Bifurcation analysis of a forced delay equation for machine tool vibrations

A machining tool can be subject to different kinds of excitations. The forcing may have external sources (such as rotating imbalance, misalignment of the workpiece or ultrasonic excitation), or it can arise from the cutting process itself (e.g., periodic chip formation). We investigate the classical one-degree-of-freedom tool vibration model, a delay-differential equation with quadratic and cubic nonlinearity, and periodic forcing. The method of multiple scales is used to derive the slow flow equations. Stability and bifurcation analysis of equilibria of the slow flow equations is presented. Analytical expressions are obtained for the saddle-node and Hopf bifurcation points. Bifurcation analysis is also carried out numerically. Sub- and supercritical Hopf, cusp, fold, generalized Hopf (Bautin), Bogdanov–Takens bifurcations are found. Limit cycle continuation is performed using MatCont. Local and global bifurcations are studied and illustrated with phase portraits and direct numerical integration of the original equation.

To better understand the machining process, the chip formation has to be modeled. Periodic chip formation was investigated in [38][39][40][41][42][43][44][45][46]. In recent works of Csernák and Pálmai [47,48], a nonlinear system of differential equations is used to model chip segmentation. They find periodic, aperiodic and chaotic behavior of the chip formation model, which provides an excitation within the machine-tool-workpiece system.
Vibration-assisted machining is another example of externally excited machine tool vibrations, where the small-amplitude, high-frequency tool displacement leads to improved surface finish and accuracy compared to conventional machining [49][50][51][52].

The model and its linear stability analysis
In this paper, a harmonically excited single-degree-offreedom machine tool vibration model of orthogonal cutting ( Fig. 1) is investigated (for derivation see [14,18]) where x is the tool displacement, ζ > 0 is the relative damping factor, p > 0 is the nondimensional cutting force, q > 0 is the coefficient of nonlinearity, A ≥ 0 is the amplitude of the forcing, and ω is its frequency. The term x τ denotes x (t − τ ), the delayed value of the position with with positive delay τ . The regenerative effect is considered with the expressions containing the dimensionless chip thickness variation x − x τ on the right-hand side of Eq. (1). The stability analysis of the x = 0 solution of the linearized equation was performed in [36,53]. The characteristic function of Eq. (2) is obtained by substituting the trial solution (2). The stability diagram in Fig. 2 is given in parametric form (see [14]) (4) where j corresponds to the jth 'lobe' and Ω > 1. At the minima ('notches') of the stability lobes, Ω, p, τ assume the particularly simple forms Along the stability boundaries, Hopf bifurcation may occur, giving rise to periodic solutions of the nonlinear retarded system [18,36].

Slow flow equations and equilibria
We approximate the solution of Eq. (1) by using the method of multiple scales. In the following we will assume that damping is small, nonlinearity and forcing are weak (see [14]) and the forcing is near-resonant, i.e., Here ε 1 is the bookkeeping parameter and σ is the detuning frequency. We express the system parameters as Substituting these into Eq. (1) we geẗ We also assume that the solution of Eq. (10) can be well approximated by the two-scale expansion where the fast and slow timescales are defined as With the differential operators time differentiation can be written as and second derivative with respect to time is Substituting the differential operators (14) and (15) into Eq. (10) and equating like powers of ε one obtains Solving Eq. (16) for x 0 (t 0 , t 1 ) yields where α(t 1 ) and β(t 1 ) are the slowly varying and amplitude and phase, respectively. We now substitute this solution into Eq. (17). To eliminate the secular terms (terms containing sin(t 0 + β(t 1 )) and cos(t 0 + β(t 1 ))), we require the following equations to hold: where we introduced Equations (19,20) are two ordinary differential equations describing the evolution of the amplitude and phase (slow flow equation). We note that Eq. (20) becomes an algebraic equation for α = 0; therefore, we only consider the α > 0 case.

Amplitude of the steady-state vibration
The fixed points of the slow flow correspond to periodic solutions of the original Eq. (1). To get the amplitude α * and the phase φ * of the steady-state vibration, i.e., the fixed points of the slow flow equations (19,20), we set the left-hand sides of Eqs. (19,20) to zero, and multiply the expressions by ε, to get back the original ζ, p, q, A, ω system parameters [see Eq. (9)] We substitute σ = (ω − 1) and solve the resulting algebraic equations (by eliminating the trigonometric terms from them). From these for the amplitude α * and for the phase φ * , we get Note that Eq. (27) is an implicit function of the equilibrium amplitude α * . The shape of the equilibrium amplitude α * as a function of ω is illustrated in Fig. 3 for different A values.

Stability of the equilibria of the slow flow
We examine the stability of the equilibrium points (α * , φ * ) of the slow flow Eqs. (19) and (20). Stability is determined by the eigenvalues where J is the Jacobian Using Eqs. (25) and (26) we eliminate cos φ * and sin φ * from the Jacobian (30) to yield where To determine the bifurcation points, the trace and the determinant of the Jacobian J will be useful where We note that tr J is a linear function and det J is a second-order polynomial of α * 2 .

Saddle-node bifurcation
A saddle-node bifurcation of an equilibria occurs when Equation (27) implicitly determines equilibria and can be written as a third-order polynomial of α * 2 Eliminating α * 2 from the simultaneous Eqs. (36) and (37) yields Provided 16 3 Equation (39) determines the two curves on the ω − A plane where saddle-node bifurcation occurs (Fig. 4).

Hopf bifurcation
Equation (37) implicitly determines equilibria. A Hopf bifurcation of equilibrium point can occur if tr J = 0 and det J > 0 (necessary conditions). To get the sufficient condition of the Hopf bifurcation, the transversality (positive root crossing velocity) and genericity (equilibrium is weakly attracting/repelling) conditions have to be fulfilled [54].
We start with From Eq. (40) we express and substitute it into Eq. (37) to get an implicit function where We eliminate α * 2 from the det J > 0 condition by substituting Eq. (41) into Eq. (34) to get the inequality as the function of ω Equation (

Numerical results of the bifurcation analysis
The angular frequency ω of the forcing was chosen as a bifurcation parameter. As in [18,55,56] we set the other parameter values at the first lobe ( j = 1, see Fig. 2) and weak nonlinearity and forcing τ = 4.676, ζ = 0.01, p = 0.5 p crit = 0.0101, The equilibrium amplitude α * as a function of ω is illustrated in Fig. 5 (this is the uppermost curve of Fig. 3). Using Eq. (39) saddle-node bifurcations occur at While at ω SN 1 a regular saddle-node bifurcation occurs, at ω SN 2 a so-called homoclinic saddle-node bifurcation [57] can be observed. At this point a limit cycle is born (see Fig. 8f, g). Using Eqs. (42) and inequality (44), Hopf bifurcation occurs at At this point the stable equilibrium point becomes unstable and a subcritical Hopf bifurcation occurs. The bifurcation diagram is shown in Fig. 6.

Phase plane and phase portraits
To illustrate the dynamical behavior of the system, we chose the following angular frequencies of the forcing The right-hand sides of Eqs. (19) and (20) are periodic functions of the phase φ with period 2π , thus (α, φ) ∈ (R 1 , S 1 ).The true phase space is a cylindrical surface, see Fig. 7. Figure 8 shows the phase portrait of the slow flow at various forcing frequencies. The filled circles denote the stable, the empty circles the unstable equilibria and the dashed lines correspond to the unstable limit cycles.
At ω = ω I the equilibrium (filled circle) is a stable spiral and the thick dashed line is a 2π -periodic unstable limit cycle (Fig. 8a). At ω = ω I I a pair of complex conjugate eigenvalues cross the imaginary axis. At this point the fixed point is weakly repelling, giving rise to a subcritical Hopf bifurcation (Fig. 8b). After the Hopf bifurcation point (ω = ω I I I ) the stable equilibrium becomes unstable (Fig. 8c) and all solutions go off to infinity. Further increasing the bifurcation parameter ω the unstable spiral equilibrium (empty circle) becomes an unstable node. In Fig. 8d at ω = ω I V the left dot (α * = 1.72) is the unstable node, the right dot (α * = 0.93) is a non-hyperbolic fixed point undergoing saddle-node bifurcation. After the saddle- node bifurcation a stable node and a saddle point is created (Fig. 8e at ω = ω V ). The stable node transforms into a stable spiral at ω = 1.0119 > ω V and the saddle point moves toward the unstable node. At ω = ω V I the unstable node and saddle point coalesce and a saddle-node homoclinic bifurcation (global bifurcation) occurs (Fig. 8f). After the saddle-node homoclinic bifurcation (ω = ω V I I ) an unstable limit cycle (thick arrowless line) is created (see Fig. 8g).
Another global bifurcation is a homoclinic saddlenode bifurcation [57]. This occurs at ω = ω V I . The coalescence of the saddle and unstable node is a homoclinic bifurcation and a "global" unstable limit cycle is born (see Fig. 10).

The ω − A plane
Now we consider the forcing frequency ω and amplitude A as two bifurcation parameters. The other system  Fig. 8 continued parameters were the same as in Eq. (45). We determined the saddle-node curves together with the cusp and Bogdanov-Takens points with MatCont [58][59][60][61] (see Fig. 11). The boundary of the gray region is the same as the analytical result (39) (see also Fig. 4).
Inside the closed wedge (filled with gray in Fig. 11) three equilibrium points of the slow flow exist, two on the boundary and one outside the wedge. At the corners of the wedge we have three cusp (CP) bifurcation points. Equations (42) and (44) determine two Bogdanov-Takens (BT) points [54]. Figure 12 shows the Hopf bifurcation curve determined by Eq. (42).  [54]. This bifurcation point separates branches of subcritical and supercritical Hopf bifurcations (left segment until GH and GH-BT segment in Fig. 12). The segment BT-BT corresponds to neutral saddle points. In a neighborhood of the GH point, fold (saddle-node) bifurcation of limit cycles occurs. To illustrate this, we chose 4 points (denoted by squared in Fig. 13), one before and three after the Generalized Hopf bifurcation point.
To trace the limit cycles we used MatCont, with the bifurcation parameter ω. At the first point A = 0.0075 the Hopf bifurcation is subcritical (Fig. 14a). After the Generalized Hopf bifurcation point the Hopf bifurcation becomes supercritical. At A = 0.0070 and A = 0.0065 the system has two limit cycles which collide and disappear in a fold bifurcation (Fig. 14b, c).
The unstable limit cycles of the subcritical Hopf bifurcations of the slow flow equations (19,20) are unstable quasi-periodic solutions of Eq. (1). Figure 17 where (52) Figure 18 shows the amplitude α * (Eq. 27) as a function of the forcing frequency ω. The solid line indicates the stable multiple scales solution (MMS Stable); the dashed line shows the unstable multiple scales solution (MMS Unstable). The stable numerical equilibrium solutions are shown with filled circles; unstable numerical solution are depicted by empty circles. The solution was deemed unstable if its magnitude grew beyond a large number. The amplitude of the unstable limit cycles is depicted by empty squares (these amplitudes were determined using MatCont and bisection method). The thin vertical lines indicate the ω i , i ∈ {I, I I, . . . , V I I } angular frequencies. The red markers correspond to the subcritical Hopf (Hopf), homoclinic saddle-node (Homoclinic S-N), global bifurcation of limit cycles (Global) and saddle-node (Saddle-Node) bifurcation points.

Conclusions
Equation (1) without forcing admits a subcritical Hopf bifurcation [18]. Forcing and Hopf bifurcation together may yield complicated motions [62]. Plaut and Hsieh [63] studied a forced 1-DOF mechanical system with delay and found periodic, chaotic and unbounded responses. Daqaq et al. investigated a harmonically forced delay system and got similar primary resonance curve [23]. The Taylor expansion of forced delay equation of machine tool vibration was investigated in [10].
Here we utilized the method of multiple scales to derive the slow flow equations for the forced delay equation Eq. (1). Equlibria of the slow flow were given by implicit algebraic equations. Analytical expressions were derived for the saddle-node and Hopf bifurcations of the slow flow equations. Bifurcation analysis of the slow flow equations has been shown for some numerical values of the parameters. Sub-and supercritical Hopf, generalized Hopf (Bautin), saddle-node, homoclinic saddle-node bifurcations were found. The bifurcations were illustrated with phase portraits and direct numerical integration of the original equation. Using MatCont, we determined the Generalized Hopf point of the system and located the local limit cycles. The analysis presented here demonstrates the rich dynamics of the system.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http:// creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.