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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. Weigo 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 BIOGRAPEIICAL 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 inprocess 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 inprocess 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 inprocess 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 runout 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 machinetool 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 toolmachine 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 nonvarying 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 runout and react to in process breakage. This subsystem 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 21. 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 runout 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 runout 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 runout 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 22. 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 n1) . .. bkX(nk) = bkX (nk) 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 23, 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. (zRe e1) ZR1 S ( These roots can then be plotted in the complex Z plane as in Figure 24. 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 zR Figure 24 PoleZero Plot for a Second Order FIR Filter. function can be written as (zle et) zle 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 26. theta equal zero, both N, and N, theta N, ,T' J Frequency Figure 26 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 (nl . +bx(nk) 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(nk) _a1 (nm 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 . .+amzm) 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 28. 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) = zle r /e, nfl rC jOIA 1+ a, z 1 +b, z je, le Figure 28 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 29 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 zle I .~~ The product of these terms leads to the simple form of the comb filter transfer function. H(z) l1 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(nk) There will be a zero at the runout 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 31. The associated magnitude response of the filter is shown in Figure 32. 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 31 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 32 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 runout 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 runout 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 runout 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 33 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 runout 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 33 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 34. The spectrum shows strong content at the tooth harmonics as would be expected for the shallow immersion cut and minor content at the runout 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 34 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 36 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 36 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 38. 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 38 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 39 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 39. 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 310A, 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 311 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 312 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 runout 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 runout 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 runout harmonics and thereby avoid filtering a valid chatter signal that may exist at one of the upper runout 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 runout 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) ZN) 1R 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 (nN) +R Y (nN) 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 41 and 42. Figure 41 shows a polezero 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 42. 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 42 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 runout 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 (NNa) 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 runout 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 43 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 44. 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 43 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 44 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 runout 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 runout 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 runout harmonic which is located at 583.3 hz. The chatter 0006 1200 0000 Hz. The spectrum of the unfiltered signal is shown in Figure 45. near the 14" runout harmonic. The chatter is seen freqiecy (hi) Figure 45 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 46 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 47 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 runout harmonic. to 575 hz. The natural frequency of the system is decreased spectra for the two filtered signals are shown in Figure 47. 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 harmrnnir 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 48 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 runout harmonics they are meant to filter, leaving the signal essentially unfiltered. FVnncy ( LLJ itJ L.. I, ..1t 1, t .1 , 4crl~li Sax)I I IM# RIW Fiscar O R Rngwcy Oj Figure 48 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 runout 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 noninteger 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 runout 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 runout 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 runout 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 runout 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 runout 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 49. 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 410. Several parameters can be varied to adjust its exact shape. X(n) Hgh Poss rlter Tooth Comb Figure 49 Flow Diagram for HighLow Pass Filtering with Combs. First, the shape of the transition region between where all the runout 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 410 Magnitude Transfer Function for HighLow 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 runout 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 runout 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 highlow 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 runout 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 runout 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 runout 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 runout 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 onehalf 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 runout comb filter. To achieve equal performance for the tooth frequency comb to that of the runout comb, the sampling frequency would have to be increased by fifty percent. By this method Figure 38 may be used to establish system requirements for a specified performance of the tooth frequency comb just as was done for the runout comb. The second subject of concern when applying the highlow pass comb system is to determine which runout harmonics are to be filtered from the spectrum. mentioned earlier, the runout 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 runout harmonics and their expected distribution in the spectrum. The first thing to note is the term "runout harmonics" is a bit misleading. tool runout, 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 runout frequency, and at frequencies separated from to the tooth frequency and its harmonics. The runout does not produce a full population of runout harmonics in the spectrum. The effects of pure runout are demonstrated in Figures 411 and 412. Figure shows the time domain vector sum force pattern for an eight tooth cutter with a runout 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 412 shows the spectrum of this force signal with the runout component and the side lobing effect. 1100 900 700 Figure 41 Force Pattern for Four Revolutions of an Eight Tooth Milling Cutter with 5 0% Runout, Radial Immersion = 20%. The existence of a full population of runout harmonics is in fact due to tooth to tooth variation or tooth throw. Figures 413 and 414 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 412 Spectrum of Force Signal from an Eight Tooth Cutter with 50% Run Radial Immersion Figure 413 = 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 runout 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 415. 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 414 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 416, 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 416 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 runout 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 417 Frequency Content of a Square Window. 5 10 15 20 25 30 Cycles per Revolution Figure 418 Frequency Content of a Single Tooth Force. The convolution of the two signal portions is shown in Figure 418, the actual values are marked with crosses and the envelope of the maximum values is also plotted. This convolution results in a series of runout harmonic ines with peak I 1 4' *t 1* 1 1* *4 f .1 the runout 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 runout 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 runout 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 419 for a 25% radial immersion cut with a four tooth cutter. The frequency content of this signal is shown in Figure 420, 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 419 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 runout harmonics spread between these zeros in a rounded fashion. This indeed the pattern that is seen from simulation of cutting with tooth throw (Figure 414 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 420 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 421. = 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 runout harmonics. In order to accomplish our filtering task, determining which runout harmonics to filter from the spectrum, it is of more interest to know the maximum possible Figure 421 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 418). This is demonstrated in Figure 422. 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 runout harmonics, :... ~ I J... 2... ~ I. .. ... L. P & 44 n , 1 .a j Cycles per Revolution Figure 422 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 runout 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 runout harmonics are plotted for 12, and 100 percent radial immersion as Figure 423. It is obvious that the shallower the radial immersion the further the runout harmonics will be spread into the spectrum. The point at which the runout 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 423 Maximum Possible Distribution of Runout 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 runout 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 runout Very high runout harmonics past the high low filter transition may be amplified and cause false triggering of the detection system. the runout 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 runout 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 runout harmonics up to the second tooth frequency harmonic, ie. for an eight tooth cutter the first 16 runout harmonics. The runout 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 runout 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 runout 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 424. The rooted Z transfer function can be written: leje) zR eje) eje) 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 424 PoleZero Plot for a Second Order IIR Notch Filter. The transfer function can be written in terms of real coefficients as follows. H(z) 12 cosO 2R cos z1+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 runout frequency that is used in the runout 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 425. 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 R0nan Frequency Frequency Figure 425 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 runout, 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, zle je1 (z1 zR1 e~e zR 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 z1+z2 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 runout 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 runout cosine argument. An efficient method for calculating the cosine values can then be formed from the trigonometric identity. COS (nO) COS nI COS  COS The values can be calculated in a loop. Hence, only the cosine value of the runout 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 426. contained in chapter 12 of Input Reference cosCnn This figure and basic explanation are Widrow and Stearns. Output Figure 426 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 runout 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 highlow 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 highlow 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 runout 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 revolutionbased instead of time based, the fourier transform of the signal is now in the perrevolution 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 perrevolution 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 antialiasing 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 subsystems 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 runout. The second subsystem 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 subsystems 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 timebased 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 timebased 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, ,N1 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 perrevolution 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 51. 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 51 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 53 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 Runout is the result of the center of the circle not being located at the center of rotation of the spindle. Runout 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 runout 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 54 Axis of Rotation Out of Plane Cutter Runout due to Eccentricity and Out of Plane Conditions. Figure 54. 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 runout '. 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 runout 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 _ runout profile is then subtracted from the measured data at each revolution. way the instrument runout is removed from the measured data. In this For all future discussion the term runout will refer to actual cutter runout as opposed to instrument runout. 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 runout, 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 56A 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 56B. 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 56 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 57A. 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~ 'ti 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 runout on the force signal can be seen in Figure 58A. Because the runout is an offset of the center of rotation, the effect is to vary the cutting force at the runout frequency. This is a pure harmonic variation so that all of the frequency content due to runout is located at the once per revolution frequency. This can be seen in Figure 58B. The effl of runout, as can be seen in Figure 59A. 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 runout 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 58 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 510 and 511 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 512 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, 510 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% 