Tao Li

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

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

On the dynamics around targeted energy transfer for vibro-impact nonlinear energy sink

Keywords: Targeted energy transfer, Impact damper, Vibro-impact nonlinear energy sink, Slow invariant manifold, Strongly modulated response, Intermittency, Chaos

A periodically forced linear oscillator with impact attachment has been studied. An asymptotical analytical method has been developed to obtain the fixed points and to analyze the transient 1:1 resonance (two impacts per cycle) of the modulated response. The influence of parameters on dynamics has been analyzed around the Slow Invariant Manifold (SIM). Five different response regimes have been observed from theoretical and numerical results. It is demonstrated that they are closely related to the topological structure and relative position of fixed points. The bifurcation, route to chaos and the efficiency of Targeted Energy Transfer (TET) with the variation of different parameters (i.e. amplitude and frequency of excitation, clearance, damping, mass ratio and restitution coefficient) have been investigated and well explained around SIM. Experimental results validate the existence of different regimes and different routes to chaos by the variation of the return map of time difference between consecutive impact moments. TET phenomenon has been analyzed for a strongly modulated response and different cases of TET have been observed and analyzed. It is clearly observed that TET depends not only

Introduction

Tuned Mass Damper (TMD) is an energy reduction device with a mass connected to main system by a linear spring conceived by Frahm [START_REF] Frahm | Device for damping vibrations of bodies[END_REF]. This device has been extensively studied and proved that it is simple and efficient but limited in the vicinity of a single frequency [START_REF] Sun | Passive, adaptive and active tuned vibration absorbersa survey[END_REF]. In contrast, nonlinear vibration absorber is proved effective to broaden the suppression bandwidth [START_REF] Roberson | Synthesis of a nonlinear dynamic vibration absorber[END_REF]. In the past approximately fifteen years, the phenomenon of Targeted Energy Transfer (TET) existing in a series of nonlinear vibration absorber called Nonlinear Energy Sink (NES) is considerably studied [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]. Many kinds of NES with different couple nonlinearities are developed and investigated: cubic NES [START_REF] Gendelman | Energy pumping in nonlinear mechanical oscillators: Part i: Dynamics of the underlying hamiltonian systems[END_REF][START_REF] Vakakis | Energy pumping in nonlinear mechanical oscillators: Part ii: Resonance capture[END_REF][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], rotational NES [START_REF] Gendelman | Dynamics of an eccentric rotational nonlinear energy sink[END_REF][START_REF] Sigalov | Resonance captures and targeted energy transfers in an inertially-coupled rotational nonlinear energy sink[END_REF], piece-wise NES [START_REF] Lamarque | Targeted energy transfer in mechanical systems by means of non-smooth nonlinear energy sink[END_REF], NES by nonlinear membrane [START_REF] Bellet | Experimental study of targeted energy transfer from an acoustic system to a nonlinear membrane absorber[END_REF][START_REF] Bellet | Enhancing the dynamic range of targeted energy transfer in acoustics using several nonlinear membrane absorbers[END_REF] and Vibro-Impact (VI) NES [START_REF] Nucera | Targeted energy transfers in vibro-impact oscillators for seismic mitigation[END_REF][START_REF] Lee | Periodic orbits, damped transitions and targeted energy transfers in oscillators with vibro-impact attachments[END_REF], in which special orbits for the occurrence of TET is observed by studying the underlying Hamiltonian system and the energy of the main system can be irreversibly transferred into attached VI NES and dissipated effectively during transient resonance captures. Actually, the main damping mechanism of VI NES is that the energy of main system is transferred and dissipated at the same moments of impact.

VI NES is also referred as impact damper and its dynamics have been extensively studied since nearly seventy years ago [START_REF] Ibrahim | Vibro-impact dynamics: modeling, mapping and applications[END_REF][START_REF] Babitsky | Theory of vibro-impact systems and applications[END_REF] since the initial study about an acceleration damper (early name of the impact damper) by Lieber [START_REF] Lieber | An acceleration damper: development, design and some applications[END_REF]. The principle is to transfer the energy of the main system to a small attached mass and to dissipate the energy by their mutual impact interactions. Although there exist different types of impact models [START_REF] Blazejczyk-Okolewska | Classification principles of types of mechanical systems with impactsfundamental assumptions and rules[END_REF], it has been proven that there is a similarity of several important vibro-impact systems under certain specific limits or constraints [START_REF] Bapat | Several similar vibroimpact systems[END_REF].

The research of impact damper has been around two main themes. The first one is about the dynamics: response regimes and stability by analytical study in [START_REF] Masri | On the stability of the impact damper[END_REF] and with experimental validation in [START_REF] Popplewell | Stable periodic vibroimpacts of an oscillator[END_REF][START_REF] Bapat | Stable periodic motions of an impact-pair[END_REF], bifurcation and chaos by numerical study with the combination use of time series, phase trajectories, bifurcation diagrams [START_REF] Peterka | More detail view on the dynamics of the impact damper[END_REF], Poincare maps [START_REF] Sung | Dynamics of a harmonically excited impact damper: bifurcations and chaotic motion[END_REF] and Lyapunov exponential [START_REF] Yoshitake | Periodic solutions, bifurcations, chaos and vibration quenching in impact damper[END_REF]. Similar study has been carried out for the impact oscillator by Pekerta [START_REF] Peterka | Bifurcations and transition phenomena in an impact oscillator[END_REF]. Based on the above mentioned work, the first point of view on the relationship between parameters and dynamics is established.

Another topic of research is concentrated on the efficiency of energy dissipation for free or forced vibration [START_REF] Bapat | Single unit impact damper in free and forced vibration[END_REF][START_REF] Brown | The impact damped harmonic oscillator in free decay[END_REF]. The influence of system parameters (e.g. mass ratio, clearance and coefficient of restitution) is considerably investigated. The finite experimental results are concentrated on response regimes by measuring the acceleration of main system [START_REF] Bapat | Single unit impact damper in free and forced vibration[END_REF] or displacement of impact damper directly [START_REF] Ekwaro-Osire | Performance of a bi-unit damper using digital image processing[END_REF]. To evaluate the efficiency of TET, the second point of view about the relationship between parameters and the energy (e.g. amplitude) of main system is constructed.

Inspired by the extensive study in the domain of TET, an asymptotic analytical method originally used for cubic NES [START_REF] Gendelman | Bifurcations of nonlinear modes of linear oscillator with strongly nonlinear damped attachment[END_REF] is improved to explain the transient TET process for linear oscillator coupled with VI NES under free excitation [START_REF] Gendelman | Analytic treatment of a system with a vibroimpact nonlinear energy sink[END_REF], i.e. inconstant amplitude and frequency spectrum. The study of VI NES was extended under periodic excitation [START_REF] Gourc | Targeted energy transfer under harmonic forcing with a vibro-impact nonlinear energy sink: Analytical and experimental developments[END_REF][START_REF] Gendelman | Dynamics of forced system with vibro-impact energy sink[END_REF] and for quenching chatter instability [START_REF] Gourc | Quenching chatter instability in turning process with a vibro-impact nonlinear energy sink[END_REF].

Slow Invariant Manifold (SIM) describing all possible fixed points and possible variation routes is obtained through this analytical method in the first order of the time scale, and the fixed points are obtained by combining the equations in the first and second order. The condition of 1:1 resonance between main system and VI NES (i.e. two impacts per cycle) required for the application of this method is relaxed to cover the more general case (i.e. non 1:1 resonance) [START_REF] Li | Dynamics of cubic and vibro-impact nonlinear energy sink: Analytical, numerical, and experimental analysis[END_REF]. Actually, this asymptotic approach describes the impact system from another point of view. However, the recent research about the impact damper as NES with this new approach is limited to study the topological structure of SIM and possible response regimes. Other aspects of its dynamics (e.g. bifurcation, chaos and optimization) should be studied further and the past work about impact damper should be understood with new horizons. Therefore, study of the dynamics from this new point of view will be the first objective of this paper. Another goal will be concentrated on the efficiency of TET. It is observed that resonance is the essence for all NES from the perspective of TET and the type of resonance decides its efficiency [START_REF] Vakakis | Nonlinear targeted energy transfer in mechanical and structural systems[END_REF]. The 1:1 resonance regime is claimed and proved to be the most efficient. Considering two impacts per cycle for VI NES, it may be the ideal candidate to illustrate the 1:1 resonance and variation of efficiency of TET with more clear and direct evidence.

The system composed of a linear oscillator and a VI NES (i.e. impact damper) will be studied in this paper. In the section two, modelling and the analytical treatments are developed. In the section three, the influence of parameters on dynamics and energy reduction is analyzed by analytical and numerical results. In the next section, the experimental results are presented. Finally, the conclusion is addressed.

Modelling and analytic treatment

System of harmonically forced Linear Oscillator (LO) attached with VI NES is presented in Fig. 1 and described by the following equation:

ẍ + ελ 1 ẋ + x = εG sin Ω τ + ε 2 λ 1 GΩ cos Ω τ ε ÿ = 0 ∀ |x -y| < b (1)
The corresponding physical parameters are expressed as follows:

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
where x, m 1 , c 1 and k 1 are the displacement, mass, damping and stiffness of the LO respectively. y and m 2 are displacement and mass of VI NES. The dots denote differentiation with respect to dimensionless time τ. b represents the clearance. x e (t) = F sin (Ω t) is the displacement imposed on the base by the shaker.

When |x -y| = b, an 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 (2)
where r is the restitution coefficient and the superscripts + anddenote 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 (3)
Substituting Eq. (3) into Eqs. ( 1) and ( 2), the equation between impacts in barycentric coordinate are given as:

v + ελ 1 v + ε ẇ 1 + ε + v + εw 1 + ε = εG sin Ω τ + ε 2 λ 1 GΩ cos Ω τ ẅ + ελ 1 v + ε ẇ 1 + ε + v + εw 1 + ε = εG sin Ω τ + ε 2 λ 1 GΩ cos Ω τ ∀|w| < b (4)
and the impact condition (2) is rewritten as:

v + = v -, w + = w -, v+ = v-, ẇ+ = -r ẇ-, for |w| = b (5)
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, . . . (6)
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 (7)
Substituting Eqs. ( 6) and [START_REF] Vakakis | Energy pumping in nonlinear mechanical oscillators: Part ii: Resonance capture[END_REF] into Eqs. ( 4) and ( 5), then equating coefficients of like power of ε gives:

(z) Π(z) M(z)
Fig. 2 Representation of the non-smooth functions Π (z) and M(z) Order ε 0 :

D 2 0 v 0 + v 0 = 0 D 2 0 w 0 + v 0 = 0, ∀|w 0 | < b (8) 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 (9)
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 + σ τ 1 ) (10)
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 have been done in [START_REF] Gourc | Targeted energy transfer under harmonic forcing with a vibro-impact nonlinear energy sink: Analytical and experimental developments[END_REF].

For v 0 , Eq. 8 and Eq. 9 simply represent an undamped harmonic oscillator case and its solution can be expressed as follow:

v 0 = C(τ 1 ) sin (τ 0 + θ (τ 1 )) (11)
For w 0 , Eq. 8 and Eq. 9 represent a 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 )) (12)
Where Π (z) is a non-smooth saw tooth function [START_REF] Pilipchuk | Some remarks on non-smooth transformations of space and time for vibrating systems with rigid barriers[END_REF][START_REF] Pilipchuk | Closed-form solutions for oscillators with inelastic impacts[END_REF]. This folded function and its derivative is depicted in Fig. 2 and are expressed as follows:

Π (z) = arcsin(sin z), M(z) = dΠ dz = sgn(cos z) (13)
According to Eq. ( 12) and ( 13), impact occurs at T 0 = π/2-η + jπ with j = 0, 1, 2, . . . The impact condition |w 0 | = b is rewritten with Eq. ( 12) as:

C cos (η -θ ) = b -B (14)
Rewriting now the inelastic impact condition (9) yields:

C (1 + r) sin (η -θ ) = 2 π B (1 -r) (15)
Combining Eqs. ( 14) and ( 15), a relation between B and C is obtained as follows:

C 2 = 1 + 4 (1 -r) 2 π 2 (1 + r) 2 B 2 -2bB + b 2 (16)
An example of a SIM with b = 1 and r = 0.6 is presented in Fig. 3. The stability of SIM is analyzed by the approach used in [START_REF] Masri | On the stability of the impact damper[END_REF] , in which the stable branch is defined by the condition that the modulus of all the eigenvalues of a certain matrix relating conditions after each of two consecutive impacts is less than unity. And it also can be evaluated by direct numerical integration of Eq. [START_REF] Vaurigaud | Targeted energy transfer with parallel nonlinear energy sinks. part i: Design theory and numerical results[END_REF].

In order to obtain the fixed points or study the evolution of motion of system on the SIM for Strongly Modulated Response (SMR), Eq. ( 10) 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 form:

w 0 =C(τ 1 ) sin (τ 0 + θ (τ 1 )) + E (τ 1 ) sin (τ 0 + ζ (T 1 )) + RFC (17)
where RFC represents the rest frequency components compared to the first two terms. E (τ 1 ) is decided by the motion of VI NES. The Eq. ( 17) is a more general and relaxed analytical description with respect to the motion of VI NES compared to the case of just 1:1 resonance (i.e. two symmetric impacts per cycle). If the case of 1:1 resonance exists, the fixed points can be analytically obtained. However, only partial duration of the whole process is in resonance for response regime of transient loading or SMR. In these two cases, the above relaxed equation is useful and efficient to explain the underlying mechanism.

Substituting Eq. ( 11), ( 12) and ( 17) into Eq. ( 10) 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) D 1 η = 1 2 G cos (η) /C - 1 2 E cos (Θ ) /C + σ (18)
where

Θ = ζ -θ η = σ τ 1 -θ (19)
Θ represents the phase difference related to that between LO and VI NES. η represents the phase difference related to that between LO and outside excitation.

3 Analytical and numerical results

Response regimes

The experimental identified parameters of the system in Table 1 (see Sec. 4) will be used. The regime is labeled by the quantity z = p/n, where p is the number of impacts and n is the number of excitation periods T during the considered time [START_REF] Peterka | More detail view on the dynamics of the impact damper[END_REF]. Five different response regimes exist with the variation of clearance under outside excitation frequency of 8.5 Hz and they are categorized by the value of z as showed in Fig. 4. The first, second and third column represents the displacement of LO, relative displacement and projection of motion (i.e. yellow lines) into SIM respectively. z can be observed from the second column combined with the first column, i.e. the total occurrence times of extreme relative displacement value (i.e. b or -b) during one period of LO. Fig. 4(a) is chosen as representation for all case with z > 2, in which z = 3 and z = 4 co-exist. In this case, the time difference between impacts is not equally distributed as showed in the middle figure and the stable projected motion is located at the right unstable branch of SIM and it is For the part z = 2, the case with two impacts per cycle occurs and its variance is governed by the blue stable branch. For the part z < 2, the case with sparse impacts occurs, its variation law is not known. This irregularity is also demonstrated by the variation of amplitude of LO in the left figure, which reveals the chaotic characteristic of SMR. Fig. 4(e) demonstrates loose impact case with z < 2, in which transient 1:1 resonance with continuous two impacts per cycle does not exist. In this area, the response is chaos. From the variation of impact number and amplitude of LO in Fig. 4, it is seen that the average value of z with long time duration decreases and the average value of amplitude of displacement (x) decreases at the first place and then increases.

Areas of SIM

From the third column of Fig. 4, it is seen that the relative position of motion projected into SIM varies with clearance.

According to the value of C 2 , 4 areas can exist. This law of variation by numerical study can also be analytically obtained and the result is showed in Fig. 5(a). With the increase of clearance, one of the fixed point represented by red circle will move along the SIM from area Z1 with z > 2 to z = 2 through a series of grazing bifurcation. In area Z2, the regime with two asymmetric impacts per cycle persists until the critical point between stable blue branch and the right unstable red branch, in which the double period bifurcation occurs. Then the regime with two symmetric impacts per cycle occurs. If the above process is continued, there exist two routes entering into chaos. The first case is showed in Fig. 5(b), the two unstable points will approach each other and then disappear. The response regime is SMR demonstrated in Fig. 4(d) and the duration time of transient 1:1 resonance part will decrease with the increase of clearance, which is numerically observed by an outside excitation frequency 8.5 Hz. Another case is demonstrated in Fig. 5(c) with an outside excitation frequency 8 Hz. Two fixed points exist in the stable branch and then disappear by collision with each other. During the evolution of this case, there exist two possible solutions, i.e. two symmetric impacts per cycle and SMR. The regime chaos in area 4 will appear after the collision of fixed points at the end of the above two cases with further increase of clearance. The above analysis process for the variation of regimes can be applied to other parameters, although the specific variation route will depend on the specific parameters. In return, the different bifurcations and routes to chaos can be explained with the above analysis approach.

Transition phenomena and optimization criteria

From the above analysis, there exist two possible transition routes to chaos with two different transition regions, which are respectively defined as hysteresis region and beat motion. The former corresponds to the Fig. 5(c) in which the response regime is ambiguous (1:1 resonance or SMR). The latter corresponds to the Fig. 5(b) characterized by the SMR of main system [START_REF] Peterka | Bifurcations and transition phenomena in an impact oscillator[END_REF]. Two boundaries can be analytically obtained by two bifurcation conditions (i.e. the bifurcation showed in Fig. 6(a) and the bifurcation demonstrated in Fig. 6(b)). The first one is defined by the contact point between the left unstable red branch and the stable blue branch and the other condition is obtained by the collision of the two fixed points. For a frequency bandwidth, the above two transition regions are obtained and showed in Fig. 6(c). A1 represents the hysteresis region, which is numerically validated by co-existence of SMR and 1:1 resonance, which has been experimentally observed in [START_REF] Bapat | Stable periodic motions of an impact-pair[END_REF]. A2 represents the beat motion region. The point P1 represents the singular point in which the continuous transition without SMR (i.e. no intermittency) exists. From the variation of amplitude of LO in Fig. 4, it is reasonable to assume that the optimal energy dissipation exists at the point where 1:1 resonance disappears, which is represented by the red cross in Fig. 6(c). Different response regimes in different areas (i.e. Z1, Z2, Z3 and Z4) are also validated by numerical results and they are represented by the squares, circle and star.

Frequency and SIM

With the variation of relative value between the amplitude of outside force and clearance (i.e. G/b), the relative position of Frequency Response Function (FRF) of the main system will vary and it will result in the change of the projected motion location in the SIM. This phenomenon has been experimentally observed in [START_REF] Gourc | Targeted energy transfer under harmonic forcing with a vibro-impact nonlinear energy sink: Analytical and experimental developments[END_REF]. For a fixed G, this relative location will change from area Z1 to Z4 (i.e. from black thick dotted curve to blue thin curve) with the increase of b as showed in Fig. 7. For any relative location, just some interval of frequency can situate in area Z3 (e.g. red fine dotted curve and green thick curve), in which the most efficient energy reduction by TET is possible, no matter in the form of resonance capture by the 1:1 resonance regimes or just one part of the process in resonance capture by SMR. This is the reason why one optimal designed parameter is not the still optimal for another frequency. The fact is that the 1:1 resonance regime for all range of frequency is not possible. Another phenomenon is about possible types of the bifurcation and route to chaos during sweep experiments (e.g. observed in [START_REF] Yoshitake | Periodic solutions, bifurcations, chaos and vibration quenching in impact damper[END_REF]). All possible cases can be explained clearly. For the case of fine blue line in Fig. 7(b), there will not exist any route to chaos. However, there will be two possible routes for the case represented by the thick black dotted line, i.e. the first route can go up along the line as follows: two symmetric impacts, then two asymmetric impacts, then increase the number of impacts until total chaos. The second route can go down along the line in the following way: two symmetric impacts, then SMR, then decrease of the duration time for the 1:1 resonance part of SMR until total chaos. The typical cases with fine red dotted line and thick green line can be a little different but can be explained in the same way. For the past research about parameter optimization for a frequency range around resonant frequency, different cases have be observed and explained. However, the study is incomplete and the goal is not clear. From Fig. 7, the optimization criterial is clear and is just to make the objected bandwidth of frequency locate in the area Z3 as more as possible. Normally, the objected bandwidth of frequency is around natural frequency of LO, therefore the case with fine red fine dotted line is the better.

Influence of mass ratio and damping

The SMR projected in SIM is showed in Fig. 8(a). More information about variation mechanism of the displacement of LO and intern displacement can be seen in Fig. 4(d). In the stable branch with thick blue curve, the good prediction (correlation between theoretical blue curve and numerical green curve) is seen and enlarged in Fig. 8(c), which cannot be obtained by the traditional analytic approach comprehensively summarized in [START_REF] Ibrahim | Vibro-impact dynamics: modeling, mapping and applications[END_REF]. For the analytical method firstly developed in [START_REF] Gendelman | Analytic treatment of a system with a vibroimpact nonlinear energy sink[END_REF], its objective is to analyze the transient resonance capture (two impacts per cycle) during free vibration and it is applied to periodically forced vibration. Its application requires small magnitude of mass ratio and damping. For the transition part between non 1:1 resonance and 1:1 resonance showed in Fig. 8(b) and the stable part showed in Fig. 8(c), the vertical value difference between every vertical yellow line is decided by the energy dissipation of impact related to mass ratio and restitution coefficient (the influence of restitution will be explained later) and that between every horizontal yellow line is decided by the energy dissipation related to the damping of LO. Large energy reduction resulted from large mass ratio and damping will result in the failure of this approach, i.e. the prediction of SIM to projected motions, although it may increase the efficiency of energy reduction. From Fig. 4(d), it is seen that the response is strongly modulated and the local maxima changes every time, which is defined as chaotic SMR in [START_REF] Gendelman | Dynamics of forced system with vibro-impact energy sink[END_REF]. To explain its variation trend, the non 1:1 resonance motion of VI NES should be considered and its average action force on the LO should be developed in Fourier series. Its small amplitude in Eq. ( 17) reveals the cause for the increase of the amplitude of main system. This way of treatment is also fairly convenient to understand experimental results which will be explained in the experimental part. In addition, a method applying limiting phase trajectories by studying the underlying Hamiltonian system is used to explain the formation of SMR in a duffing system under biharmonic excitation [START_REF] Starosvetsky | Nonstationary regimes in a duffing oscillator subject to biharmonic forcing near a primary resonance[END_REF]. Actually, it is expected that the optimal point obtained by this method could be closely related to the limit point entering into SMR from two impacts per cycle for the system with VI NES. Here, the interaction force of the impact moment is stressed here to explain the formation process.

Influence of restitution coefficient

From Eq. ( 16) and Fig. 3, there exists a minimal value for the occurrence of 1:1 resonance. When the restitution of coefficient is varied with the amplitude of LO (black dotted line) and other parameters fixed, the relative position of SIM will vary as showed in Fig. 9(a). When r = 1 showed in red fine curve, it will locate in area Z1, in which the impact number is large but with low impact impulse strength. As a result, the energy dissipation is not high. With the decrease of r, the transition will follow along the black arrow in Fig. 9(b). When r = 0.2 showed in black thick dotted curve, the impact number will be low and impact pulse may be large. Globally, it will result in low efficiency. Therefore, an intermediate value will be optimal and lead to a maximal value of E in Eq. ( 17).

Experimental results

The experimental setup is presented in Fig. 10(a). It consists of a LO, with an embedded VI NES. The whole system is embedded on 10 kN electrodynamic shaker. The displacement of the LO as well as the imposed displacement of the shaker are measured using contact-less laser displacement The cylinder and the other side cover are made of hardened steel. The parameters of the system have been identified by performing modal analysis and are summarized in Table 1. The excitation frequency is fixed to 8 Hz.

Route to chaos

The route to chaos can be judged in different ways such as those showed by the variation of response in the different columns of Fig. 4, but they require the displacement and velocity of VI NES to be known in advance, which is not allowed and limited by measure device. Another way is to use a return map of time duration between every two consecutive impacts [START_REF] Korsch | Chaos: a program collection for the PC[END_REF]. For a limited time of data acquisition with the excitation period given, the time of impact t i can be measured as showed by red stars in right subfigure of Fig. 11.

The acceleration of LO (blue curve) will change abruptly and then the time difference between these moments δt i can be calculated. Two neighboring time difference δt i can form one point (δt i , δt (i+1) ), and every two points can constitute a vector with the first point (δt i , δt (i+1) ) and the second point (δt (i+1) , δt (i+2) ).

With the variation of b, different response regimes have been observed. All left subfigures in Fig. 11(a-f) represent the time difference return map, the yellow dotted triangle can be expressed by L1: δt x + δt y = T (1/8 s) , L2: δt x = T (1/8 s) and L3 δt y = T (1/8 s). The black dotted line can be expressed by L4: δt x = δ y . All right subfigures in Fig. 11(af) represent a typical sample of the acceleration of LO, in which the red stars denote the time of impact and the value of acceleration. Fig. 11(g) demonstrates the relative position of every regime in SIM. For b = 0.3 mm, the results are showed in Fig. 11(a), there exist more than 3 impacts during one period, and the distribution of impact time is also irregular, which is judged as chaos. For b = 2.5 mm showed in Fig. 11(b), there exist three impacts every period and the impact time distribution is not equal but stable which is illustrated by the fact that all vector form in a triangle, meanwhile all points are inside L1. For b = 4 mm in Fig. 11(c), the vector is located at the L1, which means that it just exist two impacts every period. However, the absolute value of acceleration is not equal. For b = 7.5 mm in Fig. 11(d), there also exist two impacts per cycle, but this time the absolute value of acceleration is almost equal. For b = 11.5 mm in Fig. 11(e), the case of two impact per cycle and no impact appears alternatively and the duration time of 1:1 resonance is short and irregular. For b = 20 mm in Fig. 11(f), all points is in the right and up side of L1, it means that two impacts per cycle does not exist during the whole process, i.e. transient resonance captures does not exist.

In summary, the relative position in the SIM moves with the increase of clearance as observed by the study in the last section. The value of acceleration (i.e. impulse strength) is small although with many impacts during one period for small clearance. Then it augments with the two impacts per cycle, and the efficiency of TET is the largest. A further increase of clearance, instantaneous large value (i.e. strong impulse strength) can exist, but it cannot persist during the whole process. From analytical analysis, the number and strength of impact impulse can decide the value of E in Eq. ( 17) directly and it will then decide the efficiency of TET, which is validated by experimental results here.

TET during SMR

A SMR with b = 15 mm is demonstrated in Fig. 12, which is characterized by the intermittency of 1:1 resonance (i.e. two impacts per cycle) and the chaotic strongly modulated amplitude of LO. In Fig. 12(a), the acceleration of LO with the duration 5 s is demonstrated by the blue curve and meanwhile the points of the impact moments represented by red stars illustrate the time and acceleration value of impact moments.The intermittent and chaotic characteristic is demonstrated by the irregular occurrence and duration time of 1:1 resonance as showed by the distribution of impact points in Fig. 12(a) and also by transient escape in the return map of time difference showed in Fig. 12(b). In Fig. 12(c), the black curve and red curve represent the displacement (mm) of shaker and LO respectively, which is superimposed with the acceleration of LO. Here, the influence of the amplitude and frequency of impact to the energy reduction is showed by the variation of the amplitude of LO. The transient resonance capture is seen by two impacts per cycle, in which TET is possible. In the first equation of Eq. ( 18), the amplitude variation of main system at the slow time scale is decided only by the amplitude (i.e. E) which is obtained by Fourier transfer and has the same frequency as that of LO, provided that the damping of LO and outside excitation are fixed. For the decreasing period of the amplitude of LO, two impacts per cycle and large value of impact impulse strength are required to guarantee the condition of 1:1 resonance and large impact damping as showed in Fig. 12(e). Otherwise, it will increase for not enough dense impacts illustrated in Fig. 12(f) or not big enough impact impulse even with two impacts per cycle as showed in Fig. 12(d). Therefore, although 1:1 resonance (two impacts per cycle) is necessary for TET, their efficiency is different for different 1:1 resonance regimes as is showed by the difference of acceleration value for the same 1:1 resonance regime from the above experimental results and is also analytically represented by different points at the stable blue branch in Fig. 3. Or it is reasonable to suppose that the most efficient case is the one with not only 1:1 resonance but also the largest impact impulse strength in impact moments as is may be presented by the optimal curve proposed in Fig. 6, which is out the scope of this paper and need be studied further.

If we look closely, there exist small peaks of the acceleration during the period defined by no impacts period as has been used in the analysis, which is showed for the time around t = 2.7 s in Fig. 11(a). This phenomenon is caused by the small inclination of the LO which results in the location of the ball in one side rather than in the middle [START_REF] Bapat | The general motion of an inclined impact damper with friction[END_REF]. Another problem is that the time difference is not equal for two consecutive impacts per cycle as has been predicted by analytical results, which is possibly caused by the friction between LO and VI NES, the inclination of LO and the difference of the restitution coefficient at each side [START_REF] Bapat | Stable periodic motions of an impact-pair[END_REF]. Although the above two phenomena have not been considered in the analytical study, they do not influence the above obtained conclusion.

(b) (c) (d) (e) (f) (a) (g) (b) (c) (d) (f) (a) (e)

Conclusion

An asymptotic analytical approach is developed for vibrating system with VI NES. All response regimes are analyzed and organized around the SIM analytically obtained , which represents all possible 1:1 resonance points and establishes the relationship between motion of main system and VI NES. The influence of system parameters and initial conditions on the dynamics and efficiency of energy reduction is demonstrated with analytical and numerical results. Experimental results prove the existence of different regimes and different routes to chaos. TET during SMR is also experimentally demonstrated.

The general equation (i.e. Eq. ( 18)) can not only be used to solve fixed points but also explain the amplitude variation of the 1:1 resonance regimes for free and forced vibration. In addition, it can explain the complicated variation of amplitude from experimental results. Considering that the impact system (e.g. impact oscillator and impact damper) is similar from the point view of main systems with the help of interacting force presented in the form of Fourier series, the analytical method can be popularized.

That all regimes are organized around SIM provides another point of view compared to the combination of parameters and regimes for the study of dynamics or that of parameters and amplitude for the study of energy reduction efficiency used by researchers like Peterka [START_REF] Peterka | More detail view on the dynamics of the impact damper[END_REF]. Specifically, the influence of different parameters on the appearance of different response regimes can be investigated by the same criteria with the SIM as an intermediate. The optimal design parameter is decided by whether the 1:1 resonance occurs.

Another interesting phenomenon is about the SMR from experimental results. It demonstrates different efficiency of TET for different regimes with short duration. It can be imagined that it is related to the transient process if the outside excitation is interrupted abruptly and set to zero. Therefore, it can provide some proof for the optimization for free vibration system with transient TET. The complicated experimental SMR characterized by strongly modulation and chaotic impacts proves the convenient application of the extended analytical method.

Fig. 1

 1 Fig. 1 Schema of a vibro-impact NES (impact damper)

Fig. 3

 3 Fig. 3 SIM of VI NES: one stable branch in blue line and two unstable branches in red line

Fig. 4

 4 Fig. 4 Displacement of LO in the first column, relative displacement in the second column and motion of system projected into SIM in the third column for different response regimes: (a) z > 2; (b) z = 2 and asymmetric; (c) z = 2 and symmetric; (d) z = 2 and z < 2 for SMR; (e) z < 2

Fig. 5

 5 Fig. 5 Areas of SIM: (a) 4 typical areas; (b) first case in Z3; (c) second case in Z3

Fig. 6

 6 Fig. 6 Transition phenomenon: (a) saddle-node bifurcation boundary; (b) Hopf bifurcation boundary; (c) transition regions

Fig. 7

 7 Fig. 7 SIM and dynamics of FRF: (a) SIM; (b) relative position of FRF

Fig. 8

 8 Fig. 8 SIM and SMR: (a) motion projected in green lines; (b) enlarged view of transition part; (c) enlarged view of 1:1 resonance part

Fig. 9 Fig. 10

 910 Fig. 9 Influence of restitution coefficient: (a) SIMs with different r; (b) relative position for different r

Fig. 11 Fig. 12

 1112 Fig. 11 Route of variation of regimes: (a) b=0.3 mm, chaos and z > 3; (b) b=2.5 mm and z = 3; (c) b=4 mm, asymmetric resonance and z = 2; (d) b=7.5 mm, symmetric resonance z = 2; (e) b=11.5 mm, SMR and partially z = 2 ; (f) b=20 mm, chaos and z < 2; (g) relation positon of different regimes in SIM

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 • 20130449 0063 for their financial support.


Tao Li

Etienne Gourc

Sébastien Seguy

A Berlioz

Dynamics of two vibro-impact nonlinear energy sinks in parallel under periodic and transient excitations

Keywords: Vibro-impact, Targeted energy transfer, Nonlinear energy sink, Impact damper

Introduction

The excessive vibration energy of a targeted system can be dissipated by using an auxiliary device, which absorbs and dissipates such undesired energy by producing a force opposing it, either continuously such as in the case of nonlinear energy sink (NES) with continuous nonlinearity [START_REF] Mcfarland | Experimental study of nonlinear energy pumping occurring at a single fast frequency[END_REF][START_REF] Gourc | Experimental investigation and design optimization of targeted energy transfer under periodic forcing[END_REF][START_REF] Bellet | Experimental study of targeted energy transfer from an acoustic system to a nonlinear membrane absorber[END_REF][START_REF] Mattei | Nonlinear targeted energy transfer of two coupled cantilever beams coupled to a bistable light attachment[END_REF][START_REF] Benacchio | Design of a magnetic vibration absorber with tunable stiffnesses[END_REF] or segmentally by NES with piece-wise nonlinearity [START_REF] Lamarque | Targeted energy transfer in mechanical systems by means of non-smooth nonlinear energy sink[END_REF] or intermittently and instantly by impact damper [START_REF] Lieber | An acceleration damper: development, design and some applications[END_REF]. Due to its simplicity of construction, fast response at the initial stage of perturbation and effectiveness at a broadband frequency, impact damper has been studied several decades ago [START_REF] Ibrahim | Vibro-impact dynamics: modeling, mapping and applications[END_REF][START_REF] Babitsky | Theory of vibro-impact systems and applications[END_REF], no matter the case with only one impact pair [START_REF] Masri | On the stability of the impact damper[END_REF][START_REF] Bapat | Stable periodic motions of an impact-pair[END_REF][START_REF] Popplewell | Stable periodic vibroimpacts of an oscillator[END_REF][START_REF] Bapat | Single unit impact damper in free and forced vibration[END_REF][START_REF] Peterka | More detail view on the dynamics of the impact damper, Mechanics[END_REF] or multi-units with several impact-pairs [START_REF] Masri | Effectiveness of two-particle impact dampers[END_REF][START_REF] Masri | Analytical and experimental studies of multiple-unit impact dampers[END_REF][START_REF] Cempel | The multi-unit impact damper: equivalent continuous force approach[END_REF][START_REF] Bapat | Experimental investigation of controlling vibrations using multi-unit impact damper[END_REF][START_REF] Popplewell | Stable periodic motion of multiple-unit impacting mechanisms[END_REF][START_REF] Bapat | Multiunit impact damper-re-examined[END_REF].

Recently, impact damper is studied further and more clear under the context of targeted energy transfer (TET) [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] and termed as vibro-impact (VI) NES [START_REF] Nucera | Targeted energy transfers in vibro-impact 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][START_REF] Karayannis | Vibro-impact attachments as shock absorbers[END_REF]. Its mechanism of TET is revealed by studying its Hamiltonian system [START_REF] Lee | Periodic orbits, damped transitions and targeted energy transfers in oscillators with vibroimpact attachments[END_REF], and it is observed that some special orbits are responsible for the irreversible transfer of energy from a main system to an attached VI NES. Through directly studying the system with damping, a slow invariant manifold (SIM), which is firstly used for the analysis of a NES with cubic nonlinearity [START_REF] Gendelman | Bifurcations of nonlinear normal modes of linear oscillator with strongly nonlinear damped attachment[END_REF][START_REF] Gendelman | Attractors of harmonically forced linear oscillator with attached nonlinear energy sink i: Description of response regimes[END_REF], can be obtained by multiple scales method and be applied to analyze response regimes [START_REF] Gendelman | Analytic treatment of a system with a vibro-impact nonlinear energy sink[END_REF][START_REF] Gourc | Targeted energy transfer under harmonic forcing with a vibro-impact nonlinear energy sink: Analytical and experimental developments[END_REF][START_REF] Gendelman | Dynamics of forced system with vibro-impact energy sink[END_REF][START_REF] Pennisi | Vibro-impact nes: A correlation between experimental investigation and analytical description[END_REF][START_REF] Li | Dynamics of cubic and vibro-impact nonlinear energy sink: Analytical, numerical, and experimental analysis[END_REF]. As a result, transient responses and unsteady responses such as chaotic strongly modulated response (SMR) can be well explained for the first time [START_REF] Gendelman | Dynamics of forced system with vibro-impact energy sink[END_REF][START_REF] Gendelman | Forced system with vibro-impact energy sink: Chaotic strongly modulated responses[END_REF]. Dynamics, such as response regimes and bifurcations, is comprehensively re-analyzed around SIM [START_REF] Li | On the dynamics around targeted energy transfer for vibro-impact nonlinear energy sink[END_REF], more specifically, around the regime with two impacts per cycle. In [START_REF] Li | Optimization mechanism of targeted energy transfer with vibro-impact energy sink under periodic and transient excitation[END_REF], the efficiency of different response regimes around SIM is compared, and the response regime with two impact per cycle and around the entrance of SMR is found to be optimal. Based on this efficiency comparison of response regimes, a common optimization design criterion of VI NES is proposed for different excitation types, and the central idea is to make the most efficient response regime with two impacts per cycle exist or last as long as possible. Therefore, it is natural to expect the following design criterion for multi VI NES: to make each VI NES activated with this optimal response regime. About its application, any analytical study of nonlinear systems coupled with NES will be difficult, no matter this nonlinear system is a turning system [START_REF] Gourc | Quenching chatter instability in turning process with a vibro-impact nonlinear energy sink[END_REF] or a helicopter system [START_REF] Bergeot | Analysis of steady-state response regimes of a helicopter ground resonance model including a non-linear energy sink attachment[END_REF], or even a simple rod [START_REF] Georgiades | Broadband passive targeted energy pumping from a linear dispersive rod to a lightweight essentially nonlinear end attachment[END_REF]. For VI NES, a solution is to found its activation condition without considering the specific type of a targeted system. It is found that the activation of a VI NES is limited to a range of displacement amplitude of a linear system [START_REF] Li | Optimization mechanism of targeted energy transfer with vibro-impact energy sink under periodic and transient excitation[END_REF]. From the viewpoint of energy, it means that its effectiveness is not only limited by a minimal value (activation threshold) but also a maximal value. Therefore, it is natural to increase the range of effectiveness of vibration control by designing different VI NES with different activation ranges for both linear and nonlinear systems. In this way, the efficiency and robustness of TET can be improved, even at small energy levels. About multi NES, there already exist extensive studies [START_REF] Tsakirtzis | Multifrequency nonlinear energy transfer from linear oscillators to mdof essentially nonlinear attachments[END_REF][START_REF] Tsakirtzis | Complex dynamics and targeted energy transfer in linear oscillators coupled to multi-degree-of-freedom essentially nonlinear attachments[END_REF][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], and normally frequency components are used as indices to judge the possible transient or sustained resonance captures, and to identify different levels of activation.

However, this kind of index will not be used here for VI NES, since the impact number per cycle is a more direct measure. For example, the response regime with two impacts per cycle can be easily distinguished from numerical or experimental viewpoints, and actually it corresponds to 1:1 resonance. Although frequency components can be calculated by wavelet transform, it will not make the analysis easier and will not be used here.

In addition to the viewpoint of TET, the analytical study of systems coupled with multi VI NES also deserves further investigation fortwo reasons. Firstly, the research of multi VI NES is closely related to particle dampers [START_REF] Lu | Studies of the performance of particle dampers 587 under dynamic loads[END_REF]. Secondly, complicated response regimes such as intermittent beating responses [START_REF] Bapat | Experimental investigation of controlling vibrations using multi-unit impact damper[END_REF][START_REF] Popplewell | Stable periodic motion of multiple-unit impacting mechanisms[END_REF] need further analytical explanations rather than only analytical treatments of steady periodic regimes [START_REF] Masri | Analytical and experimental studies of multiple-unit impact dampers[END_REF][START_REF] Bapat | Multiunit impact damper-re-examined[END_REF].

Therefore, the objective of this paper is to generalize the optimization design criterion for one VI NES [START_REF] Li | Optimization mechanism of targeted energy transfer with vibro-impact energy sink under periodic and transient excitation[END_REF] to multiple VI NES. The focus is still around response regimes, efficiency and the relation between them. The impact number per cycle is used to characterize response regimes and corresponding TET.

The paper is organized as follows: a linear oscillator (LO) coupled with two VI NES in parallel under periodic excitation is analytically developed in Section 2.

Then, numerical validations are demonstrated. In Section 4, experimental results under periodic and transient excitations are demonstrated for different conditions.

Finally, a conclusion is addressed.

Analytical development

The considered system is presented in Fig. 1. Two VI NES in parallel are attached to a LO. The equation of motion between impacts is expressed as follows:

M d 2 x dt 2 +C dx dt + Kx = F sin Ωt m 1 d 2 y 1 dt 2 = 0, ∀|x -y 1 | < b1 m 2 d 2 y 2 dt 2 = 0, ∀|x -y 2 | < b2 (1)
The motion relation between states before and after impacts can be obtained under the condition of simplified impact theory and conservation of momentum and it can be written as:

∀|x -y 1 | = b1 or |x -y 2 | = b2 x + = x -, y 1+ = y 1-, y 2+ = y 2- M dx + dt + m 1 dy 1+ dt + m 2 dy 2+ dt = M dx - dt + m 1 dy 1- dt + m 2 dy 2- dt if |x -y 1 | = b1 dx + dt - dy 1+ dt = -R dx - dt - dy 1- dt , dy 2+ dt = dy 2- dt if |x -y 2 | = b2 dx + dt - dy 2+ dt = -R dx - dt - dy 2- dt , dy 1+ dt = dy 1- dt (2)
The subscripts + andindicate time immediately after and before impacts.

R represents the restitution coefficient of impact. The dimensionless variables as follows are introduced:

ω 2 = K M , T = ωt, λ = C (m 1 + m 2 )ω , ε = m 1 + m 2 M , G = F (m 1 + m 2 )ω 2 , X = x b1 , Y 1 = y 1 b1 , Y 2 = y 2 b1 , ∆ = b2 b1 , α i = m i m 1 + m 2 , Ω = Ω ω (3)
After substitution of Eq.( 3) in Eq.( 1), it becomes:

ẍ + ελ Ẋ + X = εG sin ΩT, εα 1 Ÿ1 = 0, εα 2 Ÿ2 = 0, ∀|X -Y 1 | < 1 and |X -Y 2 | < ∆ (4)
In the same way, after substitution of Eq.(3) in Eq.( 2), it becomes:

∀|X -Y 1 | = 1 or |X -Y 2 | = ∆ X + = X -, Y 1+ = Y 1-, Y 2+ = Y 2- M Ẋ+ + εα 1 Ẏ1+ + εα 2 Ẏ2+ = M Ẋ-+ εα 1 Ẏ1-+ εα 2 Ẏ2- if |X -Y 1 | = 1 Ẋ+ -Ẏ1+ = -R Ẋ--Ẏ1-, Ẏ2+ = Ẏ2- ifi |X -Y 2 | = ∆ Ẋ+ -Ẏ2+ = -R Ẋ--Ẏ2-, Ẏ1+ = Ẏ1- (5)
The barycentric coordinates are introduced in the following way:

V = X + εα 1 Y 1 + εα 2 Y 2 , W 1 = X -Y 1 , W 2 = X -Y 2 (6)
These coordinates correspond to the physical displacement of the center of mass and the relative displacement of VI NES. The substitution of Eq.(6) in Eq.( 4) and Eq.( 5) gives:

V + V + εα 1 W 1 + εα 2 W 2 1 + ε + ελ V + εα 1 Ẇ1 + εα 2 Ẇ2 1 + ε = εG sin ΩT Ẅ1 + V + εα 1 W 1 + εα 2 W 2 1 + ε + ελ V + εα 1 Ẇ1 + εα 2 Ẇ2 1 + ε = εG sin ΩT Ẅ2 + V + εα 1 W 1 + εα 2 W 2 1 + ε + ελ V + εα 1 Ẇ1 + εα 2 Ẇ2 1 + ε = εG sin ΩT ∀|W 1 | < 1 and |W 2 | < ∆ (7) ∀|W 1 | = 1 or |W 2 | = ∆ V + = V -, W 1+ = W 1-, W 2+ = W 2-, V+ = V- if |W 1 | = 1 Ẇ1+ = -R Ẇ1-, Ẇ2+ = Ẇ2-- εα 1 (1 + R) 1 + εα 1 Ẇ1- if |W 2 | = ∆ Ẇ2+ = -R Ẇ2-, Ẇ1+ = Ẇ1-- εα 2 (1 + R) 1 + εα 2 Ẇ2- (8)
Eqs. [START_REF] Lieber | An acceleration damper: development, design and some applications[END_REF] and ( 8) are analyzed with the multiple scales method. Solutions are found in the following form:

V (T 0 , T 1 ) = V 0 (T 0 , T 1 ) + εV 1 (T 0 , T 1 ) + . . . W 1 (T 0 , T 1 ) = W 10 (T 0 , T 1 ) + εW 11 (T 0 , T 1 ) + . . . W 2 (T 0 , T 1 ) = W 20 (T 0 , T 1 ) + εW 21 (T 0 , T 1 ) + . . . (9)
with T i = ε i T . After substitution of Eq.( 9) in Eq.( 7) and Eq.( 8), they become as follows:

• Ordre ε 0

∀|W 10 | < 1and |W 20 | < ∆ D 2 0 V 0 +V 0 = 0, D 2 0 W 10 +V 0 = 0, D 2 0 W 20 +V 0 = 0 ∀|W 10 | < 1or |W 20 | < ∆ V 0+ = V 0-, W 10+ = W 10-, W 20+ = W 20-, D 0 V 0+ = D 0 V 0- if |W 10 | = 1 D 0 W 10+ = -RD 0 W 10-, D 0 W 20+ = D 0 W 20- if |W 20 | = ∆ D 0 W 20+ = -RD 0 W 20-, D 0 W 10+ = D 0 W 10- (10) • Ordre ε 1 D 2 0 V 1 +V 1 = -α 1 W 10 -α 2 W 20 +V 0 -2D 0 D 1 V 0 -λ D 0 V 0 + G sin ΩT 0 (11)
The solution of the first equation of Eq.( 10) can be written in the following way:

V 0 (T 0 , T 1 ) = A(T 1 ) sin (T 0 + α(T 1 )) (12)
The second and third equation of Eq.( 10) represent an oscillator with vibroimpacts under periodic excitations. Like in [START_REF] Gendelman | Analytic treatment of a system with a vibro-impact nonlinear energy sink[END_REF], Their solutions can be expressed with standard trigonometric functions as follows:

W j0 (T 0 , T 1 ) = A(T 1 ) sin(T 0 + α(T 1 ))

+ 2B j π arcsin(cos(T 0 + η j (T 1 ))), j = 1, 2 (13)
Combined with the conditions of impacts given in Eq.( 10), the following relations are obtained:

sin(α -η 1 ) = 1 -B 1 A , cos(α -η 1 ) = 2B 1 δ Aπ sin(α -η 2 ) = ∆ -B 2 A , cos(α -η 2 ) = 2B 2 δ Aπ (14)
with δ = (1 -R)/(1 + R), these relations can be combined through using the trigonometric identity:

A 2 = (1 -B 1 ) 2 + 4B 2 1 δ 2 π 2 = (∆ -B 2 ) 2 + 4B 2 2 δ 2 π 2 (15)
Eq.( 15) represents one algebraic relation related to the displacement amplitude of LO and that of two VI NES, which is termed as SIM. In the first approximation order, the amplitude of every VI NES only depends on the amplitude of the main system and the performance of two VI NES is decoupled. It establishes the analytical foundation of the principle of separate activation as stated 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] for the case of cubic NES, which is also applicable for VI NES here.

Eq.( 11) in the order ε 1 is now studied. To identify secular terms, the solutions of W j0 are developed in series of Fourier. The development for one harmonic gives:

W j0 = A sin(T 0 + α) + 8B j π 2 cos(T 0 + η j ), j = 1, 2 (16)
The system is studied in the vicinity of the resonance frequency of the main system. The pulsation of excitation is expressed in the following way:

Ω = 1 + εσ (17)
After the substitution of Eqs. [START_REF] Popplewell | Stable periodic vibroimpacts of an oscillator[END_REF][START_REF] Masri | Analytical and experimental studies of multiple-unit impact dampers[END_REF] and Eq.( 17) in Eq.( 11), it comes:

D 2 0 V 1 +V 1 = -∑ 2 j=1 α j A sin(T 0 + α) + 8B j π 2 cos(T 0 + η j ) -2D 1 A cos(T 0 + α) + A sin(T 0 + α) (1 + 2D 1 α) -λ A cos(T 0 + α) + G sin(T 0 + σ T 1 ) (18)
After arrangement, the condition for the elimination of secular term is as follows:

2D 1 A = -∑ 2 j=1 8α j B j π 2 cos(α -η j ) -λ A + G sin(T 1 σ -α) 2AD 1 α = ∑ 2 j=1 8α j B j π 2 sin(α -η j ) -G cos(T 1 σ -T 1 ) (19)
The expressions of sin(αη j ) and cos(αη j ) ( j = 1, 2) given in Eq.( 14) are substituted in Eq.( 19), then the change of variable T 1 σα = θ is introduced:

2D 1 A = 16α 1 B 2 1 δ π 3 A - 16α 2 B 2 2 δ π 3 A -λ A + G sin θ 2AD 1 θ = - 8α 1 B 1 (1 -B 1 ) Aπ 2 - 8α 2 B 2 (∆ -B 2 ) Aπ 2 + G cos θ + 2Aσ (20)
The fixed points to the limit T 1 → ∞ are calculated by eliminating the derivation in Eq.( 20). The obtained equations can be combined by using the trigonometric identity. After rearrangement, one equation of the following form is obtained:

c 2 A 4 + c 1 (B 1 , B 2 )A 2 + c 0 (B 1 , B 2 ) = 0 (21)
Eq.( 21) can be expressed in the function of only A and B 1 or B 2 by using Eq.( 15). The obtained equation can be resolved for A 2 . The thereby obtained algebraic relation between A and B 1 or B 2 represents the invariant manifold of the problem in the time-scale T 1 . Subsequently, different results of numerical integration and their projections into the invariant manifold of each VI NES are presented.

Numerical results

In this part, the objective is to validate the analytical results by numerical simulations. The Eqs.( 7) and ( 8) will be used. The used parameters are showed as follows: ε = 0.01, ∆ = 0.7, R = 0.6,

α 1 = α 2 = 0.5, λ = 1 (22)
The numerical integration results with SIM for G = 0.8 and σ = 0.5 are presented in Fig. 2. In this case, the time history of V , which is an approximate description of the displacement of LO, illustrates its strongly modulated feature [START_REF] Gendelman | Dynamics of forced system with vibro-impact energy sink[END_REF][START_REF] Gendelman | Forced system with vibro-impact energy sink: Chaotic strongly modulated responses[END_REF] as displayed in Fig. 2(a). Its amplitude is chaotic strongly modulated and the relative maximal amplitude is not constant every time. In Fig. Another case is presented in Fig. 3 for G = 0.9 and σ = -0.2. In this case, the two VI NES are attracted by fixed points, namely the intersections between the blue curve and red curve, as showed in Fig. 3 (c) and (e). The main system performs steady oscillation as displayed in Fig. 3 (a). Compared to the last case, the values of A, B 1 and B 2 are constant and well predicted by the SIMs.

Actually, the ability of SIM obtained by analytical study is limited on predicting the resonance responses with two impacts per cycle of main system or other response regimes with resonant parts. Other more complicated phenomena will be demonstrated by the following experimental study.

Experimental results

In this section, experiments are done for periodic and transient excitation respectively to compare the case with one VI NES and that with two VI NES. For periodic excitation, the experimentally obtained and used parameters are displayed justed by a cylinder. The cylinder and the cover at the opposite side are made of hardened steel. The parameters of this system have been identified by performing modal analysis and are summarized in Table 1.

Single frequency excitation

The frequency of the sinusoidal excitation is slowly varied from 7.82 Hz to At first, the time history of the acceleration of LO without VI NES is showed in Fig. 5(a) as a reference. Although there exist small shakes at some of its maximal place, no impacts exist. In addition, its amplitude is highest compared to other cases coupled with VI NES. Then, the time history of the acceleration of LO with a VI NES for a clearance of 30 mm is showed in Fig. 5(b) as another reference.

This value of clearance is proved to be almost optimal in the sense of vibration control and the response regime is two impacts per cycle [START_REF] Li | Optimization mechanism of targeted energy transfer with vibro-impact energy sink under periodic and transient excitation[END_REF]. It is not so easy to distinguish impacts because the order of impact strength is close to that of the acceleration of LO, but the impact number is observable during the process of experiment. Moreover, the peaks of acceleration are evidently reduced compared to the results in Fig. 5(a).

With the addition of another ball and different combinations of clearances, the response regimes will be more complicated. However, the possible variations are always based on the above two cases. Then, the regime with four impacts per cycle is observed with the addition of a second ball with clearance b1 = 5 mm (another ball with clearance b2 = 30 mm), and both ball impact twice per cycle as showed in Fig. 5(c), and the four impact moments during one cycle are marked out in rectangles. The acceleration of LO is further decreased compared to the optimized one ball case with 30 mm clearance, and this efficiency improvement cannot be easily judged here and will be demonstrated from the viewpoint of displacement later.

When the clearance of the added ball is increased to 40 mm, the response is complicated as displayed in Fig. 5(d). The ball below with b2 = 30 mm continuously impacts twice per cycle. In contrast, the added ball impacts only once during many cycles (> 1), and a few of them are marked out in rectangles.

In addition to the above relative stable response regimes, there exist some even more complicated transient response regimes during one time history for some combinations of clearances. For the case with b1 = b2 = 30 mm, there are some periods as showed in Fig. 5(e), in which just one ball is activated with two impacts per cycle, but this activated state can alternate between these two balls.

This phenomenon is complicated, and it cannot be judged from the results showed here and can just be observed in the test site. There are also some periods, two balls impact twice per cycle as showed in Fig. 5(f) and there are even some periods there are almost no impacts as showed in Fig. 5(g).

Therefore, the strong nonlinear coupling between these two balls and LO is well observed by the complicated variations of response regimes during one time history. The above mentioned basic response regimes will be applied to explain the complication variation of efficiency, and this point is closely related to the targeted energy transfer by transient resonance captures.

In [START_REF] Li | Optimization mechanism of targeted energy transfer with vibro-impact energy sink under periodic and transient excitation[END_REF], it is observed that the optimal response regime is the one with two impacts per cycle and around the entrance of SMR. The idea of optimization design is to make efficient response regimes occur for different excitations. This idea can still apply in the optimization design of two VI NES, namely to make each VI NES activated at its best state with two impacts per cycle.

Then, the efficiency comparison of different combinations of clearances is performed here to observe the possible relation between the types of response regimes and their efficiency. The two cases without VI NES and with one optimized VI NES (30 mm clearance) are chosen as two references as showed in Fig. 6(a-b).

The length of the down clearance b2 is fixed to 30 mm and only the upper clearance b1 is varied from 5 mm to 50 mm. Here, only the time histories of displacement are demonstrated for b1 = 30 mm and b1 = 5 mm, respectively.

The former is displayed in Fig. 6(a), there are three typical areas A1, A2 and A3, which correspond to typical response regimes from Fig. 5(e) to Fig. 5(g), respectively. In area A1, there is a small decrease of amplitude compared to the one VI NES case in red curve. In area A2, both ball impact twice per cycle and the amplitude is lowest. In area A3, the occasional out of activation for both balls and the amplitude increases. In the whole process, there are many transitions between them, which results in the complicated variation of displacement amplitude of LO.

It is a direct proof of nonlinear coupling between two VI NES and LO. In addition, the efficiency is highest when both balls impact twice per cycle, which is the most effective form of transient resonance captures.

This efficient targeted energy transfer is better demonstrated by the decrease of b1 to 5 mm as showed in Fig. 6(b). This time, both VI NES impact twice per cycle and it means the permanent resonance captures.

With the excitation of LO at the beginning, different activation amplitudes of LO with different combinations of clearances are observed, namely C1, C2 and C3 in Fig. 6(a-b). With the increase of VI NES from 0 to 1, and then 2, the amplitude is evidently decreased at the activation point of VI NES. The same conclusion is obtained when the value of another clearance is decreased. In this sense, the robustness and efficiency can be improved.

Then, the average and maximal amplitude ratios between the case with VI NES and without NES are calculated for all cases during a stable time period (20-70 s), and the results are showed in Fig. 6(c). Compared to the case without VI NES and with one VI NES, the optimal case is the addition of another VI NES with a small clearance 5 mm. For other cases with large clearances, it cannot improve the efficiency and, in return, it will result in the occasional out of activation of VI NES, and meanwhile, the large displacement of LO.

According to the optimization design criterion proposed in [START_REF] Li | Optimization mechanism of targeted energy transfer with vibro-impact energy sink under periodic and transient excitation[END_REF], the design of parameters is to make the efficient response regimes appear and it means the most efficient transfer and dissipation of energy. From the above experimental results with limited combinations of clearances, this optimization design criterion still applies since both balls impacts twice in the optimal case with b1 = 5 mm.

However, the phase difference of two VI NES cannot be measured or calculated, it may be another factor related to efficiency.

Except the above combinations of clearances, experiments with the same clearance for both VI NES are also performed. The average and maximal amplitude ratios are also calculated and showed in Fig. 7. From the viewpoint of Ae, the case with b1 = b2 = 30 mm is optimal. On the contrary, the case with b1 = b2 = 15 mm is optimal in terms of Am. Generally speaking, they are less effective than the combination with b1 = 5 mm and b2=30 mm.

As a summary, the objective to find the relation between response regimes and their efficiency is accomplished in two steps. The first step is to find basic response regimes. Then, they are applied to explain the complicated variation of response. As a result, a general design criterion can also be proposed here: the design of two VI NES is to make them activated around most efficient response regimes.

Excitation with a band of frequency

Then, the band of excitation frequency is enlarged around the natural frequency of LO. The objective here is to study the influence of different combinations of clearances on the response regimes and their efficiency. The starting frequency f s and ending frequency f e during this sweep is showed in Table 1 and the acceleration is still fixed to 0.06 g.

For the above experimental configuration, the one VI NES case with a clearance 27.5 mm has been observed optimal [START_REF] Li | Optimization mechanism of targeted energy transfer with vibro-impact energy sink under periodic and transient excitation[END_REF]. Here the clearance of an added of motion for LO and both VI NES can be more complicated than the abovementioned characteristics. Except the observed relation between efficiency and the response regimes, it is desirable to obtain further information but this kind of try would be difficult.

Therefore, the following optimization design criterion is recommended. If just one ball is applied, it is recommended that VI NES should be optimized at the point of natural frequency. If two VI NES are applied to improve the robustness and increase efficiency, a smaller length of clearance should be chosen compared to the optimized clearance of the one ball case to avoid the occasional failure.

Transient excitation

Experimental setup

The same experimental device as the periodic case is used, but it is attached to a cast iron bench as showed in Fig. 9(a). The addition of a small ring bolt for prestretch will not influence the total mass of LO and its influence can be neglected.

Therefore, its parameters are nearly the same as these in Table 1. One laser sensor and one accelerometer are used to measure the displacement and acceleration of LO, respectively. The fixation of accelerometer is showed in Fig. 9 NES with a small clearance b1 = 2 mm gets out of two impacts per cycle around points C1 and C2. In zone 1, the first VI NES with a large clearance is activated with two impacts per cycle. In zone 2, the first VI NES gets out of activation and the second VI NES with a small clearance is activated with two impacts per cycle.

In the sense of the activation with two impacts per cycle, the principle of separate activation of VI NES with different clearances is observed here. This definition of activation is important, since the regime with two impacts per cycle is most efficient in vibration control [START_REF] Li | Optimization mechanism of targeted energy transfer with vibro-impact energy sink under periodic and transient excitation[END_REF].

Then, the decay rates of displacements for both cases are compared. Before point A1, there is almost no difference, since the first VI NES with large clearance is activated with two impacts per cycle and the second VI NES behaves in low efficient regime (more than two impacts per cycle). Between A1 and B1, the role of the second VI NES increases but still small. In contrast, it plays an important role between B1 and C1 since it is activated with two impacts per cycle and the first VI NES is totally out of excitation. The difference of decay rates in this period is relative large. Here, the role of separate activation in vibration control is evident, and the two VI NES with difference clearances can be effective at different ranges of displacement amplitude. More importantly, this effectiveness is related to the efficient response regime with two impacts per cycle.

During the above transition process, the first VI NES changes from two impacts per cycle to the state with no continuous periods of two impacts per cycle (i.e., permanent out of activation), the limit point is A1. It means that it plays the main role in vibration control before A1. In contrast, the second VI NES changes from more than two impacts per cycle to two impacts per cycle, and finally out of activation, it plays a main role in vibration control between B1 and C1. Between In summary, the principle of separate activation of VI NES is observed, and the effectiveness of the vibration control can be increased to a large range of displacement amplitude of LO. Moreover, the addition of VI NES can reduce impact strengths and scatter impacts though it may not increase the efficiency of vibration control.

Conclusion

A LO coupled with two VI NES in parallel is studied under periodic and transient excitations. Firstly, the system is analytically studied with the multiple scales method. SIM is obtained and fixed points can be calculated. Then, numerical simulations are performed to observe the typical response regimes and to validate the In a word, many aspects of the dynamics are further explained by the analytical, numerical and experimental studies, but other factors, such as the friction between LO and VI NES, inclinations of clearances and impact models, should be considered in the analytical treatment to improve the accuracy of explanation.

Fig. 1 .

 1 Fig. 1. Schema of a LO with two VI NES coupled in parallel.

Fig. 2 .

 2 Fig. 2. Numerical integration of Eqs. (7, 8) for G = 0.8 and σ = 0.5: (a) SMR of V ; (b) intermittent resonance of W 1 related to the first VI NES; (c) the analytically obtained SIM of the first order in blue curve and of the second order in green curve, the projection of motion for the first VI NES in red curve; (d) the constant response of W 2 related to the second VI NES; (e) the projection of motion for the second VI NES in red curve.

Fig. 3 .

 3 Fig. 3. Numerical integration of Eqs. (7, 8) for G = 0.9 and σ = -0.2: (a) steady state response of V ; (b) constant resonance of W 1 related to the first VI NES; (c) the analytically obtained SIM of the first order in blue curve and of the second order in green curve, the projection of motion for the first VI NES in red curve; (d) the constant response of W 2 related to the second VI NES; (e) the projection of motion for the second VI NES in red curve.

  2(b), the relative displacement related to VI NES 1 is showed. The dense parts denote that it realizes two impacts per cycle, and the sparse parts denote the occasional out of activation. Therefore, VI NES1 realizes intermittent response . Then, its response is projected to the SIM in red curve as showed in Fig.2(c). The intersections of the blue curve and the green curve denote the fixed points, but their stabilities are not calculated here. The overlapping parts between the blue curve and the red curve demonstrate the parts with two impacts per cycle and validate the prediction accuracy of SIM. For VI NES 2, it is activated in permanence with two impacts per cycle as showed by W 2 in Fig.2(d). Its motion is projected to SIM as showed in Fig.2(e), and the projected A and B 2 are always in right branch of SIM. It means that the displacement amplitude of VI NES 2 will always vary simultaneously with that of LO.

Fig. 4 .

 4 Fig. 4. Experimental setup: (a) global configuration; (b) detailed view of VI NES.

f s -f e 6

 6 the shaker are measured by contact-less laser displacement sensors. Their accelerations are measured by accelerometers and the impacts between VI NES and LO can be judged from the sudden changes of the acceleration of LO. A detailed view of the configuration for two VI NES is presented in Fig. 4 (b). It simply consists of two closed clearances of lengths d + 2 • b1 and d + 2 • b2, respectively, where d is the diameter of both balls (VI NES). b1 and b2 are lengths of the above clearance and the below clearance, respectively, and each can be ad-

7. 84

 84 Hz during 80 s, which can be considered almost fixed to the value 7.83 Hz. This value is closed to the natural frequency of LO ( f 0 = 7.86 Hz). The acceleration of the shaker is fixed to 0.06 g. During the whole experimental process, the time histories of displacement and acceleration of LO are recorded. With the change of the number of VI NES and different combinations of clearances b1 and b2, different periodic and transient response regimes are observed and demonstrated here. They are identified by the difference of impact numbers per cycle of LO, which can be judged from the time history of the acceleration of LO.

Fig. 5 .

 5 Fig. 5. Typical response regimes by comparison of impact numbers per cycle of the acceleration of LO and some impacts moments are marked out in rectangle: (a) no impact and no VI NES; (b) two impacts per cycle; (c) four impacts per cycle; (d) one VI NES impacts once during many cycles and another VI NES impacts twice per cycle; (e) one period of a response with at least one VI NES in the state of two impacts per cycle; (f) one period of a response with both VI NES in the state of two impacts per cycle; (g) one period of a response with both VI NES out of activation and no impact.

Fig. 6 .

 6 Fig. 6. Typical response regimes and their efficiency comparison with different combinations of clearances: (a-b) response regimes; (c) efficiency comparison by average and maximal amplitude ratios (Ae and Am).

Fig. 7 .

 7 Fig. 7. Efficiency comparison of different response regimes with the same clearance for both VI NES.

Fig. 8 .

 8 Fig. 8. Responses of LO during the sweep experiments and efficiency comparison: (a) with different b1 and b2; (b) with the same b1 and b2; (c) the efficiency comparison for two relative optimal cases.

Fig. 9 .

 9 Fig. 9. Experimental setup: (a) global configuration; (b) installation of accelerometer; (c) detailed view of VI NES.

  (b) and a detailed view of VI NES is displayed in Fig.9(c). The number of balls can be changed. The initial displacement of LO is regulated by a device and is fixed to 20 mm for all tests. The initial location of two balls are at random, and the velocities for LO and both balls are zero. Since only the stable transition process is studied and the transient process quickly disappears, the initial conditions will not influence the expected conclusion.4.2.2. Principle of separate activationWith two sets of b1 and b2, the responses of two cases are compared here. For the case with one VI NES (b2 = 20 mm), the time history of displacement of LO is represented by the red curve in Fig.10(a) and its corresponding acceleration is showed in Fig.10(b). The points A1 and A2 are related to the transition from the regime with two impacts per cycle to that without any continuous period of two impacts per cycle.For the case with two balls (b1 = 2 mm and b2 = 20 mm), the time history of displacement of LO is represented by the blue curve in Fig.10(a) and its corresponding acceleration is showed in Fig.10(c). The first VI NES with b2 = 20 mm gets out of two impacts per cycle around points B1 and B2. The second VI

Fig. 10 .

 10 Fig. 10. Response comparison between LO coupled with one VI NES and two VI NES: (a) an imposed time history of displacement; (b) a time history of acceleration with one VI NES; (c) a time history of acceleration with two VI NES; (d) a detailed view of one period of acceleration with two VI NES; (e) a detailed view of one period of acceleration with one VI NES.

Fig. 11 .

 11 Fig. 11. Principle of separate activation: (a) b1 = 5 mm and b2 = 20 mm; (b) b1 = 5 mm and b2 = 30 mm; (c) b1 = 5 mm and b2 = 40 mm.

  analytical results. Finally, experiments under different types of excitations are carried out to observe the possible response regimes and to compare the efficiency of different combinations of clearances. The asymptotic method, which has been proved feasible for the analytical study of systems coupled with one VI NES, is generalized to the case with two VI NES. Two different SIMs are obtained for two VI NES with different clearances. Since a SIM is related to an activation threshold, namely displacement amplitude of LO, the analytically obtained two different SIMs can explain the principle of separate activation. This principle can be more directly reflected by the transient excitation case. In addition to SIM, fixed points can be calculated and their po-sition can be applied to judge the types of response regimes. Compared to the former analytical methods, the variation of transient and modulated response can be further explained this time. Numerical results prove the consistency of analytical prediction of two classic response regimes, namely two impacts per cycle and SMR. More specifically about SMR, its existence and its periods with two impacts per cycle can be well predicted by SIM. Although analytical study is limited to obtain results for regimes with two impacts per cycle and SMR, or any combinations of these two regimes, they are still important. From the viewpoint of TET by resonance captures, these regimes possess the most efficient permanent or transient resonance captures. Through experiments under different excitations, different response regimes are observed and their relation to their efficiency is analyzed. Whatever types of excitation, it is more effective as long as the two VI NES are activated with two impacts per cycle or around this regime. In addition to this common law, the results are complicated from the viewpoint of each excitation type. This observation proves the relevance and effectiveness of analytical study around SIM, more precisely, around the response regime with two impacts per cycle (i.e., resonance captures). More specifically about experimental results, several points should be pointed out. Firstly , different basic periodic and transient response regimes are observed and analyzed from the experimental results under excitation with a fixed frequency. In addition, more complicated responses are observed but they can be explained by these basic response regimes. Then, they are applied to explain the complicated variation of response regimes, not only for excitation with a fixed frequency and a range frequency, but also for transient excitation. All observations here prove the important role of the regime with two impacts per cycle. Under transient excitation, this role is further demonstrated by the separate activation of VI NES of different clearances. This separate activation of VI NES with two impacts per cycle can be applied to control vibration at different levels of displacement amplitude. Sometimes, although the addition of VI NES cannot improve the efficiency of vibration control, the impacts are scattered and their strengths are reduced.

Table 1 .

 1 Experimental parameters

		Physical Parameters
	M	4.7 kg	C 3.02 Ns/m
	K 11.47 • 10 3 N/m
	m 1	32 g	m 2	32 g
	b 1	0 -50 mm	b 2 0 -50 mm
		Reduced Parameters
	ε	0.76 %	λ 1	1.91
	f 0	7.86 Hz
		Single frequency test
		f r	7.82 -7.84 Hz
	Shaker acceleration	0.06 g
		Frequency band test

Acknowledgments

The authors acknowledge the French Ministry of Science and the Chinese Scholarship Council [grant number: 201304490063] for their financial support.