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Journal ofManufacturingScience andEngineering

Technical Briefs

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Nonlinear Influence of Effective LeadAngle in Turning ProcessStability

Michael P. VoglerGraduate Research Assistant

Richard E. DeVorProfessor, Fellow ASME

Shiv G. KapoorProfessor, Fellow ASME

Department of Mechanical and Industrial Engineering,University of Illinois at Urbana-Champaign,Urbana, IL 61801

An analytical method for stability prediction incorporating thnonlinear influence of the effective lead angle in turning is pposed and validated. It is shown that as the effective lead anchanges, due to depth of cut variations on a nose radiused cuinsert, different structural modes are excited, resulting in differstability results. Experiments have been performed on adegree-of-freedom system representative of the turning of loslender bars. It is shown that chatter may be present at low deof cut, typically less than the nose radius of the insert. The pposed model is also capable of predicting the chatter presenlarger depths of cut that is typically reported in literature.@DOI: 10.1115/1.1419200#

1 IntroductionMachining instability is one of the leading causes for subop

mal process performance. The presence of chatter results icreased tool wear, decreased surface integrity and accuracychine tool damage, and unacceptable levels of noise. Eliminachatter through proper selection of cutting conditions and tgeometry is critical for the economically feasible operationmachining processes. The relation between the orientation oimportant modal directions and the resultant cutting forceknown to have an effect on the process stability@1,2#. In turning,the parameter that orients the thrust force is the effective langle. Ozdoganlar and Endres@3# showed the variation in thestability charts as the effective lead angle and the orientationthe structural mode change. As the effective lead angle or

Contributed by the Manufacturing Engineering Division for publication in tJOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript receivedJan. 2000; Revised May 2001. Associate Editor: K. Ehmann.

Journal of Manufacturing Science and EngineeringCopyright © 2

oaded 17 Oct 2007 to 140.120.80.11. Redistribution subject to ASME lic

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orientation of the structural mode changes such that the thforce becomes more aligned with the structural mode, the stabof the system decreases.

Rao and Shin@4# were the first researchers to present modeland evidence that chatter can occur during low depth of cutsthat by increasing the depth of cut, the process can be stabiliTheir model, however, is a time domain model and fails to nthe link between the low depth of cut chatter they observedthe standard governing stability equations of the turning proce

In this paper, the nonlinear effect of effective lead angle wdepth of cut is considered in the stability analysis of turning wcorner radiused tooling. It is shown that multiple regions of insbility can be present, in which different strategies to avoid tchatter are required, which can be predicted by solving the sdard system equations in a novel manner.

2 Stability FormulationThe characteristic equation for the turning process as develo

by Ozdoganlar and Endres@5# is

@12@Kd#@G#@K f #dKTw~12me2 jwCT!#Tej vCt50. (1)

where @Kd# transforms the modal displacements into the chthickness variation,@G# is the matrix of compliance transfer functions, @K f # projects the thrust force into the model coordinatedKT is the thrust specific energy linearized with respect tochip thickness,w is the width of cut,m is the overlap factor,vC isthe chatter frequency, andT is the time period for one revolutionof the workpiece. If the transformations,@Kd# and@K f #, in Eq.~1!are constant, then an analytic solution for the limiting width of ccan be obtained and the resulting stability chart easily created@5#.

However, the transformations,@Kd# and @K f # are nonlinearfunctions of the effective lead angle,gL , which is computed ac-cording to the equation developed by Fu et al.@6#

gL5

tan21S tangL1r n

d*

12sin gL

cosgLD d.r n~12sin gL!

tan21A2r n

d21 d,r n~12sin gL!

(2)

wherer n is the nose radius of the insert,d is the depth of cut andgL is the side cutting edge angle. For example,@K f # has the form@cos(gL) sin gL)]

T when the modal coordinates are the transvedirection of the workpiece and of the tool as is often the casethe turning of slender bars and tubes. When turning with nradiused tooling, the effective lead angle and equivalent widthcut are nonlinear functions of the depth of cut as shown by Endand Waldorf @7#. Since the parameters in Eq.~1! are nonlinearwhen turning with nose radiused tooling, a different solutimethod must be employed. One alternative is to linearize therameters about a single set of process conditions and creacomplete stability chart with those parameters. This approacsound when the linear approximation to the nonlinear functiongood. When cutting at depths of cut much larger than the nradius, this is the case. When cutting at depths of cut belownose radius, the parameters in Eq.~1! are highly nonlinear, result-

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MAY 2002, Vol. 124 Õ 473002 by ASME

ense or copyright, see http://www.asme.org/terms/Terms_Use.cfm

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ing in a small region in which the linearization is valid. Threnders the information obtained from a single linearization toinsufficient to characterize the process with even modest chain depth of cut.

The following solution method is undertaken in this studyavoid the difficulty just stated. First, the depth of cut and feed tneeds to be considered is determined. Then, for these conditthe chip thickness – cutting force relation is determined. Thefective lead angle for the given feed and depth of cut is calculaaccording to Eq.~2!. The transformations,@Kd# and@K f # are thendetermined based on the computed effective lead angle. Equ~1! is then solved to determine the critical widths of cut for thcombination of depth of cut and feed. The actual width of cudetermined according to the procedure outlined by EndresWaldorf @7#. The actual width of cut is then compared to thcritical widths of cut determined from Eq.~1!. If the actual widthof cut is greater than the critical depth of cut, that combinationfeed, speed and depth of cut is considered to be unstable. Owise the combination is considered stable. This procedure isconsidered for all combinations of feed, speed and depth of cuinterest.

3 Experimental SetupExperimental validation was performed on a Mori Seiki ZL-2

4-axis turning center. Aluminum 6061 tubing with dimensions220 mm length, 50.8 mm outer diameter, and 6.35 mm wall thiness was used for the workpieces. Two tools were used ininvestigation: one which has25 deg back rake,25 deg side rake,and 15 deg side cutting edge angles and another which has25deg back rake,25 deg side rake, and 0 deg side cutting edangles.

Impact hammer tests were performed on the turning toolthe workpiece to determine the transfer functions of the com

Fig. 1 Predicted and experimental stability results

Table 1 Experimental design

474 Õ Vol. 124, MAY 2002


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nents. A second order model was fit to the dynamic responsthe tool parallel to the feed direction~Z direction! and of theworkpiece perpendicular to the feed direction~X direction!. Thenatural frequency, modal mass and the modal damping ratiothe workpiece in the X direction are 686 Hz, 0.538 kg and 0.0respectively; the modal parameters of the tool in the Z directare 397 Hz, 6.55 kg, and 0.060. Since both of these naturalquencies are much higher than the spindle speeds that couutilized, the usual lobed behavior of the stability charts is nobserved.

4 Results and ValidationIn order to study the influence of the effective lead angle on

turning process stability, two levels of cutting insert nose radius0.4 mm and 0.8 mm—and two levels of side cutting edge angl0 deg and 15 deg—were chosen as listed in Table 1. The depcut was varied in increments of 0.5 mm from 0.5 mm to 3.5 mThe presence of chatter was detected by examining the frequspectra of the cutting force signal and the tool vibration signa

The results from the tests are shown in Fig. 1. The model pdicted region of stability is the range between the lower limit athe upper limit. For all four cases considered, the model pretions of the lower and the upper limits of stability compare favoably with the experimental results. For case 2, the time seriethe thrust force for the low and high depth of cut chatter ashown in Fig. 2. By looking at the measured chatter frequelisted in Table 2, it was observed that the chatter frequency atlower limit of stability is near the workpiece natural frequenand that the chatter frequency at the upper limit of stability is nthe tool natural frequency.

The explanation for this observation is as follows. As the effetive lead angle changes, the chatter frequency of the syschanges from near the workpiece natural frequency at low de

Fig. 2 Thrust force time series for low and high depth of cutchatter

Table 2 Experimental chatter frequencies

Transactions of the ASME


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of cut to near the tool natural frequency at higher depths of cutthe effective lead angle orients the thrust force more orthogonaa vibration mode, the mode’s contribution to the vibration of tsystem is reduced. In the limiting case, it will reduce the tdegree of freedom system to a one degree of freedom systemsuch a system at low spindle speeds~relative to modal naturafrequencies!, it is well known that the chatter frequency will bnear the natural frequency of the single mode. For the givendegree of freedom system, at low depths of cut the effective langle orients the thrust force more along the transverse direcof the workpiece~perpendicular to feed direction!. At higherdepths of cut, the effective lead angle approaches the side cuedge angle of the tool, which, in this case, orients the force malong the transverse direction of the cutting tool~feed direction!.In these two cases, the system may be modeled as a one degfreedom in the X direction~at low depths of cut! and a one degreeof freedom system in the Z direction~at higher depths of cut!.Therefore, it is expected that the chatter frequency would bethe workpiece natural frequency at low depths of cut and neartool natural frequency at high depths of cut.

The effect of the nose radius on the lower limit of stabilityclearly seen by comparing case 1 to case 2 and by comparing3 to case 4 in Fig. 1. The upper limit of stability is unaffectedthe change in nose radius but the lower limit changes. As the nradius increases, the lower limit of stability increases. This flows directly from the fact that the influence of the nose radiusthe effective lead angle continues to a larger depth of cut folarger value of the nose radius.

5 Improved Stability ChartThis nonlinear influence of the effective lead angle changes

look of the stability charts for many processes involving cuttiwith a corner radiused tool. An example stability chart generawith the approach explained in this paper is presented in FigThe unstable regions of the cutting process are represented ifigure with the shading and the stable regions are white. Twofeatures of this modified stability chart can be seen. First, thexists an envelope in which the process is stable as opposed tan upper limit on the depth of cut for a stable process. Secosince the important structural frequency is different at low depof cut than at higher depths of cut, the peaks on the lobes atdepth of cut are located at different spindle speeds than the lolimit of the low depth of cut limit. For example, there is a peakthe upper limit at 5700 rpm while the valley of the lower limit haits minimum near 6100 rpm.

6 ConclusionFrom this work, the following conclusions can be made

Fig. 3 Stability chart with multiple regions of instability



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• By linearizing the characteristic equation at every depthcut, multiple regions of stability can be predicted.

• The multiple regions of stability are caused by the changeffective lead angle aligning the thrust force in different drections as the depth of cut is increased.

• By reducing the nose radius of the insert, the lower limitthe envelope of stability seen for turning systems of lonslender bars can be reduced.

7 AcknowledgmentThis research was supported by the Ford Motor Company

the National Science Foundation Industry/University CooperaResearch Center for Machine Tool Systems Research at theversity of Illinois at Urbana-Champaign.

References@1# Merritt, H. E., 1965, ‘‘Theory of Self-Excited Machine Tool Chatter,’’ ASME

J. Eng. Ind.,87, pp. 447–454.@2# Tlusty, J., and Polacek, M., 1963, ‘‘The Stability of the Machine Tool Again

Self-Excited Vibration in Machining,’’ International Research in ProductioEngineering, pp. 465–474.

@3# Ozdoganlar, O. B., and Endres, W. J., 1998, ‘‘An Analytical Stability Solutifor the Turning Process with Depth-Direction Dynamics and Corner-RadiuTooling,’’ Proceedings of the ASME Symposium on Advances in ModeMonitoring, and Control of Machining Systems,DSC-64, pp. 511–518.

@4# Rao, B. C., and Shin, Y. C., 1999, ‘‘An Comprehensive Dynamic CuttiForce Model for Chatter Prediction in Turning,’’ Int. J. Mach. Tools Manu39, pp. 1631–1654.

@5# Ozdoganlar, O. B., and Endres, W. J., 1997, ‘‘A Structured Approach to Alytical Multi-Degree-of-Freedom Time-Invariant Stability Analysis for Machining,’’ Proceedings of the ASME Symposium on Predictable ModelingMetal Cutting as Means of Bridging the Gap Between Theory and PractMED-6-2, pp. 153–160.

@6# Fu, H. J., DeVor, R. E., and Kapoor, S. G., 1984, ‘‘A Mechanistic Model fthe Prediction of the Force System in Face Milling Operations,’’ASME J. EInd., 106, pp. 81–88.

@7# Endres, W. J., and Waldorf, D. J., 1994, ‘‘The Importance of Considering SEffect Along the Cutting Edge in Predicting the Effective Lead Angle fTurning,’’ Transactions of the NAMRI/SME,22, pp. 65–72.

Experimental Investigation ofPolymeric Material Removal by LowDiffraction Laser Beam

Xuanhui Lu

Y. Lawrence Yaoe-mail: [email protected]

Kai Chen

Department of Mechanical Engineering, ColumbiaUniversity, 220 Mudd Bldg., MC 4703, New York,NY 10027

Effects of improved beam quality of a low diffraction laser beaon laser material removal processes are experimentally invegated in a polymeric material. The experimental results areagreement with theoretical predictions. The results show thatlow diffraction beam has marked advantages over the Gausbeam in ablation-dominated material removal processes in teof larger depth and smaller taper at the same average powlevel. @DOI: 10.1115/1.1445156#

Contributed by the Manufacturing Engineering Division for publication in tJOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript receivedNovember 1999; Revised February 2001. Associate Editor: S. Smelser.

MAY 2002, Vol. 124 Õ 475002 by ASME


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1 IntroductionLaser beam quality plays an important role in quality a

efficiency of laser materials processing applications. Highbeam quality typically means nearly fundamental-mode oscition. Many efforts have thus been made to change high-ormodes into the fundamental modes including simple methods sas using an aperture but often at the cost of excessive poattenuation. The fundamental-mode Gaussian beam~i.e.,

Fig. 1 Schematic of resonator configuration „with a half phaseplate attached on the rear mirror inside laser cavity … to realizethe low diffraction beam

Fig. 2 Beam radius of the measured low diffraction beam andtheoretical TEM 00 mode vs. axial distance between the outputcoupler to the measurement location

476 Õ Vol. 124, MAY 2002


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TEM00-Transverse Electromagnetic Mode! has long been re-garded as an ideal beam, or diffraction-limited beam. The bequality can be described quantitatively in term ofM2 as definedby Siegman@1#. A product of the standard deviation of the beasize and that of the divergence is formed.M2 is the ratio of theproduct for a non-diffraction-limited, multi-transverse-modbeam, to that of a Gaussian beam. TheM2 for the fundamental-mode Gaussian beam is thus unity. An interesting questionwhether or not it is possible for a practical beam to have anM2

value smaller than that of the fundamental-mode Gaussian beThe concept of a low-diffraction beam havingM2,1 has beenproposed@2,3#. The low diffraction beam is based on the bounary diffraction principle. An advantage of the beam is that it cbe obtained by altering the existing resonator of a CO2 laserthrough a special phase plate implemented at the resonatormirror ~Fig. 1!. Additional details can be founded in@2,3#.

The next question is whether the low-diffraction beam, whoM2 value is smaller than that of a Gaussian beam, will transinto better quality and efficiency in laser materials processingplications, such as laser machining. Although it is generaagreed that the laser beam quality has a direct effect on machiquality, no consensus has been reached that a smallerM2 is al-ways beneficial to a machining process because the machiprocess is a complicated thermal process that could also invfluid flow and melt rejection. A beam with a smallerM2 value islikely to result in smaller hole sizes or narrower slots, which is nin favor of melt rejection. However, in an ablation-dominatedser machining process, most of the material is vaporized alminstantly and is mainly removed by vapor pressure. The low dfraction laser beam with a smallerM2 value is thus expected tohave beneficial effects on the ablative machining process.

The quality and profile of laser made holes, grooves and care obviously of importance especially in the growing microeletronic and precision medical device industry@4–6#. The quality isgenerally gauged by wall definition, extent of heat-affected zoand ability to produce features with higher aspect ratio. Laablation of polymeric materials using laser beams is a westablished process and examples are found in@7,8#. Factors oflaser beams likely to affect drilling and grooving have been stied in many reports@9,10#. This paper presents the applicationa low diffraction beam to ablation-dominated drilling and grooing processes involving a polymer material. Its beneficial effeon process quality are investigated in comparison with a Gausbeam.

Fig. 3 Intensity distributions „calculations are based on Eqs. „2… and „3…, with W0Ä1.49 mm „experimentally measured …

and Me2Ä0.3 „using Eq. „1………



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2 Characterization of the Low Diffraction BeamExperiments were carried out using a continuous-wave C2

laser with maximum average power of 12 W. The original renator of the laser system generates a fundamental mode Gaubeam~i.e., TEM00!. According to the principle described in@2,3#,an identical resonator is modified with its structure as schemcally shown in Fig. 1 to generator a low diffraction beam. Tintensity profile and divergence of beams from both resonawere measured. Because the output of the low diffraction beanot a Gaussian distribution, it is more practical to use the defition of 86.5 percent power content to measure the beam sizeorder to compare the beam quality of this new mode withGaussian mode, the equivalent beam quality factorMe

2 is definedas follows:

Me25W86.5u86.5

p

l(1)

whereW86.5 is the equivalent beam waist size with 86.5 percepower content,u86.5 is the divergence angle corresponding to t86.5 percent power content, andl is the beam wavelength.

The intensity distribution of Gaussian beamI 0G(r ), and the low

diffraction beam,I 0L(r ) at the beam waist can be written as

Fig. 4 Acrylic imprints with Gaussian „left … and the low diffrac-tion beam „right … „9W, 1 sec, both unfocused …

Journal of Manufacturing Science and Engineering


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2DwhereW0 is the Gaussian beam waist radius andI 0 is the peakintensity. The intensity distribution at the far field can be obtainby using the beam propagation law, i.e.,ABCD law @11#:

I zG~r ,z!5I 1 expF 22r 2

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wherez is the axial distance from the waist, andzr5pW02/l is the

Rayleigh range.The focal point radius for a Gaussian beam,Wf

G , is well known@12#:

WfG5

l

pWz(4)

wherel is wavelength,f is lens focal length, andWz is originalunfocused beam radius. The depth of focus for a Gaussian bhG , is briefly derived below.

According to Gaussian beam properties, its beam radius, atdistance along the beam path,z, from the waist is given from thebasic propagation equation:

W~z!5W0F11S z

zrD 2G1/2

(5)

The depth of focus is normally defined as the distance betwtwo points slightly away from the beam waist and the beam radat these points is about 5 percent above the beam waist radiussubstitutingW(z)51.05 W0 into Eq. ~5!, the depth of focus isobtained as:

h 50.64l S f D 2

(6)

Fig. 5 Theoretical „dotted line … and experimental results „solid line … of hole profiles „power Ä9 W, duration Ä1 sec, unfo-cused, acrylic …

MAY 2002, Vol. 124 Õ 477


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According to theMe2 definition~Eq. 1!, the focal point radius of

a low diffraction beam ofMe2 can be written as follows.

WfL5

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For the low diffraction beam, its Rayleigh rangezr

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Compared to the focal point radius and the focal depth oGaussian Beam, i.e., Eq.~4! and Eq.~6!, the focal point radius forthe low diffraction beam is smaller than that of the Gaussbeam, while the focal depth for the low diffraction beam is largthan that of the Gaussian beam, since theMe

2 value for the lowdiffraction beam is less than unity.

Figure 2 shows experimental results of beam radius of thediffraction beam at various distances. It is seen that the divergeangle of the low diffraction beam is smaller than that of theor

Fig. 6 The hole taper vs. ablation power „ablation durationÄ0.8 sec, focused, acrylic …

Fig. 7 Drilling depth comparison between the Gaussian beamand the low diffraction beam vs. ablation duration „focused,acrylic …

478 Õ Vol. 124, MAY 2002


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ical TEM00 mode. Based on experimentally measuredW86.5 andu86.5, and l510.6mm for the CO2 laser, the equivalent beamquality factorMe

2>0.3 is obtained.The intensity profile of the low diffraction beam in the far fiel

is experimentally measured and superposed in Fig. 3~a! with thecalculated intensity profiles of both the low diffraction and Gausian beam according to Eq.~3!. As seen, there is a good agreemebetween the experimental and calculated profiles, and thediffraction beam has a much higher central intensity and smadivergence than that of the Gaussian beam. Using Eq.~2!, thenear-field intensity profiles of both beams are plotted in Fig. 3~b!.It can be seen that the low diffraction beam in the near field ahas higher central intensity and smaller diameter than the Gaian beam.

3 Comparison of Theoretical and Experimental Re-sults

Figure 4 shows imprints made on acrylic by the Gaussian be~best achievable on the laser used! and low diffraction beam~bothunfocused! when the average power is 9 W. Acrylic is choseprimarily because of its low ablation threshold that the laser ucan reach. It is also because its removal is primarily due to abtion such that the imprint better reflects the beam shape. Althothe power level is the same, the low-diffraction beam has a higenergy intensity and a smaller beam size. Not surprisingly,hole profiles closely follow that of beams in the ablative maching process. Cross-sections of the profiles are also shown in slines in Fig. 5 to compare with a theoretically calculated ablatiprofiles shown in dotted lines.

The theoretically calculated ablation profiles are obtained baon the model by Andrews and Atthey@13#. The energy density andthe beam size for both the low diffraction beam and Gaussbeam are experimentally obtained and used in the theoremodel to predict the hole profile as shown in Fig. 5 in dotted linThe waist radius for the resonator generating the Gaussian beameasured asW051.49 mm, while for the resonator generating thlow diffraction beam the beam radius is measured asW0850.75 mm. As a result, the intensity is 129 W/cm2 at the waist forthe Gaussian beam, and 509 W/cm2 at the waist for the low dif-fraction beam. The theoretical predication agrees with experimtal results.

There is some discrepancy at the top part of the hole profiunder the condition of the low diffraction beam~Fig. 5~b!!. Thereason is that beam intensity in the theoretical model is based

Fig. 8 Drilled hole diameter comparison between the Gauss-ian beam and the low diffraction beam vs. ablation time „fo-cused, acrylic …



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Fig. 9 „a… Hole diameter and „b… depth vs. the distance from focus lens to workpiece top surface „focal lengthÄ4 cm, power Ä7 W, duration Ä0.5 sec …

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while the actual beam is obtained based on the boundary difftion principle.

The beams are then focused using a lens with a focal lengt40 mm. The CO2 laser varies at two average power levels, 7and 9.2 W. For the Gaussian beam, the resultant average pintensity is 5.223104 W/cm2 for 7 W, and 6.713104 W/cm2 for9.2 W. For the low diffraction beam, the resultant average pointensity is 2.013105 W/cm2 for 7 W, and 2.683105 W/cm2 for9.2 W.

Figure 6 shows the variation of hole taper against ablatpower for both the low diffraction beam and Gaussian beaTaper is defined as the ratio of hole diameter to hole depth~there-fore is the inverse of aspect ratio! and is one of the quality factorfor hole profile. It is seen from Fig. 6 that the hole drilled with thlow diffraction beam has significantly smaller taper values ththe hole drilled with Gaussian beam. The predicted values frthe theoretical model are also shown in the figure and are geally in agreement with the measured values. The taper valuecreases with the increasing power level for both low diffractibeam and Gaussian beam. This is because the diameter of theincreases much slower than the hole depth when the powerincreases, as seen from Figs. 7 and 8.

nal of Manufacturing Science and Engineering

17 Oct 2007 to 140.120.80.11. Redistribution subject to ASME lic

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4 Parametric Studies and Grooving ExperimentsFigure 7 shows the measured drilling depth vs. ablation time

two power levels. The hole depth drilled with the low diffractiobeam is much larger than that with the Gaussian beam becauthe higher central energy intensity at the same average polevel. The depth with the low diffraction beam is about 40 perchigher than that with the Gaussian beam under the condition u

Figure 8 shows the measured diameter of the drilled holeablation time at two power levels. It is seen that the hole diamedrilled with low diffraction beam is about 25 percent smaller ththat with Gaussian beam. In addition, with the ablation timecreasing, the drilled hole diameter with the low diffraction beaincreases slower than that with the Gaussian beam especialonger ablation times clearly because the low diffraction beamsmaller divergence and longer focal depth. The parametric stuconfirm that the low diffraction beam consistently provides betresults under different laser power and ablation time. Power leof 7 W and 9.2 W were used to avoid to be too close tomaximal power of the laser used~12 W! but be high enough toremove the material.

Equations~6! and~8! show that the low diffraction beam haslonger depth of focus than that of a Gaussian beam becauseMe

2

Fig. 10 Typical groove profiles with Gaussian beam and the low diffraction beam „powerÄ9 W, speedÄ14 mm Õsec, both focused, acrylic …

MAY 2002, Vol. 124 Õ 479


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,1 for the low diffraction beam. As seen in Fig. 9~a!, the holediameter drilled with the low diffraction beam and the Gaussbeam change from about 0.3 to 0.7 mm and 0.4 to 1.0 mm,spectively, when the distanceL between the focus lens to tosurface of the workpiece are changed from 3.4 to 4.4 cm~the focallength of the lens is 4 cm!. This verifies that the low diffractionbeam has a longer depth of focus. Obviously, whenL is around 4cm, the focal point which has the minimal beam diameter is rion the top surface of the workpiece. As a result, the hole diamis the smallest whenLi is around 4 cm. From Fig. 9~b!, it is seenthat the hole depth varies slower for the low diffraction beam thfor the Gaussian beam whenL is around 4 cm. The longer deptof focus of the low diffraction beam is desirable especially whthick section machining is concerned.

The focused low diffraction beam and Gaussian beam areplied to grooving the same material. Figure 10 compares the csections of groove profiles ablated by both beams. It is seen ththe same power level~9 W! and the grooving speed~14 mm/s!,the cross sectional profile with the low diffraction beam haslower taper or higher aspect ratio than that with the Gaussbeam. It is seen that the beneficial effects of the low diffractbeam in drilling extend to applications such as grooving and likcutting as well. These beneficial effects include a higher aspratio and lower sensitivity to focal point location. They are epected to be more significant at higher power levels. While tpaper only covers acrylic, other materials are expected to hsimilar beneficial effects when ablated by the low diffractibeam because during ablative laser machining, machined prochiefly rely on the optical beam quality. When the power intensis below the ablation threshold of a material, other factors aplay a significant role.

5 Conclusion

A low diffraction beam, which has aM2 factor smaller thanunity, is implemented with a low power CO2 laser and applied toablation-dominated drilling and grooving of acrylic. The expemental results show that the low diffraction beam produced lardepth, smaller taper and smaller hole diameter, as compareda Gaussian beam at the same average power level. This holdsfor both the unfocused and focused cases. The depth of focuthe focused low diffraction beam is also longer than the Gausbeam, indicating its suitability for processing thick sectionsmaterial. Similar results are obtained when the beam is appliegrooving applications. If the implementation of the low diffractiobeam is extended to a higher power level laser system, the amentioned beneficial effects will be more significant. For othmaterials, as long as ablation is the dominant mechanism ofterial removal, similar beneficial effects can be expected. In cawhere ablation is not dominant, the low diffraction beam is liketo offer at least some of the advantages but further studiesneeded.

AcknowledgmentsThe financial support provided via a Pao Scholarship and

lumbia University is gratefully acknowledged.

References@1# Siegman, A. E., 1993, ‘‘Defining and Measuring Laser Beam Quality,’’Solid

State Laser-New Developments and Applications, M. Inguscio and R. Wallen-stein, eds., Plenum, New York, pp. 13–28.

@2# Wang, S., Lu, X., Pan, C., and Yu, S., 1995, ‘‘A New Beam CO2 Laser,’’ Optik~Stuttgart!, 101, pp. 32.

@3# Wang, S. et al., 1995, ‘‘A New Beam Produced by CO2 laser,’’ Optik ~Stut-tgart!, 101, pp. 84.

@4# Yibas, B. S., 1987, ‘‘Study of Affecting Parameters in Laser Hole DrillingSheet Metals,’’ ASME J. Eng. Mater. Technol.,109, pp. 282–287.

@5# Olson, R. W., and Swope, W. C., 1993, ‘‘Laser Drilling with Focused GaussBeams,’’ J. Appl. Phys.,72, No. 8, pp. 3686–3696.

@6# Rohde, H., and Dausinger, F., 1995, ‘‘The Forming Process of a Through H

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s ofianofd tonoveerma-seslyare

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f

ian

ole

Drilled with a Single Laser Pulse,’’ICALEO’95, pp. 331–340.@7# Miyamoto, I., and Maruo, H., 1991, ‘‘The Mechanism of Laser Cutting, We

ing in the World,’’ Journal of the International Institute of Welding,29, No.9/10, pp. 283–294.

@8# Redmond, T. F., Lankard, J. R., Balz, J. R., Proto, G. R., and Wassick, T1993, ‘‘The Application of Laser Process Technology to Thin Film Packaing,’’ IEEE Trans. Compon., Hybrids, Manuf. Technol.,16, No. 1, pp. 6–12.

@9# Batteh, J. J., Chen, M. M., and Mazumder, J., 1998, ‘‘Scaling and NumerAnalysis of Pulsed Laser Drilling,’’ICALEO’98, B-30.

@10# Powell, J., 1993,CO2 Laser Cutting, Springer-Verlag.@11# Kogelnik, H., and Li, T., 1966, ‘‘Laser Beams and Resonators,’’ Appl. Opt.,5,

pp. 1550.@12# Collins, S. A., 1970, ‘‘Lens-System Diffraction Integral Written in Term o

Matrix Optics,’’ J. Opt. Soc. Am.,60, No. 9, pp. 1168–1177.@13# Andrews, J. G., and Atthey, D. R., 1976, ‘‘Hydrodynamic Limit to Penetrati

of a Material by a High-Power Beam,’’ J. Phys. D,9, pp. 2181–2194.

Design of Reconfigurable MachineTools

Yong-Mo MoonResearch Fellowe-mail: [email protected]

Sridhar KotaProfessore-mail: [email protected]

Dept. of Mech. Engr., NSF Engineering Research Centfor Reconfigurable Machining System, Universityof Michigan, Ann Arbor, MI 48109-2125

In this paper, we present a systematic methodology for designReconfigurable Machine Tools (RMTs). The synthesis methoogy takes as input a set of functional requirements—a set ofcess plans and generates a set of kinematically viable reconurable machine tools that meet the given design specificationspresent a mathematical framework for synthesis of machine tusing a library of building blocks. The framework is rooted in (graph theoretic methods of enumeration of alternate structuconfigurations and (b) screw theory that enables us to manipumatrix representations of motions to identify appropriate kinmatic building blocks. @DOI: 10.1115/1.1452748#

IntroductionA Reconfigurable Machine Tool~RMT! is designed to process

given family of machining features and is constructed from aof standard modules. An RMT provides a cost-effective solutto mass customization and high-speed capability. There isknown systematic method or a scientific basis for designing RM@1#. This paper presents a mathematical framework for designRMTs starting from process requirements. The key feature ofmethodology is the use of screw-theory based mathematicalresentation to transform a given description of machining taskbe performed~process planning data! into a machine tool that iscapable of performing the prescribed machining tasks. Starfrom machining operations data, a set of feasible structural cfigurations of the machine is determined using graph theory. Vous machine functions are then mapped to individual entitieseach structural configuration. Using a precompiled parameter

Contributed by the Manufacturing Engineering Division for publication in tJOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript receivedJan. 1999; Revised Aug. 2001. Associate Editor: K. Ehmann.

02 by ASME Transactions of the ASME


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library of commercially available machine modules, each functis then mapped to a feasible set of modules. This provides a skinematically feasible machine tools that provide desired motioMachining process models~not discussed in this paper! are usedto establish desired dynamic stiffness criteria.

Information needed to create machine tool designs is extrafrom CL ~cutter-location! data of parts to be machined. Note thin the design of conventional machine tools, process plansgeneratedafter the machine is designed. In RMT design, this prcess is reversed—machine tools are created to fit the function anthe performance required to perform a set of operations. That is,the machine tools are tailored to meet the requirements of a gpart ~or a family of parts! to ensure that exactly the functionalitneeded is in fact built into the machine tool. This presents a nparadigm and a new set of challenges in the design of mactools.

Although the kinematic and topological synthesis methods psented by other researchers do not directly address synthesRMTs, some of these methods are relevant to the design of RMIn this research, some traditional methods of representatiomotions and topologies~screw theory, graph theory etc.! are em-ployed to capture the characteristics of RMTs.

Motion SpecificationsThe set of operation plans provided as input to RMT des

process consists of afamily of machining features including machining operation types, cutter locations, and process plans.first step is to interpret the given information using a new motrepresentation method to:

• represent motions of required machining tasks, machtools, and machine modules,

• enable automatic module selection,• represent motion characteristics such as motion type, mo

range, and direction of the motions.

We devised a new method using dual vectors@2# which takesadvantages of convenience of HTMs and at the same time ocomes the limitations of Plu¨cker’s coordinate system. In thimethod, the motion of a screw is represented as

$W 5@M M Mm MC#~PA1«PT!$SW 1«SW 0% (1)

The first part of the dual vector representation shows the morange of the motion.M M , Mm , and MC represent maximumminimum, and current value of the motion. The second te(PA1«PT), represents thepitch of the motion.PA andPT rep-resent angular and translational pitches respectively.

The actual tool positioning motion and the tool feeding motioare computed from the operation plan. The screws corresponto positioning~translation along -Z-axis by 21.95 mm in Fig. 1!and feeding~translation along -Z-axis by 3.07 mm in Fig. 1! mo-tions are represented in dual numbers as:

M P125@0 221.95 221.95#~01«1!$~0,0,1!1«~0,0,0!%

MF015@0 23.07 23.07#~01«1!$~0,0,1!1«~0,0,0!% (2)

]

M P12 is the dual vector representation for tool motion froposition 1 to position 2. Likewise,MF01 corresponds to the specfied feeding motion. All other required motions are thus captuin the dual vector form which enables ease of comparison of tmotion characteristics.

Once individual motions~lines in Fig. 1! are computed, the nexstep is to investigate if some of these motion segments needmerged. This is necessary to avoid a redundant number of actors. By merging the motions, the minimum set of motions toperformed by the machine tool are calculated as:

M P15@0 225.02 0#~01«1!$~0,0,1!1«~0,0,0!%



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[email protected] 2115.15 0#~01«1!$~0,1,0!1«~0,0,0!%(3)

[email protected] 2,199.76 0#~01«1!$~1,0,0!1«~0,0,0!%

[email protected] 0 0#~01«1!$~0,0,1!1«~0,0,0!%

The feeding motionMF3 and the position motionM P1 ~whichhave the same dual vector representation! are to be merged.Therefore,MF31M P1 is computed.

MF31M P1 :(4)

[email protected] 225.02 0#~01«1!$~0,0,1!1«~0,0,0!%

Once the minimal set and range of motions are thus derivedtask matrix is established. In this simple case, the task invothree translational motions:

• motion alongx-axis is 124.051115.155240.00• motion alongy-axis: 199.761199.765399.52• motion alongz-axis5105.43125.025120.45.

These motions shown as variablesu1 , u2 , u3 in the task matrixT below. The initial distance between the bottom of the workpieand the tool tip in this case is 300 units.

T5F 1 0 0 u1

0 1 0 u2

0 0 1 3001u3

0 0 0 1

G (5)

The basic function structure of the RMT is determined by ttypes of operations~drilling, turning, etc.! to be performed. Eachoperation has a template that determines the core kinematic ftions. Next, these set of motion functions are to be assignevarious ‘‘entities’’ in a machine tool such as column, tool-suppetc. There are many different ways to assign these functionspending on the arrangement of the machine structure. We em

Fig. 1 Operation requirements

Fig. 2 Function Structure Graph of CNC Milling Machine

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graph theory to systematically enumerate different structural cfigurations and then assign each of the motions identified abovdifferent ‘‘entities’’ within a preferred structural configuration before selecting the final configuration.

Function Structure DesignBased on the type of machining operations to be performed

the types of tool to be used, the overall topology or structuconfiguration of the machine tool is established in this step.

The structure of a machine tool is represented by a graphcause;

• It allows for systematic enumeration of alternate structuconfigurations@1#,

• It provides a method of identification of nonisomorphgraphs which in turn provides a basis to ensure that alterstructural configurations are distinctly different from one aother @1,3#, and

• It provides a bookkeeping means to assign machine modto its various entities.

Figure 2~b! shows the graph representation of a vertical millimachine shown in Fig. 2~a!. The function structure graph in Fig2~b! served as a reference in mapping the kinematic functioAssigning each of the functions to different edges of the graphgenerate multiple solutions.

Module Selection

Module Selection with Parameterized Module Library. Inmodule selection stage, commercially available modules arelected from the module library for each of the functions~structuralas well as kinematic! mapped into the function structure grapThe module library consists of structural as well as motion mules such as bases, spindles, columns, tool-supports, indetables ~single axis rotational motion module!, and n-axis linearslides ~where n can be 1, 2 or 3! and different sizes of eachmodule-type. Additionally, the library contains a modulconnectivity matrix@Moon 2000# that indicates how each modulcan or cannot be connected to all other modules.

The first step in module selection is to compare the homoneous transformation matrices of the modules with the taskquirement matrix such that when appropriate modules are seleto meet the task requirements, the product of all module matrshould be equal to the task matrix.

M1M2M3 . . . Mn5T (6)

Where M i is the motion transformation matrix of moduleiP$1,2,3, . . . ,n%, T is the machining task matrix.

Table 1 Module library

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The first module to be selected is the spindle with adequpower index. Using the HTM of this module, the task matrixmathematically reduced to a form that indicates remaining mtions that are required to perform the machining task.

From the module library shown in Table 1, modules 29-32the available spindle modules. Of these, spindle modules 2930 satisfy the power requirement~MRR! of the given machiningtask. Since only two different tools are required~milling and drill-ing!, the spindle 311~module 29! with a 3-tool changer is selected. The transformation matrix for module 29 and the redutask matrix are:

Mn5F 0 0 1 260

0 1 0 0

21 0 0 60

0 0 0 1

G , T5F 1 0 0 u1

0 1 0 u2

0 0 1 3001u3

0 0 0 1

G(7)

TM n215M1M2M3 . . . Mn215F 0 0 21 601u1

0 1 0 u2

1 0 0 25601u3

0 0 0 1

GThis procedure is continued until the ‘‘task matrix is reduced

identity matrix.’’ In each step alternate paths are pursued to gerate alternate design configurations.

Enumeration of Alternate Designs. We start with the selec-tion of work-support module and traverse clockwise all the waythe tool in function structure graph. For each vertex and the ealong the path we can identify one or more candidate moduFor instance, the first module in the chain must have a trantional motion and the library suggests modules 15, 16, 21, 22and 24 as the candidate modules. The only module that satithe desired translational motion of2115.15<Y<124.85 is mod-ule 21. Likewise, the next module requires2199.76<X<199.76 and once again module 21 meets the desired morequirement. The next module in the chain is the ‘‘bed’’ and tmodules 10, 12 and 14 are the potential candidates.

The possible nine different configurations~Fig. 3! are the samein kinematic function structure but use a different set of machmodules.

ConclusionsMachine tool design is for the most part experience-bas

There are no known systematic methods for configuration of mchine tools, let alone reconfigurable machine tools, starting frfunctional requirements. In this paper, we attempted to lay a m

Fig. 3 Some of the proposed candidate machine tool configu-rations „all derived from a single graph …



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ematical basis for synthesis of Reconfigurable Machine ToThis paper has addressed a generation of kinematically viablelutions. The systematic design process begins with the familyproducts~in operation plan!. The methodology is rooted in thnotion of customized flexibility since it starts with a matrix reresentation of desired set of machining operations and culminin a set of machine tools. The methodology also ensures thakinematically viable and distinctly different configurations asystematically enumerated to reduce the chance of missinpromising design.

AcknowledgmentsThe authors gratefully acknowledge the financial support of

National Science Foundation’s ERC and the member compato carry out this research. The authors acknowledge the supand invaluable feedback of the industrial partners on this proincluding: Cross-Hu¨ller, Lamb Technion, R&B Machine Tool Co.Ford Motor Company and Schneider Automation.

References@1# Moon, Y., 2000, ‘‘Reconfigurable Machine Tool Design: Theory and Applic

tion,’’ Ph.D. Dissertation, The University of Michigan.@2# Moon, Y., and Kota, S., 1998, ‘‘Generalized Kinematic Modeling Method

Reconfigurable Machine Tools,’’ MECH/5946, 25th Biannual Mechanism Dsign Conference, Atlanta, GA.

@3# Yan, H.-S., and Chen, F.-C., 1998, ‘‘Configuration Synthesis of MachinCenters Without Tool Changers,’’ Mech. Mach. Theory,33, No. 1/2, pp. 197–212.

Parallel Structures and TheirApplications in ReconfigurableMachining Systems

V. Gopalakrishnan, D. Fedewa,M. G. Mehrabi, S. Kota, and N. OrlandeaDepartment of Mechanical Engineering, University ofMichigan, Ann Arbor, MI 48109

Reconfigurable Machine Tools (RMTs), assembled from macmodules such as spindles, slides and worktables are designebe easily reconfigured to accommodate new machining requments. Their goal is to provide exactly the capacity and functiality, exactly when needed. In this paper, we present a noparallely-actuated work-support module as a part of an RMTmeet the machining requirements of specific features on a faof automotive cylinder heads. A prototype of the proposed mois designed/built and experimental results regarding its perfmance are presented.@DOI: 10.1115/1.1459468#

1 IntroductionManufacturing systems can be generalized into t

categories—dedicated and flexible. Transfer lines are dedicatemachining a particular part in a high volume production, for eample, a cylinder head. Therefore they cannot produce diffeparts; this limits their application when market demands charapidly. At the other end of the spectrum is the Flexible Manufturing System~FMS!. When they came into existence, they wethought to be ideal for accommodating changes to parts. Howe

Contributed by the Manufacturing Engineering Division for publication in tJOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript receivedApril, 2000; revised August 2001. Associate Editor: K. Ehmann.



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in a number of cases these expensive systems possess waflexibility that is never used in day to day applications.

Reconfigurable Manufacturing Systems~RMSs! falls in be-tween the dedicated hard automation of the transfer lines andhigh flexibility of the machining centers@1,2#. In this regard, Re-configurable Machine Tools~RMTs! are machine tools that arebuilt from machine modules. A machine module may be a wotable, a tool holder or a spindle@3#. These modules~buildingblocks! can be put together to assemble a machining station foparticular application. In this work, parallel structure machineamong many applications@4–7#, are considered as a module oReconfigurable machining Systems~RmSs!. In this paper, wepresent a novel parallely-actuated work-support module as aof an RMT to meet the machining requirements of specific fetures on a family of parts~e.g., automotive cylinder heads!. Aworking prototype of this system is designed and built at the Uversity of Michigan; some of the experimental results obtainwhen the system is implemented in machining applicationsreported.

2 Proposed Work Support ModuleThe proposed work support module will be part of an RM

designed to machine selected features on a cylinder head. A fily of cylinder heads can be fixtured to the moving platform of thwork support module. Cylinder head features such as the vaguide holes and the spark plug holes, which require simplecompound angle~pitch and roll! tilts are chosen to demonstratthe application of the parallel mechanism as a reconfigurable wsupport module@6#. The holes are drilled by integrating the paallely actuated work support module with an RMT or a vertic3-axis CNC machining center. The pitch and roll motions~thetable is designed for delivering compound angles of 36 deg23.5 deg and cutting forces up to 2000 lbs; total legs stroke isin.! help orient the cylinder head and the vertical travel is requirto position the platform at the right altitude~to obtain the requiredtilt ! without colliding with the tool.

Using 3 DOF in this application also falls in line with one othe objectives of reconfigurability—providing just the requireamount of flexibility. Integrating the module with a convention3-axis CNC machining center enhances the capability of the mchining center as and when needed.

3 Kinematic AnalysisFigure 1 illustrates the schematic diagram of the work supp

module. It consists of a fixed base and a moving platform. Tplatform is connected to the base by three linear actuators. Thuniversal joints are used to connect the actuators to the baspoints B1 , B2 and B3 respectively. Spherical joints are usedconnect the actuators to the platform at points P1 , P2 and P3 . Aspline shaft is used to prevent rotation of the platform aroundZ-axis. It is connected to the platform by using a universal joiIn order to control the platform, the inverse kinematics of tparallel mechanism is to be solved to calculate the leg leng

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Fig. 1 Schematic diagram of the system

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Fig. 2 Block diagram of the controlled system

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(L1 , L2 and L3!, given the position~Z! and orientation of theplatform ~rotations around X and Y axis; namelya and b!. Thebase coordinate system~BCS; represented by XYZ! and the plat-form coordinate system~PCS; represented byxyz! have their ori-gin at the center of the fixed and moving plates, respectively~seeFig. 1!. The position of the origin of the PCS with respect to thBCS is defined as@px ,py ,pz#

T. In general, the coordinates of thupper platform’s leg attachment points in the BCS can be obtai@7# from the following equation:

F Xpi

Ypi

Zpi

1G5 T

BCS

PCSF xpi

ypi

zpi

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where TBCS

PCSis the homogeneous transformation of the form:

TBCS

PCS

5F cosb sina sinb cosa sinb Px

0 cosa 2sina Py

2sinb sina cosb cosa cosb Pz

0 0 0 1

G . (2)

Fig. 3 Parallel structure fixture in a drilling operation

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Therefore, the leg lengths can be obtained from:

Li5A~Xpi2XBi!21~Ypi2YBi!

21~Zpi2ZBi!2 ~ i 51,3!

(3)

where XBi , YBi and ZBi are the coordinates of the lower platformleg attachment points, namely B1-B3 ~see Fig. 1!.

4 Experimental ResultsIn order to study the performance of the system, a protot

was designed and built at the Engineering Research CenteReconfigurable Machining Systems~ERC/RMS! at the Universityof Michigan. This prototype is equipped with three slideactuators attached to upper and lower plates by using ball-scand universal joints, respectively. The on-board controller ofmachine is an INTEL-80486/DX50 micro-computer operatingWindows NT environment. Low level servo-control loops featuthe National Semiconductor LM628 precision motion controlchip, and the pulse-width modulated servo amplifiers from GaAll control codes are developed using the Borland C compiler

The block diagram of the controlled system is shown in Fig.As it is shown in this figure, based on desired position and oritation of the platform, an inverse kinematic model~as explained insection 3! is used to calculate the desired positions for each lDesired positions and the resulting errors are used by the secontrollers to generate the command signals to the motorsbrakes are applied to the motor shaft during machining opera~at each configuration!. Therefore, the structure is fairly rigid in itsfinal position/orientation and the surface finish of the machinparts are excellent. Furthermore, in the course of integration ofmodule and the CNC machining center, necessary compensawere made with regard to the offsets caused by positiorientation of the workpiece and the table relative to the CNmachine. Several tests@8# are carried out to investigate the accracy and repeatability of the system under different conditioCMM is used to measure the position and orientation of the pform. Figure 3 shows a snap shot picture of the machine in ding operation. Extensive experimental results revealed thatproximately60.05 degree accuracy~in both roll and pitch! can beachieved by using this prototype. Typical time histories of potion and velocity of a leg are depicted in Fig. 4. It can be app

Fig. 4 Typical time histories of position and velocity of a leg



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ciated that the servo control follows the command signals~veloc-ity of 14 cm/sec! very well, after an initial transient period~typicalof servo-controller performance!. This also demonstrates appropriate tuning of the servo-controller loop~PID controller! gainsand other pertinent parameters of the controller. It should be mtioned that generally speaking, the accuracy of the parallel sttures is very sensitive to the performance of the joints used in tstructure. This was also observed during the course of expments as the effects of any loose motion of the joints were amfied and directly affected the accuracy of the platform. Thethors tried to improve the accuracy of this device~whereverpossible in lieu of the fact that limited resources were availablebuilding the prototype! by designing and building some of thjoints ~e.g., universal joints!. Although they perform better thanthose commercially available, however their performance sneeds to be improved. And because of this reason, the ovaccuracy of this prototype is less than a typical CNC’s rotary ta~yaw and pitch motions!. However, it is important to notice thawith this platform, compound angles~roll and the other twoangles namely pitch and yaw! can be achieved which is not possible with the existing rotary tables.

5 ConclusionIn this paper, a 3 DOF parallel structure module was propos

to serve as a fixture for a family of cylinder heads. It possessesrequired amount of flexibility to accommodate changes andmodular to meet the objectives of RMSs. The module orientsworkpiece at simple and compound angles and can readily accmodate late changes in design. A working prototype of the syswas built and a real-time PC-based controller was designedimplemented. This prototype was used to machine some offeatures of several cylinder heads~i.e., milling and drilling opera-tions!. Experimental results demonstrating the performance ofsystem was reported.

AcknowledgmentsThe authors are grateful for the financial support of the N

Engineering Research Center for Reconfigurable Machining Stems throughout the project. They would also like to thankmember companies~for their support and feedback! and allgraduate/undergraduate students who participated in diffephases of this project.

References@1# Koren, Y., Heisel, U., Jovane, F., Moriwoki, T., Pritschow, G., Ulosy, A. G

and Brussel, H., 1999, ‘‘Reconfigurable Manufacturing Systems,’’ CIRP An2, pp. 1–13.

@2# Mehrabi, M. G., Ulsoy, A. G., and Koren, Y., 1998-b, ‘‘Reconfigurable Manfacturing Systems: Key to Future Manufacturing,’’Proceedings of the 1998Japan-USA Symposium on Flexible Automation, pp. 677–682, Otsu, Japan.

@3# Moon, Y., and Kota, S., 1998, ‘‘Generalized Kinematic Modeling Method FReconfigurable Machine Tools,’’ASME DETC 98, Atlanta, GA, paper No.MECH-5946.

@4# Gosselin, C. M., Sefrioui, J., and Richard, M. J., 1992, ‘‘On the Direct Kinmatics of General Spherical Three Degree-of-Freedom Parallel ManipulatoRobotics, Spatial Mechanisms and Mechanical Systems,45, pp. 7–11.

@5# Tsai, L. W., Walsh, G. C., and Stamper, R. E., 1996, ‘‘Kinematics of a NoThree DOF Translational Platform,’’Proceedings of the IEEE InternationaConference on Robotics and Automation, Vol. 4, pp. 3446–3451.

@6# Gopalakrishnan, G., and Kota, S., 1998, ‘‘A Parallely Actuated Work SuppModule for Reconfigurable Machining Systems,’’Proc. Of 1998 ASME DesignEngineering Technical Conference (DETC’98), Atlanta, Georgia.

@7# Lee, K. M., and Shah, D. K., 1988, ‘‘Kinematic Analysis of a Three-Degreof-Freedom In-Parallel Actuated Manipulator,’’ IEEE Trans. Rob. Autom4~3!, pp. 354–360.

@8# Fedewa, D., Mehrabi, M. G., Kota, S., and Gopalakrishnan, V., 2000, ‘‘ParaStructures and Their Applications in Machining Systems,’’Proceedings of theJapan-USA Symposium on Flexible Manufacturing Automation, Paper#2000JUSFA-13218, Ann Arbor, USA.



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Finite Element Simulation ofSegmented Chip Formation of Ti6Al4V

Martin Ba kere-mail: [email protected]

Joachim Roslere-mail: [email protected]

Carsten Siemerse-mail: [email protected]

Technical University of Braunschweig,Institut fur Werkstoffe, Langer Kamp 8,D-38106 Braunschweig, Germany

The mechanism behind chip segmentation of titanium alloystill unclear. Experimentally, different kinds of deformation paterns are observed. In this paper we perform a finite elemanalysis where chip segmentation is done by introducing crafrom the chip surface into the material. A fine mesh is used withe segments, allowing a high resolution of the occurring strgradients. Because of stress concentration at the crack tip, abatic shear bands can form and lead to a strongly segmentedwith pronounced deformation in the crack region. Dependingthe cracking parameters, different deformation patterns atcrack flanks are observed.@DOI: 10.1115/1.1459469#

IntroductionTitanium alloys like Ti6A14V produce segmented chips at

wide range of cutting speeds. The segmentation mechanism, hever, is still widely discussed. Two different mechanisms are pposed, namely adiabatic shear band formation@1–3# and thegrowth of cracks from the outer surface of the chip@4,5#.

Obikawa and Usui@4# have performed a finite element analysof segmented chip formation by implementing crack growth inthe chip. They found that, by using a fracture strain criteriochips can be formed in the simulation that agree well with thoproduced experimentally. Their study concentrated on the anaof the fully formed chip and used a rather coarse mesh~about15–20 elements! for each chip segment.

A close investigation of experimentally produced chips~seeFig. 8 below! shows different deformation patterns on the segmedge: In some cases, there is a clearly visible shear band onedge, in others, no deformation is visible and the shear bandswhere two segments meet. Any proposed segmentation menism must therefore be able to explain the occurrence of thdifferent deformation patterns.

In this work we focus on the effects that a growing crack hasthe plastic deformation of the material. In order to resolve the hgradients, a high mesh density is needed within the segments.allows to study the details of the deformation concentration alothe growing crack. If chip segmentation is governed by cragrowing into the chip, it should be possible to produce strondiffering deformation patterns by varying the crack propagatparameters. To study this question, the propagation parameterprescribed in the simulations described here and not basedphysical criterion.

Contributed by the Manufacturing Engineering Division for publication in tJOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript receivedAugust 2001. Associate Editor: M. Elbestawi.

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consists of three parts: the tool, which is modelled as a risurface, the workpiece material and the chip material. Workpiand chip material are joined along a line by tying the nodes online together and separating the nodes when a critical distancthe tool tip is reached@6#. The simulated cutting speed was chosin the high-speed regime as 10 m/s, because experimentsperformed at this speed@7#. The rake angle was 10 deg, the udeformed cutting depth 0.08 mm. The model consists of 1first-order elements with a strong mesh refinement in the shzone, see Fig. 1. The refinement was done using a linear consfor the nodes of finer elements meeting midside points on coaones@8#.

The tool consists of two separate rigid surfaces with a smradius at the rake face in order to ease guidance of the chip noThus it is ensured that nodes of the upper and the lower part omaterial do not move in the wrong direction which would lead

Fig. 1 The finite element model with a part of the chip alreadycut. A strong refinement leads to a high mesh density in theshear zone.

Fig. 2 Crack lines of the model for simulating segmentedchips. „a… Crack line for the first crack. „b… Crack line for thesecond crack, introduced after the first segment has formed.The second crack begins to grow as soon as the line reachesthe shear zone.

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strongly deformed elements and penetration of the tool intomaterial. Coulombian friction between tool and chip and betwetool and workpiece was assumed with a friction coefficient of 0which is reasonable at the high cutting speeds considered@7#.However, the heat generation caused by friction was neglectethis simulation, because due to the low thermal conductivitytitanium alloys, it will not influence the deformation caused by tadvancing crack.

The main aim of this study is not to model the details of tcrack propagation, but rather to understand the effects of a gring crack on the plastic deformation of the chip. Thereforecrack line was not calculated using a crack criterion like thproposed in@4#, which is not available at the high strain rates~ofthe order of 105/s! of interest here.~The cracking criterion givenin @4# fails for plastic deformation rates of this magnitude.!

Instead, the crack line was introduced so that the resulting csegment has a length comparable to that found experimentalis clear that this approach is not entirely satisfactory. A simulatperformed with such a preformed crack cannot answer the qtion of whether cracking is responsible for segmented chip formtion. It can, however, help to understand the mechanism of pladeformation in the vicinity of a formed crack.

The simulation proceeded as follows: In a first step, the marial was cut until the state shown in Fig. 1 and magnified in Fig.~a!, was reached. Then a remeshing was done, introducingcrack line shown in the figure. The nodes along this line were ssuccessively and the crack propagated, while the tool was cufurther into the material. The next node pair of the crack openwhen the simulation time was larger than the actual crack lendivided by the desired propagation speed so that the crack swas approximately held constant. The crack was stopped areaching the position shown in Fig. 2~b!, when a new remeshingwas done and a new crack line introduced. Two points are tonoted about the mesh shown in the figure: near the tool cutedge two elements have seemingly ‘‘hanging’’ nodes on the mline. As in the refinement region, these nodes are fixed witlinear constraint. They were introduced in order to keep the nuber of elements in the vertical direction a power of two, so thsuccessive mesh refinement as seen in Fig. 1 was possiblerather coarse size and elongated shape of the elements neabeginning of the crack is not of concern, as subsequent pladeformation will be small there. Plastic deformation indeed cocentrates near the second crack and is also smaller near theside of the chip as can be seen from Fig. 4.

The second crack began propagation as soon as its line reathe shear region. Details on the implementation can be foin @9#.

The material data for the alloy Ti6A14V were taken froplastic flow curves described in@10#. As reliable data for thestrain rate dependence of the flow curves are not yet availablestrain rate dependence has not been taken into account insimulation.

Calculations were performed using a fully thermomechanicacoupled implicit finite element code@8#. The time steps usedranged between 10211 and 1027 s, depending on the convergencof the simulation. They were chosen automatically by the usoftware. Overall, 35 timesteps were needed to form the first sment~simulation time 2.5•1026 s! and 208 timesteps for advancing the tool to the position of the second crack and the formingthis crack~simulation time 1025 s!.

The technique described here is very similar to that used in@4#.There are two main differences: Obikawa and Usui used a crcriterion based on deformation measurements, focusing onquestion of how segmented chips would form. In this work, tsegmentation is introduced a priori into the model. On the othand, the mesh density used here is much higher, thereby allogood resolution of occurring gradients in the fields and thumore detailed study of the deformation pattern.



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ResultsFigure 3 shows the equivalent plastic strain during growth

the first crack, where the crack speed was chosen as 100 m/sdeformation is strongly concentrated at the crack tip and along

Fig. 3 Equivalent plastic strain after first crack begins to grow.There is a strong concentration of deformation along the cracksides and in front of the crack.

Fig. 4 Equivalent plastic strain for two later timesteps of thesimulation. Top: After formation of the first crack. Bottom: Afterformation of the second crack. Deformation is strongly concen-trated in the crack region.



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Fig. 5 Equivalent plastic strain for a simulation with differentcrack configuration than that of the previous figures: The cracklength is only one third of the chip thickness; the crack propa-gation speed was again chosen at 100 m Õs. There is strongshear deformation along the upper crack flank.

Fig. 6 Temperature field †K‡ for the same timestep as in Fig. 5.Rapid plastic deformation occurs in front of the crack, leadingto increasing temperature.

Fig. 7 Equivalent plastic strain for a simulation with samecrack length as in Fig. 5; the crack propagation speed was re-duced to 20 m Õs. The shear deformation along the upper crackflank is even stronger, agreeing well with the experimentallyproduced chip in Fig. 8.

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crack sides. Plastic deformation is also concentrated in the szone in front of the crack tip, which therefore seems to initiaadiabatic shearing.

Figure 4 shows the equivalent plastic strain for the furthsimulation. The effect of deformation concentration is pronounalso during the growth of the second crack.

As the crack propagation parameters~speed and crack line! arechosen arbitrarily, it is important to check the influence of theparameters. Therefore a simulation was performed wheremaximum length of the crack was only one third of the chthickness while the crack speed was the same as before.

Figure 5 shows the plastic deformation for this case. The strdeformation in the crack and shear region is again very pnounced and the equivalent plastic strain on the crack flankeven higher. A plot of the temperature field~Fig. 6! shows thatrapid plastic deformation is concentrated in this zone.

If the crack propagation speed is reduced to 20 m/s~Fig. 7!,plastic deformation on the upper side of the crack is even mpronounced. This is easily understood, as a high deformationnear the crack tip will lead to stronger deformation if the crackmoves more slowly.

Comparing the calculated chips to experimentally produones~Fig. 8, orthogonal cutting conditions, cutting speed 20 mcutting depth 0.085 mm, rake angle 0 deg, experimental sedescribed in@7#!, it can be seen that there are differences inoverall deformation of the chip, which is smaller in the expementally produced chip than in the simulation. These differenare probably due to the uncertainty of the plastic flow curvesthe simulation. However, the deformation pattern near the smentation region is very similar, with a strong concentrationplastic deformation which is much stronger on the upper sidethe crack and increases along the crack~Fig. 7!.

DiscussionThe main result of this study is that even short cracks grow

into a chip can serve as a very strong means of deformation

Fig. 8 Chip segments produced under orthogonal cutting con-ditions at cutting speed 20 m Õs, cutting depth 0.085 mm, rakeangle 0 deg. In the upper figure, strong plastic deformation onthe shear edge is visible, in the lower figure, a crack with novisible deformation on the flank occurs.

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centration and can thus initiate an adiabatic shear band. Thetic deformation near the crack line is several times higher ththat in the interior region of a chip segment. Obikawa and Ustated that thermal softening seems to exert little influence onflow stress in the shear zone. Due to the higher mesh densityhere the effect of stress concentration is more clearly seen anadiabatic shear localization is stronger.

It is also found that varying the crack propagation paramehas some influence on the deformation pattern. The plastic demation in the region directly above the crack strongly variespending on the propagation parameters. Thus the experimenobserved variation of the plastic flow lines may be explained bdifference in the crack propagation parameters, which probadepend on the microstructure of the material and are theredifficult to simulate realistically.

Because strain rate hardening has not been taken into acchere, the strength of the adiabatic shear localization is oestimated in this simulation. However, it is well-known from dformation experiments that titanium alloys strongly tend to adbatic shear band formation, so the rate dependence willdestroy this effect completely. In addition, the used elementsside length of about 3mm effectively perform temperature aveaging on this length scale and are not small enough to fullysolve the shear localization. Crack induced adiabatic shear loization is therefore a plausible chip formation mechanisespecially since the deformation pattern produced in the simtion closely resembles experimentally produced chips at highting speeds. However, it cannot be concluded from the presimulation that cracking really is responsible for adiabatic cformation. If even a small crack occurs, it will trigger adiabashearing, but it may be possible that even without a crack shlocalization occurs. It is therefore important to adress the queswhether a finite element model without introduction of cracks calso form segmented chips. Such a simulation of chip segmetion by adiabatic shear is presented in@11#.

The results of this study suggest that a small crack growingthe chip can induce an adiabatic shear band. Thus it is possthat the question of segmented chip formation in titanium allowill not be decided in favor of either pure crack growth or pushear localization but that both effects contribute.

AcknowledgmentThanks to Arnold Gente for many helpful discussions. Financ

support by the Deutsche Forschungsgemeinschaft is grateacknowledged.

References@1# Hou, Z. B., and Komanduri, R., 1997, ‘‘Modeling of Thermomechanical Sh

Instability in Machining,’’ Int. J. Mech. Sci.,39, p. 1273.@2# Mercier, S., and Molinari, A., 1998, ‘‘Steady-State Shear Band Propaga

under Dynamic Conditions,’’ J. Mech. Phys. Solids,46, p. 1463.@3# Walter, J. W., 1992, ‘‘Numerical Experiments in Adiabatic Shear Band Form

tion in One Dimension,’’ Int. J. of Plasticity,8, p. 657.@4# Obikawa, T., and Usui, E., 1996, ‘‘Computational Machining of Titaniu

Alloy—Finite Element Modeling and a Few Results,’’ J. of ManufacturinScience and Engineering,118, p. 208.

@5# Vyas, A., and Shaw, M. C., 1999, ‘‘Mechanics of Saw-Tooth Chip Formatin Metal Cutting,’’ J. of Manufacturing Science and Engineering,121, p. 165.

@6# van Luttervelt, C. A., Childs, T. H. C., Jawahir, I. S., Klocke, F., and Venunod, P. K., 1998, ‘‘Present Situation and Future Trends in Modelling of Mchining Operations,’’ CIRP Ann.,47, p. 587.

@7# Hoffmeister, H.-W., Gente, A., and Weber, T., 1999, ‘‘Chip Formation at Tinium Alloys under Cutting speeds of up to 100 m/s.’’ H. Schulz et al., eProceedings of the 2nd International German and French Conference on HSpeed Machining, Darmstadt.

@8# HKS Inc., 1998, ABAQUS/Standard User’s Manual, Version 5.8, USA.@9# Baker, M., and Siemers, C., 1999, ‘‘Simulation der Lamellenspanbildung

ABAQUS/Standard,’’ABAQUS Anwendertreffen, Essen.@10# Klimanek, P., Cyrener, K., and Jenkner, K., 2000, H.-K. To¨nshoff, F. Holl-

mann, ed.,Spanen metallischer Werkstoffe bei hohen Geschwindigkei,Bonn, ISBN 3-00-006320-X.

@11# Baker, M., Rosler, J., and Siemers, C., 2000, ‘‘High Speed Chip FormationTi6Al4V,’’ Proceedings of Materials Week 2000, Munich, Germany.



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