High Frequency Correction of Dynamometer for Cutting Force Observation in Milling

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François Girardin1

e-mail: [email protected]

Didier Remond

Jean-François Rigal

Université de Lyon,CNRS INSA-Lyon,

LaMCoS UMR5259, F-69621, France

High Frequency Correction ofDynamometer for Cutting ForceObservation in MillingPiezoelectric dynamometers are widely used for cutting force measurements. Indeed, thisdevice has the largest bandwidth for this kind of measurement. Nevertheless, the behaviorof this device is not very well-known and its use is sometimes inappropriate for static andhigh frequency dynamic measurements. In this paper, a piezoelectric dynamometer isused for cutting force measurements in a milling case. Cutting forces in milling arediscontinuous by nature due to successive inward and outward movements of tool-teethon the workpiece. As a result, a bandwidth criterion based on cutting parameters isdefined in order to permit clear observation of the mean oscillation of the cutting force.The frequency response of a dynamometer is then analyzed over a wide frequency range.A 2 kHz bandwidth can be defined for an efficient correction of cutting force. The dyna-mometer appears to be exploitable for higher frequencies up to at least 16 kHz though alarge number of factors must be taken into account in the analysis. Finally, severallateral milling tests are performed by changing cutting speed, feed rate, and lubricantconditions. The correction of measurements permits highlighting certain particularities inthe cutting force signals, such as the effect of shock of inward tool-teeth strokes on theworkpiece, the specific behavior for outward tool-teeth strokes, and the effect of a lubri-cant on the variation in cutting forces. �DOI: 10.1115/1.4001538�

IntroductionIn manufacturing, cutting operations are always used for pro-

ucing final workpieces directly, or indirectly, for producing toolsnd dies. The cutting process is a very old process but it is still theubject of studies due to the introduction of new materials that areard to cut, which lead to specific forms of wear �1� or, for ex-mple, to tool-teeth failure �2�. Moreover, productivity is anotheroncern and cutting speeds are becoming increasingly fast thankso new coatings. A large number of tests have been developed inrder to control the tool wear and qualify new tools, the machin-bility of new materials �3�, and workpiece deformation �4�. Manyypes of signal are used to achieve this such as spindle poweronsumption and cutting force measurements. In both cases,nalysis bandwidth is limited in frequency and appears narrowerhen measuring power consumption than when using a dyna-ometer. The turning process is often studied due to the simplic-

ty of considering an orthogonal and continuous cutting case withonstant chip geometry and constant cutting parameters. In theilling process, however, the phenomena are more complex due

o the rotation of the milling tool and successive inward and out-ard tool-teeth strokes on the workpiece. This paper deals with

orce measurement in a milling case in view to obtaining an ex-loitable cutting force signal for milling and improving under-tanding of the operation.

The main drawback of effort measuring devices is that theirandwidth is limited. Measurements have been improved by in-reasing the rigidity of devices, working with strain gages, andiezoelectric sensors but accurate modeling requires larger band-idths in terms of frequency. Improving measurement correctionas taken two main directions: correction in the time domain or inhe frequency domain. Regarding a piezoelectric sensor similar tohat used in this paper �three dimensional information, force am-

1Corresponding author.Contributed by the Manufacturing Engineering Division of ASME for publication

n the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript receivedpril 10, 2009; final manuscript received March 19, 2010; published online June 16,
010. Assoc. Editor: Suhas Joshi.
ournal of Manufacturing Science and EngineeringCopyright © 20

om: http://manufacturingscience.asmedigitalcollection.asme.org/ on 10/07

plitude of up to 5 kN with 0.01 N resolution�, Lapujoulade �5�developed a correction in the time domain by subtracting the in-ertial influence of the upper part of the dynamometer. This methodrequires the addition of a large number of accelerometers to thesensor and the evaluation of the mass of the upper part. Moreover,it concerns only the rigid bodies in movement in the dynamom-eter. Thus, the bandwidth increases from 1 kHz to 2 kHz by cor-recting the first natural frequency of each axis.

In the frequency domain, the method is based on structuralanalysis: the frequency response as a function of the structure isfirst evaluated in order to correct the spectrum of the signal mea-sured. Castro et al. �6� analyzed the frequency response of a dy-namometer up to 4 kHz with good results for a turning case but heworked with the z axis only and did not study coupling effectsbetween different axes. Tounsi and Otho �7� focused on this cor-rection but within a bandwidth of 2 kHz. Altintas and Park �8�performed numerous studies of milling operations, working with aspindle integrated dynamometer. The response of this dynamom-eter is corrected in the frequency domain using a Kalman filterand the bandwidth is extended from 300 Hz to 1000 Hz. The sameapproach was reproduced more recently by Chae and Park �9�using a new small dynamometer for micromilling observation.The bandwidth was extended to 5 kHz but the dynamometer can-not be used for macromilling observation. These works are syn-thesized in Table 1.

In this paper, the process studied is milling where the cut sec-tion is discontinuous due to the rotation of the mill and successiveinward and outward strokes of the tool-teeth on the workpiece. Asa result, the cutting forces are variables, which make it necessaryto define the bandwidth needed for precise observation. Section 2presents a milling model that takes into account the discontinuitiesof the cut section and cutting forces. This model is then used todefine a bandwidth criterion for a clear observation of the meanoscillation of the cutting force signal. In some cases, the calcu-lated bandwidths can be higher than those advised by the manu-facturer. Consequently, Sec. 3 proposes an analysis of dynamom-eter frequency responses over a large frequency bandwidth. Amaximal bandwidth value for efficient cutting force correction is
also proposed at about 16 kHz. The method is based on the one
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roposed by Castro et al. �6� and is extended to three componentsorrections and time signal reconstruction, considering cross-talkshat are outlined to be important at high frequency. This corrections then used in the last part for observing the real shape of theutting force signal for different cutting speeds, feed rates, andubricant conditions. Certain particularities are highlighted for theutting force signals, such as the effect of the shock of the inwardtroke of the tool-teeth on the workpiece, the specific behavior ofhe outward stroke of the tool-teeth, and the unusual effect of theubricant. Finally, we provide conclusions and discuss the outlookor the use of piezoelectric dynamometers.

Milling Model and Bandwidth Criterion Definition

2.1 Model and Experimental Conditions. The model stud-ed shown in Fig. 1 concerns the lateral milling of a plate such thathe mill never works with its end. The efforts applied on the platere measured using a Galilean approach �x ,y ,z�, thus, they areocated either at the tool center or on the plate due to the uniformeed movement. The mill rotates around axis z and translates atonstant speed Vf along axis x.

Experimentally, the machining assembly illustrated in Fig. 2as composed of a machined metal plate fixed on to a dynamom-

ter �Kistler 9257A� in order to measure the cutting forces. Theata were acquired for each direction at a 50 kHz sampling rateia synchronous channels. The tests presented here were per-ormed on AISI 4340 steel plate, hardness 47 Hrc, previously

Table 1 E

Lapujoulade �5� Tounsi and Otho �7�

ime domain Xrequency domain Discreteeference Acc. Acc.olicitation Model Millingross talk No Negligibleandwidth 2 kHz 2 kHz

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Fig. 1 Model studied

Fig. 2 Experimental set-up

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studied for a turning case �10�. Cutting tests were performed with-out lubricant using a four teeth tool �APX 3000 from Mitsubishiwith AOMT 123608 PEER-M VP15TF-coated inserts�. The ex-perimental and analytical results were obtained with the param-eters given in Table 2.

The development of the analysis relies on a geometrical modelfor the cutting force. The cutting thickness hi for each tooth i isdefined by Eq. �1� and Fig. 3. Then the cutting force is given byEq. �2�. The parameter values of the model are those given inTable 3 without additional precision. They correspond to the caseof hardened steel milling.

hi�t� = �ei�t�� · �wi�t��

hi�t� = �fz · cos��i�t�� · �rect� t − ti

T� � ��t�� 1�

Ft = �i=1

nz

Kc0· � hi

h0�p

· hi · ap �2�

Experimental and theoretical cutting forces are shown in Fig.4�a�. The experimental cutting force is close to the analytic modelbut several oscillations and vibrations can be observed. The firstnatural frequency of the dynamometer used is given by the manu-facturer at around 2500 Hz �11�, thus, it corresponds to the oscil-lations observed �Fig. 4�b��.

2.2 Definition of the Bandwidth Criterion for ObservingMilling Force. The cutting force model is used to define a mini-mal bandwidth for the mean curve of the cutting force. Fouriertransform is used to define the spectrum of the model in the fre-

ting work

Castro et al. �6� Altintas and Park �8� Chae and Park �9�

Discrete Curve fitting Curve fittingForce Dynamometer Force, acc.Shaker Milling Hammer

No �10% �10%4 kHz 1 kHz 5 kHz

Table 2 Cutting parameters used

R �mm� nz

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Fig. 3 Construction of cutting depth model

Table 3 Model parameters used

Kc0�MPa� h0 �mm� p nz

650 0.4 −0.5 4

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uency domain. The analytical result for the theoretical spectrums explained in Eq. �3�. As shown in Fig. 5, the global shape islose to a cardinal sine. This is due to the transformation of func-ion wi, which defines the tool-material contact through time ando the weak effect of transformation of ei

1+p.

FT�Ft� = �i=1

nz Kc0

h0p · ap · FT�hi

1+p�

FT�Ft� = �i=1

nz Kc0. ap

h0p · FT�ei

1+p� � �e−j2�fti · sin c�fT� · �1/��f��

�3�As a result, the theoretical bandwidth of the spectrum in such ailling case is infinite. Moreover, there is no integral of the abso-

ute value of the cardinal sine function, so no limit can be fixed tohe power spectrum of the cutting force model, such as 80% ofpectrum density. Thus, a criterion has to be defined for a tempo-al signal such as the first overshoot value or time at 5% or 2%.

The pseudoperiod T of the cardinal sine is used to define aeference frequency for the cutting force spectrum. This pseudo-eriod corresponds to the duration of tool-material contact forach tooth during one revolution. The reference frequency 1 /T isxplained by Eq. �4�. In the latter, the useful cutting parametersre spindle frequency N, radial depth ap, and tool radius R. It ismportant to note that the number of teeth does not impact thisrequency.

F0 =1

T

F0 =�

��d − �e�

F0 =N · �

30 · ��/2 − a sin�1 − �ae/R��4�

The cutting force model is then filtered by removing its spec-rum at different k multiples of F0 to observe the result on theeconstructed signal. The analytical formulation of this operations detailed in Eq. �5�. The convolution of the initial force signal

0 0.005 0.01 0.015 0.020

200

400

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time (s)

TotaleffortFt(N)

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2.5 3 3.5 4

x 10−3

300

400

500

600

b)

time (s)

TotaleffortFt(N)

Ftth

Ftexp 3 oscillations

ig. 4 Theoretical and experimental cutting forces for Vc160 m/min

0 500 1000 1500 2000 25000

50

100

150

200

frequency (Hz)

|FT(Ft th)|(N)

FT(Ftth)

max(FT(Ftth)) . sinc(f.T)

Fig. 5 Frequency spectrum for the cutting force model

ournal of Manufacturing Science and Engineering


with a cardinal sine introduces oscillations on the reconstructedsignal, especially near the discontinuities, as can be seen in Fig.6�a�. A bandwidth of about ten times the reference frequencyseems to be acceptable for observing the mean curve of the cut-ting force. Indeed, the reconstructed signal represents the entry ofteeth into material quite well and the oscillations quickly becomenegligible.

Ftcut= FT−1FT�Ft� · �rect� f

2 · k . F0��

Ftcut= Ft � sin c�2 · k . F0 · t� �5�

Figure 6�b� shows that if ten times the reference frequency isused for different radial depths, the global shape of the recon-structed signal is always the same. A suitable bandwidth of tentimes the reference frequency can be thus defined for any depth.

2.3 Bandwidth Validation. For the cutting parameters ofTable 2, the reference frequency is about 200 Hz. The experimen-tal signal of the cutting force obtained with these parameters forAISI 4340 steel is shown in Fig. 7. Filtered signals at 5.F0 and10.F0 are also plotted. The latter signal is very close to the meancurve of the original signal. This confirms that a suitable band-width of 10.F0 can be considered for studying cutting forces inmilling. A precise analysis of the shape of this signal will beperformed in a further section.

Table 4 gives an indication of suitable bandwidths for differentradial depths of mill and a spindle frequency at 1000 rpm. Fullengagement corresponds to slotting, whereas few radial depth ra-tios concern finishing operations. In the first case, standard mea-surement can be enough, provided that the spindle frequency usedis not too high. Indeed, 1000 rpm can be very slow for a small

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time (s)

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radial depth : 50 %

Fig. 6 Sufficient bandwidth for frequency analysis

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Fig. 7 Bandwidth validation—case of the cutting parameter of
Table 2
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iameter mill and the bandwidth will increase to 3 kHz if 10,000pm is needed. This corresponds to a case of a 10 mm diameterool and a cutting speed of 300 m/min, which are very commonalues.

In a finishing case �ae=0.5 mm�5%� with the same cuttingarameters, the required bandwidth is about 23 kHz, which isuch higher than the theoretical dynamometer capacity ��1 kHz

11��. In spite of this theoretical limit, based on the first naturalrequency of the dynamometer, it is necessary to know the realapacity of the dynamometer in order to obtain usable informationt high frequencies. Previous study shows that dynamometer pro-ides force information of at least up to 20 kHz �12� but magni-udes and directions are not certified. The next section presents annvestigation of dynamometer response and a proposal for correc-ion in order to widen the bandwidth of correct observation ofutting forces.

Dynamometer Correction

3.1 Theory of Dynamometer Correction. Errors in force ob-ervation due to the use of dynamometer can be defined by com-aring the real effort applied to the dynamometer to the measuredffort delivered by the dynamometer �equivalent electric signal�.he aim of dynamometer improvement is to correct the influencef dynamometer. The correction theory in the frequency domain ischematized in Fig. 8.

The applied effort �F� is the real effort to which the dynamom-ter is subjected. This effort is unknown; only the signal deliveredy the dynamometer is known and called measured effort �K�. Theynamometer is considered as a linear system, thus a transferatrix �H� can be defined �13� that links applied and measured

fforts in Eq. �6�. As transfer matrix �H� is different from identityatrix, measured effort will never be equal to applied effort.Correction process consists in multiplying measured effort by

he inverse of the transfer matrix to avoid effect of dynamometereasurement. Signal of measured effort �K� is low-pass filtered

efore correction using the previous bandwidth criterion to limitxtent of computation. The result is a corrected effort �Fcor� in-ended to be as close as possible to the applied effort.

Kx�f�Ky�f�Kz�f�

� = Hxx�f� Hxy�f� Hxz�f�Hyx�f� Hyy�f� Hyz�f�Hzx�f� Hzy�f� Hzz�f�

� · Fx�f�Fy�f�Fz�f�

� �6�

A precise definition of the transfer matrix �H� has to be estab-ished for a nice correction. The methodology of transfer matrixvaluation based on conventional modal analysis is defined inext paragraph.

3.2 Material and Method for Transfer Matrix Evaluation.n order to obtain the transfer matrix �H�, an effort is applied onlylong axis j and is evaluated with a reference sensor. The compo-ents of measured effort �K� are provided by Eq. �7�. The applied

able 4 Reference bandwidths for N=1000 rpm and differentadial depths

ae /2.R 1% 5% 10% 50% 100%

10 /T �Hz� 5200 2300 1600 670 330

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Fig. 8 Theory of correction in frequency domain

31002-4 / Vol. 132, JUNE 2010


effort Fj is obtained from reference sensor and the different Ki arethe efforts measured with the dynamometer along axis i �i=x ,y ,z�. Then each column j of �H� can be obtained easily withEq. �8� for each frequency channel f in the bandwidth of interest.

Ki�f� = Hij�f� · Fj�f� �7�

Hij�f� =Ki�f�Fj�f�

�8�

In modal analysis, two main devices are used for applying aspecific effort Fj and determining structural stress. The first is animpact hammer. The repeatability of the impact is rather poor forboth power and direction while its bandwidth is limited to 10 kHz�13�. Consequently, we preferred the second solution, i.e., ashaker, which is easier to control for stress type, power, direction,and bandwidth.

The experimental set-up is shown in Fig. 9. The devices aredetailed in Table 5. A noise generator delivers specific signal tothe shaker that excites the dynamometer through a reference sen-sor and the workpiece. The reference sensor also delivers the ef-fort applied to the system. Reference sensor’s signal and dyna-mometer’s signals are then synchronously sampled with 50 kHzsampling frequency, using default anti-aliasing filter of the dataacquisition card �DAQ�. It is noted that the dynamometer is ex-cited through the workpiece so as to reproduce the milling tests asclosely as possible. Thus, this is the global cut system�workpiece+dynamometer� that is characterized and not only thedynamometer. This set-up should be done again if the workpieceis changed.

The excitation was performed using the periodic chirp functionwith a bandwidth of 1.6 kHz between 0 kHz and 3.2 kHz, and 3.2kHz from 3.2 kHz to 20 kHz, in order to achieve greater precisionon the first natural modes of the dynamometer. Over 20 kHz, theshaker was too weak to excite the system properly. Referencesensor has a unloaded mounted resonant frequency of 75 kHz �14�that is reduced to about 25 kHz considering the mass of the mov-ing part of the shaker �170 g� and the pushrod �negligible�. Thiscertifies the exact measurement of applied effort for all the ana-lyzed bandwidth, up to 20 kHz. The same tests were performedfor the three different solicitation directions by moving the shakerand the reference sensor around the workpiece.

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Fig. 9 Experimental arrangement

Table 5 List of used devices

Dynamometer Reference sensor Shaker Noise generatorKistler 9257A Dytran 1051 v2 G.W. V20B HP 35670A

Charge amplifier Conditioner Amplifier DAQKistler 5015 MCE 4114 G.W. SS100 NI 4472



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3.3 Transfer Matrix Computation. Once all the experimen-al tests have been performed in all the directions and for the fullnalysis bandwidth, transfer matrix �H� is evaluated. In order toinimize the influence of noise in acquired signals, �H� is com-

uted using spectral density power �cross-spectrum input/outputKiFj

and the autospectrum for input SFjFj� �13� rather than only

orce spectrum. The relation is then given by Eq. �9�.

Hij�f� =SKiFj

�f�

SFjFj�f�

�9�

To verify the linearity hypothesis for the dynamometer, coher-nce matrix �C� is computed as defined by Eq. �10�. The coher-nce function is equal to one if the system is stationary and linear.n the transfer function computation, the limit value for the satis-ying order of magnitude for the coherence is often defined in theiterature as being about 0.75 �13�.

Cij�f� =SKiFj

�f�

�SKiKi�f� · SFjFj

�f��10�

The data processing was done with MATLAB, using the signalrocessing toolbox to compute components of �H� and �C�. There-ore an average of ten parts of spectral density power was calcu-ated before computing the transfer and coherence function, inrder to minimize the noise and to obtain better coherence. Theesults for the Hix modulus and coherence are shown in Fig. 10.

Between 2 kHz and 16 kHz, coherence is very close to onexcept for certain peaks that correspond to resonances or antireso-ances. Over 16 kHz, all the coherence functions decrease sud-enly, probably due to the weakness of the shaker at these fre-uencies. Under 2 kHz, Cxx is about one but Cyx and Czx arerongly estimated. Moreover, there is no correlation between the

ignal measured and the effort applied at these frequencies. Localaps can be observed between 2 kHz and 16 kHz and correspondo resonances or antiresonances of the system along the differentxes.

Concerning Hix components, the experiments clearly confirmhe specifications of the dynamometer given by the manufacturer:he first natural frequency is between 2000 kHz and 2500 Hz, the

0 0.2 0.4 0.6 0.8

10−3

10−2

10−1

100

101

Fre

|Hix|(N/N)

0 0.2 0.4 0.6 0.80

0.5

1

Fre

|Cix|

Fig. 10 Transfer and coherence fu

esponse of the axis ranges from about one to 1000 Hz while the



signals are clearly independent within the same range. �Hyx� and�Hzx� are of the same order of magnitude as �Hxx�, so couplingterms must be considered. It is important to note that componentsof �H� never go below 10−1, so the signals contain valuable infor-mation for high frequencies. The same remarks can be made forother axis excitations. However, above 2 kHz the different reso-nances and antiresonances are never at the same frequency withrespect to axis stress and response. As a result, one of the levels ofone of the coherence functions is often weak, as can be seen inFig. 11. This problem may be due to incorrect dynamometer use.Nevertheless, the dynamometer appears usable for a bandwidthfrom 0 kHz to at least 16 kHz. The transfer matrix can be appliedeasily from 0 kHz to 2 kHz, and could probably be extended ifgreater care were taken to characterize the dynamometer in orderto reduce coherence problems.

3.4 Transfer Matrix Inversion and Correction. The transfermatrix must be inverted in order to correct the efforts measured. Itis important to consider that no regression has been set on thecomputed transfer matrix, so �H� is in the discrete frequency do-main. �H� is thus composed of as many matrix �H�f�� as thediscretization of frequency bandwidth is providing. Each �H�f��matrix for a discrete frequency f is basically inverted using theMATLAB function. �H−1� is also composed of all the �H−1�f�� com-puted.

1 1.2 1.4 1.6 1.8 2

x 104ncy (Hz)

1 1.2 1.4 1.6 1.8 2

x 104ncy (Hz)

|Hxx| |Hyx| |Hzx|

|Cxx| |Cyx| |Czx|

ions for a solicitation along x axis

0 2000 4000 6000 8000 10000 12000 14000 160000

0.2

0.4

0.6

0.8

1

Frequency (Hz)

|Cii|

Cxx Cyy Czz

que

que

Fig. 11 Coherence for each axes solicitation

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Transfer matrix components are in the complex domain. Some-imes only the modulus of the corrected signal is considered �6�.his is not enough if we want to rebuild a time signal, in order tobserve the exact instant of tool tooth entry into a material duringilling. The correction must be validated both on the modulus and

n the phase of the signals. An initial verification can be per-ormed on the stress signal. In the time domain, the correctedignal is often quite good, as shown in Fig. 12.

The corrected signal can occasionally be worse than the initialne, especially for high frequencies. The problem arises from theual nature of the correction: the correction of the modulus canrovide satisfying results while the correction of the phase canverapproximate. Therefore, close frequencies can be added in-tead of being subtracted. This phenomenon appears specifically

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Fx(N)

Fxapplied

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ig. 12 Time representation of applied, measured and cor-ected signals for solicitation effort

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ig. 13 Effect of correction on the cutting force-cutting param-ters of Table 2

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Fig. 14 Corrected efforts for two fe

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for the frequencies corresponding to resonances or antiresonanceswith a bad coherence factor. To avoid the influence of bad correc-tion, evaluation of Hij

−1 is limited when Cij is less 0.75. In thiscase, Hij�fu� is set to the last value of Hij�f� �f � fu� computedwith Cij�f� higher than 0.75 before inverting.

4 Exploitation of Force Correction

4.1 Validation of Correction on the Cutting Force Signal.An application of this correction was performed on the experi-mental signal of Sec. 2.3. The bandwidth needed for this experi-mental test was defined at 2060 Hz, corresponding to the upperlimit of the corrected signal spectrum �higher frequencies were setto zero�. Figure 13 shows the effect of the correction in compari-son with the experimental total force �low-pass filtered signal with2060 Hz bandwidth�. Two kinds of correction are presented here.The first signal is corrected using all the terms of the inversetransfer matrix of which only the diagonal was considered for thesecond signal �coupling terms are ignored�. There are very fewdifferences between the two signals, confirming that the couplingterms are quite negligible in this bandwidth.

The most significant difference between the original filteredsignal and the corrected signal can be observed just after the entryof the teeth in the material for about 1 ms. The corrected signal isabout 10% higher than the original one in this zone. After this, allthe signals have roughly the same shape. This appears to validatethe correction methodology since no deviation occurs. Neverthe-less, further experimental tests are required in order to interpretthis new signal force shape and confirm the efficiency of the cor-rection. This is the subject of the next subsection.

4.2 Influence of Cutting Speed and Feed Rate. Differenttests were performed in order to observe the influence of cuttingparameters on the mean curve of the cutting force. The machinedmaterial is still AISI 4340 steel, hardened to 47 HRc. In Fig. 14,the signals of the corrected total forces are plotted for the � anglerather than time, in order to compare their mean oscillations fordifferent cutting speeds. In �Fig. 14�a��, the cutting force signalsare around the same level whatever the cutting speed, confirmingthe slight influence of cutting speed if it is higher than the criticalspeed. The outward tooth stroke appears after a small oscillation,like a pause in decreasing cutting force. Moreover, the higher thecutting speed, the higher the level of this pause. Final surfaceroughness is the result of the behavior of material removal in thiszone, so understanding this behavior is important. Observing the

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ean cutting force curve constitutes the first step toward under-tanding this phenomenon. The correlation of cutting force signalith surface roughness measurements is potentially very

nteresting.Concerning the inward stroke of the tooth into the material, two

scillations can be observed. Amplitude increases with cuttingpeed. Moreover, the angle between the two successive maxi-ums increases from 0.065 rad ��0.005 rad� for a cutting speed

f 80 m/min to 0.13 rad for 160 m/min. These periods correspondo a constant frequency of about 1500 Hz, which is expected to bene of the natural frequencies of the spindle. The shock energy atool edge impact increases with cutting speed �quite similar to theorce amplitude but shorter transient state time�. As a result, theigher the cutting speed, the more the shock excites the spindleatural modes. Consequently, the oscillations observed will be theesponse to at least one of these modes.

In Fig. 14�b�, for the same cutting speed, the global shape of theorrected signals is very similar if a feed rate of 0.2 mm or 0.1m is considered. In particular, oscillations at the inward stroke

f the teeth and the break before the outgoing stroke are stillresent. Only a proportional transformation is observed. Indeed, ifhe cutting force model is considered �Eq. �2� and Table 2�, aactor of two in the feed rate is converted into a factor of 1.4 in theutting forces due to exponent p. For the inward stroke of theeeth in the workpiece, the experimental factor is about 1.25, cor-esponding to an error of 10% in the model results. This gap is notery significant but it shows the limit of this simple model with aighly variable cutting thickness. What is more, the cutting thick-ess hi decreases to zero in Fig. 3.

4.3 Application of the Correction: The Influence of Lubri-ant on Cutting Forces. The effect of lubricant is observed onorrected signals in Fig. 15 for two cutting speeds and a feed ratef 0.1 mm. The global shapes of the signals are very similar. Thepecific zones observed beforehand are still present with the lu-ricant. Nevertheless, following the two oscillations at the inwardtroke of the tooth on workpiece, cutting force is higher in thease with lubricant than without it. This may seem surprising butoth uses of lubricant require consideration. First, “cutting fluid”s used as a lubricant in order to reduce friction and consequentlyutting forces. Second, lubricant is used as a coolant in order torotect the cutting edge from high temperature and therefore fromhermal diffusion wear. Nevertheless, the mechanical strength andardness of some materials can be excellent at ambient tempera-ure and poor at medium or high temperature. As a result, they

ight be machined more easily at high temperature, despite pos-ible tool wear. This is probably the case of the steel used forhese tests: it had been thermally hardened, so it was quite difficulto machine. However, its mechanical strength falls by about 10%hen temperature is increased to 200°C �15,16�. This observation

Fig. 15 Corrected efforts for fz=0

an explain the increase in cutting force when using a lubricant.



5 ConclusionThis paper described the extension of a piezoelectric dynamom-

eter and cutting model to a milling case. This particular cuttingprocess can generate highly variable cutting forces. As a result, aspecific bandwidth was defined to achieve clear observation of themean oscillation of the cutting force in a case of lateral milling.This specific bandwidth depends on spindle speed and radialdepth ratio. It is larger if spindle speed is high and radial depth islow. Since the dynamometer bandwidth is limited, dynamic cut-ting forces are more difficult to observe for finishing lateral mill-ing than for slotting.

In order to know the real capacity of a piezoelectric dynamom-eter, a frequency analysis was performed and a transfer matrix ofthe dynamometer computed. The terms of this matrix are clearlyobservable up to 16 kHz. The axes of the dynamometer werefairly independent up to 2 kHz but couplings were very strong forhigher frequencies. Correction of the dynamometer response wasproposed with good results up to 2 kHz. In this bandwidth, thecorrection results were equivalent for full correction �consideringcoupling� or for correction with only diagonal terms �without cou-pling�. On corrected signals, the shock at the inward stroke oftool-teeth on a workpiece was observed as were several oscilla-tions caused by this shock. These were the responses of one of thevibration modes of the spindle. The cutting force signal provideda characteristic shape at the outward stroke of the tool-teeth witha break in the decrease in cutting force before a sudden fall inforce. This reveals a characteristic behavior of material removal inthis zone of tool-teeth moving outwards. It should be noted thatthe geometry of the finished surface is generated by the cutting inthis zone. Additional investigations were performed in this direc-tion to better understand the behavior studied here and improvethe milling process in industrial applications. Curiously, when us-ing a lubricant, cutting force increased rather than decreased. Thedecrease in mechanical properties observed in the tests withoutlubricant could be correlated with higher temperatures generatedin the workpiece.

To sum up, it is important to note that the method used canensure maximum measurement bandwidth since no curve fittingor regression is utilized. Moreover, whereas existing workspointed out that cross-talk components are negligible, which isconfirmed with this study for frequencies lower than 2 kHz, theyare of the same order of magnitude than direct-talk componentsfor higher frequencies. At high frequencies, dynamometer has thusto be precisely characterized using small bandwidth excitations,maybe less than the 3.2 kHz used in this study, in order to obtainproper correction of measured forces. The analysis of the responseof the dynamometer showed that it was weakened only ten timesat high frequencies at least up to 16 kHz, meaning that the infor-mation is exploitable for high frequencies and corrected if char-

mm/tooth and two cutting speeds
.1
acterization is well down. Consequently, the dynamometer could

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e used at high frequency in order to obtain information on spe-ific frequencies, such as segmentation frequencies, provided thathe appropriate parameters are taken into account.

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ened Steels: Part I: Alumina/TiC Cutting Tool Wear,” Wear, 247�2�, pp. 139–151.

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�4� Budak, E., 2006, “Analytical Models for High Performance Milling. Part I:Cutting Forces, Structural Deformations and Tolerance Integrity,” Int. J. Mach.Tools Manuf., 46�12–13�, pp. 1478–1488.

�5� Lapujoulade, F., 1997, “Measuring of Cutting Forces During Fast TransientPeriods,” Proceedings of the First French and German Conference on HighSpeed Machining, Metz, pp. 372–376.

�6� Castro, L. R., Viéville, P., and Lipinski, P., 2006, “Correction of Dynamic

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Effects on Force Measurements Made With Piezoelectric Dynamometers,” Int.J. Mach. Tools Manuf., 46, pp. 1707–1715.

�7� Tounsi, N., and Otho, A., 2000, “Dynamic Cutting Force Measuring,” Int. J.Mach. Tools Manuf., 40�8�, pp. 1157–1170.

�8� Altintas, Y., and Park, S. S., 2004, “Dynamic Compensation of Spindle Inte-grated Force Sensors,” CIRP Ann., 53, pp. 305–308.

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Documents

High Frequency Correction of Dynamometer for Cutting Force Observation in Milling