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
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Thesis:
Thesis (Ph. D.)--University of Florida, 1994.
Bibliography:
Includes bibliographical references (leaves 146-148).
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Typescript.
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Vita.
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William T. Cobb Jr.

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DIGITAL FILTERING TECHNIQUES FOR PROCESSING
SIGNALS UTILIZED IN THE UNMANNED SUPERVISION OF MACHINING












By

WILLIAM T. COBB JR.


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

1994





























To Catherine and Anastasia for their beautiful inspiration;

A bed time story sure to put anyone to sleep.













ACKNOWLEDGMENTS


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


hand.


The author also thanks Dr. Scott Smith whose friendship and guidance have aided


greatly


the throughout his


graduate


career.


Thanks


to Dr. Jose


Principe


for his


guidance in the area of digital signal processing.


The author would like to express his


appreciation


to Dr. Sencer


Yeralan and


Dr. John Ziegert for service on his advisory


committee.

Special thanks go to the members of the Machine Tool Research Center, both past


present;


their


friendship


greatly


enhanced


authors


educational


experience and life during this work. Specifically, the author would like to thank Dr. Tom


Delio, Mr. John


Frost, Mr.


Winfough,


Mr. Chris


Vierck,


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.













TABLE OF CONTENT


ACKNOWLEDGEMENTS


ABSTRACT

CHAPTERS


INTRODUCTION


THE FILTERING TASK


REVOLUTION REFERENCED COMB


Filter Notch Misplacement .
The Effects of Speed Variation


FILTER MODIFICATIONS


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 .


TOOL BREAKAGE INTRODUCTION


Synchronized Sampling and the Per Revolution Domain


Model of the Cutter and Signal due to


Cutting Force


Tooth Position


Simnal and Milline Simulation







THE ONCE PER REVOLUTION DIFFERENCE SUB SYSTEM

THE PROGRESSIVE DAMAGE DETECTION SUB SYSTEM .


CONCLUSIONS AND FUTURE WORK


Filters for Chatter Detection . .
Filters for Cutter Breakage Detection


~FERENCE LIST

BIOGRAPE-IICAL SKETCH ........._____ ........._____













Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

DIGITAL FILTERING TECHNIQUES FOR PROCESSING
SIGNALS UTILIZED IN THE UNMANNED SUPERVISION OF MACHINING

By


William


T. Cobb Jr.


December


1994


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


investigated;


an infinite


impulse


response


comb


a high


low pass


comb


filtering


technique.

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


during


transient


milling


automatic


thresholding.


second


system


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.












CHAPTER


INTRODUCTION


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


process.


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


tasks.


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

capabilities.

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


chatter.


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


system


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,


which


would not be possible without the monitoring


system.


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.


Hence,


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


teeth.


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.


normal


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









sought.


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.


That


is, the system is sensitive to slowly developing cutter damage and previously damaged

cutters.


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

piece.












CHAPTER


THE FILTERING TASK


Digital filters in chatter control are used to remove signal content due to

normal cutting and known noise sources from the content due to chatter in the cutting


process. Machine tool chatter is a self excited vibration of the tool relative to the

work piece. This self excited vibration occurs at the most dynamically flexible mode


of vibration of the system, provided steps have not been taken to stabilize a mode.

The task of chatter recognition and control is to recognize the signal due to chatter and


adjust the machining conditions to the most stable configuration.


In order to recognize


chatter, the signal content due to normal cutting and known noise sources must be


filtered from the cutting signal.


In this way the remaining content is due to chatter


and the control system can act on this signal.


In the present chatter control system at the Machine Tool Research Center at

the University of Florida, a microphone is used to sense chatter. The microphone is


nonintrusive and senses both vibration of the tool and of the work piece.


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







7

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.


signal.


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


frequency.


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


signals.


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.


8=2**x


if
ifs


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

following.


cos


II -L- a


3...


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


Y(n)


Digital filters can conveniently be separated into two classes; finite impulse


filters


( FIR filters) and infinite impulse filters ( IIR filters).


For a FIR filter the


(8 +n)











FI ter


Figure


Filter Diagram


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


series.


Thus,


Y(n) can be represented as follows:


Y(n) =boX


+biX


n-1)


. .. 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:


Y(z)


= (bo +b,


-1+b z


-2 + S


+bk -k)X(


Figure


Signal Flow Diagram for a Finite Impulse Response Filter


Y(n)


III










Sz)(z _
x(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.


Only


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


Cnr)ira


'4


*~l r I) J K.Z1 t


inws a sinle


z-R




























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
Re













N 7
"- N
.74


Figure


Graphical Magnitude Evaluation for a FIR Filter


Transfer Function.


MN2N
1


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


filter.


Because the root is on the unit circle,


when theta equals theta,


goes to


zero.


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,


,T'
J

























Frequency


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:


y(n


+ a~y


+,+a~y~n-"-r)


= 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.

follows:


This can be more compactly written as


brx(n-k)


-_a1


(n-m


A flow type diagram for an IIR filter is shown in Figure


. There is the


possibility


of an unstable filter.

delay operator Z as


The transfer function of the filter can be written in terms of the


follows:


n)














H(z)


Y(z)
x(z)


(bo +b, z


-2+
+ *


l+az


+bkz


. .+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

stable.


Figure


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) =


z-le


r---- /e,


n-fl rC jOIA


-1+ a, z


-1 +b, z


je,
-le





























Figure 2-8


rfl,


Graphical Magnitude Evaluation for an IIR Filter


Re








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.


NN2
DD,


The terms D,


and D2 are the magnitudes of the vectors from the tip of the


to the


poles.


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


filter.


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.



























Frequency


Figure 2-9


Magnitude Response of a Second Order IIR Filter.


S(pi)

Pole Radius 0.75.













CHAPTER 3
REVOLUTION REFERENCED COMB


In its present form, the chatter recognition and control system utilizes a


specialized form of a FIR comb filter.


This filtering method has been described by


Tlusty et al. [


6 ] and by Frost [ 7 ].


The comb filter is a filter with its zeros equally


spaced around the unit circle.


This equal placement of the zeros leads to a very


simple difference equation used to implement the filter.

zeros would have k terms in the rooted numerator. Thi


In general a filter with k


is leads to a numerator


polynomial with k+1


coefficients.


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


m2g


z-le


I .~~


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

function.


H(z)


l-1-


3 .k-







19

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

sample.


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.


Y(n)


= 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.


IH(z)


= |2sin(ke)











8zro
"zero


it
6k


zero


+J
6k


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.


This


The two signals used are


the sound signal from the cutting process and a once per revolution signal from the


spindle.


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


























Frequency


Figure 3-2


Magnitude


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


method.

the data,

spindle.


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:


rev)
sec


samples


2x
N


sec


where N is the number of samples per revolution.

filter notches can then be calculated


8err


The maximum misplacement of the


-= true


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

revolution signal.


8err


2xt
true


S2
N


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


err


2i+
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


used.


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


radians).


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.


2500


RPM


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.


2000


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


3000


5X00



























4000


Frequency (Hz.)


Figure 3-6
Frequency


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.


harmon

filter.


0.05
0,045
0.04


0.005


ic


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


.1~~_L.,L~_L~L..L~L


1000


2000


3000


5000








28

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


radians.


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

misplacement.



2


.c 1.5


1


o




0 10 20 30 40 50


Samples per Cycle


Figure 3-8
Misplacemen


Sever


Magnification versus Number of Samples per Cycle at Maximum Notch
t.


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


frequency,


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


speed.


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.


wave,


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

filtering.


2600


240%~


1000


3000


4000


50001


6000


sample number (samp. freq.


Figure 3-9
Frequency


= 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 *







31

Three simulations are used to first demonstrate the effect of the speed variation


alone.


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.


1.5%


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.


7.5%


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


ease




















1000 2000


4000


XW i


Frequency (Hz)


3000


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,


5000


r -
V -



~Th


2000


4000


5000









Figure


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.


7.5%


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


Variation.


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.


signal.


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.












CHAPTER 4
FILTER MODIFICATIONS


Two modifications to the revolution referenced comb are presented here.


First


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

determined.


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







36

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:


H(z)


-Z-N)


1-R


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



Im









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.


R=0.5


0.5 1


R=0.9


R=0.95


0.5 1


Figure 4-2


Magnitude


Transfer Function of IIR Comb Filters with


Various Radii


Poles.


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


case


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.


Because


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,
Signal


Fourier


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


Radius


= 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.


A) FIR


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


signal.


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


0006


1200


0000










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


25%


Radial Immersion, Natural Frequency 579 Hz.


The effect of both the FIR and IIR cc

shown in Figure 4-6 A and B, respectively.






0.06


with pole radius equal to 0.85, are


The FIR filter is seen, as expected, to










A

ter

800 100 1200 1400



matter
B



800 1000 1200 1400


Frequency (hz)









more thoroughly remove the tooth frequency components than the IIR filter.


However,


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.


A



1400




B



1400


Frequency (hz)


Figure 4-7


Chattering Force Spectra for Natural Frequency Equal to


575 Hz.


Filtered


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


filter.


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


misplacement.


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.


Figure


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


misplacement.


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

unfiltered.




















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


=0.7


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


harmonic.


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


misplaceme

sampling.


3nt.


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







47

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


harmonics.


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







48

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.


X(n)


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.


Y(n)


This may


The frequency location of this


This cutoff may be


























0.2 0.4 0.6 0.8


Frequency


Figure 4-10


Magnitude


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


change.


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







50

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


comb.


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.


This


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,


phase


shift


m (2x


where m is the number of teeth on the cutter and N is the number of samples per


revolution.


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.


This


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
N


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










phase


shift


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


percent.


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

comb.


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.


Figure


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.



1100

900

700


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


its.


-'i I I I I I fl .1 t *























Cydes per Reokuto


Figure 4-14


Throw,


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.


cos


sin,


- 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


sin,


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


in,


'masked' by a function that is equal to one during tooth engagement


and eaual to zero when the tooth is not engaged.


This


'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

immersion.


e and the


25 % radial


R..o&Mon


Figure 4-16


Single


Tooth Force and Windowing Function for


25%


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...


where


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


plotted.


This convolution results in a series of run-out harmonic


ines with peak


I 1 4' *t 1* -1 1* *4 f


.1









the run-out harmonics in the spectrum, the envelope of the harmonic peaks will be

used.






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


signal.


However,


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


tooth).


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








60

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


25%


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.



1
08
06
04
02

-02
-0.4
-06
-0.8
I i i i j -


Resudaa


Figure 4-19


Tooth


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


fashion.


This


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,


IH(z


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

harmonics.

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


Magnitude


0.2 04 06 0.8 1 (p.)
Frequency
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


immersion.


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.


This


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


frequency,


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


number.


Therefore,


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.







66

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


cases,


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:


le-je)


z-R


eje)


e-je)


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.


H(z)


1-2 cosO
-2R cos


z-1+z


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











=2%


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


2x
-
N


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.


R=0.5


R=0.7


R=0.9


R0-nan


Frequency


Frequency


Figure 4-25


Magnitude


Transfer Functions of Second Order IIR Notch Filters with


Pole Radii of 0.5, 0.7, 0.9 and 0.9


Respectively.









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


These


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-1


z-R1


e~e


z-R e


-~iQi)


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.


111(z)


cos81


-2R1 cosO1


o0+a1l


z-1+z-2
z -1 +R


z +b z
z -1a4,2
z1
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


coefficients.


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.


COS


(nO)


COS


n-I-


COS


- COS


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


Therefore,


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.


Therefore,


it is not practical at this time to implement this filtering method.


However,


with the


rapid advance of computing hardware, this may not be the case in the not too distant


future.


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


Input


Reference
cosCnn


This figure and basic explanation are


Widrow and Stearns.


Output


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


signal.


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


signal


Because the reference signal is not available without containing the desired


signal,


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


Wlk


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







74

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


signal.


This leads to simplifying advantages in the filtering but introduces other


complications.


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


speed.


That is the Nyquist frequency is now expressed as,


if
nyquist


* 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.












CHAPTER


TOOL BREAKAGE INTRODUCTION


Digital filtering techniques are applied to the area of tool breakage. It is

desired to implement a system of detecting cutter breakage in process. This system


should be responsive to slowly developing cutter damage such as the progressive


chipping of a cutter edge, and also responsive to catastrophic failure of an edge.


system is designed as two sub-systems to perform these two separate tasks.


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.


work


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







78

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.


ige.


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


sampled.


That is they are sampled in synchronization with the revolution of the


spindle.


This technique was utilized by


Altintas,


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


there


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) =


x(nT)


21
j(-) (-nk)


,1, ,N-1


where,


N is the length of the signal in points

T is the time step, (1/sample frequency)







80

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.


frequency


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


variation.


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


signal.


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










immersion.


period.


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


AXIAL









RADIAL


IMMERSION









IMMERSION


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


case


where the cutting


teeth enter the work piece with zero or small chip thickness and exit with a larger


thickness.


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














FEED


DOWN


MILLING


FEED


FEED


SLOTTING







UP MILLING


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.


This


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


Center


of Cutter


Center


of Rotation


Cutter Plane


Eccentricity


Figure 5-4


Axis of Rotation


Out of Plane


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


Figure 5-4.










Throw


Throw


Ideal


Ideal


Plane


Circle


RadiQl.


Throw


Axia


Throw


Figure


Diagram of Radial and Axial Tooth


Throw.


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

run-out.


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 ].


This


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.


F=O


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.


Vierck's


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


directions


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.


case.


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.


Revolutions


Cycles per Revolution


Figure 5-6


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


Revolution Domain


Transform.


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.


2000

1500 A

1000

500 / /

S, l /1
420 40 6

10



5


OL~ 't-i tt' _______ n^^ .-


Cycles Per Revolution


Figure


Simulated Cutting Force Signal for a Broken Eight Insert Cutter and the


Per Revolution Domain


Transform.


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


severe


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.





1500

1000

500

0 _____
S.5 1 1.5 2.5
:10 Revolution
8
6
4

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


cutter.


As more and more teeth are engaged in the cut at one time, their individual
























Cycles Per Revolution


Figure


Simulated Cutting Force Signal for an Eight Insert Cutter with


Tooth


Throw and the Per Revolution Domain


region of the spectrum.


Transform.


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


immersion.


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


cases


of entry and exit are also of concern as the cutter


breakage system must obviously handle these conditions.


As the exit


case


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














2000

1500

1000

500

0


2000

1500

1000

500

0


-edton


2 4


cyches


per revolution


revolution


cycles per revolution


2000

1500

1000

500

0


revolution


2000

1500

1000

500

0


revolution


cycles per revolution


Figure
Cutter,


5-10


Simulated Cutting Force


50%


75%


Signal and Frequency Content for an Unbroken


100%


Radial Immersion.


Ii 1*1





















revolution


revolution


revolution


cycles per revolution


revolution


Figure


cycles per revolution


mulated Force Signal and Frequency Content for a Broken Cutter,


50%


75%


100%


Radial Immersion.


25%