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Improving Cycle Time in Sculptured Surface Machining Through Force ModelingE. Budak

1(2), I. Lazoglu

2, B.U. Guzel

2

1Faculty of Engineering and Natural Sciences, Sabanci University, Istanbul, Turkey

2Department of Mechanical Engineering, Koc University, Istanbul, Turkey

AbstractIn this paper, an enhanced mathematical model is presented for the prediction of cutting force system in ballend milling of sculptured surfaces. This force model is also used as the basis for off-line feed rate schedulingalong the tool path in order to decrease the cycle time in sculptured surface machining. As an alternative forsetting a constant feed rate all along the tool path in rough machining of sculptured surfaces, resultant cuttingforces are aimed to be kept under a pre-set threshold value along the tool path by off-line scheduledpiecewise variable feed rates. In this paper, it is shown that machining time, depending on complexity ofsculptured surfaces, can be decreased significantly by scheduling feed rate along the tool path. The model istested under various cutting conditions and some of the results are also presented and discussed in thepaper.

Keywords:End Milling, Force, Feed rate Scheduling

1 INTRODUCTION

Sculptured surface machining is an important processcommonly used in various industries such as automotive,aerospace, and die/mold industries. Due to increasingcompetitiveness in the market, decreasing production timeand cost without sacrificing from part quality are becoming

more vital nowadays.Although recently new scientific studies on milling toolshave been carried out and focused on different aspects [1-6], most of the ball-end milling models are applicable toeither machining of simple and limited workpiecegeometries, such as 2 axis milling cases, but notrelatively complex sculptured surfaces, or can not detectthe cutter / workpiece engagement regions automaticallyand can not simulate multi-pass machining operations.

Unfortunately, there is a lack of scientific tools in theselection of appropriate feed rate values in the machiningof sculpture surfaces that depend on a reliable forcemodel. Therefore, conservative constant values of feedrates have been mostly used up to now. Currently and

commonly used CAD/CAM programs and NC codegenerators are based on only the geometric and volumetricanalyses, but not on the mechanics of the machiningprocesses, yet. The CAD/CAM programs, in whichsculpture surfaces are created, and the NC codegenerators suggest preset feed rate values for tool-material combinations from their databases for roughing,semi-finishing and finishing. These databases usuallycontain general and conservative values based on trial-error tests and experiences. Although the information canbe valuable, in order to avoid undesirable results such aschipping, excessive tool deflections and tool breakage,they can be quite conservative. As production engineers inmost of the cases in industry have no scientific tools basedon the mechanics of the sculpture surface machining

processes, they can not predict cutting forces, therefore,during the process planning stage, they have no choice butto be conservative in the selection of feed rate.

Additionally, setting a constant feed rate all along the toolpath will result in losses in productivity in sculpturedsurface machining where tool-workpiece engagementchanges continuously along the tool path. These changescreate potentials for using variable and increased feed ratevalues in certain sections of tool path. However, varying

feed rate should be in such a way that it should not affectthe resultant cutting force levels due to the reasonsexplained above. Therefore, knowledge on the cuttingforce levels, and having a reliable model of force systemare critical for the selection of varying feed rates along thetool path.

Therefore, in this paper, rather than setting the feed rate toa constant value all along the tool path, setting theresulting cutting force values along the tool path to a pre-set value is proposed. This appropriate pre-set value canbe selected by considering the desired surface accuracy,tool deflections, chipping, tool breakage, etc. This paperwill not discuss the selection process of the appropriateconstant force value, since it depends on the workpiece,tool, desired surface quality, etc. However, the paper

presents an enhanced model to predict the cutting forcevalues all along the cutter path for a given set of inputssuch as cutter geometry, workpiece materials, processparameters as explained in the following sections.Moreover, by using the force model predictions, in thispaper it is shown that feed rate values can be changedpiecewise along the tool path in sculptured surfacemachining in order to improve cycle time and productivity.

2 FORCE MODEL IN SCULPTURED SURFACEMACHINING

Cutting force system model consists of various modulessuch as cutter/workpiece intersection, kinematics/chipload, cutting force modules. The force system modelemploys a Boolean approach for given cutter, workpiecegeometry, process parameters and tool path to determineinstantaneous cutting forces.

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2.1 Cutter / Workpiece intersection module

In 3D sculpture surface machining, the cutter/workpieceengagement region does vary along the cutter path and ingeneral, unless some specific and very simple workpiecegeometry is machined, it is difficult to find an exactanalytical representation for the engagement region. Themodel detects and outputs the engagement regions usingits own self-sufficient algorithm for a given cutter path.

knxknc tt = )sin()sin()(

(2)

{ }wdwdwd YYXXWQDPZZ == ,,,|

(4)dzKdAKdFdzKdAKdF eccrecrcr +=+= ,

knkncc dztdA )()( =

(1)

3

Chip load and force calculations are based on thecutter/workpiece engagements; therefore the output of thismodule is very critical. The cutter and workpiece aremeshed into small elements whose projection into the X-Yplane is a square (Figure 1). Due to this meshing,automatic determination of the engagement regions alongthe tool path is possible. It should be noted that there is atrade-off between computation time and accuracy of thepredictions in the selection of grid sizes. Based on theinputs of the model (tool path, cutter and workpiecegeometry), the cutter is translated at the nodal points of themesh elements in the workspace according to the tool andworkpiece kinematics. Considering spindle speed and feedrate together with the other inputs, all the locations of

discrete points for each region of the ball-end mill duringthe machining process are calculated in terms ofX-Y-Zglobal coordinates (Figure 1). Thereafter, theinstantaneous workpiece heights at each of the X-Ycoordinates of the cutting region are also determined, andbased on the difference between the cutting edge andworkpiece height at these points, the algorithm determineswhich portion of the cutter is in contact with the workpiece.This can also be explained as the following; let D and Wbedefined as the cutter domain and workpiece surface at anyinstant of machining, respectively, and let P(Xd, Yd, Zd) bea discrete point on the cutting nose, Q(Xw, Yw, Zw) be anodal point on the workpiece surface. If at any time, thecondition,

is satisfied then the point P is said to be in cutting withpoint Q and considered to be in the engagement domain atthat instant.

Figure 1: Illustration of meshed workpiece and cutter.

In the next increment, the cutter tip travels one gridaccording to tool-path, and the workpiece height at thisQ(Xw, Yw, Zw) is set to the Zd on the workpiece meshmatrix. The simulation keeps and updates the workpiecegeometry after each cutting occurs.

2.2 Kinematics and chip load module

In order to determine the differential cutting forces at anycutter point in the engagement domain, the first step to betaken is to find the infinitesimal chip load for such a

discrete element. Thereafter, for the chip load to beevaluated, chip thickness must be known. For a ball-endmill cutter, the instantaneous undeformed chip thickness isfound as follows,

where tx [mm/tooth] is the feed-per-tooth-per-revolution,

and it is determined fromfeed rate [mm/min] /Nf. [rpm].Nf and represent the number of flutes and spindle

speed, respectively. is the cutting element rotation angle,

and is the cutting element position angle (Figure 2).

kn takes the value of 1, if the kth

discrete point on the nth

cutting edge is in engagement with the workpiece.

Otherwise, it takes the value of 0. Therefore, instantaneous

infinitesimal chip load for each discrete element that is in

contact with the workpiece can be written as the following,

here (dz)kn represents differential chip height along the

orce Module

Ac) in engagement domain,

are the radial, axial and tangential(Figure 2) cutting constants the

force components can be easily

wlongitudinal cutter axis at the k

thdiscrete point on the n

th

cutting edge.

2.3 Cutting F

For a differential chip load (dthe differential radial (dFr), axial (dF) and tangential (dFt)cutting forces can be written as follows,

where Krc, Kc and Ktc

tectct

, and Kre, Ke and Kte arerelated edge coefficients, respectively. These specificcutting energy coefficients vary based on the cutter /

workpiece material combination and vary along the cuttingedge in ball-end milling.

Once dFr, dF and dFt were obtained through use ofEquation 4, these cuttingtransformed into theX-Y-Zglobal coordinate system as thefollowing,

rX dF

dF

dF

nknkZ

Y

dFt

dFA

dF,,

=

(5)

nk

A

,

0)sin()cos(

)sin()cos()cos()cos()sin(

)cos()sin()cos()sin()sin(

=

fkf

NnN

n...1;

2)1(=

+=

= =

+

=

fN

n

K

knkpresZ

Y

X

Z

Y

X

dFdF

dF

dF

F

F

F

1 1,

(6)

where Krepresents the total number of discrete points ona cutting edge, is the cutting edge rotation angle (Figure

dzKdAKdF +=

2). One important aspect of the model to mention here isthe additional dFpres force that is added to dFz. This force isassumed to result from a constant pressure value existingover the workpiece as long as the cutter moved down into

the workpiece in the Zdirection. Its amplitude equals thisconstant pressure times the area of the cutter/workpiececontact region. From the experiments, it was observedthat value of constant pressure is mainly dominated byfeed rate. Therefore, in the calibration stage, the pressure

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(in MPa) was determined as a function of feed rate (inmm/min) for Al7039 which was used in the experiments asthe following;

3248.48767.10409.00004.0)(23 ++= ffffP

Figure 2: Illustration of the cutting force components andangular relationships.

3 FEED RATE SCHEDULING IN SCULPTUREDSURFACE MACHINING

el, thethre e components along the tool

desired constant level along the tool

feed rate and the

in the slot cutting tests.forces for constant feed rate are obtained through the use

SIMULATIONS AND EXPERIMENTAL RESULTS

t is

In the first stage of simulations, using the above mode orthogonal cutting forc

path are predicted for a given input set and a constant feedrate value. Due to various reasons such as limiting tooldeflections, tool wear, avoiding tool breakage, etc.,production engineers may set a reasonable constantresultant cutting force value that should not be violated allalong the tool path.

In the second stage, the model can be run to keep theresultant force at thepath by varying the feed rate piecewise.

In order to increase the resultant force at any instant to adesired value, a linear relation betweenresultant force has been utilized. During the calibrationstage for Al7039, it is noticed that the rate of increase inthe resultant force, Iforce, is related to the rate of increase infeed rate, Ifeed rate, through the equation

725.0725.1 = orcefeedrate II (7)f

Therefore, once the resultant

of the force model simulations, the tool path is divided intoa desired number of intervals for using different feed ratesand the maximum resultant force values are determined foreach interval. There has to be a pre-set peak force levelthat limits the value of resultant forces through the cut. I

force

is determined for all the intervals by dividing the maximumresultant force at each interval to the pre-set peak force.Therefore, once Ifeed rate values are obtained for all

intervals, these are multiplied with the constant feed rate tocome up with the new feed rates throughout the tool path.

4

In order to test the piecewise feed rate scheduling thabased on the force system model, various experimentalvalidation tests were performed on a vertical machiningcenter using Al7039 (Al 91.65%, Cr 0.15%, Mg 2.8%, Mn0.4%, Si 0.3%, Ti 0.1%, Zn 4%, Cu 0.1%, Fe 0.4%, other0.1%) workpiece material. A two-fluted ball-end mill withthe diameter of 12 mm, the nominal helix angle of 30degrees, and the projection length of 37 mm was used inthe experiments. For the tools with constant helix angle(0), the lag angle can be determined as

200 2/)tan(./)tan(. cccc zRzzrz == (8)

where zc is the distance from the tip, r is the radius of a

Intervals from tool tip [mm]

point on the cutting edge on a plane perpendicular to thecutter longitudinal axis and is the lag angle between theline which connects this point to the tip and the line whichis tangent to the cutting edge at the tip (Figure 2).Calibration tests have been performed in advance to obtain

the cutting constants and edge coefficients to be used inthe mathematical model. Since chip thickness, cuttingvelocity, and therefore the cutting coefficient values,change along the cutting edge in the ball part, the 6 mmradius cutter was divided into seven intervals along thecutter axis. For each feed rate, seven incremental slot-cutting tests, which correspond to these intervals, havebeen performed. In order to find the cutting forces of aspecific interval, the difference of cutting forces for twosubsequent intervals have been found first. The radial,axial and tangential differential forces have been plottedversus the average chip thickness per revolution-per-tooth[7] in order to obtain the cutting force (Ktc, Krc, Kc in Mpa)and edge force (Kte, Kre, Ke in N/mm) constants for therespective interval (Table 1).

0-0.5 0.5- 4-61 1-1.5 1.5-2 2-3 3-4

10478 3327 2376 1805 1404 1364 1032

Krc 6156 2191 741 398 212 173 16

Kc 110 255 433 190 143 123 168

Kte 8 9 9 7 11 4 4

Kre 24 0 7 6 12 3 8

Ke 14 19 8 13 6 3 0

Ktc

Table 1: Cutting and edge force coefficients for Al7039.

In ,

ng is

order to perform concave and convex machining casesthree sine waves each with the amplitude of 3 mm, thewavelength of 50 mm and the pick feed of 3 mm/track wereaimed to be created on 150mmx50mm blank workpiecesurface (Figure 3). In these cutting tests, spindle speedand feed rate were 1000 rpm and 150 mm/min,respectively. In the simulation, ball-part of the cutter wasdiscretized into disks of 0.1 mm height, and the forcecalculations were performed every 3.6 degrees of cutterrotation. Amplitude and waveforms of simulated forcecomponents agreed very well with the experimental datacollected at 2000 Hz (Figure 4). A zoomed view ofresultant cutting force between two maximum points of thesine waves on the workpiece was shown in Figure 5c forconstant feed rate case. Maximum resultant forceamplitude was 550 N along the 50 mm long tool path.

In order to decrease the cycle time, feed rate scheduliimplemented in the machining of the sine waves. Whilescheduling the feed rate along the tool path according to

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Equation 7, it was desired that the resultant cutting forceshould not exceed 600 N.

NC code was modified to accommodate the piecewisevarying feed rate values as shown in Figure5b.

Figure 3: Illustration of machined sculptured surface.

Figure 4: Simulated and measured force components for

During the impl the scheduled

rence between the constant and

CONCLUSIONS

resented for sculptured surface

the constant feed rate case.

ementation of machining withfeed rate values, all the other cutting conditions were keptconstant, and cutting forces were sampled at 2000 Hz. Thezoomed view of resultant force for one period of sine waveis shown in Figure5d. The resultant force amplitude was

kept under the threshold limit of 600 N along the tool path.Moreover, it was also observed from the collected data thatin this typical example, with the scheduled feed rate,machining time is decreased 41% compared to theconstant feed rate case.

The only observable diffescheduled feed rate cases was the tooth marks on themachined workpiece surface, due to the high feed values,near the maximum points of the sine waves. This maycreate problems in finishing operations. However, in therough milling case, this can be tolerated.

5

A force model was pmachining with ball-end mill. Based on this force model,while keeping resultant cutting force under a pre-setthreshold value, an off-line feed rate scheduling method

was introduced in order to decrease the cycle time in roughsculptured surface machining processes.

Figure 5: Zoomed view of experimental resultant forces forconstant and scheduled feed rate cases.

It was shown that off-line scheduling feed rate along the

tool path based on a reliable force model has a potential tosubstantially decrease cycle time in sculptured surfacemachining.

6 REFERENCES

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[2] Yang M., Park, H., 1991, The Prediction of CuttingForce in Ball-End Milling, Int. J. of Machine Tools &Manufacture, 31:45-54.

[3] Altintas, Y., Lee, P., 1996, A General Mechanics and

Dynamics Model for Helical End Mills, Annals ofCIRP, 45/1:59-64.

[4] Taunsi, N., Elbestawi, M.A., 2003, Optimized FeedScheduling in Three Axes Machining. Part I:Fundamentals of the Optimized Feed SchedulingStrategy, Int. J. Machine Tools & Manufacture, 43:253-267.

[5] Lazoglu, I., 2003, Sculpture Surface Machining: AGeneralized Model of Ball-End Milling Force System,Int. J. of Machine Tools & Manufacture, 43:453-462.

[6] Bouzakis, K.-D., Aichouh, P., Efstathiou, K., 2003,Determination of the Chip Geometry, Cutting Forceand Roughness in Free Form Surfaces Finish Millingwith Ball End Tools, Int. J. Machine Tools &Manufacture, 43:499-514.

[7] Altintas, Y., 2000, Manufacturing Automation: MetalCutting Mechanics, Machine Tool Vibrations, andCNC Design, Cambridge University Press.