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Proceedings of International Conference on Mechanical &
Manufacturing Engineering (ICME2008)
General Mechanical Engineering
Manufacturing
Automotive Technology
Heat and Fluid Technology
Industrial Engineering
International Conference on Mechanical & Manufacturing Engineering (ICME2008), 21–23
May 2008, Johor Bahru, Malaysia.
© Faculty of Mechanical & Manufacturing Engineering, Universiti Tun Hussein Onn
Malaysia (UTHM), Malaysia.
ISBN: 97–98–2963–59–2
Faculty of Mechanical and Manufacturing Engineering
Universiti Tun Hussein Onn Malaysia (UTHM)
Parit Raja, 86400 Batu Pahat, Johor, MALAYSIA
Proceedings of International Conference on Mechanical &
Manufacturing Engineering (ICME2008)
MANUFACTURING ENGINEERING
Study of Surface Roughness on Induction Hardened Steel using CBN Cutting Tool
MFG_ID_0001.pdf
Control of Blank Holder Force to Eliminate Wrinkling and Fracture in Deep-Drawing Rectangular
Components
MFG_ID_0004.pdf
Implementation of Toyota Production System (TPS) in the Production Line of A Local Automotive Parts
Manufacturer
MFG_ID_0006.pdf
Analysis of Variance on the Metal Injection molding parameters using a bimodal particle size distribution
feedstock
MFG_ID_0009.pdf
Effect of Cryogenic Cooling during Grinding of Mild Steel and Stainless Steel
MFG_ID_0011.pdf
Traveling Salesman Problem with Precedence Constraint for Manufacturing Application: A Review
MFG_ID_0013.pdf
Strain Rate and Temperature Dependence of Mechanical Pproperties and Microstructure of Biomedical
Titanium Alloy
MFG_ID_0020.pdf
Temperature Distribution and Bending Characteristics
in Plastics Laser Forming
MFG_ID_0027.pdf
EFFECT OF ELECTRODE COOLING ON THE ELECTRICAL DISCHARGE
MACHINING OF TITANIUM ALLOY
MFG_ID_0028.pdf
Optimal Lot Size of EPQ Model Considering Imperfect and
Defective Products
MFG_ID_0052.pdf
Simulation based Control System for a Flat Screen Monitor
Remanufacturing System
MFG_ID_0054.pdf
Scaling Effects In Milling Operations Of Tungsten-Copper-Composites
MFG_ID_0067.pdf
A Study on Surface Roughness of an RP Part Fabricated on
Stereolithography Apparatus
MFG_ID_0071.pdf
A Study on Dimensional Accuracy of FDM Machine Fabrication Style via DOE Technique
MFG_ID_0078.pdf
Design and Manufacturing of a Spherical Rolling Robot
MFG_ID_0083.pdf
The effect of nodularisation parameters on the quality of ductile iron
MFG_ID_0087.pdf
Development of an Artificial Neural Network Algorithm for Predicting the
Cutting Force in End Milling of Inconel 718 Alloy
MFG_ID_0089.pdf
Mathematical Model of chip Serration frequency in end milling of Inconel 718
MFG_ID_0094.pdf
Potential Application of Rapid Prototyping Techniques to Fabricate a Laminated Rapid Tooling of
Polyurethane Foam Mould
MFG_ID_0098.pdf
Influence of Micro End Milling Process Parameters on Surface Roughness
MFG_ID_0125.pdf
Fabrication of Micromold Cavity for Microreplication: A Review
MFG_ID_0127.pdf
Design of an aluminum alloy side door impact beam for passenger cars
MFG_ID_0143.pdf
The Effect of Radial Clearance between Impeller-Diffuser on Design Point Operation in a Centrifugal
Fan
MFG_ID_0152.pdf
Automotive Part Prototype Development Using Reverse Engineering Technology
MFG_ID_0155.pdf
Integrating STEP with a PC-based Open Architecture Controller (OAPC-NC) for a Milling Process
MFG_ID_0169.pdf
Design of a Reliable Stair Climbing Tracked Robot
MFG_ID_0176.pdf
Processing and properties of PA6/MMT clay nanocomposites
produced using selective laser sintering
MFG_ID_0193.pdf
Shear Deformation of Non-Crimp Fabrics
MFG_ID_0207.pdf
A Web-based Real-time Mould Machining Process Tracking System
MFG_ID_0222.pdf
A Study of Machining Error Compensation for Tool Deflection
in Micro End-Milling
MFG_ID_0228.pdf
Development of a Micro Tool Inspection and Verification System
MFG_ID_0229.pdf
Cutting Force Simulation of Nose Radius Oblique Tools
MFG_ID_0233.pdf
Optimum hydroforming preform design by shape sensitivity analysis
MFG_ID_0239.pdf
An investigation of process parameters on quality of X-shape hydroformed joint by design of experiment
and finite element method
MFG_ID_0240.pdf
Semi-automated Robotic Sculpting of Freeform Surfaces for Direct Digital Manufacture
MFG_ID_0316.pdf
The EWR of graphite and copper electrodes in electrical discharge machining (EDM) of AISI H13
harden steel
MFG_ID_0332.pdf
Experimental Study of the Effective Parameters in Polymeric Coating of Metal Powder Used as Raw
Material in Powder-based Rapid Prototyping
MFG_ID_0333.pdf
A Study of Wire Looping Formation to Improve Ball Neck Strength of Wire Bonding Process
MFG_ID_0341.pdf
Modeling of General Etching System in Wafer Fabrication
MFG_ID_0343.pdf
A study of oxidized leadframe for QFN package on the cyclic test variable temperature effect
MFG_ID_0376.pdf
The Effect of Drill Point Geometry and Drilling Technique on Tool Life when Drilling Titanium Alloy,
Ti-6Al-4V
MFG_ID_0393.pdf
Mathematical Modeling of Cutting Force in End Milling Ti-6Al4V using TiAlN Coated Carbide Tools
MFG_ID_0394.pdf
MECHANICAL MANUFACTURING AUTOMOTIVE HEAT & FLUID INDUSTRIAL
ENGINEERING ENGINEERING TECHNOLOGY TECHNOLOGY ENGINEERING
International Conference on Mechanical & Manufacturing Engineering (ICME2008), 21–23
May 2008, Johor Bahru, Malaysia.
© Faculty of Mechanical & Manufacturing Engineering, Universiti Tun Hussein Onn
Malaysia (UTHM), Malaysia.
ISBN: 97–98–2963–59–2
Faculty of Mechanical and Manufacturing Engineering
Universiti Tun Hussein Onn Malaysia (UTHM)
Parit Raja, 86400 Batu Pahat, Johor, MALAYSIA
Proceedings of International Conference on Mechanical & Manufacturing Engineering (ICME2008), 21– 23 May 2008, Johor Bahru, Malaysia.
© Faculty of Mechanical & Manufacturing Engineering, Universiti Tun Hussein Onn Malaysia (UTHM), Malaysia.
ISBN: 97–98 –2963–59–2
Mathematical Modeling of Cutting Force in End Milling Ti-6Al4V
using TiAlN Coated Carbide Tools
Mohruni, A.S.1,2*, Sharif, S.2, Noordin, M.Y.2
1
Department of Mechanical Engineering
Sriwijaya University
Indralaya - 30662 – South Sumatera
Indonesia
2
Faculty of Mechanical Engineering
Universiti Teknologi Malaysia
UTM Skudai - 81310 - Johor
Malaysia
*mohrunias@yahoo.com, mohrunias@unsri.ac.id
Abstract:
This paper deals with the development of cutting force predicted models in end milling titanium
alloy Ti-6Al4V using TiAlN coated solid carbide tools under flood conditions. The primary
machining parameters such as cutting speed, feed and radial rake angle, were used as
independent variables for factorial design of experiment coupled with response surface
methodology (RSM). Results from the 3D-response surface contour showed that the linear
model generate better results than the second order models obtained during machining this
advanced material. An optimum cutting conditions was also recognized for a particular range of
cutting force values. The models were verified by analysis of variances and were found to be
adequate.
Keywords: Cutting Force, End Milling, TiAlN Coated Carbide, Titanium Alloy, RSM.
1. Introduction
Numerous studies have shown
titanium and its alloys are difficult to machine,
regardless of the various types of cutting tools
used. This has been attributed to their low
thermal conductivity, which concentrates heat
in the cutting zone (typically less than 25%
that of steel), retention of strength at elevated
temperatures and high chemical affinity for all
cutting tool materials.
Although the cutting forces generated
are not excessively high (almost similar to
those with steel), they are confined to a small
area due to the short chip contact length which
leads to high stresses. The combination of
high stress and temperature resulted in plastic
deformation of the tool edge. Depth of cut
notching and chipping at the flank can also be
a problem with intermittent cutting operations
[1].
Knowledge of the cutting forces owing
to a predictive model is very interesting with
respect to the choice of machine tool power,
the cutting tools and the optimization of
cutting conditions for a given machining
operation. It could allow the number of long
and expensive tests to be limited and the best
tool geometry to obtain quasi-constant and
low cutting forces, which lead to a reduced
tool wear and consequently, a better tool-life,
to be found [2].
Many researchers have followed
purely experimental approaches to study the
relationship between cutting forces and
Proceedings of International Conference on Mechanical & Manufacturing Engineering (ICME2008), 21– 23 May 2008, Johor Bahru, Malaysia.
© Faculty of Mechanical & Manufacturing Engineering, Universiti Tun Hussein Onn Malaysia (UTHM), Malaysia.
ISBN: 97–98 –2963–59–2
independent cutting conditions such as feed
rate, cutting speed and depth of cut etc. This
has reflected on the increased total cost of the
study as it involved a large number of cutting
experiments. Furthermore, with this purely
experimental approach, researchers have
investigated the effect of cutting parameters
on cutting forces using machining experiments
based on a one-factor-at a-time design,
without having any idea about the behavior of
cutting forces when two or more cutting
factors are varied at the same time.
Furthermore, this approach cannot describe
and identify, with a great accuracy, the effect
of the interactions of different independent
variables on the responses when they are
varied simultaneously [3].
Recent study takes into account the
effect of simultaneous variations of three
cutting parameters such as cutting speed (V),
feed per tooth (fz) and radial rake angle ( o) on
the behavior of cutting forces by utilizing
response surface methodology (RSM). RSM is
a group of mathematical and statistical
techniques that are useful for modeling the
relationship between the input parameters
(cutting conditions) and the output variable or
response (cutting force) [4].
This method was also used by previous
researchers [5]-[9], which studied cutting
force as the machining response.
2. Mathematical Models for Cutting Force
In this study, RSM was used and the
mathematical models relating to the
machining
responses
were
developed
according to Alauddin et. al.[5].
where Fta is the calculated average tangential
force or cutting force (N), fz is the feed per
tooth (mm.tooth-1), o is the radial rake angle
(o), ’ is the experimental error and C, k, l, m
are parameters to be estimated using
experimental data.
By utilizing a natural logarithmic
transformation Equation (1) can be written in
first order polynomial as
ln Fta = ln C + k ln V + l ln f z + m ln γ + ln ε ′ (2)
which can be transformed to
y = b0 x0 + b1 x1 + b2 x2 + b3 x3 + ε
(3)
and finally can be formed as
yˆ1 = y − ε = b0 x0 + b1 x1 + b2 x2 + b3 x3
(4)
where y is the calculated average tangential
force on a natural logarithmic scale, 1 is the
natural logarithmic value of predictive
(estimated) tangential force or cutting force, x0
= 1 as a dummy variable, xi (i= 1 to 3) are the
coded variables of V, fz and o respectively, =
ln ’ and bj (j = 0 to 3) are the model
parameters to be estimated using experimental
data [5].
If the observation region is extended,
then the second order model is useful to
represent the effect of second order and
interaction components. The second order can
be extended from the first order model in
Equation (4) as
yˆ 2 = b 0 x 0 + b1 x 1 + b 2 x 2 + b 3 x 3
2.1 Postulation of the Mathematical Models
Assuming that the proposed model for
cutting force is merely depend on cutting
speed V, feed per tooth fz and radial rake o.
Other factors, which influence machining
process, are kept constant. Thus, the cutting
force model for end milling Ti-6Al4V can be
written as
(1)
Fta = CV k f zl γ mε ′
+ b12 x 1 x 2 + b13 x 1 x 3 + b 23 x 2 x 3
+ b11 x
2
1
+ b 22 x
2
2
+ b 33 x
(5)
2
3
where b values are the parameters to be
estimated using least squares method and 2 is
the predicted response on natural logarithmic
scale.
ANOVA was employed, to check the
adequacy of the predicted mathematical
model.
Proceedings of International Conference on Mechanical & Manufacturing Engineering (ICME2008), 21– 23 May 2008, Johor Bahru, Malaysia.
© Faculty of Mechanical & Manufacturing Engineering, Universiti Tun Hussein Onn Malaysia (UTHM), Malaysia.
ISBN: 97–98 –2963–59–2
2.2 Experimental Design.
In order to determine the Equation of
the response surface, several experimental
designs have been developed which attempt to
approximate the Equation using the smallest
number of possible experiments [4][10].
In this study the 2k-factorial design
using the first 8 points from design of
experiments as shown in Figure 1, was carried
out for screening test. This design is necessary
when interactions between variables are to be
investigated. Furthermore, factorial design
allow the effects of a factor to be estimated at
several levels of other factors, giving
conclusions that are valid over a range of
experimental conditions [4][10].
23 24
19
7
8
replicated star points ( = 1.4142 [9]) as
shown completely in Figure 1.
2.3 Coding of the Independent Variables
The independent variables were coded
by taking into account the capacity and
limiting cutting conditions of milling machine.
The following transforming Equation was
used.
ln xn − ln xn 0
ln xn1 − ln xn 0
x=
(5)
where x is the coded variable of any factor
corresponding to its natural xn, xn1 is the
natural value at the level +1 and xn0 is the
natural value to the base or zero level [5][9].
The coded values of the independent
variables shown in Table 1 for use in Equation
(4) and (5) were obtained from the following
Equations:
20
5
6
x1 =
13
9 10
15
14
11 12
16
3
4
17
18
x3
1
ln V − ln 144.22
ln 160.00 − ln 144.22
(6)
x2 =
ln f z − ln 0.046
(7)
ln 0.07 − ln 0.046
x3 =
ln γ o − ln 9.5
ln 13.0 − ln 9.5
2
21 22
x2
(8)
x1
Figure 1: Design of Experiment.
To investigate the effect of
nonlinearity in the observation region and to
construct an estimated errors with nc -1 (nc is
number of center points), it is useful to add
center points in screening test with 2k-factorial
design when the factorial points in the design
are not replicated [10]. Four experiments
represent added center points to the first 8
points and were repeated four times to
estimate the pure error.
An extended design of 23-factorial
design is a second order central composite
design (CCD), which easily gained by
augmentation of 23-factorial design with
where x1 is coded value of cutting speed V
corresponding to its natural value of V, x2 is
the coded value of feed per tooth
corresponding to its natural value of fz and x3
is the coded value of radial rake angle
corresponding to its natural value of o.
Table 1: Levels of Independent Variables for
Ti-6Al4V
Independent
Variables
V (m.min-1)
x1
fz (mm.tooth-1)
x2
o
o ( )
x3
Level in coded form
-
-1
0
+1
+
124.53
130.00
144.22
160.00
167.03
0.025
0.03
0.046
0.07
0.083
6.2
7.0
9.5
13.0
14.8
Proceedings of International Conference on Mechanical & Manufacturing Engineering (ICME2008), 21– 23 May 2008, Johor Bahru, Malaysia.
© Faculty of Mechanical & Manufacturing Engineering, Universiti Tun Hussein Onn Malaysia (UTHM), Malaysia.
ISBN: 97–98 –2963–59–2
3. Experimental Set-Up
A CNC MAHO 700S machining
centre was used for experimentation, while
side-milling process was conducted with a
constant axial depth of cut aa 5 mm and radial
depth of cut ae 2 mm under flood coolant with
a 6 % concentration.
The reference workpiece material of
Ti-6Al4V, which was a rectangular block of
110 mm x 55 mm x 150 mm, was used for
cutting force measurements.
The end mill was clamped to the tool
holder with a constant 22 mm overhang. The
TiAlN coated grade-K-30 solid carbide end
mills with different radial rake angle
according to design of experiment, were used
in the experiments. To avoid the influence of
tool wear, the forces data (Fx, Fy, Fz) were
recorded during the initial cut when the end
mill was still new without wear. The recording
of cutting force was carried out using multi
component force measuring system consisting
of the following elements:
• A
3-component
dynamometer
comprising of basic unit (Kistler, Type
9265B) and a screwed-on working
adapter for milling (Kistler, Type
9443B).
• A multi channel charge amplifier
(Kistler, Type 5019A).
• A data acquisition system consisting of
a personal computer (PC) equipped
with an A/D board as well as the
DynoWare software (Kistler, Type
2825 D1-2, version 2.31).
force was factor B (feed), followed by
interaction BC (feed and radial rake angle), C
(radial rake angle), interaction AB (cutting
speed and feed) and lastly factor A (cutting
speed.
Table 2: Cutting Force Fta or Fc when using
TiAlN Coated Carbide Tools.
Std
Type
V
(m.min-1)
fz
(mm.th-1)
(o)
Calc. Fc
(N)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Fact.
Fact.
Fact.
Fact.
Fact.
Fact.
Fact.
Fact.
Center
Center
Center
Center
Axial
Axial
Axial
Axial
Axial
Axial
Axial
Axial
Axial
Axial
Axial
Axial
-1
1
-1
1
-1
1
-1
1
0
0
0
0
-1.4142
-1.4142
1.4142
1.4142
0
0
0
0
0
0
0
0
-1
-1
1
1
-1
-1
1
1
0
0
0
0
0
0
0
0
-1.4142
-1.4142
1.4142
1.4142
0
0
0
0
-1
-1
-1
-1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
-1.4142
-1.4142
1.4142
1.4142
82.76
88.45
116.53
106.57
64.38
66.34
129.92
107.38
80.88
67.99
74.80
80.07
94.40
82.87
68.76
79.80
52.23
78.43
103.13
96.74
101.56
107.11
65.04
115.04
o
The analysis for the developed models
was carried out using a Design Expert 6.0
package.
4. Experimental Results and Discussion
4.1 Development of the Cutting Force Model
using 2k-Factorial Design.
The development of the cutting force
model was based on the first 12 trials of the
experiments shown in Table 2. From the main
effect plot in Figure 2, it was observed that the
most significant factor that affected the cutting
Figure 2: Main Effect of 3F1-Cutting Force
Prediction Model for TiAlN Coated Tools
Proceedings of International Conference on Mechanical & Manufacturing Engineering (ICME2008), 21– 23 May 2008, Johor Bahru, Malaysia.
© Faculty of Mechanical & Manufacturing Engineering, Universiti Tun Hussein Onn Malaysia (UTHM), Malaysia.
ISBN: 97–98 –2963–59–2
The cutting force prediction model can
be formulated as
yˆ = 4.5285− 0.02293x1 + 0.21403x2 − 0.0528x3
− 0.04705x1x2 + 0.08189x2 x3
(9)
Equation (9) shows that the cutting
force decreases with increasing cutting speed
and radial rake angle. In contrary, it increases
with increase in feed. From interaction terms,
it was observed that the combination of speed
and feed contributes to decrease in cutting
force. However, the combination of feed and
radial rake angle adversely increases the
cutting force, whilst the feed alone tends to
increase the cutting force. From the ANOVA
results in Table 3, it is evident that the 3F1model is valid for the observation region,
because the lack of fit (LOF) is not significant.
Figure 4: Response Surface of fz and o for
3F1-Cutting Force Model using TiAlN Coated
Tools.
Table 3: ANOVA for 3F1-Cutting Force
Prediction Model using TiAlN Coated Tools
Figure 5: Response Surface of V and o for
3F1-Cutting Force Model using TiAlN Coated
Tools.
Figure 3: Response Surface of V and fz for
3F1-Cutting Force Model using TiAlN Coated
Tools.
The response surface of the cutting force
distribution in relation to the interaction of
cutting speed and feed, feed with radial rake
angle and cutting speed with radial rake angle
are shown graphically in Figure 3 to 5
respectively. Results showed that with
increasing radial rake angle, the cutting force
decreases. From Figure 3, it is obvious that
increasing the cutting speed slightly increases
the cutting force at lower feed. However,
increasing cutting speed at higher feed
decreases the cutting force. Another
performance evaluation is shown in Figure 4,
whereby it showed that with an increase of
radial rake angle at lower feed, the cutting
force decreases. However at higher feed,
increasing of radial rake angle causes a slight
increase of cutting force. It was also found
that the most significant factor that influenced
the cutting force is feed. This can be seen
Proceedings of International Conference on Mechanical & Manufacturing Engineering (ICME2008), 21– 23 May 2008, Johor Bahru, Malaysia.
© Faculty of Mechanical & Manufacturing Engineering, Universiti Tun Hussein Onn Malaysia (UTHM), Malaysia.
ISBN: 97–98 –2963–59–2
from the highest slope of the feed curve. The
last response surface for investigation of
performance evaluation is shown in Figure 5.
It indicates that radial rake angle has more
significant effect than cutting speed on cutting
force. The maximum cutting force was
achieved with combination of the lowest
radial rake angle and lowest cutting speed
when end milling Ti-6Al4V using TiAlN
coated tools.
In combination of all independent
variables, the minimum cutting force can be
achieved when using the lowest feed coupled
with the highest cutting speed and highest
radial rake angle.
that the LOF of the cutting force linear CCD
model was not significant. Thus the model is
valid for end milling of Ti-6Al4V using
TiAlN coated carbide tools under wet
conditions with the following range of
respective cutting speed V, feed per tooth fz
and radial rake angle o: 130 V 160 m.min1
; 0.03 fz 0.07 mm.tooth-1; 7 o 13 (o).
Table 5: ANOVA for 1st order CCD-Cutting
Force Model using TiAlN Coated Tools
4.2 Development of the First Order Cutting
Force Model using CCD.
The same data from Table 2 for 3F1model were used in developing the first order
CCD model. According to fit and summary
test for the first order cutting force model
(Table 4), a linear model was suggested.
Table 4: Fit and Summary Test for the First
Order Cutting Force CCD Model using TiAlN
Coated Tools
Figure 6: Response Surface of V and fz for 1st
CCD-Cutting Force Model using TiAlN
Coated Tools.
The first order CCD model for cutting
force is
(10)
which can be presented in the following form
yˆ = 4.4615 − 0.02293 x1 + 0.21403x2 − 0.0528 x3
Fc = 1811.32819V −0.22086 f z0.50521γ o−0.17062 (11)
where Fc is the predicted cutting force in (N).
To validate the first order CCD cutting
force model, ANOVA was conducted and the
results are presented in Table 5. It is obvious
Figure 7: Response Surface of V and o for 1st
CCD-Cutting Force Model using TiAlN
Coated Tools.
Proceedings of International Conference on Mechanical & Manufacturing Engineering (ICME2008), 21– 23 May 2008, Johor Bahru, Malaysia.
© Faculty of Mechanical & Manufacturing Engineering, Universiti Tun Hussein Onn Malaysia (UTHM), Malaysia.
ISBN: 97–98 –2963–59–2
researchers [11][12] for low and high cutting
speeds region.
Table 6: ANOVA for 2nd order CCD-Cutting
Force Model using TiAlN Coated Tools
Figure 8: Response Surface of fz and o for 1st
CCD-Cutting Force Model using TiAlN
Coated Tools.
More information resulted in the CCD
linear cutting force model is shown by the
response surface in Figure 6 to 8. From these
graphical plots, it can be recognized that
increasing the cutting speed decreases the
cutting force slightly. Similar findings was
reported by other researchers [11][12] for the
observation region of cutting speed.
4.3 Development of the Second Order Cutting
Force Model using CCD.
A second order model was postulated to
extend the variables range in obtaining the
relationship between the cutting force and
machining variables. The model is based on
the second order CCD model for k = 3 (Figure
1) and 24 set of experimental results as given
in Table 2. The results is presented in the
following form:
Figure 9: Response Surface of V and fz for 2nd
CCD-Cutting Force Model using TiAlN
Coated Tools.
yˆ = 4.3182 − 0.042812x1 + 0.1857x2 − 0.05948x3
+ 0.04239x12 + 0.03626x22 + 0.12237x32
(12)
− 0.04705x1 x2 − 0.01721x1 x3 + 0.08189x2 x3
From ANOVA results, it was also found
that the second order CCD model can be used
as the mathematical model in the region of
observation, since the LOF is not significant
as shown in Table 6.
It was interesting to observe that when
the region was extended, the contour of
cutting force in the cutting range changes from
linear (Figure 6) to a slightly curve form
(Figure 9). This was also confirmed by other
Figure 10: Response Surface of V and o for
2nd CCD-Cutting Force Model using TiAlN
Coated Tools.
Proceedings of International Conference on Mechanical & Manufacturing Engineering (ICME2008), 21– 23 May 2008, Johor Bahru, Malaysia.
© Faculty of Mechanical & Manufacturing Engineering, Universiti Tun Hussein Onn Malaysia (UTHM), Malaysia.
ISBN: 97–98 –2963–59–2
the higher the contribution of each factor to
the response (cutting force).
ANOVA was conducted once again
using the backward elimination to find the
reduced form of Equation (12). The results are
presented in Table 7.
Table 7: ANOVA for 2nd order CCD-Cutting
Force Model using TiAlN Coated Tools.
Figure 11: Response Surface of fz and o for
2nd CCD-Cutting Force Model using TiAlN
Coated Tools.
They found that the cutting force was
very high at low cutting speed and reduced
rapidly at medium cutting speed and finally
increased slightly with further increase in
cutting speed. It was also observed in Figure 9
that there was a significant increase in cutting
force with increase in feed.
The significant findings from the
experimental result in Figure 10 as compared
to Figure 7 is that the effect of radial rake
angle in the extended observation region
(using second order model) increased
significantly than in the linear region due to
the value of the radial rake angle almost
achieved the maximum value. Nevertheless,
increasing trend when radial rake angle
increased can still be seen.
It is obvious in Figure 11 that the effect
of feed as a function of radial rake angle
decreases with increasing radial rake angle.
The maximum value was achieved when the
highest feed combined with the highest radial
rake angle.
Further observation by means of
ANOVA, was conducted to find the
significant level of each factor of the model
and to reduce the second order CCD model
into a simpler form. This method known as
backward elimination, can be used when some
of the influencing factors have “Probe>F”
larger than 0.05 confident level. Contribution
of each factor can also be found from the
value of coefficient of each factor (Equation
(12)). The larger the coefficient of each factor,
Comparing the ANOVA in reduced form
(Table 7) with ANOVA in completed form
(Table 6), the mean square error (MSE) of
ANOVA in reduced form is higher than MSE
of ANOVA in completed form.
The Equation in reduced form can be
presented as
yˆ = 4.3182 + 0.1857 x2 − 0.05948 x3 + 0.10664 x32 (13)
Equation (13) is much simpler than the
origin of Equation (12). However, the
accuracy decreased with reduced parameters
involved in the original Equation (12).
4.4 Optimum Cutting Conditions
In this optimization, the numerical
optimization was carried out based on the
best-observed cutting force model to find the
optimize value from the given constraints for
optimization, which were proposed according
to Table 8 and 9.
Proceedings of International Conference on Mechanical & Manufacturing Engineering (ICME2008), 21– 23 May 2008, Johor Bahru, Malaysia.
© Faculty of Mechanical & Manufacturing Engineering, Universiti Tun Hussein Onn Malaysia (UTHM), Malaysia.
ISBN: 97–98 –2963–59–2
Table 8: Numerical Optimization of validated
cutting force model for Fc minimum
•
The second optimum cutting condition
when cutting force is in range, is cutting
speed V = 149.57 m.min-1, feed fz = 0.042
and radial rake angle o = 9.2 (o).
Acknowledgements
The authors wish to thank the Research
Management Center, UTM and the Ministry
of Science, Technology and Innovation
Malaysia for their financial support to the
above project through the IRPA funding 0302-02-0068 PR0074/03-01- Vote no. 74545.
Table 9: Numerical Optimization of validated
cutting force model for Fc in range
5. Conclusions
• Response surface methodology (RSM) has
proved to be a successful technique that
can be used to predict the cutting force Fc
and to reveal the relationship between
independent cutting conditions and cutting
force with minimum set of trials.
• There are three appropriate prediction
models namely 3F1, 1st and 2nd order CCD
model to formulate the relationship
amongst machining parameters such as
cutting speed, feed and radial rake angle.
• The models indicate that feed is the most
significant factor, which influenced cutting
force. It increases significantly with
increasing feed in the observation region.
• The first optimum cutting condition to
achieve minimum cutting force is cutting
speed V = 133.20 m.min-1, feed fz = 0.03
mm.tooth-1 and radial rake angle o =
12.98 (o).
References
[1]. H. Niemann, Ng. Eu-gene, H. Loftus, A.
Sharman, R. Dewes and D. Aspinwall,
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[3]. K.A. Abou-El-Hossein, K. Kadirgama,
M. Hamdi and K.Y. Benyounis, Journal
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[4]. D.C. Montgomery, Design, Analysis of
Experiments, 5th ed. John Wiley & Sons,
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[5]. M. Alauddin, M.A. El-Baradie and
M.S.J. Hashmi, Journal of Materials
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[6]. H.F. Kuang and Y.C. Hung, Journal of
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[7]. M.Y. Noordin, V.C. Venkatesh, S.
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[8]. T. Radhakrishnan and U. Nandan,
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Proceedings of International Conference on Mechanical & Manufacturing Engineering (ICME2008), 21– 23 May 2008, Johor Bahru, Malaysia.
© Faculty of Mechanical & Manufacturing Engineering, Universiti Tun Hussein Onn Malaysia (UTHM), Malaysia.
ISBN: 97–98 –2963–59–2
[9]. A.S. Mohruni, S. Sharif and M.Y.
Noordin, Proceeding of Regional
Postgraduate
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Malaysia, July 2006, pp. 337-342.
[10]. R.H. Meyers and D.C. Montgomery,
Response Surface Methodology: Process
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Designed Experiments, 2nd ed. John
Wiley & Sons Inc., 2002.
[11]. E.M. Trent and P.K. Wright, Metal
Cutting, 4th ed. Butterworth and
Heinemann, 2000.
[12]. J.H. Xu, K.Q. Ren and G.S. Geng, In
Key Engineering Materials, 259-260, pp.
451-455, 2004.
Manufacturing Engineering (ICME2008)
General Mechanical Engineering
Manufacturing
Automotive Technology
Heat and Fluid Technology
Industrial Engineering
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Faculty of Mechanical and Manufacturing Engineering
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Parit Raja, 86400 Batu Pahat, Johor, MALAYSIA
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Mathematical Modeling of Cutting Force in End Milling Ti-6Al4V using TiAlN Coated Carbide Tools
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MECHANICAL MANUFACTURING AUTOMOTIVE HEAT & FLUID INDUSTRIAL
ENGINEERING ENGINEERING TECHNOLOGY TECHNOLOGY ENGINEERING
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May 2008, Johor Bahru, Malaysia.
© Faculty of Mechanical & Manufacturing Engineering, Universiti Tun Hussein Onn
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ISBN: 97–98–2963–59–2
Faculty of Mechanical and Manufacturing Engineering
Universiti Tun Hussein Onn Malaysia (UTHM)
Parit Raja, 86400 Batu Pahat, Johor, MALAYSIA
Proceedings of International Conference on Mechanical & Manufacturing Engineering (ICME2008), 21– 23 May 2008, Johor Bahru, Malaysia.
© Faculty of Mechanical & Manufacturing Engineering, Universiti Tun Hussein Onn Malaysia (UTHM), Malaysia.
ISBN: 97–98 –2963–59–2
Mathematical Modeling of Cutting Force in End Milling Ti-6Al4V
using TiAlN Coated Carbide Tools
Mohruni, A.S.1,2*, Sharif, S.2, Noordin, M.Y.2
1
Department of Mechanical Engineering
Sriwijaya University
Indralaya - 30662 – South Sumatera
Indonesia
2
Faculty of Mechanical Engineering
Universiti Teknologi Malaysia
UTM Skudai - 81310 - Johor
Malaysia
*mohrunias@yahoo.com, mohrunias@unsri.ac.id
Abstract:
This paper deals with the development of cutting force predicted models in end milling titanium
alloy Ti-6Al4V using TiAlN coated solid carbide tools under flood conditions. The primary
machining parameters such as cutting speed, feed and radial rake angle, were used as
independent variables for factorial design of experiment coupled with response surface
methodology (RSM). Results from the 3D-response surface contour showed that the linear
model generate better results than the second order models obtained during machining this
advanced material. An optimum cutting conditions was also recognized for a particular range of
cutting force values. The models were verified by analysis of variances and were found to be
adequate.
Keywords: Cutting Force, End Milling, TiAlN Coated Carbide, Titanium Alloy, RSM.
1. Introduction
Numerous studies have shown
titanium and its alloys are difficult to machine,
regardless of the various types of cutting tools
used. This has been attributed to their low
thermal conductivity, which concentrates heat
in the cutting zone (typically less than 25%
that of steel), retention of strength at elevated
temperatures and high chemical affinity for all
cutting tool materials.
Although the cutting forces generated
are not excessively high (almost similar to
those with steel), they are confined to a small
area due to the short chip contact length which
leads to high stresses. The combination of
high stress and temperature resulted in plastic
deformation of the tool edge. Depth of cut
notching and chipping at the flank can also be
a problem with intermittent cutting operations
[1].
Knowledge of the cutting forces owing
to a predictive model is very interesting with
respect to the choice of machine tool power,
the cutting tools and the optimization of
cutting conditions for a given machining
operation. It could allow the number of long
and expensive tests to be limited and the best
tool geometry to obtain quasi-constant and
low cutting forces, which lead to a reduced
tool wear and consequently, a better tool-life,
to be found [2].
Many researchers have followed
purely experimental approaches to study the
relationship between cutting forces and
Proceedings of International Conference on Mechanical & Manufacturing Engineering (ICME2008), 21– 23 May 2008, Johor Bahru, Malaysia.
© Faculty of Mechanical & Manufacturing Engineering, Universiti Tun Hussein Onn Malaysia (UTHM), Malaysia.
ISBN: 97–98 –2963–59–2
independent cutting conditions such as feed
rate, cutting speed and depth of cut etc. This
has reflected on the increased total cost of the
study as it involved a large number of cutting
experiments. Furthermore, with this purely
experimental approach, researchers have
investigated the effect of cutting parameters
on cutting forces using machining experiments
based on a one-factor-at a-time design,
without having any idea about the behavior of
cutting forces when two or more cutting
factors are varied at the same time.
Furthermore, this approach cannot describe
and identify, with a great accuracy, the effect
of the interactions of different independent
variables on the responses when they are
varied simultaneously [3].
Recent study takes into account the
effect of simultaneous variations of three
cutting parameters such as cutting speed (V),
feed per tooth (fz) and radial rake angle ( o) on
the behavior of cutting forces by utilizing
response surface methodology (RSM). RSM is
a group of mathematical and statistical
techniques that are useful for modeling the
relationship between the input parameters
(cutting conditions) and the output variable or
response (cutting force) [4].
This method was also used by previous
researchers [5]-[9], which studied cutting
force as the machining response.
2. Mathematical Models for Cutting Force
In this study, RSM was used and the
mathematical models relating to the
machining
responses
were
developed
according to Alauddin et. al.[5].
where Fta is the calculated average tangential
force or cutting force (N), fz is the feed per
tooth (mm.tooth-1), o is the radial rake angle
(o), ’ is the experimental error and C, k, l, m
are parameters to be estimated using
experimental data.
By utilizing a natural logarithmic
transformation Equation (1) can be written in
first order polynomial as
ln Fta = ln C + k ln V + l ln f z + m ln γ + ln ε ′ (2)
which can be transformed to
y = b0 x0 + b1 x1 + b2 x2 + b3 x3 + ε
(3)
and finally can be formed as
yˆ1 = y − ε = b0 x0 + b1 x1 + b2 x2 + b3 x3
(4)
where y is the calculated average tangential
force on a natural logarithmic scale, 1 is the
natural logarithmic value of predictive
(estimated) tangential force or cutting force, x0
= 1 as a dummy variable, xi (i= 1 to 3) are the
coded variables of V, fz and o respectively, =
ln ’ and bj (j = 0 to 3) are the model
parameters to be estimated using experimental
data [5].
If the observation region is extended,
then the second order model is useful to
represent the effect of second order and
interaction components. The second order can
be extended from the first order model in
Equation (4) as
yˆ 2 = b 0 x 0 + b1 x 1 + b 2 x 2 + b 3 x 3
2.1 Postulation of the Mathematical Models
Assuming that the proposed model for
cutting force is merely depend on cutting
speed V, feed per tooth fz and radial rake o.
Other factors, which influence machining
process, are kept constant. Thus, the cutting
force model for end milling Ti-6Al4V can be
written as
(1)
Fta = CV k f zl γ mε ′
+ b12 x 1 x 2 + b13 x 1 x 3 + b 23 x 2 x 3
+ b11 x
2
1
+ b 22 x
2
2
+ b 33 x
(5)
2
3
where b values are the parameters to be
estimated using least squares method and 2 is
the predicted response on natural logarithmic
scale.
ANOVA was employed, to check the
adequacy of the predicted mathematical
model.
Proceedings of International Conference on Mechanical & Manufacturing Engineering (ICME2008), 21– 23 May 2008, Johor Bahru, Malaysia.
© Faculty of Mechanical & Manufacturing Engineering, Universiti Tun Hussein Onn Malaysia (UTHM), Malaysia.
ISBN: 97–98 –2963–59–2
2.2 Experimental Design.
In order to determine the Equation of
the response surface, several experimental
designs have been developed which attempt to
approximate the Equation using the smallest
number of possible experiments [4][10].
In this study the 2k-factorial design
using the first 8 points from design of
experiments as shown in Figure 1, was carried
out for screening test. This design is necessary
when interactions between variables are to be
investigated. Furthermore, factorial design
allow the effects of a factor to be estimated at
several levels of other factors, giving
conclusions that are valid over a range of
experimental conditions [4][10].
23 24
19
7
8
replicated star points ( = 1.4142 [9]) as
shown completely in Figure 1.
2.3 Coding of the Independent Variables
The independent variables were coded
by taking into account the capacity and
limiting cutting conditions of milling machine.
The following transforming Equation was
used.
ln xn − ln xn 0
ln xn1 − ln xn 0
x=
(5)
where x is the coded variable of any factor
corresponding to its natural xn, xn1 is the
natural value at the level +1 and xn0 is the
natural value to the base or zero level [5][9].
The coded values of the independent
variables shown in Table 1 for use in Equation
(4) and (5) were obtained from the following
Equations:
20
5
6
x1 =
13
9 10
15
14
11 12
16
3
4
17
18
x3
1
ln V − ln 144.22
ln 160.00 − ln 144.22
(6)
x2 =
ln f z − ln 0.046
(7)
ln 0.07 − ln 0.046
x3 =
ln γ o − ln 9.5
ln 13.0 − ln 9.5
2
21 22
x2
(8)
x1
Figure 1: Design of Experiment.
To investigate the effect of
nonlinearity in the observation region and to
construct an estimated errors with nc -1 (nc is
number of center points), it is useful to add
center points in screening test with 2k-factorial
design when the factorial points in the design
are not replicated [10]. Four experiments
represent added center points to the first 8
points and were repeated four times to
estimate the pure error.
An extended design of 23-factorial
design is a second order central composite
design (CCD), which easily gained by
augmentation of 23-factorial design with
where x1 is coded value of cutting speed V
corresponding to its natural value of V, x2 is
the coded value of feed per tooth
corresponding to its natural value of fz and x3
is the coded value of radial rake angle
corresponding to its natural value of o.
Table 1: Levels of Independent Variables for
Ti-6Al4V
Independent
Variables
V (m.min-1)
x1
fz (mm.tooth-1)
x2
o
o ( )
x3
Level in coded form
-
-1
0
+1
+
124.53
130.00
144.22
160.00
167.03
0.025
0.03
0.046
0.07
0.083
6.2
7.0
9.5
13.0
14.8
Proceedings of International Conference on Mechanical & Manufacturing Engineering (ICME2008), 21– 23 May 2008, Johor Bahru, Malaysia.
© Faculty of Mechanical & Manufacturing Engineering, Universiti Tun Hussein Onn Malaysia (UTHM), Malaysia.
ISBN: 97–98 –2963–59–2
3. Experimental Set-Up
A CNC MAHO 700S machining
centre was used for experimentation, while
side-milling process was conducted with a
constant axial depth of cut aa 5 mm and radial
depth of cut ae 2 mm under flood coolant with
a 6 % concentration.
The reference workpiece material of
Ti-6Al4V, which was a rectangular block of
110 mm x 55 mm x 150 mm, was used for
cutting force measurements.
The end mill was clamped to the tool
holder with a constant 22 mm overhang. The
TiAlN coated grade-K-30 solid carbide end
mills with different radial rake angle
according to design of experiment, were used
in the experiments. To avoid the influence of
tool wear, the forces data (Fx, Fy, Fz) were
recorded during the initial cut when the end
mill was still new without wear. The recording
of cutting force was carried out using multi
component force measuring system consisting
of the following elements:
• A
3-component
dynamometer
comprising of basic unit (Kistler, Type
9265B) and a screwed-on working
adapter for milling (Kistler, Type
9443B).
• A multi channel charge amplifier
(Kistler, Type 5019A).
• A data acquisition system consisting of
a personal computer (PC) equipped
with an A/D board as well as the
DynoWare software (Kistler, Type
2825 D1-2, version 2.31).
force was factor B (feed), followed by
interaction BC (feed and radial rake angle), C
(radial rake angle), interaction AB (cutting
speed and feed) and lastly factor A (cutting
speed.
Table 2: Cutting Force Fta or Fc when using
TiAlN Coated Carbide Tools.
Std
Type
V
(m.min-1)
fz
(mm.th-1)
(o)
Calc. Fc
(N)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Fact.
Fact.
Fact.
Fact.
Fact.
Fact.
Fact.
Fact.
Center
Center
Center
Center
Axial
Axial
Axial
Axial
Axial
Axial
Axial
Axial
Axial
Axial
Axial
Axial
-1
1
-1
1
-1
1
-1
1
0
0
0
0
-1.4142
-1.4142
1.4142
1.4142
0
0
0
0
0
0
0
0
-1
-1
1
1
-1
-1
1
1
0
0
0
0
0
0
0
0
-1.4142
-1.4142
1.4142
1.4142
0
0
0
0
-1
-1
-1
-1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
-1.4142
-1.4142
1.4142
1.4142
82.76
88.45
116.53
106.57
64.38
66.34
129.92
107.38
80.88
67.99
74.80
80.07
94.40
82.87
68.76
79.80
52.23
78.43
103.13
96.74
101.56
107.11
65.04
115.04
o
The analysis for the developed models
was carried out using a Design Expert 6.0
package.
4. Experimental Results and Discussion
4.1 Development of the Cutting Force Model
using 2k-Factorial Design.
The development of the cutting force
model was based on the first 12 trials of the
experiments shown in Table 2. From the main
effect plot in Figure 2, it was observed that the
most significant factor that affected the cutting
Figure 2: Main Effect of 3F1-Cutting Force
Prediction Model for TiAlN Coated Tools
Proceedings of International Conference on Mechanical & Manufacturing Engineering (ICME2008), 21– 23 May 2008, Johor Bahru, Malaysia.
© Faculty of Mechanical & Manufacturing Engineering, Universiti Tun Hussein Onn Malaysia (UTHM), Malaysia.
ISBN: 97–98 –2963–59–2
The cutting force prediction model can
be formulated as
yˆ = 4.5285− 0.02293x1 + 0.21403x2 − 0.0528x3
− 0.04705x1x2 + 0.08189x2 x3
(9)
Equation (9) shows that the cutting
force decreases with increasing cutting speed
and radial rake angle. In contrary, it increases
with increase in feed. From interaction terms,
it was observed that the combination of speed
and feed contributes to decrease in cutting
force. However, the combination of feed and
radial rake angle adversely increases the
cutting force, whilst the feed alone tends to
increase the cutting force. From the ANOVA
results in Table 3, it is evident that the 3F1model is valid for the observation region,
because the lack of fit (LOF) is not significant.
Figure 4: Response Surface of fz and o for
3F1-Cutting Force Model using TiAlN Coated
Tools.
Table 3: ANOVA for 3F1-Cutting Force
Prediction Model using TiAlN Coated Tools
Figure 5: Response Surface of V and o for
3F1-Cutting Force Model using TiAlN Coated
Tools.
Figure 3: Response Surface of V and fz for
3F1-Cutting Force Model using TiAlN Coated
Tools.
The response surface of the cutting force
distribution in relation to the interaction of
cutting speed and feed, feed with radial rake
angle and cutting speed with radial rake angle
are shown graphically in Figure 3 to 5
respectively. Results showed that with
increasing radial rake angle, the cutting force
decreases. From Figure 3, it is obvious that
increasing the cutting speed slightly increases
the cutting force at lower feed. However,
increasing cutting speed at higher feed
decreases the cutting force. Another
performance evaluation is shown in Figure 4,
whereby it showed that with an increase of
radial rake angle at lower feed, the cutting
force decreases. However at higher feed,
increasing of radial rake angle causes a slight
increase of cutting force. It was also found
that the most significant factor that influenced
the cutting force is feed. This can be seen
Proceedings of International Conference on Mechanical & Manufacturing Engineering (ICME2008), 21– 23 May 2008, Johor Bahru, Malaysia.
© Faculty of Mechanical & Manufacturing Engineering, Universiti Tun Hussein Onn Malaysia (UTHM), Malaysia.
ISBN: 97–98 –2963–59–2
from the highest slope of the feed curve. The
last response surface for investigation of
performance evaluation is shown in Figure 5.
It indicates that radial rake angle has more
significant effect than cutting speed on cutting
force. The maximum cutting force was
achieved with combination of the lowest
radial rake angle and lowest cutting speed
when end milling Ti-6Al4V using TiAlN
coated tools.
In combination of all independent
variables, the minimum cutting force can be
achieved when using the lowest feed coupled
with the highest cutting speed and highest
radial rake angle.
that the LOF of the cutting force linear CCD
model was not significant. Thus the model is
valid for end milling of Ti-6Al4V using
TiAlN coated carbide tools under wet
conditions with the following range of
respective cutting speed V, feed per tooth fz
and radial rake angle o: 130 V 160 m.min1
; 0.03 fz 0.07 mm.tooth-1; 7 o 13 (o).
Table 5: ANOVA for 1st order CCD-Cutting
Force Model using TiAlN Coated Tools
4.2 Development of the First Order Cutting
Force Model using CCD.
The same data from Table 2 for 3F1model were used in developing the first order
CCD model. According to fit and summary
test for the first order cutting force model
(Table 4), a linear model was suggested.
Table 4: Fit and Summary Test for the First
Order Cutting Force CCD Model using TiAlN
Coated Tools
Figure 6: Response Surface of V and fz for 1st
CCD-Cutting Force Model using TiAlN
Coated Tools.
The first order CCD model for cutting
force is
(10)
which can be presented in the following form
yˆ = 4.4615 − 0.02293 x1 + 0.21403x2 − 0.0528 x3
Fc = 1811.32819V −0.22086 f z0.50521γ o−0.17062 (11)
where Fc is the predicted cutting force in (N).
To validate the first order CCD cutting
force model, ANOVA was conducted and the
results are presented in Table 5. It is obvious
Figure 7: Response Surface of V and o for 1st
CCD-Cutting Force Model using TiAlN
Coated Tools.
Proceedings of International Conference on Mechanical & Manufacturing Engineering (ICME2008), 21– 23 May 2008, Johor Bahru, Malaysia.
© Faculty of Mechanical & Manufacturing Engineering, Universiti Tun Hussein Onn Malaysia (UTHM), Malaysia.
ISBN: 97–98 –2963–59–2
researchers [11][12] for low and high cutting
speeds region.
Table 6: ANOVA for 2nd order CCD-Cutting
Force Model using TiAlN Coated Tools
Figure 8: Response Surface of fz and o for 1st
CCD-Cutting Force Model using TiAlN
Coated Tools.
More information resulted in the CCD
linear cutting force model is shown by the
response surface in Figure 6 to 8. From these
graphical plots, it can be recognized that
increasing the cutting speed decreases the
cutting force slightly. Similar findings was
reported by other researchers [11][12] for the
observation region of cutting speed.
4.3 Development of the Second Order Cutting
Force Model using CCD.
A second order model was postulated to
extend the variables range in obtaining the
relationship between the cutting force and
machining variables. The model is based on
the second order CCD model for k = 3 (Figure
1) and 24 set of experimental results as given
in Table 2. The results is presented in the
following form:
Figure 9: Response Surface of V and fz for 2nd
CCD-Cutting Force Model using TiAlN
Coated Tools.
yˆ = 4.3182 − 0.042812x1 + 0.1857x2 − 0.05948x3
+ 0.04239x12 + 0.03626x22 + 0.12237x32
(12)
− 0.04705x1 x2 − 0.01721x1 x3 + 0.08189x2 x3
From ANOVA results, it was also found
that the second order CCD model can be used
as the mathematical model in the region of
observation, since the LOF is not significant
as shown in Table 6.
It was interesting to observe that when
the region was extended, the contour of
cutting force in the cutting range changes from
linear (Figure 6) to a slightly curve form
(Figure 9). This was also confirmed by other
Figure 10: Response Surface of V and o for
2nd CCD-Cutting Force Model using TiAlN
Coated Tools.
Proceedings of International Conference on Mechanical & Manufacturing Engineering (ICME2008), 21– 23 May 2008, Johor Bahru, Malaysia.
© Faculty of Mechanical & Manufacturing Engineering, Universiti Tun Hussein Onn Malaysia (UTHM), Malaysia.
ISBN: 97–98 –2963–59–2
the higher the contribution of each factor to
the response (cutting force).
ANOVA was conducted once again
using the backward elimination to find the
reduced form of Equation (12). The results are
presented in Table 7.
Table 7: ANOVA for 2nd order CCD-Cutting
Force Model using TiAlN Coated Tools.
Figure 11: Response Surface of fz and o for
2nd CCD-Cutting Force Model using TiAlN
Coated Tools.
They found that the cutting force was
very high at low cutting speed and reduced
rapidly at medium cutting speed and finally
increased slightly with further increase in
cutting speed. It was also observed in Figure 9
that there was a significant increase in cutting
force with increase in feed.
The significant findings from the
experimental result in Figure 10 as compared
to Figure 7 is that the effect of radial rake
angle in the extended observation region
(using second order model) increased
significantly than in the linear region due to
the value of the radial rake angle almost
achieved the maximum value. Nevertheless,
increasing trend when radial rake angle
increased can still be seen.
It is obvious in Figure 11 that the effect
of feed as a function of radial rake angle
decreases with increasing radial rake angle.
The maximum value was achieved when the
highest feed combined with the highest radial
rake angle.
Further observation by means of
ANOVA, was conducted to find the
significant level of each factor of the model
and to reduce the second order CCD model
into a simpler form. This method known as
backward elimination, can be used when some
of the influencing factors have “Probe>F”
larger than 0.05 confident level. Contribution
of each factor can also be found from the
value of coefficient of each factor (Equation
(12)). The larger the coefficient of each factor,
Comparing the ANOVA in reduced form
(Table 7) with ANOVA in completed form
(Table 6), the mean square error (MSE) of
ANOVA in reduced form is higher than MSE
of ANOVA in completed form.
The Equation in reduced form can be
presented as
yˆ = 4.3182 + 0.1857 x2 − 0.05948 x3 + 0.10664 x32 (13)
Equation (13) is much simpler than the
origin of Equation (12). However, the
accuracy decreased with reduced parameters
involved in the original Equation (12).
4.4 Optimum Cutting Conditions
In this optimization, the numerical
optimization was carried out based on the
best-observed cutting force model to find the
optimize value from the given constraints for
optimization, which were proposed according
to Table 8 and 9.
Proceedings of International Conference on Mechanical & Manufacturing Engineering (ICME2008), 21– 23 May 2008, Johor Bahru, Malaysia.
© Faculty of Mechanical & Manufacturing Engineering, Universiti Tun Hussein Onn Malaysia (UTHM), Malaysia.
ISBN: 97–98 –2963–59–2
Table 8: Numerical Optimization of validated
cutting force model for Fc minimum
•
The second optimum cutting condition
when cutting force is in range, is cutting
speed V = 149.57 m.min-1, feed fz = 0.042
and radial rake angle o = 9.2 (o).
Acknowledgements
The authors wish to thank the Research
Management Center, UTM and the Ministry
of Science, Technology and Innovation
Malaysia for their financial support to the
above project through the IRPA funding 0302-02-0068 PR0074/03-01- Vote no. 74545.
Table 9: Numerical Optimization of validated
cutting force model for Fc in range
5. Conclusions
• Response surface methodology (RSM) has
proved to be a successful technique that
can be used to predict the cutting force Fc
and to reveal the relationship between
independent cutting conditions and cutting
force with minimum set of trials.
• There are three appropriate prediction
models namely 3F1, 1st and 2nd order CCD
model to formulate the relationship
amongst machining parameters such as
cutting speed, feed and radial rake angle.
• The models indicate that feed is the most
significant factor, which influenced cutting
force. It increases significantly with
increasing feed in the observation region.
• The first optimum cutting condition to
achieve minimum cutting force is cutting
speed V = 133.20 m.min-1, feed fz = 0.03
mm.tooth-1 and radial rake angle o =
12.98 (o).
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Proceedings of International Conference on Mechanical & Manufacturing Engineering (ICME2008), 21– 23 May 2008, Johor Bahru, Malaysia.
© Faculty of Mechanical & Manufacturing Engineering, Universiti Tun Hussein Onn Malaysia (UTHM), Malaysia.
ISBN: 97–98 –2963–59–2
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