This paper investigates the diagnosis of cutting tools in a milling operation using vibration signals and proposes a signal processing algorithm to achieve that. In the proposed algorithm, the impulse response of the measured vibration signal is firstly identified using the random decrement technique. This is then converted to a cepstrum and subtracted from the measured signal in the quefrency domain using the additive properties of cepstra. The residual signal representing the forcing function is then transformed back into the time domain using the inverse cepstrum. Finally the power spectral density is estimated, and a comparison is made between the different states of the cutting tool. For a good estimation of the force, four measurement points are used, and the identified excitation sources are then averaged. By comparing the spectra of the forcing functions, the efficiency of the method is demonstrated, and the faulty case is clearly distinguished from the fault-free case. This was not the case with the original response signals

Badaoui, M. E.

Guillet, F.

Randall, R. B.

Sghir, K. A.

Ait Sghir, K.

Rb., Randall

El Badaoui, Mohamed

Guillet, François

HAL-UJM

milling cutting tool diagnosis using comparisons of the excitation identified by cepstral techniques

University of Queensland eSpace

5th Australasian Congress on Applied Mechanics, ACAM 2007  
10-12 December 2007, Brisbane, Australia 
MILLING CUTTING TOOL DIAGNOSIS USING COMPARISONS OF THE 
EXCITATION IDENTIFIED BY CEPSTRAL TECHNIQUES 
 
K. Ait Sghir1,2, R.B. Randall2, M. El Badaoui1 and F. Guillet1 
 
1LASPI, IUT de Roanne, Jean Monnet University, 20 avenue de paris, 42334, Roanne, France 
2School of Mechanical and Manufacturing Engineering, University of New South Wales 
Sydney, NSW 2052, Australia 
 
Abstract 
 
This paper investigates the diagnosis of cutting tools in a milling operation using vibration signals and 
proposes a signal processing algorithm to achieve that. In the proposed algorithm, the impulse 
response of the measured vibration signal is firstly identified using the random decrement technique. 
This is then converted to a cepstrum and subtracted from the measured signal in the quefrency 
domain using the additive properties of cepstra. The residual signal representing the forcing function is 
then transformed back into the time domain using the inverse cepstrum. Finally the power spectral 
density is estimated, and a comparison is made between the different states of the cutting tool. For a 
good estimation of the force, four measurement points are used, and the identified excitation sources 
are then averaged.  By comparing the spectra of the forcing functions, the efficiency of the method is 
demonstrated, and the faulty case is clearly distinguished from the fault-free case. This was not the 
case with the original response signals.  
Keywords: Random decrement, complex cepstrum, blind identification, diagnosis, milling cutting tools. 
 1 Introduction 
The identification of the state of a cutting tool is important in any metal cutting process; as additional 
cost - in terms of scrapped components, machine tool breakage and unscheduled downtime - occurs 
as a result of tool defects. Tool wear and breakage are the most encountered defects in the machining 
process [1]. Several factors such as cutting conditions, workpiece surface and tool geometry make the 
condition monitoring of the tool complex; especially when developing an effective system to monitor 
the early stages of the fault in the tool-tip. Several parameters have been proposed for the condition 
monitoring. Among these are the cutting forces, vibration signatures, temperature and acoustic 
emission. For each of these parameters, several features have been tried such as the power spectral 
density (PSD), kurtosis, etc. 
Acoustic emission proved efficient for detecting the tool breakage but less suitable for monitoring the 
tool wear [2] . The measurement of cutting forces requires special mounting fixtures [2]. The vibration 
signal has been proved effective in the diagnosis of tool wear monitoring and is easy to acquire [2, 3]. 
However, the different transmission paths distort the signals at different measurement points.  
The identification of the transmission path/excitation is one of the methods that have been used for the 
diagnosis of the cutting process. In [4] a blind identification method based on the bicepstrum was used 
to estimate the characteristics of the precision turning process. In this current work a vibration signal 
approach is proposed to identify the excitation source and to monitor the tool state. 
The paper is organized as follows. In section 2, we  introduce the blind identification method and 
explain its essentials. In section 3, the experimental measurements and the ir analysis are discussed. 
Finally, the conclusion and perspectives are given in section 4. 
2. Theoretical background: 
2.1 Random Decrement technique: 
The Random-Decrement (RD) is a signal processing technique which transforms the response of a 
resonant system to random excitation, into its impulse response. It was developed by H.A Coles at 
NASA during the late 60s and early 70s [5] in order to detect space structure damage from the 
measured response. Since then, it has been applied to a wide variety of structures subjected to un-
measurable ambient excitations, to extract the modal parameters and eventually to detect failures. 
  
The technique was later given a mathematical basis. Let X(t) be a stochastic process, the RD function 
is defined as the mean value of a stochastic process on condition, T, of the process itself |6]:  
 
[ ])()()( tXXX TtXED tt +=                                                          (1)  
The condition T is the triggering condition. 
In order to estimate the conditional mean value correctly from a single observation it is necessary to 
assume that the stochastic process is not only stationary but also ergodic. In this case the RD function 
can be estimated as the empirical conditional mean value from a single realization: 
å
=
+=
N
i
txiXX i
Ttx
N
D
1
)()(
1)( tt                                                      (2)  
Where N is the number of points in the process which fulfils the triggering condition and x(t) is a 
realisation of X(t). The triggering condition used in this work is { })(0)( 1)( 1 iitx txtxT i >>= ++ , ie a 
positive going zero crossing. 
In damage failure detection the basic idea is often that an incipient failure will change the stiffness and 
the damping characteristics of the structure. In this case, the tool tip damage has little effect on the 
system properties (except possibly damping) but primarily changes the forcing function. The RD 
functions can be used to remove the different transfer functions to obtain several estimates of the 
single forcing function which can be averaged.  
2.2 Complex Cepstrum: 
Cepstral analysis is a technique that transforms a convolution to an additive relationship [7]. The 
complex cepstrum is defined by: 
))]([log()( 1 fXFFTc x
-=t                                                                (3)  
Where )( fX is the Fourier transform of x (t) and 1-FFT is the inverse Fast Fourier Transform [8].  
We apply (3) to the relationship between the input e(t) (excitation source) and output x(t) for a Single 
input Multiple output (SIMO) system: 
)(*)()( thtetx =                                                                  (4) 
Where h(t) is the impulse response of the system for that response point. 
The complex cepstrum of (4) transforms the convolution in the time domain into an addition in the 
quefrency domain (also having dimensions of time) [7] as: 
)()()( ttt hex ccc +=                                                               (5) 
As seen from (5) the source and path phenomena are additive. The excitation force can be recovered 
very easily if the system impulse response estimates are available. The cepstrum has proved very 
useful in condition monitoring and fault diagnostics of mechanical systems, especially in the case of 
operating machines where periodic excitation sources exist, such as gear meshing forces, and milling 
cutting forces Two applications are widely used in condition monitoring and fault diagnostics: 
extraction of vibration features corresponding to given mechanical faults, and recovery of the vibration 
sources and transmission paths [8].  
2.3 The schematic presentation of the algorithm:  
Figure 1 illustrates the flow chart of the blind identification process that is used in this work. The 
impulse responses )( thi of the measured vibration signals )(txi  (acceleration) are firstly identified 
using the random decrement technique (the length of the sequence is padded with zeros to have the 
same size as the original measured signal x(t)). This impulse response is then transformed into the 
quefrency domain )( t
ih
c , as is the measured signal )(txc , by means of the complex cepstrum. In this 
domain the system properties and the excitation are additive and can then be separated by a simple 
subtraction. The transfer function is subtracted from the signal and the remainder )( tiec is 
transformed back to the temporal domain )( tei by the inverse complex cepstrum. The process 
presented in the figure 1 is calculated for each measurement point and the excitation forces obtained 
are then averaged to obtain an improved estimation of the excitation force. 
  
Finally, by using PSD comparison the tool cutting state can be diagnosed.  
 
Figure 1 Schematic presentation of the identification algorithm. 
  
3 Experimental results: 
3.1 Test rig: 
A test rig to generate cutting data from a milling operation was prepared. Four accelerometers (gain 
100mv/g), were placed in three mutually perpendicular directions [x direction, y workpiece, -y, z 
direction as shown in figure 2]. An optical encoder to enable the angular sampling was installed in the 
spindle, (figure 2). It delivers a  position information (squared signal at frequency 2500*fr, where fr is 
the rotational speed) which is used as a clock for the data acquisition card. Therefore, signals were 
sampled at constant angle intervals. Note that while it was important to have a fixed number of 
samples per rotation of the cutting tool, the speed was very constant, so the rotation axis still 
corresponded to a time axis and preserved natural frequencies. 
 
 
Figure 2 Schematic presentation of the experimental setup. 
 
Experiments with the milling cutting tool were performed for one minute of milling. The face milling 
cutter had 5 unequally spaced teeth. The cutting parameters and the operating conditions were kept 
constant during the experiment (see table 1).  
)()()( ttt hex ccc +=  
Random Decrement 
And zero padding 
Complex Cepstrum 
)( thi  
)( tiec  
Inverse Complex Cepstrum )( tei  
)( t
ih
c  
)(*)()( thtetx iii =  
Complex Cepstrum 
 
  
In the experimental analysis, we have considered three tool cutting states:  without fault, with two worn 
teeth (0.2mm) and with a broken tooth.    
Table 1 Cutting conditions and angular sampling parameters. 
 
Material of the specimen  steel 
Rotation speed 650 rpm 
Feed rate 220 mm/min 
Milling cutter diameter  100mm 
Number of teeth   5 
Cutting depth 0.7mm 
Optical encoder resolution 2500 points per revolution 
Number of recorded cycles 500 
Number of samples 1250000 samples 
Averaged sampling rate 27 kHz 
Anti-aliasing filter 9 kHz 
3.2 Analysis of the results: 
Figure 3 presents three cycles of the signal from the four sensors in the fault free case. It shows the 
repetition of the peaks that correspond to each tooth when it enters in the workpiece. This repetition 
shows the cyclostationary nature of the milling vibration signal taken under angular sampling. It shows 
that the magnitude of each sensor signal is different from the others. The signal number 2 has a large 
magnitude because the sensor is arranged in the cutting force direction.  
0 1000 2000 3000 4000 5000 6000 7000 8000
-20
0
20
Accelerometer 1
A
cc
el
er
at
io
n 
m
/s
²
Angle
0 1000 2000 3000 4000 5000 6000 7000 8000
-200
0
200
Accelerometer 2
Angle
A
cc
el
er
at
io
n 
m
/s
²
0 1000 2000 3000 4000 5000 6000 7000 8000
-100
0
100
Accelerometer 3
Angle
A
cc
el
er
at
io
n 
m
/s
²
0 1000 2000 3000 4000 5000 6000 7000 8000
-50
0
50
Angle
A
cc
el
er
at
io
n 
m
/s
² Accelerometer 4
 
Figure 3 Typica l milling signals for different sensors 
 
Figure 4 presents a segment of the excitation signal identified from sensor 4 by the process presented 
in figure 1, the arrows indicate the excitation impulsion, it is seen that it consists of an impulse that 
represents the force. This is a sample of the identification procedure applied on the entire signal.  
The objective of this procedure, as indicated in the introduction, is the diagnosis of different states of 
the cutting tool. To achieve this, all identified excitation forces are averaged and then the PSD is 
applied. The identified forces were much more similar than the responses at each point. 
Figure 5 shows the PSD computed for the measured response signals averaged for all the sensors. 
Here the fault-free case and the case with a broken tooth are presented. As can be seen in this figure, 
there are many peaks which correspond to the system resonances. For these two cases presented it 
  
 
Figure 4 Typical Identified signal from sensor 4, top the identified signal, down the original signal   
 
is clear that there is no significant difference detected from the response signals.  The conclusion is 
that the direct application of PSD analysis to the averaged measurements does not give any 
information about the fault because of the distortion given by the different transfer paths.  
 
 
Figure 5 PSD of the averaged signal for different sensors, for the fault-free case and a broken tooth 
 
Figure 6 shows the power spectral density computed for the averaged identified excitation force. Note 
that the identified force spectra have a flattened characteristic that is typical of a white excitation 
because of the removal of the transfer function information. There are some peaks in the low 
frequency region, but the rest is quite uniform. 
Also it is clear that there is a big difference between two of the cases in the frequency range [2-4 kHz] 
which corresponds to the main system resonances in Fig. 5. This difference has a maximum in the 
range [2000-2500Hz], for a broken tooth of more than 10dB, which is very significant. For two worn 
teeth, the maximum difference is 6dB.  The conclusion is that this method enables making a distinction 
between faulty cases and the fault-free case. In [9] higher order statistics are used for the diagnosis, 
but the inconvenience with that method is the large computational burden. The current method does 
not need to employ higher order statistics and is therefore of considerable interest in condition 
monitoring. 
  
 
Figure 6 PSD of the average identified signal for different sensors, for the free fault, two worn teeth 
and a broken tooth 
4 Conclusion and perspective: 
In this work a blind identification method is proposed, it consists of the use of the random decrement 
for impulse response estimation and the cepstrum to separate the common excitation force from these 
different impulse responses. As seen by the results of the application to milling cutter signals, an 
ability is shown to distinguish between the faulty cases and the fault-free case. once the differences in 
the zone dominated by the system responses are removed. It has been seen that for the case of a 
broken tooth the difference in dB is more than 10dB, whereas in the case of two worn teeth the 
difference of 6 dB is just significant. On the other hand it is seen that there is no significant diffe rence 
when the measurement signals are directly used. Therefore, this method of identification can be used 
for the diagnosis of milling cutting tools.  
The perspective also exists to use other more advanced techniques that make use of the 
cyclostationary properties such as spectral correlation, and synchronous average, since the signal is 
sampled in the angular domain with a fixed number of samples per cutting cycle. 
References 
[1] D. E.Dimla Snr, Sensor signals for tool-wear monitoring in metal cutting operations- a review of 
methods, Int. J. of Machine Tools & Manufacture 40 (2000)1073-1098. 
[2] T. I. El-Wardany, D. Gao and M. A. Elbestawi,Tool condition monitoring in drilling using vibration 
signature analysis, Inter. Journal of Machine Tools & Manufacture 36, No.6, (1996) pp,687-711,. 
[3] D. E. Dimla, Snr. The Correlation of Vibration Signal Features to Cutting Tool Wear in a Metal 
Turning Operation, Int J Adv Manuf Technol (2002) 19:705713 
[4] A. Iturrospe, D. Dornfeld, V. Atxa, J. M. Abete, Bicepstrum based blind identification of the 
acoustic emission (AE) signal in precision turning. Mechanical Systems and Signal Processing, 19 
(2005) 447-466. 
[5] H. A. Coles, On-line Analysis of Random Vibrations, AIAA Paper No.68-288,AIAA/ASME 
Structures, Structural Dynamics and Materials Conference, Palm Springs,1968. 
[6].C Asmussen, Modal analysis based on the random decrement technique, Application to civil 
engineering structures, thesis 1997, University of Aalborg, Aalborg, Denmark. 
[7] A.V Oppenheim, RW. Schafer, Discrete time signal processing, Englewood Cliffs, NJ: Prentice-
Hall. 
[8] R. B. Randall, Cepstrum Analysis,  Encyclopedia of Vibration (Eds. S. Braun, D.J. Ewins, S.S. 
Rao).  Academic Press, London  (2001). 
[9] K. Ait-sghir, M. El Badaoui, M. Bakrim, F. Guillet and D.Aboutajdine, Milling cutting tools diagnosis 
with higher order cyclic spectra, 14th International Congress on Sound and Vibration, Cairns, 
Australia, 09-12 July 2007. 


Milling cutting tool diagnosis using comparisons of the excitation identified by cepstral techniques

https://hal.archives-ouvertes.fr/hal-00552211

Milling cutting tool diagnosis using comparisons of the excitation identified by cepstral techniques

Abstract

Similar works

Full text

Available Versions

HAL-UJM

University of Queensland eSpace