a unique methodology for chatter stability mapping in simultaneous machining

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Nejat Olgace-mail: [email protected]

Rifat Sipahi1

Mechanical Engineering,University of Connecticut,

Storrs, Connecticut 06269-3139

A Unique Methodology for ChatterStability Mapping inSimultaneous MachiningA novel analytical tool is presented to assess the stability of simultaneous machining(SM) dynamics, which is also known as parallel machining. In SM, multiple cutting tools,which are driven by multiple spindles at different speeds, operate on the same workpiece.Its superior machining efficiency is the main reason for using SM compared with thetraditional single tool machining (STM). When SM is optimized in the sense of maximiz-ing the rate of metal removal constrained with the machined surface quality, typical“chatter instability” phenomenon appears. Chatter instability for single tool machining(STM) is broadly studied in the literature. When formulated for SM, however, the problembecomes notoriously more complex. There is practically no literature on the SM chatter,except a few ad hoc and inconclusive reports. This study presents a unique treatment,which declares the complete stability picture of SM chatter within the mathematicalframework of multiple time-delay systems (MTDS). What resides at the core of thisdevelopment is our own paradigm, which is called the cluster treatment of characteristicroots (CTCR). This procedure determines the regions of stability completely in the do-main of the spindle speeds for varying chip thickness. The new methodology opens theresearch to some interesting directions. They, in essence, aim towards duplicating thewell-known “stability lobes” concept of STM for simultaneous machining, which isclearly a nontrivial task. �DOI: 10.1115/1.2037086�

1 Introduction and a Review of Single Tool ChatterWe present a new methodology on the stability of simultaneous

(or parallel) machining (SM), where multiple tools operate at dif-ferent spindle speeds on the same workpiece. It is obvious thatsuch an operation, as opposed to conventional single tool machin-ing �STM� �also known as serial process�, is more time efficient�1–5�. It is also known that its dynamics is considerably morecomplex. When metal removal rates are maximized, the dynamiccoupling among the cutting tools, the workpiece, and the machinetool�s� become very critical, regenerative forces become pro-nounced, etc. The dynamic stability repercussions of such settingsare poorly understood at present even in the mathematics commu-nity. In fact, there is no analytical mechanism available to assessthem and no evidence in the literature addressing the stability ofsimultaneous machining.

It is well known from numerous investigations that the conven-tional single tool machining �STM� introduces some importantstability issues when it is optimized. Similar problems result inmuch more complex form for SM. Due to the lack of a solidmathematical methodology to study the SM chatter, the existingpractice is likely to be �and it really is� suboptimal and guided bytrial-and-error or ad hoc procedures. There is certainly room formuch-needed improvement in the field.

Optimum machining aims to maximize the material removalrate, while maintaining a sufficient stability margin to assure thesurface quality. The machine tool instability primarily relates to“chatter.” As accepted in the manufacturing community, there aretwo groups of machine tool chatter: regenerative and nonregen-erative �6�. Regenerative chatter occurs due to the periodic toolpassing over the undulations on the previously cut surface, andnonregenerative chatter has to do with mode coupling among the

1Currently at Université de Technologie de Compiègne, France under Chateaubri-and Bourse.

Contributed by the Manufacturing Engineering Division for publication in theASME JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript receivedJune 18, 2004; final revision received February 15, 2005. Review conducted by:
D.-W. Cho.
Journal of Manufacturing Science and EngineeringCopyright © 20

rom: http://manufacturingscience.asmedigitalcollection.asme.org/ on 02/0

existing modal oscillations. This text evolves primarily on regen-erative type, thus we refer to it simply as “chatter” for the rest ofthe paper.

In order to prevent the onset of chatter, one has to select theoperational parameters appropriately, namely chip loads andspindle speeds. Existing studies on machine tool stability addressconventional single-tool machining processes. They are inappli-cable, however, to SM because of the substantial difference in theunderlying mathematics. Leaving the details to the later sections,we can say that simultaneous machining gives rise to a complexmathematical characterization known as “parametric quasi-polynomials with multiple delay terms.” There is no practicalmethodology, known at this point, to resolve the complete stabil-ity mapping for such constructs. This text presents a unique pro-cedure in that sense.

Machine tool chatter is an undesired engineering phenomenon.Its negative effects on the surface quality, tool life, etc. are wellknown. Starting with the early works of �7–10�, many researchersmeticulously addressed the issues of modeling, the dynamic pro-gression, structural reasoning, and stability limit aspects of thisseemingly straightforward and very common behavior. Further re-search focused on the particulars of parameter selections in ma-chining to avoid the build-up of these undesired oscillations andon the analytical predictions of chatter. Some interesting readingson this are �6,10–12�. Most commonly chatter research has fo-cused on the conventional single tool machining �STM�. The aim,again, is to increase the metal removal rate while avoiding theonset of chatter �13–15�. A natural progressive trend is to increasethe productivity through simultaneous �or parallel� machining.This process can be further optimized by determining the bestcombination of the chip loads and spindle speeds with the con-straint of chatter instability. For SM, however, multiple spindlespeeds, which cross-influence each other, create governing differ-ential equations with multiple time delay terms. Their character-istic equations are known in mathematics as “quasi-polynomialswith multiple time delays.” Multiplicity of the delays presentsenormously more complicated problems compared with the con-
ventional single-tool machining �STM� chatter.
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Let us review the basics of STM chatter dynamics for the clar-ity of the argument, borrowing some parts from �6�. Consider anorthogonal turning process for the sake of example �Fig. 1�. Theunderlying mechanism for regenerative chatter is quite simple tostate. The desired �and nominal� chip thickness, h0, is consideredconstant. The tool actually cuts the chip from the surface, which iscreated during the previous pass. The process generated cuttingforce, F, is realistically assumed to be proportional to the dynamicchip thickness, h�t�. It carries the signature of y�t�−y�t−��, wherey�t� is the fluctuating part of the chip thickness at time t �so called“offset chip thickness” from the nominal value h0�, and ��s� is theperiod of successive passages of the tool, which is equal to 60/N,N being the spindle speed �RPM�.

The block diagram in Fig. 2 gives a classical causality repre-sentation of the dynamics for this orthogonal turning operation.Nominal chip thickness, h0, is disturbed by the undulating offsetchip thickness, y. These undulations create driving forces for the ydynamics � s later during the next passage and thus the attribute“regenerative.” G�s� is the transfer function between the cuttingforce, F, and y. For the sake of streamlining the presentation, asingle-degree-of-freedom cutting dynamics is taken into accountinstead of higher-degree-of-freedom and more complex models.

Fig. 1 Orthogonal turning process

Fig. 2 Functional block diagram of chatter regeneration

Fig. 3 Typical stability lobes and

792 / Vol. 127, NOVEMBER 2005


The workpiece and its rotational axis are considered to be rigidlyfixed and the only tool flexibility is taken in the radial direction. InFig. 2 the following causal relations are incorporated

the cutting force F�t� = Cbh�t� �1�

actual chip thickness h�t� = h0 − y�t� + y�t − �� 2�

where b is the chip width which is user selected and assumedconstant, C is the cutting force constant, and � �s� is the period ofone spindle revolution, �=60/N, N �RPM�.

Assuming that the force-displacement transfer function G�s� islinear, the entire cutting mechanism described by Fig. 1 is linear.Cutting would be at equilibrium if y=0, which means an ideal cutwith no waviness. The cutting force, F, remains constant and thetool support structure �i.e., k, c� yields a static deflection through-out the cutting. This equilibrium is called “stable” or “asymptoti-cally stable,” if the loop characteristic equation of the block inFig. 2

1 + �1 − e−�s�bCG�s� = 0 �3�

has all its roots on the stable left half-plane �16,17�. This equationis transcendental, and it possesses infinitely many finite charac-teristic roots, all of which have to be taken into account for sta-bility. Although the problem looks prohibitively complex, thecomplete stability map is obtainable for single delay cases �i.e.,STM�. It is clear that the selection of b and � influences the sta-bility of the system considerably. The complete stability map ofthis system in b and � domain is the well-known “chatter stabilitylobes,” Fig. 3 �using the parameters from �18��. There are severalsuch lobes marked on this figure. They represent the �b ,�� planemapping of the dynamics with dominant characteristic root atRe�s�=−a. When a=0 �the thick line�, these curves show thewell-known stability lobes, which form the stability constraintsfor any process optimization. For instance, if we increase the re-moval rate �by increasing the chip width, b�, we ultimately invitechatter instability, which is obviously not allowed.

There are two cutting conditions under the control of the ma-chinist: �, which is the inverse of the spindle speed �60/N�, and b,the chip width. The others, i.e., C and G�s�, represent the existingcutting characteristics, which are considered to remain unchanged.The open loop transfer function G�s� typically manifests high-impedance, damped, and stable dynamic behavior.

Certain selections of b and ��=60/N� can introduce marginalstability to the system as shown in Fig. 3. At these operatingpoints the characteristic equation �3� possesses a pair of imaginaryroots, ±�i, as dominant roots. As such the complete system isresonant at �. That is, the entire structure mimics a spring-massresonator �a conservative system� at the respective frequency of

constant stability margin lobes

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marginal stability �also known as “chatter frequency”�. The desir-able operating point should lie in the shaded region, marked as“stable.”

It is desirable to select the cutting parameters �b ,N� sufficientlyaway from the chatter stability bounds. Conventional terminologyalluding to this feature is called the “stability margin.” It refers tothe a=−Re �dominant characteristic root�. The bigger the a, thehigher the “chatter rejection speed,” therefore the better the sur-face quality. A set of operating points is shown where a=5, 10,and 15, just to give an idea of the distribution of the loci where thechatter rejection speed is constant. Optimum working conditionsare reached by increasing b and N up to the physical limitationsprovided that a desirable stability margin �i.e., a� is guaranteed.Figure 3 also represents a unique declaration in the machine toolliterature as “equal stability margin” lobes. The determination ofthese curves also offers a mathematically challenging problem,which is deferred to another publication.

2 Regenerative Chatter in Simultaneous MachiningThe functional block diagram in Fig. 2 expands in dimension

by the number of tools involved in the case of simultaneous ma-chining, as depicted in Figs. 4�a� and 4�b� for two example opera-tions. The flexures are shown at one point and in the 1-D sense, tosymbolize the most complex form of the restoring forces in 3-D.Spindle speeds and their directions are also selected symbolicallyjust to describe the types of operations we consider here.

The crucial difference between the STM and SM is the cou-pling among the individual tool-workpiece interfaces, eitherthrough the flexible workpiece �as in Fig. 4�a�� or the machinetool compliance characteristics �as in Fig. 4�b��, or both. Clearlythe chip load at tool i �i=1, . . . ,n�, which carries the signature ofyi�t�−yi�t−�i�, will influence the dynamics at tool j. This is the“cross-regenerative effect,” which implies that the state of the ithtool one revolution earlier �i.e., �i s� affects the present dynamicsof the jth tool. Consequently the dynamics of tool j will reflect thecombined regenerative effects from all the tools including itself.We assume these tool-to-tool interfaces are governed under linearrelations. The overall dynamics becomes a truly cross-coupledlinear multiple time-delay system. We are fully cognizant of thetime-varying �and periodic� nature of the cutting force directions.However, as it was pointed out in the literature �e.g., �19�� we takethe fundamental elements of these forces in their Fourier expan-sions. As such, the linear-time-invariant nature is maintained inthe overall dynamics in this study.

The stability of multiple delay systems is poorly known in themathematics community. There is no simple extension of the con-ventional �STM� stability treatment to multiple spindle and con-sequently to multiple delay cases �SM�. The specific objective of

Fig. 4 Conceptual depiction of simultaneMachine tool coupled.

this paper is to deploy a new mathematical framework to address

Journal of Manufacturing Science and Engineering


the problem. A notational selection: bold capital fonts representthe vector or matrix forms of the lower case elements, such as��= ��1 ,�2�.

We present next the multi-spindle cutting tool dynamics in ageneric form �see Fig. 5�. Analogous to Fig. 2, H0�n�1� andH�n�1� represent the commanded and actual depth-of-cut �d-o-c�vectors, components of which denote the d-o-c at n individualtools. Y�n�1� is the tool displacement fluctuations vector alongthe d-o-c directions. Similarly BCdiag=diag�biCi�, i=1, . . .n, rep-resents the influence of bi �chip width� and Ci �the cuttingforce constants� vis-à-vis the tool “i.” BCdiag=diag�b1C1 ,b2C2 , . . . ,bnCn� are the same terms as in the singletool case except the multiple tool version of it. The exponentialdiagonal matrix ediag

−�s =diag�e−�1s ,e−�2s , . . . ,e−�ns�, �i�=60/Ni�, i=1, . . . ,n, represents the delay effects �i.e., the regenerative ele-ments�. Ni’s are the spindle speeds. Notice that the relation be-tween � j and Nj should go through the number of flutes in case ofmilling. Mathematically, however, this nuance introduces only ascale factor along the delays. As such, it is overlooked at thisstage, and this point is revisited in case study I. G�s��n�n� is thedynamic influence transfer function which entails all the auto- andcross-coupling effects between the cutting force vector, F�n�1�and the tool displacements in the d-o-c direction, Y�n�1�. We aremaking the assumption that these relations are all linear �as it iscommon in most fundamental chatter studies�. It is clear that G�s�may appear in much more complicated form, for instance, in mill-ing it becomes periodically time-variant matrix �19�. The past in-vestigations suggest the use of Fourier expansion’s fundamentalterm in such cases, to avoid the mathematical complexity whileextracting the underlying regenerative characteristics.

The characteristic equation of the loop in Fig. 5 is

CE�s,�,B� = det�I + G�s�BCdiag�I − ediag−�s �� = 0 �4�

which is representative of a dynamics with multiple time delays�� and multiple parameters �B� as opposed to a single tool ma-chining, where there is a single delay “�” and a single parameter,“b.” Equation �4� is a “parameterized quasi-polynomial in s with

s face milling. „a… Workpiece coupled. „b…

Fig. 5 Block diagram representing the simultaneous

ou

machining

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multiple time delays.” For a stable operation, its infinitely manycharacteristic roots should all be on the left half of the complexplane. The most general form of CE�s ,� ,B� contains terms likeexp�−� j=1

n � j� js� with � j =0 or 1 which represent the cross-talkamong the delay terms �i.e., among the tools�. For instance, thee−��1+�3�s term would indicate the cross-coupling between the re-generative effects of spindles 1 and 3.

For the exceptional case where G�s� is also diagonal, the equa-tion �4� decouples all the delay effects so that the problem reducesto n independent STM chatter problems. Otherwise the cross-talkterms will appear in Eq. �4�, which add to the complexity of theanalysis considerably. A critical point to note is that there is nopossible commensurate delay formation here, i.e., no terms willappear with e−k� js, k�2. Physically this implies that the regenera-tive effect of tool j acts on itself only once.

One can find substantial literature on the stability of systemswith only one single time delay �20–24� including the commensu-rate cases. Single-tool machining chatter dynamics is a simplesubclass of this. The published studies address the question of the“stability margin” in time delay, �max. They consider all the pa-rameters �B� to be fixed and claim that the higher delay valuesthan a certain �max would invite instability. These methods typi-cally stop there. A recent paradigm by the authors enables thedetermination of all the stable regions of � completely �includingthe first stability interval of 0��max� �25–29�. This newframework, called “cluster treatment of characteristic roots�CTCR�,” yields a complete stability picture for single delay sys-tems. Multiple time delay systems �MTDS� are, however, signifi-cantly more complex, and their parameterized forms �i.e., forvarying B matrix� bring another order of magnitude to the diffi-culty.

Again, the system depicted in Fig. 5 is asymptotically stable ifand only if all the roots of the transcendental characteristic equa-tion �4� are on the left half of the complex “s” plane. Infinitelymany such roots will have to be tracked for the desired stabilityassessment. This is obviously a prohibitively difficult task. Thereexists no methodology at present to study the stability of suchsystems. We wish to clarify this statement that investigations suchas �19� performing time domain simulations are not included sim-ply because they are not considered comparable operations. First,they are performed point-by-point in ��1 ,�2� space and thereforethe application is computationally overwhelming. Second, theyare numerical as opposed to the analytical nature of CTCR. Veryfew mathematical investigations reported on the subject also de-clare strict limitations and restrictions to their pursuits. For ex-ample, the author of �20� claims that it is very difficult to utilizePontryagin’s theorem for the stability analysis when n�1 anddet�G�s�� invites higher degree terms of “s” than one. Reference�30� handles the same problem using simultaneous nonlinearequation solvers, but is only able to treat systems with Eq. �4� ins+a+be−�1s+ce−�2s=0 construct with a ,b ,c being constants. Thisimposes, clearly, a very serious restriction. References �31,32�tackle a limited subclass of �4� with det G�s� of degree 2, butwithout a damping term, and their approach is not expandable todamped cases. They also consider the parameter vector �B� asfixed. We study in this text the most general form of �4� under theconceptual framework of the Cluster Treatment of CharacteristicRoots �CTCR� �25–28�. This is a novel attempt in mathematicsand it is undoubtedly unique for SM.

3 The CTCR MethodologyWe present the description of the CTCR methodology in the

next section, using a reduced system for n=2, in order to avoidthe notational complexity, without loss of generality. The most
general form of the characteristic equation �4� for n=2 becomes
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CE�s,�1,�2,b1,b2� = a0 + a1e−�1s + a2e−�2s + a3e−��1+�2�s = 0 �5�

where aj�s ,b1 ,b2�, j=0,1 ,2 ,3, are polynomials in s with param-eterized coefficients in b1 and b2. The highest degree of s in �5�resides within a0�s� and it has no time delay accompanying it.This characteristic equation represents the behavior of a dual-toolSM �n=2� cutting on a workpiece simultaneously. We wish torecover the stability portrait �referred to as the “lobes”� in four-dimensional ��1 ,�2 ,b1 ,b2� space. The delay terms �i�=60/Ni�’sare independent from each other. In short, the dynamics at hand isa retarded multiple time delay system, as it is known in the math-ematics �20,21,23,30–33�.

The intended stability map over the four-dimensional paramet-ric space of ��1 ,�2 ,b1 ,b2� is, in fact, analogous to the conven-tional stability lobes for a single � and single b in the space ofdual �’s and dual b’s instead. The transition, however, from singleto multiple time delays is not trivial even when the parameters�b1 ,b2� are fixed. To introduce a mathematical tool for this op-eration is the task of this section.

Let us first concentrate on the simpler stability problem whenB= �b1 ,b2�T parameters are fixed. To the best knowledge of theauthors there is no available methodology even for this simplifiedproblem. We direct the rest of the text to such systems.

Although the dynamics represented by �5� possesses infinitelymany characteristic roots, the most critical ones are those that arepurely imaginary. Obviously, any stability switching �from stableto unstable or vice-versa� takes place when the parameters causesuch purely imaginary roots. These imaginary roots display somevery interesting constructs. Based on our preliminary work, weclaim two novel propositions to summarize these peculiar fea-tures. They are analogous to the original propositions, which werestated for a single-delay dynamics �26–28�, and they were unrec-ognized earlier in the literature. These two propositions form thefoundations of “cluster treatment of characteristic roots �CTCR�”paradigm. A nontrivial transition of the CTCR from single delayto multiple delays leads to an exhaustive stability analysis tool inthe space of the time delays ��1 ,�2�. In the interest of streamliningthe discussions we provide these propositions without proof andrecommend �34,35� to the interested reader for the theoretical de-velopment. We present later some example cases to demonstratethe strengths and the uniqueness of CTCR.

Proposition I: The equation �5� can have an imaginary root onlyalong countably infinite number of hyperplanes ��1 ,�2�; �1 and�2�R+. These hyperplanes �which are simply “curves” in 2-D�are indeed offspring of a manageably small number of hyper-planes, which we will call the “kernel hyperplanes,” �0��1 ,�2�.All the hyperplanes in ��1 ,�2� are some descendants of�0��1 ,�2�.

Related to this proposition we present two explanatory remarks,which are also key to the CTCR framework:

Remark I: Kernel and offspring: If there is an imaginary root ats= ��ci �subscript “c” is for crossing� for a given set of timedelays ��0�= ��10,�20�, the same imaginary root will appear at allthe countably infinite grid points of

�� = ��10 +2

�cj,�20 +

2

�ck, j = 1,2 . . . , k = 1,2, . . . �6�

Notice that for a fixed �c the distinct points of �6� generate a gridin ��R2+ space with equidistant grid size in both dimensions.When �c is varied continuously the respective grid points alsodisplay a continuous variation, which ultimately form the hyper-planes ��1 ,�2�. Therefore, instead of generating these grid pointsand studying their variational properties one can search only forthe critical building block, “the kernel,” for j=k=0 and for allpossible �c’s, �c�R+. It is alternatively defined by min��1 ,�2��c

,��1 ,�2��R2+ for all possible �c’s. �

Remark II: The determination of the kernel. As stated earlier, if
there is any stability switching �i.e., from stability to instability or


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vice versa� it will take place at a point on ��1 ,�2� curves. There-fore we need all possible ��1 ,�2�, and the representative �c’sexhaustively. In other words, we have to determine the kernel�0��1 ,�2� along which all the imaginary roots, s=�ci, of �5� arefound. That means the description of kernel has to be exhaustive.Furthermore, any and every point ��1 ,�2� causing an s= ��ci rootmust be either on �0��1 ,�2� or a ��1 ,�2�. The determination ofsuch a kernel and its offspring is mathematically very challengingproblem. In order to solve this problem, we deploy a unique trans-formation called “the Rekasius substitution” �24�

e−�is =1 − Tis

1 + Tis, Ti � R, i = 1,2 �7�

which holds identically for s=�ci, �c�R. This is an exact sub-stitution for the exponential term, not an approximation, for s=�ci, with the obvious mapping condition of

�i =2

�c�tan−1��cTi� + j�, j = 0,1, . . . �8�

Equation �8� describes an asymmetric mapping in which Ti �dis-tinct in general� is mapped into countably infinite �i sets each ofwhich has periodically distributed time delays for a given �c withperiodicity 2 /�c. Substitution of �8� into Eq. �5� converts it fromCE�s ,�1 ,�2� to CE��s ,T1 ,T2�. Notice the slight breach of notationwhich drops b1, b2 parameters from the arguments both in CE andCE�. We next create another equation as described below:

CE�s,T1,T2� = CE��s,T1,T2��1 + T1s��1 + T2s� = �k=0

4

bk�T1,T2�sk

�9�

Since the transcendental terms have all disappeared, this equationcan now be studied much more efficiently. Notice that all theimaginary roots of CE�s ,�1 ,�2� and CE�s ,T1 ,T2� are identical,i.e., they coincide, while there is no enforced correspondence be-tween the remaining roots of these equations. That is, consideringthe root topologies

1 = �sCE�s,�1,�2� = 0,��1,�2� � R2+� ,�10�

2 = �sCE�s,T1,T2� = 0,�T1,T2� � R2�

the imaginary elements of these two topologies are identical. Inanother notation one can write 1�C0�2�C0, where C0 rep-resents the imaginary axis. It is clear that the exhaustive determi-nation of the �T1 ,T2��R2 loci �and the corresponding �c’s� fromEq. �9� is a much easier task than the exhaustive evaluation of thesame loci in ��1 ,�2��R2+ from Eq. �5�. Once these loci in�T1 ,T2� are found, the corresponding kernel and the offspring in��1 ,�2� can be determined as per �8�. �

We give a definition next, before stating the second proposition.The root sensitivities of each purely imaginary characteristic rootcrossing, �ci, with respect to one of the time delays is defined as

S�j

s s=�ci = � ds

d� j�

s=�ci

= � −�CE/�� j

�CE/�s�

s=�ci

, i = − 1, j = 1,2

�11�And the corresponding root tendency with respect to one of thedelays is given as

root tendency = RTs=�ci�j = sgn�Re�S�j

s s=�ci�� 12�

This property represents the direction of the characteristic root’scrossing when only one of the delays varies.

Proposition II: Let’s take a crossing frequency, �c, caused by
the point ��10,�20� on the kernel and its 2-D offspring


��1,�2��c= ��1j = �10 +

2

�cj, �2k = �20 +

2

�ck;

j = 0,1,2, . . . , k = 0,1,2, . . . �13�

The root tendency at this point is invariant with respect to j �or k�when k �or j� is fixed. That is, regardless of which offspring��1j ,�2k� of the kernel set ��10,�20� causes the crossing, RT s=�ci

�1

�RT s=�ci�2 � are the same for all j=1,2 , . . . �k=1,2 , . . . �, respec-

tively. That is, the imaginary root always crosses either to theunstable right half-plane �RT= +1� or to the stable left half-plane�RT=−1�, when one of the delays is kept fixed, and the other oneis skipping from one grid point to the next, regardless of the actualvalues of the time delays as long as they are derived from thesame kernel ��10,�20�.

3.1 CTCR in Brief The cluster treatment of the characteristicroots �CTCR� paradigm is based on the two propositions above. Itconsists of the following steps:

i. Among the infinitely many characteristic roots generatedby an arbitrary time delay set ��= ��1 ,�2� typically thereis not an imaginary pair. The CTCR routine requires anexhaustive detection of all those �� sets, which yieldimaginary characteristic roots, say �ci. This is the first“clustering” operation with the respective earmarking of�c’s. The thought behind this clustering is simple. Thesystem stability can possibly change only at those �� setsmarked by the clustering identifier, s=�ci. We are refer-ring to a hyperplane in �� which displays a continuum ofs=�ci root crossings. As stated earlier, one needs to deter-mine the complete kernel �0��1 ,�2�. As per proposition I,it is analytically and numerically manageable in size. This�0��1 ,�2� is the generator of the infinitely many and com-plete sets of crossing curves ��1 ,�2� according to �13�.

ii. A second “clustering” is done within the kernel,�0��1 ,�2�, utilizing the concept of invariant RT �root ten-dency�. Certain segments of the kernel �0��1 ,�2� exhibitcertain RT’s along �1 and �2 axes. These tendencies willset the stability switching regime for the offspring curves,��1 ,�2�, as per proposition II.

These steps of CTCR ultimately generate a complete stabilitymapping in ��1 ,�2� space. We next present two case studies toapply the capabilities of CTCR. The first is an experimental study,which is detailed in �19�. And the second is to demonstrate howone can deploy CTCR to the conventional single tool machiningchatter, which is a very simple problem for the new method.

4 Example Case Studies

4.1 Case Study I: Experimental Validation of CTCR onVariable Pitch Milling Cutters. There is no experimental workreported in the literature on a true “simultaneous machining” withsufficient scientific data. The authors’ group is embarking on anew program in this direction. For the moment, however, we arepleased to report the following experimental validation of CTCRon an interesting machining process: milling with variable pitchcutters. An elegant treatment of variable pitch milling is presentedin �19�. We adopt all the underlying dynamics as well as theparameters from that document, in correspondence with Altintas�the lead author�.

The practice of variable pitch cutters originates from the desireof attenuating the regenerative chatter. Instead of four equidistantflutes located around the cutting tool �4�90° as described in Fig.6�a��, variable pitch is used ��1,�2 as shown in Fig. 6�b��. Manyinteresting variations of this clever idea have been developed and
put to practice over 40 years. Expectedly an interesting dynamics
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emanates from these applications. Due to the nonuniform pitchdistribution multiple regenerative effects appear in the governingequation with multiple time delays. The delays are proportional tothe respective pitch angles.

Reference �19� presents an in-depth study on variable pitch cut-ters on an end milling process with both numerical and supportingexperimental effort. They converge in very closely matching re-sults, when it comes to the chatter boundaries. In their numericalstudy, the authors take the system characteristic equation andstudy it for a given pitch ratio. The core problem is defined brieflyas follows: A four-fluted uniform-pitch cutter is used first in mill-ing Al356 alloy. The cutter has 19.05 mm diameter and 30° helixand 10° rake angles. The stability chart indicates that this millingprocess is unstable for axial depth-of-cut a=5 mm and spindlespeed N=5000 RPM. A natural question follows: Which pitchangles should be selected for the best chatter stability marginswhen variable-pitch milling is considered?

We take this case study exactly, except we free the pitch ratio,therefore introducing a truly multiple time delay construct. Inother words, we completely relax the selection of the two “pitchangles.” When we take the cross section of our findings for agiven pitch ratio, we obtain nicely coinciding results to the refer-enced publication. The steps of this exercise and the numericalresults are given next.

The system characteristic equation is taken as ��19�, Eq �15��

det�I −1

4Kta�4 − 2�e−�1s + e−�2s��0�s�� = 0 �14�

with Kt=697 MPa, a=axial depth-of-cut �m�, �1�s�= ��1�deg� /360��60/N�RPM��, is the delay occurring due to thepitch angle �1, �2= ��2 /�1��1 is the second delay due to �2, N isthe spindle speed, and s is the Laplace variable. The matrix �0containing the transfer functions and the mean cutting directionsis defined as

�0 = � xx�xx + yx�xy xy�xx + yy�xy

xx�yx + yx�yy xy�yx + yy�yy �15�

where

�xx = 0.423, �xy = − 1.203

�yx = 1.937, �yy = − 1.576

are the fundamental Fourier components of the periodically vary-ing directional coefficient matrix. The transfer functions in �15�are populated from ��19�, Table 1� as

xx =0.08989

s2 + 159.4s + 0.77 � 107 +0.6673

s2 + 395.2s + 0.1254 � 108

+0.07655

s2 + 577.2s + 0.2393 � 108

Fig. 6 „a… Uniform pitch. „b… Variable pitch.

xy = yx = 0

796 / Vol. 127, NOVEMBER 2005


yy =0.834

s2 + 162.2s + 0.1052 � 108

Including these expressions and parameters in the characteristicequation and expanding it into a scalar expression, we reach thestarting point of the CTCR paradigm. For a=4 mm depth-of-cut,the characteristic equation of this dynamics is

CE�s,�1,�2� = 625s8 + 808,750s7 + 35068.83 � 106s6

+ 31,376.027 � 109s5 + 0.686 � 1018s4

+ 0.385 � 1021s3 + 0.565 � 1025s2 + 0.150 � 1028s

+ 0.166 � 1032 + �− 0.266 � 109s6 − 0.324 � 1012s5

− 0.127 � 1017s4 − 0.924 � 1019s3 − 0.18 � 1024s2

− 0.572 � 1026s − 0.756 � 1030��e−�1s + e−�2s�

+ �284,943.13 � 109s4 + 0.212 � 1018s3

+ 0.889 � 1022s2 + 0.253 � 1025s

+ 0.538 � 1029�e−��1+�2�s + �14,247.16 � 1010s4

+ 0.106 � 1018s3 + 0.445 � 1022s2 + 0.126 � 1025s

+ 0.269 � 1029��e−2�1s + e−2�2s� = 0 �16�

The parametric form of �16�, i.e., CE�s ,�1 ,�2 ,a� expression, isprohibitive to display due to space limitations, thus the substitu-tion of a=4 mm. Please note that all the numerical values aboveare given in their truncated form to conserve space.

Notice the critical nuance between Eqs. �4� and �14�. Theformer represents a truly two-spindle, two-cutter setting, while�14� is for a single spindle with four nonuniformly distributedflutes in a milling cutter. Mathematical expression for the charac-teristic equation becomes two time-delayed quasi-polynomials of�16� which is, in fact, more complex than �5� due to the commen-surate delay formation �i.e., e−2�1s, e−2�2s terms are present�. Toassess the stability posture of this equation is a formidable task.The authors encountered no technique in the mathematics whichcan handle this mission. We put CTCR to the test and present theresults below.

The CTCR takes over from Eq. �16� and creates the completestability outlook in ��1 ,�2� space as in Fig. 7�a�. The kernel ismarked thicker to discriminate it from its offspring. The grids ofD0 �kernel� and D1 ,D2 ,D3 , . . . �offspring sets� are marked for easeof observation. Notice the equidistant grid size 2 /� as per Eq.�6�, D0D1=D1D2=D2D3=D3D4, etc. The ��1 ,�2� delays at all ofthese sibling points impart the same �i imaginary root for �16�.Figure 7�b� shows the possible chatter frequencies of this systemfor all ��1 ,�2��R2+ for varying pitch ratios �0,�� whether theyare operationally feasible or not. It is clear that this system canexhibit only a restricted set of imaginary roots �from Fig. 7�b�,3250 rad/s �517 Hz��3616 rad/s �575 Hz��. All of thesechatter frequencies are created by the kernel and no ��1 ,�2� pointon the offspring can cause an additional chatter frequency outsidethe given set.

Figure 7�a� displays the stable �shaded� and unstable regions in��1 ,�2� space for a given axial depth-of-cut �a=4 mm�. It containssome further information as well. Each point ��1 ,�2� in this figurerepresents a spindle speed. It is obvious that, considering the re-lation �1+�2=30/N, all the constant spindle speed lines are withslope −1, as annotated on the figure. The constant pitch ratio linespass through the origin �pitch ratio=�2 /�1�.

Notice that the most desirable pitch ratios are close to �1 /�2=1 for effective chip removal purposes. Therefore very high orvery low pitch ratios are not desirable �55° ��1�90° as declaredin �19��. The pitch ratios between �2 /�1� �1.374,2.618� offerstable operation �along the segment AB on Fig. 7�a�� for
5000 RPM. The corresponding pitch angles are ��1A=75.8° ,�2A


Downloaded F

=104.2° � and ��1B=49.8° ,�2B=130.2° �. They coincide preciselywith the results declared in ��19�, Fig. 3�. Figure 7�a� provides avery powerful tool in the hands of the manufacturing engineer.One can select uniform pitch cutter and 7500 RPM speed �pointO2� as opposed to variable pitch cutter �pitch ratio 11/7� and5000 RPM �point O1� increasing the metal removal rate by 50%.Tool wear maybe worsened, however, due to the increased speed�19�. Nevertheless, choices are always good to have.

This figure is for a constant depth-of-cut �a�. A 3-D stabilityplot can be produced scanning the values of a in �a=1, . . . ,6 mm,�1�R+ ,�2�R+� domain. A cross section of this3-D plot with N=constant planes is comparable with ��19�, Fig.3�. We simply state that the matching is perfect without givingnumerical values, except for the two points, A and B, as statedabove.

We next study the chatter stability for two different settings, �a�uniform pitch cutters ��1=�2=90° � and �b� variable pitch cutters��1=70° ,�2=110° �, both with Al356 workpiece, which are alsoinvestigated numerically and experimentally in ��19�, Fig. 2�. TheCTCR results are obtained by a single cross section of the stabilitypicture in Fig. 7�a� along �2 /�1=1 and �2 /�1=11/7 lines, respec-tively, while the entire figure is refreshed for a=1, . . . ,6 mm. Theresults of the cross section are given in Figs. 8�a� and 8�b�, re-

Fig. 7 „a… Stability regions for 4 mm axial depth-of-cutoffspring: thin blue, stable zones: shaded…. „b… Chatter f

Fig. 8 „a… Comparison of CTCR „circle… and „†33



spectively, superposed on the experimental results of �19�. Thecorrespondence is again very encouraging.

These comparisons fortify the claims we made that the CTCRmethodology is a powerful tool to predict chatter in multiple timedelay dynamics.

4.2 Case Study II. How does CTCR Apply to Single ToolMachining (STM)? This is a very well-posed question by ananonymous reviewer. We gladly include a section in response tothe request. For the conventional chatter stability study �with asingle cutter, n=1�, a regenerative dynamics with one single timedelay appears. The characteristic equation �5� reduces to

CE�s,�,b� = a0�b,s� + a1�b,s�e−�s = 0 �17�

where the only delay is ��s�=60/N�RPM�, N is the spindle speed,and b is the width-of-cut. There are numerous case studies in theliterature on this problem. Just for demonstration purposes wetake the example of orthogonal turning given in ��18�, Eq. �7��,which starts from the characteristic equation

ith four-flute cutter milling on Al 356 „kernel: thick red,uencies „exhaustive… „rad/s… vs. the kernel „�1 ,�2….

, wreq

‡, Fig. 2… „square… for pitch ratio=1. „b… 11/7.

NOVEMBER 2005, Vol. 127 / 797


Downloaded F

1 + bCcos ��1 − e−�s�

ms2 + cs + k= 0 �18�

where b is the chip width and C, �, m, c, and k are the constantsrelated to the cutting dynamics. This equation can also be writtenas

CE�s,�,b� = ms2 + cs + k + bC cos ��1 − e−�s� = 0 �19�

which is in the same form as �17�. We take the parametric valuesas

C = 2 � 109 N/m2, � = 70°

m = 50 kg, c = 2 � 103 kg/s, k = 2 � 107 N/m

The equation �19� with these numerical values takes the form of

s2 + 40s + 400,000 + 13,680,805.73b − 13,680,805.73be−�s = 0

�20�

We search for the stability pockets in ��R+ space for varyingchip widths b�R+. This picture is conventionally known as the“stability lobes” for regenerative chatter. Unlike multiple delaycases this problem with a single delay is solvable using a numberof different procedures given in the literature �6,18,23�. Our aim isto show how the CTCR paradigm applies in this case, step bystep:

�i� Take “b” as a fixed parameter�ii� Use the Rekasius substitution of �7� e−�s= �1−Ts� / �1

+Ts� in �20� to obtain

CE�s,b,T� = �1 + Ts��s2 + 40s + 400,000

+ 13,680,805.73b� − 13,680,805.73b�1 − Ts�

= Ts3 + �40T + 1�s2 + �40 + �400,000

+ 27,361,611.46b�T�s + 400,000

= b3�T,b�s3 + b2�T,b�s2 + b1�T,b�s + b0 = 0

�21�

where bj�T ,b� , j=0, . . . ,3 are self-evident expressions.�iii� Search for the values of T, which render s=�i as a root for

�21�. It is best handled using the Routh’s array �16� �Table1�.

�iv� Apply the standard rules of Routh’s array dictating that theonly term in s1 row to be zero for Eq. �21� to possess apair of imaginary roots;

b2b1 − b3b0 = 0, b2 � 0 �22�

gives a quadratic equation in T for a given chip width, b,which results in at most two real roots for T. If these rootsT1,T2 are real, we proceed further. It is obvious �16� thatfor those real T values the imaginary characteristic roots of

Table 1 The Routh’s array „b and T arguments aresuppressed…

�21� will be

798 / Vol. 127, NOVEMBER 2005


sj = � ji = � b0

b2�

j

= � b1

b3�

j

, with T = Tj, j = 1,2

�23�

These are the only two imaginary roots �representing thetwo chatter frequencies� that can exist for a given “b”value. Obviously, if T1, T2 are complex conjugate numbersfor a value of b, it implies that there is no possible imagi-nary root for �20� regardless of ��R+. In other words,this depth-of-cut causes no stability switching for anyspindle speed, N�0. The same conclusion is reached ifT1, T2 are real but b0 and b2 disagree in sign.

�v� Using Eq. �8� we determine the � values corresponding tothese �T1 ,�1� and �T2 ,�2� pairs. There are infinitely many�1j and �2j , j=0,1 ,2 , . . ., respectively, according to �8�. Asper the definition of kernel, following �6�, the smallestvalues of these �1 and �2 form the two-point kernel for thiscase. For instance, for b=0.005�m�, these values are

T1 = − 0.006 142, T2 = − 0.000 303 2

�1 = 728.21 rad/s, �2 = 636.33 rad/s,

�1 = 0.0049s, �2 = 0.0093s �24�

The first “clustering” is already at hand. For b=0.005, the kernelconsists of two discrete points �as opposed to a curve in two-delaycases, such as in case I� �1=0.0049 s, �2=0.0093 s for which thecharacteristic equation has two imaginary roots. And these are theonly two imaginary roots that any ��R+ can ever produce �againfor b=0.005�. Each �1 �or �2� resulting in �1 �or �2� also de-scribes a set of countably infinite delay sets, called the “offspring”of the original �1 �or �2�. They are given by Eq. �8� �of course insingle delay domain� as

�1j = �1 +2

�1j

with the cluster identifier �1 = 728.21 rad/s, j = 1,2, . . .

�2j = �2 +2

�2j

with the cluster identifier �2 = 636.33 rad/s, j = 1,2, . . .

The second “clustering” feature �Proposition II� is slightly moresubtle: the root tendencies associated with the transitions of �defined by

�1j − � → �1 → �1j + � are all RT = + 1 �i.e.,destabilizing�

�2j − � → �2 → �2j + � are all RT = − 1 �i.e.,stabilizing�

Let’s examine this invariance property for ��1j� cluster, every el-ement of which renders the same s=�1 i characteristic root. Thedifferential form of �19� is

dCE�s,�,b� =�CE�s,�,b�

�sds +

�CE�s,�,b��

d�

= �2ms + c + bC cos ��e−�s�ds + bC cos �se−�sd�

= 0 �25�

which results in the

ds

d�= −

bC cos �se−�s

2ms + c + bC cos ��e−�s �26�

It is easy to show two features in this expression: �i� e−�s remainsunchanged for s=�1 i and �=�1j , j=0,1 ,2 , . . ., and �ii� RT1=sgn�Re�ds /d��s=�1 i� is independent of � despite the varying s
term in the denominator�, and it is +1 for all �1j , j=0,1 ,2 , . . .. The


Downloaded F

same invariance feature can be obtained for ��2 ,�2�, but withstabilizing RT2=−1. This proves the second clustering feature�i.e., Proposition II�.

We can now declare the stability posture of the system for agiven b �say 0.005 m� and the deployment of CTCR is completedfor this depth-of-cut �Fig. 9�. Notice the invariant RT from pointsC1 ,C1� ,C1� , . . . �all destabilizing� and C2 ,C2� ,C2� , . . . �all stabiliz-ing� as � increases. As a consequence, the number of unstableroots, NU, can be declared in each region very easily as sparinglyshown on the figure. Obviously, when NU=0, the cutting isstable; 0��1 stable, �1��2 unstable, �2��1+2 /�1stable, etc. stability switchings occur. We marked the stable inter-vals with thicker line style for ease of recognition. If we sweepb� �0, . . . ,30 mm�, we obtain the complete stability chart of Fig.9 in the delay domain.

Stable cutting appears below the dark curve �also known as thechatter bound�. The conventional chatter stability lobes �as the

Fig. 9 Chatter stabili

Fig. 10 Chatter s



machine tools community calls them� of Fig. 10 in the RPM do-main can readily be obtained using �=60/N coordinateconversion.

General Remark 1: Notice that for a given b=0.005�m� chipwidth there are a few pockets of spindle speeds �RPM� where thecutting is stable. This feature is very similar to the pockets in��1 ,�2� space in Fig. 7�a� as we walk on a line representing aconstant pitch ratio ��2 /�1=const�. Obviously the problem is atleast an order of magnitude more complex in the two-delay case,because the delays can influence the stability of the operation inconflicting directions. That is, increasing one delay maybe stabi-lizing, while increasing the other is destabilizing at the same op-erating point ��1 ,�2�, such as at point A of Fig. 7�a�. Increasing �implies increasing the corresponding pitch angles in the variablepitch milling. Another interesting point is that the two-delay prob-lem in case I reduces to that of one delay, if �2 /�1=n2 /n1 ,n1 ,n2

hart in delay domain
ty c
tability lobes

NOVEMBER 2005, Vol. 127 / 799


Downloaded F

=integers. For instance, when n2 /n1=11/7 the constant pitch ratioline �as marked on Fig. 7�a�� is obtained. On this line we againobserve stable-unstable-stable sequence as �1 �or �2� increases.

General Remark 2: Just to give an idea to the reader about thecomputational complexity of these operations we list the follow-ing CPU times from start to finish �including the graphical displayof the chatter stability lobes�:

case I:38.5 s

case II:0.5 s

Considering the complexity of case I, especially, and the intensityof the information which can be extracted from it, the CPU load-ing looks very attractive. All of these durations are obtained on aPC with Pentium 4, 3.2 MHz processor and 512 MB RAM.

5 ConclusionA unique procedure, CTCR, is presented for determining the

complete stability posture of regenerative chatter dynamics in si-multaneous machining �SM�. The novelty appears in the exhaus-tive declaration of the stability regions and the complete set ofchatter frequencies, which can possibly occur for the given pro-cess. For this purpose an intriguing transformation, Rekasius sub-stitution, is used which converts the common transcendental char-acteristic equation into algebraic form. The resulting tableau isunprecedented even in the mathematics community. We give apractical case study following a meticulous research effort of arespected group on variable pitch milling process �19�. A smallpart of the results of CTCR reproduces very well the findingspublished in that paper. In another case study, we also look at asimpler problem: the conventional single-tool-machining chatter.This exercise displays that the CTCR paradigm can comfortablyrecreate the simpler and well-known results of stability lobes. Nu-merous open directions await further research in utilizing thispowerful mathematical framework. We see the present work justas a prelude to many to follow.

AcknowledgmentsThe authors wish to express their special thanks to Professor

Yusuf Altintas, University of British Columbia, for his informativeassistance relevant to �19�. This research has been supported inpart by the awards from DOE �DE-FG02-04ER25656� and NSF�CMS-0439980� and �DMI-0522910�.

A patent application is pending on the method presented in thispaper.

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a unique methodology for chatter stability mapping in simultaneous machining

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