DOI 10.1007/s00170-003-1723-x

O R I G I N A L A R T I C L E

Int J Adv Manuf Technol (2004) 24: 621631

L.N. Lopez de Lacalle A. Lamikiz J.A. Sanchez M.A. Salgado

Effects of tool deection in the high-speed milling of inclined surfaces

Received: 14 September 2002 / Accepted: 1 March 2003 / Published online: 5 May 2004 Springer-Verlag London Limited 2004

Abstract The present paper looks at the dimensional errorsresulting from tool deflection in the high-speed milling of hard-ened steel surfaces. These errors are measured as the differencebetween the theoretical surface and the high-speed milling ma-chined using ball-end mills.

The effect of various factors on this dimensional error is in-vestigated. First, account was taken of the workpiece materialand the slope of surfaces; the values chosen were those normallyused in injection mould manufacturing. The workpiece materialswere of 30 and 50 HRC hardness, with slopes of 15, 30, and45. In this manner, results may thus be of utility to the mouldand die industry. The selected tools were solid ball end mills ofsintered tungsten carbide, coated with TiAIN. These were of var-ious diameters and lengths, and accordingly exhibited variousdegrees of slenderness. A great value for this latter parameter isa restraint on the potential application of the high-speed millingtechnique. This is the main reason for this work.

Tests were carried out using three machining strategies,namely, upward, downward, and z-level (horizontal), as well aswith two cutting types, downmilling (also called climb milling)and upmilling (or conventional milling). In all cases the resultingroughness was also measured. Dimensional errors in several flatslope planes were measured, comparing with those obtained bysimulation.

The results of these tests have been applied to the predictionof error in the high-speed milling of two industrial parts. Know-ledge of error magnitude may be useful when NC programs areprepared for the machining of mould complex surfaces, since itmay then be attempted to enhance accuracy.

Reference is made to various practical problems that werenecessary to resolve in order to achieve measurement errors less

L.N. Lopez de Lacalle () A. Lamikiz J.A. Sanchez M.A. SalgadoDepartment of Mechanical Engineering,University of the Basque Country ESI,c/Alameda de Urquijo s/n, 48013 Bilbao, SpainE-mail: implomal@bi.ehu.esTel.: +34-94-6014216Fax: +34-94-6014215

than 20 m in a process as complex as that of high-speed millingin three axes machining centres.

Keywords Complex surfaces High-speed machining Machining errors Moulds

Notation

ap Axial depth of cutae Radial depth of cut, similar to Pvc Cutting speedC, C1, C2 Cusp heightfz Feed per toothF, f Feed rateS Spindle speedE Youngs modulusI Second moment areaRa Mean roughnessRt Maximum roughnessF Cutting force, component perpendicular to tool axis Tool deflection, perpendicular to tool axis Surface slope Dimensional errorHSM High-speed millingEDM Electro-discharge machiningHRC Rockwell C hardnessCAM Computer-aided manufacturing

1 Introduction

In the automotive industry, and more specifically in die andmould manufacturing, a common requirement is to produce com-plex surfaces on hardened steels. Tolerances and dimensionalaccuracy are very narrow. To date, the main manufacturing pro-cess has been to use electro-discharge machining (EDM) [1], butsince 1995 it has been clear that high-speed milling (HSM) isa technology of great potential for hardened steels (> 30 HRC).

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To achieve the finishing requirements of the mould sector,HSM technology has to minimise the effect of various factors [2]producing dimensional errors. Such factors include: (1) Thosederiving from the CAM stage [4], owing to the approximation ofthe paths produced by current commercial software; (2) Thoseresulting from insufficient resolution of the CNC control loopsrelating to the choice of both feed forward and look-ahead pa-rameters;(3) The construction and stiffness of the actual machinetools, which are directly related with vibration in machining,inertial tool-path inaccuracy and thermal distortions; (4) Var-ious problems derived from the tool-clamping systems [5, 6];or e) from thermal deformation [7] of the workpiece and ofthe tools.

One of the errors produced by the cutting tools is that dueto runout and grinding errors on the tools. There is also thepossibility of wear or tool breakage [8]. And finally there areimportant errors owing to the deflection of cutting tools, whichis looked at in the present paper. In some preliminary tests andmoulds manufactured by authors, errors derived from tool deflec-tion in finishing processes exceeded 100 m. Such errors, addedto those already mentioned, is critical where tolerances in themould industry are concerned, which are commonly in the range0.050.1 mm.

For the present study a test plan was carried out by virtue ofwhich the errors resulting from tool deflection were predicted.The test variables taken into account were cutting strategy, tooldimensions, material hardness and surface slope. It was thenpossible to determine what cutting strategies should be used tominimise errors resulting from tool deflection. In some cases thebest strategies for minimising deflection may have negative as-pects, for example, it induces more dynamic problems and toolwear. In such cases, the optimal solution will be a compromise.

Included here is a more complete study of the cutting typespossible than that given in Kang et al. [3], in which cuttingtype, i.e., downmilling (climb milling) or upmilling (conven-tional milling), is not included among the test factors, and inwhich cutting speed is low in terms of HSM and the radial depthof cut ae is high for mould finishing. However, [3] provides use-ful information for a database relating to accuracy, roughness,and wear in the case of inclined surfaces machining.

The results of the test plan proposed here may serve as a ba-sis for a machining path correction system that will minimise theerror derived from tool deflection, as proposed in Seo [13] and inHascoet et al. [20].

2 Tool deection. The parameters in question

Prior to the quantitative study of tool deflection, the question tobe considered is what parameters affect it. We can assert that aslong as cutting is stable, i.e., as long as there is neither forced norregenerative vibrations (chatter), the model approaches a staticcase. So, the cutting conditions (ap, ae, vc, fz) of all tests mustavoid the influence of forced vibration due to the tooth pass-ing frequencies. And it must be a stable case with respect tochatter problems. We have measured the natural frequencies of

the spindle-tool assembly of the three axis-machining centre.A modal analysis of the high-speed machine tool, following theprocedure described in [16] has been done, obtaining the stabilitylobes of the machining operation. The value of lower frequenciesare in X axis 436, 685 and 1115 Hz and in Y axis 430, 720 and1110 Hz. These values and the small depth of cut of the finishingtests, avoid the presence of dynamic problems like chatter in alltrials.

An appealing model [1719] for the study of deflection isthat in which the tool is regarded as a cylindrical cantilever beam.Deflection then conforms to the equation

= FL3

3EI(1)

If the second moment of area I of the tool section is replaced, theresult is

= 64 F3 E

L3

D4(2)

It may thus be seen in Eq. 2 that tool deflection in the staticmodel is a function of the following three parameters:

E : Youngs modulus for the tool material.L3/D4 : Tool slenderness parameter, where D is equivalent tool

diameter and L is overhang length. These parameterswill be looked at below.

F : Cutting force perpendicular to the tool axis.The modulus of elasticity (Youngs modulus) E depends on

the tool material. The slenderness parameter depends on thegeometry of tools that are selected. The other parameter is thecomponent of the cutting force perpendicular to the tool axis inthe plane perpendicular to surface in the maximum slope line.This cutting force component depends on the feed per tooth ( fz)and the engagement conditions (ap, ae). The toolholder used wasa hydraulic HSK 63 with much more stiffness [5] than tool. In [5]a complete study about the radial stiffness of hollow shank kegel(HSK) is presented; its main conclusion is that radial stiffness ofHSK50 shanks ranges from 19 N/m at 5000 rpm to 17 N/mat 25 000 rpm. Comparing these values with those derived fromthe cantilever beam model (see Eq. 2), which values are, for thetested tools (see Table 2), 6mm 0.15 N/m, for 12mm is4.4 N/m and for 16mm is 1.6 N/m, it can be noted thatin the best case, (12mm) tool stiffness is approximately fourtimes lesser than shank.

2.1 Modulus of elasticity of tool material

Youngs modulus is a function of tool material. In the machiningof moulds and dies, the use of micrograin tungsten carbide toolssintered with 1012% cobalt binder (usually named hard metal)is very common. Manufacturers of hard metal do not usually pro-vide the value of Youngs modulus. In the present case it wasdetermined by two methods, namely an analysis of the naturalfrequencies of the material and by a compression test of a piece.In Appendix 1, a value of 6105 N/mm2 is obtained by bothmethods.

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2.2 Tool slenderness

In several works [1, 18], L/D or L3/D4 were used as the charac-teristic slenderness parameter. The latter expression was selectedsince, as may be seen in the cantilever beam model of Eq. 2, de-flection is a linear function of this slenderness parameter. HereD is the equivalent tool diameter. The equivalent diameter willbe less than the actual tool diameter, since the helicoidal shape ofthe two edges reduces the resistant section. In [16] an equivalentdiameter of 0.8 D0 is assumed, D0 being the tool diameter.

The length of the overhang L is the free length betweenthe tool-toolholder connection and the cutting point. As in high-speed milling finishing of moulds, the depth of cut is small, theforce is simplified to act at the tool tip. Thus, in this work, thedepth of cut is 0.2 mm, and is insignificant in comparison to thelength of the overhang, which exceeds 50 mm in all cases (seeTable 2).

2.3 Cutting forces

Cutting force is a function of various factors, including the toolgeometry, the work material, the geometry of the workpiece, thecutting conditions, and the sense of machining respect to sur-face. It greatly varies with the machining strategy used, both inmagnitude and direction.

Fig. 1. Relation between tool deflection and dimensional error

Fig. 2. Actual and simulated cut-ting forces for tests 3 and 5(Table 2), on 50 HRC temperedsteel

Table 1. Test cutting conditions to assess the cutting force simulation model

Experiment Material Slope () ap (mm) ae (mm) fz (mm)

1 Al 7075 15 2 8 0.052 Al 7075 15 2 8 0.053 50 HRc Steel 15 1 8 0.0134 Al 7075 30 0.5 8 0.055 50 HRc Steel 30 0.8 8 0.0196 Al 7075 45 1.5 8 0.15

The dimensional error on the surface also depends on thevalue and direction of the deflection, as it is shown in Fig. 1.Thus it may happen that with strategies conductive to greater cut-ting forces, hence also to greater deflection, deflection results insmaller errors owing to the direction of the force with respect tothe surface. For this reason it becomes necessary to draw up a testplan to determine how machining strategies affect the finishingoperation.

So that the bending-based model can be checked, cuttingforces were obtained by means of force-simulation based ona mechanistic approach [16]. In the case of slope 15,30 and45, the values obtained with specially developed simulationsoftware by the authors, described in detail in [21], were com-pared with those obtained with the aid of a dynamometer. Thisforces measurement was problematic owing to the small valueof cutting forces in finishing conditions, in virtue of which theywere sensitive to noise and other disturbances (see Discussion,point 3). For this reason our cutting forces simulation model waschecked with a larger depth of cut (Table 2) of those correspond-ing the finishing conditions.

Figure 2 shows both measured and simulated cutting forcesfor two slope-machining tests, (see cases 3 and 5 of Table 1).The cutting conditions can be observed in this table. Compar-ison between experimental data and simulation results showsthat the model is valid for different slopes, strategies andmaterials. The agreement between measured and simulatedforces, not only in terms of the values but also the shape ofthe wave, can be observed in all cases, with differences lessthan 20%. Errors in Fz are due to the electrical charge ofthe Fz channel amplifier due to isolation problems, and areinsignificant.

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Table 2. Dimensions of the tools used. Slenderness parameters

Lt Lvol (mm) D0 (mm) D Lvol/D L3vol/D4 (mm1)

100 66 6 4.8 13.75 541.6100 55 12 9.6 5.72 19.9150 112 16 12.8 8.75 52.33

This simulation force has been used to estimate deflectionerrors using Eq. 2 (see Discussion, point 2).

3 Parameters affecting roughness

In finishing it is sought to achieve not only good accuracy butalso good roughness levels. Theoretical roughness conforms tothe equation

P =

(8RRt 4R2t ) cos (3)

Fig. 3. Influence of slope variation on scallop height C; with same P,C2 > C1

Fig. 4. Left workpiece, Centre workpiece clamped on the Kistler in the machine, Right three tools

Here, P (named step, similar to ae) is the radial depth of cut,R is the tool radius, Rt is the maximum theoretical roughness(Rt is similar to C, the scallop height), and is the slope of thesurface (in the direction in which the step P is increased). FromEq. 3 it can be seen that theoretical roughness depends on tool ra-dius, radial depth of cut, and slope as it is represented in Fig. 3.However, in practice there are other factors, such as vibration,material plasticity, and of course tool deflection, and as conse-quence the predicted values by Eq. 3 may be exceeded. Hencethe interest in how roughness is affected by various machiningconditions.

In the test plan carried out, cutting conditions were selectedwith a view to ensuring that Rt would not exceed 5m. Aftertestpieces had been machined, the mean roughness, Ra, and themaximum, Rt , were measured.

4 Test plan

A test plan was carried out with all the variables involved in theProblem and is presented in Fig. 4, and explained below.

4.1 Workpieces

There are two workpiece variables that affect the machining,namely, geometry and material. The test plan was carried out ontestpieces with flat ground surfaces at three different slopes, i.e.,15, 30, and 45.

The hardness of the material has a decisive effect on tool de-flection, for it affects the magnitude of the cutting forces. Twotest parts made of hardened steels typical of the mould and dieindustry were used:- The first one was in steel F117 (P20 approx., W-No 1.2738),

hardness is 30 HRC. Used among other things for thermo-plastics and blow moulds.

- The second part was ORVAR SUPREME (H13, W-No1.2344, SKD61). Used in tools for die-casting, extrusion,plastic injection, and forging. Hardness is in excess of50 HRC.

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4.2 Tools

Where the tool is concerned, there are three factors that can con-tribute to machining errors. These are geometrical inaccuracy(bad sphericity and edge runout) due to grinding defects, wear,and deflection. A preliminary study was carried out in order toknow the precision of mills made by various manufacturers. Theresults (mean of three similar tools) for the 16mm tool, withtwo cutting edges, TiAIN coating, and helix angle 30 were asfollows:

Manufacturer A (very large firm, expensive tools), max-imum negative error: 0.011 mm, maximum positive error:0.088 mm.

Manufacturer B (large firm), maximum negative error:0.017 mm, maximum positive error: 0.009 mm.

Manufacturer C (small firm focused in special tools), max-imum negative error: 0.055 mm, maximum positive error:0.000 mm.

Tools made by manufacturer B were chosen, because theyhave enough tolerances and are often used by European mouldmakers. The characteristics were as follows: integral solidmills of hard metal (CW +10% Co) of submicrograin grade,

Fig. 5. Test plan for 50 HRC. Horizontal view

with a monolayer TiAIN coating, two flutes, helix angle 30,clearance angle 11, normal rake angle 1, maximum runout0.003 mm. The dimensions are shown in Table 2.

Here Lt is the overall tool length, Lvol is the overhang length,D0 is the nominal tool diameter, and D is the equivalent tooldiameter (calculated as 0.8 D0). The total machined length perworkpiece was sufficiently short for flank wear to be well below0.1 mm, therefore wear effect is not considered.

The toolholder was HSK63 in all cases, with hydraulicclamping and a balance level of 3 g mm. The HSK toolholderis stiffer [5], provides lower runout (less than 2 m), and hashigher repeatability than the ISO 7388 tapered shank. Hydraulictool clamping minimises tool runout in the shank-tool interface.Using a micrometer dial, the runout in the maximum tool lengthwas ensured to be less than 2m in all tests. In this way, the max-imum radial runout, including the effects of spindle, shank andtool, was less than 5 m.

4.3 Cutting conditions

The test plan was carried out using finishing cutting conditions.The following were selected:

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- Cutting speed: for all tests, effective cutting speed at tool ma-terial contact point (point A in Fig. 1) was held constant at200 m/min. Doing that, cutting speed in the nominal radiusvaries in all tests, because it depends on piece angle .

- Depth of cut ap: for the Do 6 mm tool, depths of cut of 0.1and 0.2 mm were used. For tools of Do 12 mm and of Do16 mm, they were 0.2 mm and 0.3 mm respectively.

- Radial depth of cut ae: this was set to ensure a good value forthe maximum roughness Rt , less than 5m in all cases. Inthe case of the Do 6 mm tool, a radial depth of cut of 0.1 mmwas used, while for those of Do 12 and of Do 16 mm, depthwas 0.2 mm.

- Feed per tooth fz : feed per tooth was the recommended oneby tool manufacturer. It depends on the size and stiffness ofedges, so it depends on each tool diameter, that is, for Do 60.055 mm/tooth, for Do 12 0.077 mm/tooth, and for Do 160.089 mm/tooth.

- In all cases dry machining.- Direction of machining: tests were carried out with upward

machining, downward machining, and in transversal direc-tions (horizontal). In these directions, downmilling (climbmilling) or upmilling (conventional milling) was employed.Hence, the cutting types studied were:AV-D: upward machining, downmillingAV-U: upward machining, upmillingDV-D: downward machining, downmillingDV-U: downward machining, upmillingH-D: z-level horizontal machining, downmillingH-U: z-level horizontal machining, upmillingFigure 5 shows the whole test plan for the 50 HRC testpiece,

similar to that of 30 HRC. The test plan in Fig. 5 indicates withS the spindle r.p.m., as well as the cutting conditions. The longarrow in each square indicates the direction of linear feed, whilethe short one indicates the sense of the radial depth of cut (orstep). The M, L, and P indexes are the order numbers of tests,without technical meaning.

5 Results

5.1 Errors due to tool deflection

The value of dimensional errors due to tool deflection on themachined workpiece was obtained by measurement of the work-piece on a ZEISS 850 Coordinate Measuring Machine. A relativemeasurement of the dimensional difference between two planes,that machined once, and that machined in two similar passes (seeFig. 6) was made. In this way the second pass eliminates thestock allowance left by tool deflection in the first cutting pass,machining a flat plane very near to the theoretical surface. In thisway the average error between both machined surfaces approxi-mates the actual error. With this relative measurement the effectof the workpiece misalignment on the CMM is avoided. Thiserror is always present, even if an alignment procedure based onthree reference planes in the original workpiece is carried out.This procedure was used first, but the lack of precision in the

Fig. 6. Measurement of errors in machined planes with a CMM

workpiece set-up in both the machine tool and CMM made itimpossible to discern tool deflection from set up errors.

The most significant dimensional errors in workpiece ma-chining are shown in Figs. 78. Here workpiece materials aresteels 30 and 50 HRC, with the three tools used, three slopes,and six cutting strategies (see point 4). 20% of the tests were re-peated in order to increase results reliability. Consistency in theresults suggests that the errors due to measurement proceduresin the CMM are insignificant, and so the measurement procedureshown in Fig. 6 may be regarded as suitable.

5.2 Roughness

The maximum roughness (Rt ) predicted by Eq. 3 ranges from 4to 6m, with an average value (Ra) ranging from 0.8 to 0.9m.Figure 9 shows actual mean workpiece roughness, in the case ofthree slopes and all the cutting strategies. Both mean roughness(Ra) and maximum roughness (Rt ) had been measured in a di-rection perpendicular to tool feed movement. As for Rt , this wasless than 5 m except in some tests with tool of diameter 6 mm.Mean roughness Ra reached 300% of the theoretical value, up to3m. But this value is less than the common admissible one inthe mould industry.

6 Discussion of results

In Fig. 10 theoretical errors calculated with deflection Eq. 2 areshown. Cutting force F has been estimated using the simulationsoftware developed by authors, as it was explained in Sect. 2.3.Comparing this figure with Figs. 78, the first conclusion is thatactual errors are in general greater than estimated ones. In theworse case (that is, DV-D downward-downmilling), divergenceis up to 100%. Some points can be noted:1. It should be noted that in the case of the 6 mm tool ae was

equal to 0.1 mm, while in the cases of 12 mm and 16 mmae was 0.2 mm. That is because Figs. 7c and 8c do notshow errors higher than 24 m, where the 6 mm tool is con-cerned. In some preliminary tests carried out with this tool

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Fig. 7ac. Dimensional error in 30-HRC steel, with a 12 mm ball-endmilling tool. b 16 mm ball-end milling tool. c 6 mm ball-end milling tool

at ae 0.2 mm, measurements of 70 and 80 m were taken,on slopes of 15 with 50 HRC. However, the high values ofroughness for these trials indicated that there had been toolvibration, so this value is neither reliable nor representative.

2. Errors when 15 slope were greater than those resulting froma slope 45. As an example, the greatest error with the 16 mmtool arises in a plane of 15, with a value of 20m (Figs. 7band 8b), while in the 45 plane error is 3m. This fact was

Fig. 8ac. Dimensional error in 50-HRC steel, with a 12 mm ball-endmilling tool. b 6 mm ball-end milling tool. c 6 mm ball-end milling tool

studied using simulation of cutting forces. Here, transversalcutting forces for 15 are three times greater than those for45. This greater cutting force value may be attributed to twofactors, the large length of the tool edge in contact with work-piece in 15, and the decrease of the helix angle in the toolextreme, less than the nominal angle of 30 (in tool extremeis 0).

3. If a comparison is made between deflection results obtainedwith Eq. 2, using those transversal cutting forces Fy derivedfrom simulation with experimentally measured errors, it can

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Fig. 9a,b. Mean roughness in a 30 HRC workpiece b 50 HRC workpiece

Fig. 10. Theoretical errors based in simulation of cutting forces, steel of50 HRC, at slopes of 15, 30, and 45, for the 16mm tool

be deduced that the cantilever beam model provides an ap-proximation of tool performance. Thus in the case of 16 mmdiameter tool the error computed by formula (Eq. 2) is 7m,while that measured is 9m. In the case of the 12 mm tool,that computed is 3m and the actual is 6m (Fig. 8a, 15,H-U).There are several reasons for this difference: (a) With thepresent model one can use an effective diameter between0.8 D0 and 1 D0; (b) Youngs modulus can range from4.7 to 6105 N/mm2; (c) Also, the inherent inaccuracy toforce simulation is introduced; (d) On the other hand, if cut-

ting forces are measured experimentally, for small depths apand high cutting speeds, measurement is highly problematicowing to noise and for a high sampling frequency needed;and (e) Another factor may be the HSK63 toolholder, whosestiffness is high but not infinite.Hence while it is valid qualitatively, the cantilever beammodel for tools is not exact. It can be reasonably asserted thatthe results presented here, help us better understand earliersimulation work, such as that reported by Hascoet et al. [21],which is based on the cantilever beam model.An important conclusion is the tendency for error to decreasewith slope, much as in the case of simulated cutting forces(Fig. 10).

4. Upmilling produces smaller deflection errors than down-milling. While in the case of upmilling the maximum is15 m, with downmilling one can achieve values in excessof 20m for the 6 mm tool. This is because in the upmillingcase the sense of force tends to clamp the tool into the work-piece, rather than to separate them.

5. The largest errors are always derived from transverse ma-chining (cases H).

6. Upward (AV) conduces to smaller deflection errors thandownward (DV), although in many cases DV also producesgood results.

7. The magnitude of tool deflection and the stability of cut-ting process strongly depend on the slenderness parameterL3/D4. In 6 mm tool cases, vibrations and a rapid wear oftool edges appear, producing poor roughness levels. It maybe concluded that with such slender tools, finishing operationis unreliable, unless the operator is extremely conservative inselecting the feed per tooth and axial depth of cut.

Regarding roughness, details are presented in Fig. 9 and here:8. Mean roughness (Ra) levels are higher in the case of 50 HRC

steel, which is owed to larger cutting forces in this material.9. Smaller diameter tools produce greater roughness. So, the

higher roughness was obtained in 6 mm tool tests. As can beseen in Fig. 9, there are no great differences in Ra betweentools of diameter 12 mm and 16 mm.

10. There is a little tendency for Ra to increase with slope. Thisfact is consistent with the Rt behaviour, as is derived fromEq. 3 (Rt is inverse to cos ). Mean roughness level is verymuch affected by diverse factors. These include tool deflec-tion, as is common knowledge.

7 Application to error estimation in industrialworkpieces

The above results can be applied to the machining of indus-trial workpieces in order to predict errors, optimise machiningstrategy, and ensure good roughness levels. The strategy mostconducive to accuracy, with minimum tool deflection, can thenbe chosen. At present there are few references in the literatureto the value of the dimensional error in sculptured parts, so theabove results can be very useful for mould makers.

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The error estimation was applied to two surfaces, machinedin a three-axis high-speed machining centre, with a 25 000 rpmspindle (17.5 Kw). The first (Fig. 11) was a part of an aluminiuminjection mould exhibiting a complex geometry, with small fil-let radii, deep slots, and walls that implied the use of tools witha large overhang.

To predict dimensional errors in the case of zig-zag ma-chining, errors in the cases AV-D (zig) and DV-U (zag) wereaveraged, as well as in the cases AV-U (zig) and DV-D (zag), seeTable 3. In a zig-zag strategy each pass corresponds to one of thecases, one pass is upward-downmilling the other is downward-upmilling.

The predicted values of the deflection error were interpolatedfrom those deduced from the tests. In the case of zones C andD, theoretical and actual errors are similar. However, in the planewith slope 41.7 (zone B) the measured error is larger than the pre-dicted one. In this case the presence of vibration marks on surfacerevealed a dynamic problem, in which case the beam model can-not be applied. In zone E the slot was so deep that the entire tooldiameter was involved in cutting; the tool was doing a slotting op-eration. The 2 mm tool bent and a 100 m error was observed inthe workpiece, showing marks typical of tool vibration. This caseis a limit that cannot be studied by the present method.

Fig. 11. Sector of an aluminium injection mould

Table 3. Strategies, errors predicted, and mean errors in the case of theworkpiece shown in Fig. 11

Zone Strategy Slope L3/D4 predicted measured

A Z level 90 60 No data 10mB Zig-zag 41.7 120 10 m 30 mC Zig-zag 30 120 12 m 15 mD Zig-zag 10 315 25 m 31 mE Upward 10 315 25 m 100m

Tool vibration-waves marks

in part

Fig. 12. Industrial test

Table 4. Strategies, predicted errors, and measured mean errors, in the caseof the workpiece shown in Fig. 12

Zone Strategy Slope L3/D4 predicted measured

A Z level 4050 60 3m 6mB Zig-zag 40 120 10 m 16 mC Zig-zag 30 120 12 m 18 mD Zig-zag 10 315 25 m 33 mE Upward 10 315 25 m 110 m

On the other hand, a very well known industrial test (Fig. 12)was HSMed on a 50 HRC hardened steel. Results are shown inTable 4.

This case involves a less complex geometry than the first ex-ample. The coincidence in zones B, C, and D is good. But zone Eappears to have an error greater than might be expected. Here novibration marks were observed, so that this error may be attributedto low values of the look-ahead parameters programmed in theCNC control (see Appendix 2) and to a large tool diameter for thesmall slot radius. To confirm this hypothesis, a new machining ofan aluminium workpiece, in which case cutting forces were small,were run. After the CMM measurement showed in Fig. 13, errorsfrom 32 up to 90 m in this zone were found. It is therefore reason-able to assume that errors up to 3080 m are attributable to themachine+control behaviour, not to tool deflection.

In these two examples there are other factors affecting accu-racy. However, the foregoing data afford indication of error mag-nitude and can serve as a basis for the enhancement of accuracy.Thus, the above prediction can be used to change the cutting con-ditions (Vc, fz, ap and ae) in some zones, where errors are high.

8 Conclusions

When hardened steels are HSMed, tool deflection gives rise toinaccuracy on the produced surface. The aim of this empiricalwork is focused on how tool slenderness, workpiece slope, andcutting strategy affect this deflection. It was concluded that theuser could reduce deflection by judiciously selecting tool andmilling strategy.

Empirical results show that the deflection model based ina cantilever beam is not very accurate, but is useful in order to

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Fig. 13. Errors in an aluminiumpart due to bad look ahead param-eters, measured in a CMM

have an aim of the magnitude of dimensional errors in the ma-chining of complex surfaces. See the Discussion, point 6, formore practical conclusions about the influence of strategies ondimensional errors of parts. These conclusions can be useful formould manufacturers.

Two applications have shown that the foregoing results canbe used to provide an approximate prediction of dimensionalerror in industrial pieces. In this way, the CAM programmer mayidentify his case with one of those in the database, and with thisinformation estimate the error in the machined workpiece. A realcase will not exactly coincide with a case in the database, ofcourse, and it will be necessary to make approximations.

Acknowledgement Grateful thanks are due to Israel Gago and EduardoSasia for assistance provided in the test plan machining. The workpiecein Fig. 11 was machined with the assistance of Patxi Aristimuo of theMondragon University. This article was sponsored by the projects MI-CYT DPI2002-04167-C02-02 and DPI2000-0075-p4 (Spanish Government)and UPV/EHU 145.345-TB7923/2000. Special thanks must be given to theAquitania-Basque Governments Collaboration Programme 2001 for its fi-nancial support.

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Appendix 1: Determination of the modulus of elasticity

Analysis of the tools natural frequency

The natural frequency of tools was determined with a vibrationanalyser, which was used in studying the response to an impulsetest (impact with a hammer) on a cylindrical hard metal testpiece.

631

The equation governing the vibration of a free beam is

fi = 2i

2 L2

EIm

(4)

Here fi is the natural frequency of the cylinder, and is thewavelength of the vibration. In a free beam, in the case of the firstnatural frequency, 1 is 4.73004074, L is the length of the beam,E is the modulus of elasticity of the material, l is the second mo-ment area of the beam section (circular), and m is the mass perunit length of the beam material. In the case of the analysed tool,the values were fi 546 Hz, L 0.15315 m, l 3.21699109 m4,the mass is 2.82 Kg/m (density is 14 100 kg/m3) and E results4.841011 N/m2.

Compression test

The results of the normalised test were obtained, where a test-piece of hard metal was submitted to compression. The extremehardness of the tungsten carbide, exceeding 94 HRA, makes ten-sile tests inadequate. Values were obtained for the tangent or se-cant elasticity modulus. Tangent elasticity depends on the stresslevel:

Stress < 2 kg/mm2 E = 4.8105 N/mm2Stress > 2 kg/mm2 E = 7.2105 N/mm2

In the case of secant elasticity, when stress levels are low, theYoungs modulus is 2.5105 N/mm2. It increases gradually upto 6105 N/mm2. The useful value of the elasticity modulus ap-proaches this latter value, since in the machining process toolswork in a high-stress mode. This is the same value as that re-ported in other publications [14].

Appendix 2: Concept of look-ahead

The look-ahead functions of the machine tool numerical con-trol adapt the linear feed to variations in surface geometry. Thegreater the change in direction that the tool has to follow, thelower the actual work feed. To each commercial control therecorresponds a certain nomenclature for the look-ahead parame-ters, but in all cases calculations are similar. User can vary these

Fig. 14. Influence of look ahead parameters in mean error. All tests carriedout to programmed feed lower than 10 m/min

parameters to determine the compromise between process rateand machining precision. Parameters are usually five (P1, P2, P3,P4, P5).

Look-ahead calculations are performed with regard to eachthree points determined by the numerical control interpolator.The first calculation relates to the radius:

F1 = P1Radius (5)

The second relates to the angle between the two chords betweenthe three points :

F2 = P2 cos (angle) (6)A necessary condition is that the angle be not less than P4 andnot greater than P3. Should it exceed P3, speed falls to 0 for aninstant. Should it be less than P4, there is no fall in speed. Whenthese two speeds have been calculated, the numerical control ap-plies to the lesser. If the speed corresponding to F1 is applied, thevalue applied will never be less than P5.

Figure 14 shows the relation between precision and the sever-ity of the look-ahead parameters (and ultimately the speed andmachining time), for the three-axis high-speed machining cen-tre used in our work. As can be seen in Fig. 12, the look-aheadparameters affect the machining times of the piece from 23to 12.7 min. Behaviour is not linear, as is to be expected inview of Eqs. 5 and 6. There is a zone [AB] in which a littlechange in look-ahead values has a great effect on machiningprecision.