3.1 FOA-CV-GRNN prediction mode
The common data prediction algorithm is GRNN, what’s more, GRNN has strong nonlinear mapping ability for data when there are few sample data [20–21]. After the deformation data of the right inclined rib is available, 52 groups of data are randomly selected as training samples, and the rest of the sample data are used as test samples. The key to improving the forecast accuracy of GRNN is to obtain the best input and output samples and smoothing factors, while FOA and CV can search for the optimal variables required in GRNN through iterative optimization method. Hence, in order to identify the milling distortion quickly and efficiently under different processing parameters, the combination of FOA and CV is proposed based on GRNN to predict the milling distortion under different processing parameters. The detailed flowchart of FOA-CV-GRNN deformation prediction is shown in Fig. 6.
|
Figure 6 Flowchart of deformation prediction based on FOA-CV-GRNN |
By constantly adjusting the initial parameters, the initial parameters of FOA are finally selected as 150 iterations, the population size is 20, the initial coordinate \(x\)and \(y\) are all within the [0, 1] range, the search step value of Drosophila melanogaster is 2 rand-1, and the iteration termination condition is the optimal individual concentration (Best Smell) < 10− 6. Moreover, a mathematical computing platform is used to perform 4-fold, 6-fold, 8-fold, and 12-fold cross-computations, which are then used to obtain smoothing factors, training and output sample data, and a GRNN model is constructed using the obtained data for simulation prediction. In order to verify the forecast accuracy of the proposed FOA-CV-GRNN model, the best GRNN network is established by using the best input sample, the best output sample and the smoothing factor, and the test set is input for simulation prediction. Besides, the forecast accuracy of the evaluation model is determined by Mean Square Error (MRE), Root Mean Square Error (RMSE) and R Square (\({R}^{2}\)) in the evaluation indicators. The comparison between the FOA-CV-GRNN prediction and the finite element results is shown in Fig. 7. |
(a) 4-fold Cross Validation |
(b) 6-fold Cross Validation |
(c) 8-fold Cross Validation |
(c) 12-fold Cross Validation |
Figure 7 Comparison of FOA-CV-GRNN prediction results with finite element analysis |
According to Fig. 7 and the statistical results of the mathematical calculation platform, the best Cross Validation smoothing factors of 4 folds, 6 folds, 8 folds and 12 folds are 0.5048, 0.5273, 0.4849 and 0.5357, respectively, \({\text{R}}^{2}\) is 93.11%, 97.39%, 99.68% and 96.2% respectively, and the Cross-validation calculation time of 4-fold, 6-old, 8-fold and 12-fold is 420.01s, 601.80s, 806.14s and 1250.90s, respectively. The accuracy of the 4-fold Cross Validation is the lowest and the calculation time is the shortest. The forecast accuracy of the 6-fold Cross Validation is 4.28% higher than that of the 4-fold Cross Validation, while the calculation time is increased by 181.79s. The forecast accuracy of the 8-fold Cross Validation is 2.29% higher than that of the 4-fold Cross Validation, and the calculation time is increased by 204.34s. At the 12-fold Cross Validation, the forecast accuracy begins to decline, which is 3.48% lower than that of the 8-fold Cross Validation, meanwhile, the calculation time is increased by 444.76s. It can be seen from the analysis that the relationship between folded number and the forecast accuracy is first increased and then decreased, and the folded number is positively correlated with the calculation time. In the 8-fold Cross Validation, the forecast accuracy is the highest, which can meet the use requirements. Adding more folds not only reduces the prediction accuracy, but also increases calculation time, which increases the prediction cost. Therefore, in this paper, the training samples of 8-fold Cross Validation is used as the best input sample, and the best smoothing factor is 0.4849.
3.2 Analysis of dynamic distortion and dynamic feed per tooth
During multi-toothed milling, the dynamic feed per tooth of the next tooth is affected by the milling distortion of the previous tooth. When the first tooth is cutting, the actual feed per tooth should be \(f-{\delta }_{0}\) due to the initial deformation (\({\delta }_{0}\)), the milling force generated will lead to new milling distortion (\({\delta }_{1}\)) of the workpiece. Therefore, the actual feed per tooth is \(f-{\delta }_{0}+{\delta }_{1}\). Correspondingly, the second, third and fourth teeth are the same. The effect of elastic deformation of workpiece on feed is shown in Fig. 8.
|
Figure 8 The effect of elastic deformation of workpiece on feed |
The milling force model is normally used to calculate the loading force, and then the loading force is input into the finite element model to predict milling distortion. However, this process is repeated for each tooth milling, which is inefficient. Therefore, this paper uses FOA-CV-GRNN deformation mixed prediction model to replace the above process, which can quickly update the milling parameters to predict current milling distortion. Nevertheless, the milling process is a dynamic process, the static stiffness of the workpiece at different position is different, which will affect the deformation and feed per tooth. Therefore, the static stiffness of the workpiece at different cutting positions is also one of the factors that cannot be ignored. The formula for calculating the static stiffness of the workpiece can be defined as Eq. (4):
$${K}_{s}=\frac{{F}_{x}}{x} \left(4\right)$$
Where \(x\) is the deformation of the workpiece in the feed direction under the action of the milling force.
In order to research the influence of variables (radial depth of cut \({a}_{c}\), height of cutting point from bottom surface \(h\)) on the workpiece stiffness in the milling process, the milling distortion is solved by finite element simulation, hence, the static stiffness is calculated by the milling distortion and Eq. (4). The fitted static stiffness surface is shown in Fig. 9.
|
Figure 9 Fitting surface for static stiffness |
3.3 Calculation of actual feed per tooth
According to the analysis in Sect.2.2 and the FOA-CV-GRNN deformation prediction model, it is possible to further propose an algorithm to calculate the actual feed per tooth based on the change of machining position. The algorithm flow is shown in Fig. 10.
|
Figure 10 Flowchart of actual feed per tooth algorithm based on the change of machining position |
The actual feed per tooth is a necessary condition for calculating the milling force, the method mentioned above is used to calculate the actual feed per tooth under the conditions of experiment No.1 (\(n\)=6000r/min, \(f\)=0.14mm/z, \({a}_{c}\)=4mm, \({v}_{f}\)=1680mm/min) and experiment No.2 (\(n\) =7000r/min, \(f\) =0.14mm/z, \({a}_{c}\)=3.5mm, \({v}_{f}\)=1960mm/min). Figure 11 shows the results of the calculation for the actual feed per tooth.
(a) \(n\)=6000r/min, \(f\)=0.14mm/z, \({a}_{c}\)=4mm (b) \(n\)=7000r/min, \(f\)=0.14mm/z, \({a}_{c}\) =3.5mm |
Figure 11 Calculation results of actual feed per tooth under different machining parameters |
As shown in Fig. 11, \(i\)=1, 10, and 20 represent three different treatment positions on the right inclined rib, where \(i\) =1 is a deformed output point and \(i\) =10 and 20 represent the position of the tool after it has been moved forward for 10 and 20 cycles, respectively. Besides, the curve of the actual feed per tooth as a function of height of cutter teeth is also fitted with a polynomial of order 5. According to the analysis of Fig. 11, when the height of cutter teeth is lower (less than 1mm), the actual feed per tooth when machining the right inclined rib is close to the theoretical feed per tooth. However, when the height of cutter teeth is higher, the shear plane gradually increases, leading to an increase in the milling force and thus a greater elastic deformation and a different feed per tooth.
3.4 Milling force prediction of right inclined rib
The milling force prediction model can be obtained on the basis of the actual feed per tooth, which varies mainly influenced by the undeformed chip thickness in the feed direction. Figure 12 shows the variation of the undeformed chip thickness.
|
Figure 12 Variation of theoretical and actual chip thickness |
Based on the geometric relations in Fig. 12, the instantaneous undeformed chip thickness and milling force model can be defined as Eqs. (5)-(7):
$${h}_{ucycle}\left(\phi ,z\right)=\left\{\begin{array}{c}\left[f-{\delta }_{min}\left(z\right)+{\delta }_{max}\left(z\right)\right]\text{cos}\left(\phi +{\phi }_{0}-\frac{ztan\beta }{R}\right)\\ \left({\phi }_{0}\le \phi <{\phi }_{2}or{\phi }_{2}\le \phi <{\phi }_{3},z>{z}_{1}\right)\\ 0\left({\phi }_{2}\le \phi <{\phi }_{3},0<z<{z}_{1}\right)\end{array}\left(5\right)\right.$$
$$\left\{\begin{array}{c}{F}_{t1}={\int }_{{z}_{1}\left(\phi \right)}^{{z}_{2}\left(\phi \right)}{K}_{T}\left(\phi ,z\right)\bullet {h}_{ucycle}\left(\phi ,z\right)\text{d}\phi \\ {F}_{r1}={\int }_{{z}_{1}\left(\phi \right)}^{{z}_{2}\left(\phi \right)}{K}_{R}\left(\phi ,z\right)\bullet {h}_{ucycle}\left(\phi ,z\right)\text{d}\phi \\ {F}_{a1}={\int }_{{z}_{1}\left(\phi \right)}^{{z}_{2}\left(\phi \right)}{K}_{A}\left(\phi ,z\right)\bullet {h}_{ucycle}\left(\phi ,z\right)\text{d}\phi \end{array}\right. \left(6\right)$$
$$\left\{\begin{array}{c}{F}_{t2}={\int }_{{z}_{1}\left(\phi \right)}^{{z}_{2}\left(\phi \right)}{K}_{T}\left(\phi ,z\right)\bullet {h}_{ucycle2}\left(\phi ,z\right)\text{d}\phi \\ {F}_{r2}={\int }_{{z}_{1}\left(\phi \right)}^{{z}_{2}\left(\phi \right)}{K}_{R}\left(\phi ,z\right)\bullet {h}_{ucycle2}\left(\phi ,z\right)\text{d}\phi \\ {F}_{a2}={\int }_{{z}_{1}\left(\phi \right)}^{{z}_{2}\left(\phi \right)}{K}_{A}\left(\phi ,z\right)\bullet {h}_{ucycle2}\left(\phi ,z\right)\text{d}\phi \end{array}\right.\left(7\right)$$
Where \({z}_{1},{z}_{2}\) is the height value of the contact position between the cutting edge and the workpiece.
By substituting the two adjacent feed per tooth parameters and experiment No.1 and No.2 parameters at a certain position in Fig. 11 into Eq. (6) and Eq. (7), and the milling force can be calculated through orthogonal decomposition method [22], thus the variation of the milling force in each direction over adjacent rotation periods is obtained, as shown in Fig. 13.
|
(a) Milling forces (Experiment No. 1) |
(b) Milling forces (Experiment No. 2) |
Figure 13 Theoretical result of milling forces |
As can be seen from Fig. 13, the peak values of the milling force in adjacent rotational radians are slightly different for the same milling parameters. Specifically, the X direction milling force changes in a sinusoidal pattern, gradually increases from 0 to the peak value, then decreases to the minimum value, and then returns to the 0 position, and finally fluctuates around 0, the Y-direction milling force gradually increases to the maximum value, then gradually decreases to the position near 0, and finally fluctuates around 0, the Z-direction milling force gradually increases from 0 to the peak value, and after the peak value is stable for a period of time, it gradually decreases to the 0 position, and finally remains near 0. Moreover, these phenomena are similar to the existing results [23–25], which verify the effectiveness of the modified model of feed per tooth and milling force model. In order to further analyze the peak value and trend of milling force in each cycle, the absolute value of peak value of milling force under different experimental numbers can be extracted as shown in Fig. 14.
|
Figure 14 Peak value of theoretical milling force |
As can be seen from Fig. 14, the absolute values of X, Y, Z milling forces in Period 1 of experiment No.1 are 359N, 786N, 228N, respectively, and the X, Y, Z milling forces in Period 2 of experiment No.1 are 361N, 788N, 229N, respectively. What’s more, the absolute values of X, Y, Z milling forces in Period 1 of experiment No.2 are 359.5N,779.5N, and 228.5N, respectively, and the X, Y, Z milling forces in Period 2 of experiment No.2 are 361.5N,782N, and 229.5N, respectively. It is not difficult to find that the peak value of milling force in Period 1 of each experimental number is slightly larger than that in Period 2, besides, there is a certain difference between the peak value of Y-direction milling force generated by the processing parameter of Experiment No. 1 and the Y-direction milling force generated by the processing parameter of Experiment No. 2, and there is little difference between the X and Z direction milling forces. It is not difficult to find that when the workpiece is deformed, the increase of the actual feed per tooth will cause a small increase in the peak value of the milling force in all directions after orthogonal decomposition of the multiple-cutting-edge tool, however, the fluctuation of milling force is small.