A diagnostic approach for turning tool based on the dynamic force signals

1. S. E. Oraby1
Associate Professor
Department of Production and Mechanical
Design,
Faculty of Engineering,
Suez Canal University,
Port Said, Egypt
e-mail: soraby@paaet.edu.kw
A. F. Al-Modhuf
Assistant Professor
Department of Mechanical Production
and Technology,
College of Technological Studies,
PAAET,
P.O. Box 42325,
Shuwaikh 70654, Kuwait
D. R. Hayhurst
Professor
Department of Mechanical, Aerospace
and Manufacturing Engineering,
UMIST,
P.O. Box 88, Sackville Street,
Manchester M60 1QD, UK
A Diagnostic Approach for Turning
Tool Based on the Dynamic Force
Signals
In the current work it is proposed a simple, and fast softwired tool wear monitoring
approach, based upon the features of the time series analysis and the Green’s Function
(GF) features. The proposed technique involves the decomposition of the force signals
into deterministic component and stochastic variation-carrying component. Then, only
the stochastic component is processed to detect the adequate autoregressive moving
average (ARMA) models representing the tool state at every wear condition. Models are
further reduced to form a more representative parameter, the “Green’s Function (GF).”
This reflects the dynamic behavior of the tool prior to failure and, may provide a com-
prehensive and accurate measure of the damping variation of the cutting process sub-
system at different forms of tool’s edge wear. As wear enters the high rate region, the
cutting process is forced toward the instability domain where it tends to have less damp-
ing resistance. It is also explained how a system response surface can be generated based
on its Green’s function. It is proposed that this concept can be the basis for a diagnostic
technique for use with many systems. ͓DOI: 10.1115/1.1948397͔
1 Introduction
It is universally acknowledged that the performance of any pro-
cess is usually evaluated in terms of its productivity and surface
quality. Each of these controlling criteria is affected directly, and
in different way by the tool edge wear and fracture.
The complexity of modern industrial and manufacturing sys-
tems, with their emphasis on quality, increased effectiveness and
higher productivity through automation and computerization, have
led to an increase demand for a new breed of tool monitoring
techniques. These methods are usually achieved via observing
variations in one or more of the operation responses ͑outputs͒
related to tool deformation and, consequently, exploited to inves-
tigate the aspect of tool wear monitoring and control ͓1–5͔. Al-
though many successful attempts have been reported, research is
still carried on seeking a smart integrated strategy. A robust online
wear monitoring and control technique depends, to a great extent,
on the sensitivity of the data carrier ͑machining output or re-
sponse͒ in addition to the formulation tools by which the useful
information can be isolated from the huge variation involved.
Among responses from machining operation, cutting forces are
always considered as one of the most reliable measures for tool
wear monitoring and breakage detection ͓3–5͔. Both static and
dynamic force components are frequently used. Variation in static
force signals shows a good correlation with tool wear progress in
fixed parameters machining systems ͓3,6–8͔. However, in situa-
tions where cutting parameters, especially feed and depth of cut,
are subject to on-line change, wear effects usually masked static
force trend is no longer appropriate. In such situations, it is better
to use dynamic oscillation patterns instead ͓4͔.
Among the approaches introduced to deal with force and vibra-
tion signals are the spectral analysis ͓4,9͔, the conventional aver-
aging and time series analysis ͓10–12͔. Although averaging and
filtering are simple and fast, they narrow the frequency bandwidth
thus leading to masking the effect of random disturbances such as
tool chipping and fracture ͓10͔. Many investigators employed
both the time series ARMA analysis and the dynamic data systems
͑DDS͒ technique to monitor machining performance ͓11–13͔. In
their work, Pandit and Kashou ͓12,13͔ have applied the method-
ology for tool wear monitoring through the analysis of cutting
vibration. Although these are very useful studies in this field, the
proposed technique is very complicated and time-consuming. It
requires a higher order ARMA model, ARMA ͑20,19͒, which
would increase the response time. Although a similar wear-
vibration trend was depicted by two of the authors ͓14͔ using
much simpler experimental and computational procedures, expe-
rience always indicated that vibration was not the response of
choice. Environmental noise, in addition to instantaneous varia-
tion in system dynamic features, usually acts against reaching a
firm and universal conclusion. Bandyopadhyay et al. ͓15͔ have
used the Dynamic Data Systems to analyze the dynamics of the
thrust/torque signals of drilling operations to develop a “Normal-
ized Damping Ratio” as the ratio between a worn tool and that
with no wear. Oglec and Guttermuth ͓16͔ suggested a numerical
technique based on ARMA time series analysis to represent the
tool force dynamics as affected by some controlling parameters:
depth of cut, feed rate, and axial tool position. A one-by-one pa-
rameter technique was introduced, and, claimed to be superior to
the dynamic data system methodology. However, information re-
garding tool wear has not been reported.
Altintas ͓17͔ has used time series analysis to develop a soft-
wired filter, AR͑1͒, to separate the cutting transients from a tool
breakage event in milling operation. The DDS methodology has
also been used in the detection of the tool failure and breakage
͓18,19͔.
Therefore, time series analysis can produce a significant and
comprehensive modeling technique, if a proper isolation of the
only variation-carrying component is successfully isolated from
the entire signals. Nevertheless, it can be stated that a robust data
processing technique, to analyze and to formulate the functional
relationship between tool wear and the accompanied variation in
1
On leave. Presently at the Department of Mechanical Production Technology,
College of Technological Studies, PAAET, P.O. Box 42325, Shuwaikh 70654, Ku-
wait.
Contributed by the Manufacturing Engineering Division for publication in the
ASME JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received
February 4, 2004; final revision received September 1, 2004. Associate Editor: D.-W.
Cho.
Journal of Manufacturing Science and Engineering AUGUST 2005, Vol. 127 / 463
Copyright © 2005 by ASME

2. the force signals is still debatable, and a fast, accurate, sensitive,
and economically feasible approach is still required.
In this study, the possible hidden correlation between instanta-
neous tool wear state and the corresponding variations in the sto-
chastic signals ͑residuals͒ of the dynamic force signals is investi-
gated. Special attention is devoted to the instant at which the tool
unexpectedly and catastrophically fails due to temperature soften-
ing.
In order to apply time series analysis, data or signals should be
of random stationary nature. Applying the ARMA technique to a
nonstationary time series usually leads to incorrect results. Also,
the conversion of the nonstationary data using first and second
integration is not a good strategy since some useful features may
be lost ͓20͔.
In the current study, the remaining stochastic signals are mod-
eled using autoregressive moving average ͑ARMA͒ procedures
employing the appropriate statistical criteria to examine the sig-
nificance and the adequacy of the resulting models. For each wear
level, the corresponding model is further reduced to the equivalent
“Green’s function” ͑GF͒, which determines its behavior dynamic
characteristics.
Working on only the stochastic part of the signals pattern is
considered, bearing in mind its suitability to be used in adaptive
control strategy, where cutting parameters are continuously sub-
jected to change as the need arises. Variable parameters affect
only the deterministic part of the force signals. Nevertheless, as
discussed earlier, the use of both elements of the signals are nec-
essary in the prediction and forecasting applications.
In Sec. 3, it is explained how the stochastic force component is
isolated from the data carrying signals. Also, modeling using the
time series ARMA procedures is explained. Section 3 explains
how each developed model is reduced to its equivalent “Green’s
function.” A numerical procedure is introduced to indicate how a
response can be generated from a system signals, providing its
Green’s function is known.
2 Experimental Procedures
The experimental set up and signal processing techniques,
throughout the different stages of the current work are schemati-
cally illustrated in Fig. 1. Multicoated carbide inserts ͑Sandvik
GC415-ISO P15͒ are used to turn hard and tempered alloy steel
͑En24͒ 8-in. bars. Such types of tools and workpiece material are
selected for comparison with the different tool grade ͑Sandvik
GC435͒, which was used previously ͓6͔. Dry turning is carried out
employing a rigid Colchester 1600 Mascot center lathe. Actual
workpiece rotation is directly read from a digital counter based on
pulses from the loaded spindle. Force signals are measured using
a three-component dynamometer ͓21͔ and digitized using a three-
channel ADC, Fig. 1, at a sampling period of 0.036 s, and then
they are permanently stored for further off-line analysis. The test
consists of subtests at about 2 min interval repeated until the tool
fails by plastic deformation. After machining of each subtest, nose
wear is evaluated using a three-dimensional optical microscope.
Wear history of the entire test ͑six subtests a–f͒ are shown in Fig.
2. A surface speed of 200 m/min is selected for testing to be high
enough to ensure a noticeable wear progress, and also to be well
within the practical speed range employed in modern machine
tools. Another advantage of using such an operational range is to
avoid built-up edge formation mechanism and, to be inside the
stable chatter-free machining domain.
Figure 3 shows the records of the six subtests consisting the
whole test where force signals are plotted against time sequence
considering a sampling time of 0.036 s between every two suc-
Fig. 1 Experimental set up and signal processing procedures
464 / Vol. 127, AUGUST 2005 Transactions of the ASME

3. cessive readings. As shown in Fig. 3, the static mean of force
usually increases as wear increases. It is well observed that the
tangential force component Fy, however, is not as sensitive to
progressive wear as the other feeding Fx and Fz components.
While Fy increases only about 40% of its original mean value
until the tool is plastically failed ͑subtest f, Fig. 3͒, a correspond-
ing increase of about 700% is noticed within the same period for
Fx and Fz. Also, both Fx and Fz increase about 100% of their
original value at subtest e, and around 125% at subtest f, where
wear level reaches 0.344 mm. This is expected since the Fy is
predominately determined by the cut cross section, and changes
only slightly as the tool wears while both Fx and Fz are mainly
penetrating and frictional loads, and hence are very sensitive to
wear. The frequent but irregular chipping and fracture of the edge,
subtest c ͑after Ϸ6 min͒, is observed during optical examination
of the tool. This widens the cut frictional area between the edge
and the workpiece leading to instantaneous force increase. At sub-
tests a–c, Fig. 3, Fx and Fz tend to have similar values due to the
fact that wear has almost regular pattern on both nose and flank.
At later stages, subtest d, Fz attains slightly higher rate revealing
the possibility of nose wear domination. Subsequent data, subtests
e and f, supported the idea that this is the onset of the tool soft-
ening stage where the edge started to lose bulk material. There-
fore, a combination of Fx and Fy to form the thrust component
normal to the cutting edge Fxz, ͑Fxz=Fx sin ␬+Fz cos ␬, ␬ is the
Fig. 2 Wear-time curve and cutting conditions
Fig. 3 Recorded dynamic force signals
Journal of Manufacturing Science and Engineering AUGUST 2005, Vol. 127 / 465

4. approach angle͒, may produce a better measure ͓3͔. As is practi-
cally experienced, the wear on the cutting edge does not usually
conform a uniform pattern on nose and flank areas and this leads
to a relative continuous change in the values of Fx and Fz. This
justifies the use of the resultant measure namely the thrust com-
ponent Fxz. In a previous study by one of the authors ͓4͔, different
tool failure forms are successfully monitored and assessed via
force signals using spectral analysis techniques. In this study,
same data are manipulated using more physically interpreted ap-
proach, which are the time series analysis and its associated
Green’s function.
3 Analyses and Discussion
3.1 Constituents of the Cutting Force Signals. Large num-
bers of parameters are involved in the dynamics of tool-workpiece
engagement during metal removal by machining. Variations in the
force signals are caused by variability involved in any of the in-
dividual system’s elements or in their mutual interaction. Com-
mon variability sources include machine tool, workpiece, chip for-
mation and separation mechanism, etc. and hence the first
approximation of steady state static cutting is invalidated. It is
suggested ͓10͔ that force variations can be attributed to one or
more of its pertaining subsystems: the cutting process, mechanical
structure of the machine tool, or a secondary mode system and can
be considered as follows:
Ft͑i͒ = F0͑i͒ + Xt͑i͒, ͑1͒
where; ͑F0͒ is the deterministic, or the static mean component and
͑Xt͒ is the stochastic, or the “white noise” component of the signal
Ft.
3.2 Isolation of the Deterministic Trend. As shown by Figs.
2 and 3, the tool was plastically deformed at the end of the sixth
subtest f after attaining a high level of progressive wear. Many
model structures are evaluated using nonlinear regression tech-
nique to formulate the deterministic part of the force signals. Plain
first order model produced poor results while adding exponential
term improved the model predictability through giving better sta-
tistical criteria. Also, fewer numbers of iterations were found to
converge to the final model. This is thought to be due to the
nonlinearity involved in the mean, or deterministic, part of the
force signals. The proposed model structure hence takes the fol-
lowing form:
F0͑i͒ = ␤0 + ␤1͑t͒ + ␤2e͑t͒
+ ␧ ͑2͒
where ␤’s are the coefficients of the model, ͑t͒ is the aggregated
time for a given interval and ␧ is the residuals. Nonlinear recur-
sive regression analysis is used to estimate the model’s coeffi-
cients ␤’s for each subtest using the last 1000 readings of the force
record of each subtest, Fig. 3, leading to results and significance
criteria listed in Table 1. The determination factor ͑R2
͒ initially
increases as wear increases and then, sharply drops at wear level
of 0.29 mm, which is considered as the practical entrance of the
high wear rate zone. Higher increase is found when tool enters the
softening failure zone represented by subtests e and f. The
residuals-sum of squares ͑RSS͒ varies differently at different wear
levels ͑Table 1͒. At the constant wear rate region, subtests a and b,
the ͑RSS͒ values are almost constant with relatively low R2
. At
subtest c, both R2
and RSS increase to almost twofold of their
subsequent values. This indicates that, even a deterministic pattern
is well grasped ͑R2
=71%͒; more signals fluctuations are evident
͑RSS=864534͒. At subtest d, however, signals show wider dy-
namic amplitudes since it is affected by the remaining dynamic
from preceding subtests in addition to its local dynamic consider-
ations. A poor R2
of 23% and rapidly growing RSS of 1597180
reflect the dynamic instability of subtest d.
As tool enters the softening zone, subtests e and f, a predomi-
nating dynamic nature is observed with a better isolation of the
deterministic trend. Signals for subtest e, Fig. 3, however, produce
better R2
and RSS than those for substrate f. Even both subtests
are in the softening zone, tool edge maintains solidarity in subtest
e while it is catastrophically failed at subtest f with entirely dif-
ferent non-linear trend. All these observations imply that a great
proportion of the disturbances due to tool wear is carried by the
stochastic part of the force series and an appropriate modeling
strategy may quantitatively describe their true influential relation-
ship.
3.3 Time Series Modeling Analysis of the Stochastic Com-
ponent of Force Signals. In the case of zero tool wear, residuals
after the deterministic modeling ͑␧͒ are of a random “white noise”
time series, with uncorrelated zero mean and standard deviation.
As wear develops on the cutting edge, the series is disturbed. The
emerged characteristics can be formulated using the AutoRegres-
Table 1 Results of the deterministic component modeling
Table 2 ARMA results for subtest a
466 / Vol. 127, AUGUST 2005 Transactions of the ASME

5. sion Moving Average ARMA͑n,m͒ model. The ARMA model
takes a set of data registrations and recasts it into a discrete, re-
cursive, linear stochastic format:
Xt − ␾1Xt−1 − ␾2Xt−2 − ... ... ... ... ... ... . − ␾nXt−n = at
− ␪1at−1 − ␪2at−2 − ... ... ... ... ... ... − ␪mat−m ͑3͒
where ͑Xt͒ denotes the state parameter at instant ͑t͒, ͑at͒ denotes
the residuals zero mean “white noise” term, ͑␾i͒ is the autoregres-
sive coefficients at i=1,2,3, ... ,n and ͑␪j͒ is the moving average
coefficients at j=1,2,3, ... ,m. The ARMA model usually ex-
presses the dependence of one variable on its own past values, or
the effect of some disturbances at’s on the behavior of subsequent
values of the variable. A disturbance affecting a system lasts a
certain period depending on its dynamic damping resistance. Ad-
equate ARMA͑n,m͒ is usually obtained by fitting higher-order
͑n,m͒; ͑mϾn−1͒, models and applying the checks of adequacy.
This is carried out in steps by increasing the order by two. The
model is usually judged through the reduction in the residuals-
sum of squares ͑RSS͒ and the F-value ͓10͔.
F =
ΆͫA1 − A0
S
ͬ
ͫ A0
N − r
ͬ ·= F͑S,N − r͒ ͑4͒
where ͑A0͒ is the smaller ͑RSS͒ of the ARMA͑2n+2,2n+1͒
model, ͑A1͒ is the larger ͑RSS͒ of the ARMA͑2n,2n−1͒ model,
F͑S,N−r͒ denotes the F-distribution with ͑S͒ and ͑N−r͒ degrees
of freedom, r=͑2n+2͒+͑2n+1͒=4n+3, S=number of additional
parameters in the higher-order model and N=number of observa-
tions. Estimation procedures start with n=0 which yields an
Table 3 ARMA results for subtest b
Table 4 ARMA results for subtest c
Table 5 ARMA results for subtest d
Journal of Manufacturing Science and Engineering AUGUST 2005, Vol. 127 / 467

6. ARMA͑2,1͒ model, then n=1 which gives ARMA͑4,3͒ and so on.
Procedures are terminated once an adequate model is obtained.
The ARMA modeling procedures are carried out for each sub-
test in turn using special software in association with the ͑SPSS͒
statistical computer package. Results are shown in Tables 2–7
with the adequate model in the last column. The corresponding
͑RSS͒ and F-value are at the last two rows of each table. Gener-
ally, as wear level increases, higher-order autoregressive models
are found necessary to adequately fit the data. As shown in Tables
3–7, these adequate models are AR͑2͒, ARMA͑3,1͒, ARMA͑3,3͒,
ARMA͑4,1͒, and ARMA͑4,1͒, for subtests b, c, d, e, and f, respec-
tively. Again, three levels may be distinguished: the first at subtest
b, the second at subtests c and d, and the third at subtests e and f.
While the moving average parameters ͑␪’s͒ are not affected, the
autoregressive parameters ͑␾’s͒ are always greater as the wear
level advances. This reflects the strong dependence of data on its
preceding values, where the tool wear presents a continuous
analogous and dependent disturbance to the signals. Nevertheless,
Table 6 ARMA results for subtest e
Table 7 ARMA results for subtest f
Table 8 RSS reduction due to ARMA modeling
468 / Vol. 127, AUGUST 2005 Transactions of the ASME

7. a higher-order ARMA͑4,2͒ is found adequate to fit the data of the
first subtest even though of a low wear level ͑Table 2͒. This is due
to the discontinuous nature of tool wear ͑chipping͒ associated
with the rapid initial wear rate. However, the low model param-
eters and the slight ͑RSS͒ reduction imply that it is just on the
boundary of the domain and an ARMA͑2,1͒ can be considered
adequate without sacrificing accuracy. Another exception is that a
larger number of moving average parameters result for subtest d
͑Table 5͒. As explained earlier, this is due to the dynamic effects
from both proceeding and local considerations in addition to being
just entered the high wear rate zone. The high wear rate originates
at the tool’s nose area of the second half of the subtest ͓Fig. 3͑d͔͒.
This affected the radial force component Fz only, Shortly follow-
ing that, the tool failed by thermal softening leading to a simulta-
neous increase in both the feed Fx and the radial Fz components
and, consequently, in the thrust Fxz component. Table 8 indicates
how much of the deterministic ͑RSS͒ reduces after the develop-
ment of the adequate final ARMA model. The ͑RSS͒ reduction
represents the part of the stochastic component induced by the
associated amount of tool wear and indicates a similar quantitative
Fig. 4 Tool dynamic characteristics under wear variation of Green’s function within a domain of 100 disturbances „j=1–100…
Fig. 5 Three-dimensional global view of dynamic characteris-
tics of the whole test, variation of Green’s function with 100
disturbances
Journal of Manufacturing Science and Engineering AUGUST 2005, Vol. 127 / 469

8. trend of variability between final higher order adequate model and
the ARMA͑2,1͒. For the first three subtests a, b, and c, the RSS
increased proportionally. However, at a point around the practical
critical wear ͑between 0.3 and 0.35 mm͒, the RSS suddenly drops
to its minimum value and then, significantly increases as wear
enters its final softening, where approximately 90% of variability
is attributed to wear.
4 Relationship Between Tool Wear and Dynamic
Changes in the Cutting Process Subsystem
According to the conclusion previously drawn, a cutting pro-
cess subsystem can be expressed by a second-order one-degree of
freedom dynamic system. Such a system may be expressed by an
ARMA͑2,1͒:
Xt − ␾1Xt−1 − ␾2Xt−2 = at − ␪1at−1. ͑5͒
A characteristic equation is
␭2
− ␾1␭ − ␾2 = 0 ͑6͒
with roots,
␭1,2 =
1
2 ͕␾1 ± ͱ␾1 + 4␾2͖ ͑7͒
However, machining stability is determined by satisfaction of
the following constraints:
΄
␾1
2
+ 4 ␾2 ജ 0
␾1 + ␾2 Ͻ 0
␾2 − ␾1 Ͻ 0
΅ ͑8͒
4.1 Green’s Function of the Force Signals. One of the pa-
rameters that describe the dynamics features of the system is the
Green’s function. It explains how the disturbances ͑at’s͒ affect, or
influence, the response ͑Xt͒ by expressing the response as a linear
combination of at’s. For ARMA͑2,1͒, the Green’s function ͑Gj͒
can be expressed in the characterized form
Gj = g1␭1
j
+ g2␭2
j
͑9͒
in which
g1 = ͫ␭1 − ␪1
␭1 − ␭2
ͬ and g2 = ͫ␭2 − ␪1
␭2 − ␭1
ͬ ͑10͒
Table 9 Numerical illustration of response generation of periodical disturbances using Green’s
function „subtest a… „␾1 =1.143, ␾2 =−0.151, ␪1 =0.967…
Table 10 Numerical illustration of response generation of periodical disturbances using Green’s function
„subtest b… „␾1 =0.818, ␾2 =−0.01, ␪1 =0.541…
470 / Vol. 127, AUGUST 2005 Transactions of the ASME

9. To demonstrate the dynamic characteristics throughout the
tool’s working lifetime ͑six subtests͒, Green’s function values
͑Gj͒, j=0,1, ... ,100, are considered based on Eqs. ͑7͒–͑11͒ and a
graphical representation is shown in Figs. 4͑a͒–4͑f͒. Additionally,
a 3D global representation for the same results is shown in Fig. 5.
The Green’s function starts with a unit value at j=0, implying a
steady state stable conditions. Then, its further behavior usually
relies both on the severity of the current disturbances and the
permanent impact remaining from the preceding disturbances. At
lower wear levels, subtests a, b, and c, Fig. 3, the tool dynamic
oscillations decay rapidly, reaching a maximum damping resis-
tance, or minimum amplitude. At subtest e, system shows a sig-
nificant instability level or, very low damping resistance. This
trends continues through subtest f at which tool catastrophically
fails, Fig. 3͑f͒.
Through the first three subtests, the tool’s dynamic characteris-
tics are almost unchanged especially at subtests a and b since wear
is within the low rate level and, therefore, there is no severe past
or current disturbances. As a result, oscillations die out rapidly
revealing a higher damping resistance. Throughout that interval,
the tool indicates one-sided positive oscillations due to edge
acuteness and inherent low friction. In subtest e, Figs. 3͑e͒ and
4͑e͒, the tool is set into total instability ͑two-side oscillations͒ due
to the inherent very high wear rate in addition to the remaining
disturbances was initiated in the preceding subtests. This explains
that the tool still strives to retain some of its hardness, hence
resisting the imposed fluctuating friction stresses. In subtest f,
Figs. 3͑f͒ and 4͑f͒, the edge starts to deteriorate gradually and
rapidly losing much of its material allowing an intimate contact
between the tool and the rotating workpiece, hence constraining
the tool to vibrate in one direction only. This is shown in Fig. 4͑f͒
by the undamped positive values of the Green’s function.
4.2 Response Generation Using Green’s Function. A mea-
sure, which can be used to determine the system dynamic charac-
teristics and behaviors, represented by the response amplitude
͑Xt͒, is obtained by evaluating the system’s response as a reaction
to regular disturbances ͑at͒. A dynamically stable system with no
sudden variability is expected to behave in such a way that its
output response amplitude ͑Xt͒ is with a similar pattern to the
excited force. Deviation from such a state is usually attributed to
some external effects such as tool wear or, to a change in the
system’s dynamic features.
To extract the tool’s dynamic behavior under progressive tool
wear, the system response ͑Xt͒ of the ARMA͑2,1͒ can be gener-
ated according to relation
Table 11 Numerical illustration of response generation of periodical disturbances using Green’s func-
tion „subtest c… „␾1 =1.134, ␾2 =−0.139, ␪1 =0.923…
Table 12 Numerical illustration of response generation of periodical disturbances using Green’s function
„subtest d… „␾1 =0.846, ␾2 =−0.151, ␪1 =0.999…
Journal of Manufacturing Science and Engineering AUGUST 2005, Vol. 127 / 471

10. Xt = ͚j=0
j=ϱ
Gjat−j = ͚j=0
j=t
Gt−jaj ͑11͒
In this section two cases of system behaviors, represented by ͑Xt͒,
are presented. The first is when the system is subjected to periodi-
cal disturbances, while the other is when random disturbances are
applied.
A periodical unit disturbances ͑at’s͒ has been fed into the
Green’s function and hence, the response ͑Xt͒ of the ARMA͑2,1͒
at a given interval is generated according to Eq. ͑11͒ as the sum-
mation of the product of the disturbances and the Green’s func-
tion, the last row in Tables 9–14. The dynamic characteristics of
each subtest in turn are evaluated by superimposing the system
response ͑Xt͒ on the disturbance as shown in Figs. 6͑a͒–6͑f͒. For
the first four subtests ͑a–d͒ where lower wear levels are observed,
the system response resembles the excitation signal in both mag-
nitude and direction. However, in subtests e and f, at the onset of
the tool’s softening stage, the response amplitude ͑Xt͒ varies in
such a way that system oscillates in a one-sided positive direction.
As concluded earlier, at the softening failure stage, the tool is in
tight contact with the workpiece as the friction area widens. Sig-
nificant tool edge material is lost and, hence, the sharp edge van-
ishes preventing tool-workpiece penetration mechanism to con-
tinue, and, this allows the tool to oscillate in only one direction
outward the workpiece, or in the positive vertical ͑power͒
direction.
Since, in practical machining, periodical disturbances are less
likely to represent the system, a second case is introduced where
random disturbances are assumed. Data and analysis of the case
are listed in Table 15 and graphically plotted in Fig. 7. As shown
by Figs. 7͑a͒–7͑f͒, the system behaves differently at different wear
levels, although it is subjected to the same random disturbances.
Results may be judged considering two mathematical factors: the
weight and the distribution of the area underneath response curve.
At lower wear values, Figs. 7͑a͒–7͑c͒, some facts are evident, they
are: the amplitude of the system response never exceeds the dis-
turbance amplitude and response is almost evenly distributed
around abscissa. The influence of any disturbance at instant j has
a local effect, and is not transferred to subsequent intervals reveal-
ing that there is no memory effect from previous events. For sub-
test d, Fig. 7͑d͒, response area grows on due to the conditions
described earlier. However, oscillations are evenly distributed
around the abscissa revealing that the system is still with a sig-
nificant rigidity to resist disturbances. However, at elevated wear
Table 13 Numerical illustration of response generation of periodical disturbances using Green’s function
„subtest e… „␾1 =−0.008, ␾2 =0.984, ␪1 =0.253…
Table 14 Numerical illustration of response generation of periodical disturbances using Green’s function
„subtest f… „␾1 =−0.012, ␾2 =0.939, ␪1 =−0.126…
472 / Vol. 127, AUGUST 2005 Transactions of the ASME

11. and wear rate the system has less damping resistance so that dis-
turbances occurring at some previous event are not easily forgot-
ten. For subtest e, Fig. 7͑e͒, the system behaves after j=3 as it is
affected by event at ͑j-2͒. At j=6 and at=0, the response ͑Xt͒ is
͑2.759͒. Although the preceding impact was ͑−2͒, that positive
value indicates that the system is still affected by the positive
impact at j=4. A similar response may be observed in subtest f,
Fig. 7͑f͒. Data case indicates that the machining system is dy-
namically affected not only by the instantaneous condition but
also by the past accumulated tool deformation modes throughout
its service life. When the plastic deformation zone is reached, the
system exhibits a dominating instantaneous effect that may hide
what is left in the system’s memory. This aspect may be observed
from data shown in Fig. 7͑f͒ at j=9–11 as a negative response
area.
A general overview of the last case is shown in Fig. 8, where
the response surface is generated for the whole test ͑six subtests͒.
While a stable response is noticed within the area that is charac-
Fig. 6 Response generation using Green’s function with periodical impacts
Table 15 Summary of dynamic response values at different test stages using random
disturbances
Journal of Manufacturing Science and Engineering AUGUST 2005, Vol. 127 / 473

12. terized by the conditions of subtests ͑a–c͒, higher and deeper
peaks are observed for subtests ͑e and f͒. Also, the graph clearly
shows the dynamic characteristics of subtest d.
5 Conclusions
In this work, an approach is proposed to relate the state of the
tool and its wear state with the variation encountered in the sto-
chastic stationary component ͑residuals͒ of the cutting force sig-
nals. ARMA analysis has been used to obtain significant and ad-
equate models at various levels of the progressive tool wear.
Models are post-processed using the “Green’s function” to extract
information about the tool dynamic behavior at various tool’s de-
formation and wear modes. The principal conclusions are as fol-
lows:
͑1͒ Only the random stochastic stationary part of the force sig-
nals is proven to carry most variability equivalent to the
severity of tool progressive wear.
͑2͒ The parameters of the adequate ARMA models are found to
reflect the tool state where models with higher order of
autoregressive parameters are found for higher wear levels.
͑3͒ To avoid complexity of calculation when applied in any
Fig. 7 Response generation using Green’s function with random impacts
Fig. 8 3D response surface of the whole test using Green’s
function with random impacts
474 / Vol. 127, AUGUST 2005 Transactions of the ASME

13. online monitoring technique, ARMA͑2,1͒ is recommended
to represent the cutting process subsystem with a reason-
able accuracy. Based on ARMA͑2,1͒, the dynamic charac-
teristics variation due to wear is explained through its
Green’s function. The Green’s function analysis indicates
that at low wear level, system stability is maintained. How-
ever, it manifests a different trend at elevated wear level
especially at that onset of the plastic deformation zone. An
undamped unidirectional damping resistance is observed.
͑4͒ A numerical method is introduced and discussed which can
be used to explain how the variable dynamic characteristics
of a system may be monitored using its output data provid-
ing there is a prior knowledge about its Green’s function.
The analyses clearly demonstrated the ability of the ap-
proach to accurately detect the onset of the plastic defor-
mation zone.
The proposed approach may be utilized in an integrated
monitoring and control system for implementation of a tool
change strategy in automated machining systems. System
dynamic characteristics are in process defined and, exam-
ined at regular working intervals, used to ensure an efficient
performance. As an early warning approach, system perfor-
mance may be compared at different conditions by the ac-
tivation of its Green’s function or by observing its response
behavior when some disturbances are injected into its char-
acteristics equation.
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