Drill fracture detection by the discrete wavelet transform

B.Y. Leea, Y.S. Tarngb,*

aDepartment of Mechanical Manufacture Engineering, National Huwei Institute of Technology, Yunlin, Taiwan 632, ROCbDepartment of Mechanical Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan 106, ROC

Received 7 July 1998

Abstract

This paper describes the use of induction motor current to detect the occurrence of drill fracture based on the discrete wavelet transform.

The scaling and shifting coef®cients in the discrete wavelet transform are chosen as powers of two so that a multi-level signal

decomposition of the induction motor current can be performed. Different wavelets are also considered in order to recognize drill fracture

clearly. Experimental results have indicated that drill fracture can be detected successfully by this approach. # 2000 Elsevier Science S.A.

All rights reserved.

Keywords: Drill fracture; Discrete wavelet transform; Induction motor current

1. Introduction

Detection of tool fracture in machining operations has

been considered a key technology in achieving unmanned

machining in the automated factory. To detect the occur-

rence of tool fracture correctly, the selection of an appro-

priate signal processing algorithm is very important. It is

well known that the Fourier transform is a useful signal

processing tool for transforming a signal from the time

domain to the frequency domain. Based on the Fourier

transform, a signal can be decomposed into a family of

complex sinusoids as basis functions. However, the Fourier

transform has a serious drawback. In transforming to the

frequency domain, information of the time domain is lost.

As a result, it is impossible to tell when a particular event

(e.g., tool fracture) takes place from the frequency domain.

To correct this de®ciency, the Fourier transform of a small

section of the signal at a time, called the short-time Fourier

transform, has been proposed. The use of the short-time

Fourier transform to detect tool fracture in machining

operations has been reported [1,2]. However, the precision

of the short-time Fourier transform is still limited and is

greatly dependent on the size of the window. Furthermore,

tool fracture features may not be so clear in the frequency

domain even when using the short-time Fourier transform. In

recent years, a more ¯exible approach, called the wavelet

transform [3,4], has been developed to decompose the signal

into various components for different time windows and

frequency bands using a family of wavelets as the basic

functions. As a result, the use of the wavelet transform to

detect tool fracture is better than the use of the Fourier

transform to detect tool fracture [5].

In this paper, the use of induction motor current as a

sensing signal to monitor tool fracture in drilling operations

has been studied. This is because the induction spindle

motor system is already built into a machine tool and

therefore the cost of sensor investment can be greatly

reduced. In addition, the mounting of the sensor does not

interfere with the operation of the machine tool. On the other

hand, the induction motor current signal is not so sensitive to

the occurrence of tool fracture owing to the limited band-

width of the induction spindle motor system [6]. However,

the response time for tool fracture is still acceptable based on

sensitivity analysis. To detect tool fracture accurately, the

induction motor current signal is processed by the discrete

wavelet transform [7]. Based on the discrete wavelet trans-

form, the induction motor current signal is decomposed into

a set of approximations and details of the signal. The

approximation is the low-frequency component of the signal

and the detail is the high-frequency components of the

signal. This decomposition process can be iterated so that

the induction motor current signal is broken into a hierarch-

Journal of Materials Processing Technology 99 (2000) 250±254

* Corresponding author. Tel.: �886-2-2737-6456; fax: �886-2-2737-

6460.

E-mail address: ystarng@mail.ntust.edu.tw (Y.S. Tarng).

0924-0136/00/$ ± see front matter # 2000 Elsevier Science S.A. All rights reserved.

PII: S 0 9 2 4 - 0 1 3 6 ( 9 9 ) 0 0 4 3 2 - X

ical set of approximations and details, also called a multi-

level signal decomposition [8]. The results of the multi-level

signal decomposition can be used to monitor tool fracture in

drilling operations accurately.

In the following, the discrete wavelet transform is intro-

duced in order to perform a multi-level signal decomposi-

tion. Then, the measurement of induction motor current in

drilling operations is discussed. Experimental veri®cation of

the discrete wavelet transform for the detection of drill

fracture is shown. Finally, the paper concludes with a

summary of this study.

2. Discrete wavelet transform

The wavelet transform of a signal f(t) is de®ned as the sum

over all of the time of the signal f(t) multiplied by a scaled

and shifted version of the wavelet function c(t). The coef®-

cients C(a,b) of the wavelet transform of the signal f(t) can

be expressed as

C�a; b� �Z 1ÿ1

f �t� 1���ap c

t ÿ b

a

� �dt (1)

where a and b are the scaling and shifting parameters in the

wavelet transform.

Basically, a small scaling parameter corresponds to a

compressed wavelet function. As a result, the rapidly chan-

ging features in the signal f(t), i.e. high-frequency compo-

nents, can be obtained from the wavelet transform by using a

small scaling parameter. On the other hand, low-frequency

features in the signal f(t) can be extracted by using a large

scaling parameter with a stretched wavelet function. In other

word, a small scaling value is used for local analysis, whilst a

large scaling value is used for global analysis.

For a digital signal f(k), k � 0,1,2,. . ., a discrete wavelet

transform can be used. The most commonly used discrete

wavelet transform is the scaling and shifting of parameters

with powers of two, i.e.,

a � 2j (2)

b � ka � k2j (3)

where j is the number of levels in the discrete wavelet

transform.

The coef®cients C(a,b) of the discrete wavelet transform

can be divided into two parts: one is the approximation

coef®cients and the other is the detail coef®cients. The

approximation coef®cients are the high-scale, low-fre-

quency components of the signal f(k), whilst the detail

coef®cients are the low-scale, high-frequency components

of the signal f(k). The approximation coef®cients of the

discrete wavelet transform for the digital signal f(k) at level j

can be expressed as

Aj �X1n�0

f �n�fj;k�n� �X1n�0

f �n� 1�����2 jp f

nÿ k2 j

2 j

� �(4)

where fj,k(n) is the scaling function associated with the

wavelet function cj,k(n).

Similarly, the detail coef®cients of the discrete wavelet

transform for the digital signal f(k) at level j can be expressed

as

Dj �X1n�0

f �n�cj;k�n� �X1n�0

f �n� 1�����2 jp c

nÿ k2 j

2 j

� �(5)

Based on Eqs. (4) and (5), the decomposition of the signal

f(k) can be iterated as the number of levels increases. As a

result, a hierarchical set of approximations and details can be

obtained through the multi-level signal decomposition. Use-

ful information for the signal f(k) can then be yielded from

the multi-level wavelet signal decomposition. Fig. 1 shows

an example of a two-level wavelet decomposition of the

signal f(k). First, the signal f(k) is split into an approximation

A1 and a detail D1. Then, the approximation A1 is also split

so as to obtain a second-level approximation A2 and detail

D2. As shown in Fig. 1, a wavelet decomposition tree can be

obtained through this transformation process.

3. Measurement of the induction motor current

The most often used spindle motor in the machine tool

industry is an induction motor [9]. Basically, the induction

motor consists of two components: a stationary stator and a

revolving rotor, the rotor being separated from the stator by a

small air gap. A three-phase set of voltages are applied to the

stator causing a three-phase set of currents to ¯ow. These

currents produce a rotating magnetic ®eld, which drags the

rotor along in the same direction. Therefore, the rotor

rotation speed n is equal to the rotation speed of the

induction motor. However, the rotor rotation speed is always

slightly less than the rotation speed of the magnetic ®eld ns.

The rotation speed of the magnetic ®eld, which is also called

the synchronous speed ns (rpm), can be expressed as

ns � 120f

p(6)

Fig. 1. A two-level wavelet decomposition.

B.Y. Lee, Y.S. Tarng / Journal of Materials Processing Technology 99 (2000) 250±254 251

where f is the frequency of the stator in Hertz and p the

number of poles per phase in the induction motor.

In a drilling operations, once tool fracture occurs, cutting

edges with chipping, breakage or severe deformation lose

their usefulness. If the damaged tool is still in use, an

excessive cutting force acting on the cutting edges is una-

voidable. A larger motor torque must be generated to over-

come an excessive cutting force acting on the cutting edges.

Basically, the stator current of the induction motor increases

with the motor torque. Therefore, in this paper, the stator

current of the induction motor is used as the sensing signal

for the detection of tool fracture in drilling operations.

In the experiments, a three-phase four-pole induction

spindle motor was installed in a machining center for the

machining of the S45C steel plate using a 12 mm � 140 mm

(diameter � length) twist drill. For the induction motor,

each phase of the stator current signal has the same peak-

to-peak amplitude but is displaced in time by a phase angle

of 1208. Therefore, only one of the phases of the stator

current signal was measured by a current to voltage (C/V)

sensor (LEM Module LA50-P) and recorded on a PC work-

station through a data acquisition board (DT2828). To per-

form the sensitivity analysis for the stator current, a

dynamometer (Kistler 9271A) was mounted under the work-

piece. The dynamometer signal was transmitted through a

charge ampli®er (Kistler 5007) from which the torque signal

was obtained and also recorded in the PC workstation. The

transfer function between the stator current and the torque is

shown in Fig. 2. As shown in this ®gure, a change of spindle

speed varies the frequency response a little. The bandwidth

of the transfer function is not wide and the corresponding

time constant is about 0.2 s. Therefore, the stator current

signal is not so sensitive to the torque variations due to the

limited bandwidth. Once tool fracture occurs, the use of the

stator current signal to detect tool fracture will have a little

time delay. However, the time delay is acceptable for the

detection of tool fracture in drilling operations.

4. Results and discussion

A schematic diagram of the experimental set-up for the

detection of tool fracture using the induction motor current

is shown in Fig. 3. As seen in this ®gure, the stator current

signal was measured by the current sensor and recorded on

Fig. 2. The transfer function between the stator current and the torque in

drilling.

Fig. 3. Schematic diagram for the detection of drill fracture.

252 B.Y. Lee, Y.S. Tarng / Journal of Materials Processing Technology 99 (2000) 250±254

the PC workstation. The stator current signal is further

processed by the discrete wavelet transform. To detect the

occurrence of tool fracture clearly, the discrete wavelet

transform uses Daubechies wavelets for performing the

multi-level wavelet signal decomposition: this is because

the Daubechies wavelets are suitable for the detection of a

sudden signal change at the time or frequency domain. In

this study, the names of the Daubechies wavelets are written

dbN, where N is the order of the wavelet. Fig. 4 shows the

result of a drilling test in the 10th drilling cycle with a

spindle speed of 1000 rpm and a feed of 240 mm/min. A

two-level wavelet decomposition (Fig. 1) of the stator

current signal using the Daubechies wavelets was per-

formed. Fig. 4a shows the original stator current signal in

a drilling operation with tool fracture. A constant peak-to-

Fig. 4. Experimental results for the detection of drill fracture: (a) stator

current; (b) approximation coef®cients A2 using the db1 wavelet; and (c)

approximation coef®cients A2 using the db6 wavelet (spindle

speed � 1000 rpm; feed � 240 mm/min).

Fig. 5. Daubechies wavelets: (a) db1; (b) db6.

Fig. 6. The three-dimensional approximation coef®cients in the level two (spindle speed � 1000 rpm; feed � 240 mm/min).

B.Y. Lee, Y.S. Tarng / Journal of Materials Processing Technology 99 (2000) 250±254 253

peak AC current signal was recorded at the beginning due to

the free run of the spindle. Once the drill started to engage

the workpiece at 1.6 s, the current signal gradually

increased. The peak-to-peak current signal became constant

again when the drill fully entered the workpiece. However, a

large variation of the stator current signal suddenly occurred

at 9.5 s due to tool fracture. Fig. 4b shows the approximation

coef®cients A2 (Fig. 1) using the db1 wavelet and Fig. 4c

shows the approximation coef®cients A2 using the db6

wavelet. The scaling function and wavelet function for

the Daubechies wavelets, db1 and db6, are shown in Fig.

5. It is seen that the approximation coef®cients A2 can

clearly detect the occurrence of tool fracture (Fig. 4b and c).

This is because the stator current signal in the normal cutting

states is ®ltered out and the large variation of the stator

current signal owing to tool fracture is left in the approx-

imation coef®cients A2. Furthermore, a higher order of the

Daubechies wavelets can distinguish a drill with tool frac-

ture and one without tool fracture more clearly. As shown in

Fig. 4, the use of the db6 wavelet to perform the discrete

wavelet transform is clear enough to detect the occurrence of

tool fracture. The three-dimensional approximation coef®-

cients A2 in this drilling test are shown in Fig. 6, where it is

seen that the approximation coef®cients A2 is very small

from the ®rst drilling cycle to the ninth drilling cycle.

However, the approximation coef®cients A2 become very

large in the 10th drilling cycle due to the occurrence of tool

fracture. Fig. 7 shows the experimental result for other

drilling tests using a spindle speed of 1200 rpm and a feed

of 220 mm/min. A similar result for the detection of tool

fracture from the approximation coef®cients A2 using the

db6 wavelet is also illustrated in Fig. 7. Therefore, the

approximation coef®cients A2 can be used to detect the

occurrence of drill fracture successfully in drilling, even

under different cutting conditions.

5. Conclusions

An application of the discrete wavelet transform to the

detection of drill fracture has been reported in this paper.

The discrete wavelet transform performs a two-level signal

decomposition of the induction motor current in drilling

operations. Based on the results of the signal decomposition,

the approximation coef®cients in the second-level can be

used to detect the occurrence of tool fracture clearly. It has

also been found that the use of a Daubechies wavelet with

the order of six is clear enough to detect the occurrence of

tool fracture. Experimental veri®cation has con®rmed the

effectiveness of this proposed approach for the detection of

tool fracture in drilling operations.

Acknowledgements

The authors wish to express appreciation to Associate

Professor H. S. Liu, National Huwei Institute of Technology,

for his help in the experiments.

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Fig. 7. Experimental results for the detection of drill fracture: (a) stator

current; (b) approximation coef®cients A2 using the db6 wavelet (spindle

speed � 1200 rpm; feed � 220 mm/min).

254 B.Y. Lee, Y.S. Tarng / Journal of Materials Processing Technology 99 (2000) 250±254