j,k = f t sin(p j,k ) (7)

[0034] The tangential cutting force and the radial cutting force corresponding to the cutting microelement are respectively:

[0035]

[0036] where K te , K tc , K re and K rc are the cutting force coefficients, and the cutting force in the tool coordinate system is expressed as:

[0037]

[0038] By traversing all the cutting microelements in the cutting, the instantaneous cutting force in the tool coordinate system X c -Y c is

[0039]

[0040] where g(p j,k ) is a switch function for judging whether the current moment cutting microelement is in the cutting or not, when p st ≤ p j,k ≤ p ex , the switch function is 1, otherwise it is 0, the cutting-in angle p st and the cutting-out angle p ex are determined by the modeling step, and the coordinate transformation of the instantaneous cutting force in the tool coordinate system to the global coordinate system is obtained:

[0041]

[0042] Preferably, the stability judging step comprises:

[0043] Considering the flexibility of the tool in two lateral directions in the dynamic model for stability prediction, since the static chip thickness in equation (7) does not affect the stability, only the dynamic chip thickness is included in the dynamic model, which is expressed as

[0044]

[0045] where Δx = x(t) - x(t - T) and Δy = y(t) - y(t - T) are the dynamic displacements of the tool in the global coordinate system at the previous tooth period and the current tooth period, T is the time lag and equals the tooth cutting period, i.e. T = 60 / (N t Ω);

[0046] Substituting equation (8) and equation (12) into equation (11), the total dynamic cutting force in the global coordinate system is calculated as follows:

[0047]

[0048] where δ = φ j,k + θ contains the feed direction angle θ and the instantaneous position angle φ j,k .

[0049] Thus, the dynamics model of cavity helical milling is established as:

[0050]

[0051] where M, C, K are the matrices of modal mass, modal damping and modal stiffness, respectively, q(t) = [x(t) y(t)] T is the tool displacement vector, and the coefficient matrix K c (θ, t) related to time and feed direction is expressed as:

[0052]

[0053] The stability prediction of cavity helical milling includes: traversing the curve parameter u and calculating the feed direction angle θ and the corresponding engagement angle φ en ; traversing the representative feed direction angle, for each traversed feed direction angle θ traverse , selecting the maximum engagement angle max(φ en ) corresponding to the feed direction angle θ traverse to consider the most conservative stability condition; determining the stability lobe diagram of the entire cavity helical milling process from the lowest envelope curve of the stability boundary at each representative feed direction angle.

[0054] Preferably, the optimization step comprises:

[0055] After determining the feed direction angle θ and the engagement angle φ en in the dynamics model, the cavity helical milling dynamics model represented by equation (14) is simplified as:

[0056]

[0057] Let Simultaneously, denote U(t) = [q(t)p(t)] T Equation (16) is transformed into the following state-space form:

[0058]

[0059] in

[0060]

[0061]

[0062] U(t) is a d-dimensional state vector, A is a constant matrix representing the time-invariant properties of the system, and B(t) is a periodic matrix determined by the dynamic cutting force considering the regeneration effect, satisfying B(t+T)=B(t), where T is the time period and equal to the time delay.

[0063] Depending on whether the tool is in contact with the workpiece, the time period T is divided into two stages: free vibration and forced vibration. Let t0 represent the moment the tool leaves the workpiece. f The duration of free vibration after the tool leaves the workpiece is denoted as t. c Let B(t) be the duration of the cutting time in the workpiece. When the cutting tool is not in contact with the workpiece, the term B(t) in equation (17) is equal to zero. The relationship between the state at the beginning and end of the free vibration is as follows:

[0064]

[0065] Next, we process the forced vibration period, dividing the time interval [t0+t] into... f The interval [t0+T] is divided into m-hour segments at equal distances, thus satisfying t c =Tt f =mτ, where m is a positive integer, and ε represents the division of the forced vibration period. τ Defined as t1 < t2 < ... < t m <t m+1 Where t1 = t0 + t f ,t n =t1+(n-1)τ, n=1,...,m+1, define the open interval J n =(t n ,t n+1 ), n = 1, ..., m, and use D k (ε τ Let d represent the d-dimensional vector space of a piecewise discontinuous polynomial of degree k.

[0066]

[0067] Where P k (J n ) is the interval Jn on k, v is a d-dimensional vector function of D k (ε τ ), define and Then define the jump and average at an interior node of an interval:

[0068]

[0069] Extend the definition of the jump and average at an interval endpoint:

[0070]

[0071] Take the dot product of equation (17) with the vector v and integrate over the interval J n :

[0072]

[0073] Integrate by parts the left side of equation (24) to obtain:

[0074]

[0075] Substitute equation (25) into equation (24) and sum over all intervals to obtain:

[0076]

[0077] For interior nodes, equation (22) gives:

[0078]

[0079] Combine equations (23) and (27) and note that the exact solution U(t) satisfies Equation (26) transforms to:

[0080]

[0081] Since the state variable U(t) should be continuous when switching between the free and forced vibration phases, we obtain:

[0082]

[0083] From equation (29) we further obtain:

[0084]

[0085] Combine equation (28) with equation (30) to finally obtain the equation that makes the sought solution x(t) satisfy the continuity constraint:

[0086]

[0087] Consider the solution using discontinuous piecewise quadratic polynomials, i.e. k = 2, for the local basis functions of p2(J n ), select the monomial basis functions converted from the interval (-1, 1):

[0088]

[0089] where

[0090] For each interval J n , each component of the solution vector U(t) is approximated by a quadratic polynomial, then the solution vector U(t) on the interval J n is expressed as:

[0091]

[0092] where is the coefficient vector of each basis function of this interval, combined as

[0093] For each time interval J n , each component of the d-dimensional vector v is independent of each other and satisfies Consider each element of the vector v respectively, the following equations are derived from the integral of the vector v and the solution vector U(t) on the time interval J n :

[0094]

[0095] where I is the d-dimensional unit matrix;

[0096] Similarly to equation (34), the integral terms of equation (31) are calculated to obtain the local matrix M n , N n , R n corresponding to the coefficient vector of each interval basis function;

[0097] Then, using the same piecewise quadratic polynomial expansion method and component selection method of the vector v, expand each term containing the jump and average at the nodes in equation (31) to obtain the local matrix E n , F n , G n , H n , J n , K n , L n , U n corresponding to the coefficient vector of each interval basis function;

[0098] All local matrices are calculated according to each item in formula (31), and are combined into a global matrix according to the order of the coefficient vector, and then the following discrete dynamic mapping is constructed:

[0099]

[0100] Wherein

[0101]

[0102]

[0103]

[0104]

[0105] Then, the state transition matrix Φ on a single tooth cutting period is expressed as:

[0106] Φ=(S+T+Q) -1 (W+V) (36)

[0107] Finally, the milling stability is determined by the Floquet theory, if the modulus of any eigenvalue of the state transition matrix is greater than 1, the milling process is unstable; if the modulus of the maximum eigenvalue is less than 1, the milling process is stable; if the modulus of the maximum eigenvalue is equal to 1, it is on the stability domain boundary.

[0108] According to the cutting force prediction and stability discrimination system for cavity helical milling process provided by the application, comprising:

[0109] The tool path generation module generates a smooth helical milling tool path in the form of a B-spline curve according to the known shape and size of the cavity to be machined and the set maximum span;

[0110] The modeling module performs geometric modeling on the machining engagement area at each discrete tool position of the smooth helical milling tool path based on the tool tooth trajectory arc assumption;

[0111] The cutting force prediction module establishes a cavity helical milling cutting force model based on the modeling results of the modeling module, and realizes the prediction of the instantaneous cutting force in the cavity helical milling process;

[0112] The stability judgment module establishes a helical milling dynamics equation dependent on the feed direction based on the modeling results of the modeling module, considering the dynamic cutting thickness regeneration effect, and performs stability analysis on the representative feed direction along the helical tool path, so as to determine the stability of the cavity helical milling process;

[0113] The optimization module optimizes the helical milling dynamics equation by using a time domain method based on the discontinuous Galerkin method.

[0114] Preferably, the tool path generation module comprises:

[0115] An elliptic partial differential equation with Dirichlet boundary condition is solved by finite element method, an equivalent closed field curve is generated according to the set maximum span, a transition curve is generated after the field curve is divided, a group of spiral point columns are obtained, and a smooth spiral milling tool path in the form of B-spline curve is obtained by fitting the spiral point columns by using B-spline curve.

[0116] Preferably, the modeling module comprises:

[0117] The machining engagement area is geometrically modeled by solving the intersection point of the tool and the workpiece surface, and the intersection relationship between the tool circumference with the center of the tool position point C(x C ,y C ) and the machined surface is used to divide the cavity spiral milling process into two sub-stages: in the cutting-in stage, the tool circumference intersects with the initial circumference surface of the inner boundary; in the steady stage, the tool circumference intersects with the machined surface formed by the spiral tool path;

[0118] For the cutting-in stage, the intersection point P2(x2,y2) is solved by solving the equation group of the tool circumference and the initial circumference surface:

[0119]

[0120] Where r is the tool radius, R i is the inner boundary radius;

[0121] For the steady stage, the discrete tool position point C(x C ,y C ) and the corresponding feed direction angle θ are obtained by discretizing the curve parameter u, and then the discrete machined surface profile point P1(x1,y1) is calculated:

[0122]

[0123] For a given tool position point C(u i ), the discrete machined surface profile point is calculated by formula (2), the distance between the discrete machined surface profile point and the tool position point is obtained, and the intersection point of the tool circumference and the discrete machined surface profile is determined, and the point corresponding to the maximum curve parameter in the intersection point is the intersection point P2(x2,y2);

[0124] After the intersection point P2(x2,y2) is determined by geometric modeling, the cutting-in angle φ st and the cutting-out angle φ ex are calculated, for the finish milling, the cutting-out angle is fixed as π, and the cutting-in angle is determined by the intersection point P2(x2,y2);

[0125]

[0126] Preferably, the cutting force prediction module comprises:

[0127] By discretizing the smooth helical milling tool path in the form of B-spline curve, the length of the spline curve between adjacent discrete tool positions is approximated by the straight line distance between them, thus the relationship between the motion distance l along the tool path and the curve parameter u is obtained, when the tool feed speed V and spindle speed Ω are determined, the time t(u) corresponding to each tool position and the feed per tooth f t :

[0128]

[0129]

[0130] where N t is the number of teeth of the tool;

[0131] The milling force is obtained by using the discrete summation strategy, the milling tool with helix angle effect is divided into N A circular discs along the tool axis, for each tool position, the position angle of the cutting microelement of the kth disc of the jth tooth in the tool coordinate system X c -Y c

[0132] φ j,k = (2πΩ / 60) t(u) - θ(u) + (j - 1) · 2π / N - (i - 1) · (2 · dz · tanβ / D) (6)

[0133] where dz is the axial cutting depth of the disc, β is the helix angle of the tool, and D is the diameter of the tool;

[0134] According to the circular arc approximation assumption of the tooth trace, the instantaneous undeformed thickness of the cutting microelement is determined as follows:

[0135] h j,k = f t sin(φ j,k ) (7)

[0136] The tangential cutting force and the radial cutting force corresponding to the cutting microelement are respectively:

[0137]

[0138] where K te ,K tc ,K re and K rc are the cutting force coefficients, and the cutting force in the tool coordinate system is expressed as:

[0139] ​

[0140] By traversing all the cutting micro elements in the cutting, the instantaneous cutting force in the tool coordinate system X c -Y c is

[0141]

[0142] Where g(φ j,k ) is a switch function that determines whether the cutting micro element at the current time is in the cutting, and when φ st ≤φ j,k ≤φ ex , the switch function is 1, otherwise it is 0, the cutting-in angle φ st and the cutting-out angle φ ex are determined by the modeling module, and the instantaneous cutting force in the tool coordinate system is transformed to the global coordinate system:

[0143]

[0144] Compared with the prior art, the present application has the following beneficial effects:

[0145] 1. Based on the tool tooth trajectory arc assumption, the present application geometrically models the machining engagement area at each tool position of the spiral tool path for machining the cavity, and calculates the tool tooth cutting-in and cutting-out angles;

[0146] 2. Based on the geometric modeling results, the present application establishes a cavity spiral milling cutting force model and a spiral milling dynamics equation dependent on the feed direction, and realizes accurate prediction of the instantaneous cutting force and stability determination during the cavity spiral milling process;

[0147] 3. Based on the discontinuous Galerkin method, the present application proposes a time-domain method to improve the calculation accuracy and efficiency of stability analysis. BRIEF DESCRIPTION OF DRAWINGS

[0148] Other features, objects and advantages of the present application will become more apparent from the following detailed description of non-limiting embodiments, made with reference to the accompanying drawings:

[0149] Figure 1 is a flowchart of the present application;

[0150] Figure 2 is a schematic diagram of the spiral milling tool path for rectangular cavity spiral milling in the present application;

[0151] Figure 3 , 4 , 5 is a schematic diagram of geometric modeling at different stages in the cavity spiral milling process in the present application Figure 3 is the cutting-in stage, Figure 5 is the steady state stage, Figure 4is the critical state between two stages);

[0152] Figure 6 is the transient cutting force in x, y direction in the process of cavity spiral milling in the present application;

[0153] Figure 7 is the stability blade diagram of the process of cavity spiral milling in the present application. DETAILED DESCRIPTION

[0154] The present application will be described in detail below with specific examples. The following examples will help those skilled in the art to further understand the present application, but do not limit the present application in any form. It should be noted that those skilled in the art can make several changes and improvements without departing from the concept of the present application. These are within the scope of protection of the present application.

[0155] The present embodiment provides a cavity spiral milling process cutting force prediction and stability determination method, and its flow chart is shown in Figure 1 First, according to the size of the cavity to be machined, the machining path parameters are selected, and the spiral machining tool path is generated; based on the tooth track arc assumption, the machining engagement area at each tool position is geometrically modeled, and the tooth cutting in and out angle is calculated; based on the geometric modeling results, the cavity spiral milling cutting force model is established, and the accurate prediction of the transient cutting force in the cavity spiral milling process is realized; then, considering the dynamic thickness regeneration effect, the spiral milling dynamics equation dependent on the feed direction is established, and the stability of the representative feed direction along the spiral tool path is analyzed, so as to determine the stability of the cavity spiral milling process; further, the time domain method based on the discontinuous Galerkin method is proposed to improve the calculation accuracy and efficiency of the stability analysis.

[0156] The following takes a square cavity with a spiral milling side length of 90mm as an example for specific description.

[0157] Step 1, cavity spiral machining tool path generation: given that the cavity to be machined is a square with a side length of 90mm, the cavity inner boundary radius R i = 4mm, the tool radius r = 5mm, and the maximum span set is 5mm. The finite element method is used to solve the elliptic partial differential equation of Dirichlet boundary condition, and the equivalent closed field curve is generated according to the set maximum span, and the transition curve is generated after the field curve is divided, to obtain a set of spiral point column, and then the B-spline curve is used to fit the spiral point column to obtain a B-spline curve form smooth spiral milling tool path, as shown in Figure 2 .

[0158] Step 2, modeling of machining engagement area and calculation of tooth cutting in and out angle: first, the machining engagement area is geometrically modeled by solving the intersection of the tool and the workpiece surface. According to the tool position C(x C ,yC The intersection of the tool envelope and the machined surface is shown in Fig. 2. The cavity helical milling process is divided into two sub-stages: in the cutting-in stage, the tool envelope intersects with the initial envelope surface of the inner boundary, as shown in Fig. 2(a); in the steady stage, the tool envelope intersects with the machined surface formed by the helical tool path, as shown in Fig. 2(b). The critical state between the two stages is shown in Fig. 2(c). Figure 3 Figure 5 Figure 4

[0159] For the cutting-in stage, the intersection point P2(x2, y2) is solved by the simultaneous equations of the tool envelope and the initial envelope surface:

[0160]

[0161] where r is the tool radius, R i is the inner boundary radius.

[0162] For the steady stage, by discretizing the curve parameter u, the discrete tool position C(x C ,y C ) and the corresponding feed direction angle θ can be obtained. Then the discrete machined surface profile point P1(x1, y1) can be calculated:

[0163]

[0164] For a given tool position C(u i ), the discrete machined surface profile point can be calculated by equation (2). The distance between the discrete machined surface profile point and the tool position can be obtained numerically to determine the intersection point P2(x2, y2). The point corresponding to the maximum curve parameter among these intersection points is the intersection point P2(x2, y2).

[0165] After the intersection point P2(x2, y2) is determined by the geometric modeling, the cutting-in angle φ st and the cutting-out angle φ ex can be calculated. For the finish milling process, the cutting-out angle is fixed as π, and the cutting-in angle is determined by the intersection point P2(x2, y2):

[0166]

[0167] Step 3, the cavity helical milling cutting force model is established: the blank material is AL6061, the tool parameters are selected: the number of teeth N t = 2, the helix angle is β = 25°. The cavity helical milling machining parameters are set: the spindle speed Ω = 1000 rpm, the axial depth of cut a p = 1 mm, the feed speed V = 280 mm / min. The cutting force coefficient K tc = 882.0 N / mm is obtained by fitting based on the linear regression method.​​​2 , K te = 17.1 N / mm, K nc = 419.4 N / mm 2 , K ne = 10.3 N / mm.

[0168] By discretizing the smooth helical milling tool path in the form of B-spline curve, the length of the B-spline curve between adjacent discrete tool positions can be approximated by the straight line distance between them, so that the relationship between the motion distance l along the tool path and the curve parameter u can be obtained. When the tool feed speed V and spindle speed Ω are determined, the time t(u) corresponding to each tool position and the feed per tooth f t can be obtained.

[0169]

[0170]

[0171] where N t is the number of teeth.

[0172] The milling force is obtained by the "discrete summation" strategy, which divides the milling tool with helix angle effect into N A circular discs along the tool axis. For each tool position, the position angle of the cutting microelement of the kth disc of the jth tooth in the tool coordinate system X c -Y c is

[0173] φ j,k = (2πΩ / 60)t(u)-θ(u)+(j-1)·2π / N-(i-1)·(2·dz·tanβ / D) (6)

[0174] where dz is the axial cutting depth of the disc, β is the helix angle of the tool, and D is the diameter of the tool.

[0175] According to the circular arc approximation assumption of the tooth trace, the instantaneous undeformed thickness of the cutting microelement can be determined as follows:

[0176] h j,k = f t sin(φ j,k ) (7)

[0177] The tangential cutting force and the radial cutting force corresponding to the cutting microelement are:

[0178]

[0179] where K te , K tc , K re , and K rcis the cutting force coefficient. For each cutting micro-element, the cutting force in the tool coordinate system can be expressed as:

[0180]

[0181] By traversing all the cutting micro-elements in the cutting, the instantaneous cutting force in the tool coordinate system X c -Y c is

[0182]

[0183] where g(φ j,k ) is a switch function that determines whether the current time cutting micro-element is in the cutting or not. When φ st ≤ φ j,k ≤ φ ex , the switch function is 1, otherwise it is 0. The cutting-in angle φ st and the cutting-out angle φ ex are determined by step 2. The instantaneous cutting force in the tool coordinate system is transformed to the global coordinate system to obtain the instantaneous cutting force in the global coordinate system (as shown in Figure 6 ):

[0184]

[0185] Step 4, the process of establishing the helical milling dynamics equation and stability determination: the flexibility of the tool in two lateral directions is considered in the dynamic model for stability prediction. Since the static chip thickness in equation (7) does not affect the stability, only the dynamic chip thickness is included in the dynamic model, which is expressed as

[0186]

[0187] where Δx = x(t) - x(t-T) and Δy = y(t) - y(t-T) are the dynamic displacements of the tool in the global coordinate system in the previous tooth period and the current tooth period. T is the time lag and is equal to the tooth cutting period, i.e. T = 60 / (N t Ω).

[0188] Substituting equation (8) and equation (12) into equation (11), the total dynamic cutting force in the global coordinate system is calculated as follows:

[0189]

[0190] where δ = φ j,k + θ contains the feed direction angle θ and the instantaneous position angle φ j,k . Since the terms related to the edge coefficients K te and K re are not related to the chip thickness and have no effect on the stability, the terms related to Kte and K re related terms.

[0191] Thus the dynamic model of cavity spiral milling can be established as:

[0192]

[0193] where M, C, K are the matrices of modal mass, modal damping and modal stiffness respectively. q(t) = [x(t) y(t)] T is the tool displacement vector. The coefficient matrix K c (θ, t) is expressed as:

[0194]

[0195] Different from conventional milling, the feed direction angle θ and the engagement angle φ en at different tool positions are different, which increases the complexity of the dynamic model. To solve this problem, the dynamic model is established for a representative feed direction and its corresponding maximum engagement angle, thus eliminating the changes of the feed direction angle θ and the engagement angle φ en in the dynamic model. The stability prediction of cavity spiral milling mainly includes three steps: 1) traverse the curve parameter u and calculate the feed direction angle θ and the corresponding engagement angle φ en ; 2) traverse the representative feed direction angle. For each traversed feed direction angle θ traverse , the maximum engagement angle max(φ en ) corresponding to the feed direction angle θ traverse is selected for dynamic modeling to consider the most conservative stability condition; 3) determine the stability lobe diagram of the entire cavity spiral milling process from the lowest envelope curve of the stability boundaries at each representative feed direction angle. Through modal testing, there are two main modes in the x direction. For the first order mode, the natural frequency f x,1 = 1674.27 Hz, the damping ratio ξ x,1 = 0.018, and the modal mass m x,1 = 0.0313 kg; for the second order mode, the natural frequency f x,2 = 2018.36 Hz, the damping ratio ξ x,2 = 0.015, and the modal mass m x,2 = 0.0463 kg. There are also two main modes in the y direction. For the first order mode, the natural frequency f y,1 = 1680.82 Hz, the damping ratio ξ y,1 = 0.022, and the modal mass m y,1 = 0.0307 kg; for the second order mode, the natural frequency f y,2= 1991.19 Hz, damping ratio ξ y,2 = 0.020, modal mass m x,2 = 0.0455 kg. After determining the M, C, K matrices from the modal test results, the stability lobe diagrams at each representative feed direction angle are calculated, and the final stability lobe diagram of the cavity spiral milling process is determined, as shown in FIG. 6. Figure 7

[0196] Step 5, time-domain stability analysis method based on the discontinuous Galerkin method: after determining the feed direction angle θ and the engagement angle φ in the dynamic model, en the cavity spiral milling dynamic model represented by equation (14) can be simplified as:

[0197]

[0198] Let At the same time, U(t) = [q(t)p(t)] T , equation (16) can be converted into the following state space form:

[0199]

[0200] Where

[0201]

[0202]

[0203] U(t) is a d-dimensional state vector. A is a constant matrix representing the time-invariant nature of the system, B(t) is a periodic matrix determined by the dynamic cutting force considering the regenerative effect, and satisfies B(t+T) = B(t), T is the time period and is equal to the time lag.

[0204] According to whether the tool is in contact with the workpiece, the time period T is divided into two stages of free vibration and forced vibration. Without loss of generality, t0represents the time when the tool leaves the workpiece, t f represents the length of the free vibration period after the tool leaves the workpiece, and t c is the length of the period when the tool cuts into the workpiece (the length of the forced vibration period). When the tool is not in contact with the workpiece, the B(t) term in equation (17) is equal to zero, and the relationship between the states at the beginning and end of the free vibration is:

[0205]

[0206] Next, the forced vibration period is processed, and the time period [t0+t f , t0+T] is evenly scattered into m small time zones, so that t c = T-t f ​= mτ, where m is a positive integer. The partition ε of the forced vibration period is defined as τ is defined as t1< t2<... < t m < t m+1 , where t1= t0+ τ, t f ,..., t n = t1+ (n-1)τ, n = 1,..., m+1. The open interval J n = (t n , t n+1 ), n = 1,..., m, and the d-dimensional vector space of k times piecewise discontinuous polynomials is denoted by D k (ε τ ):

[0207]

[0208] where P k (J n ) is the space of k times polynomials on the interval J n , and v is a d-dimensional vector function of D k (ε τ ). The jump and average at the interior nodes of the interval can be defined as and

[0209]

[0210] By convention, the definitions of the jump and average at the endpoints of the interval are extended as

[0211]

[0212] Taking the dot product of equation (17) with the vector v and integrating over the interval J n , we obtain

[0213]

[0214] Integrating by parts on the left side of the equation, we obtain

[0215]

[0216] Substituting equation (25) into equation (24) and summing over all intervals, we obtain

[0217]

[0218] For the interior nodes, equation (22) gives

[0219]

[0220] Combining equations (23) and (27), and noting that the exact solution U(t) satisfies​ Equation (26) can be transformed into:

[0221]

[0222] Since the state variable U(t) should be continuous when switching between the free vibration phase and the forced vibration phase, we have:

[0223]

[0224] From equation (29), we further have:

[0225]

[0226] Combining equation (28) and equation (30), we finally have the equation that makes the solution x(t) satisfy the continuity constraint:

[0227]

[0228] We consider using discontinuous piecewise quadratic polynomials for the solution, i.e., k = 2. For the local basis functions of P2(J n ), we choose monomial basis functions converted from the interval (-1, 1):

[0229]

[0230] where

[0231] For each interval J n , each component of the solution vector U(t) is approximated by a quadratic polynomial. Then, the solution vector U(t) on the interval J n can be expressed as:

[0232]

[0233] where is the coefficient vector of each basis function for this interval, which can be combined as

[0234] For each time interval J n , each component of the d-dimensional vector v is independent of each other and satisfies Considering each element of the vector v separately, we have the following equation from the integral of the vector v and the solution vector U(t) on the time interval J n :

[0235]

[0236] where I is the d-dimensional identity matrix.

[0237] The integral terms of formula (31) can be calculated in the same way as formula (34) to obtain the local matrix M corresponding to the coefficient vector of the interval basis function n , N n , R n .

[0238] Then, the same segmented quadratic polynomial expansion method and the component selection method of vector v are used to expand each term containing the jump and average at the node in formula (31) to obtain the local matrix E corresponding to the coefficient vector of the interval basis function n , F n , G n , H n , J n , K n , L n , U n .

[0239] All local matrices are calculated according to each term in formula (31), and are combined into a global matrix according to the order of the coefficient vector. Then the following discrete dynamic mapping is constructed:

[0240]

[0241] wherein

[0242]

[0243]

[0244]

[0245]

[0246] Then, the state transition matrix Φ on a single tooth cutting period can be expressed as:

[0247] Φ = (S + T + Q) -1 (W + V) (36)

[0248] Finally, the milling stability can be determined by Floquet theory. If the modulus of any eigenvalue of the state transition matrix is greater than 1, the milling process is unstable; if the modulus of the maximum eigenvalue is less than 1, the milling process is stable; if the modulus of the maximum eigenvalue is equal to 1, it is on the stability domain boundary.

[0249] The application further provides a system for predicting cutting force and judging stability in a cavity helical milling process, which can be realized by executing the process steps of the method for predicting cutting force and judging stability in the cavity helical milling process, i.e., the method for predicting cutting force and judging stability in the cavity helical milling process can be understood as a preferred embodiment of the system for predicting cutting force and judging stability in the cavity helical milling process.

[0250] A system for predicting cutting force and judging stability in a cavity helical milling process comprises:

[0251] A tool path generation module generates a smooth helical milling tool path in the form of a B-spline curve according to a known shape and size of a cavity to be machined and a set maximum span.

[0252] A modeling module performs geometric modeling of a machining engagement area at each discrete tool position of the smooth helical milling tool path based on a tool tooth trajectory circular arc assumption.

[0253] A cutting force prediction module establishes a cavity helical milling cutting force model based on the modeling result of the modeling module, and realizes prediction of transient cutting force in the cavity helical milling process.

[0254] A stability judgment module establishes a helical milling dynamics equation dependent on a feed direction based on the modeling result of the modeling module, considers dynamic cutting thickness regeneration effects, performs stability analysis on a representative feed direction along the helical tool path, and judges the stability of the cavity helical milling process.

[0255] An optimization module performs optimization by using a time domain method based on the discontinuous Galerkin method based on the helical milling dynamics equation.

[0256] Those skilled in the art know that, in addition to implementing the system and each device, module and unit thereof provided by the application in a pure computer readable program code manner, the system and each device, module and unit thereof provided by the application can also be realized in the form of logic gates, switches, application specific integrated circuits, programmable logic controllers and embedded microcontrollers by logically programming the method steps to achieve the same functions. Therefore, the system and each device, module and unit thereof provided by the application can be considered as a hardware component, and the devices, modules and units included therein for achieving various functions can also be considered as structures within the hardware component; the devices, modules and units for achieving various functions can also be considered as both software modules for implementing the method and structures within the hardware component.

[0257] The specific embodiments of the present application are described above. It needs to be understood that the present application is not limited to the specific embodiments described above, and various changes or modifications can be made by those skilled in the art within the scope of the claims, which does not affect the essential content of the present application. The embodiments of the present application and the features in the embodiments can be combined with each other at will without conflict.

Claims

1. A method for predicting cutting force and determining stability during cavity helical milling, characterized in that, include: Toolpath generation steps: Generate a smooth helical milling toolpath in the form of a B-spline curve based on the known shape and size of the cavity to be machined and the set maximum span; Modeling steps: Based on the assumption of circular arc of the cutter tooth trajectory, geometric modeling is performed on the machining meshing area at each discrete tool point of the smooth helical milling toolpath; Cutting force prediction steps: Based on the modeling results of the modeling steps, a cutting force model for cavity helical milling is established to predict the transient cutting force during cavity helical milling. Stability assessment steps: Based on the modeling results of the modeling steps, considering the dynamic thickness regeneration effect, establish the helical milling dynamic equation that depends on the feed direction, and perform stability analysis on the representative feed directions along the helical tool path to determine the stability of the cavity helical milling process. Optimization steps: Based on the dynamic equations of helical milling, optimization is performed using a time-domain method based on the discontinuous Galerkin method; The modeling steps include: Geometric modeling of the machining engagement area is performed by solving for the intersection point between the tool and the workpiece surface, based on the tool position point C(x). C ,y C The intersection relationship between the tool circumference centered at ) and the machined surface divides the cavity helical milling process into two sub-stages: in the entry stage, the tool circumference intersects with the initial circumferential surface of the internal boundary; in the steady-state stage, the tool circumference intersects with the machined surface formed by the helical tool path. For the cutting phase, the intersection point P2(x2,y2) is solved by solving the system of equations between the tool circumference and the initial circumference surface: Where r is the tool radius, R i It is the inner boundary radius; For the steady-state phase, the discrete tool position C(x) is obtained by discretizing the curve parameter u. C ,y C The discrete machined surface profile points P1(x1,y1) are then calculated using the corresponding feed direction angle θ. For a given knife point C(u) i ), calculate the discrete machined surface contour points using equation (2), obtain the discrete machined surface contour points and the distance between the discrete machined surface contour points and the tool position point, so as to determine the intersection point of the tool circumference and the discrete machined surface contour. The point with the largest corresponding curve parameter among these intersection points is the intersection point P2(x2,y2). After determining the intersection point P2(x2,y2) through geometric modeling, the angle of inclination φ is calculated. st and the cutting angle φ ex For climb milling, the exit angle is fixed at π, and the entry angle is determined by the intersection point P2(x2,y2).

2. The method for predicting cutting force and determining stability during cavity helical milling according to claim 1, characterized in that, The toolpath generation step includes: The elliptic partial differential equations of the Dirichlet boundary conditions are solved using the finite element method. Based on the set maximum span, equal closed field curves are generated. After dividing the field curves equally, transition curves are generated, resulting in a set of spiral point sequences. The spiral point sequences are then fitted with B-spline curves to obtain a smooth spiral milling toolpath in the form of B-spline curves.

3. The method for predicting cutting force and determining stability during cavity helical milling according to claim 1, characterized in that, The cutting force prediction step includes: By densely discretizing the toolpath of smooth helical milling in the form of B-spline curves, the length of the spline curve between adjacent discrete tool positions is approximated by the straight-line distance between them, thus obtaining the relationship between the motion distance l along the toolpath and the curve parameter u. Once the tool feed rate V and spindle speed Ω are determined, the time t(u) and feed per tooth f corresponding to each tool position are obtained. t : Where N t The number of cutting teeth; The milling force is determined using a discrete summation strategy, dividing the milling cutter, which has a helix angle effect, into N along the tool axis. A Each disk, for each tool position point, is located in the tool coordinate system X. c -Y c In the diagram, the position angle of the cutting element of the k-th disk of the j-th tooth of the tool is... f j,k =(2πΩ / 60)t(u)-θ(u)+(j-1)·2π / N-(k-1)·(2·dz·tanβ / D) (6) Where dz is the axial cutting depth of the disk, β is the helix angle of the tool, and D is the diameter of the tool; Based on the approximate assumption of the circular arc of the cutter tooth trajectory, the instantaneous undeformed cutting thickness of the cutting micro-element is determined as follows: h j,k =f t sin(φ j,k ) (7) The tangential and radial cutting forces corresponding to the cutting micro-element are as follows: Where K te ,K tc ,K re and K rc This is the cutting force coefficient. For each cutting micro-element, the cutting force in the tool coordinate system is expressed as: By traversing all the cutting micro-elements in the cutting process, the tool coordinate system X... c -Y c The instantaneous cutting force in is Where g(φ) j,k The switching function is used to determine whether the cutting element is cutting at the current moment, when φ is satisfied. st ≤φ j,k ≤φ ex When the switching function is active, it is 1; otherwise, it is 0. The angle of entry φ st and the cutting angle φ ex As determined by the modeling steps, the instantaneous cutting force in the tool coordinate system is transformed to the global coordinate system to obtain:

4. The method for predicting cutting force and determining stability during cavity helical milling according to claim 3, characterized in that, The stability determination step includes: In the dynamic model, the flexibility of the tool in two transverse directions is considered for stability prediction. Since the static chip thickness in equation (7) does not affect stability, only the dynamic chip thickness is included in the dynamic model, which is expressed as follows: Where Δx = x(t) - x(tT) and Δy = y(t) - y(tT) are the dynamic displacements of the tool in the global coordinate system during the previous and current cutting cycles, respectively, and T is the time delay and equal to the cutting cycle, i.e., T = 60 / (N). t Ω); Substituting equations (8) and (12) into equation (11), the total dynamic cutting force in the global coordinate system is calculated as follows: Where δ=φ j,k +θ includes the feed direction angle θ and the instantaneous position angle φ. j,k ; Therefore, the dynamic model of cavity helical milling is established as follows: Where M, C, and K are the matrices of modal mass, modal damping, and modal stiffness, respectively, and q(t) = [x(t)y(t)]. T K is the tool displacement vector, and its coefficient matrix is ​​related to time and feed direction. c (θ,t) is represented as: Stability prediction for cavity helical milling includes: traversing the curve parameters u and calculating the feed direction angle θ and the corresponding meshing angle φ. en ; Traverse the representative feed direction angles, for each traversed feed direction angle θ traverse Select the feed direction angle θ traverse The corresponding maximum engagement angle max(φ) en Dynamic modeling is performed to consider the most conservative stability conditions; the stability lobe diagram of the entire cavity helical milling process is determined by the minimum envelope curve of the stability boundary at each representative feed direction angle.

5. The method for predicting cutting force and determining stability during cavity helical milling according to claim 4, characterized in that, The optimization steps include: Determining the feed direction angle θ and engagement angle φ in the dynamic model en Then, the dynamic model of cavity helical milling represented by equation (14) is simplified to: make Let U(t) = [q(t) p(t)] T Equation (16) is transformed into the following state-space form: in U(t) is a d-dimensional state vector, A is a constant matrix representing the time-invariant properties of the system, and B(t) is a periodic matrix determined by the dynamic cutting force considering the regeneration effect, satisfying B(t+T)=B(t), where T is the time period and equal to the time delay. Depending on whether the tool is in contact with the workpiece, the time period T is divided into two stages: free vibration and forced vibration. Let t0 represent the moment the tool leaves the workpiece. f The duration of free vibration after the tool leaves the workpiece is denoted as t. c Let B(t) be the duration of the cutting time in the workpiece. When the cutting tool is not in contact with the workpiece, the term B(t) in equation (17) is equal to zero. The relationship between the state at the beginning and end of the free vibration is as follows: Next, we process the forced vibration period, dividing the time interval [t0+t] into... f The interval [t0+T] is divided into m-hour segments at equal distances, thus satisfying t c =Tt f =mτ, where m is a positive integer, and ε represents the division of the forced vibration period. τ Defined as t1 <t2<…<t m <t m+1 Where t1 = t0 + t f ,t n =t1+(n-1)τ, n=1,...,m+1, define the open interval J n =(t n ,t n+1 ), n = 1, ..., m, and use D k (ε τ Let d represent the d-dimensional vector space of a piecewise discontinuous polynomial of degree k. Among them, P k (J n ) is the interval J n A polynomial space of degree k on D, where v is a polynomial space of degree D k (ε τ The d-dimensional vector function of ) is defined. and Then define the jumps and averages at the nodes within the interval: The definitions of jumps and averages at the endpoints of the interval are extended: Take the dot product of equation (17) and vector v, and then apply the product to the interval J. n Integrating the above, we get: Integrating by parts on the left side of the equation, we get: Substituting equation (25) into equation (24) and summing over all intervals, we get: For internal nodes, from equation (22) we get: Combining equations (23) and (27), and noting that the exact solution U(t) satisfies Equation (26) is transformed into: Since the state variable U(t) should be continuous during the transition between the free vibration stage and the forced vibration stage, we get: From equation (29), we can further obtain: Combining equations (28) and (30), we finally obtain the equation that satisfies the continuity constraint for the solution x(t): Consider using a discontinuous piecewise quadratic polynomial to solve the problem, i.e., k = 2, for P2(J n For the local basis functions of (-1, 1), choose monomial basis functions derived from the interval (-1, 1): in For each interval J n Each component of the solution vector U(t) is approximated by a quadratic polynomial, and then the interval J n The solution vector U(t) on the solution vector is expressed as: in The coefficient vectors of the basis functions for this interval are combined as follows: For each time interval J n Each component of the d-dimensional vector v is independent of each other and satisfies Considering each element of vector v individually, the vector v and the solution vector U(t) are in time interval J. n Integrating over the given information yields the following equation: Where I is a d-dimensional identity matrix; Similarly to equation (34), the integral terms of equation (31) are calculated to obtain the local matrix M corresponding to the coefficient vector of the basis function in each interval. n N n R n ; Next, using the same piecewise quadratic polynomial expansion method and the same method of selecting the components of vector v, the terms in expansion (31) that contain jumps and averages at the nodes are used to obtain the local matrix E corresponding to the coefficient vector of the basis function in each interval. n F n G n H n J n K n L n U n ; Calculate all local matrices according to the terms in equation (31), and combine them into a global matrix according to the order of the coefficient vectors. Then construct the following discrete dynamic mapping: in Then, the state transition matrix Φ over a single cutting tooth cycle is expressed as: Φ=(S+T+Q) -1 (W+V) (36) Finally, milling stability is determined by Floquet theory: if the modulus of any eigenvalue of the state transition matrix is ​​greater than 1, the milling process is unstable; if the modulus of the largest eigenvalue is less than 1, the milling process is stable; if the modulus of the largest eigenvalue is equal to 1, it is at the boundary of the stable region.

6. A cutting force prediction and stability judgment system for cavity helical milling, characterized in that, include: Toolpath generation module: Generates a smooth helical milling toolpath in the form of a B-spline curve based on the known shape and size of the cavity to be machined and the set maximum span; Modeling module: Based on the assumption of circular arc of the cutter tooth trajectory, geometric modeling is performed on the machining meshing area at each discrete tool point of the smooth helical milling toolpath; Cutting force prediction module: Based on the modeling results of the modeling module, a cutting force model for cavity helical milling is established to predict the transient cutting force during cavity helical milling. Stability assessment module: Based on the modeling results of the modeling module, considering the dynamic thickness regeneration effect, a helical milling dynamic equation dependent on the feed direction is established, and a stability analysis is performed on the representative feed directions along the helical tool path to determine the stability of the cavity helical milling process. Optimization module: Based on the dynamic equations of helical milling, optimization is performed using a time-domain method based on the discontinuous Galerkin method; The modeling module includes: Geometric modeling of the machining engagement area is performed by solving for the intersection point between the tool and the workpiece surface, based on the tool position point C(x). C ,y C The intersection relationship between the tool circumference centered at ) and the machined surface divides the cavity helical milling process into two sub-stages: in the entry stage, the tool circumference intersects with the initial circumferential surface of the internal boundary; in the steady-state stage, the tool circumference intersects with the machined surface formed by the helical tool path. For the cutting phase, the intersection point P2(x2,y2) is solved by solving the system of equations between the tool circumference and the initial circumference surface: Where r is the tool radius, R i It is the inner boundary radius; For the steady-state phase, the discrete tool position C(x) is obtained by discretizing the curve parameter u. C ,y C The discrete machined surface profile points P1(x1,y1) are then calculated using the corresponding feed direction angle θ. For a given knife point C(u) i ), calculate the discrete machined surface contour points using equation (2), obtain the discrete machined surface contour points and the distance between the discrete machined surface contour points and the tool position point, so as to determine the intersection point of the tool circumference and the discrete machined surface contour. The point with the largest corresponding curve parameter among these intersection points is the intersection point P2(x2,y2). After determining the intersection point P2(x2,y2) through geometric modeling, the angle of inclination φ is calculated. st and the cutting angle φ ex For climb milling, the exit angle is fixed at π, and the entry angle is determined by the intersection point P2(x2,y2).

7. The cutting force prediction and stability judgment system for cavity helical milling according to claim 6, characterized in that, The toolpath generation module includes: The elliptic partial differential equations of the Dirichlet boundary conditions are solved using the finite element method. Based on the set maximum span, equal closed field curves are generated. After dividing the field curves equally, transition curves are generated, resulting in a set of spiral point sequences. The spiral point sequences are then fitted with B-spline curves to obtain a smooth spiral milling toolpath in the form of B-spline curves.

8. The cutting force prediction and stability judgment system for cavity helical milling according to claim 6, characterized in that, The cutting force prediction module includes: By densely discretizing the toolpath of smooth helical milling in the form of B-spline curves, the length of the spline curve between adjacent discrete tool positions is approximated by the straight-line distance between them, thus obtaining the relationship between the motion distance l along the toolpath and the curve parameter u. Once the tool feed rate V and spindle speed Ω are determined, the time t(u) and feed per tooth f corresponding to each tool position are obtained. t : Where N t The number of cutting teeth; The milling force is determined using a discrete summation strategy, dividing the milling cutter, which has a helix angle effect, into N along the tool axis. A Each disk, for each tool position point, is located in the tool coordinate system X. c -Y c In the diagram, the position angle of the cutting element of the k-th disk of the j-th tooth of the tool is... f j,k =(2πΩ / 60)t(u)-θ(u)+(j-1)·2π / N-(k-1)·(2·dz·tanβ / D) (6) Where dz is the axial cutting depth of the disk, β is the helix angle of the tool, and D is the diameter of the tool; Based on the approximate assumption of the circular arc of the cutter tooth trajectory, the instantaneous undeformed cutting thickness of the cutting micro-element is determined as follows: h j,k =f t sin(φ j,k ) (7) The tangential and radial cutting forces corresponding to the cutting micro-element are as follows: Where K te ,K tc ,K re and K rc This is the cutting force coefficient. For each cutting micro-element, the cutting force in the tool coordinate system is expressed as: By traversing all the cutting micro-elements in the cutting process, the tool coordinate system X... c -Y c The instantaneous cutting force in is Where g(φ) j,k The switching function is used to determine whether the cutting element is cutting at the current moment, when φ is satisfied. st ≤φ j,k ≤φ ex When the switching function is active, it is 1; otherwise, it is 0. The angle of entry φ st and the cutting angle φ ex Determined by the modeling module, the instantaneous cutting force in the tool coordinate system is transformed to the global coordinate system to obtain:

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