A new approach to improve noncircular turning process

Processing efficiency optimization is often conducted in production environments. For turning, however, the introduction of noncircular cross-section workpieces generates new complexity. This paper presents the kinematic analysis and efficiency optimization of turning a noncircular cross-section workpiece on the basis of ISO 10208:1991 male rope thread machining, characterized by a smooth contour. This thread can be machined, for example, with standard thread turning or using X-axis motions characteristic of noncircular objects, i.e., rope threading. In that case, selecting the proper method and machining parameter values for efficiency can be more challenging than in circular cross-section workpiece turning. The latter method avoids many tool passes but requires highly dynamic movements of the machine in the X-axis. In addition to these two methods, a hybrid method is presented that is characterized by reduced dynamics in the X-axis and more passes than the rope threading method. A description of the methods using mathematical parameters is developed to optimize the process efficiency. Numerical calculations to select a method and its associated cutting parameters are carried out for exemplary cutting edges, theoretical roughness values, tool life models, and other variables. The obtained results and the optimization algorithm of the process are presented.


À Á
Rz surface roughness (μm) f z feed rate in the Z-axis (m) r ε tool nose radius value (m) n rotational spindle speed (rpm) a M maximum tool acceleration value in the X-axis for the machine m angle that define the position of a cutting tool on the arcs (°) β 1 , β 2 angles that define the position of a cutting tool on the arcs ( o ), respectively, A 1 B 1 A 2 B 2 , β 1 ∈ (0, α), β 2 ∈ (α, 0) Units taken for calculation are as below unless otherwise stated. They may have prefixes in tables or in figures presenting results.
α inclination angle of a tangent connecting the arcs A 1 B 1 , B 2 A 2 ( Fig. 3a) (°), α ≈ 19°for the analyzed thread, and the rounded number is taken for the calculations a x1 tool acceleration value in the X-axis on arcs defined by radius R 1 m s 2 À Á a x2 tool acceleration value in the X-axis on arcs defined by radius R 2 m s 2 À Á t A1 time when the conducted analysis starts, when the tool is in point A 1 , equal to zero for further calculations (s) t A2 time recorded from the moment the tool moves from A 1 to A 2 (s) t B1 time recorded from the moment the tool moves from A 1 to B 1 (s) t B2 time recorded from the moment the tool moves from A 1 to B 2 (s) T rev spindle rotation period (s) a max x1 maximum acceleration value for t∈ t A1 ; t B1 ð Þ m s 2 À Á a max x2 maximum acceleration value for t∈ t B2 ; t A2 ð Þ m s 2 Þ À a R1 average acceleration equation in the X-axis on the arc defined by radius R 1 a R2 average acceleration equation in the X-axis on the arc defined by radius R 2 v x A 1 tool speed in the X-axis at point A 1 equal to zero m s Þ

Introduction
Noncircular turning (also known as radial contour turning) is a single-point cutting process that generates workpieces with noncircular cross sections. This turning method is frequently used in machining industrial parts such as camshafts [1] and piston heads [2][3][4][5][6][7]. The noncircular section is realized by moving a tool along the radial direction, synchronous with the workpiece rotation. The radial motion of the tool is typically implemented by a special servo tool stage. In such cases, the possibilities of efficient machining are typically limited by the ability to follow the given trajectory. Many technical solutions have been developed to ensure the best possible dynamics; some of them use voice coil actuators [6,8] or piezoelectric actuators [2,4,9]. The high-frequency synchronization of the radial motion of the turning tool with the rotation of the spindle also poses a challenge to the control system [3]. Over the years, many papers have been published about this subject [1, 2, 5-8, 10, 11]. In [8], the tracking performances of acceleration and force-feedback controllers were investigated. Wu and Chen [6] and Wu et al. [7] addressed time-varying dynamics, cutting force disturbances, and other uncertainties and proposed the concept of disturbance rejection control. Another approach by researchers Wang and Yang [2], and Hanson RD and Tsao T-C [10] focused on using repetitive control to minimize tracking error in machining an ellipse piston. A two-level control structure composed of servo control and frequency-domain learning control was used by Sun and Tsao [1]. Wu, Chen, and Wang analyzed the process for stability [11] and investigated variable spindle speed machining. Qiang, Wu, and Bing [5] studied noncircular turning also from the cutting process standpoint and proposed a spindle control with variable speed and the variable angle compensation mechanism, aiming to maintain a constant rake angle and velocity during oval piston machining. In contrast to pistons and camshafts, rope threads can typically be manufactured using both methods, i.e., noncircular turning (rope threading) and classical circular turning (standard threading). Application of the former requires only one pass of the tool, whereas the latter requires multiple tool passes performed at various diameters. This may lead to the question of which of these methods is more efficient, because high-performance turning is also an issue being studied [12][13][14]. Lee and Tarng [12] presented a sequential quadratic programming method for minimizing the production cost and maximizing its rate in multistage turning operation. In [14], Paiva et al. applied multivariate optimization to maximize the material removal rate and minimize the cutting time, costs, cycle time, and surface roughness. Additionally, they took into account the tool life, which is typically considered an important performance index when optimizing turning parameters [13,[15][16][17][18].
Another issue related to the optimization of the turning process is the surface roughness, which depends primarily on the tool geometry and preset feed rate [19][20][21][22]. To formulate the efficiency optimization issue for the rope threads, it is required to analyze the kinematics of the rope threading method, which is different from the classical turning. The analysis was made in the present paper. To optimize the machining process of the rope threads, the present paper describes the kinematic analysis of the rope threading process and presents the optimization taking into account the desired surface roughness, relationship between the nose radius and maximum tool acceleration of the tool motion, and (optionally) the chosen tool life model. The issue of possible shape errors was also discussed. Furthermore, next to the mentioned standard threading and rope threading, the paper puts forward a new hybrid threading method, which is a compromise between many tool passes characteristic for standard threading vs. singlepass machining. Obtained results and the algorithm of the machining efficiency optimization, for the given constraints and described tool life models, are presented. This paper is organized as follows. In Section 2, the theoretical background of the issue is presented. Performance optimization of rope threading without considering the tool life and with two tool life models is shown in Section 3. In Section 4, the machining methods, rope/standard/hybrid threading, are analyzed. In Section 5, an algorithm for selecting cutting parameters and threading method is presented. The calculated efficiency results are shown in Section 6. Conclusions are given in Section 7.  Table 1.
The contour is made up of arcs linked with a straight line tangent to each of them. The thread pitch is defined by P, and the thread depth is defined by h.

Machining methods of the thread
There are two typical turning methods for thread machining: standard threading and rope threading, as shown in Fig. 2. Standard thread machining requires many passes of a tool, and the feed rate is equal to the thread pitch. This method uses a relatively slow spindle speed and a relatively high Z-axis feed rate. The tool placement in the X-axis is set to be constant during the pass. The use of many positioning movements after each pass increases machining time. In contrast to the standard threading method, rope threading requires only one pass of a tool, and the feed rate in the Z-axis is smaller. However, tool movements in the X-axis must be implemented, and high accelerations are required to ensure efficient machining. For rope threading, suitable dynamic properties of a lathe drive responsible for the X-axis feed are typically the main constraint.
Both methods have their advantages and disadvantages. Standard threading requires time for tool return to avoid high dynamic movements in the X-axis. Rope threading does not need multiple passes of the tool, but its efficiency is often constrained by dynamic tool motion limitations. Therefore, a third method is proposed: hybrid threading, which achieves a favorable balance between both methods described above. In this method, the number of tool passes is less than in standard threading but more than one (as in rope threading). As a result, high accelerations of rope threading and the high feed rate in the Z-axis of standard threading are both mitigated. Detailed tool movements of the three methods are described in Section 4.

Rope threading modelled as a dynamic system
It is possible to control certain machining parameters affecting the tool acceleration in the X-axis a x and the surface roughness Rz by the input cutting parameters. Because of that, the turning machine model in rope threading can be treated as a dynamic system. In the rope threading, the input parameters affecting the mentioned a x and Rz can be distinguished as feed per revolution f z and tool tip geometry with radius r ε . The input parameter which additionally affects the a x is the rotational spindle speed n. Furthermore, the model is also time dependent as the output parameter values change during machining because of the influence of object geometry. It is assumed that for each machine, it is possible to determine the maximum acceleration of the X-axisa M . The maximum acceleration depends on the feed drive construction and the dynamic properties of the components and lathe control system. Based on the expected R z and determined a M , the input controllable parameters can be optimized.

Determination of accelerations in rope threading
Rotational movement of a workpiece can be idealized as a tool moving linearly along a thread contour because the analyzed rope thread is a single-start thread form, i.e., the pitch and lead are equal.
The tool path in the X-axis for one spindle revolution at constant speed is presented in Fig. 3a as the tool path along a profile in the Z′-axis. When the thread is rotating in the spindle, the theoretical profile of the XZ-plane cross section appears to move along the Z-axis (in the area constrained by the thread length). To describe that movement relative to the tool movement in the Z-axis, the Z′ axis is introduced. Figure 3a shows the relative movements conducted by the tool. The relative speed of this motion is designated v z′ . Depending on the directions of the spindle and feed motion of the tool in the  The Z′-axis corresponds to the angle of spindle revolution φ or time t given a constant spindle speed. The vertical axis in Fig. 3a defines displacements in the X-axis direction of the lathe. The tool path is separated from the thread contour by the value of the nose radius. Considering section A 1 A 2 , the tool accelerates in the X-axis along arcs A 1 B 1 and B 2 A 2 . In section B 1 B 2 , the tool moves along the X-axis with a constant speed. Further tool positions during machining replicate or flip this basic movement relative to the X-axis. The relation between the vectors v z′ and v x is presented in the Fig. 3b.
Accelerations on arcs vary. The tool speed at each point is the resultant of the speeds v x and v z′ and is tangent to the tool path. Therefore, the relations between speed vectors are defined by angles β 1 and β 2 which are described by β in Eq. (1), according to Fig. 3b. The speed in the X-axis can be given by Angle β 1 presented in Fig. 3a as an arc defined by radius R 1 can be written as For an angle β 2 defined as tool angular position from point B 2 on an arc defined by radius R 2 , the equation can be determined as Substituting Eqs. (2) and (3) into Eq. (1) and differentiating the result, we obtain Eqs. (4) and (6), which define  the tool acceleration in the X-axis on arcs defined by radii R 1 and R 2 , respectively: a x1 (4) and a x2 (6). The section that connects both thread arcs defined by radii R 1 and R 2 is tangent to the arcs and is a straight line, which means that tool acceleration in the X-axis in this section is zero, as described by Eq. (5).
For t ∈ (0, t B1 ) (arc A 1 B 1 ): When the directions of spindle rotations and feed f z are correctly set, the tool does not pass over the whole contour in one spindle revolution, and its value is decreased by f z . The results are presented in Fig. 4, which are schematic and may not fully reflect the real relations between dimensions of analyzed objects.
The speed values v z′ m s À Á can be given by Having substituted Eq. (7) into Eqs. (4) and (6) and having assumed angle β = α correlating with positions of maximum acceleration, we obtain equations of maximum acceleration on radii R 1 and R 2 , as shown in Eqs. (8) and (9).
Absolute accelerations on each arc are presented in Fig. 5 in the time domain. Due to jerk limitation in each feed drive system, the profile can never actually be perfectly copied due to sudden acceleration increases at the beginning and the end of each arc. It is therefore assumed that the acceleration value of each machine a M can be defined as the average acceleration on a given arc. The average accelerations on sections A 1 B 1 and B 2 A 2 are equal to the average acceleration along the whole length of arcs R 1 and R 2 , respectively.
The average acceleration on an arc defined by radius R 1 can be given by The speed v x B 1 can be determined from Eqs. (1) and (7) v The time in which the center of the tool nose travels from point A 1 to point B 1 can be determined by dividing the travelled section along axis Z′, by constant speed v z′ given by Eq. (7) After substituting Eqs. (11) and (12) into Eq. (10), we obtain the average acceleration equation in the X-axis on the arc defined by radius R 1 The time in which the center of a circle in tool nose travels from point B 2 to point A 2 can be determined by dividing the travelled section along Z′-axis by the speed constant v z′ given by Eq. (7) The average acceleration on an arc defined by radius R 2 can be found in a similar way The tool path is dependent on the tool nose radius r ε because of the distance that separates the center of a circle defining the nose from the workpiece. Figure 6a presents tool paths for different values of r ε in rope threading defined by the standard. Rope threading is possible only for r ε < R 2 . The acceleration a R1 decreases with increasing nose radius r ε , but the value of a R2 increases. Figure 6b presents the values of: The dependence is presented as the function of tool nose radius in the radius r ε ∈ (0.03e − 3, 5.00e − 3) (m). By Eqs. (13) and (15) plotted in Fig. 6b and substituting values R 1 and R 2 , the dependencies shown in Table 2 are obtained.
To ensure proper threading, the accelerations a R1 and a R2 should not exceed the acceleration value a M specified for a given machine. It was assumed that machining is most efficient when the greater value from a R1 , a R2 is equal to 2.5 Determination of the cutting width a e for a given roughness class The theoretical surface roughness depends on the cutting width a e , geometry of the cutting edge, and the machined profile. The cutting width a e , defined as the distance obtained by projecting tool points tangent with the machined surface profile on the Z-axis (Fig. 8a), should be constant for a given set of machining parameters. In rope threading, the defined cutting width a e is equal to the feed per revolution f z .
Since a e is the more general parameter, it was chosen to determine the roughness. Simulations were used to determine the maximum cutting width values a e to obtain the required roughness class for different values of the nose radius r ε . For cutting tools with a nose radius of r ε < 2e − 3 (m), the geometry from Fig. 7a was assumed, where the tool end cutting edge angle and the side cutting edge angle were 30°. For cutting tools with a larger radius r ε , the geometry from Fig. 7b was used. Further calculations adopted the simplification that the influence of the roughness on the diameter d is negligibly small and that the cutting edge during machining is tangent to the thread contour. The thread contour's inclination angle α varies during machining. Consequently, a constant cutting width a e results in varied roughness. For a given feed per revolution, the roughness Rz was defined by Eq. (17) and Fig. 8a. The case Rz 2 with a horizontal line is a theoretical, non-existent case of the analyzed thread contour due to the finite value of the radius values. In such Rz definition, the curved contour of the thread is neglected.
Equation (17) was used to determine the cutting width a e required for a given class of roughness. The calculations were repeated for different nose radii r ε . Results are presented in Fig. 8b.
The curves plotted in Fig. 8b can be used to determine the cutting width a e for a given nose radius to obtain a required roughness class.  Fig. 4 The influence of f z on the tool movements in Z′-axis. The following colors refer to the online version: Blue-thread helix; orange-part machined in one spindle revolution; green-part that had not been machined equal to f z /P; P-thread pitch. a Illustrative model. b Presentation in XZ′-plane Note that in the case of high cutting width values, the thread profile may not be achieved (Fig. 9). Therefore, it is not recommended to use high nose radius values for high Rz. When the a e value increases, the thread profile may be increasingly flattened because of the material that is not machined. Therefore, the profile accuracy can be added as an additional constraint.

A rope threading model that does not account for tool life
To obtain the maximum efficiency, the time t 1 (min), described by Eq. (18), should be as short as possible. Given a constant length l (m) of a machined thread, time reduction is equivalent to maximizing the efficiency function defined as t −1 1 , as shown in Eq. (19) In rope threading, the maximum rotational spindle speed n a max derived by a given acceleration a M can be derived from Eqs. (13) and (15) by substituting a M into a R1 and a R2 To obtain a required class of roughness Rz for exemplary tools given tool nose radius r ε , Fig. 8b can be used to read the defined cutting width values a e that are equal to f z in rope threading. After substituting the values into Eqs. (20) and (21), the value of n a max depends on: (Rz, r ε , and a M ).
Apart from machine acceleration a M , the cutting speed v c is another constraint on the rotational spindle speed n both in the upper and lower bands.
The v c values change during machining because of dimension h, which must be taken into account if these changes significantly affect the cutting speed v c .
For further calculations in the paper, we assumed v c min ¼ 30 m min À Á and v c max ¼ 100 m min À Á . The maximum rotational spindle speed instead of n a max is also constrained by v c min and v c max , which corresponds to n min and n v max , respectively. It results, that the maximum rotational spindle speed is equal to n max ¼ min n a max ; n v max f g ð22Þ and the condition n a max ≥ n min ð23Þ has to be satisfied also. The maximum rotational spindle speed n max as a function of r ε and Rz is shown in Fig. 10a. Graphs were plotted for thread diameters defined by the standard for three different lathes. Each lathe enabled machining with its particular tool acceleration characteristics in the X-axis-a M . The graphs account for limitation resulting from the cutting speed v c . Figure 10b shows t −1 1 values as a function of the selected roughness class Rz and nose radius r ε .
The data from Fig. 10b can be used to determine nose radius of a cutting tool r ε for a selected roughness class to ensure the highest efficiency of machining. The tool life is a factor to be considered when planning the mass production of threads. Equations describing it were proposed by F. W. Taylor. A tool operating at a higher cutting speed has a shorter lifetime. The minimization of the machining time t 2 accounting for the tool life relative to the cutting speed v c is given by T The optimum tool life is given by Using v c definition and Eqs. (26) and (27), we obtain the optimum rotational spindle speed n opt T v when the tool life is taken into account We emphasize that values n opt T v are not always feasible due to constraints imposed on a M or v c max . Values can be considered optimum ones if (and only if) they do not exceed n max . Therefore, the optimum rotational spindle speed that accounts for tool tip lifetime T v for a given set of machining parameters can be given as  Fig. 11. Figure 11b shows that when tool life is neglected, the most efficient machining for a roughness class of Rz = 400 (μm) occurs for r ε = 4e − 3 (m). When the tool life is taken into consideration, the most efficient machining is obtained with a nose radius of r ε = 5e − 3 (m). Differences in machining efficiency using tools with various r ε are significant, particularly between a tool with r ε = 4e − 3 (m) and a tool with r ε = 2e − 3 (m).

Accounting
Similarly, as in the previous case, we obtain optimum rotational speed, given that When considering the effect of the cutting speed and feed on the tool life, the optimum rotational spindle speed that does not exceed the permissible acceleration is given by n opt v; f ¼ min n max ; n opt T v; f n o ð32Þ Figure 12a shows graphs of n max and n opt T v; f . Efficiency values t −1 2 corresponding to the rotational spindle speed n opt T v; f are presented in Fig. 12b. 4 Analysis of machining methods: rope/standard/hybrid threading

Description of the methods
Given the number of tool passes m, three machining types can be distinguished, as shown in Fig. 13. First is rope threading, where a complete thread is machined in one tool pass (Fig. 13a). Second is standard thread machining, where the number of tool passes is equal to the pitch divided by the cutting width a e (Fig. 13b). Third is hybrid threading, where a thread is machined in more than one pass but the number of passes required is smaller than that in standard threading (Fig. 13c). The diagrams in Fig. 13 relate to the XZ-plane Table 2 Dependence between accelerations a R1 and a R2 for a given r ε Fig. 8 Theoretical roughness values and cutting width values. a Determination of the theoretical roughness. b Defined cutting width values a e for given roughness classes Rz as a function of the nose radius r ε Fig. 7 Tool geometry types. a For cutting tools with a radius r ε less than 2 (mm). b For cutting tools with a radius greater than or equal to 2 (mm) and are only illustrative (the actual geometrical relationships of the analyzed thread are not retained).

Machining in m-passes, without accounting for tool life
Equation (33) defines the time t 3 (min) required to machine a thread in m-passes without considering the tool life Inequality (34) indicates when it is beneficial to increase the number of passes by u Substituting values from Eq. (33) to Eq. (34), we obtain where Fig. 10 Plots for rope threading model that does not account for tool life. a Feasible n max values as a function of the selected roughness class Rz and nose radius r ε . b Efficiency values t −1 1 as a function of the selected roughness class Rz and nose radius r ε , for thread length l = 1 (m) Fig. 9 Defined cutting width values a e for given roughness classes Rz as a function of the nose radius r ε (2019) 104:3343-3360 Int J Adv Manuf Technol Having solved inequality (35), we obtain the condition that defines when it is favorable to increase the number of tool passes The values R ∈ {R 1 , R 2 } and sign ± depend on the r e value, according to Table 2.
Notably, neither the number of passes m nor the difference in passes u affects the final form of inequality (38).
This finding means that if inequality (38) is satisfied, it is worth increasing the number of passes as much as possible; hence, standard threading is the most efficient method. If the inequality is not satisfied, the most efficient machining is conducted in one pass, corresponding to rope threading, and each additional pass increases the machining time.
Substituting the acceleration Eqs. (13) and (15) into inequality (38), we obtain The shortest machining time in the model with neglecting the tool life is achieved for rotational spindle speed equal to n max . The most efficient machining method corresponding to the cutting width a e , rotational spindle speed n max , and other given parameter values-for the model with neglecting the tool life-is presented in Fig. 14a. The machining time t 4 (min) conducted in m-passes that accounts for tool life defined as T v or T v, f can be expressed as Substituting Eq. (33) into (40) and using the definition of n T , we obtain Similar considerations as those for t 3 (Eqs. (34), (38), and (39)) can be made for t 4 regarding the benefits of increasing the number of tool passes. The most efficient machining method, for the model which account the tool life and for the given parameters is presented in Fig. 14b as a function n opt of a e .
For areas of high rotational spindle speeds, the most efficient method is rope threading. In case of spindle speeds from lower areas, the most efficient machining method depends on the tool life consideration. When this factor is neglected, the solution is standard threading; otherwise, it can be one of the three methods. In the latter case, there is an optimum number of tool passes m opt ∈<1; P a e j k >, m opt ∈{m v ,m v , f } and optimum rotational speed n′ opt ∈ {n opt1 v , n opt1 v , f } to ensure the shortest time of machining t 4 for a given roughness class Rz(a e ).
The actual feed rate which should be entered into the lathe is f z = m opt • a e , and the actual rotational spindle speed which should be set is n′ opt (42, 43). The dependency between n opt1 and n′ opt given by the number of tool passes m opt can be derived from Eq. (37). Thus, Eq. (42) is obtained. When the calculated cutting speed v c is higher than v c max , then n′ opt is defined by Eq. (43).
ÞÁn opt P−m opt Áa e ; for πÁdÁ n′ opt 1000 ≤ v c max ð42Þ The algorithm presented below can be used to select the cutting parameters and threading method. threading. According to the model, use n max (20) or (21), n opt v (29), and n opt v, f (32). 4. Calculate the efficiency function for the selected model.
Select the nose radius r ε , ensuring the most efficient machining for a required class of roughness. Determine the cutting width a e for the selected r ε and Rz in rope threading on the basis of Fig. 8b. 5. Select a machining method (standard, hybrid, rope threading) using Eq. (39) plotted for some exact values in Fig. 14, using the calculated optimum rotational spindle speed and the cutting width a e reading in rope threading. If a e is not constrained due to the correct thread profile, take that influence into account when choosing the optimum parameter values.
(a) When rope threading is the most efficient method, perform cutting with a spindle speed of n max (20) or (21), n opt v (29), and n opt v, f (32) depending on the selected model and with feed f z and a single tool pass with m = 1.
(b) When standard threading is the most efficient method (the model neglects tool life), cut with m = a e tool passes with a rotational spindle speed n′ (37) and a feed per revolution equal to the thread pitch.
(c) When the number of tool passes has not been clearly determined (the lower area of Fig. 14b), an optimum value from interval m opt ∈ < 1; P a e j k > should be found so that the machining time t 4 defined in Eq. (41) is minimized. Cut in mopt tool passes with a rotational spindle speed of n′ opt (37) feed per revolution equal to f z = m opt • a e .

Results
The results presented in Figs. 15 and 16 were plotted for the smallest and largest thread diameters d respectively, which are given by the standard (Table 1).  Fig. 8b). The shortest times t 4v and t 4v, f obtained for diameter d1 (Fig. 15a) were almost the same for a given pair p(Rz, r ε ). In most cases, the differences between tool life models were higher for d5 (Fig. 16a). For the most pairs, especially with small values of r ε and Rzt 4v was higher than t 4v,f but for some cases with high r ε and Rzt 4v,f , values were higher than t 4v .
The optimum values of rotational spindle speeds for m = 1 in case of d1 were mostly limited by n a max (Fig. 15b). The defined n v max could be omitted in this case, because it was higher than n a max for each pair. In some cases of high Rz values, n opt v and n opt v, f do not reach the maximum limited values. It was due to tool exchange time. For both considered thread diameters, the standard threading was not the most efficient machining in any considered case (Figs. 15c and  16c). For d1-in contrast to d5-the optimum number of tool passes m v and m v, f was the same for the given pair in most cases. In d5, m v was higher than m v, f in most cases. The main limitation for n′ opt v and n′ opt v, f for d1 was n′ a max v and n′ a max v, f , respectively, where both limitations had the same values for the most number of pairs (Fig. 15d). For d5, the most n′- opt v, f values were limited by n v max . Some values of n′ opt v and n′ opt v, f for high r ε values and not very high Rz values were limited by n v max for the highest considered thread diameter.
The values of n′ opt v were not limited by n v max in any case. When choosing the optimal solution, the thread profile contour accuracy must also be considered. It may be that the results of small t 4v and t 4 v,f achieved by high Rz and high r ε values do not fulfil the geometrical requirements or that the time saved relative to other solutions is not worth the degradation of geometric correctness.

Conclusion
The paper presents a thorough analysis of the noncircular turning process on the basis of the smooth-contour ISO 10208:1991 male rope thread. The optimization process results in the selection of the number of tool passes, which defines the threading method and the cutting parameter values. In the case of rope threading or hybrid threading, the increase in the tool nose radius value increases the required tool accelerations in the X-axis. Relative to standard threading, the processing time is reduced for a smaller number of passes, but the cutting speed can be constrained by the machine tool acceleration capabilities. Surface roughness is considered along with an indication of the issue related to geometrical errors associated with excessively high cutting width values. The impact of tool life modelling is explored. The diameter of the machined thread also has an influence on the obtained values. Ultimately, algorithm-based selection of a threading method and cutting parameters is enabled. On the basis of the input parameter values and the model choice, the machining method can be selected along with the values of cutting parameters such as the number of passes, rotational spindle speed, tool nose radius, and feed per revolution in the Z-axis. The corresponding analysis shows the complexity of the machining of the rope threads mainly focusing on tool kinematics as a function of cutting parameters and tool geometry. Note that the analysis does not account for all phenomena that may influence the process; for instance, dynamic cutting forces are not taken into account.