# Modeling and Optimizing the Effects of Insert Angles on Hard Turning Performance

Authors: Pham Minh Duc; Mai Duc Dai; Le Hieu Giang
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

To analyze hard turning performance characteristics, a new mathematical model was developed for the hard turning process, and cutting force (CF), another important response for cutting machining, was also studied in the present work. The analysis of the mathematical model and experimental results revealed that thrust force (Fy) was the largest, followed by tangential force (Fz) and feed force (Fx). The resultant CF was most influenced by inclination angle (IA) with 25.02%, followed by rake angle (RA) (14.26%) and cutting edge angle (CEA) (10.04%). Increasing CEA changed the position of cutting on the tool-nose radius and increased local negative RA and correspondingly local clearance angle (CA). Meanwhile, increasing negative RA and IA resulted in larger local negative RA and CA. Moreover, local RA and local CA were the main geometric factors affecting surface roughness (SR), tool wear (TW), and CF. Increasing local negative RA resulted in higher SR and CF. In contrast, increasing local CA resulted in lower SR, TW, and CF. Under specific conditions, the positive effects of the local CA overcame the negative effects of the local negative RA, leading to a simultaneous decrease in SR and TW. The proposed novel mathematical model can be further applied to calculate local CF, cutting temperature, and TW for each cutting-edge element, to analyze and optimize the hard turning process.


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## Body

## 1. Introduction

Hard turning is an advanced technology to machine hard materials and has many advantages compared to conventional grinding. However, the main problem of hard turning is the high surface roughness (SR) of the machined components and quick tool wear (TW). Because of large differences between conventional machining and hard turning, the existing theoretical calculations for the former are usually not applicable to the latter [1].The tool geometry is crucial in hard turning because it affects the performance characteristics such as SR, TW, and CF. In our previously published study [2], we performed an experimental study on the influence of tool angles on SR and TW in hard turning. The results indicated that SR and TW were affected the most by IA. And Gökkaya and Nalbant studied the effects of the tool nose radius and cutting conditions on SR when turning with coated cemented carbide inserts [3]. A minimum SR was obtained with the highest tool-nose radius. The vibrations when using high tool-nose radii did not significantly affect SR. Therefore, high tool-nose radii were recommended for lower SR. From an experimental study on the effect of RA on CF, Günay et al. found that the main CF can be reduced by increasing positive RA and increased by increasing negative RA [4]. And Cao et al. also indicated that CF and SR decreased with increasing RA [5]. Zerti et al. used the Taguchi method to optimize CEA and insert nose radius, feed rate, cutting depth, and cutting speed in turning AISI steel [6]. The authors found that higher CEA and tool-nose radius reduced SR. Harisha et al. also used the Taguchi approach for experimental design and optimization of machining parameters, including CEA, cutting depth, feed rate, and tool nose radius to minimize CF when turning AISI 1055 hardened steel [7]. The authors found that the cutting depth and CEA were important factors on CF. When increasing the CEA, the feed force increased, while the thrust force decreased. CF increased with the feed rate. The optimal CEA was in the range of 60°–70°. Saglam et al. studied the effect of feed rate, CEA, and RA on CF components and cutting temperature in hard turning of AISI 1040 hardened steel by uncoated cemented carbide tool [8]. The authors found that higher CEA produced lower thrust force and higher feed force, and increasing the positive RA decreased CF and increased the cutting temperature. And Singh and Rao investigated the influence of tool geometries, including nose radius and effective rake angle (RA), and machining parameters, including feed rate and cutting speed on SR when hard turning AISI 52100 steel with mixed ceramic inserts [9]. The results showed that higher effective negative RA and feed rate, lower nose radius, and cutting speed increased SR. Moreover, the influence of the feed rate and the insert nose radius on SR was consistent with the theoretical equation SR=f2/32r. Neseli et al. also tried to optimize the tool geometry (tool-nose radius, RA, and CEA) on SR when turning AISI 1040 steel with Al2O3/TiC mixed ceramic inserts [10]. The most influencing factor on SR was the tool-nose radius, followed by CEA and RA. The SR increased with an increase in the CEA and negative back RA. However, the author found that higher tool-nose radii resulted in higher SR, which contradicts common expectations. Sharma et al. found that SR increases with feed rate but decreases with CEA, cutting speed and cutting depth when hard turning with coated carbide insert [11]. This means that the influences of CEA on SR in the studies of Neseli et al. and Sharma et al. were opposite, indicating that the influences of the tool geometry on conventional turning were different from the influences on hard turning. Sahoo and Sahoo investigated TW, SR, chip morphology, and CFs during hard turning of AISI 4340 steel using uncoated and multilayer-coated carbide inserts [12]. Experimental results showed that coated carbide insert was better than uncoated insert in terms of CF, SR, and TW.On the other hand, Zhou et al. found that CF and tool life are strongly influenced by chamfer angle in hard turning, and cutting force increases with an increase in chamfer angle [13]. Zhao et al. assessed the effects of cutting-edge radius on surface quality and TW during hard turning of AISI 52100 steel tests [14]. The authors found that an increase in cutting-edge radius reduced TW. Xu et al. studied the hard machining of AISI 52100 steel and found that changing tool rake surface geometry significantly reduced cutting temperatures, CF, and residual stresses, compared to the flat tool [15]. However, it also lowered the strength of the cutting edge. In milling process, the tool geometry was also included in a cutter displacements (vibrations) model developed by Wojciechowski et al. [16].Besides the tool geometry, cutting conditions such as cutting speed, cutting depth, and feed rate were shown to significantly affect the hard turning performance. For example, Suresh and Basavarajappa studied the effect of machining parameters (depth of cut, cutting speed, and feed rate) on TW and SR in hard turning AISI H13 steel (55 HRC) by coated ceramic inserts [17]. The results indicated that cutting speed mainly affected TW, while SR was most affected by feed rate. Rashid et al. found that when the feed rate was as low as 0.02 mm/rev, it was the most influencing factor on SR (99.16%). However, a low feed rate caused high TW, so the choice of feed rate must be reasonable between cost and quality [18]. In an experimental study on the hard turning of AISI 52100 steel with CBN inserts, Kumar found that the feed rate was the most important contributor to SR while cutting depth was the main factor on CF [19]. CF and SR increased with the increase in feed rate and cutting depth, while decreased with cutting speed. Khamel et al. used response surface methodology (RSM) combined with L27 Taguchi design to model the relationship between responses and cutting conditions during hard turning AISI 52100 steel [20]. Analysis of variance indicated that higher values of cutting depth, feed rate, and cutting speed reduced the tool life. The cutting speed was the most important for tool life, cutting depth was the most influential factor for CF, and feed rate was the main contributor to SR. Thrust force was the greatest component of CF. High cutting speeds resulted in high temperatures in the cutting zone, hence softened the machined materials and reduced the chip thickness and the length of tool-chip contact. The final results were decreases in SR and CF. Zerti et al. studied the effects of cutting speed, cutting depth, and feed rate on SR, CF, and rate of material removal (RMR) in hard turning AISI 420 hardened steel using Taguchi experimental design (L25) [21]. The result showed that SR was greatly affected by the feed rate, while the cutting depth had the most influence on the CF, the cutting power, and the RMR. RSM and artificial neural network (ANN) were used for modeling the machining performance with high accuracies. Chabbi et al. also used RSM and ANN to reveal the relationship between process parameters (cutting speed, feed rate, and cutting depth) and hard turning performance (SR, CF, and RMR) [22]. The determination coefficient of the ANN model was larger than that of the RSM. However, the RSM model is essential for ANOVA. Laouissi et al. compared SR, cutting power, tangential CF, and RMR when turning cast iron using coated and uncoated ceramic tools [23]. The coated tool showed lower SR and CF. The most influential factor on SR was feed rate, followed by cutting speed and cutting depth. The most influencing factor on CF was cutting depth, then feed rate, and cutting speed. RSM and desirability function (DF) were used to model and optimize the hard turning of AISI 316 steel using carbide inserts [24]. Feed rate was the major factor affecting SR, while cutting speed was the most influencing factor on TW. SR increased with higher cutting speeds and feed rates, while the TW increased with higher cutting speeds and cutting depths. The determination coefficients of the quadratic models were 90%, 93.5%, and 88.5% for power consumption, SR, and TW, respectively. Besides DF as a widely used multiobjective optimization method, different optimization techniques can be combined to increase optimal efficiency, such as multilevel strategy or combining fuzzy and sequential Taguchi methods [25, 26].From the mathematical point of view, there were only a few studies on geometrical modeling of the hard turning process. For example, Bushlya et al. proposed the chip area and tool geometry models when oblique turning with round tools [27]. Khlifi et al. modeled the turning process by the equivalent cutting-edge, which theoretically induced the same CF components as the real tool [28]. And Orra and Choudhury also used the equivalent cutting-edge concept to take into account the effects of cutting tool-nose radius [29]. In this context, equivalent angles and uncut chip thickness are defined. However, this approach is not suitable to study tool design and TW, where local data, e.g., temperature distribution at the rake face, are required. Abdellaoui and Bouzid developed a geometric model for the uncut chip area [30]. The cutting edge was decomposed into cutting-edge elements, and then the thickness of uncut chips, direction angle, and cutting depth were determined for each element. A thermomechanical model was applied for every cutting-edge element to determine cutting force components for each element. Molinari and Moufki also modeled the turning process with CEA lower than 90° [31]. To simulate the real cutting tool, its tool-nose radius was mathematically decomposed into straight cutting-edge elements with defined local geometry parameters. This approach was applied to calculate the global chip flow direction, the cutting temperature distribution, CF, and TW. Fu et al. proposed an analytical model where the real cutting-edge was decomposed into a series of infinitesimal cutting elements [32]. However, the mathematical models reported above described only the tool-nose radius as the major cutting-edge element, but did not describe the essence of hard turning and was limited in the number of elements.The literature survey above showed that most studies focused on cutting conditions, while only a few investigated the tool geometry parameters, among which the tool angle is crucial for hard turning [33]. In this study, CF, another important response for cutting machining, was studied in the present work, and we proposed a new geometrical model to explain the effects of tool angle parameters on performance characteristics of hard turning, including SR, TW, and CF.


## 2. Experimental Procedure

### 2.1. Equipment and Materials

CNC lathe BOEHRINGER DUS-400ti with a spindle power of 11 kW was used for hard turning workpieces of AISI 1055 steel (HRC 52 ± 1) with a 53 mm diameter and 130 mm length using mixed ceramic inserts composed of 70% Al2O3 and 30% TiC and coated with PVD-TiN (Figure 1). A tool holder (ISO PTGNR 1616H 16) clamped the inserts (ISO TNGA160408S01525) with RA γo = −6°, CEA κr = 91° and IA λs = −6°. Both the tool holder and the inserts were purchased from Sandvik (Sweden). The cutting conditions were based on the catalog of the tool manufacturer: v = 120 m/min, dw = 0.2 mm, and f = 0.08 mm/rev.Figure 1
Experimental setup.


### 2.2. Methodology

RSM coupled with central composite design (CCD) was used to build a quadratic mathematical model:(1)Y=a0+∑i=13aiXi+∑i=13aiiXi2+∑i<j3aijXiXj+ε,where Y is the response (SR, TW, or CF); a0 is the constant; ai, aii and aij are the coefficients of linear, quadratic and interaction terms, respectively. Xi is the coded variables (CEA κr, RA γo and IA λs), and ε is the random experimental error. The experimental design based on CCD is shown in Table 1.Table 1
Levels of input variables based on CCD of experiments.
Input parametersLevel−α−10+1+αCutting edge angle (κr, °)6066758490Rake angle (γo, °)−2−3.6−6−8.4−10IA (λs, °)−2−3.6−6−8.4−10


## 2.1. Equipment and Materials

CNC lathe BOEHRINGER DUS-400ti with a spindle power of 11 kW was used for hard turning workpieces of AISI 1055 steel (HRC 52 ± 1) with a 53 mm diameter and 130 mm length using mixed ceramic inserts composed of 70% Al2O3 and 30% TiC and coated with PVD-TiN (Figure 1). A tool holder (ISO PTGNR 1616H 16) clamped the inserts (ISO TNGA160408S01525) with RA γo = −6°, CEA κr = 91° and IA λs = −6°. Both the tool holder and the inserts were purchased from Sandvik (Sweden). The cutting conditions were based on the catalog of the tool manufacturer: v = 120 m/min, dw = 0.2 mm, and f = 0.08 mm/rev.Figure 1
Experimental setup.


## 2.2. Methodology

RSM coupled with central composite design (CCD) was used to build a quadratic mathematical model:(1)Y=a0+∑i=13aiXi+∑i=13aiiXi2+∑i<j3aijXiXj+ε,where Y is the response (SR, TW, or CF); a0 is the constant; ai, aii and aij are the coefficients of linear, quadratic and interaction terms, respectively. Xi is the coded variables (CEA κr, RA γo and IA λs), and ε is the random experimental error. The experimental design based on CCD is shown in Table 1.Table 1
Levels of input variables based on CCD of experiments.
Input parametersLevel−α−10+1+αCutting edge angle (κr, °)6066758490Rake angle (γo, °)−2−3.6−6−8.4−10IA (λs, °)−2−3.6−6−8.4−10


## 3. Modeling the Hard Turning Process

To analyze the effects of tool angle parameters on TW, SR, and CF with only the tool-nose radius engagingdw≤r1−cosKr, the main cutting edge was considered as a cutting edge element (j = 0) and the engaged part of the tool-nose radius was modeled by decomposing into straight cutting edge elements (j) so that different models of CF, cutting temperature, TW, etc., can be applied to each element. The configuration in hard turning and local tool geometry parameters are shown in Figure 2. The cutting edge elements from θA to θC are the major, while the elements from θC to θD are the minor cutting edge.Figure 2
(a) The configuration of hard turning. (b) The tool geometry of cutting-edge element (j).
(a)(b)


### 3.1. Modeling the Tool Geometry

Now, we will define the local tool geometry parameters and the chip load of each cutting-edge element. The planesPs1j, Ps−sj, and Po−oj are parallel to the tool cutting-edge plane of cutting-edge element (j) Psj, the tool cutting-edge plane Ps, and the orthogonal plane Po, respectively, as Figure 3. The plane Pc is the tool rake plane; the planes Pnj and Poj are the cutting-edge normal and orthogonal planes of cutting-edge element (j).Figure 3
Model to correlate angles of the main cutting edge and cutting-edge elements (j).
(a)(b)(c)(d)(e)(f)(g)As follows from Figure3,Local cutting-edge angle:(2)Krj=Kr−θj,atθj≤Kr,Krj=θj−Kr,atθj>Kr.Besides, we get(3)tanγoj=HL1,(4)ΔH2=L4tanλsj,ΔH3=L3tanλsj,(5)L1L2=sinπ2−θj,L3L2=cosπ2−θj,L1L5=cosπ2−θj,L4L5=sinπ2−θj,(6)tanγo−oj=tanγo=H−ΔH3L2,tanγs−sj=tanλs=H+ΔH2L5.Substituting equations (3)−(5) into equation (6), one obtains(7)tanγo=sinπ2−θjtanγoj−cosπ2−θjtanλsj,tanλs=cosπ2−θjtanγoj+sinπ2−θjtanλsj.These relationships have been derived using equation (7).Local rake angle:(8)γoj=tan−1sinπ2−θjtanγo+cosπ2−θjtanλs.Local inclination angle:(9)λsj=tan−1−cosπ2−θjtanγo+sinπ2−θjtanλs,atθj≤Kr,λsj=−tan−1−cosπ2−θjtanγo+sinπ2−θjtanλs,atθj>Kr.And local normal rake angle from local rake angle:(10)tanγnj=tanγojcosλsj.


### 3.2. Modeling Undeformed Chip Area

From Figure4, we get the engaged part in the cutting of the nose radius:(11)θA=Kr−∠AO2E=Kr−cos−1r−dwr,θC=Kr,θD=Kr+∠CO2D=Kr+sin−1f2r,θB=Kr−∠FO2E=Kr−tan−1AE−AFO2E=Kr−tan−1rsinKr−θA−fr−dw.Figure 4
The local chip thickness of each cutting-edge element.Local uncut chip thickness:Zone 1:θA≤θj<θB.(12)t1θj=r−O2X1=r−r−dwcosKr−θj.Zone 2:θB≤θj≤θD.(13)t2θj=r−O2X2=r−r2+f2−2rfcos∠O2O1X2.Using the law of sines,(14)O1O2sin∠O2X2O1=O1X2sin∠O1O2X2,fsinθj+π/2−Kr−∠O2O1X2=rsinπ−θj+π/2−Kr.The following relationship is obtained from equation (14):(15)∠O2O1X2=θj+π/2−Kr−sin−1frsinθj+π/2−Kr.Substituting equation (15) into equation (13), one obtains(16)t2θj=r−r2+f2−2rfcosθj+π2−Kr−sin−1frsinθj+π2−Kr.The corresponding local chip area is(17)dAj=tθjrdθ,with an angular increment dθ.To calculateθcj at the tool rake plane Pc, we use the coordinate transformation matrix:The basiszc,ycj,xcj is obtained from zc,yco,xco by a rotation of angle θc around zc, Figure 5(a):(18)zcycjxcj=1000cosθcjsinθcj0−sinθcjcosθcjzcycoxco,ycj=cosθcjyco+sinθcjxco.Figure 5
Relationships between systems: (a)zc,ycj,xcj and zc,yco,xco on the tool rake plane Pc; (b) xr,yrj,xrj and xr,yro,xro on the reference plane Pr; (c) znj,ycj,xrj and zr,yrj,xrj on the cutting-edge plane (j) Psj; and (d) zc,ycj,xcj and znj,ycj,xrj on the normal plane (j) Pnj.
(a)(b)(c)(d)The basisxr,yrj,xrj is obtained from xr,yro,xro by a rotation of angle θr around zr (Figure 5(b)):(19)zryrjxrj=1000cosθjsinθj0−sinθjcosθjzryroxro,yrj=cosθjyro+sinθjxro.The basisznj,ycj,xrj is obtained from zr,yrj,xrj by a rotation of angle λsj around xsj (Figure 5(c)):(20)znjycjxrj=cosλsjsinλsj0−sinλsjcosλsj0001zryrjxrj,zno=cosλsozr+sinλsoyro,yco=−sinλsozr+cosλsoyro.The basiszc,ycj,xcj is obtained from znj,ycj,xrj by a rotation of angleγnj around ycj, Figure 5(d):(21)zcycjxcj=cosγnj0−sinγnj010sinγnj0cosγnjznjycjxrj,xco=sinγnozno+cosγnoxro.By substituting equations (20) and (21) into equation (18), we get(22)ycj=−cosθcjsinλso+sinθcjsinγnocosλsozr+cosθcjcosλso+sinθcjsinγnosinλsoyro+sinθcjsinγnoxro,orycj=Azr+Byro+Cxro,withA=−cosθcjsinλso+sinθcjsinγnocosλso,B=cosθcjcosλso+sinθcjsinγnosinλso,C=sinθcjcosγno.And,yrj is the unit vector of the projection of cutting-edge element (j) on the reference plane (Pr). Therefore,(23)yrj=ycj−ycj.zrzrycj−ycj.zrzr.Substituting equations (19) and (22) in to equation (23), we get(24)cosθjyro+sinθjxro=Byro+CxroX,withX=ycj−ycj.zrzr.Then, after some algebraic manipulation we get(25)tanθj=tanθcjcosγnocosλso+tanθcjsinγnosinλso=tanθcjcosγncosλs+tanθcjsinγnsinλs,(26)θcj=tan−1tanθjcosλscosγn−tanθjsinγnsinλs,atθj<π2,θcj=tan−1tanθjcosλscosγn−tanθjsinγnsinλs+π,atθj>π2.Summary of the modeling hard turning process is as follows.Local cutting-edge angle:(27)Krj=Kr−θj,atθj≤Kr,Krj=θj−Kr,atθj>Kr.Local rake angle:(28)γoj=tan−1sinπ2−θjtanγo+cosπ2−θjtanλs.Local normal rake angle:(29)tanγnj=tanγojcosλs.Local inclination angle:(30)λsj=tan−1−cosπ2−θjtanγo+sinπ2−θjtanλs,atθj≤Kr,λsj=−tan−1−cosπ2−θjtanγo+sinπ2−θjtanλs,atθj>Kr.The engaged part in the cutting of the nose radius:(31)θA=Kr−cos−1r−dwr,θC=Kr,θD=Kr+sin−1f2r,θB=Kr−tan−1rsinKr−θA−fr−dw.Angleθcj in the tool rake plane:(32)θcj=tan−1tanθjcosλscosγn−tanθjsinγnsinλs,atθj<π2,θcj=tan−1tanθjcosλscosγn−tanθjsinγnsinλs+π,atθj>π2.Local uncut chip thickness:Zone 1:θA≤θj<θB(33)t1θj=r−r−dwcosKr−θj.Zone 2:θB≤θj≤θD(34)t2θj=r−r2+f2−2rfcosθj+π2−Kr−sin−1frsinθj+π2−Kr.The corresponding local chip area is.dAj=tθjrdθwith an angular increment dθ


### 3.3. Comparison between the Mathematical Model and Real Cutting Process

To evaluate the accuracy of the mathematical model, we compared the calculated results with the experimental results for the hard turning at various conditions. For example, consider CEAKr = 84°, RA γo = −6.03°, IA λs = −6°, tool-nose radius r = 0.8 mm, chamfer angle = 0°, cutting speed v = 120 m/min, cutting depth dw = 0.2 mm, and feed rate v = 0.08 mm/rev. The calculated engaged part of the tool-nose radius was from θA = 42.59° to θD = 86.87°, the local tool geometry parameters of the straight cutting edge element (j) (θA=42.59°) were the local CEA Krj = 41.41°, the local RA γoj = -8.47°, IA λsj = −0.34°, and the local uncut chip thickness tj  = 0. At the cutting edge element j (θB=47.18°), the maximum undeformed chip thickness was tj = 0.05 mm. The real cutting process (Figure 6) showed that the new mathematical model of the hard turning was accurate, and compared with the conventional mathematical model [30–32], our new model described the essence of the hard turning process and was not limited in the number of elements and CEA.Figure 6
The real cutting process in hard turning withKr = 84°, RA γo = −6.03°, IA λs = −6°, r = 0.8 mm, dw = 0.2 mm, and f = 0.08 mm/rev. (a) The local tool geometry and (b) the uncut chip thickness parameters of the cutting edge element (j).
(a)(b)


## 3.1. Modeling the Tool Geometry

Now, we will define the local tool geometry parameters and the chip load of each cutting-edge element. The planesPs1j, Ps−sj, and Po−oj are parallel to the tool cutting-edge plane of cutting-edge element (j) Psj, the tool cutting-edge plane Ps, and the orthogonal plane Po, respectively, as Figure 3. The plane Pc is the tool rake plane; the planes Pnj and Poj are the cutting-edge normal and orthogonal planes of cutting-edge element (j).Figure 3
Model to correlate angles of the main cutting edge and cutting-edge elements (j).
(a)(b)(c)(d)(e)(f)(g)As follows from Figure3,Local cutting-edge angle:(2)Krj=Kr−θj,atθj≤Kr,Krj=θj−Kr,atθj>Kr.Besides, we get(3)tanγoj=HL1,(4)ΔH2=L4tanλsj,ΔH3=L3tanλsj,(5)L1L2=sinπ2−θj,L3L2=cosπ2−θj,L1L5=cosπ2−θj,L4L5=sinπ2−θj,(6)tanγo−oj=tanγo=H−ΔH3L2,tanγs−sj=tanλs=H+ΔH2L5.Substituting equations (3)−(5) into equation (6), one obtains(7)tanγo=sinπ2−θjtanγoj−cosπ2−θjtanλsj,tanλs=cosπ2−θjtanγoj+sinπ2−θjtanλsj.These relationships have been derived using equation (7).Local rake angle:(8)γoj=tan−1sinπ2−θjtanγo+cosπ2−θjtanλs.Local inclination angle:(9)λsj=tan−1−cosπ2−θjtanγo+sinπ2−θjtanλs,atθj≤Kr,λsj=−tan−1−cosπ2−θjtanγo+sinπ2−θjtanλs,atθj>Kr.And local normal rake angle from local rake angle:(10)tanγnj=tanγojcosλsj.


## 3.2. Modeling Undeformed Chip Area

From Figure4, we get the engaged part in the cutting of the nose radius:(11)θA=Kr−∠AO2E=Kr−cos−1r−dwr,θC=Kr,θD=Kr+∠CO2D=Kr+sin−1f2r,θB=Kr−∠FO2E=Kr−tan−1AE−AFO2E=Kr−tan−1rsinKr−θA−fr−dw.Figure 4
The local chip thickness of each cutting-edge element.Local uncut chip thickness:Zone 1:θA≤θj<θB.(12)t1θj=r−O2X1=r−r−dwcosKr−θj.Zone 2:θB≤θj≤θD.(13)t2θj=r−O2X2=r−r2+f2−2rfcos∠O2O1X2.Using the law of sines,(14)O1O2sin∠O2X2O1=O1X2sin∠O1O2X2,fsinθj+π/2−Kr−∠O2O1X2=rsinπ−θj+π/2−Kr.The following relationship is obtained from equation (14):(15)∠O2O1X2=θj+π/2−Kr−sin−1frsinθj+π/2−Kr.Substituting equation (15) into equation (13), one obtains(16)t2θj=r−r2+f2−2rfcosθj+π2−Kr−sin−1frsinθj+π2−Kr.The corresponding local chip area is(17)dAj=tθjrdθ,with an angular increment dθ.To calculateθcj at the tool rake plane Pc, we use the coordinate transformation matrix:The basiszc,ycj,xcj is obtained from zc,yco,xco by a rotation of angle θc around zc, Figure 5(a):(18)zcycjxcj=1000cosθcjsinθcj0−sinθcjcosθcjzcycoxco,ycj=cosθcjyco+sinθcjxco.Figure 5
Relationships between systems: (a)zc,ycj,xcj and zc,yco,xco on the tool rake plane Pc; (b) xr,yrj,xrj and xr,yro,xro on the reference plane Pr; (c) znj,ycj,xrj and zr,yrj,xrj on the cutting-edge plane (j) Psj; and (d) zc,ycj,xcj and znj,ycj,xrj on the normal plane (j) Pnj.
(a)(b)(c)(d)The basisxr,yrj,xrj is obtained from xr,yro,xro by a rotation of angle θr around zr (Figure 5(b)):(19)zryrjxrj=1000cosθjsinθj0−sinθjcosθjzryroxro,yrj=cosθjyro+sinθjxro.The basisznj,ycj,xrj is obtained from zr,yrj,xrj by a rotation of angle λsj around xsj (Figure 5(c)):(20)znjycjxrj=cosλsjsinλsj0−sinλsjcosλsj0001zryrjxrj,zno=cosλsozr+sinλsoyro,yco=−sinλsozr+cosλsoyro.The basiszc,ycj,xcj is obtained from znj,ycj,xrj by a rotation of angleγnj around ycj, Figure 5(d):(21)zcycjxcj=cosγnj0−sinγnj010sinγnj0cosγnjznjycjxrj,xco=sinγnozno+cosγnoxro.By substituting equations (20) and (21) into equation (18), we get(22)ycj=−cosθcjsinλso+sinθcjsinγnocosλsozr+cosθcjcosλso+sinθcjsinγnosinλsoyro+sinθcjsinγnoxro,orycj=Azr+Byro+Cxro,withA=−cosθcjsinλso+sinθcjsinγnocosλso,B=cosθcjcosλso+sinθcjsinγnosinλso,C=sinθcjcosγno.And,yrj is the unit vector of the projection of cutting-edge element (j) on the reference plane (Pr). Therefore,(23)yrj=ycj−ycj.zrzrycj−ycj.zrzr.Substituting equations (19) and (22) in to equation (23), we get(24)cosθjyro+sinθjxro=Byro+CxroX,withX=ycj−ycj.zrzr.Then, after some algebraic manipulation we get(25)tanθj=tanθcjcosγnocosλso+tanθcjsinγnosinλso=tanθcjcosγncosλs+tanθcjsinγnsinλs,(26)θcj=tan−1tanθjcosλscosγn−tanθjsinγnsinλs,atθj<π2,θcj=tan−1tanθjcosλscosγn−tanθjsinγnsinλs+π,atθj>π2.Summary of the modeling hard turning process is as follows.Local cutting-edge angle:(27)Krj=Kr−θj,atθj≤Kr,Krj=θj−Kr,atθj>Kr.Local rake angle:(28)γoj=tan−1sinπ2−θjtanγo+cosπ2−θjtanλs.Local normal rake angle:(29)tanγnj=tanγojcosλs.Local inclination angle:(30)λsj=tan−1−cosπ2−θjtanγo+sinπ2−θjtanλs,atθj≤Kr,λsj=−tan−1−cosπ2−θjtanγo+sinπ2−θjtanλs,atθj>Kr.The engaged part in the cutting of the nose radius:(31)θA=Kr−cos−1r−dwr,θC=Kr,θD=Kr+sin−1f2r,θB=Kr−tan−1rsinKr−θA−fr−dw.Angleθcj in the tool rake plane:(32)θcj=tan−1tanθjcosλscosγn−tanθjsinγnsinλs,atθj<π2,θcj=tan−1tanθjcosλscosγn−tanθjsinγnsinλs+π,atθj>π2.Local uncut chip thickness:Zone 1:θA≤θj<θB(33)t1θj=r−r−dwcosKr−θj.Zone 2:θB≤θj≤θD(34)t2θj=r−r2+f2−2rfcosθj+π2−Kr−sin−1frsinθj+π2−Kr.The corresponding local chip area is.dAj=tθjrdθwith an angular increment dθ


## 3.3. Comparison between the Mathematical Model and Real Cutting Process

To evaluate the accuracy of the mathematical model, we compared the calculated results with the experimental results for the hard turning at various conditions. For example, consider CEAKr = 84°, RA γo = −6.03°, IA λs = −6°, tool-nose radius r = 0.8 mm, chamfer angle = 0°, cutting speed v = 120 m/min, cutting depth dw = 0.2 mm, and feed rate v = 0.08 mm/rev. The calculated engaged part of the tool-nose radius was from θA = 42.59° to θD = 86.87°, the local tool geometry parameters of the straight cutting edge element (j) (θA=42.59°) were the local CEA Krj = 41.41°, the local RA γoj = -8.47°, IA λsj = −0.34°, and the local uncut chip thickness tj  = 0. At the cutting edge element j (θB=47.18°), the maximum undeformed chip thickness was tj = 0.05 mm. The real cutting process (Figure 6) showed that the new mathematical model of the hard turning was accurate, and compared with the conventional mathematical model [30–32], our new model described the essence of the hard turning process and was not limited in the number of elements and CEA.Figure 6
The real cutting process in hard turning withKr = 84°, RA γo = −6.03°, IA λs = −6°, r = 0.8 mm, dw = 0.2 mm, and f = 0.08 mm/rev. (a) The local tool geometry and (b) the uncut chip thickness parameters of the cutting edge element (j).
(a)(b)


## 4. Analysis of the Mathematical Model and Experimental Results

### 4.1. Analysis of the Mathematical Model

Hard turning usually uses small cutting depths of 0.1-0.2 mm [34]. Based on the experimental results and the mathematical model of hard turning, we found that the cutting occurred only at the corner of tool-nose radius. Changing CEA resulted in a change in the cutting position on tool-nose radius (Figures 7(e) and 7(f) and Figure 8). Figures 7(a)–7(d) also show that, at a given cutting-edge element on the tool-nose radius, changing RA mainly caused a change in the local IA, while changing IA mainly caused a change in the local RA.Figure 7
The local IA and RA when increasing RA, IA, and CEA, with chamfer angle = 25° and cutting conditions: dw = 0.2,f = 0.08, r = 0.8. (a) Local RA of engaged cutting-edge elements as increasing RA. (b) Local RA of engaged cutting-edge elements as increasing IA. (c) Local IA of engaged cutting-edge elements as increasing RA. (d) Local IA of engaged cutting-edge elements as increasing IA. (e) Local RA of engaged cutting-edge elements as increasing CEA. (f) Local IA of engaged cutting-edge elements as increasing CEA.
(a)(b)(c)(d)(e)(f)Figure 8
The local uncut chip thickness of engaged cutting-edge elements as increasing CEA, with cutting conditions:dw = 0.2 mm, f = 0.08 mm/rev, r = 0.8 mm.Based on the mathematical model, we can determine the change of local tool angles when changing the insert angle parameters as shown in Figure7.


### 4.2. Analysis of the Experimental Results

#### 4.2.1. CF Analysis

CF was measured in three directions, including feed force (Fx), thrust force (Fy), and tangential force (Fz) using a dynamometer (Kistler, 9257B) equipped with an amplifier (Kistler, 5070A) and a computer.Table2 shows that the thrust force (Fy) was the largest CF component, the tangential force (Fz) is the middle one and the feed force (Fx) is the smallest CF component, which was in accordance with previous works Khamel et al. [20], Azizi et al. [35], and Bouacha [36]. Due to the low cutting depth and feed rate during hard turning, CFs were usually small and had negligible influences on SR. However, there was a relationship between CF and TW. As in the 13th experiment, the CF and TW were the greatest, but the SR was the smallest.Table 2
Experimental results based on CCD.
κr (°)γo (°)λs (°)Fx (N)Fy (N)Fz (N)F (N)Ra (µm)VB (µm)84−3.6−3.647.19134.9290.91169.40.27944.0466−3.6−3.645.52123.6884.44156.520.73044.8284−8.4−3.647.11122.8686.24157.330.59435.9866−8.4−3.646.02118.2383.28151.760.86632.2484−3.6−8.445.47122.6785.13156.080.72031.2466−3.6−8.441.67116.4482.68148.760.89932.0084−8.4−8.443.28117.0983.171501.00430.1266−8.4−8.442.7119.6982.76151.650.89023.5490−6.0−6.046119.8784.8153.870.83637.8060−6.0−6.041.64117.2881.14148.560.99529.2775−2.0−6.039.19128.7186.89160.160.76642.1175−10.0−6.043.07116.3781.79148.621.02027.7075−6.0−2.050.6913992.28174.380.25250.2675−6.0−10.048.26125.3387.47160.270.76722.2075−6.0−6.050.67118.6687.13155.690.87031.8575−6.0−6.051.75119.9285.59156.160.85332.0175−6.0−6.050.72120.0287.02156.680.85232.3275−6.0−6.052.91118.4786.88156.150.90632.9375−6.0−6.052.07118.4485.11154.860.91931.6375−6.0−6.050.83118.5685.45154.730.87631.92Figure9 shows the main effect and interaction effect plots for the resultant force. Figure 9(a) shows that CF decreased when the negative IA and RA increased. However, when IA increased higher than −7.2°, CF increased. Base on the mathematical model, this result was because the increase in the negative IA and RA led to higher local negative RA, which in turn increased the local CA (Figure 2(b)). Moreover, the increase due to IA was the largest (Figures 7(a) and 7(b)). Therefore, normal and friction forces on the tool rake face increased but these forces on the clearance face decreased, hence reaching a balance at IA of -7.2o with a minimum CF.Figure 9
Main and interaction effects of tool angle parameters on CF (F).
(a)(b)In our previous study, SR and TW were affected the most by IA. From Table3–Table 6, IA was also the most important contributor to the CF components (Fx, Fy, and Fz) and resultant CF (F). From the mathematical model, IA was the major factor affecting local RA (and correspondingly local CA). And, the performance characteristics (SR, TW, and CF) of hard turning were affected by these local angles.Table 3
Analysis of variance for CF (F).
SourceDFSeq SSContribution (%)Adj SSAdj MSF valueP valueModel9762.04895.52762.04884.67223.70<0.0001Linear3393.52049.33393.520131.17336.72<0.0001κr180.13210.0480.13280.13222.430.001γo1113.80114.26113.801113.80131.86<0.0001λs1199.58725.02199.587199.58755.87<0.0001Square3291.76836.57291.76897.25627.22<0.0001κr∗ κr181.98610.2866.96166.96118.740.001γo∗γo126.6173.3415.26215.2624.270.066λs∗λs1183.16522.96183.165183.16551.27<0.00012-way interaction376.7619.6276.76125.5877.160.007κr∗γo133.1204.1533.12033.1209.270.012κr∗λs120.4082.5620.40820.4085.710.038γo∗λs123.2322.9123.23223.2326.500.029Error1035.7234.4835.7233.572Lack-of-fit532.7034.1032.7036.54110.830.010Pure error53.0210.383.0210.604Total19797.772100.00Table 4
Analysis of variance for thrust CF (Fy).
SourceDFSeq SSContributionAdj SSAdj MSF valueP valueModel9656.89993.19656.89972.98915.22<0.0001Linear3322.45045.75322.450107.48322.41<0.0001κr141.8495.9441.84941.8498.720.014γo1120.39917.08120.399120.39925.100.001λs1160.20222.73160.202160.20233.40<0.0001Square3257.09836.47257.09885.69917.86<0.0001κr∗ κr120.0152.848.7598.7591.830.206γo∗γo10.9890.145.7035.7031.190.301λs∗λs1236.09333.49236.093236.09349.22<0.00012-way interaction377.35210.9777.35225.7845.370.018κr∗γo129.8074.2329.80729.8076.210.032κr∗λs118.7332.6618.73318.7333.910.076γo∗λs128.8124.0928.81228.8126.010.034Error1047.9716.8147.9714.797Lack-of-fit545.1636.4145.1639.03316.080.004Pure error52.8090.402.8090.562Total19704.871100.00Table 5
Analysis of Variance for tangential CF (Fz).
SourceDFSeq SSContribution (%)Adj SSAdj MSF valueP valueModel9140.66694.35140.66615.629618.54<0.0001Linear371.46547.9371.46523.821628.26<0.0001κr124.99016.7624.99024.990229.64<0.0001γo119.42913.0319.42919.428923.050.001λs127.04618.1427.04627.045832.08<0.0001Square357.99638.9057.99619.331922.93<0.0001κr∗ κr126.84218.0025.54125.540730.29<0.0001γo∗γo113.0238.7310.28210.281812.200.006λs∗λs118.13112.1618.13118.130621.510.0012-way interaction311.2057.5211.2053.73524.430.032κr∗γo13.8382.573.8383.83784.550.059κr∗λs15.4203.645.4205.42036.430.030γo∗λs11.9471.311.9471.94742.310.160Error108.4315.658.4310.8431Lack-of-fit54.2892.884.2890.85771.040.485Pure error54.1422.784.1420.8284Total19149.097100.00Table 6
Analysis of variance for feed CF (Fx).
SourceDFSeq SSContribution (%)Adj SSAdj MSF-valueP-valueModel9297.88695.43297.88633.09823.21<0.0001Linear338.47012.3238.47012.8238.990.003κr115.3274.9115.32715.32710.750.008γo12.4120.772.4122.4121.690.223λs120.7326.6420.73220.73214.540.003Square3256.97682.32256.97685.65960.06<0.0001κr∗ κr174.16523.7697.15397.15368.12<0.0001γo∗γo1177.83156.97181.792181.792127.46<0.0001λs∗λs14.9801.604.9804.9803.490.0912-way interaction32.4400.782.4400.8130.570.647κr∗γo11.8000.581.8001.8001.260.287κr∗λs10.3260.100.3260.3260.230.643γo∗λs10.3130.100.3130.3130.220.649Error1014.2624.5714.2621.426Lack-of-fit510.1443.2510.1442.0292.460.172Pure error54.1181.324.1180.824Total19312.148100.00The regression equations forF, Fz, Fy, and Fx in terms of the cutting angles were(35)F=−48.3+5.325κr−6.26γo+5.36λs−0.02697κr∗κr−0.1811γo∗γo+0.6273λs∗λs+0.0942κr∗γo+0.0739κr∗λs+0.296γo∗λsR2=95.52%,Fy=35.1+2.62κr−2.16γo+6.64λs−0.00976κr∗κr+0.111γo∗γo+0.712λs∗λs+0.0894κr∗γo+0.0708κr∗λs+0.329γo∗λsR2=93.19%,Fz=−39.0+3.071κr−3.18γo+0.61λs−0.01666κr∗κr−0.1486γo∗γo+0.1974λs∗λs+0.0321κr∗γo+0.0381κr∗λs+0.0857γo∗λsR2=94.35%,Fx=−171.2+5.067κr−9.53γo−0.23λs−0.03249κr∗κr−0.6250γo∗γo−0.1034λs∗λs+0.0220κr∗γo−0.0093κr∗λs−0.0344γo∗λsR2=95.43%.The determination coefficientsR2 for F, Fz, Fy, and Fx were high enough for reliable estimates.


#### 4.2.2. TW Analysis

Under our cutting conditions (0.2 mm cutting depth, 0.08 mm/rev feed rate, and 0.8 mm nose radius), the mathematical model revealed that the cutting occurred only at the tool-nose radius corner. This result was in accordance with the experimental results that the TW occurred only at the tool-nose radius corner.Figure10(a) shows that the flank wear decreased when CEA decreased from 90° to 60° and RA, IA increased in the negative direction. The reason for this effect could be due to higher local negative RA and local CA resulted when decreasing CEA (Figure 7(e)) and increasing RA and IA (Figures 7(a) and 7(b)). This increase in the local CA finally resulted in less contact and less friction between the machined surface and the flank face because of the springback of the workpiece.Figure 10
Main effect plots for (a) SR and (b) TW.
(a)(b)


#### 4.2.3. SR Analysis

From our previous investigation [2], the results showed that as the negative IA and RA increased and CEA changed from 90° to 60°, SR also increased. But IA continued to increase over −8.1°, and SR decreased (Figure 10(a)). Based on the proposed geometrical model, the reason for these results was that decreasing CEA and increasing negative IA and RA led to higher local negative RA and higher local CA (Figure 2(b)), with the increase due to IA was the largest (Figures 7(a) and 7(b)). An increase in the local negative RA resulted in a longer tool-chip contact and a higher ratio of chip compression, which caused more vibrations and subsequently higher SR. Oppositely, larger local CA reduced the tool-machined surface contact and the friction because of the springback of the workpiece [37], thus generating less vibration and lower SR. Under specific conditions, positive influences of local CA outplayed the negative influences of local negative RA, thus decreasing SR.


### 4.3. Multiobjective Optimization

Another goal of this research was to optimize the tool angle parameters to obtain multiple objectives based on SR, TW, and CF. The desirability function approach was used with the desirability function (D) defined as follows:(36)D=∏i=1ndi1/n.In the first case, it is desired to obtain target values of SR, TW, and CF. The desirability, in this case, is assigned as follows:(37)di=0,atYi<Lowi,Yi−LowiTari−Lowir,atLowi≤Yi<Tari,Highi−YiHighi−Tarir,atTari≤Yi≤Highi,0,atYi>Highi,where di is the desirability derived from responses Yi (SR, TW, and CF) and ranges from 0 to 1 (from least to most desired), n = 3 is the number of responses, and r is a parameter determining the form of di. Table 7 shows the results of multiobjective optimization for TW, SR, and CF.Table 7
Optimization results for desired SR, TW, and CF.
ResponseGoalLowerTargetUpperWeightImportanceRaTarget0.2520.7341.02011VBTarget22.20025.00050.26011FTarget148.562159.000174.37511RaVBFCompositeSolutionKrγoλsFitFitFitDesirability175−6−100.73400025.0000159.0001.00000Under optimized tool angle parameters in Table7 (CEA κr = 75°, RA γo = −6°, and IA λs = −10°), the measured SR, flank wear, and CF were Ra = 0.767 µm, VB = 22.2 µm, and F = 160.27 N. Compared to the respective Ra = 0.836 µm, VB = 37.8 µm, and F = 153.87 N, when using standard tool parameters (κr = 91°, γo = −6° and λs = −6°), SR and TW decreased by 8.3% and 41.3%, respectively, while CF increased by 4%. Because CF is not an important output when considering optimal conditions for hard turning, these improvements in SR and TW have significant practical implications. Figure 11 shows the comparison between standard and optimized local tool angles.Figure 11
Comparison between standard and optimized local tool geometric parameters: (a) local RA, (b) local IA, and (c) local CEA.
(a)(b)(c)In the second case, it is desired to attain the lowest TW, SR, and CF, so equation (38) was used.(38)di=1,atYi<Lowi,Highi−YiHighi−Lowir,atLowi≤Yi≤Highi,0,atYi>Highi.Compared to using standard cutting tool angles, the optimal tool angles in Table8 decreased SR and TW by 30.4% and 53.8%, respectively, while insignificantly increased CF by 1.7%.Table 8
Optimization results for minimum SR, TW and CF.
ResponseGoalLowerTargetUpperWeightImportanceRaMinimum0.2521.02011VBMinimum22.20050.26011FMinimum148.56174.3811RaVBFCompositeSolutionKrγoλsFitFitFitDesirability160−10−100.58156317.4606156.440.734796


## 4.1. Analysis of the Mathematical Model

Hard turning usually uses small cutting depths of 0.1-0.2 mm [34]. Based on the experimental results and the mathematical model of hard turning, we found that the cutting occurred only at the corner of tool-nose radius. Changing CEA resulted in a change in the cutting position on tool-nose radius (Figures 7(e) and 7(f) and Figure 8). Figures 7(a)–7(d) also show that, at a given cutting-edge element on the tool-nose radius, changing RA mainly caused a change in the local IA, while changing IA mainly caused a change in the local RA.Figure 7
The local IA and RA when increasing RA, IA, and CEA, with chamfer angle = 25° and cutting conditions: dw = 0.2,f = 0.08, r = 0.8. (a) Local RA of engaged cutting-edge elements as increasing RA. (b) Local RA of engaged cutting-edge elements as increasing IA. (c) Local IA of engaged cutting-edge elements as increasing RA. (d) Local IA of engaged cutting-edge elements as increasing IA. (e) Local RA of engaged cutting-edge elements as increasing CEA. (f) Local IA of engaged cutting-edge elements as increasing CEA.
(a)(b)(c)(d)(e)(f)Figure 8
The local uncut chip thickness of engaged cutting-edge elements as increasing CEA, with cutting conditions:dw = 0.2 mm, f = 0.08 mm/rev, r = 0.8 mm.Based on the mathematical model, we can determine the change of local tool angles when changing the insert angle parameters as shown in Figure7.


## 4.2. Analysis of the Experimental Results

### 4.2.1. CF Analysis

CF was measured in three directions, including feed force (Fx), thrust force (Fy), and tangential force (Fz) using a dynamometer (Kistler, 9257B) equipped with an amplifier (Kistler, 5070A) and a computer.Table2 shows that the thrust force (Fy) was the largest CF component, the tangential force (Fz) is the middle one and the feed force (Fx) is the smallest CF component, which was in accordance with previous works Khamel et al. [20], Azizi et al. [35], and Bouacha [36]. Due to the low cutting depth and feed rate during hard turning, CFs were usually small and had negligible influences on SR. However, there was a relationship between CF and TW. As in the 13th experiment, the CF and TW were the greatest, but the SR was the smallest.Table 2
Experimental results based on CCD.
κr (°)γo (°)λs (°)Fx (N)Fy (N)Fz (N)F (N)Ra (µm)VB (µm)84−3.6−3.647.19134.9290.91169.40.27944.0466−3.6−3.645.52123.6884.44156.520.73044.8284−8.4−3.647.11122.8686.24157.330.59435.9866−8.4−3.646.02118.2383.28151.760.86632.2484−3.6−8.445.47122.6785.13156.080.72031.2466−3.6−8.441.67116.4482.68148.760.89932.0084−8.4−8.443.28117.0983.171501.00430.1266−8.4−8.442.7119.6982.76151.650.89023.5490−6.0−6.046119.8784.8153.870.83637.8060−6.0−6.041.64117.2881.14148.560.99529.2775−2.0−6.039.19128.7186.89160.160.76642.1175−10.0−6.043.07116.3781.79148.621.02027.7075−6.0−2.050.6913992.28174.380.25250.2675−6.0−10.048.26125.3387.47160.270.76722.2075−6.0−6.050.67118.6687.13155.690.87031.8575−6.0−6.051.75119.9285.59156.160.85332.0175−6.0−6.050.72120.0287.02156.680.85232.3275−6.0−6.052.91118.4786.88156.150.90632.9375−6.0−6.052.07118.4485.11154.860.91931.6375−6.0−6.050.83118.5685.45154.730.87631.92Figure9 shows the main effect and interaction effect plots for the resultant force. Figure 9(a) shows that CF decreased when the negative IA and RA increased. However, when IA increased higher than −7.2°, CF increased. Base on the mathematical model, this result was because the increase in the negative IA and RA led to higher local negative RA, which in turn increased the local CA (Figure 2(b)). Moreover, the increase due to IA was the largest (Figures 7(a) and 7(b)). Therefore, normal and friction forces on the tool rake face increased but these forces on the clearance face decreased, hence reaching a balance at IA of -7.2o with a minimum CF.Figure 9
Main and interaction effects of tool angle parameters on CF (F).
(a)(b)In our previous study, SR and TW were affected the most by IA. From Table3–Table 6, IA was also the most important contributor to the CF components (Fx, Fy, and Fz) and resultant CF (F). From the mathematical model, IA was the major factor affecting local RA (and correspondingly local CA). And, the performance characteristics (SR, TW, and CF) of hard turning were affected by these local angles.Table 3
Analysis of variance for CF (F).
SourceDFSeq SSContribution (%)Adj SSAdj MSF valueP valueModel9762.04895.52762.04884.67223.70<0.0001Linear3393.52049.33393.520131.17336.72<0.0001κr180.13210.0480.13280.13222.430.001γo1113.80114.26113.801113.80131.86<0.0001λs1199.58725.02199.587199.58755.87<0.0001Square3291.76836.57291.76897.25627.22<0.0001κr∗ κr181.98610.2866.96166.96118.740.001γo∗γo126.6173.3415.26215.2624.270.066λs∗λs1183.16522.96183.165183.16551.27<0.00012-way interaction376.7619.6276.76125.5877.160.007κr∗γo133.1204.1533.12033.1209.270.012κr∗λs120.4082.5620.40820.4085.710.038γo∗λs123.2322.9123.23223.2326.500.029Error1035.7234.4835.7233.572Lack-of-fit532.7034.1032.7036.54110.830.010Pure error53.0210.383.0210.604Total19797.772100.00Table 4
Analysis of variance for thrust CF (Fy).
SourceDFSeq SSContributionAdj SSAdj MSF valueP valueModel9656.89993.19656.89972.98915.22<0.0001Linear3322.45045.75322.450107.48322.41<0.0001κr141.8495.9441.84941.8498.720.014γo1120.39917.08120.399120.39925.100.001λs1160.20222.73160.202160.20233.40<0.0001Square3257.09836.47257.09885.69917.86<0.0001κr∗ κr120.0152.848.7598.7591.830.206γo∗γo10.9890.145.7035.7031.190.301λs∗λs1236.09333.49236.093236.09349.22<0.00012-way interaction377.35210.9777.35225.7845.370.018κr∗γo129.8074.2329.80729.8076.210.032κr∗λs118.7332.6618.73318.7333.910.076γo∗λs128.8124.0928.81228.8126.010.034Error1047.9716.8147.9714.797Lack-of-fit545.1636.4145.1639.03316.080.004Pure error52.8090.402.8090.562Total19704.871100.00Table 5
Analysis of Variance for tangential CF (Fz).
SourceDFSeq SSContribution (%)Adj SSAdj MSF valueP valueModel9140.66694.35140.66615.629618.54<0.0001Linear371.46547.9371.46523.821628.26<0.0001κr124.99016.7624.99024.990229.64<0.0001γo119.42913.0319.42919.428923.050.001λs127.04618.1427.04627.045832.08<0.0001Square357.99638.9057.99619.331922.93<0.0001κr∗ κr126.84218.0025.54125.540730.29<0.0001γo∗γo113.0238.7310.28210.281812.200.006λs∗λs118.13112.1618.13118.130621.510.0012-way interaction311.2057.5211.2053.73524.430.032κr∗γo13.8382.573.8383.83784.550.059κr∗λs15.4203.645.4205.42036.430.030γo∗λs11.9471.311.9471.94742.310.160Error108.4315.658.4310.8431Lack-of-fit54.2892.884.2890.85771.040.485Pure error54.1422.784.1420.8284Total19149.097100.00Table 6
Analysis of variance for feed CF (Fx).
SourceDFSeq SSContribution (%)Adj SSAdj MSF-valueP-valueModel9297.88695.43297.88633.09823.21<0.0001Linear338.47012.3238.47012.8238.990.003κr115.3274.9115.32715.32710.750.008γo12.4120.772.4122.4121.690.223λs120.7326.6420.73220.73214.540.003Square3256.97682.32256.97685.65960.06<0.0001κr∗ κr174.16523.7697.15397.15368.12<0.0001γo∗γo1177.83156.97181.792181.792127.46<0.0001λs∗λs14.9801.604.9804.9803.490.0912-way interaction32.4400.782.4400.8130.570.647κr∗γo11.8000.581.8001.8001.260.287κr∗λs10.3260.100.3260.3260.230.643γo∗λs10.3130.100.3130.3130.220.649Error1014.2624.5714.2621.426Lack-of-fit510.1443.2510.1442.0292.460.172Pure error54.1181.324.1180.824Total19312.148100.00The regression equations forF, Fz, Fy, and Fx in terms of the cutting angles were(35)F=−48.3+5.325κr−6.26γo+5.36λs−0.02697κr∗κr−0.1811γo∗γo+0.6273λs∗λs+0.0942κr∗γo+0.0739κr∗λs+0.296γo∗λsR2=95.52%,Fy=35.1+2.62κr−2.16γo+6.64λs−0.00976κr∗κr+0.111γo∗γo+0.712λs∗λs+0.0894κr∗γo+0.0708κr∗λs+0.329γo∗λsR2=93.19%,Fz=−39.0+3.071κr−3.18γo+0.61λs−0.01666κr∗κr−0.1486γo∗γo+0.1974λs∗λs+0.0321κr∗γo+0.0381κr∗λs+0.0857γo∗λsR2=94.35%,Fx=−171.2+5.067κr−9.53γo−0.23λs−0.03249κr∗κr−0.6250γo∗γo−0.1034λs∗λs+0.0220κr∗γo−0.0093κr∗λs−0.0344γo∗λsR2=95.43%.The determination coefficientsR2 for F, Fz, Fy, and Fx were high enough for reliable estimates.


### 4.2.2. TW Analysis

Under our cutting conditions (0.2 mm cutting depth, 0.08 mm/rev feed rate, and 0.8 mm nose radius), the mathematical model revealed that the cutting occurred only at the tool-nose radius corner. This result was in accordance with the experimental results that the TW occurred only at the tool-nose radius corner.Figure10(a) shows that the flank wear decreased when CEA decreased from 90° to 60° and RA, IA increased in the negative direction. The reason for this effect could be due to higher local negative RA and local CA resulted when decreasing CEA (Figure 7(e)) and increasing RA and IA (Figures 7(a) and 7(b)). This increase in the local CA finally resulted in less contact and less friction between the machined surface and the flank face because of the springback of the workpiece.Figure 10
Main effect plots for (a) SR and (b) TW.
(a)(b)


### 4.2.3. SR Analysis

From our previous investigation [2], the results showed that as the negative IA and RA increased and CEA changed from 90° to 60°, SR also increased. But IA continued to increase over −8.1°, and SR decreased (Figure 10(a)). Based on the proposed geometrical model, the reason for these results was that decreasing CEA and increasing negative IA and RA led to higher local negative RA and higher local CA (Figure 2(b)), with the increase due to IA was the largest (Figures 7(a) and 7(b)). An increase in the local negative RA resulted in a longer tool-chip contact and a higher ratio of chip compression, which caused more vibrations and subsequently higher SR. Oppositely, larger local CA reduced the tool-machined surface contact and the friction because of the springback of the workpiece [37], thus generating less vibration and lower SR. Under specific conditions, positive influences of local CA outplayed the negative influences of local negative RA, thus decreasing SR.


## 4.2.1. CF Analysis

CF was measured in three directions, including feed force (Fx), thrust force (Fy), and tangential force (Fz) using a dynamometer (Kistler, 9257B) equipped with an amplifier (Kistler, 5070A) and a computer.Table2 shows that the thrust force (Fy) was the largest CF component, the tangential force (Fz) is the middle one and the feed force (Fx) is the smallest CF component, which was in accordance with previous works Khamel et al. [20], Azizi et al. [35], and Bouacha [36]. Due to the low cutting depth and feed rate during hard turning, CFs were usually small and had negligible influences on SR. However, there was a relationship between CF and TW. As in the 13th experiment, the CF and TW were the greatest, but the SR was the smallest.Table 2
Experimental results based on CCD.
κr (°)γo (°)λs (°)Fx (N)Fy (N)Fz (N)F (N)Ra (µm)VB (µm)84−3.6−3.647.19134.9290.91169.40.27944.0466−3.6−3.645.52123.6884.44156.520.73044.8284−8.4−3.647.11122.8686.24157.330.59435.9866−8.4−3.646.02118.2383.28151.760.86632.2484−3.6−8.445.47122.6785.13156.080.72031.2466−3.6−8.441.67116.4482.68148.760.89932.0084−8.4−8.443.28117.0983.171501.00430.1266−8.4−8.442.7119.6982.76151.650.89023.5490−6.0−6.046119.8784.8153.870.83637.8060−6.0−6.041.64117.2881.14148.560.99529.2775−2.0−6.039.19128.7186.89160.160.76642.1175−10.0−6.043.07116.3781.79148.621.02027.7075−6.0−2.050.6913992.28174.380.25250.2675−6.0−10.048.26125.3387.47160.270.76722.2075−6.0−6.050.67118.6687.13155.690.87031.8575−6.0−6.051.75119.9285.59156.160.85332.0175−6.0−6.050.72120.0287.02156.680.85232.3275−6.0−6.052.91118.4786.88156.150.90632.9375−6.0−6.052.07118.4485.11154.860.91931.6375−6.0−6.050.83118.5685.45154.730.87631.92Figure9 shows the main effect and interaction effect plots for the resultant force. Figure 9(a) shows that CF decreased when the negative IA and RA increased. However, when IA increased higher than −7.2°, CF increased. Base on the mathematical model, this result was because the increase in the negative IA and RA led to higher local negative RA, which in turn increased the local CA (Figure 2(b)). Moreover, the increase due to IA was the largest (Figures 7(a) and 7(b)). Therefore, normal and friction forces on the tool rake face increased but these forces on the clearance face decreased, hence reaching a balance at IA of -7.2o with a minimum CF.Figure 9
Main and interaction effects of tool angle parameters on CF (F).
(a)(b)In our previous study, SR and TW were affected the most by IA. From Table3–Table 6, IA was also the most important contributor to the CF components (Fx, Fy, and Fz) and resultant CF (F). From the mathematical model, IA was the major factor affecting local RA (and correspondingly local CA). And, the performance characteristics (SR, TW, and CF) of hard turning were affected by these local angles.Table 3
Analysis of variance for CF (F).
SourceDFSeq SSContribution (%)Adj SSAdj MSF valueP valueModel9762.04895.52762.04884.67223.70<0.0001Linear3393.52049.33393.520131.17336.72<0.0001κr180.13210.0480.13280.13222.430.001γo1113.80114.26113.801113.80131.86<0.0001λs1199.58725.02199.587199.58755.87<0.0001Square3291.76836.57291.76897.25627.22<0.0001κr∗ κr181.98610.2866.96166.96118.740.001γo∗γo126.6173.3415.26215.2624.270.066λs∗λs1183.16522.96183.165183.16551.27<0.00012-way interaction376.7619.6276.76125.5877.160.007κr∗γo133.1204.1533.12033.1209.270.012κr∗λs120.4082.5620.40820.4085.710.038γo∗λs123.2322.9123.23223.2326.500.029Error1035.7234.4835.7233.572Lack-of-fit532.7034.1032.7036.54110.830.010Pure error53.0210.383.0210.604Total19797.772100.00Table 4
Analysis of variance for thrust CF (Fy).
SourceDFSeq SSContributionAdj SSAdj MSF valueP valueModel9656.89993.19656.89972.98915.22<0.0001Linear3322.45045.75322.450107.48322.41<0.0001κr141.8495.9441.84941.8498.720.014γo1120.39917.08120.399120.39925.100.001λs1160.20222.73160.202160.20233.40<0.0001Square3257.09836.47257.09885.69917.86<0.0001κr∗ κr120.0152.848.7598.7591.830.206γo∗γo10.9890.145.7035.7031.190.301λs∗λs1236.09333.49236.093236.09349.22<0.00012-way interaction377.35210.9777.35225.7845.370.018κr∗γo129.8074.2329.80729.8076.210.032κr∗λs118.7332.6618.73318.7333.910.076γo∗λs128.8124.0928.81228.8126.010.034Error1047.9716.8147.9714.797Lack-of-fit545.1636.4145.1639.03316.080.004Pure error52.8090.402.8090.562Total19704.871100.00Table 5
Analysis of Variance for tangential CF (Fz).
SourceDFSeq SSContribution (%)Adj SSAdj MSF valueP valueModel9140.66694.35140.66615.629618.54<0.0001Linear371.46547.9371.46523.821628.26<0.0001κr124.99016.7624.99024.990229.64<0.0001γo119.42913.0319.42919.428923.050.001λs127.04618.1427.04627.045832.08<0.0001Square357.99638.9057.99619.331922.93<0.0001κr∗ κr126.84218.0025.54125.540730.29<0.0001γo∗γo113.0238.7310.28210.281812.200.006λs∗λs118.13112.1618.13118.130621.510.0012-way interaction311.2057.5211.2053.73524.430.032κr∗γo13.8382.573.8383.83784.550.059κr∗λs15.4203.645.4205.42036.430.030γo∗λs11.9471.311.9471.94742.310.160Error108.4315.658.4310.8431Lack-of-fit54.2892.884.2890.85771.040.485Pure error54.1422.784.1420.8284Total19149.097100.00Table 6
Analysis of variance for feed CF (Fx).
SourceDFSeq SSContribution (%)Adj SSAdj MSF-valueP-valueModel9297.88695.43297.88633.09823.21<0.0001Linear338.47012.3238.47012.8238.990.003κr115.3274.9115.32715.32710.750.008γo12.4120.772.4122.4121.690.223λs120.7326.6420.73220.73214.540.003Square3256.97682.32256.97685.65960.06<0.0001κr∗ κr174.16523.7697.15397.15368.12<0.0001γo∗γo1177.83156.97181.792181.792127.46<0.0001λs∗λs14.9801.604.9804.9803.490.0912-way interaction32.4400.782.4400.8130.570.647κr∗γo11.8000.581.8001.8001.260.287κr∗λs10.3260.100.3260.3260.230.643γo∗λs10.3130.100.3130.3130.220.649Error1014.2624.5714.2621.426Lack-of-fit510.1443.2510.1442.0292.460.172Pure error54.1181.324.1180.824Total19312.148100.00The regression equations forF, Fz, Fy, and Fx in terms of the cutting angles were(35)F=−48.3+5.325κr−6.26γo+5.36λs−0.02697κr∗κr−0.1811γo∗γo+0.6273λs∗λs+0.0942κr∗γo+0.0739κr∗λs+0.296γo∗λsR2=95.52%,Fy=35.1+2.62κr−2.16γo+6.64λs−0.00976κr∗κr+0.111γo∗γo+0.712λs∗λs+0.0894κr∗γo+0.0708κr∗λs+0.329γo∗λsR2=93.19%,Fz=−39.0+3.071κr−3.18γo+0.61λs−0.01666κr∗κr−0.1486γo∗γo+0.1974λs∗λs+0.0321κr∗γo+0.0381κr∗λs+0.0857γo∗λsR2=94.35%,Fx=−171.2+5.067κr−9.53γo−0.23λs−0.03249κr∗κr−0.6250γo∗γo−0.1034λs∗λs+0.0220κr∗γo−0.0093κr∗λs−0.0344γo∗λsR2=95.43%.The determination coefficientsR2 for F, Fz, Fy, and Fx were high enough for reliable estimates.


## 4.2.2. TW Analysis

Under our cutting conditions (0.2 mm cutting depth, 0.08 mm/rev feed rate, and 0.8 mm nose radius), the mathematical model revealed that the cutting occurred only at the tool-nose radius corner. This result was in accordance with the experimental results that the TW occurred only at the tool-nose radius corner.Figure10(a) shows that the flank wear decreased when CEA decreased from 90° to 60° and RA, IA increased in the negative direction. The reason for this effect could be due to higher local negative RA and local CA resulted when decreasing CEA (Figure 7(e)) and increasing RA and IA (Figures 7(a) and 7(b)). This increase in the local CA finally resulted in less contact and less friction between the machined surface and the flank face because of the springback of the workpiece.Figure 10
Main effect plots for (a) SR and (b) TW.
(a)(b)


## 4.2.3. SR Analysis

From our previous investigation [2], the results showed that as the negative IA and RA increased and CEA changed from 90° to 60°, SR also increased. But IA continued to increase over −8.1°, and SR decreased (Figure 10(a)). Based on the proposed geometrical model, the reason for these results was that decreasing CEA and increasing negative IA and RA led to higher local negative RA and higher local CA (Figure 2(b)), with the increase due to IA was the largest (Figures 7(a) and 7(b)). An increase in the local negative RA resulted in a longer tool-chip contact and a higher ratio of chip compression, which caused more vibrations and subsequently higher SR. Oppositely, larger local CA reduced the tool-machined surface contact and the friction because of the springback of the workpiece [37], thus generating less vibration and lower SR. Under specific conditions, positive influences of local CA outplayed the negative influences of local negative RA, thus decreasing SR.


## 4.3. Multiobjective Optimization

Another goal of this research was to optimize the tool angle parameters to obtain multiple objectives based on SR, TW, and CF. The desirability function approach was used with the desirability function (D) defined as follows:(36)D=∏i=1ndi1/n.In the first case, it is desired to obtain target values of SR, TW, and CF. The desirability, in this case, is assigned as follows:(37)di=0,atYi<Lowi,Yi−LowiTari−Lowir,atLowi≤Yi<Tari,Highi−YiHighi−Tarir,atTari≤Yi≤Highi,0,atYi>Highi,where di is the desirability derived from responses Yi (SR, TW, and CF) and ranges from 0 to 1 (from least to most desired), n = 3 is the number of responses, and r is a parameter determining the form of di. Table 7 shows the results of multiobjective optimization for TW, SR, and CF.Table 7
Optimization results for desired SR, TW, and CF.
ResponseGoalLowerTargetUpperWeightImportanceRaTarget0.2520.7341.02011VBTarget22.20025.00050.26011FTarget148.562159.000174.37511RaVBFCompositeSolutionKrγoλsFitFitFitDesirability175−6−100.73400025.0000159.0001.00000Under optimized tool angle parameters in Table7 (CEA κr = 75°, RA γo = −6°, and IA λs = −10°), the measured SR, flank wear, and CF were Ra = 0.767 µm, VB = 22.2 µm, and F = 160.27 N. Compared to the respective Ra = 0.836 µm, VB = 37.8 µm, and F = 153.87 N, when using standard tool parameters (κr = 91°, γo = −6° and λs = −6°), SR and TW decreased by 8.3% and 41.3%, respectively, while CF increased by 4%. Because CF is not an important output when considering optimal conditions for hard turning, these improvements in SR and TW have significant practical implications. Figure 11 shows the comparison between standard and optimized local tool angles.Figure 11
Comparison between standard and optimized local tool geometric parameters: (a) local RA, (b) local IA, and (c) local CEA.
(a)(b)(c)In the second case, it is desired to attain the lowest TW, SR, and CF, so equation (38) was used.(38)di=1,atYi<Lowi,Highi−YiHighi−Lowir,atLowi≤Yi≤Highi,0,atYi>Highi.Compared to using standard cutting tool angles, the optimal tool angles in Table8 decreased SR and TW by 30.4% and 53.8%, respectively, while insignificantly increased CF by 1.7%.Table 8
Optimization results for minimum SR, TW and CF.
ResponseGoalLowerTargetUpperWeightImportanceRaMinimum0.2521.02011VBMinimum22.20050.26011FMinimum148.56174.3811RaVBFCompositeSolutionKrγoλsFitFitFitDesirability160−10−100.58156317.4606156.440.734796


## 5. Conclusion

Based on the proposed mathematical model and experimental results about the influences of tool geometry (CEA, RA, and IA) on performance characteristics (SR, TW, and CF) of hard turning, the main conclusions are drawn as follows.The thrust force was the largest component of CF, followed by tangential force and feed force; and CF was too small to have meaningful effects on SR.IA was the most influential on CF (25.02%), followed by RA (14.26%) and CEA (10.04%). There were interaction effects between CEA, RA, and IA on CF. CF decreased with the increase in negative RA and IA. If IA increases higher than a certain value (λs = –7.2°, with a minimum CF), CF increases after.Increasing CEA resulted in changes in the cutting position on the tool-nose radius and hence decreased local negative RA (and correspondingly local CA). Increasing RA and IA also increased local negative RA. And, the changes of RA and IA mainly caused the changes of local IA and local RA, respectively.Local RA and CA were the main factors that directly influenced performance characteristics (SR, TW, and CF) of hard turning. Increasing local negative RA resulted in higher SR and CF. Meanwhile, increasing local CA resulted in lower SR, TW, and CF. Under certain conditions, the positive influences of local CA outplayed the negative influences of local negative RA, thus decreasing SR and TW.Local RA (and correspondingly local CA) was affected the most by IA, followed by RA and CEA. Therefore, IA is the most crucial tool angle for hard turning, followed by RA and CEA. At the same time, it is known that RA is the most influential factor in conventional turning. Compared to standard cutting tool angles, the optimal tool angles in our work significantly reduced SR and TW.The proposed new mathematical model can be combined with formulas and other mathematical models to calculate local CF, cutting temperature, and TW for each cutting-edge element and the overall hard turning process. The analytical results of the mathematical model were in accordance with the experimental results.



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*Source: 9924427-2021-06-15.xml*