Astakhov V. Tribology of Metal Cutting
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Design of Experiments in Metal Cutting Tests 317
0.11
f(mm/rev)
M
T
(Nm)
0.13 0.15 0.17 0.19
30
40
50
0.825 m/s
0.440 m/s
0.330 m/s
n = 1.150 m/s
Fig. 5.11. Influence of uncut chip thickness (the cutting feed f ) on the drilling torque.
changes from 0.11 to 0.21 mm/rev, the axial force increases by 50% at the same cutting
speed. Increasing the cutting speed from 0.33 to 1.15 m/s leads to approximately a 20%
increase in the axial force at the same feed. As was expected, the cutting torque has the
opposite variations with the cutting speed and feed.
The analysis of the records of the axial force (an example is shown in Fig. 5.12) and
drilling torque allowed the following representation of these parameters
P
a
(
t
)
=P
as
+∆P
a
(
t
)
+δP
a
(
t
)
(5.53)
M
T
(
t
)
=M
Ts
+∆M
T
(
t
)
+δM
T
(
t
)
, (5.54)
where P
as
and M
Ts
are the static and time invariant parts of the axial force and drilling
torque, respectively, ∆P
a
(
t
)
and ∆M
T
(
t
)
are the time-dependent parts of the axial force
and drilling torque due to the variation of the properties of the work material, respectively,
δP
a
(
t
)
and δM
T
(
t
)
are the time-dependent parts of the axial force and drilling torque due
to the cyclic nature of chip formation and interactions of the deformation and thermal
waves.
The experiments showed that the variation of the axial force ∆P
a
(
t
)
has the frequency
proportional to the rotational frequency of the shank. Its amplitude increases with the feed
and decreases with the cutting (rotational) speed. In the case considered, the variation
of ∆P
a
(
t
)
was in the range from 11 to 19% of P
a
. The variation ∆M
T
(
t
)
of the drilling
torque had smaller amplitude and was in the range from 10 to 15% of M
T
in the selected
318 Tribology of Metal Cutting
∆T ∆T
Axial Force P
a
Time t
dT
∆P
a
∆P
a
dP
a
P
a
Fig. 5.12. Structure of relationship P
a
(
t
)
.
ranges of the feed and speed. The variation of the axial force δP
a
(
t
)
was in the range
from 13 to 24% and that of torque δM
T
(
t
)
was in the range from 13 to 24% of P
a
. The
amplitude of δP
a
(
t
)
does not vary under these given cutting conditions while that of
∆P
a
(
t
)
varies and its maximum corresponds to the beginning of period δT .
Using these results, one may conclude that the resonance phenomenon in the tests con-
ducted did not affect the drill performance significantly so that DOE can be used in the
assigned ranges of feeds and speeds (Table 5.14).
Model. Using the constructed matrix, the model determination was carried out using the
simplified algorithm of GMDH [20,21], according to which the search for the model of
the process under investigation is carried out among those accepted for the processes in
the field considered. As discussed above, logarithmic models are normally used for tool
life tests. As such, the domain of input factors was formed as ln ¯x. Because the function
in Eq. (5.52) is unknown, the input domain was extended by introducing variables ln ¯x,
(
ln ¯x
)
−1
to the existing ¯x and
(
¯x
)
−1
. As such, the response was represented as ln ¯y. Each
vector of the input factors in this domain includes nine (first in the order) geometrical
parameters and two regime parameters and their set forms the domain of input variables
¯x
, i.e.
¯x =
x
1
,...,x
11
(5.55)
(
¯x
)
−1
=
x
12
,...,x
22
(5.56)
(
ln ¯x
)
=
x
23
,...,x
33
(5.57)
(
ln ¯x
)
−1
=
x
34
,...,x
44
(5.58)
Design of Experiments in Metal Cutting Tests 319
As such, Eq. (5.52) can be re-written as
E
%
ln ¯y
x
&
=F
¯x,(¯x)
−1
,ln ¯x,(ln ¯x)
−1
,
¯
B
+
¯
Θ, (5.59)
where E
%
ln ¯y
x
&
is the estimation of the response, F
¯x
,
¯
B
is the unknown
functional and
¯
Θ is the vector of the residual errors of the model.
The basic GMDH algorithm (COMBI) uses an input data sample as a matrix containing
N levels (points) of observations over a set of M variables. A data sample is divided
into two parts. If regularity criterion AR(s) is used, then approximately two-thirds of the
observations form the training subset N
A
, and the remaining part of observations (e.g.
every third point with same variance) form the test subset N
B
. The training subset is used
to derive estimates for the coefficients of the polynomial, and the test subset is used to
select the structure of the optimal model, that is one for which the regularity criterion
AR(s) assumes its minimum
AR
(
s
)
=
1
N
B
N
#
1
[
y
i
−¯y
(
B
)]
2
→min (5.60)
Experience shows [20] that better results are achieved if the cross-validation criterion
RRR(s) is used because it takes into account all the information available in the data set
and it can be computed without recalculation of the system for each test point. Each
point is taken successively as a test subset and then the mean of criteria is used
RRR
(
s
)
=
1
N
N
#
1
[
y
i
−¯y
(
B
)]
2
→min N
A
=N −1,N
B
=1 (5.61)
To test a model for compliance with the differential balance criterion, the input data
set is divided into two equal parts. According to this criterion, the selected model must
yield the same results on both the subsets. The balance criterion yields the only optimal
physical model solely if the input data are noisy.
To obtain a smooth curve of criterion value, which would permit one to formulate the
exhaustive-search termination rule, a full exhaustive search is performed on models
classified into groups of equal complexity. The first layer uses the information contained
in every column of the set so that the search is applied to partial descriptions of the form
y =a
0
+a
1
x
i
,i=1,...,M (5.62)
Non-linear members are considered as new input variables in data sampling. The output
variable is specified in this algorithm in advance by the experimentalist. For each model,
a system of Gauss normal equations is solved. At the second layer all model candidates
of the following form are sorted
y =a
0
+a
1
x
i
+a
2
x
j
,j=1,...,M (5.63)
320 Tribology of Metal Cutting
Fig. 5.13. Combinatorial GMDH algorithm. 1 – data sampling; 2 – layers of partial descrip-
tions complexion; 3 – form of partial descriptions; 4 – choice of best model sets for structure
identification; 5 – additional optimal model definition by discriminating criterion.
Design of Experiments in Metal Cutting Tests 321
The models are evaluated for compliance with the criterion, and the procedure is carried
out until the minimum of the criterion is found. To decrease the calculation time, it is
recommended to select at a certain (6–8) layer a set of best F variables and to use them
only in the further full sorting-out procedure. As such, the number of input variables
can be significantly increased. For extended definition of the only optimal model, the
discriminating criterion is recommended.
Figure 5.13 represents the algorithm graphically. This algorithm can be implemented
simply in the following way: an available Gauss procedure of a system of linear equations
solution is set in a loop for the generation of all possible combination of inputs and
evaluate the results on separate sub-sample (not used for learning) by error criterion.
Then, the best model among the calculated is selected. The extended combinatorial
GMDH algorithm can briefly be described as follows:
1. Input of parameters and data.
2. Preliminary data handling: norming, generation of secondary non-linear variables.
3. Selection of criterion: regularity, balance or cross-validation type and division of
input data sample into learning subset and test subset. For the regularity criterion,
it is proposed to range all the observations according to dispersion of function and
take every third point to a test subset.
4. Three loops:
– for each function,
– for each number of variables (at each layer),
– for all variable sets for a given variable number
(a) selection of criterion,
(b) estimation of coefficients on learning sample and criterion value on
test subset. Coefficients are obtained by reducing the initial data array
into quadratic array of normal equations, which are solved by the LSE
(Least-Squares Estimation) (Gauss or Choletsky) procedure,
(c) selection and saving of some best criterion values and corresponding
ensemble of variable number for each layer,
(d) end of the third loop.
At the end of the second loop:
(e) output of criterion value (as text or graph),
(f) termination of procedure on user demand,
(g) end of the second loop.
At the end of the first loop:
(h) output results:
– regularization: selection of the best models by secondary criterion among the
selected models,
322 Tribology of Metal Cutting
– calculation of forecast value and errors,
– output of several best models.
(i) end of the first loop.
5. End.
For the case considered, this algorithm allowed generating a model having RRR
(
s
)
=14%
after six rows of selection. At this stage, further complication of the model was found
unnecessary, so only the estimates for the model’s coefficients were taken into consid-
eration for further stages of selection. The model was obtained in the following form:
y =b
0
+b
1
x
5
x
7
(
x
1
ln x
3
)
−1
−b
2
x
3
(
x
2
x
6
)
−1
−b
3
x
10
x
11
(
x
1
)
−1
−b
4
x
4
ln x
6
(
x
7
)
−1
−b
5
x
2
1
(
x
6
)
−1
+b
6
ln x
6
(
x
5
x
1
)
−1
(5.64)
or transforming Eq. (5.64) into initial factors, one can obtain
T =6.7020−0.6518
α
1
ϕ
2
c
2
−0.0354
α
2
ln c
2
m
d
−0.0005
ϕ
2
1
c
2
+0.0168
ln c
2
ϕ
1
−2.8350
νf
ϕ
1
−0.5743
c
2
m
d
ln α
1
ϕ
1
(5.65)
The adequacy of the model was tested using procedure described in the Section, Adequacy
of the Model.
The mathematical model of tool life (Eq. (5.65)) indicates that tool life in gundrilling is a
complex function of not only design and process variables but also of their interactions.
The inclusion of these interactions in the model brings a new level of understanding
about their influence on tool life. For example, it is known that the approach angle of
the outer cutting edge (ϕ
1
) is considered as the most important parameter of the tool
geometry in gundrilling because it has controlling influence on tool life and on other
important output parameters [25]. Traditionally, this angle along with approach angle of
the inner cutting edge (ϕ
2
) are selected depending on the properties of the work material.
Although the contradictive influence of these angles has been observed [25], none of the
studies reveals their correlations with the cutting regime as suggested by Eq. (5.65).
5.5.3 Optimization of the tool geometry
Problems that involve the operation or the design of systems are generally of the type
to which optimization principles can be beneficially applied. Unfortunately, there are no
fundamental studies on the optimization of the tool geometry so that the tool geom-
etry given by the tool manufacturer is considered as the best. Whenever one uses
“best” or “optimal” to describe a technical system including the cutting tool, an imme-
diate question to be asked is: “The best with respect to what criteria and subject to
what limitations?” Given a specific measure of performance and a specific set of con-
straints, one can designate a system as optimal (with respect to the performance measure
Design of Experiments in Metal Cutting Tests 323
and constraints) if it performs as well as, if not better than, any other system which
satisfies the given constraints. The term suboptimal is used to describe any system,
which is not optimal (with respect to the given performance measure and constraints).
The problem of optimization of the tool geometry can be thought of as: for a given
cutting regime and process constraints, and for the selected performance measure(s)
(tool life, production rate, cost per drilled hole, for example), find a combination of
parameters of the cutting tool geometry that maximize (minimize when necessary) the
chosen performance measure(s). In practical terms, the problem is reduced to finding
the extremum of the objective function
¯
A =F
x,
¯
B
(given by Eq. (5.52)) which is a
continuous function in the domain of its existence with respect to vector ¯x =
(
¯x
1
,...,¯x
9
)
.
Such a problem can be classified as a multi-dimensional problem of optimization.
A primary goal for the selection of an optimization method is to find a method which is
adequate to the problem and its constraints and that, given good starting points, requires
only a few iterations for finding a solution. The comparison of different optimization
techniques showed that the Hook and Jeeves method [26] is the most suitable for the
problem considered. The advantages of this method are:
1. the quality of solutions improves at each successive step on the response surface,
2. it is suitable for the response surface which may have deep narrow “cavities” –
craters (thus the gradient methods are not suitable), and
3. self-acceleration.
The essence of this method can be described as follows: first, the initial or starting
point of vector ¯x =
(
¯x
1
,...,¯x
9
)
is selected (¯x
[
0
]
). By changing the components of this
vector, the vicinities of the starting point are investigated and, as a result, the direction
of increasing (decreasing) the objective function
¯
A is found. The climb (or descending)
takes place until
¯
A increases (decreases). When there is no greater (smaller) value of
¯
A
in this direction, the step of climb (or descending) is reduced. If it does not help, this
direction is abounded and a new search in the vicinity of the current point takes place.
After few changes in the search direction, the method allows to reach the optimum point.
A step of 3% of the initial value of each parameter involved in the study was used.
The optimum parameters of the tool geometry corresponding to a tool life of 497 min,
obtained using the Hook and Jeeves method, are: approach angle of the outer cutting
edge (ϕ
1
) =31.5
◦
; approach angle of the inner cutting edge (ϕ
2
) =18.0
◦
; normal flank
angle of the outer cutting edge (α
1
) =14.5
◦
; normal flank angle of the outer cutting
edge, α
2
=9.0
◦
; location distances c
1
=1.36mm, c
2
=0.9mm; location distance of the
drill point m
d
=10.4mm for cutting speed ν =0.43m/ min and feed f =0.11mm/rev.
References
[1] Mason, R.L., Gunst, R.F., Hess, J.L., Statistical Design and Analysis of Experiments with
Application to Engineering and Science, John Wiley & Sons, New York, 1989.
324 Tribology of Metal Cutting
[2] Montgomery, D.C., Design and Analysis of Experiments, John Wiley & Sons, New York,
2000.
[3] Astakhov, V.P., Osman, M.O.M., Al-Ata, M., Statistical design of experiments in metal
cutting. Part 1: Methodology, Journal of Testing and Evaluation, 25 (1997), 322–327.
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cutting. Part 2: Application, Journal of Testing and Evaluation, 25 (1997), 328–336.
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[7] Plackett, R.L., Burman, J.P., The design of optimum multifactorial experiments, Biometrica,
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[8] Astakhov, V.P., An application of the random balance method in conjunction with the Plackett-
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rate metal-deforming technological processes, Journal of Materials Processing Technology,
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indmagdig.com/fmmag/article_archives.asp?action=details&article_id=416.
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[13] Astakhov, V.P., The mechanisms of bell mouth formation in gundrilling when the drill rotates
and the workpiece is stationary. Part 1: The first stage of drill entrance, International Journal
of Machine Tools and Manufacture, 42 (2002), 1135–1144.
[14] Astakhov, V.P., The mechanisms of bell mouth formation in gundrilling when the drill rotates
and the workpiece is stationary. Part 2: The second stage of drill entrance, International
Journal of Machine Tools and Manufacture, 42 (2002), 145–152.
[15] Astakhov, V.P., High-penetration rate gundrilling for the automotive industry: system outlook,
SME Paper TPO4PUB249, 2004, pp. 1–20.
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Press, Boca Raton, USA, 1999.
[18] Kronenberg, M., Machining Science and Application. Theory and Practice for Operation and
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[21] Ivakhnenko, A.G., Polynomial theory of complex systems, IEEE Transactions on Systems,
Man, and Cybernetics, 1(4) (1971), 364–378.
Design of Experiments in Metal Cutting Tests 325
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Mathematics (SIAM), Philadelphia, PA, 1999.
CHAPTER 6
Improvements of Tribological Conditions
Any modification of tribological conditions in metal cutting is considered as an
improvement if it results in at least one of the following:
• Improved tool life according to the criterion (or criteria when applicable) selected
to evaluate this life for a given machining operation.
• Improved productivity of a given operation by allowing higher cutting speed, feed
or, sometimes, depth of cut.
• Increased efficiency of a given operation (cost per drilled hole, for example).
All methods of improvement of tribological conditions can be broadly divided into two
categories:
1. Component methods. These methods include modification of component(s) of the
cutting system prior to the cutting process to improve the tribological conditions.
Although there are a great number of these methods, the following are common:
1.1 Cutting tool
1.1.1 Coating of the cutting tools with thin layer(s) of wear-resistant and/or
friction-reducing (often referred to as tribological) materials [1–3].
1.1.2 Polishing of the rake and flank surfaces of cutting inserts to reduce
the roughness of these surfaces in order to prevent severe adhesion.
1.1.3 Improving microgeometry of the cutting wedge by special edge-
preparation honing technology, which is particularly beneficial for
tools with polycrystalline diamond (PCD) and Cubic boron nitride
(CBN) tool materials [4].
1.2 Workpiece
1.2.1 Altering the properties of the work material by introducing special
heat treatment such as isothermal annealing.
326