We study the dynamics of targeted energy transfers in suppressing chatter instability in a single-degree-of-freedom (SDOF) machine tool system. The nonlinear regenerative (time-delayed) cutting force is a main source of machine tool vibrations (chatter). We introduce an ungrounded nonlinear energy sink (NES) coupled to the tool, by which energy transfers from the tool to the NES and efficient dissipation can be realized during chatter. Studying variations of a transition curve with respect to the NES parameters, we analytically show that the location of the Hopf bifurcation point is influenced only by the NES mass and damping coefficient. We demonstrate that application of a well-designed NES renders the subcritical limit cycle oscillations (LCOs) into supercritical ones, followed by Neimark–Sacker and saddle-node bifurcations, which help to increase the stability margin in machining. Numerical and asymptotic bifurcation analyses are performed and three suppression mechanisms are identified. The asymptotic stability analysis is performed to study the domains of attraction for these suppression mechanisms which exhibit good agreement with the bifurcations sets obtained from the numerical continuation methods. The results will help to design nonlinear energy sinks for passive control of regenerative instabilities in machining.

## Introduction

In machining processes, the undesired vibration of the tool relative to the workpiece gives rise to a low-quality product. One of the most important causes of the instability in the cutting process is the so-called regenerative effect, which arises from the fact that the cutting force exerted on a tool is influenced not only by the current tool position but also by that in the previous revolution. Hence, the equation of motion for the tool is a delay differential equation, which renders even a single-degree-of-freedom (SDOF) dynamical system to be infinite-dimensional. In practical machining processes, regenerative limit cycle oscillations (LCOs) create adverse effects on machining quality, and many papers deal with the stability and bifurcations of machining processes, as well as with various passive and active means to improve their stability (see, for example, Refs. [18]).

Direct use or variations of linear/nonlinear-tuned mass dampers (TMDs, [7,8]) are probably the most popular approach to passive chatter suppression. However, even if a TMD is initially designed (tuned) to eliminate resonant response near the eigenfrequency of a primary system, the mitigating performance may become less effective over time due to aging of the system, temperature, or humidity variations, thus, requiring additional adjustment or tuning of parameters. It is only recently that passively controlled spatial/dynamic transfers of vibrational energy in coupled oscillators to a targeted point where the energy eventually localizes were studied by utilizing a nonlinear energy sink (NES, see Ref. [9] for the summary of developments); and this phenomenon is simply called targeted energy transfer (TET). The NES is basically a device that interacts with a primary structure over broad frequency bands; indeed, since the NES possesses essential stiffness nonlinearity (no linear stiffness term), it may engage in (transient) resonance capture [10] with any mode of the primary system. This is why an NES, more effective than a TMD [11,12], can be designed to extract broadband vibration energy from a primary system, engaging in transient resonance with a set of “most energetic” modes [13]. In Particular, Lee et al. [1416] applied an ungrounded NES to an aero-elastic system, and numerically and experimentally demonstrated that a well-designed NES can even completely eliminate aero-elastic instability. Three suppression mechanisms were identified; that is, recurrent burstouts and suppressions, intermediate and complete elimination of self-excited instability in the aero-elastic system. Such mechanisms were investigated by means of bifurcation analysis and complexification-averaging (CxA) technique [17].

Kalmár-Nagy et al. [3] analytically proved, by means of the center manifold theorem [18], the existence of subcritical Hopf bifurcation in an SDOF machine tool model with the regenerative cutting force. Furthermore, practical stability limit in turning process was investigated by considering contact loss issues in the regenerative cutting force [4], which can predict stable, steady-state periodic tool vibrations (or limit cycle oscillations—LCOs). Nankali et al. [19,20], proposed application of the NES system to a time-delayed machine tool system. They utilized numerical techniques to obtain basin of attraction for different TET mechanisms. Recently, Gourc et al. [21] investigated the different response regimes of a cutting tool on a lathe strongly coupled to a nonlinear energy sink. They derived the equation of the slow invariant manifold (SIM) and explained the behavior of the system by studying the location of the fixed points of the slow flow on this manifold. Moreover, Gourc et al. [22] have studied the passive control of chatter instability in turning processes using a vibro-impact nonlinear energy sink.

In this work, we present a comprehensive study of TET mechanisms in suppressing regenerative chatter instability in a turning process. For this purpose, we first review nonlinear dynamics of a SDOF machine tool model; then, perform a linear stability analysis to explore the effects of NES parameters on the occurrence of Hopf bifurcation (i.e., stability boundary on the plane of cutting depth and rotational speed of a workpiece). Furthermore, the numerical [23] and asymptotic [24] techniques are applied to properly understand the suppression mechanisms that appear by the NES. Moreover, applications of an NES to a practical machine tool model are introduced.

## SDOF Machine Tool Dynamics: Summary

Kalmár-Nagy et al. [3] studied a nonlinear dynamics of a single-degree-of-freedom machine tool model depicted in Fig. 1. The equation of motion can be expressed by a delay-differential equation with nonlinear regenerative terms retained up to cubic order

$ẍ+2ζωnẋ+ωn2x=−ω02(x−xτ)+ω028f0[(x−xτ)2−512f0(x−xτ)3]$
(1)

where $xτ≜x(t−τ)$ is the delayed variable with a time delay $τ=2π/Ω$; Ω is the rotating speed of the workpiece; $ωn=k/m$ is the (linearized) natural frequency; $ω0=k1/m$, where k1 is the cutting force coefficient; ζ is the damping factor of the machine tool; and f0 is the chip thickness at the steady-state cutting.

Introducing the following scaling transformations
(2)
we rewrite the equation of motion as
$ẍ+2ζẋ+x=−p(x−xτ)+3p10[(x−xτ)2−(x−xτ)3]$
(3)

where the differentiation is now with respect to the new time variable. Note that the parameter $p=(ω0/ωn)2=(k1/k)2$ indicate the effect of stiffness-hardening due to machining conditions.

Kalmár-Nagy et al. [3] calculated the transition curves (cf. Fig. 2(a) for $ζ=0.1$) as a function parameterized by the eigenfrequency ω, , through which the steady-state cutting loses stability through a Hopf bifurcation. Since the periodic solution that is born from the equilibrium point is unstable, this bifurcation is a subcritical one. For the damping factor $ζ=0.1$ (which will be assumed throughout this work), the stiffness ratio p has the minimum value, $pmin=2ζ(1+ζ)=0.22$, and the minimum eigenfrequency of the limit cycle oscillation (LCO) is $ωmin=1.0$ such that $p(ω)>0$ for $ω>ωmin$. Moreover, Fig. 2(b) depicts the minimum and maximum eigenfrequencies (denoted by $ωmin(n)$ and $ωmax(n)$, respectively) that the $nth$ lobe yields. Whereas, the first two or three lobes exhibit relatively broad distribution of eigenfrequencies, the rest possess harmonic components concentrated near $ω=1.1$. Note that these eigenfrequencies act as “seed” harmonic bases for the new-born periodic motion, and the triggering of a machine tool chatter appears as a result of competition between the eigenfrequency and the rotational speed of the workpiece.

## Nonlinear Energy Sink

We apply an ungrounded nonlinear energy sink (NES) to the SDOF machine tool model, as the mathematical model is depicted in Fig. 1(b). The equations of motion can be written as
$ẍ+2ζẋ+2ζ1ε(ẋ−ẏ)+x+C(x−y)3=pΔx+p310(Δx2+Δx3)εÿ+2ζ1ε(ẏ−ẋ)+C(y−x)3=0$
(4)
where rescaling similar to Eq. (2) is incorporated; that is, we have $ϵ=ms/m$ (the mass ratio), $ζ1=cs/(2mωn)$ (the damping factor of the NES), $C=ks/k$ (the stiffness ratio), $y↦5y/(12f0)$, and $t↦ωnt$. Moreover, $Δx=xτ−x$ in which $xτ=x(t−τ)$ is the tool position one revolution ago. We rewrite Eq. (4) in vector form as
$x.=Ax+Rxτ+f(x,xτ)$
(5)
where $x=(x1,x2,x3,x4)T$,
(6)

and $f=(0,0,−C(x1−x2)3+(3p/10)[(x1−x1τ)2−(x1−x1τ)3],−C(x2−x1)3/ϵ)T$.

Assuming and substituting the solution of Eq. (5) to be $x(t)=exp(λt)X$, then we obtain the eigenvalue problem typical for a delay-differential system
$(λI−A−Re−λτ)X=0$
(7)
where I is an identity matrix. For a nontrivial eigenvector X, we derive the characteristic equation as $det(λI−A−Re−λτ)=0$. That is, we write
$λ[λ3+2(ζ+ζ1+ζ1/ϵ)λ2+(1+p+4ζζ1/ϵ)λ+2ζ1(1+p)/ϵ−p(λ+2ζ1/ϵ)e−λτ]=0$
(8)
We remark that one of the eigenvalues is always zero; that is, Eq. (7) is degenerative and bifurcation analysis of the trivial equilibrium should be at least of co-dimension 2. For $λ≠0$, we seek the parameter conditions that yield two eigenvalues of purely imaginary, complex conjugates. Substitution of $λ=jω$, where $j2=−1$, into Eq. (8) and separation of real and imaginary parts yield
$−2(ζ+ζ1+ζ1/ϵ)ω2+2ζ1(1+p)/ϵ=(2ζ1p/ϵ)cos ωτ+pω sin ωτ−ω3+(1+p+4ζζ1/ϵ)ω=pω cos ωτ−(2ζ1p/ϵ)sin ωτ$
(9)
By squaring and summing both sides of the two equations in Eq. (9) we obtain
$p(ω)=G(ω)/F(ω)$
(10)
where $G(ω)=ω6−2(1−2ζ12−2ζ12/ϵ2−4ζ12/ϵ−4ζζ1−2ζ2)ω4+(1−8ζ12/ϵ2−8ζ12/ϵ+16ζ2ζ12/ϵ2)ω2+4ζ12/ϵ2$ and $F(ω)=2ω4−2(1−4ζ12/ϵ2−4ζ12/ϵ)ω2−8ζ12/ϵ2$. Also, noting that $1−cos ωτ=2 sin2(ωτ/2)$ and $sin ωτ=2 sin(ωτ/2)cos(ωτ/2)$, we rearrange Eq. (9) as
$−2(ζ+ζ1+ζ1/ϵ)ω2+2ζ1/ϵ=−(2ζ1p/ϵ)(1−cos ωτ)+pω sin ωτ=2pR sin(ωτ/2)cos(ωτ/2+ϕ)−ω3+(1+4ζζ1/ϵ)ω=−pω(1−cos ωτ)−(2ζ1p/ϵ)sin ωτ=−2pR sin(ωτ/2)sin(ωτ/2+ϕ)$
(11)
where $R=(2ζ1/ϵ)2+ω2$ and $ϕ=tan−1[2ζ1/(ϵω)]$. Then, we compute
$tan(ωτ2+ϕ)=ω3−(1+4ζζ1/ϵ)ω−2(ζ+ζ1+ζ1/ϵ)ω2+2ζ1/ϵ≜K(ω)$
(12)
Since $τ=2π/Ω$, the rotational speed Ω of the workpiece can be derived as
$Ω(ω)=πωnπ+tan−1K(ω)−ϕ$
(13)

where n is the order of the lobe in the stability chart, and $K(ω)$ and $ϕ$ are defined in Eqs. (11) and (12).

Before examining changes in the transition curves by adding the NES, we remark that the minimum eigenfrequency $ωmin$ to yield positive $p(ω)$ becomes smaller than the unity. One can easily show that ; therefore, $ωmin$ should be the critical eigenfrequency that renders $F(ω)>0$ for $ω>ωmin$. Since $F(ω)=0$ is a quadratic equation, $ωmin$ can be calculated as
$ωmin=12(b+b2−4c)$
(14)

where $b=1−4ζ12/ϵ2−4ζ12/ϵ$ and $c=−4ζ12/ϵ2$. It can be analytically shown that $ωmin=1$ in the limit of $ϵ→0$ or $ζ1→0$.

Figure 3 depicts the minimum eigenfrequency $ωmin$ with respect to the mass ratio ϵ and the damping factor ζ1 of the NES. As ϵ and ζ1 increase, $ωmin$ tends to decrease; however, it remains near unity but decreasing for small mass ratios (e.g., $ϵ<0.1$) and damping factors (e.g., $ζ1<0.1$), which is usually the case in practice. For fixed mass ratios, $ωmin$ decreases as ζ1 increases; and after a while the eigenfrequency seem to converge a constant (top of Fig. 3(b)). On the other hand, for fixed damping factors, the variation of $ωmin$ with respect to a mass ratio differs from the order of the damping factor (bottom of Fig. 3(b)). That is, for small ζ1, there exist a minimum of $ωmin$ and then it increases; but for larger ζ1 it decreases monotonically. This observation suggests not only that the frequency of the subcritical LCO when no NES is applied should be higher than unity, but also that the LCO with a frequency altered by an NES can be smaller than unity. Note that the linearized natural frequency in the rescaled nondimensional equation (4) is unity.

Now, we compute the transition curves based on Eqs. (10) and (13). Figure 4 depicts the changes of the stability boundary in (Ω, p)-plane by varying the mass ratio ϵ and fixing the other two NES parameters. The stability enhancement due to the application of an NES can be measured by directly calculating the point-wise shift amount as $Δp=(p′−p)/p×100$ (%), where p and $p′$ denote the values at the stability boundary with respect to each Ω without and with an NES, respectively. Upward shift of the stability boundary occurs more significantly near the valley than near the cusp points of the lobes, which will be useful in practical applications of chatter suppression. The shifting amount of the transition curve does not appear to be significant with a small NES mass (about 5% improvement near valley of the lobes); however, the upward shift becomes increasing monotonically as the mass ratio increases. The ranges of the eigenfrequencies at the transition curves tend to become lower as the mass ratio increases; and above certain mass ratio the eigenfrequency intervals are shifted upward (cf. Fig. 4(b) when $ϵ=0.6$). Note that the bifurcation occurring on the transition curve can be referred to as a degenerate Hopf bifurcation in delay-differential equations, because one of the eigenvalues is always zero.

We remark that, although we can delay the occurrence of Hopf bifurcations by adding NES, this is not all one can achieve with an NES as a passive broadband vibration controller. The application of an NES can also alter the topology of local bifurcations such that it produces Neimark–Sacker and saddle-node bifurcations as well as Hopf bifurcation. The former two bifurcations are essential in discussing targeted energy transfer (TET) mechanisms in suppressing any types of instabilities introduced in a dynamical system. We deal with this in more detail in Sec. 4 by utilizing a numerical continuation technique for a system of delay-differential equations.

## Bifurcation Analysis and Robustness

### TET Mechanisms.

As in the previous aero-elastic applications [14], three distinct TET mechanisms are identified in suppressing regenerative chatter instability; that is, recurrent burstouts and suppression, intermediate and complete elimination of regenerative instability (cf. see Fig. 5 for typical time history for each suppression mechanism).

The first suppression mechanism is characterized by a recurrent series of suppressed burstouts of the tool response, followed by eventual complete suppression of the regenerative instabilities. The beating-like (quasiperiodic) modal interactions observed during the recurrent burstouts turn out to be associated with Neimark–Sacker bifurcations of a periodic solution (cf. Fig. 6) and is critical for determining domains of robust suppression [16]. To investigate this mechanism in more detail, Fig. 7 depicts the displacements of both the tool and NES and their wavelet transforms. Energy exchanges between the two modes are evidenced in Fig. 7, through which a series of 1:1 transient resonance captures and escapes from resonance occurs (see Fig. 9(b).

The second suppression mechanism is characterized by intermediate suppression of LCOs, and is commonly observed when partial LCO suppression occurs. The initial action of the NES is the same as in the first suppression mechanism. Targeted energy transfer to the NES then follows under conditions of 1:1 transient resonance capture, followed by conditions of 1:1 permanent resonance capture where the tool mode attains constant (but nonzero) steady-state amplitudes. We note that, in contrast to the first suppression mechanism, the action of the NES is nonrecurring in this case, as it acts at the early phase of the motion stabilizing the tool and suppressing the LCO.

Finally, in the third suppression mechanism, energy transfers from the tool to the NES are caused by nonlinear modal interactions during 1:1 RCs. Both tool mode and the NES exhibit exponentially decaying responses resulting in complete elimination of LCOs.

### A Practical Tool Model With Contact Loss.

We note that the numerical and analytical studies for TET mechanisms above are valid only for vibrations with small amplitudes; in particular, the permanent contact model with truncated nonlinear terms cannot predict any stable steady-state periodic vibrations of high amplitudes. That is, the truncated nonlinearity in the regenerative cutting force will not predict the existence of a saddle-node bifurcation point right after contact loss occurs. The details of machine tool dynamics can be found in Kalmár-Nagy [4], where stable periodic motions are predicted. Performing numerical continuation analysis, we obtain the LCO surfaces for without and with NES being applied (Fig. 8). Still three distinct TET mechanisms for the model with contact loss are observed.

### Complexification-Averaging (CxA) Technique.

In order to study the underlying TET mechanisms, we employ the CxA method first introduced by Manevitch [17]. We introduce the new complex variables in the following:
$ψ1(t)=ẋ(t)+jωx(t)≡φ1(t)ejωtψ2(t)=ẏ(t)+jωy(t)≡φ2(t)ejωt$
(15)
where $j2=−1$. Then, denoting by $()∗$ the complex conjugate, we can express the original real variables in terms of the new complex ones
$x(t)=12jω(ψ1−ψ1∗)=12jω(φ1ejωt−φ1∗e−jωt)x(t−τ)=12jω(ψ1(t−τ)−ψ1∗(t−τ))=12jω(φ1(t−τ)ejω(t−τ)−φ1∗(t−τ)e−jω(t−τ))ẋ(t)=12(ψ1+ψ1∗)=12(φ1ejωt+φ1∗e−jωt)ẍ(t)=(φ̇1+jωφ1)ejωt−jω2(φ1ejωt+φ1∗e−jωt)$
(16)
and similar expressions can be obtained for the NES variables. Substituting into the equations of motion and averaging out the fast dynamics over $ejωt$, we obtain a set of two complex-valued modulation equations governing the slow-flow dynamics
$φ̇=F(φ,φτ)$
(17)
where $φ={φ1,φ2}T$. Expressing the slow-flow amplitudes in polar form, $φk(t)=ak(t)ejβk(t)$, where , we obtain the set of real-valued slow-flow equations such that
(18)

where $ϕ≡β1−β2$.

Finally, this set of first-order delay differential equations are solved to find $a1,a2,β1$, and β2 to yield
$x=a1ωsin(β1+ωt)$
(19)

Figure 9(a) directly compares the approximate (Eq. (19)) and exact (Eq. (4)) solutions for the tool displacement, which demonstrates a good agreement; furthermore, the nontime-like patterns (i.e., spirals) of the phase difference $ϕ$ depicts that the underlying TET mechanism for the first suppression mechanism involves a series of 1:1 transient resonance captures and escapes from resonance. Also, Fig. 9(b) depicts 1:1 resonance capture in the slow-flow phase plane ($β,β̇$) in which $β=β1−β2$.

## Analytical Study: Basins of Attraction for Suppression Mechanisms

This section includes analytical study of TET mechanisms for a SDOF machine tool. The asymptotic analysis [24,25] is applied in order to estimate domains of attraction in the parameter space for three suppression mechanisms. First step of this analysis is to rescale the equations of motion and remove nonlinearities due to regenerative forces. So that, nonlinearities of the system would correspond only to the NES. Then, we identify NES nonlinearity modal interaction with the tool. Similar to Ref. [24], we can eliminate the terms related to structural nonlinearities using following scaling:
$x→4ε3Cx, y→4ε3Cy, Δx→4ε3CΔx$
(20)
Moreover, assuming strong NES nonlinearity, we introduce $C→C/ε3$. So, Eq. (4) can be expressed as
$ẍ+2ζẋ+2ζ1ε(ẋ−ẏ)+x+43ε(x−y)3=pΔx−Pδ(4Δx2ε23C+4Δx3ε43C)εÿ+2ζ1ε(ẏ−ẋ)+43ε(y−x)3=0$
(21)
In an approximation, we can omit terms including $ϵ2$ and $ϵ3$ because their effects on the dynamics are negligible. Indeed, we eliminate nonlinearities due to regenerative forces
$ẍ+2ζẋ+2ζ1ε(ẋ−ẏ)+x+43ε(x−y)3=p(xτ−x)εÿ+2ζ1ε(ẏ−ẋ)+43ε(y−x)3=0$
(22)

We note that through this approximation only external nonlinearities due to the NES are left in the system. Further, the NES stiffness C is dropped out from the equations which indicates indecency of the suppression mechanisms from this parameter.

Utilizing the complexification-averaging method introduced in Sec. 4.3, steady-state periodic response can be analyzed. To this end, we introduce the coordinate transformation
$v=x+εyw=x−y$
(23)
Here, v and w are the physical quantities for the center of mass (with a factor of $1+ε$) and the relative displacement, respectively. Then, the equations of motion (22) become
$v̈+2ζ(v̇+εẇ1+ε)+v+εw1+ε=p[vτ+εwτ1+ε−v+εw1+ε]ẅ+2ζ(v̇+εẇ1+ε)+2ζ1(1+ε)ẇ+v+εw1+ε+43(1+ε)w3=p[vτ+εwτ1+ε−v+εw1+ε]$
(24)
Numerical simulations (Fig. 5) reveal that the steady-state responses are in the form of a fast oscillation (fast dynamics) modulated by a slowly varying envelope (slow dynamics); the dynamics can be expressed in the form of $ϕn(t)eiwt;n=1,2$ where $i2=−1$. Now, we separate fast and slow dynamics by introducing
$ϕ1eiωt=v̇+iωvϕ2eiwt=ẇ+iωw$
(25)
Substituting into Eq. (24) and performing averaging over the fast component eiwt, we obtain the slow-flow equation
$ϕ̇1=ϕ1[−iω2−ζ1+ε+(1+p)i(1+ε)2ω]+ϕ2[−εζ1+ε+ε(1+p)i(1+ε)2ω]−pi(1+ε)2ωe−iωτϕ1τ−εpi(1+ε)2ωe−iωτϕ2τϕ̇2=ϕ1[−ζ1+ε+(1+p)i(1+ε)2ω]+ϕ2[−iω2−ζε1+ε−ζ1(1+ε)+ε(1+p)i(1+ε)2ω+(1+ε)i2ω3|ϕ2|2]−pi(1+ε)2ωe−iωτ(ϕ1τ+εϕ2τ)$
(26)
Here, $ϕnτ=ϕn(t−τ) ;n=1,2$ and $ϕn∗$ is the complex conjugate of $ϕn$. Since the delay term is finite ($τ=(2π/Ω)$), we can simplify the system by considering $ϕ1τ=ϕ1;ϕ2τ=ϕ2$ as $t→∞$. Moreover, for the sake of convenience in further mathematical manipulations, we define the following variables:
$a1=p2ω(1+ε)cos(ωτ), b=p+12ω(1+ε)a2=p2ω(1+ε)sin(ωτ), h=ζ(1+ε)L1=1/ε(−h−a2), S1=−h−a2L2=1/ε(−ω2+b−a1), S2=(b−a1)$
(27)
Substituting these variables into Eq. (26) and neglecting small terms, we can write:
$ϕ̇1=εLϕ1−εSϕ2ϕ̇2=−Sϕ1+ϕ2[−iω2−ζ1+i2ω3|ϕ2|2]$
(28)
Here, $L=L1+L2i$ and $S=S1+S2i$. After a rescaling, $ϕ1→ϕ1(ω/2iS)$ and introducing a polar form, $ϕ1=Reiδ1;ϕ2=Feiδ2$, we can derive the real-valued slow-flow dynamics
$Ṙ=ε(RL1−2Fω((S12−S22)sin δ−2S1S2 cos δ))Ḟ=−ω2R sin δ−ζ1Fδ̇=ε(L2−2FωR(S12−S22)cos δ−4FS1S2ωRsin δ)−Rω2Fcos δ−F22ω3+ω2$
(29)
where R and F represent real amplitude modulations while $δ=δ=δ1−δ2$ is the phase difference. Equation (29) represents the slow dynamics of the original system (Eq. (21)). The slow dynamics is analyzed using the method of multiple scales. Based on this method, we consider different scales of time by defining $t=εnτn$, where $n=0,1,2,…$ and consequently, $(∂/∂t)=(∂/∂τ0)+ε(∂/∂τ1)+⋯$. Moreover, we introduce the perturbation-series of the variables as
$R(τ)=R0(τ0,τ1)+εR1(τ0,τ1)+⋯F(τ)=F0(τ0,τ1)+εF1(τ0,τ1)+⋯δ(τ)=δ0(τ0,τ1)+εδ1(τ0,τ1)+⋯$
(30)
Plugging Eq. (30) into Eq. (29) and matching coefficients of powers of ε, we derive subproblems governing the solution of slow dynamics. In this study, we calculate the responses for the “slow” time scale (τ0) and “super-slow” time scale (τ1). The first-order approximation is computed by considering $ε0$ subproblem
$∂R0∂τ0=0∂F0∂τ0=−ω2R0 sin(δ0)−ζ1F0∂δ0∂τ0=−R0ω2F0cos(δ0)−F022ω3+ω2$
(31)
Note that R0 is fixed with respect to slow time variable τ0, but not with respect to slower time variable τ1. No limit cycle oscillations are possible for system (31) and the only steady-state solutions are in the form of equilibrium points. Equilibrium points of the slow dynamics with respect to slow time (τ0) can be calculated as
$R̂02=F̂20{4(ζ1ω)2+([1−F̂02ω4])2}$
(32)
This equation defines a slow invariant manifold (SIM) on the plane ($R02−F02$), where $R̂0(τ1)$ and $F̂0(τ1)$ refer to the equilibrium points of the slow dynamics with respect to slow time. Depending on the values for ζ and ω, there are either one or three branches, when plotting $R̂02$ versus $F̂02$. For the case of three branches (Figs. 1012), performing linearized stability analysis on the slow dynamics with respect to slow time, reveals that the middle branch is unstable while the other two are stable. Equilibrium points of stable and unstable branches are in the form of node (stable) and saddle node, respectively. Therefore, at the leading-order approximation the dynamics will be attracted to either of stable nodes.
There is no attractor in the form of LCO for the first-order approximation of slow dynamics. So, subproblems governing higher order of ε should be considered to compute possible LCO's for the slow flow. To this end, we plug Eq. (30) into Eq. (31) and match coefficients of $ε1$ for the first equation to get
$ε:∂∂τ1R0+∂∂τ0R1=R0L1−2F0ω{(S12−S22)sin δ0−2S1S2 cos δ0}$
(33)
The other two equations of $ε1$ subproblem introduce small corrections to the shape of SIM. Similar to SIM, we apply equilibrium condition with respect to slow time ($(∂/∂τ0)=0$) into Eq. (33)
$∂∂τ1R̂0=R̂0L1−2F̂0ω{(S12−S22)sin δ̂0−2S1S2 cos δ̂0}$
(34)
This equation is satisfied for points on the SIM. Substituting $sin δ̂0$ and $cos δ̂0$ from SIM (Eq. (31)) and setting derivative of $R̂0$ equal to zero, we can find equilibrium points of the slow dynamics with respect to super-slow time (τ1). This is called super-slow flow (SSF). The intersection points of the SIM and SSF can be calculated by plugging $(∂R̂0/∂τ1)$ from Eq. (32) into Eq. (34) and solving for $(∂F̂20/∂τ1)≡f(F̂02)$
$F̂20e1=0F̂20e2=ω3(2S1S2+L1ω−2(S22−L1ξ1)(S12+L1ξ1))L1F̂20e3=ω3(2S1S2+L1ω+2(S22−L1ξ1)(S12+L1ξ1))L1$
(35)
These intersection points are depicted by letters A, B, and C in Figs. 1013. Stability of these points (by computing $f′(F̂20e)$), determines type of suppression mechanism.

First suppression mechanism: This mechanism corresponds to the existence of stable LCO on slow dynamics. It refers to the condition in which stable equilibrium point of slow dynamics $(F̂0e22)$ lies on the unstable (middle) branch of SIM, while $F̂0e12$ is unstable. In fact, the attractor is in the form of LCO rather than equilibrium point for the slow dynamics. Cycle of relaxation oscillation of slow dynamics creates a quasi-periodic oscillation in full-order system which are illustrated in Fig. 10. We remark that none of the equilibrium points of slow dynamics are complex numbers, and the trivial equilibrium point is unstable for this mechanism.

Second suppression mechanism: This mechanism corresponds to stable LCO for full dynamics. According to the slow, fast dynamics separation (Eq. (25)), second suppression mechanism refers to existence of stable nontrivial equilibrium point of slow dynamics. As derived in Eq. (35), there are two nontrivial equilibrium points $(F̂0e22,F̂0e32)$ for the slow dynamics. It can be shown that $F̂0e22$ is stable when it lies on the stable branches of SIM which indicates second suppression mechanism. Note that none of the slow dynamics equilibrium points are complex, and only the second equilibrium point is stable for this mechanism. The SIM and super-slow flow corresponding to this mechanism are shown in Fig. 11, along with the time simulation of the full dynamics.

Third suppression mechanism: This mechanism corresponds to the stable trivial equilibrium point of slow dynamics. It can be shown that if $f′(F̂20e1)<0$, then origin is the only attractor of the slow dynamics and third suppression mechanism is predicted (Fig. 12).

No suppression: In addition to the three suppression mechanisms explained above, there is possibility of no suppression. This happens when trivial equilibrium point is the only real equilibrium point of the slow dynamics; e.g., other two equilibrium points are complex. In this case, if trivial equilibrium point is unstable $(f′(F̂e02)>0)$, no attractor exists for the slow dynamics which predicts instability of the original system. Super-slow flow and SIM for this case are shown in Fig. 13.

Table 1 summarizes mathematical conditions for the SIM and SSF configuration, corresponding to different cases discussed above. Also, basin of attraction for all suppression mechanisms, in $ζ1−p$ plane for given set of parameters, is depicted in Fig. 14.

Figure 15 compares the numerical and analytical calculated basin of attractions for TET mechanisms of SDOF machine tool. Dashed lines depict boundary of suppression mechanism computed numerically utilizing DDEBIFTOOL, while solid lines correspond to asymptotic analysis results. Numerical boundaries introduce Neimark–Sacker, saddle-node, and Hopf points for the first-, second-, and third-suppression mechanism, respectively. It is obvious that the numerical and analytical results are well matched. We note at this point that the accuracy of asymptotic stability analysis dependents on the value of mass ratio (ε). The smaller the mass ratio is the more accurate basin of attraction is computed. That is because we consider ε as the perturbation parameter in our study.

As it is plotted in Fig. 15, there is a type of bifurcation we could not detect through numerical study. It is not observed from analytical study for large values of ε either. This bifurcation which is called Shilnikov homoclinic bifurcation is predicted when strongly modulated response (SMR) disappears due to the existence of an unstable equilibrium point in an SMR cycle. In other words, it occurs when an unstable equilibrium point $(F̂20e3)$ meets a point of an SMR cycle $(F̂20u)$ while the second equilibrium point $(F̂20e2)$ is unstable. Figure 16 illustrates this situation.

Domains of attraction obtained from asymptotic analysis and associated points on the bifurcation surface are plotted in Fig. 17. A similar technique can be used to compute basins of attraction for suppression mechanisms in the $ε−p$ plane. Figure 18 illustrates the bifurcation surface computed numerically (DDEBIFTOOL), and the corresponding analytical basin of attraction in the $ε−p$ plane. As seen, for small values of ε there is a good agreement between the numerical and analytical results. However, larger values of ε decrease the accuracy of the asymptotic analysis. That is because ε is used as perturbation parameter and has to be small ($ε≪1$).

## Concluding Remarks

In this paper, we studied targeted energy transfer phenomena in suppressing chatter instability in a single-degree-of-freedom machine tool system, to which an ungrounded nonlinear energy sink is connected. Two models were considered for the tool dynamics: permanent contact model and contact loss model. The limit cycle oscillation due to the regenerative instability in a tool model which appeared as being subcritical for permanent contact model were (locally) eliminated or attenuated at a fixed rotational speed of a workpiece (i.e., a delay period) by TETs to the NES. It was shown that there should be an optimal value of damping for a fixed mass ratio to shift the stability boundary for stably cutting more material off by increasing chip thickness. Also, magnitude of NES nonlinear stiffness does not have any effect on stability boundary while increasing mass ratio improves stability. Three suppression mechanisms have been identified and each mechanism was investigated numerically by time histories of displacements, and wavelet transforms and instantaneous modal energy exchanges. Furthermore, we extend the CxA analysis to perform asymptotic analysis by introducing a reduced-order model and partitioning slow-fast dynamics. The resulting singular perturbation analysis yields parameter conditions and regions for the three suppression mechanisms, which exhibit good agreement with the bifurcations sets obtained from numerical continuation methods.

## Acknowledgment

This work was supported in part by National Science Foundation of United States, Grant Numbers CMMI-0928062 (YL) and CMMI-0846783 (TKN).

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