Kalmar-nagy Nonlinear Models For Complex Dynamics In Cutting Materials

10.1098/rsta.2000.0751

Nonlinear models for complex dynamics in cutting materials ¶ s K a lma ¶ r - N a g y2 By F ra n c is C. M o o n1 a n d T a m a 1 Sibley School of Aerospace and Mechanical Engineering, 2 Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, NY 14853, USA

This paper reviews the prediction of complex, unsteady and chaotic dynamics associated with material-cutting processes through nonlinear dynamical models. The status of bifurcation phenomena such as subcritical Hopf instabilities is assessed. A new model using hysteresis in the cutting force is presented, which is shown to exhibit complex quasi-periodic solutions. In addition, further evidence for chaotic dynamics in non-regenerative cutting of polycarbonate plastic is reviewed. The authors draw the conclusion that single-degree-of-freedom models are not likely to predict lowlevel cutting chaos and that more complex models, such as multi-degree-of-freedom systems 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, going back 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 interest in modelling cutting dynamics for several reasons. First, there are now higher cutting speeds, new materials and hard-turning problems, as well as an interest in higher-precision machining. Second, advances in nonlinear dynamics in the last two decades has lent promise to the prospects of analysing more complex models attendant to material processing. Third, there is a renewed intellectual interest in both the physics and mathematics associated with material removal. One such problem is the unsteady nature of both chatter and pre-chatter, or normal machining and cutting. 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, chaotic and even random motions in cutting. In this paper we review the development of cutting-dynamics modelling in the context of new low-dimensional nonlinear models and new experimental work in material cutting. Although the linear delay model has been fairly successful in capturing the onset of the large amplitude periodic chatter, the limit-cycle behaviour itself has not been well understood. There are also other nonlinear phenomena that require more complex 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

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® c 2001 The Royal Society

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F. C. Moon and T. Kalm¶a r-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 e¬ects in cutting dynamics, (vii) induced electromagnetic voltages at the material{tool interface, (viii) fracture processes in cutting of brittle materials, (ix) fracture e¬ects in chip breakage. The length of this list serves to suggest that a single-degree-of-freedom (singleDOF) regenerative model cannot begin to predict all the important phenomena in cutting 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 viscoelasticity. Numerical results show that hysteretic cutting-force laws lead to more complex dynamics, but that one-DOF models are not su¯ cient to explain the broader range of cutting-dynamics phenomena. We also present some new experimental results in non-regenerative cutting of polycarbonate plastic that associates chaos-like dynamics with normal or `good’ cutting.

2. Nonlinear e® ects in material cutting Nonlinearity has always been recognized as an essential element in machining. For example, Doi & Kato (1956) performed some beautiful experiments on establishing chatter as a time-delay problem and also presented one of the earliest nonlinear models. Also, Tobias (1965) and Tlusty (see Tlusty & Ismail 1981) and others have considered nonlinearity in their studies. Before 1975{1980 , nonlinear dynamics analysis mainly consisted of perturbation analysis and numerical simulation. Random-like motions were not considered, even though time records of cutting dynamics clearly showed unsteady oscillations (see, for example, Tobias 1965). Since the 1980s new concepts of modelling, measuring and controlling nonlinear dynamics in material processing have appeared. The principal nonlinear e¬ects on cutting dynamics include (i) material constitutive relations (stress versus strain, strain rate and temperature), (ii) tool-structure nonlinearities, (iii) friction at the tool{chip interface, (iv) loss of tool{workpiece contact, (v) in®uence 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)

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(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{structure ®utter, rail{wheel instabilities, stick{slip friction vibrations, etc. However, the regenerative model seems to be unique to material processing systems. It appears in turning, drilling, milling, grinding and rolling operations. The ­ nite time delay introduces an in­ nite-dimensional phase space, even for single-DOF systems. Because of this unique feature, regenerative chatter problems have attracted the greatest interest among applied mathematicians (see, for example, St´ ep´an 1989; Nayfeh et al . 1998). Fascination with time-delay di¬erential equations has often overshadowed the physics of material processing. For example, in cutting physics the essential processes involve thermo-viscoplasticity and fracture mechanics. Yet most dynamic models of chatter do not include temperature as a state variable. In some brittle materials, electric and magnetic ­ elds are generated in the cutting process, yet these variables are also missing from the models. In most cutting models, the physics is hidden in a cutting-energy density factor. In the last several years several dynamic models have examined basic material nonlinearities, including thermal softening (see Davies et al . 1996; Davies 1998). Other groups have used nonlinear dynamics methodology to study cutting chatter (Moon 1994; Bukkapatnam et al. 1995a; Wiercigroch & Cheng 1997; St´e p´an & Kalm´ar-Nagy 1997; Nayfeh et al . 1998; Minis & Berger 1998; Moon & Johnson 1998). Studies of nonlinear phenomena in machine-tool operations involve three di¬erent approaches. (1) Measurement of nonlinear force{displacement behaviour of cutting or forming tools. (2) Model-based studies of bifurcations using parameter variation. (3) Time-series analysis of dynamic data for system identi­ cation.

3. Nonlinear cutting forces The fundamental origins of nonlinear dynamics in material processing usually involve nonlinear relations between stress and strain, or stress and temperature or chemical kinetics and solid-state reactions in the material. Other sources involve nonlinear geometry such as contact forces or tool{workpiece separation. There is a long history of force measurements in the literature over the past century. Many of these data are based 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 the real dynamic nature of the process. In a dynamic process, what happens when the cutting depth instantaneously decreases? Does one follow the average-force{depth curve or is there an unloading path similar to elasto-plastic unloading? Average force measurements often ­ lter out the dynamic nature of the process. Phil. Trans. R. Soc. Lond. A (2001)

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F. C. Moon and T. Kalm¶a r-Nagy 10

(a) D Fx

(b)

Fx

8 F (N)

Fx ( f0)

k1 D f D f

6 4

Kwf a

2

0

20

f (µm)

40

60

0

f0

f

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

One popular steady cutting force (F ) versus chip thickness (f ) relationship is that proposed by Taylor (1907), F = Kwf n ; (3.1) where a popular value for n is 34 (w is the chip width and K is a material-based constant). For aluminium, the value for n was found to be 0:41 (Kalm´ar-Nagy et al . 1999, ­ g. 1). Equation (3.1) is a softening force law. It is also single valued. In recent years, 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 cutting force versus material ®ow velocity in steel. They also measured the cutting forces for di¬erent tool rake angles. These relations were used by Grabec (1986, 1988) to propose a non-regenerative two-DOF model for cutting that predicted chaotic dynamics. However, the force measurements themselves are quasi-steady and were taken to be single-valued functions of chip thickness and material ®ow velocity. Below we 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 dynamic orbits, such as a jump from equilibrium to a limit cycle (Hopf bifurcation) or a doubling of the period of a limit cycle. The critical values of the control parameter at which the dynamics topology changes enable the researcher to connect the model behaviour with experimental observation in the actual process. These studies also allow one to design controllers to suppress unwanted dynamics or to change a subcritical Hopf bifurcation into a supercritical one. The phase-space methodology also lends itself to new diagnostic tools, such as Poincar´e maps, which can be used to look 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 are usually overly simplistic and not based on fundamental physics. The use of bifurcation tools is most e¬ective when the phase-space dimension is small, say, less than or equal to four. Phil. Trans. R. Soc. Lond. A (2001)

Nonlinear models for complex dynamics in cutting materials k

699

c sD l

cx

m f Fx

x (t)

Figure 2. One-DOF mechanical model and FBD.

5. Single-DOF models These models have been the principal source of nonlinear analysis, beginning with the 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 motion takes the form 1 x  + 2± !n x_ + !n2 x = ¡ ¢F; (5.1) m where !n is the natural angular frequency of the undamped free oscillating system and ± is the relative damping factor. ¢F = Fx ¡ Fx (f0 ) is the cutting-force variation . Sometimes nonlinear sti¬ness terms are added to the tool sti¬ness (Hanna & Tobias 1974). 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 chip thickness f0 , i.e. f = f0 + ¢f;

(5.2)

where ¢f = x(t¡ ½ )¡ x(t). Here, ½ is the delay time related to the angular rate « , i.e. ½ = 2º =« (that is, ½ is the period of revolution). After linearizing the cutting-force variation (¢F ) at some nominal chip thickness, the linearized equation of motion of classical regenerative chatter becomes (see, for example, St´ep´an 1989) x  + 2± !n x_ + !n 2 x =

k1 (x ¡ m

x½ );

(5.3)

where x½ denotes the delayed value of x(t). The linear stability theory predicts unbounded motion above the lobes in the parameter plane of cutting-force coe¯ cient 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 a value of k1 = 2m!n 2 ± (1 + ± ) ¹ 6:8 £ 104 N m¡ 1 . Below this value the theory predicts no sustained motion, which is counter to experimental evidence. The linear model is insu¯ cient in at least three phenomena. First, it does not predict the amplitude of the limit cycle for post-chatter. Second, the chatter is often subcritical, as shown in ­ gure 4 (Kalm´ar-Nagy et al . 1999). Finally, there is the matter of the pre-chatter vibrations, which in experiments appear to be non-steady of either a chaotic or random nature (see, for example, Johnson & Moon 2001). Phil. Trans. R. Soc. Lond. A (2001)

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k1 (N m- 1)

200 000

100 000

0

5000 (RPM)

W

10 000

Figure 3. Classical stability chart. 50 forwards sweep backwards sweep RPM vibration amplitude (µm)

40

30

20

10

0 0

0.1

0.2 chip width (mm)

0.3

0.4

Figure 4. Amplitude of tool vibration versus chip width. amplitude of oscillation

(a)

bifurcation parameter

amplitude of oscillation

(b)

bifurcation parameter

Figure 5. Supercritical and subcritical Hopf bifurcation. Phil. Trans. R. Soc. Lond. A (2001)

Nonlinear models for complex dynamics in cutting materials

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6. Subcritical chatter bifurcations If the chatter amplitude grows smoothly as the parameter increases, the instability is called supercritical (­ gure 5a). Here, continuous lines correspond to stability, while dashed lines indicate instability. However, an increase in the parameter often results in a ­ nite jump in chatter amplitude. If the parameter is then decreased, the chatter persists below the critical level of the machining parameter predicted by the linear theory. At the second critical value, the vibration amplitude drops close to zero or low-level vibrations. This condition is called a subcritical Hopf bifurcation (­ gure 5b). The value of this second critical parameter is more useful in practice, since it de­ nes a robust parameter operating range, whereas the range between the lower critical parameter and the linear critical parameter is sensitive to initial conditions and impact, knocking the tool onto the upper branch of the chatter bifurcation curve. Subcritical or hysteretic chatter amplitude behaviour was documented by Hooke & Tobias (1963) and by Kalm´ar-Nagy et al. (1999). Modern analysis of subcritical behaviour has been presented by Nayfeh et al . (1998), St´e p´an & Kalm´ar-Nagy (1997) and Kalm´ar-Nagy et al. (2001a; b). Another work is a PhD dissertation of Fofana (1993). These results depend critically on the assumption of the cutting-force{tooldisplacement nonlinearity. The analytical tools used in these studies were based on perturbation methods and on the use of centre manifold and normal form theory. Typical of the single-DOF models on which bifurcation studies have been conducted is the model of Hanna & Tobias (1974). This model was used by Nayfeh et al . (1998) using modern perturbation methods, x  + 2± x_ + p2 (x + ­ 2 x2 + ­ 3 x3 ) = p2 w(¢f ¡ ¬

2 ¢f

2

+¬

3 ¢f

3

):

(6.1)

This model incorporates both structural (­ 2 , ­ 3 ) and material nonlinearities (¬ 2 , This group was able to show that this equation exhibited a global subcritical Hopf bifurcation (initially supercritical and then turning subcritical at higher vibration amplitudes). A similar model is the work of St´e p´an & Kalm´ar-Nagy (1997), which incorporates only quadratic and cubic terms in the material nonlinearities. This equation has the form x  + 2± x_ + x = p¢f + q(¢f 2 + ¢f 3 ): (6.2) ¬

3 ).

Using centre manifold theory, this equation was shown to exhibit a subcritical Hopf bifurcation (Kalm´ar-Nagy et al. 2001a; b).

7. Quasi-periodic bifurcations Nayfeh et al . (1998) also showed in numerical simulation that the equation exhibited quasi-periodic motions and bifurcations. Similar results were found by Johnson and Moon (see Moon & Johnson 1998; Johnson & Moon 1999, 2001). Johnson used a simpler delay model with only a cubic structural nonlinearity of the form x  + ® 1 x_ + ® 2 (x + x3 ) = ¡ ® 3 x(t ¡

1):

(7.1)

Numerical simulation of this equation ­ rst revealed a periodic limit cycle in postchatter. But as the parameter ® 3 was increased, the Poincar´e map of the motion revealed secondary bifurcations of the periodic motion into a torus and period-2N Phil. Trans. R. Soc. Lond. A (2001)

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F. C. Moon and T. Kalm¶a r-Nagy 400

400

(a)

0

0

- 200

- 200

- 400

- 400

400

600

(c)

200 0

- 200 - 400 - 0.2

(d )

300 x’ (t)

x’ (t)

(b)

200 x’ (t)

x’ (t)

200

0

- 300 0 x (t)

0.2

- 600

- 0.3

Figure 6. Bifurcation sequence for Johnson’s model (®

0 x (t) 3

0.3

= 300, 1000, 2000, 4000).

tori. There is evidence that the limit of these bifurcations is a chaotic attractor. An example 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. Remarkably, the experimental results agreed exactly with the numerical simulation of the model (Johnson & Moon 1999). Experiments were also conducted by Pratt & Nayfeh (1996) using an analogue computer. Even though these models showed new bifurcation phenomena in nonlinear delay equations, experimental results on chatter itself have not exhibited such bifurcation behaviour as of this date. These results are important, however, because they show that dynamics in a fourdimensional 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 that some cutting model, with one or two degrees of freedom, will eventually predict these complex motions.

8. Hysteretic cutting-force model The above models all involve smooth continuous single-valued force functions of the chip thickness. However, there is no reason to expect that the function F (f ) be smooth 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 or elasto-plastic behaviour of the material. This phenomenon has been studied in other ­ elds, such as soil mechanics, ferroelectricity and superconducting levitation. The model presented here was inspired by past research at Cornell on chaos in elastoplastic structures (Poddar et al . 1988; Pratap et al. 1994). Phil. Trans. R. Soc. Lond. A (2001)

Nonlinear models for complex dynamics in cutting materials (b)

F

(a)

703

RHS f

D F

D f

contact loss

F D f D X1 D X0

D X2 D Xcrit

D X

f D F Figure 7. (a) Bilinear cutting-force law. (b) Hysteretic cutting-force model. RHS

RHS

contact loss

D f

contact loss

D f D X

D X

D F

D F

RHS

RHS

D f

D f D X D F

D X D F

Figure 8. Loading{unloading paths.

The idea of cutting-force hysteresis is based on the fact that the cutting force is an elasto-plastic process in many materials. In such behaviour, the stress follows a workhardening rule for positive strain rate but reverts to a linear elastic rule for decreasing strain 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 curve has been replaced with a piecewise-linear function, where the lower line is tangent to the nonlinear cutting-force relation at ¢x = 0 (­ gure 7a). The loading line and the unloading line can have di¬erent slopes (­ gure 8 shows possible loading{unloading paths). This model also includes separation of the tool and workpiece. An interesting feature of this model is the coexistence of periodic and quasi-periodic attractors below the linear stability boundary. As shown in ­ gure 9, there exists a torus `inside’ of the stable limit cycle. This could explain the experimental observation of the sudden Phil. Trans. R. Soc. Lond. A (2001)

704

F. C. Moon and T. Kalm¶a r-Nagy x’ (t) 0.3

x (t) 0.35

- 0.15

- 0.3 Figure 9. Torus inside the stable limit cycle. RHS 0.3

RHS 0.3

- 0.3

0.6 - 0.1

D x - 0.08

- 0.05

0.06

D x

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

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

9. Viscoelastic models Most of the theoretical analyses of machine-tool vibrations employ force laws that are based on the assumption that cutting is steady-state. However, cutting is a dynamic process and experimental results show clear di¬erences between steady-state and dynamic cutting. As shown by Albrecht (1965) and Szakovits & D’Souza (1976), the cutting-force{chip-thickness relation exhibits hysteresis. This hysteresis depends on the cutting speed, the frequency of chip segmentation, the functional angles of the tool’s edges, etc. (Kudinov et al . 1978). Saravanja-Fabris & D’Souza (1974) employed the describing function method to obtain linear stability conditions. In this paper, we derive a delay-di¬erential equation model that includes hysteretic e¬ects via a constitutive relation. To describe elasto-plastic materials, the Kelvin{Voigt model is often used. This model describes solid-like behaviour with delayed elasticity (instantaneous elastic Phil. Trans. R. Soc. Lond. A (2001)

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deformation and delayed elastic deformation) via a constitutive relation that is linear in stress, rate of stress, strain and strain-rate. We assume that a similar relation between cutting force and chip thickness holds, where the coe¯ cients of the rates depend on the cutting speed (through the time delay, using ¢f = x ¡ x½ ), ¢F + q0 ½ ¢F_ = k1 ¢f + q1 ½ ¢f_:

(9.1)

The usual one-DOF model is 1 ¢F: m Multiplying the time derivative of (9.2) by q0 ½ and adding it to (9.2) gives x  + 2± !n x_ + !n2 x = ¡

(9.2)

1 (¢F + q0 ½ ¢F_ ); (9.3) m which can be rewritten using (9.1) and the relation for chip-thickness variation ¢f = x ¡ x½ as ... x  + 2± !n x_ + !n 2 x + q0 ½ ( x + 2± !n x  + !n 2 x) _ =¡

q1 ½ ... q0 ½ x + (1 + 2± q0 ½ !n )x + 2± !n + q0 ½ !n 2 + x_ m k k1 + !n 2 + 1 x ¡ x½ ¡ m m

q1 ½ x_ ½ = 0: m

(9.4)

The characteristic equation of (9.4) is D(¶ ) = q0 ½ ¶

3

+ (1 + 2± q0 ½ !n )¶

2

+

2± !n + q0 ½ !n2 + !n 2 +

+

k1 ¶ ¡ m ¡

q1 ½ m

k1 ¡ ¶ e m

¶ ½

¡

q1 ½ ¶ e¡ ¶ ½ : m

(9.5)

The stability boundaries can be found by solving D(i!) = 0, Re D(i!) = ¡ ! 2 + ¬

1!

+¬

= 0;

(9.6)

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

(9.7)

2

+¬

3 k1

2

De­ ning Á = !½ , the coe¯ cients ¬ i (Á), ­ i (Á) can be expressed as ¬ ­

1

= ¡ 2q0 ± Á!n ;

1

=

2± !n ; q0 Á

¬

2

­

2

q1 Á sin Á ; m q1 (1 ¡ cos Á) = !n 2 + ; mq0 = !n 2 ¡

¬

3

­

3

1¡

cos Á m sin Á = : mq0 Á =

(9.8) (9.9)

One can eliminate k1 from (9.6, 9.7) to get ! 2 + 2® ! ¡ ¯

2

= 0;

(9.10)

where ± !n (1 ¡ cos Á + q0 Á sin Á) ; 2(¬ 3 ¡ ­ 3 ) q0 Á(sin Á ¡ q0 Á(1 ¡ cos Á)) ¬ ­ ¡ ¬ 3­ 2 2q1 Á(1 ¡ cos Á) = 2 3 = !n 2 ¡ : ­3¡ ¬ 3 m(sin Á ¡ q0 Á(1 ¡ cos Á))

® = ¯

2

¬

1­ 3

¡

¬

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

3­ 1

=

(9.11) (9.12)

706

F. C. Moon and T. Kalm¶a r-Nagy

k1 (N mm- 1)

0.90

0.45

0

500 (RPM) W

1000

Figure 11. Stability chart for the viscoelastic model, q 1 =0.

Equation (9.10) can then be solved, !(Á) =

®

2

+¯

2

¡

® :

(9.13)

And ­ nally ½ (thus « ) and k1 can be expressed as functions of ! and Á, Á 2º !(Á) ; ) « (Á) = !(Á) Á 1 2 k1 (Á) = (! ¡ ¬ 1 ! ¡ ¬ 2 ): ¬ 3 ½ (Á) =

(9.14) (9.15)

The stability chart can be drawn as a function of the real parameter Á. If q1 = 0, equation (9.4) is equivalent to that obtained by St´e p´an (1998), who calculated the cutting force by integrating an exponentially distributed force system on the rake face. The stability chart for this case is shown in ­ gure 11 (the same parameters were used as in ­ gure 3 and q0 = 0:01). Experiments also show that the chatter threshold is higher for lower cutting speeds than for higher speeds. Small values of q1 do not seem to in®uence this chart; however, for higher values of this variable, the minima of the lobes in the low-speed region decrease (in contrast to the experimental observations).

10. Chaotic cutting dynamics The time-series analysis method has become popular in recent years to analyse many dynamic physical phenomena from ocean waves, heartbeats, lasers and machine-tool cutting (see, for example, Abarbanel 1996). This method is based on the use of a series of digitally sampled data fxi g, from which the user constructs an orbit in a pseudo-M -dimensional phase space. One of the fundamental objectives of this method is to place a bound on the dimension of the underlying phase space from which the dynamic data were sampled. This can be done with several statistical methods, including fractal dimension, false nearest neighbours (FNN), Lyapunov exponents, 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 it can be used for a wide variety of applications, it contains no physics. It is dependent Phil. Trans. R. Soc. Lond. A (2001)

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on the data alone. Thus the results may be sensitive to the signal-to-noise ratio of the source measurement, signal ­ ltering, the time delay of the sampling, the number of data points in the sampling and whether the sensor captures the essential dynamics of the process. One of the fundamental questions regarding the physics of cutting solid materials is the nature and origin of low-level vibrations in so-called normal or good machining. This is cutting below the chatter threshold. Below this threshold, linear models predict no self-excited motion. Yet when cutting tools are instrumented, one can see random-like bursts of oscillations with a centre frequency near the tool natural frequency. Work by Johnson (1996) has carefully shown that these vibrations are signi­ cantly above any machine noise in a lathe-turning operation. These observation s have been done by several laboratories, and time-series methodology has been used to diagnose the data to determine whether the signals are random or deterministic chaos (Berger et al . 1992, 1995; Minis & Berger 1998; Bukkapatnam 1999; Bukkapatnam et al. 1995a; b; Moon 1994; Moon & Abarbanel 1995; Johnson 1996; Gradiµ sek et al . 1998). One of the new techniques for examining dynamical systems from time-series measurements is the method of FNN (see Abarbanel 1996). Given a temporal series of data fxi g, 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 phase space if one had access to M state variables. The method is used to determine the largest dimensional phase space in which the orbital trajectory, which threads through the ends of the discrete vectors de­ ned above, does not intersect. Thus, if the reconstructed phase space is of too low a dimension, some orbits will appear to cross and some of the points on the orbits will be false neighbours. In an ideal calculation, as the embedding dimension M increases, the number of such false neighbours goes to zero. One then assumes that the attractor has been unraveled. This gives an estimate of the dimension of the low-order nonlinear model that one hopes will be found to predict the time-series. Using data from low-level cutting of aluminium, for example, the FNN method predicts a ­ nite dimension for the phase space of between four and ­ ve (Moon & Johnson 1998). This low dimension suggests that these low-level vibrations may have a deterministic origin, such as in chip shear band instabilities or chip-fracture processes. 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 normal cutting operations may be naturally chaotic. This idea would suggest that a small amount of chaos may actually be good in machining, since it introduces many scales in 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 e¬ects. The time-history of the vibrations of the 16 cm cantilevered stylus holder is shown in ­ gure 13, along with a photograph of the cut tracks. The cut tracks appear to be fairly uniform, even Phil. Trans. R. Soc. Lond. A (2001)

708

F. C. Moon and T. Kalm¶a r-Nagy uz N

uy ux chip

V

stress gauge output

Figure 12. Non-regenerative cutting.

time

normal force (N)

Figure 13. Time-history for cutting of plastic and magni¯cation of cut surface.

poor-quality cut periodic motion

good-quality cut chaotic-looking motion

cutting velocity, V Figure 14. Stylus dead load versus cutting speed.

though the tool vibrations appear to be random or chaotic. When the cutting speed is increased, however, the cutting width becomes highly irregular, and the vibrations become more periodic looking. An FNN of the unsteady vibrations seems to indicate that the dynamics of ­ gure 13 could be captured in a four- or ­ ve-dimensional phase space, lending evidence that the motion may be deterministic chaos. A summary of these 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 metals and plastics may be deterministic chaos, there is no apparent experimental evidence for the usual bifurcations attendant to classic low-dimensional nonlinear mappings or ®ows. However, traditional explanations for this low-level noise do not seem to Phil. Trans. R. Soc. Lond. A (2001)

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­ t the observations. Claims that the noise is the result of random grain structure in the material are not convincing, since the grain size in metals is of 10{100 m m, which would lead to frequencies in the 100 kHz range, whereas the cutting noise is usually in the 1 kHz range or lower. Besides, the grain structure theory would not apply to plastics as in the above discussion of cutting polycarbonate. Another possible explanation is the shear banding instabilities in metals (see, for example, Davies et al . 1996). But the wavelengths here are also in the 10 m m range and lead to a spectrum with higher frequency content than that observed in cutting noise. One possible candidate explanation might be tool{chip friction. A friction model was used by Grabec (1986) in his pioneering paper on chaos in machining. However, in a recent paper (Gradiµ sek et al . 1998), they now disavow the chaos theory for cutting 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 and chatter? It has been known for some time how to predict the onset of chatter using linear theory (Tlusty 1978; Tobias 1965). The special tasks for nonlinear theory in cutting 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 methodology in machining. But the use of nonlinear theory acknowledges the essential dynamic character of material removable processes that in more classical theories were ­ ltered out. However, there is a need to integrate the di¬erent methods of research, such as bifurcation theory, cutting-force characterization and time-series analysis, before nonlinear dynamics modelling can be useful in practice. It is also likely that single-DOF models will not capture all the phenomena to achieve the above goals and more degrees of freedom and added state variables such as temperature will be needed. Phil. Trans. R. Soc. Lond. A (2001)

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References Abarbanel, H. 1996 Analysis of observed chaotic data. Springer. Albrecht, P. 1965 Dynamics of the metal-cutting process. J. Engng Industry 87, 429{441. Arnold, R. N. 1946 The mechanism of tool vibration in the cutting of steel. Proc. Inst. Mech. Engrs (Lond.) 154, 261{284. Berger, B., Rokni, M. & Minis, I. 1992 The nonlinear dynamics of metal cutting. Int. J. Engng Sci. 30, 1433{1440. Berger, B., Minis, I., Chen, Y., Chavali, A. & Rokni, M. 1995 Attractor embedding in metal cutting. J. Sound Vib. 184, 936{942. Bukkapatnam, S. T. S. 1999 Compact nonlinear signal representation in machine tool operations. In Proc. 1999 ASME Design Engineering Technical Conf., DETC99/VIB-8068, Las Vegas, NV, USA. Bukkapatnam, S., Lakhtakia, A. & Kumara, S. 1995a Analysis of sensor signals shows turning on a lathe exhibits low-dimensional chaos. Phys. Rev. E 52, 2375{2387. Bukkapatnam, S., Lakhtakia, A., Kumara, S. & Satapathy, G. 1995b Characterization of nonlinearity of cutting tool vibrations and chatter. In ASME Symp. on Intelligent Manufacturing and Material Processing, vol. 69, pp. 1207{1223. Davies, M. 1998 Dynamic problems in hard-turning, milling and grinding. In Dynamics and chaos in manufacturing processes (ed. F. C. Moon), pp. 57{92. Wiley. Davies, M., Chou, Y. & Evans, C. 1996 On chip morphology, tool wear and cutting mechanics in ¯nish hard turning. Ann. CIRP 45, 77{82. Doi, S. & Kato, S. 1956 Chatter vibration of lathe tools. Trans. ASME 78, 1127{1134. 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, 384{386. Grabec, I. 1988 Chaotic dynamics of the cutting process. Int. J. Machine Tools Manufacture 28, 19{32. Gradi· s ek, J., Govekar, E. & Grabec, I. 1998 Time series analysis in metal cutting: chatter versus chatter-free cutting. Mech. Sys. Signal Proc. 12, 839{854. Hanna, N. & Tobias, S. 1974 A theory of nonlinear regenerative chatter. J. Engng Industry 96, 247{255. Hooke, C. & Tobias, S. 1963 Finite amplitude instability|a new type of chatter. In Proc. 4th Int. MTDR Conf., Manchester, UK, pp. 97{109. Oxford: Pergamon. Johnson, M. 1996 Nonlinear di® erential equations with delay as models for vibrations in the machining of metals. PhD thesis, Cornell University. Johnson, M. & Moon, F. C. 1999 Experimental characterization of quasiperiodicity and chaos in a mechanical system with delay. Int. J. Bifurc. Chaos 9, 49{65. Johnson, M. & Moon, F. C. 2001 Nonlinear techniques to characterize pre-chatter and chatter vibrations in the machining of metals. Int. J. Bifurc. Chaos. (In the press.) Kalm¶ar-Nagy, T., Pratt, J. R., Davies, M. A. & Kennedy, M. D. 1999 Experimental and analytical investigation of the subcritical instability in turning. In Proc. 1999 ASME Design Engineering Technical Conf., DETC99/VIB-8060, Las Vegas, NV, USA. Kalm¶ar-Nagy, T., St¶ ep¶an, G. & Moon, F. C. 2001a Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations. Nonlinear Dynamics. (In the press.) Kalm¶ar-Nagy, T., Moon, F. C. & St¶e p¶an, G. 2001b Regenerative machine tool vibrations. Dynamics Continuous, Discrete Impulsive Systems. (In the press.) Kudinov, V. A., Klyuchnikov, A. V. & Shustikov, A. D. 1978 Experimental investigation of the non-linear dynamic cutting process. Stanki i instrumenty, 11, 11{13. (In Russian.) Minis, I. & Berger, B. S. 1998 Modelling, analysis, and characterization of machining dynamics. In Dynamics and Chaos in Manufacturing Processes (ed. F. C. Moon), pp. 125{163. Wiley. Phil. Trans. R. Soc. Lond. A (2001)

Nonlinear models for complex dynamics in cutting materials

711

Moon, F. C. 1994 Chaotic dynamics and fractals in material removal processes. In Nonlinearity and 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 classi¯cation of chatter in lathe operations. In Summary Report of a Workshop on Nonlinear Dynamics 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 a diamond cutter. IUTAM Symp. on New Application of Nonlinear and Chaotic Dynamics, Ithaca. Moon, F. & Johnson, M. 1998 Nonlinear dynamics and chaos in manufacturing processes. In Dynamics 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 chatter dynamics. 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 shear in 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 elastoplastic 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. 37th AIAA/ASME/ASCE/AHS/ACS Structures, Structural Dynamics and Materials Conf., Salt Lake City, USA. Saravanja-Fabris, N. & D’Souza, A. 1974 Nonlinear stability analysis of chatter in metal cutting. J. Engng Industry 96, 670{675. St¶ep¶a n, G. 1989 Retarded dynamical systems: stability and characteristic functions. Pitman Research Notes in Mathematics, vol. 210. London: Longman Scienti¯c and Technical. St¶ep¶a n, G. 1998 Delay-di® erential equation models for machine tool chatter. In Dynamics and chaos in manufacturing processes (ed. F. C. Moon), pp. 165{191. Wiley. St¶ep¶a n, G. & Kalm¶ar-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. & D’Souza, 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. Manufacturing Technol. 30, 299{304. Tobias, S. 1965 Machine tool vibration. London: Blackie. Wiercigroch, M. & Cheng, A. H.-D. 1997 Chaotic and stochastic dynamics of orthogonal metal cutting. Chaos Solitons Fractals 8, 715{726.

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