Digital filtering techniques for processing signals utilized in the unmanned supervision of machining


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Digital filtering techniques for processing signals utilized in the unmanned supervision of machining
Physical Description:
vii, 149 leaves : ill. ; 29 cm.
Cobb, William T., 1963-
Publication Date:


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theses   ( marcgt )
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Thesis (Ph. D.)--University of Florida, 1994.
Includes bibliographical references (leaves 146-148).
General Note:
General Note:
Statement of Responsibility:
William T. Cobb Jr.

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University of Florida
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All applicable rights reserved by the source institution and holding location.
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Full Text







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

guidance and

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


for his

guidance in the area of digital signal processing.

The author would like to express his


to Dr. Sencer

Yeralan and

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

Frost, Mr.


Mr. Chris


Mr. David Smith,

Chuck Bales, Dr. Mahesh Nallakatla

and Dr.

Wei-go Zhang.

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.








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

Notch Filters

Other Possible Filtering Methods
Cascaded Notch Filters
Adaptive Cancelers ...
Synchronous Sampling .


Synchronized Sampling and the Per Revolution Domain

Model of the Cutter and Signal due to

Cutting Force

Tooth Position

Simnal and Milline Simulation




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




T. Cobb Jr.



Chairperson: Dr. Jiri Tlusty

Major Department:

Mechanical Engineering

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


an infinite




a high

low pass




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.

previous damage

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

developed by

Tlusty, Smith and Delio [

,3,4 ].

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.

The corrective

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

as electrical

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

spindle itself.

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

cutting operations.


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

these stoppages.

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

damaged cutters.

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




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.

A full

description of the chatter control system can be found in the works by Delio, Smith

and Tlusty [

,3,4 ].

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.

no regeneration.

harmonics. The

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.

A tooth

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

two classes.

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

times pi.



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

(8 +n)

FI ter


Filter Diagram

The output of a FIR filter can be written as linear combination of the input



Y(n) can be represented as follows:

Y(n) =boX



. .. bkX(n-k) =

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

as follows:


= (bo +b,

-1+b z

-2 + S

+bk -k)X(


Signal Flow Diagram for a Finite Impulse Response Filter



Sz)(z _

Cb0 +b1

-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.

(z-Re e1)

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.

continuously from

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


Figure 2-4

Pole-Zero Plot for a Second Order FIR Filter.

function can be written as

(z-le et)

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,

and N2

for the two numerator roots.

The over-

all magnitude can then be written.

"] II

N 7
"- N


Graphical Magnitude Evaluation for a FIR Filter

Transfer Function.


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,

goes to


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,

and N,

become larger than one and the filter amplifies signals at

the higher frequencies.

H(e e)

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,

theta N,



Figure 2-6

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:


+ a~y


= box(n

+byx (n-l

. +bx(n-k)

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

+ *



. .+amz-m)

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

in the

Z plane.

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

frequency, 0,

,as the zeros but inset from the unit circle.

This is of course not a

requirement but used for illustration of the magnitude evaluation.

The transfer

function can be written in its rooted form as follows:

H(z) =


r---- /e,

n-fl rC jOIA

-1+ a, z

-1 +b, z


Figure 2-8


Graphical Magnitude Evaluation for an IIR Filter


Transfer Function.

The magnitude is again the product of the magnitude of the individual terms of

the polynomials,

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

to 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.


Figure 2-9

Magnitude Response of a Second Order IIR Filter.


Pole Radius 0.75.


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



I .~~

The product of these terms leads to the simple form of the comb filter transfer




3 .k-


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.

If the

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.

The delay

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

transfer function.

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.

In this

It can be

shown that the actual form of the magnitude can be written as follows.


= |2sin(ke)





will be attenuated by the filter.

Figure 3-1

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

of chatter.

The once per revolution signal is used to adjust the filter.

Because the

sound signal is sampled at a set sampling frequency, there is not the same number of


Figure 3-2


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


the data,


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.

This means

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

-= true

where is the true digital frequency, 0 is the calculated one from the once per

revolution signal.




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

countming system.

therefore the maximum misplacement can be calculated


N2 +N

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.

Further, Om

is the

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

spindle speed.

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* ..

If the

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.

This may

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

use of

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.

sample number

Figure 3-3

Simulated X Direction Force Signals for a Four Flute Cutter with

Radial Immersion and 10% Tooth

Throw, Sampling Frequency

10,000 Hz.



2490 RPM.

The spectrum of the 2500 RPM cut is shown in Figure 3-4.

The spectrum

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.

This is

expected as there is no notch misplacement for the case.

1000 2000 3000 4000

Frequency (Hz.)

Figure 3-4

Spectrum of Four

Tooth 5% Radial Immersion Cut, 2500 RPM,

Sampling Frequency

10,000 Hz.


Frequecy (Hz)

Figure 3-

Spectrum for Four

Tooth 5% Radial Immersion Cut, 2500 RPM,

Sampling Frequency

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




Frequency (Hz.)

Figure 3-6

Spectrum of Four

Tooth 5% Radial Immersion Cut, 2490 RPM, Sampling

10,000 Hz.

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

Frequency (Hz.)

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.

The magnification

factor of the filter is plotted verse the number of waves per cycle for maximum notch



.c 1.5



0 10 20 30 40 50

Samples per Cycle

Figure 3-8


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

65,000 Hz.

Soeed Variation

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

axial immersion.


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.

Figure 3-9

= 10,000 hz)

Speed Profile for 1.5% Speed Variation Over 30 Revolutions, Sample
10,000 Hz.

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

five percent.

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


1000 2000


XW i

Frequency (Hz)


Frequency (Hz)

Figure 3

Frequency (Hz)
Spectrum of Simulated Cutting Force Signals

for A) Steady State

Speed Variation (
Radial Immersion.

7.5% Speed Variation.

Average Speed

2500 RPM,

4 Teeth,


r -
V -






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

Frequency (Hz)

Figure 3-12

Frequency (Hz)
Spectrum of Simulated Cutting Force Signals Filtered Through the

Revolution Referenced Comb Filter

1.5% Speed Variation B)

7.5% Speed


Average Speed 2500 RPM,

4 Teeth,

Radial Immersion.

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

sound signal.

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.

The wide

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-

out harmonic.

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

following sections.

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:




z -N)

where N is again the number of samples per revolution and R is the radial location of

the poles.

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

-X (n-N) +R

Y (n-N)

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.

Figure 4-1

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.


0.5 1



0.5 1

Figure 4-2


Transfer Function of IIR Comb Filters with

Various Radii


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:

~ Rax

-ml (N-Na)

where N is the count number for the present revolution and Npt, is the count for the

previous revolution.

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.

The filtered

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,

are sharper.

In the

while for the IIR, they

The content in either filtered case is seen to be greatly reduced when

compared to the unfiltered case.

Fnqguic (1w

o 2CC tQaoe uo~ 2rn 1 4
FeRqen 9w)

frnqgiy (Ix)

Friincm y ( ig

Figure 4-3

4 Flute,


Transform of Simulated Cutting Force Signals

% Radial Immersion Sampling Frequency

B) FIR Comb Filtered

IIR Comb

10,000 Hz.

2500 RPM,

A) Unfiltered

Pole Radius = 0.7

Comb Pole


= 0.9.

F~grwqyc (1w)

Fr~qancy (tt

Figure 4-4
Comb Filter

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

similar conditions.

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.

The chatter




Hz. The spectrum of the unfiltered signal is shown in Figure 4-5.

near the 14" run-out harmonic.

The chatter is seen

freqiecy (hi)

Figure 4-5
Insert Cutter

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

Frequency (hz)

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.





Frequency (hz)

Figure 4-7

Chattering Force Spectra for Natural Frequency Equal to

575 Hz.


FIR Comb

IIR Comb.

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.

The signal

of a simulated

milling cutter with four teeth cutting at 2490 RPM, and a sampling frequency of

10.000 Hz.

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%.

Figure 4-8

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


FVnncy (

L-LJ- itJ

L.. I, ..1t 1, t .1 ,

4crl~li Sax)I

I IM# RIW Fiscar O R

Rngwcy Oj

Figure 4-8

Effects of Notch Misplacement

IIR Comb, Pole Radius


IIR Comb

\) Unfiltered Signal
Pole Radius = 0.9.

FIR Comb

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

IIR filter

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

exact shape.


Hgh Poss rlter Tooth Comb

Figure 4-9

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.


This may

The frequency location of this

This cutoff may be

0.2 0.4 0.6 0.8


Figure 4-10


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

spindle speed.

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.

Additionally, these


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

by the

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.

Rounding of

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.

Therefore, the

misplacement of the primary tooth frequency notch is,



m (2x

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

sample period.

Therefore the total phase shift of the waves due to the miscounting of

the revolution length and the rounding error can be expressed:

phase shift

m x

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

m (2x



m (3

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.

tool run-out,

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.




Figure 4-1

Force Pattern for Four Revolutions of an Eight Tooth Milling Cutter

with 5

0% Run-out, Radial Immersion

= 20%.

The existence of a full population of run-out harmonics is in fact due to tooth

to tooth variation or tooth throw.

Figures 4-13

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

Figure 4-12

Spectrum of Force Signal from an Eight Tooth Cutter with 50% Run-

Radial Immersion

Figure 4-13

= 20%, Magnitude Plotted versus Cycles Per Revolution.

Force Pattern for Four Revolutions of an Eight Tooth Milling Cutter

50% Tooth

Throw, Radial Immersion

= 20%.

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

Figure 4-14


pectrum of Force

Radial Immersion

= 20%,

gnal from Eight Tooth Cutter with 50% Tooth
Magnitude Plotted versus Cycles Per Revolution.

where KI

is the cutting stiffness of the material, a is the axial immersion, and f,

is the

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

component forces.



- 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.

= Kaff


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

This wave


'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


Figure 4-16


Tooth Force and Windowing Function for


Radial Immersion.

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

-~~ 4

length of the window.

The zeros are located at the frequencies of 1/T,

2/T, 3/T...


T is the time duration of the square window.

Cyde pmr Ruvt~a1

Figure 4-17

Frequency Content of a Square Window.

5 10 15 20 25 30
Cycles per Revolution

Figure 4-18

Frequency Content of a Single

Tooth Force.

The convolution of the two signal portions is shown in Figure 4-18,

the actual

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

composition filter'

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.

Because a

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 -


Figure 4-19


Throw Force Pattern for Leading High Tooth 25% Radial

Immersion Four

Tooth Cutter.

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

Figure 4-20

Frequency Content of Tooth

Throw Force Pattern for 25% Radial

Immersion Four

Tooth Cutter.

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

Figure 4-21


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

Figure 4-18).

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

- .4

I n I

_ -rl

Rad. Im.=12%

Rad. Im.=50%

10 20 30

Rad. Im.=25%

Rad. Im.=100%

10 20 30

Figure 4-23

Maximum Possible Distribution of Run-out Harmonics in the Cutting

Force Spectrum for Radial Immersions of 12,

50 and 100%.

Frequency in Cycles

Per Revolution.

immersion of the cut is known, the transition point for the high low filter scheme can

be chosen.

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

t n

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.

In low

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.

The run-out

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.

In such


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

sampling system.

Notch Filters

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

Z plane.

A zero-

pole plot of a second order notch filter is shown as Figure 4-24.

The rooted

Z transfer

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

Figure 4-24

Pole-Zero Plot for a Second Order IIR Notch Filter.

The transfer function can be written in terms of real coefficients as follows.


1-2 cosO
-2R cos


The digital frequency, 0, is calculated based on the type of signal to be filtered.

If the

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 filter.

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.







Figure 4-25


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.

are: 1


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

function of,

z-le je1




z-R e


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

* Il

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.



-2R1 cosO1


z -1 +R

z +b z
z -1a4,2
z +~

It can be seen that three coefficients must be calculated for each filter,

specifically b1

and a2.

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


Iditionally, the

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.

This is

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.


with the

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.

Adaptive Cancelers

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.


Figure 4-26

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.

W1, k+1

# k+l


P EkXlk

C Ek

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.

The 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

W~ k+2


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.

Synchronous Sampling

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.

The present

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.

revolution 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

the processing.

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

frequency domain.

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,


* Spd

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

detection system.

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.



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.

The sub-

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.

In this

way the remaining signal is determined to be due to abnormal conditions such as

cutter breakU

be present.


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.

The synchronous

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

the samples,

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

frequency domain.

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.

The fourier

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,

X(kAf) =


j(-) (-nk)

,1, ,N-1


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

cutting signal.

It should be realized that these are used for illustration of the

processing and that the signal is never transformed into this domain.

All processing

and detection is carried out in the time, or more properly, the revolution domain.

Model of the Cutter and Signal due to

Tooth Position

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.

Figure 5-1

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








Figure 5-3

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


of Cutter


of Rotation

Cutter Plane


Figure 5-4

Axis of Rotation

Out of Plane

Cutter Run-out due to Eccentricity and Out of Plane Conditions.

Figure 5-4.












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

'instrument run-out

'. 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.

which measure

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.

As the

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.

In this

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

ing Simulation

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,

F =bfK8

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.

3 F,

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.


Thesis [14].

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

Figure 5-6

Simulated Cutting Force Signal for an Eight Insert Cutter and the Per

Revolution Domain


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

broken tooth,

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.


1500 A


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.

Thus, a


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.




0 _____
S.5 1 1.5 2.5
:10 Revolution

0. n t

Cycles Per Revolution

Figure 5-8

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












2 4


per revolution


cycles per revolution













cycles per revolution



Simulated Cutting Force



Signal and Frequency Content for an Unbroken


Radial Immersion.

Ii 1*1




cycles per revolution



cycles per revolution

mulated Force Signal and Frequency Content for a Broken Cutter,




Radial Immersion.