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

Tamás Insperger

Lionel Arnaud
email: lionel.arnaud@enit.fr

Gilles Dessein
email: gilles.dessein@enit.fr

Grégoire Peigné

Suppression of period doubling chatter in high-speed milling by spindle speed variation

Keywords: Stability, milling, spindle speed variation, period doubling, chatter, surface roughness 3 B

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 [START_REF] Altintas | Analytical prediction of stability lobes in milling[END_REF]and led to the development of the stability lobes theory. Since then several improved modeling and analysis techniques have appeared including detailed analysis of the governing time-periodic delay-differential equation and time domain simulations, see, for example [START_REF] Bayly | Stability of interrupted cutting by temporal finite element analysis[END_REF][START_REF] Merdol | Multi frequency solution of chatter stability for low immersion milling[END_REF][START_REF] Paris | Surface shape prediction in high speed milling[END_REF][START_REF] Insperger | On the higher-order semi-discretizations for periodic delayed systems[END_REF].

These techniques are used to construct the so-called stability lobe diagrams that help in selecting the spindle speeds and the axial depths of cut associated with a chatter free machining. These stability diagrams present the stability boundaries that separate stable machining from chatter.

Usually, below the stability boundaries, the process is stable, i.e., any disturbance is settles.

Above the stability boundaries, the process is unstable, i.e., any disturbance tend to increase and chatter develops. Close to the stability boundaries, the rate of change of the vibration amplitudes is small (i.e., the exponential time constant is large), consequently, the identification of stable machining or chatter might be not so clear as for parameter points far from the stability boundaries. For instance, in case of a machining operation with parameters just slightly above the stability boundaries, the operation may be terminated before chatter fully develops [START_REF] Seguy | On the stability of high-speed milling with spindle speed variation[END_REF]. In many practical cases, the choice of the optimal speed is difficult because many coupled parameters interact non-linearly in a production setting [START_REF] Seguy | Toolpath dependent stability lobes for the milling of thin-walled parts[END_REF].

The main source of chatter vibrations is a subtle shift between the vibrations of the workpiece (or tool) and the tooth passing frequency. This phase shift leads to a chatter frequency slightly lower or higher than a multiple of the tooth passing frequency. In the particular case of "period doubling chatter", there is a perfect synchronism between the half tooth passing frequency and the chatter frequency.

One possible way to suppress chatter is the application of variable tool pitches [START_REF] Altintas | Analytical stability prediction and design of variable pitch cutters[END_REF][START_REF] Budak | An analytical design method for milling cutters with nonconstant pitch to increase stability, Part I: Theory[END_REF]. Another technique to reduce chatter vibrations is the continuous spindle speed variation. Both techniques are based on the disturbance of the regenerative effect by creating varying tooth passing periods. Compared to variable pitch cutters, spindle speed variation can effectively be used in a wider spindle speed range, since the frequency and the amplitude of the speed variation can easily be adjusted in CNC machines even during the machining process. The idea of spindle speed variation became a focus of interest in the 1970's. [START_REF] Takemura | Active suppression of chatter by programmed variation of spindle speed[END_REF] presented the first simple model to study the stability of variable speed machining; they predicted a significant shift of the stability lobes to higher depth of cuts, but the experimental tests showed only small improvements. [START_REF] Sexton | A stability analysis of single point machining with varying spindle speed[END_REF] developed a more realistic model, and they found some improvements in the stability properties for low spindle speeds. Moreover, they showed that the presence of transient vibrations may drastically reduce these gains [START_REF] Sexton | An investigation of the transient effects during variable speed cutting[END_REF].

The study of the stability for variable speed machining requires a special mathematical analysis to compute stability lobes. Tsao et al. (1993) have developed a model taking the angular coordinates as variables instead of time. [START_REF] Insperger | Stability analysis of turning with periodic spindle speed modulation via semidiscretization[END_REF] showed that the semidiscretization method can effectively be used for the stability analysis of turning at variable speed. They showed that the critical depths of cut can be increased for low speeds, but for the high-speed domain, no improvement was found. Experimental results were presented by [START_REF] Al-Regib | Programming spindle speed variation for machine tool chatter suppression[END_REF] showing gains on surface roughness for turning at low speed. Recently, Zhang et al. (2009) presented a systematic stability analysis based on a model of a non-linear delay differential equation. Experimental validations in turning show reduction of the displacement and improvements in the surface roughness at low speed [START_REF] Sexton | An investigation of the transient effects during variable speed cutting[END_REF][START_REF] Al-Regib | Programming spindle speed variation for machine tool chatter suppression[END_REF]Zhang et al., 2009).

The modeling of variable spindle speed milling is more complex than that of turning, since the speed variation frequency and the tooth passing frequency interact and the resulting system is typically quasi-periodic. Still, there are mathematical techniques to determine approximate dynamic properties. [START_REF] Sastry | Floquet theory based approach for stability analysis of the variable speed face-milling process[END_REF] used Fourier expansion and applied the Floquet theory to derive stability lobe diagrams for face milling. They obtained some improvements for low spindle speeds. Recently, Zatarain et al. (2008) presented a general method in the frequency domain to the problem, and showed that varying spindle speed can effectively be used for chatter suppression. Another approach is to use time domain simulation [START_REF] Altintas | In-process detection and suppression of chatter in milling[END_REF][START_REF] Bediaga | Reducción de la inestabilidad en cortes interrumpidos en fresado a alta velocidad mediante variación de la velocidad del husillo[END_REF] that makes it possible to obtain more detailed information like the amplitude of vibrations, the chip thickness or the cutting forces. Simulations were sometimes made at high speed [START_REF] Bediaga | Reducción de la inestabilidad en cortes interrumpidos en fresado a alta velocidad mediante variación de la velocidad del husillo[END_REF], but to the best knowledge of the authors, experimental validations were only made at low spindle speed [START_REF] Lin | The effects of variable speed cutting on vibration control in face milling[END_REF][START_REF] Altintas | In-process detection and suppression of chatter in milling[END_REF]Zatarain et al., 2008).

In this paper, the stability of variable speed milling is analyzed in the high-speed domain, for spindle speeds corresponding to the first flip (period doubling) lobe. Theoretical stability predictions are obtained using the semi-discretization method based on [START_REF] Insperger | On the higher-order semi-discretizations for periodic delayed systems[END_REF], and the results are confirmed by experiments. The structure of the paper is as follows. First the model of the process is presented. Then triangular and sinusoidal spindle speed modulations are compared, and the selection of the optimal amplitude and frequency is presented. Experimental verifications are provided with detailed analysis of the machined surface. Finally, the paper is concluded.

B MECHANICAL MODELLING

Variation of the spindle speed

In the literature, mostly sinusoidal, triangular or square-wave modulations are considered [START_REF] Lin | The effects of variable speed cutting on vibration control in face milling[END_REF]. Here, the sinusoidal and the triangular variations shown in Figure 1 are compared and analyzed. We assume that the spindle speed variation is periodic at period Tv with a mean value N0 and an amplitude Na, that is,

0 ( ) ( ) ( ) v a N t N t T N N S t    
, where ( ) (

) v S t S t T  
is the shape function. For a triangular modulation the shape function is defined as:

              v v v v v v v v v T T t T if T T t T T t if T T t t S ) , mod( 2 / / ) , mod( 4 3 2 / ) , mod( 0 / ) , mod( 4 1 ) ( . (1)
Here, mod(t,Tv) denotes the modulo function. For a sinusoidal modulation   S t is defined as:

  2 sin v t S t T         . (2)
In order to normalize amplitude and frequency variation, the parameters RVA and RVF are introduced:

0 a N RVA N  , ( 3
) 0 0 60 60 v v f RVF N T N   . ( 4
)
RVA represents the ratio of the amplitude Na and the mean value N0, it is always less than 1. In order to have reasonable cutting conditions, its value was limited at 0.3. This represents a variation of 30% in the spindle speed and a variation of 30% in the feed by tooth at the same time due to the constant feed velocity. RVF is the ratio of the variation frequency fv and the average spindle frequency N0.

For given spindle speed acceleration, different shape functions result in different maximum amplitude. Figure 1 presents sinusoidal and triangular speed modulations for a fixed maximal spindle acceleration. It can be seen that the triangular modulation provides larger amplitude than the sinusoidal one for a given maximal spindle acceleration.

Mechanical model

A schematic diagram of the milling process is shown in Figure 2. The structure is assumed to be flexible in the x direction that is perpendicular to the feed.

The governing equation for the single degree of freedom oscillator model is:

( ) ( ) ( ) ( ) x mx t cx t kx t F t      , ( 5
)
where m is the modal mass, c is the damping, k is the stiffness and ( )

x F t is the x component of the cutting force. According to the linear cutting law [START_REF] Insperger | Stability of up-milling and downmilling, part 1 : alternative analytical methods[END_REF], the x component of the force is given by: 1 ( ) ( cos sin ) ( )

z x p R j T j j j F t A K K h t           , ( 6
)
where Ap is the axial depth of cut, j  is the angular position of the jth cutting edge and KT and KR are the specific tangential and radial cutting coefficients. The chip thickness is expressed by:

    ( ) ( ) sin ( ) ( ( )) cos j j z j j h t g t f x t x t t        , ( 7
)
where, fz is the feed per tooth, ( ) x t is the current position of the tool and (

j
g t is a screen function, it is equal to 1 when the tooth j is cutting, otherwise it is equal to 0:

1 ( ) 0 st j e x j if g t otherwise            , ( 8
)
where st  and ex  is the start and exit angles of the tooth j.

Stability analysis

Stability of the milling process with variable spindle speed can be estimated theoretically via the analysis of the governing Equations ( 5), ( 6) and ( 7). These equations imply the delaydifferential equation in the form:

( ) ( ) ( ) ( ) ( ( )), t t t t t t     x A x B u  (9)
where

  0 1 0 ( ) ( ) , ( ) , ( ) , ( ) ( ) ( ) ( ( )) ( ( )) 0 , p p x t t t t H t A H t A k c x t m m m m t t x t t                                     x A B u  (10) with 1 ( ) ( ) cos sin cos z j R j T j j j H t g t K K            . ( 11
)
If the modulation period Tv is rationally related to the mean time delay 0 = 60/(zN0), i.e., pTv=q0 with p and q being relative primes, then the system is purely periodic at period T=pTv=q0. For this case, the Floquet theory of delayed systems can be applied with the principal period T. However, if Tv/0 is not a rational number, then the system is quasi-periodic, and the Floquet theory does not apply. A triangular shape variation with p=4 and q=15 is shown in Figure 3. For this example, the amplitude variation is 20% and the RVF=0.8. The evolution of the function H(t) is also presented. The principal period is T=4Tv=150, i.e., 15 tooth passes spaced with variable delays (t) occur during the period of four speed modulations period.

In the current analysis, parameters are selected so that Tv and 0 are rationally related and the Floquet theory can be applied. This assumption may seem restrictive, since these two parameters are in general not rationally related, and the system is therefore quasi-periodic. Still, this assumption will be used in the analysis. The explanation is that for any pair of Tv and 0, one can give a rationally related pair v T  and 0

 (i.e., 0 v pT q  
 with p and q being integers) that are close enough to the original pair Tv and 0. From practical point of view, it is reasonable to assume that the behaviour of the two systems are similar, although the mathematical theorems for the stability analysis of the quasi-periodic systems are more difficult than they are for the periodic system with v T  and 0  . According to the Floquet theory, the stability of a time-periodic system is determined by the eigenvalues of the associated monodromy operator. If all the eigenvalues are in modulus less than 1, then the system is asymptotically stable. In general, this operator is infinite dimensional with infinitely many eigenvalues, but it can be approximated by finite dimensional matrices. Here, we apply the semi-discretization according to [START_REF] Insperger | On the higher-order semi-discretizations for periodic delayed systems[END_REF] to derive stability properties. This method was previously validated by time domain simulations for milling processes by [START_REF] Seguy | From the spindle speed selection to the spindle speed variation for chatter control in thin wall milling: modelling and experiments[END_REF].

SELECTION OF OPTIMAL PARAMETERS

Optimal area

Stability lobes can be constructed numerically by scanning the cutting conditions (spindle speed and axial depth of cut) for a couple of (RVA, RVF) parameters. Figure 4 presents the lobes for constant, triangular and sinusoidal spindle speeds at the region of the first flip and first Hopf lobes. The critical depth of cut can be seen to be increased by speed variation for some ranges of spindle speeds, but for some other ranges, the critical depth of cut is less than that of the constant spindle speed. For example, a cutting process with an axial depth of cut of 1 mm and a spindle speed of 9100 rpm -that is unstable for constant spindle speed -can be stabilized by a speed variation. Globally, the critical depth of cut can essentially be increased in the area of the first flip lobe by spindle speed variation. We did not test all cases to make a complete generalization, but we have given the various simulations that seem to be quite general.

Comparison of the shape variation

The effectiveness of the spindle speed variation is tested on the flip lobe area (N0 = 9100 rpm)

with a triangular and sinusoidal shape variation. Figure 5 shows a contour plot presenting the maximal depth of cut available without chatter for the amplitude (vertical axis) and frequency (horizontal axis) with an average spindle speed of 9100 rpm. At constant speed, the maximal depth of cut is 0.5 mm. It can be seen that the triangular shape always increases the critical depth of cut Ap for any RVA and RVF values. For some domains, even Ap = 2.4 mm can be achieved that corresponds to 380% improvements. As mentioned earlier, the choice of the frequency and the amplitude variation is limited by the spindle dynamics. We measured that within the range of variation (0-4 Hz), the maximum measured acceleration of the spindle is 100 rev/s 2 , i.e., 6000 rpm/s. Considering the limits of the dynamic spindle for the triangular shape, the optimal choice is to use low frequency modulation with high amplitude. Such a point is denoted by point C.

A similar plot is determined for a sinusoidal shape in Figure 6. Like the triangular shape, the sinusoidal shape always increases the critical depth of cut. For some areas, an improvement of 500% can be achieved (Ap = 3 mm). This simulations show that, for the same frequency and amplitude parameters, the sinusoidal shape is more effective than the triangular shape. However, considering the spindle acceleration, then the sinusoidal shape would give only a maximum improvement of 300% (compare the spindle limits in Figures 5 and6). Thus, for the machinist, a triangular speed modulation would be more effective.

It can be seen that the dependence of the maximum depth of cut on the modulation frequency is not so strong as on the modulation amplitude. Application of a large RVF coupled with a low RVA does not provide any improvement in the depth of cut at all. In order to explore more precisely the potential of spindle speed variation, a similar diagram was determined for large modulation frequencies with a triangular shape (see Figure 7). The frequency variation was increased up to the mean tooth passing frequency 1/0. For RVF = 1, the spindle makes one complete variation period during a single tooth pass. In fact this case corresponds to a very fast variation and is not relevant for practical applications. Figure 7 shows that the improvements in the high frequency domain are low. It can be seen that for lower frequencies there are domains bounded by the specific lines in the form RVA = k × RVF, where the depth of cut significantly increases and reaches 2.5-3.5 mm (that corresponds to 400-600% improvements). However, for the higher frequencies the maximum depth of cut remains around 0.5 mm, except for some punctual areas, where, again, large depths of cut (about 2.5-3.5 mm) can be achieved. Overall, it can be stated that it is not effective working in the high frequency domains, since the resulting gains are modest compared to the constant spindle speed case. Based on the above numerical studies, it can be concluded that the triangular modulation allows -for a given spindle dynamics -the maximal gain for the machinist. For this reason, only triangular shaped modulation experiments will be shown here. In Figure 5, it was shown that the critical depth of cut can essentially be increased around the first flip lobe. Furthermore, the improvements were found to depend mostly on the amplitude of the speed variation, and dependence on frequency was modest, even for the high frequency range.

EXPERIMENTAL WORK

The cutting tests were conducted on a 3 axis high-speed milling center (Huron, KX10). The tool was an inserted mill with three teeth, 25 mm diameter without helix angle. The average feed per tooth was 0.1 mm/tooth. The spindle speed modulation was controlled by a sub-program using a synchronous function (Siemens 840D). According to manufacturers, spindle speed variation has no negative effect on the spindle's and the controller's lifetime.

A removable part in aluminium alloy (2017A) was machined with a radial depth of cut (Ae) of 2 mm. For such a small radial immersion, the number of cutting teeth engaged to the workpiece is changing between 0 and 1. Figure 8 shows the experimental set-up. The workpiece was fixed to a flexure that was compliant in the x direction in order to assure the single degree of freedom system. The tool was considered to be rigid compared to the flexure. The vibrations of the part were measured by a laser velocimeter (Ometron, VH300+). Filtering was achieved with a high pass filter, Butterworth 14 order type, with a 50 Hz cutoff frequency. This numerical filtering was followed by a numerical integration with the ode45 solver (Simulink ®). The measurement of the static displacements is not possible using a velocimeter signal, especially for machining conditions where laser beam may be crossed by particles. This filtering is coherent with the chatter phenomenon studied, mainly over 100 Hz. A 3D profilometer (Optical Profil, WYKO NT 1100) was used to measure the surface roughness.

The dynamic parameters of the system were obtained by hammer impact test. The cutting force coefficients were determined according to [START_REF] Seguy | Toolpath dependent stability lobes for the milling of thin-walled parts[END_REF]. The dynamic parameters and the cutting force coefficients are summarized in Table 1.

Constant spindle speed test

In order to verify the model, a series of tests at a constant speed has been conducted. The experimental results have been superimposed onto a plot of the theoretical stability predictions in Figure 9. Cutting tests were declared stable if the 1/rev sampled position of the part approached a steady constant value [START_REF] Bayly | Stability of interrupted cutting by temporal finite element analysis[END_REF] this information was also correlated by frequency analysis. Stable cutting tests are denoted by circle while unstable tests by crosses. The predicted behaviour of the system agrees well with the experiments. The classical Hopf lobes were just checked by some focused tests, while, the area of period doubling chatter at the 1st flip lobe was explored using a finer resolution of the spindle speed.

High frequency spindle speed variation

Cutting tests were conducted at spindle speed of 9100 rpm with RVA = 0.005, RVF = 0.15 (fv = 22.75 Hz) and a depth of cut of 1 mm. In this range of variation (20-25 Hz), the maximum acceleration of the spindle was measured: 70 rev/s 2 , i.e., 4200 rpm/s. The results are presented in Figure 10. Test (A) refers to the variable speed machining (see point A in Figure 7 and Figure 9), test (B) to the constant spindle speed machining (note that these cutting parameters are in the first flip lobe, see point B in Figure 4). It can be seen that both cutting tests are unstable in accordance with the theoretical predictions. The envelopes of the measured displacements are approximately the same for both the constant and the varying spindle speeds, and both machining operations have generated similarly poor surface roughness on the workpiece. These cases demonstrate that high frequency of the spindle speed modulation has a minor effect on the performance.

Low frequency spindle speed variation

In this section, cutting tests are presented where the frequency of the speed modulation is low, about 1 Hz. Consider the machining process with spindle speed 9100 rpm and depth of cut 1 mm.

For constant spindle speed, this process is unstable (see Figure 4). For high frequency spindle speed variation with low amplitude (RVA = 0.005, RVF = 0.15), the chatter still develops, i.e. the system is still unstable. Here, we apply the spindle speed variation according to point C in Figure 5. The corresponding parameters are RVA = 0.2, RVF = 0.0046875 (fv = 0.71 Hz). Based on the theoretical predictions in Figure 5, the corresponding critical depth of cut is about 2 mm, thus the machining operation with depth of cut 1 mm is predicted to be stable. Figure 11 presents the experimental results obtained for the cutting test with low frequency spindle speed variation that confirms the theoretical prediction.

At constant spindle speed (see Figure 12), chatter was clearly identified (Test B). The amplitude of the vibrations was about 0.07 mm. While the application of high frequency spindle speed variation with RVA = 0.005 and RVF = 0.15 does not result in any improvement in the stability at all (see Figure 10), low frequency modulation of the spindle speed with RVA = 0.2 and RVF = 0.0046875 stabilizes the process and reduces the amplitudes of the vibrations to 0.01 mm (see Figure 11).

The feed per tooth was 0.1 mm for all machining operations. Without vibrations, the three fluted tool would leave a machined profile of pitch 0.3 mm due to the runout (since the profile is generated by the tooth with the maximum radial runout). For the constant speed cutting in Figure 12, this pitch is 0.6 mm, which refers to the period doubling phenomena, i.e., this mark is left after two complete rotation of the tool. The corresponding surface roughness was 3.7 μm. This phenomenon is clearly presented on the 3D view of the machined surface in Figure 12.

During the tests with low frequency spindle speed variation, no chatter was observed (see Figure 11). In this case, the pitch is 0.3 mm, i.e., the profile was generated during a single rotation of the tool. The surface roughness was reduced to 1.75 μm that corresponds to 50% improvement. The 3D view in the Figure 11 shows a smoother machined surface, as well.

CONCLUSIONS

In this paper, variable spindle speed machining, on a single DOF system, was investigated for high-speed milling at the first flip lobes. Based on both numerical simulation and from experiments using specific cutting operation parameters, the following points can be concluded:

(1) The semi-discretization method is an effective and reliable method for the prediction of stability properties for milling operations with spindle speed variation.

(2) Triangular and sinusoidal speed modulations were compared in order to find the optimal technique to suppress chatter. It was found that the sinusoidal shaped modulation is more effective than the triangular one, for the same amplitude and frequency parameters. However, for the same spindle dynamics the triangular shape allows larger modulation amplitude. In fact, the triangular shape allows the maximal gain for the machinist.

(3) Spindle speed variation technique was analyzed for different variation amplitudes and frequencies. It was found that the critical depth of cut increases mainly with the amplitude of the variation, while the frequency of the variation has no significant effect on the dynamic behaviour of the process even in the case of high frequency variation reaches RVF≈1. The amplitude of the speed variation is more important from stability point of view than its frequency.

(4) The stabilization effect of spindle speed variation was confirmed by experiments. Chatter was effectively suppressed by applying low frequency spindle speed variation, while the process was unstable both for constant speed variation and for high frequency spindle speed variation.

(5) The surface roughness of the machined workpiece was improved when the spindle speed variation was applied.

(6) Finally, it should be noted that in practice it might be easier to change the spindle speed to stable parameter domains rather than applying spindle speed variation, which is highly power consuming. However, on a complex part with multiple modes, the structure of the stability diagram is complex and usually unknown for the machinist. Find a stable "window" in the stability lobe diagram is not a trivial task, if it's possible. In these cases, the application of spindle speed variation may be more practical than searching for stable spindle speeds. Furthermore, the spindle speed is often limited by other technological factors. In this context, spindle speed variation is an extra tool in addition to the traditional methods that can be used in the complex optimization of machining processes.

NOMENCLATURE

LIST OF TABLE

  Fig. 1: Typical triangular and sinusoidal shape variation.

Fig. 2 :

 2 Fig. 2: Mechanical model of the milling process.

Fig. 3 :

 3 Fig. 3: Principal period determination.

Fig. 4 :

 4 Fig. 4: Stability diagrams for variable speed milling with RVA = 0.3 and RVF = 0.003.

Fig. 6 :

 6 Fig. 6: Parametric study for sinusoidal shape for N0 = 9100 rpm and Ae = 2 mm.

Fig. 7 :

 7 Fig. 7: Parametric study for high frequency triangular shape for N0 = 9100 rpm and Ae = 2 mm.

Fig. 8 :

 8 Fig. 8: Experimental set-up.

Fig. 9 :

 9 Fig.9: Experimental test at constant spindle speed.

Fig. 10 :

 10 Fig. 10: Two machining unstable for Ap = 1 mm and N0 = 9100 rpm.

Figure 11 :

 11 Figure 11: Chatter suppression by spindle speed variation for Ap = 1 mm and N0 = 9100 rpm.

Table 1 :

 1 Flexure modal parameters and cutting coefficients.

	k [N/m]	f [Hz]	ξ [%]	KT [MPa]	KR [MPa]
	3.2×10 6	222.5	0.50	700	140

ACKNOWLEDGMENTS

This work was partly supported by the European Union Interreg IIIa AEROSFIN, by the Midi-Pyrénées Region Project "Complex workpiece machining in aeronautical context", and by the János Bolyai Research Scholarship of the HAS and by the HSNF under grant no. OTKA K72911.

The authors would also like to thank Vincent WAGNER for his help on data processing.


Etienne Gourc
email: gourc@insa-toulouse.fr

Guilhem Michon
email: guilhem.michon@isae.fr

S Ébastien Seguy

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

Experimental Investigation and Design Optimization of Targeted Energy Transfer under Periodic Forcing

In this paper, the dynamic response of a harmonically forced Linear Oscillator (LO) strongly coupled to a

Etienne Gourc

Paper VIB-13-1091 1

Nonlinear Energy Sink (NES) is investigated both theoretically and experimentally. The system studied comprises a LO with an embedded, purely cubic NES. The behavior of the system is analyzed in the vicinity of 1 : 1 resonance.

The complexification-averaging technique is used to obtain modulation equations and the associated fixed points.

These modulation equations are analyzed using asymptotic expansion to study the regimes related to relaxation oscillation of the slow flow called Strongly Modulated Response (SMR). The zones where SMR occurs are computed using a mapping procedure. The Slow Invariant Manifolds (SIM) is used to derive a proper optimization procedure.

It is shown that there is an optimal zone in the forcing amplitude-nonlinear stiffness parameter plane, where SMR occurs without having a high amplitude detached resonance tongue. Two experimental setups are presented. One is not optimized and has a relatively high mass ratio (≈ 13%) and the other one is optimized and exhibits strong mass asymmetry (mass ratio ≈ 1%). Different frequency response curves and associated zones of SMR are obtained for various forcing amplitudes. The reported experimental results confirm the design procedure, and the possible application of NES for vibration mitigation under periodic forcing.

Introduction

Over the past decade, it has been demonstrated that the addition of small mass, with a strong nonlinear attachment to a linear system may give rise, under transient loading, to localization and irreversible transfer of energy, also called pumping.

It has been shown that the pumping phenomenon can be explained by studying the nonlinear normal modes of the undamped system [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]. More recent studies have introduced a suitable asymptotic procedure based on the invariant manifold approach to include damping force [START_REF] Gendelman | Bifurcations of nonlinear normal modes of linear oscillator with strongly nonlinear damped attachment[END_REF]. Addition of a Nonlinear Energy Sink (NES) drastically changes the dynamic response of the whole system, and may be beneficial to vibration mitigation. Energy pumping under transient loading has been widely studied theoretically [START_REF] Quinn | Efficiency of Etienne Gourc Paper VIB-13-1091 16 targeted energy transfers in coupled nonlinear oscillators associated with 1:1 resonance captures: Part i[END_REF][START_REF] Sapsis | Efficiency of targeted energy transfers in coupled nonlinear oscillators associated with 1:1 resonance captures: Part ii, analytical study[END_REF][START_REF] Vakakis | Isolated resonance captures and resonance capture cascades leading to single-or multi-mode passive energy pumping in damped coupled oscillators[END_REF][START_REF] Gourdon | Nonlinear energy pumping under transient forcing with strongly nonlinear coupling: Theoretical and experimental results[END_REF] and experimentally [START_REF] Gourdon | Nonlinear energy pumping under transient forcing with strongly nonlinear coupling: Theoretical and experimental results[END_REF][START_REF] Mcfarland | Experimental study of non-linear energy pumping occurring at a single fast frequency[END_REF][START_REF] Kerschen | Experimental demonstration of transient resonance capture in a system of two coupled oscillators with essential stiffness nonlinearity[END_REF]. More recently, differents configurations of NES and their effect on the damping behavior of a structure have been studied in [START_REF] Sapsis | Effective stiffening and damping enhancement of structures with strongly nonlinear local attachments[END_REF]. In addition to transient loading, systems with NES under periodic forcing have also been studied. Steady state response (response with almost constant amplitude) was studied in [START_REF] Jiang | Steady state passive nonlinear energy pumping in coupled oscillators: Theoretical and experimental results[END_REF][START_REF] Malatkar | Steady-state dynamics of a linear structure weakly coupled to an essentially nonlinear oscillator[END_REF] for a grounded NES. The use of a NES in the field of acoustics has been studied experimentally in [START_REF] Bellet | Experimental study of targeted energy transfer from an acoustic system to a nonlinear membrane absorber[END_REF][START_REF] Mariani | Toward an adjustable nonlinear low frequency acoustic absorber[END_REF]. The use of a piecewise linear NES has been studied in [START_REF] Lamarque | Targeted energy transfer in mechanical systems by means of non-smooth nonlinear energy sink[END_REF]. The Complexification-Averaging (CX-A) technique [START_REF] Manevitch | The description of localized normal modes in a chain of nonlinear coupled oscillators using complex variables[END_REF] has been used to derive modulation equation and compute the fixed points. These regimes have also been analytically and experimentally studied for an embedded NES in [START_REF] Gourc | Experimental investigation and theoretical analysis of a nonlinear energy sink under harmonic forcing[END_REF]. The possibility of using NES in the presence of gravity has been investigated in [START_REF] Savadkoohi | Vibratory energy exchange between a linear and a nonsmooth system in the presence of the gravity[END_REF]. It has been demonstrated that in addition to weak quasiperiodic response, which is related to a Hopf bifurcation of the slow flow, systems with NES can exhibit more complex mechanisms for vibration mitigation. These regimes are related to relaxation oscillations and are not related to slow flow fixed points. When the system exhibits the latter type of response (often called Strongly Modulated Response (SMR)), the amplitude of modulation is comparable to the amplitude of the response itself.

SMR regimes have been studied in detail in [START_REF] Starosvetsky | Response regimes of linear oscillator coupled to nonlinear energy sink Etienne Gourc Paper VIB-13-1091[END_REF]20]. A NES design methodology has been proposed in [21] and the result was compared to numerical simulations. This paper aims to provide experimental developments of energy pumping under periodic forcing and also a NES design procedure. The next section is devoted to the theoretical treatment of the equation of motion. In the third section, two

Etienne Gourc

Paper VIB-13-1091 2 different experiments and design procedures are presented. The experimental measurements are compared to theoretical prediction. The last section presents the concluding remarks.

Theoretical developments

The theoretical developments presented herein are based on [START_REF] Starosvetsky | Response regimes of linear oscillator coupled to nonlinear energy sink Etienne Gourc Paper VIB-13-1091[END_REF]21]. The system studied in this paper is composed of a harmonically excited Linear Oscillator (LO) strongly coupled to a Nonlinear Energy Sink (NES) (see Fig. 1) and is described by the following equation of motion:

m 1 d 2 x 1 dt 2 + c 1 dx 1 dt + c 2 dx 1 dt - dx 2 dt + k 1 x 1 + k 2 (x 1 -x 2 ) 3 = k 1 x e + c 1 dx e dt (1)
m 2 d 2 x 2 dt 2 + c 2 dx 2 dt - dx 1 dt + k 2 (x 2 -x 1 ) 3 = 0 (2)
[Fig. 1

about here.]

Where x 1 , m 1 , c 1 , k 1 and x 2 , m 2 , c 2 , k 2 are the displacement, mass, damping and stiffness of the LO and the NES respectively. All the physical parameters, including the damping of the LO, are taken into account. The imposed harmonic displacement x e is expressed as follow:

x e = G cos Ωt (3)

Fixed points

After rescaling, system (1,2) is reduced to a more convenient form:

ẍ1 + ελ 1 ẋ1 + ελ 2 ( ẋ1 -ẋ2 ) + x 1 + εK (x 1 -x 2 ) 3 = εF cos Ωτ -ε 2 λ 1 Fω sin Ωτ (4)
Etienne Gourc Paper VIB-13-1091

ε ẍ2 + ελ 2 ( ẋ2 -ẋ1 ) + εK (x 2 -x 1 ) 3 = 0 (5)
Where the dots denote differentiation with respect to τ and the following parameters are defined:

{τ, ω 1 , ω 2 , ε, λ 1 , λ 1 , Ω, K, F} = ω 1 t, k 1 m 1 , k 2 m 2 , m 2 m 1 , c 1 m 2 ω 1 , c 2 m 2 ω 1 , Ω ω 1 , ω 2 2 ω 2 1 , G ε
It should be noticed that the rescaled equations are identical for imposed force or displacement up to O (ε). New variables are introduced as follow:

v = x 1 + εx 2 , w = x 1 -x 2 (6)
As the system is studied in the vicinity of the 1 : 1 resonance where both oscillators oscillate at the excitation frequency Ω, it is convenient to introduce the following complex variables [START_REF] Manevitch | The description of localized normal modes in a chain of nonlinear coupled oscillators using complex variables[END_REF]:

φ 1 e iΩτ = v + iΩv, φ 2 e iΩτ = ẇ + iΩw (7)
Introducing Eq. (6,7) into (4,5), and keeping only terms containing e iΩτ yields to the following slow modulated system:

φ1 + iΩ 2 ϕ 1 + ελ 1 2 (1 + ε) (ϕ 1 + εϕ 2 ) - i (ϕ 1 + εϕ 2 ) 2Ω (1 + ε) - εF 2 - iε 2 λ 1 FΩ 2 = 0 (8) φ2 + iΩ 2 ϕ 2 + ελ 1 2 (1 + ε) (ϕ 1 + εϕ 2 ) - i (ϕ 1 + εϕ 2 ) 2Ω (1 + ε) + λ 2 (1 + ε) 2 ϕ 2 - 3iK (1 + ε) 8Ω 3 ϕ 2 2 |ϕ 2 | - εF 2 - iε 2 λ 1 FΩ 2 = 0 (9)
Etienne Gourc Paper VIB-13-1091 4

A detuning parameter σ representing the nearness of excitation frequency to the reduced natural frequency of the LO is introduced as follow:

Ω = 1 + εσ (10)
Fixed points of Eq. [START_REF] Mcfarland | Experimental study of non-linear energy pumping occurring at a single fast frequency[END_REF][START_REF] Kerschen | Experimental demonstration of transient resonance capture in a system of two coupled oscillators with essential stiffness nonlinearity[END_REF] correspond to the periodic solutions of system [START_REF] Quinn | Efficiency of Etienne Gourc Paper VIB-13-1091 16 targeted energy transfers in coupled nonlinear oscillators associated with 1:1 resonance captures: Part i[END_REF][START_REF] Sapsis | Efficiency of targeted energy transfers in coupled nonlinear oscillators associated with 1:1 resonance captures: Part ii, analytical study[END_REF]. They are computed by equating the derivatives to zero, yielding a system of complex algebraic equations. After algebraic operations, this system is expressed in a more convenient form:

φ1 = φ2 = 0 ⇒ ϕ 1 (τ) = ϕ 10 , ϕ 2 (τ) = ϕ 20 ϕ 10 = iεϕ 20 (1 + ε) (1 + εσ) - ε 2 λ 1 ϕ 20 1 + ε + εF + iε 2 λ 1 F (1 + εσ) i (1 + εσ) + ελ 1 1 + ε - i (1 + ε) (1 + εσ) (11)
α 3 Z 3 20 + α 2 Z 2 20 + α 1 Z 20 + α 0 = 0, Z 20 = |ϕ 20 | 2 (12)
Coefficients α i (i = 1..3) are given in Appendix A. To study the stability of these fixed points, small perturbations are introduced:

ϕ 1 = ϕ 10 + ρ 1 , ϕ 2 = ϕ 20 + ρ 2 (13)
Keeping only linear terms with respect to ρ i (i = 1..2), taking the complex conjugate and putting the resulting system into matrix form, the stability of the fixed points is then deduced by analyzing the root of the characteristic equation. If a real root crosses the left-half complex plane, it corresponds to a Saddle-Node (SN) bifurcation and if a pair of complex Etienne Gourc Paper VIB-13-1091 5

conjugates crosses the left-half complex plane, it corresponds to a Hopf bifurcation of the slow flow.

Asymptotic analysis

As the case of small mass ratio (ε << 1) is studied here, Eq. (8,9) are analyzed using a perturbation method. Multiple time scales are introduced as follows:

ϕ i = ϕ i (τ 0 , τ 1 , . . .) , d dτ = ∂ ∂τ 0 + ε ∂ ∂τ 1 + . . . , τ k = ε k τ, k = 0, 1, . . . (14)
Substituting Eq. ( 10) and ( 14) into Eq. [START_REF] Mcfarland | Experimental study of non-linear energy pumping occurring at a single fast frequency[END_REF][START_REF] Kerschen | Experimental demonstration of transient resonance capture in a system of two coupled oscillators with essential stiffness nonlinearity[END_REF] and equating coefficients of like power of ε yields:

ε 0 : ∂ ∂τ 0 ϕ 1 = 0 ∂ ∂τ 0 ϕ 2 + λ 2 2 ϕ 2 + i 2 (ϕ 2 -ϕ 1 ) - 3iK 8 ϕ 2 2 |ϕ 2 | = 0 (15)
ε 1 : ∂ ∂τ 1 ϕ 1 + i 2 (ϕ 1 -ϕ 2 ) + iσϕ 1 + λ 1 2 ϕ 1 - F 2 = 0 ∂ ∂τ 1 ϕ 2 + i 2 (ϕ 1 -ϕ 2 ) + iσ 2 (ϕ 1 + ϕ 2 ) + λ 1 2 ϕ 1 + λ 2 2 ϕ 2 - 3iK (1 -3σ) 8 ϕ 2 2 |ϕ 2 | - F 2 = 0 (16)
The first equation of [START_REF] Lamarque | Targeted energy transfer in mechanical systems by means of non-smooth nonlinear energy sink[END_REF] gives:

∂ϕ 1 ∂τ 0 = 0 ⇒ ϕ 1 = ϕ 1 (τ 1 , . . .) (17)
Substituting Eq. ( 17) into the second equation of [START_REF] Lamarque | Targeted energy transfer in mechanical systems by means of non-smooth nonlinear energy sink[END_REF], fixed points Φ (τ 1 ) depend only on time scale τ 1 and obey the following algebraic equation:

Etienne Gourc Paper VIB-13-1091

λ 2 2 Φ + i 2 Φ - i 2 ϕ 1 - 3iK 8 Φ 2 |Φ| = 0 (18) Φ = lim τ 0 →∞ φ 2
Equation ( 18) is solved by taking Φ (τ 1 ) = N 2 e iδ 2 :

|φ 1 | 2 = λ 2 2 Z 2 + Z 2 - 3K 2 Z 2 2 + 9K 2 16 Z 3 2 Z 2 (τ 1 ) = N 2 (τ 1 ) 2 (19)
The number of solutions to Eq. ( 19) depends only on the parameter λ 2 . The roots of the derivative of the right hand side of Eq. ( 19) are computed to find the critical value of λ 2 :

Z 2,i = 4 2 ± 1 -3λ 2 2 9K i = 1, 2 (20)
Therefore, if λ 2 < 1/ √ 3, two roots and a pair of saddle-node bifurcation exist and does not exist otherwise. In fact, Eq. ( 19) represents the Slow Invariant Manifold (SIM) of the problem. In the case λ 2 > 1/ √ 3, the SIM is monotonous. On the othe hand, if λ 2 < 1/ √ 3, the SIM admits extrema and is divided into two stable branches and one unstable branch. An illustration of the SIM is given in Fig. 2 where K = 100 and λ 2 = 0.2. Such a SIM structure may give rise to relaxation oscillations. In effect, the slow flow may rise on the first stable branch until reaching the fold point Z 21 . The slow flow will jump onto the upper stable branch of the SIM to the landing point Z 2u . The amplitude of the slow flow will decrease slowly until reaching the second fold point Z 22 to jump back to the lower stable branch on point Z 2d . To investigate this possibility, the behavior of Eq. ( 16) on the SIM is analyzed.

[Fig. 2 about here.] Introducing Eq. ( 18) into the first equation of ( 16) yields:

Etienne Gourc Paper VIB-13-1091

∂ ∂τ 1 2i - λ 2 2 Φ - i 2 Φ + 3iK 8 Φ 2 |Φ| +2i i 2 + σi + λ 1 2 - λ 2 2 Φ - i 2 Φ + 3iK 8 Φ 2 |Φ| - i 2 Φ - F 2 = 0 (21)
By expressing Φ(τ 1 ) in polar coordinates, the equations governing the evolution of N 2 and δ 2 with respect to time scale τ 1 are obtained:

∂N 2 ∂τ 1 = f 1 (N 2 , δ 2 ) g (N 2 ) ∂δ 2 ∂τ 1 = f 2 (N 2 , δ 2 ) g (N 2 ) (22)
where

f 1 (N 2 , δ 2 ) = -9λ 1 K 2 N 5 2 + 24λ 1 KN 3 2 -12FKN 2 2 cos δ 2 -16 λ 2 + λ 1 + λ 2 2 λ 1 N 2 + 16F cos δ 2 + 16λ 2 F sin δ 2 (23) f 2 (N 2 , δ 2 ) = -54K 2 σ -27K 2 N 4 2 + (96Kσ + 12K -24λ 2 λ 1 K) N 2 2 + 36KFN 2 sin δ 2 -16λ 2 2 -32σ -32σλ 2 2 + 16λ 2 F cos δ 2 -16F sin δ 2 N 2 (24) g (N 2 ) = 54K 2 N 4 2 -96KN 2 2 + 32 + 32λ 2 2 (25)
It has been demonstrated [20] that Eq. ( 22) admits two types of fixed point. The first type is referred to as ordinary fixed points and is found for f 1 = f 2 = 0 and g = 0. These fixed points corresponds to those computed in Eq. ( 12) if the term at O(ε 2 ) is neglected. The others are referred to as folded singularities and are found for f 1 = f 2 = g = 0. The system

f 1 = f 2 = 0 is rewritten in matrix form: Etienne Gourc Paper VIB-13-1091    a 11 a 12 a 21 a 22       sin δ 2 cos δ 2    =    b 1 b 2    (26)
with

a 11 =16λ 2 F, a 12 = -12FKN 2 2 + 16F, a 21 = 36FKN 2 2 -16F N2 , a 22 = 16λ 2 F N 2 , b 1 =9λ 1 K 2 N 5 2 -24λ 1 KN 3 2 + 16N 2 λ 1 + λ 2 + λ 2 2 λ 1 , b 2 = 27K 2 N 5 2 (1 + 2σ) -12KN 3 2 (1 -2λ 1 λ 2 + 8σ) +16N 2 λ 2 2 + 2σλ 2 2 + 2σ /N 2 (27)
Ordinary fixed points are found by solving Eq. ( 26) for sin δ 2 and cos δ 2 , and assuming that the determinant does not vanish. It can be noticed that det(a) = 8F 2 g/N 2 , so that, when eliminating f 1 and g, the condition f 2 = 0 is automatically satisfied by Eq. ( 26), thus obtaining the expression of the folded singularities:

∆ i, j = γ i ± arccos N 2,i 16λ 1 -24λ 1 KN 2 2,i + 9λ 1 K 2 N 4 2,i + 16λ 2 +16λ 1 λ 2 2 / 4F 9K 2 N 4 2,i -24KN 2 2,i + 16 + 16λ 2 2 , ( 28
)
γ i = arcsin   4λ 2 9K 2 N 4 2,i -24KN 2 2,i + 16 + 16λ 2 2  
A condition for the excitation amplitude for the existence of folded singularities on the lower and upper fold is obtained Etienne Gourc Paper VIB-13-1091

N 2,i 16λ 1 -24λ 1 KN 2 2,i + 9λ 1 K 2 N 4 2,i + 16λ 2 + 16λ 1 λ 2 2 4F 9K 2 N 4 2,i -24KN 2 2,i + 16 + 16λ 2 2 ≤ 1 (29)
Yielding:

F ≥ F ic = N 2,i 16λ 1 -24λ 1 KN 2 2,i + 9λ 1 K 2 N 4 2,i + 16λ 2 +16λ 1 λ 2 2 / 4 9K 2 N 4 2,i -24KN 2 2,i + 16 + 16λ 2 2 (30)
However, the condition on Eq. ( 30) is necessary but not sufficient to guarantee the stability of SMR regimes. Under certain conditions, the slow flow may be attracted to another stable response. This mechanism of annihilation of SMR is explained in detail in [22]. To access this possibility, a procedure of 1D mapping has been developed in [20]. The principle consists in following the slow flow during one cycle of relaxation, with initial conditions inside the interval

[∆ 11 , ∆ 12 ].
The procedure is illustrated in Fig. 3 and consists of four steps described below:

1. A starting point is chosen on the lower fold (see Eq. ( 20)) with a phase inside the interval [∆ 11 , ∆ 12 ]. The landing points on the upper fold are then computed using Eq. ( 18) and the invariant property of the SIM (see Fig. 2 for the corresponding notation):

- 1 2 λ 2 Φ 2,1 - 1 2 iΦ 2,1 + 3 8 iKΦ 2 2,1 |Φ 2,1 | = - 1 2 λ 2 Φ 2,u - 1 2 iΦ 2,u + 3 8 iKΦ 2 2,u |Φ 2,u | (31)
2. Equation ( 22) is numerically integrated with Φ 2u as initial conditions, until reaching the upper fold line.

3. The landing point Φ 2d on the lower fold is computed in the same way as in the first step.

4. Equation ( 22) is numerically integrated again, until reaching the lower fold line.

[Fig. 3

about here.]

This procedure is repeated for various starting points inside the interval [∆ 11 , ∆ 12 ]. Finally, if at the end of step 4, all the points return inside this interval, the SMR cycle is stable. On the other hand, if the slow flow goes through the basin of attraction of a stable fixed point, the SMR cycle is unstable.

Etienne Gourc

Paper VIB-13-1091 10

In the following section, two different experiments will be presented. The main difference between these experiments is the value of the mass ratio (ε). For the first one ε = 12.9% and for the second one ε = 1.2%. Moreover, the first experiment is subject to harmonic forcing, and the second one to an imposed displacement. As mentioned previously, in the case of base excitation, a term related to the damping of the LO is present (see Eq. ( 4)); this is not the case for imposed force.

However, this term is of O(ε 2 ) and does not have that much influence on the behavior of the whole system. In both cases, the displacement is measured using contact-less laser sensors. Raw signals are recorded using a digital oscilloscope and a band pass filter is applied to correct biases and suppress high frequency noise. The cubic stiffness has been implemented

geometrically with two linear springs that extend axially and are free to rotate. The force-displacement relationship (given in Eq. ( 32)), expanded in Taylor series, is shown to be approximately cubic in nature.

f = 2k l u + 2u (P -k l l) √ l 2 + u 2 ≈ 2P l u + k l -P l 3 u 3 + O u 5 ( 32
)
Where u is the displacement, k l is the linear spring stiffness, l the initial length of the spring, and P is the pre-stress force. Experimentally, P must be kept as small as possible. Precision rail guides are used for all guidance. For each test, the Root Mean Square (RMS) value of the absolute displacement of the LO (x) and the relative displacement between the NES and the LO (w) are plotted versus the frequency of excitation. RMS values are used in order to highlight the benefits of SMR cycles.

First experiments with ε = 12.9%

The first experimental fixture built to investigate the behavior of a single-degree-of-freedom oscillator strongly coupled to a NES under excitation is depicted in Fig. 4. It consists of a main mass (LO) grounded by means of a linear spring, and connected to an electrodynamic shaker. The nonlinear oscillator is embedded on this main mass. Both masses are connected by means of an essential cubic stiffness.

[Fig. 4

about here.]

As shown in Fig. 4, the main system receives the electrodynamic force directly from the modal shaker. This force is constant whatever the response of the system. It has therefore been considered as the input excitation force, and the mass and stiffness of the shaker are considered together with the LO. The exciter force is obtained by measuring the current delivered by the power amplifier. The nonlinear stiffness value used in the theoretical analysis has been obtained with a nonlinear least square cubic polynomial fitting of the experimental curve.

[Table 1 about here.]

The parameters identified on the experimental setup and used for the calculations are given in Tab. 1.

Etienne Gourc

Paper VIB-13-1091

The aim of the experimental tests is to obtain the nonlinear Frequency Response Function (FRF) of the system around the 1 : 1 resonance. To this end, the displacement signals of both the LO and the NES have been recorded for increasing and decreasing frequency varying from 5 Hz to 20 Hz.

Figure 5 show the nonlinear response curves for the NES and the LO respectively, for a forcing amplitude of 2.7 N. Thin lines correspond to stable periodic motion, and thick lines refer to unstable region of periodic solutions. "SN" and "Hopf" indicate the location of the Saddle-Node and Hopf bifurcation points obtained analytically using Eq. ( 13). In addition to the classical resonance curve, a secondary resonance curve with a stable upper branch is observed.

Those figures also display the measured frequency response of both oscillators, where "o" and " * " denote periodic and quasiperiodic regimes respectively, and arrows show the jumps and the direction of evolution of the frequency. For an increasing frequency, the motion takes place on the stable upper branch before jumping onto the lower branch at 12.5 Hz. Then, the amplitude of motion increases again when following the main resonance curve. At this time, weak quasiperiodic regime takes place. A time measurement of such a regime is depicted in Fig. 6. This clearly shows a modulated response which match well with a Hopf bifurcation. However, unlike the analytical prediction, motion of the oscillators becomes periodic again before jumping back onto the lower curve. Nevertheless, the frequency at which the jump occurs is approximately the same in both cases.

[Fig. 5 Comparing experimental and theoretical results, it is interesting to notice that there is a qualitative agreement even if differences occur with the amplitude, and instability zones, that are mainly due to the uncertainty in the characterization of the nonlinear stiffness and damping. In this experiment, no SMR has been observed. As it has been shown in [22], in addition to the previously mentioned mechanism of annihilation of SMR, when the mass ratio increases above a given threshold ε cr2

(not computed here), neither stable SMR nor unstable limit cycles are possible, and only stable weak modulated response is observed. These considerations motivated the design of a new experiment.

NES optimization

The previously presented experiment has two major drawbacks:

1. The mass ratio was too high to allow SMR response.

2. A high amplitude detached resonance tongue appeared on the left of the main backbone branch.

In this section, a design procedure is proposed to eliminate this detached resonance tongue and to allow SMR response at the same time, considering a system with a small mass ratio (ε ≈ 1%). To study the conditions for the appearance of the detached resonance tongue, boundaries separating single and triple solutions in Eq. ( 12), corresponding to Saddle-Node bifurcation, are analyzed. Saddle-Node bifurcation arises when a real root of the characteristic polynomial leaves the left-half complex plane. By setting the root to 0, the resulting equation is written in the following form:

Etienne Gourc Paper VIB-13-1091

v 2 Z 2 20 + v 1 Z 20 + v 0 = 0 (33)
Coefficients v i are not given here due to their length. Eliminating Z 20 from Eq. ( 33) yields:

Z 20 = -v 1 ± v 2 1 -4v 2 v 0 2v 2 (34)
Substituting Eq. (34) into Eq. ( 12) and solving for F produces the boundary separating single from multiple solutions in the plane of parameters (F, σ). An illustration is given in Fig. 7.

[Fig. 7 about here.] Fig. 7 highlights that there is a narrow zone (F 1c < F < F SN ) where SMR may be possible (F > F 1c ) and where no high amplitude detached resonance tongue exists. This zone is optimal for passive control of vibration using a NES. Taking arbitrary values for ε, λ 1 and λ 2 , the boundaries for optimal NES sizing are plotted in the plane of parameters (F, K) in Fig. 8. The nonlinear stiffness K must be chosen inside the interval between the solid and the dashed curve.

[Fig. 8 about here.]

In Figure 9 the evolution of the width of the zone of SMR as a function of the forcing amplitude is presented. It is observed, that this zone is larger when the forcing amplitude is close to the boundary F SN , which is interesting from a vibration mitigation point of view. In the next section the experimental setup designed using the proposed criterion is presented.

[Fig. 9 about here.]

Second experiments with

ε = 1.2%
The second experiment is depicted in Fig. 10. Here, the whole setup is embedded on a 10 kN elecrodynamic shaker.

Feedback position control of the electrodynamic shaker ensures a constant excitation amplitude (especially during SMR regimes).

[Fig. 10 about here.]

The moving masses of the LO and the NES are: m 1 = 4.178kg and m NES = 0.042kg respectively. As the mass of the NES is very small, the inertia of the springs is no longer negligible and has to be considered. In a rough approximation, considering the spring as a beam and neglecting axial inertia, the kinetic energy of the NES is written as follows:

Etienne Gourc Paper VIB-13-1091

T NES = l 0 0 ρ S x l 0 ẏ 2 dx + 1 2 m 2 ẏ2 ( 35
)
Where ρ S = m S /l 0 is the mass density of the spring. Thus the moving mass of the NES is m 2 = 2m S /3 + m NES . The natural frequency and the viscous damping factor of the main linear system are estimated by performing modal analysis without the NES. The damping coefficient of the linear guide of the NES has also been estimated by removing the nonlinear stiffness, adding a linear spring and performing modal analysis. The friction in the spring's attachment is neglected. The parameters are summarized in Tab. 2.

[Table 2 about here.]

The nominal excitation amplitude is fixed to define the NES stiffness (G = 0.25mm; F = 0.02). The sizing curves corresponding to the physical parameters of the system are presented in Fig. 11.

[Fig. 11 about here.]

The red horizontal line corresponds to the dimensionless nominal forcing amplitude, and the two black dots correspond to the excitation amplitude for which the trials have been performed, for the chosen stiffness (K = 1874). Both the measured force-displacement relationship and the cubic fitting are presented in Fig. 12 for the designed NES.

[Fig. 12 Experimentally, energy pumping through SMR is observed for this forcing amplitude and time response is presented in Fig. 14 corresponding to "Point A" in Fig. 13. It is clear that this regime is related to relaxation oscillation. For this forcing amplitude, no detached resonance curve is observed.

[Fig. 14 about here.] Fig. 15 shows the obtained frequency response functions for G = 0.325mm. Energy pumping still occurs, but high vibration amplitudes before the natural frequency are also observed. This is in accordance with the analytical predictions in Fig. 11. The width of SMR zone is also larger when the excitation amplitude increases, as reported theoretically in Fig. 9.

Etienne Gourc

Paper VIB-13-1091 14

[Fig. 15 about here.]

The previous results highlight discrepancies between theoretical prediction and experimental measurements in the SMR zone. Numerical simulations have revealed that this zone is sensitive to the damping of both LO and NES, which may explain the difference in its width. It is also observed that, in the both cases, SMR zones shift to the right-hand-side. This is certainly due to a nonlinearity induced in the linear spring anchorage of the LO. However, the behavior observed experimentally shows that energy pumping, under harmonic excitation, is possible without having high amplitude detached resonance tongue. Despite the small mass ratio, the NES induces significant changes in the behavior of the main linear system and the frequency response curve of the LO is flattened. Qualitative behavior of the system is fully explained by the theoretical study.

Conclusion

In this paper, the dynamic response of a 2 dof system comprising a harmonically excited Linear Oscillator strongly coupled to a Nonlinear Energy Sink was investigated theoretically and experimentally.

The complexification-averaging thechnique was used to derive modulation equations. The modulation equations were analyzed using asymptotic expansion. The zone of existence of stable strongly modulated response was determined using a mapping procedure.

The first experiment presented was not optimized. This experiment had a relatively high mass ratio (ε = 12.9%). For this case, the so-called strongly modulated response was not observed, but the obtained frequency response function match fairly well with the theoretical prediction (fixed points computed using the complexification averaging technique). A design procedure was then proposed to optimize the system. The aim of this procedure was to avoid the detached resonance curve and allow SMR response at the same time. The improved experiment with ε = 1.2% and the associated measurements were presented. Frequency response functions were plotted for two different excitation amplitudes. At both excitation amplitudes, energy pumping by means of strongly modulated response was observed. A good correlation between theoretical and experimental result was observed.

The experiment also showed that it is possible to avoid the detached resonance curve and still have energy pumping, validating the design methodology.

The use of NES under harmonic excitation is a useful vibration mitigation device due to the lack of preferential linear frequency. However, a proper design procedure must be conducted to avoid spurious high amplitude response.

α 0 = -(1 + ε) 2 ε 2 λ 2 1 (1 + εσ) 2 + 1 F 2 (1 + εσ) 4 /Γ α 1 = (1 + ε) 2 2λ 2 λ 1 (1 + εσ) 4 + λ 2 2 λ 2 1 (1 + εσ) 2 + (1 + εσ) 4 + σ (εσ + 2) 2 (1 + εσ) 2 + εσ 2 + 2σ + (1 + εσ) 2 λ 2 1 (1 + εσ) 2 + σ 2 (εσ + 2) 2 /Γ α 2 = -3 (1 + ε) 2 K λ 2 1 (1 + εσ) 2 + σ (εσ + 2) × (1 + εσ) 2 + εσ 2 + 2σ / 2Γ (1 + εσ) 2 α 3 = 9K 2 (1 + ε) 2 64 (1 + εσ) 6 With Γ = 4λ 2 1 (1 + εσ) 2 + 4 (1 + εσ) 2 + εσ 2 +

  about here.] [Fig. 6 about here.]

  about here.] The first results carried out at the nominal forcing amplitude (G = 0.25mm) are presented in Fig. 13. The analytical frequency response function is presented in blue, and the experimental one in green. The gray dashed line and the red one represent the theoretical and experimental zone where SMR occurs. The thin black line corresponds to the theoretical FrF of the LO without NES. Theoretically at the resonance frequency, the RMS amplitude of the LO without NES is x RMS = 0.106 which is significantly reduced by the NES [Fig. 13 about here.]

Fig. 1 .

 1 Fig. 1. Schema of the 2 dof system comprising a LO and a NES

Fig. 2 .

 2 Fig. 2. Exemple of SIM for K = 100, λ 2 = 0.2

Fig. 3 .

 3 Fig. 3. Illustration of the mapping procedure for K = 100, σ = 1, F = 0.15, ε = 0.01, λ 1 = 0.1, λ 2 = 0.2

Fig. 4 .

 4 Fig. 4. First experimental set-up (ε = 12.9%)

Fig. 5 .

 5 Fig. 5. Experimental (Green) and analytical (Blue) frequency response curve of the LO and the NES for the first experiments and F = 2.7 N ("o": experimental periodic oscillation, " * ": experimental modulated oscillation, thin line: analytical stable fixed points, thick line: analytical unstable fixed point)

Fig. 6 .

 6 Fig. 6. Experimental measure of weak quasi-periodic response for F = 2.7 N, f = 14.5 Hz, ε = 12.9%

Fig. 7 .

 7 Fig. 7. Boundary of the Saddle-Node bifurcation for K = 100, ε = 0.01, λ 1 = 0.1, λ 2 = 0.2

Fig. 8 .

 8 Fig. 8. Critical forcing amplitude as a function of the nonlinear stiffness ε = 0.01, λ 1 = 0.1, λ 2 = 0.2

Fig. 9 .

 9 Fig. 9. Zone of SMR as a function of the forcing amplitude for K = 1000, ε = 0.01, λ 1 = 0.1, λ 2 = 0.2

Fig. 10 .

 10 Fig. 10. General view of the second experimental set-up (ε = 1.2%)

Fig. 11 .

 11 Fig. 11. Design curve corresponding to physical parameters

Fig. 12 .

 12 Fig. 12. Force-displacement relationship of the designed NES

Fig. 13 .

 13 Fig. 13. Experimental (Green) and analytical (Blue) frequency response curve of the LO and the NES for the second experiments and G = 0.25mm (vertical red line : analytically determined zone of SMR, vertical dashed line: experimentally determined zone of SMR, thin black line: theoretical FrF without NES)

Fig. 14 .

 14 Fig. 14. Experimental measurement of SMR with the second experiment with G = 0.25mm, f = 8.5Hz, ε = 1.2% (Point A in Fig. 13)

Fig. 15 .

 15 Fig. 15. Experimental (Green) and analytical (Blue) frequency response curve of the LO and the NES for the second experiments and G = 0.325mm (vertical red line : analytically determined zone of SMR, vertical dashed line: experimentally determined zone of SMR)

Table 1 .

 1 2σ 2 with harmonic forcing and frequency detuning". J. Sound Vib., 315(3), pp. 746 -765.[20] Starosvetsky, Y., and Gendelman, O. V., 2008. "Strongly modulated response in forced 2dof oscillatory system with essential mass and potential asymmetry". Physica D, 237[START_REF] Bellet | Experimental study of targeted energy transfer from an acoustic system to a nonlinear membrane absorber[END_REF], pp. 1719 -1733. [21] Starosvetsky, Y., and Gendelman, O. V., 2008. "Attractors of harmonically forced linear oscillator with attached nonlinear energy sink. ii: Optimization of a nonlinear vibration absorber". Nonlinear Dyn., 51, pp. 47-57. Parameters of the first experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 Parameters of the second experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Schema of the 2 dof system comprising a LO and a NES . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2 Exemple of SIM for K = 100, λ 2 = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Design curve corresponding to physical parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 12 Force-displacement relationship of the designed NES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parameters of the first experiment

	[22] Starosvetsky, Y., and Gendelman, O. V., 2010. "Bifurcations of attractors in forced system with nonlinear energy sink:
	the effect of mass asymmetry". Nonlinear Dyn., 59(4), pp. 711-731.
	List of Tables
	1
	List of Figures
	1

3 Illustration of the mapping procedure for K = 100, σ = 1, F = 0.15, ε = 0.01, λ 1 = 0.1, λ 2 = 0.2 . . . . . . 23 4 First experimental set-up (ε = 12.9%) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5 Experimental (Green) and analytical (Blue) frequency response curve of the LO and the NES for the first experiments and F = 2.7 N ("o": experimental periodic oscillation, " * ": experimental modulated oscillation, thin line: analytical stable fixed points, thick line: analytical unstable fixed point) . . . . . . . . . . . . . . 25 6 Experimental measure of weak quasi-periodic response for F = 2.7 N, f = 14.5 Hz, ε = 12.9% . . . . . . 26 7 Boundary of the Saddle-Node bifurcation for K = 100, ε = 0.01, λ 1 = 0.1, λ 2 = 0.2 . . . . . . . . . . . . . 27 8 Critical forcing amplitude as a function of the nonlinear stiffness ε = 0.01, λ 1 = 0.1, λ 2 = 0.2 . . . . . . . 28 9 Zone of SMR as a function of the forcing amplitude for K = 1000, ε = 0.01, λ 1 = 0.1, λ 2 = 0.2 . . . . . . 29 10 General view of the second experimental set-up (ε = 1.2%) . . . . . . . . . . . . . . . . . . . . . . . . . . 30 11 (Point A in Fig. 13) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 15 Experimental (Green) and analytical (Blue) frequency response curve of the LO and the NES for the second experiments and G = 0.325mm (vertical red line : analytically determined zone of SMR, vertical dashed line: experimentally determined zone of SMR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Etienne Gourc Paper VIB-13-1091 18

Table 2 .

 2 Parameters of the second experiment

		Physical Parameters
	m 1	4.178 kg	m 2	0.052 kg
	k 1 1.12 * 10 4 N/m c 1 3.97 Ns/m
	c 2	0.36 Ns/m
		Reduced Parameters
	ε	1.2%	λ 1	1.45
	λ 2	0.13

Coefficients of Eq. ( 12):

Etienne Gourc

Paper VIB-13-1091