Point Laser

8/8/2019 Point Laser

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Journal ofManufacturing

Science andEngineering

Technical Briefs

Point Laser Triangulation Probe

Calibration for Coordinate Metrology

Kevin B. SmithAssistant Professor, Brigham Young University,

Department of Electrical and Computer Engineering,

Provo, UT 84602

Yuan F. ZhengProfessor, The Ohio State University, Department of

Electrical Engineering, Columbus, OH 43210

Point Laser Triangulation (PLT) probes have significant advan-tages over traditional touch probes. These advantages includethroughput and no contact force, which motivate use of PLT

probes on Coordinate Measuring Machines (CMMs). This docu-ment addresses the problem of extrinsic calibration. We present a

precise technique for calibrating a PLT probe to a CMM. Thisnew method uses known information from a localized polyhedronand measurements taken on the polyhedron by the PLT probe todetermine the calibration parameters. With increasing interest inapplying PLT probes for point measurements in coordinate me-trology, such a calibration method is needed.

S1087-13570001703-2

1 Introduction

Point Laser Triangulation PLT probes have a number of sig-nificant advantages over touch trigger and contact scanning probesthat make them very useful in coordinate metrology. Some mainadvantages include zero contact force, small foot print, largerange of allowable stand-off distances, and high bandwidth.

To apply Point Laser Triangulation PLT probes to coordinatemetrology, the probes must be calibrated using a fast, readily au-tomated, and accurate method to meet the needs of the industry.Existing calibration methods for CMM probes, such as touch trig-ger and contact scanning probes, are not adequate for calibratingPLT probes. Unlike other probes, the extrinsic calibration of PLT

probes requires the additional parameters of an approach vectorand an orientation vector as well as a translational vector. Currentapproaches do not model the added information provided by PLTprobes, nor do they address the probes unique operating con-straints, such as sensor-to-surface orientation.

Much important work has been done in the calibration of posi-tioning systems and sensors. Research in the calibration of posi-tioning systems extends to applications in robotics 1 3, CNCmachines tools 4 6, and Coordinate Measurement Machines 7.

Issues dealing with the intrinsic calibration of a noncontact LI-DAR system 8 and the extrinsic calibration of noncontact sen-sors i.e., cameras and laser line scanners for integration on ro-bots 9,10 and CMMs 11 have also been studied. As in ourapproach, these calibration methods primarily use parameter esti-mation techniques.

The extrinsic calibration of a noncontact light-striping probeon a CMM is presented in 12. The modeling is based on askewed frame representation, and the unknowns are solved for byminimizing an augmented objective function. The technology of alight-striping probe is very similar to that of a PLT probe; how-ever, the light-striping probe produces 2D data points rather that1D and therefore uses different calibration parameters.

One extrinsic calibration method for PLT probes on CMMsproposed to date 13 uses crosshairs to visually position the PLTprobe laser beam to known coordinates. The disadvantages of thismethod are that 1 the calibration process cannot be automated, 2the accuracy of the calibration is a function of the skill of theoperator, and 3 the accuracy of the calibration parameters islower than the accuracy of the PLT probe displacementmeasurement.

Our proposed calibration method overcomes these disadvan-tages. This method can be readily automated. Its accuracy is in-dependent of operator skill level, and it yields accurate calibrationparameters. The basic ideas for this method are extensions of theparameter estimation methods cited above and of part localization,14,15.

In our approach, a calibration artifact is used that has a known

geometry and is localized i.e., the artifacts local reference frameis known relative to the CMM reference frame. The artifact apolyhedron is measured with the PLT probe mounted on andtranslated by the CMM. By fitting the measured points to knowninformation, the calibration parameters can be estimated.

The rest of the paper is organized as follows. The calibrationparameters of a touch probe are presented and then compared withthe calibration parameters of a PLT probe. The proposed calibra-tion method is then presented, along with the least squares solu-tion. The approach is then analyzed from a sensitivity study, usingthe condition number to determine optimal configurations for theartifact, probe orientation, and number and location of points tomeasure. A simulation analysis is used to confirm the analysis.Finally, the effectiveness of the calibration is verified by use ofcomparative analysis.

2 Touch Probe Calibration Parameters

The calibration parameters of a touch probe include a tip sphereradius, rT , and a translational vector, tT , that extends from theCMM quill to the center of the touch probe tip see Fig. 1. Whena part is measured with a touch probe, the probe tip is moved tocontact the workpiece surface. At this contact location, the CMMquill position, Pq , is determined from the x, y, z linear scales ofthe CMM. The point, PT , on the workpiece at which the probe tipmakes contact is given by

Contributed by the Manufacturing Engineering Division for publication in the

JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received

April 1997; revised Oct. 1999. Associate Technical Editor: C. H. Menq.

582 Vol. 122, AUGUST 2000 Copyright 2000 by ASME Transactions of the ASME

Downloaded 21 Aug 2007 to 164.107.166.202. Redistribution subject to ASME license or copyright, see http://www.asme.org/terms/Terms_Use.c


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PTPqtTrTnw , (1)

where nw is the normal to the workpiece surface at the point ofcontact.

3 PLT Probe Calibration Parameters

A PLT probe has two additional calibration parameter vectorsnot required by a touch probe. The calibration parameters of aPLT probe include a translational vector, tp , an approach vector,ap , and an orientation vector, op .

The translational vector, tp , is defined from a known locationon the CMM quill typically the center of a previously calibratedtouch probe to a reference position on the laser beam of the PLTprobe. The approach vector, ap , is parallel to the laser beam andpoints toward the workpiece away from the PLT probe. The ori-entation vector, op , is perpendicular to the laser beam and isdirected from the laser beam toward the optical axis of the receiv-

ing lens.An absolute position point, Pp , on the workpiece is calculated

from the PLT probes measured displacement value, , and othercalibration parameters by

PpPqtpap . (2)

The orientation vector, o, is not used in this calculation, but isneeded in path-planning algorithms to optimize sensor-to-surfaceorientations and to operate in collision-free regions. Since the ori-entation vector does not directly affect the accuracy of the PLTprobe measurement, it can be approximated by visual inspectionand is not part of the method presented here.

4 Proposed Calibration Method

The main steps in the proposed PLT probe calibration methodare to 1 localize and calibrate the polyhedron artifact; 2 measurethe facets of the polyhedron by recording the PLT probes dis-placement value, i j , and the CMMs position, Pq , at each mea-surement; and 3 find the PLT probe calibration parameters thatbest fit the measurements to the polyhedron.

For this calibration method, a polyhedron with three facets isrecommended for the artifact; although, more facets can be usedsee Fig. 2. The polyhedrons m facets, Sj(n,d), j1, . . . , m aredefined by their corresponding normals, nj , and minimum dis-tances, dj , from the origin as referenced by the CMM to thefacet.

5 Deriving the Calibration Parameters

The measurement error or residual, i j , of the ith point nor-mal to the facet is given by

i jnjPi jdj , (3)

nj Pqi jtpi japdj . (4)

By construction, the only unknowns in Eq. 4 are tp , and ap .Since the equation is linear in unknowns, they can be solved forusing the following simple linear least squares approach. Let f bethe sum of the square of all the residuals, that is, f

j1m

i1

kj i j2 , where kj is the number of measured points on

facet j. The calibration parameters can then be determined byfinding values for a and t that minimize f. The function f is aminimum when the partials of f with respect to the unknowns

ax ,ay ,az ,tx ,ty ,tz are all equal to zero. Writing this system inmatrix notation with h ax ,ay ,az ,tx ,ty ,tz

T results in

GhB, (5)

where

G nj ,i j j1

m

i1

kj

g i j , (6)

g i jMi jV 1 6x6VMi j , (7)

B

nj ,dj , Pqi j ,i j

j1

m

i1

kj

b i j , (8)

b i jMi jnx j ,ny j ,nz j ,nx j ,ny j ,nz jT djnjPqi j . (9)

The 66 matrix, Mi j , has i j ,i j ,i j,1,1,1 on the main diag-onal and 0 elsewhere. The 66 matrix V has

nx j ,ny j ,nz j ,nx j ,ny j ,nz j on the main diagonal and 0 elsewhere.

The 66 matrix, 16x6 , has all elements equal to 1.As shown, the calibration parameters can be determined by

solving Eq. 5 which is linear. The matrix G is called the dis-placement matrix because it is a function of the displacementmeasurements. The conditions for which a solution exists i.e., Gis invertible will now be examined.

Fig. 1 The calibration parameters for a touch probe include atranslational vector, tT , and tip radius, rT . The calibration pa-rameters for a PLT probe include a translational vector, tp , anapproach vector, ap , and an orientation vector, o.

Fig. 2 A polyhedron with three facets is proposed as the

calibration artifact

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6 Constraints from the Displacement Matrix

A constraint on the displacement measurements and the facetnormals is found by examining the conditions needed for the dis-placement matrix to be invertible. Carrying out the second sum inEq. 6 and applying elementary row operations the top three rowscan be written in the form

knxnynznx ny nz nx ny nzj , (10)

and the bottom three rows can be written in the form

nxnynznx ny nz knx kny knz j (11)

where

ji1

kj

i j2 , (12)

ji1

kj

i j . (13)

Thus the rank of Dj is

rankDj 1, if kjjj2

2, if kjjj2

.(14)

From this information, the rank of G is bounded by

rank G rank j1m

Dj min 6,j1m

rankDj . (15)Using only three facets i.e., m3 requires that the rank of D1D2D32 i.e., requiring that kjjj

2. Conversely, if kjjj

2, more than three facets i.e., m6 are required for G to befull rank.

In addition, if i jCj for all i, where Cj is a constant, then

from Eqs. 12 and 13, jkjCj2 and jkjCj . Under this con-

dition, kjjj2 and the rank of Dj is 1. Also, if m3, then G is

not invertible. As a practical consequence, the PLT probe must bepositioned so that i j is not constant while a facet is beingmeasured.

Further, the rank(D1D2D3)6 requires that therank(n1 ,n2 ,n3)3 i.e., the facet normals must be linearlyindependent.

In summary, the sufficient constraints from the displacementmatrix are as follows: 1 at least three facets that have linearlyindependent normal vectors must be used, 2 each facet must bemeasured at least twice, and 3 the displacement values measuredon a single facet cannot all be the same.

7 Calibration Parameter Sensitivity

For the calibration parameters, ap and tp , to be determinedaccurately, they need to be sensitive to the PLT probe displace-ment measurement and the CMM quill position which improvesthe accuracy of the calibration while the effects of random noiseerrors can be minimized.

The calibration parameters are determined by minimizing theresiduals in Eq. 4. The sensitivity of these residuals to the PLTprobe measurement is given by

i ji j

njap1. (16)This sensitivity increases as the angle between the approach

vector, ap , and the facet normal, n, approaches 180 deg. There-fore, the facet normals should be aligned as close as possible tothe PLT approach vector.

The sensitivity of residuals, i j , to the CMM quill position,Pqi j , is given by

i j

Pxqi jnx , (17)

i j

Pyqi jny , (18)

i j

Pzqijnz . (19)

Equations 18 and 19 show that the sensitivity of the residu-als to changes in CMM quill position along an axis is equal to the

coordinate value of the facet normal vector in that axis. Therefore,the facet unit normal should ideally point along the CMM axisthat has the highest resolution. However, if all three of the CMMaxes have equal linear resolutions then the facet normal can pointin any direction without changing the sensitivity of the residualsto CMM position changes.

8 Condition Number Analysis

The sensitivity of parameter errors, h, resulting from the er-rors G and B in G and B respectively are characterized by thewell-known condition number, (G). This number is defined as

(G)G G11. In our application, a small condition num-ber corresponds to the solution being less sensitive to data error.

By examining, G, one finds that the four parameters whichaffect the condition number include: 1 the number of measure-

ments taken on each facet, Kj , 2 the threshold value of i j , 3the range value ofi j , and 4 the facet angle, .

To reduce the condition number as much as possible, one has tomake an effort to optimize the following items: 1 the number ofmeasurements taken on each facet should be as large as realisti-cally possible, 2 the threshold value of i j should be set smallbecause a large threshold value will result in a large conditionnumber, and 3 the facet angle should be close to zero.

While it is feasible to achieve item 1 and 2, item 3 cannot befully optimized because of limitations on allowable sensor-to-surface orientations. So a trade-off between the magnitude of(G) and the expected accuracy of the PLT probe must be made.

To address this trade-off, PLT probe manufacturers specify amaximum allowable approach angle, max . The approach angle is

calculated from cos1(nj(a)) and must be less than or

equal to max for the PLT probe to meet its performance specifi-cations. The smallest block angle, min , possible under this con-straint is realized when the calibration artifact is positioned so thatthe approach angle is the same for all facets. Using plane geom-

etry, this angle is given by mintan1(tan(t)/2), where t

cos1(cos(max)/2)max/2.

9 Calibration Experiment

An experiment was designed and conducted to validate the pro-posed PLT probe calibration method.

The main steps in the validation process are as follows: 1calibrate a CMM touch probe; 2 secure the polyhedron artifact inthe working zone of the CMM; 3 use the touch probe to localizeand calibrate the polyhedron artifact; 4 calibrate the PLT probe

using the proposed calibration method; 5 move the artifact to anew location in the CMM working zone, change its orientation,and secure its position; 6 measure the artifact facets at the newlocation with the PLT probe; 7 install and calibrate the touchprobe; 8 measure the artifact with the touch probe in its newlocation; and 9 compare the facet normals and minimum dis-tances obtained from the PLT probe with those obtained from thetouch probe.

The parameters nj and dj were obtained using the CMMs stan-dard plane measuring and datuming routines for touch probe mea-surements. The displacement measurements, i j , and the corre-sponding CMM quill positions, Pqi j , were obtained whilemeasuring the facets with the PLT probe.

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A method for determining the facet normal vectors, np j , andcorresponding minimum distances, dp j , measured by the PLTprobe in the moved validation position step #5 needed to bedeveloped. Since the calibration parameters were previously de-termined, a point, Ppij , measured by the PLT probe is determinedfrom Eq. 2. The relationship between np j , dp j , and Ppij is givenby

Ppijnp jdp j . (20)

Using the change of variable, p j1/dp jnp j , and letting the ma-trix Z be defined as

Zj

Pxp 1 Pyp 1 Pzp 1

Pxp 2 Pyp 2 Pzp 2

]

Pxp 16 Py p16 Pzp 16 j, (21)

the system of equations for the points measured on a facet inmatrix form is

Zjp j 1 161 . (22)

This least squares problem has the unique solution

p jZjTZj

1ZT 1 161 . (23)

Finally, the desired parameters can be determined from p j by

np jp j

p j(24)

and

dp j1

p j. (25)

For purposes of comparison, the validation experiment was per-formed using two commercially available PLT probes referencedas PLTA and PLTB in this document. The CMM used to calibratethe PLT probes was a Sheffield RS-30. The touch probe used wasa Renishaw TP-2 touch probe with a 1 mm tip diameter. The

polyhedron artifact with three facets, depicted in Fig. 2, wasmanufactured from an aluminum block with a facet angle of60 deg.

To quantify the performance of the touch probe and two PLTprobes independent of the calibration method, a 5-arcmin angleblock was measured. In this test, the probes were mounted verti-cally and the block was measured 100 times at evenly spacedintervals over a length of 20 mm. A line was fit to the measuredpoints; and the following were calculated: the range of residualsROR, the standard deviation of the residuals SDR, and anglebetween theoretical line and fitted line ABL. The results areshown in Table 1.

In this experiment, the touch probe performed significantly bet-ter than both laser probes. These probes were then used in thevalidation experiment explained above; the results are shown inTables 2, 3, and 4.

Table 2 reports the angle, n , between the facet normal nideal ,measured by the touch probe, and np , measured by the PLTprobe. The angle, n , is calculated by

ncos1 npnideal. (26)

Table 3 reports the difference, d , between the minimum dis-tance from the facet to the origin determined by the touch probe,dj ,ideal , and by the PLT probe, dp j ; that is,

ddj ,idealdp j . (27)

Table 4 reports the range, r , of the difference between theleast-squares minimum facet distance, dp j , and the minimumfacet distance calculated from individual facet points measured bythe PLT probe. That is,

r

range dj

dpij j i

1,2, . . . ,16. (28)The variation in experimental results between the two PLT

probes tested is consistent with the measurement accuracy of therespective PLT probes. The validation experiment results confirmthe high accuracy of this proposed PLT probe calibration method.

10 Summary

The proposed PLT probe calibration method is an accurate cali-bration method that can execute autonomously and provide a lin-ear solution. A mathematical and experimental analysis of theproposed calibration method revealed the following design con-straints.

Table 1 Performance of touch probe and two PLT probes

Table 2 The angle between the facet normal nideal and n, wherenideal was measured by the touch probe and n was measured bythe PLT probe.

Table 3 The difference between the minimum facet distance tothe origin determined by the touch probe and by the PLT probe

Table 4 The range of minimum distance residuals from thePLT probe measurements




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1 The condition number, (G), should be small.2 The polyhedron should be positioned during the calibration

so that the angle between the facet normal vectors and thePLT laser beam is as small as possible.

3 Certain conditions must be satisfied for the inverse of the

displacement matrix, G1, to exist:

At least three facets must be measured.

The facet normal vectors must span R3 i.e.,Rank(n1 ,n2 , . . . , nm)3).

When only three facets are measured, at least two measure-

ments per facet must be takeni.e., kj

2 for j1,2, . . . , m where m3.

The PLT probe displacement values must be controlled, asshown in Eq. 14.

This proposed calibration method was validated experimentallyto provide an accurate coordinate calibration for a PLT probe. Theaccuracy of the calibration is limited principally by the accuracyof the PLT probe.

Acknowledgments

This research was funded in part by the Engineering ResearchCenter for Net Shape Manufacturing ERC/NSM at The Ohio

State University, by the Office of Naval Research under grantN00014-90-J-1516, and by the National Institute of Standards andTechnology NIST, an agency of the U.S. Department ofCommerce.

References

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configurations for robot calibration based on observability measure, Int. J.

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accuracy, Machine Tool Task Force.

5 Ni, J., 1997, CNC machine accuracy enhancement through real-time errorcompensation, Trans. ASME, J. Manuf. Sci. Eng., 119, pp. 717725.

6 Lee, E. S., Suh, S. H., and Shon, J. W., 1998, Comprehensive method for thecalibration of volumetric positioning accuracy of CNC-machines, Int. J. Adv.

Manufac. Technol. Springer-Verlag London Ltd., London Engl., 119, No. 1,pp. 4349.

7 Chen, J., and Chen, Y. F., 1987, Estimation of coordinate measuring machineerror parameters, Proceedings of IEEE International Conference on Robotics

and Automation San Francisco, pp. 10111016, Apr. 710.8 Chen, Y. D., and Ni, J., 1993, Dynamic calibration and compensation of a

3-D laser radar scanning system, IEEE Trans. Rob. Autom., 9, No. 3, pp.

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9 Agin, G. J., 1985, Calibration and use of a light stripe range sensor mountedon the hand of a robot, Proceedings of the IEEE International Conference on

Robotics and Automation, St. Louis.

10 Chen, C., H., and Kak, A. C., 1987, Modeling and calibration of a structured

light scanner for 3-D robot vision, Proceedings of the 1987 IEEE Interna-tional Conference on Robotics and Automation, CH2413-3., pp. 807815.

11 Agin, G. J., and Highnam, P. T., 1987, Movable light-strip sensor for obtain-ing three-dimensional coordinate measurement, Proceedings of the SPIE In-

ternational Tech. Sym., San Diego, CA, pp. 326333, Aug. 2127.

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Sci. Eng., 118, pp. 595603.

13 Hu, W. 1995, 3-D surface digitizing using coordinate measuring machinesCMM with laser displacement sensors LDS, Masters thesis, Dept. ofElectrical Engineering, The Ohio State University.

14 Sahoo, K. C., and Menq, C. H., 1991, Localization of 3-D objects havingcomplex sculptured surfaces using tactile sensing and surface description,

ASME J. Eng. Ind., 523, pp. 8592.15 Gunnarsson, K. T., and Prinz, F. B., 1987, CAD model-based localization of

parts in manufacturing, IEEE Comput., 20, pp. 6674.

Simultaneous Optimization of

Machining Parameters for Dimensional

Instability Control in Aero Gas

Turbine Components Made of Inconel

718 Alloy

B. K. SubhasScientist F

Ramaraja BhatScientist B

K. RamachandraScientist G

Gas Turbine Research Establishment, Bangalore,

India-560 093

e-mail: [email protected]

H. K. BalakrishnaDean,

Sir M. Visvesvarayya Institute of Technology,

Bangalore, India-562 157

Aero gas turbine components made of Inconel 718 superalloy re-vealed significant dimensional instability after machining. The di-mensional instability is a manifestation of alterations in residualstresses and microstructure, which are influenced by machining

parameters. This paper presents a simultaneous optimizationtechnique used to control the dimensional instability in turningoperation. Empirical equations are established for predicting sur-

face residual stresses, surface finish, dimensional instability andtool life using response surface methodology. Experimental re-

sults show a strong correlation between residual stresses and di-mensional instability. A desirability function approach is used tooptimize the multiple responses and machining parameters arederived for practical applications. Inconel 718 test specimen and

jet engine components machined with optimal cutting parametersshow dimensional instability within the acceptable toleranceband. S1087-13570001704-4

1 Introduction

Modern aero gas turbine engine components are complex intheir shape, thin walled and call for very close dimensional toler-ances and good surface integrity. Inconel 718 superalloy is exten-sively used for jet engine components. This nickel-based superal-

loy is not only difficult to machine but also revealed dimensionalinstability after machining. The dimensional instability is achange in dimension with respect to time without doing any fur-ther work on the component 1. The dimensional instability ofcomponents poses problems during assembly. Residual stressesand microstructure changes in a machined surface region causedimensional instability 1. Dimensional instability can be con-trolled by selection of proper machining parameters that influencethe residual stresses and microstructure changes in a machined



Oct. 1997; revised Feb. 2000. Associate Technical Editor: K. Sutherland.




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surface region. The selection of machining parameters also de-pends on other responses such as required tool life, surface finish,material removal rate, etc. Therefore, the machining operation hasto be optimized considering several process responses.

Derringer and Suich 2 presented a problem involving simul-taneous optimization of several response functions that dependupon the number of process variables. This approach is based ontransforming response variables into desirability functions 3,which can be optimized by univariable techniques. The desirabil-ity function is a dimensionless number, varying between 0 forcompletely undesirable level and 1 for completely acceptable

level of process quality. The geometric mean of desirabilities forseveral responses is a dimensionless number that represents over-all quality of the process. Thus, optimization problems are re-duced to problems of optimizing single function. That is, overalldesirability has to be optimized with respect to process variables.This paper presents a desirability function approach for simulta-neous optimization of turning operations. Empirical equations aredeveloped for predicting responses by response surface methodol-ogy 4. Response surface methodology involves statistical designof experiments that reduce the number of experiments required.The prediction equations for the responses are derived in terms ofindependent machining parameters by regression analysis.

In this paper, dimensional instability of machined component,tool life, surface finish and material removal rate are consideredfor optimizing turning parameters. The validity of optimal ma-chining parameters to control the dimensional instability is experi-mentally verified on test specimens and actual gas turbine enginecomponents.

2 Methodology

The steps involved in the proposed methodology are:

i Selection of process variables and responses.ii Design of statistical experiment.iii To determine empirical equations for predicting responses

by response surface methodology and check the adequacyof equations.

iv To compute optimal machining parameters using desirabil-ity function approach.

2.1 Selection of Process Variables and Responses. The

optimization technique is developed here for a turning operationsince most of the critical rotating gas turbine components are ma-chined by turning. Speed, feed, depth of cut, tool rake angle andtool nose radius are the variables that significantly affect residualstresses and microstructure in a machined surface region 5.Soluble oil 1:20 cutting fluid and micrograin grade carbideKennametal K68 cutting tool material were selected based onearlier experimental investigation 6. Table 1 shows the maxi-mum and minimum values of cutting parameters used inexperiments.

Considering the surface integrity and accuracy requirement ofprecision rotating gas turbine components and efficiency of turn-ing operation, optimization is done for six process responses,namely, circumferential and longitudinal residual stresses, dimen-sional instability, surface finish, tool life and material removal

rate. Therefore, empirical equations are developed for these sixresponses. The material removal rate is calculated using,

MRR1000fd (1)

The prediction equations for other five responses are deter-mined by statistical design of experiment.

2.2 Statistical Design of Experiment. The influence ofprocess variables on response functions was studied at two levels.

A half replicate 25 factorial experiment was designed that in-volved only sixteen trials. The treatment combinations of fivevariables and corresponding experimental conditions are shown inTable 2 and Table 3. Error variances for the responses were esti-mated by carrying out four repeated experiments Trial No. 17 to20 in Table 3 at the central values of variables.

2.3 Experimental Procedure. Cylindrical test specimensID 55 mmOD 76 mmlength 70 mm made of Inconel 718were stress relieved and aged hardness 44 HRC. The specimenwas held in a mandrel to avoid clamping pressure while turning.Turning trials were conducted in a HMT H-26 lathe 7.5 kW. Themachined specimens were inspected at predetermined locations ina Zeiss Mauser 3D coordinate measuring machine. The dimen-sional instability was calculated as a maximum absolute differ-

ence between dimensions measured immediately after machiningand final stable dimensions. Residual stresses were determined byhole drilling strain gage method 7. The machining operation wasinterrupted periodically and tool flank wear was measured using atool makers microscope. Time taken to reach a value of 0.18 mmmaximum flank wear was the criterion for tool life. Surface finishwas measured using a portable perth-o-meter. Results of the ex-periments are given in Table 3.

Table 1 Cutting parameters used for machining

Table 2 Treatment combinations of process variables

Table 3 Experimental data




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2.4 Equations for Predicting Process Responses. Follow-ing equation is postulated for the responses:

yb0b ix ib i jx ixj ; i ,j1,2,3,4,5 and ij (2)

Equation 2 includes direct linear and interaction effects of thevariables. For convenience, two levels of each independent vari-able are coded into 1low and 1high. Transformation equa-tions used for coding the variables are given in Table 4. Thesixteen coefficients required for Eq. 2 were estimated by themethod of least squares.

BXX

1

XY (3)

where, X is the matrix of coded variables and Y is the vector ofmeasured responses.

Predicting equations for all the responses are obtained in theform of Eq. 2, which involves sixteen terms. These equationswere further simplified to include only statistically significantterm of variables and their interactions 5. The ratio of meansquare effect to error variance was compared with F-statistics totest the significance of each effect at 95 percent confidence level.The prediction equations are thus simplified and involves fewerterms as follows:

c138.64571.8569643.4365f286.8036d2.3259

109.2222r60.0156f15.6297d174.7535dr (4)

l1.88360.044752.8845f13.2834d0.4828

5.8073r3.4268f0.6195d9.2915dr (5)

T14.39110.203930.9241f0.10521.1720r

0.5473f (6)

DIMI8.97660.216126.7825f33.5055d2.2595

18.7364f0.4259d0.0302122.2237f d

4.7463f2.3614d (7)

Ra1.01070.01728.8607f0.7035d0.00311.2789r

0.1766f0.0246r0.1874f (8)

2.5 Adequacy of Prediction Equations. Equations 48are valid when , f, d, and r vary within the minimum andmaximum levels used in the experiments Table 1. Further, theseequations were checked for adequacy by analysis of variance. Theratio of lack of fit mean square to pure error mean square iscompared with F-statistic. The pure error mean square for theresponses is estimated by repeated tests carried out at central val-ues of the variables. The predicted equations are adequate since

lack of fit is not significant at 95 percent confidence level Table5.

2.6 Correlation between Residual Stresses and Dimen-sional Instability. The coefficient of correlation between re-sidual stresses and dimensional instability are calculated from theresults of experiments presented in Table 3. The correlation coef-ficient between c and D IM I is 0.9602; l and D IM I is0.9658that is, residual stresses and dimensional instability arehighly correlated, and by controlling dimensional instability re-sidual stresses are also controlled. Therefore, the response func-tions, c and l are not considered for optimization.

2.7 Simultaneous Optimization. The machining processresponses were transformed into desirability functions 2. Thedesirability functions derived for the responses are given below:

d10 TTmin

TTminTmaxTmin

TminTTmax1 TTmax

(9)

d2

0 M RR MRRmin

M RRMRRminMRRmaxMRRmin

M RR minMRRMRRmax1 M RR MRRmax

(10)

d30 D IM IDIMImin , D IM IDIMImax

D IM IDIMIminDIMIcDIMImin

D IM IminDIMIDIMIc D IM IDIMImaxDIMIcDIMImax

D IM IcDIMIDIMImax(11)

Table 4 Transformation equations for variables

Table 5 Analysis of variance for adequacy test




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d40 RaRa max

RaRa maxRa minRa max Ra minRaRa max1 RaRa min

(12)

One-sided transformation is used for the desirability of tool life,actual metal removal rate, surface finish; and two-sided transfor-mation for dimensional instability. Overall composite desirabil-ity of the process is computed by geometric mean of individualdesirabilities.

D d1d2d3d4 1/4 (13)

Maximizing this composite desirability gives the best overallperformance for machining.

2.8 Optimization Criteria. In this study, tolerance bandsfor the jet engine components are classified into four groups.

Group No. I - 510 microns rangeGroup No. II - 1120 microns rangeGroup No. III - 2130 microns rangeGroup No. IV - 3150 microns range

Only few components require 510 microns tolerance, which isachieved by either jig boring or grinding. Since the present studyis limited only to turning operation, machining parameters areoptimized for Group II to IV tolerance bands. Upper and lowerlimits of the tolerance group fix the maximum and minimum val-ues for acceptable dimensional instability. The D IM Ic is the mid-value of the tolerance band, where d31. The constraints on toollife and surface finish are selected based on practical experience.

d10 at T5 min and d11 at T20 min. Similarly, d40 at

Ra0.8 m and d41 at Ra0.2 m, M RR min and M RR maxwere calculated as follows:

MRRmin1000minfmindmin (14)

MRRmaxPmE

1.25pCtCr(15)

Following values were used in Eq. 15: Pm is 7500 W, p is1.03 W/mm3/min, E is 0.7, Ct and Cr are 1.2 6. With theseconstraints, desirabilities for the responses are computed by vary-ing , f, d, and r in discrete steps. A computer program iswritten for computing individual and composite desirabilities. Thecomposite desirabilities for different combinations of processvariables were numerically compared. The program prints optimalresponses, variables and corresponding desirabilities upon reach-

ing maximum value of D. Output of the computer program issummarized in Table 6.

3 Experimental Validation of Optimal Parameters

1120 micron tolerance band is the most critical for turningoperation. The validity of optimal machining parameters derivedfor this range are verified by carrying out tests on cylindrical testspecimen and actual jet engine components.

3.1 Validation on Test Specimen. Table 7 shows dimen-sional details for the test specimen machined with 10 m/min,f0.04 mm/rev, d0.5 mm, 6 deg and r1.2 mm. Thechanges in dimensions are less than 10 microns, which are wellwithin the predicted value. Repeated tests carried out with these

Table 6 Optimal parameters

Fig. 1 Dimensional instability in compressor disc

Table 7 Dimensional instability in test specimen




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machining parameters showed 5 percent scatter in dimensionalinstability, but all were within the required tolerance band.

3.2 Validation on Compressor Disc. Effect of optimal ma-chining parameters on dimensional instability in compressor discsis shown in Fig. 1. The locating diameter 400 0.02/0.00 mmwas machined using the optimal machining parameters derived for1120 micron tolerance group. The diameter maintained at 4000.010/0.00 mm during machining has changed to 4000.015/0.00 mm after 360 hours. This change in dimension iswithin the acceptable tolerance band.

4 Conclusion

1 Empirical equations for predicting surface residual stresses,dimensional instability, surface finish and tool life were derivedby response surface methodology.

2 Experimental results have shown that residual stresses anddimensional instability are highly correlated. Therefore, dimen-sional instability, tool life, surface finish and material removal ratewere considered for optimization.

3 A simultaneous optimization method based on desirabilityfunction approach was used. The optimal cutting parameters arederived to control dimensional instability within 1120, 2130and 3150 micron tolerance bands in turning precision aero gasturbine engine components made of Inconel 718.

4 Dimensional instability was within the acceptable toleranceband for the test specimen and actual jet engine components thatare machined with optimal machining parameters obtained by si-multaneous optimization.

Nomenclature

rake angle, deg.b i , b i j regression coefficients

B vector of regression coefficientsCr rake angle correction factorCl chip thickness correction factor

d depth of cut, mmd1 desirability of tool life

d2 desirability of material removal rated3 desirability of dimensional instabilityd4 desirability of surface roughnessD composite desirability

DIMI dimensional instability, mDIMIc most desirable dimensional instability, m

E transmission efficiency of the drivef feed, mm/rev

ID inner diameter, mmMRR metal removal rate, mm3/min

OD outer diameter, mmp power per unit metal removal rate, W/mm3/min

Pm lathe motor power, Wr tool nose radius, mm

Ra surface finish, m

c

circumferential residual stress MPal longitudinal residual stress MPaT tool life, min cutting speed, m/min

x 1 coded value for x 2 coded value for fx 3 coded value for dx 4 coded value for x 5 coded value for rX matrix of coded variables

X transpose of Xy estimated response

Y vector of measured responses

Subscripts

max indicates maximum valuemin indicates minimum value

References

1 Marschall, C. W., and Maringer, R. E., 1977, Dimensional Instability-Anintroduction, International Series of Material Science and Technology, Per-

gamon Press, Vol. 22.2 Derringer, G., and Suich, R., 1980, Simultaneous Optimization of Several

Response Variables, J. Quality Technol., 12, No. 4, pp. 214219.

3 Harrington, Jr., E. C., 1965, The Desirability Function, Ind. Quality Con-

trol, pp. 494498.4 Wu, S. M., 1964, Tool Life Testing by Response Surface Methodology, Part1 and 2, ASME J. Eng. Ind., pp. 105115.

5 Devarajan, N. et al., 1984, Experimental Method of Predicting ResidualStress due to Turning in Stainless Steel, J. Exp. Tech., 8, pp. 2226.

6 Subhas, B. K., 1983, Some Experimental Studies in Machining of Superal-loys as Applied to Gas Turbine, M. S. Research thesis, J.N.T.U., Hyderabed.

7 Measurement of Residual Stress by Hole Drilling Strain Gauge Method,1993, Technical Note No. T.N.503-4, Measurement Group Inc.

Edge Radius Variability and Force

Measurement Considerations

Roy J. Schimmel1

Graduate Research Assistant

Jairam Manjunathaiah2

Graduate Research Assistant

William J. EndresAssistant Professor, Mem. ASME

Department of Mechanical Engineering and Applied

Mechanics, University of Michigan, Ann Arbor, MI 48109

A new, noncontact instrument, based on white light interferom-etry, is used to measure the edge radii of cutting tools with mea-surement errors of less than 3 m. Edges of several commercialcutting inserts are measured and compared. It is found that theradius of the hone varies along the length of the edge in a para-bolic manner. The difference between the edge radius at the cen-ter of the edge and the radius at the start of the corner can be aslarge as 25 m (0.001 in). The variation between the edges on aninsert and across inserts in a batch of tools can be as high as 25m (0.001 in). Statistically significant variations are also seen inthe corner radius region in which much cutting occurs in turning,boring and face milling processes. Orthogonal cutting tests withtools of measured edge radius in the zone of cut indicate that themachining forces, especially the thrust force component, are sen-sitive to changes in edge radius on the order of measured varia-tions. S1087-13570001603-8

Introduction

Edge hones are commonly used as an edge preparation in manyoperations, like interrupted cutting, machining of hard materials,etc., where increased edge strength is desired. Edge hones in the

1Currently with General Motors.2Currently with Lamb Technicon Machining Systems, Warren, MI.



July 1997; revised Oct. 1999. Associate Technical Editor: S. Kapoor.




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range of 75 m 0.003 in to 125 m 0.005 in are commerciallyavailable for heavy duty machining of hardened steels 35 Rcand cast irons. The edge preparations on commercially availableinserts are generally prescribed as a range. For example, edgepreparation A is specified to have a hone ranging from 10 m0.0005 in to 80 m 0.003 in. Two of the processes that arecommonly used to obtain edge hones are honing by nylon brushesimpregnated with silicon carbide, and honing by abrasive entrain-ment in an air-stream. In the brush honing process consideredhere, cutting edges are polished when the inserts, mounted on arotary carrier, are slowly rotated about their inscribed-circle axis

while being fed through the rotating brushes. By varying the timeand depth of contact between the brush and the cutting edge,different edge hones can be obtained. The variation of an edgefeature from its nominal value, both along an edge and edge-to-edge, is not well understood. Large tolerances on the edge honecould lead to significant variations in machining forces when cut-ting with different inserts of the same nominal specification. Theaim of the reported work is to study edge hone variability oncommercial inserts and the relative effects on machining forcesand model calibration data. It is notthe aim of this paper to modelthe effects of the edge radius on the cutting process, nor is it tomodel the brush honing process.

Edge Radius Measurements

Accurate measurement of the edge radius is a fairly difficult

task that has been addressed in some detail in only a few studies1,2. In this study, we describe the use of a new optical techniquebased on white light interferometry see Sasmor and Caber 3 fora detailed review of optical measurement techniques. TheWYKO measurement system used here, which is based on thephysics of white light phase-shift interferometry, combines accu-rate optics, axis movement and a computational software interfaceto make measurements with vertical resolution as good as 2 nm0.002 m 4.

Three sets of TPG432 uncoated carbide inserts sets A, B andC were requested from a vendor to have corresponding nominaledge radii specified at the center of the edge of 50.8 m 0.002in, 100.6 m 0.004 in and 152.4 m 0.006 in. After goingthrough quality checks at the vendors facility, the inserts weremeasured independently by the research team. Care was taken toreduce sources of measurement errors. A small fixture that heldthe insert at the proper and constant orientation relative to theoptics of the system, together with an automated positioning table,ensured that measurements on different inserts and edges were atidentical locations along the edge. Repeatability tests showed thatthe estimated profiles were within 50 nm 0.05 m of each other.Details of scan data processing are given in 5.

Variability Between Edges and Inserts. Eight inserts fromeach set were chosen to evaluate the variability across edges andinserts. Comparing the nominal specifications to the measureddata indicated that the edge-center-point means were smaller thanthe prescribed nominal values, in this case by about 15 m0.0007 in for the sets A and B, and by about 5 m 0.0002 infor set C. This would indicate that it is difficult to manufacture theinserts to a tight tolerance on the mean for tools with a smaller

edge radius within 35 percent of the nominal for set A as com-pared to within 3 percent for set C. The variation about the meanis about 6 m 0.00025 in for sets A and B, and 14 m 0.0005in for set C, which is approximately 10 percent of the nominalvalues for all sets. Only 50 percent of the measured values were ina range of 10 m 0.0004 in for sets A and B, and 25 m 0.001in for set C.

A general nested linear analysis of variance was conducted inwhich the edge radius was modeled as equal to a tool effect plusa nested effect of edge on tool. It was found that both these vari-ables were statistically significant at a P-level of 0.024 on tool and0.014 on edge both of set B. Sets A and C were significant ateven smaller P-values. In other words, between inserts there ex-

isted a significant shift in insert mean the average of the edge-center-point measurements on each of the three edges of an insertand, furthermore, there existed an edge-to-edge mean shift oneach insert.

Placing 95 percent confidence bands on the insert means fornine inserts of set A demonstrated that if an insert with nominaledge radius of 50.8 m 0.002 in is procured commercially, theactual edge radius at a specified point on the edge could be off byas much as 22 m 0.0009 in. This error is perhaps indicative ofwhy most manufacturers specify the edge preparations in rangesof values.

Variability Along an Edge. The second important issue toinvestigate is whether or not the location along the cutting edgeaffects the edge radius in a statistically significant manner. Twoinserts from every set were chosen and measurements were madeon four of the six edges at five zones on each edge correspondingto 7.8 mm, 5.2 mm, 0.7 mm, 3.4 mm and 6.7 mm from thecenter of the edge. These points were chosen such that one pointcorresponded approximately to the center of the edge with theother points distributed on either side. Using the automated posi-tioning table, the locations of these positions were maintained tobe the same for all the measured edges. The variation of edgeradius along the edge is shown in Fig. 1. It is obvious that theedge radius at the center of the edge is consistently lower ascompared to the radius at points closer to the corner 7.8 mm and

6.7 mm locations. The R2 values for the individual fits were

about 60 percent. Analysis of variance showed that edge locationindeed was a significant factor. It also showed that the edge-number label was insignificant.

Variability Around Insert Corner. While variability alongthe straight portion of an insert edge may impact straight-edgedorthogonal cutting tests, most real-world applications of cuttinginserts involve cutting, at least in part, on the corner radius of theinsert. Therefore, force prediction models being developed forsuch applications, which extend the models being formulated forstraight-edged orthogonal cutting, should account for edge radiusvariation around the corner, if it exists. With this in mind, theedge radius around the corner was measured on several insertsfrom set A to evaluate the relative variation in the corner region ascompared to the straight lead-edge region studied above.

A fixture was fabricated to permit three measurements on eachcornerone at each of the two tangent points where the cornermeets the two straight edges, and one at the apex of the corner.The statistical model employed to analyze these data was a gen-eral linear model with the corners as nested within insert for thesame reason edge number was nested within insert in the earlieranalysis. The two possible two-factor interactions were also

Fig. 1 The variation of edge radius along the four cuttingedges of inserts from set A 50.8 m 0.002 in, set B 101.6m 0.004 in, and set C 152.4 m 0.006 in




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evaluated in this model. Results of the statistical evaluation showinsert and position to be statistically significant, at P-values of0.035 and 0.009, respectively, and cornerinsert as well as thetwo interactions to be insignificant. This reveals that the edgeradius does vary with position around any given corner and fur-thermore that there is a difference between inserts, although varia-tion between corners of a single insert is statistically indistin-guishable from random errors.

Force Measurement Considerations

Orthogonal cutting experiments were conducted on a J & LCNC lathe by end cutting of tubes of three materials: gray castiron, 2024 aluminum and commercially pure zinc. A wall thick-ness equal to 4.83 mm 0.19 in was used to maintain plane strainconditions. A Kistler piezoelectric dynamometer was used tomonitor three machining force components. To reduce the varia-tion of the edge radius along the width of cut, which could lead tomisleading force measurements, all the cutting tests were per-formed at the center of the edge where the edge radius had beenmeasured prior to the experiment and where the variation gradientis lowest across the width of cut. It can be seen from Fig. 1 thatthe maximum variation in edge radius across a 5 mm 0.2 inwidth of cut that is equally distributed about the edge center-pointis only about 6 m 0.0004 in for inserts from set C.

Each material was cut at several feeds by two inserts whosehones were described by the vendor as 50 m 0.002 in and 100

m 0.004 in. Since the significance of the effect of the edgeradius increases at smaller uncut chip thickness, feeds were se-lected such that the ratio of uncut chip thickness to edge radius(h/rn) varied between 0.5 and 5. This represents the range com-monly seen in the operations of practical interest hard turning andfinishing operations. The edge radius effects seen here would bemore pronounced at even lower uncut chip thicknesses.

Graphs of the cutting and thrust forces normalized by thewidth of cut versus the uncut chip thickness h, for all three ma-terials, are presented in Figs. 2 and 3. As expected, the effect ofthe edge radius is visibly larger on the thrust force than on thecutting force. It can be seen that intercepts a qualitative measureof edge/ploughing contributions are higher when the materialsare being cut with tools of higher edge radius. Even for zincwhere the edge radius effect seems to be quite small, 95 percent

confidence bands on the force measurements lie far apart band-to-band separation of 2.5 to 3 times the confidence bands , whichstatistically supports the significance of the edge radius effect onthe forces. If it is assumed that there is a linear variation of ma-chining forces with edge radius, then it can be shown that thepredicted force for an insert with edge radius of 75.6 m 0.003in would lie well between the confidence bands of sets A and B.

This proves that the increase in forces are of a higher order thanthe experimental error in the force measurements.

The variability along an edge raises an important consequenceregarding force measurements made for force model calibrationvia straight-edged orthogonal cutting tests. If the machining forcesare assumed to have some portion that is proportional to the edgeradius, then the machining force must vary parabolically along thelength of the cutting edge. Arbitrary or randomized selection ofzones along the cutting edge will result in measurement of ma-chining forces that appear to exhibit noise. Machining force datais meaningful only when all the tests are run with tools of thesame edge radius. In Fig. 4, cutting forces that were predictedusing a simple force model, based on data collected here, areshown for three situationscontrolled tests in which a single edgeradius is used, randomized tests along the cutting edge, and non-randomized tests. It can be seen that randomized and nonrandom-

ized tests lead to lower R2 values and, more importantly, an in-

correct slope and/or intercept. Hence, we can conclusively statethat it is important to measure and maintain the edge radius at thecutting zone in order to obtain accurate force data.

Conclusions

It was shown that there is a statistically significant parabolicvariation of edge radius along a cutting edge, which is presumablydue to the geometry of the brush honing process. Statisticallysignificant variation of the edge-center-point mean across edgesand inserts about 25 m was also observed, which is attributedto the difficulty of controlling the honing process. Variation wasobserved in the corner radius region as well, being statistically

Fig. 2 Variation of the cutting force component with uncutchip thickness for cast iron, aluminum and zinc at various edgeradii

Fig. 3 Variation of the thrust force component with uncut chipthickness for cast iron, aluminum and zinc at various edge radii

Fig. 4 Increased errors observed when different parts of thecutting edge are used for calibration testing




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significant around the corner and from insert to insert. It was alsoshown that the machining force components are sensitive tochanges in edge radius on the order of the measured edge varia-tion. This machining force sensitivity is of a higher order thanexperimental error noise in the measurement of forces. Hence,accurate prediction of force magnitude and direction, for honedtools, may likely require consideration of hone variation aroundthe corner and along the lead edge. Furthermore, care must betaken to avoid introducing hone variation by changing edges oradjusting the cutting-zone location along the edge when conduct-ing cutting tests, which is a typical approach to avoiding excessive

wear evolution over the duration of a set of tests.This study also further supports the fact that edge radius playsan important role in process mechanics, particularly at conditionsof low h/rn . Work that addresses the effects of the edge radius onthe cutting process and the development of a cutting model thatalleviates the sharp tool assumption are in progress.

Acknowledgments

The authors would like to thank Mr. Glenn Novak of the Poly-mers Department, General Motors Research Center, for the use ofthe measurement system. We also thank Dr. Don Cohen of Michi-gan Metrology for his help in optimizing the system for our use.

References

1 Albrecht, P., 1960, New developments in the theory of metal cutting pro-cess, J. Eng. Ind., 82, pp. 348358.

2 Nakayama, K., and Tamura, K., 1968, Size effect in metal cutting force, J.Eng. Ind., 90, pp. 119126.

3 Sasmor, J., and Caber, P., 1996, InterferometerHandles the rough jobs,Photonics Spectra, 30, pp. 9398.

4 Caber, P. J., 1993, Interferometric profiler for rough surfaces, Appl. Opt.,32, pp. 34383441.

5 Schimmel, R. S., Manjunathaiah, J., and Endres, W. J., 1997, An experimen-tal investigation of the variability of edge hones and their effects on machining

forces, G. J. Weins, ed., Manufact. Sci. Eng., MED 6-2, pp. 261268.


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