3 Int. J. Machining and Machinability of Materials, Vol. 4, No. 4, 8 On the robustness of stable turning processes Zoltan Dombovari* Department of Applied Mechanics, Budapest University of Technology and Economics, Budapest, Hungary E-mail: dombo@mm.bme.hu *Corresponding author R. Eddie Wilson Department of Engineering Mathematics, University of Bristol, England, UK E-mail: re.wilson@bristol.ac.uk Gabor Stepan Department of Applied Mechanics, Budapest University of Technology and Economics, Budapest, Hungary E-mail: stepan@mm.bme.hu Abstract: Self-excited non-linear vibrations occurring in the machining processes are investigated in this paper. Our treatment applies analytical techniques to a one Degree of Freedom (DOF but strongly non-linear mechanical model of the turning process. This tool enables us to describe and analyse the highly non-linear dynamics of the appearing periodic motions. Using normal form calculations for the Delay-Differential Equation (DDE model, we prove that the low-amplitude vibrations are unstable all along the stability lobes due to the subcriticality of Hopf bifurcations. This means that self-excited vibrations of the machine tool may occur below the stability boundaries predicted by the linear theory. Consequently, stable stationary cutting may not be robust enough for external perturbations close to the linear stability limits determined during the parameter optimisation of turning processes. Robustness is characterised by the amplitude of unstable oscillations along the stability lobes for non-linear cutting force characteristics having essential inflection points against chip thickness. Keywords: orthogonal cutting; Hopf bifurcation; subcritical; bi-stable. Reference to this paper should be made as follows: Dombovari, Z., Wilson, R.E. and Stepan, G. (8 On the robustness of stable turning processes, Int. J. Machining and Machinability of Materials, Vol. 4, No. 4, pp.3 334. Biographical notes: Zoltan Dombovari received his MSc from the Budapest University of Technology and Economics Collaborating with Bristol University in 6. Currently, he is a PhD student in the Department of Applied Mechanics at the Budapest University of Technology and Economics. Copyright 8 Inderscience Enterprises Ltd.
On the robustness of stable turning processes 3 R. Eddie Wilson was educated at the University of Oxford (Lincoln College where he received a BA in Mathematics. He also received a PhD in Applied Mathematics from the same institution, and spent two years as a Postdoctoral Research Fellow in the Department of Mathematics at the University of Surrey. He joined the Department of Engineering Mathematics at the University of Bristol as a Lecturer in and he is currently in a Reader position in this department. Gabor Stepan received his PhD in 98 from the Technical University of Budapest. He has held visiting positions at the Universities in England, Denmark, the Netherlands and France and a Fulbright Fellowship at CalTech, Pasadena. Currently, he is the Head of the Department of Applied Mechanics at the Budapest University of Technology and Economics and a Member of the Hungarian National Academy of Sciences. Among several other journals, he serves on the Editorial Board of the Phil Trans of the Royal Society. Introduction Nowadays, high speed cutting comes to the front due to high quality and economic requirements. The corresponding machine tools are expensive compared to the usual machines because of their complex structure. The so-called regenerative chatter is not just harmful for these machines, but it also has a negative effect on the machined surface quality. The regenerative effect causes a kind of self-excited high-frequency vibration originated in the chip formation process (Tlusty and Spacek, 954; Tobias, 965. The stability of the regenerative vibrations has been studied extensively during the last few decades (Altintas and Budak, 995; Stepan, 989. The study of the corresponding non-linear vibrations started in the last years only (Insperger et al., 3; Mann et al., 3; Stepan and Kalmar-Nagy, 997; Szalai et al., 4; Zatarain et al., 6. These results led to the final conclusion that unstable vibrations exist in the vicinity of the otherwise stable stationary cutting in the stable parameter regions. These unstable vibrations separate two attractors in the sense of dynamical systems. Apart from the stable stationary cutting, there must exist another attractor, that is, a large amplitude stable oscillation outside the unstable vibration. This large amplitude non-linear vibration is stable in dynamical sense, but these vibrations are of little interest from technology view-point. However, the amplitudes of the unstable vibrations are important: they define that region in the state space also called domain of attraction, or basin where the stable cutting process is not sensitive for perturbations like non-homogeneous workpiece material. The larger the amplitude of the unstable oscillation is, the more robust the cutting process is. In this paper, we prove the existence of the unstable vibrations for a large set of cutting force characteristics. Also, the amplitudes of these oscillations are calculated in the parameter space of the technological parameters like cutting speed and chip width.
3 Z. Dombovari, R.E. Wilson and G. Stepan Model construction In order to investigate the fundamental effects of non-linearity during cutting, we have to apply as simple model as possible. Our one Degree of Freedom (DOF orthogonal cutting model (Figure might be far from the real cutting conditions, but it will help to follow analytically the influence of the cutting parameters on the dynamical behaviour. The desired chip thickness h equals the feed per revolution and the chip width w corresponds to the depth of cut. We disregard the geometric non-linearities of the machine tool structure since their influences are usually far smaller than those of cutting force characteristics. Thus, the equation of motion of the chosen model assumes the form: qt ( + κωn qt ( + ωn qt ( = Fx ( t ( m where ω n and κ are the natural angular frequency and the damping ratio of the essential vibration mode described by the general coordinate q that refers to the tool position. We can express these parameters with the modal mass m, stiffness k and the damping factor b. k b ωn =, κ = m mω n F x (t is the essential component of the actual cutting force. Dot refers to time derivative. Figure Orthogonal cutting with regenerative effect. Cutting force characteristics To build up the dynamical model, we need a simple expression of the cutting force characteristics. There are a number of traditional, empirical formulas for the cutting force depending on the chip width and thickness. The most popular and generally applied one is the power-law (Taylor, 97 ν F ( h = Kwh, K, ν x + K and ν are empirical parameters. The exponent ν may vary from /5 (Kalmar-Nagy and Pratt, 999 through 3/4 (Kienzle, 957 to 4/5 (Tlusty and Spacek, 954. The origin of
On the robustness of stable turning processes 33 these power-law expressions is in the linear optimisation techniques in the parameter space of the logarithms of the cutting parameters w and h. There is a less frequently used expression of the cutting force, which is essentially a cubic polynomial curve fitted on the experimental data (Shi and Tobias, 984 x ( 3 F ( h = w ρ h+ ρ h + ρ h, ρ ( 3,,3 (e.g. ρ = 6.96 9 [N/m ], ρ = 5.446 3 [N/m 3 ], ρ 3 =.3769 7 [N/m 4 ]; (Shi and Tobias, 984. Originally, these measurement data were obtained for full immersion milling with a face mill with even number (z = 4 of teeth. For this case, the cutting process nearly corresponds to the orthogonal cutting since the parametric excitation can be averaged in the system. From dynamical view-point, there are essential differences between the presented empirical interpretations of the cutting forces. The power-law in Figure (a has a vertical tangent at the origin, where the tool just touches the surface of the workpiece, and it cannot have inflection point. Also, the bifurcation calculations require Taylor series expansion of the power-law function at the stationary cutting. In turn, the Tobias cubic curve Figure (b has a finite gradient at the origin and an inflection at h ρ inf = (3 3 ρ3 while it has more parameters and so greater class of functions can be identified this way. Figure (a and (b show the curves of the power-law and the Tobias cubic curve cutting force characteristics. Equation of motion As shown in Figure, the so-called regenerative effect arises during the chip separation. Through the relative vibration between the tool and the workpiece, the tool leaves its motion pattern on the workpiece that excites the system after one revolution of the workpiece. The actual chip thickness can be expressed as a function of the present and the delayed motions of the tool ht ( = qt ( τ qt ( + h (4
34 Z. Dombovari, R.E. Wilson and G. Stepan where h is the theoretical chip thickness and τ = π/ω is the time delay which is equal to the time period of one revolution of the workpiece of angular velocity Ω. The chip thickness variation has the form χ( t = ht ( h = qt ( τ qt ( The power series of the cutting force with respect to the chip thickness variation is given by F ( h( t = F + k χ( t + k χ ( t + k χ ( t + x 3 x 3 l F + kl =, l l! h x l h Note that in the case of Tobias cubic curve, this is a finite polynomial function of third degree. Let us consider the perturbed system around the steady state solution with the new coordinate x(t defined by q(t = q + x(t. With substitution into (, we obtain q = Fx /( mωn for the equilibrium, which is the static deformation of the tool. Thus, we can obtain the perturbed equation of motion: where 3 xt ( + κωnxt ( + ωnxt ( = kχ ( t + kχ ( t + k3χ ( t m χ( t = x( t τ x( t ( is the chip thickness variation. After the introduction of the dimensionless time we obtain a more general form ( ( + κ ( + ( + ( = ( τ + η ( ( τ ( x t x t w x t w x t x t x t 3 ( ( τ ( + w η x t x t 3 t = ω t n where prime is the derivative with respect to t and k w =, k = w + h + 3 h mω ( ρ ρ ρ 3 n and the coefficients of the square and cubic terms in the non-linear excitation are ρ + 3ρ h ρ η =, η = 3 3 3 ρ + ρh + 3ρ3h ρ + ρh + 3ρ3h (5 By abuse of notation, we drop the tilde, thus, the equation of motion is ( 3 x ( t + κx ( t + ( + wxt ( = w xt ( τ + η ( xt ( τ xt ( + η ( xt ( τ xt ( (6 3
On the robustness of stable turning processes 35 Repeat this equation of motion for the delayed time t τ to obtain ( ( xt ( xt ( ( xt ( xt ( 3 x ( t τ + κx ( t τ + ( + w x( t τ = w x( t τ + η τ τ + η τ τ 3 (7 and subtract (6 from (7. Then the non-linear equation of motion for the chip thickness variation will be simplified to ( ( ( 3 3 3 χ ( t + κχ ( t + ( + w χ( t = w χ( t τ + η χ ( t τ χ ( t + η χ ( t τ χ ( t where much less number of non-linear terms will show up in the algebraic calculations. In first order Delay-Differential Equations (DDE form, we have y ( t = Ly( t + Ry( t τ + g( y(, t y ( t τ (8 where y ( t = col( y( t, y( t = col( χ( t, χ ( t and the linear non-delayed and delayed coefficient matrices and the non-linear term are given by L =,, ( w κ R = w + gy ( ( t, y( t τ wη y t y t ( ( τ ( 3 3 + wη3( y( t τ y( t =.3 Operator formulation In order to investigate the non-linear DDE, we transform the system into the space of continuously differentiable functions by the shift, u ( θ = u( t + θ:[ τ,], θ [ τ,] t In this space, the whole system can be rewritten as an Operational Differential Equation (OpDE ( y = Ay + F y (9 t t t with the linear operator A defined by o yt ( θ, if θ [ τ, A t ( θ = ( Lyt( + Ryt( τ, if θ = ( y and the non-linear operator F defined by F, if θ [ τ,] ( θ = g( yt(, yt( τ, if θ = ( ( ( yt
36 Z. Dombovari, R.E. Wilson and G. Stepan It can be shown that (9 corresponds to (8 since the two kinds of derivates are equivalent o d d ut = ut( θ = u ( t + θ, θ [ τ,] dθ dt 3 Linear stability Consider only the linear part y t = A yt of (9 with the exponential trial solution: y ( θ = b( θ, θ [ τ,], λ ( t e λt The trivial solution of (9 is exponentially stable if the characteristic exponents λ k, that is, the eigenvalues of the linear operator A satisfy Re λ k <, k + With the substitution of the general solution into the linear equation, we get a boundary value problem b o ( θ = λb ( θ ( L λi b( + Rb ( τ = With the trial solution b(θ = B exp(λθ, this leads to an eigenvalue problem leading to the characteristic function λτ ( L λi Re D( λ = det + λ κλ w = + + + we λτ The damping factor is fixed for a certain machine tool, so the stability chart will be constructed for the variable parameters w and τ, which are determined by the chip width and the cutting speed. The stability boundaries for these parameters are expressed as a function of the dimensionless vibration frequency ω after the substitution λ = iω :π w( ω = ( ω + 4 ( ω κ ω ω τω ( = jπ+ arctan, j ω κω Then the dimensionless angular velocity Ω of the workpiece is calculated as (3 (4 Ω ( ω = π τω ( (5 The jth parametric function formed from (4 to (5 is called jth lobe. The stability limit consists of the sections of the lobes in the (w, Ω plane beneath, which the cutting process is asymptotically stable (Stepan, 989; Tobias, 965 (see Figure 3.
On the robustness of stable turning processes 37 Figure 3 The linear stability chart (a and the vibration frequency at the stability limits (b; κ =. 4 Non-linear investigation At the stability limits, there are two complex conjugate critical characteristic roots. We will choose the dimensionless chip width w as a bifurcation parameter and follow the motion of these critical roots in the complex plane as w increases through the stability limit. 4. Overview of the Hopf bifurcation calculus We briefly present here the procedure of the Hopf bifurcation calculation following the algorithm given in Dombovari (6 and Stepan and Kalmar-Nagy (997. 4.. Variation of critical eigenvalues According to the Hopf bifurcation theory, we need the derivates of the critical eigenvalues with respect to the bifurcation parameter w, which comes from the implicit differentiation of characteristic function (3: where d λ ( ω γ n( ω γω ( = Re = d w λ= iω w( ω γ d ( ω ( ( γ ω = κ τ ω ω + κ ω + + τ ω ω (6 ( 4 ( 4 ( n 4 4 3 ( 8 κτ ( ω ( ω ( 3ω 8κ ( ω ( τ ( ω ω ( ω 6 ω τ ( ω ( ω γ ( ω = 6 κτ ( ωω + 3 κτωω ( ω d + + + + ( + + (7
38 Z. Dombovari, R.E. Wilson and G. Stepan 4.. Centre manifold At the linear stability limit, the Hopf bifurcation can be studied on a two-dimensional centre manifold embedded in the infinite dimensional phase space. The tangent subspace of the centre manifold at the origin is spanned by the real and imaginary part of the critical eigenvectors of the linear operator A (Figure 4. Figure 4 Tangent subspace and centre manifold at the steady state The critical eigenvectors are calculated from ( A s( θ = i ωs( θ, s( θ = s ( θ + is ( θ R I Substituting the operator A according to (, and solving the corresponding boundary value problem, we obtain the real and the imaginary part of the eigenvectors s cosωθ sinωθ R( θ = and I( θ = ωsinωθ ωcosωθ s (8 Since the base of the tangent space defined by (8 is not orthogonal, to make the projection, we need its reciprocal base spanned by n R and n I, which satisfy the adjoint problem ( n = i n n = nr + ini A ( ϑ ω ( ϑ, ( ϑ ( ϑ ( ϑ (9 where ϑ [, τ] and the adjoint operator A is defined as d A u ( ϑ if ϑ (, τ d ϑ ( u ( ϑ = L u( + R u( τ if ϑ = The orthonormality of the critical eigenvectors are prescribed by the conditions ( n s ( n s, =,, = R R R I where the scalar product is defined by * * ( uv, = u( θ + τ Rv( θd θ + u( v ( τ
On the robustness of stable turning processes 39 The solution of the linear boundary value problem (9 provides the critical normed adjoint eigenvectors nr( ϑ = b( ω cos ωϑ b( ω sin ωϑ ϑ [, τ ] n ( ϑ = b ( ωsin ωϑ+ b ( ωcosωϑ I where the coefficient vectors b (ω and b (ω are not presented here. With the help of the new coordinates z ( t = col( z(, t z( t we can decompose the phase space in the following way: y t ( θ = z( t sr ( θ + z ( t s I ( θ + vt ( θ = ( T z( θ + vt ( θ, where the transformation operator is [ ] ( T z( θ = s ( θ s ( θ z( t R and the corresponding projector operator will be T p consequently z (, (, T u = n u n u t R t I t ( t = T p y t I With the new coordinates, we can express the transformed OpDE as ( t ( p z ( t Jb z( t T F z(, t v = + t( θ o t( θ v A v G zv, t ( where ( p ( zv, = ( zv, ( zv, G F T T F t t t and the Jordan block at the critical parameters assumes the form J b ω = ω The centre manifold is bent, so its second degree approximation with the new coordinates z(t is vt ( θ v( θ, z( t = h ( θ z ( t + h ( θ z ( t z ( t + h ( θ z ( t ( 3 where the unknown coefficient functions are calculated from a linear boundary value problem, again. During the lengthy calculations, the restriction of the non-linear operator F ( on the centre manifold requires the following substitution of the non-linear function
33 Z. Dombovari, R.E. Wilson and G. Stepan gy ( ( t, y( t τ gz ( ( t, v (, z( t η + η ( a ( z z + ( a z v ( τ, z z v (, z 3 3 3( a ( z z = w( ω a z w ω ω w ω z κωz and η, η 3 come from (5. ( = ( (( + ( + 4. Poincaré-Ljapunov constant The first two scalar equations of ( describe the flow on the centre manifold, where the Hopf bifurcation takes place. Its second and third order terms assume the form ( (, (( (, ( p p T F z t vt T F z t v z t j k a,,,3 jkz( t z( t jk j+ k= = j k b,,,3 jkz( t z( t jk j+ k= The Poincaré-Ljapunov Constant (PLC can be expressed directly with the help of the coefficients from ( (Stepan 989 ( ω = + + 8ω + + + 8ω + ( 3 a3 + a + b + 3 b3 8 ( a a ( a b b ( b b ( a a b If ( ω >, then the Hopf bifurcation is subcritical, otherwise, it is supercritical. Since the PLC can be calculated along the lobes as a function of the vibration frequency ω, we can follow the criticality of the Hopf bifurcation along the lobes. If we substitute the coefficients a jk, b jk, ( we obtain δn ( ω η + δn( ω η3 ( ω = ( ω (3 ( ωγ ( ω u d d ( ( where ( ( 4 4 ( ω 4 ( ω ( ( ( 4 ( δ ω κω κ τ ω ω κ ω ω 5 4 4 n( = 6 48 ( + 48 + 384 κτωω ( + 48 κτωω ( 3 + 64κω ω + 7ω 3κω ω 4ω (4
On the robustness of stable turning processes 33 n d n ( ( ( δ ( ω = 3 u ( ω γ ( ω ω u ( ω = 36κ ω + ω 4ω d (5 and γ n (ω, γ d (ω can be found in (6 and (7. We can determine the first harmonic component of the arising periodic motion analytically. Its amplitude can be expressed as a function of the bifurcation parameter w: γω ( r( ω, w = ( w w( ω ( ω δ ( ω w = 3 δ ω η δ ω η w( ω n n( + n( 3 (6 4.3 The criticality of the Hopf bifurcation It has great technical relevance to what kind of limit cycle exists in the vicinity of the stability boundaries, that is, whether the arisen Hopf bifurcation is subcritical or supercritical. In the subcritical case, an unstable oscillation exists around the stable stationary cutting, which can still lead to chatter for perturbations larger than the amplitude of the unstable oscillation. Consequently the sign of (3 refers to the nature of the stability. Since the chatter frequencies ω (, along the stability boundaries, w(ω, γ d (ω and u d (ω are positive in (3 in accordance with (4, (7 and (5. Thus, the sense of the bifurcation depends only on the nominator of the PLC, more exactly on the signs of δ n (ω defined in (4 and η 3 defined in (5: δ ( ω η + δ ( ω η > n n 3 This way, we get the condition δ ( ω η > η (7 n 3 δn( ω Substituting (5 into the condition (7, the problem leads to a second order polynomial condition with respect to the theoretical chip thickness h : ( ( 3 δ ω n ρ ρ n h h δ ω ρ + + + + > δn( ω 3 ρ3 3ρ3 3 δn( ω ρ3 This condition is always satisfied, if it has no real root for h, that is, if the discriminant D satisfies ρ δ ( ω D = ρ + < n 3 ρ 3 3 ρ3 δn( ω Since ρ 3 > for all reasonable cutting force characteristics ( and the last factor is always positive for all ω (,, κ [,, we are left with the simple condition 3ρ ρ > ρ 3
33 Z. Dombovari, R.E. Wilson and G. Stepan This condition is equivalent with the condition of the positive gradient at the inflection point (3 since F x ρ ( h w h inf = ρ > 3ρρ3 > ρ 3 ρ3 Thus, if the gradient of the cubic cutting force function at the inflection point is positive, then the Hopf bifurcation is subcritical. Otherwise, the subcriticality is not proven mathematically, but again, having negative derivative at a possible inflection point on the cutting force characteristic is physically unreasonable. 5 Robustness of stable cutting The unstable period one branch determined by the Hopf calculation gives us the good approximation (6 of the amplitude r of the chip thickness variation χ as a function of the chip width values w close enough to the critical value w(ω from below. Figure 5 clearly shows that the otherwise stable cutting process is much more robust at the left side of the lobes than it is at their right side. The parabolas are flat; they have large curvature at the right side of the lobes, so the stationary cutting is more sensitive to the perturbations there. It is safer to tune the parameters to the left side of the lobes, these regions are smaller, though, with respect to the cutting speeds. This conclusion is similar to the one obtained for the power law by Wahi and Chatterjee (5. Figure 5 Structure of periodic orbits; j =, κ =. The subcriticality of the Hopf bifurcation means that the unstable limit cycle separates two coexisting stable motions, the desired stable stationary cutting and an undesired large amplitude chatter that is also stable in the sense of dynamical systems theory. This is why this region is called the region of bi-stability. Since we proved that the Hopf bifurcation is subcritical along the lobes, the bi-stable regions are located below these lobes, covering a large region of the parameter domain of stable stationary cutting.
On the robustness of stable turning processes 333 According to (6, the first harmonic component of the limit cycle can be calculated as a projection of the real orbit to the subspace tangent to the centre manifold (Figure 4: z ( t r( ω, wcosωt z( t r( ω, wsinωt t ( θ v This way, the actual chip thickness is given by (4: ht ( = χ( t + h = yt,( + h = z( t + h = r( ω, wcos( ωt + h The tool leaves the surface of the workpiece when the chip thickness becomes zero: ht ( = r( ω, w = h (8 Expressing the bifurcation parameter w from (8, we get the so-called bi-stable limit where the periodic orbits satisfy the switching condition: 3 n( n( 3 wbs ( w( h δ ω η + δ ω η ω = ω δn ( ω Figure 6 shows the bi-stable regions as dark strips in the grey stable cutting domain, just below the white unstable parameter domain. With the appropriate choice of the parameters η, η 3 from (6, the results are shown for both 3/4 law and for the Tobias cubic curve. While the dimensionless linear stability boundary is not affected at all by the applied cutting force characteristic function, the size of the bi-stable region strongly depends on it. Clearly, this bi-stable region is much larger if the cutting force characteristics follow the cubic curve rather than the power-law. Figure 6 Both panels show the stability chart with the bi-stable region in case of the 3/4 powerlaw (a and Tobias cubic curve (b; h =.6 [mm], κ =. 6 Conclusion During the design of the technological parameters, it is much more convenient and common to use the power-law approximation of the real measured cutting force characteristics. From the view-point of the stability prediction of the stationary cutting, the linearisation of this power-law is still appropriate. However, from non-linear
334 Z. Dombovari, R.E. Wilson and G. Stepan vibrations view-point, the use of the power-law approximation has great deficiencies. As we showed, the prediction of the bi-stable parameter domain is important if we want to secure stable stationary cutting robust enough for perturbations. However, this domain cannot be predicted correctly, in a conservative way, if we use the power-law approximation. We proved that the Tobias cubic curve approximation of the cutting force characteristics leads to subcritical Hopf bifurcations along all the stability limits in the same way as it was shown for the power-law by Kalmar-Nagy and Pratt (999 and Wahi and Chatterjee (5. Namely, the stable cutting process is more robust against perturbations originated in non-homogeneous workpiece material at the left side of the lobes than it is at their right side. In the meantime we also showed that the bi-stable domain can be substantially larger for more realistic cutting force characteristics involving inflection points than the ones predicted by the power-law. Acknowledgement This research was supported by the Hungarian Nation Sciences Foundation under grant no. OTKA F4738 and OTKA T43368. References Altintas, Y. and Budak, E. (995 Analytical prediction of stability lobes in milling, CIRP, 448..., pp.357 36. Dombovari, Z. (6 Bifurcation analysis of a cutting process, MSc Thesis, BME, Hungary. Insperger, T., Stepan, G., Bayly, P.V. and Mann, B.P. (3 Multiple chatter frequencies in milling processes, Journal of Sound and Vibration, Vol. 6, pp.333 345. Kalmar-Nagy, T. and Pratt, R.J. (999 Experimental and analytical investigation of the subcritical instability in metal cutting, ASME, Nevada, USA. Kienzle, O. (957 Spezifische Schnittkräfte bei der Metallbearbeitung (in German, Werkstattstechnik und Maschinenbau, Vol. 47. Mann, B.P., Bayly, P.V., Davies, M.A. and Halley, J.E. (3 Limit cycles, bifurcations, and accuracy of the milling process, Journal of Sound and Vibration, Vol. 77, pp.3 48. Shi, H.M. and Tobias, S.A. (984 Theory of finite amplitude machine tool instability, International Journal of Machine Tool Design and Research, Vol. 4, pp.45 69. Stepan, G. (989 Retarded Dynamical Systems, London: Longman. Stepan, G. and Kalmar-Nagy, T. (997 Nonlinear regenerative machine tool vibrations, ASME, California, USA. Szalai, R., Stepan, G. and Hogan, S.J. (4 Global dynamics of low immersion high-speed milling, Chaos, Vol. 4, pp.69 77. Taylor, F.W. (97 On the art of cutting metals, Transaction of ASME, Vol. 8, pp.3 35. Tlusty, J. and Spacek, L. (954 Self-excited vibrations on machine tools (in Czech, Nakl CSAV, Prague. Tobias, S.A. (965 Machine Tool Vibrations, London: Blackie. Wahi, P. and Chatterjee, A. (5 Regenerative tool chatter near a Codimension hopf point using multiple scales, Nonlinear Dynamics, Vol. 4, pp.33 338. Zatarain, M., Muñoa, J., Peigné, G. and Insperger, T. (6 Analysis of the influence of mill helix angle on chatter stability, Annals of the CIRP, Vol. 55, No..