New Indicators Based on Cyclostationarity Approach for Machining Monitoring M. LAMRAOUI1,2 , M. THOMAS 1 , M. EL B ADAOUI2 , I. ZAGHB ANI1 ,V. SONGMÉN É1 1
École de Technologie Supérieur de, 1100 Notre Dame Ouest, Montréal, H1C1K3 Québec, Canada 2
LASPI, Université Jean Monnet de St Etienne, 20 avenue de paris, 42334, Roanne, France
[email protected] [email protected] [email protected]
Abstract In manufacturing, the current evolution towards productivity improvement and cost effectiveness has led to the adoption of high speed machining. However, the use of high speed can cause a loss of work-piece quality due to dynamic problems, such as tool wear and self-excited vibrations that produce chatter. Most of the known chatter and tool wear detection methods are based on the assumption of stationarity. On this basis, scalar time descriptors and spectral analysis are often employed to extract information on the cutting tool health. However, most rotating machineries undergo non-stationary operations. In fact, even under cutting conditions of pseudo-constant operations (speed, torque, temperature…), repetitive shocks and friction due to the action of the tool and the work-piece during machining exhibit non stationary phenomena during each cycle of the machine. The study of the cyclostationarity of the vibratory signals in milling operations allows for taking into account the random effect that can be produced between each tool revolution. The statistical properties of machine-tool vibratory signals are periodic with regard to the basic cycle. This periodicity is inferred by the cyclic operation of the machine. New vectorial indicators based on the property of cyclostationarity for identifying the angular position where specific phenomena can occur are presented in this paper. These angular indicators were employed for the detection of slight chatter and tool wear. For the tested conditions, the use of the angular power, the angular kurtosis , the angular spectrum and the Wigner-Ville Spectrum (WVS) of the residual part of signal showed their efficiency for the detection of the tool wear and the detection of chatter in the machining operation. The results reveal that tool wear reduces the angular Kurtosis, but increases its power and the projection of the WVS presents a good mean to analyze the tool wear especially for broken tooth. In addition, the angle-frequency of Wigner-Ville representation of the residual signal shows that the energy corresponding to the tooth passing decreases when a chatter phenomenon happens. The effect of tool wear and the number of broken teeth on the excitation of structure resonances appears on the Wigner-Ville Spectrum. Angular power and kurtosis spectrum are also used for analyzing chatter phenomena. By machining in the unstable region, chatter is produced that results in flat angle kurtosis and flat angle power, such as a pseudo (white) random signal with flat spectrum. Since a vectorial feature is difficult to manipulate for an efficient diagnosis, new scalar indicators were developed. The Kurto Angular Power (KAP) and the Kurto Angular Kurtosis (KAK) were used to analyze and detect chatter phenomena. These news scalars indicators are calculated from the angular features in order to evaluate the severity of damage. These new indicators are useful for an intelligent monitoring of the chatter and tool wear in high speed machining.
Keywords: Monitoring / Machining / Cyclostationarity / Chatter / Tool wear / WVS / KAP / KAK.
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1. Introduction 1.1. Chatter and tool wear For some combination of spindle speed and depth of cut, chatter may arise and vibration level can be come very high. These vibrations can lead to undesired phenomena and poor surface quality. Moreover, the machine tool wears out rapidly and noise is produced. An experienced machine tool operator can often detect a chatter phenomenon, due to the noise produced during cutting, and the characteristic surface finish. Some factors like the need for automation of damage detection and the avoidance of human error have motivated research into more advanced detection methods. Consequently, different approaches have been developed; Sims [1] makes some attempts to classify these various techniques. Chatter affects the surface finish and gives a high roughness. Consequently, one of the most obvious methods to diagnose or illustrate it is through visual inspection of the surface finish. The surface average roughness can be measured using profilometers or white light interferometers [2]. Others techniques based on Fast Fourier Transform (FFT) are used to determine the information in frequency domain of the measurement signal, such as the power spectral density with Welch's technique [3]. This technique can be applied to a variety of measurement signals, such as workpiece or tool acceleration/velocity/displacement, or force reading from a dynamometer. A microphone can be also be used to detect and control chatter phenomenon [4]. The Fourier analysis techniques have been deployed by researchers and software producers. The authors in [5] used a microphone and laptop-based software to identify chatter frequencies. Li et al [6] have proposed a frequency domain approach based on the crosscoherence of two perpendicular acceleration signals to detect wear and chatter. The authors in [7] used frequency descriptors to analyze the stability in robotic high speed machining of aluminum alloys. A major drawback with the majority of frequency domain techniques is that they offer no information in the time domain. Consequently, the time-frequency analysis comes to reconcile the advantages of both the frequency and time analysis by characterizing the signal in time and frequency. It brings a solution to separate near vibratory signatures that regain themselves in frequency or in time, but it asks for a compromise between the frequency resolution and the temporal resolution. In time domain, Nayfeh and Balanchandran [8] analyze the stability of milling operation using a Poincaré section and the bifurcation diagram is so called to identify the transition from stable cutting to chatter. Insperger in [9] analyze the stability of system studying a differential equation modeling machine tool. Bayly et al [10] used the variance indicator in time domain to determine stability boundary. Smith and Tlusty [11] propose the peak to peak forces method to identify the limits of stability. Some of the earliest work use the wavelets for chatter problems, this approach reduce the problem of time versus frequency resolution [12]. Advanced techniques are used also, such as multi-layer perceptions or back propagation neural network [13] and Information Theory [14]. Another critical problem met in machining is the tool wear/failure, which increases the production cost and degrades the product quality. Another method used for the chatter and the tool wear detection that is exploited little in the literature [15, - 17] is based on the cyclostationarity property of the mechanical signals of machining. The following section explains this approach and present briefly the state of the art.
1.2. Cyclostationarity signal vibrations of rotating machines The diagnosis of the cutting tool health and the used of chatter detection methods become so essential. In the machining domain, global indicators and spectrum analysis are often used to extract the information about the wear of the cutting tools or about the stability or not of milling operation. However these methods suppose that the signals stemming from the machine are stationary. By definition, stationary signals are presenting physical phenomena which maintain a constant statistical behavior in time. 2
The rotating machines are governed by mechanisms which evolve cyclically. Consequently, for stable functioning (speed, pressure, temperature, driving cycle, period of the reducer), the physical parameters which describe the generation of the vibrations undergo periodic behavior, such as the strengths of excitations distr ibuted periodically or the strengths of excitation led by repetitive impacts. However, the rotating machinery cannot undergo stationary operations, so even under conditions of constant operations (velocity, couple, and temperature). Successions of the phenomena take place in the cycle of the machine and so release energies, such as shocks due to the work of metals by the cutting tool in manufacturing. It is recognized actually that the majority of the mechanical signals are intrinsically non-stationary because of the evolutionary phenomena which generate them and that main of the defects can be detected in this non-stationarity part of signal. Consequently, even if it simplifies the treatments, the hypothesis of the stationarity is not able to reveal all the required information. The cyclostationarity analysis allows for showing this information for cyclic and repetitive phenomena which are particular cases of no stationarity. It will then be possible in this context, to add to the classic indicators, an additional dimension which translates its cyclic evolution. The first studies on the cyclostationarity date back to 1950s with the works of Bennett [18] and Gladyshev [19, 20]. Recently, the domain knew an increasing interest, mainly because of its applications in telecommunication. The cyclostationarity allowed for improving the precision and the reliability of algorithms existing in noisy environment. It also opened the way to new perspectives concerning problems previously treated [21, 22] and allowed to obtain interesting results in blind separation [23- 25] and identification in experimental modal analysis [26]. Studies have already put in evidence the cyclostationarity of the signals of combustion engines [27, 28] and gears [29]. The recognition of the cyclostationarity of the signals of rotating machines allows for taking into account their natural stationary behavior by designing new tools of treatment, more effective and more precise than those traditionally based on the hypothesis of stationarity. The innovation lies on an additional dimension related to angular variables which describe the evolution of the machine behavior. The objective of this article is to propose news methods for machining monitoring. Indicators for chatter and tool wear detection are introduced and applied in high speed milling operation; these indicators are based on the notion of cyclostationarity. The mathematic background deepened on the cyclostationarity will be able to be found in reference [30]. The paper is organized as follows: after applying the principles of cyclostationarity on slot milling vibration signals to develop new indicators, we present the set-up, data acquisition and the cutting conditions selected in milling operation. Then, to illustrate the methods proposed in this paper, an application to industrial vibration signals recorded in a slot milling operation is provided and discussed in the fourth section. The last section presents the conclusions.
2. Experime ntation 2.1. Experime ntal set up The objective of this section is to present the different steps to achieve the tests of slot milling on a block of aluminum A7075-T6 of size [194mm 133mm 50mm] (#8 in Fig. 1). The experimental setup used in this section is shown in Fig. 1. All tests in the present paper were conducted on the same machine tool which is a CNC milling machine with a spindle Kessler made by SIEMENS; the model is Huron K2X10. The characteristics of the milling machine are illustrated in Tab. 1. We used the structure of test in Fig. 1 to achieve two sets of milling tests: the first aims together to validate the developed method that will be retailed in the following section for the detection of chatter while the second aims together to study the detection of the tool wear. Samples in aluminum # 7075-T6 # were fixed on the structure; some slot milling has been achieved with a 2 and 3 flutes micro grain solid carbide end mills. The characteristics of the tools used for the tests of machining are illustrated in Tab. 2.
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Fig. 1. Schematization of the experimental slot milling setup ( 1) accelero meter#1 (PCB352C22) (2) non-rotative part of the spindle (3) holder ( 4) three (two) flute carbide end mill ( 5) Kistler dynamo meter [9255-B] ( 6) accelero meter#2 (PCB352C22) (7) accelero meter#3 (PCB352C22) (8) workpiece AA7075-T6. Arrows indicate the data transfer to the data acquisition system.
Tab. 1. Characteristics of the milling machine. Controller Maximum speed of rotation(rpm) Displacement (mm) Maximum feed rate (m/min)
Tab. 2. Geometry of the tool (ISO9001-2000). Symbols Terminology 𝐵(°) Tool cutting edge inclination (Helix angle) 𝐴𝑝 (𝑚𝑚) Maximum depth of cut 𝐷(𝑚𝑚) Tool diameter 𝐿(𝑚𝑚) Tool length 𝑍 Number of teeth
Siemens 840 D 28000 1000 x 800 x 500 mm 60
Geometry and tolerances 30 38.1 25.4 101.6 2 /3
2.2. Data acquisition The different steps realized to acquire data are illustrated in Fig. 2. Firstly, the cutting tool was mounted on the spindle. Three accelerometers (PCB352C22) were placed, two were placed at the free end of the nonrotative part of the spindle in x-y direction and the third accelerometer was placed on workpiece in cutting direction (#1 ,#6 and #7 in Fig. 1); for all tests, we use two kinds of tool, 2 and 3 flutes micrograin solid carbide end mills.
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Secondly, this step consists in choosing the cutting conditions (speed of rotation, depth of cut) from the simulated Stability Lobe Diagram (SLD). To determine the SLD, we need to know the cutting coefficients and the transfer function (Fig. 2b) measured on the Huron machine tool from a tap test (Fig. 2a). These data must be used to determine the SLD (Fig. 2b). The algorithm can be found in references [29, 30]. The SLD defines the stable and unstable regions accordingly with the spindle speed (frequency) and depth of cut for a certain width of cut (Fig. 2c). The SLD can then be used to determine what the optimal cutting conditions are. Thus the stability lobes allow for an efficient way for assessing the cutting conditions that we use to realize our tests (tab. 3, 4) in order to analyze the chatter phenomena and tool wear. The feed rate is 0.03 mm/tooth for all tests. The different results of this step are shown in Fig. 2. Thirdly, the acceleration signals for every couple of spindle speed and depth of pass chosen from SLD were acquired with a data acquisition system. This system contains card acquisition #Data Translation DT 9837 # and a laptop with software developed by BETAVIB (Fig. 2.d). The sampling frequency selected for all tests is 48 kHz. A table dynamometer Kistler 9255-B (#5 in Fig. 1) was used for measuring the cutting forces in the same time.
2.3. Chatter tests The first tests concern the study of the chatter phenomenon. The parameters of the design of experiments for machining are selected from SLD (Fig. 2b) and are presented in Tab. 3. We affected a series of test cuts using 2 and 3 flutes end mills during which we recorded the time signals of acceleration se nsors and dynamometer. We varied the depth of cut (from 2 to 6 mm) for every speed of rotation (from 3000 to 26000 rpm) to be able to analyze the stable and unstable cuts. The stability of the operation of the machining depends on the selected conditions of cut.
2.4. Tool wear/ failure tests The second tests concern the study of tool wear, the first step consists in making tests of machining (#Test I) in stable cuts conditions with healthy teeth. The figure (Fig. 3.a) represents a microscopic photograph of the tool used in this test. The chosen parameters are illustrated in the table 4 (#Test I in Tab. 4). The previous tool was used therefore for several tests of machining in different cuts conditions, and a tooth of the tool has presented a flunk wear. Then we used it to make tests of machining (#Test II in Tab. 4) while taking the same conditions of #Test I. To achieve the third test (#Test III Tab in. 4), we took the tool used in the (#Test II) with a flunk wear and created a second wear while breaking the second tooth. To achieve the fourth test (#Test IV Tab in. 4), we took the tool used in the (#Test III) with two broken teeth and created a third wear while breaking the third tooth.
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Fig. 2. The different steps considered to acquire the data. (a:Step1) Tap test with PCB 086c04 hammer, steel tip, PCB accelero meter (b:step2) Transfer function: Real-imaginary parts, Magnitude, Coherence (c:step3) Selection of cutting conditions from stability lobe diagrams, SLD determined for regular t wo and three fluted cutter in slot milling of A7075T6; Hu ron milling mach ine, Flexib le spindle-tool-holder system and rigid workpiece. (d:step4) Acquisition of data by BETA VIB system and dynamometer.
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Tab. 3. Summary of the cutting conditions for the chatter tests. Case
Test #
Axi al depth (mm)
Radial depth (mm)
Cutting s peed (m/min)
Feed (mm/tooth)
Revoluti on speed (rpm)
Flute Number
𝑰
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
2 3 5 6 3 5 6 2 3 6 3 6 2 3 5 6 2 4 2 4 2 4
25,4 25,4 25,4 25,4 25,4 25,4 25,4 25,4 25,4 25,4 25,4 25,4 25,4 25,4 25,4 25,4 25,4 25,4 25,4 25,4 25,4 25,4
239,3823 239,3823 239,3823 239,3823 398,9705 398,9705 398,9705 638,3528 638,3528 638,3528 797,941 797,941 957,5292 957,5292 957,5292 957,5292 1276,7056 1276,7056 1755,4702 1755,4702 2074,6466 2074,6466
0,03 0,03 0,03 0,03 0,03 0,03 0,03 0,03 0,03 0,03 0,03 0,03 0,03 0,03 0,03 0,03 0,03 0,03 0,03 0,03 0,03 0,03
3000 3000 3000 3000 5000 5000 5000 8000 8000 8000 10000 10000 12000 12000 12000 12000 16000 16000 22000 22000 26000 26000
2/3 2/3 2/3 2/3 2/3 2/3 2/3 2/3 2/3 2/3 2/3 2/3 2/3 2/3 2/3 2/3 2/3 2/3 2/3 2/3 2/3 2/3
𝑰𝑰 𝑰𝑰𝑰 𝑰𝑽 𝑽
𝑽𝑰 𝑽 ⋀
The table (Tab. 4) summarizes the script of tool wear tests and the figure (Fig. 3) represents the microscopic photographs of the tool taken by a microscope VHX-500 F in the different tests.
Tab. 4. Summary of the cutting conditions for the tool wear tests. Case
Test #
Axi al depth (mm)
Radial depth (mm)
Cutting s peed (m/min)
Feed (mm/tooth)
Revoluti on speed (rpm)
Flute Number
𝑰
1 2 3 4 5 6 7 8 9 10 11 12 13 14
2 3 3 5 2 3 4 5 2 3 3 2 3 3
25,4 25,4 25,4 25,4 25,4 25,4 25,4 25,4 25,4 25,4 25,4 25,4 25,4 25,4
239,3823 239,3823 398,9705 398,9705 239,3823 239,3823 398,9705 398,9705 239,3823 239,3823 398,9705 239,3823 239,3823 398,9705
0,03 0,03 0,03 0,03 0,03 0,03 0,03 0.03 0,03 0,03 0,03 0,03 0,03 0,03
3000 3000 5000 5000 3000 3000 5000 5000 3000 3000 5000 3000 3000 5000
3 3 3 3 3 3 3 3 3 3 3 3 3 3
𝑰𝑰
𝑰𝑰𝑰 𝑰𝑽
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Fig. 3. Microscopic photographs taken in the different tool wear tests. (a) the new tool used to realize the first test #Case I- (b) Flunk wear tool used in -#Case II- (c) Tool with a flunk tool and a broken tooth, this tool was used to realize the -# Case III- (d ) Tool with a flunk wear and two b roken tooth, it used to realize the -# Case IV-
3. Results and discussion 3.1. First Application: analyse of chatter detection 3.1.1. Experime ntal selected conditions:
Figure 4 shows the stability lobes diagrams simulated by frequency domain method [31, 32], and the conditions selected which are analysed (small black circle).
Fig. 4. Stability lobes diagrams determined for regular two and three fluted cutter in slot milling of AA7075-T6; Huron milling mach ine, flexib le spindle-tool-holder and rig id work-p iece.
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The diagram of stability lobes achieved with the tool with 2 teeth predicts a weaker stability limit for the set of spindle speeds [3000, 10000, 16000] rpm, a few more for the set [5000, 8000] rpm and a big limit of stability for the set [12000, 22000, 26000] rpm (see blue curve Fig. 4) while the diagram of stability lobes achieved with the tool with 3 teeth predicts a weaker stability limit for the set of spindle speeds [3000, 5000, 10000, 12000, 22000, 26000] rpm and a big limit for the set [8000, 16000] rpm (see red curve Fig. 4).
3.1.2. Cyclostationary analysis: a) Basic: Figure 5 shows the acceleration signals in X and Y direction for a spindle speed of 50 Hz (3000 rpm) and depth of 3 mm. We can distinguish three zones in the acceleration signal: the spindle rotations without cutting, the cutting part, and the spindle rotation without cutting. In this section, only the cutting part has been analyzed. Acceleration signals reflect a cyclic and repetitive phenomenon which occurs during the operation of machining.
Fig. 5. Acceleration signals and zoo ms (a) co mp lete accelerat ion signal in Y d irection (b) zoom shows the acceleration signal in Y direction (c) co mplete acceleration signal in X direction (d) zoo m shows the acceleration signal in Y direction.
The analysis of signal acquired at 3000 rpm and depth of 3mm (see Tab.3) has for vocation to illustrate the basis tools of the cyclostationarity. The registration of 35 s of machining is illustrated to the figure 6.a (a zoom) that reveals a random behaviour with a related periodic structure. The spectral analysis of the signal (Fig. 6b.) confirms this observation through a continuous spectral density and exempt of spectral stripes. The cyclic power of the signal represented in red to the figure 6.a permits to reveal the presence of a periodicity very distinctly in the signal. The spectral correlation of signal (CS) and the Fourier coefficients corresponding are represented to the figure 6.c and d, that indicate a fundamental to 50 Hz and two harmonics to 100 Hz and 200 Hz: it is the frequencies of rotation of the spindle ( measurements made in Canada).
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Fig. 6. (a) Acceleration signal of #test1 with a spindle speed of 3000 rp m and a depth of 3 mm and 2 flute, cyclic power with red color (d) Power spectral density with resolution of 1.3 Hz and FFT length of 48000 (c) Spectral correlations (d) Fourier coefficient of power cyclic accord ing to the cyclic frequency alpha.
b) Angular analysis and estimation of the angular statistics The acquisition with angular sampling is particularly suitable for realizing statistics on the cycles of the vibratory signals, by synchronizing them over the period of the basic cycle [33]. Another procedure requires to re-sampling the signal in batch mode, based on techniques more or less elaborated according to the order of interpolation required on the signal of an encoder acquired together with the temporal signal. Techniques of real-time re-sampling were also proposed [30]. In this paper, we used the method detailed in [14, 15 and 16] because we do not possess the information of optical encoder to make the angular re-sampling. The angle of rotation θ of the cycling machine is considered as a generic variable, because x(θ) θ∈Z is a stochastic process containing an important number of cycles and x (θ) is a particular realization w. The signal is cut in K consecutive blocks of length Θ, . The mean, the variance, the power and the angular kurtosis are then calculated by (“Eq. 1”, “Eq.2”, “Eq.3” and “Eq.4”) respectively:
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] 𝐾Θ
𝑚𝑥 𝜃
𝑉𝑥 𝜃 ] 𝐾Θ =
𝑃𝑥 𝜃
𝐾𝑢𝑟𝑥 𝜃
] 𝐾Θ
=
1 𝐾
1 𝐾
=
1 𝐾
𝐾−1
𝑥[𝑚𝑜𝑑 𝜃 + 𝑘Θ, 𝐾Θ
(1)
𝑘=0
𝐾−1
(𝑥[𝑚𝑜𝑑 𝜃 + 𝑘Θ, 𝐾Θ ] − 𝑚𝑥 𝜃 ] 𝐾Θ )2 𝑘 =0
] 𝐾Θ
=
1 𝐾
(2)
𝐾−1
(𝑥[𝑚𝑜𝑑 𝜃 + 𝑘Θ, 𝐾Θ ])2
(3)
𝑘=0
𝐾−1
𝑥 𝑚𝑜𝑑 𝜃 + 𝑘Θ, 𝐾Θ
− 𝑚𝑥 𝜃 ] 𝐾Θ
4
𝑘 =0
(𝑉𝑥 𝜃 ] 𝐾Θ )2
(4)
where 𝑚𝑜𝑑(𝑎, 𝑏) is the rest of the whole division of a by b. The function 𝑚𝑜𝑑 allows for defining the angular average for each 𝜃.
This operation is schematized on “Fig. 7”.
Fig. 7. Calculat ion of the angular statistics.
Because the angular average defined by “Eq.1” was extracted from a periodic component which admits series of Fourier, it can also be written when the number of blocks is infinite: +∞
mx θ ] KΘ = ℘ x(u) =
PΘ x(u) e2πjk θ /Θ k =−∞
(5)
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with PΘ x(u) = limW →∞
1 W/2 x W −W/2
u e −2πjku/Θ du , the Fourier coefficient at the cyclic frequency
and α = 1/Θ and ℘ ∙ an operator of cyclic averaging.
The periodic component is then generated by the series of Fourier. The immediate average thus appears as a filter selecting only the frequency 1/Θ and its harmonics. The cyclic average of the power ℘ x(θ) 2 represents periodic fluctuations in the energy. It is represented itself by its coefficients of Fourier which translates the intensity of the periodic components of the energy with the cyclic frequencies. It is thus interesting to decompose these quantities also into frequency, by means of the spectral correlation SCxα . It is a frequency function with two variables: the frequency f and the cyclic frequency α. The angular kurtosis, as defined by “Eq.4”, supplies coefficients at every angle θ. They are normalized with regard to the cyclic energy of signal. This tool is going to allow for detecting the angular positions where the kurtosis takes high or low values, and thus to detect the source of an impact. If the angular kurtosis is periodic, we say that x(θ) represented by the coefficients of Fourier is cyclostationary at the order 4, and thus is possible. This tool may thus be used for the monitoring of the cutting tool wear in high speed milling [15 -17].
c) Angular statistics for chatter detection Angular statistics estimated in last section are used to analysis the chatter phenomenon. The set up for this first application is explained in the experimental set up section. Tab.3 recapitulates all tests used to achieve this objective. Figure 8 represents the results after synchronization of the signal in the case of #test1 (spindle speed = 3000 rpm, depth = 3mm and flute number = 2); the number of points by period of rotation of the spindle is 960 points, every block contains 10 cycles of rotation.
Fig. 8. blocks of #test 1 after synchronization, (10 cycles by block).
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After having synchronized all these blocks between them, the average, the variance, the power and the angular kurtosis as defined by formulae (“Eq.1”, “Eq.2”, “Eq.3” and “Eq.4”) may be calculated. Figure 9 represents the signal of #test 1, with its average, power and angular kurtosis. The angular average presents the periodic contribution of the vibratory signal. In that case, we can say that the average signal is cyclostationary at order 1. This type of cyclostationarity arises from a macroscopic phenomenon of determinist nature (the passage of teeth on the workpiece), whereas the angular power reports periodic fluctuations in the energy, in that case the signal is cyclostationary at order 2. This power informs us about the energy produced by a tooth into the workpiece.
Fig. 9. Angular statistics of stable contact -#test 2- (spindle speed = 3000 Rp m, depth = 3 mm,flute number = 2) (a) bloc of 10 cycles (b) angular average shows 5 spindle rotation(c) angular power shows the tooth -workpiece contact (d) Peak to peak (e) angular kurtosis shows the 2 flute contact with the workpiece.
In the stable cuts, we can easily identify the passage of flutes in workpiece, the same observation is shown when we used a three flute tool. Angular power reflects the energy produced during the passage of teeth. The angular power is a periodic function so it admits a Fourier decomposition. The kurtosis angular allows for identifying the passage of teeth in workpiece, it is periodic in the stable conditions.
d) Angular power and kurtosis spectrum Figure 10 represents the angular average spectrum, power and kurtosis angular for stable and unstable tests (#test2 and #test 4 in Tab.3 case I). The power and kurtosis angular spectrum are used also for distinguishing the stable case from the instable case (Fig. 10 and 11). 13
Fig. 10. (a) Angular average spectrum o f stable and instable cuts conditions (b) Power and kurtosis angular -blue- (c) Power and kurtosis angular -red-.
The angular power and kurtosis are very sensitive to shocks, and allow for finding the angular position at which an event can occur. Every shock is corresponding to the passage of teeth on the workpiece. The angular power and kurtosis for the second case (#test 4) don’t present a characteristic pattern of shocks, but rather a pseudo-random shape.
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Fig. 11. (a, c) Angular power spectrum and angular ku rtosis spectrum for stable and instable cuts (b) Angular power for stable and instable cuts (c) Angular ku rtosis for stable and unstable cuts.
e) Wigne r-Ville Spectrum In this part, the periodic contribution of the residual part is analyzed. To analyze the residual part (the signal less the angular average) the use of SWV or the Wigner-Ville spectrum seems natural because of the nature of the cyclostationary process. Figure 12 shows the SWV of the residual part of five cycles, with a Hanning window and a frequency resolution of 60 Hz. The signals were selected in the conditions of #test1 and #test4 (see Tab.3 case I). A certain number of remarks impose themselves: 1) At every cycle (960 points) we can easily identify the passage of the two teeth marked by one strong dissipations of power if the conditions of machining are stable (Fig. 12a), these visible marks disappear if the conditions of machining are unstable (Fig. 12b). 2) The advantage of this presentation of SWV is the absence of the interfering cross terms and the good visualization of results.
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Fig. 12. (a,b) W igner-Ville spectrum fo r stable and unstable cuts conditions (c, d) Projection of SW V on the angular axis with zoo ms of SW V.
3) The analysis of SWV makes to take out better the influence of the structure resonances. 4) This representation of SWV seems useful for the analysis of chatter phenomenon. However, this tool presents a certain number of problems for the interpretation of results and the necessary calculation time. 5) The analysis of the projection of SWV seems interesting, this approach is not innovative since it is already evoked in the works of J.Antoni [26] for the treatment of the engine thermal, but it is introduced the first time for the monitoring of high speed machining. Figures (12.c, 12.d) clearly show the appearance of the marks due to the passages of the teeth in the stable case but not in the unstable case. Figure 13 shows a polar representation of the projection of SWV in the two studied cases.
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Fig. 13. Polar diagrams of W igner-Ville spectrums projection (a) in stable conditions milling -#test2 case1 in Tab.3(b) in unstable conditions milling -#test4 case1 in Tab.3-.
f) New descriptors: Kurto Angular Powe r and Kurto Angular Kurtosis The indicators presented in the cyclostationary treatment section (angular statistics, angular statistics spectrums, SWV, SWV projection...) are vectorial quantities. The specialists in maintenance prefer to use scalar quantities and the objective of this part is to introduce new scalar indicators extracted from the presented vectorial quantities: a) Kurto Angular Kurtosis (KAK) and the Kurto Angular Power (KAP) are merely the kurtosis of kurtosis angular and the kurtosis of angular power: We apply this method on the previously analyzed tests (#test1, #test4 case I in Tab.3). Figure 14 presents the histograms of the distributions of angular kurtosis and angular power and their fitted distributions. The KAK and the KAP are then computed from these vectorial quantities. We note an increase in the KAK and KAP due to the instability of the machining.
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Fig. 14. Angular kurtosis and angular power h istograms for stable and instable cuts condition with the fitted distrib ution (a) stable milling -#test2 case I in Tab.3- (b) unstable milling -#test4 case I in Tab.3-, KAK is the Kurto Angular Kurtosis and KAP is the Kurto Angular Power.
To validate the KAK and the KAP, we are going to follow their evolution according to the depth of cut and the speed of rotation. These conditions were selected from Tab. 4. Table 4 Case
Test #
Axi al depth (mm)
Radial depth (mm)
Cutting s peed (m/min)
Feed (mm/tooth)
Revoluti on speed (rpm)
Flute Number
𝑰𝑰𝑰
8 9 10 13 14 15 16
2 3 6 2 3 5 6
25,4 25,4 25,4 25,4 25,4 25,4 25,4
638,3528 638,3528 638,3528 957,5292 957,5292 957,5292 957,5292
0,03 0,03 0,03 0,03 0,03 0,03 0,03
8000 8000 8000 12000 12000 12000 12000
2 2 2 2 2 2 2
𝑽
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Figure 15 represents the results of the KAK and KAP.
Fig. 15. Kurto Angular Kurtosis and Power for some cuts conditions in X and Y direct ion (a) KAK for X and Y d irection (b) KAP for X and Y direction.
We note that there is an evolution in the two indicators (KAK and KAP) in proportion with the depth of pass, the KAK and KAP are less sensitive to the variations of the speed or to the variations of the depth of cut in the Y direction. These two indicators present a complementary study for the analysis of chatter phenomenon. In this section, news tools based on a property of the signals acquired that is the cyclostationarity were presented. New vectorial tools are proposed as the angular power, the angular kurtosis, the angular power spectrum, the angular kurtosis spectrum, the Wigner-Ville spectrum of residual signal (SWV) and the SWV projection and new scalar tools as the Kurto Angular Kurtosis and the Kurto Angular Power. The combination of these tools presents a good detector of chatter phenomenon.
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3.2. Second application: analyse of tool wear detection
The objective of this second application is to analyze the wear phenomenon and propose some new tools to detect it. Tab.4 and Figure 3 (Fig. 16.a) summarize the achieved tests. Tab.3 includes four cases of tests, the first case serves as a reference (case I) and the others are tests of wear. Fig. 16 represents the temporal signals of two speeds of rotation (3000 and 5000 rpm) for the four cases (reference: without wear, wear-1: flunk wear, wear-2: flunk wear and a broken tooth, wear-3: flunk wear and two broken tooth).
Fig 16. Accelerations signals of different cases (a) microscopic images of reference tool and different tool wear (b) accelerations signals for 3000 rp m of spindle speed and 2 mm for depth (c) accelerations signals for 5000 rp m of spindle speed and 3 mm for depth.
We analyse the signals in frequency domain (Figure 17), the amplitude of spindle speed harmonics of the power spectral density increase with wear, this is due to the increase of energy produced by each tooth in milling operation.
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Fig. 17. Power spectral densities of reference test and different test of tool wear for the case of 3000 rp m of spindle speed and 2 mm o f depth (a) PSD of reference signal without wear (b) PSD o f signal with flunk wear (c) PSD of signal with flunk wear and one broken tooth (d) PSD of signal with flunk wear and two bro ken tooth. Frequency resolution is 1.46 Hz.
The angular analysis shows the difference between these cases. Fig. 18 shows the angular averages estimated on 5 revolutions, the angular average reveals the shocks corresponding to the passages of the three tooth in the healthy case (Fig. 18.a), these points of shock begin to disappear once the teeth begin to wear out (Fig. 18.b, c, d). The same remarks may be made if we analyse the angular power. We analyse the residual part (the signal less the angular average) by using of Wigner-Ville spectrum (Fig. 19). Figure 19.a represents the SWV of signal references in the healthy case without wear; the passage of the three teeth appears well and reveals again only a resonance of the structure. On the figure 19.b, we note that one of the shocks due to the passage of the worn-out tooth produced a weak energy in relation to the two other teeth; this observation is confirmed by the projection of the SWV (Fig. 20.b). The analysis of the SWV of the third case reveals three resonances of the structure and the effect of passage of the teeth begins to disappear (Fig. 19.c). In the last case, we used a complete worn-out tool (Fig. 19.d). The very high frequencies that are excited and the projection of the SWV reveal that there is a problem on all teeth (Fig. 20.d).
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Fig. 18. Angular average and angular power to analyse the tool wear estimated on 5 revolutions (a) Angular average and angular power for reference case without wear (b) Angular average and angular power for the first case of wear (b) Angular average and angular power for the second case of wear (c) Angular average and angular power for the third case of wear.
Figure 20 represents the SWV projection of different cases. In the case healthy, we see that the rotational period is predominant due to the balancing (Fig. 20.a). On the figure 20.b, the passage of the three teeth can easily identify the worn-out tooth, the problem of flunk wear is on the tooth 2 where the time of shock is a little big (B in 20.b). On the figure 20.c, the passage of the two teeth can identify the two shocks; the tooth 1 is healthy, the tooth 2 carries the flunk wear and the tooth 3 is broken; therefore the energy cleared by this tooth is very weak. On the figure 20.d, the third tooth is broken.
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Fig. 19. Wigner-Ville spectrum for different cases with projection of SW V (a) SW V for the reference case (b) SW V for the first case -flunk wear- (c) SW V for the second case -flunk wear and a broken tooth-(d) SW V for the third case flunk wear and two broken tooth.
Fig. 20. SW V Project ion estimated on 5 revolutions for the reference case and the tool wear cases with a microscopic picture (a) reference case without wear (b) SW V Projection of tool with a flunk wear (c) SW V Projection of tool with a flunk wear and one broken tooth (d) SW V Projection of tool with a flunk wear and two broken tooth.
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5. Conclusions This study showed the importance to use cyclostationarity analysis of vibratory signals in machining monitoring. We were interested in this paper in two big problems frequently met in machining that is the chatter and the tool wear. To reach our objective, we achieved two sets of tests. The first test was aimed for analyzing the chatter phenomenon and the second test has for goal to study the wear phenomenon. Through these two applications, different methods were used to analyze and to detect the chatter and the tool wear. Finally, some new scalar and vectorial indicators based on a cyclostationarity property are proposed. Angular statistics permit to develop new temporal vectorial indicators (angular kurtosis and angular power) or spectral (angular kurtosis and angular power spectrums) that can be uses by themselves for the detection of chatter. The use of Wigner-Ville spectrum was efficient for this type of cyclostationary signals, its projection on the axis of time reflect well each event and can indicate the presence or not of chatter. New scalar indicators (the Kurto Angular Kurtosis and the Kurto Angular Power) have been developped for the detection of chatter. For the tool wear, we tested the efficiency of the cyclostationary tools on different types of wear (broken tooth, flunk wear), and we showed their efficiency for the diagnosis. The cyclostationary tools permit to distinguish between the chatter and the wear.
6. Acknowledgments The authors would like to thank the CRSNG-RDC program of the Council of Research in Natural Sciences and Engineering of Canada, CRIAQ, Aeronautic Bombardier and Pratt and Whitney Canada Inc. for their financial support in this research.
7. References 1-
Neil D. Sims. Machining Dynamics. Springer Series in Advanced Manufacturing, 2009, 85-115.
2-
Schmitz, TL, Medicus, K, and Dutterer, B. Exploring once–per–revolution audio signal variance as a chatter indicator. Machining Science and Technology, 2002. 6(2): p. 215–233.
3-
Welch, PD. The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms. IEEE Trans. Audio Electroacoust., 1967. AU–15: p. 70–73.
4-
Delio, T, Tlusty, J, and Smith, S. Use of audio signals for chatter detection and control. Journal of Engineering for Industry, Transactions of the ASME, 1992. 114(2): p. 146–157.
5-
MetalMax Harmonizer. Manufacturing Laboratories, Inc, Las Vegas, 2004.
6-
Li, XQ, Wong, YS, and Nee, AYC. Tool wear and chatter detection using the coherence function of two crossed accelerations. International Journal of Machine Tools and Manufacture. 1997. 37(4): p. 425–435.
7-
Zaghbani I., Lamraoui M., Songméné V., Thomas M. and El Badaoui M., Oct 2010. Robotic High Speed Machining of Aluminum Alloys, the 4th edition of the International Conference on High Speed Machining (ICHSM’2010), Harbin, China, 19p.
25
8-
Nayfeh, A and Balachandran, B, Applied Nonlinear Dynamics: Analytical,Computational and Experimental Methods. 1994: Wiley.
9-
T.Insperger. Stability analysis of Periodic Delay Differential Equation Modeling Machine Tool Chatter., 2002, PHD thesis, Budapest University of Technology and Economics.
10-
Bayly, PV, Mann, BP, Schmitz, TL, Peters, DA, Stepan, G, and Insperger, T. Effects of radial immersion and cutting direction on chatter stability in end milling. ASME International Congress and Exposition. 2002. 13: p. 351–363.
11-
Smith, S and Tlusty, J. Update on High–Speed Milling Dynamics. Journal of Engineering for Industry, 1990. 112: p. 142–149.
12-
Khraisheh, MK, Pezeshki, C, and Bayoumi, AE. Time series based analysis for primary chatter in metal cutting. Journal of Sound and Vibration, 1995. 180(1): p. 67–87.
13-
Tarng, YS and Chen, MC. Intelligent sensor for detection of milling chatter. Journal of Intelligent Manufacturing, 1994. 5(3): p. 193–200.
14-
Gradisek, J, Govekar, E, and Grabec, I. Using coarse–grained entropy rate to detect chatter in cutting. Journal of Sound and Vibration, 1998. 214(5): p. 941–952.
15-
Lamraoui M., Thomas M., El Badaoui M. et Zaghbani I., Juin 2010. Le kurtosis angulaire comme outil de diagnostic du broutage et d’usure des outils de coupe, Vibration, Choc et bruit (VCB2010), Lyon, France, article AC-29, 21 p.
16-
Lamraoui M., Thomas M., El Badaoui M., Zaghbani I. et Songméné V., Oct 2010, Le Kurtosis angulaire, un nouvel outil de diagnostic. Proceedings of the 28th CMVA, Québec, Vol 2, pp 217-240.
17-
Lamraoui M., M. Thomas , M. El Badaoui , I. Zaghbani and V. Songméné, March 2011, The angular Kurtosis and power: new features for machining monitoring, Proceedings of the XIV International Symposium on Dynamic Problems of Mechanics (DINAME 2011), ISSN 2179-9601, Fleury, A. T., Kurka, P. R. G. (Editors), ABCM, São Sebastião, Brazil, pp. 494-511.
18-
Bennett W., 1958. Statistics of regenerative digital transmission, Bell Systems Technical Journal, 37: 1501-1542.
19-
Gladyshev E., 1961. Periodically correlated random sequences, Soviet Mathematics Doklady, 2: 385388.
20-
Gladyshev E., 1963. Periodically and almost-periodically correlated random processes with continuous time parameter, Theory of Probability and Its Applications, 8: 173-177.
21-
Gardner W. A. et al, 1989. Frequency-Shift Filtering Theory for Adaptive Co-Channel Interference Removal, 23rd Asimolar Conference on Signals, Systems and Computers. 562-567.
22-
Gardner W. A. Et al, 1993. Exploitation of Cyclostationarity for Identifying the Volterra Kernels of Nonlinear Systems, IEEE transactions on information theorie, 39 (2): 535-542.
23-
Antoni J. and F. Guillet, 2005. Blind separation of convolved cyclostationary processes, Signal Processing, 85: 51-66.
26
24-
Sghir K.A.; M. El Badaoui ; M. Thomas; I. Zaghbani; V. Songméné ; A. Lakis; N. Mureithi, Identification aveugle paramétrique de la fonction de réponse impulsionnelle basée sur la cyclostationnarité de second ordre à partir des réponses seulement, CIRI 2009, Mai 2009, Reims, 16 p.
25-
Sghir K. A., M. El Badaoui, M. Thomas, F. Guillet, M. Bakrim and D. Aboutajdine, July 2009b. Parametric blind identification of the transfer function from vibration measurements based on second order cyclostationnarity, 16th International congress on Sound and Vibration, Krakow (Poland), 8 p.
26-
Antoni J. and P. W. Ha. 2004. A Consistent Estimator for Frequency Response Functions with Input and Output Noise, IEEE Instrumentation and Measurement, 53 (2): 457-465.
27-
Antoni J. et al, 2002a. Effective vibration analysis of IC Engines using Cyclostationarity. Part I: A Methodology for Condition Monitoring, Journal of Sound and Vibration, 257 (5): 815-837.
28-
Antoni J. et al, 2002b. Effective vibration analysis of IC Engines using Cyclostationarity. Part II: New Results on the Reconstruction of the Cylinder Pressure, Journal of Sound and Vibration, 257 (5): 839856.
29-
Antoni J., Bonnardot F., Raad A. and Elbadaoui M., 2004. Cyclostationnary modelling of rotating machine vibration signals. Mechanical Systems and Signal Processing 18, 1285-1314.
30-
Bonnardot F., 2004. Comparaison entre les analyses angulaires et temporelles des signaux vibratoires de machines tournantes. Etude du concept de cyclostationnarité floue, Thèse, Institut National Polytechnique de Grenoble.
31-
Altintas Y, Budak E., 1995, Analytical prediction of stability lobes in milling, Annals of the CIRP, Vol 44(1), pp 357-362.
32-
Budak E., Altintas Y., 1998, Analycal prediction of chatter stability in milling- Part 1: general formulation, Transaction of the ASME, Vol 120,pp 22-30.
33-
Bonnardot F. et al, 2004. Unsupervised Angular Resampling and Noise Cancellation for Planetary Bearing Fault Diagnosis, International Journal of Acoustics and Vibration, Vol 9(2).
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