A Digital Twin Method for Thermal Expansion Deformation at the Contact Position between an End Milling Cutter and a Workpiece

Abstract

The present invention belongs to the technical field of end milling machining, and particularly relates to a digital twin method for thermal expansion deformation at the contact position between an end milling cutter and a workpiece. It is characterized in that physical parameters of the end milling workpiece are introduced into a finite element model to obtain time series data of the temperature and thermal expansion deformation of the temperature monitoring points. The MLR method is introduced to construct an empirical mathematical model, and after further optimization, the final model of the end milling process is obtained to realize the prediction and control of the temperature and deformation of the monitoring points. The beneficial effects of the present invention are as follows: 1) It solves the problem of real-time monitoring of the thermal expansion deformation at the contact position between the cutter and the workpiece surface during the end milling process of the workpiece; 2) It solves the problem of various disturbance factors during the end milling process; 3) It is beneficial to improve the working efficiency of predicting the thermal expansion deformation at the contact position between the cutter and the workpiece surface during the end milling process and improve the quality of the machined surface of the end milling workpiece.

Description

Technical Field

[0001] The present invention belongs to the technical field of end milling machining, and particularly relates to a digital twin method for thermal expansion deformation at the contact position between an end milling cutter and a workpiece. Background Art

[0002] The surface finish and dimensional accuracy of an end-milled workpiece are crucial for determining the quality of the part. The quality of the workpiece is affected by many factors, such as tool deflection, vibration, and thermal expansion of the workpiece. Among them, environmental factors emphasize the importance of thermal problems. The heat generated during the cutting process mainly acts on the workpiece, tool, and chips, and the resulting thermal deformation has an important impact on the machining accuracy and surface quality of the workpiece. Therefore, evaluating the temperature and thermal deformation of the workpiece has an important impact on controlling machining accuracy and improving part quality.

[0003] Chinese invention with application number 201711366436.2 discloses a temperature prediction method for the cutting area of a lead screw in whirling milling. First, establish transient undeformed chip thickness, width, and area models for the first and second stages of lead screw whirling milling cutting; then, model the transient heat source area in the cutting area of lead screw whirling milling: mainly including the shear transient heat source width model and area model in the first deformation zone and the tool-chip contact friction transient heat source area model in the second deformation zone; finally, establish a temperature model for the cutting area of lead screw whirling milling, and substitute the established models regarding the undeformed chip and the heat source area into the temperature model for solution, and finally obtain the transient temperature distribution of the workpiece, chip, and tool in the cutting area. There have been many years of research on the process simulation combining thermal measurement and finite element method. By combining the temperature history curve of finite element simulation with the sensitivity evaluation of process changes, appropriate measurement points can be determined. However, the current disadvantage of finite element analysis is that real-time calculation is very time-consuming. In traditional small-batch end milling production, the workpiece shapes and machining conditions are often different. Therefore, it is difficult to equivalent finite element analysis into a simplified model and apply it to the thermal analysis of various workpiece shapes to accurately predict the temperature at the tool contact position. Summary of the Invention

[0004] The purpose of the present invention is to provide a digital twin method for thermal expansion deformation at the contact position between an end milling cutter and a workpiece, to overcome the deficiencies of the prior art, solve the problem that process disturbances occur during end milling of a machined workpiece, thermal expansion deformation occurs at the contact position between the tool and the workpiece surface, resulting in difficulty in directly measuring the workpiece deformation, and achieve accurate prediction of the temperature at the contact position between the workpiece and the tool during end milling.

[0005] To achieve the above technical objectives, the present invention is implemented by the following solutions.

[0006] A digital twin method for thermal expansion deformation at the contact position between an end milling cutter and a workpiece, characterized in that physical parameters of the workpiece in end milling are introduced into a finite element model to obtain time series data of the temperature and thermal expansion deformation at temperature monitoring points, the MLR method is introduced to construct an empirical mathematical model, and after further optimization, the final optimized empirical mathematical model in the end milling process is obtained to realize the prediction and control of the temperature and deformation at the monitoring points, specifically including the following steps:

[0007] 1) Determine the physical parameters of the workpiece in end milling and set boundary conditions according to the machining process parameters;

[0008] 2) Adopt the finite element (FEM) calculation method to establish a finite element model of the workpiece to be machined, and complete the numerical simulation of the finite element model of the end milling process;

[0009] 3) Extract the time series values of the temperature at the temperature monitoring points and the time series data of the thermal expansion deformation at the contact position between the cutter and the workpiece surface, complete the simulation calculation of the entire end milling process, and output the time series data of the temperature at each temperature monitoring point and the time series data of the thermal expansion deformation at the contact position between the cutter and the workpiece;

[0010] 4) Use multiple linear regression (MLR) combined with the FEM calculation output data to establish an initial empirical mathematical model of the time series data of the temperature at the monitoring point position and the time series data of the thermal expansion deformation at the contact position between the cutter and the workpiece, and solve the goodness of fit R of the initial empirical mathematical model 2 Compare with the standard value of the goodness of fit to complete the first verification;

[0011] 5) Use the Akaike information criterion (AIC) to exclude the monitoring points with the same variation law of the temperature time series data;

[0012] 6) Judge the contribution degree of each temperature monitoring point according to the p value, exclude the monitoring points with lower contribution degree, determine the number of temperature measurement points in the actual end milling process based on the optimized empirical mathematical model, and solve the goodness of fit R of the optimized empirical mathematical model 2 Compare with the standard value of the goodness of fit to complete the second verification and obtain the optimized empirical mathematical model.

[0013] Compared with the prior art, the beneficial effects of the present invention are as follows: 1) It solves the problem that it is difficult to monitor the thermal expansion deformation at the contact position between the tool and the workpiece surface in real time during the end milling process of the workpiece in the prior art; 2) It solves the problem that it is difficult to consider various disturbance factors during the end milling process when simply predicting the thermal expansion deformation at the contact position between the tool and the workpiece surface by using the finite element method. 3) The use of AIC and p-value judgment criteria can further optimize the number and position of temperature monitoring points, which is beneficial to improving the working efficiency of predicting the thermal expansion deformation at the contact position between the tool and the workpiece surface during the end milling process. 4) Based on the constructed optimized empirical mathematical model and combined with the actual temperature monitoring time series data during the end milling process, the thermal expansion deformation value at the contact position between the tool and the workpiece surface during the end milling process is predicted, and the optimization of the end milling process parameters is guided in real time during the end milling process, thereby improving the quality of the machined surface of the end milled workpiece. Description of the Drawings

[0014] Figure 1 Flow chart of the steps of the embodiment of the present invention;

[0015] Figure 2 Geometric model and machining path of the end milled workpiece in the embodiment of the present invention (the blue line represents the machining tool path);

[0016] Figure 3 Positions of the temperature monitoring points of the end milled workpiece in the embodiment of the present invention;

[0017] Figure 4 Temperature time series data at the positions of the temperature monitoring points during the end milling process in the embodiment of the present invention;

[0018] Figure 5 Thermal expansion deformation time series data at the contact position between the tool and the workpiece surface during the end milling process in the embodiment of the present invention;

[0019] Figure 6 Optimization process of the constructed empirical mathematical model by the AIC and p-value judgment criteria in the embodiment of the present invention;

[0020] Figure 7 Result comparison before and after the optimization of the empirical mathematical model in the embodiment of the present invention;

[0021] Figure 8 Error analysis before and after the optimization of the empirical mathematical model in the embodiment of the present invention. Specific Embodiment

[0022] In order to make the objectives, technical solutions and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0023] A digital twin method for thermal expansion deformation at the contact position between an end milling cutter and a workpiece in the present invention introduces the physical parameters of the end milling workpiece into a finite element model to obtain the time series data of the temperature and thermal expansion deformation at the temperature monitoring points. The multiple linear regression (MLR) method is introduced to construct an empirical mathematical model, and after further optimization, the final optimized empirical mathematical model for the end milling process is obtained to realize the prediction and control of the temperature and deformation at the monitoring points. The specific steps are as follows:

[0024] 1) According to the flowchart of the present invention, see Figure 1 ;

[0025] 2) Determine a simple geometric model of the workpiece and end milling process, see Figure 2 . The diameter of the end milling cutter is 16 mm, the feed rate is 0.00125 m / s, and the tool path spacing is 10 mm;

[0026] 3) Adopt the finite element method (FEM) calculation method to establish a finite element model of the workpiece to be machined, and its physical parameters are shown in Table 1. Since this method focuses on finish machining, the volume reduction of the workpiece during the machining process is ignored, and the machining process is modeled as a heat conduction problem with a moving heat source;

[0027] Table 1 Physical parameters

[0028] Physical parameter name Parameter value Density 7850 kg / m3 Heat source (W) 573 Thermal conductivity (W / (m×°C)) 58 Convective heat transfer coefficient (W / (m2×°C)) 550 Initial temperature (°C) 10.8 Ambient temperature (°C) 10.8 Elastic modulus (GPa) 206 Poisson's ratio 0.3 Coefficient of linear expansion (°C-1) 14.8×10-6

[0029] Set the boundary conditions according to the machining process parameters. For the finite element numerical simulation of thermal analysis, the top surface of the workpiece is the loading area, and the others are heat convection boundaries. For the finite element numerical simulation of the structure, the bottom surface is the constrained area.

[0030] According to Figure 3 the positions of the temperature monitoring points shown, Figure 3 Figures (a) and (b) in Figure 4 each show the schematic diagrams of the positions of the temperature monitoring points at the upper and lower ends of the workpiece. Extract the finite element calculation temperature time series data results of these points, see Figure 4 Figures (a) and (b) in Figure 5 each show the temperature time series data measured at the temperature monitoring points at the upper and lower ends of the workpiece. Extract the finite element calculation data results of the time series of thermal expansion deformation in the contact area between the cutter and the workpiece, see

[0031] 4) Use the multiple linear regression (MLR) method. Regard the time series temperature data of 20 monitoring points (all monitoring points) as input variables (T ch1 (t), T ch2 (t), …, T chn (t) are independent variables), and regard the time series data of the thermal expansion deformation in the contact area between the cutter and the workpiece surface (i.e., the heat source area) as the output variable (D热源 (where \(t\) is the dependent variable). In addition, the coefficient of determination (\(R^{2}\)) 2 is used to judge the fitting degree of the regression equation curve to the observed values.

[0032] The equation of the initial empirical mathematical model is as follows:

[0033] D 热源 \(t = \beta_{0}+\beta_{1}T(t)+\beta_{2}T(t)+\cdots+\beta_{n}T(t)\) (1) CH1 (t)+\beta_{2}T CH2 (t)+\cdots+\beta n T CHn (t)(1)

[0034] where \(\beta_{0}\) is a real constant, \(\beta_{i}\) i is the coefficient of the \(i\)-th term of the regression equation, and \(T(t)\) CHi is the temperature-time series data of the \(CH_{i}\)-th monitoring point. Through fitting, the coefficients in the initial formula equation can be determined as shown in Table 2. The \(R^{2}\) 2 value of the initial empirical mathematical model is 0.7381, which is compared with the standard value of 0.6 for the coefficient of determination . Obviously, the \(R^{2}\) 2 value of the initial empirical mathematical model is relatively large, so the first verification is completed, and the following optimization process is carried out.

[0035] Table 2 Coefficients of each item of the initial empirical mathematical model

[0036]

[0037]

[0038] 5) Use the Akaike information criterion (AIC) and p-value judgment to optimize each variable term of the initial empirical mathematical model, and eliminate the variable terms with low contribution degrees, that is, delete the variable terms corresponding to the larger AIC values and p-values (variable terms with low contribution degrees), as shown in Figure 6 . Figure 6 (a) shows the results of the coefficients of each item of the initial empirical mathematical model, Figure 6 (b) shows the AIC values obtained by solving the remaining independent variable terms when each independent variable term in the initial empirical mathematical model is deleted separately, Figure 6 (c) shows the p-values corresponding to each independent variable term after deleting the independent variable terms with lower AIC values, Figure 6 (d) shows the coefficients of each item of the optimized empirical mathematical model after deleting the independent variable terms corresponding to the larger p-values and the corresponding p-values.

[0039] When the change trends of independent variables are similar, a collinearity problem occurs, which is the main reason for the low efficiency of the initial empirical mathematical model. And AIC is usually used to solve the collinearity problem. To eliminate the collinearity problem of the initial model, the variable terms of the monitoring points are deleted one by one, and the AIC values are calculated using the variable terms of other monitoring points. The AIC values are as shown in Figure 6 (b).

[0040] By comparing the AIC values, the AIC value of the model with the temperature variable of the monitoring point deleted alone is greater than that of the model without deletion (100.617), indicating that the process of deleting the temperature variable of the monitoring point alone is beneficial to improving the efficiency of the model. Therefore, monitoring points 13, 14, 10, and 11 are reasonably deleted in the optimized model.

[0041] 6) Use MLR to fit the optimized monitoring point temperature variable model, as shown in Figure 6 (c). The R value of the optimized model is 0.7316, which is close to the value of 0.7381 of the initial model. However, the p value corresponding to the independent variable term is greater than 1×10 2 , indicating that the model is still inefficient. Therefore, the independent variable terms corresponding to the larger p values (greater than 1×10 -4 ) are deleted one by one from largest to smallest, and the equation is refitted using the MLR method. This optimization process is carried out ten times, and the R values of the empirical mathematical model after each optimization are 0.729, 0.7196, 0.7152, 0.7127, 0.7123, 0.7103, 0.676, 0.6177, 0.6047, and 0.6038 respectively. The final model consists of six independent variable terms. At this time, the R value of the empirical mathematical model is 0.6038, which is greater than the predetermined -4 standard value of 0.6, and the second verification is completed. The expression of the final optimized empirical mathematical model is as follows: 2 D 2 (t) = 31.4 - 11.74T 热源 (t) + 1.36T CH2 (t) + 5.76T CH6 (t) + 0.75T CH7 (t) + 0.18T CH12 (t) + 0.89T CH16 (t) (2)

[0043] The error analysis of the optimized empirical mathematical model and the initial empirical mathematical model with the finite element calculation results is shown in Figure 7 Figure 7 Figure 7

[0044] Figure 8 As shown in (a) of, it is the comparison of the thermal expansion deformation value results of the tool and workpiece contact position calculated by the initial empirical mathematical model and the finite element numerical simulation calculation.​​​​Figure 7 In Figure (b), it is a comparison of the results of the thermal expansion deformation values at the tool-workpiece contact position calculated by the optimized empirical mathematical model and the finite element numerical simulation. Obviously, while the optimized model significantly improves the temperature monitoring efficiency, it also ensures the prediction accuracy of thermal deformation.

[0044] In the embodiment, error analysis is performed on the initial empirical mathematical model and the optimized empirical mathematical model constructed for 10 monitoring points in the upper and lower parts, as shown in Figure 8 , Figure 8 In Figure (a), it is an error analysis of the thermal expansion deformation in the tool-workpiece contact area calculated by the empirical mathematical model constructed with 10 detection points at the lower end of the workpiece and the finite element calculation results. Figure 8 In Figure (b), it is an error analysis of the thermal expansion deformation in the tool-workpiece contact area calculated by the empirical mathematical model constructed with 10 detection points at the upper end of the workpiece and the finite element calculation results. Figure 8 In Figure (c), it is an error analysis of the thermal expansion deformation in the tool-workpiece contact area calculated by the optimized empirical mathematical model and the finite element calculation results. Obviously, the error of the optimized empirical mathematical model is smaller than that of the initial empirical mathematical models constructed for 10 monitoring points in the upper and lower parts respectively. And Figure 8 (d) In it, the R 2 value at the low position represents the initial empirical mathematical model constructed for 10 monitoring points in the lower part, the R 2 value at the high position represents the initial empirical mathematical model constructed for 10 monitoring points in the upper part, and the R 2 value at the final position represents the optimized empirical mathematical model. Obviously, the R 2 value of the optimized empirical mathematical model is the largest and the fitting degree is the highest.

[0045] The above are only the preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, and improvements made within the principles of the present invention shall be included within the protection scope of the present invention.

Claims

1. A digital twin method for thermal expansion deformation at the contact position between an end milling cutter and a workpiece, characterized in that Introduce the physical parameters of the workpiece in end milling into the finite element model to obtain the time series data of the temperature and thermal expansion deformation of the temperature monitoring points. Introduce the MLR method to construct an empirical mathematical model, and further optimize it to obtain the final empirical mathematical model of the end milling process, realizing the prediction and control of the temperature and deformation of the monitoring points. The specific steps are as follows: 1) Determine the physical parameters of the workpiece in end milling and set the boundary conditions according to the machining process parameters; 2) Adopt the finite element (FEM) calculation method to establish the finite element model of the workpiece to be machined, and complete the numerical simulation of the finite element model of the end milling process; 3) Extract the temperature time series values of the temperature monitoring points and the time series data of the thermal expansion deformation at the contact position between the tool and the workpiece surface, complete the simulation calculation of the entire end milling process, and output the temperature time series data of each temperature monitoring point and the time series data of the thermal expansion deformation at the contact position between the tool and the workpiece; 4) Use multiple linear regression (MLR) combined with FEM to calculate the output data, establish an initial empirical mathematical model for the temperature time series data at the monitoring point location and the thermal expansion deformation time series data at the tool-workpiece contact location, and solve the goodness of fit R of the initial empirical mathematical model 2 with the standard value of the goodness of fit for the first verification; 5) Use the Akaike information criterion (AIC) to exclude the monitoring points with the same variation law of the temperature time series data; 6) Determine the contribution degree of each temperature monitoring point according to the p-value, exclude the monitoring points with lower contribution degrees, determine the number of temperature measurement points in the actual end milling process based on the optimized empirical mathematical model, and solve the goodness of fit R of the optimized empirical mathematical model 2 and the goodness of fit is compared with the standard value to complete the second verification and obtain the optimized empirical mathematical model.

2. The digital twin method for thermal expansion deformation at the contact position between an end milling cutter and a workpiece according to claim 1, characterized in that In step 4), when R in the verification model 2 is relatively small, re-determine the number and location of the temperature monitoring points, and return to step 3).

3. A digital twin method for thermal expansion deformation at the contact position between an end milling cutter and a workpiece according to claim 2, characterized in that, When re-determining the number and position of the temperature monitoring points, it is necessary to avoid the contact area between the machining and the fixture and find the temperature monitoring area.

4. A digital twin method for thermal expansion deformation at the contact position between an end milling cutter and a workpiece according to claim 1, characterized in that, The physical parameters include any one or a combination of two or more of density, heat source, thermal conductivity, convective heat transfer coefficient, initial temperature, ambient temperature, elastic modulus, Poisson's ratio, and linear expansion coefficient.

5. A digital twin method for thermal expansion deformation at the contact position between an end milling cutter and a workpiece according to claim 1, characterized in that The equation of the initial empirical mathematical model is as follows: D 热源 D(t) = β0 + β1T CH1 D(t) + β2T CH2 D(t) + … + β n T CHn D(t).

6. The digital twin method for thermal expansion deformation at the contact position between an end milling cutter and a workpiece according to claim 1, characterized in that The expression of the optimized empirical mathematical model is as follows: D 热源 (t) = 31.4 - 11.74T CH2 (t) + 1.36T CH6 (t) + 5.76T CH7 (t) + 0.75T CH12 (t) + 0.18T CH16 (t) + 0.89T CH17 (t).

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