Paper about tool life

International Journal of Machine Tools & Manufacture 40 (2000) 1709–1733

Calculation of optimum cutting conditions for turningoperations using a machining theory

Q. Meng, J.A. Arsecularatne*, P. MathewSchool of Mechanical and Manufacturing Engineering, The University of New South Wales, Sydney 2052, Australia

Received 17 January 2000; accepted 17 March 2000

Abstract

A method is described for calculating the optimum cutting conditions in turning for objective criteriasuch as minimum cost or maximum production rate. The method uses a variable flow stress machiningtheory to predict cutting forces, stresses, etc. which are then used to check process constraints such asmachine power, tool plastic deformation and built-up edge formation. A modified form of Taylor tool lifeequation where the constants are determined using the machining theory has been employed in predictingtool life for the optimisation procedure. The obtained results indicate that the described method is capableof selecting the appropriate cutting conditions. 2000 Elsevier Science Ltd. All rights reserved.

Keywords:Optimum cutting conditions; Turning operations; Machining theory

1. Introduction

For a machining process such as turning, the cutting conditions play an important role in theefficient use of a machine tool. Because of the high cost of numerically controlled (NC) machines,compared with their conventional counterparts, there is an economic need to operate thesemachines as efficiently as possible in order to obtain the required payback. Since the cost ofturning on these machines is sensitive to the cutting conditions, optimum values have to be determ-ined before a part is put into production. This need is even greater in the case of rough machiningsince a greater amount of material is removed thus increasing possible savings. The optimumcutting conditions in this context are those which do not violate any of the constraints that mayapply on the process and satisfy the economic criterion.

Procedures reported so far to determine the optimum cutting conditions are various nomograms

* Corresponding author. Tel.:+612-9385-5698; fax:+612-9663-1222.E-mail address:[email protected] (J.A. Arsecularatne).

0890-6955/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved.PII: S0890-6955(00)00026-2

1710 Q. Meng et al. / International Journal of Machine Tools & Manufacture 40 (2000) 1709–1733

Nomenclature

Cs side cutting edge angle (deg)d depth of cut (mm)f feed (mm/rev)F frictional force at tool-chip interface (N)FC orthogonal force component in direction of cutting (N)FT orthogonal force component normal toFC acting in plane normal to cutting edge (N)FS shear force onAB (N)FR force component normal toFC andFT (N)i inclination angle (deg)kAB shear flow stress onAB (N/mm2)kchip shear flow stress in chip at tool-chip interface (N/mm2)le length of cutting edge (mm)N normal force at tool-chip interface (N), spindle speed (rev/min)Nbreak speed at which the power-speed characteristic of the machine changes (rev/min)Prq power required (W)r radius of cutting (m)re tool nose radius (mm)R resultant force in orthogonal chip formation model (N)t2 machining time (min)t3 tool change time (min)tc cut thickness (mm)tch chip thickness (mm)T tool life (min)Tint average temperature along tool-chip interface (°C)Tmod velocity modified temperature (K)Topt optimum tool life (min)U cutting speed (m/min)Uopt optimum cutting speed (m/min)w width of cut (mm)W volume of metal removed (cm3)x operating cost of machine ($/min)y cost of cutting edge ($)a tool rake angle (deg)an tool normal rake angle (deg)f shear angle (deg)hc chip flow angle (deg)hO chip flow angle due to the effect of the nose radius measured from the normal to the side

cutting edge on the reference plane (deg)sf average normal stress on tool flank-work interface (N/mm2)sN average normal stress on tool rake face (N/mm2)tf average shear stress on tool flank-work interface (N/mm2)tint average shear stress on the tool-chip interface (N/mm2)tmax maximum shear stress in the tool (N/mm2)* star sign to indicate angles associated with equivalent cutting edge

1711Q. Meng et al. / International Journal of Machine Tools & Manufacture 40 (2000) 1709–1733

[1], graphical techniques [2], performance envelope [3], linear programming [4], geometrical pro-gramming [5], search procedures [6,7] and approaches based on classical mathematical optimis-ation [8,9]. Although many of these works consider the machine power as the only process con-straint, some of the recent research workers have been able to consider constraints such as chipcontrol [6,7], maximum allowable force [6–9], etc. From the above work it is clear that in orderto determine the optimum cutting conditions, one has to estimate the tool life and cutting forceswith a reasonable degree of accuracy since many of the constraints that may apply on the processare influenced by these parameters. For a practical machining situation, since no adequate machin-ing theory is available to predict the tool life and cutting forces, one is compelled to rely onempirical equations to predict these parameters. However, these empirical equations involve anumber of constants which are not readily available. Furthermore these constants depend on manyfactors thus requiring a huge amount of data for a general workshop situation. To obtain andmanage such a huge amount of data is an extremely difficult task. Therefore an alternative to thisempirical approach that can be used to predict cutting forces, tool life, etc. will be of great value.

This paper presents a method by which cutting forces, tool life, etc. and subsequently optimumcutting conditions can be predicted in turning with oblique nose radius tools using as a basis theorthogonal machining theory developed by Oxley [10] and his co-workers. This theory, whichtakes account of variations in work material flow stress with strain, strain-rate and temperature,has been applied with considerable success in predicting cutting forces, temperatures, etc. froma knowledge of the work material properties and the cutting conditions.

2. Determination of optimum cutting conditions

Part programming, which involves preparation of control instructions for NC machine tools, isa time consuming task for complex parts and therefore, was one of the first manufacturing activi-ties where computers were employed (since the 1950s). In early computer-aided part programmingsystems, although the computer does the bulk of the calculations involved it was still necessaryto define part geometry, tool motions, etc. which is a tedious procedure for complex parts. Further-more, even if a few of the available systems incorporate some cutting technology, the level oftechnology advancement, particularly in determining cutting conditions is clearly lagging and isanother deterrent in achieving goals in productivity improvements. In order to overcome theseproblems, computer automated systems are being developed which are designed so that part pro-grams can be prepared with minimum manual intervention. They cater for all the stages involvedin generating part programs which include: determine the volume of material to be removed;sequence the operations; select the tools; determine the optimum cutting conditions; and generatethe tool paths. The system TECHTURN [7] is such a programming system for NC turning centres.

Once the operation planning and the appropriate tools have been determined, success of themachining process depends on the cutting conditions which are required to satisfy an economiccriterion such as minimum cost or maximum production rate. In determining the cutting con-ditions, parameters that must be taken into account include type of operation, whether roughingor finishing; machine tool parameters such as available power, speed and feed ranges and rigidityof the spindle bearing system; cutting tool parameters which include tool material, geometry andthe tool cost; work piece characteristics such as work material properties, geometry, tolerances


and surface finish requirements. From the work described in [7] it is clear that in determiningthe optimum cutting conditions one has to estimate the tool life and cutting forces with reasonableaccuracy since many of the constraints are influenced by these parameters. The tool life andcutting forces are expressed using empirical equations which involve a number of constants. Theseconstants which are affected by work material, tool material and tool geometrical parameters suchas nose radius, rake angle and cutting edge angle, are not readily available. It has been shownthat the amount of required data can be huge for a general workshop situation [11]. To obtainand manage such a huge amount of data is an extremely difficult task. When machining a batchof components, an on-line method devised to obtain the necessary force data using the motorarmature current signals of an NC turning centre is described in [11]. However difficulties wereencountered in this work when calculating the cutting conditions for the first component in thebatch. Furthermore it was necessary to remove a substantial amount of material to obtain reliableconstants. For tool life, data derived utilising those provided by the tool manufacturers were used.These tool life data are not accurate and are intended to be general guides since they often donot coincide with a specific combination of work and tool.

Due to the aforementioned difficulties with the empirical approach the machining theory [10]has been used in this work to predict cutting forces, stresses, etc. in determining the optimumcutting conditions. Another feature of this work is that the extended Taylor tool life equationconstants which are needed for the optimisation procedure have been determined using the mach-ining theory thus eliminating the need for a huge amount of experimental work.

3. Economics of a single-pass turning operation

This section considers the economics of a single pass turning operation and is based on thework described in [7]. The cost of an operation can be expressed as

cost5xHt21t3St2TDJ1ySt2

TD (1)

The cost per cutting edgey for a turning tool with an indexable insert can be calculated as fol-lows

y5cost of holder

4001

cost of insert0.75×number of cutting edges

Although the work set-up time is also sometimes included in Eq. (1), when considering theeffect of cutting conditions on cost, it can be neglected.

Using a modified form of extended Taylor equation, the tool life can be expressed as

T5At

Ubttctc wdt

(2)

where tc andw are cut thickness and width of cut with reference to the equivalent cutting edge


(the equivalent cutting edge which, as will be seen later, is a single straight cutting edge, isdetermined using the chip flow direction and can be used in place of the actual straight and roundcutting edges of an oblique nose radius tool). This form of tool life equation was used because,as will be seen later, the constantsAt, bt, ct, anddt do not depend on a number of tool geometricalparameters thus substantially reducing the amount of empirical data required.

The volume of metal removed is given by

W5t2Utcw (3)

In developing the necessary equations only the minimum cost criterion is considered here. Theresults for the maximum production rate criterion can be deduced by settingx=1 andy=0.

From Eq. (1), Eq. (2) and Eq. (3) the cost per unit volume of metal removed (that is specificcost) can be obtained as

costW

5x

Utcw1

(xt3+y)Ubt−1tct−1c

Atw1−dt(4)

By simplifying Eq. (4) further

costW

51

UtcwHx1

(xt3+y)Ubttctc wdt

AtJ (5)

By eliminatingU in Eq. (5) using Eq. (2)

costW

51

A1/btt t(bt−ct)/bt

c w(bt−dt)/btHxT1/bt1

xt3+yT(bt−1)/bt

J (6)

According to Eq. (4), for given values ofU and tc the specific cost decreases continuously asw increases. Thus a given amount of material can be removed most economically by using themaximum possible width/depth of cut. From Eq. (6) the optimum value ofT for minimum specificcost can be obtained as

Topt5(bt21)St31yxD (7)

For given values ofw andT, it can be shown that material can be removed more economicallyby using a highertc and low cutting speed [7]. For given values ofw and tc the optimum cuttingspeed is given by

Uopt5S At

Topttctc wdt

D1bt (8)


4. Optimisation procedure

In turning, the cutting conditions that need to be optimised are depth of cut, feed and cuttingspeed. The method adopted here, in order to determine the optimum cutting conditions is similarto that used in [6,7] in that it uses a direct search procedure on the depth-feed (d-f) plane andoptimises one pass at a time when more than one pass is required to remove the total depth(multi-pass situation). However one major improvement in the present method is in using themachining theory in place of the empirical equations used in [6,7] to determine the forces, etc.in checking the process constraints. Thed-f plane is defined by the minimum and maximumvalues of depth of cut and feed for the operation, tool and work material. Alternatively thesevalues can be specified by the user. To start with, thed-f plane is divided into a grid, say 10×20,as shown in Fig. 1. Some of the grid points will not be feasible because of the constraints. Thefeasible region is usually separated from the non-feasible region by a curve, as shown in Fig. 1.The specific cost/time of machining at each of the feasible points can be determined. However,the point at which the cost/time is a minimum always lies on the boundary separating the feasibleand non-feasible regions [6,7]. Therefore, it is not necessary to consider all the points on thed-f plane. The search procedure starts from point O in Fig. 1 and the steps involved are

1. Determine the equivalent cutting edge geometry for the given oblique nose radius tool, depthof cut and feed as discussed in section 6.3. Then determinetc and w with reference to theequivalent cutting edge.

2. For a grid point defined by (di, fj) the optimum cutting speed is calculated using Eq. (8).Topt is

Fig. 1. Region for optimisation and search for the optimum point.


the tool life calculated using the appropriate objective criterion (i.e. minimum cost or maximumproduction rate). If the user does not have the relevant cost and time data (required for theeconomic criterion) then he/she can specify the tool life directly.

3. The optimum cutting speed calculated in step 2 is checked for the tool plastic deformation andpower constraints discussed in sections 5.1 and 5.2. If it violates these constraints then thepoint may still be feasible but a sub-optimum speedU0 is calculated to satisfy these constraints.If it is non-feasible (eg. point 1 in Fig. 1) then the point with the next lower depth but withthe feed on the same line (point 2) is tested and the method returns to step 1.

4. The optimum/sub-optimum speed is checked for built-up edge (BUE) constraint. If it violatesthis constraint or the resulting tool life is unacceptably high then the point becomes non-feas-ible. If it is non-feasible then the point with the next lower depth but with the feed on thesame line is tested and the method returns to step 1.

5. If it is feasible (point 3) the specific cost/time of machining is calculated for this point usingthe appropriate values ofd, f and U. The point on the grid with the same depth but a higherfeed is considered next (point 4) and the method returns to step 1.

6. Steps 1 to 5 are repeated until the first non-feasible point on the lowest depth line or point Mis met. The optimum depth of cut, feed and cutting speed are given by the point at which thespecific cost/time is a minimum.

If the total depth to be removed is greater than the optimum depth determined in step 6 (whichwould normally be the case) then steps from 1 to 6 should be repeated and the optimumd, f andU for each pass should be determined until the sum of the optimum depths of cut equals orexceeds the total depth to be removed. If the latter occurs these optimum depths should then bemodified in order to remove the exact total depth. This depth modification process will be furtherdiscussed in section 7.3.

5. Constraints

5.1. Tool plastic deformation

Cutting tools have a tendency to deform plastically under the influence of the high compressivestresses and temperatures encountered during machining at high speeds and feeds. Plastic defor-mation of the tool changes the geometry of the cutting edge which in turn causes acceleratedrates of tool wear, resulting in a decrease in tool life and in the machined surface quality. Plasticdeformation can also cause catastrophic tool failure which can damage the component, the tooland/or machine tool and thus interrupt the machining process substantially. In obtaining the opti-mum cutting conditions it is important to determine the conditions that cause tool plastic defor-mation and avoid them.

In order to predict the conditions giving plastic deformation of the cutting edge it is necessaryto calculate the maximum shear stresstmax in the region of the tool adjacent to the cutting edge.It has been shown that [12]tmax can be determined using Eq. (9).


tmax512(tf2tint)1

12

F(sf−sN)1p2(tint−tf)G2

!F(sf−sN)1p2(tint−tf)G2

1(tint1tf)2

112

(tint1tf)2


1(tint1tf)2

, whenF(sf2sN)1p2(tint2tf)G$0

tmax512(tf2tint) (9)

212

F(sf−sN)1p2(tint−tf)G2


1(tint1tf)2

212

(tint1tf)2


1(tint1tf)2

,

whenF(sf2sN)1p2(tint2tf)G,0

When considering plastic deformation of a tool working at high temperatures, the failure cri-terion can be taken as

|tmax|$|scomp

2 | (10)

wherescomp is the uniaxial compressive strength of the tool material at the onset of plastic defor-mation. In practice, it is assumed that whentmax reaches a value ofs0.05/2, i.e. when

|tmax|$|s0.05

2 | (11)

the tool starts to deform plastically wheres0.05 is the 5% proof stress for the tool material deter-mined from high temperature uniaxial compression test data such as that given by Trent [13].Thustmax and hence the cutting edge plastic deformation conditions can be determined ifsN, tint,sf, tf andTint are known. For a given set of cutting conditions and tool geometrysN, tint andTint

are determined using the machining theory. The tool flank-work interface stressessf and tf aredetermined using the empirical equations which are given in the Appendix.

5.2. Machine tool torque/power

The power speed characteristics of a d.c. motor drive are shown in Fig. 2. In the speed range0 to Nbreak1 available torque is a maximum and is given by


Fig. 2. Power-speed characteristics of the machine.

Tmmax5Pbreak1

2pNbreak1

(12)

For given values ofdi, fj andU if the required torque to machine the component is greater thanTmmax the grid pointdi, fj becomes non-feasible. If it is feasible the power required to machinethe component is calculated as

Prq5FcU/60 (13)

The cutting conditionsdi, fj andU are feasible if

Prq#Pav

Pav is the maximum available power at spindle speedU/2pr. If Prq.Pav andN.Nbreak1 then thepoint di, fj may still be feasible but a new value forU should be calculated so thatPrq=Pav. Atcutting speedU=2prNbreak1, if Prq.Pbreak1 then the grid point becomes non-feasible.

5.3. Minimum and maximum tool life values

It is assumed that tool life equation (Eq. (2)) is valid forT only in the rangeTmin#T#Tmax.For given tc, w and U the tool life is calculated using Eq. (2). If this value of tool life is lessthan Tmin, cutting speed is reduced so thatT=Tmin. If, on the other hand, tool life is greater thanTmax, di, fj combination is considered to be non-feasible.


5.4. Minimum and maximum depths/feeds for tool and work piece

A given di, fj combination must satisfy

dtmin#d#dtmax

ftmin#f#ftmax

wheredtmin andftmin depend on the operation, tool and work material [7]. In accordance with toolmanufacturers’ recommendationsdtmax and ftmax are calculated as follows

dtmax523le

ftmax50.8re

If the stock (total depth) to be removed is less thandtmax thendtmax=stock. Alternativelydtmax

and ftmax can be specified by the user. The region for optimisation in Fig. 1 is obtained using theabove values ofdtmin, ftmin, dtmax and ftmax.

5.5. Built-up edge formation

For plain carbon steels it has been shown [10] that when the values ofTmod at the tool-chipinterface are higher than (approximately) 700 K it follows that the layer of chip material adjacentto the interface that has the highest temperature in the chip is the weakest; therefore deformationshould occur in this layer. However, as a result of dynamic strain ageing this is not true for valuesof Tmod in the range (approximately) 500 K,Tmod,700 K, and for such cases the chip will gener-ally be weaker some distance from the interface where the temperature is lower. Noting this andthe experimental results obtained from machining tests it was proposed that [10]

1. if Tmod .700 K then there would be no built-up edge but that for lower values there will be,2. even ifTmod ,700K there will be no built-up edge ifTint.1,000 K.

Using the experimental results obtained from bar turning tests it has been shown that the built-up edge range can be predicted exceptionally well using the above two criteria. These experimentswere carried out under orthogonal conditions [10] and under oblique conditions using nose radiustools [14].

Thus for a given depth of cut, feed and cutting speed in order to consider the above constraintsit is necessary to estimate the cutting forces, temperatures, stresses, etc. The following sectiondescribes the methods used to calculate the above parameters in oblique machining with noseradius tools.

6. Prediction of cutting forces, temperatures, etc. in oblique machining

Using extensive experimental results it has been shown that the orthogonal machining theory[10] can be extended to predict the chip flow direction and cutting forces in oblique machining


with nose radius tools by introducing the concept of an equivalent cutting edge [15]. It has alsobeen shown that predicted tool temperatures can be used to determine the built-up edge formationrange [14] and tool life [16] with reasonable accuracy. The method used in the present work todetermine the equivalent cutting edge and subsequently cutting forces, temperatures, etc. whichis based on this previous work, is now described.

6.1. Chip flow direction and equivalent cutting edge

In order to predict the chip flow direction the method adopted for nose radius tools with non-zero rake and inclination angles is as follows. The chip flow angle due to the effect of the noseradius is determined first by assuming a tool with zero rake and inclination angles irrespectiveof their actual values. The equivalent cutting edge for this case is taken to be at right angles tothe chip flow direction. The line representing this equivalent cutting edge is now projected ontothe face of the tool with non-zero rake and inclination angles with the projected line assumed torepresent the equivalent cutting edge for the actual tool. Since this method is discussed in detailin reference [15] what follows is only a brief review of the method.

6.2. Chip flow angle due to the effect of nose radius

In the method described in [15] the chip is treated as a series of elements of infinitesimal width.The frictional force component for each element changes in magnitude as well as direction. Thesefrictional force components are summed up in order to find the resultant and it is assumed thatthis resultant coincides with the chip flow direction. In this way the resultant chip flow angle dueto the nose radius effect,V̄0 can be determined from the relation

V̄05tan−11EsinV0dA

EcosV0dA2 (14)

where dA is the area of the cut element andV0 is the angle a chip element makes with theoutward radial direction. By integrating the numerator and denominator of Eq. (14) over the entirearea of cut section, the chip flow angleV̄0 is determined. These relations are given in refer-ence [15].

It is also possible to define the chip flow angle with reference to the normal to the straightside cutting edge of the tool. As depicted in Fig. 3, if this angle is denoted byh0, it can be relatedto V̄0 by the relationship

h05p2

2Cs2V̄0 (15)


Fig. 3. Equivalent cutting edge and tool angles.

6.3. Modified tool angles and equivalent cutting edge

Using three dimensional geometric analysis the equation forh90, which is the projection ofh0

on the tool rake face plane as shown in Fig. 3, is obtained as follows

h905cos−1S seci−tanitanh0tanan

{(tani−tanh0tananseci)2+sec2h0}12

D (16)

The same technique andh90 are then used to obtain the equations for the equivalent cutting

edge normal rake angle,a∗n, inclination anglei* and side cutting edge angleC∗

s which aregiven below

i∗5sin−1(cosh90sini2sinh9

0sinancosi)

a∗n5sin−1Ssech9

0sini−sini∗

tanh90cosi∗ D (17)

C∗s5Cs1h0


For a typical nose radius tool, modified tool angles associated with the equivalent cutting edgeand the general tool angles are shown in Fig. 3.

In summary, a tool with a nose radiusre and side cutting edge angleCs, inclination angleiand normal rake anglean can be replaced by a tool having a single straight cutting edge, i.e. theequivalent cutting edge, with a side cutting edge angleC∗

s , inclination anglei* and normal rakeanglea∗

n. Note that the cut thicknesstc (Fig. 4) and width of cutw needed in the calculationsare determined from the feedf and depth of cutd in terms of the equivalent cutting edge angleCs

*, i.e. tc=fcosC∗s , w=d/cosC∗

s .

6.4. Cutting forces, temperatures and stresses in oblique machining

Once the geometry of the equivalent cutting edge is known, the method described in [15] canbe applied to determine the cutting forces, temperatures, etc. The main assumption made indeveloping the oblique theory is based on the experimental observation that for a given normalrake angle and other cutting conditions, the force component in the direction of cutting,FC, andthe force component normal to the direction of cutting and machined surface,FT, are nearlyindependent of the cutting edge inclination angle. Therefore it is assumed thatFC andFT can bedetermined from the orthogonal machining theory by assuming zero inclination angle irrespectiveof its actual value and with the rake angle in the orthogonal theory taken asa∗

n. The actual forces

Fig. 4. Model of orthogonal chip formation used in the analysis.


on the oblique tool can then be determined in terms ofFC andFT from the tool geometry. It isimportant to note that the force in the direction of cutting remains equal toFC. As it is FC whichdetermines the power expended in chip formation, it can be seen that with this model, the obliquetool temperature can be taken as being equal to the temperature calculated from the orthogonaltheory. Considering the relatively small inclination angles used with carbide tools it is alsoassumed that the uniform tool rake face (normal and shear) stress distributions determined fromthe orthogonal theory are applicable to oblique machining.

6.5. The orthogonal machining theory

In predicting the cutting forces, temperatures, stresses, etc. a shortened version of the orthogonalmachining theory as described by Mathew [17] is used. The machining theory is based on a modelof chip formation derived from analyses of experimental chip formation flow fields. Since thetheory has been described in detail elsewhere [10,17] only a brief outline is given here. The modelused in developing the theory (Fig. 4) assumes the chip formation process to take place underplane strain, steady-state conditions. The plane AB, near the centre of the chip formation zoneand the tool-chip interface are both assumed to be directions of maximum shear stress andmaximum shear strain-rate. The plastic zone in the chip adjacent to the tool-chip interface isrepresented in the theory by a parallel-sided boundary layer across which the velocity changesfrom zero at the interface to the chip velocity at its outer boundary. In order to predict cuttingforces, temperatures, etc. it is first necessary to determine the so called shear anglef which canbe seen to define the geometry of Fig. 4 for given values of tool rake anglea and cut thicknesstc. This is achieved in the following way. The resultant forceR (Fig. 4) transmitted across ABis calculated for a range of values off by analysing the stresses along AB. In making the stressanalysis account is taken of the variations in flow stress with strain, strain-rate and temperature.OnceR is known it is resolved along the tool-chip interface to give the frictional forceF. Bydividing F by the interface area the average value of the interface shear stresstint is determined.In calculating the interface area it is assumed that the normal stress is constant along the interfacewith the tool-chip interface boundary layer extending over the full interface. For the same rangeof values off the temperatures and strain-rates in the boundary layer are calculated and used todetermine the average value of shear flow stress in the chip along the interfacekchip. The solutionfor f is taken as the value which makestint=kchip as the assumed model of chip formation is thenin equilibrium.

So far the machining theory has mainly been applied to making predictions for steel workmaterials with the required flow stress properties obtained from high speed compression testsmade over a wide range of temperatures. The thermal properties used in calculating temperatureshave been determined from well established empirical equations. The theory has been applied tothe prediction of cutting forces, temperatures, stresses etc. for wide ranges of cutting conditionsand steels [10]. Good agreement has been shown between predicted and experimental results.


7. Results and discussion

7.1. Determination of tool life equation constants

In order to estimate the tool life the extended Taylor equation (Eq. (18)) is normally used. Inthis equation the influence of operator controllable variablesd, f andU on tool life are indepen-dently considered. Although this type of equation can be used to obtain good estimates of toollife, one major disadvantage is that the constantsAt1, bt1, ct1, anddt1 depend on many parameterssuch as work material, tool material and tool geometrical parameters that include rake angle, noseradius and side cutting edge angle. Thus every time any one of the above parameters changes anew set of constants will be required.

T5At1

Ubt1f ct1ddt1(18)

In the modified form of Taylor equation (Eq. (2)), cut thicknesstc and width of cutw referredto equivalent cutting edge are used in place off and d. Since the tool geometrical parameterssuch as nose radius and side cutting edge angle have been taken into account in determining theequivalent cutting edge, the constantsAt, bt, ct, anddt do not depend on these parameters. Thiswill considerably reduce the amount of tool life data required. However these constants willdepend on work material, tool material and tool rake angle. Due to the above reason Eq. (2) hasbeen used in estimating the tool life in the present work. What follows is a brief description ofthe method used to obtain the tool life equation (Eq. (2)) constants for the two carbide grades(uncoated P25 and coated P15) used in the present work.

For the uncoated P25 grade tool material in order to obtain the constants of Eq. (2) the approachused was as follows. Taking account of the observation that when machining with cutting tooltemperatures above 800°C, which would generally be the case for carbide tools in the normalcutting speed range for turning operations, the main wear mechanism is diffusion which is atemperature controlled rate process [10]. Therefore, if the appropriate tool temperature can bedetermined and the relationship between tool life and temperature is known then it should bepossible to predict tool life. When the tool life is defined in terms of flank wear, the relationshipbetween tool life,T and tool flank temperature,Tf can be written as [16]

T5AT−Bf (19)

where A and B are constants. It has also been shown thatTf could be determined from therelation [16]

Tf50.9Tint (20)

whereTint is the interface temperature determined using the machining theory and the temperaturesare in kelvins.

First the constantsA and B in Eq. (19) were determined from a small number of tool liferesults obtained for near-orthogonal conditions. In these experiments nose radius tools with zeroinclination angles were used. For the same cutting conditions the corresponding values ofTf were


determined from the machining theory. When plotted on a log(Tf) versus log(T) graph the resultsfell close to a straight line and the corresponding equation was

T51035.323T−11.872f (21)

In order to obtain the constantsAt, bt, ct, and dt, 15 combinations of width of cut/cutthickness/cutting speed values were first selected from rangesw=1.5 to 4.5 mm,tc=0.1 to 0.4 mmand U=100 to 200 m/min, respectively. For each combination of cutting conditionsTint valueswere determined from machining theory. The rake angle of the actual tool which was26° wasused in calculations. The correspondingTf values and tool life values were then determined fromEq. (20) and (21). Finally the values ofw, t1, U andT were used to determine the constantsAt,bt, ct, and dt using the multiple linear regression analysis provided in the SPSS package [18].These constants are given below

At5281104bt52.56319ct51.66511dt<0.0

As noted earlier the constantsAt, bt, ct, anddt will depend on work material, tool material andtool rake angle. However for a given tool material to account for variations in rake angle andwork material composition, additional experimental data will not be required since these variationscan be taken into account in calculatingTint using the machining theory. Only when the toolmaterial changes will it be necessary to obtain a new set of constantsA andB for Eq. (19) whichcan be done using a relatively small number of machining tests. Thus with the described methodfor determining the constantsAt, bt, ct, anddt reliance on experimental data has been minimised.

For the coated P15 grade tool, since the constants of Eq. (18) can be obtained from [7] theconstantsAt, bt, ct, anddt were determined using a different procedure. For 15 combinations ofdepth of cut/feed/cutting speed values selected from rangesd=1.5–4.5 mm,f=0.1–0.4 mm andU=100–300 m/min, respectively, tool life values were determined using Eq. (18). The constantsAt1, bt1, ct1, anddt1 used were those given in [7]. Then for each combination, using depth of cut,feed and tool geometrytc andw values with reference to the equivalent cutting edge were determ-ined. The values ofw, tc, U and T were then used to determine the constantsAt, bt, ct, and dt

using the SPSS package and are given below.

At5278,351,778bt53.0 ct51.2 dt50.22

7.2. Worked examples

In this section the optimisation procedure discussed so far is illustrated with examples. Thecost rate of the machine used is assumed to be $150 per hour. The maximum depthdtmax isassumed to be 2.5 mm. The maximum feedftmax when calculated as described in section 5.4 is0.64 mm/rev. The minimum values for depth and feed,dtmin and ftmin are assumed to be 0.5 mmand 0.2 mm/rev, respectively. These values ofdtmin, ftmin, dtmax andftmax define thed-f planes (Fig.5 and Fig. 8) for the optimisation procedure. The tool life equation (Eq. (2)) constants used aregiven in section 7.1. All the remaining data used in these examples are given in the Appendix.


Fig. 5. Example 1: feasible and non-feasible grid points on the boundary.

7.2.1. Example 1It is assumed that a 50 mm diameter AISI-1045 steel bar has to be machined using a P25 grade

carbide tool (Holder: PDNNR2525; Insert: DNGA150408). The search for the optimum cuttingconditions is confined to thed-f plane and the optimisation procedure starts from the top leftcorner as shown in Fig. 5.

Fig. 6a shows the variation of optimum cutting speed at considered grid points while Fig. 6bshows the variation of specific cost at these grid points. These results were obtained using theoptimisation procedure discussed above with minimum cost as the economic criterion. The opti-mum tool life calculated is 6.12 min. At the first grid point (d=2.5 andf=0.2 mm/rev) the optimumcutting speed corresponding to a 6.12 min tool life is 209 m/min. The maximum speed that canbe used without violating plastic deformation constraint is 390 m/min. Therefore this constraintis satisfied. The estimated power requirement at speed 209 m/min is 4.33 kW. Since the availablepower at that speed is 11.15 kW power constraint is also satisfied. The calculated interface tem-perature is 1,180 K. Since it is above 1,000 K, BUE constraint is also satisfied. The specific costof machining at this grid point is 26,198 $/m3. The grid points at which the calculations areperformed are clearly labelled in Fig. 5; the specific costs corresponding to these grid points areshown in Fig. 6b. The encircled grid point results in the optimum depth/feed/speed combination,that is the position at which the specific cost is a minimum. It can also be seen that for gridpoints from O to N (Fig. 5) depth remains constant at 2.5 mm and feed increases from 0.2 to0.64 mm/rev. With increase in feed both the unconstrained optimum cutting speed (Fig. 6a) andthe specific cost of machining (Fig. 6b) decreases continuously. This clearly shows that, at a givendepth, it is much more economical to machine using a high-feed/low-speed combination than alow-feed/high-speed combination.

Fig. 7a and b show the results obtained using the optimisation procedure discussed above withmaximum production rate as the economic criterion. The optimum tool life calculated is 3.126


Fig. 6. Example 1: (a) optimum cutting speeds at feasible points; (b) specific cost values at feasible points.


Fig. 7. Example 1: (a) optimum cutting speeds at feasible points; (b) specific production time values at feasible points.


min. Since this is lower than the minimum preset value (5 min), a tool life of 5 min is selected.Once again none of the grid points lying on the boundary ONM (Fig. 5) were found to violateany of the constraints considered. The encircled grid point at which depth and feed are maximumand speed is minimum represents the optimum point since these cutting conditions result in theminimum production time (Fig. 7b). For grid points from O to N with increase in feed both theunconstrained optimum cutting speed (Fig. 7a) and specific production time (Fig. 7b) decreasecontinuously. This confirms that, at a given depth, it is much more economical to machine usinga high-feed/low-speed combination than a low-feed/high-speed combination. Note that at a givengrid point (depth/feed combination) the optimum speed for minimum cost criterion is lower thanthat for maximum production rate criterion. This is due to the fact that tool life for minimumcost criterion is greater than the tool life for maximum production rate criterion.

7.2.2. Example 2It is assumed that a 200 mm diameter AISI-1045 steel bar has to be machined using a tool

with a P15 grade-coated carbide insert (Holder: PCLNR2525; Insert: CNMG120408). Thed-fplane used in the search for the optimum cutting conditions is shown in Fig. 8.

Fig. 9a and b show the results obtained using the optimisation procedure with maximum pro-duction rate as the economic criterion. The optimum tool life calculated is 3.99 min. Since thisis lower than the minimum preset value (5 min), a tool life of 5 min is selected. At the first gridpoint (d=2.5 andf=0.2 mm/rev) the optimum cutting speed corresponding to a 5 min tool life is699 m/min. The maximum speed that can be used without violating the plastic deformation con-straint is 390 m/min. Therefore, the optimum speed cannot be used; instead the sub-optimumvalue of 390 m/min is selected. With this relatively low sub-optimum value still the power con-straint cannot be satisfied. Therefore the grid point becomes unfeasible. The grid point correspond-ing to the next lower depth is also unfeasible due to power constraint. At the third grid point

Fig. 8. Example 2: feasible and non-feasible grid points on the boundary.


Fig. 9. Example 2: (a) optimum cutting speeds at feasible points; (b) specific production time values at feasible points.


considered power constraint can be satisfied. All the grid points that are considered by the optimis-ation procedure and the point corresponding to the optimum depth/feed combination are shownin Fig. 8. In this example all the unfeasible points are due to power constraint. At each feasiblepoint the corresponding optimum cutting speed and specific production time are given in Fig. 9aand b. Note that, at a given depth and feed, the optimum cutting speeds are much higher due tosuperior wear characteristics of the coated tool used in this example than the corresponding speedsfor the previous one (Fig. 7a) which are for an uncoated P25 grade tool.

7.3. Discussion

For a given tool/workpiece material combination, the optimum cutting conditions determinedusing the described procedure were, in general, found to be less conservative than those rec-ommended by the tool manufacturers. The cutting conditions recommended by the tool manufac-turers have to be conservative since they do not take into account process constraints such asmachine power, tool plastic deformation, etc. The given results show that the described optimis-ation procedure which uses the machining theory can be used to determine the optimum cuttingconditions in rough turning. Although the procedure described in section 4 does not specificallyconsider single pass or multi-pass situation it can be used to determine the optimum cuttingconditions for either of the above two situations. In cases where the optimum depth for the firstpass is less than the total depth to be removed, a single pass operation is still possible if themaximum feasible depth on thed-f plane is equal to the total depth. The corresponding feed,speed and machining cost/time for the single pass operation can be obtained from thed-f plane(eg. Fig. 8), speed graph (eg. Fig. 9a) and specific cost/time graph (eg. Fig. 9b), respectively. Insuch cases the user can compare the cost/time for the single pass and multi-pass situations (thelatter of which is discussed below) and choose the more economical option. In the case of multi-pass turning the procedure described in section 4 should be repeated and the optimumd, f andU for each pass should be determined until the sum of the optimum depths equals or exceeds thetotal depth to be removed. If the latter occurs these optimum depths should then be modified inorder to remove the exact total depth. These modified depths should be maintained, as far aspossible, close to their optimum values. Furthermore the geometry of the component should betaken into consideration. For example, if the machining profile is stepped, care should be takento ensure that in any one pass, long lengths of the component are not machined with small depths,as this results in uneconomical machining. At the same time, it should be ensured that none ofthe other constraints, such as minimum depth of cut for the tool, are violated. Finally for eachmodified depth (and possible secondary depths in each pass due to stepped machining profile)the optimumf andU should be redetermined using the correspondingd-f plane, optimum cuttingspeed graph and specific cost/time graph. A similar procedure was used in the work described in[7] to modify the initial depths and to recalculate the optimumf andU. It is possible to developa computer program to achieve the above depth modification process automatically. It is expectedto develop and incorporate such a procedure in future work.

The originality of the present approach is in using the machining theory to determine the forces,stresses, temperatures, etc. in checking the process constraints and to determine the constants inmodified Taylor tool life equation. At this stage analysis has to be restricted to plane face toolswhich may limit the practical value of the work. In particular chip breaking constraint needs to


be incorporated into the optimisation procedure so that the method can have wider application.This will require the machining theory to be extended so that cutting forces, tool life, etc. forchip breaker tools can be predicted. As a first step towards the prediction of cutting forces forthese tools a method has been developed [19] by which the cutting forces can be predicted inorthogonal machining with restricted contact tools. The method uses orthogonal machining theory[10] to predict the cutting forces and natural tool-chip contact length for the equivalent plane facetool. These parameters are then used together with suitable empirical equations to predict forcesunder restricted contact conditions. Further work has been planned for extending this approachto predict cutting forces, tool life, etc. in machining with commercial chip breaker tools. Oncecompleted this work should offer an alternative, far more efficient approach to selecting cuttingconditions when using tools with chip breakers than the empirical methods widely used at presentfor this purpose.

8. Conclusions

In this work it has been shown that a modified form of the extended Taylor tool life equationcan be used to predict the tool life in turning. The required constants for this equation weredetermined using a method based on machining theory. The approach used should greatly reducethe experimental work needed in collecting tool life data as it allows variations in work materialproperties and tool geometry to be allowed for independently of experiments. It was also shownthat, in determining the optimum cutting conditions for economic criteria such as minimum costand maximum production rate, the machining theory can be used to check the process constraintssuch as tool plastic deformation, machine power and BUE formation.

Acknowledgements

The authors gratefully acknowledge the valuable help rendered by Professor P.L.B. Oxley dur-ing the course of this work. Thanks are also due to the Australian Research Council for finan-cial support.

Appendix A. Data for examples

Work materials: nominal AISI-1045 plain carbon steel bars of chemical composition 0.45% C,0.01% P, 0.66% Mn, 0.053% Si, 0.031% S, 0.01% Ni, 0.02% Cr, 0.01% Mo, 0.01% Cu and0.001% Al.

Tool materials: chemical composition of P25 grade tool insert was 7.6–8.4% TiC+3.5% Co+11–12% TaC. Same chemical composition was assumed for P15 grade-coated insert. The high tem-perature shear strength curve for these tools were obtained using the method described in [12].

The tool flank-work interface shear stresstf and normal stresssf are obtained using the empiri-cal relations:


sf50.0528FT

tc

tf50.0671FC

tc

wheretc, FC andFT are cut thickness, cutting force and feed force, respectively.Power-speed characteristics of the machine

Nbreak1=1000,Pbreak1=11.3 kW;Nbreak2=3000,Pbreak2=10.4 kW;Nbreak3=4500,Pbreak3=6.9 kW

t3=2 min

Tmin=5 min

Tmax=100 min

x=1.67 $/min

y=3.2 $

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