# Analysis and mathematical model of the circumferential accuracy of the groove cut on the surface of rotation

Open Access
Review

## Abstract

An analysis is made of the technology of vortex cutting method of groove for sealing rings provided on the lateral surfaces of rotation of a part of the sealing system and the dimensional chains thus formed, as well as accuracy issues by applying probability theories and dimensional chains. Features of the developed technology are presented; the factors influencing the accuracy of the circular diameter of the groove are analyzed, and mathematical dependencies are given for their determination; the mathematical model of the accuracy of the diameter of the groove is derived. The technological dimension chain, which is formed in the vortex cutting method of groove, is given its properties, and the new linear–circular dimensional chain, which is not found in the literature, and its mechanism of formation are presented. The features of the linear–circular dimensional chain and the management of the functional relation between the linear and circular feeds are used to determine the technological capabilities of the vortex method on the lateral surface of the groove. Also, families of possible single- and double-symmetrical forms of the grooves are identified. It is recommended to use the results obtained, when analyzing the technology of cutting of grooving and other similar surfaces with a vortex method, when solving accuracy problems, including the derivation of a mathematical model of accuracy.

## Keywords

Vortex method Groove Dimension chain Accuracy

## List of symbols

$$f_{t}$$

Circular feed

$$f_{m}$$

Linear feed

$$f{}_{r}$$

Vertical feed

$$N_{s}$$

Rotational motion

$$T_{m}$$

Norms of the time

$$d_{\text{t}}$$

Distance between the tool base surfaces of the tool head

$$b_{o}$$

Distance between the base surface and the cutting edge of the tool

$$D$$

Inner diameter of the groove

$$D_{0}$$

Nominal diameter of the surface of rotation

$$a$$

Growth of the groove diameter, in the cross section

$$D_{\text{t}}$$

Size of the tool’s position at the tool head

$$L$$

Size of linear displacement of the workpiece

$$B$$

Length of the chord

$$\alpha$$

Angle of rotation of the workpiece

$$D_{m}$$

Outer diameter of the groove

$$b_{\text{t}}$$

Width of the tool

$$\delta_{{\text{D}_{0} }}$$

Initial error, resulting from the complex effect of tolerance of the diameter $$D_{0}$$ and the error of the angle of turn of the workpiece

$$D_{\text{max} }$$

Maximum diameter of the groove to be cut

$$D_{\text{min} }$$

Minimum diameter of the groove to be cut

$$\delta_{a}$$

Initial error resulting from the turn of the workpiece

$$\Delta \alpha$$

Allowable angular error of turning the workpiece

$$\alpha_{\text{min} }$$

Half of the minimum angle of the turn of the workpiece

$$T_{{\text{D}_{0} }}$$

Tolerance of the diameter of the surface of rotation

$$\delta_{L}$$

Initial error in the diameter $$D$$ in the section with the angular position, expressed by the angle $$\beta$$, associated with the error of only linear feed $$f_{m}$$

$$\delta_{{f_{t} }}$$

Initial error in the diameter $$D$$ in a section with an angular position, expressed by the angle $$\beta$$, associated with the error of only circular feed $$f_{t}$$

$$\Delta L_{x}$$

Permissible error of the drive of the linear feed

$$\Delta f_{t}$$

Permissible error of the drive of the circular feed

$$\beta$$

Angular position of the groove size

$$t$$

Coefficient determining the percentage of risk of getting a reject when groove cutting

$$\lambda_{1} ,\,\lambda_{2} ,\,\,\lambda_{3} ,\,\,\lambda_{4}$$ and $$\lambda_{5}$$

Coefficients that take into account the distribution forms of the corresponding primary errors

$$\delta_{{\text{w}}}$$

Error related to the accuracy of manufacture and tool wears

$$\delta_{{\text{D}_{\text{t}} }}$$

Error associated with the position of the tool in the head

$$T_{\text{w}}$$

Tolerances for tool manufacture and wear

$$\Delta f_{t}$$

Formed the errors of the device parameters

$$T_{\text{d}}$$

Tolerance of the diameter $$D$$

$$f_{m\text{max} }$$, $$f_{m\text{min} }$$

Largest and smallest values of the linear feed, respectively, provided by the drive of the machine when it installed

$$T_{{\text{D}_{\text{t}} }}$$

Tolerance of the diameter $$D_{\text{t}}$$

## 1 Introduction

The types of machine-building equipment are continuously growing, and technological possibilities of machining with CNC are expanding. As a result, favorable conditions are created for the use in engineering structures and their parts used in any field of human activity, surfaces that are relatively sophisticated in technology, but more effective in performing the assignment. For forming of such as surfaces, advanced and efficient manufacturing methods and technologies are developed and applied [1, 2, 3, 4].

The development of a theoretical basis to ensure the required accuracy during machining relies on the dimensional analysis of technological process. This is one of the main directions to achieve the formation of efficient surfaces [5, 6, 7, 8]. The analysis of accuracy formation of the closing link when processing is exercised out with application of the theory of dimensional chains. Technological dimensional chains are reflections of dimensional links acting between elements of a technological system while performing a technological operation.

The analysis of the dimension chains and the control of their constituent parts makes it possible to effectively solve a wide range of constructive and technological problems.

Applying the dimensional analysis methods allowed researchers to solve various tasks associated with dimensional links between elements and parameters of technical systems [9, 10, 11], as well as with output and input parameters of technological processes [12, 13, 14, 15, 16], and it can be said in any branch of technology and technology.

According to the above, dimensional analysis of the accuracy of the closing links formed when the vortex cutting of the rings of sealing grooves provided on the lateral surfaces of rotation allows one to scientifically take into account the technological capabilities of the processing method and the elements of the technological process and to develop of highly efficient processes.

A similar surface is used in the construction of one of the main parts of the sealing system—split shell the insert of SK 50 × 50 type cylindrical cock used in the oil industry (Fig. 1). On the curvilinear surface of the rotation, there is provided a groove for an elastic sealing element with a circular cross section, which has a special profile along the circumference perpendicular to the conical surface of the rotation. Obviously, the required condition for the operation of such a groove with complex design causes the designer to designate numerous quality parameters to the process of its formation: Fig. 1 Construction of the groove on the rotation side surface
• The accuracy of diameter of the groove’s positioning surface,

• The accuracy of the width of the groove,

• The accuracy of coaxiality of the groove’s hole transition,

• The accuracy of the radius of rounding of the bottom of the groove,

• The accuracy of the outer and inner circles of the groove,

• The accuracy of the perpendicularity of the axes of symmetry of the local groove profiles to the conical surface of rotation,

• The accuracy of roundness of groove circles,

• Groove surface roughness, which should be provided, as well as economic efficiency.

The kinematics of any surface processing method is based on the geometrical formation mechanism of that surface. From this point of view, the processing of the groove on existing machine tools is possible in two variants, depending on the production volume:
1. 1.

Cutting of the groove is carried out by a shaped dimensional tool (for example, end mill). Its profile has the form of a cross section of the groove. The tool rotates around its own axis. The workpiece is transmitted three-way linked movements—turning (feeds in the directions of two horizontal axes and circular feed—turning of the workpiece around its own axis) [1, 2, 3]. In this case, the processing process imposes on the technological operation two constraints of the first level (the shape and diameter of the tool) and one constraints of the second level (three-way constraints movement feed of the workpiece .

2. 2.

Cutting of the groove is carried by tools such as lathe tool (offset tool), which the major cutting edges has the form of a cross section of the groove and rotates around the axis of the groove circumferences. The workpiece is transmitted two-way linked movements (linear and circular feeds) [18, 19, 20]. In this case, the machining process imposes on the technological operation three constraints of the first level (the shape and the diameter of the tools, their distance from the rotation axis) and one double-sided constraints of the first level (linear and circular feeds) .

The second option is relatively more technological and simple than the first. This is more affordable in terms of processing quality and performance [19, 20]. The second option machining is named by us, vortex machining grooving. Two different grooving technologies were developed and tested in the Sabunchu Production Association and the Sardarov Machine-Building Plant. In one of them, the constraint-linked movements of circular and linear feeds were ensured with the help of the mechanical drive of machine (machining with CNC machines), and in the other with the special fixture (machining on universal milling and boring machines).

The basis for the effective organization of any technological operation is the development and management of the theoretical basis for ensuring the quality of processing and productivity [21, 22]. The features of the kinematic communications between the elements of the technological system play an important role, to ensure them [3, 22, 23, 24, 25]. Therefore, the study of the theoretical basis of the formation of accuracy when cutting a groove vortex method is an urgent task.

Below are the questions of providing circumferential accuracies of grooves along its circumferences, cut by the vortex method.

To solve the problem, theoretical bases are used to ensure accuracy in the manufacture of machine parts, dimensional chains and vector theories [3, 4, 26, 27].

## 2 Methods

### 2.1 Features of the formation of the accuracy of the circumferential dimensions of the groove

Cut grooves by the vortex method on the lateral surfaces of rotation are realized as follows [19, 20]: The tool head (1), for groove cutting, is installed in the spindle of the milling head, and the workpiece (3) is installed on the chuck of the machine tool CNC (or the second technology, on the fixture). After that the tool (2) and the workpiece (3), they are brought into the required interrelations for processing, and the technological system is adjusted (Fig. 2). Fig. 2 Scheme of influence of some factors on the formation of groove circular accuracy

Cutting grooves on the side surfaces is performed by a continuously, with a rotating ($$N_{s}$$) tools, for two passes, first with the vertical feed ($$f{}_{r}$$) of the workpiece (3) until the required depth of the groove is reached (Fig. 2a). Rotary tools (2) remove material from the workpiece (3), with intermittent, at each cut movement. When the tool reaches the required groove depth, the vertical feed stops. Then, the linear feed (feed rate, $$f_{m}$$) of the workpiece (3), coordinated with its turn-circular feed ($$f{}_{t}$$), is reported (Fig. 2b) until the required processing length is equal to $$D$$ [19, 20].

Thus, when the workpiece (3) is rotated through an angle of 2α and, accordingly, its linear displacement $$L$$ is equal to $$D$$, ($$L = D$$), the groove is formed completely.
$$\sin \alpha = \frac{D}{{D_{0} }};\quad \alpha = \arcsin \frac{D}{{D_{0} }}.$$

The main technological issue of the groove cutting is the definition and implementation of functional links between circular and linear feeds.

To solve this problem and derive a mathematical model of the circular feed, the conditions of equality of the norms of the time $$T_{m}$$ used for turning and linear displacement of the workpiece are applied:
$$T_{m} = \frac{{\mathop {AC}\limits^{ \cup } }}{{f_{t} }} = \frac{L}{{f_{m} }};\quad {\text{Then}}\;\;\;\frac{{\pi \cdot D_{0} \cdot 2\alpha }}{{360 \cdot f_{t} }} = \frac{D}{{f_{m} }},$$
From the latter equation, a mathematical model of the circular feed of the workpiece is obtained $$f_{t} = F(f_{m} )$$, and it expresses its functional connection with the linear feed :
$$f_{t} = \frac{{\pi \cdot D_{0} \cdot f_{m} \cdot \arcsin \frac{D}{{D_{0} }}}}{180 \cdot D}$$
(1)

To realize this connection, the design of the technological means can be elaborated which is manufactured and introduced into production .

A very limited number study has been carried out in the area of the cut of the groove, provided on the lateral surfaces of rotation by the vortex method [18, 19]. The studies carried out in this direction are mainly intended for the authors of this work. The vortex method is also used for processing other surfaces (e.g., thread, etc.). However, of the groove cutting with the help of the developed vortex method differs sharply from cutting other surfaces (e.g. threaded, etc.) by the vortex method, both in the mechanism of surface formation and in the mechanism of ensuring their quality (accuracy).

In the studies carried out, the issues of ensuring the accuracy of grooves were solved in a general form (in terms of functional and structural requirements for means). However, the results of the study, based on the analysis of the technological dimensional chain, formed during the cutting, allow in a proper way, to predict the accuracy of the output cutting parameters and to determine the directions for improving the efficiency of the method. Technological dimensional chains formed during the cutting of grooves by a vortex method have special schemes and features of vector relationships. In the literature, there is no information on this type of dimensional chains [3, 4]. It is advisable to use the theory of dimensional chains and the theory of vectors in the study of the accuracy of grooves cut by a vortex method [3, 26, 27].

On the other hand, the accuracy of the circumferences of the groove is closely related to the mechanism of their formation along the contour. However, there is no information on the shape of grooves that can be formed by cutting them with a vortex method. In fact, it is impossible to deny the possibility of using analogical designs in the future, with high efficiency, in various technical means.

### 2.2 Technological possibilities of groove cutting by a vortex method

The groove cutting with a vortex method on the surfaces of rotation is a progressive technology.

Since this method is new, information about its technological capabilities is limited. Nevertheless, the current level of development of equipment and technology creates ample opportunities for use in the design of surface parts, which until now were considered not technologically, but more efficient from the point of view of operation. Therefore, it is of interest to investigate the technological possibilities of cutting groove by a vortex method. As, thus, there is an opportunity to use progressive, already technological designs of grooves, etc., in the design of parts and devices for the purpose, when design them.

The design of the tool head, during cutting groove, is a reflection of the dimensional relationships, inherent in the groove. The tool distance from the axis of rotation is the diameter of the circumference of the groove. This size forms the diameter of the circumference of the groove in the axial-longitudinal section, during the groove cutting process (Fig. 2). The feature of the method ensures high stability of this size.

When the groove is cut with a vortex method, the functional relationship between the linear and circular feeds plays a special role in the formation of groove dimensions on the cross section, perpendicular to the axis of the liner. By controlling this functional relation, it is possible to control the design (contour shape) of the groove and the accuracy of its dimensions. Figure 3 shows some forms of grooves that can be cut by a vortex method. All shapes on the side surfaces of the rotation are formed by the vertex method, and there is a symmetric form that is perpendicular to the axis of the cross section (Fig. 3). However, the symmetrical form of the groove regarding the plane is perpendicular to the axis of the surface of rotation are divided, two groups: symmetrical (Fig. 3a, b) and asymmetrical (Fig. 3c, d). Therefore, the first group includes a double symmetric and the second group includes a single symmetric form of the groove. The projections on the horizontal plane that passes through the axis of the surface of the rotation of the double symmetric grooves are an oval form (Fig. 3a, b). The left and the right sides of the horizontal plane that passes through the axis of surface of the rotation in the single symmetrical form of the grooves are different from each other (Fig. 3c, d). The left and the right semi-parts of the projections of grooves on the horizontal plane are semicircular, semi-oval with a small radius, semi-oval with a large radius, etc. Fig. 3 Designs of groove that can be cutting by the method of vortex

When the groove is cut, the dimension $$D_{1}$$ is provided by the tool dimension, and the size $$D_{2}$$ is the functional relationship between the linear and circular feeds $$f_{t} = F\left( {f_{m} } \right)$$.

We consider some characteristic variants of the connection:
1. A.
The groove design is symmetrical with respect to the plane passing through the symmetry axis of the part:
1. 1.

The diameter of the groove in the longitudinal section is $$D_{1} = D$$ and the diameter of the groove in the cross (perpendicular to the axis of the detail) section $$D_{2} = \left( {D + a} \right)$$. In this case, $$L = (D + a)$$; the relationship between linear and circular feeds is expressed as follows:

$$f_{t} = \frac{{\pi \cdot D_{0} \cdot f_{m} \cdot \arcsin \frac{D + a}{{D_{0} }}}}{180 \cdot (D + a)}$$
(2)

In this case, the groove, cut by the vortex method, has the shapes as shown in Fig. 3a.

2. 2.

The diameter of the groove in the longitudinal section is $$D_{1} = D$$, and the diameter of the groove in the cross section is $$D_{2} = \left( {D - a} \right)$$. In this case, $$L = (D - a)$$; the relationship $$f_{t} = F\left( {f_{m} } \right)$$ will be:

$$f_{t} = \frac{{\pi \cdot D_{0} \cdot f_{m} \cdot \arcsin \frac{D - a}{{D_{0} }}}}{180 \cdot (D - a)}$$
(3)

In this case, the groove, cut by the vortex method, has the shapes as shown in Fig. 3b.

2. B.

The design of the groove is asymmetrical, with respect to the plane passing through the axis of symmetry of the part:

In this case, the functional relationship between the linear and circular feeds $$f_{t} = F\left( {f_{m} } \right)$$ differs for the left and right parts of the groove. If the groove diameter in the longitudinal section is equal to $$D_{1} = D$$; and:
1. 3.

The diameter of the right side (with respect to the axial plane) of the groove is equal to $$D_{21} = D$$, and the diameter of the left side of the groove (with respect to the axial plane) is equal to $$D_{22} = \left( {D + 0.5a} \right)$$ (Fig. 3c). In this case, the relationship between the linear and circular feeds is determined for the right-hand side by formula (1) and for the left-hand side by formula (2), and $$L = \left( {D + 0.5a} \right)$$.

2. 4.

The diameter of the right side of the groove is equal to $$D_{21} = D$$, and the diameter of the left side of the groove is equal to $$D_{22} = \left( {D - 0.5a} \right)$$ (Fig. 3d). In this case, the relationship between the linear and circular feeds is determined for the right-hand side by formula (1) and for the left-hand part by formula (3), and $$L = (D - 0.5a)$$.

3. 5.

The diameter of the right side of the groove is equal to $$D_{1} = D$$, and the diameter of the left side of the groove is equal to $$D_{21} = \left( {D + 0.5a} \right)$$ and of the right side of the groove is equal to $$D_{22} = \left( {D - 0.5a} \right)$$. In this case, the relationship between the linear and circular feeds is determined for the right-hand side by formula (2) and for the left-hand part by formula (3).

Therefore, to cut the groove with different sizes in the right and left side of the groove, different relationships between the linear and circular feeds, respectively, should be provided.

### 2.3 Vector dimension chains of diametrical groove sizes and their analysis

When the is groove cut with a vortex method, the dimension $$D$$ and its circumferential accuracy have special meanings, both technological and functional considerations. Therefore, for the development of an efficient process, cutting a groove with a vortex method and the full use of the technological capabilities of the method, the derivation of the mathematical model of the groove diameter and its analysis are of paramount importance [22, 23, 24].

Figure 3 shows a diagram of the formation of the diameter of the inner circumference of the groove. Analysis of the processing circuit shows that, in theory, the diameter of the internal circumference of the groove is a function of three equal and overlapping dimensions (the size of the tool’s position in the head $$D_{{\text{t}}}$$; the size of linear displacement of the workpiece $$L$$; and the length of the chord B, formed by the circumference of the surface of rotation, when the workpiece is rotated through an angle of $$2\alpha$$; $$B = AC$$), as well as the diameter of the base surface of the workpiece, on which the groove is located (Fig. 2).
$$D = F(D_{\text{t}} ,\alpha ,L,D_{0} )$$
(4)
For arguments in (4): because of the error in placing the cutting tool into the tool head, $$D_{\text{tmin}} \le D_{\text{t}} \le D_{\text{tmax}}$$; due to the total error of the machine drive, $$L_{\text{min} } \le L \le L_{\text{max} }^{{}}$$, and $$(2\alpha )_{\text{min} } \le 2\alpha \le (2\alpha )_{\text{max} }$$; because of the scattering of the diameter of the surface of rotation of numerous workpieces, according to the tolerance on them, $$D_{0\text{min} } \le D_{0} \le D_{0\text{max} }$$. Analysis of the groove cutting scheme and formula (4) shows that in the formation of the dimension $$D$$, dimensions $$D_{\text{t}}$$, $$\alpha = F(f_{t} )$$ and $$L$$ participate in both their own scalar values and directions. Therefore, the dimensional chain, which is the closing dimension $$D$$, is expressed in vector form:
$$\vec{D} = \vec{D}_{\text{t}} + \vec{B} + \vec{L}$$
(5)

The last vector equation makes it possible to determine the diameter of the internal “circumference” of the groove along the entire contour, (which may have different meanings in different sections) formed during the cutting process and analyze it; compare the sizes, obtained in two characteristic sections [passing through the axes of the liner and the passage hole (in the future longitudinal section) and passing through the axis of the passage hole perpendicular to the axis of the liner (in the future transverse section)]. In Eq. (5), the nominal values of the vectors $$\vec{B}$$ and $$\vec{L}$$ are equal, and their directions are opposite. The accuracy of size $$D$$ is affected only by their errors.

The analysis of the vector-dimensional chain shows that the mechanisms of the formation of dimensions ($$D$$ and $$D_{m}$$) differ in the longitudinal and transversal sections of the groove, but obey the general law. However, on the longitudinal section, the circular accuracy of the circumference is not affected either by circular or linear feeds. Indeed, their ($$\vec{B}$$ and $$\vec{L}$$) directions on the corresponding plane are perpendicular to the size $$D$$. To mean, $$\vec{D} = \vec{D}_{\text{t}}$$. The shaping of the longitudinal sectional dimensions is only achieved by the static dimensions of tool $$b_{\text{t}}$$ and of fixture $$D_{\text{t}}$$ (Fig. 2b), and the accuracy of the cross-sectional dimensions is affected by the kinematic-dimensional relationships of the technological system, linear and circular feeds. The latter are the closing dimensions of numerous dimensional relationships, as well as dynamic. Thus, their range of variation is wide and they are formed by both static and kinematic and dynamic dimensional constraints. As the groove is formed, numerous dimensional relationships, acting at the beginning of cutting, a wide range of variation and complex character gradually changing, their number and range of variation decrease. The conditions mechanism of the influence of technological factors on the accuracy of output parameters softens. When the workpiece is rotated through an angle of $$90^\circ$$ (which corresponds to the longitudinal section of the groove), the number of influencing factors becomes minimal, and the conditions for forming the groove are the most favorable. In the continuation of the groove cutting, the dimensional constraint that affects the dimensions being formed changes in the opposite direction, their number increases, and the mechanism of their influence on accuracy, etc., gradually becomes more complicated. The accuracy of groove formation is complicated.

Thus, static dimensional relationships of diameters are formed from vector equations (5).
$$D = D_{\text{t}} \quad {\text{and}}\quad D_{m} = D_{\text{t}} + b_{\text{t}} + b_{\text{t}}$$
(6)

In the longitudinal section of the workpiece, the diametrical dimensions of the groove are formed by static constraint: the width of the tools and their distance from the axis of rotation, which are called the tool size and fixture size . Naturally, the dynamic constraints: the wear of cutting tools, the elastic and thermal deformations of the workpiece and tools also affect the accuracy of the closing dimension in expressions (6).

In the cross section, the constraints of diametrical dimensions of the groove are formed below the following relative movements and dimensions of the tool and workpiece:
• the rotation of tool, its distance from the axis and width, ($$D_{\text{t}}$$ and $$b_{\text{t}}$$),

• moving the workpiece in a direction perpendicular to its own axis, ($$L$$),

• turn the workpiece around its own axis, angle of rotation ($$2\alpha$$ and the dimensions that affect it $$B$$ and $$D_{0}$$).

In this section, the diameter of the circumference of the groove is formed not only with the already mentioned static constraints, but also by the kinematic-dimensional constraints inherent in the rotation of the workpiece when cutting the groove. Hence, from vector Eq. (5), we obtain the dimensional chains of the groove circle diameters on the cross section (Fig. 2),
$$D = L - D_{0} \cdot \sin \alpha + D_{\text{t}}$$
(7)
$$D_{m} = L - D_{0} \cdot \sin \alpha + D_{\text{t}} + 2b_{\text{t}}$$
(8)

### 2.4 Analysis of errors influencing the accuracy of the groove circumference, mathematical models of total errors

As the analysis of last expressions (7) and (8) shows, each dimensional chain has three components (linear displacement, chord and tool size) whose nominal dimensions are equal. Since two of them are directed against each other, they do not influencing the nominal sizes of the closing dimensions. However, their changes and deviations from the nominal cause changes in the dispersion of the sizes $$D$$ and $$D_{m}^{{}}$$. In order to avoid the complication of Fig. 2, their effect on the dimensions of $$D$$ and $$D_{m}^{{}}$$ is not shown. The initial error, resulting from the complex action of tolerance of diameter $$D_{0}$$ and the error of the angle of turn of the workpiece $$2\alpha$$, is determined by the formula (Fig. 2):
$$\delta_{{\text{D}_{0} }} = D_{\text{max} } - D_{\text{min} } = \delta_{\alpha } + 2\delta_{{\text{D}_{0} }} = \frac{{\pi D_{0} \cdot \Delta \alpha }}{360} + T_{{\text{D}_{0} }} \cdot \sin \alpha_{\text{min} }$$
(9)
The mechanisms of influence of technological factors $$L$$ and $$f_{t}$$ on the size $$D$$ are similar and depend on the angular position of size of the groove (Fig. 4, for the required size $$D_{x}$$ angle β). In accordance with the mechanism of grooving cutting by a vortex method, the formation of the groove diameter $$D$$ in any of its sections is realized with two positions of the cutting tool, relative to the workpiece. Fig. 4 Top view of groove shaping scheme
Therefore, in a mathematical model of accuracy of size $$D$$, double values of the errors of the parameters $$L$$ and $$f_{t}$$ should be taken into account (Fig. 4):
$$\delta_{L} = 2\Delta L_{x} \cdot \sin \beta \quad {\text{and}}\quad \delta_{{f_{t} }} = 2\Delta f_{t} \cdot \sin \beta$$
(10)
Thus, taking into account that the numerous factors influencing the accuracy of the diameter are random, the total error of diameter $$D$$ is determined by the probability method [3, 26, 28]:
$$\delta_{{\text{D}}} = t\sqrt {\lambda_{1} \cdot \delta_{{\text{D}_{0} }}^{2} + \lambda_{2} \cdot \delta_{L}^{2} + \lambda_{3} \cdot \delta_{{f_{t} }}^{2} + \lambda_{4} \cdot \delta_{{\text{D}_{\text{t}} }}^{2} + \lambda_{5} \cdot \delta_{{\text{w}}}^{2} }$$
(11)
(a risk percentage of a reject is $$0.27\%$$, $$t = 3$$) .
Analysis of the origin of all primary errors shows that they obey basically the Gaussian law [$$\lambda_{1} = \lambda_{2} = \lambda_{3} = \lambda_{5} = 1/9$$ (except for tool wear, $$\lambda_{4} = 1/3$$)]. Taking into account the values of the coefficients in expression (11), we obtain:
$$\delta_{{\text{D}}} = \sqrt {\delta_{{\text{D}_{\text{o}} }}^{2} + \delta_{L}^{2} + \delta_{{f_{t} }}^{2} + \delta_{{\text{D}_{\text{t}} }}^{2} + 3\delta_{\text{w}}^{2} }$$
If we take into account dependences (9) and (10) in the latter, we obtain a mathematical model for the error of the internal diameter $$D$$ with the groove cutting on CNC machines:
$$\delta_{{\text{D}}} = \sqrt {\left( {\frac{{\pi D_{0} \cdot \Delta \alpha }}{360} + T_{{\text{D}_{0} }} \cdot \sin \alpha } \right)^{2} + 2\left( {\left( {\Delta L_{x} \sin \beta } \right)^{2} + \left( {\Delta f_{t} \sin \beta } \right)^{2} } \right) + T_{{\text{D}_{\text{t}} }}^{2} + 3T_{\text{w}}^{2} }$$
(12)

If we take β = 0 (or 180°) in (9), we obtain formulas for determining the error of diameter D in the cross section of the groove and for $$\beta = 90^\circ$$ (or 270°) in the longitudinal section.

The above error $$\delta_{{\text{D}}}$$ is made up of two components $$\delta_{L}$$ (and, by the same way $$\delta_{{f_{t} }}$$—errors), and they have an independent character in the process of cutting the groove (resulted in right and left). Therefore, formulas (12) and (13), they were considered as an initial error by an independent effect.

The error in the diameter of the outer circumference of the groove is determined by the formula:
$$\delta_{{\text{D}_{m} }} = \sqrt {\delta_{{\text{D}_{0} }}^{2} + \delta_{L}^{2} + \delta_{{f_{t} }}^{2} + \delta_{{\text{D}_{\text{t}} }}^{2} + 2 \cdot 3\delta_{\text{w}}^{2} }$$

By accepting the tolerance of the tool width to a symmetrical form, it is participating twice with 0.5 part of it in internal diameter of groove and its other 0.5 in external diameter of the groove.

If we take into account the values of the primary factors errors, we get:
$$\delta_{{\text{D}_{m} }} = \sqrt {\left( {\frac{{\pi D_{0} \cdot \Delta \alpha }}{360} + T_{{\text{D}_{0} }} \cdot \sin \alpha } \right)^{2} + 2\left( {\left( {\Delta L_{x} \sin \beta } \right)^{2} + \left( {\Delta f_{t} \sin \beta } \right)^{2} } \right) + T_{{\text{D}_{\text{t}} }}^{2} + 6T_{\text{w}}^{2} }$$
(13)

Formula (13) is the mathematical model of the error the diameter of the outer circumference of the groove, when it is cut on CNC machines.

When the cutting of the groove is carried out on the boring machine, to provide a functional relationship $$f_{t} = F(f_{m} )$$ between linear and circular feeds, a special device is used. The functional relationship is carried out by a rack-and-pinion transmission. Rack–pinion is installed in the device fixture, on the workpiece axis. The (gear) rack is fixed to the column of the machine, motionless. When a linear feed is communicated in workpiece by means of the table of the machine (on which the device is mounted), the rack–pinion returns around its axis, due to the engagement with the (gear) rack. The workpiece returns, according to law, provided in the construction of the device. An analogous way, for this case, was to derive a mathematical model of the errors of the external and internal circumferences of the groove. Cutting a groove with a vortex method with one tool makes it possible to reduce the machine time by eight times, compared to a end fill cutter. When machining with two tools, the machine time is reduced by a further 1.5 times. In this case, all qualitative characteristics of the groove being cut are also within the required limit .

Examination of expression, (12) and (13), one obtains complete information on the formation of diametrical groove precision.

According to the theory basis of the production technology of machines, the tolerance of the closing dimension (mean, $$T_{\text{D}} = 0.8\,{\text{mm}}$$) must be distributed between its components [3, 4, 28].

### 2.5 Practical application

The received mathematical model (12) is applied when developing the process of cutting a flute by a vortex method. Here, its application at design of the equipment and mechanical operation is emphasized. In this case, in the mathematical model (12), the quantities $$\Delta \alpha$$ and $$\Delta f_{t}$$ formed the errors of the device parameters, which are the constituent links of the derived dimensional chains $$\Delta \alpha$$ and $$\Delta f_{t}$$.

According to dependency (12), the necessary conditions for ensuring the diametrical accuracy of the groove during its formation are:
\begin{aligned}&\left( {\frac{{\pi D_{0} \cdot \Delta \alpha }}{360} + T_{{\text{D}_{0} }} \cdot \sin \alpha } \right)^{2} + 2\left( {\left( {\Delta L_{x} \sin \beta } \right)^{2}}\right. \\&\quad \left.{+ \left( {\Delta f_{t} \sin \beta } \right)^{2} } \right) + T_{{\text{D}_{\text{t}} }}^{2} + 3T_{\text{w}}^{2} \le T_{\text{d}}^{2}\end{aligned}
(14)
The left side of the inequality (14) is the total error of the diameter of the inner circumference of the groove. Analysis of inequality (14) reveals that its left side gets its extreme value at $$\beta = 0^\circ$$ and $$\beta = 180^\circ$$, that is, as distinct and above the cross section of the part. Then,
$$\left( {\frac{{\pi D_{0} \cdot \Delta \alpha }}{360} + T_{{\text{D}_{0} }} \cdot \sin \alpha } \right)^{2} + 2\left( {\left( {\Delta L_{x} } \right)^{2} + \left( {\Delta f_{t} } \right)^{2} } \right) + T_{{\text{D}_{\text{t}} }}^{2} + 3T_{\text{w}}^{2} \le T_{\text{d}}^{2}$$
(15)
Using basic formula (1), $$\Delta f_{t}$$ is determined:
$$\Delta f_{t} = \frac{{\pi D_{0\text{max} } \cdot f_{m\text{max} } \cdot \arcsin \frac{{D_{\text{min} } }}{{D_{\text{max} } }}}}{{D_{\text{min} } }} - \frac{{\pi D_{0\text{min} } \cdot f_{m\text{min} } \cdot \arcsin \frac{{D_{\text{max} } }}{{D_{\text{min} } }}}}{{D_{\text{max} } }}$$
(16)

In formulas (15) and (16), the quantities $$D_{0}$$, $$T_{{\text{D}_{0} }}$$, $$D_{\text{max} }$$, $$D_{\text{min} }$$, $$D_{0\text{max} }$$, $$D_{0\text{min} }$$ and $$\alpha$$ were determined according to the size as shown in Fig. 1.

Their values accepted according to technical characteristics of the machine, with increase by 20%, considering its state. Therefore, for $$T_{{\text{D}_{\text{t}} }}$$, we have adopted $$IT\,\,7$$ for dimension, ISO 286-2.

Thus, in inequality (15), $$\Delta \alpha$$ and $$T_{\text{w}}$$ are unknown, and $$\Delta \alpha$$ elements adaptation is, in turn, the closing-unit dimensional chain of the corresponding sizes (distance from an axis of a pinion wheel to the average line of tooth of a rack, a step of tooth of a rack, dividing diameter of the pinion, etc.).

After analyzing the results obtained from inequality (15), and taking into account production experience, quantitative values were assigned to $$\Delta \alpha$$ and $$T_{\text{w}}$$. These will represent further work on the production of the “Inserts” as shown in Fig. 1 and ensures the efficiency of the process.

From this point of view, ensuring the accuracy of the inner diameter of the groove is not as simple as expected at first glance. Investigating the process of cutting the groove on the details from the insert in the Scientific and Production Association “Sabunchu,” it was found that the accuracy is ensured only with the provision of high-precision adjustment of the technological system. Such work takes quite a while. Consequently, the tolerance foreseen for the adjustment was rather hard. Therefore, it is very difficult to maintain the required tolerance. To improve the efficiency of groove cutting, technological measures taken were developed and proposed. For example, using the mathematical models (12) and (13), a rational distribution of the tolerance of the closing dimension between its components was defined and applied. The process of adjusting the technological system for performing the operation in the enterprise is facilitated, and the formatting of the groove has been improved.

## 3 Conclusions

Cutting grooves on rotating surfaces with a vortex method ensures high processing efficiency. The essence of the work is that the groove is cut with rotatable tools whose rotational axes are perpendicular to the surface of rotation, while ensuring a functional connection between circular and linear feeds of the workpiece.

Cutting a groove with a vortex method with one tool leads to a reduction in the machine time by eight times, compared to the treatment with a end mill. When processing with two tools, the machine time is reduced by another 1.5 times. In this case, all qualitative parameters of the groove being cut are also within the required limit.

Two variants of the vortex method cut grooving have been developed:
• A functional connection was established between the circular and linear feeds of the workpiece, and a “family” of circumferential grooves was determined, which are technological when they are cut by vortex meth. In the future, such grooves can be used in engineering.

• With the use of the theory of dimensional chains and the theory of vectors, technological vector-dimensional chains for outer and inner diameters of circumferences of the groove were compiled. The factors influencing the accuracy of diametrical dimensions of the groove are revealed; the mechanism of their influence on accuracy is determined, and a functional connection between them is established, analyzed ensuring accuracy.

• A new type of dimensional chains was revealed: “linear–circular,” which till now is not known to science. Mathematical models of the accuracy of diametrical sizes of the outer and inner circumferences of grooves cut on the lateral surfaces of rotation are given, and they are recommended for use in predicting the expected accuracy in similar processing processes.

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