RU2807258C1 - Method for determining optimal cutting conditions for cnc machines - Google Patents

Abstract

FIELD: metalworking.

SUBSTANCE: invention can be used for automatically assigning cutting modes for CNC machines, calculated by optimization based on previously obtained data on the value of the cutting EMF during a test pass. The method includes measuring the thermoelectromotive force by pre-processing the part under vibration-free cutting conditions in the speed range above the build-up zone in the tool-part pair, and determining the optimal values of the spindle speed and cutting tool feed is carried out using the simplex method, taking into account the limitations of the machine and the cutting capabilities of the tool, determined based on the values of thermoelectromotive force, while the numerical values of the spindle rotation speed and cutting tool feed are determined using the corresponding mathematical formulas.

EFFECT: use of the invention makes it possible to increase the accuracy of processing and reduce the time of technological preparation of the processing process.

1 cl, 6 dwg, 3 ex

Description

The invention relates to mechanical engineering and is intended for the automatic assignment of cutting modes calculated by optimization based on previously obtained data on the value of the cutting EMF through a test pass.

A known system for adaptive control of the cutting process, including a numerical control (CNC) system connected to actuators, a mode optimality meter, temperature and cutting speed sensors. This system allows you to adjust the spindle speed based on the temperature in the cutting zone measured by the thermoEMF method. (Copyright certificate SU 1009717, IPC B23Q 15/12, published 04/07/1981)

The disadvantage of this system is that the spindle rotation speed changes based on the obtained data on the temperature in the cutting zone, directly during the processing process. Reducing the cutting speed directly during the processing process can lead to deterioration in the quality of the machined surface and negatively affect the cutting properties of the tool. This method of regulating the spindle rotation speed also involves intervention in the machine components, which is not applicable in an industrial environment, since it deprives the consumer of the guarantee for the machine.

The prototype of this invention is a method for determining the permissible cutting speed during machining of a part with a carbide tool (RF patent for invention 2063307, IPC B23B 25/06, published on July 10, 1996), which consists of preliminary short-term processing of various grades of metal with carbide tools of different brands, measuring the thermoelectromotive force, and the permissible cutting speed is determined using the measured value of the thermoelectromotive force and the operating parameters of the technological process.

The disadvantage of this method is that it is applicable only when using carbide tools, which limits the scope of its application. This method also does not take into account the features of the machine equipment used, and the calculation is carried out only for cutting speed.

The objective of this invention is to develop a method for calculating optimal cutting conditions for a specific contact pair of tool - workpiece material, by determining the mutual properties of the contact pair through the value of thermoEMF during machining.

The technical result is an increase in processing accuracy, a reduction in the time of technological preparation of production.

The technical result is achieved by using a method for determining optimal cutting conditions for CNC machines, including measurement of thermoelectromotive force, through pre-processing of the part under vibration-free cutting conditions in the speed range above the built-up zone in the tool-part pair and calculation of the desired parameter, and determining the optimal values of the rotation speed and feeding of the cutting tool is carried out using the simplex method, taking into account the following constraint parameters:

- the minimum and maximum feed of the cutting tool, permissible kinematics of the machine,

- the lowest and highest spindle speed, permissible kinematics of the machine,

- the highest technologically permissible spindle speed,

- the highest feed allowed by the requirements for the roughness of the machined surface,

- parameters determined based on the values of thermoelectromotive force:

- cutting capabilities of the tool,

- power of the electric drive of the main movement of the machine,

- strength of the cutting tool,

- rigidity of the cutting tool,

- processing accuracy,

and the numerical values of the spindle rotation speed and cutting tool feed are determined by the formulas:

,

where n is the spindle speed, rpm,

e is the base of the natural logarithm,

X _1opt - the value of the coefficient of the objective function with a variable characterizing the rotation frequency, obtained by the simplex method,

,

where S is the cutting tool feed, mm/min;

e is the base of the natural logarithm;

X _2opt - the value of the coefficient of the objective function with a variable characterizing the feed, obtained by the simplex method.

Calculation of optimal cutting conditions using the simplex method, taking into account the imposed conditions, allows us to take into account the physical and chemical properties of a specific contact pair of tool - workpiece material, which allows us to take into account the spread of physical and chemical properties due to the peculiarities of foundry production. Since each initial workpiece may have a shape error or surface defect, and the modes specified by the technological process are calculated on the basis of average tabular data, which does not allow taking into account the real parameters of the material being processed. Also, the calculation of optimal modes is carried out taking into account the technical characteristics of the equipment, eliminating the possibility of assigning modes that are not achievable on a specific machine. The claimed method makes it possible to obtain the most accurate numerical values of rotation speed and feed, since the calculation uses the values of the thermoelectromotive force of each specific tool-workpiece pair, and also takes into account the capabilities of the machine and the cutting capabilities of the tool, which allows increasing the processing accuracy of each part.

Automatic determination of optimal cutting conditions using the simplex method and automatic entry of them into the CNC of the machine allows you to reduce the time of technological preparation of production, as well as eliminate the risk of errors due to the human factor. Calculation of optimal cutting conditions by a microcontroller, after a trial run, allows you to eliminate the human factor, as well as quickly and accurately determine the required values using the simplex method and promptly transfer them to the CNC of the machine, which will allow you to quickly and independently start the technological cycle. Also, the implementation of this method does not require intervention in machine components and program code, which allows it to be used on machines that have a manufacturer’s warranty.

In fig. 1 shows a block diagram of the implementation of the claimed method.

In fig. Figure 2 shows a block diagram of the operating algorithm of the claimed method.

In fig. Figure 3 shows a block diagram of the simplex method algorithm.

In fig. 4-6 shows a graphical definition of optimal cutting conditions.

The method for determining optimal cutting conditions for CNC machines is implemented as follows.

A processing program is entered into the machine numerical control device (CNC) 1 to obtain a part of a given shape and size. Before the processing cycle, the power source 2 supplies voltage to the converter board 3 and the microcontroller (MK) 4, then the part is preprocessed under vibration-free cutting conditions in the speed range above the build-up zone. Before the start of pre-processing, the CNC of machine 1 sends to MK 4 a command to start measuring the thermoelectromotive force (thermoEMF) of cutting, which serves to assess the physical and mechanical properties of the contact pair tool 5 and part 6. During pre-processing of MK 4, using an analog-to-digital converter ( ADC) 7 and the measuring circuit (when processing a cutter using a tool 5 according to patent RU No. 201939, IPC B23B 25/06, published on January 21, 2021, the measuring circuit is included in the design of the cutter itself, and when using a tool that does not contain a measuring circuit, the functions of the latter perform an ADC and a conductor mounted on a cutter or workpiece and connected to the network), a tool 5 and a workpiece 6 connected to the contact pair, and measure the value of the cutting thermoEMF. Then MK 4 confirms the receipt of thermoEMF data during pre-processing and the machine moves the working element to the starting point of the part processing cycle. Based on the obtained thermoEMF value, cutting modes are optimized in MK 4.

When optimizing cutting conditions, the following technical restrictions are imposed on the system, taking into account the thermoEMF data obtained during preliminary processing:

1) The smallest cutting tool feed S allowed by the kinematics of the machine. ,

Where - minimum feed rate of the cutting tool of the machine, mm/rev.

2) The highest cutting tool feed S allowed by the kinematics of the machine.

,

Where - maximum feed rate of the cutting tool of the machine, mm/rev.

3) The lowest spindle speed n allowed by the kinematics of the machine.

,

Where - minimum machine spindle speed, rpm.

4) The highest spindle speed n allowed by the kinematics of the machine.

,

Where - maximum machine spindle speed, rpm.

5) Cutting capabilities of the tool.

,

where A and k are the coefficients of the regression model of the dependence of cutting conditions on the thermoEMF of pretreatment;

T - accepted tool life, min;

m is an indicator of relative resistance;

t - cutting depth, mm;

S - cutting tool feed, mm/rev;

E - thermoelectromotive force, mV;

- Pi;

x _v , y _v - empirical exponents;

d - diameter of the workpiece, mm;

n - spindle rotation speed, rpm.

6) The highest technologically permissible spindle speed n.

,

where v _technical - technologically permissible cutting speed, m/min;

d - diameter of the workpiece, mm.

7) Power of the electric drive of the main movement of the machine.

where A and k are the coefficients of the regression model of the dependence of processing conditions on the thermoEMF of pretreatment;

x _z , n _z , y _z - empirical exponents;

- power of the electric motor of the main movement of the machine, kW;

n - spindle rotation speed, rpm;

S - cutting tool feed, mm/rev;

E - thermoelectromotive force, mV;

t - cutting depth, mm;

d - diameter of the workpiece, mm;

- Pi;

- Efficiency of the transmission mechanism from the engine to the tool.

8) Durability of the cutting tool.

Where - tensile strength of the cutter holder material;

B, H - width and height of the holder section;

- the distance of the cutter from the point of application of the circumferential force;

n - spindle rotation speed, rpm;

S - cutting tool feed, mm/rev;

x _z , n _z , y _z - empirical exponents;

A and k are the coefficients of the regression model of the dependence of cutting conditions on the thermoEMF of pretreatment;

E - thermoelectromotive force, mV;

t - cutting depth, mm;

d - diameter of the workpiece, mm;

- Pi;

- safety factor.

9) Rigidity of the cutting tool.

where n is the spindle rotation speed, rpm;

S - cutting tool feed, mm/rev;

x _z , n _z , y _z - empirical exponents;

B, H - width and height of the holder section, mm;

- distance of the cutter from the point of application of the circumferential force, mm;

A and k are the coefficients of the regression model of the dependence of cutting conditions on the thermoEMF of pretreatment;

E - thermoelectromotive force, mV;

t - cutting depth, mm;

d - diameter of the workpiece, mm;

- Pi;

10) Processing precision.

where _KZh is the stiffness coefficient of the part, depending on the method of fastening

n - spindle rotation speed, rpm;

S - cutting tool feed, mm/rev;

x _y , n _y , y _y - empirical exponents;

- allowance for size, mm;

E _control - modulus of elasticity of the material of the part, kg/mm ² ;

d _pr - reduced diameter of the stepped part;

_Ktr - tubularity coefficient, equal to the ratio of the diameter of the hole in the part to the reduced diameter of the part;

A and k are the coefficients of the regression model of the dependence of cutting conditions on the thermoEMF of pretreatment;

E - thermoelectromotive force, mV;

t - cutting depth, mm;

- Pi;

d - diameter of the workpiece, mm.

When fixing a part in a chuck or on a mandrel in a cantilever manner

,

where L _zag is the length of the part protruding from the cartridge, mm;

x _p - distance from the right end of the part to the place of application of force (to the cutter), mm.

When securing a part in a chuck or on a mandrel with the free end pressed by the center of the tailstock or with the free end supported on the rest

,

where L _zag is the length of the part protruding from the cartridge, mm;

x _p - distance from the right end of the part to the place of application of force (to the cutter), mm.

When fixing a part in centers without a steady rest

,

where L _zag is the length of the part protruding from the cartridge, mm;

x _p - distance from the right end of the part to the place of application of force (to the cutter), mm.

When fixing the part in the centers with a steady rest in the middle of the span

,

where L _zag is the length of the part protruding from the cartridge, mm;

x _p - distance from the right end of the part to the place of application of force (to the cutter), mm.

11) The highest cutting tool feed S allowed by the requirements for the roughness of the machined surface.

,

Where - the maximum permissible feed of the cutting tool, ensuring the required roughness class of the machined surface.

For fine processing:

,

where R _z is the specified surface roughness, μm;

r - radius of curvature at the tip of the cutter, mm.

For finishing/semi-finishing and roughing:

,

where C _H is a coefficient characterizing the processing conditions;

H _max - maximum height of surface microroughness, microns;

φ - main plan angle;

φ ₁ - auxiliary main plan angle;

r - radius of curvature at the tip of the cutter, mm;

t - cutting depth, mm;

y, u, x, z - empirical exponents.

Next, the objective function and the system of constraints are linearized by logarithm and then the optimal cutting conditions are determined.

To solve the problem of determining optimal cutting conditions, a simplex method is used, which implements a rational search of basic feasible solutions in the form of a final iterative process that necessarily improves the value of the objective function at each step.

First, among the variables of the problem, an initial basis is selected from m variables, (X ₁ ,...,X _m ), which must have non-negative values at a time when the remaining (nm) free variables are equal to 0. In the simplest case, the basis variables can be take m variables, each of which is included in only one constraint, with a positive sign, and the free terms of the constraints must be positive. In this way, the initial simplex table is compiled.

Then the existence of a solution is checked. To do this, consider all columns of the initial simplex table with m variables, (X ₁ ,...,X _m ), for which the coefficient in the row representing the objective function is <0 (when finding the maximum), or >0 (when finding the minimum ). If there is at least one column for which all coefficients A _i,j < 0, then the problem has no solution, because the set of feasible solutions is not limited and the objective function increases (decreases) without limit. If there is no solution, the MC sends an error message to the CNC and the processing cycle ends.

If the problem has a solution, then a new simplex table is formed. To do this, a free variable from a string representing the objective function of the problem is introduced into the basis. When finding the maximum, select a column with a negative element in the row representing the objective function, and when finding a minimum, choose a column with a positive one. The variable corresponding to this column (let the selected column be the k-th) is entered into the new basis. To select the basis variable to be derived from the basis (selecting a resolving row), it is necessary to look through all the positive elements of the selected column (A _i,k ). The element A _i,k is called the resolving element. For all resolving elements, find the ratio B _i / A _i,k (B _i is the free term) and select the line that corresponds to the minimum value of this ratio (let it be the i-th line). The variable corresponding to the i-th row is derived from the basis. If there are several identical relations, any row is taken.

Then the new k-th row of the new simplex table is filled in, into which the elements of the old (previous) i-th row are written, divided by the resolving element A _i,k . After filling the k-th line, the remaining lines are filled. To do this, the k-th row is multiplied sequentially by such numbers that after adding it with each row of the initial simplex table in the k-th column, we get zero everywhere (except for the value in the new k-th row).

After compiling a new simplex table, the resulting solution is checked for optimality. If in the line representing the objective function there is a coefficient <=0 (when finding the maximum), or >=0 (when finding the minimum), then the solution can be improved. Then the process of forming a new simplex table and checking the resulting solution for optimality is repeated.

If in the line representing the objective function, all coefficients (except for the free term) >=0 (when finding the maximum), or <= 0 (when finding the minimum), then the optimal solution is obtained (points of optimal cutting conditions are determined): X=(B ₁ ,...,B _m ,0,...,0), F=C ₀ . E

After determining the optimal solution, the numerical values of the spindle speed and cutting tool feed are determined by the formulas:

,

where n is the spindle rotation speed, rpm;

e is the base of the natural logarithm;

X _1opt - the value of the coefficient of the objective function with a variable characterizing the rotation frequency.

,

where S is the cutting tool feed, mm/min;

e is the base of the natural logarithm;

X _2opt - the value of the coefficient of the objective function with a variable characterizing the feed.

Then MK 4 sends numerical values of the spindle speed and cutting tool feed, and the CNC of machine 1 reads data from the buffer and writes these values to variables that are referenced in the processing program as storing cutting modes, according to which the CNC of machine 1 begins the work cycle processing the part.

The method for determining optimal cutting conditions for CNC machines is illustrated by the following examples.

Example 1

A workpiece (ШХ15) with length L = 100 mm and diameter D = 100 mm with elastic modulus E is processed_control= 21516 kg/mm² , fixed in the chuck, with a cutter (T5K10), having the following characteristics: main angle φ = 45°; auxiliary angle φ₁= 45°; accepted durability period T = 90 min; bending strength of the cutter holder material = 5 kg/mm²; cross-sectional width of the cutter holder B = 20 mm; cutter holder section height H = 20 mm; safety factor of the holder material= 1.4; radius of curvature at the tip r = 1.5 mm, on a machine with the following parameters: minimum permissible spindle speed n_min = 12.5 rpm; maximum permissible spindle speed n_min = 2000 rpm; minimum permissible cutting tool feed S_min = 0.1 mm/rev; maximum permissible cutting tool feed S_max = 2 mm/rev; machine drive power N_y = 10 kW; Efficiency from the drive to the working element = 0.8. Processing parameters: cutting depth t = 1.0 mm; thermoEMF of pre-treatment E = 11.3 mV; distance to start processing from the right end of the part x₀ = 1 mm; distance of the end of processing from the right end of the part x₀₊₁ = 102 mm; tool stroke length L_p.x.= 103 mm; allowance for size = 1.0 mm; number of parts processed simultaneously r_R= 1 piece; technologically permissible cutting speed V_tech= 1.4 m/s; protrusion of the cutter from the tool holder l_vr = 30 mm; accepted machine load factor = 0.75. Pre-processing of the part was carried out at V = 100 m/min, S = 0.1 mm/rev, t = 1 mm.

Next, the objective function and the system of constraints were linearized by logarithm.

Limitation 1. Minimum cutting tool feed allowed by the kinematics of the machine.

Limitation 2. The highest cutting tool feed allowed by the kinematics of the machine.

Limitation 3. The lowest spindle speed allowed by the kinematics of the machine.

Limitation 4. The highest spindle speed allowed by the kinematics of the machine.

Limitation 5. Cutting capabilities of the tool.

Limitation 6. Highest technologically permissible cutting speed.

Limitation 7. Power of the electric drive of the main movement of the machine

Limitation 8. Durability of the cutting tool.

Limitation 9. Rigidity of the cutting tool.

Limitation 10. Rigidity of the cutting tool.

Limitation 11. The highest cutting tool feed allowed by the requirements for the roughness of the machined surface.

Mathematical model of the cutting process, expressed by a set of restrictions in the form of a system of inequalities:

Optimal cutting conditions were determined using the simplex method, implementing a rational search of basic feasible solutions, in the form of a final iterative process that necessarily improves the value of the objective function at each step.

Based on the results of compiling simplex tables and checking the resulting solutions for optimality, the numerical values of the spindle rotation speed and cutting tool feed were determined using the formulas:

rpm

mm/rev

Example 2

A workpiece (steel 40) with length L = 100 mm and diameter D = 300 mm with elastic modulus E is processed_control= 20000 kg/mm² , fixed in the chuck, with a cutter (T15K6), having the following characteristics: main angle φ = 45°; auxiliary angle φ₁= 45°; accepted durability period T = 90 min; bending strength of the cutter holder material = 5 kg/mm²; cross-sectional width of the cutter holder B = 20 mm; cutter holder section height H = 20 mm; safety factor of the holder material= 1.4; radius of curvature at the tip r = 1.5 mm, on a machine with the following parameters: minimum permissible spindle speed n_min = 12.5 rpm; maximum permissible spindle speed n_min = 2000 rpm; minimum permissible cutting tool feed S_min = 0.1 mm/rev; maximum permissible cutting tool feed S_max = 2 mm/rev; machine drive power N_y = 10 kW; Efficiency from the drive to the working element = 0.8. Processing parameters: cutting depth t = 2 mm; thermoEMF of pre-treatment E = 8.5 mV; distance to start processing from the right end of the part x₀ = 1 mm; distance of the end of processing from the right end of the part x₀₊₁ = 102 mm; tool stroke length L_p.x.= 103 mm; allowance for size = 1.5 mm; number of parts processed simultaneously r_R= 1 piece; technologically permissible cutting speed V_tech= 1.4 m/s; protrusion of the cutter from the tool holder l_vr = 30 mm; accepted machine load factor = 0.75. Pre-processing of the part was carried out at V = 100 m/min, S = 0.1 mm/rev, t = 1 mm.

Next, the objective function and the system of constraints were linearized by logarithm.

Limitation 1. Minimum cutting tool feed allowed by the kinematics of the machine.

Limitation 2. The highest cutting tool feed allowed by the kinematics of the machine.

Limitation 3. The lowest spindle speed allowed by the kinematics of the machine.

Limitation 4. The highest spindle speed allowed by the kinematics of the machine.

Limitation 5. Cutting capabilities of the tool.

Limitation 6. Highest technologically permissible cutting speed.

Limitation 7. Power of the electric drive of the main movement of the machine.

Limitation 8. Durability of the cutting tool.

Limitation 9. Rigidity of the cutting tool.

Limitation 10. Rigidity of the cutting tool.

Limitation 11. The highest cutting tool feed allowed by the requirements for the roughness of the machined surface.

Mathematical model of the cutting process, expressed by a set of restrictions in the form of a system of inequalities:

Optimal cutting conditions were determined using the simplex method, implementing a rational search of basic feasible solutions, in the form of a final iterative process that necessarily improves the value of the objective function at each step.

Based on the results of compiling simplex tables and checking the resulting solutions for optimality, the numerical values of the spindle speed and cutting tool feed were determined using the formulas:

rpm

mm/rev

Example 3

A workpiece (40X) with a length L = 100 mm and a diameter D = 300 mm with a modulus of elasticity E _control = 21822 kg/mm ² is processed, fixed in a chuck, with a cutter (T15K6) having the following characteristics: main angle φ = 45 °; auxiliary angle φ ₁ = 45°; accepted durability period T = 90 min; bending strength of the cutter holder material = 5 kg/mm ² ; cross-sectional width of the cutter holder B = 20 mm; cutter holder section height H = 20 mm; safety factor of the holder material = 1.4; radius of curvature at the tip r = 1.5 mm, on a machine with the following parameters: minimum permissible spindle speed n _min = 12.5 rpm; maximum permissible spindle speed n _min = 2000 rpm; minimum permissible cutting tool feed S _min = 0.1 mm/rev; maximum permissible cutting tool feed S _max = 2 mm/rev; machine drive power N _y = 10 kW; Efficiency from the drive to the working element = 0.8. Processing parameters: cutting depth t = 1.0 mm; thermoEMF of pre-treatment E = 10.2 mV; distance to start processing from the right end of the part x ₀ = 1 mm; distance of the end of processing from the right end of the part x ₀₊₁ = 102 mm; tool stroke length L _p.x. = 103 mm; allowance for size = 1.0 mm; number of parts processed simultaneously r _R = 1 piece; technologically permissible cutting speed V _tech = 1.4 m/s; protrusion of the cutter from the tool holder l _BP = 30 mm; accepted machine load factor = 0.75. Pre-processing of the part was carried out at V = 100 m/min, S = 0.1 mm/rev, t = 1 mm.

Next, the objective function and the system of constraints were linearized by logarithm.

Limitation 1. Minimum cutting tool feed allowed by the kinematics of the machine.

Limitation 2. The highest cutting tool feed allowed by the kinematics of the machine.

Limitation 3. The lowest spindle speed allowed by the kinematics of the machine.

Limitation 4. The highest spindle speed allowed by the kinematics of the machine.

Limitation 5. Cutting capabilities of the tool.

Limitation 6. Highest technologically permissible cutting speed.

Limitation 7. Power of the electric drive of the main movement of the machine.

Limitation 8. Durability of the cutting tool.

Limitation 9. Rigidity of the cutting tool.

Limitation 10. Rigidity of the cutting tool.

Limitation 11. The highest cutting tool feed allowed by the requirements for the roughness of the machined surface.

Mathematical model of the cutting process, expressed by a set of restrictions in the form of a system of inequalities:

Optimal cutting conditions were determined using the simplex method, implementing a rational search of basic feasible solutions, in the form of a final iterative process that necessarily improves the value of the objective function at each step.

Based on the results of compiling simplex tables and checking the resulting solutions for optimality, the numerical values of the spindle rotation speed and cutting tool feed were determined using the formulas:

rpm

mm/rev

Thus, the use of a method for determining optimal cutting conditions for CNC machines, including measurement of thermoelectromotive force, through preliminary processing of the part under vibration-free cutting conditions in the speed range above the built-up zone in the tool-part pair and calculation of the optimal values of the spindle speed and cutting tool feed, using the simplex method, taking into account the limitations of the machine and the cutting capabilities of the tool, determined based on the values of thermoelectromotive force, allows to increase the accuracy of processing and reduce the time of technological preparation of production.

Claims (20)

A method for determining optimal cutting conditions for CNC machines, including measuring the thermoelectromotive force by pre-processing the part under vibration-free cutting conditions in the speed range above the built-up zone in the tool-part pair and determining the required parameters of the cutting mode, characterized in that the determination of the optimal spindle speed values and feeding of the cutting tool is carried out using the simplex method, taking into account the following limitation parameters: - the smallest and largest feed of the cutting tool allowed by the kinematics of the machine, - the lowest and highest spindle speed allowed by the kinematics of the machine, - the highest technologically permissible spindle speed, - the highest feed allowed by the requirements for the roughness of the machined surface, and - parameters determined from the values of thermoelectromotive force, which are used as cutting capabilities of the tool, power of the electric drive of the main movement of the machine, cutting tool strength, cutting tool rigidity, specified processing accuracy, in this case, the numerical values of the spindle rotation speed and cutting tool feed are determined by the formulas $$n=e^{x_{1OPT}}$$ , where n is the spindle speed, rpm, e is the base of the natural logarithm, X _1opt - the value of the coefficient of the objective function with a variable characterizing the rotation frequency, obtained by the simplex method, $$S=\frac{e^{x_{2OPT}}}{100}$$ , where S is the cutting tool feed, mm/min, e is the base of the natural logarithm, X _2opt - the value of the coefficient of the objective function with a variable characterizing the feed, obtained by the simplex method.