DIGITAL FILTERING TECHNIQUES FOR PROCESSING
SIGNALS UTILIZED IN THE UNMANNED SUPERVISION OF MACHINING
WILLIAM T. COBB JR.
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
To Catherine and Anastasia for their beautiful inspiration;
A bed time story sure to put anyone to sleep.
The author would like to express his sincere gratitude to Dr. Jiri Tlusty for his
support during this work.
Tlusty provided an unexcelled example of
how engineering practice should be based on solid basic understanding of the task at
The author also thanks Dr. Scott Smith whose friendship and guidance have aided
the throughout his
to Dr. Jose
guidance in the area of digital signal processing.
The author would like to express his
to Dr. Sencer
Dr. John Ziegert for service on his advisory
Special thanks go to the members of the Machine Tool Research Center, both past
experience and life during this work. Specifically, the author would like to thank Dr. Tom
Delio, Mr. John
Mr. David Smith,
Chuck Bales, Dr. Mahesh Nallakatla
Finally, the author would like to thank
his family whose constant support and
encouragement has sustained him through the long task of his graduate education.
TABLE OF CONTENT
THE FILTERING TASK
REVOLUTION REFERENCED COMB
Filter Notch Misplacement .
The Effects of Speed Variation
Infinite Impulse Comb . . . .
Results for the Infinite Impulse Comb Filter
The Effect of Notch Misplacement .. .. .
Conclusions for IIR Comb
High Pass Low Pass with Combs ..... .
Conclusions for High Low Pass Comb Filter
Other Possible Filtering Methods
Cascaded Notch Filters
Adaptive Cancelers ...
Synchronous Sampling .
TOOL BREAKAGE INTRODUCTION
Synchronized Sampling and the Per Revolution Domain
Model of the Cutter and Signal due to
Simnal and Milline Simulation
THE ONCE PER REVOLUTION DIFFERENCE SUB SYSTEM
THE PROGRESSIVE DAMAGE DETECTION SUB SYSTEM .
CONCLUSIONS AND FUTURE WORK
Filters for Chatter Detection . .
Filters for Cutter Breakage Detection
BIOGRAPE-IICAL SKETCH ........._____ ........._____
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
DIGITAL FILTERING TECHNIQUES FOR PROCESSING
SIGNALS UTILIZED IN THE UNMANNED SUPERVISION OF MACHINING
T. Cobb Jr.
Chairperson: Dr. Jiri Tlusty
The application of digital filtering techniques is applied to the task of unmanned
supervision of machining.
Unmanned supervision of machining is desirable to free the
operator from constantly overseeing the machining process and to expand his capability
to detect in-process cutter breakage and control chatter.
The filtering task is presented
in two areas; the filtering of a time synchronous sound signal used in the detection of
chatter, and filtering a rotation synchronous force or displacement signal in the detection
of in-process cutter breakage.
In the area of filtering for the detection of chatter, the use and limitations of a
revolution referenced finite impulse response comb filter is investigated.
The issue of
filter notch misplacement due to filtering the revolution synchronous signal utilizing a
time based sampling is addressed, and system design parameters are established for a
frequency range of interest versus sampling frequency.
during transient spindle speeds is also presented. Two al
The performance of the filter
ternative filtering techniques are
In the area of in-process cutter breakage, two filtering sub systems are presented.
The first is based on a once per revolution difference comb filter to detect catastrophic
in process breakage.
The sub system utilizes a moving average filter to suppress signals
utilized to detect slowly progressing cutter damage and the existence of
to the cutter.
The sub system is designed to be insensitive to cutter run-out and relies on
a calibration cut to establish threshold levels.
The purpose of this dissertation is to demonstrate the application of digital
filtering techniques to the unmanned supervision of machining.
Filters are applied to
two different areas of specific interest; filtering of a sound signal for use in the
detection and correction of chatter, and to the detection of milling cutter breakage in
It is of interest to automate these systems for several reasons.
implementation of unmanned milling frees the operator from simply watching the CNC
machine during the cutting process, this liberates the operator's time, allowing one
operator to operate several machines at once and relieves him of tedious supervision
In addition, the automated system is designed to increase the abilities of the
operator to detect and correct machining problems that were previously beyond his
Filtering is applied in the detection and correction of chatter during machining
operations. Chatter is the unstable self excited vibration of the cutting tool relative to
the work piece. Operation of the machine in a chattering condition leads to
unacceptable finish, out of tolerance parts and possible damage to the tool or machine.
The mechanism of chatter is controlled by the dynamic stiffness of the machine-tool-
work piece system.
It has been shown by
Tlusty [ 1
] that chatter is strongly
milling cutter teeth.
This leads directly to chatter stability being strongly dependent on
the spindle speed of the cutting machine.
of stability versus the spindle speed. Bec
The stability of the operation exhibits lobes
cause of this, it is of interest to operate the
machine in one of its highly stable lobes thus reducing the instability and allowing
higher stable metal removal rates.
An automated control system to do this has been
Tlusty, Smith and Delio [
The system known as the chatter
recognition and control, CRAC, utilizes a signal from the cutting process that contains
frequency information from the chattering cut. This signal may be a force signal, a
displacement signal, or any other representative input. In its present form the CRAC
system utilizes a sound signal from the cutting process. This signal is used as a matter
of convenience for instrumenting the process.
As the sound signal contains
information from the vibration of both the tool-machine system and the work piece, it
can be used to detect chatter of either the tool or the work piece.
action for both these conditions is the same.
Further, the microphone is not sensitive
to its placement depending on the mode of vibration; for instance, if an accelerometer
is used it must be located away from the nodes of vibration in order to sense the
As the mode of vibration is not previously known, the placement of the
accelerometer can not be a priori determined. Further, placement of the accelerometer
on the machine structure may completely isolate it from chatter of the work piece.
The chatter recognition and control
designed to automatically
recognize chatter and direct the cutting machine to its stable speeds.
To do this
signal of a stable cut and that of the chatte
process is made up of many components.
The signal from the cutting
Some are normal content of the signal from
a stable cut.
There are also noise components due to various sources such
equipment and bearing or transmission sources and
chattering of the process.
the signal component due to the
The chatter control system recognizes chatter as a spectral
peak that exceeds a predetermined background threshold that is not due to normal
cutting signals or known noise sources.
It is thus the task of the filtering system to
remove the content due to normal cutting and all known noise sources from the signal
so that the only content remaining will be that of the chatter.
Therefore any spectral
peak in the filtered signal that exceeds the threshold is determined to be that of chatter
and the control system acts on this signal.
By recognizing and controlling chatter, the system allows the operator to
correct a chattering condition,
would not be possible without the monitoring
Although an experienced operator can recognize chatter by its distinctive
sound and the surface left by the cutter, it
is not possible to properly correct the
cutting conditions without knowing the frequency content of the cutting signal.
the use of this system not only allows less manned supervision of the cutting process,
it allows enhanced performance that would not be possible without the system.
Filtering is also applied to the detection of milling cutter breakage.
desirable to detect milling cutter breakage in process.
If the cutter edges break during
the cutting process, the surface produced can deteriorate giving an unacceptable finish.
a tooth can overload the subsequent teeth, leading to rapid failure of the remaining
This can produce damage or destroy the cutter,
work piece or machine tool
Thus it is important to quickly detect the breakage of a cutting edge
and stop the cutting process before damage can spread.
The signal due to tool
breakage must also be definitively distinguished from other signals due to
This must be done to avoid false alarms from the detection system.
False alarms lead to stoppage of the cutting process and inefficiencies associated with
Further, if the system consistently issues false alarms it will become
recognized as unreliable and will be disconnected and unused.
The system to detect cutter breakage utilizes a force or displacement signal in
order to detect breakage.
The displacement signal is that of the forced displacement of
the spindle relative to the machine tool frame.
The displacement signal is used to
represent the force on the cutter. It must be understood that this is not the true force
signal, but the force signal filtered through the displacement versus force transfer
function of the individual tool and spindle combination.
The signal used in the cutter
breakage detection system is synchronously sampled; that is the signal is sampled in
synchronization with the spindle rotation.
Each sample is acquired at the same
position during the spindle rotation on each revolution.
This synchronizing leads to
great simplicity in the filtering of the signal and non-varying coefficients for digital
filters that are applied to it.
In the detection of cutter breakage it is, as in chatter, necessary to separate the
The task of the filtering is to remove the signal content due to normal
operation from those signals that are indicative of cutter breakage.
The other major
task of the cutter breakage system is to establish a threshold which delineates between
breakage and nonbreakage of the cutter.
the cut, such as the radial immersion. 1
This threshold varies based on parameters of
[he overall system is based on two sub-
systems. One is designed to eliminate signal content due to cutter run-out and react to
in process breakage.
This sub-system is insensitive to slow damage such as chipping
or beginning the cutting process with a previously damaged cutter.
The second sub-
system is slower reacting but is sensitive to the absolute condition of the cutter.
is, the system is sensitive to slowly developing cutter damage and previously damaged
The cutter breakage system frees the operator from watching and listening for
With the system in place it is practical to push the cutter at higher
loads and longer in to the cutter life.
If the system is reliable, the cutter can be used
more aggressively without the danger of a tooth failure destroying the cutter or work
THE FILTERING TASK
Digital filters in chatter control are used to remove signal content due to
normal cutting and known noise sources from the content due to chatter in the cutting
process. Machine tool chatter is a self excited vibration of the tool relative to the
work piece. This self excited vibration occurs at the most dynamically flexible mode
of vibration of the system, provided steps have not been taken to stabilize a mode.
The task of chatter recognition and control is to recognize the signal due to chatter and
adjust the machining conditions to the most stable configuration.
In order to recognize
chatter, the signal content due to normal cutting and known noise sources must be
filtered from the cutting signal.
In this way the remaining content is due to chatter
and the control system can act on this signal.
In the present chatter control system at the Machine Tool Research Center at
the University of Florida, a microphone is used to sense chatter. The microphone is
nonintrusive and senses both vibration of the tool and of the work piece.
description of the chatter control system can be found in the works by Delio, Smith
and Tlusty [
The signal gathered contains content due to the normal cutting,
noise from other sources such as electrical or bearing passing, and the signal from the
chatter vibration if it is present.
The signal content due to normal cutting conditions
Because this periodic striking is not harmonic in nature, in fact it may resemble
a series of pulses, the signal contains frequency content at the tooth striking frequency
and its harmonics.
It can be shown that the regenerative feedback mechanism of
variable chip thickness that causes the instability of chatter is defeated by operating
the machine such that the tooth frequency is equal to the natural frequency of the
mode of chatter of the machine [
Further, it can be shown that the mechanism is
defeated if there is an integer number of vibration waves between each successive
tooth of the milling cutter.
tooth to tooth.
In this way the phase of vibration does not change from
This constant phase produces a constant thickness chip, thus causing
Because of this, chatter cannot occur at the tooth frequency or its
before, it is always desirable to filter the tooth frequency and its
harmonics from the signal collected to sense chatter.
There is also content in the milling signal due to slight tooth throws.
throw is the misplacement of a cutting tooth off the ideal circle of the cutting tool.
This is shown in Figure 2-1.
The misplaced tooth inputs a once per revolution pulse
Clti- of Pfolotio
Slo1 Cutter Cc,-e
*- *,agh 1Ihro.
'^- C~l ^Ir' lfmfr -
i~' Cutte boyr Bb
into the cutting signal.
of the cutter.
The duration of this pulse is dictated by the radial immersion
The once per revolution pulse contributes frequency content at the once
per revolution frequency known as the run-out frequency and its harmonics.
shorter the pulse,
that is the lower the radial immersion, the farther the power will be
spread into the harmonics.
This signal content must also be filtered from the chatter
The run-out harmonics are not stable milling conditions as was the case with
the tooth frequency harmonics.
Therefore, chatter can occur at these frequencies.
thus desirable to remove only those run-out harmonics necessary to avoid false
triggering of the chatter detection system and leave the remaining harmonics unfiltered
to avoid filtering of a valid chatter signal.
Other signals that must be filtered from the chatter signal can be separated into
Frequency stationary signals, such as electrical noise, remain at a set
frequency no matter the operational conditions of the machine tool.
There is also
noise that varies in frequency but is located at a set multiple of the spindle rotation
Signals such as this arise from various sources such as air passing over a
bearing or the noise from a gear train attached to the spindle.
These signals must be
removed from the chatter detection signal to avoid false triggering.
A short introduction to digital filtering basics is included here.
This is not
meant to be a comprehensive introduction, but only to introduce those concepts
necessary in understanding the filters implemented and the reasons behind them.
complete introduction to digital filtering and general digital signal processing is
included in Strum and Kirk, Discrete Systems and Digital Signal Processing
Much of this basic introduction is condensed from this text.
Digital signal processing is based on the processing of digitally sampled
These signals are commonly sampled at equal time intervals, the time interval
is dictated by the sampling frequency, f, .
The digital frequency,
0, of a signal is
defined as the ratio of the frequency of the signal to the sampling frequency times two
By utilizing the digital frequency of a signal the individual sampled values can be
attained by simply indexing the sample number.
For example, the sampled values of a
cosine wave of frequency, f, sampled at a frequency f, can be evaluated from the
II -L- a
It is obvious that the Nyquist frequency, f
, (the maximum frequency that can be
resolved) has a digital frequency of t.
A simple representation of a digital filter is shown in Figure 2-2.
The input of
the filter is represented by the series of samples X(n) and the output is represented by
Digital filters can conveniently be separated into two classes; finite impulse
( FIR filters) and infinite impulse filters ( IIR filters).
For a FIR filter the
The output of a FIR filter can be written as linear combination of the input
Y(n) can be represented as follows:
. .. bkX(n-k) =
where k is the order of the filter and the b's are the filter coefficients.
commonly represented by the flow diagram shown as Figure 2-3,
This series is
where the D
represents a delay of one sample period and the b's are again the filter coefficients.
a delay operator, Z, is defined the series can be written in terms of this delay operator
= (bo +b,
-2 + S
Signal Flow Diagram for a Finite Impulse Response Filter
-1+bz -2+ .. +bkz
The roots of the Z polynomial in the numerator of the transfer function are
know as the 'zeros'
of the transfer function.
It can be shown that for real input and
real output, these roots will either be real or occur in complex conjugate pairs.
rooted transfer function can be written as follows.
ZR1 S (
These roots can then be plotted in the complex Z plane as in Figure 2-4.
a single pair of complex conjugate roots are shown in the figure.
figure is the unit circle of radius one.
The circle in the
The frequency response of the filter transfer
function may be evaluated by replacing the delay operator Z with
digital frequency at which the response is to be evaluated.
where O is the
By varying theta
zero to pi, the continuous Fourier transform of the impulse
response of the filter may be evaluated.
By evaluating the function at discrete
locations with spacing equal to two pi divided by the number of points in the
transform, the discrete fourier transform is evaluated.
The section of the transform
past the Nyquist frequency of pi will be a mirror image of the section below pi, thus it
is only necessary to evaluate the function from zero to pi.
The evaluation of the frequency response can be thought of in graphical terms
hv imaoinno nihbta ic hrnn ninn in the -rnmnleY 7. nflnf
*~l r I) J K.Z1 t
inws a sinle
Pole-Zero Plot for a Second Order FIR Filter.
function can be written as
z-le -je )
The frequency response is evaluated by replacing Z with a unit vector at the
desired frequency, e'e The magnitude of the freque
product of the magnitude of each the numerator terms.
;ncy response is then the
The magnitude of a numerator
term is the length of the vector from the tip of the e& vector to the root for that term.
This is represented in the figure as N,
for the two numerator roots.
all magnitude can then be written.
Graphical Magnitude Evaluation for a FIR Filter
The magnitude of the
For low frequencies near
will be less than one, thus the overall magnitude is
less than one and signals in this frequency range are attenuated.
As theta approaches
becomes very small and this term dominates the overall response of the
Because the root is on the unit circle,
when theta equals theta,
The filter completely removes any signal content at this frequency.
There is a
zero in the magnitude, hence the term zero for the root.
As the frequency is further
increased, both N,
become larger than one and the filter amplifies signals at
the higher frequencies.
The complete frequency response can be pictured this way.
frequency response from zero to pi is plotted as Figure 2-6.
theta equal zero, both N, and N,
Magnitude Response of a Second Order FIR Filter.
Infinite impulse response filters depend both on the input and past output of the
filter to form the output series.
The filtering action can be written as follows:
where the terms are as previously mentioned and the a's represent the coefficients
placed on the past output values of the filter.
This can be more compactly written as
A flow type diagram for an IIR filter is shown in Figure
. There is the
of an unstable filter.
delay operator Z as
The transfer function of the filter can be written in terms of the
(bo +b, z
The roots of the denominator polynomial are known as the 'poles' of the filter.
The poles must lie with in the unit circle in the Z plane in order for the filter to be
Signal Flow Diagram for an Infinite Impulse Response Filter.
Just as with the FIR filter, the filter response for the IIR filter may be pictured
A diagram for a two pole, two zero filter is shown as Figure 2-8.
zeros are again located on the unit circle and the poles are located at the same digital
,as the zeros but inset from the unit circle.
This is of course not a
requirement but used for illustration of the magnitude evaluation.
function can be written in its rooted form as follows:
n-fl rC jOIA
-1+ a, z
-1 +b, z
Graphical Magnitude Evaluation for an IIR Filter
The magnitude is again the product of the magnitude of the individual terms of
with the numerator product now divided by the denominator product
instead of one.
The terms D,
and D2 are the magnitudes of the vectors from the tip of the
The magnitude response is shown as Figure 9, as the frequency approaches the
frequency of the zero, the magnitude still goes to zero as was the case with the FIR
But, because the vector D,
is also shrinking, the magnitude is less effected
until the frequency closely approaches the zero.
This results in a much sharper or
higher quality notch in the frequency response function.
Magnitude Response of a Second Order IIR Filter.
Pole Radius 0.75.
REVOLUTION REFERENCED COMB
In its present form, the chatter recognition and control system utilizes a
specialized form of a FIR comb filter.
This filtering method has been described by
Tlusty et al. [
6 ] and by Frost [ 7 ].
The comb filter is a filter with its zeros equally
spaced around the unit circle.
This equal placement of the zeros leads to a very
simple difference equation used to implement the filter.
zeros would have k terms in the rooted numerator. Thi
In general a filter with k
is leads to a numerator
polynomial with k+1
In order to implement this, the coefficients must be
calculated and then each multiplied by the appropriate sample in the input series and
then summed to form the out put sample.
Neglecting the evaluation of the
coefficients, this leads to k+l multiplies and k additions for each out put sample.
the comb filter with the zeros located on the unit circle, the terms of the factored
numerator can be written
The product of these terms leads to the simple form of the comb filter transfer
It is seen that in order to implement the kth order comb filter, only a single subtraction
of the sample k delays back, from the present sample, is required to form the out put
The action of the comb filter is easily understood in the time domain.
filter is applied to a signal from a rotating cutting tool, and an integer number of
samples is collected each revolution,
by subtracting the sample one revolution back
from the present sample any signal that repeats each revolution will be canceled.
This is represented by the simple difference equation used to implement the filter.
= X(n) -X(n-k)
There will be a zero at the run-out frequency and each of its harmonics.
number or filter order is equal to the number of samples taken each revolution.
The zero plot of a 15i order comb filter is shown as Figure 3-1. The
associated magnitude response of the filter is shown in Figure 3-2. It can be seen that
the equally spaced zeros produce equally spaced notches in the magnitude of the filter
Also the signal between these zeros is distorted.
If a signal is
exactly in between two zeros it will be amplified by a factor of two.
This is due to
the subtracted wave at these points being out of phase with the present signal.
way the subtraction adds the two waves producing constructive interference.
It can be
shown that the actual form of the magnitude can be written as follows.
will be attenuated by the filter.
Zero Plot of a 15th Order Comb Filter.
The actual implementation of the comb filter is done using two signals.
system is referred to as the Revolution Referenced Comb.
The two signals used are
the sound signal from the cutting process and a once per revolution signal from the
The sound signal is of course the signal that is to be filtered for the detection
The once per revolution signal is used to adjust the filter.
sound signal is sampled at a set sampling frequency, there is not the same number of
Transfer Function of 15' Order Comb Filter.
revolution at any given speed.
signal from the spindle. The
This is accomplished by using the once per revolution
number of samples acquired between successive once per
revolution signals is counted by the controlling program.
In this way the number of
samples per revolution is known and hence the proper delay for the filter is known.
Due to the method of the setting of the filter delay and the signal being
sampled at a constant frequency, there is a misplacement of the filter notches in this
Because the spindle speed is in no way synchronized with the sampling of
there is not normally an integer number of samples per revolution of the
However, the comb filter must operate with an integer delay number ( a non
integer delay would require interpolation between the sampled values).
that the true delay must be rounded to the nearest integer.
In fact, due to the method
To count the delay number the controlling program counts the number of
samples acquired between edges of the once per revolution signal.
If the once per
revolution edge comes just previous to a sample being taken, the sample period was
indeed in that revolution.
However, it would not be counted by the system as the
sample was not acquired in that revolution.
rounded down to the integer value. It can
By this mechanism the delay is always
be seen that in the limit case the counted
delay would be one sample period shorter than the actual revolution of the spindle.
The digital frequency of the run-out signal can be calculated as follows:
where N is the number of samples per revolution.
filter notches can then be calculated
The maximum misplacement of the
where is the true digital frequency, 0 is the calculated one from the once per
The true noninteger number of sample periods in the revolution is
counted number of samples per revolution.
N., and N is the
By the limiting condition imposed by the
therefore the maximum misplacement can be calculated
It is seen that the misplacement of the filter notches is strongly dependent on
the number of samples per revolution, that is, the ratio of the spindle run-out
frequency to the sampling frequency.
Obviously if there are many samples on a wave
the misplacement of the subtracted wave by a sample period will have little effect,
while if only a few samples are available the misplacement by one represents a
significant phase shift and will greatly disturb the filtering action.
misplacement of the primary notch at the run-out frequency.
harmonics are misplaced by a multiple of this error. This is
The notches at the higher
due to having fewer and
fewer samples on each wave as the frequency increases.
From the previous discussion it is seen that in order to diminish the
misplacement of the filter notches, the highest possible sampling frequency should be
The data must be sampled at this high frequency, it is not sufficient to sample
the once per revolution signal at a higher frequency to better resolve the spindle speed.
Unless some sort of interpolation scheme is used, the comb filter must work on integer
numbers of delays therefore requiring the high speed sampling of the data.
For a set sampling frequency and a set range of interest in the frequency
response of the machine tool, the misplacement of the filter notches is independent of
For example, if the spindle speed is such that there are ten harmonics
first), then the misplacement of the notch for the tenth harmonic is 10* ..
spindle speed is increased to ten times its original speed, the misplacement of the
primary notch is now ten times the misplacement of the original primary notch, and
therefore has the same misplacement as the previous higher harmonic notch.
not give the exact same effect in practice, because the power in the primary and the
harmonics are not necessarily equal.
Filter Notch Misplacement
The performance of the revolution referenced comb is presented with emphasis
on two areas of concern, the effect of notch misplacement and the effect of speed
variation on the filtering.
The effect of notch misplacement is investigated through the
simulated cutting signals.
Simulated cutting force signals are shown in Figures
3-3 A and B.
The signal in A is from a four tooth cutter operating at 2500 RPM
a sampling frequency of 10,000 Hz.
This gives exactly
240 samples per revolution,
therefore there is no misplacement of the filter notches.
similar cutter operating at 2490 RPM.
The signal in B is from a
This is gives 240.9 samples per revolution.
both cases the revolution referenced filter would filter the signal with a delay count of
This produces a misplacement of the primary run-out notch of
97.8 xl0 -6
radians in case B (the maximum possible for a 240 sample delay being 108.6 x 106
Therefore case B
has nearly the maximum misplacement of the filter
notches that can be.
Both simulated cuts are run with very shallow radial immersion,
tooth throw of ten percent of the feed per tooth is included to introduce the runout
harmonics into the spectrum.
The x direction force is plotted.
Simulated X Direction Force Signals for a Four Flute Cutter with
Radial Immersion and 10% Tooth
Throw, Sampling Frequency
The spectrum of the 2500 RPM cut is shown in Figure 3-4.
shows strong content at the tooth harmonics as would be expected for the shallow
immersion cut and minor content at the run-out harmonics due to the tooth throw.
The spectrum of the signal filtered through a 240 delay comb filter is shown in Figure
The filter almost completely removes al
the content of the signal.
expected as there is no notch misplacement for the case.
1000 2000 3000 4000
Spectrum of Four
Tooth 5% Radial Immersion Cut, 2500 RPM,
Spectrum for Four
Tooth 5% Radial Immersion Cut, 2500 RPM,
Figure 3-6 shc
10,000 RPM, Filter through a 240 Delay FIR Comb Filter.
,ws the spectrum of the 2490 RPM cut. The spectrum is
essentially the same as that of the 2500 RPM cut except for a slight frequency shift
due to the different spindle speed. The sign,
al is then filtered through the 240 delay
Spectrum of Four
Tooth 5% Radial Immersion Cut, 2490 RPM, Sampling
seen that the misplacement of the filter notches has caused the filter to leave a
significant portion of the signal.
Further, past a certain frequency in the spectrum the
ines are not filtered out but in fact amplified by the revolution referenced
00 iJ.j 1
S3000 3000 4000 50C
The performance of the filter at this limit case with maximum notch
misplacement can be investigated, in order to establish minium performance criteria.
The filter action is of course to subtract the wave from the previous revolution from
that of the present.
by one sample. TI
sample period. Th
In the limit case the number of samples in a wave is miscalculated
lerefore, the subtracted wave will have a phase shift equal to one
is can be evaluated in terms of the number of samples per wave.
there are m samples per wave the phase shift in the limit case will be equal to 2t/m
The results of these calculations are shown in Figure 3-8.
factor of the filter is plotted verse the number of waves per cycle for maximum notch
0 10 20 30 40 50
Samples per Cycle
Magnification versus Number of Samples per Cycle at Maximum Notch
al points of this plot are of interest. With only two samples per wave the
phase shift is it or 180 degrees; when the two out of phase waves are subtracted the
4 4 S
Therefore any wave with fewer than six samples per cycle may be magnified by the
revolution referenced filter.
This is the lower limit performance of the filter, that is
with the spindle speed such that there is a one sample miscalculation of the spindle
speed. If the spindle is operating at a speed with less notch misplacement, ie. closer to
an integer number of samples per revolution, the performance will
improve with the
limit being at an integer number of samples or synchronous sampling.
These results can be checked against the previously plotted signal for a spindle
speed of 2490 RPM.
For a sampling frequency of 10,000 Hz,
the frequency of a
wave with six samples per cycle is 1666.7 Hz.
By comparing the plot of the
unfiltered to the filtered signal it can be seen that the content above this
fewer than six samples per wave, is in fact amplified by the filter.
This can also be used to set the minimum required sampling frequency for a set
filtering action. For example, if the frequency content of a signal must be attenuated
by a factor of two (magnification factor equal to 0.5) through a frequency of 5000 hz.,
there must be
13 samples or more per wave.
Therefore to have
13 samples per wave
at 5000 Hz, the sampling frequency must be at least
The Effects of
The effects of spindle speed variation on the filtering action are demonstrated
through the use of a time domain cutting simulation that allows nonconstant spindle
The speed variation is used to simulate spindle speed sag upon entering a cut.
variation and the number of revolutions of the cutter to be simulated.
It also allows
input of other cut information, such as the number of teeth on the cutter, radial and
The speed variation is then set equal to a full cycle of a cosine
with length equal to the entire simulation time, and amplitude equal to the
percent speed variation times the average s
the average speed and the speed variation.
The overall speed is then the sum of
The simulation also generates a series of
the integer number of samples per revolution for use in the revolution referenced
sample number (samp. freq.
= 10,000 hz)
Speed Profile for 1.5% Speed Variation Over 30 Revolutions, Sample
This speed profile is shown in Figure 3-9.
RPM and a speed variation of 1.5% is applied, a
The average speed is set to 2500
nd the simulation is run for 30
revolutions of the cutter with a sampling frequency of 10,000 Hz.
average speed variation of 0.1% per revolution.
This gives an
Similar speed variations have been
I fl I S -n fl IS1 1 I 1 V *
Three simulations are used to first demonstrate the effect of the speed variation
The first is a steady state cut with a spindle speed of 2500 RPM, the second is
a simulation with a 1.5% speed variation as shown in the previous figure, and the third
has a large speed variation of 7.5
% or 0.5% per revolution.
The effect of the speed
variation on the spectral content is shown by plotting the magnitude of the Fourier
transform of the signals.
The window of the transform is 1024 points long, therefore
encompassing approximately 4.25 revolutions of the cutter.
The data window
transformed is selected between points
1001 and 2024 of each simulation.
It can be
seen in the previous figure that this data is located during the period of maximum
speed variation for the simulation.
Each of the simulations is run with four cutting
teeth and a radial immersion of
The very low radial immersion is used
to assure that there will be frequency content throughout the spectrum.
Figures 3-10A, B and C show the transformed data for the steady state,
speed variation and 7.5% speed variation, respectively.
In the steady state milling
condition the tooth frequency harmonics are clear and sharp, but as the speed variation
increases, there is a blurring of the spectral peaks.
This is particularly apparent in the
speed variation case, in which the upper harmonics become so blurred as to be
lost and the lower harmonics widden their peaks as compared to the steady state case.
The action of the revolution referenced filter is shown in Figure 3-11
A and B.
figure shows the spectrum of the filtered signal for the 1.5 and 7.5% speed variations.
The figures are plotted to the same scale as the unfiltered signals for
Spectrum of Simulated Cutting Force Signals
for A) Steady State
Speed Variation (
7.5% Speed Variation.
The filter uses the count signal generated by the simulation to update its
delay value once per revolution.
Therefore, the action of the filter on the simulated
signal is identical to that of the actual revolution referenced filter.
The filter shows excellent performance on the 1.5% speed variation case,
eliminating all noticable peaks.
speed variation case still has some low
frequency content, but it has been greatly attenuated.
The performance of the filter in
Spectrum of Simulated Cutting Force Signals Filtered Through the
Revolution Referenced Comb Filter
1.5% Speed Variation B)
Average Speed 2500 RPM,
the higher frequency ranges should diminish first, as the phase of these signals would
unfiltered spectra, the natural blurring of the high frequency content due to the
averaging effect of the Fourier
Transform effectively removes these signals.
Although simple to implement and very effective, the revolution reference
comb filter has two main draw backs.
First, because of the comb action with the run-
out as the primary signal, all of the harmonics of the run-out are filtered from the
As previously mentioned in the introduction, these are not necessarily
stable milling speeds.
Therefore the filter can inadvertently filter a valid chatter
Recently it has been shown by Smith and Winfough [ 8 ] that for a four flute
cutter, one of these notches falls such that chatter at the most unstable speed is filtered
The second area of concern is the relatively wide filter notches.
notches filter what may be a valid chatter signal that is near a run-out harmonic.
revolution referenced comb attenuates one third of the frequency spectrum to half
power or lower,
when compared to the power of the maximum transmitted signal.
Two modifications to the revolution referenced comb are presented here.
the addition of poles to the filter to make an IIR comb filter which narrows the filter
notches and reduces the likelihood of filtering a valid chatter signal that is near a run-
The second modification splits the signal into two signals and performs
separate filtering functions on each signal portion, then recombines them.
This is done
to avoid filtering all of the run-out harmonics and thereby avoid filtering a valid
chatter signal that may exist at one of the upper run-out harmonics, while still
removing the lower harmonics to avoid false triggering of the system. The specific
detail and the results of the application of the two modifications is presented in the
Further the existence of the run-out harmonics in the spectrum is
explained and consequently, the number of harmonics that should be filtered is
Infinite Impulse Comb
The first modification to the revolution referenced comb is the addition of poles
to the filter to make an IIR comb.
This is done to narrow the filter notch width to
avoid filtering of a chatter signal that is positioned in the frequency domain near one
The revolution referenced (FIR) comb is modified by placing a series of poles
at the same frequency locations as the zeros of the FIR comb but slightly inset from
the unit circle.
The transfer function of this filter can be written as follows:
where N is again the number of samples per revolution and R is the radial location of
The radius of the poles must of course be less than one for stability.
difference equation used to implement this can be written:
-X (n-N) +R
It is seen that the implementation requires only one more multiplication and one more
addition per output sample than the FIR comb.
The effect of the added poles is shown in Figures 4-1 and 4-2.
shows a pole-zero plot of the IIR comb filter with the poles inset along the radial lines
through the zeros.
The magnitude response transfer function is shown as Figure 4-2.
Four curves are plotted,
with radial locations of 0, 0.5, 0.9 and 0.95.
The curve for
radial location zero is the same transfer function as the FIR comb.
As the radius is
increased it can be seen that the filter notch widths can be narrowed or sharpened, thus
attenuating less of the surrounding signal.
Additionally the pass bands, between the
zeros, are flattened, giving a much more even amplification of the transmitted signal.
Transfer Function of IIR Comb Filters with
A concern of implementing the IIR comb with its narrow notches is the effect
of spindle speed transients on the filtering action.
The spindle speed may sag when
entering a cut, and when this happens, the run-out frequency and its harmonics also
the IIR filter may allow the decreasing frequency to pass.
If this proves to be the
in practice, the IIR filter can be modified to include a term to vary the radial location
of the poles based on the amount of spindle speed variation.
The radial location can
be based on a simple linear formula as follows:
where N is the count number for the present revolution and Npt, is the count for the
The slope is the sensitivity to speed variation, and R sets the
narrowest notch width,
which is used during steady state cutting.
By this method,
when there was a speed transient the poles would be moved in towards the origin,
thereby increasing the width of the filter notches.
In the limit case the poles would be
allowed to move to the origin (a limit would be needed to keep the radial location
greater than or equal to zero) which would give the same transfer function and
performance as the present FIR filter.
As the speed stabilized, the poles would move
out, narrowing the filter notches thereby attenuating less of the signal.
Although the filter notch widths will be reduced by the addition of the poles,
two problems of the FIR comb are still present in the IIR comb.
The filter still
eliminates all the run-out harmonics, possibly filtering a valid chatter signal.
the notches have been sharpened this is somewhat less likely with the IIR comb.
There is still
a misplacement of the zeros and poles due to the discrete delay required
by the comb filter.
Results for the Infinite Impulse Comb Filter
The performance of the Infinite Impulse Response comb filter is first
demonstrated on simulated stable milling cuts.
The signal used in demonstrations of
the revolution referenced comb is also used here for direct comparison.
That signal is
of a four tooth cutter, milling at 2500 RPM at 5% radial immersion (to assure
frequency content throughout the spectrum) and the sampling frequency is 10,000 Hz.
The Fourier transform of the unfiltered signal, the signal filtered through the
FIR comb, and the signal filtered through IIR combs with pole radii of 0.7 and 0.9, are
shown in Figure 4-3 A, B, C and D, respectively.
Note that for direct comparison the
filtered signals are plotted on the same scale as the unfiltered signal.
signal from the FIR comb and the IIR comb with pole radius of 0.7 are replotted on an
expanded scale to better reveal their form in Figure 4-4.
It can be seen in the figures that the FIR comb with its wider notches more
effectively removes the tooth frequency harmonic components from the signal.
expanded plot the peaks are seen to be spread by the FIR filter,
while for the IIR, they
The content in either filtered case is seen to be greatly reduced when
compared to the unfiltered case.
o 2CC tQaoe uo~ 2rn 1 4
Friincm y ( ig
Transform of Simulated Cutting Force Signals
% Radial Immersion Sampling Frequency
B) FIR Comb Filtered
Pole Radius = 0.7
Expanded Frequency Plot of Filtered, Simulated Force Signals
B) IIR Comb Filter, Pole Radius = 0.7.
The effect of the IIR comb on an unstable cut with the chatter frequency very
close to a run-out harmonic is now compared to the performance of the FIR comb in
The effects are demonstrated on a simulated unstable milling cut
with a single degree of freedom.
The natural frequency of the mode of the system
was varied to enable placing the chatter near a run-out harmonic of the cutting force
The simulated cut is for an eight tooth cutter operating at 2500 RPM with
percent radial immersion.
The sampling frequency for the signal is 10,000 hz.
natural frequency of the chatter mode was first chosen equal to 579 hz., resulting in
chatter very near the
14 run-out harmonic which is located at 583.3 hz.
Hz. The spectrum of the unfiltered signal is shown in Figure 4-5.
near the 14" run-out harmonic.
The chatter is seen
Spectrum of simulated Force Signal Unstable Milling, 2500 RPM, 8
Radial Immersion, Natural Frequency 579 Hz.
The effect of both the FIR and IIR cc
shown in Figure 4-6 A and B, respectively.
with pole radius equal to 0.85, are
The FIR filter is seen, as expected, to
800 100 1200 1400
800 1000 1200 1400
more thoroughly remove the tooth frequency components than the IIR filter.
the FIR filter also suppresses the chatter signal more than the IIR filter: the IIR filter
leaves more than twice the power in the chatter signal.
Chattering Force Spectra for Natural Frequency Equal to
The performance of the filters is demonstrated as the chatter frequency moves
further from the run-out harmonic.
to 575 hz.
The natural frequency of the system is decreased
spectra for the two filtered signals are shown in Figure 4-7.
FIR filter still slightly suppresses the chatter signal.
The chatter signal in this case is
now completely out of the IIR filter notch and is in fact slightly amplified when
compared to the unfiltered signal.
This slight improvement of the IIR filter as
compared to the FIR is only seen in the very limited situation of the chatter being
rlncliv 21 1 nipr1 xnith a rin-.nuit h-armrnnir
The Effect of Notch Misplacement
Because of the narrow notch width of the IIR comb as compared to that of the
FIR comb, the effect of notch misplacement will be much more prevalent for the IIR
The effect of notch misplacement is investigated using the same simulated
signal as used to demonstrate the effect on the FIR comb.
of a simulated
milling cutter with four teeth cutting at 2490 RPM, and a sampling frequency of
This gives just under 241 samples per revolution, such that the filter will
be operated with a delay equal to 240 samples and have nearly maximum notch
In order to assure frequency content at the tooth harmonics through
out the spectrum, the cut is simulated to have radial immersion of 5%.
shows the spectra of the simulated signal.
A is the unfiltered signal,
B is the
signal filtered through the FIR comb, and C and D are filtered through IIR comb with
pole radii of
0.7 and 0.9 respectively.
The IIR combs are both seen to perform poorly under the condition of notch
The spectrum of the filtered signal from the comb with pole radius of
0.9 is almost identical to the unfiltered signal.
being greater than the notch width of the filter.
This is due to the notch misplacement
Hence, the filter notches completely
miss the run-out harmonics they are meant to filter, leaving the signal essentially
L.. I, ..1t 1, t .1 ,
I IM# RIW Fiscar O R
Effects of Notch Misplacement
IIR Comb, Pole Radius
\) Unfiltered Signal
Pole Radius = 0.9.
Conclusions for IIR Comb
Due to its narrow notch width, the IIR comb filter was shown to give an
advantage when filtering a signal with the chatter component very close to a run-out
The narrow notch width, however, introduces a serious disadvantage to the
as compared to the FIR comb.
Due to the narrow notch width, the effect of
notch misplacement, from using a non-integer number of samples in a revolution, is
much more severe.
The notch misplacement can cause the IIR filter to be completely
ineffective in filtering the tooth and run-out harmonics in the present system, thus it is
not recommended for use.
Steps could be taken to reduce or eliminate the problem of notch misplacement,
thereby allowing the use of the IIR filter and gaining its advantages.
To reduce the
notch misplacement in the present system, the sampling frequency may be increased.
This increase in sampling frequency means there are more samples per spindle
revolution; therefore, miscounting the revolution by one sample causes a smaller notch
The notch misplacement is directly proportional to the duration of the
Thus doubling the sampling frequency will reduce the notch misplacement
by a factor of two.
This would allow the use of a IIR filter with notch width equal to
one half of the FIR notch width while still attaining the same performance for tooth
and run-out harmonic removal.
The narrower notches would filter less content around
each harmonic, giving improved performance when the chatter was closely located by
The problem of notch misplacement may be completely eliminated by changing
the sampling system.
The system may be changed from a time based sampling system
to a revolution synchronized sampling system.
By synchronizing the sampling with
the spindle rotation, there is always an integer number of samples in the revolution (a
method to do this is explained under other filtering methods at the end of this section).
With an integer number of samples there is no notch misplacement and the full
advantage of the narrow notch width of the IIR filter can be realized.
High Pass Low Pass with Combs
In order to avoid the filtering of all the run-out harmonics and possibly filtering
a valid chatter signal, a two path filtering scheme is proposed.
to credit Russ Walters for first suggesting this filtering method.
The author would like
By first passing the
sampled signal through both a high pass and a low pass filter, the signal can
effectively be split into two signals.
other, the high frequency range. Tl
One carrying the low frequency range, and the
ie low frequency signal can then be filtered with
the presently used FIR comb or the proposed IIR comb to remove the run-out and its
The high frequency signal can be filtered through a similar comb with the
delay shortened by dividing the original delay by the number of teeth on the cutter
(this new delay must be rounded to the nearest integer).
The new comb removes the
tooth frequency and its harmonics from the high frequency signal.
may now be recombined through addition.
The two signals
In this way, the run-out and its harmonics
are removed up to the frequency of the cut off of the low and high pass filters, and the
tooth frequency harmonics are removed throughout the entire frequency range.
A signal flow diagram of this process is shown in Figure 4-9.
The upper path
is through the low pass filter and revolution referenced comb, the lower path is
through the high pass filter and tooth harmonic filter. The figure shows the transfer
function of each path up to the point at which it is shown. The overall transfer
function is reproduced as Figure 4-10.
Several parameters can be varied to adjust its
Hgh Poss rlter Tooth Comb
Flow Diagram for High-Low Pass Filtering with Combs.
First, the shape of the transition region between where all the run-out
harmonics are removed and the area where only the tooth harmonics are removed, may
be shaped by adjusting the filter parameters of the low and high pass filters.
be limited by the relative phase of the signals.
transition is positioned with the cutoff frequency of the filters.
The frequency location of this
This cutoff may be
0.2 0.4 0.6 0.8
Transfer Function for High-Low Pass Comb Filter System.
set frequency they are stationary with respect to the spindle speed.
That is, the filter
coefficients may be calculated at the start of the program and are not affected by the
If the filter is set to remove a set number of run-out harmonics, the
filter coefficients must be recalculated for a change in spindle speed.
It is not
proposed to vary these with a sag in speed, only during a commanded spindle speed
By varying the order of the filters the width of the transition between the two
filter regions may be adjusted.
A low order filter will have a broader transition region.
The sharpness of the notches in each comb filter can be varied independently.
Therefore, the notches applied to the tooth harmonics can have their poles at a lower
radial location than those of the run-out filter.
Because these harmonics are higher in
frequency region, the misplacement of the zeros is more severe.
Two main concerns must be addressed in the implementation of the high-low
comb filtering method.
The first concern is that of the effect of
rounding the delay
number in setting the tooth harmonic comb.
To set the delay used in the tooth
harmonic filter, the delay used in the run-out filter must be divided by the number of
teeth on the cutter.
The number of samples collected in one revolution (the delay
number used in the run-out filter),
will most probably not be evenly divisible
number of teeth on the cutter; this number must be rounded to be used in the new
The effect of this rounding on the location of the zeros must be investigated.
The second concern is which run-out harmonics to remove from the signal.
The issue of rounding the count number will be addressed first.
the delay number produces a notch misplacement in addition to the misplacement from
using an integer delay number to approximate the revolution length.
misplacement can be expressed as a phase shift between the two waves, that is the
present wave and the one from the previous tooth period that is being subtracted to
perform the filtering.
As previously addressed, the phase shift due to miscounting the
revolution length is at a maximum with a shift of one sample period.
misplacement of the primary tooth frequency notch is,
where m is the number of teeth on the cutter and N is the number of samples per
For a particular notch, this is the same phase shift and therefore notch
run-out comb would have the same misplacement as the primary notch for a tooth
frequency comb of a four tooth cutter.
When applying the tooth frequency comb an additional phase shift is
introduced due to the rounding.
The tooth frequency comb is essentially a first
difference method, therefore the delay used must be equal to the tooth period.
delay is obtained by dividing the number of samples per revolution by the number of
teeth on the cutter.
In order to perform the filtering without the use of interpolation of
the data points, this
delay must be an integer.
The rounding of the delay number to
the nearest integer value may produce, at the limit, a phase shift of one-half of a
Therefore the total phase shift of the waves due to the miscounting of
the revolution length and the rounding error can be expressed:
where the first term is due to the revolution miscount and the second term is due to
the rounding error.
It can be seen that the rounding may improve the placement or
further misplace the filter notch, depending on the direction of the rounding.
However, because the misplacement direction due to the miscounting of the revolution
length is not known, for the limit condition it must be assumed that the rounding will
further misplace the filter notches.
Therefore the limit notch misplacement with
rounding will be
This is the phase shift for maximum miscounting of the revolution, miscount
by one sample,
and maximum rounding error.
This is a fifty percent increase in the
misplacement of the filter notches when compared to the corresponding notches of a
run-out comb filter.
To achieve equal performance for the tooth frequency comb to
that of the run-out comb, the sampling frequency would have to be increased by fifty
By this method Figure 3-8 may be used to establish system requirements for
a specified performance of the tooth frequency comb just as was done for the run-out
The second subject of concern when applying the high-low pass comb system
is to determine which run-out harmonics are to be filtered from the spectrum.
mentioned earlier, the run-out harmonics (not including the tooth harmonics), are
possible chatter frequencies.
Therefore, only those harmonics which may cause false
triggering during stable cutting conditions should be filtered.
To make this
determination, it is first required to understand the source of the run-out harmonics and
their expected distribution in the spectrum.
The first thing to note is the term "run-out harmonics" is a bit misleading.
when defined as an eccentricity of the tool, produces a sinusoidal
variation of the chip load at a once per revolution frequency.
This variation will result
in content in the spectrum at the run-out frequency, and at frequencies separated from
to the tooth frequency and its harmonics.
The run-out does not produce a full
population of run-out harmonics in the spectrum.
The effects of pure run-out are demonstrated in Figures 4-11 and 4-12.
shows the time domain vector sum force pattern for an eight tooth cutter with a
run-out of 50
of the chip load.
The cut is simulated for 20 % radial immersion.
The figure shows the vector sum force pattern for four revolutions of the cutter.
Figure 4-12 shows the spectrum of this force signal with the run-out component and
the side lobing effect.
Force Pattern for Four Revolutions of an Eight Tooth Milling Cutter
0% Run-out, Radial Immersion
The existence of a full population of run-out harmonics is in fact due to tooth
to tooth variation or tooth throw.
and 4-14 show a vector sum force
signal and its spectrum, respectively, for an eight tooth cutter with 50% tooth throw.
This is defined as random radial variation of the tooth position with the maximum
Cyds. pm Rr.'vboo
Spectrum of Force Signal from an Eight Tooth Cutter with 50% Run-
= 20%, Magnitude Plotted versus Cycles Per Revolution.
Force Pattern for Four Revolutions of an Eight Tooth Milling Cutter
Throw, Radial Immersion
This distribution of the run-out harmonics is controlled by the radial immersion
of the cut and distinctive force pattern introduced by tooth throw.
is convenient to first look at the form of the force on the cutter.
milling cutter with the tooth forces is shown in Figure 4-15. The
decomposed into its tangential, F,, and radial, FR, force componei
To illustrate this it
A diagram of a
; tooth force is
-'i I I I I I fl .1 t *
Cydes per Reokuto
pectrum of Force
gnal from Eight Tooth Cutter with 50% Tooth
Magnitude Plotted versus Cycles Per Revolution.
is the cutting stiffness of the material, a is the axial immersion, and f,
feed per tooth.
The radial force is assumed to be proportional to the tangential force
and can be expressed as follows:
wt I 1
These tooth forces can be projected into the X and Y
coordinate system to give the
- F, sinf
+ F, COS
The magnitude of the vector sum of the forces can be expressed as the square root of
the sum of the squares of the force components.
This force signal is a scaled and rectified sine wave. The scaling comes from the
magnitude of the radial force relative to the tangential force. Because the cutter is
only engaged from a possible position of 0 to
180 degrees, the vector sum force can
be represented by a windowed sine wave as presented by Walters[ 9 ].
It should be
noted that the vector sum and the tangential force have the same form and hence will
have similar spectral properties.
The force due to a single tooth is presented as a sine
wave, scaled by a factor to account for the radial force component,the cutting stiffness,
axial immersion and feed per tooth.
= c Ksaf
'masked' by a function that is equal to one during tooth engagement
and eaual to zero when the tooth is not engaged.
'masking operator' is simply a
tooth engagement. This operation is shown in Figure 4-16, the sine wav
windowing function are shown, then the product of the two is plotted for
e and the
25 % radial
Tooth Force and Windowing Function for
This windowing in the time domain is a multiplication operation equivalent to
convolution in the frequency domain.
Therefore, the frequency content of the single
tooth force pattern is equal to the frequency content of the sine wave convolved with
the content of the windowing function.
The frequency content of the sine wave is of
course a single line at the run-out frequency, and its mirror image at the negative of
length of the window.
The zeros are located at the frequencies of 1/T,
T is the time duration of the square window.
Cyde pmr Ruvt~a1
Frequency Content of a Square Window.
5 10 15 20 25 30
Cycles per Revolution
Frequency Content of a Single
The convolution of the two signal portions is shown in Figure 4-18,
values are marked with crosses and the envelope of the maximum values is also
This convolution results in a series of run-out harmonic
ines with peak
I 1 4' *t 1* -1 1* *4 f
the run-out harmonics in the spectrum, the envelope of the harmonic peaks will be
Walters showed that for an ideal cutter with identical teeth, these frequency
patterns for the individual tooth could be combined through the use of the 'force
to produce the overall cutting force signal.
When all the teeth are
equal, this filter produces zeros at all multiples of the run-out frequency and poles at
the tooth frequency and its harmonics.
In this way the force composition filter works
as a comb filter, leaving only the tooth frequency and its harmonics in the composed
with tooth throw the teeth are no longer identical and there is
another component added into the final signal.
the run-out and its harmonics
This added component is the source of
n the spectrum.
The pattern of tooth throw is not just a single high or low tooth.
high tooth will remove more material from the cut than normal, the following tooth
will remove less than normal by this same amount.
leading low tooth.
The inverse being true for a
Therefore the pattern introduced by tooth throw is an increase and
then a corresponding decrease of the force on successive teeth (assuming a high lead
This pattern is superimposed on the ideal force pattern for equal teeth.
Further, there is a throw between each successive tooth pair to make up the random
variation of the teeth within the limits of the throw.
Each of these throws is
superposition in the frequency domain, hence the effect of the throw can be presented
separately and then imposed on the force of the
deal cutter to generate the complete
cutting force pattern.
The high low type force pattern of tooth throw is shown in Figure 4-19 for a
radial immersion cut with a four tooth cutter.
The frequency content of this
signal is shown in Figure 4-20, with the peaks of the harmonics marked.
I i i i j -
Throw Force Pattern for Leading High Tooth 25% Radial
The frequency content is seen to have zeros at the tooth frequency and its
harmonics, and the run-out harmonics spread between these zeros in a rounded
indeed the pattern that is seen from simulation of cutting with tooth
throw (Figure 4-14 at the start of this section) or due to cutter breakage which is
equivalent to a tooth throw equal to the chip load.
The position of the zeros is
determined by the tooth spacing, ie. the number of teeth on the cutter and always
produces zeros at the tooth frequency and its harmonics.
Cycles pr Revlon
Frequency Content of Tooth
Throw Force Pattern for 25% Radial
Alternatively, the formation of the high low pattern from the single tooth force
pattern can be thought of as passing the single tooth force through a filter which
delays the signal by the tooth period and then subtracts it from the original signal.
This is the action of a FIR comb filter with delay equal to the tooth period.
transfer function of such a filter was previously shown to be,
where is the digital tooth frequency.
shown in Figure 4-21.
= I2sin (kO)
The magnitude of the transfer function is
The zeros of this function are located at the tooth frequency
and its harmonics, and the maximum magnification is equal to two.
This spectrum can
be convolved with that of the single tooth to produce the expected pattern of run-out
In order to accomplish our filtering task, determining which run-out harmonics
to filter from the spectrum, it is of more interest to know the maximum possible
0.2 04 06 0.8 1 (p.)
Transfer Function of 15"h Order Comb Filter.
tooth throw is combined, neglecting phase, an envelope of the maximum possible
content is obtained.
That is, the magnitude of the frequency of the positive tooth force
is added to that of the negative.
Further, the component due to the following tooth is
simply the negative of a time shifted version of the positive force on the high tooth.
Therefore the magnitude of the frequency content is the same for each component.
The overall magnitude of the frequency content is thus two times the magnitude due to
a single high tooth (the frequency content due to a single high tooth was shown as
This is demonstrated in Figure 4-22.
The figure shows two times the
magnitude of the frequency content for of a single high tooth for
25 % radial
Also plotted is the content due to tooth throw of the same amplitude for a
four and an eight tooth cutter at the same radial immersion.
The line for two times
the single tooth is seen to form a maximum envelope for the run-out harmonics,
:... ~ I J... 2... ~ I. .. ... L. P &- -44-
-n -, 1 .a j
Cycles per Revolution
Figure 4-22 Frequency Content Two Times Single High Tooth, 25% Radial
Immersion and Tooth Throw for Four and Eight Tooth Cutters.
The envelope of the distribution of the maximum possible run-out harmonic
peaks can thus be defined by the radial immersion.
This envelope is scaled by the
actual amount of tooth throw but the form is determined only by the radial immersion.
The envelope of the maximum possible run-out harmonics are plotted for 12,
and 100 percent radial immersion as Figure 4-23.
It is obvious that the shallower the radial immersion the further the run-out
harmonics will be spread into the spectrum.
The point at which the run-out harmonic
amplitude is 1/8 of the maximum envelope height comes at approximately the 20th run-
out harmonic for the 12% radial immersion case.
This same point is located at the
4 4 -- -4* 44 4 it
I n I
10 20 30
10 20 30
Maximum Possible Distribution of Run-out Harmonics in the Cutting
Force Spectrum for Radial Immersions of 12,
50 and 100%.
Frequency in Cycles
immersion of the cut is known, the transition point for the high low filter scheme can
The radial immersion of the cut is generally not known.
If a minimum radial
immersion during the process is known, the transition point can be based on this value.
The high and low pass filter coefficients are calculated based on the spindle speed in
revolutions per second, multiplied by the harmonic number that is to be filtered.
gives the cut off frequency of the filters.
A further complication comes from the fact that this is the distribution of the
*~ I, I .. I I
displacement versus force transfer function.
Therefore, harmonics of the run-out that
lie near resonances of this transfer function are greatly amplified.
most difficulties during very shallow radial immersion cuts, where
harmonics are spread almost completely throughout the spectrum.
This presents the
e the run-out
Very high run-out
harmonics past the high low filter transition may be amplified and cause false
triggering of the detection system.
the run-out harmonics, as
The only solution in such cases is to filter all of
presently the practice.
Conclusions for High Low Pass Comb Filter
The form of the high low pass comb filtering system is by nature more
applicable to cases with a large number of cutting teeth, such as face milling.
speed milling the tooth frequency is substantially lower than the expected chatter
ie. for stability the cutter is being operated in the higher number lobes.
The run-out harmonic number is equal to the number of teeth times the tooth harmonic
with the cutter being operated such that the chatter can be
expected around the upper tooth harmonics, ie. the third or forth, the power in the run-
out harmonics will be smal
in this area.
The cut off frequency for the high low pass
filter may be set to filter the run-out harmonics up to the second tooth frequency
harmonic, ie. for an eight tooth cutter the first 16 run-out harmonics.
filter would then not be active in the region of the chatter, eliminating the possibility
of filtering out the valid chatter signal.
The two path system is not applicable to the case of high speed milling with a
low tooth count.
In this case the cutter is being operated with the tooth frequency
near the chatter frequency,
in the region of the first stability lobe.
Because the chatter
is expected near the tooth frequency and there is a small number of teeth, the chatter
will be in the area of the low number run-out harmonics.
Therefore it would be
impossible to locate a useful cut off frequency for the high low system below the area
of the expected chatter.
and as is presently the practice, all the run-out
harmonics must be filtered.
The possibility of filtering the chatter may be reduced by
narrowing the notches of the filter by going to the IIR comb.
However, this would
require that the issue of notch misplacement be corrected by going to a synchronous
As mentioned in the introduction, there are commonly other signals contained
in the collected sound signal.
These signals are due to noise sources such as electrical
transformers, bearing noise or some other drive system noise. The signals may be
stationary in frequency or related in some way to the spindle speed. Generally there
are only a few noise sources and the signals are removed by placing individual notch
filters at their respective frequencies.
A second order notch filter is formed by a
complex conjugate zero pair and a similar pair of poles located at the desired digital
frequency; the pole is just radially inside the zero in the complex
pole plot of a second order notch filter is shown as Figure 4-24.
function can be written:
where R is the radial location of the pole.
the zeros are located on the unit circle.
Theta is the desired digital frequency and
Pole-Zero Plot for a Second Order IIR Notch Filter.
The transfer function can be written in terms of real coefficients as follows.
The digital frequency, 0, is calculated based on the type of signal to be filtered.
signal is not dependent on the spindle speed, 0 can be calculated from the definition
of the digital frequency, namely
If the signal is based on the spindle speed, can be calculated as a ratio of the
run-out frequency that is used in the run-out comb
where a is the desired ratio.
The radius of the pole is set to adjust the band width of
The radius is commonly around 0.9 or greater.
A figure showing the effect
of varying radial pole location is shown as Figure 4-25.
A very sharp filter notch can
be used, as the frequency of the noise can normally be accurately determined.
Transfer Functions of Second Order IIR Notch Filters with
Pole Radii of 0.5, 0.7, 0.9 and 0.9
Other Possible Filtering Methods
Three other possible filtering methods that show promise in this area will be
discussed, although it is not proposed to fully investigate them in this work.
other techniques are described only as areas of future research.
The three techniques
) a series of cascaded notch filters, 2) multiple adaptive cancelers, and 3) using a
phase lock loop and frequency synthesis to provide synchronous sampling.
Cascaded Notch Filters
The first technique presented is the use of a series of cascaded notch filters to
remove the run-out, the tooth frequency and any number of desired harmonics of each.
This system utilizes second order notch filters similar to those presently used to
remove a few noise sources from the sound signal.
A notch filter with the transfer
is formed for each signal component that is to be filtered.
The signal is successively
passed through each of the filters until all the components are removed.
To pass the
signal through multiple second order filters may seem inefficient, as opposed to
convolving the coefficients of the filters together and then passing the signal through
rt .II _~_.. _1 ,I:,, nccrri, ,,,,,i,:,,~
down into a series of second order filters for implementation.
To implement these
filters their coefficients must be calculated, and must be done for each revolution in
order to track spindle speed variation.
The transfer function in terms of real coefficients can be written.
z -1 +R
z +b z
It can be seen that three coefficients must be calculated for each filter,
This also involves calculating a cosine for each set of
For the run-out and any harmonics (including the tooth frequency and its
harmonics), the arguments of the cosines are conveniently related in that they are
multiples of the run-out cosine argument.
An efficient method for calculating the
cosine values can then be formed from the trigonometric identity.
The values can be calculated in a loop. Hence, only the cosine value of the run-out
frequency need be calculated directly.
An issue that would need to be investigated is
A aI AI* a.. n .A 1f aL 4l a A AA nr a: -IC .J ... 4k IIaI( a A a1 aU s* aI:l nA 4l n 1*a a1a. ka 1. an C
The main advantage of using the cascade notch filters is that the zeros of the
filter are located by the coefficients of the filter instead of simply by the delay as in
the comb filter.
By locating the filters with the coefficients, they may be continuously
placed as opposed to the discrete placement of the delay located filter.
there is not the problem of the misplacement of the filter notches. Ad
specific signals to be filtered can be individually selected allowing any number of
harmonics to be filtered and the filter for each to be selectively tuned for sharpness
and overall attenuation. The disadvantage of the notch filter approach is that it is very
computationally expensive. Considering a high speed milling application with a
frequency range of interest of six thousand hertz, implementing the notch filters in the
present DSP hardware would require approximately a 70 percent duty cycle.
neglecting the calculation of the coefficients, the further transforming of the signal into
the frequency domain and comparing peaks to a threshold to detect chatter.
it is not practical at this time to implement this filtering method.
rapid advance of computing hardware, this may not be the case in the not too distant
A final concern that would need to be investigated is the effect of changing
the filter coefficients during the processing at a speed change.
The effects of this time
variant filter are not fully understood by the author.
The second alternative filtering technique is the use of multiple adaptive
cancelers, and a general text on adaptive signal processing,
Signal Processing by Widrow and Steams [
is contained in Adaptive
The application of adaptive
interference cancelers as notch filters is best described through the use of a flow
diagram of such a filter.
A diagram showing a filter designed to cancel a single
interference frequency is shown as Figure 4-26.
contained in chapter 12 of
This figure and basic explanation are
Widrow and Stearns.
Signal Flow Diagram for Adaptive Notch Filter.
The primary input, d(n), is the signal that is to have the interference filtered
from it. The reference input is a signal that represents the interfering signal slated to
be removed. In general this may be any interfering signal and is normally a measured
However, in the present case this reference signal would be the run-out
frequency or one of its harmonics, and the primary input would be the measured sound
Because the reference signal is not available without containing the desired
the reference signal would be internally generated.
That is, a digital reference
signal and a 90 degree phase shifted reference signal are used, representing a sine-
cosine pair at the desired frequency.
an adaptive filter weight,
The sine and cosine wave are each multiplied by
w, and w2 respectively, and then added together to form the
output wave y. These weights are adaptively controlled by a least mean square
adaptive algorithm. By changing the weights relative to each other, the phase of the
output wave can be varied. By adjusting the weights in the same direction, the
magnitude of the output wave can be varied. This outp
the primary input signal forming the output of the filter
ut wave is then subtracted from
The least mean square
algorithm adapts the weights to minimize the power in this output signal, thereby
providing maximum cancellation of the reference input in the primary input. 1
formulation for up dating the filter weights is given by the following equations.
Where the subscript k is the sample number and I is the adaptation constant.
diagram presented is the filter for a single harmonic.
A similar filter would be used
for each of the harmonics to be removed.
There are several concerns about the application of these cancelers to the
filtering of the chatter signal.
The first is the speed of adaptation.
speed is controlled by the constant Ip.
introduces noise due to the adaptation I
A large value allows rapid adaptation but
process. Too large of a value of p leads to an
one over the value of the maximum eigenvalue of the correlation matrix of the primary
signal to insure stability.
As a rule of thumb, the authors recommend that a value of
less than one tenth of the trace of this matrix be used.
Because there are multiple
harmonics to be removed from the sound signal this leads to large eigenvalues due to
the upper harmonics.
Thus, in order to assure stability of the filters small values for t
will be required.
This leads to a slow adapting filter.
Further, the calculating of
reference signals and then implementing a filter for each harmonic is computationally
expensive. Indeed, the reference of each will have to be adjusted with the spindle
speed to effectively filter during speed variations.
The final alternative filtering technique is to utilize the present comb filter or
the proposed high-low pass and comb filtering technique on a synchronized sampled
This leads to simplifying advantages in the filtering but introduces other
Synchronized sampling is sampling the signal with a set number of
samples per revolution of the spindle.
A more complete explanation of this is
contained in the tool breakage detection chapters of this report.
synchronized sampling system receives its trigger signal from an encoder that is
connected to the spindle.
The encoder has limitations as to the maximum operating
speed and cannot be used in a high speed system. This shortcoming may be avoided
by using an electronic synchronized triggering system. This system would consist of a
may simply be a once per revolution edge, as used in the present system, from a
reflectance type probe.
A complete discussion of the use of phase lock loops and their
use in frequency multiplication is contained in
The Art of Electronics by Horowitz
and Hill [ 11
This system uses a reference frequency of digital edges and generates
a signal that is at n times the reference frequency.
be digitally controlled by the supervision computer.
times per revolution can be generated. Further, the
The value of the counter, n, can
In this way a trigger signal of n
number of samples per revolution
can be controlled from the supervision computer.
Utilizing a synchronously sampled signal leads to advantages in the filtering
just as it does in the tool breakage processing.
Because the sampling is tied to the
spindle speed the filter notches stay properly placed during spindle speed variations.
The high-low pass filters are now designed in the per revolution domain (see tool
breakage), instead of the time domain.
In this way they are stationary in the per-
That is, for a set number of run-out harmonics to be removed, the
coefficients of the filters are constant with a varying spindle speed.
Thus they only
have to be calculated at start up instead of recalculated at each speed change as in the
time based system.
The use of a synchronously sampled signal introduces some disadvantages to
Because the signal is revolution-based instead of time based, the
fourier transform of the signal is now in the per-revolution domain instead of the
The chatter signal is at a set frequency, thus if the spindle speed is
the chatter signal will be blurred over several lines in the transformed domain. The
extent of this blurring will depend on the level of spindle speed variation. How
strongly this will effect the chatter detection would have to be investigated. Of course,
when chatter is detected its frequency would be known in the cycles per-revolution
domain and the spindle speed during this detection must be known in order to
calculate the new command speed for the spindle in RPM.
In setting up a synchronous sampling system, consideration would have to be
given to the fact that the band width of the detection system now changes with spindle
That is the Nyquist frequency is now expressed as,
where N is the number of samples per revolution and Spd is the spindle speed in
revolutions per minute.
Thus N must set high enough to have sufficient band width at
low speed and low enough not to overload the computations or data acquisition
hardware at the maximum spindle speed.
The value may be changed for different
spindle speed ranges or the data down sampled if the range of spindle speed warrants.
A single analog anti-aliasing filter may be used with the system.
The cutoff frequency
of the filter should be set just above the maximum frequency of interest of the chatter
The number of samples per revolution should then be set such that,
at the lowest spindle speed the nyquist frequency is not less than the cutoff frequency
of the filter.
TOOL BREAKAGE INTRODUCTION
Digital filtering techniques are applied to the area of tool breakage. It is
desired to implement a system of detecting cutter breakage in process. This system
should be responsive to slowly developing cutter damage such as the progressive
chipping of a cutter edge, and also responsive to catastrophic failure of an edge.
system is designed as two sub-systems to perform these two separate tasks.
system to detect progressive cutter damage is configured to detect both progressive
damage and the starting of the cutting process with a damaged edge.
not detect cutter wear, unless that wear is isolated to a single tooth.
This system will
Further, the sub-
system is designed to reject the force or displacement signal due to cutter run-out.
The second sub-system is designed to detect sudden failure of a cutting edge and
quickly stop the cutting process in order to prevent further damage to the cutter,
piece or machine.
By separating these two tasks, it is possible to utilize approaches
that have been previously implemented by others.
The previous implementations have
suffered from being required to detect both failure modes, and this requirement
prevented either system from being optimized for its specific task.
Both tool breakage sub-systems operate on a synchronously sampled
displacement signal from the machine tool spindle.
The parameters that describe the
Hence the common features will be discussed first and then the individual systems and
their performance will be described.
The background of each subsystem, and their
previous implementation will be covered with the individual discussions.
The task of the filtering in a cutter breakage detection system is to remove,
from the cutting signal,
the content that is due to normal cutting conditions.
way the remaining signal is determined to be due to abnormal conditions such as
Content due to other abnormal conditions, such as chatter, may also
These signals may cause the cutter breakage system to false trigger.
These issues will not be discussed here.
Instead, the discussion will start with a model
of the cutter and cutting process and in this way the sources and form of the signal
content can be explained.
Once the signal content is understood,
with both a broken
and unbroken cutter, the filtering to expose the cutter breakage feature will be covered.
Synchronized Sampling and the Per Revolution Domain
The signals used in the detection of cutter breakage are all synchronously
That is they are sampled in synchronization with the revolution of the
This technique was utilized by
Yellowley and Tlusty [ 12 ], and
leads to great simplification of the processing of the signal.
sampling is accomplished by using an encoder attached to the spindle.
configuration the encoder produces two signals.
In the present
One signal is a once per revolution
pulse and the other is a pulse train of 120 lines equally spaced around the revolution
data is collected at the same points during each revolution of the spindle.
One hundred and twenty lines are used because it is integrally divisible by eight, six,
four, three and two.
This means that for cutters with these numbers of teeth
will be an integer number of samples collected during each tooth period.
In discussion of the application and design of digital filters, it is convenient to
look at the response of the filter and its effects on the processed signals in the
When processing time-based signals, the signals are fourier
transformed into the frequency domain and the impulse response of the filter can also
be transformed directly into the frequency domain.
However, the synchronously
sampled signal is not a time-based signal but a rotation based signal.
transform of these rotation based signals casts the signal in a per revolution domain as
opposed to the frequency domain.
Similarly the impulse response of filters based in
the rotation domain can be projected into the per revolution domain.
For further illustration, the discrete fourier transform of a time based signal
may be written,
N is the length of the signal in points
T is the time step, (1/sample frequency)
Therefore, x(nT) represents the sampled time signal and X(kAf) is the transformed
frequency series, where Af is the frequency spacing and equal to sampling frequency
divided by the number of points, N.
In the rotation base system the sampling
in samples per revolution as opposed to samples per second or hertz.
Specifically, our sampling frequency is 120 samples per revolution.
The samples are
located at 1/120 revolution separation and the resolution of the fourier transformed
series is 120/N cycles per revolution.
The Nyquist criteria dictates that the maximum
resolved signal is not above 60 cycles per revolution.
Obviously, for a constant spindle speed the per-revolution domain can be
directly mapped into the frequency domain by simply scaling the per revolution axis.
The scaling factor will be different for each speed and vary during spindle speed
The advantage of the synchronously sampled signal is that the filters
designed in the per revolution domain have constant coefficients even with non
constant spindle speed.
In contrast, filters used on a time based series which must
have their coefficients recalculated to adapt to a changing spindle speed.
Throughout the following discussions of the cutter breakage detection schemes,
the per revolution response of the filters will be shown along with the content of the
It should be realized that these are used for illustration of the
processing and that the signal is never transformed into this domain.
and detection is carried out in the time, or more properly, the revolution domain.
Model of the Cutter and Signal due to
A figure of a typical inserted cutter is shown in Figure 5-1.
The main cutter
body supports individual cutting inserts that may be individually indexed or replaced.
All of the discussion pertaining to cutter breakage will be restricted to cutters with
integrally spaced teeth (when the space between each sequential tooth pair is the same
as all others on the cutter).
Either type of cutter, the indexable or the integral cutter,
is normally mounted to a tool holder that is held in the spindle taper.
Replacable Insert Face Mill.
The basic cutting parameters and the terms used to describe them will now be
discussed, including errors of tooth position and their effects on the cutting force
Axial immersion is defined as the axial depth of the cut, the depth of the cut
along the direction of the rotation axis of the spindle.
immersion perpendicular to the axis of rotation. Rad
The radial immersion is the
ial immersion is often expressed
The chip load is the distance of travel of the center of the cutter per tooth
A diagram depicting these parameters is shown as Figure
Radial and Axial Immersion.
The mode of milling is defined by the relative position of the work piece to the
direction of the cutter path and its rotation.
Up milling is the
where the cutting
teeth enter the work piece with zero or small chip thickness and exit with a larger
How much larger is dictated by the chip load and the radial immersion.
down milling, the teeth enter the work piece with some chip thickness and exit with a
reduced or zero chip thickness.
The slotting case can be defined as neither up nor
down milling, as the teeth enter and exit the work piece with zero chip thickness.
milling, down milling and slotting are diagramed in Figure
Milling Mode Diagram.
the location of the milling teeth.
There are two main errors of position of the teeth
that effect the cutting signal for the detection of cutter breakage.
out of the cutter.
There may be run-
Run-out is the result of the center of the circle not being located at
the center of rotation of the spindle.
Run-out can also be produced by the plane of the
circle not being normal to the axis of rotation of the spindle.
Both of these conditions
lead to a varying force signal that has strong content at the once per revolution
-~ - n -- -11 -Y -~l ~ C -I 4. -nr -r ~.A -- a *Ct r
As the effect of either form of run-out is the same for the cutter breakage
schemes, the two will not be differentiated in further discussion.
error of position is that of tooth throw.
teeth off the ideal circle,
plane of the circle. The
The second main
Radial tooth throw is the misplacement of the
while axial throw is the misplacement of the teeth out of the
signal due to either type of tooth throw is periodic in nature
but not harmonic.
because the signal content is spread on the once per
revolution frequency and its harmonics.
type signal generated from a broken cuttu
and cutter breakage is only a matter of rr
It will be shown later that this is the same
er tooth. In fact, the difference in tooth throw
laanitude. A severe tooth throw will have the
same effect as a broken cutter on the cutter breakage detection system, as well as its
effect on the actual milling operation.
Radial and axial throw will be collectively
referred to as tooth throw and are depicted in Figure
Axis of Rotation
Out of Plane
Cutter Run-out due to Eccentricity and Out of Plane Conditions.
Diagram of Radial and Axial Tooth
Another error in the collected signal which is not truly present in the cutting
force is the error of
'. In its present form, the cutter breakage
detection system, as implemented at the Machine
Tool Research Center at the
University of Florida, utilizes the displacement of the spindle relative to the machine
tool frame as representative of the cutting force.
This displacement is measured by
using inductive eddy current probes mounted on the spindle housing,
the gap between themselves and a target ring mounted on the spindle.
This signal is
not the true cutting force signal but the force signal filtered through the displacement
versus force transfer function of the spindle tool combination.
The instrument run-out
arises from the target ring not being truly round or centered on the spindle.
spindle rotates, any eccentricity of the ring produces a harmonic displacement signal at
the once per revolution frequency. Addition;
/ A .. i1 -" A -I
ally, any unroundness produces higher
I _! i i _1 i *I i _
run-out profile is then subtracted from the measured data at each revolution.
way the instrument run-out is removed from the measured data.
For all future
discussion the term run-out will refer to actual cutter run-out as opposed to instrument
The Cutting Force Signal and Mi
The modeled cutting force signal used in this study is based on the
'Instantaneous Rigid Force Model' as presented by Smith and Tlusty in [ 13 ].
model assumes the cutter to be rigid and hence, does not allow cutter displacement to
participate in the evaluation of the cutting forces.
displacement is considered.
will be noted.
In this way no regenerative
When regenerative effects or 'dynamics' are included, it
In the instantaneous rigid force model, the tangential force is
considered proportional to the area of the chip being removed.
F, is proportional to the chip width b,
The tangential force,
the instantaneous chip thickness f, and the
cutting stiffness of the work piece material K,
The Radial force is assume to be proportional to the tangential force.
The total force on the cutter is the vector sum of the individual force on the teeth.
The simulation used in this work was written by Chris Vierck
on a simulation routine written by Carlos Zamudio.
14 ] and based
The simulation allows for various
radial and axial immersions, spindle
speeds and tooth numbers.
Simulations can be
carried out for steady state and entry and exiting conditions.
Provisions are available
to simulate sudden cutter breakage, slow chipping and to, include cutter run-out, tooth
throws and random noise content in the signal.
A complete description and listing of
the program can be referenced in Mr.
The Cutting Force Signal
In order to understand the cutter breakage detection schemes it is first
necessary to understand the form of the cutting signal itself.
In its present
implementation, the cutter breakage detection system utilizes displacement signals
from the spindle in two orthogonal directions.
the overall force or displacement of the spindle
The cutter breakage system works on
. hence the vector sum of the two
used by the detection system.
Because of this, the remaining work will
refer to the force or displacement with the understanding that this is the vector sum of
the two directions.
Figure 5-6A shows a picture of the simulated cutting signal for an ideal
unbroken milling cutter. The cut is of shallow immersion with only one tooth engaged
with the workpiece at a time. As each tooth engages, the force rises as the chip width
increases and then drops as the tooth exits the work piece. The signal is seen to be
this force signal is shown in Figure 5-6B.
The cutter is an eight tooth cutter for this
The signal contains spectral content only at the steady state, the tooth frequency
(eight cycles per revolution) and its harmonics.
Cycles per Revolution
Simulated Cutting Force Signal for an Eight Insert Cutter and the Per
The signal simulated for an idealized broken cutter is shown in Figure 5-7A.
For the idealized broken cutter, all of the teeth are considered identical except the
which is completely absent.
Because the broken tooth does not cut, the
following tooth must cut twice as much material as normal, leading to an increased
force spike for that tooth.
The frequency content of this signal is shown in Figure
It can be seen that there is now spectral content at the once per revolution
frequency and its harmonics.
The region below the tooth frequency, in this case eight
cycles per revolution, has the most content and is referred to here as 'breakage region',
this term being borrowed from
Tarng and Tlusty [
This is the content that is to
be isolated in order to detect cutter breakage.
500 / /
S, l /1
420 40 6
OL~ 't-i tt' _______ n^^ .-
Cycles Per Revolution
Simulated Cutting Force Signal for a Broken Eight Insert Cutter and the
Per Revolution Domain
The effect of cutter run-out on the force signal can be seen in Figure 5-8A.
Because the run-out is an offset of the center of rotation, the effect is to vary the
cutting force at the run-out frequency.
This is a pure harmonic variation so that all of
the frequency content due to run-out is located at the once per revolution frequency.
This can be seen in Figure 5-8B. The effl
of run-out, as can be seen in Figure 5-9A.
ect of tooth throw is very different than that
As each tooth varies slightly in its position,
the signal is periodic but not harmonic at the rotation frequency.
This leads to
cutter signal by the scale.
As the effects of tooth throw and cutter breakage are the
same, there is no reason to try and differentiate them.
tooth throw is
simply considered tooth breakage.
A further concern due to tooth throw is that the
higher harmonics of the run-out may be significantly amplified by the spindle's transfer
function and, due to the low sampling frequency, be aliased into the breakage signal.
This will lead to significant content in the breakage region, even for a good cutter with
minor tooth throws.
This condition makes the separation of good and bad cutters very
difficult, and will be addressed further in a later section.
S.5 1 1.5 2.5
0. n t
Cycles Per Revolution
Simulated Cutting Force Signal for an Eight Insert Cutter with Runout
and the Per Revolution Domain Transform.
The form of the cutting force signal changes with the radial immersion of the
As more and more teeth are engaged in the cut at one time, their individual
Cycles Per Revolution
Simulated Cutting Force Signal for an Eight Insert Cutter with
Throw and the Per Revolution Domain
region of the spectrum.
The content in the higher harmonics is reduced as the broken
tooth and the following tooth are engaged for a longer period with increasing radial
Figures 5-10 and 5-11 show the simulated signal for an eight tooth cutter
and the spectral content for 1/4,
1/2, 3/4 and full radial immersion.
The figures show
the signal for an unbroken and a broken cutter, respectively.
The transient milling
of entry and exit are also of concern as the cutter
breakage system must obviously handle these conditions.
As the exit
is a mirror
image of the entry case, only the entry case will be discussed here and it will be
understood that the exit shows similar features.
It can be seen in Figure 5-12 that as
the cutter encounters the work piece, the teeth first enter into cutting near the
maximum chip thickness.
later steady state cutting.
The duration of the engagement is shorter than during the
This leads to a force spike that is essentially the same
cycles per revolution
cycles per revolution
Simulated Cutting Force
Signal and Frequency Content for an Unbroken
cycles per revolution
cycles per revolution
mulated Force Signal and Frequency Content for a Broken Cutter,