Available online at ScienceDirect. Procedia IUTAM 22 (2017 )

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1 Available online at ScienceDirect Procedia IUTAM (17 ) IUTAM Syposiu on Nonlinear and Delayed Dynaics of Mechatronic Systes Dynaics of cutting near double bifurcation Taás G. Molnár a,, Zoltán Dobóvári a,taás Insperger b,gábor Stépán a a Departent of Applied Mechanics, Budapest University of Technology and Econoics, H-1111 Budapest, Hungary b Departent of Applied Mechanics, Budapest University of Technology and Econoics and MTA-BME Lendület Huan Balancing Research Group, H-1111 Budapest, Hungary Abstract Bifurcation analysis of the orthogonal cutting odel with cutting force nonlinearity is presented with special attention to double bifurcations. The noral for of the syste in the vicinity of the double point is derived analytically by eans of center anifold reduction. The dynaics is restricted to a four-diensional center anifold, and the long-ter behavior is illustrated on siplified phase portraits in two diensions. The topology of the phase portraits reveal the coexistence of periodic and quasi-periodic solutions, which are coputed by approxiate analytical forulas. c 17 The Authors. Published by by Elsevier B.V. B.V. This is an open access article under the CC BY-NC-ND license ( Peer-review under responsibility of organizing coittee of the IUTAM Syposiu on Nonlinear and Delayed Dynaics of Peer-review Mechatronic under Systes. responsibility of organizing coittee of the IUTAM Syposiu on Nonlinear and Delayed Dynaics of Mechatronic Systes Keywords: achine tool vibrations, nonlinearity, bifurcation, double bifurcation, center anifold reduction 1. Introduction The theory of surface regeneration has been widely used since the 195 s to explain self-excited vibrations in etal cutting 1,. According to the theory, the vibrations of the achine tool copy onto the workpiece during cutting and hence affect the vibrations at the subsequent cut. Therefore, regenerative achine tool vibrations (or achine tool chatter) can be described by differential equations with tie delay. The equations are typically nonlinear due to nonlinear cutting force expressions 3,4,5,6. It is well-known that tie delay often leads to the instability of dynaical systes. In the nonlinear odels of achining (turning), the stability of stationary cutting is lost via bifurcation. bifurcation can be investigated either by analytical ethods, such as the center anifold reduction 7,8,9 or the ethod of ultiple scales 1,11,1,orby nuerical bifurcation analysis 13. At the intersections of the stability boundaries associated with bifurcation, codiension-two double bifurcations take place. Double bifurcation gives rise to rich dynaics and coplex behavior in dynaic systes 14,15,16,17,18,19,,1,,3. The double bifurcation in orthogonal cutting has been analyzed in 4 by the ethod of ultiple scales. Here, we dedicate this paper to the analysis of the double bifurcation in cutting by eans of center anifold reduction. Corresponding author. E-ail address: olnar@.be.hu The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license ( Peer-review under responsibility of organizing coittee of the IUTAM Syposiu on Nonlinear and Delayed Dynaics of Mechatronic Systes doi:1.116/j.piuta

2 14 Taás G. Molnár et al. / Procedia IUTAM ( 17 ) a) b) x(t) x(t t).8 w 1 w w 1 w chatter w 1 w h(t) h F tool feed x c k W.6 w.4. j=4 j=3 double j= j=1 workpiece w stable W Fig. 1. Single-degree-of-freedo echanical odel of orthogonal cutting (a); the corresponding stability lobe diagra (b).. Mechanical odel, stability, and bifurcations Consider the single-degree-of-freedo echanical odel of orthogonal cutting shown in Fig. 1(a). The workpiece is assued to be rigid, rotating with diensionless angular velocity Ω. The tool is assued to be copliant in the feed direction, and is odeled by a ass-spring-daper syste. The diensionless equation governing the tool s otion relative to the workpiece can be given in the for 9 ẍ(t) + ζẋ(t) + x(t) = w ( (x(t τ) x(t)) + η (x(t τ) x(t)) + η 3 (x(t τ) x(t)) 3 ), (1) where t is the diensionless tie (scaled by the natural angular frequency of the syste), x is the diensionless tool position (scaled by the feed h per revolution), and ζ is the daping ratio. The right-hand side of Eq. (1) yields the x-directional coponent of the cutting force variation, which is proportional to the diensionless chip width w and is a cubic polynoial of the diensionless chip thickness variation x(t τ) x(t) 3. Paraeters η and η 3 denote diensionless quadratic and cubic cutting-force coefficients, whereas τ = π/ω is called regenerative delay, which is equal to the period of workpiece rotations. The equivalent first-order representation of the second-order syste (1) reads y (t) = Ly(t) + Ry(t τ) + g(y t ). () The vector y of state variables, the linear and retarded coefficient atrices L and R, and the nonlinear ter g are defined by [ [ [ [ x(t) 1 y(t) =, L =, R =, g(y ẋ(t) (1 + w) ζ w t ) = w ( η (y 1 (t τ) y 1 (t)) + η 3 (y 1 (t τ) y 1 (t)) 3), (3) where subscript 1 refers to the first coponent of a vector, y 1 (t) = x(t)..1. bifurcation The equilibriu x(t) of Eq. (1) corresponds to stationary cutting with unifor prescribed chip thickness h. Machine tool vibrations are associated with the loss of stability of this equilibriu. The linear stability of the solution x(t) is deterined by the real part of the characteristic exponents of syste (), which are the roots of the characteristic equation D(λ) = det ( λi L Re λτ) = λ + ζλ w ( 1 e λτ) =. (4) The equilibriu related to stationary cutting is linearly stable provided that all the infinitely any characteristic exponents are located in the left half of the coplex plane. Along the stability boundaries of the syste, a pair of

3 Taás G. Molnár et al. / Procedia IUTAM ( 17 ) coplex conjugate characteristic exponents λ = ±iω (i = 1,ω ) lies on the iaginary axis. Correspondingly, the real and iaginary parts of D(iω) = give the stability boundaries (or stability lobes) in the for 9 ( ω 1 ) + 4ζ ω w H (ω) = ( ω 1 ), τ H( j,ω) = ( ( ω )) 1 π jπ arctan, Ω H ( j,ω) = ω ζω τ H ( j,ω), (5) where j Z is the lobe nuber. The stability boundaries are often depicted in the plane (Ω, w) of the angular velocity and the chip width, resulting in so-called stability lobe diagras, see Fig. 1(b) for ζ =.. Along the stability boundaries (5), a bifurcation takes place note that a fold bifurcation cannot happen in syste (1). For the analysis of the bifurcation, the reader is referred to 8,9,11,1. According to 9, the sense of the bifurcation is subcritical at each point of the stability lobes (5) for real-life cutting-force paraeters η, η 3. The subcritical bifurcation gives rise to an unstable periodic solution in the vicinity of the equilibriu. The approxiate angular frequency of the periodic solution is ω... Double bifurcation Hereinafter we use previous results 9 for the bifurcation, and extend the to analyze the double bifurcation in syste (1). According to Fig. 1(b), the stability boundaries associated with bifurcation intersect at soe points. At the intersection point of two lobes with lobe nubers j 1 and j, two periodic solutions are born siultaneously with angular frequencies ω 1 and ω, respectively. In such points, a double bifurcation takes place, which even gives rise to a quasi-periodic solution with two angular frequencies ω 1 and ω 5. The double bifurcation points and the two angular frequencies are given by w dh = w H (ω 1 ) = w H (ω ) and Ω dh =Ω H (ω 1 ) =Ω H (ω ) and Eq. (5). In what follows, we analyze the double bifurcation by deterining the noral for of the syste by eans of center anifold reduction, in order to find approxiate expressions for the arising periodic and quasi-periodic solutions. We choose Ω and w as bifurcation paraeters for the analysis of this codiension-two bifurcation. 3. Center anifold reduction and noral for calculations The noral for of syste (1) in the vicinity of the double bifurcation point (Ω dh, w dh ) can be deterined by eans of the ethod of ultiple scales 4, or by eans of center anifold reduction. Here, we use the latter for noral for analysis. The theory of center anifold reduction is discussed in 7, whereas its application to and double bifurcations can be found for exaple in 8,9,14,17,18,1,. The ain idea behind center anifold reduction is the following. The phase space of the tie-delay syste (1) is infinite-diensional, as it has infinitely any characteristic exponents satisfying Eq. (4). At the double bifurcation point, four characteristic exponents ±iω 1, are located on the iaginary axis, while the other infinitely any exponents lie in the left-half plane. Therefore, in the vicinity of the double bifurcation, there exists a fourdiensional center anifold ebedded in the infinite-diensional phase space, which attracts the solutions of the tie-delay syste. The flow on this center anifold deterines the long-ter dynaics of the syste. The long-ter behavior can be investigated by restricting the dynaics to the four-diensional center anifold, and analyzing a fourdiensional syste of ordinary differential equations instead of the infinite-diensional delay-differential equation Restriction to the center anifold The restriction to the center anifold can be done according to the decoposition theore in Chapter 7.3 of the book by Hale 7. In order to carry out the decoposition, we rewrite Eq. () in operator differential equation for ẏ t (θ) = ( Ay t ) (θ) + ( F (yt ) ) (θ), (6) where y t H:[ τ, R, y t (θ) = y(t + θ) is the state of the syste in the Hilbert space H of continuously differentiable vector-valued functions. The linear and the nonlinear operators A, F : H Hare defined by u (θ) if θ [ τ, ), { if θ [ τ, ), (Au) (θ) = (F (u)) (θ) = (7) Lu() + Ru( τ) ifθ =, g(u) ifθ =.

4 16 Taás G. Molnár et al. / Procedia IUTAM ( 17 ) Furtherore, we introduce operator A : H H, which is forally adjoint to operator A relative to a certain bilinear for. The foral adjoint satisfies (v, Au) = (A v, u) for any pair of u H:[ τ, R and v H : [,τ R, where H is the adjoint space, and operation (, ):H H R indicates a bilinear for. According to Eqs. (3.1) and (3.3) in Chapter 7.3 of Hale s book 7, the definition of the foral adjoint and the bilinear for becoe v (ϕ) if ϕ (,τ, (A v) (ϕ) = L H v() + R H v(τ) ifϕ =, (u, v) = u H ()v() + τ u H (ϕ)rv(ϕ τ)dϕ, (8) where the superscript HofR, L, and u refers to conjugate transpose. The four-diensional center anifold is tangent to the eigenfunctions (infinite-diensional eigenvectors) s k (θ) and s k (θ) (k = 1, ) of operator A at (Ω dh, w dh ). Whereas restriction to the center anifold can be perfored using the eigenfunctions n k (ϕ) and n k (ϕ)(k = 1, ) of operator A at (Ω dh, w dh ). The eigenfunctions are defined by ( ) ( As k (θ) = iω k s k (θ), ) As k (θ) = iω k s k (θ), (9) ( A n k) (ϕ) = iω k n k (ϕ), ( A n k) (ϕ) = iω k n k (ϕ), (1) where over-bar denotes coplex conjugate. After solving the boundary value probles (9) and (1), the eigenfunctions are obtained in the for s k (θ) = s k R (θ) + isk I (θ), nk (ϕ) = n k R (ϕ) + ink R (ϕ), [ s k R (θ) = cos(ω k θ) ω k sin(ω k θ) [ s k I (θ) = sin(ω k θ) ω k cos(ω k θ), n k R (ϕ) = p k +q k, n k I (ϕ) = p k +q k [ (ζ pk + ω k q k ) cos(ω k ϕ)+(ω k p k ζq k ) sin(ω k ϕ) p k cos(ω k ϕ) q k sin(ω k ϕ) [ ( ωk p k +ζq k ) cos(ω k ϕ)+(ζ p k +ω k q k ) sin(ω k ϕ) q k cos(ω k ϕ) + p k sin(ω k ϕ) where the constants p k and q k are given by Eq. (A.1) in Appendix A. Note that the eigenfunctions satisfy the orthonorality conditions (n 1 R, s1 R ) = 1, (n1 R, s1 I ) =, (n R, s R ) = 1, and (n R, s I ) =. Using these eigenfunctions and the decoposition theore in Hale s book 7, the phase space of the tie-delay syste (6) can be decoposed into stable and center subspaces as y t (θ) = z 1 (t)s 1 R (θ) + z (t)s 1 I (θ) + z 3(t)s R (θ) + z 4(t)s I (θ) + y tn(θ), (1) where z 1 (t), z (t), z 3 (t) and z 4 (t) are coordinates aligned with the center anifold that describe the long-ter dynaics of the tie-delay syste (1), while y tn (θ) represents the infinite-diensional stable subsyste transverse to the center anifold. According to the decoposition theore, the forulas of these coponents read z 1 (t) = (n 1 R, y t), z (t) = (n 1 I, y t), z 3 (t) = (n R, y t), z 4 (t) = (n I, y t), y tn (θ) = y t (θ) z 1 (t)s 1 R (θ) z (t)s 1 I (θ) z 3(t)s R (θ) z 4(t)s I (θ). (13),, (11) Differentiating Eq. (13) with respect to tie, using Eqs. (6), (1), (9), and oitting the arguents t, θ we get ż 1 ż ż 3 ż 4 ẏ tn = ω 1 O ω 1 O ω O ω O o o o o A z 1 z z 3 z 4 y tn n 1 R ()F () n 1 I ()F () + n R ()F (), n I ()F () G(y t ) G(y t ) = F (y t ) F () ( n 1 R ()s1 R + n1 I ()s1 I + n R ()s R + n I ()s I ) (14) where o : R His a zero operator, O : H R is a zero functional, subscript indicates the second coponent of vectors, and F () shortly indicates the second coponent of F (y t )atθ = with Ω=Ω dh and w = w dh. Note that in Eq. (14), the four-diensional center subsyste is already decoupled on the linear level fro the infinite-diensional stable subsyste.

5 Taás G. Molnár et al. / Procedia IUTAM ( 17 ) In order to decouple the nonlinear ters in the equations for z 1, z, z 3, z 4, the dynaics ust be restricted to the four-diensional center anifold y tn (θ) = y CM tn (z 1, z, z 3, z 4 )(θ). Second-order approxiation of the center anifold allows us to decouple the nonlinear ters up to third order in Eq. (14): y CM tn (z 1, z, z 3, z 4 )(θ) 1 ( h11 (θ)z 1 + h (θ)z + h 33(θ)z 3 + h 44(θ)z 4 + h 1 (θ)z 1 z + h 13 (θ)z 1 z 3 + h 14 (θ)z 1 z 4 + h 3 (θ)z z 3 + h 4 (θ)z z 4 + h 34 (θ)z 3 z 4 ), (15) where the coefficients h n (θ)(, n = 1,, 3, 4, n) can be calculated by solving a boundary value proble defined by Eqs. (14) and (15). Restricting the dynaics to the center anifold, using the second-order approxiation (15) of the center anifold and the third-order approxiation of the nonlinearities in Eq. (14), we get a four-diensional decoupled set of ordinary differential equations in the for ż 1 ż = ż 3 ż 4 ω 1 ω 1 ω ω z 1 z z 3 z 4 G 1 (z 1, z, z 3, z 4 ) G (z 1, z, z 3, z 4 ) + G 3 (z 1, z, z 3, z 4 ) G 4 (z 1, z, z 3, z 4 ). (16) Here, G 1,,3,4 are functions with purely quadratic and cubic nonlinearities. Hence Eq. (16) gives a third-order approxiation of the center subsyste, which is suitable for noral for calculations and bifurcation analysis Noral for in the vicinity of the double point The four-diensional syste (16) can be written in polar for with aplitudes, and phase angles θ 1, θ as 5 ṙ 1 =μ 1 + a 11 r a 1, θ 1 = ω 1 + c c 1, ṙ =μ + a a r 3, θ = ω + c c, (17) where the constants a n and c n (, n = 1, ) can be obtained by the forulas given in 6. Accordingly, periodic and quasi-periodic orbits arise, which can be approxiated by cos(ω 1 t) + cos(ω t). In this paper, we focus only on the aplitude of the arising periodic and quasi-periodic solutions, hence we copute and only, for which the cubic coefficients a n are listed in Eq. (A.) of Appendix A. The coefficients μ 1 and μ of the linear ters are μ 1 = γ 11 ŵ + γ 1 ˆΩ, μ = γ 1 ŵ + γ ˆΩ, (18) where ˆΩ =Ω Ω dh and ŵ = w w dh are the bifurcation paraeters shifted to the double point. Whereas γ k1 and γ k (k = 1, ) are the root tendencies obtained by iplicit differentiation of Eq. (4): ( ) λ γ k1 = Re w λ=iωk = p kv k + q k W k p k + q k with paraeters listed in Eq. (A.1) of Appendix A. ( ) λ, γ k = Re Ω λ=iωk τ dh = w dh π ω q k (V k + 1) p k W k k p k +, (19) q k 4. Results and discussion The analysis of the polar-for syste (17) can be found in 5. Accordingly, the long-ter dynaics near the double point is deterined by the paraeters a = a 11 / a 11, b = a 1 / a, c = a 1 / a 11, d = a / a, and A = ad bc. The values of the bifurcation paraeters, the chatter frequencies, the root tendencies, and the noral for coefficients at the first five double bifurcation points are listed in Tab. 1 for η = and η 3 = , which are diensionless paraeters corresponding to easured cutting-force coefficients 3 and feed per revolution h = 5 μ. The coefficients b and c have also been coputed nuerically by DDE-Biftool 13, and they agree with the analytical results (see b nu, c nu ). According to the table, a, b, c, and d are always positive, whereas A > holds for the first

6 18 Taás G. Molnár et al. / Procedia IUTAM ( 17 ) a).3 = 1 1 b) x. = w.1 r1 r = d = c b 1 a W x. x Fig.. Topology of the siplified two-diensional phase portraits (a); its interpretation in the infinite-diensional state space (b). and A < for all other double points. Based on the coefficients a, b, c, d, A, and the bifurcation paraeters Ω, w, different topologies can be observed in siplified phase portraits depicted in two diensions along (, ). The possible topologies are discussed in 5. Case Ia of the book by Guckenheier and Holes 5 applies for the first double point, and case Ib for all others. Fig. (a) shows the siplified phase portraits at the third double point (at the intersection of the third and the fourth lobe). According to 5, the lines μ 1 =, μ = approxiate the lobes (indicated by blue and red), where periodic solutions are born. In the siplified phase portraits, these periodic solutions are represented by the equilibriu points (r p 1, ) and (, rp ), respectively. Whereas lines μ = cμ 1 /a and μ = dμ 1 /b are the approxiations of torus bifurcation branches (shown by black), where a quasi-periodic solution represented by (r qp 1, rqp ) is born fro one of the periodic solutions ((r p 1, ) or (, rp ), respectively). The aplitude of these solutions is given by r p 1 (ŵ, ˆΩ) = γ 11ŵ + γ 1 ˆΩ 1 (ŵ, (a 1 γ 1 a γ 11 ) ŵ + (a 1 γ a γ 1 ) ˆΩ ˆΩ) = r p (ŵ, ˆΩ) = a 11, r qp γ 1ŵ + γ ˆΩ a, r qp (ŵ, ˆΩ) =, a 11 a a 1 a 1 (a 1 γ 11 a 11 γ 1 ) ŵ + (a 1 γ 1 a 11 γ ) ˆΩ. a 11 a a 1 a 1 The interpretations of the two-diensional siplified phase portraits is shown in Fig. (b). The equilibria (, ), (r p 1, ), (, rp ), and (rqp 1, rqp ) correspond to the trivial equilibriu, to two periodic solutions, and to the quasi-periodic () Table 1. Bifurcation paraeters, chatter frequencies, root tendencies, and noral for coefficients at the double points. j 1 j ω 1 ω Ω dh w dh γ 11 γ 1 γ 1 γ j 1 j a 11 a 1 a 1 a a b c d A b nu c nu

7 Taás G. Molnár et al. / Procedia IUTAM ( 17 ) a) 8 b) W= w=.15 r 6 4 Torus = d b 1 = 1= w r 6 4 Torus = d b 1 Torus = c 1= W a 1 Fig. 3. Bifurcation diagras near the double point as a function of the chip width w (a); as a function of the spindle speed Ω (b). solution, respectively, in the infinite-diensional phase space of the original tie-delay syste. For the sake of a siple illustration, the infinite-diensional phase portrait is represented by a three-diensional one in Fig. (b) using axes x, ẋ, and x. Here, x stands for the reaining infinite diensions, and x has no physical eaning, since it is introduced for illustration purposes only. Therefore, only the qualitative behavior and not the actual (infinitediensional) solutions are shown in the figure. Finally, Fig. 3 shows the aplitude r = + of the arising periodic and quasi-periodic solutions according to Eq. () as a function of the bifurcation paraeters Ω and w. The bifurcation diagras are coputed for Ω =.337 and w =.15 as shown by the dotted lines in Fig. (a). The solutions related to (r p 1, ), (, rp ), and (rqp 1, rqp ) are shown by blue, red, and black colors, respectively. bifurcations are indicated in Fig. 3(a), where the periodic solutions are born when decreasing w. A torus bifurcation point can also be seen, where the quasi-periodic solution is born fro the periodic solution related to (, r p ). These bifurcation points are also indicated by dots in Fig. (a). In Fig. 3(b), two torus bifurcation points can be observed. When increasing Ω, the quasi-periodic solution is born fro the periodic solution related to (, r p ), and then it vanishes by colliding with the periodic solution of (rp 1, ). Note that the bifurcation diagras in Fig. 3 are not valid at large aplitudes due to loss of contact between the tool and the workpiece. When loss of contact occurs, Eq. (1) becoes no longer valid and the periodic solutions disappear 9. Siilarly, we expect the quasi-periodic solution to vanish at loss of contact. Locating the point where the tool loses contact is out of scope of this paper, but we reark that it has an iportant role in deterining the global stability boundaries of the equilibriu. Acknowledgeents This work has been supported by the ÚNKP-16-3-I. New National Excellence Progra of the Ministry of Huan Capacities. This work has received funding fro the János Bolyai Research Scholarship of MTA (BO/589/13/6). This work was supported by the Hungarian National Science Foundation under grant OTKA-K The research leading to these results has received funding fro the European Research Council under the European Union s Seventh Fraework Prograe (FP/7-13) / ERC Advanced Grant Agreeent n Appendix A. An exaple appendix The auxiliary paraeters used in the paper are V 1, = cos(ω 1, τ dh ) 1, W 1, = sin(ω 1, τ dh ), p 1, = ζ + τ dh ( 1 + wdh ω 1,), q1, = ω 1, (1 + ζτ dh ), R 1, = 1 cos(ω 1, τ dh ), S 1, = sin(ω 1, τ dh ), R 3,4 = 1 cos((ω ± ω 1 )τ dh ), S 3,4 = sin((ω ± ω 1 )τ dh ), K 1, = wr 1, ( 4ω 1, 1), L 1, = ws 1, + 4ζω 1,, K 3,4 = wr 3,4 ( (ω ± ω 1 ) 1 ), L 3,4 = ws 3,4 + ζ(ω ± ω 1 ). (A.1)

8 13 Taás G. Molnár et al. / Procedia IUTAM ( 17 ) The expressions of the cubic noral for coefficients read a 11 = 1 w (V1 + W 1 )η K 1R 1 + L 1 S 1 p 1 V 1 + q 1 W 1 K1 + L 1 p K 1S 1 L 1 R 1 q 1 K1 + L 1 a 1 = w (V + W )η K 3R 3 + L 3 S 3 K3 + + K 4R 4 + L 4 S 4 L 3 K4 + p 1V 1 + q 1 W 1 L 4 p 1 + q 1 + K 3S 3 L 3 R 3 K3 + K 4S 4 L 4 R 4 L 3 K4 + q 1V 1 p 1 W 1 L 4 p q w(v + W )η 3 1 q 1 V 1 p 1 W 1 p q 4 w(v 1 + W 1 )η p 1 V 1 + q 1 W p 1 +, q 1 p 1 V 1 + q 1 W 1 p 1 + q 1 whereas a 1 and a can be obtained by interchanging ω 1 and ω in the forulas of a 1 and a 11, respectively., (A.) References 1. Tobias, S.A., Fishwick, W.. Theory of regenerative achine tool chatter. The Enginee958;:199 3, Tlusty, J., Polacek, M.. The stability of the achine tool against self-excited vibration in achining. In: ASME Production Engineering Research Conference. Pittsburgh; 1963, p Shi, H.M., Tobias, S.A.. Theory of finite aplitude achine tool instability. Int JMachToolDR1984;4(1): Endres, W.J., Loo, M.. Modeling cutting process nonlinearity for stability analysis - application to tooling selection for valve-seat achining. In: 5th CIRP International Workshop on Modeling of Machining. West Lafayette, IN, USA;, p Ahadi, K., Isail, F.. Experiental investigation of process daping nonlinearity in achining chatter. Int J Mach Tool Manu 1; 5(11): Altintas, Y.. Manufacturing Autoation - Metal Cutting Mechanics, Machine Tool Vibrations and CNC Design, Second Edition. Cabridge: Cabridge University Press; Hale, J.. Theory of Functional Differential Equations. New York: Springer; Stépán, G., Kalár-Nagy, T.. Nonlinear regenerative achine tool vibrations. In: Proc of DETC 97, ASME Design and Technical Conferences. Sacraento, CA, USA; 1997, p Dobóvári, Z., Wilson, R.E., Stépán, G.. Estiates of the bistable region in etal cutting. P Roy Soc Lond A-Math Phy 8;464: Nayfeh, A.H., Mook, D.T.. Nonlinear Oscillations. New York: Wiley; Nayfeh, A.H.. Order reduction of retarded nonlinear systes the ethod of ultiple scales versus center-anifold reduction. Nonlinear Dyna 8;51(4): Nandakuar, K., Wahi, P., Chatterjee, A.. Infinite diensional slow odulations in a well known delayed odel for cutting tool vibrations. Nonlinear Dyna 1;6(4): Engelborghs, K., Luzyanina, T., Roose, D.. Nuerical bifurcation analysis of delay differential equations using DDE-BIFTOOL. ACM T Math Software ;8(1): Capbell, S.A., Bélair, J., Ohira, T., Milton, J.. Liit cycles, tori, and coplex dynaics in a second-order differential equation with delayed negative feedback. J Dyna Differential Equations 1995;7(1): Stépán, G.. Modelling nonlinear regenerative effects in etal cutting. P Roy Soc A-Math Phy 1;359(1781): Xu, J., Chung, K.W., Chan, C.L.. An efficient ethod for studying weak resonant double bifurcation in nonlinear systes with delayed feedbacks. SIAM J Appl Dyn Syst 7;6(1): Guo, S., Chen, Y., Wu, J.. Two-paraeter bifurcations in a network of two neurons with ultiple delays. J Differ Equations 8; 44(): Ma, S., Lu, Q., Feng, Z.. Double bifurcation for Van der Pol-Duffing oscillator with paraetric delay feedback control. J Math Anal Appl 8;338(): Wang, W., Xu, J.. Multiple scales analysis for double bifurcation with 1:3 resonance. Nonlinear Dyna 11;66(1): Bazsó, C., Chapneys, A.R., Hős, C.J.. Bifurcation analysis of a siplified odel of a pressure relief valve attached to a pipe. SIAM J Appl Dyn Syst 14;13(): Qesi, R., Babra, M.A.. Double bifurcation in delay differential equations. Arab J Math Sci 14;(): Shen, Z., Zhang, C.. Double hopf bifurcation of coupled dissipative Stuart-Landau oscillators with delay. Appl Math Coput 14; 7: Ding, Y., Cao, J., Jiang, W.. Double hopf bifurcation in active control syste with delayed feedback: application to glue dosing processes for particleboard. Nonlinear Dyna 16;83(3): Wahi, P., Chatterjee, A.. Regenerative tool chatter near a codiension point using ultiple scales. Nonlinear Dyna 5; 4(4): Guckenheier, J., Holes, P.. Nonlinear Oscillations, Dynaical Systes, and Bifurcations of Vector Fields. New York: Springer; Knobloch, E.. Noral for coefficients for the nonresonant double bifurcation. Phys Lett A 1986;116(8):