A method for five-axis milling dynamics modeling and stability analysis considering process damping

Abstract

The application discloses a kind of five-axis milling dynamics modeling and stability analysis method considering process damping, comprising the following steps: 1) construct the five-axis machining coordinate conversion equation of workpiece, and determine tool-workpiece contact area;2) combined with tool posture, calculate tool instantaneous undeformed chip thickness and shear force;3) consider the process damping effect when high strength and toughness metal is cut, analyze the extrusion volume of tool tip blunt circle and relief surface extruding workpiece material, calculate the ploughing force and dynamic cutting force of tool;4) based on the dynamic cutting force of tool, establish the five-axis milling dynamics model of workpiece;5) use the five-axis milling dynamics model of workpiece to analyze the tool stability during workpiece milling process.The application can be used to guide the stable cutting parameter selection of high-end equipment with high strength and toughness metal special-shaped curved surface parts.

Description

Technical Field

[0001] This invention relates to the field of five-axis CNC machining of complex curved parts, specifically a method for five-axis milling dynamics modeling and stability analysis that considers process damping. Background Technology

[0002] Deep-cavity, irregularly shaped curved surface critical components are widely used in high-end equipment fields such as energy, power, and aerospace. Due to the limited space between adjacent workpieces, long tool overhangs are used in five-axis milling to avoid machining interference, resulting in a typical rigid workpiece-weakly rigid tool system. To meet the performance requirements of high-end equipment, critical components are made of high-strength, tough, and difficult-to-machine metals. The dynamic variation in the thickness of the undeformed chips during cutting, coupled with drastic fluctuations in dynamic cutting forces, leads to chatter, damaging workpiece quality and reducing tool life. Furthermore, the process damping effect is significant when machining high-strength, tough metals; neglecting process damping in the selection of cutting parameters results in low production efficiency. Therefore, researching five-axis milling dynamics modeling and stability analysis methods that consider process damping is of great significance for achieving high-quality machining of irregularly shaped curved surface parts and promoting the development of high-end equipment manufacturing in fields such as energy, power, and aerospace.

[0003] While existing technologies have analyzed the impact of tool posture on cutting stability, they have not considered the damping during the cutting process of high-strength and tough materials, the tool-workpiece meshing conditions on the tool path for machining irregular curved surfaces, and the asymptotic stability limit. This results in the selection of a smaller limiting depth of cut during low-speed cutting processes, leading to low machining efficiency. Summary of the Invention

[0004] The purpose of this invention is to provide a method for five-axis milling dynamics modeling and stability analysis considering process damping, comprising the following steps:

[0005] 1) Construct the five-axis machining coordinate transformation equations for the workpiece and determine the tool-workpiece contact area;

[0006] 2) Calculate the instantaneous undeformed chip thickness and shearing force based on the tool posture;

[0007] 3) Considering the process damping effect during metal cutting, analyze the extrusion volume of the workpiece material by the blunt circle of the tool tip and the flank face, and calculate the plowing force and dynamic cutting force of the tool;

[0008] 4) Based on the instantaneous undeformed chip thickness, shearing force, tool plowing force, and dynamic cutting force, a five-axis milling dynamic model of the workpiece is established;

[0009] 5) The stability of the cutting tool during the workpiece milling process is analyzed using a five-axis milling dynamics model of the workpiece.

[0010] Furthermore, the cutting tool is a milling cutter, and the cutting edge includes a rounded portion and a cylindrical portion, with a tool radius of R. DThe radius of the blade arc is R.

[0011] Furthermore, the steps for constructing the five-axis machining coordinate transformation equations for the workpiece include:

[0012] 1.1) Let the machine tool coordinate system be O. M -X M Y M Z M Let the workpiece coordinate system be O. w -X w Y w Z w The machine tool coordinate system and workpiece coordinate system are fixed coordinate systems; the X-axis of the workpiece coordinate system is set. w Axis, Y w Axis and Z w The axes are respectively related to the X coordinate system of the machine tool. M Axis, Y M Axis and Z M The axes are in the same direction;

[0013] Let the tool coordinate system O be... t -X t Y t Z t The origin is located at the center of the bottom of the tool, Z t The direction is upward along the center line of the tool axis, X t Axis, Y t The axes are respectively related to the X coordinate system of the workpiece. w Axis, Y w The axes are in the same direction;

[0014] Let the feed coordinate system be O. F -X F Y F Z F ;X F Y F and Z F This indicates the feed axis, cross feed axis, and surface normal axis; the feed coordinate system is a motion coordinate system.

[0015] 1.2) Construct the transformation equation from the tool coordinate system to the feed coordinate system. Right now:

[0016]

[0017] Where α and β are the pitch angle and side tilt angle of the milling cutter during the machining process, respectively;

[0018] 1.3) Construct the transformation matrix between the workpiece coordinate system and the feed coordinate system. Right now:

[0019]

[0020] In the formula, F x F y F z For feed coordinate system X F The unit vectors along the axial directions are respectively related to the workpiece coordinate system X. w Axis, Y w Axis and Z w The cosine of the angle between the axes, C x C y C z For feed coordinate system Y F The unit vectors along the axial directions are respectively related to the workpiece coordinate system X. w Axis, Y w Axis and Z w The cosine of the angle between the axes, N x N y N z For feed coordinate system Z F The unit vectors along the axial directions are respectively related to the workpiece coordinate system X. w Axis, Y w Axis and Z w The cosine of the angle between the axes;

[0021] 1.4) Constructing the machine tool coordinate system O M -X M Y M Z M To the workpiece coordinate system O w -X w Y w Z w Transformation matrix Right now:

[0022]

[0023] Where, θ B θ C These represent the rotation angles of the rotary axes of the five-axis milling machine; B and C represent the rotary axes of the five-axis milling machine.

[0024] 1.5) Construct the five-axis machining coordinate system transformation equations for the workpiece. Right now:

[0025]

[0026] 1.6) Use Unigraphics NX software to generate the contact surface between the workpiece and the tool at the current tool position during the milling process, and perform Boolean operations on the workpiece 3D model and the feasible contact surface of the tool to obtain the tool-workpiece contact area;

[0027] Starting from the origin of the tool coordinate system, generate planes with equal spacing along the tool axis. Intersect these planes with the tool-workpiece contact area to obtain the intersection lines of the planes with the tool-workpiece contact area at different axial heights.

[0028] Project the intersection line onto the workpiece coordinate system X. t O t Y t The plane is used to obtain the projected curve segment; the line connecting the endpoint of the projected curve segment and the origin of the workpiece coordinate system is measured and intersected with the Y-axis. t The included angle between the axes is the entry angle and exit angle of the tool-workpiece contact area at that axial height at the tool position point. The entry angle and exit angle are used to determine whether the tool cutting edge is in a cutting state during the machining process.

[0029] Furthermore, the steps for calculating the instantaneous undeformed chip thickness and shear force include:

[0030] 2.1) Establish the tool geometry model; in the tool geometry model, the tool radius is R. D The radius of the cutting edge arc is R, and the local radius of the tool is r. i ;

[0031] Wherein, the local radius r of the tool i As shown below:

[0032]

[0033] In the formula, i = 1, 2, ..., M s M s To increase the axial cutting depth a p The number of discrete infinitesimal elements along the axial direction; parameter Δz = a p / M s ;

[0034] 2.2) Calculate the radial contact angle between the workpiece and the cutting tool during the cutting process. Right now:

[0035]

[0036] In the formula, j = 1, 2, ..., N t N t S represents the number of teeth on the cutting tool. ω Main spindle speed; θ β The helix angle of the cutting tool; The tooth angle is equal in pitch. t represents time;

[0037] 2.3) Calculate the axial contact angle κ between the workpiece and the tool during the cutting process. i and the unit outward normal vector n of the blade envelope at the point of contact. i (t), that is:

[0038]

[0039]

[0040] 2.4) Calculate the instantaneous, undeformed static chip thickness during the cutting process. With dynamic chip thickness Right now:

[0041]

[0042]

[0043]

[0044] In the formula, τ T The angle between the teeth of the milling cutter The resulting cutting time delay; d represents the relative vibration displacement between the tool and the workpiece in the tool coordinate system; f t x is the feed per tooth cutting parameter; tcs (t)-x tcs (t-τ T ), y tcs (t)-y tcs (t-τ T ), z tcs (t)-z tcs (t-τ T The numbers () represent the relative vibration displacements of the tool and workpiece on the X, Y, and Z axes in the tool coordinate system, respectively.

[0045] 2.5) Considering the tool posture change and regenerative chatter effect during the cutting process, calculate the instantaneous undeformed chip thickness h at the micro-element of the cutting edge. j,i (t), that is:

[0046]

[0047] 2.6) Calculate the infinitesimal shear force. Right now:

[0048]

[0049] In the formula, q = ra, ta, ax represent the radial, tangential, and axial directions, respectively; K qc is the shear force coefficient in three directions; db is the width of the infinitesimal cutting edge; g(·) is the window function used to determine whether the milling cutter cutting edge is in a cutting state;

[0050] The window function g(·) is shown below:

[0051]

[0052] When the cutting teeth enter the workpiece, the tool angle φ st The tool angle φ when the cutting teeth cut the workpiece ex They are shown below:

[0053]

[0054]

[0055] In the formula, a r Where D is the radial depth of cut and a is the tool diameter. r / D represents the radial penetration rate.

[0056] Furthermore, the plowing force of the blade As shown below:

[0057]

[0058] In the formula, K qp These are the plowing force coefficients in three directions;

[0059] Wherein, the extrusion volume V of the i-th infinitesimal element on the j-th cutting edge at time t. j,i (t) is shown below:

[0060] V j,i (t)=S j,i (t)db (18)

[0061] In the formula, Let be the compression area of ​​the i-th micro-element on the j-th cutting edge at time t;

[0062] Static pressing area and dynamic pressing area They are shown below:

[0063]

[0064]

[0065] In the formula, r e It is the radius of the cutting edge blunt circle; β s It is the material separation angle; γ is the cutting edge clearance angle; l w v is the length of the wear band on the flank of the tool. j,i It is the cutting speed of the i-th cutting element of the j-th cutting edge in the direction parallel to the workpiece surface;

[0066] Vibration speed of the cutting tool As shown below:

[0067]

[0068] In the formula, This indicates that the i-th cutting element of the cutting edge at time t is located in the tool coordinate system X. w Y w and Z w The derivative of the axial displacement.

[0069] Furthermore, the dynamic cutting force of the tool is as follows:

[0070]

[0071] In the formula, a xx a xy a xz a yx a yy a yz a zx a zy a zz c is the shear force coefficient; xx c xy c xz c yx c yy c yz c zx c zy c zz This is the plowing force coefficient. The dynamic cutting force of the tool in the X, Y, and Z directions.

[0072] Furthermore, the steps for establishing a five-axis milling dynamics model of the workpiece include:

[0073] 4.1) The milling system is simplified into a mass-spring-damped system with multiple modes in the X and Y directions. The dynamic equations of this mass-spring-damped system are as follows:

[0074]

[0075] In the formula, M, C, and K are the mass matrix, damping matrix, and stiffness matrix of the cutting tool in physical space, respectively. q(t) and q(t) represent the acceleration vector, velocity vector, and displacement vector of the tool vibration in physical space, respectively; q(t) = [x tcs (t) y tcs (t)] T F(t) is the dynamic cutting force vector acting on the tool; x tcs (t), y tcs (t) represents the displacement of the tool vibration in the X and Y directions in physical space;

[0076] 4.2) Construct the mass matrix M, damping matrix C, and stiffness matrix K of the tool in physical space, i.e.:

[0077]

[0078] In the formula, M xx M xy M yx M yy C represents the mass of the cutting tool in physical space. xx C xy C yx C yy For the damping of the cutting tool in physical space; K xx K xy K yx K yy The stiffness of the cutting tool in physical space;

[0079] 4.3) By using modal coordinate transformation to convert the physical space displacement vector to modal space, we obtain:

[0080] Q(t) = U -1 q(t) (25)

[0081] In the formula, U is the modal matrix. Q(t) represents the tool displacement vector in modal space, Q(t) = [q x q y ] T ;q x q y This represents the displacement of the tool in the X and Y directions within the modal space;

[0082] 4.4) Normalize the tool displacement vector Q(t) in modal space at the tool tip and establish the cutting dynamics equations decoupled in modal space, resulting in:

[0083]

[0084] In the formula, the mass vector M m =diag(m m,x ,m m,y Damping vector C m =diag(c m,x ,c m,y Stiffness vector K m =diag(k) m,x ,k m,y ), vector F m (t)=U T F(t); m m,x m m,y c represents the mass parameter in modal space. m,xc m,y k represents the damping parameter in modal space. m,x k m,y These are the stiffness parameters in modal space;

[0085] 4.5) Construct a simplified equation for the dynamic cutting force of the tool, namely:

[0086]

[0087] In the formula, H j (t), P j (t) represents the shear force coefficient matrix and the plowing force coefficient matrix; Q(t-τ) T ) represents the tool displacement vector in modal space;

[0088] 4.6) Substitute formula (27) into formula (25) to construct the five-axis milling dynamics model of the workpiece, that is:

[0089]

[0090] Furthermore, the steps for analyzing tool stability during workpiece milling using a five-axis milling dynamics model include:

[0091] 5.1) Order The five-axis milling dynamics model of the workpiece is transformed into state-space equations, resulting in:

[0092]

[0093] In the formula, h xx (t), h xy (t), h yx (t) and h yy (t) represents the dynamic cutting force coefficient; matrix matrix matrix T is the period;

[0094] 5.2) The cutting time delay τ T Discretize the equation into m parts, and solve by integrating both sides of the state-space equation to obtain the following result.

[0095]

[0096] Where, τ0=τ T / m, x(kτ0) is the state value at time kτ0; x(t) is the solution to the state-space equation; ξ is the time;

[0097] 5.3) Using the interpolation quadrature formula (30) with equidistant nodes, the discrete dynamic mapping equation is obtained, namely:

[0098]

[0099] In the formula, the matrix matrix matrix I is a unit vector; a p t is the axial depth of cut; f t Ms τ represents time.

[0100] 5.4) Construct the state transition matrix Φ1 between adjacent moments of the cutting tooth, i.e.:

[0101] Φ1=(I-C1-a p τD1 / 2) -1 (-a p τD1 / 2+E) (32)

[0102] 5.5) Calculate the eigenvalues ​​of the state transition matrix Φ1 at adjacent moments of the cutting tooth; if the modulus of all eigenvalues ​​is less than 1, the tool motion tends to be stable; if the modulus of all eigenvalues ​​is equal to 1, the tool motion is in a critical stable state; if the modulus of any eigenvalue is greater than 1, the machining system is in a chattering state.

[0103] Furthermore, the workpiece is an irregularly shaped curved surface part.

[0104] Furthermore, the workpiece material is a high-strength, tough, and difficult-to-machine metal, including but not limited to stainless steel and titanium alloys.

[0105] The technical effects of this invention are undeniable. This invention analyzes the influence of tool posture on the instantaneous thickness of undeformed chips during machining curved parts, calculates the five-axis cutting shear force, and considers the damping effect in the cutting process of high-strength and tough metals. It analyzes the cutting edge blunt circle and the volume of the tool face pressing into the workpiece surface, calculates the plowing force, and thus establishes a five-axis cutting dynamics model for high-strength and tough material irregular curved parts. Furthermore, it uses numerical integration to analyze cutting stability and calculates the leaf-shaped diagram of the five-axis cutting stability of high-strength and tough material irregular curved parts. This model can be used to guide the selection of stable cutting parameters for irregular curved parts used in high-end equipment, increasing cutting parameters while ensuring stable cutting. This method is convenient to implement and has significant effects. Attached Figure Description

[0106] Figure 1 Flowchart for five-axis milling dynamics modeling and stability analysis considering process damping.

[0107] Figure 2 This is a diagram of the contact area between the workpiece and the cutting tool when a round nose end mill is used to cut a curved surface.

[0108] Figure 3 This is a schematic diagram of the instantaneous, undeformed cut thickness. Where h... s h represents the static chip thickness. dThis represents the dynamic chip thickness.

[0109] Figure 4 This is a schematic diagram of the compression area on the flank face of the cutting tool. Among them, For dynamic pressing area, For the static pressed-in area, l w This refers to the length of the wear band on the back face of the cutting tool.

[0110] Figure 5 This is a leaf-shaped diagram illustrating the cutting stability of an embodiment of the present invention. Detailed Implementation

[0111] The present invention will be further described below with reference to embodiments, but it should not be construed that the scope of the present invention is limited to the following embodiments. Various substitutions and modifications made based on ordinary technical knowledge and common practices in the art without departing from the above-described technical concept of the present invention should be included within the scope of protection of the present invention.

[0112] Example 1:

[0113] See Figures 1 to 5 A method for five-axis milling dynamics modeling and stability analysis considering process damping includes the following steps:

[0114] 1) Construct the five-axis machining coordinate transformation equations for the workpiece and determine the tool-workpiece contact area;

[0115] 2) If the cutting edge of the tool is in a cutting state, calculate the instantaneous undeformed chip thickness and shearing force based on the tool posture;

[0116] 3) Considering the process damping effect during metal cutting, analyze the extrusion volume of the workpiece material by the blunt circle of the tool tip and the flank face, and calculate the plowing force and dynamic cutting force of the tool;

[0117] 4) Based on the plowing force and dynamic cutting force of the cutting tool, a five-axis milling dynamics model of the workpiece is established;

[0118] 5) The stability of the cutting tool during the workpiece milling process is analyzed using a five-axis milling dynamics model of the workpiece.

[0119] Example 2:

[0120] A method for five-axis milling dynamics modeling and stability analysis considering process damping, with the same technical content as Embodiment 1, further wherein the tool is a milling cutter, the cutting edge includes a rounded portion and a cylindrical portion, and the tool radius is R. D The radius of the blade arc is R.

[0121] Example 3:

[0122] A method for five-axis milling dynamics modeling and stability analysis considering process damping, with technical content the same as any one of Embodiments 1-2, further comprising the following steps for constructing the five-axis machining coordinate transformation equations of the workpiece:

[0123] 1.1) Let the machine tool coordinate system be O. M -X M Y M Z M Let the workpiece coordinate system be O. w -X w Y w Z w The machine tool coordinate system and workpiece coordinate system are fixed coordinate systems; the X-axis of the workpiece coordinate system is set. w Axis, Y w Axis and Z w The axes are respectively related to the X coordinate system of the machine tool. M Axis, Y M Axis and Z M The axes are in the same direction;

[0124] Let the tool coordinate system O be... t -X t Y t Z t The origin is located at the center of the bottom of the tool, Z t The direction is upward along the center line of the tool axis, X t Axis, Y t The axes are respectively related to the X coordinate system of the workpiece. w Axis, Y w The axes are in the same direction;

[0125] Let the feed coordinate system be O. F -X F Y F Z F ;X F Y F and Z F This indicates the feed axis, cross feed axis, and surface normal axis; the feed coordinate system is a motion coordinate system.

[0126] 1.2) Construct the transformation equation from the tool coordinate system to the feed coordinate system. Right now:

[0127]

[0128] Where α and β are the pitch angle and side tilt angle of the milling cutter during the machining process, respectively;

[0129] 1.3) Construct the transformation matrix between the workpiece coordinate system and the feed coordinate system. Right now:

[0130]

[0131] In the formula, F x F y F z For feed coordinate system X F The unit vectors along the axial directions are respectively related to the workpiece coordinate system X. w Axis, Y w Axis and Z w The cosine of the angle between the axes, C x C y C z For feed coordinate system Y F The unit vectors along the axial directions are respectively related to the workpiece coordinate system X. w Axis, Y w Axis and Z w The cosine of the angle between the axes, N x N y N z For feed coordinate system Z F The unit vectors along the axial directions are respectively related to the workpiece coordinate system X. w Axis, Y w Axis and Z w The cosine of the angle between the axes;

[0132] 1.4) Constructing the machine tool coordinate system O M -X M Y M Z M To the workpiece coordinate system O w -X w Y w Z w Transformation matrix Right now:

[0133]

[0134] Where, θ B θ C These represent the rotation angles of the rotary axes of the five-axis milling machine; B and C represent the rotary axes of the five-axis milling machine.

[0135] 1.5) Construct the five-axis machining coordinate system transformation equations for the workpiece. Right now:

[0136]

[0137] 1.6) In Unigraphics NX software, execute the "Insert" - "Tool" command. The system will pop up the "Create Tool" dialog box. Enter the geometric parameters of the selected machining tool, such as tool radius, cutting edge radius, number of teeth, helix angle and length, to create a 3D model of the tool and a 3D model of the workpiece. Set the tool displacement parameters. Use the "Intersection" command in the feature options of the drawing design window to generate the contact surface between the workpiece and the tool at the current tool position during the milling process. Perform Boolean operation between the 3D model of the workpiece and the feasible contact surface of the tool to obtain the tool-workpiece contact area.

[0138] Starting from the origin of the tool coordinate system, generate planes with equal spacing along the tool axis. Intersect these planes with the tool-workpiece contact area to obtain the intersection lines of the planes with the tool-workpiece contact area at different axial heights.

[0139] Project the intersection line onto the workpiece coordinate system X. t O t Y t The plane is used to obtain the projected curve segment; the line connecting the endpoint of the projected curve segment and the origin of the workpiece coordinate system is measured and intersected with the Y-axis. t The included angle between the axes is the entry angle and exit angle of the tool-workpiece contact area at the tool position point along that axial height. These entry and exit angles are used to determine whether the tool cutting edge is in a cutting state during the machining process. If the tool cutting edge is in a cutting state, then step 2 is executed.

[0140] Example 4:

[0141] A method for five-axis milling dynamics modeling and stability analysis considering process damping, with technical content the same as any one of embodiments 1-3, further comprising the steps of calculating the instantaneous undeformed chip thickness and shear force, including:

[0142] 2.1) Establish the tool geometry model; in the tool geometry model, the tool radius is R. D The radius of the cutting edge arc is R, and the local radius of the tool is r. i ;

[0143] Wherein, the local radius r of the tool i As shown below:

[0144]

[0145] In the formula, i = 1, 2, ..., M s M s To increase the axial cutting depth a p The number of discrete infinitesimal elements along the axial direction; parameter Δz = a p / M s The radius r corresponding to i·Δz < R i Let r be the radius of the rounded portion, i·Δz≥R.i Let be the radius of the cylindrical portion.

[0146] 2.2) Calculate the radial contact angle between the workpiece and the cutting tool during the cutting process. Right now:

[0147]

[0148] In the formula, j = 1, 2, ..., N t N t S represents the number of teeth on the cutting tool. ω Main spindle speed; θ β The helix angle of the cutting tool; The tooth angle is equal in pitch. t represents time;

[0149] 2.3) Calculate the axial contact angle κ between the workpiece and the tool during the cutting process. i and the unit outward normal vector n of the blade envelope at the point of contact. i (t), that is:

[0150]

[0151]

[0152] 2.4) Calculate the instantaneous, undeformed static chip thickness during the cutting process. With dynamic chip thickness Right now:

[0153]

[0154]

[0155]

[0156] In the formula, τ T The angle between the teeth of the milling cutter The resulting cutting time delay; d represents the relative vibration displacement between the tool and the workpiece in the tool coordinate system; f t x is the feed per tooth cutting parameter; tcs (t)-x tcs (t-τ T ), y tcs (t)-y tcs (t-τ T ), z tcs (t)-z tcs (t-τ T The numbers () represent the relative vibration displacements of the tool and workpiece on the X, Y, and Z axes in the tool coordinate system, respectively.

[0157] 2.5) Considering the tool posture change and regenerative chatter effect during the cutting process, calculate the instantaneous undeformed chip thickness h at the micro-element of the cutting edge. j,i (t), that is:

[0158]

[0159] 2.6) Calculate the infinitesimal shear force. Right now:

[0160]

[0161] In the formula, q = ra, ta, ax represent the radial, tangential, and axial directions, respectively; K qc λ is the shear force coefficient in three directions; db is the width of the micro-element cutting edge; db = Δz / sinκ(z); κ(z) is the axial contact angle of the micro-element; g(·) is the window function used to determine whether the milling cutter cutting edge is in a cutting state;

[0162] The window function g(·) is shown below:

[0163]

[0164] When the cutting teeth enter the workpiece, the tool angle φ st The tool angle φ when the cutting teeth cut the workpiece ex They are shown below:

[0165]

[0166]

[0167] In the formula, a r Where D is the radial depth of cut and a is the tool diameter. r / D represents the radial penetration rate.

[0168] Example 5:

[0169] A method for five-axis milling dynamics modeling and stability analysis considering process damping, with technical content the same as any one of embodiments 1-4, further including the plowing force of the tool. As shown below:

[0170]

[0171] In the formula, K qp These are the plowing force coefficients in three directions;

[0172] Wherein, the extrusion volume V of the i-th infinitesimal element on the j-th cutting edge at time t. j,i (t) is shown below:

[0173] V j,i (t)=Sj,i (t)db (18)

[0174] In the formula, Let be the compression area of ​​the i-th micro-element on the j-th cutting edge at time t;

[0175] Static pressing area and dynamic pressing area They are shown below:

[0176]

[0177]

[0178] In the formula, r e It is the radius of the cutting edge blunt circle; β s It is the material separation angle; γ is the cutting edge clearance angle; l w v is the length of the wear band on the flank of the tool. j,i It is the cutting speed of the i-th cutting element of the j-th cutting edge in the direction parallel to the workpiece surface;

[0179] Vibration speed of the cutting tool As shown below:

[0180]

[0181] In the formula, This indicates that the i-th cutting element of the cutting edge at time t is located in the tool coordinate system X. w Y w and Z w The derivative of the axial displacement.

[0182] Example 6:

[0183] A method for five-axis milling dynamics modeling and stability analysis considering process damping, with technical content the same as any one of embodiments 1-5, further wherein the dynamic cutting force of the tool is as follows:

[0184]

[0185] In the formula, a xx a xy a xz a yx a yy a yz a zx a zy a zz c is the shear force coefficient; xx c xy c xz c yx c yy cyz c zx c zy c zz This is the plowing force coefficient. The dynamic cutting force of the tool in the X, Y, and Z directions.

[0186] Example 7:

[0187] A method for five-axis milling dynamics modeling and stability analysis considering process damping, with technical content the same as any one of embodiments 1-6, further comprising the following steps for establishing a workpiece five-axis milling dynamics model:

[0188] 4.1) The milling system is simplified into a mass-spring-damped system with multiple modes in the X and Y directions. The dynamic equations of this mass-spring-damped system are as follows:

[0189]

[0190] In the formula, M, C, and K are the mass matrix, damping matrix, and stiffness matrix of the cutting tool in physical space, respectively. q(t) and q(t) represent the acceleration vector, velocity vector, and displacement vector of the tool vibration in physical space, respectively; q(t) = [x tcs (t) y tcs (t)] T F(t) is the dynamic cutting force vector acting on the tool; x tcs (t), y tcs (t) represents the displacement of the tool vibration in the X and Y directions in physical space;

[0191] 4.2) Construct the mass matrix M, damping matrix C, and stiffness matrix K of the tool in physical space, i.e.:

[0192]

[0193] In the formula, M xx M xy M yx M yy C represents the mass of the cutting tool in physical space. xx C xy C yx C yy For the damping of the cutting tool in physical space; K xx K xy K yx K yy The stiffness of the cutting tool in physical space;

[0194] 4.3) By using modal coordinate transformation to convert the physical space displacement vector to modal space, we obtain:

[0195] Q(t) = U -1 q(t) (25)

[0196] In the formula, U is the modal matrix. Q(t) represents the tool displacement vector in modal space, Q(t) = [q x q y ] T ;q x q y This represents the displacement of the tool in the X and Y directions within the modal space;

[0197] 4.4) Normalize the tool displacement vector Q(t) in modal space at the tool tip and establish the cutting dynamics equations decoupled in modal space, resulting in:

[0198]

[0199] In the formula, the mass vector M m =diag(m m,x ,m m,y Damping vector C m =diag(c m,x ,c m,y Stiffness vector K m =diag(k) m,x ,k m,y ),vector m m,x m m,y c represents the mass parameter in modal space. m,x c m,y k represents the damping parameter in modal space. m,x k m,y These are the stiffness parameters in modal space; Let F be a vector m Elements of (t);

[0200] 4.5) Construct a simplified equation for the dynamic cutting force of the tool, namely:

[0201]

[0202] Among them, the shear force coefficient matrix Plowing force coefficient matrix Q(t-τ T ) represents the tool displacement vector in modal space;

[0203] 4.6) Substitute formula (27) into formula (25) to construct the five-axis milling dynamics model of the workpiece, that is:

[0204]

[0205] Example 8:

[0206] A method for five-axis milling dynamics modeling and stability analysis considering process damping, with technical content identical to any one of embodiments 1-7, further comprising the following steps for analyzing tool stability during workpiece milling using a workpiece five-axis milling dynamics model:

[0207] 5.1) Order The five-axis milling dynamics model of the workpiece is transformed into state-space equations, resulting in:

[0208]

[0209] In the formula, h xx (t), h xy (t), h yx (t) and h yy (t) is the dynamic cutting force coefficient; coefficient

[0210] T is the period;

[0211] 5.2) The cutting time delay τ T Discretize the equation into m parts, and solve by integrating both sides of the state-space equation to obtain the following result.

[0212]

[0213] Where, τ0=τ T / m, x(kτ0) is the state value at time kτ0; x(t) is the solution to the state-space equation; ξ is the time;

[0214] 5.3) Using the interpolation quadrature formula (30) with equidistant nodes, the discrete dynamic mapping equation is obtained, namely:

[0215]

[0216] In the formula, the matrix matrix matrix I is a unit vector; a p t is the axial depth of cut; f , τ represents time.

[0217] 5.4) Construct the state transition matrix Φ1 between adjacent moments of the cutting tooth, i.e.:

[0218] Φ1=(I-C1-a p τD1 / 2) -1 (-a p τD1 / 2+E) (32)

[0219] 5.5) Calculate the eigenvalues ​​of the state transition matrix Φ1 at adjacent moments of the cutting tooth; if the modulus of all eigenvalues ​​is less than 1, the tool motion tends to be stable; if the modulus of all eigenvalues ​​is equal to 1, the tool motion is in a critical stable state; if the modulus of any eigenvalue is greater than 1, the machining system is in a chattering state.

[0220] Example 9:

[0221] A method for five-axis milling dynamics modeling and stability analysis considering process damping, with the same technical content as any one of embodiments 1-8, further wherein the workpiece is an irregular curved surface part.

[0222] Example 10:

[0223] A method for five-axis milling dynamics modeling and stability analysis considering process damping, with the same technical content as any one of Embodiments 1-9. Furthermore, the workpiece material is a high-strength and tough, difficult-to-machine metal, including but not limited to stainless steel and titanium alloy.

[0224] Example 11:

[0225] A method for five-axis milling dynamics modeling and stability analysis considering process damping, with the same technical content as any one of Embodiments 1-10, further wherein the milling cutter is a round nose milling cutter.

[0226] Example 12:

[0227] A method for five-axis milling dynamics modeling and stability analysis considering process damping is presented below:

[0228] First, the coordinate system transformation relationship for five-axis machining is constructed, a round nose end mill model is established, and the tool-workpiece contact area is analyzed and extracted. The instantaneous undeformed chip thickness and shearing force are calculated by combining the tool posture during surface machining. Second, considering the process damping effect during the cutting of high-strength and tough metals, the extrusion volume of the workpiece material by the blunt circle of the tool tip and the flank face is analyzed, and the plowing force and total cutting force are calculated. Thus, a five-axis milling dynamic model for high-strength and tough material irregular curved surfaces is established. Then, the five-axis cutting stability is analyzed by numerical integration, and a cutting stability lobe diagram is generated, realizing the five-axis milling dynamic modeling and stability analysis of high-strength and tough material irregular curved surface parts.

[0229] The specific steps of the method are as follows:

[0230] Step 1: Establish the five-axis machining coordinate system transformation relationship and analyze and extract the tool-workpiece contact area.

[0231] Let the machine tool coordinate system be O. M -X M Y M Z MThe machine tool coordinate system is a fixed coordinate system, and its coordinate axes and origin are set by the machine tool manufacturer; the workpiece coordinate system is denoted as O. w -X w Y w Z w The workpiece coordinate system is also a fixed coordinate system. The X-axis of the workpiece coordinate system is set. w Axis, Y w Axis and Z w The axes are respectively related to the X coordinate system of the machine tool. M Axis, Y M Axis and Z M The axes are aligned. The cutting edge of a ball nose end mill consists of a rounded portion and a cylindrical portion, with a tool radius of R. D The radius of the cutting edge arc R, the tool coordinate system O t -X t Y t Z t The origin is located at the center of the bottom of the milling cutter, Z t The direction is upward along the center line of the tool axis, X t Axis, Y t The axes are respectively related to the X coordinate system of the workpiece. w Axis, Y w The axes are in the same direction. Feed coordinate system O F -X F Y F Z F It is a motion coordinate system, starting from the feed axis X. F Cross feed axis Y F and the surface normal axis Z F Composition, transformation from tool coordinate system to feed coordinate system Calculate according to formula (1),

[0232]

[0233] Where α and β are the pitch angle and side tilt angle of the milling cutter during the machining process, respectively.

[0234] The displacements of each axis in the feed coordinate system within the workpiece coordinate system are determined based on the toolpath file generated by the computer-aided manufacturing software post-processing. F =[F x F y F z ] T y C =[C x C y C z ] T , z N =[N x N y N z ] TThe transformation matrix between the workpiece coordinate system and the feed coordinate system is:

[0235]

[0236] For a five-axis milling machine with two rotary axes B and C, the machine coordinate system O M -X M Y M Z M To the workpiece coordinate system O w -X w Y w Z w The transformation matrix is Calculate according to formula (3)

[0237]

[0238] Where, θ B θ C These represent the rotation angles of the milling machine's rotary axis.

[0239] Therefore, the transformation relationship between the machine tool coordinate system and the tool coordinate system is calculated according to formula (4):

[0240]

[0241] In the five-axis milling process of irregular curved parts, the tool-workpiece contact area changes with the tool posture. Computer-aided manufacturing software is used to generate the contact surface between the workpiece and the tool at the current tool position during the round nose milling process. Boolean operations are performed on the workpiece 3D model and the feasible tool contact surface to obtain the tool-workpiece contact area. Planes with equal spacing are generated along the tool axis, starting from the origin of the tool coordinate system. The intersection of these planes with the tool-workpiece contact area is obtained at different axial heights. The intersection lines are then projected onto the workpiece coordinate system X. t O t Y t Plane, the line connecting the endpoint of the measured projection curve segment to the origin of the workpiece coordinate system and the Y-axis. t The included angle of the axis is the entry angle and exit angle of the tool-workpiece contact area at the tool position point at that axial height.

[0242] Step 2: Calculate the instantaneous undeformed chip thickness and, considering process damping, calculate the dynamic cutting force.

[0243] The instantaneous undeformed cutting thickness is the distance measured along the normal to the tool's outer contour between the current cutting edge's trajectory and the previous cutting edge's trajectory. To calculate the instantaneous undeformed cutting thickness, a tool geometry model is first established, with a radius of R for the round nose end mill. D The radius of the cutting edge arc R, and the local radius r of the round nose end mill. i Calculate according to formula (5).

[0244]

[0245] i = 1, 2, ..., M s M s To increase the axial cutting depth a p The number of discrete infinitesimal elements along the axial direction, Δz = a p / M s .

[0246] Radial contact angle between workpiece and tool during cutting process Calculate according to formula (6),

[0247]

[0248] Where j = 1, 2, ..., N t N t S represents the number of cutting teeth. ω Spindle speed, θ β Tool helix angle, The tooth angle between teeth with equal pitch is...

[0249] The axial contact angle κ between the workpiece and the cutting tool during the cutting process i According to formula (7), the unit outward normal vector of the blade envelope surface at the blade contact point is calculated according to formula (8).

[0250]

[0251]

[0252] The instantaneous undeformed chip thickness during the cutting process consists of the static chip thickness and the dynamic chip thickness. The static chip thickness is calculated according to formula (9).

[0253]

[0254] f t This represents the feed per tooth as a cutting parameter.

[0255] The relative vibration displacement between the tool and the workpiece in the tool coordinate system is calculated according to formula (10), and the dynamic chip thickness caused by the tool-workpiece vibration is calculated according to formula (11).

[0256]

[0257]

[0258] Where, τ T The angle between the teeth of the milling cutter The resulting cutting time delay

[0259] The instantaneous undeformed chip thickness at the micro-element of the cutting edge under the changes in tool posture and regenerative chatter effect during the cutting process is calculated according to formula (12).

[0260]

[0261] During the cutting process, the workpiece material is divided into two parts at the separation point in front of the cutting edge of the tool. One part of the material slides along the rake face of the tool, generating a shearing effect and generating shearing force. The other part of the material is squeezed by the blunt rounded tip and the flank face, resulting in a plowing effect and generating plowing force. According to the linear cutting force model, the micro-element shearing force is calculated according to formula (13).

[0262]

[0263] Where q = ra, ta, ax represent the radial, tangential, and axial directions, respectively, and K qc Here, is the shear force coefficient in three directions, db is the width of the infinitesimal cutting edge, db=Δz / sinκ(z), and g(·) is the window function used to determine whether the milling cutter cutting edge is in a cutting state, calculated according to formula (14).

[0264]

[0265] Where, φ st φ represents the angle of the cutting tool when the cutting teeth enter the workpiece. ex The cutting angle when the cutting teeth cut the workpiece is indicated by formulas (15) and (16).

[0266]

[0267]

[0268] a r Where D is the radial depth of cut and a is the tool diameter. r / D represents the radial penetration rate.

[0269] Under the assumption of small amplitude vibration, the plowing effect generated by the blunt circle of the tool tip and the flank face pressing against the workpiece material is analyzed. The plowing force acting on the micro-element of the cutting edge is proportional to the pressing area. The plowing force of the micro-element is calculated according to formula (17):

[0270]

[0271] Among them, K qp Let Vj,i(t) be the plowing force coefficient in three directions, and Vj,i(t) be the extrusion volume of the i-th micro-element on the j-th cutting edge at time t, calculated according to formula (18).

[0272] V j,i (t)=S j,i (t)db (18)

[0273] Among them, S j,i (t) represents the compression area of ​​the i-th micro-element on the j-th cutting edge at time t, including the static compression area. and dynamic pressing area

[0274] The static pressing area is the area enclosed by the auxiliary horizontal line parallel to the cutting speed, the blunt circle of the cutting edge, and the flank face of the tool, calculated according to formula (19).

[0275]

[0276] Where, r e It is the radius of the cutting edge blunt circle, β s It is the material separation angle, and γ is the cutting edge clearance angle, which is the angle between the clearance face and the cutting plane in the orthogonal plane of the tool.

[0277] The dynamic pressing area is the area of ​​the approximate triangular region enclosed by the separation point trajectory, the auxiliary horizontal line, and the tool's back face. It is calculated according to formula (20), which correlates the plowing force with the tool's vibration speed.

[0278]

[0279] Among them, l w v is the length of the wear band on the flank face of the tool. j,i It is the cutting speed of the i-th cutting element of the j-th cutting edge in the direction parallel to the workpiece surface. It is the vibration velocity of the cutting tool, calculated according to formula (21).

[0280]

[0281] The total extrusion area is calculated according to formula (22).

[0282]

[0283] Therefore, the cutting force on the i-th infinitesimal element of the j-th cutting edge at time t is calculated according to formula (23).

[0284]

[0285] The transformation matrix for converting the infinitesimal cutting force to the tool coordinate system is calculated according to formula (24), and the total cutting force of the round nose end mill is obtained by summing all the infinitesimal cutting forces according to formula (25).

[0286]

[0287]

[0288] The dynamic cutting force is obtained by subtracting the static cutting force from the total cutting force. The dynamic cutting force on the milling cutter in three directions is calculated according to formula (26) in the tool coordinate system.

[0289]

[0290] Among them, a xx a xy a xz a yx a yy a yz a zx a zy a zz With c xx c xy c xz c yx c yy c yz c zx c zy c zz This is a parameter composed of the shear force coefficient and the plowing force coefficient.

[0291] Step 3: Establishment of a five-axis milling dynamic model for high-strength and tough material curved surfaces.

[0292] The machining process of deep-cavity irregular curved surface parts involves long tool overhangs, forming a weakly rigid tool-rigid workpiece system. The tool extends along the Z-axis... t The stiffness in the directional direction is much greater than that in the X direction. t With Y t The stiffness in the X and Y directions simplifies the milling system into a mass-spring-damped system with multiple modes in the X and Y directions. The dynamic equation of the system is Equation (27).

[0293]

[0294] Where M, C, and K are the mass matrix, damping matrix, and stiffness matrix of the cutting tool in physical space, respectively. Let q(t) and q(t) represent the acceleration vector, velocity vector, and displacement vector of the tool vibration in physical space, respectively. q(t) = [x tcs (t) y tcs (t)] T F(t) is the dynamic cutting force vector acting on the tool. The first-order mode of the tool is analyzed, and the mass matrix, damping matrix, and stiffness matrix of the tool in physical space are constructed according to formula (28).

[0295]

[0296] According to formula (29), the physical space displacement vector is transformed into the modal space using modal coordinate transformation.

[0297] Q(t) = U -1 q(t) (29)

[0298] Where U is the mode matrix. Q(t) represents the tool displacement vector in modal space, Q(t) = [q x q y ] T .

[0299] The modal shape vectors (i.e., the tool vibration shape eigenvectors in modal space) are normalized at the tool tip, and the cutting dynamics equations for modal space decoupling are established according to formula (30).

[0300]

[0301] Among them, M m =diag(m m,x ,m m,y ), C m =diag(c m,x ,c m,y ), K m =diag(k) m,x ,k m,y ),

[0302] Because the tool is in Z t The stiffness in the axial direction is much greater than that in the X direction. t With Y t Since the stiffness is in two directions, the dynamic cutting force along the tool axis can be ignored. Equation (26) can be simplified to Equation (31), and then denoted as Equation (32).

[0303]

[0304]

[0305] in,

[0306] Substituting equation (32) into equation (30), we obtain equation (33) for the cutting dynamics of a five-axis round nose end mill considering process damping in modal space.

[0307]

[0308] Step 4: Numerical integration method analysis of five-axis milling stability

[0309] To reduce the order of the cutting dynamics equation (33), let Transform the cutting dynamics equation (33) into a state-space form.

[0310]

[0311] Among them, coefficient

[0312] h xx (t), h xy (t), h yx (t) and h yy (t) represents the dynamic cutting force coefficient.

[0313] The cutting time delay τ T Discretize into m parts, integrate both sides of equation (34), and the solution to equation (34) is:

[0314]

[0315] Where, τ0=τ T / m, x(kτ0) is the state value at time kτ0.

[0316] Using the simplified quadrature formula (35) with equidistant nodes, the discrete dynamic mapping can be obtained as shown in equation (36) based on the discrete time points.

[0317]

[0318] Among them, matrix

[0319] The state transition matrix Φ1 between adjacent moments of the cutting tooth is expressed as follows:

[0320] Φ1=(I-C1-a p τD1 / 2) -1 (-a p τD1 / 2+E) (37)

[0321] According to Floquet's theory, if the modulus of all eigenvalues ​​of the state transition matrix is ​​less than 1, the motion of the machining system tends to be stable; if the modulus of the eigenvalues ​​of the state transition matrix is ​​equal to 1, the motion of the machining system is at critical stability; if the modulus of the eigenvalues ​​of the state transition matrix is ​​greater than 1, the machining system is in a chattering state. Ultimately, this achieves the stability assessment of five-axis milling of irregular curved parts considering process damping.

[0322] Example 13:

[0323] An application of the five-axis milling dynamics modeling and stability analysis method considering process damping described in Embodiment 12 is as follows:

[0324] In the five-axis milling process of high-strength and tough metal irregular curved surface parts, the drastic fluctuations in dynamic cutting force can easily lead to cutting chatter during machining, which restricts the improvement of machining quality of difficult-to-machine irregular curved surface parts for high-end equipment. To select stable cutting parameters for five-axis milling, the five-axis milling dynamics modeling and stability analysis method considering process damping, as described in Example 12, is adopted. The overall process is attached. Figure 1 As shown.

[0325] Step 1: Establish the transformation relationship of the five-axis machining coordinate system and analyze and extract the tool-workpiece contact area. The machine tool coordinate system is a fixed coordinate system, and its coordinate axes and origin are set by the machine tool manufacturer. The workpiece coordinate system is also a fixed coordinate system, and its three coordinate axes are in the same direction as the three coordinate axes of the machine tool coordinate system. Calculate the transformation matrix from the tool coordinate system to the feed coordinate system according to formula (1), calculate the transformation matrix between the workpiece coordinate system and the feed coordinate system according to formula (2), calculate the transformation matrix from the machine tool coordinate system to the workpiece coordinate system of the five-axis machine tool with two rotation axes of B and C according to formula (3), calculate the transformation relationship matrix between the machine tool coordinate system and the tool coordinate system according to formula (4), and use computer-aided manufacturing software to generate the workpiece and the feasible contact surface of the tool at the tool position point of interest during the round nose milling cutter cutting process, as shown in the attached figure. Figure 2 As shown, Boolean operations are performed on the feasible contact surfaces of the workpiece and the tool to obtain the tool-workpiece contact area.

[0326] Step 2: Calculate the instantaneous undeformed chip thickness and calculate the dynamic cutting force considering process damping. Establish a geometric model of a round nose milling cutter with a length of 120mm, a radius of 10mm, and a cutting edge radius of 3mm. Calculate the local radius of the round nose milling cutter using formula (5). Take the spindle speed as 2000 rpm. Calculate the radial contact angle between the workpiece and the cutter during the cutting process using formula (6). Calculate the axial contact angle between the workpiece and the cutter using formula (7). Calculate the unit outward normal vector of the cutting edge envelope surface in the micro-element using formula (8). Calculate the static chip thickness using formula (9). Calculate the relative vibration displacement between the cutter and the workpiece in the cutter coordinate system using formula (10). Calculate the dynamic chip thickness using formula (11). Calculate the instantaneous undeformed chip thickness of the cutting edge micro-element during five-axis milling using formula (12). (See attached figure) Figure 3As shown. The radial and tangential shear force coefficients are taken as 1100 kPa and 2100 kPa respectively. According to the linear cutting force model, the micro-element shear force is calculated according to formula (13). The cutting edge is judged to be in the cutting state according to formula (14). Climb milling is adopted. The angle when the cutter tooth cuts into the workpiece is calculated according to formula (15). The angle when the cutter tooth cuts out of the workpiece is calculated according to formula (16). The radial and tangential plowing force coefficients are taken as 600 kPa and 300 kPa respectively. The micro-element plowing force is calculated according to formula (17). The extrusion volume of the micro-element is calculated according to formula (18). The blunt radius of the cutting edge is taken as 30 μm. The material separation angle is 50 degrees. The back angle of the cutting edge is 15 degrees. The static pressing area is calculated according to formula (19). The dynamic pressing area is calculated according to formula (20). The cutting speed of the cutting micro-element in the direction parallel to the workpiece surface is calculated according to formula (21). The total pressing area is calculated according to formula (22). As shown in the appendix. Figure 4 As shown. The instantaneous cutting force of the micro-element is calculated according to formula (23), the transformation matrix for converting the micro-element cutting force to the tool coordinate system is calculated according to formula (24), and the total cutting force of the round nose milling cutter is obtained by summing all the micro-element cutting forces according to formula (25). Then, the dynamic cutting force of the tool coordinate system is calculated according to formula (26).

[0327] Step 3: Establish the dynamic model of five-axis milling of high-strength and tough material curved parts. The dynamic equation of the system is established according to formula (27). The first mode of the system is analyzed. The mass matrix, damping matrix and stiffness matrix of the tool in physical space are constructed according to formula (28). According to formula (29), the displacement vector in physical space is transformed into modal space by modal coordinate transformation. The modal mode vector is normalized at the tool tip. The dynamic equation of decoupled modal space is established according to formula (30). The natural frequencies of the tool in the X and Y directions are 2.6kHz and 2.5kHz, respectively. The modal masses are 0.02kg and 0.02kg, respectively. The damping ratios are 7% and 10%, respectively. The dynamic cutting force calculation formula of the tool coordinate system is simplified to formula (31), and then recorded as formula (32). Substituting formula (32) into formula (30), the cutting dynamic equation (33) of five-axis milling of round nose end mill considering process damping in modal space is obtained.

[0328] Step 4: Numerical integration method to analyze the stability of multi-axis cutting. The cutting dynamics equation is transformed into a state-space form (34). The delay time is discretized, and the integral of both sides of equation (34) is obtained to form equation (35). Equation (35) is simplified by using an interpolation formula with equidistant nodes. The discrete dynamic mapping can be obtained according to the sampling time point as shown in equation (36). The state transition matrix of adjacent cutting moments is obtained according to equation (37). According to Floquet theory, if the modulus of all eigenvalues ​​of the state transition matrix is ​​less than 1, the motion of the machining system tends to be stable; if the modulus of the eigenvalues ​​of the state transition matrix is ​​equal to 1, the motion of the machining system is at critical stability; if the modulus of the eigenvalues ​​of the state transition matrix is ​​greater than 1, the machining system is in a chattering state. The spindle speed is scanned from low to high to determine the cutting stability at different cutting depths and a cutting stability lobe diagram is generated, as shown in the appendix. Figure 5 As shown, the stability judgment of five-axis milling of irregular curved surface parts is realized.

[0329] The five-axis milling dynamics modeling and stability analysis method considering process damping in this invention can realize five-axis milling dynamics modeling and stability analysis of deep cavity structure irregular curved surface parts made of high strength and toughness materials. The implementation method is convenient and can be applied to the selection of efficient and stable cutting parameters for milling irregular curved surface parts for high-end equipment.

Claims

1. A method for five-axis milling dynamics modeling and stability analysis considering process damping, characterized in that, Includes the following steps: Step 1) Construct the five-axis machining coordinate transformation equations for the workpiece and determine the tool-workpiece contact area; Step 2) Calculate the instantaneous undeformed chip thickness and shear force based on the tool posture; Step 3) Considering the process damping effect during metal cutting, analyze the extrusion volume of the workpiece material by the blunt circle of the tool tip and the flank face, and calculate the plowing force and dynamic cutting force of the tool; Step 4) Based on the instantaneous undeformed chip thickness, shearing force, tool plowing force, and dynamic cutting force, establish a five-axis milling dynamics model of the workpiece; Step 5) Analyze the tool stability during the workpiece milling process using the workpiece five-axis milling dynamics model.

2. The method for five-axis milling dynamics modeling and stability analysis considering process damping according to claim 1, characterized in that, The cutting tool is a milling cutter, with a cutting edge comprising a rounded portion and a cylindrical portion, and a tool radius of R. D The radius of the blade arc is R.

3. The method for five-axis milling dynamics modeling and stability analysis considering process damping according to claim 1, characterized in that, The steps to construct the five-axis machining coordinate transformation equations for a workpiece include: Step 1.1) Denote the machine tool coordinate system as... Let the workpiece coordinate system be . The machine tool coordinate system and workpiece coordinate system are fixed coordinate systems; the X-axis of the workpiece coordinate system is set. w Axis, Y w Axis and Z w The axes are respectively related to the X coordinate system of the machine tool. M Axis, Y M Axis and Z M The axes are in the same direction; Describe the tool coordinate system The origin is located at the center of the bottom of the tool, Z t The direction is upward along the center line of the tool axis, X t Axis, Y t The axes are respectively related to the X coordinate system of the workpiece. w Axis, Y w The axes are in the same direction; Let the feed coordinate system be... ;X F Y F and Z F This indicates the feed axis, cross feed axis, and surface normal axis; the feed coordinate system is a motion coordinate system. Step 1.2) Construct the transformation equation from the tool coordinate system to the feed coordinate system. ,Right now: （1） in, , These are the rake angle and side rake angle of the milling cutter during the machining process; Step 1.3) Construct the transformation matrix between the workpiece coordinate system and the feed coordinate system. ,Right now: （2） In the formula, , , For feed coordinate system X F The unit vectors along the axial directions are respectively related to the workpiece coordinate system X. w Axis, Y w Axis and Z w The cosine of the angle between the axes, , , For feed coordinate system Y F The unit vectors along the axial directions are respectively related to the workpiece coordinate system X. w Axis, Y w Axis and Z w The cosine of the angle between the axes, , , For feed coordinate system Z F The unit vectors along the axial directions are respectively related to the workpiece coordinate system X. w Axis, Y w Axis and Z w The cosine of the angle between the axes; Step 1.4) Construct the machine tool coordinate system To the workpiece coordinate system Transformation matrix ,Right now: （3） in, , These represent the rotation angles of the rotary axes of the five-axis milling machine; B and C represent the rotary axes of the five-axis milling machine. Step 1.5) Construct the five-axis machining coordinate system transformation equations for the workpiece. ,Right now: （4） Step 1.6) Use Unigraphics NX software to generate the contact surface between the workpiece and the tool at the current tool position during the milling process, and perform Boolean operation on the workpiece 3D model and the feasible contact surface of the tool to obtain the tool-workpiece contact area; Starting from the origin of the tool coordinate system, generate planes with equal spacing along the tool axis. Intersect these planes with the tool-workpiece contact area to obtain the intersection lines of the planes with the tool-workpiece contact area at different axial heights. Project the intersection line onto the workpiece coordinate system X. t O t Y t The plane is used to obtain the projected curve segment; the line connecting the endpoint of the projected curve segment and the origin of the workpiece coordinate system is measured and intersected with the Y-axis. t The included angle between the axes is the entry angle and exit angle of the tool-workpiece contact area at the tool position point at that axial height; the entry angle and exit angle are used to determine whether the cutting edge of the tool is in a cutting state during the machining process.

4. The method for five-axis milling dynamics modeling and stability analysis considering process damping according to claim 1, characterized in that, The steps for calculating the instantaneous undeformed chip thickness and shear force include: Step 2.1) Establish the tool geometry model; in the tool geometry model, the tool radius is R. D The radius of the cutting edge arc is R, and the local radius of the tool is r. i ; Wherein, the local radius r of the tool i As shown below: （5） In the formula, M s To increase the axial depth The number of discrete infinitesimal elements along the axial direction; parameters ; Step 2.2) Calculate the radial contact angle between the workpiece and the tool during the cutting process. ,Right now: （6） In the formula, N t This refers to the number of teeth on the cutting tool. Main spindle speed; The helix angle of the cutting tool; The tooth angle is equal in pitch. t represents time; Step 2.3) Calculate the axial contact angle between the workpiece and the tool during the cutting process. and the unit outward normal vector of the blade envelope at the point of contact. ,Right now: （7） （8） Step 2.4) Calculate the instantaneous, undeformed static chip thickness during the cutting process. With dynamic chip thickness ,Right now: （9） （10） （11） In the formula, The angle between the teeth of the milling cutter The resulting cutting time delay; ;d represents the relative vibration displacement between the tool and the workpiece in the tool coordinate system; This refers to the feed per tooth as a cutting parameter. , , These represent the relative vibration displacements of the tool and workpiece along the X, Y, and Z axes in the tool coordinate system, respectively. Step 2.5) Considering the tool posture changes and regenerative chatter effect during the cutting process, calculate the instantaneous undeformed chip thickness at the micro-element of the cutting edge. ,Right now: （12） Step 2.6) Calculate the infinitesimal shear force. ,Right now: （13） In the formula, These represent the radial, tangential, and axial directions, respectively. These are the shear force coefficients in three directions; The width of the micro-element cutting edge; It is a window function used to determine whether the milling cutter cutting edge is in a cutting state; Among them, the window function As shown below: （14） Tool angle when the cutting teeth cut into the workpiece The angle of the cutting tool when the cutting teeth cut the workpiece They are shown below: （15） （16） In the formula, a r D is the radial depth of cut, and D is the tool diameter. This represents the radial penetration rate.

5. The method for five-axis milling dynamics modeling and stability analysis considering process damping according to claim 4, characterized in that, Plowing force of the blade As shown below: （17） In the formula, These are the plowing force coefficients in three directions; Wherein, the extrusion volume of the i-th infinitesimal element on the j-th cutting edge at time t. As shown below: （18） In the formula, Let be the compression area of ​​the i-th micro-element on the j-th cutting edge at time t; Static pressing area and dynamic pressing area They are shown below: （19） （20） In the formula, It is the radius of the cutting edge blunt circle; It is the material separation angle; It is the rake angle of the cutting edge; The length of the wear band on the flank face of the tool; It is the cutting speed of the i-th cutting element of the j-th cutting edge in the direction parallel to the workpiece surface; Vibration speed of the cutting tool As shown below: （21） In the formula, , , This indicates that the i-th cutting element of the cutting edge at time t is located in the tool coordinate system X. w Y w and Z w The derivative of the axial displacement.

6. The method for five-axis milling dynamics modeling and stability analysis considering process damping according to claim 1, characterized in that, The dynamic cutting forces of the tool are shown below: （22） In the formula, , , , , , , , , This is the shear force coefficient; , , , , , , , , This is the shear force coefficient of the plow; d represents the relative vibration displacement between the tool and the workpiece in the tool coordinate system; , , The dynamic cutting force of the tool in the X, Y, and Z directions; , , This indicates that the i-th cutting element of the cutting edge at time t is located in the tool coordinate system X. w Y w and Z w The derivative of the axial displacement; N t This represents the number of teeth on the cutting tool.

7. The method for five-axis milling dynamics modeling and stability analysis considering process damping according to claim 1, characterized in that, The steps to establish a five-axis milling dynamics model of a workpiece include: Step 4.1) Simplify the milling system into a mass-spring-damped system with multiple modes in the X and Y directions. The dynamic equations of this mass-spring-damped system are as follows: （23） In the formula, M, C, and K are the mass matrix, damping matrix, and stiffness matrix of the cutting tool in physical space, respectively. , and These represent the acceleration vector, velocity vector, and displacement vector of the tool vibration in physical space, respectively. F(t) is the dynamic cutting force vector acting on the tool; , This represents the displacement of the tool vibration in the X and Y directions in physical space; Step 4.2) Construct the mass matrix M, damping matrix C, and stiffness matrix K of the tool in physical space, that is: （24） In the formula, , , , The mass of the cutting tool in physical space; , , , Damping of the cutting tool in physical space; , , , The stiffness of the cutting tool in physical space; Step 4.3) Use modal coordinate transformation to transform the physical space displacement vector to modal space, resulting in: （25） In the formula, U is the modal matrix. Q(t) represents the tool displacement vector in modal space. ; , This represents the displacement of the tool in the X and Y directions within the modal space; Step 4.4) Normalize the tool displacement vector Q(t) in modal space at the tool tip and establish the cutting dynamics equations decoupled in modal space, resulting in: （26） In the formula, the mass vector Damping vector stiffness vector ,vector ; , These are mass parameters in modal space; , These are the damping parameters in modal space; , These are the stiffness parameters in modal space; Step 4.5) Construct a simplified equation for the dynamic cutting force of the tool, namely: （27） In the formula, , These are the shear force coefficient matrix and the plowing shear force coefficient matrix; This represents the tool displacement vector in modal space; Step 4.6) Substitute formula (27) into formula (25) to construct the five-axis milling dynamics model of the workpiece, that is: （28）。 8. The method for five-axis milling dynamics modeling and stability analysis considering process damping according to claim 7, characterized in that, The steps for analyzing tool stability during workpiece milling using a five-axis milling dynamics model include: Step 5.1) Let , The five-axis milling dynamics model of the workpiece is transformed into state-space equations, resulting in: （29） In the formula, T represents the period; the matrix... ;matrix ;matrix ; , , and This refers to the dynamic cutting force coefficient. Step 5.2) The cutting time lag Discretize the equation into m parts, and solve by integrating both sides of the state-space equation to obtain the following result. （30） in, , for The state value at any given time; This is a solution to the state-space equations; For a moment; Step 5.3) Using the interpolation quadrature formula (30) with equidistant nodes, the discrete dynamic mapping equation is obtained, i.e.: （31） In the formula, the matrix ,matrix ;matrix I is a unit vector; This refers to the axial depth of cut. , , Indicates time; Step 5.4) Construct the state transition matrix between adjacent cutting moments of the cutting tool. ,Right now: （32） Step 5.5) Calculate the state transition matrix between adjacent cutting moments of the cutting tooth. The eigenvalues ​​are: if the modulus of all eigenvalues ​​is less than 1, the tool motion tends to be stable; if the modulus of all eigenvalues ​​is equal to 1, the tool motion is in a critical stable state; if the modulus of any eigenvalue is greater than 1, the machining system is in a chattering state.

9. The method for five-axis milling dynamics modeling and stability analysis considering process damping according to claim 1, characterized in that, The workpiece is an irregularly shaped curved surface part.

10. A method for five-axis milling dynamics modeling and stability analysis considering process damping as described in claim 1, characterized in that, The workpiece material is a high-strength, tough, and difficult-to-machine metal, including but not limited to stainless steel and titanium alloy.

Images