A method for modeling cutting forces in side milling of end mills considering the deformation of thin-walled parts.

Abstract

A method for modeling cutting forces in side milling of end mills considering the deformation of thin-walled parts is disclosed. This invention belongs to the field of metal processing, specifically relating to a method for modeling cutting forces in side milling of end mills. The purpose of this invention is to solve the problems of low machining accuracy, poor machining quality, and low machining efficiency in existing cutting force models. The process is as follows: 1. Establish a computational coordinate system; 2. Set parameters; 3. Calculate the theoretical cutting thickness; 4. Obtain the actual cutting thickness; 5. Calculate the tangential force, radial force, and axial force; 6. Obtain the cutting force in the Cartesian coordinate system; 7. Let l = l + 1, repeat steps 3 to 6 until the cutting force on all cutting micro-elements of tooth j is obtained; 8. Let j = j + 1, repeat steps 3 to 7 until the cutting force on all cutting micro-elements of all teeth is obtained; 9. Let φ = φ + Δφ, repeat steps 3 to 8 until φ = 2π, obtaining the cutting force at all tool rotation angles of all cutting micro-elements of all teeth.

Description

Technical Field

[0001] This invention belongs to the field of metal processing, specifically relating to a method for modeling cutting forces in side milling of end mills that takes into account the deformation of thin-walled parts. Background Technology

[0002] In modern manufacturing, thin-walled parts are widely used in aerospace, automotive, electronics, and medical device industries due to their advantages such as light weight, high strength, and high material utilization. However, thin-walled parts are prone to deformation during machining, especially during side milling with end mills. In this process, the lower rigidity of thin-walled parts makes them significantly susceptible to cutting forces, easily leading to reduced machining accuracy and deteriorated surface quality. Therefore, effectively predicting and controlling the deformation of thin-walled parts during machining has become a pressing technical challenge.

[0003] Currently, research on deformation during side milling of thin-walled parts mainly focuses on the modeling and prediction of cutting forces. Traditional cutting force models typically ignore the deformation of thin-walled parts, failing to accurately reflect the mechanical behavior during actual machining. This simplification can lead to significant errors in certain situations, thus affecting machining quality. Therefore, cutting force modeling methods that consider the deformation of thin-walled parts have become a key research focus.

[0004] The proposed method for modeling cutting forces in side milling of end mills that takes into account the deformation of thin-walled parts not only provides theoretical support and technical guarantee for the high-precision machining of thin-walled parts, but also helps to further promote its application in the machining of other complex structural parts, providing new ideas and methods for efficient and precise machining in the manufacturing industry. Summary of the Invention

[0005] The purpose of this invention is to address the problem that existing cutting force models typically ignore the deformation of thin-walled parts, resulting in low machining accuracy, poor machining quality, and low machining efficiency. Therefore, this invention proposes a cutting force modeling method for end mill side milling that considers the deformation of thin-walled parts.

[0006] The specific process of a cutting force modeling method for side milling of end mills that considers the deformation of thin-walled parts is as follows:

[0007] Step 1: Establish the computational coordinate system; the specific process is as follows:

[0008] The origin of the coordinate system is the center of the bottom of the tool.

[0009] Set the tool feed direction to the X-axis; the same direction as the tool feed direction is the positive direction.

[0010] Set the direction perpendicular to the tool surface as the Y-axis, with the direction closest to the tool surface as positive;

[0011] Set the tool axis direction to the Z-axis, with the direction pointing towards the tool holder as the positive direction;

[0012] Set the radial contact angle of the tool to φ, with the clockwise rotation angle of the tool as positive;

[0013] Step 2: Set the cutter diameter, tooth angle, helix angle β, and cutter tooth cutting element height dz;

[0014] Set the total number of cutting teeth of the tool to J;

[0015] Set the radial contact angle of the tool to 0 ≤ φ ≤ 2π;

[0016] The cutting edge of the spatial helix of each tooth of the tool is discretized into L cutting micro-elements along the tool axis;

[0017] Step 3: At any time t, the instantaneous radial contact angle at the l-th cutting element point P on the cutter tooth j is φ. j,l (t), utilizing the instantaneous radial contact angle φ j,l Calculate the theoretical cutting thickness h(φ) corresponding to the l-th cutting micro-element on the cutter tooth j. j,l );

[0018] Step 4: Based on the theoretical cutting thickness h(φ) corresponding to the l-th cutting micro-element on the cutter tooth j calculated in Step 3. j,l ), to obtain the actual cutting thickness h'(φ) corresponding to the l-th cutting micro-element on the cutter tooth j. j,l );

[0019] Step 5: Based on the actual cutting thickness h'(φ) corresponding to the l-th cutting micro-element on the cutter tooth j j,l Calculate the actual tangential force dF acting on the height dz of the l-th cutting element of the cutter tooth j. t (φ j,l ), radial force dF r (φ j,l ) and axial force dF a (φ j,l );

[0020] Step 6: Use a rotation matrix to convert the actual tangential force dF obtained in Step 5. t (φ j,l ), radial force dF r (φ j,l ), axial force dF a (φ j,l Transform the coordinates to the X, Y, and Z directions in the Cartesian coordinate system to obtain the actual cutting force dF in the X direction of the Cartesian coordinate system. X (φ j,l Cutting force dF in the Y direction of the Cartesian coordinate system Y (φ j,l Cutting force dF in the Z direction of the Cartesian coordinate system Z (φ j,l );

[0021] Step 7: Let l = l + 1, and repeat steps 3 to 6 until the cutting force F on all cutting micro-elements of the cutter tooth j in the X, Y, and Z directions in the actual Cartesian coordinate system is obtained. X (φ j ), F Y (φ j ), F Z (φ j );

[0022] Step 8: Let j = j + 1, and repeat steps 3 to 7 until the cutting force F on all cutting elements of all cutter teeth in the X, Y, and Z directions in the actual Cartesian coordinate system is obtained. X (φ), F Y (φ), F Z (φ);

[0023] Step 9: Let φ = φ + Δφ, and repeat steps 3 to 8 until φ = 2π to obtain the cutting force at all tool rotation angles of all cutting micro-elements in the X, Y, and Z directions in the actual Cartesian coordinate system.

[0024] Δφ is the angular step size.

[0025] The beneficial effects of this invention are as follows:

[0026] This invention provides a cutting force modeling method for side milling of end mills that considers the deformation of thin-walled parts. By establishing a model of the force and deformation relationship of thin-walled parts during the cutting process, and combining theories such as mechanics of materials, cutting mechanics, and finite element analysis, this method systematically analyzes and predicts the deformation of thin-walled parts during side milling and its impact on cutting forces. This model allows for more accurate prediction of the deformation and cutting force distribution of thin-walled parts, optimization of machining parameters, and improvement of machining accuracy, quality, and efficiency.

[0027] This invention belongs to the field of metal processing and discloses a method for modeling cutting forces in side milling with end mills that considers the deformation of thin-walled parts. Based on the milling force-deflection coupling law during thin-walled part machining and the changes in the tool contact area caused by workpiece deformation during milling, this method uses Kirchhoff's theory to calculate the deformation values ​​caused by dynamic milling forces at different cutting positions, accurately calculates the dynamic cutting thickness of each tooth, and incorporates the cutting thickness into a micro-element milling force model to establish a flexible milling force prediction analytical model. This method for modeling cutting forces in side milling with end mills that considers the deformation of thin-walled parts can provide a theoretical basis for optimizing reasonable tool structure matching and recommending process parameters for thin-walled part machining.

[0028] This invention systematically analyzes and predicts the deformation of thin-walled parts during side milling and its impact on cutting forces. It accurately analyzes the instantaneous changes in cutting thickness caused by the deformation of the thin-walled part and calculates the cutting forces and workpiece deformation values ​​at different tool feed positions. This provides a theoretical basis for optimizing machining deformation, thereby achieving a milling method to control machining errors. Those skilled in the art only need to obtain the workpiece, tool structure, machining process parameters, and end mill cutting force coefficient to quickly determine the cutting forces and workpiece deformation values ​​at different tool feed positions, thereby optimizing machining process parameters and improving workpiece machining quality and efficiency. Attached Figure Description

[0029] Figure 1 This is a flowchart of the milling force calculation considering workpiece deformation in an embodiment of the present invention. Ds is the critical stiffness of the workpiece under cutting force deformation; D is the tool diameter; β is the tool helix angle; S is the spindle speed; F is the feed rate; a p axial depth of cut; a e denoted as radial cutting width; E is the elastic modulus; μ is Poisson's ratio; Size(l*w*h) is the workpiece size (l length, w width, h height); the j-th cutting edge is the cutting tooth j;

[0030] Figure 2 The following is a geometric structure and micro-element diagram of the end mill in the embodiment of the present invention: σ1 is the tool length tolerance; σ2 is the cutting edge axial length tolerance; σ3 is the tool diameter tolerance; l is the cutting edge axial length; L is the tool length; D is the tool diameter; dF r dF is the radial cutting force of the infinitesimal element. t φ is the tangential cutting force of the micro-element; β is the tool helix angle; dz is the axial length of the cutting micro-element; φ1 is the tooth angle 1; φ2 is the tooth angle 2;

[0031] Figure 3 This is a schematic diagram illustrating how workpiece deformation changes the angle of entry of the micro-element in an embodiment of the present invention, φ. st dφ is the angle of entry for the (l+1)th infinitesimal element; dφ is the difference between the actual angles of entry for the lth and (l+1)th infinitesimal elements; Δφ′ is the difference between the theoretical and actual angles of entry for the (l+1)th infinitesimal element after deformation; O is the coordinate origin; the X-axis is the tool feed direction; the Y-axis is the direction perpendicular to the tool surface; a e The theoretical radial cut width; a e ' is the actual radial cutting width; Δδ' is the radial deformation of the workpiece;

[0032] Figure 4 This is a schematic diagram of the instantaneous cutting thickness change after deformation in an embodiment of the present invention, φ st φ' is the current cutting edge approach angle. stΔδ is the angle of entry for the next cutting edge; Δδ is the radial deformation of the workpiece; w is the direction of tool rotation; O′ is the origin of the coordinate system for the next cutting edge entering the workpiece; X′ is the X-axis of the coordinate system for the next cutting edge entering the workpiece; Y′ is the Y-axis of the coordinate system for the next cutting edge entering the workpiece; f z This refers to the feed per tooth.

[0033] Figure 5 This is a partial coordinate system diagram of the unit in the embodiment of the present invention. 2b is the width of the rectangular unit; 2a is the length of the rectangular unit; x is the X-axis of the coordinate system; o is the origin of the coordinate system; y is the Y-axis of the coordinate system; o o η is the origin of the local coordinate system; ξ is the vertical coordinate axis of the local coordinate system; θ is the horizontal coordinate axis of the local coordinate system. xi Let θ be the bending moment in the x-direction at point i; yi The bending moment in the y direction at point i; w i The displacement is in the z-direction; i is the element node.

[0034] Figure 6 This is an adaptive distribution rule diagram of the nodal cutting force in an embodiment of the present invention, f Y The cutting force allocated to the longitudinal node height position on the cutting edge; f a0 For finite element calculation elements, the position of node a0 is equivalent to that of node f; a1 For finite element calculation elements, the position of node a1 is equivalent to that of node f; au-1 For the finite element calculation element a u-1 Calculate the equivalent effect of node position; f au For the finite element calculation element a u Calculate the equivalent force of node positions; j is the blade j; a u-1 The node at the longitudinal u-1 position on the cutting edge and the node in the left finite element calculation element a u-1 Calculate the distance between nodes; b u-1 For b u-1 The node at the longitudinal u-1 position on the cutting edge and the node in the left finite element calculation element b u-1 Calculate the distance between nodes; j-1 is the blade j-1; f b0 For finite element calculation elements, b0 is used to calculate the node positions, which is equivalent to f. b1 For finite element calculation, the calculation of node positions in b1 is equivalent to that in the element; f bu-1 For the finite element calculation element b u-1 Calculate the equivalent effect of node position; f bu For the finite element calculation element b u Calculate the equivalent effect of node position; z The vertical height of the finite element calculation unit. Detailed Implementation

[0035] Specific Implementation Method 1: The specific process of this implementation method for modeling the cutting force of end mill side milling considering the deformation of thin-walled parts is as follows:

[0036] Step 1: Establish the computational coordinate system; the specific process is as follows:

[0037] The origin of the coordinate system is the center of the bottom of the tool.

[0038] Set the tool feed direction to the X-axis; the same direction as the tool feed direction is the positive direction.

[0039] Set the direction perpendicular to the tool surface as the Y-axis, with the direction closest to the tool surface as positive;

[0040] Set the tool axis direction to the Z-axis, with the direction pointing towards the tool holder (upward) as the positive direction;

[0041] Set the radial contact angle of the tool to φ, with the clockwise rotation angle of the tool as positive;

[0042] Step 2: Set the cutter diameter, tooth angle, helix angle β, and cutter tooth cutting element height dz;

[0043] Set the total number of cutting teeth of the tool to J;

[0044] Set the radial contact angle of the tool to 0 ≤ φ ≤ 2π;

[0045] The cutting edge of the spatial helix of each tooth of the tool is discretized into L cutting micro-elements along the tool axis (the total number of cutting micro-elements on each tooth is L);

[0046] Step 3: At any time t, the instantaneous radial contact angle at the l-th cutting element point P on the cutter tooth j is φ. j,l (t), utilizing the instantaneous radial contact angle φ j,l Calculate the theoretical cutting thickness h(φ) corresponding to the l-th cutting micro-element on the cutter tooth j. j,l );

[0047] Step 4: Based on the theoretical cutting thickness h(φ) corresponding to the l-th cutting micro-element on the cutter tooth j calculated in Step 3. j,l ), to obtain the actual cutting thickness h'(φ) corresponding to the l-th cutting micro-element on the cutter tooth j. j,l );

[0048] Step 5: Based on the actual cutting thickness h'(φ) corresponding to the l-th cutting micro-element on the cutter tooth j j,l Calculate the actual tangential force dF acting on the height dz of the l-th cutting element of the cutter tooth j. t (φ j,l ), radial force dF r (φ j,l ) and axial force dF a (φj,l );

[0049] Step 6: Use a rotation matrix to convert the actual tangential force dF obtained in Step 5. t (φ j,l ), radial force dF r (φ j,l ), axial force dF a (φ j,l Transform the coordinates to the X, Y, and Z directions in the Cartesian coordinate system to obtain the actual cutting force dF in the X direction of the Cartesian coordinate system. X (φ j,l Cutting force dF in the Y direction of the Cartesian coordinate system Y (φ j,l Cutting force dF in the Z direction of the Cartesian coordinate system Z (φ j,l );

[0050] Step 7: Let l = l + 1, and repeat steps 3 to 6 until the cutting force F on all cutting micro-elements of the cutter tooth j in the X, Y, and Z directions in the actual Cartesian coordinate system is obtained. X (φ j ), F Y (φ j ), F Z (φ j );

[0051] Step 8: Let j = j + 1, and repeat steps 3 to 7 until the cutting force F on all cutting elements of all cutter teeth in the X, Y, and Z directions in the actual Cartesian coordinate system is obtained. X (φ), F Y (φ), F Z (φ);

[0052] Step 9: Let φ = φ + Δφ, and repeat steps 3 to 8 until φ = 2π to obtain the cutting force at all tool rotation angles of all cutting micro-elements in the X, Y, and Z directions in the actual Cartesian coordinate system.

[0053] Δφ is the angular step size.

[0054] Specific Implementation Method Two: This implementation method differs from Specific Implementation Method One in that, in step three, the instantaneous radial contact angle φ at the l-th cutting micro-element point P on the cutter tooth j at any given time t is... j,l The process of obtaining (t) is as follows:

[0055]

[0056]

[0057] Where φ0(t) is the initial value of the tool contact angle, φ p φ p,j ψ is the tooth angle. j,l Let dz be the hysteresis angle at point P on the l-th cutting element of tooth j, and β be the height of the l-th cutting element of tooth j. j is the helix angle of the cutting tooth j, R is the tool radius; t is time;

[0058] φ0(t)=πnt / 30

[0059] n is the tool rotation speed;

[0060] If the cutting teeth of the tool are equally divided, φ p =2π / J;

[0061] If the cutting teeth of the tool are not unequally divided

[0062] The other steps and parameters are the same as in Specific Implementation Method 1.

[0063] Specific Implementation Method Three: This implementation method differs from Specific Implementation Method One or Two in that, in step four, the theoretical cutting thickness h(φ) corresponding to the l-th cutting micro-element on the cutter tooth j calculated in step three is... j,l ), to obtain the actual cutting thickness h′(φ) corresponding to the l-th cutting micro-element on the cutter tooth j. j,l );

[0064] The expression is:

[0065] h'(φ j,l )=h(φ j,l )-Δδ' j,l

[0066] In the formula: Δδ' j,l It is the change in cutting thickness caused by elastic deformation;

[0067] The theoretical cutting thickness h(φ) corresponding to the l-th cutting micro-element on the cutting tooth j j,l The expression for ) is:

[0068] h(φ j,l )=f z ×sin(φ j,l (t))

[0069] In the formula: f z The feed per tooth is t, and the tool rotation time is t.

[0070] The change in cutting thickness Δδ' caused by elastic deformation j,l The solution process is as follows:

[0071]

[0072] Other steps and parameters are the same as in specific implementation method one or two.

[0073] Specific Implementation Method Four: This implementation method differs from Specific Implementation Methods One to Three in that the deformation amount δ in the radial cutting width direction is... el The solution process is as follows:

[0074] The deformation value of the cutting tooth under the action of cutting force was obtained by Kirchhoff theory and finite element method.

[0075] 1) Divide the workpiece plane into N unit cells. The stiffness calculation formula for each unit cell is as follows:

[0076] [k] e =∫∫[B] T [D][B]dXdY

[0077]

[0078]

[0079] In the formula: [k] e Here is the stiffness matrix for each element, [B] is the intermediate variable, [D] is the workpiece stiffness, the superscript T indicates transpose, and the superscript e indicates element;

[0080] [N] is the shape function matrix, i.e., a function of the unit position coordinates;

[0081] E is the elastic modulus of the material, h is the thickness of the workpiece, and μ is Poisson's ratio;

[0082] 2) Calculate the equivalent nodal force of each element based on its load strength; the formula is:

[0083] [F] e =∫∫[N] T qdXdY

[0084] In the formula: q represents the load strength of each element; [F] e Equivalent nodal forces for each unit cell;

[0085] 3) Combine the stiffness matrix [k] of all unit cells. e And equivalent nodal forces [F] e The deformation δ of the l-th cutting element on the cutter tooth j in the radial cutting width direction is obtained. el ;

[0086] The calculation equation is:

[0087]

[0088] Where: δ el N represents the deformation in the radial cutting width direction, and N is the total number of unit cells in the workpiece plane.

[0089] The other steps and parameters are the same as those in one of the specific implementation methods one to three.

[0090] Specific Implementation Method Five: This implementation method differs from Specific Implementation Methods One to Four in that the equivalent nodal force [F] is... e The solution process is as follows:

[0091] Specific implementation method four explains the equivalent nodal force [F] e With deformation δ el The relationship between these factors is explained here, illustrating the equivalent nodal force [F] during the cutting process. e How to calculate it, and then substitute it into the fourth specific implementation method to obtain the deformation amount δ el .

[0092]

[0093] In the formula: a t b t The scaling factor for the equivalent forces at the nodes;

[0094] fa t (j) represents the equivalent effect of the mesh nodes on the left side of the cutting edge j of the blade, fb t (j) represents the equivalent effect of the grid node on the right side of the cutting edge j of the blade;

[0095] f z The feed per tooth, J×f z J represents the length of the rectangular surface swept by the cutter in one revolution, and J is the total number of cutter teeth.

[0096] u represents the number of longitudinal grids that the cutting edge j of the cutter tooth enters the workpiece;

[0097] j represents the cutting edge of the cutting tooth, f Y (j) represents the cutting force distributed at the longitudinal node of the cutting edge j of the cutter tooth;

[0098] F Y (j) represents the total milling force (F) at the moment when the cutting edge j of the cutter tooth enters the workpiece during each cutting cycle. Y (j) is dF Y,j,l (φ), dF Y,j,l (φ) represents the cutting force on the l-th cutting element of the cutter tooth j in the Y direction in the Cartesian coordinate system.

[0099] Based on the total milling force F at the moment the cutting edge j of the cutter tooth enters the workpiece during each cutting cycle. Y (j), obtain the equivalent force fa of the mesh node to the left of the blade j.t (j) and the right-side mesh node of the blade j have the same effect fb t (j);

[0100] The equivalent force of the left mesh node of the blade j is fa t (j) and the right-side mesh node of the blade j have the same effect fb t (j) Constituting equivalent nodal forces [F] e .

[0101] The cutting edge j is the j-th cutting edge.

[0102] like Figure 5 As shown, the iterative contact relationship between the tool and the workpiece meshing boundary distributes the load evenly at the discrete unit nodes in real-time contact.

[0103] For rectangular elements, a local coordinate system is introduced, such as Figure 5 As shown, the origin of the local coordinate system is taken at the centroid of the rectangle, and the axes are parallel to the global coordinate axes x and y, respectively.

[0104] The other steps and parameters are the same as those in one of the specific implementation methods one to four.

[0105] Specific Implementation Method Six: This implementation method differs from Specific Implementation Methods One to Five in that, in step five, the actual cutting thickness h'(φ) corresponding to the l-th cutting micro-element on the cutter tooth j is used. j,l Calculate the tangential force dF acting on the height dz of the l-th cutting element of the cutter tooth j. t (φ j,l ), radial force dF r (φ j,l ) and axial force dF a (φ j,l The formula is as follows:

[0106]

[0107] In the formula:

[0108] K te K re K ae These are the tangential, radial, and axial tillage force coefficients on the cutting micro-element, respectively.

[0109] K tc K rc K ac These are the tangential, radial, and axial shear force coefficients on the cutting micro-element, respectively.

[0110] K te,j,l K is the tangential plowing force coefficient on the l-th cutting element with thickness dz on the cutter tooth j. re,j,lK is the radial tillage force coefficient on the l-th cutting element with thickness dz on the cutter tooth j. ae,j,l denoted as the axial tillage force coefficient on the l-th cutting micro-element with thickness dz on the cutter tooth j;

[0111] dz j,l Let be the l-th cutting micro-element with thickness dz on the cutter tooth j;

[0112] K tc,j,l K is the tangential shear force coefficient on the l-th cutting element with thickness dz on the cutter tooth j. rc,j,l K is the radial shear force coefficient on the l-th cutting element with thickness dz on the cutter tooth j. ac,j,l Let be the axial shear force coefficient on the l-th cutting micro-element with thickness dz on the cutter tooth j.

[0113] The other steps and parameters are the same as those in one of the specific implementation methods one to five.

[0114] Specific Implementation Method Seven: This implementation method differs from Specific Implementation Methods One through Six in that, in step six, the actual tangential force dF obtained in step five is transformed using a rotation matrix. t (φ j,l ), radial force dF r (φ j,l ), axial force dF a (φ j,l Transform the coordinates to the X, Y, and Z directions in the Cartesian coordinate system to obtain the actual cutting force dF in the X direction of the Cartesian coordinate system. X (φ j,l Cutting force dF in the Y direction of the Cartesian coordinate system Y (φ j,l Cutting force dF in the Z direction of the Cartesian coordinate system Z (φ j,l The expression is:

[0115]

[0116]

[0117]

[0118] In the formula: T(φ) j,l ) is the rotation matrix, g(φ) j,l φ represents the step function of the current cutting state of the cutting unit. s ′ t , φ e ′ x ...

[0119] Since end mills have a multi-tooth structure, each cutting edge has spatial helical geometry. A unit step function is used to represent whether the current cutting element participates in cutting.

[0120] The other steps and parameters are the same as those in one of the specific implementation methods one to six.

[0121] Specific Implementation Method Eight: This implementation method differs from one of Specific Implementation Methods One to Seven in that the cutting angle φ of the tool is... s ′ t The updated formula is:

[0122]

[0123] In the formula: a e This is the radial (Y direction) cutting width.

[0124] The other steps and parameters are the same as those in any of the specific implementation methods one to seven.

[0125] Specific Implementation Method Nine: This implementation method differs from Specific Implementation Methods One through Eight in that, in step seven, l = l + 1 is set, and steps three through six are repeated until the cutting force F on all cutting micro-elements of the cutter tooth j in the X, Y, and Z directions in the actual Cartesian coordinate system is obtained. X (φ j ), F Y (φ j ), F Z (φ j The expression is:

[0126]

[0127] Among them, F X (φ j F represents the cutting force on all cutting elements of the cutter tooth j in the X direction in the Cartesian coordinate system. Y (φ j F represents the cutting force on all cutting elements of the cutter tooth j in the Y direction in the Cartesian coordinate system. Z (φ j ) represents the cutting force on all cutting micro-elements of the cutter tooth j in the Z direction in the Cartesian coordinate system.

[0128] The other steps and parameters are the same as those in one of the specific implementation methods one to eight.

[0129] Specific Implementation Method Ten: This implementation method differs from Specific Implementation Methods One through Nine in that, in step eight, j = j + 1, and steps three through seven are repeated until the total cutting force F on all cutting micro-elements of all cutter teeth in the X, Y, and Z directions in the actual Cartesian coordinate system is obtained. X (φ), F Y (φ), FZ (φ); the expression is:

[0130]

[0131] Among them, F X (φ) represents the total cutting force on all cutting elements of all cutter teeth in the X direction in the Cartesian coordinate system; F Y (φ) represents the total cutting force on all cutting elements of all cutter teeth in the Y direction in the Cartesian coordinate system; F Z (φ) represents the total cutting force on all cutting micro-elements of all cutter teeth in the Z direction in the Cartesian coordinate system.

[0132] The other steps and parameters are the same as those in any of the specific implementation methods one to nine.

[0133] During actual machining, the milling force causes the workpiece to deflect, resulting in a smaller cutting layer thickness in the tool-workpiece contact area, and thus a smaller actual cutting thickness h'. j,l (φ) Due to the generation of deformation δ, h' j,l (φ)<h j,l (φ), the thickness of each cutting layer changes, thus affecting the cutting force. Therefore, the milling force model needs to be corrected. The actual cutting thickness affected by deformation is:

[0134] h' j,l (φ)=h j,l (φ)-Δδ' j,l

[0135] In the formula: Δδ' is used to calculate the dynamic cutting thickness. j,l It is the change in cutting thickness caused by elastic deformation.

[0136] This causes the tool-workpiece contact relationship to change as the cutting process progresses. This is due to the deformation δ in the radial cutting width direction. el The effect of this decreases the cutting contact angle, which in turn leads to a smaller approach angle φ for the (l+1)th infinitesimal element. st If there is a delayed entry into the workpiece, with a lag angle of Δφ', such as... Figure 6 As shown. Taking the climb milling process as an example, the actual entry angle φ of the (l+1)th infinitesimal element. st The tangent angle is φ ex ':

[0137]

[0138] The formula for calculating the engagement angle of the boundary engagement element under the influence of statically induced deformation is as follows:

[0139]

[0140] In the formula: To solve for the workpiece surface fp The deformation of the Nth cutting edge at the feed position caused by the N-1 cutting edge in the cutting step, a e Radial cutting width; The entry angle of the entry element is taken into account for cutting deformation.

[0141] The deformation value under the action of the cutting force of the cutting tooth is obtained by using Kirchhoff's theory and the finite element method. The deformation of the thin plate under the action of the cutting tooth is solved according to Kirchhoff's theory. First, three assumptions are defined to transform the three-dimensional problem into a two-dimensional planar problem:

[0142] a. Before deformation, a straight line perpendicular to the mid-surface of a thin-walled part remains a straight line after deformation, and this straight line is still perpendicular to the bent mid-surface.

[0143] b. The normal stress σ acting on a plane parallel to the mid-surface, outside the contact area between the milling cutter and the thin-walled part. z τ xz τ zy σ on the cross section of the mid-plane x σ y τ xy The value is very small, so the resulting strain can be ignored; σ x σ y σ z Let τ represent the stress in the X, Y, and Z directions in the Cartesian coordinate system, respectively. xy τ xz τ zy These represent the shear stresses in the X, Y, and Z directions, respectively, in the Cartesian coordinate system.

[0144] c. To simplify the model and calculation, it is assumed that the mid-surface of the thin-walled part has no expansion or contraction deformation, and the expansion and contraction displacements u0 and v0 of the mid-surface of the thin-walled part do not exist.

[0145] u0 is the displacement component in the X direction under the Cartesian coordinate system, and v0 is the displacement component in the Y direction under the Cartesian coordinate system.

[0146] Therefore, the deformation w of the thin-walled part is independent of the coordinate z, but is a function w(x, y) of the coordinates x and y. Combining this with the assumption that the bending deformation displacement components of plates a and c are...

[0147] In the formula, u, v, and w represent the displacement components in the X, Y, and Z directions of the Cartesian coordinate system, respectively.

[0148] After dividing the thin plate into a finite number of element bodies and discretizing the structure, certain assumptions must be made about the displacement distribution within the element bodies to derive the relationship between nodal displacements and the displacements of any point within the element body:

[0149]

[0150] In the formula: {w} is the displacement vector of any point within the element, {δ} e It is the displacement matrix of the element nodes, with the superscript e denoteing the element, and [N] is called the shape function matrix;

[0151] A node is a point in the mesh generated by the finite element method (mesh vertex), while a point is any point on the plane.

[0152] By deriving the relationship between strain components and displacement components on a line segment of the infinitesimal element using the infinitesimal element assumption, the following formula is established:

[0153]

[0154] {w}=[N]{δ} e Substitution Let the strain be expressed in terms of deflection w, and the strain {ε} of the element be obtained:

[0155]

[0156]

[0157] The other three strains are the normal strain ε z Shear strain γ xz Shear strain γ yz ;

[0158] The remaining three strain components are obtained based on assumption a, where a: ε z =0,γ xz =0,γ yz =0

[0159] In the formula, u, v, and w represent the displacement components in the X, Y, and Z directions of the Cartesian coordinate system, respectively.

[0160] ε x For normal strain, ε y For normal strain, γ xy For shear strain; z(κ) is an intermediate variable;

[0161] Based on assumption b of the thin plate approximation theory, the stress component σ is neglected in the physical equations. z τ xz τ zy Derive the relationship between element stress and nodal displacement:

[0162]

[0163] In the formula: E is the elastic modulus of the material, μ is Poisson's ratio; {σ} is the element stress;

[0164] In the relationship between internal moment and stress:

[0165]

[0166] In the formula: {M} is the inner moment, M x Let M be the bending moment in the X direction of the Cartesian coordinate system. y Let M be the bending moment in the Y direction of the Cartesian coordinate system. xy Let the torque be in the X and Y directions of a Cartesian coordinate system. It is half the unit height;

[0167] According to the formula With formula Further integration simplifies to:

[0168] {M}=[D][B]{δ} e

[0169] In the formula:

[0170]

[0171] The finite element equation for thin plate bending is derived using the principle of virtual work, with displacement as the unknown quantity. It is assumed that virtual displacement {w} occurs within the element. * The corresponding virtual displacement of the nodes is {δ}. *} e The resulting virtual deformation is {χ * Then the equation for the virtual work done by bending a thin plate is:

[0172]

[0173] In the formula: q is the load strength on the element, and N is the total number of elements;

[0174] The equation {w}=[N]{δ} is used. e 、{w}=[N]{δ} e and {w}=[N]{δ} e Substitution have to:

[0175]

[0176] Since the virtual displacement matrix of nodes is independent of coordinates x and y, the ({δ) in the above equation... *} e ) T It can be moved before the integral sign. Since the imaginary shift is arbitrary, the matrices multiplied from both sides of the equation should be equal:

[0177]

[0178] In the formula, the element stiffness [k] e The formula is

[0179] [k] e =∫∫[B] T [D][B]d x d y

[0180] The equivalent nodal force generated by the load is

[0181] [F] e =∫∫[N] T qdxdy

[0182] Combining the stiffness matrices and equivalent nodal forces of all elements, the finite element equation for thin plate bending is obtained as follows:

[0183]

[0184] From the system of equations The deflection values ​​at each node of the thin plate are obtained by solving the solution.

[0185] The system of equations was obtained based on Kirchhoff.

[0186] Using the finite element method from the system of equations The deflection values ​​at each node of the thin plate (the change in cutting thickness caused by elastic deformation) are obtained from the solution.

[0187] During the calculation, to ensure the continuity of the deflection and its derivative at each element's node, the nodal displacement is specified by the deflection and the first-order partial derivatives at the node in the x and y directions. The displacement matrix of node i is as follows:

[0188]

[0189] In the formula: {δ i Let w be the displacement matrix of node i. i Let θ be the Z-axis displacement component at node i. xi Let θ be the rotation angle along the X-axis at node i. yi The rotation angle along the Y-axis at node i;

[0190] Its corresponding nodal force array is represented as:

[0191] {F i}=[Q i M θxi M θyi ] T

[0192] In the formula: Q i For nodal forces, M θxi M is the torque in the X direction at node i. θyi Let {F} be the torque in the Y direction at node i. i} represents the nodal force array;

[0193] For rectangular elements, a local coordinate system is introduced, such as Figure 4 As shown, the origin of the local coordinate system is taken at the centroid of the rectangle, and the ξ and η axes are parallel to the global coordinate axes x and y, respectively. The coordinate transformation relationship between the local and global coordinate systems is:

[0194] x = x0 + aξ, y = y0 + bη

[0195] In the formula:

[0196] In the formula: (x i ,y i Let be the coordinates of node i, where i = 1, ..., 4;

[0197] The coordinates of the node in the local coordinate system are (ξ) i η i Their values ​​are ±1.

[0198] Since each node of a rectangular element has 3 displacement components, there are a total of 12 nodal components. The displacement mode and rotation angle of the element nodes are represented as follows:

[0199]

[0200] In the formula: α1, α2, α3, α4, α5, α6, α7, α8, α9, α 10 α 11 α 12 θ is a polynomial parameter; x Let θ be the turning angle. y For the corner;

[0201] The local coordinates (ξ) of the four nodes of the rectangular element i ,η i Substitute the values ​​into the above equations to solve for α1~α. 12 Values, from α1 to α 12 Substituting back into the equation and rearranging, we get:

[0202]

[0203] In the formula:

[0204] [N] i =[N i N xi N yi (i = 1, 2, 3, 4)

[0205]

[0206] The formula Substitution The element strain is then:

[0207] {ε}=z[B]{δ} e =z[B i ]{δ} e

[0208] In the formula:

[0209]

[0210] In the formula: [B i ] represents an intermediate variable;

[0211] The formula Japanese style Substitution formula [k] e =∫∫[B] T [D][B]d x d y Find [k] i,j ] Element stiffness matrix.

[0212] This invention may have other embodiments. Without departing from the spirit and essence of this invention, those skilled in the art can make various corresponding changes and modifications according to this invention, but these corresponding changes and modifications should all fall within the protection scope of the appended claims.

Claims

1. A method for modeling cutting forces in side milling of end mills considering the deformation of thin-walled parts, characterized in that: The specific process of the method is as follows: Step 1: Establish the computational coordinate system; the specific process is as follows: The origin of the coordinate system is the center of the bottom of the tool. Set the tool feed direction to the X-axis; the same direction as the tool feed direction is the positive direction. Set the direction perpendicular to the tool surface as the Y-axis, with the direction closest to the tool surface as positive; Set the tool axis direction to the Z-axis, with the direction pointing towards the tool holder as the positive direction; Set the radial contact angle of the tool to φ, with the clockwise rotation angle of the tool as positive; Step 2: Set the cutter diameter, tooth angle, helix angle β, and cutter tooth cutting element height dz; Set the total number of cutting teeth of the tool to J; Set the radial contact angle of the tool to 0 ≤ φ ≤ 2π; The cutting edge of the spatial helix of each tooth of the tool is discretized into L cutting micro-elements along the tool axis; Step 3: At any time t, the instantaneous radial contact angle at the l-th cutting element point P on the cutter tooth j is φ. j,l (t), utilizing the instantaneous radial contact angle φ j,l Calculate the theoretical cutting thickness h(φ) corresponding to the l-th cutting micro-element on the cutter tooth j. j,l ); Step 4: Based on the theoretical cutting thickness h(φ) corresponding to the l-th cutting micro-element on the cutter tooth j calculated in Step 3. j,l ), to obtain the actual cutting thickness h′(φ) corresponding to the l-th cutting micro-element on the cutter tooth j. j,l The expression is: h′(φ j,l )=h(φ j,l )-Δδ′ j,l In the formula: Δδ′ j,l It is the change in cutting thickness caused by elastic deformation; The theoretical cutting thickness h(φ) corresponding to the l-th cutting micro-element on the cutting tooth j j,l The expression for ) is: h(φ j,l )=f z ×sin(φ j,l (t)) In the formula: f z The feed per tooth is t, and the tool rotation time is t. The change in cutting thickness Δδ′ caused by elastic deformation j,l The solution process is as follows: Where: δ el This represents the deformation in the radial cutting width direction; Step 5: Based on the actual cutting thickness h′(φ) corresponding to the l-th cutting micro-element on the cutter tooth j j,l Calculate the actual tangential force dF acting on the height dz of the l-th cutting element of the cutter tooth j. t (φ j,l ), radial force dF r (φ j,l ) and axial force dF a (φ j,l ); Step 6: Use a rotation matrix to convert the actual tangential force dF obtained in Step 5. t (φ j,l ), radial force dF r (φ j,l ), axial force dF a (φ j,l Transform the coordinates to the X, Y, and Z directions in the Cartesian coordinate system to obtain the actual cutting force dF in the X direction of the Cartesian coordinate system. X (φ j,l Cutting force dF in the Y direction of the Cartesian coordinate system Y (φ j,l Cutting force dF in the Z direction of the Cartesian coordinate system Z (φ j,l ); Step 7: Let l = l + 1, and repeat steps 3 to 6 until the cutting force F on all cutting micro-elements of the cutter tooth j in the X, Y, and Z directions in the actual Cartesian coordinate system is obtained. X (φ j ), F Y (φ j ), F Z (φ j ); Step 8: Let j = j + 1, and repeat steps 3 to 7 until the cutting force F on all cutting elements of all cutter teeth in the X, Y, and Z directions in the actual Cartesian coordinate system is obtained. X (φ), F Y (φ), F Z (φ); Step 9: Let φ = φ + Δφ, and repeat steps 3 to 8 until φ = 2∏ to obtain the cutting force at all tool rotation angles of all cutting micro-elements in the X, Y, and Z directions in the actual Cartesian coordinate system. Δφ is the angular step size.

2. The method for modeling cutting force in side milling of end mills considering the deformation of thin-walled parts according to claim 1, characterized in that: In step three, the instantaneous radial contact angle φ at the l-th cutting element point P on the cutter tooth j at any time t is... j,l The process of obtaining (t) is as follows: Where φ0(t) is the initial value of the tool contact angle, φ p φ p,j ψ is the tooth angle. j,l Let dz be the hysteresis angle at point P on the l-th cutting element of tooth j, and β be the height of the l-th cutting element of tooth j. j is the helix angle of the cutting tooth j, R is the tool radius; t is time; φ0(t)=πnt / 30 n is the tool rotation speed; If the cutting teeth of the tool are equally divided, φ p =2π / J; If the cutting teeth of the tool are not unequally divided 3. The method for modeling cutting force in side milling of end mills considering the deformation of thin-walled parts according to claim 2, characterized in that: The deformation δ in the radial cutting width direction el The solution process is as follows: 1) Divide the workpiece plane into N unit cells. The stiffness calculation formula for each unit cell is as follows: [k] e =∫∫[B] T [D][B]dXdY In the formula: [k] e Here is the stiffness matrix for each element, [B] is the intermediate variable, [D] is the workpiece stiffness, the superscript T indicates transpose, and the superscript e indicates element; [N] is the shape function matrix, i.e., a function of the unit position coordinates; E is the elastic modulus of the material, h is the thickness of the workpiece, and μ is Poisson's ratio; 2) Calculate the equivalent nodal force of each element based on its load strength; the formula is: [F] e =∫∫[N] T qdXdY In the formula: q represents the load strength of each element; [F] e Equivalent nodal forces for each unit cell; 3) Combine the stiffness matrix [k] of all unit cells. e And equivalent nodal forces [F] e The deformation δ of the l-th cutting element on the cutter tooth j in the radial cutting width direction is obtained. el ; The calculation equation is: Where: δ el N represents the deformation in the radial cutting width direction, and N is the total number of unit cells in the workpiece plane.

4. The method for modeling cutting force in side milling of end mills considering the deformation of thin-walled parts according to claim 3, characterized in that: The equivalent nodal force [F] e The solution process is as follows: In the formula: a t b t The scaling factor for the equivalent forces at the nodes; fa t (j) represents the equivalent effect of the mesh nodes on the left side of the cutting edge j of the blade, fb t (j) represents the equivalent effect of the grid node on the right side of the cutting edge j of the blade; f z The feed per tooth, J×f z J represents the length of the rectangular surface swept by the cutter in one revolution, and J is the total number of cutter teeth. u represents the number of longitudinal grids that the cutting edge j of the cutter tooth enters the workpiece; j represents the cutting edge of the cutting tooth, f Y (j) represents the cutting force distributed at the longitudinal node of the cutting edge j of the cutter tooth; F Y (j) represents the total milling force at the moment when the cutting edge j of the cutter tooth enters the workpiece during each cutting cycle of the tool; Based on the total milling force F at the moment the cutting edge j of the cutter tooth enters the workpiece during each cutting cycle. Y (j), obtain the equivalent force fa of the mesh node to the left of the blade j. t (j) and the right-side mesh node of the blade j have the same effect fb t (j); The equivalent force of the left mesh node of the blade j is fa t (j) and the right-side mesh node of the blade j have the same effect fb t (j) Constituting equivalent nodal forces [F] e .

5. The method for modeling cutting force in side milling of end mills considering the deformation of thin-walled parts according to claim 4, characterized in that: In step five, the actual cutting thickness h′(φ) corresponding to the l-th cutting micro-element on the cutter tooth j is used. j,l Calculate the tangential force dF acting on the height dz of the l-th cutting element of the cutter tooth j. t (φ j,l ), radial force dF r (φ j,l ) and axial force dF a (φ j,l The formula is as follows: In the formula: K te K re K ae These are the tangential, radial, and axial tillage force coefficients on the cutting micro-element, respectively. K tc K rc K ac These are the tangential, radial, and axial shear force coefficients on the cutting micro-element, respectively. K te,j,l K is the tangential plowing force coefficient on the l-th cutting element with thickness dz on the cutter tooth j. re,j,l K is the radial tillage force coefficient on the l-th cutting element with thickness dz on the cutter tooth j. ae,j,l denoted as the axial tillage force coefficient on the l-th cutting micro-element with thickness dz on the cutter tooth j; dz j,l Let be the l-th cutting micro-element with thickness dz on the cutter tooth j; K tc,j,l K is the tangential shear force coefficient on the l-th cutting element with thickness dz on the cutter tooth j. rc,j,l K is the radial shear force coefficient on the l-th cutting element with thickness dz on the cutter tooth j. ac,j,l Let be the axial shear force coefficient on the l-th cutting micro-element with thickness dz on the cutter tooth j.

6. The method for modeling cutting forces in side milling of end mills considering the deformation of thin-walled parts according to claim 5, characterized in that: In step six, the actual tangential force dF obtained in step five is transformed using a rotation matrix. t (φ j,l ), radial force dF r (φ j,l ), axial force dF a (φ j,l Transform the coordinates to the X, Y, and Z directions in the Cartesian coordinate system to obtain the actual cutting force dF in the X direction of the Cartesian coordinate system. X (φ j,l Cutting force dF in the Y direction of the Cartesian coordinate system Y (φ j,l Cutting force dF in the Z direction of the Cartesian coordinate system Z (φ j,l The expression is: In the formula: T(φ) j,l ) is the rotation matrix, g(φ) j,l φ′ represents the step function of the current cutting state of the cutting unit. st ,φ′ ex ...

7. The method for modeling cutting force in side milling of end mills considering the deformation of thin-walled parts according to claim 6, characterized in that: The cutting angle φ′ of the tool st The updated formula is: In the formula: a e This represents the radial cutting width.

8. The method for modeling cutting forces in side milling of end mills considering the deformation of thin-walled parts according to claim 7, characterized in that: In step seven, let l = l + 1, and repeat steps three through six until the cutting force F on all cutting micro-elements of the cutter tooth j in the X, Y, and Z directions in the actual Cartesian coordinate system is obtained. X (φ j ), F Y (φ j ), F Z (φ j The expression is: Among them, F X (φ j F represents the cutting force on all cutting elements of the cutter tooth j in the X direction in the Cartesian coordinate system. Y (φ j F represents the cutting force on all cutting elements of the cutter tooth j in the Y direction in the Cartesian coordinate system. Z (φ j ) represents the cutting force on all cutting micro-elements of the cutter tooth j in the Z direction in the Cartesian coordinate system.

9. A method for modeling cutting forces in side milling of end mills considering the deformation of thin-walled parts, as described in claim 8, is characterized in that: In step eight, let j = j + 1, and repeat steps three through seven until the total cutting force F on all cutting elements of all cutter teeth in the X, Y, and Z directions in the actual Cartesian coordinate system is obtained. X (φ), F Y (φ), F Z (φ); the expression is: Among them, F X (φ) represents the total cutting force on all cutting elements of all cutter teeth in the X direction in the Cartesian coordinate system; F Y (φ) represents the total cutting force on all cutting elements of all cutter teeth in the Y direction in the Cartesian coordinate system; F Z (φ) represents the total cutting force on all cutting micro-elements of all cutter teeth in the Z direction in the Cartesian coordinate system.

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