T Li

D Qiu
email: qiu@insa-toulouse.fr

S Seguy

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

Activation characteristic of a vibro-impact energy sink and its application to chatter control in turning

Keywords: Vibro-impact, Optimal design, Activation, Nonlinear system, Targeted energy transfer, Nonlinear energy sink, Impact damper

The ultimate goal of this paper is to propose a procedure for the optimal design of a Vibro-Impact (VI) Nonlinear Energy Sink (NES) to control the vibration of any possible linear or nonlinear main systems. To this end, the activation characteristic of VI NES at a range of displacement amplitude of a main system is generalized from linear systems to nonlinear systems. It is theoretically proved and experimentally observed that this activation characteristic is almost independent of frequency, which provides direct proof for the effectiveness of VI NES in a broad frequency bandwidth. In terms of vibration control, this feature is very attractive and builds a bridge between linear and nonlinear systems. Then it is applied for the design of VI NES attached to nonlinear systems. In this way, the design of VI NES for a nonlinear system is simplified to the optimal design for a linear system, which is designed to be similar to this target nonlinear system. Because the latter can be analytically calculated, the proposed method is feasible from a quantitative perspective. Finally, this activation characteristic and a proposed design method are applied to control chatter in a turning process, and results prove its feasibility.

Introduction

In engineering, clearance is common for structures such as linkage, gear train and joint. Impacts come into being when two objects contact, and can result in energy transfer and dissipation. This phenomenon is applied to vibration control since seventy years ago [START_REF] Lieber | An acceleration damper: development, design and some applications[END_REF] and an auxiliary device is termed as impact damper. Later, there are extensive studies around impact damper and its dynamics as a typical vibro-impact system [START_REF] Ibrahim | Vibro-impact dynamics: modeling, mapping and applications[END_REF][START_REF] Babitsky | Theory of vibro-impact systems and applications[END_REF]. The focus here is put on some recent studies since they are tightly related to the work of this paper.

Recently, impact damper is re-examined 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 called Vibro-Impact (VI) Nonlinear Energy Sink (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]. The main advance comes from the analytical study of underlying Hamiltonian system [START_REF] Lee | Periodic orbits, damped transitions and targeted energy transfers in oscillators with vibro-impact attachments[END_REF] and the application of multiple scales method [START_REF] Gendelman | Analytic treatment of a system with a vibro-impact nonlinear energy sink[END_REF]. Consequently, special orbits that lead to TET are found from the former, and a Slow Invariant Manifold (SIM) that controls the variation of resonance captures are obtained from the latter. Some following analytical and numerical studies reveal further information about the complicated dynamics of this VI system, especially Strongly Modulated Response (SMR) [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 bifurcation analysis from the perspective of impact time difference [START_REF] Li | On the dynamics around targeted energy transfer for vibro-impact nonlinear energy sink[END_REF].

In addition to the explanation of its dynamics, there are also many researches about its characteristic as a damper. In general, two aspects of study results are obtained [START_REF] Karayannis | Vibro-impact attachments as shock absorbers[END_REF][START_REF] Lee | Periodic orbits, damped transitions and targeted energy transfers in oscillators with vibro-impact attachments[END_REF], namely activation characteristic and parameter optimization. In terms of activation, VI NES is observed to respond fast at an initial stage when a main system is perturbed, and it is effective in a broad frequency range from the results of frequency spectrum analysis. With respect to parameter design, many suggestions are proposed, for example a medium clearance is better. Around these two aspects, it is found that the impact number per cycle of a main system matters for energy transfer and dissipation [START_REF] Li | Dynamics of cubic and vibro-impact nonlinear energy sink: Analytical, numerical, and experimental analysis[END_REF]. Then, the efficiency of different response regimes with different impact number per cycle is compared in [START_REF] Li | Optimization mechanism of targeted energy transfer with vibro-impact energy sink under periodic and transient excitation[END_REF], and it is found that the limit between response with permanent two impacts per cycle and that with intermittent two impacts per cycle (SMR) is optimal. Its essence behind is transient resonance captures with two impacts per cycle. This conclusion also holds for a linear system coupled with two VI NES in parallel [START_REF] Li | Dynamics of two vibro-impact nonlinear energy sinks in parallel under periodic and transient excitations[END_REF]. In [START_REF] Li | Optimization mechanism of targeted energy transfer with vibro-impact energy sink under periodic and transient excitation[END_REF], there is another interesting phenomenon, just a range of clearance will be effective for a fixed outside excitation. Equivalently, a fixed clearance will only be effective in a displacement amplitude range of a main system [START_REF] Li | Dynamics of two vibro-impact nonlinear energy sinks in parallel under periodic and transient excitations[END_REF].

This feature is very interesting in terms of vibration control and is very special as a NES. In [START_REF] Viguié | Nonlinear vibration absorber coupled to a nonlinear primary system: a tuning methodology[END_REF], this kind of characteristic is also tried to be designed for a nonlinear vibration absorber but in a general way. Its basic philosophy of design can be reflected by the response regime with two impacts per cycle of systems coupled with VI NES.

Whether this effective activation of VI NES only depends on the displacement amplitude and has nothing to do with frequency? If the answer is positive, it could be applied as a design criterion for VI NES to control the vibration of nonlinear systems at some displacement levels. Its benefit is evident since it maybe impossible to get reliable results from analytical studies of nonlinear systems coupled with VI NES, and the above-mentioned idea may provide a feasible solution.

The difficulty of the application of NES to control vibration is already observed. For example, VI NES is attached to a cutting tool to quench its chatter in [START_REF] Gourc | Quenching chatter instability in turning process with a vibro-impact nonlinear energy sink[END_REF]. Experimental results demonstrate that an appropriately designed VI NES can effectively reduce the vibration of a cutting tool. However, its analytical development is based on a simplified equation and still has a distance to predict real responses, which is the same case for a turning system coupled with a cubic NES [START_REF] Gourc | Chatter control in turning process with a nonlinear energy sink[END_REF]. The problem is the same for a helicopter [START_REF] Gourc | Post-critical analysis of ground resonance phenomenon: effect of stator asymmetry[END_REF][START_REF] Bergeot | Analysis of steady-state response regimes of a helicopter ground resonance model including a non-linear energy sink attachment[END_REF]. Considering the difficulty of analytical study for any nonlinear systems coupled with VI NES, the possibility to apply its activation characteristic and to simplify its design for nonlinear systems will be explored.

The paper is organized as follows. In Section 2, a theoretical analysis for the activation characteristic of VI NES is presented. In Section 3 and 4, this activation characteristic is validated by numerical and experimental results from different linear and nonlinear main systems, respectively. In Section 5, a design procedure of VI NES is proposed and applied to control chatter in turning. Finally, conclusion is addressed.

Analytical treatment

In essence, the analytical development here is similar to that of former studies [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] Li | Dynamics of cubic and vibro-impact nonlinear energy sink: Analytical, numerical, and experimental analysis[END_REF], but some important information of SIM is neglected and will be further analyzed.

Analytical formulation

[Fig. 1

about here.]

A LO under periodic excitation and coupled with a VI NES is showed in Fig. 1. Its motion between impacts is described by the following equation:

ẍ + ελ 1 ẋ + x = εG sin Ωτ + ε 2 λ 1 GΩ cos Ωτ ε ÿ = 0 ∀ |x -y| < b (1)
Parameters are expressed as follows:

ε = m 2 m 1 , ω 0 2 = k 1 m 1 , f 0 = ω 0 2π , 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 LO, respectively. y and m 2 represent displacement and mass of VI NES, respectively. Dots denote differentiation with respect to dimensionless time τ. b represents the clearance. x e (t) is the displacement imposed on the base by a shaker. εG sin Ωτ and ε 2 λ 1 GΩ cos Ωτ represent the force contribution of displacement and that of velocity by outside excitation, respectively. The latter term is conserved here to demonstrate its physical meaning and is neglected during the following analysis because of its small magnitude.

When |x -y| = b, impacts occur. The relation between after and before impact is obtained under the hypothesis of simplified shock theory and the condition of 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 + andindicate the time immediately after and before impact. New variables representing the displacement of the center of mass and the internal displacement of VI NES are introduced:

v = x + εy, w = x -y (3)
Substituting Eq. (3) into Eqs. ( 1) and ( 2), the equation between impacts in barycentric coordinate is given as:

v + ελ 1 v + ε ẇ 1 + ε + v + εw 1 + ε = εG sin Ωτ + ε 2 λ 1 GΩ cos Ωτ ẅ + ελ 1 v + ε ẇ 1 + ε + v + εw 1 + ε = εG sin Ωτ + ε 2 λ 1 GΩ cos Ωτ ∀|w| < b (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 simplified natural frequency of LO is introduced:

Ω = 1 + εσ (7)
Substituting Eqs. ( 6) and (7) into Eqs. ( 4) and ( 5), equating coefficients of like power of ε and only conserving the first two orders:

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)
where D 0 represents partial derivative to time τ 0 . For order ε 1 , only the term related to LO is conserved and will be used later. Combining the first order and the second one, both SIM and fixed points could be obtained. From the analysis of order ε 0 , v 0 represents an ideal undamped harmonic oscillator expressed as follows:

v 0 = C(τ 1 ) sin (τ 0 + θ (τ 1 )) (11)
where C(τ 1 ) and θ (τ 1 ) are its amplitude and phase, respectively. From the standpoint of w 0 , Eq. 8 and Eq. 9 represent a harmonically forced impact oscillator with symmetric barrier. For the response regime (1 : 1 resonance) with two impacts 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 B(τ 1 ) and η(τ 1 ) are displacement amplitude and phase of VI NES, respectively. Π(z) is a non-smooth sawtooth function [START_REF] Pilipchuk | Closed-form solutions for oscillators with inelastic impacts[END_REF]. This folded function and its derivative are depicted in Fig. 2 and expressed as follows:

[Fig. 2 about here.]

Π(z) = arcsin(sin z), M(z) = dΠ dz = sgn(cos z) (13)
According to Eqs. ( 12) and [START_REF] Li | On the dynamics around targeted energy transfer for vibro-impact nonlinear energy sink[END_REF], impact occurs at T 0 = π/2η + jπ with j = 0, 1, 2, . . .. The impact condition |w 0 | = b is rewritten with Eq. ( 12) as:

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

C (1 + r) sin (η -θ ) = 2 π B (1 -r) (15)
Combining Eqs. ( 14) and [START_REF] Li | Optimization mechanism of targeted energy transfer with vibro-impact energy sink under periodic and transient excitation[END_REF], 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 SIM described by Eq. ( 16) with b = 1 and r = 0.6 is presented in Fig. 3. The stability of SIM is analyzed by an asymptotic approach used in [START_REF] Gourc | Targeted energy transfer under harmonic forcing with a vibro-impact nonlinear energy sink: Analytical and experimental developments[END_REF] that is originally applied in [START_REF] Masri | On the stability of the impact damper[END_REF][START_REF] Bapat | Stable periodic motions of an impact-pair[END_REF]. By this stable analysis method, the stable branch is defined by the condition that the modulus of all the eigenvalues of a certain matrix relating two consecutive impacts is less than unity. This stability analysis can also be accomplished by direct numerical integration of Eqs. ( 8) and ( 9).

[Fig. 3

about here.]

In order to obtain fixed points or study the non-stationary evolution of the motion of the system on the SIM, i.e., the analysis of transient resonance captures under transient excitation and transient resonance captures of 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:

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 components in addition to the simplified natural frequency of LO. The component C (τ 1 ) is decided by the motion of LO and E (τ 1 ) is totally related to the motion of VI NES. The value of E (τ 1 ) is related to the periodic impact force. The Eq. ( 17) is a more general and relaxed analytical description with respect to the motion of VI NES compared to former studies [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] that consider only 1:1 resonance with two symmetric impacts per cycle.

Substituting Eqs. [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 [START_REF] Viguié | Nonlinear vibration absorber coupled to a nonlinear primary system: a tuning methodology[END_REF] into Eq. ( 10) and eliminating terms that produce secular terms give:

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 LO and VI NES. η represents the phase difference related to LO and outside excitation.

The fixed points can be obtained by equating the left side of Eq. ( 18) to zero and then combining it with Eq. ( 16). In this way, the fixed points (number and position) can be obtained. Compared to the classic non-asymptotic method, the functionality of this asymptotic method is twofold. Firstly, the position of points can be used to judge the type of response regime. Secondly, values can be precisely calculated for 1:1 resonance, which is related to optimal response regime [START_REF] Li | Optimization mechanism of targeted energy transfer with vibro-impact energy sink under periodic and transient excitation[END_REF]. Meanwhile, the corresponding equivalent force E between LO and VI NES can be used to analyze their underlying dynamic performance.

Analysis of SIM

For a fixed point in SIM as showed in Fig. 3, C and B represent the displacement amplitude related to LO and VI NES, respectively. After a dimensionless process, these two parameters remain the same. However, the frequency of LO is normalized and the velocity of VI NES is also scaled. It means that one point of SIM actually represents different response regimes which have the same ratio between C and B but possess different frequencies. Different points of SIM represent different responses with different ratios of displacement amplitude and they together represent all possible response regimes. Here, one important point that different systems coupled with a same VI NES can have the same SIM is neglected during all former studies, because these former studies are carried out for different systems case by case.

In [START_REF] Li | Optimization mechanism of targeted energy transfer with vibro-impact energy sink under periodic and transient excitation[END_REF], it is found that a VI NES with a specific clearance will only be effective in a range of displacement amplitude of LO. Actually, this is related to the blue stable branch of SIM in Fig. 3. Theoretically, VI NES with a fixed clearance should be activated at a fixed displacement amplitude interval for different LO with different natural frequencies. Then, whether it still holds for nonlinear systems with varying frequency when its energy changes. If a considered nonlinear system at different energy levels can be equivalent to different linear systems with different frequencies, the above activation characteristic can still be used. The essence behind is to find the activation condition of VI NES. In this way, the difficulty of analytical treatment of system coupled with VI NES during the design process can be avoided.

To design VI NES for vibration control of a nonlinear system, actually it is difficult to get the full information of this nonlinear system, let alone to treat this coupled system analytically. However, its range of work displacement amplitude is normally known in advance or can be obtained from a design objective. From the above analysis, it is exactly what needed for the optimal or relatively optimal design of VI NES.

A design procedure for VI NES coupled to nonlinear systems

For the design of a VI NES for the vibration control of a nonlinear system, the idea is simple and just to find a similar linear system to this nonlinear system, then to optimally design VI NES for this linear system. However, there are some requirements to assure VI NES still works for target nonlinear system. Firstly and at least, the amplitude of this linear system should be the same as that of nonlinear system. Secondly and unnecessarily, the frequency of this linear system should be designed the same as that of nonlinear system. If just a linear system is used to approximate this nonlinear system working in a range of displacement amplitude, usually only the amplitude range can be approximated by this linear system and the condition with same frequency cannot be met. This requires a prerequisite that VI NES will be activated at a range of amplitude and its activation is independent of frequency. This point will be investigated in the following section.

As a summary, the activation of VI NES as a damper is special from two perspectives. Firstly, a VI NES with a fixed clearance will only be effective in a range of displacement amplitude. It means that different clearances will be activated at different amplitudes of displacement as observed from experimental transient excitation results [START_REF] Li | Dynamics of two vibro-impact nonlinear energy sinks in parallel under periodic and transient excitations[END_REF]. This characteristic is very important in the domain of vibration control. The second characteristic is about its effectiveness in a broad frequency range. Former observations are based on the frequency spectrum analysis of a transient or periodic response, and this way is indirect. This time, its theoretical base is well demonstrated from the analysis of SIM.

Numerical observations

To guarantee the above-mentioned design mechanism of VI NES for nonlinear systems, several aspects about activation characteristics of VI NES will be checked in this section. Firstly, critical points (i.e., p0, p1 and p2) of SIM should have the same values for different linear or nonlinear systems with different frequencies under some same conditions. For this purpose, three linear systems with different natural frequencies and one Duffing system with cubic nonlinearity are numerically studied. Moreover, whether an optimal clearance of VI NES for one system is still optimal for other systems is investigated. Secondly, the proportional activation of VI NES with different clearances is experimentally examined.

LO and Duffing systems

Eq. ( 1) is modified to Eq. ( 20) in order to include a cubic term described by α, and a proportional factor β is introduced to regulate linear stiffness. The variation of β will be reflected in parameter λ 1 . For moments of impact, Eq. (2) still applies. The meaning of other parameters is the same as those in the last section. Eqs. ( 20) and (2) will be combined for numerical simulation, and the following parameters are fixed except specially pointed out: ε = 0.84, r = 0.6.

ẍ + ελ 1 ẋ + x + αx 3 = εG sin Ωτ + ε 2 λ 1 GΩ cos Ωτ ε ÿ = 0 ∀ |x -y| < b (20)
where

ω 0 2 = β k 1 m 1

Free vibration of LO with different natural frequencies

The following parameters and initial conditions are used: G = 0, α = 0, b = 0.05, x 0 = 0.1, ẋ0 = 0, y 0 = b, ẏ0 = 0. The following three values β are consecutively chosen: 1, 0.75 and 0.5. The objective is to create three different linear systems.

The results are showed in Fig. 4. For β = 1, the displacement of LO and relative displacement are displayed in Figs. 4 (a) and (b), respectively. Then, two points and their values are marked out in Figs. 4 (c-d). They correspond to special points p0 and p2 in Fig. 3. Because the transition from regime with two asymmetric impacts per cycle to the symmetric case is difficult to distinguish, only p0 and p2 are used in the following results. In Fig. 4(e), corresponding value x of p0 and p2 for these three different β is compared. These two broken lines are the line fitting of numerical results. It is observed that the value x is almost the same for these three values of β .

[Fig. 4 about here.] When parameters and initial conditions are fixed as: G = 0, β = 1, α = 0, x 0 = 0.02, ẋ0 = 0, y 0 = b, ẏ0 = 0, different values of b are chosen to compare its efficiency for a same main system and b = 0.05 is found to be optimal. Then this optimal value of b is applied to other systems with different β . The results are demonstrated in Fig. 5. The time history of displacement for β = 1 is displayed in Fig. 5(a) and x related to p2 is marked out. The comparison between three different β is showed in Fig. 5(b) . All three values are around 0.0052, which results in almost the same decay inclination for these three systems.

[Fig. 5 about here.]

Free vibration of different Duffing systems

For systems with cubic nonlinearity, the following parameters and initial conditions are used: G = 0, α = 250000, β = 1, x 0 = 0.02, ẋ0 = 0, y 0 = b, ẏ0 = 0. Different values of b are chosen to compare their efficiency. x related to p2 are compared and showed in Fig. 6. It is observed that x for p2 is proportional to b, and this characteristic will also be demonstrated in the following experimental results.

[Fig. 6

about here.]

As a summary, it is numerically observed the proportional activation of VI NES to different clearances and the independence of its activation to the frequency of main systems.

Experimental observations

The objective of this section is to verify the activation of VI NES from different experimental results under different excitations.

Periodic excitation

For periodic excitation, experiments for a LO coupled with a VI NES are the same as these in [START_REF] Li | Optimization mechanism of targeted energy transfer with vibro-impact energy sink under periodic and transient excitation[END_REF], in which specific information about experimental configuration and parameters can be found. Here, just the time history of displacement is demonstrated for a single frequency excitation at the first place, and for a range of frequency at the second place.

Single frequency excitation

The whole time histories of displacement of LO for three different b is showed in Fig. 7 (a-c) , respectively. Judging from the time range 20 -70 s, SMR occurs at this stable area, and the point of minimal amplitude related to p2 is marked out at the same time moment for all three of them. Its value denoted by Y is marked out. The relation between different b and its corresponding minimal amplitude is represented by blue circles in Fig. 7(d) and then is linearly interpolated. The consistency between the red broken line and circles proves the proportional activation characteristic to some extent considering that the number of points is limited. However, this activation feature could be further observed from other viewpoints.

[Fig. 7 about here.]

Excitation with a band of frequency

To observe the proportional activation characteristic of VI NES, the amplitude of LO related to the limit point p2 in SIM, i.e., the corresponding response regime limit between SMR and two impacts per cycle, is recorded for different values of b during the sweep process.

[Fig. 8

about here.]

The displacement amplitudes of LO are marked out for different b and showed in Fig. 8(a) and (b). The shift of resonance peak between different experimental results comes from the difference of the starting record time. The starting or ending points of the regime with two impacts per cycle are marked out for different b and their specific judgment requires the data of acceleration. This judgment process will be demonstrated for the analysis of the following transient experimental results. It should be pointed out here that these marked points are enough to coarsely describe the difference caused by b, though it is difficult to accurately locate these points. Moreover, these two points entering into and leaving from the regime with two impacts per cycle is visible and distinguishable during the experimental process. As has been done before, these points are fitted by a red broken line as showed in Fig. 8(c). These points almost locate in a line and prove again the proportional activation characteristic.

Transient excitation

The objective of transient experiments with different linear and nonlinear systems is to study whether the activation of VI NES is related to frequency.

Experimental configuration

[Fig. 9 about here.]

In addition to the above experimental results under periodic excitation, three linear systems with different stiffness and one Duffing system are experimentally studied under transient excitation. The objective here is to observe the dependence of the activation behavior of VI NES on frequency. More specifically, it is to test that a VI NES with a same clearance will be excited at a same displacement range of a main system with different frequencies.

[Table 1

about here.]

The same mass and damping of oscillator, same VI NES as showed in Fig. 9 will be used during the whole experimental process. Only the number of springs is changed from 4 to 3 or 2, or even the direction of springs is changed to create a Duffing system. The oscillator is installed to a cast iron bench, and its displacement and acceleration are measured by an acceleration sensor and a displacement sensor, respectively. The LO is detailed in Fig. 9(a) and its stiffness can be regulated by modifying the number of springs in its two sides. A detailed view of the fixation of springs is displayed in Fig. 9(b), in which two springs are used. If only one spring is used, this spring will be attached to middle holes. During the experimental process, 4, 3 and 2 springs are consecutively applied. Their stiffness can be calculated according to the value of 4 springs as showed in Table 1. The device to regulate displacement is showed in Fig. 9(c) and the initial displacement of LO is set to around 20 mm for all tests. An enlarged view of VI NES is demonstrated in Fig. 9(d). Its initial location is random for all tests. The specific location is not so important, since only the stable process will be considered. In addition, the initial velocities of LO and VI NES are zero. A Duffing system is showed in Fig. 9(e). It should be pointed out that only the stiffness is different for the above four systems.

[Fig. 10 about here.]

Experimental results

The response for LO with 4 springs and b = 5 mm is demonstrated in Fig. 10 to illustrate the process to identify typical values related to two limit points p0 and p2. The first limit point is that between the regime with three impacts per cycle and that with two impacts per cycle, and the second is the one between the regime with two impacts per cycle and that with less than two impacts per cycle. The displacement is showed in Fig. 10(a), the first limit is between two points: the local maximum point A2 and minimum point A3. Similarly, the second limit point is between two other points: the local maximum point B2 and minimum point B3. The identification of these four points can be obtained from special points at the time history of acceleration in Fig. 10(b) and this identification is also used for the former periodic experimental results. Two periods are enlarged and displayed in Fig. 10(c) and (d). Point A1 is the last impact of periods with three impacts per cycle. Point B1 is the last impact of the regime with two impacts per cycle. According to values of A1 and B1, special points A2, A3, B2 and B3 can be identified, and then corresponding displacement amplitudes of this main oscillator can be obtained, as showed in Fig. 10(a).

[Fig. 11

about here.]

For different main systems with same displacement amplitude, its energy is bigger for a larger stiffness. For free vibration, there exist more oscillations during the decay process for the case with larger stiffness. Therefore, it will be clearer to see the transition process of different transient response regimes with a larger stiffness for the same VI NES, since the transition process of transient response is finer.

[Fig. 12 about here.] Then, amplitudes related to these two limit points p0 and p2 for four different main systems are compared and showed in Fig. 12. As showed in Fig. 12(a), the first three values in the horizontal axis represent LO with 4, 3 and 2 springs, respectively. The last value represents a Duffing system with 4 springs. The meaning of the horizontal axis is the same for other three subfigures. Two clearances b = 5 mm and 10 mm are chosen and results are showed in Fig. 12(a-b) and Fig. 12(c-d), respectively. Broken lines are curve fittings of experimental data to show its variation trend. In Fig. 12(a), the results are obtained from positive amplitude of displacement, A2 and A3 illustrate the first and second limit points, respectively. The values of x are almost the same compared to its absolute value, which provides a direct proof that the activation characteristic of VI NES is independent of frequency. A small increase of value with the decrease of the number of springs is reasonable because the representation of transient responses by measured data becomes coarse with the decrease of stiffness. As for the comparison between LO with 2 springs and Duffing system with 4 springs, the difference is not significant because that their stiffness are close and other unclear factors like friction may play a critical role in deciding this value.

The same conclusion can be obtained from the analysis of points A3 and B3 in the negative side of displacement as showed in Fig. 12(b). Moreover, results showed in Fig. 12(c-d) for b = 10 mm also support the above analysis.

In summary, activation characteristic of VI NES is examined from experimental viewpoints, and there are some credible results for its proportional activation. In addition, the dependence of its activation on frequency is explored by a transient experiment. Although there are some results credible to some extent, they are not so ideal and it results from the fact that the frequency difference between these four systems is not large enough, meanwhile limit points p0 and p2 can not be truly accurately obtained for these four systems with low frequency.

Design of VI NES for chatter control

In this section, a design procedure of VI NES based on its activation characteristic will be applied to control the chatter of a cutting tool during a turning process. A simplified model and corresponding experimental parameters in [START_REF] Gourc | Quenching chatter instability in turning process with a vibro-impact nonlinear energy sink[END_REF] will be used. The application of VI NES has been experimentally observed to be efficient in quenching an unstable cutting in this paper. It is demonstrated that the bifurcation diagram is complex [START_REF] Nayfeh | Analysis of the cutting tool on a lathe[END_REF] and corresponding response regimes are also complex. Two typical cases will be chosen to demonstrate this optimization procedure. One case corresponds to an unstable cutting possessing a steady state response as well as a stable cutting with zero amplitude. The other case corresponds to an unstable cutting characterized by beating response.

Model of a cutting tool coupled with VI NES

[Fig. 13

about here.]

A simplified model of a cutting tool coupled with a VI NES is represented in Fig. 13. The cutting tool is supposed to only vibrate in the feed direction and the workpiece is considered rigid. The corresponding equation of motion between impacts is written as follows:

m 1 ẍ + c 1 ẋ + k 1 x + k 2 x 3 = F x F x = p(ρ 1 ∆h 1 + ρ 2 ∆h 2 + ρ 3 ∆h 3 ) ε ÿ = 0 ∀ |x -y| < b (21)
[Table 2 about here.] Related parameters are expressed as follows:

ω 0 2 = k 1 m 1 , f 1 = ω 0 2π , µ 1 = c 1 2m 2 ω 0 ( 22
)
where x, m 1 , c 1 , k 1 and k 3 are the displacement, mass, damping, coefficient of linear stiffness and coefficient of cubic stiffness of the cutting tool, respectively. y and m 2 are displacement and mass of VI NES. Dots denote the differentiation with respect to time t. ∆h is decided by the current displacement and the displacement trajectory left by the last pass. b represents the clearance. When |x -y| = b, impacts occur. The relation between after and before impact is obtained under the hypothesis of simplified shock theory and the condition of momentum conservation:

x + = x -, y + = y - m 1 ẋ+ + m 2 ẏ+ = m 1 ẋ-+ m 2 ẏ-, ẋ+ -ẏ+ = -r ẋ--ẏ-, for |x -y| = b (23)
[Fig. 14 about here.]

Design of VI NES for different cases

Physical and cutting parameters are showed in Table 2. Only the cutting width h 0 and clearance b will be varied. At first, the system is not coupled with VI NES and the cutting width is varied to see the bifurcation process. When h 0 = 0.11 mm, a case with two fixed points is observed denoted as case 1 and another fixed point with beating response is located with h 0 = 0.16 mm and is denoted as case 2. The response regimes for these two cases without and with VI NES will be demonstrated.

Case 1

For h 0 = 0.11 mm, the following initial conditions are fixed and used: ẋ0 = 0, y 0 = b, ẏ0 = 0. Only initial displacement x 0 is changed. When x 0 is chosen to a large enough value such as 0.20 mm and 0.01 mm, x will be attracted to a same steady state, which corresponds to an unstable cutting as showed in Fig. 14(a) and (b), respectively. If the initial condition is small enough, the response will decay to zero as displayed in Fig. 14(c).

To design b for VI NES, the analytical results in Section 2 will be used. At first, a LO close to this cutting tool, i.e., with the same displacement amplitude, is created. In addition, the corresponding LO will possess other same characteristics as the cutting tool as many as possible. The frequency of outside excitation for this created linear oscillator is fixed to the experimentally obtained frequency f 0 and its amplitude will be chosen in order that the displacement amplitude of LO will be the same as that of the cutting tool. Then b is designed to make the target displacement amplitude of LO locate in p2 of the corresponding SIM under periodic excitation and in p1 for transient excitation. For the cutting tool, its final steady state, i.e., a steady non-zero or zero amplitude, is applied to decide which points will be chosen.

[Fig. 15

about here.]

For the first two initial conditions, response of the cutting tool is periodic and its amplitude is 5.41 * 10 -3 mm and b is chosen to p2. For the third initial condition, response is transient and its displacement amplitude at the starting point 2.199 * 10 -3 mm is chosen to make it locate at p1. The effect with VI NES is demonstrated in Fig. 15. For the first two initial conditions, the unstable state is improved to a stable state and its vibration is completely controlled as showed in Fig. 15(a-b). Meanwhile, its transient process of energy dissipation is accelerated. This is the same case for the third initial condition as showed in Fig. 15(c).

Case 2

[Fig. 16 about here.]

Compared to case 1, a beating response occurs for h 0 = 0.16 mm as represented by the blue curve in Fig. 16(a). Its amplitude of displacement is not steady. If the relative maximal value of point T1 is chosen as the target to create LO, the estimated amplitude will be high, and the response should be designed according to p1 with a relative high clearance for VI NES. In return, it will be a little small for the relative minimal value of point T2, the response should be designed according to p2.

When the relative minimal value Y = 3.208 * 10 -3 mm of T2 is chosen, the p1 is used as the targeted point. The result is showed in Fig. 16(a) and the displacement of the cutting tool in red curve is almost zero in this case. When the relative maximal value Y = 1.14 * 10 -2 mm of point T1 is chosen with p2 as the target. The result in red curve is showed in Fig. 16(c) and the displacement of cutting tool is greater with this b. The displacement of VI NES y for these two designed values are showed in Figs. 16(b) and (d), respectively. For the first value of b, VI NES is much more activated at the beginning. Once vibration is decreased to a little value, VI NES is no longer activated. For the second value in Fig. 16 (d), VI NES is also activated, but its vibration is not reduced as the first value and VI NES is just occasionally activated.

In summary, the design procedure proposed for VI NES is feasible and effective for unstable cuttings during turning process. It demonstrates the feasibility of relatively optimal design of VI NES for vibration control of nonlinear systems.

Conclusion

IIn this paper, SIM obtained from analytical development of LO coupled with VI NES is further analyzed. Activation characteristic of VI NES as a damper is analyzed from SIM and applied for design of VI NES. Then, activation characteristic of VI NES is numerically and experimentally validated. Finally, a proposed design procedure of VI NES is applied to control the chatter during a turning process.

Although SIM has been greatly analyzed during the analytical treatment of system coupled with VI NES, its relation to frequency is not clearly explained before. It is found that VI NES with a clearance will be activated in a fixed range of displacement amplitude of a main system. Moreover it will not be affected by the frequency of this main system. Two highlights are obtained. Firstly, the effectiveness of VI NES in a broad range of frequency of a main system can be viewed from another direct viewpoint, namely analytical viewpoint, and this idea is different from traditional perspectives. Secondly, each NES could possess its own characteristic and need special attention.

The design procedure of VI NES for the chatter control of a turning process is just applied to demonstrate the way to design VI NES for vibration control of nonlinear systems. Therefore, this section appears short and needs further study later.

In general, about the activation characteristic of VI NES obtained from analytical analysis, the experimental results are general credible and prove the analytical results. But they are not so ideal, for example, when Duffing system is created, the stiffness of cubic term is so week that a significant change of frequency to assure comparison cannot be obviously observed. Appendices -should appear before the list of references and labelled A, B, C, (please see GFA for further information regarding equations, figures and tables in the appendices.

  Experimental setup: (a) linear systems with 4, 3 or 2 springs ; (b) the fixation of springs; (c) the regulation device of initial displacement; (d) an enlarged view of VI NES; (e) the Duffing system with 4 springs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 10 LO with 4 springs and b = 5 mm: (a) the time history of displacement; (b) the time history of acceleration; (c) an enlarged view of the time history of acceleration around point A1; (d) an enlarged view of the time history of acceleration around point B1. . . . . . 34 11 Duffing system with 4 springs and b = 5 mm: (a) the time history of displacement; (b) the time history of acceleration. . . . . . . . 35 12 Amplitude of the main structure around points A2, A3, B2 and B3 for three linear systems with 4, 3 and 2 springs and a Duffing system with four springs and these four systems are presented by 1,2,3 and 4 in the horizontal axis, respectively: (a) A2 and B2 with b = 5 mm; (b) A3 and B3 with b = 5 mm; (c) A2 and B2 with b = 10 mm; (d) A3 and B3 with b = 10 mm. . . . . . . . . . 36 13 Schema of a cutting tool system coupled with a VI NES. . . . . . 37 14 Displacement of the cutting tool with different initial conditions for h 0 = 0.11 mm without VI NES: (a) x 0 = 0.20 mm; (b) x 0 = 0.01 mm; (c) x 0 = 0.006 mm. . . . . . . . . . . . . . . . . . . . 38 15 Displacement of the cutting tool coupled with an effective VI NES for different initial conditions at h 0 = 0.11 mm: (a) x 0 = 0.20 mm and b = 2.2880 * 10 -2 mm; (b) x 0 = 0.01 mm and and b = 2.2880 * 10 -2 mm; (c) x 0 = 0.006 mm and b = 2.4604 * 10 -3 mm. 39 16 Response comparison of the cutting tool without and with VI NES for x 0 = 0.08 mm and h 0 = 0.16 mm: (a) x and b = 1.3517 * 10 -2 mm; (b) y and b = 1.3517 * 10 -2 mm; (c) x and b = 1.2632 * 10 -2 mm; (d) y and b = 1.2632 * 10 -2 mm. . . . . . . . . . . . . . . . 40

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

 1234 Fig. 1. Schema of a LO coupled with a VI NES under periodic excitation.

Fig. 5 .

 5 Fig. 5. Application of an optimal b from one specific LO to two other linear systems with different frequencies: (a) the time history of displacement of LO with β = 1; (b) x related to p2 for different β .

Fig. 6 .

 6 Fig. 6. x related to p2 and obtained from the response of a Duffing system with different b for VI NES.

Fig. 7 .

 7 Fig. 7. Proportional activation characteristic of VI NES reflected by p2 of SIM: (a) b = 32.5 mm; (b) b = 35 mm; (c) b = 40 mm.

Fig. 8 .Fig. 9 .Fig. 10 .Fig. 11 .

 891011 Fig. 8. Proportional activation characteristic of VI NES reflected by p2 of SIM from sweep experiment: (a) b = 10 mm, 15 mm, 20 mm and 25 mm; (b) b = 27.5 mm, 30 mm, 35 mm and 40 mm; (c) linear relation between b and x.

Fig. 12 .Fig. 13 .Fig. 14 .Fig. 15 .Fig. 16 .

 1213141516 Fig. 12. Amplitude of the main structure around points A2, A3, B2 and B3 for three linear systems with 4, 3 and 2 springs and a Duffing system with four springs and these four systems are presented by 1,2,3 and 4 in the horizontal axis, respectively: (a) A2 and B2 with b = 5 mm; (b) A3 and B3 with b = 5 mm; (c) A2 and B2 with b = 10 mm; (d) A3 and B3 with b = 10 mm.

Figures -if there

  Figures -if thereare multi-parts to a figures each part is labelled (a) (b) (c) etc. and the labels defined in the figure caption. Figures -Colour can be used for the on-line version. Figures are reproduced in black and white in the printed journal and must therefore be readable in both colour and black & white. (N.B. charges apply for production of colour figures in the printed journal)

Table 1 .

 1 Experimental parameters[START_REF] Li | Optimization mechanism of targeted energy transfer with vibro-impact energy sink under periodic and transient excitation[END_REF]

		Physical Parameters
	m 1	4.7 kg	c 1 3.02 Ns/m
	k 1 11.47 * 10 3 N/m m 2	32 g
	b	0 -50 mm
		Reduced Parameters
	ε	0.76 %	λ 1	1.91
	f 0	7.86 Hz

Table 2 .

 2 Simulation parameters for a cutting tool coupled with VI NES

		Physical Parameters
	m 1	3.1 kg	µ 1	3 %
	f 1	99.4 Hz	m 2	32 g
	r	0.6
		Cutting Parameters
	p	0.1 mm	ρ 1	6109.6 * 10 6 Nm -2
	ρ 2 -54141.6 * 10 9 Nm -2 ρ 3 203.769 * 10 12 Nm -2
	s	1800 rpm

Acknowledgments

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

Acknowledgements appear in a separate section after the conclusion section but before references.

All abbreviations, in both the abstract and main body of document, are defined once only, the first time they appear in the text. (N.B. The Abstract is treated as an independent text, where references are given in full and abbreviations and symbols, if used, are properly defined.)

List of Tables

Experimental parameters [START_REF] Li | Optimization mechanism of targeted energy transfer with vibro-impact energy sink under periodic and transient excitation[END_REF] . . . . . . . . . . . . . . . . . . . . 23 2 Simulation parameters for a cutting tool coupled with VI NES . . 24

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Donghai Qiu
email: qiu@insa-toulouse.fr

S Ébastien Seguy
email: sebastien.seguy@insa-toulouse.fr

Manuel Paredes
email: manuel.paredes@insa-toulouse.fr

Tuned nonlinear energy sink with conical spring: design theory and sensitivity analysis

This paper is devoted to the study of a Nonlinear Energy Sink (NES) intended to attenuate vibration induced in a harmonically forced linear oscillator (LO) and working under the principle of Targeted Energy Transfer (TET). The purpose motivated by practical considerations is to establish a design criterion that first ensures that the NES absorber is activated and second provides the optimally tuned nonlinear stiffness for efficient TET under a given primary system specification. Then a novel NES design yielding cubic stiffness without a linear part is exploited. To this end, two conical springs are specially sized to provide the nonlinearity. To eliminate the linear stiffness, the concept of a negative stiffness mechanism is implemented by two cylindrical compression springs. A small-sized NES system is then developed. To validate the concept, a sensitivity analysis is performed with respect to the adjustment differences of the springs and an experiment on the whole system embedded on an electrodynamic shaker is studied. The results show that this type of NES can not only output the expected nonlinear characteristics, but can also be tuned to work robustly over a range of excitation, thus making it practical for the application of passive vibration control.

Introduction

With the faster, lighter and more sophisticated mechanical products designed nowadays, vibration mitigation devices are required to be more rigorous than ever before [START_REF] Jiang | Improving the bspline method of dynamically-compensated cam design by minimizing or restricting vibrations in high-speed cam-follower systems[END_REF]. To ensure the performance of the system designed, different types of vibration control methods (e.g. smart structure, passive and active vibration absorber) have been exploited in the recent decades [START_REF] Kim | Active vibration control and isolation for micromachined devices[END_REF][START_REF] Trimble | A device for harvesting energy from rotational vibrations[END_REF][START_REF] Okwudire | Reduction of torque-induced bending vibrations in ball screw-driven machines via optimal design of the nut[END_REF][START_REF] Moeenfard | Modeling geometric nonlinearities in the free vibration of a planar beam flexure with a tip mass[END_REF]. Among them, the Tuned Mass Damper (TMD), a linear absorber with the passive control method, has been widely applied. When attached to a primary system, this absorber can be tuned to suppress the dynamic vibration of the structure and its amplitude at resonance condition. However, the combined system has two degrees of freedom, which means that there are two natural frequencies corresponding to two resonance peaks, making this absorber effective over only a narrow band of excitation frequencies [START_REF] Vakakis | Targeted Energy Transfer in Mechanical and Structural Systems[END_REF].

To overcome this limitation, an innovative nonlinear absorber named Nonlinear Energy Sink (NES) has been proposed recently [START_REF] Vakakis | Energy pumping in nonlinear mechanical oscillators: part ii: resonance capture[END_REF]. This type of absorber is characterized by a secondary mass strongly coupled via a nonlinear stiffness to the primary system that needs to be protected (see Fig. 1). Because of the strong (non-linearizable) nonlinearity, irreversible Targeted Energy Transfer (TET) from the main structure to the secondary mass is achieved, enabling the NES to be effective in a broad band of frequencies [START_REF] Lee | Passive non-linear targeted energy transfer and its applications to vibration absorption: a review[END_REF]. Mastering the nonlinearity is a key element for obtaining the optimum performance of a NES. Depending on the type of nonlinearity, a NES can be classified as a cubic NES [START_REF] Gourdon | Nonlinear energy pumping under transient forcing with strongly nonlinear coupling: Theoretical and experimental results[END_REF][START_REF] Gourc | Experimental investigation and design optimization of targeted energy transfer under periodic forcing[END_REF], a vibro-impact NES [START_REF] Gourc | Targeted energy transfer under harmonic forcing with a vibro-impact nonlinear energy sink: analytical and experimental developments[END_REF], a piece-wise NES [START_REF] Lamarque | Targeted energy transfer in mechanical systems by means of nonsmooth nonlinear energy sink[END_REF] or a rotational NES [START_REF] Sigalov | Resonance captures and targeted energy transfers in an inertially-coupled rotational nonlinear energy sink[END_REF]. As far as the cubic NES is concerned, it has been shown that this configuration is most effective at moderate-energy regimes [START_REF] Vakakis | Targeted Energy Transfer in Mechanical and Structural Systems[END_REF]. However, in practice, it is difficult to obtain cubic stiffness without a linear part. In our recent approaches, the essential cubic stiffness was mostly achieved by adopting the construction of two linear springs with no pretension. Due to self-geometric nonlinearity, the springs stretch in tension, thus creating the cubic force [START_REF] Gourdon | Nonlinear energy pumping under transient forcing with strongly nonlinear coupling: Theoretical and experimental results[END_REF][START_REF] Gourc | Experimental investigation and design optimization of targeted energy transfer under periodic forcing[END_REF]. However, this classical type of device cannot effectively profit from spring compression and extension, and the result is a large vertical structure attached to the main system. Addition of a relatively weak nonlinear stiffness existing at the beginning extension, leads to the whole cubic term being approximated to a linear term. Therefore, implementing a cubic stiffness element in practice is still an important issue if the applications of NES are to be broadened.

Recently, physical determination of the nonlinear forcedisplacement characteristics have been mainly based on nonlinear springs or variable stiffness mechanisms. Jutte and Kota [START_REF] Jutte | Design of nonlinear springs for prescribed load-displacement functions[END_REF][START_REF] Jutte | Design of single, multiple, and scaled nonlinear springs for prescribed nonlinear responses[END_REF] developed a nonlinear spring that enabled individual beam segments to vary their effective stiffness as the spring deformed. Wu and Lan [START_REF] Wu | Linear variablestiffness mechanisms based on preloaded curved beams[END_REF] proposed a linear variable stiffness mechanism with preloaded curved beams. Sönmez and Tutum [START_REF] Sönmez | Introduction to compliant long dwell mechanism designs using buckling beams and arcs[END_REF][START_REF] Sönmez | A compliant bistable mechanism design incorporating elastica buckling beam theory and pseudo-rigid-body model[END_REF] introduced a compliant bistable mechanism design by changing the relative stiffness of flexible beams. Chen and Lan [START_REF] Chen | An adjustable constant-force mechanism for adaptive end-effector operations[END_REF] designed an adjustable constant force mechanism for adaptive end effector operations. For these mechanisms, a key issue to be satisfied for a NES is that the linear part of the force-displacement characteristics should be very small so as to ensure that the NES can adapt itself to the frequency of any primary system [START_REF] Al-Shudeifat | Nonlinear energy sinks with nontraditional kinds of nonlinear restoring forces[END_REF].

With this in mind, the main objective of the work presented here is to provide a generalized methodology to design a novel NES with nonlinear spring and variable stiffness mechanism, so that it can be tuned over a range of excitation. The structure of the article is as follows: the first section is devoted to the design criterion for a cubic NES under a given primary system specification. The next section is dedicated to the theoretical design, including the design of conical spring and negative stiffness mechanism, and identification of the NES system. In the third section, a sensitivity analysis is carried out with respect to the adjustment differences of the springs and experimental verifications are studied. Finally, some concluding remarks are proposed.

Design criterion for cubic NES 2.1 Dynamic modeling

The dynamic modeling presented here is based on References [START_REF] Gourc | Experimental investigation and design optimization of targeted energy transfer under periodic forcing[END_REF][START_REF] Gendelman | Attractors of harmonically forced linear oscillator with attached nonlinear energy sink i: Description of response regimes[END_REF][START_REF] Starosvetsky | Attractors of harmonically forced linear oscillator with attached nonlinear energy sink ii: Optimization of a nonlinear vibration absorber[END_REF]. The system of a harmonically excited linear oscillator (LO) strongly coupled with a cubic NES is illustrated in Fig. 1. The objective here is to apply the asymptotic method used in the above papers to obtain a possible design optimization criterion, and thus find the best tuned parameters of the NES. The equations of motion are as follows:

m 1 ẍ + k 1 x 1 + c 1 ẋ + c 2 ( ẋ -ẏ) + k 2 (x -y) 3 = k 1 x e + c 1 ẋe m 2 ÿ + c 2 ( ẏ -ẋ) + k 2 (y -x) 3 = 0
(1) where x, m 1 , c 1 , k 1 and y, m 2 , c 2 , k 2 are the displacement, mass, damping and stiffness of the LO and the cubic NES respectively and the dots denote differentiation with respect to time. The imposed harmonic displacement x e is expressed as: x e = G cos(ωt).

After rescaling, the system of equations ( 1) can be reduced to the dimensionless form:

ẍ + x + ελ 1 ẋ + ελ 2 ( ẋ -ẏ) + εK (x -y) 3 = εF cos Ωτ ε ÿ + ελ 2 ( ẏ -ẋ) + εK (y -x) 3 = 0 (2)
where the term containing ε 2 is so small that can be eliminated. The corresponding physical parameters are expressed as follows:

ε = m 2 m 1 , ω 0 2 = k 1 m 1 , K = k 2 m 2 ω 0 2 , λ 1 = c 1 m 2 ω 0 , λ 2 = c 2 m 2 ω 0 , F = G ε , Ω = ω ω 0 , τ = ω 0 t (3)
New variables representing the displacement of the center of mass and the internal displacement of the cubic NES are introduced as follows:

v = x + εy, w = x -y (4)
Substituting Eqs. (4) into Eqs. (2):

         v + ελ 1 v + ε ẇ 1 + ε + v + εw 1 + ε = εF cos Ωτ ẅ + ελ 1 v + ε ẇ 1 + ε + v + εw 1 + ε + λ 2 (1 + ε) ẇ +K (1 + ε) w 3 = εF cos Ωt ( 5
)
The system is studied in the vicinity of the 1:1 resonance, where both the LO and the NES execute the timeperiodic oscillations with identical frequency Ω. To obtain the analytical periodic solution, two new complex variables are introduced:

φ 1 e iΩτ = v + iΩv, φ 2 e iΩτ = ẇ + iΩw (6)
Substituting Eqs. (6) into Eqs. [START_REF] Moeenfard | Modeling geometric nonlinearities in the free vibration of a planar beam flexure with a tip mass[END_REF] and keeping only the secular term containing e iΩτ yields the following slowly modulated system:

       φ1 + iΩ 2 φ 1 + ελ 1 (φ 1 +εφ 2 ) 2(1+ε) -i(φ 1 +εφ 2 ) 2Ω(1+ε) -εF 2 = 0 φ2 + iΩ 2 φ 2 + ελ 1 (φ 1 +εφ 2 ) 2(1+ε) -i(φ 1 +εφ 2 ) 2Ω(1+ε) + λ 2 (1+ε)φ 2 2 - 3iK(1+ε)φ 2 2 φ2 8Ω 3 -εF 2 = 0 (7)

Analytical treatment

The activation of energy pumping is efficient for the targeted energy transfer of NES. To predict the mechanism, a topological structure of a slow invariant manifold (SIM) is used in the following form:

N 2 10 = (1 + λ 2 2 )N 2 20 - 3K 2 N 4 20 + 9K 2 16 N 6 20 ( 8
)
where N 10 , N 20 represent the amplitude of φ 1 and φ 2 on a slow time scale. The two extrema of SIM are described as:

N 2,i = 2 3 (2 ± 1 -3λ 2 2 )/K i = 1, 2 (9)
A strongly modulated response (SMR) and the corresponding projection in the SIM are presented in Fig. 2. The quasi-periodic response with slow variation of the amplitudes of both oscillators is observed. In this regime, the procedure of TET can be classified as follows: (1) nonlinear beating, where a small amplitude of NES corresponds to the growth of LO amplitude; (2) transient resonance capture, where the NES irreversibly extracts the energy from the LO;

(3) escape from resonance capture, the NES crosses the bifurcation and is quickly attracted to the low branch of SIM, which leads to a jump down for the energy of the NES. This SMR regime demonstrates the irreversible targeted energy transfer from LO to NES [START_REF] Starosvetsky | Strongly modulated response in forced 2dof oscillatory system with essential mass and potential asymmetry[END_REF], which suppresses energy more efficiently than a steady state response.

With these two extrema of SIM (N 21 and N 22 ) located at the boundary of SMR and stable periodic response, the excitation threshold of the SMR can be obtained and written as:

G i c = εN 2,i (9λ 1 K 2 N 4 2,i -24λ 1 KN 2 2,i + 16(λ 1 + λ 2 + λ 1 λ 2 2 )) 4 9K 2 N 4 2,i -24KN 2 2,i + 16 + 16λ 2 2 ( 10
)
The detailed description of Eq. ( 8) and Eq. ( 10) are given in [START_REF] Gourc | Experimental investigation and design optimization of targeted energy transfer under periodic forcing[END_REF][START_REF] Gendelman | Attractors of harmonically forced linear oscillator with attached nonlinear energy sink i: Description of response regimes[END_REF]. When the excitation amplitude is with the interval [G 1c , G 2c ], SMR can be produced. Yet this is not sufficient to ensure the activation of SMR regimes; another interacting factor is the excitation frequency. To explain its influence, the stability of the frequency response function (FRF) is analyzed. Here, a detuning parameter σ representing the nearness of the excitation frequency ω to the reduced natural frequency of the LO is introduced:

Ω = 1 + εσ (11)
The fixed points of the FRF correspond to the periodic solutions of the system. Under the 1:1 resonance hypothesis, the solutions of fixed points can be obtained by equating the derivatives of Eq. ( 7) in the following form: By introducing Eq. ( 12) into Eq. ( 7), a system of complex algebraic equations is obtained. After several algebraic operations, the system is deduced as:

φ1 = φ2 = 0 ⇒ φ 1 (τ) = φ 10 , φ 2 (τ) = φ 20 (12)
             α 3 Z 3 20 + α 2 Z 2 20 + α 1 Z 20 + α 0 = 0, Z 20 = |φ 20 | 2 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) (13
) where coefficients α i depend on the system parameters and excitation parameters. The first equation of Eq. ( 13) is a cubic polynomial that can be resolved analytically, so as to obtain the fixed points. To determine the stability, small perturbations are introduced as follows:

φ 1 = φ 10 + ρ 1 , φ 2 = φ 20 + ρ 2 (14)
By substituting Eq. ( 14) into Eq. ( 7), the linearization model with the Jacobian matrix is obtained:

    ρ1 ρ2 ρ1 ρ2     =     M 11 εM 21 0 0 M 22 M 22 0 M 24 0 0 M11 ε M21 0 M24 M21 M22         ρ 1 ρ 2 ρ1 ρ2     (15)
where

                   M 11 = -i(1+ε) 2 -ελ 1 2(1+ε) + i 2(1+ε)(1+εσ) M 21 = -ελ 1 2(1+ε) + i 2(1+ε)(1+εσ) M 22 = 3i(1+ε)Kφ 20 φ20 4(1+εσ) 3 -λ 2 (1+ε) 2 + iε 2(1+ε)(1+εσ) -i(1+εσ) 2 -ε 2 λ 1 2(1+ε) M 24 = 3i(1+ε)Kφ 2 20 8(1+εσ) 3 (16)
By computing the root of the polynomial characteristic equation, the stability of the fixed points is deduced. If a real root crosses the left-half complex plane, the fixed point is unstable. Fig. 3 compares the FRF between the system with a cubic NES and without a NES. The maximum amplitude of the axial displacement of the primary system is plotted on the vertical axis as a function of the amplitude G and frequency σ of the excitation. With the addition of a NES, the normal model of the LO becomes nonlinear and varies for different types of energy input. When the energy is low and not sufficient to activate the NES, the resonance peak of the LO does not vanish completely (see curve a). For a relatively higher excitation, as shown in curves b and c, the energy pumping of the NES is activated in the unstable area. SMR regimes are possibly produced. As the excitation amplitude is increased still further to G = 0.23 mm (see curve d), the band of frequency for SMR becomes larger. However a high amplitude detached resonance tongue appears on the left of the main backbone branch, which reduces the efficiency of the control and can be dangerous for the system.

Criterion for efficient energy pumping

From the FRF analysis, the aim of a design criterion should be first to ensure that energy pumping of the NES absorber is activated and second to avoid the resonance tongue under a given primary system. For this, the effect of nonlinear stiffness K and excitation G is analyzed. The maximum amplitude of the FRF is calculated and presented in Fig. 4(a). As the nonlinear stiffness is larger, the resonance tongue occurs more easily at low excitation. In Fig. 4(b), as the excitation is fixed, an optimal nonlinear stiffness located at the critical resonance tongue position can be found, where the amplitude of the LO is minimum. In Fig. 4(c), as the nonlinear stiffness is fixed, the maximum amplitude of the LO will not be larger than the value at this critical point. Thus, the critical position for the appearance of the resonance tongue is studied. Fig. 5 shows the multiplicity of the periodic solution of the system in Eq. [START_REF] Sigalov | Resonance captures and targeted energy transfers in an inertially-coupled rotational nonlinear energy sink[END_REF]. Where the boundary separating single and triple solutions corresponds to the saddle-node bifurcation, G 1c and G 2c represent the threshold of the SMR, G sn represents the boundary where three periodic solutions occur in the left of the main resonance frequency. When the excitation is inside the zone [G 1c , G sn ], no resonance tongue occurs and the SMR is probably produced. Therefore, once the parameters of the NES are fixed, the value of G sn can be determined as the maximum amplitude of excitation, so as to avoid the occurrence of a resonance tongue.

In Fig. 6, the critical excitation amplitudes are presented as a function of the nonlinear stiffness. As can be seen, the width of the SMR zone decreases as nonlinear stiffness in- creases. The characteristic points a, b, c and d at K = 3000 show good agreement with the behavior of the system in Fig. 3, which demonstrates that the critical amplitudes of excitation can predict the response regimes well at different excitations.

Based on the above discussion, for a given primary system, the optimal design for a NES system is dependent on the maximum amplitude of excitation. The design parameter should be chosen to avoid a detached resonance tongue in the vicinity of the natural frequency and allow strongly modulated response at the same time. An illustration for choosing the tuned parameter of a cubic NES is given in Table 1, here the mass ratio, stiffness and damping of the LO, and the damping of the NES are fixed. The corresponding tuned parameters of nonlinear stiffness K are listed for different maximum amplitudes of excitation (using the G sn curve of Fig. 6). With this value, the amplitude of the LO is minimum and no resonance tongue exists in a range of frequencies. The next section presents the design of a tuned NES with the proposed design criterion. a constant pitch and a constant coil diameter were adopted. The detailed design of the conical springs has been achieved in [START_REF] Rodriguez | Analytical behavior law for a constant pitch conical compression spring[END_REF][START_REF] Paredes | Analytical and experimental study of conical telescoping springs with nonconstant pitch[END_REF] so that the springs do not buckle at large deflections [START_REF] Patil | Comparison of cylindrical and conical helical springs for their buckling load and deflection[END_REF]. Considering the strong nonlinearity and lower installation height, the shape of a telescoping spring was used [START_REF] Qiu | A novel design of cubic stiffness for a nonlinear energy sink (nes) based on conical spring[END_REF].

The force characteristic of a conical spring with a constant pitch can be separated into a linear and a nonlinear part. To distinguish the two phases, three particular points are introduced, as shown in Fig. 7: point O corresponds to the spring free state, point T is the transition point where the nonlinear behavior starts, and point C represents the state of maximum compression.

In the linear phase (from point O to point T), the largest coil is free to deflect like the other coils, so the forcedisplacement relation is linear and the stiffness is expressed as:

R = Gd 4 2n a (D 2 1 +D 2 2 )(D 1 +D 2 ) (17)
In the nonlinear regime (from point T to point C), the first elementary part of the largest coil has reached its maximum physical deflection. It starts to be a non-active element of the spring. During the second compression regime, the number of active coils decreases continuously, leading to a gradual increase of the spring stiffness. The forcedisplacement (F-u) relation can be described by:

u(F) = 2FD 4 1 n a Gd 4 (D 2 -D 1 ) [(1 + ( D 2 D 1 -1) • n f n a ) 4 -1] +(L a -L s )(1 - n f n a ) (18)
where d, n a , n f , L a , L s , G represent the wire diameter, number of active coils, number of free coils, initial active length, solid length of active coils and shear modulus of elasticity, respectively, and D 1 and D 2 represent the mean diameter of the smallest and largest coils. A detailed description of Eq. ( 17) and Eq. ( 18) can be found in [START_REF] Rodriguez | Analytical behavior law for a constant pitch conical compression spring[END_REF].

To benefit from the nonlinear performance of the conical spring, a symmetrical connecting type of spring is proposed, as shown in Fig. 8(a). However, this configuration has the piecewise stiffness of a linear and a nonlinear part. To skip the linear phase, a method of pre-compressing the spring at the transition point is proposed and is presented in Fig. 8(b). By changing the initial origin point, the behaviors of two conical springs can belong one to the linear and one to the nonlinear regime simultaneously. By combining the two spring curves, the composed stiffness curve is obtained (see Fig. 8(d)). It can be clearly observed that the new curve is smooth and no longer piecewise.

To analyze the internal polynomial components, the method of polynomial fitting was used, and the new forcedisplacement relation was written as:

F = a 1 u + a 2 u 2 + a 3 u 3 + O(u 4 ) (19)
Because of the superposition of linear and nonlinear parts, the linear term of Eq. ( 19) was hard to eliminate, yet it was possible to make the value of the square term a 2 u 2 small.

After optimizing the parameters of the conical spring (the mean diameters D 1 and D 2 ), the polynomial components were obtained and are presented in Fig. 9. It can be observed that the curve of the cubic and linear terms is close to the original curve F(u), which means that the contribution of the square term was small enough to be almost neglected.

Negative stiffness mechanism

To eliminate the proposed linear term, adding a new term having the negative stiffness in the translational direction seemed to be a way forward. For this, a negative stiffness mechanism was implemented from two cylindrical compression springs [START_REF] Harne | Concise and high-fidelity predictive criteria for maximizing performance and robustness of bistable energy harvesters[END_REF]. To obtain the required negative stiffness, the ends of the springs are able to rotate and the initial axes of the springs are perpendicular to the main motion axis, as shown in Fig. 10.

Based on the Taylor expansion, the force-displacement relation of pre-compression at the length l p can be expressed as:

f = 2k l p l • u -k (l+l p ) l 3 • u 3 (20)
By adding the force of two conical springs to the force of negative stiffness mechanism, the composed force of the NES system is obtained:

F = (a 1 -2k l p l ) • u + (a 3 + k (l+l p ) l 3 ) • u 3 (21)
As can be seen from Eq. ( 21), if a 1 = 2kl p /l, the linear component can be counterbalanced by the negative stiffness mechanism. In this case, only the pure cubic term of the equation will be left, and its coefficient will be larger with the addition of two linear springs. 4 Identification of NES system Based on the proposed methods, a small-sized NES system providing strongly nonlinear stiffness was designed and the assembly drawing is presented in Fig. 11. The component parts are spherical plain bearings, a linear guide, two conical springs, two linear springs and a NES mass. It is important to highlight that the distance between each spring and the NES mass is adjustable so that a suitable force shape can be reached.

To obtain the performance of the conical spring, an identification study was performed. The manufactured conical spring and the measuring equipment are presented in Fig. 12. The force gauge has a 500 N capacity with 0.1% accuracy and 0.04 N resolution, the handle allows a stroke of 2 mm per revolution, the displacement transducer with digital display has a resolution of 0.01 mm. The results of tests on five conical springs are presented in Fig. 13. As expected, correspondence is good between the experimental results and the theoretical curve.

The NES system can be divided in several parts: a part attached to the primary system, a NES part and the springs. The NES part includes the mass of the sliding part, the spherical plain bearing, and the support base of the linear and conical springs. For dynamic calculations, as the NES mass (m a ) is very small, the inertia of the springs is not negligible and has to be considered. To a rough approximation, considering the spring as a beam and neglecting axial inertia, the kinetic energy of the NES mass and the linear spring is written as follows:

T NES = l 0 0 ρ s ( x l 0 ẏ) 2 dx + 1 2 m 2 ẏ2 (22)
where ρ s = m s /l 0 is the mass density of the spring. Thus the effective mass of a linear spring is calculated:

ms = m s /3 (23)
For the conical spring, the effective mass of mc can be expressed as [START_REF] Yamamoto | Spring's effective mass in spring mass system free vibration[END_REF]:

mc = 2m c 1 10 (1-β 10 )-1 3 β 4 (1-β 6 )+ 1 2 β 8 (1-β 2 ) (1-β 4 ) 2 (1-β 2 ) , β = D 2 D 1 (24)
Thus, the total effective mass of the NES for dynamic equations is obtained: Fig. 14 illustrates an installed NES system with linear springs of large mean diameter. Here, it is important to note that a linear spring with small mean diameter has a tendency to buckle when its deflection is large. Therefore, linear springs with large mean diameter were chosen. The final spring test for the NES system is shown in Fig. 15. The NES is held by a ring so that it can be connected to the internal load cell of the force gauge. With this experimental setup, the force can be measured in both compression and extension. Fig. 16 shows the force-displacement relation of the NES that was designed. As can be seen, the experimental curve and the theoretical cubic curve corresponded well. Thus it can be concluded that combining conical springs and a negative stiffness mechanism is a feasible way to produce pure cubic stiffness. Enough pre-compressed length should be reserved in both the conical and the linear springs to enable the anticipated nonlinearity to be obtained. Moreover, the effective mass of the springs and buckling of the negative stiffness mechanism should not be neglected during the design procedure.

m 2 = m a + 2 ms + 2 mc (25)

Sensitivity analysis

A sensitivity analysis of NES system is addressed in this section, considering the precision of the installation and quantifying the uncertainty [START_REF] Opgenoord | Variance-based sensitivity analysis to support simulation-based design under uncertainty[END_REF]. The performance of the NES relies on the given primary system specification, which is presented in Table 2. Here, the maximum excitation amplitude is fixed as G = 0.205 mm, and the system parameters are related to the reduced parameters of Table 1. Thus the tuned parameter of nonlinear stiffness can de determined as K = 3000, with the physical value of k 2 = 3 × 10 6 N/m 3 . To obtain this value, both the conical springs and the cylindrical compression springs need to be pre-compressed to the anticipated length, with St = 24.8 mm and lp = 11.9 mm. However, it is hard to control the coefficients a 0 , a 1 , a 2 and a 3 at the anticipated value since differences exist in practice. To analyze their influence, differences of the adjustment length for conical and linear springs were selected as the variables, defined as γ i and η i , respectively (see Fig. 17). Then the real adjustment lengths are:

S t,i = St,i + γ i , l p,i = lp,i + η i , i = 1, 2 (26)
Thus, the system of equations ( 2) can be written in the following form: where the corresponding parameters are expressed as:

ẍ + x + ελ 1 ẋ + ελ 2 ( ẋ -ẏ) + εζ • f (x -y) = εF cos Ωτ ε ÿ + ελ 2 ( ẏ -ẋ) -εζ • f (x -y) = 0 (27)
f (x -y) = a 0 + a 1 (x -y) + a 2 (x -y) 2 + a 3 (x -y) 3 ζ = 1/(m 2 ω 2 0 ) (28)
Based on Eq. ( 27), a sensitivity analysis for the stiffness of the NES is summarized in Table 3, where the variables γ i and η i are selected in the interval [-1, 1] mm, the objective x m is extracted from the maximum amplitude of the LO in the numerical FRF. As can be seen, in the vicinity of the frequency of the resonance, no resonance tongue occurs, and all the maximum amplitudes of the LO are smaller than 12 mm. In comparison with the maximum amplitude of the LO without NES (36.2 mm), the amplitude of the LO is decreased by more than 67% with the help of NES. Another factor that needs to be considered is the damping of the NES. With the variation of the adjustment lengths, this value could also be changed. A sensitivity analysis for the NES damping is presented in Table 4, where the stiffness of the NES is fixed and the damping c 2 is selected in the variation interval [-15, 15]%. The maximum amplitudes of LO are close to each other, which means that the NES is not sensitive to differences of damping.

Three particular examples of Table 3 were extracted and the corresponding nonlinear force versus the response of LO and NES were calculated. The results are presented in Fig. 18, where positive stiffness, negative stiffness and un-Table 3. Sensitivity analysis for the stiffness of NES Adjustment parameters (mm) x m (mm) 9.5 9.6 9.7 9.7 9.8 9.8 9.9 symmetrical stiffness are observed, according to the example. In Fig. 18 (a) and (d), as the pre-compressed lengths η i are smaller than the anticipated value, the stiffness curve is no longer pure cubic and a positive linear stiffness is produced. It is interesting to observe that the amplitude of the LO is lower here than in the other cases, which indicates that adding a small positive stiffness may help with vibration mitigation under a certain excitation. In Fig. 18 (b) and (e), the force curve has two stable equilibria and one unstable equilibrium. Between the two stable equilibria, the stiffness is negative and a bistable NES is obtained. In this case, the amplitude of the LO is increased, and the range of excitation amplitude for efficient TET is broader [START_REF] Romeo | Dynamics of a linear oscillator coupled to a bistable light attachment: Numerical study[END_REF]. This suggests that, as the springs of this prototype are fixed for engineering applications, adjusting the pre-compressed length would provide an alternative way to increase the band in which excitation amplitude is robust. In Fig. 18 (c) and (f), the equilibrium of the force curve is no longer located at the center but shifts to the left. Although the stiffness curve is unsymmetrical, the energy pumping between NES and LO is still activated, leading to a significant decrease of the amplitude of the LO. Therefore, this type of NES can work robustly with differences in the adjustment lengths.

γ 1 γ 2 η 1 η 2 x m γ 1 γ 2 η 1 η 2 x m 0

Experimental verifications

The aim of the experimental tests was to obtain the nonlinear frequency response function of the system around the 1:1 resonance, so as to verify the efficiency of the designed NES. The experimental setup is presented in Fig. 19 and consists of an LO with an embedded NES. For the LO, a weakly nonlinear behavior exists but does not affect the purpose of the tests. For the setup, the periodic excitation is provided by a 10 kN electrodynamic shaker. The raw signals are recorded using a digital oscilloscope and a bandpass filter is applied to correct biases and suppress high frequency noise. The displacement of the LO and NES, and the acceleration of the shaker are measured by two contactless laser displacement sensors and an accelerometer, respectively.

The frequency response function of LO obtained with the sweeping frequency test is presented in Fig. 20 (a), where the thick blue and the thin red lines represent the response of the LO with and without NES, respectively. It can be seen that the original peak of the FRF has vanished and the detached resonance tongue no longer exists. The RMS value of LO amplitude is obviously decreased with the vibration mitigation of the NES. In the response of the LO, the dynamical flow for different values of the frequency might be attracted either to a smooth zone, or alternatively, to a number of discontinuities (range of non-monotonicity). The for- Where the displacements of the LO increase and decrease alternately with the cyclical activation and deactivation of the NES. In this process, targeted energy transfer (TET) occurs and the energy is irreversibly dissipated from the LO to the NES. Thus it can be concluded that this NES can produce energy pumping and is efficient to protect the primary system in a large band of resonance frequencies.

Conclusion

In this paper, the design theory and sensitivity analysis of a tuned NES attached to a harmonically forced linear oscillator is investigated. Firstly, a design criterion intended to find the tuned parameter of nonlinear stiffness for a given primary system specification is proposed. The aim of this criterion is to avoid the detached resonance tongue in the vicinity of the resonance frequency and allow SMR at the same time. To this end, a combined method with slow invariant manifold (SIM), threshold of SMR and stability of the fixed points is studied theoretically. Frequency response functions (FRF) are calculated when the different parameters (i.e. amplitude and frequency of excitation, nonlinear stiffness) are varied and explain the behavior of the NES well. As a result, an optimal nonlinear stiffness is obtained, so as to tune the NES be activated and produce the efficient TET. Secondly, a novel NES design leading to a strongly cubic stiffness is presented. Two conical springs are specifically sized to provide the polynomial components with only linear and cubic terms. To eliminate the linear term, a concept of negative stiffness mechanism is implemented by two cylindrical compression springs. A small-sized NES system providing strongly nonlinear stiffness is developed. To validate the concept, identification of the force-displacement relation is performed. A good correlation between the theoretical and the experimental result is observed. Finally, a sensitivity analysis is carried out with respect to adjustment differences of the springs and an experiment involving the whole system embedded on an electrodynamic shaker is performed. The results show that this type of NES can avoid the resonance tongue and still provide energy pumping with the differences of stiffness and damping. Furthermore, by adjusting the pre-compressed length of springs, this NES can be tuned to work robustly over a range of excitation, which makes it preferable for the application of passive vibration control.

Fig. 1 .

 1 Fig. 1. Schematic of a harmonically excited LO coupled with a NES

Fig. 2 .

 2 Fig. 2. Cubic NES under periodic forcing with parameters K = 3000, λ 1 = 0.6, λ 2 = 0.3, ε = 0.01, G = 0.2 mm and initial conditions x 0 = 0, ẋ0 = 0, y 0 = 0 and ẏ0 = 0. (a) time-displacement of LO and NES. (b) SIM structure and trace between LO and cubic NES: red curve represents the transient projected motion.

Fig. 3 .

 3 Fig. 3. Frequency response function (FRF) of LO with cubic NES (points) and without NES (thin line) in different types of excitation: (a) G = 0.08 mm; (b) G = 0.13 mm; (c) G = 0.18 mm; (d) G = 0.23 mm. The blue points and the red crosses represent stable and unstable fixed points respectively.

Fig. 4 .

 4 Fig. 4. Maximum amplitude of LO with the variation of (a) nonlinear stiffness and excitation amplitude; (b) nonlinear stiffness; (c) excitation amplitude. Each point is extracted from the maximum amplitude of FRF.

Fig. 5 .

 5 Fig. 5. Evolution of the multiplicity of periodic solutions for the system with parameters K = 3000, λ 1 = 0.6, λ 2 = 0.3, ε = 0.01.

Fig. 6 .

 6 Fig. 6. Critical excitation amplitude as a function of the nonlinear stiffness, λ 1 = 0.6, λ 2 = 0.3, ε = 0.01.

Fig. 7 .

 7 Fig. 7. Force characteristic of conical spring , T represents the transition point between linear and nonlinear phase.

FFig. 8 .

 8 Fig. 8. State and corresponding force of two conical springs (green and blue): (a)(c) at original length; (b)(d) pre-compressed at the transition point. The red curve represents the composed force.

Fig. 9 .

 9 Fig. 9. Polynomial components of the two optimized conical springs

Fig. 10 .Fig. 11 .

 1011 Fig. 10. Schematic of NES system: (a) negative stiffness mechanism (b) conical spring (c) the composed system

Fig. 12 .Fig. 13 .

 1213 Fig. 12. Conical spring manufactured and measuring equipment

Fig. 14 .Fig. 15 .

 1415 Fig. 14. NES system with linear springs of large mean diameter

Fig. 16 . 2 g 1 h 2 hFig. 17 .

 1621217 Fig. 16. Force-displacement relation of the designed NES

Fig. 18 .

 18 Fig. 18. The nonlinear NES force versus the response of LO and NES with G = 0.205 mm, σ = -0.2: (a)(d) positive linear stiffness,γ 1 = γ 2 = 1, η 1 = η 2 = -1; (b)(e) negative linear stiffness, γ 1 = γ 2 = -1, η 1 = η 2 = 1; (c)(f) unsymmetrical stiffness, γ 1 = 1, γ 2 = -1, η 1 = η 2 = 0.

Fig. 19 .

 19 Fig. 19. Experimental setup: (a) global view of the system and (b) detailed view of LO and NES

Fig. 20 .

 20 Fig. 20. Experimental results: (a) frequency response curve of LO with (blue) and without (red) the designed NES; (b) Strongly Modulated Response of LO (blue) and NES (green).

Table 1 .

 1 Parameters of the tuned NES

	Reduced parameters
	ε	1%	λ 1	0.6
	λ 2	0.3	K	K sn
	K as a function of G
	G (mm)	K	G (mm)	K
	0.291	1500	0.205	3000
	0.252	2000	0.191	3500
	0.226	2500	0.178	4000
	3 Theoretical design
	3.1 Conical spring design
	Due to its self-nonlinearity, a conical spring has the ad-
	vantage of providing variable spring rates and varying nat-
	ural frequencies. For this work, two conical springs with

Table 2 .

 2 Parameters of the designed NES

		Tuned parameters
	G	0.205 mm	K	3000
		System parameters
	m 1	5 kg	m 2	50 g
	k 1 1 × 10 5 N/m k 2	3 × 10 6 N/m 3
	c 1	4 Ns/m	c 2	2 Ns/m
		Anticipated parameters
	St	24.8 mm	lp	11.9 mm
	a 0	0 N	a 1	0 N/m
	a 2	-4.4 N/m 2	a 3 2.97 × 10 6 N/m 3

Table 4 .

 4 Sensitivity analysis for the damping of NES

	Adjustment parameters
	c 2 (Ns/m) 1.7 1.8 1.9 2.0 2.1 2.2 2.3