Rotational turning is one of the newly used methods in hard turning operations. Its complex kinematical relations and helical edged tool makes the study of this method more difficult than ordinary turning. The aim of this paper is to describe a case of turning with a mathematical model

Sztankovics, I.

Library of National Technical University "Kharkiv Polytechnic Institute"

264 
UDK 621.91 
 
I. Sztankovics, PhD. student; J. Kundrák, Prof. Dr. habil, 
Miskolc, Hungary 
 
MATHEMATICAL DESCRIPTION OF MACHINING  
EXTERNAL CYLINDRICAL SURFACE IN CASE  
OF ROTATIONAL TURNING  
 
Rotational turning is one of the newly used methods in hard turning operations. Its complex 
kinematical relations and helical edged tool makes the study of this method more difficult than ordinary 
turning. The aim of this paper is to describe a case of turning with a mathematical model. 
 
1. Introduction 
The opportunity of machining hardened surfaces (where hardness is above 45 
HRC) with defined edge geometry tools has been proved to be a significant 
advance. These machining methods have become available with the appearance of 
super hard CBN tools. Besides their hardness, which is near that of diamond, their 
chemical properties allow them to cut steel. This is the two main reasons wherefore 
that became the alternative for machining hardened surfaces. In this methods a 
typical saw-tooth like chip formation mechanism is can be seen [1]. There are 
several opportunities to carry out research in this mechanism, for example FEM 
analysis. However, in case of FEM we can see that at high speed the morphology of 
the chip formation mechanism changes [2]. 
Hardening heat treatments are applied mostly for the increase of toughness 
and wear resistance values. Therefore one of the main scopes of hard turning is the 
machining of tooth wheels. Previously precision machining of these parts could 
only be solved by grinding [3]. The reachable machining accuracy in hard turning 
is suitable to skip the grinding process [4]. For a better understanding of these 
methods there are research studies to determinate the processes in the surface [5] 
The main disadvantages of hard turning are the followings: the passive force 
is very high during the process due to the application of a tool with negative rake 
angle, furthermore a micro-thread is generated on the surface of the workpiece 
because of the kinematic relations during the method. One opportunity to solve 
these difficulties is the application of the combined process. Another option is to 
change the kinematic relation by using other turning methods. One of these can be 
rotational turning [6] which belongs to the group of tangential feeds [7]. 
In rotational turning the cutting edge of the tool is a helical curve. The axis of 
this helix and the axis of the workpiece must be parallel. The chip formation is 
caused because the tool rotates with a very slow speed during the ordinary rotation 
speed of the workpiece [8]. 
The study of the rotational turning method is not easy when we use the helical 
edged tool. However the chip formation mechanism is very similar to the oblique 
265 
cutting [9]. The chip flow angle depends on technological and material structural 
parameters [10]. Furthermore, in regard of the cutting force, the normal rake angle 
of the tool’s cutting edge is essential [11]. Two dimensional mathematical 
equations [12] or elastic-plastic finite element models [13] can be used to describe 
the cutting process.  
To describe the cutting process a properly chosen and defined kinematic 
model can be applied. For example such method is able to define the geometrical 
relations of toothwheels [14] or worm gear drives [15]. There is a research to apply 
the Monge projection for the description of the kinematic model [16]. The method 
presented by Perepelica [17,18] is able to define simple geometrical relations [19] 
during cutting without using of homogeneous coordinates. The model we described 
earlier [20] has proved promising. This paper describes the kinematic relations in a 
special case of the model: machining external cylindrical surface with rotational 
turning. 
 
2. DEFINING THE TRANSFORMATIONAL EQUATIONS AND THE 
COORDINATE-SYSTEMS 
The essence of the method described by Perepelica [17] is to define the 
equation of the machined surface with the transformed equation of the cutting edge 
and the relative movements of the workpiece and the tool. To do this the coordinate 
systems must be properly chosen and the equations describing the relations 
between these system must be written down. 
 
Determination of the coordinate systems 
Figure 1. shows the relative position of the workpiece and the tool’s cutting 
edge, the main movements to be described and the chosen coordinate systems. The 
model contains the case with no feed in axial direction (vt = 0 condition must be 
fulfilled). One of the main aspects in the definition of the coordinate systems was to 
make the systems’ z axes parallel to the axis of the machine’s spindle. The 
remaining axes were drawn on order that the three axes would make the coordinate 
system right-handed. Another condition is that both the workpiece and the tool 
must have one fixed and one moving coordinate system. In general it can be said 
that the moving systems describe the main movements and the fixed systems 
describe the side motions of the tool and the workpiece. For a better distinction the 
axes of the moving systems are marked with Greek letters. 
 
The symbols of Figure 1. are the followings: 
Kt – fixed coordinate system (t.) 
Ktm – moving coordinate system (t.) 
Kw – fixed coordinate system (w.) 
Kwm – moving coordinate system (w.) 
1 – intersection point of the edge and 
ωw – angular frequency (w.) 
ωt – angular frequency (t.) 
vt – axial feed rate of the tool 
aw – centre distance 
lm – initial distance between moving 
266 
the surface and standing coordinate systems 
 
Figure 1 – Kinematical model of rotational turning 
 
Defining the transformational equations 
After the determination of the coordinate systems, the transformational 
equations describing the relations between them can be written down. A general 
attribute of these equations is that they contain a rotation matrix and a linear 
translation vector [17, 18]. If the vector equation is given in the initial system the 
form of the equation can be written down on the other system. Three 
transformational equations are needed because there is four coordinate systems. 
The equations are the followings: 
 





















 

tvl
tt
tt
tm
tmtt
tt
tmttmtmtt 0
0
100
0cossin
0sincos
,, rtrRr 

 
(1) 
 






















0
0
100
010
001
,,
w
ttwttww
a
rtrRr  (2) 
 























m
www
ww
wwmwwwmwm
l
tt
tt
0
0
100
0cossin
0sincos
,, rtrRr 

  (3) 
267 
The above equations can be summed up with the solution of the following 
equation: 
 
  
wwmtwwwmtmttwwwmtmtmttwwwm
wwmtwtmttmtmttwwwmwwmwwwmwm
,,,,,,,,,
,,,,,,,,
ttRtRRrRRR
tttrRRRtrRr


  (4) 
 
Accordingly the form of the vector equation (defined in the tool’s moving 
coordinate system) is the following in the moving coordinate system of the 
workpiece: 
 
 


























tv
ta
ta
tttt
tttt
t
ww
ww
tmwtwt
wtwt
wm 



sin
cos
100
0)cos()sin(
0)sin()cos(
rr  (5) 
 
3. DEFINITION OF THE MACHINED SURFACE EQUATION 
The proper equation of the cutting edge is necessary to define of the machined 
surface. The vector equation of a helical curve with rt diameter and 90-αt pitch 
angle is the following: 
  

 eeer p
t
t
ptptptm
r
rr
)tan(
)sin()cos()(   (6) 
 
Describing the two-parametric equation of the surface 
The equation of the surface can be obtained if the equation written down in 
the moving coordinate system of the workpiece is rotated around the ζ axis by the 
following equation: 
    

 wms rr









 

100
0cossin
0sincos
,  (7) 
 
If we write the equation of the edge (6) into the defined transformation 
equation (5), then we do the previous multiplication (7), we will get the two-
parametric equation of the surface: 
  














tvr
tattr
tattr
tptt
wwwtpt
wwwtpt
ps




)cot(
)sin()](sin[
)cos()](cos[
,r  (8) 
268 
Describing two-dimensional (one-parametric) equation of the surface 
After defining the two-parametric equation, the one-parametric equation can 
only be defined with the determination of a side-condition. This condition is that 
the resultant vector must be in the [ζ,ξ] plane, that is the η component must be zero. 
 
 0)sin()](sin[  tattr wwwtptrs   (9) 
 
The υ parameter of the two-parametric equation (8) can be expressed from 
Equation 9. in function of ζ: 
 
 











tattr
tattr
wwwtpt
wwwtpt
p



cos)](cos[
sin)](sin[
arctan)(  (10) 
 
The resulting equation helps to write down the two-parametric equation in the 
form of the one-parametric equation. We can use this equation to describe the 
machined surface during the cutting. In Figure 2. some of the cases (40°,45° and 
50° pitch angle) can be seen. The hyperboloid shape can be easily observed. 
 
 








































tatt
r
tv
r
tatt
r
tv
r
tatt
r
tv
r
wwwt
tt
t
t
wwwt
tt
t
t
wwwt
tt
t
t











cos)(
)cot(
cos
sin)(
)cot(
sin
arctan)(
))(cos()()(
)cot(
cos)(
 
 (11) 
 
 
Figure 2 – Curves in case of different pitch angles (40°, 45°, 50°) 
269 
SUMMARY 
Rotational turning is a method of complex kinematics and it requires complex 
tools for machining. The mathematical description of the cutting process is not an 
easy task as can be seen from the described equations. In general we can say that 
the exact mathematical model can be solved with the defined description. Other 
important geometrical attributes of the surface (2D and 3D surface roughness 
parameters) and the topography can be determined when other parameters are taken 
into consideration. There is also an opportunity to observe the complex shaped 
cross section of the chip and we can continue to further study the process. 
 
ACKNOWLEDGEMENT 
The work was presented by the support of the Hungarian Scientific Research 
Fund (Number of Agreement: OTKA K 78482), which the authors greatly 
appreciate. The research work has been realised as part of project TÁMOP-4.2.1.B-
10/2/KONV-2010-0001 – in the frame of New Hungary Development Plan –, by 
the support of European Union, with the co-finance of European Social Found. 
 
REFERENCES: [1] Szabó G: A forgácsleválasztás jellegzetességeinek vizsgálata keményesztergálás 
esetén. GÉP Terméktervezők és Termékfejlesztők XXVII. Szemináriuma Vol. 9-10/2, pp., 2011, 80-83; 
[2] Szabó G: Research of the Mechanism of Plastic Strain in Case of Tempered Steel in Hard Turning. 
Hungarian Journal of Industrial Chemistry Vol. 38/2, 2010, pp. 163-167; [3] Karpuschewski B, 
Knoche H-J, Hipke M: Gear finishing by abrasive processes, CIRP Annals 57, 2008, pp. 621-640; [4] 
Kundrák J, Karpuschewski B, Gyáni K, Bana V: Accuracy of hard turning, Journal of Material 
Processing Technology 202(1-3) , 2008, pp. 328-338; [5] Bartarya G, Choudhury S K: State of the art 
in hard turning. International Journal of Machine Tools & Manufacture 53, 2012, pp. 1-14; [6] J. G. 
Weisser: Patent Anmeldung von Werkzeugmaschinenfabrik, St. Georgen, Swarzwald, Deutschland, 
2004; [7] Sztankovics I: Rotational Turning in Precision Finishing. Hungarian Journal of Industrial 
Chemistry Vol. 39/2, 2011, pp. 309-312; [8] Kundrák J, Gyáni K, Deszpoth I, Szabó S: 
Keményesztergálás rotációs előtolású eljárással. XXVI. microCAD International Scientific Conference: 
Section L: Production Engineering and Manufacturing Systems, Miskolc, Magyarország, 29-30. March 
2012, Paper L21; [9] Shaw M C: Metal Cutting Principles. Oxford University Press, New York, 2005, 
p. 651; [10] DeVries W R: Analysis of Material Removal Processes. New York, 1992, p. 254; [11] 
Komanduri R, Lee M, Raff L M: The significance of normal rake in oblique machining. International 
Journal of Machine Tools & Manufacture 44, 2004, pp 1115-1124; [12] Fang N: An improved model 
for oblique cutting and its application to chip-control research. Journal of Materials Processing 
Technology 79, 1998, pp. 79-85; [13] Lin Z-C, Lin Y-Y: A study of an oblique cutting model. Journal 
of Materials Processing Technology 86, 1999, pp. 119-130; [14] Litvin F L, Fuentes A: Gear Geometry 
and Applied Theory (second ed.). Cambridge University Press, 2004, p. 800; [15] Dudás I: The Theory 
and Practice of Worm Gear Drives. Penton Press, London, 2000, p. 314; [16] Balajti Zs: Practical 
Application of the Monge Projection in the Production Process of Drive Pairs. Annals of MteM for 2005 
& Proceedings of the 7th International Conference Modern Technologies in Manufacturing 6-8. October 
2005, pp. 35-38.; [17] Perepelica B A: Osznovü Teoriji proektyiroványija rézsusih insztrumentov. 
Harkov, 1974, p. 55; [18] Perepelica B A: Otobrazsenija affinnogo prosztransztva v teorii 
formoobrazovanija poverhnosztej rezaniem. Harkov, 1981, p. 152; [19] Vasilko K: Kinematics Analysis 
of the movement of cutting tool against the workpiece and creation of new schemes. Manufacturing and 
industrial engineering, Kosice, 2010, pp.8-15; [20] Sztankovics I, Kundrák J: Mathematical Model of 
Machined Surfaces in Rotational Turning and Skiving. International Conference on Tools, Miskolc, 
2012, pp. 83-88. 
Поступила в редколлегию 15.06.2012 


Mathematical description of machining external cylindrical surface in case of rotational turning

eNTUKhPIIR Electronic National Technical University "Kharkiv Polytechnic Institute"; Institutional Repository

264 UDK 621.91  I. Sztankovics, PhD. student; J. Kundrák, Prof. Dr. habil, Miskolc, Hungary  MATHEMATICAL DESCRIPTION OF MACHINING  EXTERNAL CYLINDRICAL SURFACE IN CASE  OF ROTATIONAL TURNING   Rotational turning is one of the newly used methods in hard turning operations. Its complex kinematical relations and helical edged tool makes the study of this method more difficult than ordinary turning. The aim of this paper is to describe a case of turning with a mathematical model.  1. Introduction The opportunity of machining hardened surfaces (where hardness is above 45 HRC) with defined edge geometry tools has been proved to be a significant advance. These machining methods have become available with the appearance of super hard CBN tools. Besides their hardness, which is near that of diamond, their chemical properties allow them to cut steel. This is the two main reasons wherefore that became the alternative for machining hardened surfaces. In this methods a typical saw-tooth like chip formation mechanism is can be seen [1]. There are several opportunities to carry out research in this mechanism, for example FEM analysis. However, in case of FEM we can see that at high speed the morphology of the chip formation mechanism changes [2]. Hardening heat treatments are applied mostly for the increase of toughness and wear resistance values. Therefore one of the main scopes of hard turning is the machining of tooth wheels. Previously precision machining of these parts could only be solved by grinding [3]. The reachable machining accuracy in hard turning is suitable to skip the grinding process [4]. For a better understanding of these methods there are research studies to determinate the processes in the surface [5] The main disadvantages of hard turning are the followings: the passive force is very high during the process due to the application of a tool with negative rake angle, furthermore a micro-thread is generated on the surface of the workpiece because of the kinematic relations during the method. One opportunity to solve these difficulties is the application of the combined process. Another option is to change the kinematic relation by using other turning methods. One of these can be rotational turning [6] which belongs to the group of tangential feeds [7]. In rotational turning the cutting edge of the tool is a helical curve. The axis of this helix and the axis of the workpiece must be parallel. The chip formation is caused because the tool rotates with a very slow speed during the ordinary rotation speed of the workpiece [8]. The study of the rotational turning method is not easy when we use the helical edged tool. However the chip formation mechanism is very similar to the oblique 265 cutting [9]. The chip flow angle depends on technological and material structural parameters [10]. Furthermore, in regard of the cutting force, the normal rake angle of the tool’s cutting edge is essential [11]. Two dimensional mathematical equations [12] or elastic-plastic finite element models [13] can be used to describe the cutting process.  To describe the cutting process a properly chosen and defined kinematic model can be applied. For example such method is able to define the geometrical relations of toothwheels [14] or worm gear drives [15]. There is a research to apply the Monge projection for the description of the kinematic model [16]. The method presented by Perepelica [17,18] is able to define simple geometrical relations [19] during cutting without using of homogeneous coordinates. The model we described earlier [20] has proved promising. This paper describes the kinematic relations in a special case of the model: machining external cylindrical surface with rotational turning.  2. DEFINING THE TRANSFORMATIONAL EQUATIONS AND THE COORDINATE-SYSTEMS The essence of the method described by Perepelica [17] is to define the equation of the machined surface with the transformed equation of the cutting edge and the relative movements of the workpiece and the tool. To do this the coordinate systems must be properly chosen and the equations describing the relations between these system must be written down.  Determination of the coordinate systems Figure 1. shows the relative position of the workpiece and the tool’s cutting edge, the main movements to be described and the chosen coordinate systems. The model contains the case with no feed in axial direction (vt = 0 condition must be fulfilled). One of the main aspects in the definition of the coordinate systems was to make the systems’ z axes parallel to the axis of the machine’s spindle. The remaining axes were drawn on order that the three axes would make the coordinate system right-handed. Another condition is that both the workpiece and the tool must have one fixed and one moving coordinate system. In general it can be said that the moving systems describe the main movements and the fixed systems describe the side motions of the tool and the workpiece. For a better distinction the axes of the moving systems are marked with Greek letters.  The symbols of Figure 1. are the followings: Kt – fixed coordinate system (t.) Ktm – moving coordinate system (t.) Kw – fixed coordinate system (w.) Kwm – moving coordinate system (w.) 1 – intersection point of the edge and ωw – angular frequency (w.) ωt – angular frequency (t.) vt – axial feed rate of the tool aw – centre distance lm – initial distance between moving 266 the surface and standing coordinate systems  Figure 1 – Kinematical model of rotational turning  Defining the transformational equations After the determination of the coordinate systems, the transformational equations describing the relations between them can be written down. A general attribute of these equations is that they contain a rotation matrix and a linear translation vector [17, 18]. If the vector equation is given in the initial system the form of the equation can be written down on the other system. Three transformational equations are needed because there is four coordinate systems. The equations are the followings:   tvltttttmtmtttttmttmtmtt 001000cossin0sincos,, rtrRr  (1)  00100010001,,wttwttwwartrRr  (2)  mwwwwwwwmwwwmwmltttt001000cossin0sincos,, rtrRr   (3) 267 The above equations can be summed up with the solution of the following equation:    wwmtwwwmtmttwwwmtmtmttwwwmwwmtwtmttmtmttwwwmwwmwwwmwm,,,,,,,,,,,,,,,,,ttRtRRrRRRtttrRRRtrRr  (4)  Accordingly the form of the vector equation (defined in the tool’s moving coordinate system) is the following in the moving coordinate system of the workpiece:   tvtatatttttttttwwwwtmwtwtwtwtwm sincos1000)cos()sin(0)sin()cos(rr  (5)  3. DEFINITION OF THE MACHINED SURFACE EQUATION The proper equation of the cutting edge is necessary to define of the machined surface. The vector equation of a helical curve with rt diameter and 90-αt pitch angle is the following:    eeer pttptptptmrrr)tan()sin()cos()(   (6)  Describing the two-parametric equation of the surface The equation of the surface can be obtained if the equation written down in the moving coordinate system of the workpiece is rotated around the ζ axis by the following equation:      wms rr 1000cossin0sincos,  (7)  If we write the equation of the edge (6) into the defined transformation equation (5), then we do the previous multiplication (7), we will get the two-parametric equation of the surface:   tvrtattrtattrtpttwwwtptwwwtptps)cot()sin()](sin[)cos()](cos[,r  (8) 268 Describing two-dimensional (one-parametric) equation of the surface After defining the two-parametric equation, the one-parametric equation can only be defined with the determination of a side-condition. This condition is that the resultant vector must be in the [ζ,ξ] plane, that is the η component must be zero.   0)sin()](sin[  tattr wwwtptrs   (9)  The υ parameter of the two-parametric equation (8) can be expressed from Equation 9. in function of ζ:   tattrtattrwwwtptwwwtptpcos)](cos[sin)](sin[arctan)(  (10)  The resulting equation helps to write down the two-parametric equation in the form of the one-parametric equation. We can use this equation to describe the machined surface during the cutting. In Figure 2. some of the cases (40°,45° and 50° pitch angle) can be seen. The hyperboloid shape can be easily observed.   tattrtvrtattrtvrtattrtvrwwwtttttwwwtttttwwwtttttcos)()cot(cossin)()cot(sinarctan)())(cos()()()cot(cos)(  (11)   Figure 2 – Curves in case of different pitch angles (40°, 45°, 50°) 269 SUMMARY Rotational turning is a method of complex kinematics and it requires complex tools for machining. The mathematical description of the cutting process is not an easy task as can be seen from the described equations. In general we can say that the exact mathematical model can be solved with the defined description. Other important geometrical attributes of the surface (2D and 3D surface roughness parameters) and the topography can be determined when other parameters are taken into consideration. There is also an opportunity to observe the complex shaped cross section of the chip and we can continue to further study the process.  ACKNOWLEDGEMENT The work was presented by the support of the Hungarian Scientific Research Fund (Number of Agreement: OTKA K 78482), which the authors greatly appreciate. The research work has been realised as part of project TÁMOP-4.2.1.B-10/2/KONV-2010-0001 – in the frame of New Hungary Development Plan –, by the support of European Union, with the co-finance of European Social Found.  REFERENCES: [1] Szabó G: A forgácsleválasztás jellegzetességeinek vizsgálata keményesztergálás esetén. GÉP Terméktervezők és Termékfejlesztők XXVII. Szemináriuma Vol. 9-10/2, pp., 2011, 80-83; [2] Szabó G: Research of the Mechanism of Plastic Strain in Case of Tempered Steel in Hard Turning. Hungarian Journal of Industrial Chemistry Vol. 38/2, 2010, pp. 163-167; [3] Karpuschewski B, Knoche H-J, Hipke M: Gear finishing by abrasive processes, CIRP Annals 57, 2008, pp. 621-640; [4] Kundrák J, Karpuschewski B, Gyáni K, Bana V: Accuracy of hard turning, Journal of Material Processing Technology 202(1-3) , 2008, pp. 328-338; [5] Bartarya G, Choudhury S K: State of the art in hard turning. International Journal of Machine Tools & Manufacture 53, 2012, pp. 1-14; [6] J. G. Weisser: Patent Anmeldung von Werkzeugmaschinenfabrik, St. Georgen, Swarzwald, Deutschland, 2004; [7] Sztankovics I: Rotational Turning in Precision Finishing. Hungarian Journal of Industrial Chemistry Vol. 39/2, 2011, pp. 309-312; [8] Kundrák J, Gyáni K, Deszpoth I, Szabó S: Keményesztergálás rotációs előtolású eljárással. XXVI. microCAD International Scientific Conference: Section L: Production Engineering and Manufacturing Systems, Miskolc, Magyarország, 29-30. March 2012, Paper L21; [9] Shaw M C: Metal Cutting Principles. Oxford University Press, New York, 2005, p. 651; [10] DeVries W R: Analysis of Material Removal Processes. New York, 1992, p. 254; [11] Komanduri R, Lee M, Raff L M: The significance of normal rake in oblique machining. International Journal of Machine Tools & Manufacture 44, 2004, pp 1115-1124; [12] Fang N: An improved model for oblique cutting and its application to chip-control research. Journal of Materials Processing Technology 79, 1998, pp. 79-85; [13] Lin Z-C, Lin Y-Y: A study of an oblique cutting model. Journal of Materials Processing Technology 86, 1999, pp. 119-130; [14] Litvin F L, Fuentes A: Gear Geometry and Applied Theory (second ed.). Cambridge University Press, 2004, p. 800; [15] Dudás I: The Theory and Practice of Worm Gear Drives. Penton Press, London, 2000, p. 314; [16] Balajti Zs: Practical Application of the Monge Projection in the Production Process of Drive Pairs. Annals of MteM for 2005 & Proceedings of the 7th International Conference Modern Technologies in Manufacturing 6-8. October 2005, pp. 35-38.; [17] Perepelica B A: Osznovü Teoriji proektyiroványija rézsusih insztrumentov. Harkov, 1974, p. 55; [18] Perepelica B A: Otobrazsenija affinnogo prosztransztva v teorii formoobrazovanija poverhnosztej rezaniem. Harkov, 1981, p. 152; [19] Vasilko K: Kinematics Analysis of the movement of cutting tool against the workpiece and creation of new schemes. Manufacturing and industrial engineering, Kosice, 2010, pp.8-15; [20] Sztankovics I, Kundrák J: Mathematical Model of Machined Surfaces in Rotational Turning and Skiving. International Conference on Tools, Miskolc, 2012, pp. 83-88. Поступила в редколлегию 15.06.2012 

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Mathematical description of machining external cylindrical surface in case of rotational turning

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