Titanium alloy thin-walled workpiece end milling process force-induced deformation prediction method considering tool wear

Abstract

The invention relates to a method for predicting force-induced deformation of a titanium alloy thin-wall part in an end milling machining process by considering tool wear, and belongs to the technical field of cutting machining. The method comprises the following steps: firstly, constructing an annular cutter milling thin-wall part overall milling force model considering cutter wear; on the basis, a force-induced deformation model of the material removal effect in the end milling machining process of the thin-walled workpiece is established based on a small deflection deformation theory. The model can correct element stiffness matrixes at different feeding positions after materials are removed in the milling process of the thin-wall part, and rapid extraction of force-induced deformation of the thin-wall part is achieved. Finally, an iterative calculation method is adopted, and accurate prediction of cutting force and machining deformation at different feeding positions in the end milling machining process of the thin-walled workpiece is achieved. According to the method, the influence of cutter abrasion on the milling force when titanium alloy type difficult-to-machine materials are milled is considered, and the force-induced deformation predicted value after cutter abrasion in end milling of the titanium alloy thin-wall part can be accurately obtained.

Description

Technical Field

[0001] The invention belongs to the technical field of cutting processing, and in particular to a method for predicting force-induced deformation during end milling of titanium alloy thin-walled parts taking into account tool wear. Background Art

[0002] Titanium alloys are widely used in thin-walled aerospace engine components due to their high specific strength and excellent high-temperature resistance. However, due to their weak rigidity, high machining difficulty, and high cutting forces, thin-walled titanium alloy components are prone to significant machining deformation and severe tool wear, ultimately affecting material removal accuracy and workpiece surface quality.

[0003] Force-induced deformation is an unavoidable problem during the milling of weakly rigid thin-walled parts. The patent "An Iterative Prediction Method and System for Thin-Walled Part Deformation Based on Birth-Death Cells, CN118536320B" uses birth-death cell technology to determine the deformation after material removal, avoiding the tedious step of re-meshing. The paper "Predictive Modeling of Chatter Stability Considering Force-Induced Deformation Effect in Milling Thin-Walled Parts[J]. International Journal of Machine Tools and Manufacture, 2018, 135: 38-52" simulates the material removal effect by modifying the cell stiffness based on volumetric changes using a correction factor. This approach achieves accurate calculation of force-induced deformation through an iterative strategy. While traditional finite element software methods for calculating deformation offer some reliability, they require a detailed pre-demarcation of the cutting area and determination of the milling force loading locations based on parameters such as workpiece material properties and cutting conditions. This process is time-consuming and its adaptability to working conditions needs improvement. Furthermore, existing methods ignore the critical factor of tool wear when constructing milling force models. Especially when milling difficult-to-machine materials such as titanium alloys, tool wear can lead to a significant increase in milling forces. Traditional cutting force models may not accurately reflect the changes in milling forces caused by tool wear, which in turn affects the prediction of force-induced deformation in thin-walled parts. Therefore, it is urgent to develop a method for predicting force-induced deformation during end milling of thin-walled titanium alloy parts that considers the effects of tool wear. Summary of the Invention

[0004] Aiming at the problems of machining deformation and tool wear in the end milling process of titanium alloy thin-walled parts, the purpose of this invention is to propose a force-induced deformation prediction method for the end milling process of titanium alloy thin-walled parts taking into account tool wear, and to provide theoretical support for the research on machining deformation of weak rigidity thin-walled parts.

[0005] To achieve the above object, the present invention provides the following technical solutions:

[0006] First, a global milling force model for thin-walled parts milling with an annular cutter was constructed, taking tool wear into account. Based on this, a force-induced deformation model after material removal during end milling of thin-walled parts was established based on the theory of small deflection deformation. This model can correct the element stiffness matrix at different feed positions during the milling of thin-walled parts as material is removed, enabling rapid extraction of force-induced deformation of thin-walled parts. Finally, an iterative calculation method was used to accurately predict the cutting force and machining deformation at different feed positions during end milling of thin-walled parts.

[0007] Step 1: Modeling the overall milling force considering tool wear.

[0008] When the tool is worn, friction will occur between the tool's flank face and the workpiece surface, which will cause an increase in the overall milling force. Therefore, a flank wear force model is constructed to decompose the force along the flank face into the tangential force dF tw,j,k , radial force dF rw,j,k and axial dF aw,j,k , and define the wear coefficients in the three directions as K tw ,K rw and K aw Calculate the tangential force dF for each axial height element t,j,k , radial force dF r,j,k and axial force dF a,j,k . Through the coordinate transformation matrix T j,k , and finally the overall milling force considering tool wear in three directions of the tool coordinate system is obtained.

[0009] Step 2: Construct a force-induced deformation model.

[0010] Based on the small deflection deformation theory, the unit stiffness matrix of titanium alloy thin-walled parts at different feed positions is calculated. The unit stiffness matrix is ​​superimposed using the direct stiffness method to obtain the overall stiffness matrix K of the thin-walled parts. Then, the force-induced deformation of the thin-walled parts is calculated to obtain the force-induced deformation of the thin-walled parts after loading.

[0011] Step 3: Process parameter update strategy.

[0012] Considering the material removal effect during thin-walled part milling, the element stiffness matrix at the feed position mentioned in step 2 needs to be adjusted. By directly adjusting the element stiffness matrix of the thin-walled part's machining area, the change in the overall stiffness of the thin-walled part caused by the change in the part's thickness after material removal is corrected, allowing for the rapid extraction of the displacement matrix of the deformed thin-walled part. The overall stiffness of the thin-walled part changes continuously with the feed direction. Assuming the total volume removed by the milling cutter at the current feed position is H units, the overall stiffness of the thin-walled part is the sum of the element stiffness of the unmachined area of ​​the thin-walled part and the element stiffness of the remaining material at the current feed position. The overall stiffness matrix is ​​then updated. This allows the displacement matrix of the thin-walled part after material removal to be calculated.

[0013] Step 4: Iterative calculation of force-induced deformation.

[0014] Based on the milling force model mentioned in step 1 and the force-induced deformation model mentioned in step 2, an iterative calculation of the force-induced deformation of thin-walled parts is performed considering material removal. Specifically, it includes: inputting cutting parameters, tool parameters, and thin-walled part parameters; calculating the initial overall stiffness matrix of the thin-walled part; discretizing the feed position and initializing the feed position; calculating the initial milling force to obtain the initial deformation of the workpiece; calculating the change in milling force after machining deformation and determining whether it converges. If convergence is achieved, the cutting parameters are updated. If not, the above process is repeated; after completing machining at the current feed position, proceed to the next position and update the corresponding stiffness matrix of the thin-walled part based on the material removal situation at the current feed position; after completing all machining, the final machining deformation and milling force are output.

[0015] Beneficial effects of the present invention: Force-induced deformation is an inevitable problem in the processing of flexible thin-walled parts. In addition, for titanium alloy thin-walled parts, severe tool wear will lead to a significant increase in milling force. The original cutting force model cannot accurately reflect the changes in actual milling force, affecting the accuracy of machining deformation prediction. Therefore, a new method for predicting force-induced deformation in end milling of titanium alloy thin-walled parts considering tool wear is proposed. This method can correct the unit stiffness matrix after material removal at different feed positions during the milling process of thin-walled parts, and realize the rapid extraction of force-induced deformation, thereby avoiding the complex mesh re-division and stiffness modification steps in the traditional method of using finite element software. This method can accurately predict the cutting force and deformation in the milling process of thin-walled parts, which is of great significance for optimizing machining technology and improving machining quality. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0017] Figure 1 A schematic diagram of the overall milling force considering tool wear provided by the present invention;

[0018] Figure 2 A schematic diagram of the displacement of rectangular unit nodes provided by the present invention;

[0019] Figure 3 Schematic diagram of the force-induced deformation of a thin-walled part provided by the present invention, wherein (a) is a schematic diagram of the force-induced deformation of the thin-walled part, and (b) is a schematic diagram of the material removal of the thin-walled part;

[0020] Figure 4 A flow chart of iterative calculation of force-induced deformation provided by the present invention;

[0021] Figure 5 The milling force verification provided by the present invention, wherein (a) is a comparison diagram of the measured milling force and the predicted milling force of No. 1, (b) is a comparison diagram of the measured milling force and the predicted milling force of No. 2, (c) is a comparison diagram of the measured milling force and the predicted milling force of No. 3, and (d) is a comparison diagram of the measured milling force and the predicted milling force of No. 4;

[0022] Figure 6 This is the force-induced deformation verification provided by the present invention, where (a) is a comparison diagram of the measured deformation and the predicted deformation of No. 1, (b) is a comparison diagram of the measured deformation and the predicted deformation of No. 2, (c) is a comparison diagram of the measured deformation and the predicted deformation of No. 3, and (d) is a comparison diagram of the measured deformation and the predicted deformation of No. 4. DETAILED DESCRIPTION

[0023] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0024] The purpose of the present invention is to provide a method for predicting force-induced deformation during end milling of titanium alloy thin-walled parts taking into account tool wear, so as to accurately predict the milling force and machining deformation during the milling process of titanium alloy thin-walled parts.

[0025] In order to make the above-mentioned objects, features and advantages of the present invention more obvious and easy to understand, the present invention is further described in detail below with reference to the accompanying drawings and specific embodiments.

[0026] The present invention proposes a method for predicting force-induced deformation during the end milling process of titanium alloy thin-walled parts, taking into account tool wear. First, an overall milling force model for thin-walled parts milled by annular cutters, taking into account tool wear, is constructed. On this basis, a force-induced deformation model of the material removal effect during the end milling process of thin-walled parts is established based on the small deflection deformation theory. This model can correct the unit stiffness matrix after material removal at different feed positions during the milling process of thin-walled parts, thereby realizing the rapid extraction of force-induced deformation of thin-walled parts. Finally, an iterative calculation method is used to achieve accurate prediction of cutting force and machining deformation at different feed positions during the end milling process of thin-walled parts.

[0027] Step 1. Modeling of the overall milling force considering tool wear.

[0028] When end milling titanium alloy thin-walled parts with an integral annular milling cutter, when the tool is worn, friction will occur between the tool back face and the workpiece surface, which will lead to an increase in the overall milling force. T X T Y T Z T Under this condition, the cutting depth is discretized into m parts, and the tangential force dF of the kth element acting on the jth cutting edge at time t is t,j,k , radial force dF r,j,k , axial force dF a,j,k The expression is:

[0029]

[0030] In the formula, g(φ j,k (t)) is the window function, K tc ,K rc ,K ac are the tangential, radial and axial shear force coefficients, K te ,K re ,K ae are the tangential, radial and axial cutting edge force coefficients respectively, h j,k (t) is the thickness of the undeformed chip.

[0031] Furthermore, dF tw,j,k is the microelement tangential wear force, dF rw,j,k is the microelement radial wear force, dF aw,j,k is the microelement axial wear force, and its expression is:

[0032]

[0033] Where K tw ,K rw ,K aw are the tangential, radial and axial wear force coefficients respectively, and VB is the flank wear width.

[0034] Furthermore, the expressions of the microelement axial height dz, the microelement cutting edge width db and the microelement cutting edge length ds are:

[0035]

[0036] Where a p is the cutting depth, κ j,k is the axial contact angle, R z is the effective radius, R z The first derivative of .

[0037] The thickness of the undeformed chip is:

[0038]

[0039] Indicates the feed per tooth;

[0040] The axial contact angle is the angle between point Q and the tool axis, and its expression is:

[0041]

[0042] Where r is the radius of the circular cutter corner, z is the distance from point Q to point X T O T Y T Distance on the surface.

[0043] Effective radius R z is the distance from point Q to the tool axis, and its expression is:

[0044]

[0045] The window function is used to determine whether the Q point on the cutting edge participates in cutting. Its expression is:

[0046]

[0047] Where, φ en is the cutting angle, φ ex To cut out the corners.

[0048] The expression of radial contact angle is:

[0049]

[0050] Where n is the rotation speed, β is the helical angle, φ p is the tooth angle, φ p =2π / N t , N t is the number of tool teeth.

[0051] Furthermore, the micro-element milling force in the tool coordinate system is:

[0052]

[0053] Where, T j,k (t) is the transformation matrix.

[0054]

[0055] Therefore, the overall milling force after tool wear in the tool coordinate system is:

[0056]

[0057] Step 2. Construct a force-induced deformation model.

[0058] In thin-walled parts W X W Y W Z W In this paper, the deformation prediction of thin-walled parts is carried out based on the small deflection theory of thin plates. It is assumed that after the thin-walled parts are bent, the extrusion deformation in the thickness direction of the plate is negligible, that is, the normal strain ε perpendicular to the mid-plane is z =0; the middle surface of the plate only undergoes bending deformation, that is, γ yz =γ zx =0; it is perpendicular to the mid-surface of the thin plate before deformation and remains perpendicular to the deformed mid-surface after deformation.

[0059] Furthermore, based on the above assumptions, the displacement component w of each point after the thin-walled part is bent and deformed is only related to x and y, and has nothing to do with z. The deflection of the thin-walled part is:

[0060]

[0061] The relationship between strain components and displacement is:

[0062]

[0063] Where, ε x and ε y are the normal strains in the X and Y directions, γ xy For X W O W Y W Shear strain on the plane, u and v are the displacement components in the X and Y directions, respectively.

[0064] The relationship between stress and strain is:

[0065]

[0066] Where D p is the coefficient matrix.

[0067]

[0068] Where E is the elastic modulus of the thin-walled material, and μ is the Poisson's ratio of the thin-walled material.

[0069] The curvature can be expressed as:

[0070]

[0071] Therefore, the stress-strain relationship can be expressed as:

[0072]

[0073] The internal forces of thin-walled parts are bending moment and torque, and their expressions are:

[0074]

[0075] Among them, M x , M y M are the bending moments in the X and Y directions respectively. xy For X W O W Y W Torque in the plane, t1 is the thickness of the thin-walled part.

[0076] The elastic coefficient matrix D is:

[0077]

[0078] Furthermore, in order to calculate the deformation of thin-walled parts more accurately, the rectangular thin-walled parts can be discretized into M e (m e ×n e ) rectangular units, consisting of m e length units and n e Each rectangular element e has four nodes, and each node has three degrees of freedom, namely deflection w i , angle θ xi ,θ yi . Then the displacement of the i-th node is:

[0079]

[0080] The nodal displacement vector of the rectangular element e is:

[0081]

[0082] The displacement function is defined by a polynomial with 12 undetermined coefficients. The displacement component in the coordinate system of the thin-walled part, i.e., the deflection, is expressed as:

[0083]

[0084] Where x represents the x-direction coordinate of the rectangular unit, and y represents the y-direction coordinate of the rectangular unit;

[0085]

[0086] Where L is a 1×12 function matrix, and α is a matrix consisting of 12 undetermined coefficients.

[0087] In summary, the displacement vector of the i-th node can be rewritten as:

[0088]

[0089] The nodal displacement matrix of the rectangular element can be rewritten as:

[0090]

[0091] Where C is a 12×12 matrix.

[0092] Therefore, the deflection can be rewritten as:

[0093]

[0094] Where N is the shape function matrix of the thin-walled unit bending, which is a 1×12 matrix.

[0095]

[0096] N i is the shape function matrix of the i-th node, which is a 1×3 matrix.

[0097]

[0098] Where,

[0099]

[0100] Where a and b are the rectangular unit size parameters, x i Indicates the x-direction coordinate of the i-th node, y i Indicates the y-coordinate of the i-th node.

[0101] Therefore, the curvature can be rewritten as:

[0102]

[0103] Where B is the strain-displacement matrix of the rectangular element, which can be expressed as:

[0104]

[0105] Elastic strain energy U of rectangular element e for:

[0106]

[0107] Where V is the volume of the rectangular unit.

[0108] Integrating along the thickness Z direction yields:

[0109]

[0110] Where S is the area of ​​the rectangular unit.

[0111] Stiffness matrix K of rectangular element e It can be expressed as:

[0112]

[0113] Among them, K e It is a 12×12 matrix.

[0114] Therefore, the total strain energy of the thin-walled part is:

[0115]

[0116] Where,

[0117]

[0118] Where K is the overall stiffness matrix of the thin-walled part, which is 3(m e +1)(n e +1)×3(m e +1)(n e +1) matrix. In order to construct the overall stiffness matrix K of the thin-walled part, it is necessary to obtain the element stiffness matrix K e The direct stiffness method is used to superimpose the unit stiffness matrices to obtain the overall stiffness matrix of the thin-walled part.

[0119] After the thin-walled part is loaded, the total potential energy can be expressed as:

[0120]

[0121] Where W is the work done by P, and P is the external force applied to the node, i.e., the force F in the Z direction. Z The matrix composed of .

[0122] Based on the principle of minimum potential energy, there are extreme conditions , which is converted into the extreme value condition of a general multivariate function. Therefore, the matrix of the thin-walled part after force-induced deformation can be solved:

[0123]

[0124] One end of the thin-walled part is fixed, and the three ends are free. The coordinate system of the thin-walled part is O W X W Y W Z W The fixed boundary conditions are:

[0125]

[0126] Step 3. Process parameter update strategy for thin-walled part milling. When machining weakly rigid thin-walled parts, the milling force causes deformation of the tool and workpiece. These deformations further affect the milling force. Therefore, a process parameter update strategy for thin-walled part milling force-induced deformation is proposed. This strategy directly adjusts the element stiffness at the feed position to correct for changes in the overall stiffness of the thin-walled part caused by the thickness change after material removal. The strategy also updates the cutting parameters at the feed position, thereby gradually correcting and optimizing the calculation results.

[0127] 3.1 Machining deformation affects cutting parameters, resulting in limitations in the milling force model. W O W Z W On the plane, the updated cutting depth is:

[0128]

[0129] Where, represents the deformation of the plate at position (x,y).

[0130] 3.2 Time-Varying Update Method for the Stiffness Matrix of Thin-Walled Parts After Material Removal. Due to material removal, the thickness of thin-walled parts changes, affecting their overall stiffness. To more accurately predict deformation, the stiffness matrix of the thin-walled part at different feed positions must be updated. This method directly modifies the element stiffness matrix of the thin-walled part region after material removal, thereby adjusting the overall stiffness matrix of the thin-walled part for more accurate deformation calculation.

[0131] The following formula is used to calculate the element stiffness of thin-walled parts after material removal:

[0132]

[0133] Where t2 is the thickness of the thin-walled part after material removal in the current path.

[0134] The overall stiffness of the thin-walled part changes continuously with the feed direction. Assuming that the total volume removed by the milling cutter at the current feed position is H units, the overall stiffness of the thin-walled part is the sum of the unit stiffness of the unprocessed area of ​​the thin-walled part and the unit stiffness of the remaining material at the current feed position. The updated stiffness matrix is:

[0135]

[0136] Therefore, the matrix of the thin-walled part after material removal due to force deformation is obtained:

[0137]

[0138] Step 4. Iterative calculation of force-induced deformation. Based on the milling force model mentioned in step 1 and the force-induced deformation model mentioned in step 2, an iterative calculation method of force-induced deformation of thin-walled parts considering material removal is proposed. Figure 5 shown.

[0139] 4.1 Iterative calculation process of force-induced deformation, including:

[0140] Input cutting parameters, tool parameters and thin-walled part parameters; calculate the initial overall stiffness matrix of the thin-walled part;

[0141] Discrete feed position and initialize feed position;

[0142] Calculate the initial milling force to obtain the initial deformation of the workpiece;

[0143] Calculate the change in milling force after machining deformation and determine whether it converges (ε=1N). If convergence is achieved, update the cutting parameters. If not, repeat the above process.

[0144] After finishing the machining at the current feed position, it proceeds to the next position and updates the stiffness matrix of the thin-walled part according to the material removal at the current feed position;

[0145] After all machining is completed, the final machining deformation and milling force are output.

[0146] 4.2 Calibration of milling force coefficient and wear force coefficient

[0147] Because the effective radius of the annular cutter varies with axial height, the influence of geometric dimensions must be considered during the milling force coefficient identification process. A 16 mm diameter annular cutter with a 3 mm corner radius was used, and the workpiece material was TC4. A dynamometer (Kistler 92) was used to measure milling forces, and a coordinate measuring machine was used to measure machining deformation of thin-walled parts. The cutting force coefficient must be determined under rigid workpiece conditions, so the milling force coefficient was calibrated using slot milling of a titanium alloy block. The cutting parameters are shown in Table 1.

[0148] Table 1 Cutting parameters

[0149]

[0150] The cubic polynomial fitting of the cutting force and edge force coefficients is obtained:

[0151]

[0152] The milling force will increase with tool wear and is approximately linear in the stable wear stage. p =0.6mm, f t =0.05mm / r, n=1000r / min, and record the milling force increment of VB=0-0.1mm respectively to obtain the wear force coefficient.

[0153]

[0154] 4.3 Solution and verification of force-induced deformation

[0155] The specific dimensions of the thin-walled part are 94mm×52mm×5mm, and the cutting parameters are spindle speed n=1000r / min, feed per tooth f t =0.05mm / r, cutting width a e =1mm. Using the down milling method, the lengths of milling paths No.1, No.2, No.3, and No.4 to the cantilever end of the thin-walled part are 16mm, 12mm, 8mm, and 4mm respectively. No.1 and No.2 represent the paths of the new tool milling the first layer of thin-walled parts, and the cutting depth is a p =0.5mm; No.3 and No.4 represent the path of milling the second layer with a worn tool (VB=0.05mm), with a cutting depth of a p =0.6mm.

[0156] Furthermore, the milling force prediction and measured results are compared. Figure 6 As shown in the figure, the predicted curve is a solid line and the measured curve is a dotted line. y The average peak error of (No.1-No.2) is 7.84%, F z The average peak error of (No.1-No.2) is 5.49%. After tool wear, the force F y The average peak error of (No.3-No.4) is 5.93%, F z The average peak error of (No.3-No.4) is 6.14%. Figure 6 As shown in the figure, the average error of the predicted deformation (No.1-No.2) is 11.81%. After tool wear, the average error of the predicted deformation (No.3-No.4) is 14.46%. Therefore, the predicted value is in good agreement with the measured value, thereby verifying the accuracy of the force-induced deformation prediction method proposed in the present invention. The above results show that the present invention takes into account the influence of tool wear on the milling force when milling titanium alloys and other difficult-to-machine materials, and can accurately obtain the predicted value of force-induced deformation after tool wear in the end milling of titanium alloy thin-walled parts.

[0157] This document uses specific examples to illustrate the principles and implementation methods of the present invention. The above examples are only intended to help understand the method and core concept of the present invention. At the same time, those skilled in the art will find that the specific implementation methods and application scopes may vary based on the concept of the present invention. In summary, the contents of this specification should not be construed as limiting the present invention.

Claims

1. A method for predicting force-induced deformation during end milling of titanium alloy thin-walled parts considering tool wear, characterized in that: Here are the steps: Step 1: Modeling the overall milling force considering tool wear; When the tool is worn, friction will occur between the tool's flank face and the workpiece surface, which will cause an increase in the overall milling force; Therefore, the flank wear force model is constructed to decompose the force along the flank into the tangential force dF tw,j,k , radial force dF rw,j,k and axial dF aw,j,k , and define the wear coefficients in the three directions as K tw ,K rw and K aw ; Calculate the tangential force dF of each axial height element t,j,k , radial force dF r,j,k and axial force dF a,j,k ;, through the coordinate transformation matrix T j,k , and finally the overall milling force considering tool wear in three directions of the tool coordinate system is obtained; Step 2: Construct a force-induced deformation model; Based on the small deflection deformation theory, the unit stiffness matrix of titanium alloy thin-walled parts at different feed positions is calculated. The unit stiffness matrix is ​​superimposed using the direct stiffness method to obtain the overall stiffness matrix K of the thin-walled parts. Then, the force-induced deformation of the thin-walled parts is calculated to obtain the force-induced deformation of the thin-walled parts after loading. Step 3: Process parameter update strategy; Considering the material removal effect during the milling of thin-walled parts, it is necessary to adjust the unit stiffness matrix at the feed position mentioned in step 2. The unit stiffness matrix of the thin-walled part processing area is directly adjusted to correct the change in the overall stiffness of the thin-walled part caused by the change in the thickness of the thin-walled part after material removal, and then quickly extract the displacement matrix of the thin-walled part after deformation. The overall stiffness of the thin-walled part changes continuously with the feed direction. Assuming that the total removal volume of the milling cutter at the current feed position is H units, the overall stiffness of the thin-walled part is the sum of the unit stiffness of the unprocessed area of ​​the thin-walled part and the unit stiffness of the remaining material at the current feed position, and the overall stiffness matrix is ​​updated. Then, the displacement matrix of the thin-walled part after material removal can be obtained. Step 4, iterative calculation of force-induced deformation; Based on the milling force model mentioned in step 1 and the force-induced deformation model mentioned in step 2, an iterative calculation of the force-induced deformation of thin-walled parts considering material removal is performed; specifically, the following steps are performed: inputting cutting parameters, tool parameters and thin-walled part parameters; calculating the initial overall stiffness matrix of the thin-walled part; discretizing the feed position and initializing the feed position; calculating the initial milling force to obtain the initial deformation of the workpiece; calculating the change in milling force after machining deformation and determining whether it converges. If convergence is achieved, the cutting parameters are updated. If not, the above process is repeated; after completing machining at the current feed position, proceed to the next position and update the corresponding stiffness matrix of the thin-walled part according to the material removal at the current feed position; after completing all machining, the final machining deformation and milling force are output.

2. The method for predicting force-induced deformation during end milling of titanium alloy thin-walled parts considering tool wear according to claim 1, characterized in that: The specific steps in step 1 are: When an integral annular milling cutter is used to perform end milling on titanium alloy thin-walled parts, when the tool is worn, friction will occur between the tool flank and the workpiece surface, which will lead to an increase in the overall milling force. T X T Y T Z T Under this condition, the cutting depth is discretized into m parts, and the tangential force dF of the kth element acting on the jth cutting edge at time t is t,j,k , radial force dF r,j,k , axial force dF a,j,k The expression is: , where g(φ j,k (t)) is the window function, K tc ,K rc ,K ac are the tangential, radial and axial shear force coefficients, K te ,K re ,K ae are the tangential, radial and axial cutting edge force coefficients respectively, h j,k (t) is the thickness of the undeformed chip; Furthermore, dF tw,j,k is the microelement tangential wear force, dF rw,j,k is the microelement radial wear force, dF aw,j,k is the microelement axial wear force, and its expression is: , where K tw ,K rw ,K aw are the tangential, radial, and axial wear force coefficients, respectively, and VB is the flank wear width; Furthermore, the expressions of the microelement axial height dz, the microelement cutting edge width db and the microelement cutting edge length ds are: , where a p is the cutting depth, κ j,k is the axial contact angle, R z is the effective radius, R z The first derivative of ; The thickness of the undeformed chip is: , Indicates the feed per tooth; The axial contact angle is the angle between point Q and the tool axis, and its expression is: , where r is the radius of the circular cutter corner, z is the distance from point Q to point X T O T Y T Distance on the surface; Effective radius R z is the distance from point Q to the tool axis, and its expression is: , the window function is used to determine whether the Q point on the cutting edge participates in cutting, and its expression is , where φ en is the cutting angle, φ ex To cut out the corners; The expression of radial contact angle is: , where n is the rotation speed, β is the helical angle, φ p is the tooth angle, φ p =2π / N t , N t is the number of tool teeth; Furthermore, the micro-element milling force in the tool coordinate system is: , where T j,k (t) is the transformation matrix; , therefore, the overall milling force after tool wear in the tool coordinate system is:

3. The method for predicting force-induced deformation during end milling of titanium alloy thin-walled parts considering tool wear according to claim 1, characterized in that: Furthermore, step 2 is as follows: In thin-walled parts W X W Y W Z W In this paper, the deformation prediction of thin-walled parts is carried out based on the small deflection theory of thin plates. It is assumed that after the thin-walled parts are bent, the extrusion deformation in the thickness direction of the plate is negligible, that is, the normal strain ε perpendicular to the mid-plane is z =0; the middle surface of the plate only undergoes bending deformation, that is, γ yz =γ zx =0; perpendicular to the mid-plane of the thin plate before deformation and still perpendicular to the mid-plane after deformation; Furthermore, based on the above assumptions, the displacement component w of each point after the thin-walled part is bent and deformed is only related to x and y, and has nothing to do with z. The deflection of the thin-walled part is: , the relationship between strain component and displacement is: , where ε x and ε y are the normal strains in the X and Y directions, γ xy For X W O W Y W Shear strain on the plane, u and v are the displacement components in the X and Y directions respectively; The relationship between stress and strain is: , where D p is the coefficient matrix; , where E is the elastic modulus of the thin-walled material, and μ is the Poisson's ratio of the thin-walled material; The curvature can be expressed as: , therefore, the relationship between stress and strain can be expressed as: , the internal forces of thin-walled parts are bending moment and torque, and their expressions are: , where M x , M y are the bending moments in the X and Y directions respectively; M xy For X W O W Y W Torque in the plane, t1 is the thickness of the thin-walled part; The elastic coefficient matrix D is: Furthermore, in order to calculate the deformation of thin-walled parts more accurately, the rectangular thin-walled parts can be discretized into M e (m e ×n e ) rectangular units, consisting of m e length units and n e width units; each rectangular unit e has four nodes, and each node has three degrees of freedom, namely deflection w i , angle θ xi ,θ yi ; then the displacement of the i-th node is: , the node displacement vector of the rectangular element e is: , the displacement function is defined by a polynomial with 12 undetermined coefficients. The displacement component in the thin-walled workpiece coordinate system, that is, the deflection, is expressed as: , where x represents the x-direction coordinate of the rectangular unit, and y represents the y-direction coordinate of the rectangular unit; , where L is a 1×12 function matrix and α is a matrix consisting of 12 undetermined coefficients. In summary, the displacement vector of the i-th node can be rewritten as: , the nodal displacement matrix of the rectangular element can be rewritten as: , where C is a 12×12 matrix; Therefore, the deflection can be rewritten as: , where N is the shape function matrix of the thin-walled unit bending, which is a 1×12 matrix; , N i is the shape function matrix of the i-th node, which is a 1×3 matrix; , where , where a, b are the rectangular unit size parameters, x i Indicates the x-direction coordinate of the i-th node, y i Indicates the y-direction coordinate of the i-th node; Therefore, the curvature can be rewritten as: , where B is the strain-displacement matrix of the rectangular element, which can be expressed as: , the elastic strain energy U of the rectangular element e for: , where V is the volume of the rectangular unit; Integrating along the thickness Z direction yields: , where S is the area of ​​the rectangular unit; Stiffness matrix K of rectangular element e It can be expressed as: , where K e is a 12×12 matrix; Therefore, the total strain energy of the thin-walled part is: , where , where K is the overall stiffness matrix of the thin-walled part, which is 3(m e +1)(n e +1)×3(m e +1)(n e +1) matrix; In order to construct the overall stiffness matrix K of the thin-walled part, it is necessary to obtain the element stiffness matrix K e Perform assembly operations; use the direct stiffness method to superimpose the unit stiffness matrix to obtain the overall stiffness matrix of the thin-walled part; After the thin-walled part is loaded, the total potential energy can be expressed as: , where W is the work done by P, P is the external force loaded on the node, that is, the force F in the Z direction Z The matrix composed of Based on the principle of minimum potential energy, there are extreme conditions , which is converted into the extreme value condition of a general multivariate function; therefore, the matrix of the thin-walled part after force-induced deformation can be solved: , one end of the thin-walled part is fixed, the three ends are free, and the coordinate system of the thin-walled part is O W X W Y W Z W The fixed boundary conditions are:

4. The method for predicting force-induced deformation during end milling of titanium alloy thin-walled parts considering tool wear according to claim 1, characterized in that: Step 3 is as follows: 3.1 Machining deformation affects cutting parameters, resulting in limitations in the milling force model; When the workpiece is deformed, W O W Z W On the plane, the updated cutting depth is: , where represents the deformation of the thin plate at the (x, y) position; 3.2 Time-varying update method for the stiffness matrix of thin-walled parts after material removal: Due to material removal, the thickness of the thin-walled part will change, which in turn affects the overall stiffness of the thin-walled part. To more accurately predict deformation, it is necessary to update the stiffness matrix of the thin-walled part at different feed positions; Directly modify the element stiffness matrix of the thin-walled part area after material removal, thereby adjusting the overall stiffness matrix of the thin-walled part to more accurately calculate the deformation; The following formula is used to calculate the element stiffness of thin-walled parts after material removal: , where t2 is the thickness of the thin-walled part after material removal under the current path; The overall stiffness of the thin-walled part changes continuously with the feed direction. Assuming that the total volume removed by the milling cutter at the current feed position is H units, the overall stiffness of the thin-walled part is the sum of the unit stiffness of the unprocessed area of ​​the thin-walled part and the unit stiffness of the remaining material at the current feed position. The updated stiffness matrix is: , therefore, the matrix of the thin-walled part after the material is removed and the force-induced deformation is obtained:

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