oni

o

Downlodetermining a feed rate that varies along the path, taking advan-tage of the shape of the path, the engagement of the tool, and thecapabilities of the cutting machine used. This procedure consistsof solving a trajectory optimization problem of a type arising inflight control. The algorithms use a smooth parametric represen-tation of the path and various machining constraints. This proce-dure can be used to enhance machine performance with any toolpath.

As verified by computational and aluminum-cutting experi-ments, this tool-path method saves up to 30% machining timecompared to conventional methods. It also greatly extends toollife when machining hard metals, as we saw in a titanium-cuttingexperiment. For the future, we expect considerable reductions of

parallel-offset paths! or consisted of many straight-line segments~zig-zag or raster scan!. Held @8# provides a reader many refer-ences to the generation of such tool paths as well as a good start-ing reference on many mathematical aspects of pocket machining.Aspects of parallel-offset and zig-zag paths also can be found in@915# and in the many citations in Dragomatz and Mann @16#.Parallel-offset paths are widely used in commercial CAM soft-ware packages ~cf. @17,18#!.

In a high-speed machining setting, by contrast to conventionalmachining, the best path must account for machine dynamics. It isa smoother, generally longer path, and away from the pocketboundary it sometimes resembles the part shape only vaguely.Such a path turns out to be very advantageous in many conven-tional machining situations as well. The advantages increase withthe ratio of the machine stopping distancethe distance requiredto stop a toolto the pocket length, as we have observed in simu-lations and experiments. From simple dynamics equations, the

*Corresponding author.Contributed by the Manufacturing Engineering Division for publication in the

JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript receivedApril 2003. Associate Editor: M. Davies.

Journal of Manufacturing Science and Engineering NOVEMBER 2003, Vol. 125 709Copyright 2003 by ASMEMichael B. Bieterman*e-mail: michael.bieterman@boeing.com

Donald R. Sandstrome-mail: don.sandstrom@mailaps.org

Mathematics & Computing Technology,The Boeing Company,

Seattle, WA 98124-2207

A Curvilifor PockeA novel curvilinear toets. The method usesvalue problem definedsmooth low-curvatureboundary. This morphmetals and of machinivariable feed-rate optpath, tool-engagementperformance for any t

1 IntroductionCan mathematical methods used in the design and control of

aerospace vehicles also be applied to the manufacture of aero-space parts? At The Boeing Company, as described in this article,the answer is yes. We describe a method to improve the generationof tool paths for use in pocket machining.

In pocket machining, material is removed from stock, layer bylayer, until pockets are formed and a manufactured part emerges.The tool path for a layer of a pocket is the centerline path alongwhich a toolan endmillis fed as its rotating teeth cut the ma-terial. Most machining of aerospace parts can be thought of aspocket machining. This is not as surprising as it might seem atfirst: An aerospace vehicle needs to be both strong and light.Pocket machining, with the support structure it leaves betweenpockets, helps achieve these properties.

In the aerospace industry ~and elsewhere!, tool-path generationis a part of NC ~numerical control! programming, in which aperson provides a part description and feed-rate requirements to acomputer aided manufacturing software package. The CAM pack-age then produces a file with path and feed-rate data that arepostprocessed into a form a controller can use to drive a machinetool. Conventionally, the data are specified pointwise. More andmore commonly, however, splines are used for the smooth repre-sentation and compression of data that vary along a tool path.

Our objective was to improve upon what CAM packages cur-rently deliver by developing a fairly general tool-path generationmethod that reduces the time required to machine a pocket. Themain part of the method, as described below, uses geometry inputand a partial differential equation ~PDE! that arises in many engi-neering design applications to generate a path. The path genera-tion is the main focus of this paper.

The second ~supporting! part of the method is a procedure foraded 19 Jan 2009 to 220.227.156.156. Redistribution subject to ASnear Tool-Path Methodt Machining

l-path generation method is described for planar milling of pock-the solution of an elliptic partial differential equation boundaryon a pocket region. This mathematical function helps morph a

spiral path in a pocket interior to one that conforms to the pocketing leads to substantial reductions of tool wear in cutting hardg time in cutting all metals, as experiments described here show. A

mization procedure is also described. This procedure incorporates, and machine constraints and can be applied to maximize machineol path. @DOI: 10.1115/1.1596579#

machine spindle wear and tear over time as paths generated by thenew method are used. Clearly, there are potential economic ben-efits in such results.

In this paper, we describe the tool-path method and experimen-tal results for 3-axis machining of pockets that are convex ~or nottoo nonconvex!, that are free of islands or pad-up regions, and thatare machined by removing constant-depth layers ~planar, or so-called 2 1/2 D milling!. The vast majority of pockets of interest atBoeing have these properties, and this is the target class of pock-ets for the method. We briefly point out how the shape of a pocketaffects the methods applicability. We also briefly indicate how themethod could be modified and applied when some of these pocketrestrictions are removed.

The present tool-path method was first described in the techni-cal reports @1,2# and later in @3#. Many details and results of themethod not described here can be found in these references, in@46#, and in Zelinski @7#.

2 Defining the ProblemOur investigations were performed in the context of a high-

speed machining project. Such a setting involves high spindlespeeds for rotating tools and high axis-drive velocity andacceleration/deceleration capabilities for feeding tools. How highis high-speed machining? A relevant answer for machining is a lotlike that for high-speed computing: high enough to change under-lying assumptions and paradigms.

One of the pocket-machining paradigms changed by high-speedmachining is that machine dynamics can be ignored when a toolpath is generated. Conventionally, dynamics have been prettymuch ignored; the best path was the shortest, and it either re-flected the part ~pocket boundary! shape in its entirety ~so-calledME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

Downlomachine stopping distance D5V2/2A , where V and A are themaximum ~axis-drive! velocity and acceleration/deceleration ofthe machine, respectively.

Figure 1 shows a conventional parallel-offset tool path used tomachine a rounded-cornered pentagonal pocket. In generatingsuch paths, the pocket boundary ~dashed line in the figure! isoffset inward a fixed distance, and the process is then repeated.The pocket is typically machined outward. Starting at the pocketcenter, the tool moves sequentially along each path orbit about thecenter and then goes on to the next orbit.

The main problem with this and related conventional tool pathsis that they inherit any corners or regions of high local curvaturein the part. The result is a limit on how fast a tool can be fed.Coming into a corner, the tool has to decelerate to rest, or nearrest, only to accelerate as it comes out of the corner. Why limit thefeed rate of a high-speed machine in this manner?

The path corners can be rounded slightly, which helps a bit.However, all of a tools acceleration capability would still be con-centrated in tight-radius corners, rather than being distributedalong the path where it could decrease total machining and pro-gramming time.

Cornering also greatly contributes to tool wear when hard met-als are cut. It is not known exactly why this is the case. Somemechanical wear occurs when a tool hits a corner. Perhaps moreimportantly, however, the increase in temperature that occurs incorners as the tools engagement with the stock suddenly in-creases could contribute significantly to tool wear.

It is believed that sudden changes in tool engagement due tocornering also can increase machine spindle wear over time.Many productivity losses resulting from the use of conservativefeeds and speeds to avoid such wear are probably due in part tothe presence of corners in tool paths.

Practicing machinists and NC programmers know what to do tohelp solve these problems: Introduce sweeping curves over muchof the path. What evidently was not known before in the NCprogramming and CAM communities was exactly how to do thisautomatically for fairly general pockets in a way that requiresmuch less machining time.

3 Solution: Slow Morphing into Part ShapeOne approach would be to pose and attempt a formal solution

to an optimization problem. This sounds good, but the tool-pathmachining time minimization problem with path, feed-rate, accel-eration, and other types of machine and geometry constraints ismuch harder for our target class of pockets than even the travelingsalesman problem, whose number of possible paths increases rap-idly with the number of locations to be visited.

Wang and Stori @19# recently described an interesting local it-erative refinement approach for improving existing parallel-offsettool paths in pocket machining. This method has yielded smoothpaths in formally seeking to keep tool engagement close to an a

Fig. 1 Conventional pocket-machining tool path

710 Vol. 125, NOVEMBER 2003aded 19 Jan 2009 to 220.227.156.156. Redistribution subject to ASpriori given value while trying to minimize a measure of pathcurvature. ~See Stori and Wright @20# for an earlier description ofconstant engagement tool-path generation.!

Our approach is to obtain a very good curvilinear tool path fora pocket, a tool, and all machines by spiraling between contoursof a well-chosen scalar mathematical function on the pocket. Byspiraling in this manner, the path is morphed from a very smoothshape in the pocket center to the shape of the part on the pocketboundary.

Figure 2 shows a curvilinear tool path generated with thismethod for the pentagonal pocket of Fig. 1. Unlike the conven-tional path pictured in Fig. 1, the spiral curvilinear tool path haslow, nearly constant curvature near the pocket center and slowlychanges into the part shape as it gets closer to the part boundary.

Figure 3 is a recent photograph of an aluminum test part withfour pockets ~and with coolant and chips strewn about! beingmachined with the curvilinear tool-path method. To some extent,in the slight impressions left on the pocket floors, the figure showsthe similar morphing of the spiral tool paths. In these cutting tests,the nearly constant hum heard while machining the interior ofeach pocket layer was the most convincing evidence for manypracticing NC programmers and machinists of the quality of thecurvilinear tool-path generation method.

Solving an Elliptic PDE Boundary Value Problem. Themathematical function used to morph the curvilinear path in thepresent approach is the approximate solution of a scalar ellipticsecond order partial differential equation ~PDE! boundary valueproblem that appears in many engineering design analyses ~cf.@2126#!. The idea of using such a function was not derived di-rectly from any machining model. Rather, it came from our expe-rience with other physical applications and the knowledge ofmathematical properties of solutions for these.

Fig. 2 Curvilinear tool path for pocket machining

Fig. 3 Recent curvilinear tool-path test

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well ~e.g., replacing lu by a positive constant in Eq. ~1! yields avery similar solution-cf. @4#!. We chose the eigenvalue problem

DownloThe PDE problem is defined on a two-dimensional pocket re-gion whose boundary is offset inward, by one tool radius, from thepart. We solve the eigenvalue problem approximately for the La-placian

22u5lu , (1)subject to Dirichlet boundary conditions u50 and the normaliza-tion max(x,y)u(x,y)51 for the principal eigenfunction u ~corre-sponding to the smallest of the positive eigenvalues $l j% j51,).We use a PDE like Eq. ~1! because this kind of equation hasproperties that can be exploited to generate nearly optimal curvi-linear spiral paths.

Elliptic PDE operatorsthose PDE operators whose order iseven and whose even-ordered terms have the same signhavenice smoothing and positivity properties, which are just what weneeded in our tool-path generation problem. The constant-valuecontoursor level setsof the principal eigenfunction u haveprogressively less local maximum curvature as the constant variesfrom zero to one ~i.e., as the distance inward from the boundaryincreases!. The positivity, or maximum principal, for the PDEenables one to generate a tool path as we dothat is, by begin-ning at the unique location where the scalar-valued function utakes its maximum value and then continuously spiraling outwarduntil u takes its minimum value of zero on the boundary of theregion. The tool-path methods applicability depends on there be-ing only one location where the PDE solution has a relative maxi-mum, which is a property of convex and a great many nonconvexpockets. Some of the theoretical PDE properties we utilized aredescribed in Courant and Hilbert @21#.

Except for a handful of simple pocket shapes, one does nothave available the analytical solution of the PDE, and one mustnumerically ~approximately! solve the problem. This entails gen-erating a discretization of the pocket region into a great manytriangles, reducing a system of approximate local PDEs to a largesparse system of linear equations via the finite element method,and solving this system ~cf. @2325#!. For the tool paths describedin this paper, all PDEs were approximately solved with the finiteelement method implemented in the MATLAB PDE Toolbox soft-ware package @25#. A typical solution using 5,00015,000 trianglegrid points requires on the order of a minute of workstation com-puting time.

Figure 4 shows the PDE solution contours that were used toguide the generation of a curvilinear tool path for a test pocket.The pocket boundary is shown by the dashed line. Note how theshape of the contours changes from that of a smooth three-lobedcurve in the center of the pocket to that of the boundary as thesolution value decreases from one to zero.

Our initial choice of solving an eigenvalue problem ~with to-be-determined ~constant! eigenvalue l in Eq. ~1! was not critical.Other elliptic PDEs with appropriate positivity would work as

Fig. 4 Contours used to guide curvilinear tool path for apocket

Journal of Manufacturing Science and Engineeringaded 19 Jan 2009 to 220.227.156.156. Redistribution subject to ASpartly because of physical wave-guide analogies that came tomind ~cf. @26#!, and also because of the possibility that our resultscould be extended to much more complex parts. For such parts, itprobably would be preferable to decompose the pockets intosmaller pieces rather than to use a single spiral path that repeat-edly traversed some parts of a pocket. One way to decompose apocket automatically would be to use nodal lines (u50 contours!of one of the higher-order Laplacian eigenfunctions ~cf. @1,4,21#!.

A simple modification of Eq. ~1! and the boundary conditionsalso could be used to generate contours guiding a curvilinear toolpath in a case where a pocket included a single island or pad-upregion, or when each part of a decomposed pocket with manyislands contained just one island. Here, Eq. ~1! could be solvedwith a constant right hand side, and with the boundary conditionsu50 on the pocket boundary, as before, and u51 on the closedcurve offset from the island into the pocket by one tool radius. Forsome pockets, including those with very high aspect ratios, therecould be an advantage in creating a thin virtual island in the centerof a pocket and then applying this modification.

Spiraling Between PDE Solution Contours. The spiralingthat takes place is between appropriately spaced contours of u andcontinues, orbit by orbit, until all material is removed. The spac-ing is determined by a user-specified maximum stepover, or widthof cut. This process is carried out as follows.

We parameterize the solution contours and the discrete tool pathby the angle with the horizontal of the line connecting any pathpoint with the center of the pocket, which is taken to be the pointwhere u51. ~Typically, on the order of 180 points are used torepresent each orbit of the discrete tool path.! This winding angleparameterization limits somewhat the shape of the pockets thatcan be treated, as any ray from the center may only intersect thepocket region boundary once. ~Alternative parameterizations ofthe solution contours could somewhat extend the range of pocketsamenable by the present tool-path method.! The initial angle istaken to be that corresponding to a direction in which u varies theleast. Each orbit, starting with this angle, consists of a 360 spiral~linearly interpolating with the winding angle! between two con-tours corresponding to u5C1 ~at the point where the previousorbit ended! and u5C12DC ~where the present orbit ends!. Thevalue DC is determined iteratively so that the maximum value ofthe width of cut during the orbit is about equal to the user-specified value.

As described to this point, the orbit-by-orbit spiraling algorithmcould be improved because no look-ahead is employed to opti-mize paths, and some wasteful orbits could occur ~e.g., at thepocket boundary!. To eliminate such effects and improve the al-gorithm, we generate many candidate curvilinear tool paths ~e.g.,10! for a given pocket, tool diameter, and specified maximumwidth of cut. To generate each candidate path, we take a valuea

arclength-like spline parameter s in this representation increasesmonotonically from a value of zero at the beginning of the path to

Downlogions of high curvature into the interior of the tool path somewhat,as do conventional parallel-offset methods more dramatically.Propagating corners in this manner is usually undesirable, but thisequidistance property could be exploited, either throughout apocket or for a chosen sector of it, if one wanted to combine thecurvilinear path technology with a parallel-offset approach.

Figure 5 shows a curvilinear path corresponding to the pocketand contours pictured in Fig. 4. Some of the bunching of the pathorbits at the two concave pocket corners could be alleviated withadditional path smoothing, but not without further propagating theshape of the pocket corners into the interior of the path.

Path Representation. Discrete curvilinear paths can lead tomuch bigger machine controller input files than with conventionalpaths, since the latter often consist of long straight-line andcircular-arc section movements. Nevertheless, we have success-fully cut metal via curvilinear paths by specifying point-to-pointmovements and variable feed rates with more than a thousandgoto points per pocket layer. Fewer path points can be used toreduce file sizes, but the best way to compress such data tenfold ormore is via splines. The discrete tool path can be replaced with atwice continuously differentiable parametric spline curve that is fitto the discrete path, as was described in @2#. We have used suchspline representations of path and feed-rate data to cut metal.

4 Variable Feed-Rate OptimizationThe parametric spline path representation is also used in a feed-

rate optimization method we have developed and applied. Thiswas first described in @2#, where the many details and results notdescribed here can be found. The method uses various machiningconstraints to set a variable feed rate along the path. Many aspectsof this method are similar to the more recently developed methodsdescribed in @2729#. Each of these descriptions offers someunique contributions and insights and the interested reader mightwish to see all. Variable feed-rate optimization was used in theresults we describe in this paper.

Use in Present Tool-Path Comparisons. In the machiningtime comparisons presented, the variable feed rate was chosen tominimize machining time subject to limits on maximum compo-nent axis-drive velocity and acceleration/deceleration, V and A,respectively. Machining times from computer simulations usingthese variable feed rates are very close ~,1%! to those one wouldobserve in cutting tests using a single specified maximum feedrate and a controller with a good look-ahead capability.

In the tool-wear experiments described here, a variable feedrate was set to keep material removal rate approximately constant,subject to a limit on feed rate to keep the chip load under control.

Feed-Rate Optimization Formulation. As was done in Betts@30#, the discrete tool path is replaced with a twice continuouslydifferentiable parametric spline curve that is fit to this path. The

Fig. 5 Curvilinear tool path for a pocket

712 Vol. 125, NOVEMBER 2003aded 19 Jan 2009 to 220.227.156.156. Redistribution subject to ASone at the end. Letting s depend ~monotonically! on time t, wehave the vectors of position r, velocity r, and acceleration r de-fined as functions of s or t via

r5$x~s~ t !!,y~s~ t !!%; r5rsv; r5rssv21rsa; (2)where nonnegative v5 s having units of sec21 is the arc speed anda5 s having units of sec22 is the arc acceleration ~which can bepositive or negative!. The arc speed v vanishes at s50 and s51. Now, following @2#, since s depends monotonically on time t,we can express t as a function of s via

t~s !5E0

s

~1/v~s!!ds . (3)

and use s as the independent variable for all functions in theoptimization formulation.

The optimization problem is to maximize v subject to theboundary conditions on v at s50 and s51, the ordinary differ-ential equation

~v2!s52a (4)describing the dynamics, the acceleration constraint

2A

where F is the feed rate limit in constraint 6 and the subscript idenotes the value at s . The stages in the algorithm are the fol-

Table 1 Maximum axis-drive velocity V & acceldecel Aused in the computational experiment. The machine stopping

Downloilowing.

1. v i is set to Fi ~its limit! for i52, . . . ,n21.2. Moving left to right ~for increasing i!, if a i computed in Eq.

~7! results in a violation of one of the acceleration compo-nent bounds 5, a i is adjusted and v i11 is computed againfrom Eq. ~7! unless this new value would exceed Fi11 . Atthe end of this stage each Fi is reset as min$Fi ,vi%.

3. Next, the same algorithm as in the stage 2 immediatelyabove is applied from right to left ~decreasing i!, where v iand not v i11 is possibly adjusted for each i.

4. Lastly, nodal arc acceleration values $ai% are obtained byaveraging and further smoothing the midpoint values $a i%,new midpoint accelerations are interpolated from the nodalvalues, and Equation ~4! is used with the new midpoint val-ues to reset $v i%.

The final step where local acceleration values are averaged andsmoothed results in some increase in machining time, but it isimportant in avoiding discontinuous acceleration specifications,which would usually occur otherwise ~cf. Smith et al. @29#!.

5 Comparative Tool-Path ResultsWe now describe results of experiments that quantify the ben-

efits of reducing machining time and tool wear with curvilineartool paths.

Reduction of Machining Time. Five rounded-cornered po-lygonal pockets from a Boeing aircraft production part and twotool sizes were considered in computational experiments. The ex-periments were designed to assess machining-time reductionswith the curvilinear tool-path method for a wide variety of cuttingmachines. For each pocket and tool, two paths ~of comparablelength! were generated: One was a conventional path consisting ofstraight-line and circular-arc segments; the other was a curvilinearpath generated with the present method. The internal corners ofthe conventional path were rounded so that the tool was not forcedto stop at each ~but slowed down naturally because of centripetalacceleration considerations to a feed rate of AAR , where A is theaxis-drive acceleration limit and R is the local path corner radius,which was 0.229 cm ~0.09 in! for all the conventional paths!. Themaximum width of cut for both paths was slightly less than thediameter of the tool minus twice the tool corner radius. With theconventional paths, this width of cut occured at the internalrounded corners.

Performance on many types of machines was simulated by car-rying out the computational experiments with 14 different combi-nations of limits on machine axis-drive velocity V andacceleration/deceleration A. These 14 cases are summarized inTable 1. The key machine parameter is the machine stopping dis-tance D. As was indicated in the machine-stopping-distance scal-ing observations in the previous section and as will be shownshortly by example, the relative benefit of using curvilinear pathsdepends on V and A only through D. The lowest case numbers inthe table have parameters that correspond to slower, more conven-tional machines. The higher case numbers correspond to eitherhigh-speed machines or heavy conventional machines with verylow axis-drive acceleration limits.

Detailed results are shown for one pocket and one tool; excerptsof other results are summarized. The pocket was approximately 14cm by 15 cm in size. The tool had diameter 1.27 cm ~0.5 in! andcorner radius 0.076 cm ~0.03 in!. For each pair of V and A limits,machining times were calculated for both the conventional andcurvilinear paths via the variable feed-rate optimization procedureoutlined earlier.

Figures 6 and 7 show the conventional and curvilinear toolpaths used in the experiment. The curvilinear path was about 260cm in length and 2% shorter than the conventional path. Like the

Journal of Manufacturing Science and Engineeringaded 19 Jan 2009 to 220.227.156.156. Redistribution subject to AScurvilinear path, the conventional path is itself a spiral path withrelatively smooth orbit-to-orbit transition. In both figures, thedashed line represents the actual pocket boundary.

Figure 8 shows the relative machining time savings of the cur-vilinear tool path of Fig. 7 compared to the path of Fig. 6 for the14 cases of Table 1. One can see in the figure for machine stop-ping distance D of about 2.3 cm that three data points are plottedpractically on top of one another. These points correspond toTable 1 Cases 57, which have disparate V and A values, andsupport the contention that the relative time savings depend on Vand A only through D. Figure 8 shows savings of about 15%29%, with larger savings for larger values of machining parameterD. This trend is typical; as D increases, so does the benefit ofproperly managing tool-path curvature as in the present method.

distance D is the distance needed to stop when going topspeed.

Case V ~in/min! ~cm/s! A ~g!D5

V2/2A ~cm!

14 2000 84.67 0.50 7.3213 800 33.87 0.10 5.8412 500 21.17 0.05 4.5711 1200 50.80 0.30 4.3910 2000 84.67 1.00 3.669 1000 42.33 0.30 3.058 1200 50.80 0.50 2.647 320 13.55 0.04 2.346 500 21.17 0.10 2.295 800 33.87 0.26 2.254 1000 42.33 0.50 1.833 300 12.70 0.05 1.632 500 21.17 0.20 1.141 300 12.70 0.10 0.81

Fig. 6 Conventional tool path for a pocket

Fig. 7 Curvilinear tool path for a pocket

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other was a conventional path with rounded internal corners~0.76-cm ~0.3-in! radius! like the path pictured in Fig. 6, except

DownloThese comparisons depend on the amount of internal path cor-ner rounding that was used in the conventional path. Had no suchcorner rounding been employed, then the savings of about 15.5%and 28.5% in Cases 1 and 14, respectively, would have increasedto about 22% and 34%, respectively, as we computed in separatesimulations. Alternatively, with our feed-rate optimizationmethod, we estimated that the savings would decrease to about12% and 23% for Cases 1 and 14, respectively, if it were possibleto double the conventional-path corner radii. The limits on suchradii evidently depend on factors including the tool diameter andthe local part curvature ~e.g., the part corner radii!.

Table 2 summarizes the experimental setup that was used for allfive pockets and two tool sizes. The table shows the sizes andshapes of the pockets and lists the ~peak! machining time reduc-tions with the curvilinear path method for one pair of V and A~Case 14 in Table 1 with D57.32 cm). Clearly, much machiningtime can be saved for such pockets with the present method. Onecan expect a machining time reduction in the range of 10%30%,generally. Factors affecting relative performance of the curvilinearpath method include internal-path length and the length, aspectratio, and angle sizes of a pocket.

Reduction of Tool Wear. A Titanium-cutting experiment wasconducted to assess whether the present curvilinear tool pathscould substantially increase tool life. The hypothesis was that thiswould be the case, due to the smoothly varying tool engagementand elimination of most cornering.

In the experiment, a test part consisting of a 45.7 cm ~18 in! by71.1 cm ~28 in! by 10.2 cm ~4 in! thick Ti 6Al-4V Beta annealedtitanium block was drilled with a 5.08-cm ~2.0-in! entrance holeand faced off with two rectangular-pocket machining paths. Onepath was curvilinear and very similar to that pictured in Fig. 7; the

Fig. 8 Results for computational experiment

Table 2 Maximum machine-time reduction case 14 with cur-vilinear paths for 5 pockets and 2 tool diameters*

*Results for all 14 cases for this pocket and tool are shown in Fig. 8.

714 Vol. 125, NOVEMBER 2003aded 19 Jan 2009 to 220.227.156.156. Redistribution subject to ASthat more abrupt orbit-to-orbit transitions occurred in the titanium-cutting test.

In the baseline conventional-path case, a 5.08-cm-diameter~2.0-in! 5-flute indexable mill with high-grade carbide inserts wasrun at 210 RPM, 1.27-cm ~0.5-in! axial engagement, 3.30-cm~1.3-in! nominal width of cut, and 0.21-cm/s ~5.0-in/min! nominalfeed rate, resulting in a 0.012-cm ~0.0048-in! chip load. Details ofthe curvilinear-path case were identical, except that the width ofcut varied smoothly and resulted in a path about 20% longer. Thefeed rate used in this case was varied so that the material removalrate was about constant and equal to that used with the conven-tional path along its straight segments, except that the chip load inthe curvilinear case was not allowed to exceed that in the othercase by more than 50%. This resulted in a machining time slightlyhigher than in the baseline case.

The results were as follows. For the conventional path, fourtools had to be used to remove a layer from the block. Tools werechanged out as, or just before, they failed. The lifetimes of tools14 were 15 min, 22 min, 24 min, and the balance of 18 min,respectively. The fourth tool probably could have lasted another 9or 10 minutes. The earlier tools failed sooner, we believe, becausethey encountered more path corners per unit material removed asthe path spiraled out.

In the curvilinear case, three tools were used, with lifetimes of39 min and 29 min for the first and second tools, respectively, andthe balance of 12 min for the third tool, which probably couldhave lasted another 15 or 20 minutes. The first tool removed abouthalf of the material. The curvilinear run was repeated and nearlyidentical results were obtained. Tool lifetimes then were 40 min,30 min, and a balance of 10 min for tools 13, respectively. Webelieve the earlier tools lasted longer in the curvilinear case be-cause of the sweeping internal path curves that only conform to apocket boundary shape near the boundary.

These results show that tool life in cutting titanium can begreatly increased ~nearly doubled! by using the present curvilinearpaths instead of conventional paths with tight-radius corners.

6 ConclusionsA novel curvilinear tool-path generation method was described

for planar milling of pockets. The method uses the solution of anelliptic partial differential equation boundary value problem de-fined on a pocket region. This mathematical function helps morpha smooth low-curvature spiral path in a pocket interior to one thatconforms to the pocket boundary. The morphing eliminates inter-nal tight-radius corners in conventional paths and leads to majorbenefits. One benefit is machining time reduction. Computationalcomparisons showed that up to 30% machining time is saved withthe present method. Savings are largest with parameters charac-teristic of high-speed machines or of heavy conventional ma-chines with low acceleration/deceleration limits. A second benefitis reduction of tool wear in cutting hard metals. A titanium-cuttingexperiment was described that showed that tool life could benearly doubled by using the present curvilinear paths. A thirdbenefit we expect to see over time is reduced machine spindlewear and tear. This expectation is based partly on aural observa-tions made during cutting with the methods smooth tool engage-ment, as was noted briefly here.

A variable feed-rate optimization procedure was also described.This procedure incorporates path, tool-engagement, and machineconstraints and can be applied to maximize machine performancefor any tool path.

AcknowledgmentsAmong the many people who contributed to the present work

are Thomas A. Grandine and Jan H. Vandenbrande of BoeingMathematics & Computing Technology; Professor Paul K. Wright

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and his group in the Mechanical Engineering Department at theUniversity of California at Berkeley, where metal-cutting demon-stration of some of the benefits of our tool-path method was firstperformed; Keith G. MacKay of Boeing who helped define andcarrying out the machining time experiments described here; andBoeings Stanley J. Pence and David E. West, who helped defineand carry out the present titanium-cutting tests.

References@1# Bieterman, M. B., and Sandstrom, D. R., 1996, A Strategy for Prescribing

High Speed Pocket Machining Tool Paths, Technical Report SSGTECH-96-020, The Boeing Company.

@2# Bieterman, M. B., and Sandstrom, D. R., 1998, Curvilinear Spiral Tool PathTrajectories for High Speed Pocket Machining, Technical Report SSGTECH-98-031, The Boeing Company.

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