# Tool Wear Detection Based on Duffing-Holmes Oscillator

Authors: Wanqing Song; Shen Deng; Jianguo Yang; Qiang Cheng
Journal: Mathematical Problems in Engineering
 (2008)

---

## Abstract

The cutting sound in the audible range includes plenty of tool wear information. The sound is sampled by the acoustic emission (AE) sensor as a short-time sequence, then worn wear can be detected by the Duffing-Holmes oscillator. A novel engineering method is proposed for determining the chaotic threshold of the Duffing-Holmes oscillator. First, a rough threshold value is calculated by local Lyapunov exponents with a step size 0.1. Second, the exact threshold value is calculated by the Duffing-Holmes system in terms of the law of the golden section. The advantage of the method is low computation cost. The feasibility for tool condition detection is demonstrated by the 27 kinds of cutting conditions with sharp tool and worn tool in turning experiments. The 54 group data sampled as noisy are embedded into the Duffing-Holmes oscillator, respectively. Finally, one chaotic threshold is determined conveniently which can distinguish between worn tool or sharp tool.


---

## Body

## 1. Introduction

Tool
wear is a complex phenomenon occurring in metal cutting processes. A worn tool
adversely affects the surface finish of the work piece and therefore there is a
need to detect tool wear which alerts the operator to the tool wear state,
thus, avoiding undesirable product quality. However, accurately determining
cutting conditions remains difficult.Acoustic emission based on tool condition monitoring has been
available for approximately 17 years, most of them use analog root mean square
of the signal to monitor tool wear or detect breakages. Damodarasamy and Raman [1]
combined the radial force, feed force, and AE to model the tool flank wear in a
turning operation. Wanqing et al. [2] used a wavelet
transform and fractal algorithm to capture the features of the AE signals. Yao et al. [3] used a fuzzy neural
network to describe the relation between the monitoring features, which are
derived from wavelet-based AE signals, and the tool wear condition. The data
processing methods have shown acoustic emission signal power to increase with
tool wear owing to increased friction effect [4].Nearly years,
chaotic oscillator is used widely to detect weak period signal [5–8]. The weak
signal detection is a central problem in the general field of signal processing
and the use of chaos theory in weak signal detection is also a topic of
interest in chaos control. At present, however, this research is mainly theory
and simulation, engineering practice is a few examples. The phase transforms of
Duffing-Holmes oscillator are sensitive to periodic signal and periodic
interference signals which have larger angular frequency difference from the
referential signal, but immune to the random noisy [5, 9]. Since tool wear is a
gradual processing during the turning conditions, the cutting sound is composed
of periodic signals and a large amount of periodic
interference signals and the random noise. Of course, the frequency and amplitude
of these signals also are changing gradually along with tool wear except of the
random noise. Therefore, the tool wear processing belongs to detect weak periodic
signals in strong noisy and very appropriately by Duffing-Holmes oscillator.Machining
tests were carried out on HL-32 NC turning center. This lathe does not have a
tailstock. Tungsten carbide finishing tool was used to turn free machining mild
steel. The work material was chosen for ease of machining, allowing for
generation of surfaces of varying quality without the use of cutting
fluids. The experiment equipments are
shown in Figure1. The piezoelectric AE sensor (CAE-150) was mounted on the
tool holder. A light coating of petroleum jelly was applied under the sensor to
ensure good acoustic emission coupling. Because of high impedance of the
sensor, it must be directly connected to a buffer amplifier. Low-frequency
noise components, which are inevitably present in AE signal, cannot represent
the tool’s condition and hence useless. Therefore, those components should be
eliminated (highpass filtered) at the earliest possible stage of signal processing
to enable usage of full amplitude range of the equipment. The filtered signals
were sampled at 4 MHz using a digital storage oscillograph to a PC, see Figure 1.
All test data were processed and analyzed by using the Matlab software.Figure 1
AE
measurement in metal cutting.In the experiment, according to
the cutting conditions which are presented in Table1, a sharp tool and a worn
tool was used, respectively.Table 1
Experimental cutting conditions.
No.SpeedDepth of cutFeedNo.SpeedDepth of cutFeedr/minmmmm/rr/minmmmm/r1150010.11510000.20.05215000.50.11680010.05315000.20.1178000.50.054100010.1188000.20.05510000.50.119150010.02610000.20.12015000.50.02780010.12115000.20.0288000.50.122100010.0298000.20.12310000.50.0210150010.052410000.20.021115000.50.052580010.021215000.20.05268000.50.0213100010.05278000.20.021410000.50.05The data sampled by AE, 54 group data, are merged
into Duffing-Holmes equation as an exterior perturbation of the chaotic system,
respectively. Then, with tool wear, the gradual change sound signal under the
background of strong noise can be detected by identifying the phase space trajectory.
In terms of the results from theoretical calculation, it is proved that there
is a huge difference in the phase space trajectories between the chaotic state
and the periodic state, and this difference can be used as the evidence in the
chaotic system for the detection of tool wear signal based on Duffing
oscillator. Meanwhile, Lyapunov exponents are adopted as threshold value evaluated
roughly for chaotic critical state, the law of golden section to determine the threshold
is proposed and the threshold in chaotic critical state is evaluated more
accurately. Melniko’s function also can be used to calculate the threshold for
chaos, but Melniko’s function only determines the threshold from order to
chaos, but Lyapunov exponents can determine the threshold from chaos to order [7, 10, 11].
We describe a means for tool wear whether or not a system is chaotic. When the
tool is sharp, the Duffing-Holmes oscillator is chaos in state space trajectory,
when the tool is wear, the Duffing-Holmes oscillator takes on periodic trajectory from chaos to order in state space.


## 2. Principle Detecting Weak Signal Based on Duffing-Holmes Oscillator

The
Duffing-Holmes is the second differential equation containing the item of the
power five, which can be motivated by exterior stimulations to engender
oscillation movement and then generate chaotic trajectory or periodic
trajectory; its dynamic equation is as follows:(2.1)x′′(t)+0.5x′(t)−x3(t)+x5(t)=rsin(t)+(input), where 0.5 denotes the ratio of
damping, rsin(t) is the forced periodic terms, which is the
reference signal and as an internal signal, −x3(t)+x5(t) term is the nonlinear recovery force in system
1, the kinematical state of the system mainly depends on this recovery force
term rsin(t). Input terms are the
signal measured which is imported to the dynamic system as the supplement of
special parameters of chaotic oscillator; we can adjust the amplitude r of the
reference signal to the special value as in the chaotic critical state. The value
is called threshold value in the chaotic system 1. If a weak periodic signal is
merged into system 1, so long as the threshold is adjusted appropriately, the
behavior of the Duffing-Holmes will be changed dramatically from chaotic states
to periodic states. For example, let input terms be f(t)=0.2sin(t),
then the Duffing-Holmes equation is (2.2)x′′(t)+0.5x′(t)−x3(t)+x5(t)=rsin(t)+f(t), when f(t) contains a weak white noisy, that is, f(t)=0.2sin(t)+0.01rand, rand is a random white noisy (0∼1), input terms f(t) are a low-amplitude periodicsignal with white noisy, then the
Duffing-Holmes equation is (2.3)x′′(t)+0.5x′(t)−x3(t)+x5(t)=rsin(t)+f(t), when one weak periodic noisy signal is merged into input terms f(t), that is, (2.4)f^i=fi,i≠16+128k,f^16+128k=f15+128k+0.04,k=0,1,2,3,…, then (2.5)x′′(t)+0.5x′(t)−x3(t)+x5(t)=rsin(t)+f(t). Let dynamical system 2, 3, and 4 initial point x′(0)=0,x(0)=0,
then set threshold rd=0.52544 as the critical state for the system
2, integrated with Runge-Kutta method of fourth order with a fixed step
size t=0.01 second. Total time is 16 seconds. The
phase space in systems 2 and 3 takes on periodic state
trajectory, but phase space in system 4 is chaotic trajectory.When a strong noise[0.2sin(t)] without white noisy or with white noisy is added to the
Duffing-Holmes system, both systems 2 and 3 take on the periodic state. It
means the random noisy is not influenced on the state of the dynamic system. Once the strong noisy
contains a weak periodic noise signal, the behaviors of system 4 is changed
immediately from a large-scale periodic state to a chaotic state. The temporal
waveform of f(t), the phase orbit, and the temporal waveforms of systems 3 and
4 are shown in Figures 2 and 3. In other words, the Duffing-Holmes takes on
some immunity to random noisy [12] and strong sensitivity to some weak periodic
signal. Since 0.01 rand term is too
small, it is not obvious in the temporal waveform.Dynamic character with weak periodic noisy signal,y(t)=x′(t).
(a)(b)(c)Dynamic character without weak periodic noisy signal,y(t)=x′(t).
(a)(b)(c)


## 3. Threshold Calculated Based on Lyapunov Exponents

Lyapunov exponents are frequently computed measure for
the characteristic of chaotic dynamics [10, 11, 13]. It describes a method for
diagnosing whether or not a system is chaotic. To confirm the existence of the
weak periodic signal to be detected and the amplitude of the signal, we need to
define a proper index for denoting the change in the states of the chaos
detection system. The index should be sensitive to a weak periodic signal, but insensitive
to the random noise from the viewpoint of statistical characteristics. Thus,
the dynamic properties of a certain system are reflected statistically by
Lyapunov exponents which are described as follows [14–16].Dynamic systemx′′(t)+0.5x′(t)−x3(t)+x5(t)=rcos(t) is transformed below: (3.1)y(t)=x′(t),y′(t)=−0.5y(t)+x3(t)−x5(t)+rcos(t).To a two-dimensional planex(t),y(t)=x′(t),
two Lyapunov exponents can be solved in system 5. When the system is in the
large-scale periodic state, both of the two Lyapunov exponents are negative. When
the system is in the chaotic state, at least one of the two Lyapunov exponents
of the system is positive which has behaviors of the chaos. Therefore, the detection
system is established on the basis of Lyapunov exponents.Let
initial conditionx(0)=1,x′(0)=1,
with about typical 30 points in the region r=[0.5,1] chosen to calculate the
Lyapunov exponents (LE), the computation precision of r is two digits after the
decimal dot, see Table 2. LE curve are plotted in Figure 4. r=0.70, system 5
takes on the chaotic state, and r=0.78, system 5 takes on the periodic state. They
are shown in Figures 5 and 6.Table 2
Lyapunov exponents in
Duffing-Holmes.
No.rMax LEMin LENo.rMax LEMin LE10.520.185−0.685160.74−0.18527−0.3147320.540.11914−0.61914170.75−0.22179−0.2782130.560.12314−0.62314180.76−0.23059−0.2694140.580.15765−0.65765190.77−0.23088−0.2691250.60.00782−0.50782200.78−0.22217−0.2778360.620.13867−0.63867210.8−0.24112−0.2588870.640.15349−0.65349220.82−0.23407−0.2689380.660.17321−0.67321230.84−0.23449−0.2655190.680.16921−0.66921240.86−0.24647−0.25353100.70.17056−0.67056250.88−0.24773−0.25227110.710.17226−0.67226260.9−0.24394−0.25606120.720.19317−0.69317270.92−0.22882−0.27118130.730.185−0.685280.94−0.24693−0.25307140.7330.164−0.664290.96−0.24271−0.25729150.734−0.20655−0.29345300.98−0.23848−0.26152Figure 4
The relational
curve of LE andr.r=0.70, system character and LE.
(a)(b)(c)r=0.78, system character and LE.
(a)(b)(c)Obviously, LE changed from positive to negative
correspond to regionr=[0.733,0.734] based on the chaotic system extreme
sensitivity to parameters changed. If the threshold r is equal to 0.733, because
computation precision of r is only three effective digits after decimal dot, the
sensitivity from chaos to periodic is not enough. Above, computation cost
spends about 3 hours for typical 30 point sets of r with Matlab. In order to improve sensitivity of system 5, however, if the
computation precision of r is risen 4 digits after decimal dot, namely,
r=[0.5200,0.9800], time interval 0.01 second and 1000 steps, the computation
cost will spend about 30 hours with Matlab. The more high sensitivity is, the
more long computation time is.


## 4. Threshold Computation Combined the Law of Golden Section with Lyapunov Exponents

First, rough region of the system thresholdr
is estimated by Lyapunov
exponents with computation precision to be one digit after decimal dot, the calculating process only spends
about 40 minutes in the region r=[0,1] with step size 0.1. Whatever any kinds
of weak external signal merged, the region of r=[0.7,0.8] is always sensitivity region changed
from chaotic state to large periodic state in system 5. Since the law of the golden
section can search optimizing solution quickly [12], the threshold value is
determined by the golden section accurately in the region r=[0.7,0.8]. The
Duffing-Holmes oscillator is below: (4.1)x′′(t)+0.5x′(t)−x3(t)+x5(t)=rcos(t). Let initial condition x(0)=1,x′(0)=1,
the computation precision of threshold value is six digits after the decimal
dot in system 6. The method is as follows:(1)
because
0.7 corresponds to chaotic state and 0.8 corresponding periodic state,r=0.75 is the
middle value between 0.7 and 0.8.(2)
becauser=0.75 corresponds
to periodic states, the region of r is [0.7,0.75]. Then, r is accumulated from
0.7 to 0.75 with the step 0.01 up to 0.71 which corresponds to chaotic state and
0.72 which corresponds to periodic state. 0.715 is the middle value between
0.71 and 0.72.(3)
becauser=0.715 corresponds
to chaotic state, the region of r is taken [0.715,0.72]. Then, r is accumulated
from 0.715 to 0.72 with the step 0.001 up to r=0.717 which corresponds to
chaotic state and 0.718 which corresponds to periodic state, 0.7175 is the
middle value between 0.717 and 0.718.(4)
becauser=0.7175 corresponds
to periodic state, the region of r is [0.717,0.7175]. Then, r is accumulated
from 0.717 to 0.7175 with the step 0.0001 up to 0.7173 which corresponds to chaotic
state and 0.7174 which corresponds to periodic state. 0.71735 is the middle
value between 0.7173 and 0.7174.(5)
becauser=0.71735 corresponds
to periodic state, the region of r is [0.7173,0.71735]. Then, r is accumulated
from 0.7173 to 0.71735 with the step 0.00001 up to 0.71732 which corresponds to
chaotic state and 0.71733 which corresponds to periodic state. 0.717325 is the
middle value between 0.71732 and 0.71733.(6)
becauser=0.717325
corresponds to periodic state, the region of r is [0.717325,0.71733]. Then, r
is accumulated from 0.717325 to 0.71733 with the step 0.000001 up to 0.717329
which corresponds to chaotic state and 0.717330 which corresponds to periodic
state.(7)
final, the threshold value calculated is
0.717329. When a weak periodic signal is merged into system 6, the system takes
on the large-scale periodic state. Calculating process is shown in Table3.Table 3
Thresholdr based on the golden section in the region r=[0.7,0.8].
Chaos,r1Phase planePeriodic,r2Phase planeGolden section,rPhase plane0.70.8r1+r220.710.720.750.7170.7180.7150.71730.71740.71750.717320.717330.717350.7173290.7173300.717325The computation processing only spends about 10 minutes
for computation precision to be six digits after the decimal dot. The method
has important meaning for engineering practice. 30th steps calculated yield the
search optimization threshold value. This is the most amounts of the point sets
in the case.


## 5. Experiment Work

The sound signal of sharp
tool sampled by AE as an initial condition is merged into the Duffing-Holmes
system 6 which is in the chaotic critical state, (its phase plane changes from
the chaotic state to the large-scale periodic state), the movement state of the
system will transit immediately from the chaotic state to the large-scale periodic
state. The simulation of systems
3 an 4 above has only one input signal, however, for this practice engineering,
since the sharp tool and wear tool have 27 groups data, respectively, see systems 7 and 8, the
threshold in the both the systems must satisfy to distinguish sharp tool and
wear tool in 54 group data. The dynamic system 6 is transformed to systems 7 and 8. When the
data of sharp tool are embedded to the chaotic system 7, the phase space is
chaotic state; however, when the data of wear tool are embedded to the chaotic
system 8, phase space change is the large-scale periodic state. The method
based on the change of the dynamic behaviors of a chaotic system (chaotic state,
periodic state) has been proposed for recognizing, where there exists a signal
to be detected in a system, and greatly immune to the random noise of arbitrary
zero average value with unknown probability distribution. The threshold valuer
should firstly be determined in system 7, which is the critical problem of wear
signal chaotic detection. The algorithm to determine the threshold value, using
Lyapunov exponents method based on the golden section is detailed as follows: (5.1)x′′(t)+0.5x′(t)−x3(t)+x5(t)=rcos(t)+sharp_1,x′′(t)+0.5x′(t)−x3(t)+x5(t)=rcos(t)+wear_1. Since the signal amplitude merged is too bigger than interior
perturbation force r, the signal sampled is decreased 100 times, thus, signals
embedded to Duffing-Holmes are weak perturbation noisy, see systems 9 and 10. The interior
perturbation force r is still main signal in the dynamic systems 9 and 10. Sharp_1
signal and wear_1 are shown in Figures 7 and 8
in time domain ， the initial
condition is [0,0] in systems 9 and 10 ， frequency sampled
is 0.001 second. We set up a chaotic oscillator sensitive to weak periodic signals
based on the Duffing-Holmes equation (5.2), and poising the system at its
critical state (5.2)x′′(t)+0.5x′(t)−x3(t)+x5(t)=rcos(t)+0.01sharp_1,(5.3)x′′(t)+0.5x′(t)−x3(t)+x5(t)=rcos(t)+0.01wear_1.Figure 7
Waveform of sharp tool in first condition.Figure 8
Waveform of wear tool in first condition.Here, we meet a problem. When the computation
precision of the thresholdr is not appropriate, dynamic system 10 is not
stable. In other words, perhaps one group data is chaotic and another group data is periodic state in all wear tools. Behavior
of the dynamic system is changed with the threshold difference, see Figure 4.
In order to decrease computation cost with Matlab, we fix a step size 0.1 in the region r=[0,1], the system trends
of dynamic behavior can be get roughly, this computing process spends about 40
minutes. In fact, the region r=[0.7,0.8] is the region from chaotic state to periodic
state for Duffing-Holmes oscillator, no matter what any exterior weak periodic
signals are merged into the system, the r=[0.7,0.8] can be used directly as
ruler.If
we took the critical value of each group data as the threshold value, we would
get 27 difference threshold values. However, we must get one threshold value
for all 27 group data. For the reason, the range of the threshold value will be
enlarged, that is, the threshold value will be decreased. We take the minimum threshold
value in all 27 group sharp tool data. Using the law of the golden section in
the regionr=[0.7,0.8] for each group data of sharp tool, their critical value
are calculated, see Table 4. Obviously, minimum is 0.723710.Table 4
The critical value for 27 group
sharp tool data.
No.rNo.rNo.r10.724896100.724276190.7237120.724633110.724203200.72387330.724716120.724249210.72381940.723738130.723154220.72394550.724714140.724004230.72378260.724467150.724051240.72375970.724533160.723862250.72397580.724526170.724141260.72398290.724561180.723814270.72379When
the amplitude of interior perturbation forcer is equal to 0.723710, System 9 is
critical state from chaotic to periodic, one of 27 groups data of sharp tool is
merged to system 9, the system shows chaotic state, and when one of 27 groups
of wear tool is merged to system 10, the system shows larger scale periodic
state. For first group data, system 9 and system 10 are showed Figures 9 and 10.Sharp_1
phase plane and time domain.
(a)(b)Wear_1 phase plane and time domain.
(a)(b)The computation precision is six digits after decimal
dot for the threshold determined accurately, it is enough sensitivity for distinguish
wear tool or sharp tool. Of course, the difference engineering problem may
choose the difference computation precision for the threshold value in chaotic
system model.Finally, 0.723710 is a threshold value detected wear
tool. All 54 group data sampled is merged to system 9, respectively, time
domain map of each group data sampled, system phase plane, time domain map of
system state; they are shown in Figure11.Figure 11
Map of time domain to be detected signal,
phase plane, and state of time domain on system 9.


## 6. Conclusion

Currently, Duffing-Holmes
oscillator is the area of most intense research activity for developing weak
signal detected. The method which was described in this paper can be used as a
valuable tool for the tool condition monitoring. In comparison to conventional weak
signal detected, the advantages of tool wear detected based on Duffing-Holmes
oscillator were shown. Compared to the Lyapunov exponents calculated
determining the threshold of system chaotic critical state, the law of the golden
section spends the less time and useful engineering meaning. The computation precision
of the threshold can be calculated conventionally to satisfy the sensitivity of
wear tool detected.For the future development of the presented techniques in laboratory, several 10 approaches are to be tested. For example, relationship between the computation precision of the threshold and sensitivity of the chaotic critical state for difference engineering problem. Since Runge-Kutta method of fourth order is one kind of approximate solution method for dynamic equation, a difference time-step size will impact the computation precision for the threshold value.



---

*Source: 510406-2008-09-28.xml*