CN113189948B - Method for optimizing processing technological parameters of sheet parts by considering processing precision reliability - Google Patents
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Abstract
The invention discloses a method for optimizing the processing technological parameters of sheet parts in consideration of the reliability of processing precision, which comprises the following steps: on the basis of analyzing the machining deformation characteristics in the machining process of thin plate parts, a model of average deformation error, machining time and machining precision reliability is established. And then constructing a thin plate part processing process parameter optimization model which takes the average deformation error and the processing time as targets and takes the time-varying reliability and the process conditions as constraints, and then performing double-cycle optimization solution by adopting a meta-heuristic optimization algorithm. And finally, performing multi-objective optimization simulation based on a specific example, and realizing balance between the average deformation error and the processing time for guiding actual production of an enterprise.
Description
Technical Field
The invention relates to the field of mechanical cutting machining, in particular to a numerical control machining process parameter optimization method.
Background
With the development of manufacturing technology, higher and higher requirements are put on the processing quality of parts. The thin plate type parts are widely applied due to the advantages of compact structure, light weight, high strength-to-weight ratio and the like. However, such parts have the outstanding characteristics of thin thickness, low rigidity and the like, and are easy to generate large processing deformation in the manufacturing process, so that the service performance and the processing efficiency of products are reduced. Therefore, the research and the optimized control of the machining deformation of the sheet part have important significance. Scholars at home and abroad make a great deal of research on the optimization control of machining deformation errors of thin plate parts. And part of scholars achieve the purpose of controlling machining errors by using a real-time error compensation method based on a machining deformation prediction model. However, the compensation means requires a large amount of code modification in actual production, and the processing complexity increases. Therefore, some scholars achieve control of machining deformation errors by optimizing process parameters. However, the improved process parameters may fall into the failure region and fail to satisfy the process accuracy reliability. Therefore, the method considers uncertainty factors such as dynamic errors of workpiece machining deformation and machine tool motion errors and the like, and carries out reliability analysis of the machining precision of the thin plate parts and optimization design of milling process parameters. Firstly, a finite element method is utilized to simulate the machining process of a thin plate, a machining deformation error prediction model is built based on simulation data, and a milling machining precision reliability model considering the machining deformation dynamic error is built. And secondly, establishing a technological parameter optimization design model taking the machining efficiency and the average machining deformation error as optimization targets and taking the expected machining precision reliability and the machine tool machining condition as constraints, and solving by using a multi-objective optimization algorithm to obtain an optimal technological parameter combination meeting the expected machining precision reliability.
Disclosure of Invention
The invention provides a method for optimizing the processing technological parameters of sheet parts in consideration of the reliability of processing precision.
The technical scheme adopted for achieving the purpose of the invention is that the method for optimizing the processing technological parameters of the thin plate type parts considering the processing precision reliability comprises the following steps:
step 1: analyzing the machining deformation characteristics in the machining process of the thin plate parts, and establishing an average deformation error, machining time and machining precision reliability model;
step 2: establishing a thin plate part processing process parameter optimization model which takes average deformation error and processing time as targets and time-varying reliability and process conditions as constraints;
and step 3: and performing double-loop optimization solution by adopting a meta-heuristic optimization algorithm.
Preferably, the mean deformation error model in step 1 is modeled as follows:
wherein Y is mean Is the average deformation error, N is the number of sampling points, Y is the machining deformation, t i At the ith machining time, p (t) i ) For real-time machining position, determined by the process scheme, n is the spindle speed, f is the feed speed, a p Is the axial cutting depth, a e Is the radial depth of cut.
1) As shown in FIG. 1, the workpiece has a size of a × b × h and the rigidity K of any point C C Can be expressed as the formula:
K C ＝k(E,C,a,b,h)
wherein E is the elastic modulus; c is a processing position; and a, b and h are respectively the designed length, width and thickness of the workpiece.
During the machining process, the deformation of the thin plate parts is mainly caused by Z-direction milling force F Z The Z-direction cutting force is:
F Z ＝h(n,f,a p ,a e )
the working deformation Y can be expressed as:
since machining is a dynamic process, machining position can be expressed as a function of machining time p (t) { x (t), y (t) }.
Thus, the real-time machining deformation can be expressed as a function of the process parameters and the machining position:
Y(S,p(t))＝g(n,f,a p ,a e ,p(t))
wherein Y (S, p (t)) is the real-time processing deformation; g (-) is a functional relation, S ═ n, f, a p , a e The parameters of the milling process are represented; and p (t) represents a real-time processing position determined by a process scheme.
The machining deformation prediction of the thin plate parts is complex, and the machining deformation prediction is carried out by adopting a finite element method and a Gaussian regression process.
Based on ABAQUS simulation analysis software, the functions of dynamic loading of cutting force, dynamic removal of materials, automatic extraction of node displacement and the like are realized. The main simulation process is shown in fig. 2 to obtain the actual deformation distribution of the workpiece under different process parameter combinations;
and establishing a machining deformation prediction model under different process parameter combinations by utilizing a Gaussian process regression algorithm.
The machining deformation prediction model can be expressed as:
Y(S,p(t))＝g T (n,f,a p ,a e ,p(t))w+ε
wherein w ═ w 1 ,w 2 ,…,w k ] T A weight vector representing the model; g can be regarded as a Gaussian process distribution function, and if the mean function is m (-) and the covariance function is k, g-GP (m ([ S, p (t))]),k([S,p(t)],[S,p(t)]')); epsilon is Gaussian noise, epsilon-N (0, sigma) n 2 )。
Processing deformation errors corresponding to the technological parameters at all times are used as training samples D train ＝[X train ,Y train ]＝[(S,p(t)) train ,Y train ]Test set D test ＝[X test ,Y test ]。
Preferably, the time model is processed in step 1:
wherein V is the material removal volume, M V For the material removal rate, V is the material removal area a ═ a × b, and the axial depth of cut a p Determining that a is the design length of the workpiece, b is the design width of the workpiece, M V From the feed rate f, the axial depth of cut a p And radial depth of cut a e And (6) determining.
Preferably, the accuracy reliability model is processed in step 1:
wherein R (t) 0 ,t m ) For the reliability of the machining precision, Pr {. is probability, p f (t 0 ,t m ) At a time t 0 ,t m ]Machining precision model failure rate of inner thin plate, X ═ Deltaz Z ,△β A ,△β C ] T ，S＝[n, f,a p ,a e ] T P (t) is the real-time machining position,. DELTA.z Z For the movement error of Z-axis of numerically-controlled machine tool, delta beta A For the movement error of the A axis of a numerically controlled machine tool, Delta beta C Is the motion error of the C axis of the numerical control machine tool.
The method is based on the multi-body system kinematics hypothesis and the homogeneous coordinate transformation method, comprehensively considers the machining deformation dynamic error and the numerical control machine tool motion error, and establishes a machining precision reliability model. Fig. 3 is a topological structure of an AC type five-axis numerical control machine tool, where 0 is a machine tool body, 1 is an a axis, 2 is a C axis, 3 is a workpiece, 4 is an X axis, 5 is a Y axis, 6 is a Z axis, 7 is a spindle, and 8 is a milling cutter.
The two motion transmission chains are respectively 0-1-2-3 and 0-4-5-6-7-8. Using the coordinate system of the machine tool body as a reference system and using O 0 And (4) showing. Principal axis coordinate system O 1 Relative to O 0 Is the offset vector of P 1 ＝[x 1 ,y 1 ,z 1 ] T (ii) a The tool coordinate system is arranged at the center point O of the lower end of the tool 2 Relative to O 1 Is P 2 ＝[0,0,z 2 ] T (ii) a Coordinate system of the workpiece relative to O 0 Is P 3 ＝[x 3 ,y3,z 3 ] T (ii) a Let the motion errors of the X, Y, Z, A, C axes be Δ x X 、△ y Y 、△z Z 、△β A 、△β C Corresponding to initial movement positions x 0 ,y 0 ,z 0 ,a 0 ,c 0 。
Because the processing precision of the thin plate is mainly influenced by the Z-direction processing error of the machine tool, the processing error die of the thin plateForm E Z ：
E Z ＝x 3 (sin(a 0 )-Δβ A cos(a 0 ))(sin(c 0 )+Δβ C cos(c 0 )) +x'(-Δβ A cos(a 0 )+sin(a 0 ))(Δβ C cos(c 0 )+sin(c 0 )) +y 3 (sin(a 0 )-Δβ A cos(a 0 ))(cos(c 0 )-Δβ C sin(c 0 )) +y'(-Δβ A cos(a 0 )+sin(a 0 ))(-Δβ C sin(c 0 )+cos(c 0 )) +z 3 (cos(a 0 )+Δβ A sin(a 0 ))+Y(S,p(t))-Δz Z -z +z'(cos(a 0 )+Δβ A sin(a 0 ))-z 1 -z 2 -z 0
Setting the maximum allowable machining error of the workpiece machining process as E 0 The machining precision reliability model of the sheet member can be derived:
E(X,Y(S,p(t)),t)＝E 0 -|E Z |
wherein X [. DELTA.z ] Z ,△β A ,△β C ] T ；S＝[n,f,a p ,a e ] T And p (t) is the real-time machining position.
According to the time-varying reliability theory, failure events occur when the extreme state function is greater than the threshold of any time node. At time [ t 0 ,t m ]The machining precision model failure rate of the thin plate is p f (t 0 ,t m ) Therefore, the machining accuracy reliability is R (t) 0 ,t m )＝1-p f (t 0 ,t m )。
In the formula, Pr {. cndot } is a probability.
The invention solves the machining precision reliability model of the sheet part by using a time-varying reliability analysis method based on a single-cycle Kriging surrogate model (SILK), and mainly comprises the following three steps.
Preferably, the time-varying reliability and the process conditions in step 2 are respectively as follows:
1) time-varying reliability constraints:
①
R min is the minimum allowable machining precision reliability value in the machining process.
2) And (3) process condition constraint:
②n min ≤n≤n max ,n min and n max Respectively representing the minimum and maximum rotation speeds allowed by the numerically-controlled machine tool milling cutter;
③f min ≤f≤f max ,f min and f max Respectively representing the minimum and maximum feeding speeds allowed by the numerical control machine tool;
④
a p min and a p max Respectively representing the minimum and maximum axial cutting depths allowed by the numerical control machine tool;
⑤
a e min and a e max Respectively representing the minimum and maximum radial cutting depths allowed by the numerical control machine tool;
⑥P c ≤ηP c max eta effective coefficient of numerical control machine power, P c max Rated power for the numerical control machine;
⑦F c ≤F c max ，F c max the maximum cutting force of the numerical control machine tool.
Preferably, the two-cycle optimization solving process in step 3 is as follows:
the optimization design with the machining precision reliability as the constraint is a double-cycle calculation process, which comprises an optimization optimizing process of an outer cycle and reliability analysis of an inner cycle. As a new meta-heuristic optimization algorithm, a Multi-objective water circulation algorithm (MOWCA) simulates the water circulation process in nature, has higher calculation efficiency and solution accuracy in solving the Multi-objective optimization problem, and is not repeated in the section limited by the core ideas thereof. Therefore, in order to ensure the calculation precision and efficiency, the MOWCA is introduced to solve the optimization process, and the SILK time-varying reliability analysis method is used for evaluating the processing precision reliability of the thin plate.
In each iteration solution, the reliability constraint corresponding to each initial solution in the current solution set needs to be solved. Firstly, obtaining a processing deformation Y by using a Gaussian process regression model; then, converting the machining position coordinates into machining time representation, and converting the machining deformation into an independent random variable Z; secondly, combining a machine tool machining error training agent model, if the precision requirement is not met, updating the sample and retraining until the requirement is met; finally, solving to obtain a time-varying reliability constraint function R (0, t) corresponding to each moment i ) And the method is used for checking whether each individual in the current solution set meets the requirements of time-varying reliability and processing conditions.
In summary, the invention is based on SILK time varying reliability analysis and MOWCA and performs optimization design on the processing technological parameters, and the specific solving flow is shown in FIG. 4.
Drawings
FIG. 1 sheet metal parts
FIG. 2 simulation flow of cutting of a thin plate
FIG. 3 is a schematic diagram of a certain AC type five-axis numerical control machine tool body structure
FIG. 4 algorithm solving flow chart
FIG. 5 processing Experimental conditions
FIG. 6 is a view of a machined workpiece
FIG. 7 prediction result of machining deformation
FIG. 8 is a graph showing a change in the reliability of machining accuracy
FIG. 9 distribution of the working deformations of the different variants
FIG. 10 distribution of the working deformations of the different variants
Detailed Description
To verify the effectiveness of the proposed method, a typical thin sheet piece machining example was designed. As shown in fig. 5, the experimental equipment is a MAZAK VARIAXIS j-500/5 type five-axis numerical control machine tool, and the basic parameters are as follows: the rated power is 11.5kW, and the effective power coefficient is 0.8; the diameter of the corresponding cutter is phi 10mm, the material is hard alloy, and the number of teeth is 4; the size of the workpiece to be processed is 70 multiplied by 32 multiplied by 5mm, and the processing area is 2240mm 2 The material is ZL114A, and the elastic modulus is 7.1 multiplied by 10 4 Mpa, poisson's ratio 0.33; the machining process adopts Z-shaped feed, and the tool path is shown in figure 6. Because a large amount of training data is needed for establishing the machining deformation prediction model, the method is very expensive to obtain through an experimental method and is not beneficial to engineering practice. Therefore, ABAQUS is used for cutting simulation, a machining deformation prediction model is established, and machining precision reliability analysis is carried out. And then, carrying out process parameter reliability optimization design. And finally, verifying the effectiveness of the obtained optimization result by using a processing experiment.
During the machining process, the rotating speed n of the main shaft, the feeding speed f and the cutting depth a p And cutting width a e All are variable, in order to ensure that the path of feed of each experiment is consistent, the section a e Setting to be 8mm, adding n, f and a p As a controllable factor of the experiment. First, L is selected 16 (3 4 ) Orthogonal table experimental method a simulation experiment was arranged, and the levels of the factors are shown in table 1. And then, establishing a finite element model for cutting simulation and extracting dynamic distribution data of the machining deformation along with the machining position under each group of technological parameters. Wherein, each set of cutting simulation experiment is processed to the design size of 5mm, the non-processing areas of the two end surfaces of the part are fixed by rigid surfaces, and the bottom planes of the two end surfaces are supported by special cushion blocks; in addition, the actual clamp clamping stress was measured by a static strain gauge to be about 0.75Mpa and taken into account as a boundary condition in the analysis process.
Table 1 simulation experiment each factor level
| Level of factor | n(r/min) | f(mm/min) | a p (mm) |
| 1 | 2500 | 200 | 0.15 |
| 2 | 3000 | 240 | 0.20 |
| 3 | 3500 | 280 | 0.25 |
| 4 | 4000 | 320 | 0.30 |
(2) Predicted result of working deformation
Table 2 lists the mean and maximum values of the processing deformation under each set of experimental parameters, based on which the gaussian process regression prediction model was programmed and trained using MATLAB software. To ensure the training effect of the gaussian process regression model, the training data set for the model was 1216 entries, containing 16 sets of process parameters, each set containing 76 sample points on the path of the run. In addition, in order to verify the prediction accuracy of the machining deformation error model, n is 3800r/min, f is 220mm/min, and a is p The simulation data of 0.17mm is used as the verification data set, and 70 sample points are used. The comparison result of the predicted value and the simulation value is shown in fig. 7, the variation trend of the simulation experiment value is consistent with that of the model predicted value, and the simulation experiment value and the model predicted value are uniformly distributed in a 95% confidence interval, which shows that the model has a better prediction effect on the machining deformation.
TABLE 2 mean and maximum values of the processing deformation for each set of parameters
(3) Machining accuracy reliability analysis
According to the structural parameters of the experimental numerical control machine tool, the offset of each phase of the machine tool relative to a reference coordinate system can be obtained: p 1 ＝(-100.102,-35.258,308.901) T ，P 2 ＝(0,0,168.451) T ，P 3 ＝(-22.561,-15.258,150) T (ii) a Tool location point coordinate P t (0,0, 0). Determining the tool motion track p (t) in the workpiece coordinate system as x according to the determined process scheme w ＝f[mod(t,70/f)]– 36.021,y w ＝a e [t\(70/f)]–13，z w ＝-a p (where,% is the remainder operator, \\ is the quotient operator); the machining error of the workpiece in the Z direction is allowed to be 0.040 mm; each motion error distribution of the machine tool is delta x X ～N(0,0.00125 2 ),△y Y ～N(0,0.00141 2 ),△z Z ～N(0, 0.00168 2 ),△β A ～N(0,0.000012 2 ),△β C ～N(0,0.000012 2 ). And obtaining a machining precision reliability model based on the conditions and the thin plate machining error model.
Converting the processing position into a processing time representation, and calculating the technological parameter combination of n 3800r/min, f 220mm/min and a based on a Kriging time-varying reliability analysis model p Machining accuracy reliability curve at 0.17mmThe results are shown in FIG. 8.
As can be seen from fig. 8, the machining deformation amount shows a change law with a large fluctuation with time of machining, which is caused by a difference in rigidity between different machining positions. Obviously, the machining accuracy reliability is reduced from the 66.4 th s, and gradually decreases as the machining time is accumulated; when the machining deformation reaches a certain degree, the reliability is sharply reduced, and when t is 70.1s, the reliability is only 94.4%, and the machining quality is difficult to ensure. Therefore, the machining deformation has a significant influence on the machining accuracy reliability of the thin plate. Therefore, it is necessary to optimally control the machining deformation to satisfy the desired machining accuracy reliability.
TABLE 3 optimization results
The comprehensive optimization scheme is multi-objective optimization comprehensively considering average processing deformation, only the efficiency scheme is optimized, only the deformation is optimized to be single-objective optimization, and the experience scheme is determined according to the experience of workshop technicians.
As can be seen from table 3, when only the machining efficiency (machining time) is taken as an optimization target, the larger cutting amount increases the material removal rate, thereby reducing the machining time, but the larger cutting amount increases the cutting force, thereby aggravating the machining deformation of the workpiece, and increasing the machining error. When the average machining deformation is only taken as an optimization target, the average machining deformation is reduced by 23.81%, but the machining time is increased by 39.47%. It follows that a single solution that optimizes the average deformation or efficiency of the working process has drawbacks. However, on the premise of satisfying the requirement of machining accuracy reliability, the machining time can be reduced as much as possible when the amount of machining deformation is reduced by balancing the two objectives of machining deformation and machining efficiency.
When the machining deformation and the machining efficiency are comprehensively considered, although the machining time is increased by 9.56% compared with the machining time only with the optimized efficiency, the average deformation is reduced by 6.12%, and similarly, the machining time is reduced by 21.45% compared with the machining time only with the optimized deformation increased by 23.21%; compared with the empirical scheme, the average processing deformation is reduced by 21.14% through comprehensive optimization, and the processing efficiency is improved by 4.18%. Figure 9 shows the distribution of the processing deformation for the combined optimization, optimization efficiency and empirical scheme. As can be seen from the figure, the maximum tooling distortion for the empirical plan is 0.0399mm, while the maximum tooling distortion for the combined optimization plan is up to 0.0323mm, which is a 18.80% reduction. Therefore, the processing deformation can be obviously improved after comprehensive optimization.
Fig. 10 shows a variation curve of the processing reliability in each case. As can be seen from the figure, the empirical scheme exceeds the expected minimum machining precision reliability, and the machining quality is difficult to ensure; the minimum machining precision reliability of only optimizing the machining efficiency is 97.12%, although the requirement of the expected machining precision reliability is met, the minimum machining precision reliability is only 0.12% higher than the expected minimum value, and the reliability margin is small; and the minimum processing precision reliability of the comprehensive optimization scheme reaches 98.21%, and compared with the only optimization deformation, the minimum processing precision reliability is slightly reduced, but a larger margin still exists.
In conclusion, the comprehensive optimization scheme is superior to the optimization efficiency only, the optimization deformation only and the experience scheme, and the coordination optimization of the average deformation and the processing efficiency is realized.
In order to verify the reliability of the optimization result, experimental verification is performed by using the combination of the process parameters of the scheme 1 and the scheme 4. And after the processing is finished, measuring by using a three-coordinate measuring instrument. During the measurement, sampling is carried out every 4mm along the processing path, and 76 sampling points are counted and kept consistent with the simulation process. The average of all samples was then calculated and the final result is given in table 4.
The result shows that the measured quantity is smaller than the deformation quantity obtained under the simulation condition. On one hand, the measured value of the deformation after the machining is actually the plastic deformation (residual deformation) in the state without cutting force, but not the actual deformation error in the state of the machining process; on the other hand, measurement errors and tool errors in the experiment also cause the experimental result to have deviation. In general, the relative error of the two groups of experimental results is about 10%, and the simulation result can be considered to accurately reflect the real machining deformation error.
TABLE 4 comparison of simulation and Experimental results
| Optimization scheme | Simulation result | Results of the experiment | |
| Scheme | |||
| 1 | 0.0138mm | 0.0122mm | 11.59 |
| Scheme | |||
| 4 | 0.0175mm | 0.0159mm | 9.14％ |
Claims (5)
1. The method for optimizing the processing technological parameters of the sheet part in consideration of the processing precision reliability is characterized by comprising the following steps of:
step 1: analyzing the machining deformation characteristics in the machining process of the thin plate parts, and establishing an average deformation error, machining time and machining precision reliability model;
step 2: establishing a thin plate part processing process parameter optimization model which takes average deformation error and processing time as targets and time-varying reliability and process conditions as constraints;
and step 3: performing double-cycle optimization solution by using a meta-heuristic optimization algorithm, specifically using a multi-target water circulation algorithm;
the optimization solving process comprises the following steps:
the optimization design with the processing precision reliability as the constraint is a double-cycle calculation process, which comprises an optimization optimizing process of an outer cycle and reliability analysis of an inner cycle; in each iteration solution, the reliability constraint corresponding to each initial solution in the current solution set needs to be solved; firstly, obtaining a processing deformation Y by using a Gaussian process regression model; then, converting the machining position coordinates into machining time representation, and converting the machining deformation into an independent random variable; secondly, combining a machine tool machining error training agent model, if the precision requirement is not met, updating the sample for retraining until the requirement is met; finally, solving to obtain a time-varying reliability constraint function R (0, t) corresponding to each moment i ) The system comprises a current solution set, a time-varying reliability acquisition unit, a processing unit and a processing unit, wherein the current solution set is used for acquiring a current solution set;
2. the method for optimizing the processing parameters of the thin plate type part considering the reliability of the processing precision as claimed in claim 1, wherein: average deformation error model in step 1:
wherein, Y mean The average deformation error is determined by the process scheme, N is the number of sampling points, Y is the machining deformation, ti is the ith machining time, p (ti) is the real-time machining position, N is the main shaft rotating speed, f is the feeding speed, a p Is the axial cutting depth, a e Is the radial cutting depth;
3. the method for optimizing the processing parameters of the thin plate type part considering the reliability of the processing precision as claimed in claim 1, wherein: processing a time model in the step 1:
wherein V is the material removal volume, M V For the material removal rate, V is the material removal area a ═ a × b, the axial depth of cut a p Determining that a is the design length of the workpiece, b is the design width of the workpiece, M V From the feed rate f, the axial depth of cut a p And radial depth of cut a e Determining;
4. the method for optimizing the processing parameters of the thin plate type part considering the reliability of the processing precision as claimed in claim 1, wherein: processing precision reliability model in step 1:
wherein R (t) 0 ,t m ) For the reliability of the machining precision, Pr {. is probability, p f (t 0 ,t m ) At a time t 0 ,t m ]Machining precision model failure rate of inner thin plate, X ═ Deltaz Z ,△β A ,△β C ] T ，S＝[n,f,a p ,a e ] T P (t) is the real-time machining position,. DELTA.z Z For the movement error of Z-axis of numerically-controlled machine tool, delta beta A For the movement error of the A axis of a numerically controlled machine tool, Delta beta C The motion error of the C axis of the numerical control machine tool is obtained;
5. the method for optimizing the processing parameters of the thin plate type part considering the reliability of the processing precision as claimed in claim 1, wherein: in the step 2, the time-varying reliability and the process conditions are respectively as follows:
1) time-varying reliability constraints:
①
R min the minimum allowable machining precision reliability value in the machining process is obtained;
2) and (3) process condition constraint:
②n min ≤n≤n max ,n min and n max Respectively representing the minimum and maximum rotation speeds allowed by the numerically-controlled machine tool milling cutter;
③f min ≤f≤f max ,f min and f max Respectively representing the minimum and maximum feeding speeds allowed by the numerical control machine tool;
④
a p min and a p max Respectively representing the minimum and maximum axial cutting depths allowed by the numerical control machine tool;
⑤
a e min and a e max Respectively representing the minimum and maximum radial cutting depths allowed by the numerical control machine tool;
⑥
eta is the effective coefficient of numerical control machine tool power, P c max Rated power for the numerical control machine;
⑦
F c max the maximum cutting force of the numerical control machine tool.
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