Kalmar-Nagy Nonlinear Models for Complex Dynamics in Cutting Materials

8/14/2019 Kalmar-Nagy Nonlinear Models for Complex Dynamics in Cutting Materials

1/17

10.1098/rsta.2000.0751

Nonlinear models for complex dynamics

in cutting materials

B y F r a n c is C. M o o n1 a n d T a m a s K a l m a r - N a g y2

1Sibley School of Aerospace and Mechanical Engineering, 2Department ofTheoretical and Applied Mechanics, Cornell University, Ithaca, NY 14853, USA

This paper reviews the prediction of complex, unsteady and chaotic dynamics asso-ciated with material-cutting processes through nonlinear dynamical models. Thestatus of bifurcation phenomena such as subcritical Hopf instabilities is assessed. Anew model using hysteresis in the cutting force is presented, which is shown to exhibitcomplex quasi-periodic solutions. In addition, further evidence for chaotic dynamicsin non-regenerative cutting of polycarbonate plastic is reviewed. The authors drawthe conclusion that single-degree-of-freedom models are not likely to predict low-level cutting chaos and that more complex models, such as multi-degree-of-freedomsystems based on careful cutting-force experiments, are required.

Keywords: metal cutting; nonlinear dynamics; chaos

1. IntroductionThe study of cutting of materials is an old problem by modern standards, goingback a century to research in both Europe and North America. The work of Taylor(1907) is a prominent example. In recent years, there has been a resurgence of inter-est in modelling cutting dynamics for several reasons. First, there are now highercutting speeds, new materials and hard-turning problems, as well as an interest inhigher-precision machining. Second, advances in nonlinear dynamics in the last twodecades has lent promise to the prospects of analysing more complex models atten-dant to material processing. Third, there is a renewed intellectual interest in both

the physics and mathematics associated with material removal. One such problemis the unsteady nature of both chatter and pre-chatter, or normal machining andcutting. This phenomenon, which has been documented in a number of laboratories,has led to a search for new models that can predict complex, quasi-periodic, chaoticand even random motions in cutting.

In this paper we review the development of cutting-dynamics modelling in the con-text of new low-dimensional nonlinear models and new experimental work in materialcutting. Although the linear delay model has been fairly successful in capturing theonset of the large amplitude periodic chatter, the limit-cycle behaviour itself has notbeen well understood. There are also other nonlinear phenomena that require morecomplex models than the classic linear chatter equation. A partial list includes

(i) unsteady chatter vibrations of the cutting tool,

(ii) subcritical Hopf bifurcation dynamics,

Phil. Trans. R. Soc. Lond. A (2001) 359, 695{711

695

c 2001 The Royal Society


2/17

696 F. C. Moon and T. Kalmar-Nagy

(iii) pre-chatter chaotic or random-like small amplitude cutting vibrations,

(iv) cutting dynamics in non-regenerative processes,

(v) elasto-thermoplastic workpiece material instabilities,

(vi) hysteretic eects in cutting dynamics,

(vii) induced electromagnetic voltages at the material{tool interface,

(viii) fracture processes in cutting of brittle materials,

(ix) fracture eects in chip breakage.

The length of this list serves to suggest that a single-degree-of-freedom (single-DOF) regenerative model cannot begin to predict all the important phenomena incutting dynamics. However, any new model should be judged on how successful it is

in encompassing the above dynamic problems.Our own contributions here are modest. After reviewing the current status, we

discuss two new one-dimensional models, which include hysteresis and viscoelasti-city. Numerical results show that hysteretic cutting-force laws lead to more complexdynamics, but that one-DOF models are not sucient to explain the broader rangeof cutting-dynamics phenomena. We also present some new experimental results innon-regenerative cutting of polycarbonate plastic that associates chaos-like dynamicswith normal or `good cutting.

2. Nonlinear eects in material cutting

Nonlinearity has always been recognized as an essential element in machining. Forexample, Doi & Kato (1956) performed some beautiful experiments on establish-ing chatter as a time-delay problem and also presented one of the earliest nonlinearmodels. Also, Tobias (1965) and Tlusty (see Tlusty & Ismail 1981) and others haveconsidered nonlinearity in their studies. Before 1975{1980, nonlinear dynamics analy-sis mainly consisted of perturbation analysis and numerical simulation. Random-likemotions were not considered, even though time records of cutting dynamics clearlyshowed unsteady oscillations (see, for example, Tobias 1965). Since the 1980s new

concepts of modelling, measuring and controlling nonlinear dynamics in materialprocessing have appeared.

The principal nonlinear eects on cutting dynamics include

(i) material constitutive relations (stress versus strain, strain rate and tempera-ture),

(ii) tool-structure nonlinearities,

(iii) friction at the tool{chip interface,

(iv) loss of tool{workpiece contact,

(v) inuence of machine drive unit on the cutting ow velocity.

There are at least four types of self excited machining dynamics.

Phil. Trans. R. Soc. Lond. A (2001)


3/17

Nonlinear models for complex dynamics in cutting materials 697

(i) Regenerative or time-delay models.

(ii) Coupled mode chatter.

(iii) Chip-instability models.

(iv) Negative damping models.

These instabilities parallel other unstable relative motion such as uid{structureutter, rail{wheel instabilities, stick{slip friction vibrations, etc. However, the regen-erative model seems to be unique to material processing systems. It appears in turn-ing, drilling, milling, grinding and rolling operations. The nite time delay introducesan innite-dimensional phase space, even for single-DOF systems. Because of thisunique feature, regenerative chatter problems have attracted the greatest interestamong applied mathematicians (see, for example, Stepan 1989; Nayfeh et al. 1998).

Fascination with time-delay dierential equations has often overshadowed thephysics of material processing. For example, in cutting physics the essential processesinvolve thermo-viscoplasticity and fracture mechanics. Yet most dynamic models ofchatter do not include temperature as a state variable. In some brittle materials,electric and magnetic elds are generated in the cutting process, yet these variablesare also missing from the models. In most cutting models, the physics is hidden in acutting-energy density factor. In the last several years several dynamic models haveexamined basic material nonlinearities, including thermal softening (see Davies et al.1996; Davies 1998).

Other groups have used nonlinear dynamics methodology to study cutting chat-ter (Moon 1994; Bukkapatnam et al. 1995a; Wiercigroch & Cheng 1997; Stepan &

Kalmar-Nagy 1997; Nayfeh et al. 1998; Minis & Berger 1998; Moon & Johnson 1998).Studies of nonlinear phenomena in machine-tool operations involve three dierent

approaches.

(1) Measurement of nonlinear force{displacement behaviour of cutting or formingtools.

(2) Model-based studies of bifurcations using parameter variation.

(3) Time-series analysis of dynamic data for system identication.

3. Nonlinear cutting forces

The fundamental origins of nonlinear dynamics in material processing usually involvenonlinear relations between stress and strain, or stress and temperature or chemicalkinetics and solid-state reactions in the material. Other sources involve nonlineargeometry such as contact forces or tool{workpiece separation. There is a long historyof force measurements in the literature over the past century. Many of these data arebased on an assumption of a steady process. Thus, in cutting-force measurements,the speed and depth of cut are xed and the average force is measured as a function

of steady material speed and cutting depth. However, this begs the question as to thereal dynamic nature of the process. In a dynamic process, what happens when thecutting depth instantaneously decreases? Does one follow the average-force{depthcurve or is there an unloading path similar to elasto-plastic unloading? Averageforce measurements often lter out the dynamic nature of the process.



4/17


20 40 600

2

4

6

8

10

f0

F

(N)

f(m)

Fx

Fx (f0)

DFx k1Df

Df

Kwfa

f0

(a)

(b)

Figure 1. (a) Experimental force in the feed direction for aluminium. (b) Assumed power-lawdependence of lateral cutting force on chip thickness.

One popular steady cutting force (F) versus chip thickness (f) relationship is thatproposed by Taylor (1907),

F = Kwfn; (3.1)

where a popular value for n is 34

(w is the chip width and K is a material-basedconstant). For aluminium, the value for n was found to be 0:41 (Kalmar-Nagy et al.1999, g. 1). Equation (3.1) is a softening force law. It is also single valued. In recentyears, more complete studies have been published, such as Oxley & Hastings (1977).In this work, they present steady-state forces as functions of chip thickness, as well

as cutting velocity for carbon steel. For example, they measured a decrease of cuttingforce versus material ow velocity in steel. They also measured the cutting forcesfor dierent tool rake angles. These relations were used by Grabec (1986, 1988)to propose a non-regenerative two-DOF model for cutting that predicted chaoticdynamics. However, the force measurements themselves are quasi-steady and weretaken to be single-valued functions of chip thickness and material ow velocity. Belowwe will propose a hysteretic force model of F(f) which is not single valued.

4. Bifurcation methodology

Bifurcation methodology looks for dramatic changes in the topology of the dynamicorbits, such as a jump from equilibrium to a limit cycle (Hopf bifurcation) or a dou-bling of the period of a limit cycle. The critical values of the control parameter atwhich the dynamics topology changes enable the researcher to connect the modelbehaviour with experimental observation in the actual process. These studies alsoallow one to design controllers to suppress unwanted dynamics or to change a sub-critical Hopf bifurcation into a supercritical one. The phase-space methodology alsolends itself to new diagnostic tools, such as Poincare maps, which can be used tolook for changes in the process dynamics (see, for example, Johnson 1996; Moon &

Johnson 1998).The limitations of the model-based bifurcation approach are that the models areusually overly simplistic and not based on fundamental physics. The use of bifurca-tion tools is most eective when the phase-space dimension is small, say, less thanor equal to four.



5/17


m

x (t)

k c

fFx

sDl cx

Figure 2. One-DOF mechanical model and FBD.

5. Single-DOF models

These models have been the principal source of nonlinear analysis, beginning withthe work of Arnold (1946) and Doi & Kato (1956). Figure 2 shows the one-DOF

model and the corresponding free-body diagram (FBD). The equation of motiontakes the form

x + 2! n _x + !2n

x = 1

mF; (5.1)

where ! n is the natural angular frequency of the undamped free oscillating systemand is the relative damping factor. F = FxFx(f0) is the cutting-force variation.Sometimes nonlinear stiness terms are added to the tool stiness (Hanna & Tobias1974). However, in practice, the tool holder is very linear, even in a cantilevered

boring bar. The chip thickness is often written as a departure from the steady chipthickness f0, i.e.

f = f0 + f; (5.2)

where f = x(t)x(t). Here, is the delay time related to the angular rate , i.e. = 2= (that is, is the period of revolution). After linearizing the cutting-forcevariation (F) at some nominal chip thickness, the linearized equation of motion ofclassical regenerative chatter becomes (see, for example, Stepan 1989)

x + 2! n _x + !2

nx =

k1m (x x); (5.3)

where x denotes the delayed value of x(t).The linear stability theory predicts unbounded motion above the lobes in the

parameter plane of cutting-force coecient k1 versus , as shown in gure 3 (here,k1 is the slope of the cutting-force law at the nominal feed f0). The parameters = 0:01, ! n = 580 rad s

1, m = 10 kg were used here. The lobes asymptote to avalue of k1 = 2m!

2n

(1 + ) 6:8 104 N m1. Below this value the theory predictsno sustained motion, which is counter to experimental evidence. The linear model

is insucient in at least three phenomena. First, it does not predict the amplitudeof the limit cycle for post-chatter. Second, the chatter is often subcritical, as shownin gure 4 (Kalmar-Nagy et al. 1999). Finally, there is the matter of the pre-chattervibrations, which in experiments appear to be non-steady of either a chaotic orrandom nature (see, for example, Johnson & Moon 2001).



6/17


5000 10 0000

100 000

200 000

W (RPM)

k1(N

m-

1)

Figure 3. Classical stability chart.

0 0.1 0.2 0.3 0.40

10

20

30

40

50

chip width (mm)

forwards sweep

backwards sweep

RPMv

ibration

amplitude(m)

Figure 4. Amplitude of tool vibration versus chip width.

bifurcation parameter

amplitudeof oscillation

bifurcation parameter

amplitudeof oscillation

(a) (b)

Figure 5. Supercritical and subcritical Hopf bifurcation.



7/17


8/17


400

200

0

-200

-400

x(t)

(a) (b)

(c) (d)

400

200

0

-200

-400

x(t)

400

200

0

-200

-400

x(t)

600

300

0

-300

-600

x(t)

-0.2 0 0.2

x (t)

-0.3 0 0.3

x (t)

Figure 6. Bifurcation sequence for Johnsons model (3 = 300, 1000, 2000, 4000).

tori. There is evidence that the limit of these bifurcations is a chaotic attractor. Anexample of these bifurcations is shown in gure 6.

Experiments were also conducted by Johnson using an electromechanical delay

system whose equations of motion were similar to the chatter model above. Remark-ably, the experimental results agreed exactly with the numerical simulation of themodel (Johnson & Moon 1999). Experiments were also conducted by Pratt & Nayfeh(1996) using an analogue computer. Even though these models showed new bifurca-tion phenomena in nonlinear delay equations, experimental results on chatter itselfhave not exhibited such bifurcation behaviour as of this date.

These results are important, however, because they show that dynamics in a four-dimensional phase space can be predicted by a second-order nonlinear delay equation.Experiments at several laboratories have reported complex chatter vibrations with

an apparent phase-space dimension of between four and ve. So there is hope thatsome cutting model, with one or two degrees of freedom, will eventually predict thesecomplex motions.

8. Hysteretic cutting-force mo del

The above models all involve smooth continuous single-valued force functions of thechip thickness. However, there is no reason to expect that the function F(f) besmooth and single valued when the underlying physics involves plastic deformation

in the cutting zone. Hysteresis may be due to Coulomb friction at the tool face orelasto-plastic behaviour of the material. This phenomenon has been studied in otherelds, such as soil mechanics, ferroelectricity and superconducting levitation. Themodel presented here was inspired by past research at Cornell on chaos in elasto-plastic structures (Poddar et al. 1988; Pratap et al. 1994).



9/17


DX0

Df

(a) (b)

DX2 DXcrit DX

RHS

DF

f contact loss

F

DX1

f

F

DF

Df

Figure 7. (a) Bilinear cutting-force law. (b) Hysteretic cutting-force model.

Df

DX

RHS

DF

contact loss

Df

DX

RHS

DF

contact loss

Df

DX

RHS

DF

Df

DX

RHS

DF

Figure 8. Loading{unloading paths.

The idea of cutting-force hysteresis is based on the fact that the cutting force is anelasto-plastic process in many materials. In such behaviour, the stress follows a work-hardening rule for positive strain rate but reverts to a linear elastic rule for decreasingstrain rate. A possible macroscopic model of such behaviour is shown in gure 7b(here, RHS corresponds to the right-hand side of (5.1)). Here, the power-law curvehas been replaced with a piecewise-linear function, where the lower line is tangent tothe nonlinear cutting-force relation at x = 0 (gure 7a). The loading line and the

unloading line can have dierent slopes (gure 8 shows possible loading{unloadingpaths). This model also includes separation of the tool and workpiece. An interestingfeature of this model is the coexistence of periodic and quasi-periodic attractors belowthe linear stability boundary. As shown in gure 9, there exists a torus `inside ofthe stable limit cycle. This could explain the experimental observation of the sudden



10/17


x(t)

-0.15

-0.3

x (t)0.35

0.3

Figure 9. Torus inside the stable limit cycle.

0.3

-0.1

-0.3 0.6-0.08

-0.050.06

0.3

RHS

D x

RHS

Dx

Figure 10. Hysteresis loops for periodic and quasi-periodic motions.

transition of periodic tool vibration into complex motion. Figure 10 shows hysteresisloops for the observed behaviour.

9. Viscoelastic models

Most of the theoretical analyses of machine-tool vibrations employ force laws that arebased on the assumption that cutting is steady-state. However, cutting is a dynamicprocess and experimental results show clear dierences between steady-state anddynamic cutting. As shown by Albrecht (1965) and Szakovits & DSouza (1976), thecutting-force{chip-thickness relation exhibits hysteresis. This hysteresis depends onthe cutting speed, the frequency of chip segmentation, the functional angles of thetools edges, etc. (Kudinov et al. 1978). Saravanja-Fabris & DSouza (1974) employed

the describing function method to obtain linear stability conditions. In this paper,we derive a delay-dierential equation model that includes hysteretic eects via aconstitutive relation.

To describe elasto-plastic materials, the Kelvin{Voigt model is often used. Thismodel describes solid-like behaviour with delayed elasticity (instantaneous elastic



11/17


deformation and delayed elastic deformation) via a constitutive relation that is linearin stress, rate of stress, strain and strain-rate.

We assume that a similar relation between cutting force and chip thickness holds,where the coecients of the rates depend on the cutting speed (through the timedelay, using f = x x),

F + q0 _F = k1f + q1 _f : (9.1)The usual one-DOF model is

x + 2! n _x + !2nx =

1

mF: (9.2)

Multiplying the time derivative of (9.2) by q0 and adding it to (9.2) gives

x + 2!n _x + !2n

x + q0(...x + 2! nx + !

2n

_x) = 1

m(F + q0 _F); (9.3)

which can be rewritten using (9.1) and the relation for chip-thickness variation

f = x x as

q0...x + (1 + 2q0 ! n)x + 2!n + q0 !

2n

+q1

m_x

+ !2n

+k1m

x k1m

x q1

m_x = 0: (9.4)

The characteristic equation of (9.4) is

D() = q0 3

+ (1 + 2q0 !n)2

+ 2!n + q0 !2

n +

q1

m

+ !2n

+k1m

k1m

e q1

me: (9.5)

The stability boundaries can be found by solving D(i!) = 0,

Re D(i!) = !2 + 1! + 2 + 3k1 = 0; (9.6)

Im D(i!) = !2 + -1! + -2 + -3k1 = 0: (9.7)

Dening = !, the coecients i(), -i() can be expressed as

1 = 2q0! n; 2 = !2n

q1 sin

m; 3 =

1 cos

m(9.8)

-1 =2! nq0

; -2 = !2n

+q1(1 cos )

mq0; -3 =

sin

mq0: (9.9)

One can eliminate k1 from (9.6, 9.7) to get

!2 + 2! 2 = 0; (9.10)

where

= 1-3 3-12(3 -3)

= !n(1 cos + q0 sin)q0(sin q0(1 cos ))

; (9.11)

2 =2-3 3-2

-3 3= !2

n 2q1(1 cos )

m(sin q0(1 cos )): (9.12)



12/17


500 10000

0.45

0.90

W (RPM)

k1(

Nmm-

1)

Figure 11. Stability chart for the viscoelastic model, q1 =0.

Equation (9.10) can then be solved,

!() = 2 + 2 : (9.13)

And nally (thus ) and k1 can be expressed as functions of ! and ,

() =

!()) () =

2!()

; (9.14)

k1() =1

3(!2 1! 2): (9.15)

The stability chart can be drawn as a function of the real parameter . If q1 = 0,equation (9.4) is equivalent to that obtained by Stepan (1998), who calculated thecutting force by integrating an exponentially distributed force system on the rakeface. The stability chart for this case is shown in gure 11 (the same parameterswere used as in gure 3 and q0 = 0:01). Experiments also show that the chatterthreshold is higher for lower cutting speeds than for higher speeds. Small values ofq1 do not seem to inuence this chart; however, for higher values of this variable, theminima of the lobes in the low-speed region decrease (in contrast to the experimentalobservations).

10. Chaotic cutting dynamics

The time-series analysis method has become popular in recent years to analyse manydynamic physical phenomena from ocean waves, heartbeats, lasers and machine-toolcutting (see, for example, Abarbanel 1996). This method is based on the use of aseries of digitally sampled data fxig, from which the user constructs an orbit ina pseudo-M-dimensional phase space. One of the fundamental objectives of thismethod is to place a bound on the dimension of the underlying phase space fromwhich the dynamic data were sampled. This can be done with several statistical

methods, including fractal dimension, false nearest neighbours (FNN), Lyapunovexponents, wavelets and several others.However, if model-based analysis can be criticized for its simplistic models, then

nonlinear time-series analysis can be criticized for its assumed generality. Although itcan be used for a wide variety of applications, it contains no physics. It is dependent



13/17


on the data alone. Thus the results may be sensitive to the signal-to-noise ratio of thesource measurement, signal ltering, the time delay of the sampling, the number ofdata points in the sampling and whether the sensor captures the essential dynamicsof the process.

One of the fundamental questions regarding the physics of cutting solid materialsis the nature and origin of low-level vibrations in so-called normal or good machin-ing. This is cutting below the chatter threshold. Below this threshold, linear modelspredict no self-excited motion. Yet when cutting tools are instrumented, one cansee random-like bursts of oscillations with a centre frequency near the tool naturalfrequency. Work by Johnson (1996) has carefully shown that these vibrations are sig-nicantly above any machine noise in a lathe-turning operation. These observationshave been done by several laboratories, and time-series methodology has been usedto diagnose the data to determine whether the signals are random or deterministicchaos (Berger et al. 1992, 1995; Minis & Berger 1998; Bukkapatnam 1999; Bukkap-atnam et al. 1995a; b; Moon 1994; Moon & Abarbanel 1995; Johnson 1996; Gradisek

et al. 1998).One of the new techniques for examining dynamical systems from time-series mea-

surements is the method of FNN (see Abarbanel 1996). Given a temporal series ofdata fxig, one can construct an M-dimensional vector space of vectors, (x1; : : : ; xM),(x2; : : : ; xM+1), etc., whose topological properties will be similar to the real phasespace if one had access to M state variables. The method is used to determinethe largest dimensional phase space in which the orbital trajectory, which threadsthrough the ends of the discrete vectors dened above, does not intersect. Thus, ifthe reconstructed phase space is of too low a dimension, some orbits will appear tocross and some of the points on the orbits will be false neighbours. In an ideal calcu-lation, as the embedding dimension M increases, the number of such false neighboursgoes to zero. One then assumes that the attractor has been unraveled. This gives anestimate of the dimension of the low-order nonlinear model that one hopes will befound to predict the time-series.

Using data from low-level cutting of aluminium, for example, the FNN methodpredicts a nite dimension for the phase space of between four and ve (Moon &Johnson 1998). This low dimension suggests that these low-level vibrations mayhave a deterministic origin, such as in chip shear band instabilities or chip-fractureprocesses. Minis & Berger (1998) have also used the FNN method in pre-chatter

experiments on mild steel and also obtained a dimension between four and ve.These experiments and others (Bukkapatnam et al. 1995a; b) suggest that normalcutting operations may be naturally chaotic. This idea would suggest that a smallamount of chaos may actually be good in machining, since it introduces many scalesin the surface topology.

11. Non-regenerative cutting of plastics

Complex dynamics can also occur in non-regenerative cutting. An example is shown

in gures 12{14 for a diamond stylus cutting polycarbonate plates on a turntable(Moon & Callaway 1997). The width of the cut was smaller than the turning pitch,so that there was no overlap and no regenerative or delay eects. The time-historyof the vibrations of the 16 cm cantilevered stylus holder is shown in gure 13, alongwith a photograph of the cut tracks. The cut tracks appear to be fairly uniform, even



14/17


chip

uz uy

ux

V

N

Figure 12. Non-regenerative cutting.

time

stressgaugeou

tput

Figure 13. Time-history for cutting of plastic and magnication of cut surface.

poor-quality cut

periodic motion

good-quality cut

chaotic-looking motion

cutting velocity, V

normalforce(N)

Figure 14. Stylus dead load versus cutting speed.

though the tool vibrations appear to be random or chaotic. When the cutting speedis increased, however, the cutting width becomes highly irregular, and the vibrationsbecome more periodic looking. An FNN of the unsteady vibrations seems to indicatethat the dynamics of gure 13 could be captured in a four- or ve-dimensional phasespace, lending evidence that the motion may be deterministic chaos. A summary ofthese experiments is shown in gure 14 in the parameter plane of stylus dead load

versus cutting speed of the turntable.In spite of the evidence from time-series analysis that normal cutting of metalsand plastics may be deterministic chaos, there is no apparent experimental evidencefor the usual bifurcations attendant to classic low-dimensional nonlinear mappingsor ows. However, traditional explanations for this low-level noise do not seem to



15/17


t the observations. Claims that the noise is the result of random grain structurein the material are not convincing, since the grain size in metals is of 10{100 mm,which would lead to frequencies in the 100 kHz range, whereas the cutting noiseis usually in the 1 kHz range or lower. Besides, the grain structure theory wouldnot apply to plastics as in the above discussion of cutting polycarbonate. Anotherpossible explanation is the shear banding instabilities in metals (see, for example,Davies et al. 1996). But the wavelengths here are also in the 10 mm range andlead to a spectrum with higher frequency content than that observed in cuttingnoise.

One possible candidate explanation might be tool{chip friction. A friction modelwas used by Grabec (1986) in his pioneering paper on chaos in machining. However,in a recent paper (Gradisek et al. 1998), they now disavow the chaos theory forcutting and claim that the vibrations are random noise (see also Wiercigroch &Cheng 1997).

So this controversy remains about the random or deterministic chaos nature of the

dynamics of normal cutting of materials.

12. Summary

One may ask what is the unique role of nonlinear analysis in the study of cutting andchatter? It has been known for some time how to predict the onset of chatter usinglinear theory (Tlusty 1978; Tobias 1965). The special tasks for nonlinear theory incutting research include

(i) predicting steady chatter amplitude,

(ii) providing understanding of subcritical chatter,

(iii) explaining pre-chatter low-level chaotic vibrations,

(iv) predicting dynamic chip morphology,

(v) providing new diagnostics for tool wear,

(vi) determining control models for chatter suppression,

(vii) providing clues to better surface precision and quality.

Certainly, many or all of these goals were the basis of traditional research method-ology in machining. But the use of nonlinear theory acknowledges the essentialdynamic character of material removable processes that in more classical theorieswere ltered out. However, there is a need to integrate the dierent methods of

research, such as bifurcation theory, cutting-force characterization and time-seriesanalysis, before nonlinear dynamics modelling can be useful in practice. It is alsolikely that single-DOF models will not capture all the phenomena to achieve theabove goals and more degrees of freedom and added state variables such as temper-ature will be needed.



16/17


17/17


Moon, F. C. 1994 Chaotic dynamics and fractals in material removal processes. In Nonlinearityand chaos in engineering dynamics (ed. J. Thompson & S. Bishop), pp. 25{37. Wiley.

Moon, F. C. & Abarbanel, H. 1995 Evidence for chaotic dynamics in metal cutting, and clas-sication of chatter in lathe operations. In Summary Report of a Workshop on NonlinearDynamics and Material Processes and Manufacturing (ed. F. C. Moon), pp. 11{12, 28{29.Institute for Mechanics and Materials.

Moon, F. C. & Callaway, D. 1997 Chaotic dynamics in scribing polycarbonate plates with adiamond cutter. IUTAM Symp. on New Application of Nonlinear and Chaotic Dynamics,Ithaca.

Moon, F. & Johnson, M. 1998 Nonlinear dynamics and chaos in manufacturing processes. InDynamics and chaos in manufacturing processes (ed. F. C. Moon), pp. 3{32. Wiley.

Nayfeh, A., Chin, C. & Pratt, J. 1998 Applications of perturbation methods to tool chatterdynamics. In Dynamics and chaos in manufacturing processes (ed. F. C. Moon), pp. 193{213. Wiley.

Oxley, P. L. B. & Hastings, W. F. 1977 Predicting the strain rate in the zone of intense shearin which the chip is formed in machining from the dynamic ow stress properties of the work

material and the cutting conditions. Proc. R. Soc. Lond. A 356 , 395{410.Poddar, B., Moon, F. C. & Mukherjee, S. 1988 Chaotic motion of an elastic plastic beam. ASME

J. Appl. Mech. 55, 185{189.

Pratap, R., Mukherjee, S. & Moon, F. C. 1994 Dynamic behavior of a bilinear hysteretic elasto-plastic oscillator. Part II. Oscillations under periodic impulse forcing. J. Sound Vib. 172,339{358.

Pratt, J. & Nayfeh, A. H. 1996 Experimental stability of a time-delay system. In Proc. 37thAIAA/ASME/ASCE/AHS/ACS Structures, Structural Dynamics and Materials Conf., SaltLake City, USA.

Saravanja-Fabris, N. & DSouza, A. 1974 Nonlinear stability analysis of chatter in metal cutting.

J. Engng Industry 96, 670{675.Stepan, G. 1989 Retarded dynamical systems: stability and characteristic functions. Pitman

Research Notes in Mathematics, vol. 210. London: Longman Scientic and Technical.

Stepan, G. 1998 Delay-dierential equation models for machine tool chatter. In Dynamics andchaos in manufacturing processes (ed. F. C. Moon), pp. 165{191. Wiley.

Stepan, G. & Kalmar-Nagy, T. 1997 Nonlinear regenerative machine tool vibrations. In Proc.1997 ASME Design Engineering Technical Conf. on Vibration and Noise, Sacramento, CA,paper no. DETC 97/VIB-4021, pp. 1{11.

Szakovits, R. J. & DSouza, A. F. 1976 Metal cutting dynamics with reference to primary chatter.J. Engng Industry 98, 258{264.

Taylor, F. W. 1907 On the art of cutting metals. Trans. ASME 28, 31{350.Tlusty, J. 1978 Analysis of the state of research in cutting dynamics. Ann. CIRP 27, 583{589.

Tlusty, J. & Ismail, F. 1981 Basic non-linearity in machining chatter. CIRP Ann. ManufacturingTechnol. 30, 299{304.

Tobias, S. 1965 Machine tool vibration. London: Blackie.

Wiercigroch, M. & Cheng, A. H.-D. 1997 Chaotic and stochastic dynamics of orthogonal metalcutting. Chaos Solitons Fractals 8, 715{726.

S A ( )
http://lucia.catchword.com/nw=1/rpsv/0022-460X%5E28%5E29172L.339[aid=982909,doi=10.1006/jsvi.1994.1179]http://lucia.catchword.com/nw=1/rpsv/0960-0779%5E28%5E298L.715[aid=982895,doi=10.1016/S0960-0779%5E2896%5E2900111-7]http://lucia.catchword.com/nw=1/rpsv/0022-460X%5E28%5E29172L.339[aid=982909,doi=10.1006/jsvi.1994.1179]

Documents

Kalmar-Nagy Nonlinear Models for Complex Dynamics in Cutting Materials