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									International Journal of Production Technology and Management (IJPTM), ISSN 0976 – 6383(Print),
International Journal of Production Technology
ISSNManagement (IJPTM), ISSN 0976 – 6383(Print)
and 0976 – 6391(Online) Volume 1, Number 1, January - February (2011), © IAEME
ISSN 0976 – 6391(Online) Volume 1
Number 1, January- February (2011), pp. 56-64                            ©IAEME


                                      Jacob Elias
                              Narayanan Namboothiri V.N
              Division of Mechanical Engineering, School of Engineering,
       Cochin University of Science and Technology, Cochin, Kerala-682022, India

                                     Rajesh V.G
        Model Engineering College, Thrikkakkara, Cochin, Kerala – 682022, India


Metal cutting is complex nonlinear dynamical process. Cutting signals from turning
operation exhibit a low dimensional chaos. When the depth of cut increases, with all
other cutting parameters remaining constant, creation of chatter occurs. A recurrence plot
based methodology is used to find the point of transition from normal cutting to chatter
cutting. In this method two signals, one input signal (current) and one output signal (tool
vibration) are recorded simultaneously at a constant sampling rate. A time series is
generated from the recorded values and recurrence plot is prepared for the input signal
and the output signal. This recurrence plot can be quantified using Recurrence
Quantification analysis (RQA). Variations in the roughness of machined surface created
by virtue of chatter, manifests as changes in the statistical properties of speckle images of
the surface when examined frame by frame along the axis of the machined part. A
significant parameter of such images, the frame wise average intensity value is extracted
separately and arranged in sequence for constructing the time series. Since this time
series is found to be non-stationary in nature, the nonlinear time series analysis
methodology of RQA is used for analyzing the time series. The result obtained is
compared to that of recurrence plot methodology. The experiment is repeated with
different work piece materials. The present study ascertains that this methodology is
capable of recognizing the transition from regular cutting to the chatter cutting
irrespective of the work piece material and the type of signal.

Keywords: Cutting signals, Recurrence Plot, Speckle image, Time Series Analysis

International Journal of Production Technology and Management (IJPTM), ISSN 0976 – 6383(Print),
ISSN 0976 – 6391(Online) Volume 1, Number 1, January - February (2011), © IAEME


Machining vibrations, also called chatter, correspond to the relative movement between
the work piece and the cutting tool. The vibrations result in waves on the machined
surface. This affects typical machining processes, such as turning, milling and drilling,
and atypical machining processes, such as grinding. Chatter results in poor surface
quality, unacceptable inaccuracy excessive noise, tool wear, machine tool damage,
reduced metal removal rate waste of materials and waste of energy.

        A new method called recurrence plots (RP), based on nonlinear data analysis has
become popular in the last decade. It is relevant for short, noisy and non-stationary data.
These features are indeed the crucial advantage of RPs. As deduction of information by
visually examining the RPs is more subjective, it was then developed into recurrence
quantification analysis (RQA). Here the number and duration of recurrences of a dynamic
system presented by its phase space trajectory are quantified. The Features extracted from
the RPs by RQA, contain information about the system. These features are called the
RQA variables and it can be used for characterizing a dynamic system.
        In the present work, the RQA methodology has been used in the analysis of an
input signal (current) and an output signal (vibration) for machining mild steel. The
signals are captured in a lathe under specified conditions of chatter, and investigated the
sensitivity of the extracted RQA variables to the transition from chatter free to chatter
cutting that had taken place during the process. It has been reported by Jacob et. al [1]
that by carrying out RQA analysis on the input signal, the point at which the onset of
chattering began could be determined. The abrupt changes that are seen in various RQA
variable values are suggestive of the onset of chatter. In this paper the possibility of doing
the same with an output signal and comparing it with the input signal is explored.


2.1 Experimental Setup
        An MS rod of 30 mm diameter is used as the test specimen in the present study.
This is converted into conical work piece with small end diameter of 20 mm and the
larger end diameter of 30 mm which has 150 mm axial length. The cutting has been
carried out in a heavy duty lathe of PSG make. The work involves reducing the tapering
section into a rod of 20 mm uniform diameter in a single pass of the cutting tool. A
CNMG 120408PM cutting tool with standard tool holder is used for this purpose. A feed
rate of 0.1mm/rev. at 360 rpm are set as the fixed cutting parameters whereas the depth of
cut steadily increases along the axis of the specimen at a constant rate owing to the
geometry of the section; the depth of cut varying from 0 mm to the maximum of 10 mm
at the 150 mm axial distance from the smaller end (Figure 1).

International Journal of Production Technology and Management (IJPTM), ISSN 0976 – 6383(Print),
ISSN 0976 – 6391(Online) Volume 1, Number 1, January - February (2011), © IAEME

        Figure 1 Test specimen

With the continuous increase in depth of cut a progressive reduction in surface finish is
expected due to increase in chip thickness. Keeping the spindle speed and feed rate
constant the chatter is to begin at some increased depth of cut or chip thickness. When the
chatter has begun, a drastic reduction in surface finish must result due to non-uniform
chip thickness.
        During the experiments two sensors are used. One is a lathe drive motor current
sensor and the other is an accelerometer for picking tool vibrations. The sensor
measurements are made simultaneously.

        The data acquisition system for drive motor current uses a 3 phase line current
sensor to measure the current drawn by the lathe drive motor. The sensor consists of a
current transformer (CT) having an output range of ±5 volts. The analog voltage signal
from the output of the CT is sent to DAC NI PCI 6221 through NI SHC68-68-EPM and
SCB 68 for converting it to the digital domain. The sampling rate is fixed at 500Hz which
guaranteed that the whole frequency domain contained in the signal is covered. The
digitized data is recorded in the PC hard drive using NI LabVIEW. Figure 3 shows the
set up for current measurement. Using this setup a time series of the input current is

Figure 2 Experimental set up used for                   Figure 3 Experimental set up used for
drive motor current recording                           speckle image recording

International Journal of Production Technology and Management (IJPTM), ISSN 0976 – 6383(Print),
ISSN 0976 – 6391(Online) Volume 1, Number 1, January - February (2011), © IAEME

An ADXL-150 accelerometer sensor is used to pick up the vibration of the cutting tool.
It is placed on the tool holder near its tail end to measure vibration in the feed direction.
The resulting output voltage signal, due to vibration, is amplified and passed through a
low pass filter having a cut-off frequency of 1 kHz. Here the sampling rate is fixed at 10
kHz. The data acquisition system for vibration signals used a different channel of the
same DAC NI PCI 6221 and similarly other peripherals along with NI LabVIEW.
Another time series with vibration is obtained from this.

The variations in the surface finish are recorded in speckle images captured in sequence
from point to point along the axis of the rod. Speckle photographs of the machined
surface are taken using Helium -Neon laser with a maximum of 10mW power out at
633nm. A spatial filter lens of 10mm focal length and 25µ pinhole size were used in
conjunction with it. The spatial filter output was collimated using an objective lens of
5cm focal length. The laser beam after filtering and collimation fall over the machined
surface at an angle of 45 degrees. The incident light which scatters off the surface has
been focused using a lens of focal length 10cm and the resulting subjective speckle
pattern was recorded. A Sony make Charge Coupled Device (CCD) array of VGA type,
1/3 inch, with 8bit mono recording and with 7 micrometer pixel size is used in the
recording. The exposure and the gain of the CCD were adjusted to ensure that the
recorded intensity lies well below the saturation value. For obtaining the maximum
contrast speckle pattern the position of the focusing lens and CCD array was adjusted.
The light distribution on the CCD was viewed through an IBM compatible computer.
Keeping the laser source position fixed, the machined sample was moved horizontally
along its axial direction in short steps with pauses in between using a micro-translator.
The test specimen axis and the line of the laser beam were designed to be contained in the
same horizontal plane so that effects due to the surface curvature are minimized. The test
specimen movement and the capturing of the speckle images are synchronized using
Motion Control and NI Vision Development Module software through PCI 6221 DAQ
Card. The speckle images of 480 x 640 sizes with 0 to 255 gray levels recorded at a rate
of one for each pause of micro translator. They were stored in a personal computer hard
disk for later processing. The speckle images of the surface were taken for the entire axial
length. The experimental set up used for speckle image recording is shown in Figure 3.

2.2 Time series construction
The recorded speckle patterns have been statistically analyzed to obtain some significant
parameter, which would vary with the surface roughness. It has been already established
that the parameters contrast ratio and surface roughness has an inverse relation between
them. [3] In the present study, the mean gray level value of the histogram of the speckle
is taken into consideration as it denotes the intensity of the speckle.
The mean grey level of the histogram is defined as
                                             i =0 i
                                                   f yi
                                     µ=                                                           (1)
                                              i =0
Where f i is the number of pixels having yi gray levels

International Journal of Production Technology and Management (IJPTM), ISSN 0976 – 6383(Print),
ISSN 0976 – 6391(Online) Volume 1, Number 1, January - February (2011), © IAEME

The mean gray level values of all the speckle images have been calculated and arranged
as per the order of capturing to get a spatial time series for offline analysis. The so
obtained time series is further analyzed for recurrence and subsequent quantification.

In this following section, the approach used in the study has been described which is
based on phase-space reconstruction, the recurrence plot, and the recurrence-
quantification analysis.

2.3 Phase space reconstruction
Delay coordinate reconstruction is the standard first step in most non-linear time series
analysis algorithms and proceeds by forming the multidimensional state space trajectory
of the signal. Takens [4] proved a theorem that is the firm basis of the methodology of
delays. Since only one variable is considered at a time, the delay coordinate approach is
used in the present analysis. Given a time series x(1), x(2), x(3),..........x(N) one can define
points X (i) in an m -dimensional state space as

     X (i) = [ x(i), x(i +τ ), x(i + 2τ ),.......x(i + (m −1)τ )]                                 (2)

for i = 1, 2,3,...., N − (m − 1)τ where i are time indices, τ, a time lag and m represents the
embedding dimension. Time evolution of X (i) denotes a trajectory of the system, and the
space, which this trajectory evolves in, is called the phase space. The parameter τ plays
an important role in proper reconstruction of phase space and had been carefully chosen.
After the transients were over, the evolution of the trajectory of the system settled
typically near a subset of an m-dimensional space and is named as an attractor [5].

2.4 Selecting the minimum embedding dimension

The purpose of the time-delay embedding was to unfold the trajectory in a sufficiently
large state space. If the embedding dimension had been too low, some neighbor points
might be close to each other due to the projection from some higher dimension down to
their lower dimension. Whereas, a too high an embedding dimension could lead to
excessive computation during evaluation of parameters. As Franca and Savi [6] indicates,
the method of false nearest neighbors is insensitive to noise; it is utilized here for the
determination of the embedding dimension.

        By checking the neighborhood of points embedded in projection manifolds of
increasing dimension, the algorithm eliminates 'false neighbors'. This indicates that points
apparently lying close together due to projection have been separated in higher
embedding dimensions [7]. A natural criterion for catching embedding errors appears to
be the increase in distance between two neighbored points is large when going from
dimension m to m+1. This criterion has been stated by designating as a false nearest
neighbor any neighbor for which the following is valid.

International Journal of Production Technology and Management (IJPTM), ISSN 0976 – 6383(Print),
ISSN 0976 – 6391(Online) Volume 1, Number 1, January - February (2011), © IAEME

                                          1/ 2
                 Rm+1 (i, r) − Rm (i, r) 
                   2              2
                                                     x(i + mτ ) − x(ir + mτ )
                          2                    =                              > Rtol            (3)
                        Rm (i, r)                         Rm (i, r)
Here i and ir are the times corresponding to the neighbor and the reference point,
respectively. Rm denotes the distance in phase space with embedding dimension m and
Rtol   is the tolerance threshold.

2.5 Selecting the time lag
For choosing the time lag, τ, in the present study, the non linear correlation function of
average mutual information is used, which, again as Franca and Savi [8] indicates, has no
noise sensitivity. Fraser et. al [9] have established that delay corresponds to the first local
minimum of the average mutual information function I (τ ) which has been defined as
                                                         P(x(i), x(i +τ )) 
                I (τ ) =   ∑P(x(i), x(i +τ ))log       2                    
                                                         P(x(i))P(x(i +τ )) 

where P( x(i)) , is the probability of the measure x(i) , P( x(i + τ )) is the probability of the
measure x(i + τ ) and P( x(i), x(i + τ )) is the joint probability of the measure of x(i) and
x (i + τ ) .   Plotting I (τ ) versus τ makes it possible to identify the best value for the time
delay, this is related to the first local minimum

The time lag value is located at the first local minimum of the average mutual
information function and it is selected as 7. The embedding dimension is selected by false
nearest neighbors algorithm to be 6.


3.1 Analysis based on Recurrence Plots
A recurrence plot is a way to visually investigate the multi dimensional phase space
trajectory through a two-dimensional representation. Recurrence of states of the system,
in the meaning that states are arbitrarily close after some time, is a well-known property
of deterministic dynamical systems and is typical for nonlinear or chaotic systems. An RP
has been derived from the distance plot, which is a symmetric NxN matrix where a point
(i, j ) represents some distance between coordinates X (i ) and X ( j ) on the phase space
trajectory. Thresholding the distance plot at a certain cut-off value transforms it into an
RP which shows all the recurrent points as black spots. In the present analysis recurrence
plots are constructed applying L2 norm in distance calculations. The threshold ε has been
chosen by analyzing the measure of recurrence point density [10] as percentage of
maximum distance. For the time series under study the calculated value of ε is 41.

International Journal of Production Technology and Management (IJPTM), ISSN 0976 – 6383(Print),
ISSN 0976 – 6391(Online) Volume 1, Number 1, January - February (2011), © IAEME

3.2 Recurrence Quantification Analysis
The RQA is a tool based on the statistical description of the parallel lines distribution
among the RP [11]. Measures of complexity are defined using the recurrence point
density and diagonal line structures in the recurrence plot. These measures provide a
qualitative description of the dynamics underlying the time series that is studied. In the
original definition Eckman et al [12] used a fixed number of neighbours for determining
recurrences. In the present analysis we use a fixed value for the threshold ε due to which
the RP is symmetric across the central diagonal, called the line of identity (LOI).
Attention has been focused on the diagonal and vertical structures in the RP since from
those stem the recurrence variables or quantifications. As the recurrence plot is
symmetrical across the central diagonal, all quantitative feature extractions take place
within the upper triangle in the RP [10], excluding the long diagonal, which provides no
unique information and lower triangle, and gives only redundant information.
Eight statistical values could be derived from an RP using RQA. These are:
     (1) Percent recurrence
     (2) Percent determinism
     (3) Linemax
     (4) Entropy
     (5) Trend
     (6) Percent Laminarity
     (7) Vmax and
     (8) Traptime

Turning of the designed conical workpiece can produce chatter and chatter free cutting.
Upto certain length the cutting is chatter free and beyond this point onset of chatter
occurs. By checking the machined workpiece under a profilometer this can be ascertained
from the poor surface finish. The constructed time series obtained from the sequential
speckle images (Figure 4) taken over the entire axial length of the test specimen therefore
contains this information about chatter-onset and the subsequent chatter-cutting. Using
this time series of mean gray level value of the histogram of the speckle, a recurrence plot
is constructed with the above chosen parameters for phase-space reconstruction and the
threshold radius (Figure 5). The RP axes are in time units. While plotting, a colour
scheme is selected to represent different recurrence points that are lying at different
radius within the threshold. As it could be seen there is a transition taking place at some
point of time, which in our case is a spatial point along the axial length of test specimen.
This point could be taken as the location where onset of chatter takes place. A recurrence
plot is plotted for the input parameter (current) and output parameter (vibration) from the
time series obtained from the current transformer and the accelerometer.

International Journal of Production Technology and Management (IJPTM), ISSN 0976 – 6383(Print),
ISSN 0976 – 6391(Online) Volume 1, Number 1, January - February (2011), © IAEME

                   Figure 4 Time series chart from mean gray level value

                                    Figure 5 Recurrence Plot

In the present investigation, recurrence plot prepared for the the input parameter (current)
and output parameter (vibration) are compared with the recurrence plot prepared from the
speckle images. By carrying out RQA analysis the point at which the onset of chattering
began could be determined. The abrupt changes that are seen in various RQA variable
values from Epoch No. 60 are suggestive of the onset of chatter. It has been found all
except the two RQA variables; Traptime and Laminarity exhibits this trend by producing
a sudden change in their values. Hence it can be concluded that any one signal can
faithfully reproduce the onset of chatter. The present work is an offline analysis of the
sample. It is possible to extent this technique for online detection of chatter. Compared to
the conventional chatter detection methods, this method is inexpensive and is a non-
contact and non-destructive technique.

International Journal of Production Technology and Management (IJPTM), ISSN 0976 – 6383(Print),
ISSN 0976 – 6391(Online) Volume 1, Number 1, January - February (2011), © IAEME


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