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INTERACTION BETWEEN MULTIPLE MODESIN MILLING PROCESSESJokin Munoa a , Zoltan Dombovari b , Iker Mancisidor a , Yiqing Yang c

& Mikel Zatarain aa Ideko-IK4 Technological Center , Elgoibar , Spainb Department of Applied Mechanics , Budapest University ofTechnology and Economics , Budapest , Hungaryc School of Mechanical Engineering and Automation, BeihangUniversity , Beijing , PR ChinaPublished online: 06 May 2013.

To cite this article: Jokin Munoa , Zoltan Dombovari , Iker Mancisidor , Yiqing Yang & Mikel Zatarain(2013): INTERACTION BETWEEN MULTIPLE MODES IN MILLING PROCESSES, Machining Science andTechnology: An International Journal, 17:2, 165-180

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INTERACTION BETWEEN MULTIPLE MODES IN MILLINGPROCESSES

Jokin Munoa1, Zoltan Dombovari2, Iker Mancisidor1, Yiqing Yang3, andMikel Zatarain1

1Ideko-IK4 Technological Center, Elgoibar, Spain2Department of Applied Mechanics, Budapest University of Technology and Economics,Budapest, Hungary3School of Mechanical Engineering and Automation, Beihang University, Beijing, PR China

& The productivity of many industrial cutting processes is limited by high amplitude chatter vibra-tions. An optimization technique based on the use of the stability lobes helps to increase the pro-ductivity of these processes, improving the life of machine elements and reducing the tool wear aswell. The best-known lobes correspond to Hopf bifurcations. However, in case of interrupted cutting,additional lobes appear due to period doubling or flip bifurcation. When the system has more thanone dominant vibration mode, important variations can appear in stability due to interactionbetween modes. The basic mathematics for the appearance of these new lobes are shown in this article.The frequency domain study shows that lobes related to flip bifurcation are a special case of theinteraction between modes. The results of these interactions are verified by comparison withsemi-discretization method and time domain simulations, respectively.

Keywords chatter, milling, stability

INTRODUCTION

In metal cutting, self-excited vibrations can limit productivity and putprocess safety at risk, giving rise to poor surface quality or even machinetool component failure. In industry, the use of stability diagrams has pro-vided a practical way to select the optimum conditions. The regenerativeeffect was defined as the main cause of machine-tool chatter by Tobias(Tobias and Fishwick, 1958) and Tlusty (Tlusty and Polacek, 1963) almostsimultaneously. Later, Merritt (1965) presented the problem as a feedbackloop, clarifying the problem from an engineering point of view.

Address correspondence to Jokin Munoa, Ideko-IK4 Technological Center, Arriaga Kalea, 2, 20870,Elgoibar, Basque Country, Spain. E-mail: jmunoa@ideko.es

Machining Science and Technology, 17:165–180Copyright # 2013 Taylor & Francis Group, LLCISSN: 1091-0344 print=1532-2483 onlineDOI: 10.1080/10910344.2012.747935

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In milling processes, the governing equation is a time periodic piecewisesmooth delay differential equation (DDE) (Hale, 1977; Stepan, 1989). Stabilityanalysis is therefore much more complex than those representing continuousregenerative cutting processes. Nevertheless, significant progress has beenmade in the study of milling stability during the last decades.

In 1995, Altintas and Budak presented a semi-analytical method forchatter analysis in frequency domain. Later, Budak and Altintas (1998)worked out this solution by considering several terms of in the Fourierdevelopment of the directional matrix.

The single frequency approach has been shown to be very precise, but incase of low immersionmilling, the existence of additional stability lobes relatedto period doubling or flip bifurcation was found. Davies et al. (2002) used adiscrete map tomodel highly interruptedmilling processes, where the engage-ment is infinitesimal and the cutting process is modelled as an impact. Insper-ger and Stepan (2000) used an approximation method called Fargue-method.Later, they developed the semi-discretization (SD) technique (Insperger andStepan, 2002), and Bayly et al. (2002) developed the temporal finite elementsmethod, obtaining similar results. Merdol and Altintas (2004) showed that themulti-frequency (MF) method is also able to represent accurately the perioddoubling instability phenomenon. At the same time, Corpus and Endres(2004) studied lobes related to flip bifurcations using an analytical approach.

Several authors have shown that some lens-shaped instability regions appearfor period doubling zones of third and higher order in interrupted turning(Szalai and Stepan, 2006). By using the multi-frequency approach, Zatarainet al. (2006) showed that the tool helix produces the transformation of the addedlobes (period doubling) into instability islands. Insperger et al. (2006) arrived tothe same conclusion by using the semi-discretization method. Later, Patel et al.(2008) obtained the same results by means of temporal finite element method.

In industrial applications, such as steel face milling, it is common tofind situations with more than one dominant mode at different frequen-cies. In this work, the interactions between different modes are consideredand their effect on the stability chart is studied using the multi-frequency(MF) approach. Moreover, semi-discretization (SD) (Insperger and Stepan,2011) method and time domain milling simulations (TDS) are used to vali-date the results. The objective of this article is to analyze the zero-orderapproximation (ZOA) performance for the general case, considering morethan one dominant mode. New inaccuracies of the ZOA have been found.

FREQUENCY DOMAIN MILLING STABILITY MODEL IN MODALCOORDINATES

The characteristic equation for milling stability analysis in frequencydomain was developed by Budak and Altintas (1998). Their approach was

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based on Cartesian displacements between the tool and the workpiece. Inorder to describe the regenerative effect, the dynamic cutting force is con-sidered, and the relation between the force and the vibrations is developed.For face milling operations, the dynamic cutting force can be given in thefollowing form (Altintas, 2003; Munoa et al., 2005):

fF ðtÞg ¼ Ktb

sin j

� �½AðtÞ�fDr ðtÞg ð1Þ

where

fDrðtÞg ¼ frtðtÞ � rtðt � sÞg � frwðtÞ � rwðt � sÞg ð2Þ

Here, Dr(t) is the relative regenerative vibration between the tool (t) andthe workpiece (w), b is the depth of cut (see Figure 1), Kt is the tangentialcutting coefficient, s is the tooth passing period, j is the lead angle and[A(t)] is the Cartesian directional factor matrix. The directional factormatrix concentrates the projection of the cutting force onto the modedirection and the projection of the vibration onto the chip thickness.The Cartesian approach leads to the use of the corresponding directionalcoefficient matrix [A(t)].

The modal approach is more convenient for the present study and, forthis purpose, the force expression has to be transformed into the modalspace. Therefore, the relative normalized modal transformation matrix[Q], the modal displacement fgg and the modal force fPg are introducedas

fDr ðtÞg ¼ ½Q �fDgðtÞg ð3Þ

fPðtÞg ¼ ½Q �TfF ðtÞg ð4Þ

where fDgðtÞg ¼ fgðtÞ � gðt � sÞg; ½Q � ¼ ½Qt� � ½Qw�; ½Qt� and ½Qw� are thelocal modal transformation matrices of the tool and the workpiece. Hence,the dynamic milling force can be expressed in modal coordinates definingthe modal directional factor matrix [B(t)]:

fP ðtÞg ¼ � Ktb

sin j

� �½BðtÞ�fDgðtÞg ð5Þ

where

½BðtÞ� ¼ �½Q �T½AðtÞ�½Q � ð6Þ

In milling, the dynamic cutting force is periodic at tooth passingperiod s. In case of unstable stationary cutting, the vibration has a

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frequency (xc) close to one of the dominant natural frequencies andsome modulations xc,k¼xcþkX related to the tooth passing frequencyX ¼ 2p=s. The frequency of these modulations depends on the numberof flutes (Z) and the spindle speed (n). The next formulation can beconsidered for the modal milling force and vibration:

fP ðtÞg ¼X1k¼�1

fPkge jðxcþkXÞt ð7Þ

and

fgðtÞg ¼X1r¼�1

fgkge jðxcþrXÞt ð8Þ

Considering this frequency pattern and operating, the regenerative termcan be rewritten as

fDgg ¼ ð1� e�jxcsÞfgg: ð9Þ

The modal directional matrix is also time periodic and, consequently, adiscrete Fourier development is possible, thus

½BðtÞ� ¼X1l¼�1

½Bl � e j l X t ¼ Z

2p

X1l¼�1

½bl � e j l X t ¼ !

ð10Þ

Taking into account all the different developments in Equations (5)and operating with the different harmonics as a product of Equations(7), (8), and (10), the next expression is obtained

X1k¼�1

fPkg e jðx cþk XÞt ¼� Kt b Z

2 p sin j

� �ð1� e�jxc sÞ

X1k¼�1

X1r¼�1

½bk�r �fgrg !

e jðx cþkXÞt :

ð11Þ

The dynamic modal forces and displacements can be related using thedynamic properties of the mechanical structure. Therefore, consideringthe relative frequency response function (FRF) of the system [U] betweentool (t) and workpiece (w), the next expression can be written for eachharmonic component:

fgkg ¼ ½Uðxc þ kXÞ�fPkg ð12Þ

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where

½UðxÞ� ¼ diagN

i¼1

1

mi ½ðx2n;i � x2Þ � 2jnixn;ix�

!ð13Þ

Note that, xn,i, ni, and mi, are, respectively, the natural frequency, the damp-ing ratio, and the reflected modal mass of the ith mode associated to thetool and the workpiece. Following Budak and Altintas’s (1998) develop-ment, it is possible to obtain a closed loop formulation. The main equationrelates different harmonics of the modal displacement gk taking intoaccount the directional factor matrices (10) for each harmonic and FRFmatrices (12) evaluated at different harmonics. Therefore;

fgkg ¼ � KtbZ

2p sinj

� �ð1� e�jxcsÞ

X1r¼�1

½Uðxc þ kXÞ�½Bk�r �fgrg ð14Þ

Finally, the stability problem results in an infinite dimensional matricialexpression, that is,

..

.

fg�hg...

fg0g...

fghg...

8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>;

¼� Kt b Z

2p sinj

� �ð1� e�jxc s Þ

. .. ..

. ... ..

.. . .

� � � ½U�h� � � � ½0� � � � ½0� � � �... . .

. ... . .

.

� � � ½0� � � � ½U0� � � � ½0� � � �... . .

. ... . .

. ...

� � � ½0� � � � ½0� � � � ½Uh� � � �

. . . ... ..

. ... . .

.

26666666666666664

37777777777777775

. .. ..

. ... ..

.. . .

� � � ½b0� � � � ½b�h� � � � ½b�2h� � � �... . .

. ... . .

.

� � � ½bh� � � � ½b0� � � � ½b�h� � � �... . .

. ... . .

. ...

� � � ½b2h� � � � ½bh� � � � ½b0� � � �

. . . ... ..

. ... . .

.

26666666666666664

37777777777777775

..

.

fg�hg...

fg0g...

fghg...

8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>;

ð15Þ

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where fgkg are the amplitudes of vibration of all the considered modesfor the kth modulated frequency, and [Uk]¼ [U(xcþ kX)] and [Bk] theFRF matrix and the modal directional factor matrix evaluated at the kth

modulated chatter frequency.In a theoretical basis, the size of thematrices is infinite, but in practice the

FRF takes very small values for frequencies far from the natural frequencies.Therefore, the system can be truncated without noticeable loss of accuracy.

The solution of this equation results in an eigenvalue problem wherethe obtained eigenvalues are related to the spindle speeds X=Z and depthof cuts b. Different numerical methods have been proposed to obtain themulti-frequency solution (Budak and Altintas, 1998; Merdol and Altintas,2004). If only the zeroth-order term is considered, a fast semi-analyticalsolution is possible (Altintas and Budak, 1995).

For high tooth passing frequencies X, even a low matrix size in (15)produces vibration frequencies xc � X with negative values. It must benoted that any vibration at frequency x can be considered as the sum oftwo complex vibrators, at frequency x and at its complex conjugate at�x. Therefore, when a spectrum contains some energy at frequency x,the energy is shared among frequencies at x and �x.

A formulation based on Cartesian displacements can also be used(Budak and Altintas, 1998), which may be useful because FRFs are oftenmeasured in Cartesian coordinates. Practically, when the number of con-sidered modes is bigger than three, the modal approach gives rise to biggermatrix dimensions and larger calculation times. Next, the inaccuracies ofthe ZOA are analyzed, considering basically the MF analysis and comparingthe results with SD and time domain milling simulations.

STABILITY LOBES WITH SEVERAL DOMINANT MODES

In general, a system with multiple modes refers more reliably to theotherwise high dimensional dynamics of a machine tool than a dynamicalmodel with a single mode. In machine dynamics, a two-mode model is ableto represent the complexity of a case with higher number of modes; whereas,single mode models do not capture the interactions between modes. There-fore, the results obtained for two modes offer a good view of dynamicallycomplex machine tool structures by means of an affordable mathematicalload. Most of the results can be extrapolated to the case of multiple modes.For this reason, a mechanical system with two dominant modes has beenposed in this section. An example will be considered in order to describethe different inaccuracies of the ZOA (see Table 1) in a clearer way.

In a single mode system, the angles between mode and milling force,and between mode and chip thickness direction are really important. If

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systems with two modes are considered, there are more important para-meters: the projection of both modes onto the chip thickness and cuttingforce, the difference between the frequencies of both modes and the differ-ences between their damping ratios and modal stiffness. Therefore, it iscomplex to make a dimensionless study of the system. Two modes in perpen-dicular directions have been chosen with different natural frequencies avoid-ing the most special case when the modes are identical in their dynamicparameters. Remark that, in modal space, perpendicular modes are not aspecial case. With the chosen example the presence of the double periodchatter and mode couplings is assured and, luckily, some stable islands exist.

Comparison between ZOA, Multi-frequency Solution, andSemi-discretization

Several simulations have been carried out using different down millingradial immersions (see Figure 2). Finally, the case of 25% radial engage-ment (RE) is studied. The dynamics and the cutting coefficients have beenchosen according to standard conditions for steel face milling with univer-sal milling machines (see Table 1). A straight-fluted tool with four teeth anda diameter of D¼ 50mm has been selected to make the stability calcula-tions. The lead angle of the tool is j¼ 90�.

If the results between the ZOA and the multi-frequency methods arecompared, it is clear that the results diverge in several zones of the lobe dia-grams as the cutting force becomes more interrupted (see Figure 2(a)–(c)).If a linear model is considered, a slotting operation with four flutesproduces a constant cutting force. In this case, there is no harmonics inthe dynamic cutting force and in the modal directional matrix [B]; thus,the ZOA solution is exact. On the other hand, when the engagement isreduced or even when the slotting is performed using a tool with less than

TABLE 1 Simulation Parameters

Tool

Diameter, D(mm) Number of flutes, Z Helix angle, g(deg) Lead angle, j(deg)50 4 0 90

Cutting conditions & coefficients

Kt (N=mm2) Kr (N=mm2) Sense Feed Direction (f)2000 600 Down Milling (1,0,0)

Dynamic Parameters

i xn,i (Hz) ni (%) ki (N=mm) Orientation (v)1 45 4 30 (1,0, 0)2 60 4 30 (0,1,0)

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four flutes, discrepancies among methods arise. Two different effects arecreating these deviations in the case of a dynamic system with two or moremodes: the period-doubling instability (flip bifurcation) and the modeinteraction.

FIGURE 1 Schematic milling model.

FIGURE 2 Stability charts in case of down milling processes with parameters taken from Table 1.

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The semi-discretization method shows the presence of stable isolatedzones for engagements of 25% and 50% (see Figure 2(b),(c)). The multi-frequency method also shows points limiting this stable region, but it isimpossible to completely identify a stable zone due to the fact that theMF solution directly obtains the stability limits as a parametric functionof the vibration frequency.

Period Doubling Instability

When angular immersion is small, an additional family of stability lobesappears, producing an increase of stability in a small area and a reductionin a larger zone. The main characteristic of these lobes can be recognizedin its special spectral pattern in the frequency diagram (see Figure 3(a)),where the vibration (chatter) frequencies related to these additional lobesfollow straight lines. Hence, chatter frequency has a direct relationship withcutting frequency.

xc ¼ ðm=2ÞX; wherem ¼ 1; 3; 5; . . . ð16Þ

If one point of this additional lobe is considered (see point C in Figure 3and Figure 4) and studied in frequency domain, it can be seen that boththe zeroth-order term and one of its harmonics (in this case �1 harmonic)

FIGURE 3 (a) The frequency plot of the critically stable solutions using grayscale showing the strengths ofthe harmonics (Dombovari et al., 2011). (b) The linear stability of milling process (cf. Table 1).

Mode Interactions in Milling 173

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are exciting the same mode. Operating in double period chatter, the regen-erative term of Equation (14) has the next form:

1� e�jxcs ¼ 1� e�jpm ð17Þ

Therefore, the regenerative term in case of double period chatter is a realnumber (Corpus and Endres, 2004).

1� e�jxcs ¼ 2; m ¼ �1;�3;�5; . . . ;1� e�jxcs ¼ 0; m ¼ �2;�4;�6; . . ..

�ð18Þ

Only the odd harmonics can create this effect due to the fact that the evenharmonics eliminate the regenerative term.

In the high-speed zone, the system can be truncated considering onlythe 0 and �1 harmonics, and the case of the double period chatter in pointC can be studied accurately. For the main flip lobe, only harmonics 0 and�1 are relevant, and they both have equal amplitudes and different phases.Assuming that only one mode affects the stability in the period doubling

FIGURE 4 The panels show the effect of the mode interactions (see Figure 3). In each panel the fre-quency response functions (upper panel), the harmonics of the directional factors, and the unit circleswith the critical eigenvalues are presented.

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instability in (15), the resulting equation is

g�1

g0

� �¼ � Kt b Z

p sinj

� �U�1b0 U�1b�1

U0 b1 U0b0

� �g�1

g0

� �ð19Þ

where U0 and U�1 are FRF at frequencies xc and xc � X, while b0, b1 andb�1 are the harmonics of the directional factor. It is remarkable that at per-iod doubling stability boundaries the U�1 is the complex conjugate of U0.In addition, b1 and b�1 are also conjugates.

Considering all the special characteristics of the period doubling insta-bility, it is straightforward to predict the spindle speed region zone affectedby this phenomenon. The tooth passing frequency should be two times thenatural frequency divided by an odd number, similarly as in (18)

X ¼ 2xn

m; n ¼ 120xn

Z � m ; m ¼ 1; 3; 5; ::: ð20Þ

In our case study, this formula predicts the beginning of the double periodchatter at 1800 r=min for the first harmonic and the second mode (60Hz).The Flip bifurcation is not visible for the rest of the possible combinations(see 2n2 in Figure 3).

If the stability borders related to period doubling bifurcation are ana-lyzed using the semi-discretization method (Insperger and Stepan, 2002),the unit circle shows that the instability is created when the dominanteigenvalue crosses the unit circle following the real axis (see point C inFigure 4). This means a possible periodic motion loses its stability by alter-nately in- and out-circling the stationary solution, that is, it flips in every rev-olution. The linear stability limits related to the period doubling (flip)bifurcation can have open or closed forms (Szalai and Stepan, 2006;Zatarain et al., 2006), although always independent from linear stabilitylimits related to Hopf-bifurcation. The multi-frequency approach must beapplied in frequency domain to capture these lobes.

Mode Interactions

When the system has two (or more) strong modes, another similareffect can happen. In some regions two modes start interfering and, con-sequently, this effect increases or reduces the linear stability dependingon the sense of the interaction (see point B in Figures 3 and 4). Thedynamic cutting force and the directional factor should have strong compo-nents to create this effect and, therefore, an interrupted milling processis necessary.

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In the case of mode interaction, there is not a special distribution of thechatter frequencies or the stability boundaries. Hence, the borders can onlybe obtained using pure numerical methods. Anyway, considering that thechatter frequency is always very close to the natural frequencies, it is poss-ible to predict the range of milling frequencies considering the physicalmeaning and different harmonics.

X ¼ xn;1 þ xn;2

m;

X ¼xn;1 � xn;2

�� ��m

;

m ¼ 1; 2; 3; 4; 5; . . . ð21Þ

Note that (21) can be generalized for more than two modes and the spin-dle speed can be calculated as n ¼ 60X=Z. The point B in Figure 3 repre-sents this phenomenon when formula (21) shows that the first harmonic iscreating interference around 1575 r=min, taking into account the sum ofboth modes (see n1þn2 in Figure 3).

If the study is extended using the semi-discretization method, it can beseen that this effect is related with Hopf bifurcation and the lobe-like regionis actually connected to the ordinary lobe that corresponds to point A inFigure 3. Clearly, this region is a part of an ordinary Hopf lobe and it is sepa-rated from the flip lobes. Also, ZOA is not accurate enough to capture thesespecial areas of the linear stability charts. In fact, these interactions are morecomplex than the double period chatter. It changes the usual lobes introdu-cing new shapes, reduces the location and values of the minimum stabilityand changes considerably the position of the most stable spindle speeds.

Traditionally, a stability diagram is formed by a number of lobes withthe same value for the minimum depth of cut, but with different locationof the spindle speed related to the maximum stability. Due to mode inter-action, this minimum can change for lobes of different order. For instance,in the case study the minimum for the first lobe (point B) differs from thesecond lobe (point E) or the third one (point F).

When mode interaction happens, the chatter vibration grows with twomodulated chatter frequencies close to both dominant natural frequencies(xn,1 and xn,2). The time domain simulations confirm this chatter fre-quency pattern and the stability limit (see point C in Figure 5). In otherregions (point A), the harmonics are far from other modes, therefore,ZOA defines the right stability boundary.

It is important to mention that the vibration (chatter) frequencies thatare calculated by multi-frequency and by semi-discretization only deal withthe frequencies that appear at the stability border when the system loses itsstability. Later, the vibration grows till some parts of the cutting edge lose itscontact with the workpiece. Then, the dynamics changes drastically thatresults frequency changes and appearance of new frequencies, too (see

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Figure 5(d)). The complexity of mode interactions increases with thenumber of significant modes, but the behavior is the same and the multi-frequency and semi-discretization methods are able to capture these effects.

Stable Isolated Zone

Because the semi-discretization provides information about the localbehavior of the stationary cutting process, the stable and unstable areascan be identified easily. The multi-frequency solution, on the otherhand, only finds the borders where the system has non-hyperbolic behavior(Guckenheimer andHolmes, 1983). Theoretically, it is impossible to point outthe stable areas using MF solution only, although in machine tool vibration,one can predict that the cutting process with zero depth of cut is always stable.In this manner, the only difficulty is the identification of possible stable islands,which requires further investigations. Figure 3 shows a case, where a stableisland exists, which was identified by means of semi-discretization method.Following the roots of multi-frequency in Figure 3, it is possible to realizethat this stable island also appears in the results of multi-frequency solution.

The island encircled by borders is related to Hopf and flip bifurcations,which are mostly originated by the mode interaction phenomenon. Similarphenomena have been found in the literature (Munoa et al., 2009; Sellmeierand Denkena, 2011). In the case of Sellmeier and Denkena (2011), theirexistence has been related to the introduction of variable helix angles inthe tool, whilst here, the stable island is caused by simple conventional tooland has a reachable axial depth of cut level from machining point of view.

FIGURE 5 Spectra of different time-domain simulations in different spindle speeds (see points A, B, C,and D in Figure 3).

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In fact, these isolated stables zones are theoretically the highest stablezones, and consequently themost suitable conditions to optimize production.Therefore, it is important to confirm the existence of these zones throughexperimental testing. However, these zones did not appear in time-domainsimulations if the non-linearity related to the possibility that the tool maylose contact with the workpiece is considered when the tool starts cutting.To explain this phenomenon some control points, namely a, b, and d (cf.Figure 3), were used checking the correctness of the linear stability and thetime domain solver. However, the simulation of point c (see Figure 6(a))shows that the stable sense of the island does not appear in the reality.

In Figure 6 one can realize how sensitive the stable sense of the station-ary solution in the isolated zone at point c is. These solutions were simulatedby using the correct, but slightly perturbed stationary cutting solution as aninitial function. The results show that for small perturbation the system isattracted by the linear equilibrium, but for higher perturbation the systemis pushed out and it tends to chatter. Therefore, the stable island has a littleattraction zone, which means that the stable sense might vanish or beunreachable in real circumstances. This phenomenon is similar to casesaddressed by Dombovari et al. (2008) or Bachrathy et al. (2011), but herethe system is originally linear. Therefore, the isolated zone cannot be usedin real cutting conditions to increase the material removal rate.

CONCLUSIONS

The stability of a milling process with several dominant modes has beenstudied using a two mode system as an example. The predictions of ZOA,multi-frequency method, semi-discretization method and time domainsimulations have been compared. The precision of the ZOA is reduced

FIGURE 6 (a) Time domain simulations at point c with different perturbation levels. (b) Bifurcationstructure along n (see Figure 3(b)). (HB: non-hyperbolic points related to Hopf bifurcation).

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when the cutting force becomes interrupted. These inaccuracies have twosources: the double period chatter and variations due to mode interaction.When the number of flutes is low, these effects appear in considerableengagements. The multi-frequency and semi-discretization methods leadto the same exact solution in all cases but the multi-frequency methodhas problems to determine stable isolated zones.

As it is well known, the double period chatter adds another family oflobes in the stability chart. Mathematically it is related to the flip bifurcation,and it happens when the zeroth-order term and one harmonic act on thesamemode. In this case, the chatter frequency is always related to tooth pass-ing frequency. This characteristic can help to build these lobes very fast.

Finally, the variations produced by the mode interactions have beenaddressed. In the frequency domain, the phenomenon is similar to the flipbifurcation, but in general it does not result in an independent family oflobes. Pure numerical methods are necessary to describe this effect; there-fore, it is really difficult to develop fast prediction methods. Approximatedformulas have been proposed to define the region affected by the modeinteraction.

The cases considered show that the mode interaction can create isolatedstable zones confirmed by different methods. However, as it was showed, thislinearly stable island might not exist in real cutting processes due to thepossible tiny attraction zone, concluding that any kind of perturbationcan destabilize the cutting process within the theoretically stable island.

ACKNOWLEDGEMENTS

The authors would like to thank Professor Yusuf Altintas and Dr. DorukMerdol from the Manufacturing Automation Laboratory of the UBC. Thisresearch was partially supported by the EUFP7Dynxperts (260073=FP7-2010--NMP-ICT-FoF) and the New Hungary Development Plan (Project ID:TAMOP-4.2.1=B-09=1=KMR-2010-0002) and Hungarian Scientific ResearchFoundation OTKA Grant no. K83890.

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