Etienne Gourc
email: etienne.gourc@gmail.com

Sébastien Seguy
email: sebastien.seguy@insa-toulouse.fr

Guilhem Michon
email: guilhem.michon@isae.fr

Alain Berlioz
email: alain.berlioz@univ-tlse3.fr

Brian P Mann
email: brian.mann@duke.edu

Quenching chatter instability in turning process with a vibro-impact nonlinear energy sink

Keywords: Chatter, instability, nonlinear energy sink, turning

Introduction

The surface quality of parts produced by machining operations is strongly affected by chatter. Chatter increases the surface roughness, tool wear and reduces the spindle lifespan. A possible model for this phenomenon is to consider the primary chatter. In this non resonant case, the coupling between two orthogonal cutting modes imply a pair of non-linear ordinary differential equations [START_REF] Wiercigroch | Chaotic vibration of a simple model of the machine tool-cutting process system[END_REF][START_REF] Warmiński | Approximate analytical solutions for primary chatter in the non-linear metal cutting model[END_REF]. Another modelization -improved and more used -is to consider the regenerative chatter. In this case, the instability is induced by the time-delay between two consecutive passages of the cutting teeth. Due to small disturbances, the lathe tool exhibit damped oscillations and the surface roughness is undulated. For consecutive workpiece revolutions, the chip thickness will be modulated.

!"#$ % "#&&!' (! )!0!$!)#(1! !2!"(% # 3!!$ 4)( !56&#$!' 37 83# #$' Fishwick [START_REF] Tobias | Theory of regenerative machine tool chatter[END_REF] and is mainly responsible for chatter instability. Since these works, many researchers have improved the knowledge by the well known stability lobe representation and its application to special cases [START_REF] Altintas | Analytical prediction of stability lobes in milling[END_REF][START_REF] Gourc | Chatter milling modeling of active magnetic bearing spindle in highspeed domain[END_REF][START_REF] Khasawneh | Investigation of period-doubling islands in milling with simultaneously engaged helical flutes[END_REF][START_REF] Patel | Uncharted islands of chatter instability in milling[END_REF][START_REF] Mann | Stability of up-milling and down-milling, part 2: experimental verification[END_REF][START_REF] Mousseigne | Predicting the dynamic behaviour of torus milling tools when climb milling using the stability lobes theory[END_REF]10]. A detailed nonlinear analysis of chatter instability on turning processes has been carried out in [START_REF] Nayfeh | Analysis of the cutting tool on a lathe[END_REF] using the method of multiple scales.

Various techniques for chatter mitigation have been investigated. These different strategies may be divided in two categories, namely active and passive control. Concerning active vibration control strategies, a variable spindle speed was used in milling process to disturb the time delay [START_REF] Seguy | Suppression of period doubling chatter in high-speed milling by spindle speed variation[END_REF]. The use of actively controlled piezzo-electric tools has been studied for turning processes [START_REF] Matsubara | Vibration suppression of boring bar by piezoelectric actuators and lr circuit[END_REF], also the use of a piezoelectric tool holder was investigated theoretically and experimentally in [START_REF] Brecher | Chatter suppression with an active workpiece holder[END_REF]. Passive vibration strategies have also been investigated. Tuned mass damper has been widely studied for machining operations. Specific tuning procedure was developed for the case of chatter [START_REF] Sims | Vibration absorbers for chatter suppression: a new analytical tuning methodology[END_REF]. A nonlinear tuned mass damper with coulomb friction was analyzed for the case of turning in [START_REF] Wang | Design and implementation of nonlinear tmd for chatter suppression: An application in turning processes[END_REF] and milling in [START_REF] Wang | Feasibility study of nonlinear tuned mass damper for machining chatter suppression[END_REF], while multiple tuned mass damper were applied in [START_REF] Yang | Optimization of multiple tuned mass dampers to suppress machine tool chatter[END_REF].

In the past decade, it has been demonstrated that the addition of a small mass, strongly non-linear oscillator, called a nonlinear energy sink (NES), may lead to targeted energy transfer. In this context, a NES is used to mitigate any unwanted disturbance introduced in a primary system by efficiently transferring and eliminating energy from the main system to the NES. In [START_REF] Vakakis | Energy pumping in nonlinear mechanical oscillators: part iiresonance capture[END_REF], it was demonstrated that the main phenomena allowing targeted energy transfer is based on the 1 : 1 resonance capture. When the primary system is subjected to harmonic forcing, the passive control acts through relaxation cycle, referred as strongly modulated response (SMR) [START_REF] Starosvetsky | Strongly modulated response in forced 2dof oscillatory system with essential mass and potential asymmetry[END_REF]. This behavior has been confirmed experimentally in [START_REF] Gourc | Experimental investigation and design optimization of targeted energy transfer under periodic forcing[END_REF]. All these studies dealt with NES with cubic stiffness, however, this type of NES may not be suitable for practical applications, mainly due to its size. To overcome this problem, different types of NES have been proposed. Lamarque and co-workers analyzed a NES with piecewise linear stiffness [START_REF] Lamarque | Targeted energy transfer in mechanical systems by means of non-smooth nonlinear energy sink[END_REF][START_REF] Savadkoohi | Vibratory energy exchange between a linear and a nonsmooth system in the presence of the gravity[END_REF]. Vibro-impact NES for seismic mitigation have been studied numerically in [START_REF] Nucera | Targeted energy transfers in vibro-impact oscillators for seismic mitigation[END_REF] and experimentally in [START_REF] Nucera | Application of broadband nonlinear targeted energy transfers for seismic mitigation of a shear frame: Experimental results[END_REF]. More recently, an analytical procedure based on the invariant manifold approach has been developed for a vibro-impact NES under impulse loading [START_REF] Gendelman | Analytic treatment of a system with a vibro-impact nonlinear energy sink[END_REF].

The aforementioned research on NES concern the passive control of linear system. The passive control of instability with NES is a growing interest. Reference [START_REF] Gendelman | Bifurcations of self-excitation regimes in a van der pol oscillator with a nonlinear energy sink[END_REF] investigated the mitigation of limit cycle oscillations in a Van der Pol oscillator using a NES with cubic stiffness. It was shown analytically that the system exhibits three control mechanisms, namely: suppression, stabilization and passive control through SMR. The case of a rigid airfoil in an air flow has been treated theoretically and experimentally in [START_REF] Lee | Suppression aeroelastic instability using broadband passive targeted energy transfers, part 1: Theory[END_REF][START_REF] Lee | Suppressing aeroelastic instability using broadband passive targeted energy transfers, part 2: experiments[END_REF]. A similar case has been analyzed by Vaurigaud [START_REF] Vaurigaud | Passive control of aeroelastic instability in a long span bridge model prone to coupled flutter using targeted energy transfer[END_REF], where a long bridge in which flutter may occur is considered. The use of a NES has also been investigated in [START_REF] Viguié | Using passive nonlinear targeted energy transfer to stabilize drill-string systems[END_REF] to passively control instabilities that may occur during drilling operations for oil and gas. The passive control of chatter instability, using a NES with cubic stiffness, has been explored in [START_REF] Gourc | Delayed dynamical system strongly coupled to a nonlinear energy sink: application to machining chatter[END_REF]. The same three mechanism, as those found for the Van der Pol oscillator [START_REF] Gendelman | Bifurcations of self-excitation regimes in a van der pol oscillator with a nonlinear energy sink[END_REF], have been evidenced. A refined model taking into account the loss of contact of the tool during high amplitude oscillations has been studied in [START_REF] Nankali | Suppression of machine tool chatter using non-b9TCPB CTCB@d S9T{ 9TE |BUGCCe9T@S UV Ft`hs}`hl YiDDW[END_REF].

The goal of the present paper is to investigate theoretically and experimentally the potential benefit of a vibro-impact NES to passively control chatter instability in a turning process. The system is composed of a flexible lathe tool on which a NES is embed-

m 1 k 1 c 1 q(t) h(t)
workpiece rotation p9@ABC DE FGHCIPQ9G BCRBCSCTQPQ9UT UV QHC QABT9T@ RBUGCSSW ded. The structure of the paper is as follows. The second section describes a model of a turning process. The linearized system is analyzed in section 3. Section 4 provides a theoretical analysis of the coupled system and a description of the response regimes. Section 5 describes of the experimental setup. The tuning of the vibro-impact NES and time history of the experimental measurements are also presented. Section 6 contains concluding remarks and discussion.

Modelization of turning process

The uncoupled system consists of a flexible lathe tool which is assumed to vibrate only in its first bending mode, perpendicular to the cutting direction. It has been shown that the bending mode in the direction parallel to the cutting direction play an insignificant role [START_REF] Nankali | Targeted energy transfer for suppressing regenerative instabilities in a 2-dof machine tool model[END_REF]. The workpiece is assumed to be rigid. A schematic representation of this system is given in Fig. 1 and the corresponding equation of motion is written as follows

m 1 d 2 q dt 2 + c 1 dq dt + k 1 q = F c (1)
where q represent the tool tip displacement, m 1 , k 1 and c 1 represent the modal mass, stiffness and damping of the first bending mode, respectively. The cutting force takes into account the regenerative effect and is expressed in polynomial form as follows [START_REF] Shi | Theory of finite amplitude machine tool instability[END_REF]

F c (∆h(t)) = p ρ 1 h(t) + ρ 2 h(t) 2 + ρ 3 h(t) 3 (2)
where p is the depth of cut, h(t) represents the instantaneous chip thickness and ρ i (i = 1..3) are the specific cutting coefficients obtained from experimental measurements. Typical values of these coefficients for steel cutting are given in [START_REF] Dombovari | On the global dynamics of chatter in the orthogonal cuttingmodel[END_REF]. The instantaneous chip thickness may be expressed as follows

h(t) = h 0 + ∆h(t) (3)
here, h 0 is the nominal chip thickness without vibration and h(t) represents the chip thickness variation which depends on the instantaneous position of the tool tip, and the position of the tool tip from the previous workpiece rotation, as illustrated in Fig. 1. It is expressed as

Xh(t) = q(t -τ ) -q(t) (4)
where q(t -τ ) is the delayed position of the tool. τ is the time delay between two successive tool passages, which corresponds to the period of one workpiece rotation τ = 2π/Ω (Ω is the workpiece angular speed). When the cutting process is stable, q(t) = q 0 . Substituting Eq. ( 3) into (1) gives

k 1 q 0 = p ρ 1 h 0 + ρ 2 h 2 0 + ρ 3 h 3 0 (5)
During unsteady regimes, the displacement is the sum of the steady component q 0 and an unsteady component u(t)

q(t) = q 0 + u(t) (6)
Substituting Eqs. [START_REF] Gourc | Chatter milling modeling of active magnetic bearing spindle in highspeed domain[END_REF][START_REF] Khasawneh | Investigation of period-doubling islands in milling with simultaneously engaged helical flutes[END_REF] into (1) give

m 1 d 2 u dt 2 + c 1 du dt + k 1 u = p α 1 ∆ h + α 2 ∆ h2 + α 3 ∆ h3 (7)
where

∆ h = u(t -τ ) -u(t), α 1 = 2ρ 2 h 0 + 3ρ 3 h 2 0 + ρ 1 , α 2 = 3ρ 3 h 0 + ρ 2 et α 3 = ρ 3 .
The following changes of variables are introduced

T = ω 1 t, ω 2 1 = k 1 m 1 , 2µ 1 = c 1 m 1 ω 1 , ψ = pα 1 m 1 ω 2 1 , η 1 = pα 2 ψm 1 ω 2 1 , η 2 = pα 3 ψm 1 ω 2 1 (8)
After substitution of ( 8) into [START_REF] Patel | Uncharted islands of chatter instability in milling[END_REF] we obtain

ü + 2µ 1 u + u = ψ ∆ h + η 1 ∆ h2 + η 2 ∆ h3 (9)
where the dot represents the differentiation with respect to the non-dimensional time T .

Analysis of the linearized system

After removing the nonlinear terms in the cutting law by setting η 1 = η 2 = 0 in Eq. ( 9), the linearized equation of motion of the cutting tool is given by

ü + 2µ 1 u + u + ψ (u(T -τ ) -u(T )) = 0 (10)
The solution of (10) can be written by

u = u 0 e (γ+iω)t ( 11
)
where γ is the grow or decay rate, ω is the pulsation of the oscillations and u 0 depends on the initial conditions. After substituting Eq. ( 11) into (10), we obtain

ψ 1 -e (γ+iω)τ + (γ + iω) 2 + 2µ 1 (γ + iω) + 1 = 0 (12)
After splitting Eq. ( 12) into real and imaginary parts, the following equations were obtained: p9@ABC YE `aPIRbC UV SQPc9b9Qd bUcCS e9P@BPIW ψ (1 -cos(ωτ )e -γτ ) + γ 2 -ω 2 + 2µ 1 γ + 1 = 0 ψ sin(ωτ )e -γτ + 2ω (γ + µ 1 ) = 0 [START_REF] Matsubara | Vibration suppression of boring bar by piezoelectric actuators and lr circuit[END_REF] When γ > 0, the amplitude of the oscillations growth exponentially and the system is unstable. On the contrary, when γ < 0, the amplitude of the oscillations decay with time and the turning process is stable. In order to evaluate the stability boundary, we set γ = 0 in Eq. ( 13)

0 0.5 1
ψ cos(ωτ ) = ψ -ω 2 + 1 ψ sin(ωτ ) = -2µ 1 ω (14)
Using trigonometric identity in [START_REF] Brecher | Chatter suppression with an active workpiece holder[END_REF], the pulsation of the bifurcated solution (also called chatter frequency) is obtained as

ω = 1 + ψ -2µ 2 1 ± -4µ 2 1 + (ψ -2µ 2 1 ) 2 (15)
It is possible to prove that the bifurcation (γ, ψ, ω) = (0, ψ c , ω c ) is a Hopf bifurcation [START_REF] Nayfeh | Analysis of the cutting tool on a lathe[END_REF]. For practical machining applications, the stability boundary is often plot in the space of parameters (Ω, ψ) and is called the stability lobe diagram. An example of such a diagram is depicted in Fig. 2. The turning process is stable for cutting conditions under the lobes and is unstable otherwise.

Passive control of chatter instability using a vibro-impact NES

In this section, the dynamics of the lathe tool coupled with the vibro-impact NES is investigated. A schematic representation of the coupled system is given in Fig. 3. The corresponding dimensionless equations of motion between impact, after eliminating the static deflection are written as follow

ü + 2µ 1 u + u = ψ ∆ h + η 1 ∆ h2 + η 2 ∆ h3 ǫ 2 ÿ = 0, ∀|ru -y| < ∆ ( 16
)
where y represents the absolute displacement of the free mass of the vibro-impact NES, ǫ 2 = m 2 /m 1 ≪ 1 represent the mass ratio between the modal mass of the tool and the

m 1 k 1 c 1 q(t) y(t) h(t)
workpiece rotation m 2 p9@ABC fE FGHCIPQ9G BCRBCSCTQPQ9UT UV QHC RPSS9gC GUTQBUb UV GHPQQCB 9TSQPc9b9Qd AS9T@ P g9cBUh9IRPGQ i`FW mass of the vibro-impact NES, ∆ represents the gap between the free mass and the rigid walls and r is an influence coefficient depending of the position of the vibro-impact NES on the tool. The impacts are modelled using the Newton coefficient of restitution on velocities and the condition of conservation of momentum as follow

u + = u -, y + = y - r u+ -ẏ+ = -R (r u--ẏ-) u+ + ǫ 2 ẏ+ = u-+ ǫ 2 ẏ- for |ru -y| = ∆ (17)
here, the subscripts + and -represent the time immediately after and before impact and R is the coefficient of restitution (R ∈ [0, 1]).

Asymptotic analysis

The following algebraic developments have been carried out using the computer algebra software Maple. Equations ( 16) and ( 17) are analyzed using the method of multiple scales. New coordinates are introduced as follow

v = u + ǫ 2 y, w = ru -y (18)
where v and w represents the displacement of the center of mass and the internal displacement of the vibro-impact NES, respectively. After substituting Eqs. [START_REF] Yang | Optimization of multiple tuned mass dampers to suppress machine tool chatter[END_REF] into Eqs. [START_REF] Wang | Design and implementation of nonlinear tmd for chatter suppression: An application in turning processes[END_REF][START_REF] Wang | Feasibility study of nonlinear tuned mass damper for machining chatter suppression[END_REF], it follows

v + 2µ 1 v + ǫ 2 ẇ 1 + rǫ 2 + v + ǫ 2 w 1 + rǫ 2 = ψ v τ + ǫ 2 w τ 1 + rǫ 2 - v + ǫ 2 w 1 + rǫ 2 +η 1 v τ + ǫ 2 w τ 1 + rǫ 2 - v + ǫ 2 w 1 + rǫ 2 2 + η 2 v τ + ǫ 2 w τ 1 + rǫ 2 - v + ǫ 2 w 1 + rǫ 2 3 rv - ẅ 1 + rǫ 2 = 0, ∀|w| < ∆ (19) v + = v -, w + = w - ẇ+ = -R ẇ-, v+ = v- for |w| = ∆ (20) !)! v τ = v(t -τ
) and w τ = w(t -τ ). Since secular terms first appear at O(ǫ 3 ), a detuning parameter representing the vicinity of ψ to ψ c is introduced as follow [START_REF] Nayfeh | Analysis of the cutting tool on a lathe[END_REF]

ψ = ψ c + ǫ 2 σ ( 21
)
The solutions of Eqs [START_REF] Vakakis | Energy pumping in nonlinear mechanical oscillators: part iiresonance capture[END_REF][START_REF] Starosvetsky | Strongly modulated response in forced 2dof oscillatory system with essential mass and potential asymmetry[END_REF] are expanded in power series as

v(T ; ǫ) = ǫv 1 (T 0 , T 1 , T 2 ) + ǫ 2 v 2 (T 0 , T 1 , T 2 ) + ǫ 3 v 3 (T 0 , T 1 , T 2 ) + . . . w(T ; ǫ) = ǫw 1 (T 0 , T 1 , T 2 ) + ǫ 2 w 2 (T 0 , T 1 , T 2 ) + ǫ 3 w 3 (T 0 , T 1 , T 2 ) + . . . ( 22
)
where T n = ǫ n T (n = 0, 1, . . .). The time delay is considered to be of O( 1), which is rather natural if the behavior close to the order 0 lobe is analyzed. In this case, the delayed position of the tool and the vibro-impact NES are written as

v(T -τ ; ǫ) = ǫv 1 (T 0 -τ, T 1 , T 2 ) + ǫ 2 (v 2 (T 0 -τ, T 1 , T 2 ) -τ D 1 v 1 (T 0 -τ, T 1 , T 2 )) +ǫ 3 v 3 (T 0 -τ, T 1 , T 2 ) + τ 2 2 D 2 1 v 1 (T 0 -τ, T 1 , T 2 ) -τ D 2 v 1 (T 0 -τ, T 1 , T 2 ) -τ D 1 v 2 (T 0 -τ, T 1 , T 2 ) + . . . w(T -τ ; ǫ) = ǫw 1 (T 0 -τ, T 1 , T 2 ) + ǫ 2 (w 2 (T 0 -τ, T 1 , T 2 ) -τ D 1 w 1 (T 0 -τ, T 1 , T 2 )) +ǫ 3 w 3 (T 0 -τ, T 1 , T 2 ) + τ 2 2 D 2 1 w 1 (T 0 -τ, T 1 , T 2 ) -τ D 2 w 1 (T 0 -τ, T 1 , T 2 ) -τ D 1 w 2 (T 0 -τ, T 1 , T 2 ) + . . . (23)
Substituting Eqs. [START_REF] Lamarque | Targeted energy transfer in mechanical systems by means of non-smooth nonlinear energy sink[END_REF][START_REF] Savadkoohi | Vibratory energy exchange between a linear and a nonsmooth system in the presence of the gravity[END_REF] into Eqs. [START_REF] Vakakis | Energy pumping in nonlinear mechanical oscillators: part iiresonance capture[END_REF][START_REF] Starosvetsky | Strongly modulated response in forced 2dof oscillatory system with essential mass and potential asymmetry[END_REF] and equating coefficients of like power of ǫ, it comes

ǫ 1 : D 2 0 v 1 + 2µ 1 D 0 v 1 + v 1 -ψ c (v 1τ -v 1 ) = 0 D 2 0 w 1 -rD 2 0 v 1 = 0, ∀|w 1 | < ∆ v 1+ = v 1-, w 1+ = w 1- D 0 v 1+ = D 0 v 1-, D 0 w 1+ = -RD 0 w 1-, for |w 1 | = ∆ (24) ǫ 2 : D 2 0 v 2 + 2µ 1 D 0 v 2 + v 2 -ψ c (v 2τ -v 2 ) = -2µ 1 D 1 v 1 -2D 0 D 1 v 1 -ψ c τ D 1 v 1τ + ψ c η 1 (v 1τ -v 1 ) 2 , ∀|w 2 | < ∆ v 2+ = v 2-, w 2+ = w 2- D 0 v 1+ + D 0 v 2+ = D 0 v 1-+ D 0 v 2-, D 0 w 1+ + D 0 w 2+ = -R (D 0 w 1-+ D 0 w 2+ ) , for |w 2 | = ∆ (25) q 3 : D 2 0 v 3 + 2µ 1 D 0 v 3 + v 3 -ψ c (v 3τ -v 3 ) = 2µ 1 (rD 0 v 1 -D 1 v 2 -D 2 v 1 -D 0 w 1 ) +2ψ c η 1 (v 1 -v 1τ ) (D 1 v 1τ -v 2τ + v 2 ) -ψ c η 2 (v 1 -v 1τ ) 3 + ψ c r (v 1 -v 1τ ) +ψ c w 1τ -w 1 + τ 2 2 D 2 1 v 1τ -τ D 1 v 2τ -D 2 v 1τ + v 1 r -w 1 -2D 0 D 1 v 2 -2D 0 D 2 v 1 -D 2 1 v 1 + σ (v 1τ -v 1 ) , ∀|w 3 | < ∆ v 3+ = v 3-, w 3+ = w 3- D 0 v 1+ + D 0 v 2+ + D 0 v 3+ = D 0 v 3-+ D 0 v 2-+ D 0 v 3-, D 0 w 1+ + D 0 w 2+ + D 0 w 3+ = -R (D 0 w 1-+ D 0 w 2+ + D 0 w 3+ ) , for |w 3 | = ∆ (26)
The other equations at order ǫ 2 and ǫ 3 are not given because they are not used in the following analysis.

Order ǫ 1

The solution of the first equation of ( 24) may be written as follow [START_REF] Nayfeh | Analysis of the cutting tool on a lathe[END_REF]

v 1 = Ae iωcT0 + ∞ m=1 A m e (γm+iωm)T0 + cc (27)
Where ω c is the critical chatter frequency corresponding to σ = 0, obtained using 15.

Close to the stability boundary, all the remaining roots γ m ± iω m have negative real part and decay with time. Thus, the long time behavior is expressed by

v 1 = Ae iωcT0 + cc ( 28
)
where cc denotes the complex conjugate of the preceding terms. Substituting (28) into the second equation of ( 24), gives

D 2 0 w 1 + rω 2 c Ae iωcT0 + Āe -iωcT0 , ∀|w 1 | < ∆ w 1+ = w 1-, D 0 w 1+ = -RD 0 w 1-, for |w 1 | = ∆ (29)
This equation corresponds to a harmonically forced vibro-impact oscillator. Between impact (i.e. when |w 1 | < ∆), the solution of ( 29) is given by

w 1 = r Ae iωcT0 + Āe -iωcT0 + BT 0 + C (30)
Like 1 : 1 resonance capture between the lathe tool and the vibro-impact NES is assumed, we consider solutions with two symmetric impacts per cycle. The periodicity conditions on a half period are written as follow

w 1 π ω c = -w 1 (0), D 0 w 1+ π ω c = -D 0 w 1+ (0) (31)
The solution between impact given by Eq. ( 30) together with the periodicity conditions in Eq. ( 31) are used to recover the integration constants B and C. At time T 0 = 0, it is assumed that the free mass of the vibro-impact NES is in contact with the left side of the cavity

w 1 (0) = ∆ (32)
Substituting Eq. ( 30) into the first periodicity condition in [START_REF] Viguié | Using passive nonlinear targeted energy transfer to stabilize drill-string systems[END_REF] yields

B = - 2Cω c π (33)
After substituting Eq. ( 30) into the second periodicity condition in Eq. ( 31) and in [START_REF] Gourc | Delayed dynamical system strongly coupled to a nonlinear energy sink: application to machining chatter[END_REF], the following relations are obtained

rω c i A -Ā (1 + R) + B (1 -R) = 0 r A + Ā + C -∆ = 0 ( 34
)
Substituting relation ( 33) into [START_REF] Nankali | Targeted energy transfer for suppressing regenerative instabilities in a 2-dof machine tool model[END_REF] and introducing polar form A = ae iα it follows

sin α = Cδ raπ , cos α = - C -∆ 2ra ( 35
)
where δ = (R -1)/(R + 1). Using trigonometric identity in (35) yields

a 2 = C 2 δ 2 r 2 π 2 + (C -∆) 2 4r 2 (36)
Equation ( 36) represents the slow invariant manifold (SIM) of the problem at time scale T 1 when the vibro-impact NES oscillate with two symmetric impacts per period of oscillation of the lathe tool [START_REF] Gendelman | Analytic treatment of a system with a vibro-impact nonlinear energy sink[END_REF]. The SIM admit an extremum computed by vanishing the derivative of the right hand side of [START_REF] Dombovari | On the global dynamics of chatter in the orthogonal cuttingmodel[END_REF] with respect to C

C min = π 2 ∆ 4δ 2 + π 2 , a 2 min = ∆ 2 δ 2 r 2 (4δ 2 + π 2 ) ( 37
)
Equation ( 37) is of particular interest since it represents the minimum amplitude of the lathe tool which allow the vibro-impact NES to vibrate in regimes with two symmetric impacts per cycle. The analysis of the stability of the SIM is not trivial due to the discontinuities in the velocities and requires analyzing the Poincaré section of the perturbed system [START_REF] Yue | Symmetry and bifurcations of a two-degree-of-freedom vibro-impact system[END_REF][START_REF] Lenci | Periodic solutions and bifurcations in an impact inverted pendulum under impulsive excitation[END_REF]. The perturbed solutions of Eq. ( 29) are written as follow

w1 = Ãe iωcT0 + Āe -iωcT0 + B1 T 0 + C 1 , 0 < T 0 < t 1 w1 = Ãe iωcT0 + Āe -iωcT0 + B2 T 0 + C 2 , t 1 ≤ T 0 < t e (38)
where t 1 = π/ω c + ∆t 1 , t e = 2π/ω c + ∆t 1 + ∆t 2 and à takes into account a phase perturbation as

à = ae iα+∆α (39)
The initial conditions of the perturbed system are given by

w1 (0) = ∆, D 0 w1 (0) = D 0 w 1 (0) + ∆ ẇ (40)
and the periodicity conditions of the perturbed system are expressed as w1 (t 1 ) = -w1 (0), D 0 w1 (t 1+ ) = -D 0 w1 (0 + ), w1 (t e ) = w1 (0), D 0 w1 (t e+ ) = D 0 w1 (0 + ) (41)

Substituting the solutions of the perturbed system given in Eq. ( 38) into (40,41) and solving for B i and C i (i = 1, 2), it comes

B 1 = 2raω c sin (α + ∆α) + D 0 w 1 (0) + ∆ ẇ C 1 = -2ra cos (α + ∆α) + ∆ B 2 = 2raω c (1 + R) sin (α + ∆α + ω c t 1 ) -RB 1 C 2 = -2t 1 raω c (1 + R) sin (α + ∆α + ω c t 1 ) + t 1 (1 + R)B 1 + C 1 (42)
The computed Poincaré section for solutions with two symmetric impacts per cycle is obtained

∆ ẇ′ = D 0 w1+ (t e ) -D 0 w 1 (0 + ) ≡ f 1 (∆ ẇ, ∆α) ∆α ′ = ∆α + ∆t 1 + ∆t 2 ≡ f 2 (∆ ẇ, ∆α) (43)
The stability of the SIM is determined by the location of the eigenvalues of the Jacobian matrix with respect to the unit circle. The Jacobian matrix of the linearized system is written as follow

J =    ∂f 1 ∂∆ ẇ ∂f 1 ∂∆α ∂f 2 ∂∆ ẇ ∂f 2 ∂∆α    (44)
It is observed that ∆ ẇ′ = ∆ ẇ′ (δ ẇ, ∆α, ∆t 1 , ∆t 2 ) and that the temporal variables ∆t i , i = 1, 2 are also dependent of the perturbations ∆t i = ∆t i (∆ ẇ, ∆α). Thus, the derivatives in (44) are expressed in the following manner

∂f i ∂u j = ∂f i ∂u j + ∂f i ∂∆t 1 ∂∆t 1 ∂u j + ∂f i ∂∆t 2 ∂∆t 2 ∂u j (45)
The derivatives ∂∆t i /∂u j , i, j = 1, 2 are computed with the help of the implicit function theorem; to this end, new functions expressing the periodicity conditions of the perturbated system are introduced as

h (∆ ẇ, ∆α, ∆t 1 ) = w1 π ω c + ∆t 1 + ∆ = 0 g (∆ ẇ, ∆α, ∆t 1 , ∆t 2 ) = w1 π ω c + ∆t 1 + ∆t 2 -∆ = 0 (46)
Using (46), supposing that (∂h/∂(∆t 1 )) (0,0,0) = 0 and (∂g/∂(∆t 2 )) (0,0,0,0) = 0 the derivatives are expressed by [START_REF] Yue | Symmetry and bifurcations of a two-degree-of-freedom vibro-impact system[END_REF][START_REF] Luo | Periodic motions and global bifurcations of a two-degree-of-freedom system with plastic vibro-impact[END_REF]

∂∆t 1 ∂u j = - ∂h ∂u j / ∂h ∂∆t 1 ∂∆t 2 ∂u j = - ∂g ∂u j + ∂g ∂∆t 1 ∂∆t 1 ∂u j / ∂g ∂∆t 2 (47)
An example of SIM is depicted in Fig. 4 for R = 0.65, r = 0.5 and ∆ = 0.15. The SIM is composed of two branches, the left branch is completely unstable and the right one is 10 Blue (red) lines denote stable (unstable) solutions. SN and SB stand for saddle node and symmetry breaking, respectively.

partially stable. When the amplitude of the oscillation increases, a symmetry breaking bifurcation occurs and the regimes of motion with two symmetric impacts per cycle is destroyed.

4.1.2. Order ǫ 2 , ǫ 3 Equation (25) at O(ǫ 2
) is first analyzed. Substituting the solution of v 1 (28) into ( 25) gives

D 2 0 v 2 + 2µ 1 D 0 v 2 + v 2 + ψ c (v 2 -v 2τ ) = -2µ 1 -2iω c -ψ c τ e -iωcτ D 1 Ae iωcT0 +A 2 e 2iωcT0 1 -e -iωcτ 2 + 2A Ā 1 -e -iωcτ (48)
The condition of elimination of secular terms in (48) reads

-2µ 1 -2iω c -ψ c τ e -iωcτ D 1 A = 0 → A(T 1 , T 2 ) = A(T 2 ) ( 49
)
which means that the slow modulation amplitude A does not depends on time scale T 2 .

Taking into account (49), the particular solution of v 2 is given by

v 2 = ψ c η 1 A 2 Ξ 2 e 2iωcT0 Γ -1 + 2A ĀΞ + cc (50)
where Ξ and Γ are constants defined as follow

Γ = -4ω 2 c + 4iµ 1 ω c + 1 + ψ c e 2iωcτ -ψ c Ξ = 1 -e iωcτ (51)
Equation ( 26) at O(ǫ 3 ) is now analyzed. In order to identify terms that produce secular terms, the expression of w 1 given in Eq. ( 30) is developed in Fourier series as

w 1 = 1 2 (∆ -C) + iδ π C e iωcT0 + ∞ k=1 4C π 2 (2k -1) 2 e (2k-1)iωcT0 + cc (52)
u3((u($0 vwx y% #$' y $(8 y 01!

D 2 0 v 3 + 2µ 1 D 0 v 3 + v 3 -ψ c (v 3τ -v 3 ) = D 2 A -ψ c τ e -iωcτ -2µ 1 -2iω c +µ 1 ω c i(C -∆) + 2δ π C + 2irA - 8i π 2 C -2ψ 2 c η 2 1 ĀA 2 Ξ 2 e iωcτ + 1 e iωcτ -1 2 Γe 2iωcτ +3ψ c η 2 A 2 Ā e iωcτ -1 3 e 2iωcτ + 1 + ψ c 2 -∆ + 2rA - 8C π 2 - 2iCδ π + C 1 -e -iωcτ -2σA 1 -e iωcτ e iωcT0 + cc + N ST (53)
where N ST denotes terms that do not produce secular terms. Secular terms are eliminated from (53) if

D 2 A -ψ c τ e -iωcτ -2µ 1 -2iω c +µ 1 ω c i(C -∆) + 2δ π C + 2irA - 8i π 2 C -2ψ 2 c η 2 1 ĀA 2 Ξ 2 e iωcτ + 1 e iωcτ -1 2 Γe 2iωcτ +3ψ c η 2 A 2 Ā e iωcτ -1 3 e 2iωcτ + 1 + ψ c 2 -∆ + 2rA - 8C π 2 - 2iCδ π + C 1 -e -iωcτ
-2σA 1 -e iωcτ = 0 (54)

Writing A in polar form A = ae iα and splitting into real and imaginary parts, the equation governing the evolution of the amplitude of the lathe tool with respect to time scales T 2 is obtained as

D 2 a = c 1 a + c 2 a 3 + C (c 3 cos α + c 4 sin α) + c 5 cos α + c 6 sin α (55)
where the coefficients c i are complicated (which do not need to be explicitly stated at this stage of the analysis in order to follow the procedures involved in the expansion). Since the behavior of the system on the stable branch of the SIM is analyzed, the trigonometric relations given in Eq. ( 35) are substituted into (55) and the obtained equation is multiplied by a(T 2 ), yielding

D 2 N = c ′ 1 N + c ′ 2 N 2 + c ′ 3 C + c ′ 4 C 2 + c ′ 5 ( 56
)
where N = a 2 . Fixed points are computed by vanishing the derivative in (56) and are given by

N = -c ′ 1 ± c ′2 1 -4c ′ 2 (c ′ 3 C + c ′ 4 C 2 + c ′ 5 ) 2c ′ 2 (57)
Equation ( 57) constitutes an algebraic relation between the amplitude of the oscillations of the lathe tool and the vibro-impact NES. Therefore it can be viewed as the SIM of the problem at time scale T 3 when T 2 → ∞ and the fixed points of the system are obtained graphically at the intersection of the SIM given in Eq. ( 36) and the curves (57), see [START_REF] Gendelman | Bifurcations of self-excitation regimes in a van der pol oscillator with a nonlinear energy sink[END_REF]. p9@ABC E PE D E D Fst PQ Q9IC SGPbCS T 1 in blue (red) for stable (unstable) branch and T 3 in green for σ = 1. Pink + represents an unstable fixed points. Yellow lines represents the projection of the numerical integration on the SIM. (b): Result of numerical integration

Description of different response regimes

Different response regimes may be explained by studying the location of the fixed points on the SIM. The physical parameters of the primary system and the vibro-impact NES are given by ǫ 2 = 0.01, µ 1 = 0.03, R = 0.65, r = 0.8, ∆ = 2.10 -5

(58)

The specific cutting coefficients ρ i for the machining of steel are given in [START_REF] Dombovari | On the global dynamics of chatter in the orthogonal cuttingmodel[END_REF] ρ 1 = 6109.6 10 6 N m -2 , ρ 2 = -54141.6 10 9 N m -3 , ρ 3 = 203769 10 12 N m -4 , h 0 = 1 10 -4 m (59)

The numerical integration is carried out on the zero order lobe for ψ c = 0.4 (see Fig. 2). The step by step integration has been performed using the matlab dde23 algorithm together with the function event for the detection of the successive impacts.

Analog relaxation cycles

The SIM for σ = 1 depicted in Fig. 5(a). It is observed that an unstable fixed points is located on the unstable branch on the SIM while another unstable fixed point (not displayed here) is located on the unstable part of the right branch of the SIM (above the symmetry breaking bifurcation point). In this case, if the amplitude of the oscillations of the lathe tool became higher than the critical value a min given in Eq. ( 37), the slow flow will be able to be attracted by the stable branch of the SIM. In this case, the vibro-impact NES will be synchronized with the lathe tool and will oscillate with two symmetric impacts per cycle. During the synchronized regimes, the energy of the system will be dissipated by successive impacts (R < 1) and the amplitude of the oscillations of the lathe tool will decrease until reaching the singular point a min . At this point, the vibro-impact NES will escape the resonance capture and the amplitude of motion of the primary system will increase again, and so on.

The results of numerical integration depicted in Fig. 5(b) confirm the theoretical predictions; the amplitude of the oscillations of the lathe tool increase while the vibroimpact NES perform chaotic oscillations until the flow is attracted to the stable branch 13 p9@ABC E PE D E D Fst PQ Q9IC SGPbCS T 1 in blue (red) for stable (unstable) branch and T 3 in green for σ = 4.6. Pink + represents an unstable fixed points. Yellow lines represents the projection of the numerical integration on the SIM. (b): Result of numerical integration of the SIM. Then, the amplitude of the primary system quickly decrease until the vibroimpact NES escape resonance capture. These cycle may be viewed as the analogy of the SMR regimes occurring in NES with cubic stiffness nonlinearities, but with a significant difference. In the case of NES with cubic stiffness, these cycles corresponds to jumps between the two stable branches of the SIM [START_REF] Starosvetsky | Strongly modulated response in forced 2dof oscillatory system with essential mass and potential asymmetry[END_REF], while in this case they corresponds to successive synchronization and escapes of the synchronized regimes.

Limit of the passive control of chatter

When the value of σ is increased, the unstable fixed point located on the right branch of the SIM comes down along this branch as seen in Fig. 6(a) for σ = 4.6. If during the relaxation cycle, when the slow flow jump to the stable branch of the SIM, the landing point is above the saddle point, the slow flow will be repelled to higher amplitude and the system is no more controlled. The result of numerical integration, presented in Fig. 6(b), illustrates the theoretical prediction; after two relaxation cycle, the slow flow jump back to the stable branch of the SIM in the vicinity of the saddle point and the amplitude of the oscillation growth. In addition, it can be observed on the numerical integration, that when the amplitude of the oscillation growth sufficiently, the oscillation becomes unsymmetric (around t = 400), which is in agreement with the stability analysis of the SIM.

Experimental analysis

In order to validate experimentally the efficiency of a vibro-impact NES to passively controlling the chatter instability in turning, an experimental setup has been built.

Experimental setup

The trials have been realized on a Cazeneuve lathe (CT210) and the full experimental setup is depicted in Fig. 7. The machining operations have been carried out on a 40mm diameter, XC38 steel bar hold in the mandrel and the tail-stock. The cutting tool is a p9@ABC E `aRCB9ICTQPb SCQARW P bUcPb g9C c eCQP9bCe g9C UV QHC cUB9T@ cPB 9QH QHC CIcCeeCe vibro-impact NES.

250mm long boring bar which has been softened in one direction close to the tool holder to favour the bending mode in the cutting direction. In addition, a mass has been added on the tool tip in order to reduce the natural frequency of the first bending mode. The same feed-rate h 0 = 0.1mm/tr is used for all the trials which consist in different turning passes with different spindle speed and depth of cut.

The oscillations of the tool tip have been measured using a laser vibrometer, and the specific cutting coefficients are those from [START_REF] Dombovari | On the global dynamics of chatter in the orthogonal cuttingmodel[END_REF]. The dynamics characteristics of the tool have been obtained by hammer testing and are summarized in Table 1. It can be noticed that the natural frequency is still high in comparison to other NES application, which is an interesting feature. At first, experiments were performed without the vibro-impact NES in order to compare the behavior to the one identified theoretically on the linear analysis. The determination of the stable or unstable character of the process may not be straightforward due to the presence of strong forced vibration which can be due to an eccentricity of the workpiece, or when the spindle speed is close to the natural frequency of the lathe tool. In such case, the direct time series of the measurement may not be sufficient to identify chatter vibrations. To this end, simple signal processing method may be applied, such as re-sampling the signal at the spindle speed frequency [START_REF] Mann | Stability of up-milling and down-milling, part 2: experimental verification[END_REF]; in this case, a stable behavior will be represented by a straight line, because the relative position of the tool and the workpiece is always the same at each tool-path, whereas an unstable behavior will present a disorder. Another method consist in analyzing the pseudo-Poincaré section of the measured signal. A stable behavior will present an attractor of type point, whereas an unstable behavior will present an attractor of type circle. An example of the analysis of an unstable trial is depicted in Fig. 8. The comparison between the stability lobes obtained theoretically and those obtained experimentally is presented in Fig. 9. The stable (unstable) cutting conditions are represented by green circle (cross). The theoretical and experimental results are in agreement even if a shift on the left of the stability boundary is observed.

Sizing of the vibro-impact NES

Only the gap of the vibro-impact NES may be precisely adjusted during the fabrication process. The coefficient of restitution R may be varied using different materials couple for the ball and the cover. Here, a bearing ball and steel cover were used. Typical values of the amplitude of the oscillation measured during unstable trials without the vibro-impact NES were around 0.15mm. The activation level of the vibro-impact NES computed with the help of Eq. [START_REF] Yue | Symmetry and bifurcations of a two-degree-of-freedom vibro-impact system[END_REF] has been fixed to 50% of this value, which gives a gap ∆ = 0.34mm. After fabrication, the measured gap was ∆ exp = 0.32mm. In order to check whether the activation level fit the theoretical prediction and to estimate the coefficient of restitution, the vibro-impact NES alone has been embedded on an electrodynamic shaker. In effect, this configuration corresponds to an harmonically forced vibro-impacts oscillator as in Eq. ( 29). So, it is possible to identified experimentally the stable branch of the SIM, which is very interesting for the sizing procedure. A base displacement at frequency 99.4Hz, which correspond to the chatter frequency, is imposed to the vibro-impact NES and the successive impacts of the free mass on the cover are identified using accelerometers. Raw signals of two measurements are presented in Fig. 10. It is observed that in Fig. 10(a), for a = 7.66.10 -5 , that the vibro-impact NES exhibit two symmetric impacts per cycle, that is, the time between two consecutive impacts is the same, while in Fig. 10(b), for a = 1.81.10 -4 , there is still two impacts per cycle, but the time between two consecutive impacts differ, indicating asymmetric response.

The theoretical and experimental SIM are depicted in Fig. 11. Where the green circle corresponds to the experimental measurements, the continuous pink line denotes the experimentally identified activation level and the dashed pink line indicate the amplitude at which first asymmetric solutions have been observed. The value of coefficient of restitution of impact has been estimated by fitting the experimental results with the theoretical expression of the SIM given in Eq. [START_REF] Dombovari | On the global dynamics of chatter in the orthogonal cuttingmodel[END_REF]; which give R = 0.6. Effectively, using coefficients of restitution from the literature for a plane-ball, steel-steel contact which is around 0.95 yields to erroneous results.

The experimentally identified activation level is slightly higher than the theoretical predictions, which is certainly due to the simplified model of the behavior of the free mass of the vibro-impact NES which do not capture its complex dynamics. The results are however in satisfactory agreement, and asymmetric solutions have also been observed, consistently with theoretical predictions. . . p9@ABC DYE lUIRPB9SUT UV QHC cCHPg9UB eAB9T@ ATSQPcbC IPGH9T9T@ URCBPQ9UT 9QHUAQ PTe 9QH g9cBUh impact NES for s = 1800rpm and p = 0.1mm (ψ = 0.12 and Ω = 0.307).

Passive control of chatter with a vibro-impact NES

In order to analyze experimentally the efficiency of the vibro-impact NES to mitigate chatter instability on turning process, trials with the vibro-impact NES embedded on the lathe tool on the unstable zone have been carried out. Two trials with and without vibro-impact NES are depicted in Fig. 12 for s = 1800rpm and p = 0.1mm (ψ = 0.12 and Ω = 0.307).

It is observed that the presence of the vibro-impact NES changes drastically the behavior of the system. For the trial without vibro-impact NES, a constant high amplitude was measured whereas for the trial with vibro-impact NES, modulated response with moderate amplitude is observed. The measured modulated response is very similar to the analog relaxation cycle described theoretically. The successive impacts of the ball of the vibro-impact NES were clearly audible, however, the ambient noise due to machining operation did not allow us to measure the impact of the free-mass of the vibro-impact NES with the accelerometer. This behavior is however very promising, since a reduction of almost 50% on the vibration amplitude is observed.

mn opqrstuvpq

This paper investigated the possibility of controlling the chatter instability, which may occur during machining operations, using a NES. Due to practical reasons, a vibroimpact type NES was preferred to the classic NES with cubic stiffness.

The coupled system has been analyzed using the method of multiple scales. At the first order of approximation, the expression of the SIM has been obtained. The stability analysis of its different branches was also performed. At the next order of approximation, the fixed points of the system are obtained at the intersection of the slow and super-slow invariant manifolds.

The analysis of the different response regimes allow us to reveal a control mechanism which may be viewed as the analogy of the SMR response observed in the case of NES with cubic stiffness. In this case, the vibro-impact NES enter in successive resonance capture with the lathe-tool and the energy is dissipated during these synchronous motion.

An experimental setup which consist in a lathe tool with an embedded vibro-impact NES has been presented. First trials without vibro-impact NES were performed to confirm the stable or unstable nature of the process for different cutting conditions. The activation level of the vibro-impact NES was chosen in accordance with these trials. Notice that the natural frequency of the tool (which is close to the chatter frequency) is 99.4 Hz which is particularly high for NES application.

Due to the complicated real dynamics of the vibro-impact NES an original experimental procedure has been designed to validate the tuning of the NES. This procedure enable to identify experimentally the stable branch of the SIM.

Finally, two trials without and with the vibro-impact NES have been compared an promising results were obtained with a significant reduction of the amplitude of the tool. A deeper experimental investigation is left for further studies.

  rE Fst UV QHC GUARbCe SdSQCI PQ D E D BCSUTPTGC VUB RPBPICQCBS R = 0.65, r = 0.5 and ∆ = 0.15.

  E TPbdS9S UV PT ATSQPcbC QB9PbS PQ s = 1800rpm, p = 0.1mm (corresponding to ψ = 0.12, Ω = 0.307). (a) Blue lines : time history, green dots : re-sampled signal at the spindle frequency; (b) pseudo-Poincaré section; (c) frequency spectrum. gE hHCUBCQ9GPb PTe CaRCB9ICTQPb SQPc9b9Qd GHPBQW @BCCT GBUSS PTe G9BGbC BCRBCSCTQS ATSQPcbC PTe stable trials, respectively.

  DiE h9IC ICPSABCICTQ VUB QHC CaRCB9ICTQPb 9eCTQ9jGPQ9UT UV QHC SQPcbC cBPTGH UV QHC Fst BCe triangles corresponds to the detected impacts for (a) a = 7.66.10 -5 and (b) a = 1.81.10 -4 .

  DDE hHCUBCQ9GPb PTe CaRCB9ICTQPb FstW kCe PTe cbAC b9TCS BCRBCSCTQS QHC ATSQPcbC PTe SQPcbC branch of the SIM, respectively. Green circles represents the measurements points, continuous pink lines represents the experimentally determined level of activation and pink dotted lines the amplitude at which oscillations are no more symmetric

Table 1 :

 1 Characteristics of the experimental setup

	m 1	3.1 kg	µ 1 3 %
	f 1 99.4 Hz m 2 32 g
	r	0.8
	5.2. Validation of the stability lobe without NES


Tao Li

S Ébastien Seguy

Alain Berlioz
email: alain.berlioz@univ-tlse3.fr

Dynamics of cubic and vibro-impact Nonlinear Energy Sink (NES): analytical, numerical and experimental analysis

This paper is devoted to study and compare dynamics of primary Linear Oscillator (LO) coupled to cubic and Vibro-Impact (VI) Nonlinear Energy Sink (NES) under transient and periodic forcing. The classic analytical procedure combining the approach of invariant manifold and multiple scales is extended from the analysis of steadystate resonance to other regimes, especially Strongly Modulated Response (SMR). A general equation governing the variation of motion along the Slow Invariant Manifold (SIM) is obtained. Numerical results show its convenience to explain the transition from steady-state response to SMR and the characteristics of SMR for periodic forcing.

Targeted energy transfer under transient forcing can also be well understood. Experimental results from LO coupled to VI NES under periodic forcing confirm the existence of SMR and its properties (e.g. chaotic). They also verify the feasibility of the general equation to explain complicated case like SMR in experiments.

Introduction

Nonlinear Energy Sink (NES) is a small attachment fixed to primary system with essential nonlinearity. It is proved that it can lead to irreversible energy transfer from primary system to NES and rapid dissipation of this localized energy. This phenomenon is called non-linear energy pumping or passive Targeted Energy Transfer (TET) which has been extensively studied during the last fifteen years [START_REF] Lee | Passive non-linear targeted energy transfer and its applications to vibration absorption: a review[END_REF][START_REF] Vakakis | Nonlinear targeted energy transfer in mechanical and structural systems[END_REF].

The initial studies of NES are concentrated on a NES with cubic nonlinearity. In the first two papers dedicated to study the dynamics of TET [3,[START_REF] Vakakis | Energy pumping in nonlinear mechanical oscillators: Part ii: Resonance capture[END_REF], the NES is supposed with cubic nonlinearity which can be realized with two strings with pretensions. However, NES can possess other forms of nonlinearity. NES with general non-polynomial nonlinearity is studied in [START_REF] Gendelman | Targeted energy transfer in systems with non-polynomial nonlinearity[END_REF]. NES with piecewise nonlinearity is analyzed in [START_REF] Lamarque | Targeted energy transfer in mechanical systems by means of non-smooth nonlinear energy sink[END_REF]. For seismic mitigation, a Vibro-Impact (VI) NES is studied numerically and experimentally in [START_REF] Nucera | Targeted energy transfers in vibroimpact oscillators for seismic mitigation[END_REF][START_REF] Nucera | Application of broadband nonlinear targeted energy transfers for seismic mitigation of a shear frame: Experimental results[END_REF]. It is realized though combining linear spring and vibro-impact elements and studied further in [9]. A simpler model of VI NES is presented in [START_REF] Gendelman | Analytic treatment of a system with a vibro-impact nonlinear energy sink[END_REF] and the VI NES is realized by a mass which can move freely in a cavity of the primary system and impact the system in two sides. The study of VI NES is extended from transient forcing to periodic forcing with experimental validation in [START_REF] Gourc | Targeted energy transfer under harmonic forcing with a vibro-impact nonlinear energy sink: Analytical and experimental developments[END_REF]. It is also studied analytically and numerically under periodic forcing in [12]. It is also demonstrated that a simple eccentric rotator can be used as NES [13,[START_REF] Sigalov | Resonance captures and targeted energy transfers in an inertially-coupled rotational nonlinear energy sink[END_REF]. In addition, a NES realized with a nonlinear membrane in acoustic system is studied in [START_REF] Cochelin | Experimental evidence of energy pumping in acoustics[END_REF][START_REF] Bellet | Experimental study of targeted energy transfer from an acoustic system to a nonlinear membrane absorber[END_REF] and multiple NESs in parallel are investigated in [START_REF] Vaurigaud | Targeted energy transfer with parallel nonlinear energy sinks. part i: Design theory and numerical results[END_REF][START_REF] Savadkoohi | Targeted energy transfer with parallel nonlinear energy sinks, part ii: theory and experiments[END_REF].

A classic analytical method combining the approach of invariant manifold and the multiple scales is introduced to the study of cubic NES in [START_REF] Gendelman | Bifurcations of nonlinear normal modes of linear oscillator with strongly nonlinear damped attachment[END_REF]. This method is generalized to NES with non-polynomial nonlinearity in [START_REF] Gendelman | Targeted energy transfer in systems with non-polynomial nonlinearity[END_REF], which is illustrated by two examples involving softening (non-polynomial) and piecewise-linear (non-analytic).

In the series of paper [START_REF] Gendelman | Analytic treatment of a system with a vibro-impact nonlinear energy sink[END_REF]12,[START_REF] Gendelman | Attractors of harmonically forced linear oscillator with Tao Li Paper VIB-15-1415 15 attached nonlinear energy sink i: Description of response regimes[END_REF][START_REF] Starosvetsky | Strongly modulated response in forced 2dof oscillatory system with essential mass and potential asymmetry[END_REF], SIM is obtained with the above method and proved to be a good tool to describe the quantitative relation between primary system and NES. Moreover, it represents the set of all possible fixed points under 1:1 resonance and its topologic structure can be used to judge the existence of TET and SMR. Recently, it is founded that the topological structure of SIM is different for NES with piece-wise nonlinearity [START_REF] Lamarque | Targeted energy transfer in mechanical systems by means of non-smooth nonlinear energy sink[END_REF], cubic NES [START_REF] Gourc | Experimental investigation and design optimization of targeted energy transfer under periodic forcing[END_REF] and VI NES [START_REF] Gourc | Targeted energy transfer under harmonic forcing with a vibro-impact nonlinear energy sink: Analytical and experimental developments[END_REF]. In [12], the authors assume that the special structure of SIM is the cause of the unique SMR for VI NES and it is named as chaotic SMR. In this same paper, this phenomenon is also claimed to exist for a rotational NES [13,[START_REF] Sigalov | Resonance captures and targeted energy transfers in an inertially-coupled rotational nonlinear energy sink[END_REF]. However, this method is limited at the condition of 1:1 resonance. Moreover, although the existence of SMR can be judged from the structure of SIM obtained through this method, the actual variation of SMR cannot be explained further by it.

In one side, whether there exists one better analytical method to overcome the limit of the hypothesis (i.e. 1:1 resonance) to explain all possible SMRs of different NESs mentioned above or even more complicated regimes. In another side, whether this new or improved method can explain the more complicated results during the application of NES, for example, the complicated results during the application of VI NES to quench the chatter instability in turning process [START_REF] Gourc | Quenching chatter instability in turning process with a vibro-impact nonlinear energy sink[END_REF].

In this paper, the LO separately coupled to cubic NES and VI NES is studied. The structure is as follows: Section 2 is devoted to the analytical treatment of the governing equation of motion. Section 3, numerical results is used to verify analytical developments. In the next section, experimental results are demonstrated. Finally, concluding remarks are addressed.

Tao Li

Paper VIB-15-1415 2

A Linear Oscillator (LO) coupled to cubic NES and VI NES separately is studied under transient and periodic forcing respectively. The objective is to get the SIM when LO and NES are in 1:1 resonance. The theoretic developments presented herein are based on the method used in [START_REF] Gourc | Experimental investigation and design optimization of targeted energy transfer under periodic forcing[END_REF] for cubic NES and [START_REF] Gourc | Targeted energy transfer under harmonic forcing with a vibro-impact nonlinear energy sink: Analytical and experimental developments[END_REF] for VI NES.

Cubic NES

System of harmonically forced LO attached with cubic NES presented in Fig. 1 is described by the following equation of motion:

ẍ + ελ 1 ẋ + x + ελ 2 ( ẋ -ẏ) + εK (x -y) 3 = εG sin Ωτ + ε 2 λ 1 GΩ cos Ωτ ε ÿ + ελ 2 ( ẏ -ẋ) + εK (y -x) 3 = 0 (1)
The corresponding physical parameters are expressed as follows:

ε = m 2 m 1 , ω 0 2 = k 1 m 1 , K = k 2 m 2 ω 0 2 , λ 1 = c 1 m 2 ω 0 , λ 2 = c 2 m 2 ω 0 , G = F ε , Ω = ω ω 0 , τ = ω 0 t
where x, m 1 , c 1 , k 1 and y, m 2 , c 2 , k 2 are the displacement, mass, damping and stiffness of the LO and the cubic NES respectively, the dots denote differentiation with respect to τ.

[Fig. 1

about here.]

New variables representing the displacement of the center of mass and the internal displacement of the cubic NES are introduced as follows:

v = x + εy, w = x -y (2)
Substituting Eq. (2) into Eqs. (1):

Tao Li Paper VIB-15-1415 3 v + ελ 1 v + ε ẇ 1 + ε + v + εw 1 + ε = εG sin Ωτ + ε 2 λ 1 GΩ cos Ωτ ẅ + ελ 1 v + ε ẇ 1 + ε + v + εw 1 + ε + λ 2 (1 + ε) ẇ + K (1 + ε) w 3 = εG sin Ωt + ε 2 λ 1 GΩ cos Ωt (3)
In the context of energy pumping, the mass ratio ε is supposed to be small (≈ 1%). In this case, Eq. ( 3) may be analyzed by multiple scales method with respect to this small parameter. Multiple scales are introduced in the following form:

v(τ; ε) = v 0 (τ 0 , τ 1 , . . .) + εv 1 (τ 0 , τ 1 , . . .) + . . . , w(τ; ε) = w 0 (τ 0 , τ 1 , . . .) + εw 1 (τ 0 , τ 1 , . . .) + . . . , τ k = ε k τ, k = 0, 1, . . . (4)
A detuning parameter σ representing the nearness of the forcing frequency Ω to the reduced natural frequency of the LO is introduced:

Ω = 1 + εσ (5)
Substituting Eqs. ( 4) and ( 5) into Eqs. (3) and equating coefficients of ε 0 and ε 1 :

ε 0 : ∂ 2 ∂τ 0 2 v 0 + v 0 = 0 ∂ 2 ∂τ 0 2 w 0 + λ 2 ∂ ∂τ 0 w 0 + v 0 + Kw 3 0 = 0 (6)
Tao Li Paper VIB-15-1415

ε 1 : ∂ 2 ∂τ 0 2 v 1 + v 1 = -2 ∂ 2 ∂τ 0 ∂τ 1 v 0 -λ 1 ∂ ∂τ 0 v 0 + v 0 + F cos (Ω τ 0 ) -w 0 (7)
The following complex variables are introduced under supposition of 1:1 resonance to obtain SIM:

v 0 = 1/2 A (τ 1 ) e iτ 0 + 1/2A (τ 1 ) e -iτ 0 A (τ 1 ) = N 1 (τ 1 ) e iθ 1 (τ 1 ) w 0 = 1/2 B (τ 1 ) e iτ 0 + 1/2B (τ 1 ) e -iτ 0 B (τ 1 ) = N 2 (τ 1 ) e iθ 2 (τ 1 ) (8)
Substituting Eq. ( 8) into Eqs. ( 6) and the equation of the SIM presented in Fig. 2 can be obtained. As to the stability, the analysis is already developed in [START_REF] Gendelman | Attractors of harmonically forced linear oscillator with Tao Li Paper VIB-15-1415 15 attached nonlinear energy sink i: Description of response regimes[END_REF].

N 1 2 = 1 + λ 2 2 Z - 3K 2 Z 2 + 9K 2 16 Z 3 Z = N 2 2 (9)
[Fig. 2

about here.]

Concerning the study of non 1:1 resonance (SMR), the following formula is introduced, among which the component of w containing the same frequency with the LO (frequency of outside forcing for periodic forcing or natural frequency of LO for transient forcing) can be developed in Fourier series and is supposed in the following form:

w 0 = 1/2 E (τ 1 ) e iτ 0 + 1/2E (τ 1 ) e -iτ 0 + 1/2 A (τ 1 ) e iτ 0 + 1/2A (τ 1 ) e -iτ 0 + RFC E (τ 1 ) = N 3 (τ 1 ) e iθ 3 (τ 1 ) (10)
Tao Li Paper VIB-15-1415

where RFC represents the rest frequency components and E (τ 1 ) is directly decided by the motion of cubic NES.

Substituting the first two equations of Eqs. [START_REF] Nucera | Application of broadband nonlinear targeted energy transfers for seismic mitigation of a shear frame: Experimental results[END_REF] and Eqs. [START_REF] Gendelman | Analytic treatment of a system with a vibro-impact nonlinear energy sink[END_REF] into the first equation of ( 7) and eliminating the secular terms, the following equation governing the variation of the slow variation of LO with respect to τ 1 can be obtained:

d dτ 1 N 1 = - 1 2 λ 1 N 1 - 1 2 N 3 sin (Θ) + 1 2 G sin (η) (11)
where

Θ = θ 3 -θ 1 η = τ 1 σ -θ 1 (12)
Θ represents the phase difference between relative displacement and that of LO. η represents the phase difference between outside forcing and motion of LO.

VI NES

System of harmonically forced LO attached with VI NES presented in Fig. 3 is described by the following equations:

ẍ + ελ 1 ẋ + x = εG sin Ωτ + ε 2 λ 1 GΩ cos Ωτ ε ÿ = 0 ∀ |x -y| < b (13)
where

ε = m 2 m 1 , ω 0 2 = k 1 m 1 , τ = ω 0 t, λ 1 = c 1 m 2 ω 0 , Ω = ω ω 0 , G = F ε [Fig. 3 about here.]
The meaning of notations is the same as that for system with Cubic NES and b represents a parameter related to the length of cavity.

Tao Li

Paper VIB-15-1415 6

When |x -y| = b, a impact occurs. The state of the system after impact is obtained using the simplified shock theory and the condition of total momentum conservation:

x + = x -, y + = y - ẋ+ + ε ẏ+ = ẋ-+ ε ẏ-, ẋ+ -ẏ+ = -r ( ẋ--ẏ-) , for |x -y| = b ( 14
)
where r is the restitution coefficient of impact and the superscripts + anddenotes time immediately after and before impact. New variables representing the displacement of the center of mass and the internal displacement of the VI NES are introduced as follows:

v = x + εy, w = x -y (15)
Substituting Eq. ( 15) into Eqs. ( 13) and ( 14), the equation between impact in barycentric coordinate are given as:

v + ελ 1 v + ε ẇ 1 + ε + v + εw 1 + ε = εG sin Ωτ + ε 2 λ 1 GΩ cos Ωτ ẅ + ελ 1 v + ε ẇ 1 + ε + v + εw 1 + ε = εG sin Ωτ + ε 2 λ 1 GΩ cos Ωτ ∀|w| < b (16)
And the impact condition ( 14) is rewritten as:

v + = v -, w + = w -, v+ = v-, ẇ+ = -r ẇ-, for |w| = b (17)
Multiple scales are introduced in the following form:

Tao Li Paper VIB-15-1415 v(τ; ε) = v 0 (τ 0 , τ 1 , . . .) + εv 1 (τ 0 , τ 1 , . . .) + . . . , w(τ; ε) = w 0 (τ 0 , τ 1 , . . .) + εw 1 (τ 0 , τ 1 , . . .) + . . . τ k = ε k τ, k = 0, 1, . . . (18)
A detuning parameter (σ) representing the nearness of the forcing frequency Ω to the reduced natural frequency of the LO is introduced:

Ω = 1 + εσ (19)
Substituting Eqs. ( 18) and ( 19) into Eqs. ( 16), ( 17) and equating coefficients of like power of ε gives:

Order ε 0 :

D 2 0 v 0 + v 0 = 0 D 2 0 w 0 + v 0 = 0, ∀|w 0 | < b (20) v + 0 = v - 0 , w + 0 = w - 0 , D 0 v + 0 = D 0 v - 0 , D 0 w + 0 = -rD 0 w - 0 , for |w 0 | = b (21)
Order ε 1 :

D 2 0 v 1 + v 1 = -2D 0 D 1 v 0 -λ 1 D 0 v 0 -w 0 + v 0 +G sin (τ 0 + στ 1 ) (22)
Here, the SIM will be obtained through the first order and the fixed points will be obtained by combining the first order and the second order as what have been done for cubic NES.

Tao Li

Paper VIB-15-1415

The first equation of system [START_REF] Gendelman | Attractors of harmonically forced linear oscillator with Tao Li Paper VIB-15-1415 15 attached nonlinear energy sink i: Description of response regimes[END_REF], taking into account Eq. ( 21) simply represents an undamped harmonic oscillator and its solution can be expressed as follow:

v 0 = C(τ 1 ) sin (τ 0 + θ(τ 1 )) (23)
The second equation of system [START_REF] Gendelman | Attractors of harmonically forced linear oscillator with Tao Li Paper VIB-15-1415 15 attached nonlinear energy sink i: Description of response regimes[END_REF] with ( 21) represents an harmonically forced impact oscillator with symmetric barrier.

Under the assumption of 1 : 1 resonance (i.e. motion with two symmetric impact per cycle ), its solution can be searched in the following form:

w 0 = C(τ 1 ) sin (τ 0 + θ(τ 1 )) + 2 π B(τ 1 )Π (τ 0 + η(τ 1 )) (24)
Where Π(z) is a non-smooth zig-zag function [START_REF] Pilipchuk | Some remarks on non-smooth transformations of space and time for vibrating systems with rigid barriers[END_REF]. This folded function and its derivative is depicted in Fig. 4 and are expressed as follows:

Π(z) = arcsin(sin z), M(z) = dΠ dz = sgn(cos z) (25)
[Fig. 4 about here.]

According to Eq. ( 24) and [START_REF] Gourc | Etude du contrôle passif par pompage énergétique sous sollicitation harmonique: Analyses théoriques et expérimentales[END_REF], impact occurs at T 0 = π/2η + jπ with j = 0, 1, 2, . . .. The impact condition |w 0 | = b is rewritten with Eq. ( 24) as:

C cos (η -θ) = b -B (26)
Rewriting now the inelastic impact condition (21) yields:

C (1 + r) sin (η -θ) = 2 π B (1 -r) (27)
Tao Li Paper VIB-15-1415 9

Combining Eqs. ( 26) and ( 27), a relation between the slow variables B and C presented in Fig. 5 is obtained as follow:

C 2 = 1 + 4 (1 -r) 2 π 2 (1 + r) 2 B 2 -2bB + b 2 (28)
[Fig. 5 about here.]

The analysis of the stability is fully developed in [START_REF] Gourc | Etude du contrôle passif par pompage énergétique sous sollicitation harmonique: Analyses théoriques et expérimentales[END_REF].

In order to obtain the fixed points or study the evolution of the system on the SIM for SMR, Eq. ( 22) at the next order of approximation is analyzed. To identify terms that produce secular terms, the function of w 0 is expanded in Fourier series in the following way:

w 0 = C(τ 1 ) sin (τ 0 + θ(τ 1 )) + E (τ 1 ) sin (τ 0 + ζ (T 1 )) + RFC ( 29
)
where RFC represents the rest frequency components. E (τ 1 ) is decided by the motion of VI NES.

Substituting Eq. ( 23), ( 24) and (29) into Eq. ( 22) and eliminating terms that produce secular terms gives:

D 1 C = -1 2 λ 1 C -1 2 E sin (Θ) + 1 2 G sin (η) (30)
where

Θ = ζ -θ η = στ 1 -θ (31)
Θ and η have the same meaning as defined in Eq. ( 12).

Remarks

A general governing equation of motion is obtained, specifically Eq. ( 11) for cubic NES and Eq. (30) for VI NES.

Therefore, although the topological structure of SIM is different, the underlying governing equation of motion in the slow time scale is the same. In another word, though the specific variation of motion for NES described by SIM is different, the Tao Li Paper VIB-15-1415 10

Fourier series development for NES and its variation may have some similarities showed in the general governing equation.

In short, the classic method is extended by relaxing the condition of 1:1 resonance. It can be used to explain the variation of complicated response (e.g. SMR) in which the variation of motion (LO and NES) is not steady.

In this way, the first order (e.i. ε 0 ) can be used to obtain SIM which represents the set of fixed points for all possible initial conditions and parameters. The second order (e.i. ε 1 ) is used to get slower invariant manifold and combined with the first order to obtain fixed points for specific parameters and conditions.

Numerical Analysis

In this part, the objective is to show the important role of SIM to control the variation of motion of system under transient and periodic forcing by numerical simulations, based on which some critical characteristics of SMR can be detected and explained by the general governing equation obtained in the last part.

The simulations will be based on the same LO coupled to cubic and VI NES separately under transient (G = 0) and periodic (G = 0) forcing with the following fixed parameters: λ 1 = 1.43, ε = 0.84% for cubic and VI NES, λ 2 = 0.13 for cubic NES.

It has to be pointed out that v ≈ x considering v = x + εy with ε = 0.84% in the following analysis.

Numerical Results for Cubic NES

The simulation will be based on the Eq. ( 3). For transient forcing, different types of response can exist by varying the parameters. Herein, just the regime (TET excited) is showed in Fig. 6. The history of response can be categorized into two parts, that is A and B showed in Fig. 6 (a). The TET is excited in the part A, during which the displacement of x decreases with an almost linear slope and the amplitude of cubic NES is much bigger than that in the unexcited part B as showed in Fig. 6 (b). In Fig. 6 (d), the motion of system is projected to SIM and it can predict the motion in the part A as well. However, it cannot predict the part B with escape from the 1:1 resonance.

[Fig. 6 about here.]

The above phenomena can be clearly explained by Eq. [START_REF] Gourc | Targeted energy transfer under harmonic forcing with a vibro-impact nonlinear energy sink: Analytical and experimental developments[END_REF]. During the whole process, the value of outside force is zero. For part A, the decrease of N 1 (amplitude of v) at the slow time scale is decided by the value of N 3 (large amplitude of cubic NES) and the slope is almost straight because of almost constant value of N 3 . In part B, the value of N 3 becomes very small, N 1 (i.e. damping) plays the main role for its reduction.

For periodic forcing, just the results of SMR are demonstrated in Figs. 7 and8 .

The response can be categorized into two parts (A and B) according to the increase and decrease of the amplitude of v showed in Fig. 7(a) . In the part A, the amplitude of y and w is small compared to that in the part B demonstrated respectively in Fig. 7(b) and (c). In Fig. 7(d), the part A is divided into three small parts. For sub-part A2, 1:1 resonance exists and the value of y is still small. For parts A1 and A3, there do not exist 1:1 resonance. Globally, the SIM predicts the 1:1 resonance parts A2 and B well. It is difficult to explain SMR as stated in [START_REF] Gendelman | Attractors of harmonically forced linear oscillator with Tao Li Paper VIB-15-1415 15 attached nonlinear energy sink i: Description of response regimes[END_REF], especially about the parts connecting these TRCs in which the hypothesis 1:1 resonance does not hold. With the general governing Eq. ( 11) developed in the former part, all parts can be explained to some extent. For part B, the left part Eq. ( 11) will be less than zero because of large value of N 3 and fixed phase difference which means the decrease of amplitude of LO in the slow time scale. For part A, the small value of N 3 with varying phase difference will result in the positive value of left part of Eq. ( 11) that leads to the increase of amplitude of LO, in which the outside force plays the main role.

Numerical Results for VI NES

Numerical simulations for VI NES are based on Eqs. ( 13) and ( 14). The b related to the length of cavity is varied to detect different regimes.

For transient forcing, different regimes are observed. One typical response with TET activated is showed in Fig. 9.

The difference for VI NES from cubic NES is that the former is piece-wise. However, the similar two parts are observed and can be explained by the general governing Eq. (30) as have been done for cubic NES.

[Fig. 9 about here.]

For periodic forcing, just a SMR similar to that of cubic NES is showed in Fig. 10. It is studied in detail in [START_REF] Gourc | Etude du contrôle passif par pompage énergétique sous sollicitation harmonique: Analyses théoriques et expérimentales[END_REF] and is named as chaotic SMR in [12]. It is seen that the local maximum value is different every time in Fig. 10(a) and Fig. 11(a), which is different from that of cubic NES. The 1:1 resonance can be observed from the phase difference between v and y demonstrated in Fig. 11(b). Moreover, the slow variation of the phase difference shows the move between different 1:1 resonance responses. In Fig. 11(c)(d), the frequency variation of v and y is obtained from HT by excluding the big noise that represents the impact moments.

Globally, the law of the variation for frequency and amplitude is similar to these of cubic NES, therefore it is can be explained in the same way.

[Fig. 10

Remarks

The general governing equation is used to explain the non 1:1 resonance for both cubic NES and VI NES, specifically the case under transient forcing and the SMR under periodic forcing. The above analysis process demonstrates its facility to understand the variation of non 1:1 resonance for numerical results. In the next part, its ability to explain the experimental results will be showed.

Experimental verifications

[Table 1 about here.]

The experimental setup is presented in Fig. 12(b). It consists of a LO, with an embedded VI NES. The whole system is embedded on 10kN electrodynamic shaker. The displacement of the LO as well as the imposed displacement of the shaker are measured using contact-less laser displacement sensors. The acceleration of the LO is mearsured by a accelerometer. A detailed view of the VI NES is presented in Fig. 12(a). It simply consists of a closed cavity of length d + 2b, where d is the diameter of the ball. Each cover is made of hardened steel. The parameters of the system have been identified by performing modal analysis and are summarized in Table 1.

[Fig. 12

about here.]

A chaotic SMR about VI NES is showed in Fig. 13. The forcing amplitude is fixed to 0.423mm and the forcing frequency is fixed to 8Hz. The blue line in Fig. 13(a) represents the acceleration of LO and the green cross dots represent the times and the values of the impact moments. It is seen that the LO and VI NES are in 1:1 resonance by two impacts during one period of LO except some intervals with no impact during one period of LO, specifically the intervals around t and t ∈ {0.4s, 2.4s, 4.7s}. Another phenomenon is that the value of the acceleration in the impact moment for 1:1 resonance change continuously.

[Fig. 13 alternatively rather than remains constant. An enlarged view shows that if the value of acceleration is big enough, it will decrease, or it will increase when there is not enough number of impacts or the value of acceleration in the impact moment is not big enough.

In Eq. ( 30), the value of G in Eq. ( 30) is directly related to that of acceleration in the impact moment and the density of impact. The amplitude of LO will change according to that of G. This equation can explain the above experimental phenomenon well.

Tao Li

Paper VIB-15-1415 13

This paper is dedicated to generalize the classic analytical method to better understand the non 1:1 resonance response regimes from numerical study and the complicated experimental results. Firstly, a general analytical method extended from classic method is obtained through relaxing the condition of 1:1 resonance and it is applied to cubic and VI NES. Then the developed analytical method is used to explain the excited TET under transient forcing and the SMR under periodic forcing for these two NES. Finally, a chaotic SMR from the experimental results is used to demonstrate the feasibility of the general analytical method.

A general governing equation of motion is obtained and its ability to explain the non 1:1 resonance is demonstrated by two examples in numerical study, namely excited TET in transient case and the SMR in the periodic case. Its facility applied to the complicated experimental results is validated by its ability to explain the SMR detected in experiments.

In addition, the experimental results confirm the existence of the SMR encountered for VI NES and provide much more information than in [START_REF] Gourc | Targeted energy transfer under harmonic forcing with a vibro-impact nonlinear energy sink: Analytical and experimental developments[END_REF]. The amplitude variation of LO and time variation of impact moment give experimental proof to chaotic properties of SMR defined in [12]. Moreover, since the 1:1 resonance is directly related to the TET, the experimental results show that VI NES is an ideal candidate to illustrate this characteristic. For VI NES, whether the system is in 1:1 resonance is judged from the impact directly rather than using waveform transform.

Although this method can extend the ability of classic analytical method, it still cannot obtain analytical results for the complicated regimes like SMR or experimental phenomena. For example, it could be known from this method that the case with more force in the moment of impact and 1:1 resonance between LO and VI NES is better, but how to ensure this by precise analytical calculation is still not clear.

  here.] More detailed characteristics (e.g. frequency and amplitude) can be obtained in the frequency domain by using the Hilbert Transform (HT) and Wavelet Transform (WT) function in Matlab[START_REF] Poularikas | Handbook of formulas and tables for signal processing[END_REF]. The result is showed in Fig.8. Specifically, the envelope of v and y is showed in Fig.8(a), through which the value change of v and y is clearly demonstrated. In Fig. 8(b), two intervals of 1:1 transient resonance capture (TRC) is clearly observed by almost fixed phase difference between v and y and they are interrupted by escape of resonance. In Fig. 8(c)(d)(e)(f), the 1:1 resonance part of the same frequency between v and y is clearly illustrated and the resonance frequency is showed by the red dotted line.

  about here.] [Fig. 11 about here.] Tao Li Paper VIB-15-1415 12

  about here.] In Fig.13(b), the displacement of LO, the acceleration of LO and the displacement of forcing are superimposed. One enlarged view of Fig.13(b) is showed in Fig.13(c). It is observed that the displacement of LO increases and decreases

2 Fig. 1 .Fig. 2 . 1 LOFig. 3 .Fig. 4 .

 212134 Fig. 1. Representation of the LO coupled to a cubic NES.

Fig. 5 .Fig. 6 .Fig. 7 .Fig. 8 .Fig. 9 .Fig. 10 .Fig. 11 .Fig. 12 .Fig. 13 .

 5678910111213 Fig. 5. SIM of VI NES: one stable branch in blue line and two unstable branches in red line.

Table 1 .

 1 Parameters of the experiment

		Physical Parameters
	m 1	4.168 kg	c 1 3.02 Ns/m
	k 1 11.47 * 10 3 N/m m 2	32 g
	b	11.5 mm	r	0.6
		Reduced Parameters
	ε	0.76%	λ	1.80

Acknowledgements

The authors acknowledge the French Ministry of Science and the Chinese Scholarship Council under grant N • 201304490063 for their financial support.