Astakhov V. Tribology of Metal Cutting

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Design of Experiments in Metal Cutting Tests 277

(B94.55M-1985) suggests conducting the one-variable-at-a-time test. When it comes to

acquisition of test results, the only calculation of the conﬁdence interval limits is required

to carry out and report. As a result, so that only the inﬂuence of cutting speed on the

tool life can be distinguished for a given machine (static and dynamic stiffness, spindle

runout, accuracy of motion etc.), workpiece parameters (metallurgical state, dimensions,

holding method, etc.), cutting tool material and cutting tool design.

The design of experiments (DOEs) technique allows signiﬁcant improvement in the

methodology of machining tests. DOE is the process of planning of an experiment so

that appropriate data will be collected, which are suitable for further statistical analyses

resulting in valid and objective conclusions. There are a number of different method-

ologies of DOE; so one should select the appropriate methodology depending upon the

objective of the test and the resources available.

This chapter aims to discuss the basic DOE used in metal cutting tests. Particular attention

is paid to least formalized stages of DOE, where the most important decisions affecting

the test outcome are made.

5.2 DOE in Machining: Terminology and Requirements

5.2.1 Terminology

The objective of DOE is to ﬁnd the correlation between the response (for example, the

cutting force, tool life, etc.) and the factors included (the parameters of the cutting pro-

cess taken into consideration, for example, the cutting speed, feed, depth of cut, etc.).

All the factors included in the experiment are varied simultaneously. The inﬂuence of

unknown or non-included factors is minimized by properly randomizing the experiment.

Mathematical methods are used not only at the ﬁnal stage of the study, when the eval-

uation and analysis of the experimental data are conducted, but also throughout all the

stages of DOE, i.e. from the formalization of a priori information till the decision-making

stage. This allows answering of important questions: “What is the minimum number of

tests that should be conducted? Which parameters should be taken into consideration?

Which method(s) is (are) better to use in the evaluation and analysis of experimental

data? [1,2].”

The problem of mathematical model selection for the object under investigation requires

the formulation of clear objective(s) of the study. This problem occurs in any study,

but the mathematical model selection in DOE requires the quantitative formulation of

the objective(s). Such an objective is called the response, which is the result of the

process under study or its output. The process under study may be characterized by

several important output parameters but only one of them should be selected as the

response.

The response must satisfy certain requirements. First, the response should be the effective

output in terms of reaching the ﬁnal aim of the study. Second, the response should be

easily measurable, preferably quantitatively. Third, the response should be a single-

valued function of the chosen parameters.

278 Tribology of Metal Cutting

Input process variables are called factors. In DOE, it is necessary to take all the essential

factors into consideration. Unconsidered factors change arbitrarily and increase the error

of the tests. Even when a factor does not change arbitrarily but is ﬁxed at a certain level,

a false idea about the optimum can be obtained because there is no guarantee that the

ﬁxed level is the optimum one.

The factors can be quantitative or qualitative but both should be controllable. Practically,

it means that the chosen level of any factor can be setup and maintained during the tests

with certain accuracy. The factors selected should affect the response directly and should

not be a function of other factors. For example, the cutting temperature cannot be selected

as a factor because it is not a controllable parameter. Rather, it depends on other process

parameters like the cutting speed, feed, depth of cut, etc.

The factor combinations should be compatible, i.e. all the required combinations of the

factors should be physically realizable on the setup used in the study. For example, if a

combination of cutting speed and feed results in drill breakage, then this combination

cannot be included in the test. Often, chatter occurs at high cutting regimes that limits

the combinations of the regime parameters.

5.2.2 Mathematical model

One of the important stages in DOE is the selection of the mathematical model.

Mathematically, the problem of DOE can be formulated as follows: deﬁne the estimation

E of the response surface which can be represented by a function

{

}

= φ

(

,...,x

)

, (5.1)

where y is the process response (for example, cutting temperature, tool life, surface ﬁnish,

cutting force, etc.), x

,i= 1, 2,...,k are the factors varied in the test (for example, the

cutting edge angle, cutting speed, feed, etc.).

The mathematical model represented by Eq. (5.1) is used to determine the gradient, i.e. the

direction in which the response changes faster than in any other. This model represents

the response surface, which is assumed to be continuous, two times differentiable, and

having only one extremum within the chosen limits of the factors.

In general, a particular kind of mathematical model is initially unknown due to insufﬁcient

knowledge of the phenomenon considered. Thus, a certain approximation of this model

is needed. Experience shows [2] that a power series or polynomial can be selected as

an approximation. The accuracy of such an approximation would depend upon the order

(power) of the series. To reduce the number of tests at the ﬁrst stage of the experimental

study, a polynomial of the ﬁrst order or a linear model is sufﬁciently suitable. Such a

model is successfully used to calculate the gradient of the response, thus, to reach the

stationary region. When the stationary region is reached then a polynomial containing

terms of the second, and sometimes, the third, order may be employed.

Design of Experiments in Metal Cutting Tests 279

Experience shows [3,4] that a model containing linear terms and interactions of the ﬁrst

order can be used successfully in metal cutting. Such a model can be represented as

{

}

= β

(5.2)

The coefﬁcients of Eq. (5.2) are to be determined from the tests. Using the experimen-

tal results, one can determine the regression coefﬁcients b

, b

and b

, which are the

estimates for the theoretical regression coefﬁcients β

, β

and β

. Thus, the regression

equation constructed using the test results has the following form:



y = b

, (5.3)

where



y is the estimate for E{y}.

5.2.3 Pre-process decisions

Each factor selected for the DOE study has a certain global range of variation. Within this

range, a local sub-range to be used in DOE is to be deﬁned. Practically, the limits of each

included factors should be set. To do this, one should use all available information such as

experience, results of the previous studies, expert opinions, etc. Using this information,

the approximate combination of the factors included that gives the best result should

be deﬁned. Mathematically, the deﬁned combination can be thought of as a point in

the multi-dimensional factorial space. The coordinates of this point are called the basic

(zero) levels of the factors, and the point itself is termed as the zero point [1–4].

The interval of factor variation is the number which, when added to the zero level, gives

the upper limit and, when subtracted from the zero level, gives the lower limit. The

numerical value of this interval is chosen as the unit of a new scale of each factor. To

simplify the notations of the experimental conditions and procedure of data analysis, this

new scale is selected so that the upper limit corresponds to +1, lower to −1 and the basic

level to 0. For the factors having continuous domains, a simple transformation formula

is used

˜x

−˜x

∆˜x

, (5.4)

where x

is a new value of the factor, ˜x

is the true value of the factor, ˜x

is the true

value of the zero level of the factor, ∆˜x

is the interval of factor variation (in true units)

and i is the number of the factor.

As a result, the origin of the factorial space is shifted into a new position corresponding

to zero levels of the included factors. With respect to this new coordinate system, the

upper level of each factor corresponds to “+1” and the lower level to “−1.” When two

factor levels are used in DOE, the design plan is designated as 2

, where k is the number

of factors involved [1–3].

280 Tribology of Metal Cutting

As shown, the selection of the limits of factors is the least formalized stage in DOE, so

it is carried out using the experience and information available. In such a selection, the

setting accuracy of factors at the chosen levels and the degree of inﬂuence of each factor

on the response should be considered. One of the major objectives of such considerations

is to avoid a situation where the selected interval of variation of each factor is not wide

enough to detect the true inﬂuence of each factor on the response.

At the stage of pre-process decisions, it is very useful to have at least an approxi-

mate idea about the curvature of the response surface. When this surface is far from a

plane, the correlation between the accuracy of factors setting (requires increasing a scale

unit) and the curvature of the response surface (requires reducing a scale unit) becomes

signiﬁcant.

5.2.4 Basic requirements to test conditions

DOE requires special attention to be paid to the accuracy of the testing procedure. A rel-

atively high variation occurred during the metal cutting tests, and low reproducibility of

these tests force the experimentalist to increase the number of repetitions at the same

point of the design matrix. Therefore, the ﬁrst and foremost requirement is to assure

minimum possible variation of the response in the test.

Although there are a number of sources of errors and inaccuracies during a metal cutting

test [5], it is instructive to point out some general sources in particular for practically all

metal cutting tests:

• Errors associated with the workpiece can be classiﬁed as the work material, mount-

ing and clamping, and dimension related. The work material-related inaccuracies

and errors stem from the variation in the mechanical, physical and microstructural

variations of the work material. To reduce this source of errors, it is advisable to use

the workpieces having the same mechanical and metallurgical properties preferably

from the single manufacturing batch. Moreover, these properties should be tested

and recorded prior to the test. When it is not possible, these properties should be

requested from the steel supplier. The hardness of each workpiece should be mea-

sured prior to the test and the test should be conducted only using the workpiece

where its variation does not exceed ±10%. To reduce the mounting and clamping

errors, it is necessary to assure the mounting and clamping of the workpiece with

the same accuracy (runout) and clamping force. The dimension-related error should

be accounted when workpieces of various diameters and lengths are used. This issue

is discussed in Chapter 4.

• Errors associated with the cutting tool can be classiﬁed as the tool material, tool

holder and tool mounting (clamping) related. The tool material (cutting insert)

selected for the DOE should be from the same manufacturing batch. If it is feasible,

these tools should be calibrated prior to the tests.

• Errors associated with the experimental setup. Properly calibrated experimental

equipments and apparatuses should be used. The test conditions should always

be within the calibration ranges.

Design of Experiments in Metal Cutting Tests 281

• Errors associated with environment. To reduce the inﬂuence of slow-changing

conditions (temperature of the cutting ﬂuid, tool wear, room temperature, etc.),

the test should be conducted in a randomized sequence.

5.3 Screening Test

As mentioned above, probably the weakest link in DOE implementation is a set of pre-

process decisions. Often, such decisions rely on experience, available information and

expert opinions, and thus they are highly subjective. Even a small inaccuracy in the pre-

process decisions may affect the output results dramatically. Therefore, the pre-process

stage of DOE should be more formalized.

Normally, any machining test includes a great number of independent variables. In the

testing of gundrills, for example, there are a number of tool geometry variables

(the number of cutting edges, rake angles, ﬂank angles, cutting edge angles, inclina-

tion angles, side cutting edge back taper angle, etc.) and design variables (cutting ﬂuid

hole shape, cross-sectional area and location, proﬁle angle of the chip removal ﬂute,

shoulder dub-off shape and location, number and location of the supporting pads, radial

relief, length of the cutting tip, the shank length and diameter, etc.) that affect drill

performance. However, when many factors are used in DOE, the experiment becomes

expensive and time consuming. Therefore, there is always a dilemma. On one hand, it is

desirable to take into consideration only a limited number of essential factors carefully

selected by the experts. On the other hand, even if one essential factor is missed, the

ﬁnal statistical model may not be adequate to the process under study.

Unfortunately, there is no simple and feasible way to justify the decisions made at the

pre-process stage about the number of essential variables prior to the tests. If a mistake is

made at this stage, it may show up only at the ﬁnal stage of DOE when the corresponding

statistical criteria are examined. Obviously, it is too late then to correct the test results

by adding the missed factor.

The theory of DOE offers few ways to deal with such a problem [1,2]. The ﬁrst relies

on the collective experience of the experimentalist(s) and the research team in the deter-

mination of signiﬁcant factors. The problem with such an approach is that one or more

factors could be signiﬁcant or not, depending on the particular test objectives and con-

ditions. For example, the back taper angle in gundrills is not a signiﬁcant factor in

drilling soft materials or cast irons, but it becomes highly signiﬁcant in machining hard

titanium alloys and martensitic stainless steels. A second way is to use screening DOE.

This method appears to be more promising in terms of its objectivity. Various screening

DOEs are used when a great number of factors are to be investigated using relatively

small number of tests. This kind of test is conducted to identify the signiﬁcant factors

for further analysis.

Fractional factorial DOE is commonly used for screening DOE [2]. Using this method,

the experimentalist should be fully aware that it cannot detect any interactions among the

factors involved. Unfortunately, this simple fact is misunderstood in metal cutting where

such DOE has been used to study the interactions between the variables [6]. Therefore,

282 Tribology of Metal Cutting

if any factor interactions could be signiﬁcant, this test should not be used. The author’s

experience, however, shows that there are a number of signiﬁcant interactions between

the process parameters in metal cutting so that this DOE can hardly be acceptable in

metal cutting.

Plackett and Burman [7] developed a special class of fractional factorial experiments that

includes interactions. When this kind of DOE (referred to as the Plackett–Burman DOE)

is conducted properly using a completely randomized sequence, its distinctive feature

is high resolution. Despite a number of disadvantages (for example, mixed estimation

of regression coefﬁcients), this method utilizes high-contrast diagrams for the factors

included in the test as well as for their interactions of any order. This advantage of the

Plackett–Burman DOE is very useful in screening tests.

This section presents a simple methodology of screening DOE to be used in metal

cutting tests [8]. The method, referred to as the sieve DOE, has its foundation in the

Plackett–Burman design ideas, an oversaturated design matrix and the method of random

balance. The proposed sieve DOE allows the experimentalist to include as many factors

as needed at the ﬁrst phase of the experimental study and then to sieve out the non-

essential factors and interactions by conducting a relatively small number of tests. It is

understood that no statistical model can be produced in this stage. Instead, this method

allows the experimentalist to determine the most essential factors and their interactions

to be used at the second stage of DOE.

5.3.1 Background

The proposed sieve DOE includes the method of random balance. This method utilizes

oversaturated design plans where the number of tests is fewer than the number of factors

and thus has a negative number of degrees of freedom [9]. It is postulated that if the

effects (factors and their interactions) taken into consideration are arranged as a decay-

ing sequence (in the order of their impact on the variance of the response), this will

approximate a ranged exponential-decay series. Using a limited number of tests, the

experimentalist determines the coefﬁcients of this series and then, using the regression

analysis, estimates the signiﬁcant effects that have a high contrast in the noise ﬁeld

formed by the insigniﬁcant effects.

The initial linear mathematical model, that included k number of factors (effects), has

the following form:

y = a

+···+a

k−1,k

k−1

+δ, (5.5)

where a

is the absolute term often called the main effect, a

(i = 1,k)are the coefﬁcients

of linear terms, a

(i = 1,...,k−1;j = i +1,...,k)are the coefﬁcients of interaction

terms and δ is the residual error of the model.

Let us distinguish l as the number of insigniﬁcant factors so (k − l) is the number of

signiﬁcant effects (including generally both factors and interactions) can be distinguished.

Design of Experiments in Metal Cutting Tests 283

Because the insigniﬁcant effects do not have a strong impact on the response y, the

model can be simpliﬁed further to

y = a

+···+b

h−l

+ε (5.6)

where

ε = b

h−l+1

h−l+2

+···+b

+δ (5.7)

and thus

{ε}=b

h−l+1

}+b

h−l+2

}···+b

}+σ

{δ} (5.8)

In these equations, (k − l) signiﬁcant effects are distinguished from the total number of

effects k and thus l effects are considered as nose having no important impacts on the

response y.

It is obvious that the residual variance of the model represented by Eq. (5.8) is greater

than that in Eq. (5.5), i.e. σ

{

}

>σ

{

}

and that the estimates of the coefﬁcients of

this model are mixed. Therefore, the sensitivity of the random balance method is low.

However, this method is characterized by the great contrast of essential effects, which

could be distinguished easily on the noisy ﬁelds formed by other effects.

5.3.2 Pre-process decisions

As mentioned above, pre-process decisions include the selection of factors and their

ranges. The greater the scope (cost and time) of the experiment, the greater the attention

to be paid at this stage, verifying results with independent experts in the ﬁeld. Unfortu-

nately, this has not been the case in metal cutting studies, where a great number of large,

midsize and even small companies (cutting tool users and suppliers, carbide manufac-

turers and other tool material suppliers, work material suppliers, etc.) conduct their own

experimental studies, paying little attention to this important ﬁrst stage. As a result, the

recommendations for the machining regimes to be used are very wide that makes them

impractical.

The factors taken into consideration can be of a qualitative or a quantitative nature. They

and their ranges should justify the following requirements:

• Controllability. Practically, it means that the selected levels of each factor can be

set and maintained within a reasonable range during the tests.

• Independence. Each of the selected factors cannot be a function of other factors

or their combination and thus affects the process directly. For example, the cutting

284 Tribology of Metal Cutting

temperature cannot be considered as a factor because it depends on other factors and

cannot be set and kept at a desired level independent of other cutting parameters.

• Compatibility. This requirement means that all the required (by DOE) combinations

of the chosen factors should be realizable on the test setup. For example, exces-

sive tool and/or workpiece vibration, tool shank strength or cutting edge breakage,

machine limitation in terms of speed, or feed, cutting ﬂuid pressure normally are

not realizable, and this limits compatibility.

These requirements set strict limits on the ranges of factors.

In addition, the accuracy of setting each included factor and the accuracy of the mea-

surement of the response should also be the subjects of special care. The rule of thumb

here is to measure all the factors and the responses with the same accuracy. To assure

this, a metrological matrix of the experiment should be an inherent part of any DOE in

metal cutting. The metrological matrix includes the list of measuring equipments and

apparatuses, their measuring ranges, the nominal measuring errors and their variations

within these ranges, the calibration data, etc. Particularly in metal cutting, static and

dynamic calibrations and error assessments are mandatory [5,10]. Unfortunately, it has

become customary not to report these data when presenting the experimental results in

metal cutting.

To simplify further consideration of the sieve DOE, it is reasonable to present a practical

example. Consider the use of such DOE to determine the essential factors affecting tool

life in gundrilling.

5.3.3 Basic gundrill components and geometry

Gundrilling is a highly developed and efﬁcient technique for producing deep holes in a

wide variety of materials from plastics, such as ﬁberglass and teﬂon, to high-strength

metals, like P-20 and Inconel. The process also enables size, location and straightness

accuracy where tight tolerances and ﬁne ﬁnishes are critical [11–15].

However, successful gundrilling requires complete understanding and integration of the

gundrilling system, which includes everything related to the operation: the cutting tool,

machine, ﬁxtures and accessories, workpiece, cutting ﬂuid, programming, control and

operator skill. Optimum performance is achieved when the combination of cutting speed,

feed, tool geometry, carbide grade and cutting ﬂuid parameters is selected properly.

This selection depends upon the hardness, composition and structure of the workpiece,

deep-hole machine conditions and the quality requirements to the drilled holes.

In terms of tool type, the straight-ﬂute gundrill is the most common (Fig. 5.1). It has a

solid- or brazed-carbide tip, depending on the tool’s diameter, with an internal cutting

ﬂuid channel running through its driver, shank and tip. Gundrill manufacturers have

adopted various shapes for the cutting ﬂuid passage in the tip: either one or two circular

holes or a single kidney-shaped hole. Standard gundrills produce holes from 2 to 50 mm

in diameter (D

) and up to 100 D

deep in one pass, with custom length roughly

doubling this magnitude.

Design of Experiments in Metal Cutting Tests 285

Driver

Tubular Shank

V-Flute

Chip removal

channel

Tip (Tool Head)

Two-Hole Tip

Kidne

-Hole Ti

Round-Hole Ti

Overall Length

Flute Length

Internal coolant

supply channel

Fig. 5.1. Common gundrill and its components.

The design and geometry parameters of a commonly used gundrill are shown in Fig. 5.2.

The gundrill consists of a drill body having a shank 1 and a tip 2. The tip is made up

of a hard wear-resistant material such as tungsten carbide. The other end of the shank

incorporates an enlarged driver 3 having the machine-speciﬁc design. The shank is of

tubular shape having an elongated passage 4 extending over its entire length and connects

to the drilling ﬂuid supply passage 5 in the driver. The shank has a V-shaped ﬂute 6 on

its surface which serves as the chip removal passage. The shank length depends mainly

on the depth of the drilled hole as well as on the lengths of the bushing and its holder,

chip box, etc.

The tip is larger in diameter than the shank that prevents the shank from coming into

contact with the walls of the hole being drilled. Flute 7 on the tip, which is similar in

shape to ﬂute 6, extends along the full length of the tip. This ﬂute is bounded by side

faces 8 and 9 known as the cutting face and side face, respectively. The depth of this ﬂute

is such that the cutting face 8 extends past the axis 10 (distance c) of the tip, which is

also the axis of the drill body. The angle ψ

between the side and cutting faces is known

as the proﬁle angle of the tip, which is usually equal or close to the V-ﬂute proﬁle of the

shank.

The terminal end of the tip is formed with the approach cutting edge angles ϕ

and

of the outer 11 and inner 12 cutting edges, respectively. These cutting edges meet

at the drill point P. The location of P (deﬁned by the distance m

in Fig. 5.2) can be

varied for optimum performance depending on the work material and the ﬁnished hole

speciﬁcations. One common point grind calls for the outer angle, (ϕ

), to be 30

◦

and

the inner angle (ϕ

), to be 20

◦

. The geometry of the terminal end largely determines

the shape of the chips and the effectiveness of the cutting ﬂuid, the lubrication of the

tool, and removal of the chips. The process of chip formation is also governed by other

cutting parameters such as the cutting speed, feed rate, work material, etc.

286 Tribology of Metal Cutting

Ø12.1

Fig. 5.2. Gundrill geometry.

The ﬂank surface 13 having normal clearance (ﬂank) angle α

is 8–20

◦

is applied to

the outer cutting edge 11 and the ﬂank surface 14 having normal clearance (ﬂank) angle

equal to α

(normally α

is 8–12

◦

) is applied to the inner cutting edge 12. To assure

drill-free penetration, i.e. to prevent the interference of the drill’s ﬂanks with the bottom

of the hole being drilled, the auxiliary ﬂank 15 and shoulder dub-off 16 are ground. Their

location and geometry are uniquely deﬁned for a given gundrill.

5.3.4 Design matrix: tool life test of gundrills

The test conditions were as follows:

Machine – a special deep-hole gundrilling machine was used. The drive unit was

equipped with a programmable AC converter to offer variable speed and feed rate control.

The machine contained a high-pressure cutting ﬂuid delivery system capable of delivering