maximization of chatter-free material removal rate … · zero-order approximation, i.e., retaining...

MAXIMIZATION OF CHATTER-FREE MATERIAL REMOVALRATE IN END MILLING USING ANALYTICAL METHODS

A. Tekeli and E. Budak & Faculty of Engineering and Natural Sciences, SabanciUniversity, Tuzla, Istanbul, Turkey

& Chatter vibrations in milling, which develop due to dynamic interactions between the cuttingtool and the workpiece, result in reduced productivity and part quality. Various numerical andanalytical stability models have been considered in the previous publications, where mostly the stab-ility limit of axial depth of cut is emphasized for chatter-free cutting. In this paper an analyticalstability model is used, and a simple algorithm to determine the stability limit of radial depth ofcut is presented. It is shown that, for the maximization of chatter-free material removal rate, radialdepth of cut is of equal importance with the former. A method is proposed to determine the optimalcombination of depths of cut, so that chatter-free material removal rate is maximized. The appli-cation of the method is demonstrated on a pocketing example where significant reduction in themachining time is obtained using the optimal parameters. The procedure can easily be integratedto a CAD=CAM or virtual machining environment in order to identify the optimal milling con-ditions automatically.

Keywords Chatter Vibrations, Milling Stability, Material Removal Rate (MRR),Pocketing, Depth of Cut

INTRODUCTION

Milling has a widespread use in variety of applications from die andmold machining to manufacturing of complex aerospace parts. Pro-ductivity and surface quality in milling processes have direct effects on cost,production lead time and quality of machined parts. Chatter is one of themajor limitations for productivity and part quality even for high speed andhigh precision milling machines. In this paper, applications of analyticalmilling stability models for increased material removal rate will be pre-sented and demonstrated through examples.

Chatter vibrations develop due to dynamic interactions between thecutting tool and workpiece, and result in poor surface finish and reduced

Address correspondence to Alper Tekeli, Sabanci University, 34956, Tuzla, Istanbul, Turkey. E-mail:[email protected]

Machining Science and Technology, 9:147–167Copyright # 2005 Taylor & Francis Inc.ISSN: 1091-0344 print/1532-2483 onlineDOI: 10.1081/MST-200059036

tool life. Tlusty et al. (1) and Tobias (2) identified the most powerful sourceof self-excitation, which is associated with the structural dynamics of themachine tool and the feedback between the subsequent cuts on the samecutting surface resulting in regeneration of waviness on the cutting sur-faces, and thus modulation in the chip thickness (3). Under certain con-ditions the amplitude of vibrations grows and the cutting system becomesunstable. Although chatter is always associated with vibrations, in fact it isfundamentally due to instability in the cutting system. In a particular mill-ing process, for certain cutting speed and width of cut, there is a limitingaxial depth of cut above which the system becomes unstable, and chatterdevelops. Chip thickness, or feedrate, has very little, if any, affect on thestability limit. Additional operations, mostly manual, are required to cleanthe chatter marks left on the surface. Thus, chatter vibrations result inreduced productivity, increased cost and inconsistent product quality.

The stability analysis of milling is complicated due to the rotating tool,multiple cutting teeth, periodical cutting forces and chip load directions,and multi-degree-of-freedom structural dynamics, and has been investi-gated using experimental, numerical and analytical methods. In the earlymilling stability analysis, Tlusty (3) used his orthogonal cutting model con-sidering an average direction for the cut. Later, however, Tlusty et al. (4)showed that the time domain simulations would be required for accuratestability predictions in milling. Sridhar et al. (5, 6) performed a compre-hensive analysis of milling stability which involved numerical evaluationof the dynamic milling system’s state transition matrix. Minis et al. (7, 8)used Floquet’s theorem and the Fourier series for the formulation of themilling stability, and numerically solved it using the Nyquist criterion.Budak (9) developed a stability method which leads to analytical determi-nation of stability limits. The method was verified by experimental andnumerical results, and demonstrated to be very fast for the generation ofstability lobe diagrams (9–11). This method was also applied to the stabilityof ball-end milling (12), and it was extended to three dimensional stabilityanalysis (13).

The application of the fundamental stability theory to different practi-cal cases has been considered in several publications. One of the notableones is on the affect of tool dynamics on the stability limit (14) wherethe tool length is adjusted to improve stable depth of cut. The special caseof low immersion milling has been investigated in several studies whereadded lobes were also presented (15–18). Another method of chatter sup-pression in milling is through the use of variable pitch cutting tools. Theeffectiveness of variable pitch cutters in suppressing chatter vibrations inmilling was first demonstrated by Slavicek (20). Opitz et al. (21) andVanherck (22) analyzed the stability for a given pitch variation pattern.Tlusty et al. (23) also analyzed the stability of milling cutters with special

148 A. Tekeli and E. Budak

geometries such as irregular pitch or serrated edges, using numerical simu-lations. These studies mainly concentrated on the effect of pitch variationon the stability limit. Altintas et al. (24) adapted the analytical milling stab-ility model to the case of variable pitch cutters which can be used morepractically to analyze the stability with these cutters. Budak (25, 26) recentlydeveloped an analytical method for the design of pitch angles for givenchatter frequencies and spindle speeds. He demonstrated that these cutterscan improve the stability significantly even at low cutting speeds that cannotbe obtained by regular cutting tools. All of these studies demonstratedincreased stability with specially designed milling tools, however designand production of these cutters may not be feasible for every application.

In general, the stability analysis of milling starts with identification ofstructural dynamics which is used to generate stability diagrams. Stabilitydiagrams have been extensively used in high speed milling applications,to utilize the large stability pockets (27). Structural dynamics in milling sys-tems have traditionally been identified using experimental modal analysis.However, receptance coupling and substructure analysis (RCSA) wasrecently used by Schmitz et al. (28, 29) to combine the end mill dynamicswith the machine dynamics, using analytical beam modes for the tool andmeasured tool holder-spindle transfer function. Kivanc and Budak (30)developed more accurate analytical models for the end mill dynamics,and demonstrated applications of these models to different cases of endmill-tool holder combinations.

In general, the stability limit in terms of axial depth of cut is predictedusing stability diagrams for given machining conditions. Although thisensures that the maximum chatter-free axial depth of cut is used, it doesnot guarantee maximum chatter-free material removal rate. For certaincombinations of axial depth of cut and width of cut, the material removalrate is maximized, which is investigated in this paper. Also, in some cases itis much more practical to determine the stability limit in terms of the widthof cut, e.g., if the axial depth of cut is fixed to improve the surface finisheliminating tool marks and mismatches due to multiple passes. Themethod for generation of stability diagrams in terms of width of cut is pre-sented and demonstrated by examples. The methods presented in thispaper can be used for optimization of milling conditions for improved sur-face finish and maximized chatter-free material removal rate.

CHATTER STABILITY LIMIT

In this section, the analytical stability model of Budak (9–11) isreviewed as it will be used in the rest of the analysis. After an expressionfor the dynamic chip thickness under the effect of vibrations is derived,it can be used to formulate the dynamic milling forces. The Fourier series

Chatter-Free Material Removal Rate in End Milling 149

expansions for the periodically varying directional coefficients and theFloquet’s theorems are used to obtain the stability limit (9–11). Usingzero-order approximation, i.e., retaining only the first term in the Fourierseries expansion, the axial stability limit is obtained as in Equation (1). Forvery low radial immersion milling applications, added lobes may arise in thestability diagram as shown by Davies et al. (15), Corpus and Endres (18–19),and Merdol and Altintas (17). However, the zero order solution has beenshown to provide accurate stability predictions for most of the practicalmilling applications (11–13), which form the focus of this paper.

alim ¼ � 2pKI

NKtðjþ 1 j= Þ ð1Þ

where j ¼ KR=KI . The eigen values can be obtained from

K ¼ � 1

2a0a1 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia21 � 4a0

q� �ð2Þ

where

a0K2 þ a1Kþ 1 ¼ 0

a0 ¼ GxðiwcÞGyðiwcÞðaxxayy � axyayxÞ ð3Þ

a1 ¼ axxGxðiwcÞ þ ayyGyðiwcÞ

Transfer functions are expressed as Gx ¼ Gxc þ Gxw where Gxc and Gxw

are the cutter and workpiece transfer functions in the x direction, respect-ively. The directional milling coefficients are given as

axx ¼1

2cos 2h� 2Krhþ Kr sin 2h½ �/ex

/st

axy ¼1

2� sin 2h� 2hþ Kr cos 2h½ �/ex

/st

ayx ¼1

2� sin 2hþ 2hþ Kr cos 2h½ �/ex

/st

ayy ¼1

2� cos 2h� 2Krh� Kr sin 2h½ �/ex

/st

ð4Þ

Spindle speed is related to the tooth period as follows

n ¼ 60=NT ð5Þ


The speed corresponding to a considered chatter frequency can befound from

wcT ¼ eþ 2kp ð6Þ

where the phase difference between two subsequent waves is given by

e ¼ p� 2u ð7Þ

where

u ¼ tan�1 KI

KR¼ tan�1 1

jð8Þ

The unknowns in these equations are the chatter frequency, wc , and theaxial depth of cut alim for the defined number of teeth N, start and exitangles /st , /ex (which define the radial depth of cut B), spindle speed n,and milling force constants Kt and Kr . Note that, the radial depth of cut,B, is defined as follows:

B=R ¼ 1� cosð/exÞ ðup-milling Þ ð9Þ

B=R ¼ 1þ cosð/stÞ ðdown-milling Þ ð10Þ

A normalized form of the radial depth of cut, b ¼ B=2R is used in the rest ofthe formulation for simplicity and generalization. Thus, b is unitless, and itmay only have values in the range of [0, 1].

As the characteristic determinants have imaginary and real parts, twounknowns, i.e., wc and alim or wc and blim , can be solved. Consequently, stab-ility diagrams can be generated, in terms of either alim vs. spindle speed (for afixed blim) or blim vs. spindle speed (for a fixed alim). The common practice is toexpress stability diagrams in terms of alim vs. spindle speed, and algorithms forthat are shown in (9). Here, a simple iterative algorithm to generate stabilitydiagrams in terms of blim vs. spindle speed is presented. The importance ofidentifying blim is twofold. First, in some roughing cases, a is fixed, so themaximum available b should be used to attain maximum productivity.Second, maximum MRR can only be achieved by optimizing both a and b.

The proposed algorithm is as follows:

1. Set the system and process parameters, which are outlined in Table 1.2. Select an alim for which the stability diagram, blim vs. spindle speed, will be

generated.3. Select aprobable chatter frequency range(wc min,wc max). (Chatter vibrations

occur at frequencies that are close to the natural frequencies of the system.)4. Select the step size (Dw) for the frequency loop.


5. Frequency loop:For wc ¼ wc min :Dw :wc max

i. Calculate Gx, Gy for given modal values of m, k, and wn.ii. By scanning the full range (0 to 180�) of exit angles (in the case of

up-milling) or start angles (in the case of down-milling) calculateaxx , axy, ayx , ayy, a0, a1, and the eigenvalues using the Equations (2–4).

iii. Determine /ex or /st for which the alim calculated using Equation (1)equals the selected value of alim in Step 2.

iv. Calculate blim using Equations (9) or (10).

6. Calculate the spindle speeds corresponding to the chatter frequencyusing the Equations (5–8) for k ¼ 1, 2, 3, 4, 5 . . .

CHATTER-FREE STABLE PAIRS OF AXIAL AND RADIALDEPTHS OF CUT

As explained in the previous section, either of the chatter free axial andradial depths of cut can be determined using the stability model when theother is specified. However, the maximum chatter free material removalrate may only be attained for a certain combination of radial and axialdepths of cut, which depends on the machining conditions analyzed anddemonstrated in this and the next sections. Throughout the analysis, simu-lations for different cases were carried out. One of those was the systemconsidered by Weck et al. in (31). For that system, the parameters are givenin Table 2.

Figures 1 and 2 show the stability diagrams generated using the analyti-cal method.

One importance of identifying blim is clearly seen in Figure 2. Such acase shows that, there may be a possibility of increasing the productivityby increasing the spindle speed or the axial depth of cut without sacrificing

TABLE 1 Parameters for the Milling System

wnx Natural frequency in the x directionwny Natural frequency in the y directionkx Stiffness in the x directionky Stiffness in the y directionfx Damping ratio in the x directionfy Damping ratio in the y directionN Number of flutesKt Tangential cutting force coefficientKr Radial cutting force coefficient/st ¼ 0 Start angle of a tooth (for up-milling)/ex ¼ 0 Exit angle of a tooth (for down-milling)


blim . That is another reason why the maximumMRR can only be achieved byoptimizing both a and b. If we focus on a specific spindle speed where stab-ility limits are the highest, i.e., in the lobes, we see that, no decrease in b isnecessary for some increase in a, however, after a certain point, a negativelysloped relation exists between the stable limits of axial and radial depths ofcut as represented in Figure 3.

EXPERIMENTAL VALIDATION OF CHATTER STABILITY LIMIT

Before continuing with the analysis, experimental validation of theabove analysis is presented in this section in order to make sure the rest

FIGURE 1 Stable limit of axial depth of cut vs. spindle speed.

TABLE 2 Parameters for the Milling System as used inWeck et al. (31)

wnx 600 Hzwny 660 Hzkx 5600 kN=mky 5600 kN=mfx 0.035fy 0.035N 3Kt 600 mPaKr 0.07/st 0 (up-milling)


of the study is based on a sound analytical foundation. In the first part,validity of the stability diagram in the form of blim vs. spindle speed for a givenalim is investigated, whereas the second part is concerned with the relationbetween blim and alim at a given spindle speed, more specifically, thesensitivity of the stable limits of axial and radial depths of cut with respectto each other.

FIGURE 2 Stable limit of radial depth of cut vs. spindle speed.

FIGURE 3 Stable limit of radial depth of cut vs. stable limit of axial depth of cut.


The experiments are carried out on a 2-DOF end milling system. Thealuminum workpiece is assumed to be rigid whereas the modal analysisrevealed the dynamic parameters in Table 3 for the 4-fluted HSS end millwith 16mm diameter and 85mm gauge length at the free end of the cutter.

Chatter tests using this tool were conducted on a high-speed machiningcenter shown in Figure 4.

In the first part, stability lobes (Figure 5) in terms of the radial depth ofcut are obtained with a simulation using the algorithm presented in earlier2 for the case of down-milling. (Kt ¼ 796 MPa, Kr ¼ 0.21, feed rate ¼ 0.025

FIGURE 4 High-speed machining center used for the chatter tests.

TABLE 3 Modal Parameters for the End Mill Used in Experiments

Direction x y

wn1 (Hz) 989 1001wn2 (Hz) 1202 1209wn3 (Hz) 1469 1505k1 (kN=m) 27372 30346k2 (kN=m) 7185 5178k3 (kN=m) 3364 3305f1 0.0222 0.0242f2 0.0327 0.0451f3 0.0231 0.0337


mm=tooth). The simulations are carried out for a fixed axial depth of cut of1.5mm.

The experimental results were in very good agreement with the predic-tions as shown in Figure 5. The chatter was evident especially on the fin-ished surfaces as shown in Figure 6. As it can be seen from the stabilitydiagram and the experimental results in Figure 5, the chatter-free widthof cut can be maximized using the proper spindle speeds. In this appli-cation, width of cut of 4.8mm could be obtained at 11,000 rpm as opposedto 10,000 rpm where chatter was experienced. Similarly, for 1.5mm axialdepth of cut, a chatter-free width of cut of about 6.4mm can be obtainedat a speed close to 12,000 rpm instead of using width of less than 3mmat 4,000 rpm resulting in more than 6 times increase in chatter-free materialremoval rate as shown in Figure 5.

FIGURE 5 Stable limit of radial depth of cut vs. spindle speed (experimental validation, a ¼ 1.5mm).

FIGURE 6 Milled surfaces with chatter (left and middle: for 12,000 rpm and 10,000 rpm, respectively)and no chatter (right: 11,000 rpm ).


The second part demonstrates the sensitivity of blim and alim with respectto each other at a given spindle speed. The limit of stability for pairs of vary-ing depths of cut is obtained by simulation and shown in Figure 7 as thesolid curve. Here, spindle speed is set at 11,000 rpm. As seen in the figure,experimental results are again in agreement with the predictions, thoughsome error is present most probably due to the relatively unreliable natureof the modal analysis measurement.

These examples demonstrate that the analytical chatter model can beused in order to determine chatter-free radial depth of cut for the caseswhere the axial depth is predefined and in order to see the sensitivity ofthe stable limits of depths of cut with respect to each other.

MAXIMIZATION OF CHATTER-FREE MATERIAL REMOVAL RATE

Since material removal rate (MRR) is proportional to the multiplicationof the axial and radial depths of cut, it is interesting to find out at whichcombination of axial and radial depths of cut, the maximum value ofMRR may be achieved.

MRR ¼ a�b�n�N �ft ð16Þ

where, a is the axial depth of cut, b is the radial depth of cut, n is the spindlespeed, N is the number of flutes, and ft is the feed per revolution per tooth.

FIGURE 7 Stable limit of radial depth of cut vs. stable limit of axial depth of cut at 11,000 rpm(experimental validation).


In general, effect of ft on stability is small and can be neglected. Therefore,a normalized value of MRR is used hereafter:

MRR� ¼ MRR

ff¼ a�b�n�N ð17Þ

As a result of simulations and analyses of different cases, it is found out thatdifferent situations regarding maximum MRR � may occur.

In the preceding figures, MRR � value for an axial depth of cut is calcu-lated using the blim corresponding to that alim . The procedure presentedearlier is followed to obtain the blim values corresponding to alim values. Sub-sequently, using Equation 16, the MRR � value is obtained as shown next.

MRR� ¼ alim:blim:n:N ð18Þ

It is seen that in some cases MRR� reaches a maximum, then starts todecrease (Figure 9), and in some others it continually increases. Through-out this research, it is realized that the difference stems from conditionsregarding the machine tool dynamics and the feed direction more thananything else. More specifically, everything else being constant, if the natu-ral frequencies of the cutter and workpiece system are different in the xand y directions significantly, e.g., more than 10%, and the milling is inthe direction of lower natural frequency, then it is more likely to see a situ-ation as in Figure 8. This phenomenon is explained in more detail in thenext section.

FIGURE 8 Maximum MRR vs. stable limit of axial depth of cut (Case 1).


EFFECT OF MACHINE TOOL DYNAMICS AND FEED DIRECTIONON CHATTER-FREE MRR

Whether the MRR � curve has a peak or not clearly depends on therelation between the stable limits of axial and radial depths of cut (whichmay be represented as in Figure 3), due to the fact that MRR � involvesthe multiplication of the two. A peak may occur depending on the rate ofdecrease of limits with respect to each other. Higher rate of decrease createsgreater multiplicative effect on MRR �, and thus the possibility of a peak.

Interestingly, the natural frequencies of the cutter and workpiece systemhave a significant effect on the rate of decrease of limits with respect to eachother. It is difficult to verify this effect mathematically due to the complexity ofthe stability equations, or attach a physical meaning to it. However, the effect isclearly seen in the simulations of different cases. For the system of Weck et al.(31), two different cases are simulated (Table 4). The effect of machine tooldynamics and feed direction are seen in the resulting Figures 10–13.

In both cases, the dotted curves represent the situation where the natu-ral frequencies in x and y directions are at least 10% different, and cuttingis in the direction of the lower natural frequency (cutting is in x directionin the above cases). The solid curves in both cases correspond to the situ-ation where the natural frequencies are closer. As seen in Figure 10 andFigure 12, the rate of decrease of limits with respect to each other is muchhigher when the natural frequencies in x and y directions are far from eachother. Consequently, MRR � curves corresponding to those cases have amaximum value at a certain combination of alim and blim .

FIGURE 9 Maximum MRR vs. stable limit of axial depth of cut (Case 2).


FIGURE 10 Stable limit of radial depth of cut vs. stable limit of axial depth of cut (Case 1a, 1b).

TABLE 4 Parameters for Case 1 and Case 2

Case 1a Case 1b Case 2a Case 2b

wnx (Hz) 620 600 1775 1700wny (Hz) 640 660 1825 1900

FIGURE 11 Maximum MRR vs. stable limit of axial depth of cut (Case 1a, 1b).


APPLICATIONS: DETERMINING THE OPTIMAL PAIR OFDEPTHS OF CUT FOR THE MINIMUM POCKETING TIME

Pocketing is a very common operation in milling, e.g., in die, mold, andairframe production, etc. Pocketing time simply depends on the materialremoval rate and the dimensions of the pocket. Both should be considered

FIGURE 12 Stable limit of radial depth of cut vs. stable limit of axial depth of cut (Case 2a, 2b).

FIGURE 13 Maximum MRR vs. stable limit of axial depth of cut (Case 2a, 2b).


simultaneously to calculate the minimum pocketing time. As explained inthe previous sections, if the dynamics of the machine tool is known, theoptimal depths of cut for the maximum MRR � can be easily determined.However, those obtained values for the maximum MRR � might not bethe optimal for the case of a pocketing.

In order to find the minimum pocketing time, to think in terms ofnumber of passes (nop) is more convenient. The term ‘‘pass’’ stands forone cutting pass across the pocket length (Figure 14).

Total pocketing time (TPT) can be roughly expressed as:

TPT ¼ nop �wp

fð19Þ

That is to say, minimizing total pocketing time is equivalent to minimiz-ing number of passes since we keep f (feed rate) constant throughout theanalysis. Note that the affect of feedrate on the stability limit is minimalwhich is neglected. A proper feedrate value should be used consideringthe other constraints such as the tool capacity and the surface finishrequirement. Number of passes can then be expressed as:

nop ¼ ceildpa

� ��ceil

lpb

� �ð20Þ

where dp is the pocket depth, lp is the pocket length, ceil is the round-upfunction. The ‘‘ceil ’’ function is used, because even if the remaining partfor the final pass is smaller than the geometry determined by axial andradial depths of cut, the time necessary for that final pass does not change.

FIGURE 14 Pocketing geometry.


It is because of the ‘‘ceil ’’ functions that maximizing MRR � might not bethe best for minimizing nop.

The steps of the proposed method are shown in Figure 15. Once thepocket geometry and the appropriate tool are identified, workpiece andcutter dynamics are to be measured, so that the pairs of stable limits of axialand radial depths of cut can be determined using simulations. It is thenstraightforward to find the minimum nop numerically using Equation(20) after a table that presents the pairs of stable limits of axial and radialdepths of cut is formed (as in Tables 5 and 6).

Next two cases that illustrate the method are presented. In one of thecases, MRR � behaves as the dotted curve in Figure 10 (Case 1a,wnx ¼ 620Hz, wny ¼ 640Hz), in the other as the solid curve in Figure 10(Case 1b, wnx ¼ 600Hz, wny ¼ 660Hz).

Results of the optimization are presented in Tables 7 and 8. The firsttwo columns of the tables show the required pocket depth and length.The next three column-sets present the resulting number of passes (nop)for three different methods. The first method, Conventional 1, stands forchoosing a high radial depth of cut close to full slotting, 0.8 in thiscase (Full slotting is not investigated since it is assumed that the processplanner would like to prevent accelerated tooth wear from abrasion and

FIGURE 15 Overview of the proposed method.


the tendency for the chip to stick to the tooth, both of which might happendue to the thin chip section at the entry of the teeth (32)). The secondmethod, Conventional 2, stands for cutting with half immersion. The thirdcolumn set, Optimal, presents the results for choosing the optimal pair ofdepths of cut according to the method presented in this paper. Subse-quently, the last two columns show the percentage improvements in pock-eting time attained by the optimal combination.

For example, let us look at the case where the pocket depth is equal to8mm, in Table 7. If the first method, Conventional 1, is employed, then theradial depth of cut is chosen as 0.8, as explained in the previous paragraph.

TABLE 6 Pairs of Stable Limits of Axial and Radial Depths of Cut (Case 1b)

alim (mm) blim (mm)

4.00 1.006.00 0.738.00 0.4710.00 0.3512.00 0.2814.00 0.2318.00 0.16

TABLE 5 Pairs of Stable Limits of Axial and Radial Depths of Cut (Case 1a)

alim (mm) blim (mm)

4.00 1.006.00 0.838.00 0.6510.00 0.5212.00 0.4414.00 0.3820.00 0.27

TABLE 7 Optimal Depths of Cut for the Minimum Pocketing Time (Case 1a)

PocketConventional 1

b ¼ 0.8Conventional 2

b ¼ 0.5 Optimal Improvement

Depth (mm) Length=D a (mm) nop a (mm) nop a (mm) b nop Over conv. 1 Over conv. 2

4 10 4 13 4 20 4 0.8 13 0 356 10 6 13 6 20 6 0.8 13 0 358 10 4-4 26 8 20 8 0.65 16 38 20

10 10 5-5 26 10 20 10 0.52 20 23 012 10 6-6 26 6-6 40 12 0.44 23 11 4214 10 5-5-4 39 7-7 40 14 0.38 27 30 3220 10 5-5-5-5 52 10-10 40 20 0.27 38 27 5


Here, it is assumed that the process planner knows the corresponding stablelimit of axial depth of cut to a certain extent by experience. Consequently,axial depth of cut is assumed to be chosen as 4mm, which is quite close tothe chatter limit when b is equal to 0.8. Accordingly, the 8mm deep pocketwill be cut in two steps, 4mm and 4mm (in depth), as denoted in the thirdcolumn. The resulting number of passes is 26 (calculation shown below):

nop ¼ ceil8

4

� ��ceil 10

0:8

� �

Likewise, if half-immersion was preferred, resulting number of passes wouldbe 20. Lastly, in the optimal case, an axial depth of 8mm and a radial depthof 0.65 are to be used, leading to 16 passes, which results in 38% reductionin pocketing time with respect to Conventional 1, and 20% reduction withrespect to Conventional 2. The results are similar for the Case 1b, althoughafter a certain pocket depth, it is seen that using lower axial depths of cutyields lower pocketing times, which is expected since in Case 1b, as opposedto Case1a, maximum MRR � value (Figure 11) decreases after some point asexplained in the previous sections.

In conclusion, the optimal combination of axial and radial depths ofcut, which gives the minimum pocketing time, might be quite differentthan those determined in the conventional ways, i.e., arbitrarily picking aradial depth of cut between 50% and 80% immersion. It is shown thatby using the optimal combination, significant saving, up to around 40%reduction in pocketing time may be achieved.

CONCLUSIONS

Chatter-free material removal rate in milling can be improved using theanalytical stability model. In milling stability analysis, axial depth of cut has

TABLE 8 Optimal Depths of Cut for the Minimum Pocketing Time (Case 1b)

PocketConventional 1

b ¼ 0.8Conventional 2

b ¼ 0.5 Optimal Improvement

Depth (mm) Length=D a (mm) nop a (mm) nop a (mm) b nop Over conv. 1 Over conv. 2

4 10 4 13 4 20 4 0.8 13 0 356 10 3-3 26 6 20 6 0.72 14 46 308 10 4-4 26 4-4 40 8 0.47 22 15 4510 10 5-5 26 5-5 40 5-5 0.8 26 0 3512 10 4-4-4 39 6-6 40 6-6 0.72 28 28 3014 10 5-5-4 39 7-7 40 7-7 0.56 36 7 1018 10 5-5-4-4 52 6-6-6 60 6-6-6 0.72 42 19 30


mostly been used as a limiting factor. The material removal rate, on theother hand, depends on both axial and radial depth of cuts, and thus bothlimits have to be considered in order to maximize productivity. In thispaper, a method is presented for the determination of the chatter limitin terms of radial depth of cut. This procedure is also used in order todetermine the optimal axial and radial depth combination for maximumchatter-free material removal rate. It is shown that the maximum materialremoval rate is obtained for a certain radial depth of cut, which dependson the relative values of milling system’s modal frequencies in feed and nor-mal directions. The application of the method is demonstrated on a pock-eting example where significant reduction in the machining time (around40%) is obtained using the optimal parameters compared to the intuitive,and most commonly used, milling conditions. The method can be used toidentify the optimal chatter-free milling conditions for given system trans-fer functions in any milling application and can be easily integrated to aCAD=CAM or virtual machining environment in order to identify the opti-mal milling conditions automatically.

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