Chipping of cutting tools usually occurs when a tool is exposed to
cyclical stresses, i.e. in intermittent machining processes such as
milling. It is a strictly nonlinear process, and sensor signals
demonstrate its stochastic behaviour. A signal analysis (Lessard, 2006)
in the frequency domain could result in relations connecting fast
process changes and signal changes. Since such relations are not known a
priori, it is necessary to perform estimation. Methods to be used for
the spectra estimation could be parametric and nonparametric (Lijoi et
al., 2007). While parametric methods are based on a process model,
nonparametric methods are based on the calculation of autocorrelation
function, and the calculation of the Fourier transform. Two problems
should be pointed out here: the amount of available data is not
unlimited, and the data is often corrupted by noise, or contaminated
with an interfering signal. Hence, the goal of a spectral estimation
based on a finite set of data is to describe the distribution (over
frequency) of the power contained in a signal. Among possible
nonparametric methods for the estimation of power spectral density of a
random signal, a periodogram and the MATLAB Signal Processing Toolbox
(MATLAB user's guide) are applied study.
A periodoram is based on the fact that the power spectral density
of a stationary random process is a Fourier transform of the
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
The autocorrelation function of an ergodic process in the limited
interval n=[0, N-1], k=[0, N-1] is calculated as a finite sum, (Marie,
2002) eq. 2:
[??].sub.x](k) = 1/N [N-1-k.summation over n-0]_x(n +
With the discrete Fourier transform of autocorrelation function we
obtain the estimation of power spectral density, which is a periodogram
defined in eq. 3:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
To express the periodogram directly from the process x(n), we
multiply x(n) with the rectangular window [[omega].sub.R](n), thus
limiting x(n) to interval [0, N-1]. The result is the process x(n)
defined in interval [0, N-1] and its autocorrelation function. The
application of the Fourier transform results in the final expression for
the periodgram, eq. 4, where [X.sub.N]([e.sup.j[omega]]) is the
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
discrete Fourier transform of the process [x.sub.N](n).
A periodogram can be used for the spectral estimation of the signal
structure. Some of the properties of the periodogram are: omission of
the spectrum, resolution, partiality, and consistency, .
The approximation of the power spectrum by a periodogram requires
the application of time windows. The basic shape of the window is
rectangular, but there is a number of other set and changeable window
shapes . In this study, Welch's method is used to improve the
periodogram. The method is based on the division of a set of data into
segments (which may overlap). Subsequently, a modified periodogram is
calculated for each segment (it is possible to use different windows for
each segment, i.e. a modified periodogram). Finally, the mean value of
the obtained estimates of the power spectral density is calculated.
3. EXPERIMENT PLANNING
The method of nonparametric estimation of the spectrum power is
applied for the estimation of the tool chipping process in the milling
of CK 45 steel. The parameters of the machining process were constant.
The process of milling was carried out to the point when the tool became
worn. In one of the performed experiments, tool chipping was noted and
the current, acoustic, and force signals were recorded. In each pass,
the tool wear parameter VB was measured. Forces were measured by a
Kistler three-component force sensor. An acoustic sensor was put close
to the site where the milling process was carried out, and signals of
the current were taken directly from the control unit (Mulc et al.,
2004), Fig. 1.
[FIGURE 1 OMITTED]
In order to reveal the signal structure, a detailed analysis of the
structure of force, current and acoustic signals was carried out. Figure
2 shows the diagrams of forces in which the occurrence of tool edge
chipping can be easily noted. The change is abrupt and hard to detect in
the time domain.
[FIGURE 2 OMITTED]
Changes in the signal of the current are similar to those in force
signals: they are visible but they do not enable us to draw the right
conclusion on the degree of wear.
The power contained in the acoustic signal increases significantly
with the degree of wear of the tool and it exhibits periodic behaviour.
The influence of the signal stochastic behaviour makes a timely
determination of the basic state more difficult. A periodogram was used
in this study to perform a spectral analysis of the signal. A
coefficient that expresses the area below the periodogram curve has been
selected. The results of the experiment are given in Tab. 1.
[FIGURE 3 OMITTED]
The analysis of the results shows that the occurrence of chipping
could be recorded by means of power coefficients, and thus, the chipping
process could be stopped before the tool breakage.
The force signal in the y-axis direction and the current signal in
the x-axis direction exhibit the biggest change in the power coefficient
of the spectral range. Therefore, a more detailed analysis of the
periodogram structure for these two signals has been performed. The
frequency range which reacts to changes in the cutting edge wear and to
phenomena caused by abrupt changes in chipping or by tool breakage has
to be determined. The diagrams in Fig. 3 show changes in the current
spectrum in the x-axis direction. One can see that the frequency
structure of the current signal is maintained with small changes in the
fall in amplitude when the tool cutting edge is worn, while the
frequency structure of the signal is disturbed when tool chipping and
tool breakage occur. At the same time, the power contained in the signal
is increased in the frequency range with a higher degree of wear, which
can be a good indicator for the initialization of chipping.
A successful design of the monitoring process of machining when
tool chipping occurs requires the knowledge of recorded spectra of
signals and the spectra of their power. In this study, a nonparametric
estimate of the power spectrum in the monitoring process of cutting tool
edge chipping is given. A periodogram was used, and the area below the
curve was used as a coefficient of comparison. The biggest changes in
the coefficient, which can be used for the detection of the
initialization of the tool chipping, occurred on the force signal in the
y-axis and on the current signal in the x-axis direction. This proves
that a periodogram can be a satisfactory estimator in particular
situations. The periodogram can be improved by different procedures of
window optimization (Thomson, 1998). There is not "the best"
method; rather, the selection of "the best" method depends on
the signal and on the estimation parameter. In addition, a relatively
large set of data is required for the application of nonparametric
methods in order to obtain as good results as with parametric methods.
Further research conducted by means of spectral analysis estimation
would follow a process of a more detailed description of the machining
process and an analysis of different conditions of the chipping process.
Lessard, C. S. (2006). Signal processing of random physiological
signals, Morgan & Claypool, ISBN: 9781598290387, Texas A&M
Lijoi, A; Mena, R. & Prunster, I. (2007). A Bayesian
nonparametric method for prediction in EST analysis, BMC Bioinformatics
Mulc, T.; Udiljak, T.; Cus, F. & Milfelner, M. (2004).
Monitoring Cutting-Tool Wear Using Signals from the Control System.
Journal of Mechanical Engineering, 50 (2004), 12; 568-579, ISSN:
Thomson D. J.: Multiple-Window Spectrum Estimates, Proceedings of
Highlights of Statistical Signal and Array Processing, pp. 344--347,
IEEE SP Magazine, June 1998., Portland, OR, USA
Signal Processing Toolbox for use with MATLAB, User's Guide,
Version 7.10, The MathWorks, Inc., 2010, http://www.
Maric, V. (2002). Nonparametric spectrum estimation (in
Croatian),Available from: http://spus.zesoi.fer.hr/projekt/
2001_2002/maric/estimacija.htm, Accessed: 2010-05-17
Tab. 1. Power coefficient in the frequency range
Area below the frequency spectrum
PSD Power Worn tool
spectral density Sharp tool VB=0.6 mm Chipping
Force Fx 1,9800 0,5488 1,4380
Fy 3,0410 0,9609 8,2550
Fz 0.5081 0,1570 1,0546
Current Ix 0,1385 0,1340 4,7485
Iy 11,6380 21,2650 9,3290
Is 131,9700 151,5500 105,9000
Acoustic AE 0,0506 2,1707 0,7911
AERMS 4,9107 25,0000 23,6650