Evaluation of cutting force uncertainty components in turning

International Journal of Machine Tools & Manufacture 41 (2001) 719–730

Evaluation of cutting force uncertainty components inturning

D.A. Axinte, W. Belluco, L. De Chiffre*

Department of Manufacturing Engineering, Technical University of Denmark, DTU, Building 425, DK 2800,Lyngby, Denmark

Received 26 July 2000; accepted 18 September 2000

Abstract

A procedure is proposed for the evaluation of those uncertainty components of a single cutting forcemeasurement in turning that are related to the contributions of the dynamometer calibration and the cuttingprocess itself. Based on an empirical model including errors from both sources, the uncertainty for a singlemeasurement of cutting force is presented, and expressions for the expected uncertainty vs. cutting para-meters are proposed. This approach gives the possibility of evaluating cutting force uncertainty componentsin turning, for a defined range of cutting parameters, based on few experiments. 2001 Elsevier ScienceLtd. All rights reserved.

Keywords:Turning; Cutting force; Multicomponent dynamometer; Measurement uncertainty

1. Introduction

Despite the permanent importance of cutting force measurements in machining studies, so farpublished results involving cutting force measurements have not been associated with an evalu-ation of the measurement uncertainty. A number of papers have been published concerning thecalibration of metal cutting dynamometers [1–4], but no estimation of uncertainty has been pro-posed for actual measurements. It is well known that during the cutting process, parameters suchas cutting speed, feed rate and depth of cut often present a deviation from the programmed values.This paper proposes a procedure to account for both calibration and process errors in the uncer-tainty estimation for the specific situation of single cutting force measurements. Fig. 1 schematisesthe influence parameters considered in this work; contributions to the measurement uncertainty

* Corresponding author. Tel.:+45-4525-4760; fax:+45-4593-0190.E-mail address:[email protected] (L. De Chiffre).

0890-6955/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved.PII: S0890-6955(00)00095-X

720 D.A. Axinte et al. / International Journal of Machine Tools & Manufacture 41 (2001) 719–730

Nomenclature

ap depth of cutDaci error in voltage evaluation due to averaging and rounding procedure in the

software on channelsi=1…3Dapk force error due to variation of depth of cut,k=x,y,zDdi error in voltage evaluation due to the limitation of drift compensation procedure

on channelsi=1…3Dfk force error due to variation of feed rate,k=x,y,zDti error in voltage due to the temperature variation affecting the charge amplifier of

the dynamometer on channelsi=1…3f feed rateF force vectorFR resultant cutting forceFX feed cutting forceFY reaction cutting forceFZ main cutting forceN voltage error vector due to calibration procedureNi voltage error due to calibration procedure on channelsi=1…3S calibration matrixv cutting speedV vector of output voltages of the dynamometerVi output voltage of the dynamometer on channelsi=1…3xi,j exploitation matrix coefficients,i=1…3, j=1…3X exploitation matrix (=S−1)UFX,Y,Z,R expanded uncertainty of cutting forcesFX, FY, FZ andFR, respectively

from workpiece, tool and machine are not considered. It has to be underlined that other errors inthe budget uncertainty of single cutting force measurements could be taken into account usingdifferent experimental setups.

2. Experimental details

The three-component tool-post turning dynamometer for the CNC lathe adopted in this workis shown in Fig. 2. The turning dynamometer is a dedicated instrument, designed and manufac-tured at DTU, based on piezoelectric cells [11–13]. The output charges are converted into voltagesignals by charge amplifiers. The dynamometer can measure forces on three Cartesian axes.Charge amplifiers’ output voltages are digitised using a personal computer, equipped with a 16-bit A/D converter and software developed in this laboratory. The software handles real timeprocessing, displaying, recording, drift correction and analysis of acquired data. In Table 1 thecharacteristics of the measuring system are shown.

721D.A. Axinte et al. / International Journal of Machine Tools & Manufacture 41 (2001) 719–730

Fig. 1. Cutting force uncertainty components taken into account in the present model.

Fig. 2. Three-dimensional turning dynamometer mounted on the CNC-lathe.

3. Calibration of the dynamometer

Static calibration of the dynamometer was performed to obtain the calibration matrix accordingto Eq. (1)

{ V} 1{ N} 5[S]·{F} (1)


Table 1Characteristics of the turning dynamometer measuring system

Component Characteristics

Measuring transducer 4 pcs. 9251A Kistler piezoelectric cellsAnalogue signal amplifier 3 Kiag Swiss type 5023 charge amplifiersDigital measuring instrument 16 bit A/D converter NIATMIO 16 XE-50Laboratory software Original program developed with Labview 5.0Conditions for data acquisition Sampling frequency: 50 Hz; trigger:FZ=50 N; evaluating

window: sample 500–1000

The calibration load was applied on a dummy tool holder having a hardened bearing ball insteadof the tool. To reduce parasitic load components, the calibration loads pass through a point whichis equivalent to the tip of the cutting insert during the cutting experiments. For applying the loadin each direction, the dynamometer was mounted on a base having ground orthogonal planes, andloaded using a universal testing machine in compression mode. In Table 2 the calibration plan ispresented. Each loading step was applied for 15 s, and was followed by discharge of the dyna-mometer. The calibration procedure was carried out rotating the dynamometer by 180° aroundthe load axis and was repeated three times.

Calibration was performed using the same acquisition board used to acquire data during theexperiments, thus including the whole measurement chain, from the input (forces) to the datadigitised by the acquisition board (voltages).

Signal drift, which is a typical characteristic of piezoelectric-based transducers, was subtractedfrom the signal assuming linear drift/time behaviour. Data were analysed for linear regressionusing the least squares method, and the coefficients of calibration matrixS and of the error termN were determined, as well as their minimum and maximum values for 95% confidence level.

To obtain the force vector from Eq. (1) the inversionX=S−1 has to be made. The 95% confi-dence intervals of the coefficients of theX matrix (called in the followingxij, i=1…3, j=1…3)were found by computer simulation. The inversion of a large number of matricesS, with coef-ficients obtained as random numbers normally distributed according to regression results (samemean and standard deviation) was implemented, and the coefficients of theX matrices thusobtained were analysed for mean value and 95% confidence interval.

Table 2Calibration plan

Load direction Maximum load (N) Minimum load (N) Increment of load (N)

FX, FY 3000 0 200FZ 9000 0 500


Table 3Characteristics of general setup

Workpiece material 316L stainless steel with 147 HV20±1.5 SDWorkpiece shape Cylindrical bar, 80–200 mm diameter, 800 mm lengthMachine tool VDF Boehringer PNE 480 L CNC latheTool holder PTGNL2525M16, ISO 5608:1995Insert code TNMG160408QM, ISO 1832:1991Insert type Sandvik Coromant GC215 gradeCutting geometry cn=91°; an=8°; gn=26°; ln=26°; re=0.8 mmCutting fluid Vegetable oil RV-20 cSt

4. Experiments

Details of the general setup are presented in Table 3. To obtain the extended cutting force law,the cutting parameters,ap, f andv, were varied using a two level factorial design. The maximumlevel (+) and the minimum levels (2) for the choice of cutting parameters are presented in Table 4.

Based on experimental data, an empirical equation for computing cutting forcesFx, Fy, Fz vs.cutting parameters is presented in Eq. (2). The significance of each cutting parameter in Eq. (2)could be evaluated using linear regression analysis of experimental data, after taking logarithms.Cutting speed was not found significant in Eq. (2), and a simplified Eq. (3) was used.

FX,Y,Z5Cx,y,z·aax,y,zp ·f bx,y,z·vcx,y,z (2)

FX,Y,Z5Dx,y,z·agx,y,zp ·f hx,y,z (3)

A term including material hardness variability is missing from Eqs. (2) and (3), because an experi-mental plan including the same material with different hardness would be more troublesome toimplement. To provide a uniform hardness profile along the cross section of the tested workpiece,the strain hardened shoulder was removed (Fig. 3).

5. Cutting force uncertainty components

A partial uncertainty budget was carried out in the spirit of GUM [5,6] and taking into accountother published works [7–10]. The uncertainty contributions from the acquisition system and the

Table 4Cutting parameters

ap (mm) f (mm/rev) v (m/min)

Levels 2 + 2 + 2 +

Values 1.5 2.0 0.2 0.35 220 245


Fig. 3. Hardness profile of the workpiece before (diameter 240 mm) and after shoulder removal (diameter 216 mm).

process are treated separately, since output data of the acquisition system and the calibrationprocedure results provide an uncertainty expressed in voltage units, while the actual measurementsexpress the value of the uncertainty in force units.

5.1. Contributions of the calibration procedure and acquisition system on the uncertaintybudget

5.1.1. Contribution of calibration procedure (Ni)The uncertainty contribution of the calibration procedure was evaluated for each channel

(Ni, i=1…3) taking into account the following influences:

O The universal testing machine gave a load standard uncertainty of 1.3 N. The machine hadbeen calibrated with a force transducer traceable to reference force standard. Converting theload into voltage, a value of 2.5×1023 V was obtained.

O The rotational effect, which corrects inclination of Cartesian axes of the dynamometer withreference to those of the testing machine, was expressed using a classical rotational matrix.Since this contribution is obtained multiplying the rotational matrix with a force vector, thereal forces obtained at different calibration loads were fed.

5.1.2. Contribution of temperature (Dt)Following catalogue data for 9251A Kistler piezoelectric cells and assuming the reference tem-

perature 25°C with a standard uncertainty 5°C, the voltage error with standard uncertainty of3.36×1024 V was found.

5.1.3. Contribution of drift compensation procedure (Dd)The laboratory software allows an automated drift correction assuming linear distribution drift

vs. time. Analysing raw data files acquired during experiments, the standard uncertainty stemmingfrom drift compensation procedure was 6×1023 V.


5.1.4. Contribution of averaging and rounding procedure in the software (Dar)The error connected with averaging and rounding procedure of software has a standard uncer-

tainty equal to 5×1026 V.

5.2. Contribution of process on uncertainty budget

5.2.1. Contribution of depth of cut (Dap)To evaluate the mean value and its standard deviation, measurements of workpiece diameter

were performed before and after each cutting pass. Based on Eq. (4), the error due to the variationof depth of cut was evaluated for single cutting force experiments performed in each point of theexperimental matrix.

5.2.2. Contribution of feed rate (Df)The mean value and its standard deviation were evaluated starting from the following measure-

ments:

O the machining distance was read from the control monitor of the CNC lathe, estimating astandard uncertainty of 1µm;

O the cutting time was evaluated with a standard uncertainty of 0.02 s using the cutting forcediagram vs. time identifying the tool engagement/disengagement time.

Based on Eq. (3), the error due to the variation of feed rate was evaluated for cutting forcemeasurement.

In Eqs. (4)–(6) the models for evaluating cutting force uncertainties, taking into account thecalibration errors and cutting parameter errors, are presented.

FX5x11(V11N11Dt11Dd11Dac1)1x12(V21N21Dt21Dd21Dac2)1x13(V31N31Dt31Dd3 (4)

1Dac3)1Dapx1Dfx

FY5x21(V11N11Dt11Dd11Dac1)1x22(V21N21Dt21Dd21Dac2)1x23(V31N31Dt31Dd3 (5)

1Dac3)1Dapy1Dfy

FZ5x31(V11N11Dt11Dd11Dac1)1x32(V21N21Dt21Dd21Dac2)1x33(V31N31Dt31Dd3 (6)

1Dac3)1Dapz1Dfz

The parameters figuring in Eqs. (4)–(6) were treated as follows:

xij, exploitation matrix coefficients—type B normally distributed quantities with mean and stan-dard deviations obtained from numerical inversion of matrixS, as described above;


Vi, mean output voltages—type B normally distributed quantities with mean and standard devi-ation obtained from the raw data file acquired during one cutting experiment;Ni, Dti, Ddi, Daci, voltage corrections—type B rectangular distributed quantities centred in zerowith half-width interval obtained as previously described;Dapk, Dfk, cutting parameters’ errors—type B rectangular distributed quantities centred in zerowith half-width interval obtained through Eq. (3).

This model, derived for single measurements, is intended to provide uncertainty evaluation witha limited number of experiments, to be used in the experimental design of more extensive cam-paigns.

Based on the componentsFX, FY, FZ and their uncertainties, the uncertainty for the resultantforce was obtained using Eq. (7).

FR5(F2X1F2

Y1F2Z)0.5 (7)

6. Results and discussion

The model presented above was applied for all the experimental points of the factorial design,and the uncertainty value, when different cutting parameters were used, was expressed as a func-tion of the cutting parameters.

Sensitivity analysis allowed us to conclude that the calibration error (Ni) on the channel wherethe force is measured was the dominant contribution in the uncertainty budget followed by theerror due to the feed rate variation (Dfk), and the error due to the variation of depth of cut (Dapk).For example, when the uncertainty budget of the main cutting forceFZ was evaluated throughoutall the experimental points, the maximum contributions ofN3, Dfz, Dapz from UFZ were 61%,37%, and 14%, respectively. Therefore, the influence cumulative error of cutting parameters wasas important as error calibration in the uncertainty budget of single cutting force measurements.These remarks underline our first supposition that the uncertainty in force measurements is alsoaffected by process parameters. In machining force measurements, any work reporting only thetransducer uncertainty after static calibration is likely to state measurement results that are affectedin reality by much higher uncertainty, which is not quantified. As an example, in Table 5 theuncertainty budget of the main cutting forceFZ for 1.5 mm depth of cut and 0.2 mm/rev feed rate,calculated using Eq. (6), is presented where ‘Corr. coeff’ is the correlation coefficient between theresult and the input quantity and ‘Index’ is the input contribution on output uncertainty budget.Cutting force expanded uncertainties (UFX,Y,Z) can be related linearly to cutting forces (FX,Y,Z) asin Eq. (8).

UFX,Y,Z,R5EX,Y,Z,R1GX,Y,Z,R·FX,Y,Z,R (8)

In Table 6 the coefficientsEX,Y,Z,R, GX,Y,Z,R, the coefficients of determination (R2) and the standarderrors (SE) are presented.

In many cases, the experimenter does not know beforehand the cutting force values when


Table 5Uncertainty budget: main cutting forceFZ for ap=1.5 mm; f=0.2 mm/rev

Quantity Value Standard Sensitivity Uncertainty Corr. coeff. Indexuncertainty coefficient contribution

x31 1.120 N/V 0.175 N/V 20.778 20.136 N 20.01 0.0%v1 20.777729 V 945×1026 V 1.12 0.00106 N 0.00 0.0%N1 0.0 V 0.00866 V 1.12 0.00970 N 0.00 0.0%Dt1 0.0 V 194×1026 V 1.12 217×1026 N 0.00 0.0%Dd1 0.0 V 28.9×1026 V 1.12 32.3×1026 N 0.00 0.0%Dac1 0.0 V 2.89×1026 V 0.0 0.0 N 0.0 0.0%x32 2.300 N/V 0.250 N/V 0.529 0.132 N 0.01 0.0%v2 0.528869 V 811×1026 V 2.30 0.00187 N 0.00 0.0%N2 0.0 V 0.00670 V 2.30 0.0154 N 0.00 0.0%Dt2 0.0 V 194×1026 V 2.30 446×1026 N 0.00 0.0%Dd2 0.0 V 28.9×1026 V 2.30 66.4×1026 N 0.00 0.0%Dac2 0.0 V 2.89×1026 V 0.0 0.0 N 0.0 0.0%x33 975.500 N/V 0.525 N/V 0.762 0.400 N 0.03 0.1%v3 0.761622 V 873×1026 V 976 0.852 N 0.07 0.5%N3 0.0 V 0.00958 V 976 9.35 N 0.78 60.9%Dt3 0.0 V 194×1026 V 975 0.189 N 0.02 0.0%Dd3 0.0 V 28.9×1026 V 975 0.0282 N 0.00 0.0%Dac3 0.0 V 2.89×1026 V 976 0.00282 N 0.00 0.0%Dapz 0.0 N 1.65 N 1.00 1.65 N 0.14 1.9%Dfz 0.0 N 7.25 N 1.00 7.25 N 0.60 36.6%FZ 743.3 N 12.0 N

Value Expanded uncertainty Coverage factor

ResultFZ 743.3 N 24 2

Table 6Results for statistical analysis ofUFX,Y,Z vs. FX,Y,Z

Expanded uncertainty Coefficients Statistics

E (N) G R2 SE

UFX 5.92 0.044 0.95 1.73UFY 17.71 0.060 0.98 1.11UFZ 17.51 0.009 0.98 0.57UFR 15.88 0.010 0.97 0.91

cutting parameters are programmed. For this reason a linear dependence of force expanded uncer-tainties vs. cutting parameters is proposed as in Eq. (9).

UFX,Y,Z,R5Hx,y,z,R1Kx,y,z,R·ap1Lx,y,z,R·f (9)


In Table 7 the coefficientsHx,y,z,R, Kx,y,z,R, Lx,y,z,R, the coefficients of determination (R2) and thestandard errors (SE) are presented. It should be underlined that Tables 6 and 7 must be appliedin the range of tested cutting values only.

Eq. (9) is useful in that it allows the prediction of cutting force expanded uncertainties, giventhe combination of cutting parameters. On the other hand, if Eq. (8) is used, it is necessary toobtain beforehand the values of cutting forces with Eq. (3) in order to feed necessary data. Inthis way the error of Eq. (3) would be added to the error associated with Eq. (8).

In Fig. 4(a) and (b) plots of the main cutting force (FZ) and its expanded uncertainty (UFZ)vs. cutting parameters are shown.

7. Summary of the procedure

The following steps can be followed for evaluation of the calibration uncertainty:

O determination of the coefficients and 95% confidence intervals of the calibration matrixS andof the error termN;

O calculation of the coefficients ofX=S−1 and their 95% confidence intervals by computer simul-ation;

O evaluation of the uncertainty contributions due to loading system (testing machine), driftcompensation procedure, variation of temperature, and acquisition board.

The evaluation of process-associated uncertainty contributions can be carried out as follows:

O performing cutting force measurements following a 23 factorial design;O regression analysis to express cutting force vs. cutting parameters;O evaluation of the uncertainty contributions in cutting force evaluation due to the variability of

cutting parameters.

8. Conclusion

A procedure is proposed for the evaluation of those uncertainty components in the specificsituation of a single cutting force measurement in turning that are related to the contributions

Table 7Results for statistical analysis ofUFX,Y,Z vs. ap and f

Expanded uncertainty Coefficients Statistics

H (N) K (N/mm) L (N rev/mm) R2 SE

UFX 0.61 6.42 52.41 0.99 0.8UFY 18.37 2.00 53.30 0.91 0.4UFZ 11.09 4.10 33.66 0.99 0.5UFR 10.10 4.00 40.00 0.99 0.7


Fig. 4. Dependencies of main cutting forceFZ (a) and its expanded uncertaintyUFZ (b) vs. tested cutting parameters(ap=1.5; 2 mm;f=0.2; 0.35 mm/rev).

of the dynamometer calibration and the cutting process itself, not considering those uncertaintycontributions coming from workpiece, tool and machine. A model for the expression of cuttingforces uncertainty in different cutting conditions was found. The model enables the prediction ofthe uncertainty of cutting force measurements for a defined range of cutting parameters, basedon calibration data and measurements collected in few experiments ran using a factorial design.

The results showed that the contributions in the uncertainty budget from single cutting forcemeasurements are balanced between the calibration error on the channel where the force is meas-ured and the cumulative error due to the variation of cutting parameters. Being responsible forapproximately 50% of the uncertainty in single force experiments, the variation of cutting para-meters may be responsible for the scatter of cutting force measurements found in the literature.


It is suggested that an expression for the measurement uncertainty should always be provided incutting force measurements, with account being taken of process-related contributions, accordingto the proposed model. Furthermore, the model enables the prediction of the uncertainty of cuttingforce measurements for a defined range of cutting parameters, based on few experiments and datafrom calibration.

Acknowledgements

The authors are grateful to Dr. S.N. Rasmussen, Teknologi Institute, Denmark, for providingthe software ‘GUM Workbench’. This work was carried out under funding no. 9801683 from theDanish Research Agency and no. NON97-DTU-3 by the Danish Strukturdirektoratet.

References

[1] A. Bray, G. Barbato, R. Levi, Theory and Practice of Force Measurement, Academic Press, 1990.[2] R. Levi, Multicomponent calibration of machine-tool dynamometers, Trans. ASME, J. Eng. Ind. 11 (1972)

1067–1072.[3] A. Sawla, Guidance for the determination of the best measurement capability of force calibration machines and

uncertainty of calibration results of force measuring devices, PTB-Mitteilungen 104 (4) (1994) 237–242.[4] A. Bray, G. Barbato, F. Franceschini, E. Xhamo, Rotational and end effects: a model for uncertainty evaluation

in force measurements by means of dynamometers, Measurement 17 (4) (1996) 279–286.[5] Guide of the Expression of Uncertainty in Measurement, International Organization for Standardization, 1993.[6] International Vocabulary of Basic and General Terms in Metrology, International Organization for Standardiz-

ation, 1993.[7] R.H. Dieck, Measurement uncertainty models, ISA Trans. 36 (1) (1997).[8] C. Xu, A practical model for uncertainty evaluation in force measurements, Meas. Sci. Technol. 9 (1998)

1831–1836.[9] C. Ferrero, New trends in multi-component calibration, in: ISMCR ’98: Proceedings of the Eighth International

Symposium on Measurement and Control in Robotics, 1998, pp. 69–74.[10] C. Ferrero, A six-component dynamometer to measure parasitic load components on deadweight force standard

machines, in: Recent Advances in Weighing Technology and Force Measurement. 10th International Conferenceof the IMEKO Technical Committee TC-3 on Measurement of Force and Mass, 1984, pp. 199–204.

[11] J.H. Frederiksen, K.B. Pedersen, Investigation and construction of a cutting force dynamometer. DTU, M.Sc.Thesis, 1984 (in Danish).

[12] J.H. Frederiksen, K.B. Pedersen, Design and Calibration of Dynamometer for CNC Lathe, 14th. NAMRC, 1986,pp. 183–188.

[13] D.A. Axinte, W. Belluco, Calibration procedure for 3D turning dynamometer, DTU, Department of ManufacturingEngineering, Report no. MM99.47, 1999.

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Evaluation of cutting force uncertainty components in turning