nonlinear models for complex dynamics in cutting...

101098rsta20000751

Nonlinear models for complex dynamicsin cutting materials

By Francis C M o o n1 a n d Tam paraa s K a lm paraa r-Nagy2

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 Introduction

The 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 boththe 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 laboratorieshas 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 695711

695

creg 2001 The Royal Society

696 F C Moon and T Kalmparaar-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 enotects in cutting dynamics

(vii) induced electromagnetic voltages at the materialtool interface

(viii) fracture processes in cutting of brittle materials

(ix) fracture enotects 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 isin encompassing the above dynamic problems

Our own contributions here are modest After reviewing the current status wediscuss 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 sumacr cient 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 `goodrsquo cutting

2 Nonlinear ereg ects in material cutting

Nonlinearity has always been recognized as an essential element in machining Forexample Doi amp 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 amp Ismail 1981) and others haveconsidered nonlinearity in their studies Before 19751980 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 newconcepts of modelling measuring and controlling nonlinear dynamics in materialprocessing have appeared

The principal nonlinear enotects on cutting dynamics include

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

(ii) tool-structure nonlinearities

(iii) friction at the toolchip interface

(iv) loss of toolworkpiece contact

(v) inreguence of machine drive unit on the cutting regow velocity

There are at least four types of self excited machining dynamics

Phil Trans R Soc Lond A (2001)

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 reguidstructureregutter railwheel instabilities stickslip 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 shy nite time delay introducesan inshy nite-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 Stepacutean 1989 Nayfeh et al 1998)

Fascination with time-delay dinoterential 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 materialselectric and magnetic shy 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 amp Cheng 1997 Stepan ampKalmar-Nagy 1997 Nayfeh et al 1998 Minis amp Berger 1998 Moon amp Johnson 1998)

Studies of nonlinear phenomena in machine-tool operations involve three dinoterentapproaches

(1) Measurement of nonlinear forcedisplacement behaviour of cutting or formingtools

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

(3) Time-series analysis of dynamic data for system identishy cation

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 toolworkpiece 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 measurementsthe speed and depth of cut are shy xed and the average force is measured as a functionof 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-forcedepthcurve or is there an unloading path similar to elasto-plastic unloading Averageforce measurements often shy lter out the dynamic nature of the process



20 40 600

2

4

6

8

10

f0

F (

N)

f (microm)

Fx

Fx ( f0)

D Fx k1 D f

D f

Kwf a

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 (31)

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 041 (Kalmacutear-Nagy et al 1999 shy g 1) Equation (31) is a softening force law It is also single valued In recentyears more complete studies have been published such as Oxley amp Hastings (1977)In this work they present steady-state forces as functions of chip thickness as wellas cutting velocity for carbon steel For example they measured a decrease of cuttingforce versus material regow velocity in steel They also measured the cutting forcesfor dinoterent 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 regow 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 ampJohnson 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 enotective when the phase-space dimension is small say less thanor equal to four



m

x (t)

k c

f

Fx

s D l 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 amp Kato (1956) Figure 2 shows the one-DOFmodel and the corresponding free-body diagram (FBD) The equation of motiontakes the form

x + 2 plusmn n _x + 2n x = iexcl

1

mcentF (51)

where n is the natural angular frequency of the undamped free oscillating systemand plusmn is the relative damping factor centF = Fx iexcl Fx(f0) is the cutting-force variationSometimes nonlinear stinotness terms are added to the tool stinotness (Hanna amp Tobias1974) However in practice the tool holder is very linear even in a cantileveredboring bar The chip thickness is often written as a departure from the steady chipthickness f0 ie

f = f0 + centf (52)

where centf = x(t iexcl frac12 ) iexcl x(t) Here frac12 is the delay time related to the angular rate laquo iefrac12 = 2 ordm =laquo (that is frac12 is the period of revolution) After linearizing the cutting-forcevariation (centF ) at some nominal chip thickness the linearized equation of motion ofclassical regenerative chatter becomes (see for example Stepacutean 1989)

x + 2 plusmn n _x + 2n x =

k1

m(x iexcl x frac12 ) (53)

where x frac12 denotes the delayed value of x(t)The linear stability theory predicts unbounded motion above the lobes in the

parameter plane of cutting-force coemacr cient k1 versus laquo as shown in shy gure 3 (herek1 is the slope of the cutting-force law at the nominal feed f0) The parametersplusmn = 001 n = 580 rad siexcl1 m = 10 kg were used here The lobes asymptote to avalue of k1 = 2m2

n plusmn (1 + plusmn ) sup1 68 pound 104 N miexcl1 Below this value the theory predictsno sustained motion which is counter to experimental evidence The linear modelis insu cient 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 shy gure 4 (Kalmacutear-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 amp Moon 2001)



5000 10 0000

100 000

200 000

W (RPM)

k 1 (

N m

-1)

Figure 3 Classical stability chart

0 01 02 03 04

0

10

20

30

40

50

chip width (mm)

forwards sweep

backwards sweep

RP

M v

ibra

tion

am

plit

ude

(microm

)

Figure 4 Amplitude of tool vibration versus chip width

bifurcation parameter

amplitudeof oscillation



(a) (b)

Figure 5 Supercritical and subcritical Hopf bifurcation



6 Subcritical chatter bifurcations

If the chatter amplitude grows smoothly as the parameter increases the instability iscalled supercritical (shy gure 5a) Here continuous lines correspond to stability whiledashed lines indicate instability However an increase in the parameter often resultsin a shy nite jump in chatter amplitude If the parameter is then decreased the chatterpersists below the critical level of the machining parameter predicted by the lineartheory At the second critical value the vibration amplitude drops close to zero orlow-level vibrations This condition is called a subcritical Hopf bifurcation (shy gure 5b)The value of this second critical parameter is more useful in practice since it deshy nesa robust parameter operating range whereas the range between the lower criticalparameter and the linear critical parameter is sensitive to initial conditions andimpact knocking the tool onto the upper branch of the chatter bifurcation curve

Subcritical or hysteretic chatter amplitude behaviour was documented by Hookeamp Tobias (1963) and by Kalmar-Nagy et al (1999) Modern analysis of subcriticalbehaviour has been presented by Nayfeh et al (1998) Stepan amp Kalmar-Nagy (1997)and Kalmar-Nagy et al (2001a b) Another work is a PhD dissertation of Fofana(1993) These results depend critically on the assumption of the cutting-forcetool-displacement nonlinearity The analytical tools used in these studies were based onperturbation methods and on the use of centre manifold and normal form theory

Typical of the single-DOF models on which bifurcation studies have been con-ducted is the model of Hanna amp Tobias (1974) This model was used by Nayfeh etal (1998) using modern perturbation methods

x + 2 plusmn _x + p2(x + shy 2x2 + shy 3x3) = p2w(centf iexcl not 2centf2 + not 3centf3) (61)

This model incorporates both structural (shy 2 shy 3) and material nonlinearities ( not 2not 3) This group was able to show that this equation exhibited a global subcriti-cal Hopf bifurcation (initially supercritical and then turning subcritical at highervibration amplitudes)

A similar model is the work of Stepan amp Kalmar-Nagy (1997) which incorporatesonly quadratic and cubic terms in the material nonlinearities This equation has theform

x + 2plusmn _x + x = pcentf + q(centf2 + centf3) (62)

Using centre manifold theory this equation was shown to exhibit a subcriticalHopf bifurcation (Kalmacutear-Nagy et al 2001a b)

7 Quasi-periodic bifurcations

Nayfeh et al (1998) also showed in numerical simulation that the equation exhibitedquasi-periodic motions and bifurcations Similar results were found by Johnson andMoon (see Moon amp Johnson 1998 Johnson amp Moon 1999 2001) Johnson used asimpler delay model with only a cubic structural nonlinearity of the form

x + reg 1 _x + reg 2(x + x3) = iexcl reg 3x(t iexcl 1) (71)

Numerical simulation of this equation shy rst revealed a periodic limit cycle in post-chatter But as the parameter reg 3 was increased the Poincare map of the motionrevealed secondary bifurcations of the periodic motion into a torus and period-2N



400

200

0

- 200

- 400

xrsquo (

t)(a) (b)

(c) (d )

400

200

0

- 200

- 400

xrsquo (

t)

400

200

0

- 200

- 400

xrsquo (

t)

600

300

0

- 300

- 600xrsquo

(t)

- 02 0 02x (t)

- 03 0 03x (t)

Figure 6 Bifurcation sequence for Johnsonrsquos model ( reg 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 shy gure 6

Experiments were also conducted by Johnson using an electromechanical delaysystem 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 amp Moon 1999) Experiments were also conducted by Pratt amp 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 equationExperiments at several laboratories have reported complex chatter vibrations withan apparent phase-space dimension of between four and shy ve So there is hope thatsome cutting model with one or two degrees of freedom will eventually predict thesecomplex motions

8 Hysteretic cutting-force model

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 deformationin 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 othershy elds 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)



D X0

D f

(a) (b)

D X2 D Xcrit D X

RHS

D F

f contact loss

F

D X1

f

FD F

D f

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

D f

D X

RHS

D F

contact loss

D f

D X

RHS

D F

contact loss

D f

D X

RHS

D F

D f

D X

RHS

D F

Figure 8 Loadingunloading 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 shy gure 7b(here RHS corresponds to the right-hand side of (51)) Here the power-law curvehas been replaced with a piecewise-linear function where the lower line is tangent tothe nonlinear cutting-force relation at centx = 0 (shy gure 7a) The loading line and theunloading line can have dinoterent slopes (shy gure 8 shows possible loadingunloadingpaths) 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 shy gure 9 there exists a torus `insidersquo ofthe stable limit cycle This could explain the experimental observation of the sudden



xrsquo (t)

- 015

- 03

x (t)035

03

Figure 9 Torus inside the stable limit cycle

03

- 01

- 03 06- 008

- 005006

03RHS

D x

RHS

D x

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 dinoterences between steady-state anddynamic cutting As shown by Albrecht (1965) and Szakovits amp DrsquoSouza (1976) thecutting-forcechip-thickness relation exhibits hysteresis This hysteresis depends onthe cutting speed the frequency of chip segmentation the functional angles of thetoolrsquos edges etc (Kudinov et al 1978) Saravanja-Fabris amp DrsquoSouza (1974) employedthe describing function method to obtain linear stability conditions In this paperwe derive a delay-dinoterential equation model that includes hysteretic enotects via aconstitutive relation

To describe elasto-plastic materials the KelvinVoigt model is often used Thismodel describes solid-like behaviour with delayed elasticity (instantaneous elastic



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 holdswhere the coe cients of the rates depend on the cutting speed (through the timedelay using centf = x iexcl x frac12 )

centF + q0 frac12 cent _F = k1centf + q1 frac12 cent _f (91)

The usual one-DOF model is


1

mcentF (92)

Multiplying the time derivative of (92) by q0 frac12 and adding it to (92) gives

x + 2plusmn n _x + 2n x + q0 frac12 (

x + 2 plusmn n x + 2

n _x) = iexcl1

m(centF + q0 frac12 cent _F ) (93)

which can be rewritten using (91) and the relation for chip-thickness variationcentf = x iexcl x frac12 as

q0 frac12x + (1 + 2 plusmn q0 frac12 n ) x + 2 plusmn n + q0 frac12 2

n +q1 frac12

m_x

+ 2n +

k1

mx iexcl k1

mx frac12 iexcl q1 frac12

m_x frac12 = 0 (94)

The characteristic equation of (94) is

D( para ) = q0 frac12 para 3 + (1 + 2 plusmn q0 frac12 n ) para 2 + 2 plusmn n + q0 frac12 2n +

q1 frac12

mpara

+ 2n +

k1

mpara iexcl iexcl

k1

meiexcl para frac12 iexcl

q1 frac12

mpara eiexcl para frac12 (95)

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

Re D(i) = iexcl 2 + not 1 + not 2 + not 3k1 = 0 (96)

Im D(i) = iexcl 2 + shy 1 + shy 2 + shy 3k1 = 0 (97)

Deshy ning Aacute = frac12 the coemacr cients not i(Aacute) shy i(Aacute) can be expressed as

not 1 = iexcl 2q0 plusmn Aacute n not 2 = 2n iexcl

q1Aacute sin Aacute

m not 3 =

1 iexcl cos Aacute

m(98)

shy 1 =2 plusmn n

q0Aacute shy 2 = 2

n +q1(1 iexcl cos Aacute)

mq0 shy 3 =

sin Aacute

mq0Aacute (99)

One can eliminate k1 from (96 97) to get

2 + 2 reg iexcl macr 2 = 0 (910)

where

reg =not 1shy 3 iexcl not 3shy 1

2( not 3 iexcl shy 3)=

plusmn n (1 iexcl cos Aacute + q0Aacute sin Aacute)

q0Aacute(sin Aacute iexcl q0Aacute(1 iexcl cos Aacute)) (911)

macr 2 =not 2shy 3 iexcl not 3shy 2

shy 3 iexcl not 3

= 2n iexcl 2q1Aacute(1 iexcl cos Aacute)

m(sin Aacute iexcl q0Aacute(1 iexcl cos Aacute)) (912)



500 10000

045

090

W (RPM)

k 1 (N

mm

-1)

Figure 11 Stability chart for the viscoelastic model q1 =0

Equation (910) can then be solved

(Aacute) = reg 2 + macr 2 iexcl reg (913)

And shy nally frac12 (thus laquo ) and k1 can be expressed as functions of and Aacute

frac12 (Aacute) =Aacute

(Aacute)) laquo (Aacute) =

2 ordm (Aacute)

Aacute (914)

k1(Aacute) =1

not 3(2 iexcl not 1 iexcl not 2) (915)

The stability chart can be drawn as a function of the real parameter Aacute If q1 = 0equation (94) 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 shy gure 11 (the same parameterswere used as in shy gure 3 and q0 = 001) Experiments also show that the chatterthreshold is higher for lower cutting speeds than for higher speeds Small values ofq1 do not seem to inreguence 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 statisticalmethods including fractal dimension false nearest neighbours (FNN) Lyapunovexponents wavelets and several others

However if model-based analysis can be criticized for its simplistic models thennonlinear 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



on the data alone Thus the results may be sensitive to the signal-to-noise ratio of thesource measurement signal shy 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-nishy cantly 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 amp Berger 1998 Bukkapatnam 1999 Bukkap-atnam et al 1995a b Moon 1994 Moon amp Abarbanel 1995 Johnson 1996 Gradimicroseket 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 deshy ned 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 shy nite dimension for the phase space of between four and shy ve (Moon ampJohnson 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 amp Berger (1998) have also used the FNN method in pre-chatterexperiments on mild steel and also obtained a dimension between four and shy veThese 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 shownin shy gures 1214 for a diamond stylus cutting polycarbonate plates on a turntable(Moon amp Callaway 1997) The width of the cut was smaller than the turning pitchso that there was no overlap and no regenerative or delay enotects The time-historyof the vibrations of the 16 cm cantilevered stylus holder is shown in shy gure 13 alongwith a photograph of the cut tracks The cut tracks appear to be fairly uniform even



chip

uz uy

ux

V

N

Figure 12 Non-regenerative cutting

time

stre

ss g

auge

out

put

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

poor-quality cutperiodic motion

good-quality cutchaotic-looking motion

cutting velocity V

norm

al f

orce

(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 shy gure 13 could be captured in a four- or shy ve-dimensional phasespace lending evidence that the motion may be deterministic chaos A summary ofthese experiments is shown in shy gure 14 in the parameter plane of stylus dead loadversus 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 regows However traditional explanations for this low-level noise do not seem to



shy 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 10100 m mwhich 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 exampleDavies et al 1996) But the wavelengths here are also in the 10 m m range andlead to a spectrum with higher frequency content than that observed in cuttingnoise

One possible candidate explanation might be toolchip friction A friction modelwas used by Grabec (1986) in his pioneering paper on chaos in machining Howeverin a recent paper (Gradimicrosek et al 1998) they now disavow the chaos theory forcutting and claim that the vibrations are random noise (see also Wiercigroch ampCheng 1997)

So this controversy remains about the random or deterministic chaos nature of thedynamics 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 shy ltered out However there is a need to integrate the dinoterent methods ofresearch 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



References

Abarbanel H 1996 Analysis of observed chaotic data Springer

Albrecht P 1965 Dynamics of the metal-cutting process J Engng Industry 87 429441

Arnold R N 1946 The mechanism of tool vibration in the cutting of steel Proc Inst MechEngrs (Lond) 154 261284

Berger B Rokni M amp Minis I 1992 The nonlinear dynamics of metal cutting Int J EngngSci 30 14331440

Berger B Minis I Chen Y Chavali A amp Rokni M 1995 Attractor embedding in metalcutting J Sound Vib 184 936942

Bukkapatnam S T S 1999 Compact nonlinear signal representation in machine tool operationsIn Proc 1999 ASME Design Engineering Technical Conf DETC99VIB-8068 Las VegasNV USA

Bukkapatnam S Lakhtakia A amp Kumara S 1995a Analysis of sensor signals shows turningon a lathe exhibits low-dimensional chaos Phys Rev E 52 23752387

Bukkapatnam S Lakhtakia A Kumara S amp Satapathy G 1995b Characterization of nonlin-earity of cutting tool vibrations and chatter In ASME Symp on Intelligent Manufacturingand Material Processing vol 69 pp 12071223

Davies M 1998 Dynamic problems in hard-turning milling and grinding In Dynamics andchaos in manufacturing processes (ed F C Moon) pp 5792 Wiley

Davies M Chou Y amp Evans C 1996 On chip morphology tool wear and cutting mechanicsin macrnish hard turning Ann CIRP 45 7782

Doi S amp Kato S 1956 Chatter vibration of lathe tools Trans ASME 78 11271134

Fofana M 1993 Nonlinear dynamics of cutting process PhD thesis University of Waterloo

Grabec I 1986 Chaos generated by the cutting process Phys Lett A 117 384386

Grabec I 1988 Chaotic dynamics of the cutting process Int J Machine Tools Manufacture28 1932

Gradimiddotsek J Govekar E amp Grabec I 1998 Time series analysis in metal cutting chatter versuschatter-free cutting Mech Sys Signal Proc 12 839854

Hanna N amp Tobias S 1974 A theory of nonlinear regenerative chatter J Engng Industry 96247255

Hooke C amp Tobias S 1963 Finite amplitude instability|a new type of chatter In Proc 4thInt MTDR Conf Manchester UK pp 97109 Oxford Pergamon

Johnson M 1996 Nonlinear direg erential equations with delay as models for vibrations in themachining of metals PhD thesis Cornell University

Johnson M amp Moon F C 1999 Experimental characterization of quasiperiodicity and chaosin a mechanical system with delay Int J Bifurc Chaos 9 4965

Johnson M amp Moon F C 2001 Nonlinear techniques to characterize pre-chatter and chattervibrations in the machining of metals Int J Bifurc Chaos (In the press)

Kalmparaar-Nagy T Pratt J R Davies M A amp Kennedy M D 1999 Experimental and ana-lytical investigation of the subcritical instability in turning In Proc 1999 ASME DesignEngineering Technical Conf DETC99VIB-8060 Las Vegas NV USA

Kalmparaar-Nagy T Stparaepparaan G amp Moon F C 2001a Subcritical Hopf bifurcation in the delayequation model for machine tool vibrations Nonlinear Dynamics (In the press)

Kalmparaar-Nagy T Moon F C amp Stparaepparaan G 2001b Regenerative machine tool vibrationsDynamics Continuous Discrete Impulsive Systems (In the press)

Kudinov V A Klyuchnikov A V amp Shustikov A D 1978 Experimental investigation of thenon-linear dynamic cutting process Stanki i instrumenty 11 1113 (In Russian)

Minis I amp Berger B S 1998 Modelling analysis and characterization of machining dynamicsIn Dynamics and Chaos in Manufacturing Processes (ed F C Moon) pp 125163 Wiley



Moon F C 1994 Chaotic dynamics and fractals in material removal processes In Nonlinearityand chaos in engineering dynamics (ed J Thompson amp S Bishop) pp 2537 Wiley

Moon F C amp Abarbanel H 1995 Evidence for chaotic dynamics in metal cutting and clas-simacrcation of chatter in lathe operations In Summary Report of a Workshop on NonlinearDynamics and Material Processes and Manufacturing (ed F C Moon) pp 1112 2829Institute for Mechanics and Materials

Moon F C amp Callaway D 1997 Chaotic dynamics in scribing polycarbonate plates with adiamond cutter IUTAM Symp on New Application of Nonlinear and Chaotic DynamicsIthaca

Moon F amp Johnson M 1998 Nonlinear dynamics and chaos in manufacturing processes InDynamics and chaos in manufacturing processes (ed F C Moon) pp 332 Wiley

Nayfeh A Chin C amp Pratt J 1998 Applications of perturbation methods to tool chatterdynamics In Dynamics and chaos in manufacturing processes (ed F C Moon) pp 193213 Wiley

Oxley P L B amp Hastings W F 1977 Predicting the strain rate in the zone of intense shearin which the chip is formed in machining from the dynamic deg ow stress properties of the workmaterial and the cutting conditions Proc R Soc Lond A 356 395410

Poddar B Moon F C amp Mukherjee S 1988 Chaotic motion of an elastic plastic beam ASMEJ Appl Mech 55 185189

Pratap R Mukherjee S amp Moon F C 1994 Dynamic behavior of a bilinear hysteretic elasto-plastic oscillator Part II Oscillations under periodic impulse forcing J Sound Vib 172339358

Pratt J amp Nayfeh A H 1996 Experimental stability of a time-delay system In Proc 37thAIAAASMEASCEAHSACS Structures Structural Dynamics and Materials Conf SaltLake City USA

Saravanja-Fabris N amp DrsquoSouza A 1974 Nonlinear stability analysis of chatter in metal cuttingJ Engng Industry 96 670675

Stparaepparaan G 1989 Retarded dynamical systems stability and characteristic functions PitmanResearch Notes in Mathematics vol 210 London Longman Scientimacrc and Technical

Stparaepparaan G 1998 Delay-direg erential equation models for machine tool chatter In Dynamics andchaos in manufacturing processes (ed F C Moon) pp 165191 Wiley

Stparaepparaan G amp Kalmparaar-Nagy T 1997 Nonlinear regenerative machine tool vibrations In Proc1997 ASME Design Engineering Technical Conf on Vibration and Noise Sacramento CApaper no DETC 97VIB-4021 pp 111

Szakovits R J amp DrsquoSouza A F 1976 Metal cutting dynamics with reference to primary chatterJ Engng Industry 98 258264

Taylor F W 1907 On the art of cutting metals Trans ASME 28 31350

Tlusty J 1978 Analysis of the state of research in cutting dynamics Ann CIRP 27 583589

Tlusty J amp Ismail F 1981 Basic non-linearity in machining chatter CIRP Ann ManufacturingTechnol 30 299304

Tobias S 1965 Machine tool vibration London Blackie

Wiercigroch M amp Cheng A H-D 1997 Chaotic and stochastic dynamics of orthogonal metalcutting Chaos Solitons Fractals 8 715726



(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 enotects in cutting dynamics

(vii) induced electromagnetic voltages at the materialtool interface

(viii) fracture processes in cutting of brittle materials

(ix) fracture enotects 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 isin encompassing the above dynamic problems

Our own contributions here are modest After reviewing the current status wediscuss 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 sumacr cient 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 `goodrsquo cutting

2 Nonlinear ereg ects in material cutting

Nonlinearity has always been recognized as an essential element in machining Forexample Doi amp 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 amp Ismail 1981) and others haveconsidered nonlinearity in their studies Before 19751980 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 newconcepts of modelling measuring and controlling nonlinear dynamics in materialprocessing have appeared

The principal nonlinear enotects on cutting dynamics include

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

(ii) tool-structure nonlinearities

(iii) friction at the toolchip interface

(iv) loss of toolworkpiece contact

(v) inreguence of machine drive unit on the cutting regow velocity

There are at least four types of self excited machining dynamics


















20 40 600

2

4

6

8

10

f0

F (

N)

f (microm)

Fx

Fx ( f0)

D Fx k1 D f

D f

Kwf a

f0

(a)(b)



F = Kwfn (31)








m

x (t)

k c

f

Fx

s D l cx


5 Single-DOF models



1

mcentF (51)


f = f0 + centf (52)



k1







5000 10 0000

100 000

200 000

W (RPM)

k 1 (

N m

-1)


0 01 02 03 04

0

10

20

30

40

50

chip width (mm)

forwards sweep

backwards sweep

RP

M v

ibra

tion

am

plit

ude

(microm

)






(a) (b)



















400

200

0

- 200

- 400

xrsquo (

t)(a) (b)

(c) (d )

400

200

0

- 200

- 400

xrsquo (

t)

400

200

0

- 200

- 400

xrsquo (

t)

600

300

0

- 300

- 600xrsquo

(t)

- 02 0 02x (t)

- 03 0 03x (t)









D X0

D f

(a) (b)

D X2 D Xcrit D X

RHS

D F

f contact loss

F

D X1

f

FD F

D f


D f

D X

RHS

D F

contact loss

D f

D X

RHS

D F

contact loss

D f

D X

RHS

D F

D f

D X

RHS

D F





xrsquo (t)

- 015

- 03

x (t)035

03


03

- 01

- 03 06- 008

- 005006

03RHS

D x

RHS

D x













1

mcentF (92)




n _x) = iexcl1




n +q1 frac12

m_x

+ 2n +

k1

mx iexcl k1


m_x frac12 = 0 (94)



q1 frac12

mpara

+ 2n +

k1

mpara iexcl iexcl

k1


q1 frac12







q1Aacute sin Aacute

m not 3 =

1 iexcl cos Aacute

m(98)

shy 1 =2 plusmn n

q0Aacute shy 2 = 2


mq0 shy 3 =

sin Aacute

mq0Aacute (99)


2 + 2 reg iexcl macr 2 = 0 (910)

where






shy 3 iexcl not 3





500 10000

045

090

W (RPM)

k 1 (N

mm

-1)







2 ordm (Aacute)

Aacute (914)

k1(Aacute) =1
















chip

uz uy

ux

V

N


time

stre

ss g

auge

out

put




cutting velocity V

norm

al f

orce

(N

)









12 Summary












References
































































20 40 600

2

4

6

8

10

f0

F (

N)

f (microm)

Fx

Fx ( f0)

D Fx k1 D f

D f

Kwf a

f0

(a)(b)



F = Kwfn (31)








m

x (t)

k c

f

Fx

s D l cx


5 Single-DOF models



1

mcentF (51)


f = f0 + centf (52)



k1







5000 10 0000

100 000

200 000

W (RPM)

k 1 (

N m

-1)


0 01 02 03 04

0

10

20

30

40

50

chip width (mm)

forwards sweep

backwards sweep

RP

M v

ibra

tion

am

plit

ude

(microm

)






(a) (b)



















400

200

0

- 200

- 400

xrsquo (

t)(a) (b)

(c) (d )

400

200

0

- 200

- 400

xrsquo (

t)

400

200

0

- 200

- 400

xrsquo (

t)

600

300

0

- 300

- 600xrsquo

(t)

- 02 0 02x (t)

- 03 0 03x (t)









D X0

D f

(a) (b)

D X2 D Xcrit D X

RHS

D F

f contact loss

F

D X1

f

FD F

D f


D f

D X

RHS

D F

contact loss

D f

D X

RHS

D F

contact loss

D f

D X

RHS

D F

D f

D X

RHS

D F





xrsquo (t)

- 015

- 03

x (t)035

03


03

- 01

- 03 06- 008

- 005006

03RHS

D x

RHS

D x













1

mcentF (92)




n _x) = iexcl1




n +q1 frac12

m_x

+ 2n +

k1

mx iexcl k1


m_x frac12 = 0 (94)



q1 frac12

mpara

+ 2n +

k1

mpara iexcl iexcl

k1


q1 frac12







q1Aacute sin Aacute

m not 3 =

1 iexcl cos Aacute

m(98)

shy 1 =2 plusmn n

q0Aacute shy 2 = 2


mq0 shy 3 =

sin Aacute

mq0Aacute (99)


2 + 2 reg iexcl macr 2 = 0 (910)

where






shy 3 iexcl not 3





500 10000

045

090

W (RPM)

k 1 (N

mm

-1)







2 ordm (Aacute)

Aacute (914)

k1(Aacute) =1
















chip

uz uy

ux

V

N


time

stre

ss g

auge

out

put




cutting velocity V

norm

al f

orce

(N

)









12 Summary












References

















































m

x (t)

k c

f

Fx

s D l cx


5 Single-DOF models



1

mcentF (51)


f = f0 + centf (52)



k1







5000 10 0000

100 000

200 000

W (RPM)

k 1 (

N m

-1)


0 01 02 03 04

0

10

20

30

40

50

chip width (mm)

forwards sweep

backwards sweep

RP

M v

ibra

tion

am

plit

ude

(microm

)






(a) (b)



















400

200

0

- 200

- 400

xrsquo (

t)(a) (b)

(c) (d )

400

200

0

- 200

- 400

xrsquo (

t)

400

200

0

- 200

- 400

xrsquo (

t)

600

300

0

- 300

- 600xrsquo

(t)

- 02 0 02x (t)

- 03 0 03x (t)









D X0

D f

(a) (b)

D X2 D Xcrit D X

RHS

D F

f contact loss

F

D X1

f

FD F

D f


D f

D X

RHS

D F

contact loss

D f

D X

RHS

D F

contact loss

D f

D X

RHS

D F

D f

D X

RHS

D F





xrsquo (t)

- 015

- 03

x (t)035

03


03

- 01

- 03 06- 008

- 005006

03RHS

D x

RHS

D x













1

mcentF (92)




n _x) = iexcl1




n +q1 frac12

m_x

+ 2n +

k1

mx iexcl k1


m_x frac12 = 0 (94)



q1 frac12

mpara

+ 2n +

k1

mpara iexcl iexcl

k1


q1 frac12







q1Aacute sin Aacute

m not 3 =

1 iexcl cos Aacute

m(98)

shy 1 =2 plusmn n

q0Aacute shy 2 = 2


mq0 shy 3 =

sin Aacute

mq0Aacute (99)


2 + 2 reg iexcl macr 2 = 0 (910)

where






shy 3 iexcl not 3





500 10000

045

090

W (RPM)

k 1 (N

mm

-1)







2 ordm (Aacute)

Aacute (914)

k1(Aacute) =1
















chip

uz uy

ux

V

N


time

stre

ss g

auge

out

put




cutting velocity V

norm

al f

orce

(N

)









12 Summary












References

















































5000 10 0000

100 000

200 000

W (RPM)

k 1 (

N m

-1)


0 01 02 03 04

0

10

20

30

40

50

chip width (mm)

forwards sweep

backwards sweep

RP

M v

ibra

tion

am

plit

ude

(microm

)






(a) (b)



















400

200

0

- 200

- 400

xrsquo (

t)(a) (b)

(c) (d )

400

200

0

- 200

- 400

xrsquo (

t)

400

200

0

- 200

- 400

xrsquo (

t)

600

300

0

- 300

- 600xrsquo

(t)

- 02 0 02x (t)

- 03 0 03x (t)









D X0

D f

(a) (b)

D X2 D Xcrit D X

RHS

D F

f contact loss

F

D X1

f

FD F

D f


D f

D X

RHS

D F

contact loss

D f

D X

RHS

D F

contact loss

D f

D X

RHS

D F

D f

D X

RHS

D F





xrsquo (t)

- 015

- 03

x (t)035

03


03

- 01

- 03 06- 008

- 005006

03RHS

D x

RHS

D x













1

mcentF (92)




n _x) = iexcl1




n +q1 frac12

m_x

+ 2n +

k1

mx iexcl k1


m_x frac12 = 0 (94)



q1 frac12

mpara

+ 2n +

k1

mpara iexcl iexcl

k1


q1 frac12







q1Aacute sin Aacute

m not 3 =

1 iexcl cos Aacute

m(98)

shy 1 =2 plusmn n

q0Aacute shy 2 = 2


mq0 shy 3 =

sin Aacute

mq0Aacute (99)


2 + 2 reg iexcl macr 2 = 0 (910)

where






shy 3 iexcl not 3





500 10000

045

090

W (RPM)

k 1 (N

mm

-1)







2 ordm (Aacute)

Aacute (914)

k1(Aacute) =1
















chip

uz uy

ux

V

N


time

stre

ss g

auge

out

put




cutting velocity V

norm

al f

orce

(N

)









12 Summary












References
































































400

200

0

- 200

- 400

xrsquo (

t)(a) (b)

(c) (d )

400

200

0

- 200

- 400

xrsquo (

t)

400

200

0

- 200

- 400

xrsquo (

t)

600

300

0

- 300

- 600xrsquo

(t)

- 02 0 02x (t)

- 03 0 03x (t)









D X0

D f

(a) (b)

D X2 D Xcrit D X

RHS

D F

f contact loss

F

D X1

f

FD F

D f


D f

D X

RHS

D F

contact loss

D f

D X

RHS

D F

contact loss

D f

D X

RHS

D F

D f

D X

RHS

D F





xrsquo (t)

- 015

- 03

x (t)035

03


03

- 01

- 03 06- 008

- 005006

03RHS

D x

RHS

D x













1

mcentF (92)




n _x) = iexcl1




n +q1 frac12

m_x

+ 2n +

k1

mx iexcl k1


m_x frac12 = 0 (94)



q1 frac12

mpara

+ 2n +

k1

mpara iexcl iexcl

k1


q1 frac12







q1Aacute sin Aacute

m not 3 =

1 iexcl cos Aacute

m(98)

shy 1 =2 plusmn n

q0Aacute shy 2 = 2


mq0 shy 3 =

sin Aacute

mq0Aacute (99)


2 + 2 reg iexcl macr 2 = 0 (910)

where






shy 3 iexcl not 3





500 10000

045

090

W (RPM)

k 1 (N

mm

-1)







2 ordm (Aacute)

Aacute (914)

k1(Aacute) =1
















chip

uz uy

ux

V

N


time

stre

ss g

auge

out

put




cutting velocity V

norm

al f

orce

(N

)









12 Summary












References

















































D X0

D f

(a) (b)

D X2 D Xcrit D X

RHS

D F

f contact loss

F

D X1

f

FD F

D f


D f

D X

RHS

D F

contact loss

D f

D X

RHS

D F

contact loss

D f

D X

RHS

D F

D f

D X

RHS

D F





xrsquo (t)

- 015

- 03

x (t)035

03


03

- 01

- 03 06- 008

- 005006

03RHS

D x

RHS

D x













1

mcentF (92)




n _x) = iexcl1




n +q1 frac12

m_x

+ 2n +

k1

mx iexcl k1


m_x frac12 = 0 (94)



q1 frac12

mpara

+ 2n +

k1

mpara iexcl iexcl

k1


q1 frac12







q1Aacute sin Aacute

m not 3 =

1 iexcl cos Aacute

m(98)

shy 1 =2 plusmn n

q0Aacute shy 2 = 2


mq0 shy 3 =

sin Aacute

mq0Aacute (99)


2 + 2 reg iexcl macr 2 = 0 (910)

where






shy 3 iexcl not 3





500 10000

045

090

W (RPM)

k 1 (N

mm

-1)







2 ordm (Aacute)

Aacute (914)

k1(Aacute) =1
















chip

uz uy

ux

V

N


time

stre

ss g

auge

out

put




cutting velocity V

norm

al f

orce

(N

)









12 Summary












References

















































xrsquo (t)

- 015

- 03

x (t)035

03


03

- 01

- 03 06- 008

- 005006

03RHS

D x

RHS

D x













1

mcentF (92)




n _x) = iexcl1




n +q1 frac12

m_x

+ 2n +

k1

mx iexcl k1


m_x frac12 = 0 (94)



q1 frac12

mpara

+ 2n +

k1

mpara iexcl iexcl

k1


q1 frac12







q1Aacute sin Aacute

m not 3 =

1 iexcl cos Aacute

m(98)

shy 1 =2 plusmn n

q0Aacute shy 2 = 2


mq0 shy 3 =

sin Aacute

mq0Aacute (99)


2 + 2 reg iexcl macr 2 = 0 (910)

where






shy 3 iexcl not 3





500 10000

045

090

W (RPM)

k 1 (N

mm

-1)







2 ordm (Aacute)

Aacute (914)

k1(Aacute) =1
















chip

uz uy

ux

V

N


time

stre

ss g

auge

out

put




cutting velocity V

norm

al f

orce

(N

)









12 Summary












References






















































1

mcentF (92)




n _x) = iexcl1




n +q1 frac12

m_x

+ 2n +

k1

mx iexcl k1


m_x frac12 = 0 (94)



q1 frac12

mpara

+ 2n +

k1

mpara iexcl iexcl

k1


q1 frac12







q1Aacute sin Aacute

m not 3 =

1 iexcl cos Aacute

m(98)

shy 1 =2 plusmn n

q0Aacute shy 2 = 2


mq0 shy 3 =

sin Aacute

mq0Aacute (99)


2 + 2 reg iexcl macr 2 = 0 (910)

where






shy 3 iexcl not 3





500 10000

045

090

W (RPM)

k 1 (N

mm

-1)







2 ordm (Aacute)

Aacute (914)

k1(Aacute) =1
















chip

uz uy

ux

V

N


time

stre

ss g

auge

out

put




cutting velocity V

norm

al f

orce

(N

)









12 Summary












References

















































500 10000

045

090

W (RPM)

k 1 (N

mm

-1)







2 ordm (Aacute)

Aacute (914)

k1(Aacute) =1
















chip

uz uy

ux

V

N


time

stre

ss g

auge

out

put




cutting velocity V

norm

al f

orce

(N

)









12 Summary












References

























































chip

uz uy

ux

V

N


time

stre

ss g

auge

out

put




cutting velocity V

norm

al f

orce

(N

)









12 Summary












References




















































12 Summary












References

















































References
















































nonlinear models for complex dynamics in cutting...

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