Analytical Model of Thread Turning Insert for Workpieces with Different Machinability

1. Introduction

The modern process of research of hydrocarbon [1], geothermal [2], water [3] and soil resources [4], as well as environmental [5], recreational monitoring [6] and scientific research of subsoil and development of underground infrastructure of large cities [7] is in the focus of public attention. The key technological tool in these processes is drill strings [8], long elastic systems with a large number of threaded connections [9], which ensure mechanical integrity and tightness of the columns in difficult well conditions [10]. Therefore, increased requirements are imposed on the strength of the connection material [11] and at the same time on its accuracy [12]. At the same time, the main advantage of threaded connections, namely the possibility of multiple assembly and disassembly, sometimes turns into a source of failures: self-unscrewing under the influence of vibration-cyclic loads [13], violation of the tightness of sealing areas [14], accelerated wear, etc. [15]. As a result, downtime, maintenance costs and the risk of accidents increase. One of the key factors limiting the effectiveness of deep and directional drilling is uncontrolled dynamics of the drill string [16]: torsional vibrations [17], as well as axial and transverse vibrations [18]. Dynamic loads accelerate the wear of bits [19] and downhole equipment [20], worsen the operation of threaded connections [21], and affect the resource of engines and ground systems [22]. A well-known approach to solving this problem is based on the use of special vibration-proof devices [23]: drilling shock absorbers and dampers [24], elastic spindles [25], downhole motors, etc. [26]. This improves the operating modes of threaded connections, but does not eliminate the root causes of the problems. The operating environment of drill strings is characterized by abrasiveness, corrosive aggressiveness, high pressures and temperatures, as well as the action of long-term cyclic loads [27,28]. The increase in the depth and complexity of well trajectories places special demands on materials, smart structures and modern technologies for designing and manufacturing connections [29] - from the level of microgeometry to controlled residual stresses and roughness parameters [30], which determine tightness [31] and sensitivity to fatigue failure [32]. From the point of view of continuum mechanics [33], threaded profiles contain sharp-ended stress concentrators [34,35]. The stress-strain state of pipes and cylindrical shells near such concentrators in contact has been studied in detail in [35,36], which is directly relevant for the analysis of the thread root [35] and first thread turns [36,37].

All of the above operational requirements and conditions pose the corresponding tasks of modernization of machining for manufacturers of heavily loaded threads. Therefore, a significant analysis of the machining process of parts that are: mass-produced and are functionally responsible is needed. For example, the improvement of machine tools proves that it is possible to significantly increase productivity without actually losing accuracy [38], and the applied modular approach enhances optimization in the design of machine tools [39]. In addition, a significant contribution to the intensification of parts manufacturing is a scientifically based technological route, which allows significantly optimizing the process of creating threaded parts [40,41], as well as effective management of the life cycle of production assets [42]. However, the first priority is the modernization of cutting tools themselves: from the methods of their application to their geometric parameters [43].

2. Review of the Thread Turning Achievements and Problems. Aim of Research

Studies on thread cutting in AISI 304L stainless steel indicate that an important argument for reducing cutter wear is the use of a combined infeed method (incremental feed) [44]. In [45], a significant increase in cutter stability in the process of turning a critical buttress thread from high-chromium stainless steel was proven by using a positive rake angle of 12°. To increase the accuracy of threads, it is important to control profile deviations caused by deformations [46]. In studies [47,48], modeling the tool trajectory during thread turning, taking into account the influence of vibration, made it possible to increase the stability of the cutting edge profile. The aim of the study [49] is to increase the productivity of turning high-quality threaded aluminum parts. To achieve this goal, the front surface of the cutter was supplemented with micro-holes using a laser to retain solid lubricating micro and nano materials in them. Vector analysis predicting tangential and radial deviations of conical screw surfaces [50] shows possible ways to improve thread turning processes. A large number of input technological parameters were taken into account in a scientific study of the surface condition of the external thread on an aviation part made of the difficult-to-machine material Incolonel 718 [51]. In [52], tests of turning titanium alloy threads showed the effective effect of a cryogenic environment on reducing the amount of tool blade wear. The study [52] also concerned the development of a mathematical model of cutter wear to determine the expected tool life. Measurement of the profile of the resulting thread and determination of its accuracy using a special device was proposed in [53]. In contrast to measurement, in the analytical work [54], an algorithm for calculating the conical thread profile depending on the geometric parameters of the blade of a turning thread cutter with deviations is presented. The geometric three-dimensional thread turning simulator developed on the basis of FreeCAD [55] can serve as a check or supplement to the analytical algorithm proposed in the work [54]. These results show the formation of a convex or concave thread profile instead of the expected rectilinear one. Based on the kinematics of the process of forming a conical helical surface, a method is proposed that enables the accurate formation of the screw pitch as a function of the angle of rotation of the part [56]. The study [57] presents an analysis of the movements of the cutting blade of the cutter during the process of turning a conical thread. The formation of helical linear surfaces and the accuracy of worm manufacturing are closely related to the influence of geometric parameters, including the angle of inclination of the helix. Therefore, approximate forms of mathematical models of helicoid accuracy are often used [58]. Measuring the accuracy of the lead screw thread pitch by the traditional method and by the method that takes into account the axial displacement of the shaft indicates a higher accuracy of the latter [59]. This is important for precision threads. The measuring and testing base of mechanical processing increasingly uses advanced thermoelectric and photovoltaic materials [60,61]. In [62], it is proposed to design a variety of cutting tools, including threaded ones, automatically using the developed surface generation method. Thread turning is a special type of cutting that requires in-depth research using systems for measuring and analyzing cutting forces [63]. Turning is always accompanied by high temperatures at the tip of the cutter blade. In [64], the cutting of low-alloy hardened steel in dry and minimum-lubrication conditions (MQL) was investigated. As the source [65] proves, the size of both the wear crater and the side wear of the cutting part directly depend on the duration of the cutting process with the insert. Therefore, the quality of the cutting insert for thread turning depends on the operating conditions: technological parameters (speed, feed, depth), cutting methods (profile, generator, combined), methods and types of lubrication and cooling, additional influencing factors (forced vibrations, cryogenic environments), as well as on the design of the tool, which takes into account all the listed factors.

Global manufacturers of cutting tools offer carbide inserts for thread turning only with a zero rake angle (Figure 1a) [66]. However, one of the most influential factors that increase the efficiency of thread turning is the use of a non-zero rake angle of the cutter. Scientific and technical studies prove its effectiveness, especially in the manufacture of parts from difficult-to-machine materials [44,45] (Figure 1b and Figure 1c).

According to primary sources [67,68], the value of the rake angle, together with the cutting speed, has a significant impact on the stability of the cutter during turning (Figure 2).

Despite the fact that standard thread cutters have only zero values of rake angles, recommendations on the values of rake angles for thread turning in workpieces made of various materials are widespread in the metalworking information space (Table 1).

However, it is obvious that the practice of using cutters with a non-zero rake angle is associated with significant possible deviations of the resulting threads in terms of average diameter, pitch, and especially profile [54,55]. In addition, in the case of large-pitch threads (i.e., when the helix has a large lead angle), there is a need to use a non-zero inclination angle of the cutting edge, which increases the deviation of the thread profile [58].

Therefore, the purpose of the study is to analytically determine the profile of a carbide insert for high-precision thread turning with minimal profile deviations depending on the geometric parameters of the thread and the tool (rake angle and cutting edge inclination angle).

3. Profile Accuracy of the Large Pitch Thread for Heavy Using Conditions

Threads with large helix angles, which are most often used in heavily loaded and critical screw pairs, can be divided into three groups according to their profile: conical threads for drill strings of a triangular profile, lead threads for kinematic pairs of a trapezoidal profile (for example, lead screws for metal-cutting machines), dynamic lead screws with a wedge-shaped buttress thread (for example, for hydraulic presses and jacks).

3.1. Profile of Conical Lock Thread According to the API and GOST Standards

Rotary drill string is a threaded drill pipe designed to transmit high torque from the ground rotor to the drill bit at the bottom of the string. In addition, a high-pressure flushing solution (up to 20 MPa) is supplied inside the string from top to bottom. An additional operational requirement is the need for multiple screwing and unscrewing of the pipes during the string lowering and lifting operations to replace a worn bit or for repair. All this together imposes high requirements on the accuracy of the tool-joint conical thread, especially its profile. The strength of the connection, its tightness and unhindered multiple screwing depend on this.

According to ANSI/API standards [69] and GOST standard [70] (Figure 3), such threads are made with nominal pitches P: 4.233 mm, 5.080 mm, 6.35 mm. Diameters d_2t (pitch diameter) have values from 30 to 203 mm at the larger base of the cone. The thread flank angle (half profile angle) ϕ/2 =30°± 45 ' (Figure 3).

To determine the lead angle of the thread helix, the formula is used:

$$\psi =arctan{\displaystyle \frac{P}{\pi d_2}}$$

(1)

Pitch P = 4.233 mm is only possible for threads of small pitch diameters of 27 - 41 mm. Therefore, according to the formula, the largest value $\psi =\mathrm{2,86}°$. Further values of the lead angle are less than 1.87° (pitch diameter from 59.8 mm).

3.2. Trapezoidal Thread Profile According to ISO, ASME and GOST Standards

The main purpose of trapezoidal threads is rotational -translational kinematic pairs with significant axial loads and increased requirements for accuracy of movements. This imposes high requirements for the accuracy of the pitch and flank angle, as well as the strength and wear resistance of the lead screw and nut.

According to GOST [71]: H=1.866P, h₃=0.5P. According to GOST [71] and ISO [73]: flank angle ϕ=30°, root width k=0.366P, root width along the pitch diameter is 0.5P (Figure 4). According to ASME [72]: root width k=0.3707P, root width along the pitch diameter is also 0.5P (Figure 4).

The accuracy of the profile angle ϕ=29° (Figure 4) of the trapezoidal thread is regulated by the ASME standard [72] and is dependent on the number of threads per inch (Table 2). That is, at Threads/In = 3 (i.e., thread pitch P≈8 mm), the accuracy of the flank angle is 22 min.

To determine the largest lead angle of a trapezoidal thread according to the ISO standard, one should select, according to [73], the nominal and pitch diameters of the thread of the smallest value and simultaneously with the largest pitch P. Table 3 contains data that can be used to determine the lead angle of the thread according to formula (3.1).

Therefore, according to formula (1), the largest lead angle in a trapezoidal thread with a major diameter of 24 mm and a pitch 8 mm is 7.26°.

3.3. Buttress Thread Profile According to the ASME Standard

The main purpose of buttress threads is to withstand high axial loads in one direction, for example, industrial jacks, casing pipes for oil and gas wells. With significant axial loads and increased requirements for accuracy of movements, which imposes high requirements for the accuracy of the pitch and flank angle, as well as the strength and wear resistance of the screw and nut.

According to the ANSI B1.9–1973 standard [74]: H=0.89064P, h=0.6P, r₁=0.07141P, f=0.14532P, F=0.16316P, ϕ₂=45°, ϕ₁=7°.

Figure 5. Buttress thread pattern according to ANSI standard [74]. Symbols: d – major diameter, d ₁ – minor diameter, d₂ – pitch diameter, P – pitch, H – height of sharp-V thread, h – basic height of thread engagement, r₁ – root radius (theoretical), f – crest truncation, F – crest width.

Figure 5. Buttress thread pattern according to ANSI standard [74]. Symbols: d – major diameter, d ₁ – minor diameter, d₂ – pitch diameter, P – pitch, H – height of sharp-V thread, h – basic height of thread engagement, r₁ – root radius (theoretical), f – crest truncation, F – crest width.

The accuracy of the flank angles of the thread according to the ANSI standard [74] is represented by the tolerances for angle deviation: 4 min for ϕ₂ = 45° and 5 min for ϕ₁ = 7° (Table 4).

Therefore, according to formula (1), the largest lead angle in the buttress thread with a nominal diameter of 1.5´ (≈38 mm) and a pitch of P≈8 mm is 3.83°.

3.4. Comparison of Requirements for the Accuracy of the Thread Profile Angle and the Lead Angle

So nominally the most demanding in terms of accuracy of flank angles is the buttress thread according to the standard [74] with a deviation of 4 min, which is δ=4/(45*60) = 0.15% of the nominal value (Table 4). The trapezoidal thread according to the ASME standard [72] with a deviation of 18 min can be considered less accurate, i.e.: δ=18/(60*14.5)=2%. The least demanding profile accuracy is the tool-joint thread with a deviation of 45 min, which is δ=45/(60*30)=2.5% of the nominal value of the profile angle (Figure 3).

Table 5. Summary of maximum data for buttress, trapezoidal and tool-joint threads in terms of profile angle deviation and lead angle.

Parameters	Tool-joint	Trapezoidal	Buttress
Maximum permissible relative deviation of the profile angle δ	2.5%	2%	0.15%
Maximum thread lead angle ψ (maximum helix lead angle)	2.86°	7.26°	3.83°

Table 5. Summary of maximum data for buttress, trapezoidal and tool-joint threads in terms of profile angle deviation and lead angle.

Parameters	Tool-joint	Trapezoidal	Buttress
Maximum permissible relative deviation of the profile angle δ	2.5%	2%	0.15%
Maximum thread lead angle ψ (maximum helix lead angle)	2.86°	7.26°	3.83°

It should be noted that buttress threads (P=5.08 mm) as well as triangular tapered threads (P=3.175 mm) are also widely used in the oil and gas industry, for example for connections of well casing pipes. However, despite the high mechanical loads and requirements for corrosion resistance, these threads have quite large diameters (114-508 mm) and therefore, with similar pitch values [75], have a significantly lower lead angle than those used in highly loaded threads [76], for example in powerful jacks.

4. Modern Approaches to Ensuring the Accuracy Form of the Production of Large-Step Threads Made by Turning

4.1. Providing Directrix and Generatrix Lines When Turning Helical Surfaces

All modern threads contain two helical surfaces, with a constant pitch P. Figure 3 and Figure 4, 5 show a blue line cp, which is the directrix of one of these two helical surfaces. Points 2 and 3 belong to this directrix and are at the same time profile points of the threads: tool-joint, trapezoidal, buttress. Other profile points of these threads 1 and 4 belong to the generatrix of the second helical surface. Since lathes are widely used to manufacture the specified threaded and helical surfaces, considerable attention of manufacturers and researchers is focused on thread cutters and their ability to process difficult-to-machine materials, which include high-strength and corrosion-resistant steels [77]. Of course, such abilities are developed due to the use of modern composite materials for coatings of cutting threaded inserts [78]. At the same time, the actual geometric parameters of the cutters and their movements during thread turning are very influential factors, especially for difficult-to-machine materials [45].

Usually, in the case of threading on a lathe, the cutter is installed in such a way that its cutting edge 4-1-2-3 is aligned with the axial plane of the thread [66,76] (Figure 6). Each point of the cutting edge performs an absolute movement along the trajectory of the helix with a pitch P. Points 2 and 3 belong to one helical surface and two boundary helices of this surface: the blue helix with the smallest radius r₂ (point 2) and the red helix with the largest radius R₃ (point 3). All other points forming the helical surface are located between points 2 and 3. Another helical surface is formed by the section of the cutting edge located between points 1 and 4. The absolute movement of the edge points along the helix is ensured by the constant rotational movement of the workpiece around the thread axis X and the translational movement of the cutting edge points along the axis X. In this case, the module of translational movement $\left|\overrightarrow{S_t}\right|$(mm per rotate) is equal to the step size P (mm).

Provided that the plane of the cutting edge of the tool is completely aligned with the axial plane of the thread, profiles of the edge and the thread will coincide, and accordingly their profile angles ϕ will coincide (Figure 7a).

Theoretically, the helical surface is formed by a truncated cone (guide cone), which rotates with a constant circular speed around its axis X and moves translationally along this axis. If the cutting edge 3-2 of the cutter lies in the axial plane of the guide cone - the plane X_p-X_c -c-p, then a regular closed cone is formed in the case of radial feed $\overrightarrow{S_r}$ (Figure 7c). The straight-line generator of the cone pc will coincide with the edge 2-3 when the cutter completes the radial feed $\overrightarrow{S_r}$. Such a guide cone provides a theoretically correct closed helicoid (otherwise Archimedes' screw), which defines most fastening threads and all buttress threads and trapezoidal lead screws (Figure 7b). The profile of the guide cone is the same in all axial planes, including ZX. The directrix cp in the ZX plane is described by the equation (Figure 7d):

$$z=-x·\mathrm{c}tg\left({\displaystyle \frac{\phi }{2}}\right)$$

(1)

4.2. Deviation of the Cutting Edge of the Thread Cutter from the Generating Helical Surface Under the Condition of Using the Cutting Edge Inclination Angle λ

In practice, leading manufacturers of cutting tools often recommend using a special inclination of the cutting insert at an angle λ to the thread axis and supply special shims for this. This inclination angle λ, which is equal to the lead angle, guarantees the same clearance angles of the right and left sides of the insert [66,76] (Figure 8a). This is done to maintain the same operating conditions of the left (1-4) and right (2-3) cutting edges, which helps to increase the tool life. For versatility, the cutting edge profile of such carbide inserts 4-1-2-3 is made identical to the profile of the corresponding thread (Figure 8b). The diagram of installing the cutter together with the holder with a rotation to an angle λ (Figure 8c) graphically explains the difference compared to installation without rotation to this angle (Figure 7b). However, such an installation is not possible in a conventional lathe toolholder.

However, it is obvious that the plane where the cutting edge 2-3 is located, in the case of its inclination at an angle λ to the axial plane of the thread (Figure 9a), will not coincide with the generator of the guide cone cp. (Figure 8c), which is always located in the axial plane (lime) (Figure 9b).

4.3. Deviation of the Cutting Edge of the Thread Cutter from the Generating Helical Surface Under the Condition of Using the Rake Angle γ

A number of studies suggest the use of a non-zero rake angle for threaded tools [45,54,55,67,68], which significantly expands the capabilities of the cutter when processing various materials (Figure 2, Table 1).

If the value of the rake static angle at the apex of the thread cutter is not zero, but the angle of inclination of the cutting edge has a zero value, then the front plane intersects the generating cone of the thread parallel to its axial plane (Figure 10). The value k is the distance from the plane of the section of the guide cone with the plane of the cutting edge of the cutter 2-3-4-1 to the axial plane. eg is the line of intersection of the plane of the cutting edge with the guide cone. Obviously, the curve eg is not equal to the line cp, but it is a real generator of the guide cone and must contain the cutting edge points between points 2 and 3 (Figure 10).

5. Analytical Calculation of the Points of the Hyperbolic Profile of the Cutting Edge with a Non-Zero Values of the Rake Angle and the Inclination Angle at the Nose

5.1. Analytical Calculation of the Hyperbolic Profile of the Cutting Edge as a Function of the Rake Angle

In fact, when a conical surface is intersected by a plane parallel to the axis of the cone, a hyperbola is formed [79,80]. So, Figure 11 presents a spatial scheme for the formation of a hyperbolic curve ge, which should be the shape of the cutting edge of the cutter to create one of the two helical surfaces of the thread with a pitch P, a profile angle of the thread ϕ and the height of the initial triangle H.

Therefore, the shaded area in the figures corresponds to the hyperbolic section of the rake plane with the surface of a truncated cone, which is the parallelism surface of a triangular, trapezoidal, or buttress thread [81].

The axes x₂ and x₁ lie, respectively, in the planes of the major and minor bases of the truncated cone.

If the cutting edge is not located in the axial plane of the truncated cone, which is given by the trace f_β (Figure 12), passing through the thread axis, then a cutter with a triangular profile, which should correspond to the thread profile during transverse feed, will reproduce not a triangular, but a hyperbolic thread profile [80].

Therefore, if we consider the section of the thread as the surface of a truncated cone (Figure 11 and Figure 12.) with a large radius R, a small radius r and an angle at the nose, then as a result of the intersection of this cone with the rake plane, which is given by the trace f, a hyperbola will be obtained [80].

Hyperbolic section of a cone with axis OX and generatrix Or with the rake plane marked by the shaded area (Figure 12.). Hyperbola O₁ emq asymptotically approaches the line Op. An arbitrarily chosen point m at the intersection of the axial plane and the cone corresponds to a point n at the intersection of the rake plane and the cone. They have a common coordinate x_n on the OX axis. The vertex O₁ of the hyperbola is distant from the vertex of the cone by a distance a along the OX axis, and the point Q is distant by a distance b along the OY axis.

To calculate the deviation of the hyperbola from the asymptote, depending on the distance from the vertex of the cone O along its axis, we use the following formula [80]

$$\left|mn\right|={\displaystyle \frac{ab}{x_{n }+\sqrt{{x_n}^2-a^2}}}$$

(2)

We obtain the dependence of the value of a on the values of the angles $\beta$and $\gamma$ and the radius r. To do this, we determine the value of k, which is marked on the projection plane W (Figure 12). From the right triangle O"O ₁ "Q":

$$k=\left|O´´Q´´\right|tan\beta$$

Using the projection plane H, we obtain the expression

$$k=a·tan\beta$$

Since the value k is equal to the length of the segment O₁´O´ on the projection plane V (Figure 12), then from the right triangle O₁´O´e we have the equation

$$\left|O_1^{\text{'}}{O}^{\text{'}}\right|=k=\left|O_1^{\text{'}}e\right|sin\gamma =rsin\gamma$$

Therefore:

$$k=rsin\gamma$$

(3)

and accordingly, since |Q₁'O'|=|Q₁''O'|=k=rsinγ:

$$a={\displaystyle \frac{r·\sin \gamma }{\mathrm{tg}\beta }};$$

From the same constructions we obtain the dependence

$$b=a·\mathrm{tg}\beta =r·\sin \gamma$$

As a result, we obtain the dependence of the product $ab$ on the values of the angles $\beta$and $\gamma$ and the radius r:

$$ab={\displaystyle \frac{{\left(r·\sin \gamma \right)}^2}{\mathrm{tg}\beta }}$$

(4)

Since point n is taken as an arbitrary point of the cone, and point m is the corresponding point of the hyperbola, then using the concept of an arbitrary i -th point, we will take its coordinates as x_i along the OX axis and r_i along the OY axis. Thus, the asymptote formula will look like this:

$$x_i={\displaystyle \frac{r_i}{\tan \beta }}={\displaystyle \frac{r_i}{\tan \left(\frac{\pi }{2}-\frac{\phi }{2}\right)}}={\displaystyle \frac{r_i}{\cot \frac{\phi }{2}}}$$

(5)

We replace the notations: $\left|mn\right|$ by ${ ∆}_i$, as well as $x_n$ by $x_i$ and after substituting expressions (4, 5) into formula (2), we obtain the formula

$${∆}_i={\displaystyle \frac{{\left(r·\sin \gamma \right)}^2/\mathrm{tg}\beta }{\frac{r_i}{\mathrm{tg}\beta }+\sqrt{{\left(\frac{r_i}{\mathrm{tg}\beta }\right)}^2-{\left(\frac{r·\sin \gamma }{\mathrm{tg}\beta }\right)}^2}}}={\displaystyle \frac{r^2·{\sin }^2\gamma }{r_i+\sqrt{{r_i}^2-{\left(r·\sin \gamma \right)}^2}}}$$

(6)

The obtained equation (6) indicates the functional dependence of the magnitude of the displacement of ${∆}_{і}$the hyperbolic profile of the cutting edge of the thread cutter at an arbitrary i-point on the value of the rake static angle $\gamma$ at the vertex point e and the radius of the cone at an arbitrary point r_i and the small radius r of the truncated cone.

An arbitrary point on the hyperbola profile can be given in parametric form:

$$\begin{gathered} y_i=r_i-{∆}_i \\ x_i={\displaystyle \frac{r_i}{\cot \frac{\phi }{2}}} \end{gathered}$$

So, we describe the profile of the hyperbola as follows:

$$y_i=x_i·\mathrm{c}\mathit{tg}\left({\displaystyle \frac{\phi }{2}}\right)-{\displaystyle \frac{r^2·{\sin }^2\gamma }{x_i·\mathrm{c}tg\left(\frac{\phi }{2}\right)+\sqrt{{\left(x_i·\mathrm{c}tg\left(\frac{\phi }{2}\right)\right)}^2-{\left(r·\sin \gamma \right)}^2}}}$$

(7)

where:

r_i – thread radius at a given point of the profile,

$x_i={\displaystyle \frac{r_i}{\cot \frac{\phi }{2}}}$, – coordinates of the thread profile (2-3) or (1-4) in Figure 4;

ϕ – profile angle of the thread

5.2. Implementation of in Python Using the Example of Analyzing the Process of Turning a Trapezoidal Thread of Diameter 24 mm with a Pitch of 8 mm Using a Cutter with a Rake Angle of γ=50°.

The obtained hyperbola functions (equation (7)) and asymptotes (equation (1)) make it possible to make their graphical display and graph-analytical comparison. Figure 13 presents the diagrams of the specified functions made by the authors in Python [82], as well as the diagram of linear interpolation of the hyperbola by the two extreme points e and g. Only in order to clearly distinguish visually between the hyperbola and its asymptote, the value of the front angle γ=50 ° was used. For example, the diagrams (in Figure 13) were simulated based on the data of the trapezoidal thread according to ISO 2904 and GOST9484-81 with a nominal diameter of 24 mm and a pitch of 8 mm (Figure 4). The diagram in Figure 13, for the sake of clarity of its perception, has been copied into Figure 14, where it is commented on with data and terms from Figure 11 and Figure 12. The calculations of the geometric parameters of a trapezoidal thread with a nominal diameter of 24 mm and a pitch of 8 mm, necessary for constructing the diagrams, are presented in Table 6.

5.3. Analysis of the Convexity (Concavity) Arrow of a Hyperbola and Its Maximum and Minimum Deviations from the Asymptote

Visual scaling of the thread profile and cutting edge of the thread cutter in Figure 15 was performed to determine:

the values of the maximum Δmax and minimum Δmin deviations of the simulated hyperbolic curve emg (which includes the model of the cutting edge profile of the thread cutter) from the asymptote sr (the given cutting edge profile, which is equal to the profile of the given thread).

the angles of the standard cutting edge profile (ϕ/2) (Figure 15a, Figure 14) and the angle of of the interpolation line eg (ϕ₁/2) (Figure 15a, Figure 14).

deviation values |Am|= si of given points m of the obtained hyperbolic profile of the cutting edge of the thread cutter emg from points A of the interpolation line eg.

profile angle ϕ₂ /2 of the simulated cutting edge intended for forming the straight-sided section of the thread profile 2-3, which belongs to the asymptote (Figure 15b, Figure 14).

5.3.1. The Formulas for Calculating the Maximum Deviations Δmax and Minimum Deviations Δmin

Deviations of the simulated hyperbolic curve emg (which includes the model of the cutting edge profile) from the asymptote cp (a given cutting edge profile equal to the profile of the given thread) can be determined based on formula (6):

$${∆}_i={\displaystyle \frac{r^2·{\sin }^2\gamma }{r_i+\sqrt{{r_i}^2-{\left(r·\sin \gamma \right)}^2}}}$$

According to Figure 12 and Figure 15, to determine Δmax, the following substitution should be made r_i=r:

$${∆}_{\mathit{max}}={\displaystyle \frac{r^2·{\sin }^2\gamma }{r+\sqrt{{\left(r\right)}^2-{\left(r·\sin \gamma \right)}^2}}}=r-{\displaystyle \frac{r·{\sin }^2\gamma }{1+cos\gamma }}$$

(8)

According to Figure 12 and Figure 15, to determine Δmin, the substitution should be made r_i=R:

$${∆}_{\mathit{min}}=R-{\displaystyle \frac{r^2·{\sin }^2\gamma }{R+\sqrt{R^2-{\left(r·\sin \gamma \right)}^2}}}$$

(9)

5.3.2. Determination of the Angles of the Interpolation Lines eg and wf.

Proflie angles ${\displaystyle \frac{\mathsf{\phi }}{2}},{\displaystyle \frac{1}{2}},2/2$ can be determined by Figure 15 and formulas in Table 7:

5.3.3. Definition of the Deviation (Concavity Value) of Hyperbola emg by Software-Analytical Method

For the analytical definition of the deviation (concavity value) of the hyperbola emg, it is necessary to find the maximum distance between the points of the hyperbola and the corresponding points of the interpolation line |Am|= si . (Figure 15), that is, find the maximum value of the difference between Y coordinates:

$$\left|{\mathrm{s}}_i\right|=y_{\mathit{int}}-y$$

(13)

where:

y is determined by formula (7) for a given array of points with coordinates x_i for i={x_c, … x_p};

y_int – for the same array of points i={x_c ,… x_p} using Figure 16 we determine using the method [80] the interpolation line eg , i.e. we create a line through the point e(x_c ,y_e) and we determine the point g (x_p ,y_g) by the formula:

$$y_{int}={\displaystyle \frac{(x-x_c)\left(y_g-y_e\right)}{\left(x_p-x_c\right)}}+y_e$$

(14)

where formulas for calculating point coordinates x_c ,x_p ,y_g ,y_e are listed in Table 7.

Using Python based on the algorithm built on the basis of formulas (7), (13-14), an application [82] was created to determine the maximum possible distance si :

5.3.4. Definition of the Arrow of Convexity (Concavity Value) of the Hyperbola Section emg, Which Corresponds to the Cutting Edge of WF by the Program in Python

Based on Figure 15 and using the Python, an application [82] was developed that allows calculating the distances between the hyperbola and the straight interpolation line drawn through the points W, F of the cutting edge of the modeled cutter for machining the threaded part 2-3. The program allows for a rectilinear approximation of the hyperbola section emg.

Figure 16 shows the distribution graphs of these distances: between the hyperbola and the interpolation line WF (upper curves), between the hyperbola and the approximation line (lower curves).

5.4. Analytical Calculation of Linear Interpolation at the Two Extreme Points of the Hyperbolic Profile of the Cutting Edge as a Function of the Rake Angle at the Nose and the Inclination Angle Simultaneously

To ensure high productivity of the thread cutter, the cutting edge plane (i.e., the rake plane) should be placed not only at the rake angle $\gamma$, but also at the cutting edge inclination angle λ, as recommended by tool manufacturers [66,76].

From the point of view of the well-known theory of conic sections [80], a hyperbola will be obtained under the conditions:

• if we cross the conical surface with a plane that is parallel to the axis of the cone. In Figure 17a , the plane f intersects the conical surface at a distance k from the axis and is located parallel to the axis. The value of k depends on the value of the rake angle according to formula (3);
• if we cross the conical surface at an angle λ whose value is greater than 0° and less than the angle at the vertex of the cone β (Figure 17b);
• if we cross the conical surface with a plane parallel to the generator of the cone, we will obtain a parabolic profile (Figure 17c).

That is, provided that $0°<\lambda <({\displaystyle \frac{\pi }{2}}-{\displaystyle \frac{\phi }{2}})°$ the cross-section of this triangular thread and the plane of the rake surface with a double slope is hyperbolic.

It is possible to determine the coordinates of the extreme points of the intersection of the plane uAv with the surface of a truncated cone with height ${\displaystyle \raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ as the lines of intersection of the hyperbola with circles with radii r and R (Figure 18).

Therefore, the indicated shaded region is bounded by a hyperbolic curve formed by the intersection of the rake flat surface of the thread cutter, whose parameters $\lambda \ne 0$ and $\gamma \ne 0$ the parallelogram surface of the triangular thread, i.e., with a cone of height h and an angle at the vertex β.

It is possible to determine the coordinates of the extreme points of the intersection of the plane uAv with the surface of a truncated cone with height ${\displaystyle \raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ as the lines of intersection of a hyperbola with circles of radii r and R (Figure 18).

In the plane of a circle with radius r (plane y0x₂) – coordinates on the v axis:

$$v_{min}=\sqrt{r^2-k^2}$$

(15)

In the same plane we have the minimum value of the coordinate along the u axis:

u_min =0

In the plane of a circle with radius R (the plane that is parallel to y0₁x₁ and contains the x₂ axis) – the coordinates on the v axis are:

$$v_{max}=\sqrt{R^2-{\left(k-{\displaystyle \frac{P}{2}}\mathit{tan}\lambda \right)}^2}$$

(16)

where:

according to the formula (3) $k=r\mathit{sin}\gamma$In the same plane we have the maximum value of the coordinate along the u axis:

$$u_{max}={\displaystyle \frac{P}{2\mathit{cos}\lambda }}$$

(17)

Determining the profile angle of the cutting edge of the cutter (according to the diagram in Figure 19):

$${\phi }_2=2arctan\left({\displaystyle \frac{u_{max}}{\left(v_{max}-v_{min}\right)}}\right)$$

(18)

6.1. Results of Calculation of the Value of the Profile Angles of the Cutting Edge of the Modeled Thread Cutters with Interpolated Cutting Edges According to Formulas (12) and (18) (Table 9)

As can be seen from the table, increasing the thread pitch to 8 mm requires a significant change in the angle λ , which in turn significantly requires a change in the profile angle φ of the cutting edge of the cutter to a value of 30.141°, in contrast to 29.969° when not using the inclination angle λ=7 °.

6.2. Model of the Cutting Edge Profile with Interpolated Straight Sides

Based on the obtained values ​​ϕ₃ (18) , as well as Δmax (8) and Δmin (9), it is possible to construct a high-precision profile of the initial theoretical triangle of the thread cutter (Figure 20).

The values ​​H, h3, P are in accordance with the standards [71,72,73,74].

The value flat1=|1-2|ּ cosλ, where |1-2| is determined based on standards [71,72,73,74].

The value δ is determined as a result of the graphical construction of the shelf TS , the length of which is equal to flat1.

7. Discussions

The data in Table 8 actually indicate that the convexity of the simulated cutting edge of 0.002 mm is technologically imperceptible at the value of the rake angle γ=20 °, and the convexity of 0.00056 mm at the value of the rake angle γ=10 ° is practically unreproducible. Therefore, taking into account these data, as well as the recommendations for the range of the rake angle within 0 ... 25 °, the authors believe that the modernization of the cutting edge profile of the threading cutter should be ensured by interpolation at two extreme points. This formed the basis for calculating the linear interpolation at two extreme points of the hyperbolic profile of the cutter cutting edge as a function of the rake angle at its apex and the angle of inclination of the cutting edge simultaneously, i.e. the application of formulas (15)-(18).

8. Conclusions

Turning cutting tools for the production of threaded parts from critical materials, especially parts with high quality (oil and gas too joints with triangle threads and buttress threads profile, jack screws and lead screws with a trapezoidal thread) require, to ensure durability, non-zero values ​​of the geometric parameters: the rake angle γ and the cutting edge inclination λ, which needs their high-precision profiling.

The most technologically advanced approach is to replace the curvilinear hyperbolic profile, which is theoretically responsible for the non-zero value of the rake angle γ and the edge inclination angle λ by straight profile based on interpolation beyond the two extreme points of the hyperbolic profile.

In the case of the need to manufacture large-pitch threads, there is a need to significantly increase the inclination angle to λ=7°, which leads to the need for a significant change in the cutting edge profile, which consists in changing the profile angle and the length of the cutting part of the tool.

The analytical approach to modelling the profile of a high-precision thread is based on functions, where the arguments are the thread parameters: diameter, pitch, as well as the geometric parameters of the cutter: rake angle and cutting edge inclination angle.