Mathematical Model and Simulation of Cutting Layer Geometry in Orthogonal Turn-Milling with Zero Eccentricity

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1.Jiangsu Key Laboratory of Engineering Machinery Detection and Control，Xuzhou University of Technology，Xuzhou 221111，P.R.China；2.College of Mechanical and Electrical Engineering，Nanjing University of Aeronautics and Astronautics，Nanjing 210016，P.R.China；3.Industrial Center，Nanjing Institute of Technology，Nanjing 211167，P.R.China

（Received 2 March 2020；revised 20 April 2020；accepted 5 June 2020）

Abstract:Orthogonal turn-milling is a high-efficiency and precision machining method.Its cutting layer directly affects chip formation，cutting forces，and chatter，and further affects tool life，machining quality，etc.We studied The cutting layer geometry（CLG）in orthogonal turn-milling with zero eccentricity（OTMZE）is studied to explore orthogonal turn-milling cutting layer formation process.OTMZE principles of motion and formation processes are analyzed statically without considering kinetic influences.Mathematical models of the entrance and exit angles，cutting thickness，and cutting depth are established.In addition，these models are validated experimentally and some influences of cutting parameters on the tool cutting layer are analyzed.The results show that OTMZE cutting layer formation can be divided into two stages，chip shapes are nearly consistent with the simulated CLGs，and the most influencial parameter in affecting the cutting layer is found to be the tool feed per revolation of workpiece fa，followed by the ratio of the tool and workpiece speeds λ and the cutting depth ap.These models and results can provide theoretical guidance to clarify formation processes and quantitatively analyze changes in cutting layer geometry during OTMZE.In addition，they offer theoretical guidelines for cutting forces and chatter.

Key words：orthogonal turn-milling；zero eccentricity；cutting layer geometry；mathematical model；forming process

0 Introduction

Orthogonal turn-milling can achieve high-efficiency and high-precision machining by combining workpiece rotation with rotation and axial movement of a milling cutter along the workpiece.It is widely used to machine difficult-to-cut materials such as Ti alloys，nickel alloys，and stainless steels as well as special structural components （e.g.，crankshafts，slender rods，turbine blades，and engine casings）［1-2］.

Chip formation，cutting forces，and chatter are important orthogonal turn-milling research topics that affect machining efficiency，machining quality，and tool life［3-5］.Orthogonal turn-milling differs from turning and milling such that the cutting layer geometries（CLGs）of their steps are completely different.Understanding the geometries and mathematical models of orthogonal turn-milling cutting layers is useful when researching chip formation，cutting forces，and chatter.

Zhu et al.established mathematical models of orthogonal turn-milling CLGs based on two conditions of zero eccentricity and eccentricity.The researchers simulated and predicted chip shapes，cutting forces，and chatter［6-8］.Yan and Sun et al.analyzed orthogonalturn-milling cutting chatterby building mathematical CLG models that considered eccentricity［9-10］.Karagüzel，Kara，and Crichigno et al.established a mathematical CLG model with zero eccentricity and a simulated orthogonal turn-milling cutting force［11-14］.Yan et al.created a mathematical CLG model with zero eccentricity using a helical milling cutter and predicted the corresponding cutting force［15］.

Although there are reports on orthogonal turnmilling CLGs，severalproblemsremain to be solved：（i）detailed analyses of cutting layer formation are lacking；（ii）the mathematical CLG models built thus far are complex；（iii）the validated CLG methods are relatively simple；（iv）there is a lack of analysis of the influences of cutting parameters on CLGs.

CLGs from orthogonal turn-milling with zero eccentricity（OTMZE） aremorecomplexthan those applicable to orthogonal turn-milling with eccentricity.Thus，we research OTMZE CLG mathematical modeling and simulation.This work investigates the principles of motion behind OTMZE.We analyze the CLG formation process and establish CLG mathematical models.In addition，we perform experiments to confirm mathematical model accuracy by comparing the shapes，volumes，and maximum cutting depths of various simulated CLGs to those from actual chips.Finally，we simulate and analyze the influences of cutting parameters on CLGs.Thus，the aim of this paper is to provide a theoretical guide to the cutting force and chatter in orthogonal turn-milling.

1 Cutting Layer Formation During OTMZE

1.1 Cutting layer motion principle

In orthogonal turn-milling，we assume that the workpiece is at rest and the tool moves spirally along the axis of the workpiece（xw）.The three axesxw，yw，andzware shown in Fig.1，whererwis the workpiece radius，rtthe tool radius，nwthe workpiece speed，ntthe tool speed，andapthe cutting depth.The intersected portion of the two tool positions and the workpiece produced when the tool moves from position 1 to position 2 can be viewed as the OTMZE CLG.Thus，the layer cutting process is analyzed statically without considering kinetic influences.

During the OTMZE cutting process，the side cutting edges corresponding to two separate tool positions cut the forming contour of the machined workpiece surface at two linesBB'andCC'that meet at lineDD'.The edges respectively cut the workpiece at two arcsCDandBDand the area from arcCDturning right into arcBDis formed by side cutting edge.The bottom cutting edges corresponding to the two tool positions meet at lineAA'.LineAA'is at the right of arcBD.The area from arcBDturning right into lineAA'is formed by the bottom cutting edge.The above analyses indicate that the side and bottom cutting edges participate in simultaneous cutting.Thus，the entrance and exit angles of the side and bottom cutting edges must be considered during analysis of OTMZE.

In the cutting process of OTMZE，the corresponding rotation angle of the workpiece isφzwhile a milling cutter blade is rotated as shown in Fig.1.The angleφzcan be expressed as

whereλis the ratio of the tool and workpiece speeds （i.e.，λ=nt/nw）andZthe number of teeth on the tool.

In Fig.1，θis the forming contour angle of the machined workpiece surface.

wherefais the tool feed per revolution of workpiece.

The line through the center point of the tool position is perpendicular to the workpiece forming contour.In this analysis，ψis the angle between the perpendicular line and the center line of the tool position（direction ofxw）andfzis the distance that the tool moves along the forming contour of the workpiece when the workpiece is rotated the angle（φz）.They can be expressed as

1.2 Cutting layer formation process

The fillet radius of the tool used in orthogonal turn-milling is usually less than 0.8 mm.Thus，the influence of the tool fillet radius on the cutting layer geometry is negligible and can be ignored.To simplify the calculation，the fillet radius of the tool is not considered in our analyses.

The cutting layer formation process involves parameters such as the entrance and exit angles，cutting depth， and cutting thickness， as shown in Fig.2.During OTMZE cutting，the side cutting edge cuts into the workpiece from pointCfirst.Next，the side and bottom cutting edges cut into the workpiece from pointBsimultaneously.Then，the bottom cutting edge cuts the workpiece from pointsAandA'，and finally the side cutting edge cuts out the workpiece from pointD.The entrance and exit angles corresponding to the above process are respectivelyφst，C，φst，B，φex，A，φex，A'，andφex，D.

In Fig.2，φiis the dynamic contact angle of toothias it cuts the workpiece.The cutting thickness and depth of the side cutting edge of toothiatφiarehp（φi）andap（φi），respectively.Whilehf（φi）andaf（φi）are the cutting thickness and depth of the bottom cutting edge of toothiatφi，respectively.

Fig.2 OTMZE cutting parameters

The various OTMZE tool entrance and exit angles allow one to divide the cutting layer into four stages：φst，C≤φi≤φst，B，φst，B＜φi≤φex，A'，φex，A'＜φi≤φex，D，andφex，D＜φi≤φex，A.The differences between the anglesφex，A'，φex，D，andφex，Aare negligible during cutting，so they can be replaced by a single angleφex，D.Finally，the cutting layer can be simplified to two stages ofφst，C≤φi≤φst，Bandφst，B＜φi≤φex，D，as shown in Fig.3.

Fig.3 OTMZE cutting layer cross-sections

During theφst，C≤φi≤φst，Bstage，the side cutting edge forms the cutting layer shown in Fig.3.The left side of the cutting layer cross-section is a plumb line that represents the intersection of the side cutting edge from position 2 of tool and the cutaway plane a-a.To the right is another plumb line that represents the intersection of the workpiece forming the contour and the cutaway plane a-a.The bottom is a horizontal line that represents the intersection of the plane a-a and the bottom cutting edge from position 2 of the tool.The top is an arc that represents the intersection of the cylindrical workpiece surface and the plane a-a.The arc is short enough that it can be considered equivalent to a straight line.

In theφst，B≤φi≤φex，Dstage，the side and bottom edges cut simultaneously.Together，they form the cutting layer shown in Fig.3.The cutting layer consists of two parts：the part formed by the side cutting edge on the left and the part formed by the bottom cutting edge on the right.

To the left of the cross-section of the cutting layer formed by the side cutting edge is a plumb line that represents the intersection of the side cutting edge from position 2 of the tool and the cutaway plane a-a.To the right of that is a line that represents the intersection of the side cutting edge from position 1 of the tool and the plane a-a.This is an oblique line with a smallφzthat can be treated as a plumb line.The bottom is a horizontal line that represents the intersection of the bottom cutting edge from position 2 of the tool and the plane a-a.The top is an arc that can be treated as a straight line，as in the above analysis of theφst，C≤φi≤φst，Bstage.

To the left of the cross-section of the cutting layer formed by the bottom cutting edge is a line that represents the intersection of the envelope region produced when moving from position 1 to point 2 of the tool and the cutaway plane a-a.This line is common with the right side of the cutting layer produced by the side cutting edge.To the far right is a plumb line that represents the intersection of the workpiece forming contour and the plane a-a.The bottom is a horizontal line that represents the intersection of the bottom cutting edge from position 2 of the tool and the plane a-a.This line is common with that from the bottom of the cutting layer produced by the side cutting edge.The top is a line that represents the intersection of the bottom cutting edge from position 1 of the tool and the plane a-a.This is an oblique line with a negligibly smallφzsuch that it can be treated as a horizontal line.That is，the top of the cross-section of the cutting layer formed by the bottom cutting edge can be regarded as parallel to the bottom of the layer.

2 Mathematical Model of OTMZE CLG

The above analyses indicate that the cutting layer angle ranges formed by the side and bottom cutting edges areφst，C≤φi≤φex，Dandφst，B≤φi≤φex，D， respectively.The mathematical models of tool entrance and exit angles during OTMZE can be expressed as

The cutting thicknesses of the cutting layers formed by the side and bottom cutting edges［hp（φi）andhf（φi）］are

wherey1，y2，andy3are variables that are defined as follows：y1=（rt－fa.cosψ）/sin（φi－ψ），y2=［rt2－（fz.sin（φi－ψ））2］1/2，andy3=（rw－ap）·tan（0.5φz）.

The cutting depths of the cutting layers formed by the side and bottom cutting edgesap（φi） andaf（φi）are

3 Experimental Tests

The machine tool used for experiments is a Mazak Integrex 200-IVST.The experimental cutting tool includes a toolholder and an insert，as shown in Fig.4.The specification of the insert is ISO standard R390-11 T3 08E-PLW 1130（Sandvik），and the rake and clearance angles are 16°and 12°，respectively.The specification of the tool holder is ISO standard R390-020A22-11M and the cutting diameter is 20 mm.

Fig.4 Dimensions of the insert and tool holder used in turnmilling

The OTMZE cutting parameters are as follows：rw=40 mm，rt=10 mm，ap=1 mm，fa=3 mm/r，Z=1，andλ=50.CLGs and tool relationships simulated using the OTMZE CLG mathematical model described in Eqs.（5）—（7）via software UG and Matlab are shown in Fig.5（a）.Actual OTMZE chips are measured using a digital measuring microscope（ISM-DL301-Y），as shown in Figs.5（b）and 5（c）.

Fig.5 OTMZE CLG experimental test

Although the chip is substantially deformed，its shape appears generally consistent with the simulated CLG upon analysis.As the tool moves from position 1 to position 2，the contours of the cutting layer formed by the side cutting edges are the surfacesBB'D'DandCC'D'D.The bottom cutting edges formed the surfacesB'AD'andC'AD'.The linesBB'，CC'，DD'，andAD'coincided with the simulated CLG.The results confirm that the CLG mathematical models in this paper are correct.

The cutting depth and cutting layer volume are two importantcutting parametersreflecting the CLG.The cutting depth used in turning and milling is constant，while that of turn-milling changes.Thus，the maximum cutting depthapmax（the vertical distance from pointBto the surfaceC'AD'in Fig.5）can be compared to the simulated CLG and chip measurements shown in Fig.6（a）.The chip volume can not be measured directly.Instead，an electronic scale with a precision of 0.01 g is used to measure the chip weight and the chip volume is calculated indirectly using the material density of the workpiece.The chip volume from the simulated CLG is contrasted with the measured quantity in Fig.6（b）.

Fig.6 OTMZE chip simulation and experimental results

In the theoretical case，the maximum cutting depthapmaxdoes not exceed the cutting depthap.In Fig.6（a），theapmaxmeasured on a chip exceeds theap（ap=1 mm） caused by machining deformation.Theapmaxvalues from the CLG simulation are 0.92 mm，0.92 mm，and 0.91 mm，while theapmaxvalues measured from chips are 0.99 mm，1.01 mm，and 1.02 mm whenfa=1 mm/r，3 mm/r，and 5 mm/r，respectively.The forecasting errors are 7.61%，9.78%，and 12.09%，respectively.

In Fig.6（b），the simulated chip volumes are 2.95 mm3，12.47 mm3，and 22.39 mm3while the measured volumes are 3.21 mm3，13.63 mm3，and 24.58 mm3whenfa=1 mm/r，3 mm/r，and 5 mm/r，respectively.The forecasting errors are 8.81%，9.3%，and 9.78%，respectively.

The above comparisons and analyses of results show that the simulated and measured values are quite similar.In addition，the simulated CLG coincides with the geometry of actual chip.This demonstrates that the mathematical modeling of CLG performed in Section 2 is accurate and feasible.

4 OTMZE Cutting Layer Analyses

The side and bottom cutting edges participate simultaneously in cutting and form cutting layers during OTMZE.To analyze OTMZE cutting layer changes further，data is generated using Eqs.（5）—（7）and cutting thicknesses and cutting layer depths are determined and analyzed using the data.

4.1 Effect of parameters on cutting layer formed by the side cutting edge

The cutting thicknesses and depths formed by a side cutting edge withφiarehp（φi） andap（φi），respectively，as shown in Fig.7.Asφiincreases，hp（φi） increases from zero to its maximum and then decreases from its maximum to zero，whileap（φi）increases to its maximum.

As shown in Fig.7（a），given thatap=1.5 mm，andλ=80，φst，C1，φst，C2，andφst，C3are the dynamic contact angles of toothiat pointCof the cutting workpiece（i.e.the entrance angle of the side cutting edge） whenfa=3 mm/r，5 mm/r，and 7 mm/r，respectively.The entrance angle of the side cutting edge at pointC φst，Cdecreases asfaincreases.

Fig.7 Effect of parameters on cutting layer formed by side cutting edge(rw=40 mm,rt=10 mm,Z=1)

Here，φst，B1，φst，B2，andφst，B3are the dynamic contact angles of toothiat pointBof the cutting workpiece whenfa=3 mm/r，5 mm/r，and 7 mm/r，respectively.The entrance angle of the side cutting edge at pointB φst，Bis also the entrance angle of the bottom cutting edge.We note thatφst，Bdecreases asfaincreases.

Here，φex，Dis the dynamic contact angle of toothiat pointDof the cutting workpiece（i.e.the exit angle of the side and bottom cutting edges）.The change inφDasfaincreases is quite small.We note thatφst，Candφst，Bdecrease andφex，Dis nearly unchanged asfaincreases，so the range of the cutting layer grows.

It can be seen from Fig.7（a）that the maximum ofhp（φi）increases but the maximum ofap（φi）is unchanged asfaincreases.Moreover，the rising trend ofhp（φi） from zero to its maximum becomes steeper，while the descending trend ofhp（φi） from its maximum to zero and the rising trend ofap（φi） are unchanged asfaincreases.The steeperhp（φi） curve indicates that the cutting layer formed only by the side cutting edge（i.e.the cutting layer from pointCto pointB）shrinks.

As shown in Fig.7（b），φst，Cis unchanged andφst，Bandφex，Ddecrease asλincreases whenap=1.5 mm，andfa=5 mm/r.This shows that both the cutting layer formed by only the side cutting edge and its range decrease.The maximum ofhp（φi）decreases and the maximum ofap（φi） is unchanged asλincreases.Moreover，the rising trends ofhp（φi）andap（φi） are unchanged and the decreasing trend ofhp（φi）becomes flatter asλincreases.

As shown in Fig.7（c），whenfa=5 mm/r，andλ=80，φst，Cis unchanged and the changes inφst，Bandφex，Dwith increasingapare quite small.This shows that both the cutting layer formed only by the side cutting edge and its range are nearly unchanged.The maximum，rising trend，and decreasing trend ofhp（φi）are nearly unchanged.Moreover，ap（φi）increases withapand the rising trend ofap（φi）changes little.

4.2 Effects ofparameterson cutting layer formed by bottom cutting edge

The cutting thicknesses and depths formed by the bottom cutting edge of a tool withφiarehf（φi）andaf（φi），respectively，as shown in Fig.8.Asφiincreases，hf（φi）decreases from its maximum to zero andaf（φi）increases from zero to its maximum.

As shown in Fig.8（a），φst，Bdecreases andφex，Dis nearly unchanged asfaincreases.Thus，the cutting layer formed by only the bottom cutting edge enlarges.Moreover，the maxima ofhf（φi）increase slightly and those ofaf（φi） increase substantially，the decreasing trend ofhf（φi）shrinks，and the rising trend ofaf（φi）is nearly unchanged.

As shown in Fig.8（b），φst，Bandφex，Dboth decrease asλincreases whenap=1.5 mm，andfa=5 mm/r.This shows that the cutting layer formed by only the bottom cutting edge may not change.Moreover，the maxima ofhf（φi） andaf（φi） decrease asλincreases.In addition，the decreasing trend ofhf（φi）shrinks and the rising trend ofaf（φi）begins to change.

As shown in Fig.8（c），φst，Bandφex，Dremain constant asapincreases whenfa=5 mm/r，andλ=80.This shows that the cutting layer formed by only the bottom cutting edge remains unchanged.Moreover，the curves ofhf（φi）andaf（φi）remain unchanged.This indicates thataphas no influence on the cutting layer formed by the bottom cutting edge.

Fig.8 Effects of parameters on cutting layer formed by bottom cutting edge(rw=40 mm,rt=10 mm,Z=1)

5 Conclusions

（1）Based on the principles of motion and variation in the OTMZE tool entrance and exit angles，the cutting layer can be divided into two stages：φst，C≤φi≤φst，Bandφst，B＜φi≤φex，D.The cutting layer is formed by the side cutting edge in the former stage and by both the side and bottom cutting edges in the latter stage.The prerequisites of the OTMZE CLG mathematical models are as follows：The entrance and exit angles of the side and bottom cutting edges（φst，C，φst，B，andφex，D）；the cutting thicknesses of the cutting layers formed by the side and bottom cutting edges［hp（φi） andhf（φi）］；and the cutting depths of the cutting layers formed by the side and bottom cutting edges［ap（φi）andaf（φi）］.

（2）The three-dimensional geometrical cutting layer shape is simulated using the above mathematical model.Tool position relationships during formation of the shape are discussed to validate the accuracies of the forming process analysis and CLG mathematical models in OTMZE.Moreover，the following experimental chip and simulated CLG data are compared to validate mathematical model accuracy：（i）Shape and profile，（ii）maximum cutting depthapmax，and（iii）volume.Contrasting experimental and analytical results show that the chip shapes are mostly consistent with the simulated CLGs.The errors in the simulatedapmaxand volume do not exceed 12.09%and 9.78%，respectively.

（3）Asφiincreases，hp（φi）increases from zero to its maximum and then decreases back to zero，ap（φi）increases to its maximum，hf（φi）decreases from its maximum to zero andaf（φi）increases from zero to its maximum.Asfaincreases，the cutting layer range becomes larger，while the cutting layers formed by the side and bottom cutting edges become smaller and larger，respectively.The maximum ofhp（φi） increases while that ofap（φi） remains unchanged，the maximum ofhf（φi）increases slightly，and that ofaf（φi） increases substantially.Whenλincreases，the range of the cutting layer shrinks.In addition，the cutting layer formed by only the side cutting edge shrinks while that formed by the bottom cutting edge does not change.In addition，the maxima ofhp（φi），hf（φi），andaf（φi）decrease while that ofap（φi） is unchanged.Asapincreases，the cutting layer range and the cutting layers formed by only the side and bottom cutting edges，respectively，are nearly unchanged.In addition，hp（φi）is nearly unchanged，ap（φi）increases，andhf（φi）andap（φi）are unchanged.