A method for analyzing and predicting the surface topography of peripheral milling

Abstract

The present invention relates to the technical field of milling machining simulation, and specifically discloses a method for analyzing and predicting the surface topography of peripheral milling. First, the contact mode between the workpiece and the fixture is equivalently transformed into a linear spring-damper system, and the vibration differential equation of the workpiece during the milling process is derived by using the energy method; through the coordinate transformation method, a modal analysis solution method for the vibration differential equation with constraint conditions is proposed to realize the prediction of the deviation of the workpiece position; then, the motion trajectory equation of the cutting edge during the peripheral milling process is established, and combined with the position deviation of the workpiece and the motion trajectory equation of the cutting edge, a surface topography discretization algorithm for the workpiece surface topography is proposed. The results show that this method can accurately predict the surface topography and roughness of peripheral milling.

Description

Technical Field

[0001] The present invention relates to the technical field of milling machining simulation, and particularly to a method for analyzing and predicting the surface topography of peripheral milling. Background Art

[0002] In recent years, difficult-to-machine materials have been increasingly widely used in the aviation industry. The quality and machining efficiency of parts made of difficult-to-machine materials are issues that must be considered in the machining manufacturing industry. The surface topography characteristics of the workpiece after cutting will affect the physical, mechanical properties and service life of the part. The surface quality of the machined part is generally measured by two important indicators: machining accuracy and surface topography. The surface topography of the machined workpiece refers to the various microscopic geometric forms of different shapes and sizes remaining on the workpiece surface due to factors such as tool and workpiece vibration during the machining process, and it is one of the important indicators for measuring the surface quality of the part. Machining is a dynamic process, which will inevitably cause the workpiece to vibrate during machining, resulting in the position deviation of the workpiece, and further affecting the formation of the machined surface topography.

[0003] Currently, most of the simulations of the surface topography of the milling workpiece basically start from the perspective of the tool. There are few studies on simulating the surface topography problem of milling machining by combining the influence of the clamping layout vibration and the tool structure. Summary of the Invention

[0004] The purpose of the present invention is to provide a method for analyzing and predicting the surface topography of peripheral milling to solve the problems raised in the above background art.

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

[0006] A method for analyzing and predicting the surface topography of peripheral milling includes the following steps:

[0007] Step 1: Establish a workpiece vibration model;

[0008] Step 2: Obtain the vibration differential equation of the workpiece by the energy method;

[0009] Step 3: Decouple and solve the vibration differential equation;

[0010] Step 4: Solve the cutting edge motion trajectory equation;

[0011] Step 5: Determine the simulation method of the peripheral milling surface topography;

[0012] Step 6: Simulate the peripheral milling surface topography by the discrete method in MATLAB.

[0013] As a further solution of the present invention: In step one, Fj is the resultant external force. The i-th positioning element is regarded as a spring-damping system ignoring mass in the normal direction ni = [nxi, nyi, nzi]T and two tangential directions ti = [txi, tyi, tzi]T, ai = [axi, ayi, azi]T. The stiffness and damping coefficients of the spring-damping elements in the three directions are kni, kti, kai and cni, cti, cai respectively. rw = [xw, yw, zw]T and Θw = [αw, βw, γw]T are the position and orientation of the workpiece coordinate system XwYwZw relative to the global coordinate system XYZ respectively. rw = [xw, yw, zw]T is the coordinate of any point P on the workpiece in the workpiece coordinate system. The attitude r = [x, y, z]T of point P at any moment in the global coordinate system can be expressed as:

[0014] r = r w + T(Θ w )r w (22);

[0015] s and c are sine sin and cosine cos respectively.

[0016] As a further solution of the present invention: The specific steps in step two are as follows:

[0017] S01: Solve the kinetic energy T of a single particle P on the workpiece;

[0018]

[0019] In the formula where is the linear velocity of the workpiece, is the angular velocity of the workpiece, and the mass matrix m is the mass of the workpiece, J x = ∫(y 2 + z 2 )dm, J y = ∫(x 2 + z 2 )dm, J z = ∫(y 2 + x 2 )dm are the moments of inertia, J xy = ∫xydm, J yz = ∫yzdm, J zx = ∫zxdm, and (x, y, z) are the coordinates of point P in the workpiece coordinate system;

[0020] S02: Solve the potential energy U of the workpiece:

[0021]

[0022] In the formula is the workpiece position offset, stiffness matrix is the local stiffness of the contact between the ith positioning element and the workpiece, where the tensile and compressive stiffness Torsional stiffness Bending stiffness and k xβi =k xi z i , k xγi =k xi y i , k yγi =k yi x i , k yαi =k yi z i , k zαi =k zi y i , k zβi =k zi x i , k βγi =k xi y i z i , k γαi =k yi z i x i , k αβi =k zi y i x i ;

[0023] S03: Calculate the energy consumption D of the workpiece-fixture system:

[0024]

[0025] Damping Matrix The local contact damping between the i-th positioning element and the workpiece is Tensile and compressive damping Torsional damping Bending damping and C xβi =C ix z i , C xγi =C ix y i , C yγi =C iy x i , C yαi =C iy z i, C zαi =C iz y i , C zβi =C iz x i , C βγi =C ix y i z i , C γαi =C iy z i x i , C αβi =C iz y i x i ;

[0026] S04: Solve vibration differential equations;

[0027] S05: Set the constraints of the vibration differential equation.

[0028] As a further solution of the present invention: the calculation method of S04: solving the vibration differential equation is as follows:

[0029] Differentiating the system's kinetic energy, potential energy, and energy loss separately yields relationships between acceleration, displacement, and velocity.

[0030] Differentiating the velocity in equation (2) and then differentiating the time yields

[0031]

[0032] Differentiating the workpiece position offset in equation (3) yields

[0033]

[0034] Finally, differentiating the velocity in equation (4) yields:

[0035]

[0036] According to Newton's second law, the vibration differential equation of the workpiece in the clamping layout system can be obtained, namely:

[0037]

[0038] Where x is the position offset of the workpiece, and They represent the velocity and acceleration of the workpiece respectively, and F(t) is the resultant external rotation of the system.

[0039] As a further solution of the present invention: specifically in S05: the purpose of workpiece clamping is to apply a clamping force to the workpiece to ensure that the position of the workpiece relative to the tool obtained during positioning remains unchanged. Therefore, the support reaction force of the i-th positioning element in Equation (8) should satisfy

[0040] k N ψdq w ≥0 (30)

[0041] wherein (x i ,y i ,z i ) is the position of the positioning element in the workpiece coordinate system, k n =[k ni ,0,0]。

[0042] As a further solution of the present invention: the specific steps in the third step are as follows:

[0043] S01: Convert the coordinates in the vibration differential equation to modal coordinates and decouple them. Let AN be the regular mode shape matrix of Equation (8), and Xp be the principal coordinates of the vibration system. Then there is Using the principal coordinates for linear transformation, we can obtain Denote Then Equation (8) can be further transformed into:

[0044]

[0045] The decoupled form of Equation (10) is

[0046]

[0047] wherein, 1≤i≤6,

[0048] S02: Solve each principal coordinate and obtain the solution of the equation. Each principal coordinate can be solved separately using the Duhamel integral, that is

[0049]

[0050] Therefore, the solution of the vibration differential equation is:

[0051] dq w =A N X p

[0052] s.t.

[0053] k N ψdq w ≥0 (34)。

[0054] As a further solution of the present invention: The specific steps for solving the cutting edge motion trajectory equation in step four are as follows:

[0055] S01: Solve the coordinates of any point on the cutter tooth in the tool coordinate system. The schematic diagram of the peripheral milling cutting edge motion is shown in the appendix. Figure 2 As shown, the milling cutter mills the plane ABCD of the workpiece. XYZ and XwYwZw are the global coordinate system and the workpiece coordinate system respectively, while XcYcZc and XsYsZs are the tool coordinate system and the spindle coordinate system respectively. For convenience, when establishing the coordinate system, XYZ should coincide with XwYwZw. R is the radius of the milling cutter, γ is the helix angle, e is the tool eccentricity, w is the rotational angular velocity, and α is the initial phase angle. If the tool has z cutter teeth, the initial angle of the j-th cutter tooth can be expressed as:

[0056]

[0057] The coordinates of any point Pt on the j-th cutting edge of the milling cutter in XcYcZc are:

[0058]

[0059] S02: Solve the coordinates of any point on the cutter tooth in the spindle coordinate system. Assume that the Xs axis of XsYsZs coincides with the Xc axis of XcYcZc. Then the position and direction of XcYcZc relative to XsYsZs at any time t are The coordinates of point Pt in XsYsZs should be:

[0060]

[0061] When the tool mills in the conventional direction, w takes a positive value; when milling in the climb milling direction, w takes a negative value.

[0062] S03: Solve the coordinates of any point on the cutter tooth in the spindle coordinate system. At any time t, the spindle moves Vft along the feed direction Y. If XsYsZs coincides with XYZ at the initial moment, then at time t, the position and direction of XsYsZs relative to XYZ are rs = [0, Vft, 0]T, Θs = [0, 0, 0]T. The coordinates of point Pt on the j-th cutting edge in XYZ are

[0063]

[0064] As a further solution of the present invention: The specific steps for determining the simulation method of the peripheral milling surface topography in step five are as follows:

[0065] S01: Determine the position change of any point on the workpiece during the machining process. At time t, if the position of the workpiece changes by dqw due to the vibration of the clamping layout, the position change of any point on the workpiece should be:

[0066] dr = Edq w (39)

[0067] In the formula:

[0068]

[0069] S02: Determine the cutting position. When the point Pt on the cutting tooth is within the range of the cutting area ABCD and along the direction of the depth of cut, if and only if the coordinates of the Pt point are between the theoretical cutting plane and the plane to be cut, it indicates that the Pt point participates in cutting. As shown in the appendix Figure 3 As shown, according to equations (1, 17, 18), the coordinates rP = [xP, yP, zP] of point P on the cutting plane should be

[0070]

[0071] In the formula:

[0072]

[0073] As a further solution of the present invention: The specific simulation in step six is as follows: S01: Discretize the plane to be cut. The workpiece is evenly divided into m×n grids with step sizes ΔY and ΔZ respectively, and the coordinate values of each grid node are stored in a matrix as X(i, j), Y(i, j), Z(i, j), where i = 1, 2,..., m + 1, j = 1, 2,..., n + 1;

[0074] S02: Discretize the cutting time and the cutting edge. Set the time discretization step size Δt and the number of discrete points l of the cutting edge according to the workpiece grid accuracy. Generally, take Δt = min{ΔY, ΔZ} / wR, l = hT / min{ΔY, ΔZ}, where hT is the effective length of the cutting part of the milling cutter, to ensure that the projection of the discrete microelement of the cutting edge in the machining plane does not exceed the grid spacing of the workpiece, and at most one workpiece grid point is swept through within a unit time step;

[0075] S03: Initialize the cutting time t = 0;

[0076] S04: Calculate the clamping layout vibration at the current time, and obtain the coordinates r(i, j) = [X(i, j), Y(i, j), Z(i, j)]T of each grid node on the cutting plane;

[0077] S05: Initialize the cutting edge number j = 1;

[0078] S06: Initialize the discrete point number k = 1;

[0079] S07: Determine whether the discrete point is within the machining area. According to Equation (17), calculate the coordinate values (xk, yk, zk) of the discrete point Pk on the cutting edge at the current time, and determine whether it is within the plane to be machined. If so, go to S08; otherwise, go to S10.

[0080] S08: Search for the grid node of the plane to be machined that is closest to the discrete point on the cutting edge. When the position (yk, zk) of the discrete point coincides with or is closest to the position (Y(i, j), Z(i, j)) of the grid point, then the discrete point corresponds to the grid point.

[0081] S09: Determine the cutting condition of the discrete point on the cutting edge. Compare the relationship between the nearest grid node X(i, j) and the xk of the discrete point Pk on the cutting edge. If xk < X(i, j), it means that the tool has cut into the workpiece, and update the stored value of X(i, j) with the value of xk; otherwise, do nothing.

[0082] S10: Determine whether it is the last discrete point. If so, go to S11; otherwise, calculate the next discrete point k = k + 1, and go to S07.

[0083] S11: Determine whether it is the last cutting edge. If so, go to S12; otherwise, calculate the next cutting edge j = j + 1, and go to S06.

[0084] S12: Determine whether it is the last moment. If so, end the calculation process, and the stored X(i, j), Y(i, j), Z(i, j) form the cutting surface topography; otherwise, calculate the next moment, and go to S04.

[0085] Compared with the prior art, the beneficial effects of the present invention are as follows: This application accurately predicts the surface topography and roughness of peripheral milling. By combining theoretical analysis, modeling methods, and experimental testing means, comprehensively considering the influence of workpiece vibration caused by clamping layout and tool structure on the surface topography, the purpose of predicting the milling surface topography and roughness is finally achieved, effectively ensuring the machining quality. Brief Description of the Drawings

[0086] Figure 1 is the workpiece vibration model established by the present invention;

[0087] Figure 2 is the schematic diagram of the movement of the peripheral milling cutting edge;

[0088] Figure 3 Schematic diagram of surface formation in the coordinate plane XY;

[0089] Figure 4 is the "3-2-1" clamping scheme diagram;

[0090] Figure 5It is a diagram of the workpiece position change in the x - direction without considering damping;

[0091] Figure 6 It is a diagram of the workpiece position change in the x - direction considering damping;

[0092] Figure 7 It is a three - dimensional topography diagram of climb milling without spindle eccentricity;

[0093] Figure 8 It is an experimental and simulation diagram of climb milling without spindle eccentricity;

[0094] Figure 9 It is an experimental and simulation diagram of down milling with spindle eccentricity. Detailed implementation mode

[0095] Example 1

[0096] Appendix Figure 4 It is a simplified diagram of milling a groove on the surface of a cuboid workpiece. The outer contour dimensions of the workpiece are 220mm×122mm×112mm. The Young's modulus and Poisson's ratio of the workpiece are Ew = 70GPa and νw = 0.334 respectively; the gravity of the workpiece is Fgrav = [0, 59.73N, 0]T, and the position of the centroid is rgrav = [110mm, 56mm, 61mm]T. The machining parameters of the cutting tool used for milling the groove are shown in Table 1. The machining force is Fmach = [-131N, -55N, 232N]T N, the machining torque is Mmach = [0, 2.77Nm, 0]T, the dynamic external load is F(t) = [801N, 114.73N, -872N, -9.1024Nm, 22.08 - 0.3867t Nm, -0.0917t - 0.3242Nm]T, and the cutting time t is 0 ≤ t ≤ 132. There are clamping forces F1 = 640N and F2 = 670N at r1 = [110mm, 60mm, 0]T and r2 = [220mm, 60mm, 60mm]T. The positions and unit normal vectors of each positioning element are shown in Table 2. The Young's modulus and Poisson's ratio of the positioning element are Ef = 207GPa and νf = 0.292 respectively, and the damping coefficient cf = 7000Ns / m. Perform peripheral milling topography simulation on one side of the workpiece;

[0097] Table 1 Peripheral milling machining parameters;

[0098]

[0099]

[0100] Table 2 Positions and normal vectors of positioning elements;

[0101]

[0102] A method for analyzing and predicting the surface topography of peripheral milling, comprising the following steps: Step 1: Calculate the stiffness matrix, mass matrix, and damping matrix based on data such as the workpiece weight and the elastic modulus of the positioning element as follows:

[0103]

[0104]

[0105]

[0106] Step 2: Decouple and solve the vibration differential equation of the workpiece. The specific steps are as follows:

[0107] S01: Obtain the normal mode matrix AN through the eig(K, M) function in MATLAB.

[0108] S02: Decouple to obtain 6 second-order differential equations as follows:

[0109]

[0110] where 1 ≤ i ≤ 6;

[0111] S03: Use the Duhamel integral to solve each principal coordinate respectively, that is

[0112]

[0113] S04: The solution of the vibration differential equation is;

[0114] dq w =A N X p

[0115] s.t.

[0116] k N ψdq w ≥0 (48)

[0117] The position offset of the workpiece changes with time as shown in the appendix Figure 5 , Figure 6 s shown;

[0118] Step 3: Set the milling parameters in MATLAB according to Table 2. The number of tool teeth j = 3, the tool radius R = 4 mm, the helix angle γ = 30°, the rotational speed w = 300π / 30 (positive for down milling, negative for up milling), the feed speed Vft = 310 / 60 mm / s, and the eccentricity e = 0.01 (e = 0 when eccentricity is not considered);

[0119] Step 4: Set the mesh size of the discrete surface of the workpiece as ΔY = ΔZ = 0.05 mm, and evenly divide the milling surface into 120×120 meshes. Use matrices X(i,j), Y(i,j), and Z(i,j) to store the coordinate values of each mesh node;

[0120] Step 5: Calculate the discrete time interval Δt = min{ΔY, ΔZ} / wR = 3.98×10-4 s, and the number of discrete points on the cutting edge l = hT / min{ΔY, ΔZ} = 120;

[0121] Step 6: Simulate the surface topography of peripheral milling. The specific implementation steps are as follows:

[0122] S01: Initialize the cutting time t = 0.

[0123] S02: Calculate the clamping layout vibration at the current time. According to the formula dr = Edq w Obtain the coordinate r(i,j) = [X(i,j), Y(i,j), Z(i,j)]T of each mesh node on the cutting plane.

[0124] S03: Initialize the cutting edge number j = 1.

[0125] S04: Initialize the discrete point number k = 1.

[0126] S05: Determine whether the discrete point is within the machining area. According to the formula Calculate the coordinate values (xk, yk, zk) of the discrete point Pk on the cutting edge at the current time, and determine whether it is within the plane to be machined. If so, go to S06; otherwise, go to S08.

[0127] S06: Search for the mesh node on the plane to be machined that is closest to the discrete point on the cutting edge. When the position (yk, zk) of the discrete point coincides with or is closest to the position (Y(i,j), Z(i,j)) of the mesh point, then the discrete point corresponds to the mesh point.

[0128] S07: Determine the cutting condition of the discrete point on the cutting edge. Compare the relationship between the nearest mesh node X(i,j) and xk of the discrete point Pk on the cutting edge. If xk < X(i,j), it means that the tool has cut into the workpiece, and update the stored value of X(i,j) with the value of xk; otherwise, do nothing.

[0129] S08: Determine whether it is the last discrete point? If so, go to S09; otherwise, calculate the next discrete point k = k + 1, and go to S05.

[0130] S09: Determine whether it is the last cutting edge? If so, go to S10; otherwise, calculate the next cutting edge j = j + 1, and go to S04.

[0131] S10: Determine whether it is the last moment? If so, end the calculation process, and the stored X(i,j), Y(i,j), and Z(i,j) form the cutting surface topography. The three-dimensional topography map of climb milling without spindle eccentricity is shown in the appendix Figure 7 ; otherwise, calculate the next moment t = t + Δt, and go to S02;

[0132] Step Seven: Compare the experimental and simulation topography results. The experimental and simulation diagrams of climb milling without spindle eccentricity are shown in the appendix Figure 8 ; the experimental and simulation diagrams of up milling with spindle eccentricity are shown in the appendix Figure 9 . By comparing the actual machined surface and the simulation topography results, it can be seen that whether there is spindle eccentricity or not, the calculation results of the surface topography discrete algorithm are in good agreement with the actual machining situation;

[0133] Step Eight: Compare the measured roughness with the roughness of the climb milling and up milling simulation surfaces. The measured roughness values of climb milling and up milling are 4.838 μm and 5.438 μm respectively. Considering the influence of workpiece vibration caused by the clamping layout and the tool structure on the surface topography, the surface roughness errors of climb milling and up milling are 4.845 μm and 5.463 μm respectively, and the surface roughness errors are only 1.45% and 3.65%.

Claims

1. A face milling surface topography analysis and prediction method, characterized in that It includes the following steps: It includes the following steps: Step 1: Establish a workpiece vibration model; Step 2: Obtain the vibration differential equation of the workpiece by the energy method; Step 3: Decouple and solve the vibration differential equation; Step 4: Solve the cutting edge motion trajectory equation; Step 5: Determine the simulation method for the peripheral milling surface topography; Step 6: Simulate the peripheral milling surface topography by the discrete method in MATLAB; The specific steps for solving the cutting edge motion trajectory equation in Step 4 are as follows: S01: Solve the coordinates of any point on the cutting tooth in the tool coordinate system. The milling cutter mills the plane ABCD of the workpiece. XYZ and XwYwZw are the global coordinate system and the workpiece coordinate system respectively, while XcYcZc and XsYsZs are the tool coordinate system and the spindle coordinate system respectively. For convenience, when establishing the coordinate system, XYZ should coincide with XwYwZw. R is the radius of the milling cutter, γ is the helix angle, e is the tool eccentricity, w is the rotational angular velocity, and α is the initial phase angle. If the tool has z cutting teeth, the initial angle of the jth cutting tooth can be expressed as: The coordinates of any point Pt on the jth cutting edge of the milling cutter in XcYcZc are: S02: Solve the coordinates of any point on the cutter tooth in the spindle coordinate system. Assume that the Xs axis of XsYsZs coincides with the Xc axis of XcYcZc. Then, the position and orientation of XcYcZc relative to XsYsZs at any time t are The coordinates of point Pt in XsYsZs should be: When the tool mills with the feed direction, w takes a positive value; when milling against the feed direction, w takes a negative value. S03: Solve the coordinates of any point on the cutting tooth in the spindle coordinate system. At any moment t, the spindle moves Vft along the feed direction Y. If XsYsZs coincides with XYZ at the initial moment, then at the moment t, the position and direction of XsYsZs relative to XYZ are rs = [0, Vft, 0]T, Θs = [0, 0, 0]T. The coordinates of point Pt on the jth cutting edge in XYZ are:

2. The method for analyzing and predicting the surface topography of peripheral milling according to claim 1, wherein The specific steps for determining the simulation method for the peripheral milling surface topography in Step 5 are as follows: S01: Determine the position change of any point on the workpiece during the machining process. At the moment t, if the position of the workpiece changes by dqw due to the vibration caused by the clamping layout, the position change of any point on the workpiece should be: dr = Edq w (18) In the formula: S02: Judge the cutting position. The point Pt on the cutting tooth is within the cutting area ABCD. Along the direction of the depth of cut, when and only when the coordinates of point Pt are between the theoretical cutting plane and the to-be-cut plane, the coordinates rP = [xP, yP, zP] of point P on the cutting plane should be: In the formula 3. The peripheral milling surface topography analysis and prediction method according to claim 1, characterized in that The specific simulation in Step 6 is as follows: S01: Discretize the to-be-cut plane. The workpiece is evenly divided into m×n grids with step sizes △Y and △Z respectively. The coordinate values of each grid node are stored in a matrix as X(i, j), Y(i, j), Z(i, j), where i = 1, 2,..., m + 1, j = 1, 2,..., n + 1; S02: Discretize the cutting time and the cutting edge. Set the time discretization step size △t and the number of discrete points 1 of the cutting edge according to the workpiece grid accuracy. Generally, take △t = min{△Y, △Z} / wR, 1 = hT / min{△Y, △Z}, where hT is the effective length of the cutting part of the milling cutter, to ensure that the projection of the discrete microelement of the cutting edge in the machining plane does not exceed the grid spacing of the workpiece, and at most one workpiece grid point is swept through within a unit time step; S03: Initialize the cutting time t = 0; S04: Calculate the clamping layout vibration at the current time to obtain the coordinates r(i, j) = [X(i, j), Y(i, j), Z(i, j)]T of each grid node on the cutting plane; S05: Initialize the cutting edge number j = 1; S06: Initialize the discrete point number k = 1; S07: Determine whether the discrete point is within the machining area. According to Equation (17), calculate the coordinate values (xk, yk, zk) of the discrete point Pk on the cutting edge at the current time, and determine whether it is within the plane to be machined. If so, go to S08; otherwise, go to S10; S08: Search for the grid node on the plane to be machined that is closest to the discrete point on the cutting edge. When the position (yk, zk) of the discrete point coincides with or is closest to the position (Y(i, j), Z(i, j)) of the grid point, then the discrete point corresponds to the grid point; S09: Determine the cutting condition of the discrete point on the cutting edge. Compare the relationship between the X(i, j) of the closest grid node and the xk of the discrete point Pk on the cutting edge. If xk < X(i, j), it means that the tool has cut into the workpiece, and update the stored value of X(i, j) with the value of xk; otherwise, do nothing; S10: Determine whether it is the last discrete point. If so, go to S11; otherwise, calculate the next discrete point k = k + 1 and go to S07; S11: Determine whether it is the last cutting edge. If so, go to S12; otherwise, calculate the next cutting edge j = j + 1 and go to S06; S12: Determine whether it is the last moment. If so, end the calculation process, and the stored X(i, j), Y(i, j), Z(i, j) form the cutting surface topography; Otherwise, calculate the next moment and go to S04.

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