Abstract—The paper is mainly aimed at building a method to monitor tool wear condition based on analysis of chaotic characteristics of acoustic emission (AE) signal generated during tool cutting process. First, according to the AE signals from different tool cutting periods, multiple chaos descriptions, such as reconstructing the strange attractor track and Poincare map, computing correlation dimension and the max Lyapunov exponent, are used to prove the existing of chaotic characteristics in the series. Then, use the least square method to fit a curve to the computed characteristics points, including the correlation dimension and the max Lyapunov exponent. Furthermore, the developing trend of these two parameters is discussed. The results show that chaotic phenomena exist in the acoustic emission signal, and the chaotic characteristics, like correlation dimension and the max Lyapunov exponent, will change with the development of tool wear process. Therefore relationship analysis between chaotic characteristics of AE series and the tool wear condition may provide a new path for online monitoring of tool cutting process.

I. INTRODUCTION S an important way of mechanical machining, cutting machining urgently require the reliable online real-time monitoring for burst breakage of cutting tool and the

state of tool wear, to enhance the automation lever of mechanical industry and improve product quality. But due to the complex nature of the machining processes, it is difficult to find a universally successful monitoring method. Recent attempts have concentrated on the development of the methods which monitor the cutting processes indirectly. Acoustic emission (AE) is one of the most effective indirect methods for sensing tool wear. AE is a sound wave or, more properly, a stress wave that travels through a material as the result of some sudden release of strain energy. The major advantage of using AE to monitor tool condition is that the frequency range of the AE signal is much higher than that of the machine vibrations and environmental noises, and does not interfere with the cutting operation.

Many signal processing methods have been used to analyze AE signals, with the aim to extract the features for testing or monitoring. These methods include time series

This research is supported by the project (60804025) of the National

Natural Science Foundation of China. It is also supported by the project (2008555) of education department of Liaoning province. All the support is appreciated.

Jianhui Xi is with the School of Automation, Shenyang Aerospace University, 37 Daoyi South Street, Shenyang, 110136, China (phone: +86-13998152567; e-mail: xjhui_01@ 163.com).

Wenlan Han is with the School of Automation, Shenyang Aerospace University, Shenyang, (e-mail: 3h3wtly@163.com).

Yanmei Liu is with the School of Automation, Shenyang Aerospace University, Shenyang, (e-mail: lymlczlymlcz@163.com).

analysis [1], fast Fourier transform (FFT) [2], Gabor transform (or window (local) Fourier transform), neural networks [3], and wavelet transforms [4-6]. In recent years, some methods have been proved effective under certain conditions. However no matter what monitoring methods are used, it is necessary to extract features sensitive to tool wear condition. But during the process of tool cutting, the factors, which directly or indirectly affect the tool wear condition, are of uncertainty. Therefore, the feature extraction from AE signals is very difficult.

Continuous chaos, under the name of deterministic nonperiodic flow, has been described as internal stochastic behaviors of nonlinear system. Some research indicated that the tool wear is non-linear, stochastic and dissipative. The AE signal, from the different process or the same process under different condition, has certain similarity as well as displaying obvious difference. So it is possible to apply chaos theory to tool wear condition monitoring. The current study mainly concentrates on the fractal dimension computation and evolution analysis, such as Satish T S, Akhles L [7] analyze the change of fractal dimension to estimate cutting tool wear state in the research of optimal cutting tool chatter control, Wang Zhongmin, etc [8] also apply fractal dimension to do on-line monitoring, determining the condition of cutting tool from the change of fractal dimension. This paper will analyze and describe multi-characteristics of AE signals from different cutting periods in the view of chaos. First, strange attractor reconstruction in the phase space qualitatively describe the existence of chaotic phenomena and the spatial characteristics of AE signals. Then, computation of chaotic characteristics—correlation dimension and the largest Lyapunov exponent, quantitatively describe chaotic characteristics of the series. On this basis, the internal relationship between the chaotic characteristics and tool wear state is studied.

II. CHARACTERISTICS OF ACOUSTIC EMISSION SIGNAL Acoustic emission signal is the weak stress wave

produced by material deformation during machining process, which is a physical phenomenon [9]. It is an indirect monitoring method and widely used in nondestructive testing. When the cutting tool is in different states, the characteristics of acoustic emission signals of cutting tool could be analyzed to detect small breakage or micro fissure [10][11].

The data of the paper is from the machining process of numerical control lathe, the work piece material is GH648 used in aviation engine, and material of the tool is KC5010. Data acquisition is realized by PXWAE Acoustic Emission

Relationship Analysis between Chaotic Characteristics of Acoustic Emission Signal and Tool Wear Condition

Jianhui Xi, Wenlan Han, Yanmei Liu

A

612

Third International Workshop on Advanced Computational Intelligence August 25-27, 2010 - Suzhou, Jiangsu, China

978-1-4244-6337-4/10/$26.00 @2010 IEEE

Detection System designed by Beijing Pengxiang company, the sensor is PXR30-type AE sensor of the company. The data sampling frequency is 2MHZ.

The data amount is too large, so here just part of them is plotted. The AE series from both the early and the middle

stage of tool cutting are showed in figure 1 and figure 2. For the convenience of computation, the data is normalized. It can be seen that the data is highly nonlinear and there are no obvious difference between early data and middle data, which results in the difficulties of extracting features.

III. QUALITATIVE DESCRIPTIONS OF CHAOTIC CHARACTERISTICS

A. The Phase-space Reconstruction of AE Signals The embedding theorem proposed by Takens pointed out

that an appropriate embedding dimension m can be found for one-dimensional time series x(ti), (i = 1,2,…,n). If d is the dimension of the original strange attractor, then may take m≥2d+1, the attractor of the time series must be recovered in the embedded space. The phase-space reconstruction is the basis of calculation of chaotic characteristic parameters. It is generally based on delay coordinates method, using time-lapsed sample from x(ti), τ is delay time, τ = k* tΔ , k ∈ Z. The original time series are transformed into an m-dimensional phase space [12]. That is

)}(,),(),({ )1( ττ −++= miiii txtxtxX

From above introduction we can know that the key point of reconstruction is the selection of τ and m. The optimal τ is calculated by self-correlation function [13] in the paper. To the time series x(ti), (i=1,2,…, n). The formula for self-correlation function is as follows:

12

( ( ) )( ( ) )1( )

( ) ( )

n

i ii

x t x tC

n x

τ

τμ μτ

τ σ

−

+=

− −=

−

∑ (1)

μ is the mean and σ is the standard deviation of the time series, then:

( )( )( )n

txx

n

ii∑

=

−= 1

2

2μ

σ (2)

Self-correlation function represent the interrelation degree

and similarity of the movement at subscript i and τ+i The paper take τ to be the best delay time which is corresponding to the correlation function dropped below

e11− times of the initial value. And the value of m takes big

as possible. The strange attractor of AE signals was reconstructed in

the three-dimensional space. The tracks are shown in figure 3, in which (a) is corresponding to the phase space of the AE signals in the early period of cutting, and (b) is corresponding to the middle period. It can be seen that, (i) The phase diagram of the cutting tool have obvious fractal structure and strange attractor; (ii) The strange attractor of

0 500 1000 1500-1

-0.5

0

0.5

1

Point number

Am

plitu

de

afte

r na

rmal

izat

ion

Fig. 2. The AE signal from the middle stage of tool cutting

0 500 1000 1500-1

-0.5

0

0.5

1

Point number

Am

plitu

de

afte

r na

rmal

izat

ion

Fig. 1. The AE signal from the early stage of tool cutting

613

the middle period has contracting tendency compared with the early period. Therefore, with the development of tool cutting, while the tool wear is serious, the chaotic characteristic is also changed.

B. Poincare Map

As using phase space reconstruction to describe the high-dimensional space is difficult. It can be further studied by Poincare image method, which transforms trajectories of the system into intersection points of the trajectory and the cross-section (Poincare section). When the phase trajectory is a closed curve, and the image on the Poincare section is a spot, the system presents as periodic motion; When the phase trajectory is a self-intersecting closed curve, the Poincare image is contracted to a limited number of points or a closed curve, system movement corresponds to quasi-periodic motion; When the phase trajectory gradually fill in some certain areas of the phase-space, the image in the Poincare section is shown to fill in some regions, the system is in a chaotic state [14].

Figure 4 is Poincare map respectively corresponding to the early and middle periods of cutting on the base of the phase space reconstruction. The figures show that, the points of intersection fill up specific regions gradually from outside to inside, which proves that chaos exists in the AE signals

from another view angle, and the characteristics changes along with the change of tool wear state, it is necessary to quantitatively analyze and find out the relationship.

IV. QUANTITATIVE DESCRIPTIONS OF CHAOTIC CHARACTERISTICS

Correlation dimension and the largest Lyapunov exponent are selected as indicators of Chaos [15] [16]. Correlation dimension D, proposed by the Grassberger P and Procaccia I [17], is a kind of fractal dimension, which displays infinite levels of self-similar structure that is the strange attractor's geometric properties. The largest Lyapunov exponent indicates the average spread rate of two initial orbits close to each other. It is a kind of quantitative descriptions of initial sensitivity of chaotic motion. The value being a positive number indicates that the movement of the system is chaotic. The data should be pre-processing normalized for simplifying computation in the following chaotic characteristics calculation of AE signals.

A. Calculation of Correlation Dimension Assuming two points in the phase space

( 1)

( 1)

( , , , )

( , , , )i i i i m

j j j j m

X x x x

X x x xτ τ

τ τ

+ + −

+ + −

=

=

Set rij as Euclidean distance between two points

jiij XXr −= (3)

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

X

Y

(a) Poincare map of the early stage AE signal

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

X

Y

(b) Poincare map of the middle stage AE signal

Fig. 4. Poincare map

-1-0.5

00.5

1

-1

-0.5

0

0.5

1-1

-0.5

0

0.5

1

XY

Z

(a) Phase diagram of the early stage AE signal

-1-0.5

00.5

1

-1

-0.5

0

0.5

1-1

-0.5

0

0.5

1

XY

Z

(b) Phase diagram of the middle stage AE signal Fig. 3. The reconstructing phase-diagram of AE signals of the cutting tool

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Given the critical distance r, the correlation integral Cm(r) is the density of points around a specific coordinate Xi within a distance r

∑≠

−−=ji

jim XXrHN

rC )(1)( 2 (4)

N is the number of total points in phase space, and H(·) is Heaviside function:

⎩⎨⎧

≥<

=0100

)(xx

xH (5)

Cm(r) portrays the focusing degree of the phase space points relative to a parameter point within the distance r. Therefore, as r increases in an appropriate scope, the correlation integral mD

m rrC ∝)( , it is

rrCD m

rm ln)(lnlim

0→= (6)

Select different r within a certain range, separately calculate the corresponding Cm(r), then substitute the result into equation (6), Dm can be fitted out. For a definite chaos system, Dm will tend to a saturation value with increasing of m, which is the correlation dimension D.

For a set of AE data from early stage of cutting, take r as 0.38, 0.46, 0.55, 0.63, 0.71, 0.79, 0.87, 0.96, m changes in [4, 40]. Figure 5 shows the relation between (r)Cm and rln . With m increasing, the slope of curve is also increasing, and becoming saturated gradually. Correlation dimension can be obtained by calculating the average of

dlnr / (r)dlnCm from 35 to 40 dimensions. The correlation dimension of middle cutting period can be calculated similarly, the results included in table 1.

B. Calculation of The Largest Lyapunov Exponent

The methods, such as the Wolf method [18], the Jacobean method, the p-norm method [19], and the small data method [20] are main methods of calculating the largest Lyapunov

exponent [15]. Wolf proposed the evolution of line path to directly estimate the largest Lyapunov exponent, {Y(t)} is the reconstructed phase space of the time series {x(t)} based on selecting embedding dimension and delay time. The schematic diagram of Wolf method is shown as figure 6.

Take the initial point )(tY 00 , supposes L0 as the distance of Y0(t0) and the nearest neighbor point Y(t0). Track evolution of the two points over time, the distance of two points is larger than a specified value: ε > 0,

ε>−=′ )()( 1010 tYtYL

Until the moment t1, reserve Y(t1), find another point Y1(t1) near Y(t1), which make

ε<−= )()( 1111 tYtYL

And the angle is as small as possible. Continue above process, until Y(t) arrives the end point N of the time sequence, the total number of iterations M is obtained by tracking the evolution process, the largest Lyapunov exponent is:

∑=

′

−=

M

i i

i

m LL

tt 00max ln1λ (7)

As the existence of a certain degree noise , divide the data

within 1 second into 10 groups for weakening the interference effect as well as simplifying calculation , then calculate the characteristics value of each group, take the mean of the 10 groups’ as the characteristics corresponding to the time spot. The max Lyapunov exponents in the early and midterm periods of cutting are calculated, the results included in table 1, which shows that the AE signals have fractal dimension and the positive largest Lyapunov exponent, which quantitatively prove that the movement of cutting tool is chaotic motion; Compare the chaotic characteristic parameter of two periods, it can be seen characteristic parameters of different periods are not same, it verifies the conclusions of qualitative analysis--the parameters of chaos are changing with cutting tool damaging.

Fig. 6. Phase trajectory

-2 -1.5 -1 -0.5-5

-4

-3

-2

-1

0

lnC

lnr Fig. 5. Calculation of correlation dimension

615

C. The Trend Analysis of Correlation Dimension and The Largest Lyapunov Exponent

Based on the above results, the largest Lyapunov exponent and the correlation dimension of 16 sets of data, selected from different periods of cutting, are calculated in order to further analyze the relationship between the evolution of chaotic characteristics of AE signals and the cutting tool wear state, Then use least-squares method analysis the trend of calculation results [21], the method is based on the smallest square sum of error as the criterion, being to estimate the system linear model according to known data. The basic idea is to choose the estimate which make square sum of difference between the model output and measured output achieve minimum.

Figure 7 shows the fitting results of the correlation dimension trend. Correlation dimension has an advantage compared to other features of the AE signals, as it reflects the irregular degree of the current signals [8]. From figure 7, it can be seen that as time increases, correlation dimension change from the sharp rise in the beginning to a slowly increase subsequently, which correspond with cutting tool damage trend, that is a sharp damage of cutting tool of the beginning, entering a stable period of tool damage after a certain period. Apply least squares to analyze the 16 points of the correlation dimension by one-dimensional linear regression analysis. The significance of regression equation obtained by fitting is tested at the level of 0.01; the result shows that the equation is significant. It can be inferred, with the passage of time, the degree of tool damage increasing, the correlation dimension also corresponding increase monotonically.

Figure 8 corresponds to the largest Lyapunov exponent

calculation. Similarly, it is analyzed by a linear regression and tested by the significance of the regression equation, the result of which shows that the regression equation does not have significance. It can be concluded that the max Lyapunov exponent does not change monotonously over time. The tendency of cutting tool wear thought by Literature [22] is that the speed of attrition is great in the beginning, and then slows down, after a period of time, the speed is getting large once more. The tendency of maxλ reflects the speed change of cutting tool wear to a certain extent. So the relevance should be analyzed thoroughly using non-linear method.

V. CONCLUSIONS (1) Chaotic phenomenon exists during the cutting process,

which can be proved from the qualitative and the quantitative as follows: strange attractor exists in the reconstructed phase space of AE signals; The Poincare maps show that the points of intersection fill up specific regions gradually from outside to inside; The correlation dimension is fractal dimension, the largest Lyapunov exponent is a positive value.

(2) The chaotic characteristics change over cutting processing, which acts as follows: strange attractor in phase space and the Poincare maps both have contraction tendency over time or with the tool wear deepening; Correlation dimension and the largest Lyapunov have similar trend separately with the tool wear volume and speed.

The next step necessary to study is: how to accurately describe the internal relations among the correlation dimension, maxλ and the state of cutting tool, then achieving a higher degree of accuracy online monitoring and prediction of tool wear.

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0 20 40 60 80 1000.025

0.03

0.035

0.04

0.045

t/s

Lya

Fig. 8. Trend of the largest Lyapunov exponent

0 20 40 60 80 1001.5

2

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Cr

Fig. 7. Trend of correlation dimension

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EARLY AND MID-TERM OF CUTTING

maxλ D

Early 0.0311 2.311 Mid-term 0.0445 2.7403

616

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