D Qiu

S Seguy

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

Design criteria for optimally tuned vibro-impact nonlinear energy sink

Keywords: Vibro-impact, Nonlinear energy sink, Optimal design, Activation energy, Targeted energy transfer, Passive vibration control

This paper proposes the design criteria for optimally tuned Vibro-Impact (VI) Nonlinear Energy Sink (NES) to control vibration under periodic and transient excitation. Firstly, a generalized dimensionless model of a two degrees of freedom (DOF) system comprising a harmonically excited linear oscillator (LO) strongly coupled to a VI NES is investigated. Bifurcation analysis and efficiency of Targeted Energy Transfer (TET) around the Slow Invariant Manifold (SIM) are studied with the variation of clearance. As a result, the optimal clearances for periodic and transient excitation are calculated from two transition points of the SIM, respectively. Then the procedure is extended to the case of multiple VI NESs in parallel with the LO. Two principles of additivity and separate activities of VI NESs are verified theoretically. Finally, experiments involving the whole system embedded on an electrodynamic shaker are performed. The results show that the design criteria can not only predict the efficient TET at resonance frequency, but can also achieve an optimal performance in a range of frequencies. Furthermore, the criteria can be straightforward for the application of multiple VI NESs, so as to make VI NESs work robustly under different types of excitation.

Introduction

Vibrating systems with clearance between the moving parts, such as linkages, gear trains, pinned connections and joints, are frequently encountered in engineering fields. Impacts occur when the vibration amplitudes of some parts are greater than the clearance, leading to high energy being transferred and dissipated in a transient manner. This phenomenon is attractive for vibration control, and a corresponding device, called an impact damper, was developed in 1945 [START_REF] Lieber | An acceleration damper: development, design and some applications[END_REF]. Over the past decades, impact damper and its dynamics as a typical vibro-impact system have been extensively studied [START_REF] Ibrahim | Vibro-impact dynamics: modeling, mapping and applications[END_REF][START_REF] Babitsky | Theory of vibro-impact systems and applications[END_REF][START_REF] Afsharfard | Suppressing forced vibrations of structures using smart vibroimpact systems[END_REF] and it has been demonstrated that in a range of frequencies, an impact damper can perform better efficiency than a classical damper. With this advantage, various applications of impact damper can be found in turbine blades, machine tools and tall flexible structures [START_REF] Dimentberg | Random vibrations with impacts: a review[END_REF][START_REF] Afsharfard | Free vibration analysis of nonlinear resilient impact dampers[END_REF][START_REF] Zhang | Impact dynamics of flexible-joint robots[END_REF].

Recently, impact damper has been re-examined from the viewpoint 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 is then referred to as 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]. In the present work, the mechanism of TET is revealed by 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 it is observed that some special orbits in the frequency-energy plot are responsible for the irreversible energy transfer from a primary system to an attached VI NES. Inspired by the study of TET, a method of multiple scales originally used for NES with cubic nonlinearity has been improved to explain the transient TET process for a vibrating system with a VI NES [START_REF] Gendelman | Analytic treatment of a system with a vibro-impact nonlinear energy sink[END_REF]. Consequently, a Slow Invariant Manifold (SIM) describing all possible fixed points and possible variation routes is obtained [START_REF] Gourc | Quenching chatter instability in turning process with a vibro-impact nonlinear energy sink[END_REF][START_REF] Gendelman | Dynamics of forced system with vibro-impact energy sink[END_REF].

As far as design criteria of VI NES, there are two main research themes, namely activation characteristic and parameter optimization. With respect to the activation, it is experimentally found that VI NES with a fixed outside excitation is activated in a fixed interval of clearance, and also a VI NES with a fixed clearance is activated in a fixed interval of displacement amplitude of a linear primary system [START_REF] Li | On the dynamics around targeted energy transfer for vibro-impact nonlinear energy sink[END_REF]. Theoretically, the above two describe the same thing, namely VI NES as a vibration absorber will only be effective in a range of some parameter, no matter this parameter is displacement amplitude of the primary system, or amplitude of the excitation, or energy. In [START_REF] Li | Activation characteristic of a vibroimpact energy sink and its application to chatter control in turning[END_REF], the activation characteristic of VI NES generalized from linear systems to nonlinear systems is studied. It shows that the activation characteristic is independent of frequency, so the design of a VI NES for a nonlinear system can be simplified to the optimal design of a linear system. In [START_REF] Viguié | Nonlinear vibration absorber coupled to a nonlinear primary system: a tuning methodology[END_REF], this kind of design characteristic is explored for a nonlinear vibration absorber, but in a general way. The basic philosophy of this design is reflected in the relation of activation energy and clearance of VI NES.

In terms of parameter optimization, it is found that the regime with two impacts per cycle plays an optimal role, no matter in permanent form or intermittent form [START_REF] Li | Optimization mechanism of targeted energy transfer with vibro-impact energy sink under periodic and transient excitation[END_REF]. Based on this idea, a design procedure is proposed, which is similar in essence to the case of a NES with cubic nonlinearity [START_REF] Nguyen | Design criteria for optimally tuned nonlinear energy sinks-part 1: transient regime[END_REF]. For periodic excitation, the objective is to tune the response at the boundary between two symmetrical impacts per cycle and Strongly Modulated Response (SMR). For transient excitation, the target is to make the free response start at the regime with two impacts per cycle and ensure that its duration is as long as possible. In [START_REF] Li | Dynamics of two vibro-impact nonlinear energy sinks in parallel under periodic and transient excitations[END_REF], this optimal mechanism is extended to a linear system coupled with two VI NES in parallel. However, the optimal values of clearance are obtained by trial and errors in [START_REF] Li | Optimization mechanism of targeted energy transfer with vibro-impact energy sink under periodic and transient excitation[END_REF][START_REF] Li | Dynamics of two vibro-impact nonlinear energy sinks in parallel under periodic and transient excitations[END_REF]. If the primary system and the excitation are changed, the clearance will no longer be optimal. Thus, ways of ensuring the optimal response by precise analytical calculation for the clearance need to be further studied.

Unlike NES with cubic nonlinearity, VI NES is capable of passively absorbing and dissipating significant portions of the energy of a primary system, at sufficiently fast time scales, which render it suitable for applications like seismic mitigation [START_REF] Lee | Periodic orbits, damped transitions and targeted energy transfers in oscillators with vibro-impact attachments[END_REF]. However, for both the linear and nonlinear system, the activation of VI NES is limited to a range of excitation. This means that a fixed clearance will only be effective in a given range of displacement amplitudes of a primary system. Using multiple VI NESs is an alternative way to improve the robustness and the feasibility of this approach is proved in the case of multiple NESs with cubic nonlinearity [START_REF] Boroson | Optimization under uncertainty of parallel nonlinear energy sinks[END_REF][START_REF] Vaurigaud | Targeted energy transfer with parallel nonlinear energy sinks. part i: Design theory and numerical results[END_REF]. In [START_REF] Li | Dynamics of two vibro-impact nonlinear energy sinks in parallel under periodic and transient excitations[END_REF], two VI NESs, one with a medium clearance and one with a small clearance, are proposed to be optimal for a given excitation, and experimental results demonstrate their efficiency. Yet, how to adapt the design from a single VI NES to multiple VI NESs is still not clear, neither is the corresponding calculation.

Therefore, the first objective of this paper is to provide a precise analytical calculation of clearance for a single VI NES, and the second is to establish the optimal design relation for passing from a single VI NES to multiple VI NESs, so as to obtain VI NESs that work robustly under different systems and different types of excitation. The paper is organized as follows. Section 2 and Section 3 present the asymptotic analysis of a single VI-NES and the optimal design criteria to tune the clearance for periodic and transient excitation. In Section 4, targeted energy transfer of multiple VI NESs in parallel is studied. In Section 5, details of the experiments performed on the whole system embedded on an electrodynamic shaker are presented. Finally, some conclusions are proposed.

Asymptotic analysis of a single VI-NES

Dynamic modelling

The dynamic modelling presented here is based on [START_REF] Li | Activation characteristic of a vibroimpact energy sink and its application to chatter control in turning[END_REF] and [START_REF] Gourc | Targeted energy transfer under harmonic forcing with a vibro-impact nonlinear energy sink: Analytical and experimental developments[END_REF]. The system of a harmonically excited linear oscillator (LO) attached to a VI NES is illustrated in Fig. 1. The objective here is to apply the asymptotic method used in the above papers to deduce the generalized dimensionless model, so as to obtain the optimal design criteria for a VI NES intended for any targeted primary system. The system is excited by the base, and the equations of motion between impacts are described as follows:

M ẍ + c 1 ẋ + k 1 x = k 1 x e + c 1 ẋe m ÿ = 0, ∀ |x -y| < b (1)
where x, M, c 1 and k 1 are the displacement, mass, damping and stiffness of the LO, respectively. y and m are the displacement and mass of VI NES. b represents the clearance, which can be adjusted by the length of cavity. The imposed harmonic displacement x e is expressed as:

x e = G cos(ωt) (2)
When |x -y| = b, an impact occurs. The state of the model after impact is obtained by using the simplified shock theory:

x + = x -, y + = y - ẋ+ -ẏ+ = -r ẋ--ẏ- ( 3
)
where r is the restitution coefficient and the superscripts + anddenote time immediately after and before impact. By virtue of the condition of total momentum conservation, it gives:

M ẋ+ + m ẏ+ = M ẋ-+ m ẏ- (4)
Then the following changes of variables are introduced:

ε = m M , ω 0 2 = k 1 M , τ = ω 0 t, λ = c 1 mω 0 , Ω = ω ω 0 , F = G εb (5)
The variables of the displacements are dimensioned by:

x = Xb, y = Y b (6)
After rescaling, the system of equation ( 1) can be reduced to the dimensionless form:

Ẍ + ελ Ẋ + X = εF sin Ωτ + ε 2 λ FΩ cos Ωτ ε Ÿ = 0, ∀ |X -Y | < 1 (7)
When |X -Y | = 1, an impact occurs, and the equations (3) and (4) are written as:

X + = X -, Y + = Y - Ẋ+ + ε Ẏ + = Ẋ-+ ε Ẏ - Ẋ+ -Ẏ + = -r Ẋ--Ẏ -, for |X -Y | = 1 (8)
Two new variables representing the displacement of the center of mass and the internal displacement of the VI NES are introduced (both of them are dimensionless):

V = X + εY, W = X -Y (9)
Substituting Eq. ( 9) into Eqs. ( 8) and [START_REF] Zhang | Impact dynamics of flexible-joint robots[END_REF], the equation between impacts in barycentric coordinates becomes:

V + ελ V + ε Ẇ 1 + ε + V + εW 1 + ε = εF sin Ωτ Ẅ + ελ V + ε Ẇ 1 + ε + V + εW 1 + ε = εF sin Ωτ, ∀|W | < 1 ( 10
)
where the term containing ε 2 is very small and can be neglected. The impact condition (8) can be rewritten as:

V + = V -, W + = W -, V + = V -, Ẇ + = -r Ẇ -, for |W | = 1 (11)
Then multiple scales are introduced in the following forms:

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, . . . ( 12
)
The system is studied in the vicinity of the 1:1 resonance, where both the LO and the VI NES execute time periodic oscillations with identical frequency. Thus, a detuning parameter σ representing the nearness of the forcing frequency Ω to the reduced natural frequency of the LO is introduced:

Ω = 1 + εσ (13)
Substituting Eqs. ( 12) and ( 13) into Eqs. ( 10) and [START_REF] Nucera | Application of broadband nonlinear targeted energy transfers for seismic mitigation of a shear frame: Experimental results[END_REF], then equating coefficients of like power of ε gives:

Order ε 0 :

D 2 0 V 0 +V 0 = 0 D 2 0 W 0 +V 0 = 0, ∀|W 0 | < 1 (14)
V + 0 = V - 0 , W + 0 = W - 0 , for|W 0 | = 1 D 0 V + 0 = D 0 V - 0 , D 0 W + 0 = -rD 0 W - 0 (15)
Order ε 1 :

D 2 0 V 1 +V 1 = -2D 0 D 1 V 0 -λ D 0 V 0 -W 0 +V 0 + F sin (τ 0 + σ τ 1 ) D 2 0 W 1 +V 1 = -2D 0 D 1 V 0 -λ D 0 V 0 -W 0 +V 0 + F sin (τ 0 + σ τ 1 ) ∀|W 0 | < 1 (16)
where D 0 represents partial derivative with respect to time τ 0 . From the first equation of system [START_REF] Gendelman | Analytic treatment of a system with a vibro-impact nonlinear energy sink[END_REF], the solution of V 0 (in slow time scale) can be deduced as follows:

V 0 = A(τ 1 ) sin (τ 0 + θ (τ 1 )) (17)
where A(τ 1 ) and θ (τ 1 ) represent the amplitude and phase of the LO, respectively. For W 0 (in slow time scale), Eqs. ( 14) and (15) represent a harmonically forced impact oscillator with symmetric barrier. Under the assumption of 1 : 1 resonance, its solution can be sought in the following form:

W 0 = A(τ 1 ) sin (τ 0 + θ (τ 1 )) + 2 π C(τ 1 )Π (τ 0 + η(τ 1 )) (18)
where C(τ 1 ) and η(τ 1 ) represent the amplitude and phase of VI NES, respectively. Π(z) is a non-smooth saw tooth function [START_REF] Pilipchuk | Closed-form solutions for oscillators with inelastic impacts[END_REF]. Its folded function is expressed as follows:

Π(z) = arcsin(sin z), M(z) = dΠ dz = sgn(cos z) (19)
As can be observed from Eqs. ( 18) and [START_REF] Viguié | Nonlinear vibration absorber coupled to a nonlinear primary system: a tuning methodology[END_REF], impact occurs at τ 0 = π/2η + jπ with j = 0, 1, 2, . . . Substituting the above equations in the impact condition Eq. ( 15) yields:

cos (η -θ ) = 1 -C A , sin (η -θ ) = 2CΓ πA (20)
where Γ = (1r)/(1 + r). By combining the above two equations with trigonometric identity, the expression for a Slow Invariant Manifold (SIM) is obtained:

A 2 = (1 -C) 2 + 4C 2 Γ 2 π 2 (21)

Analytical treatment of Slow Invariant Manifold (SIM)

An illustration of the SIM is presented in Fig. 2. Unlike for a cubic NES, the SIM has a topological structure composed of two branches. It can be observed that the left branch is unstable and only a part of the right branch is stable (between the points T 1 and T 2 ). The stability of the SIM can be evaluated by direct numerical integration of Eqs. ( 14) and [START_REF] Gourc | Quenching chatter instability in turning process with a vibro-impact nonlinear energy sink[END_REF].

Depending on the different positions of the fixed point in SIM [START_REF] Li | On the dynamics around targeted energy transfer for vibro-impact nonlinear energy sink[END_REF], five types of response regimes are obtained: (1) chaos with no duration of two impacts per cycle, where no fixed point occurs and the energy is too low to activate the energy pumping; (2) chaos by intermittency, i.e., Strongly Modulated Response (SMR), with the fixed point located on the left unstable branch of the SIM; (3) two symmetric impacts per cycle, while the fixed point is located on the stable branch; (4) two asymmetric impacts per cycle, the fixed point starts to pass the transition point T 2 and stays on the right unstable branch; (5) chaos with an infinite number of impacts per cycle, i.e., the regime with more than two impacts per cycle, where the fixed point remains higher than in the above cases.

Knowing the variation mechanism of the response regimes, it would be interesting to investigate the efficiency of each regime and the corresponding threshold for periodic excitation, so as to guide the design of VI NES. In [START_REF] Li | Optimization mechanism of targeted energy transfer with vibro-impact energy sink under periodic and transient excitation[END_REF], it is demonstrated that the boundary between response with permanent two impacts per cycle and that with intermittent two impact cycle (SMR) is optimal, and the critical point is located at the extreme T 1 of the SIM. By setting the derivative of the right hand side of Eq. ( 21) to zero, its expression can be obtained as follows:

C 1 = π 2 π 2 + 4Γ 2 , A 2 1 = 4Γ 2 π 2 + 4Γ 2 (22)
Here the value C 1 corresponds to the minimum amplitude of the LO, where 1:1 Targeted Energy Transfer (TET) is allowed. However, the precise analytical calculation of the cavity length (i.e., b) needed to ensure response at the critical point is still not clear and bifurcations of the system at the next order of approximation need to be analysed.

Bifurcation analysis

By introducing Eqs. ( 17) and ( 18) into the first equation of system ( 16) and eliminating the secular term, the asymptotic stability of the fixed points of the stable branch with respect to time scale τ 1 is studied in the following form:

D 1 A = F 2 sin(σ τ 1 -θ ) - 4C π 2 sin (η -θ ) - λC 2
D 1 θ = - F 2A cos (σ τ 1 -θ ) + 4 π 2 A B cos (η -θ ) (23)
Substituting Eqs. [START_REF] Li | Optimization mechanism of targeted energy transfer with vibro-impact energy sink under periodic and transient excitation[END_REF] into Eq. ( 23) and introducing ρ = σ τ 1θ , the expressions governing the evolution of the amplitude A and the phase ρ are obtained:

∂ A ∂ τ 1 = f 1 (A, ρ) g(A) , ∂ ρ ∂ τ 1 = f 2 (A, ρ) g(A) (24)
where 25)

f 1 = Fπ 3 (1 -A) sin (ρ) + 2 FA Γ π 2 cos (ρ) -4 Γ 2 π A 2 λ -16 A 2 Γ -π 3 λ (A -1) 2 (
f 2 = -2 FΓ π sin (ρ) -Fπ 2 cos (ρ) + 8 Γ 2 A σ + 2 π 2 A σ + 2 Γ π λ -2 π 2 σ + 8 A (26)
g = 8 Γ 2 A + 2 π 2 A -2 π 2 (27)
By equating the derivative of the right-hand side of Eq. ( 24) to zero, two kinds of fixed point are calculated. The first is referred to as an ordinary fixed point located on the stable branch of SIM, and it also satisfies the conditions f 1 = f 2 = 0 and g = 0. The other corresponds to the folded singularity (i.e., T 1 ). In this case the derivative of Eq. ( 21) is related to Eq. ( 27), so it can be found that g = 0. Based on this, the system f 1 = f 2 = 0 will be discussed and is rewritten in the following matrix form:

α 11 α 12 α 21 α 22 sin ρ cos ρ = β 1 β 2 ( 28
)
where

         α 11 = Fπ 3 (1 -A), α 12 = 2FAπ 2 Γ, α 21 = -2FπΓ, α 22 = -Fπ 2 β 1 = 4 Γ 2 π A 2 λ + 16 A 2 Γ + π 3 λ (A -1) 2 β 2 = -8 Γ 2 A σ -2 π 2 A σ -2 Γ π λ + 2 π 2 σ -8 A (29)
By solving Eq. ( 28) for sin ρ and cos ρ, ordinary fixed points can be obtained by assuming that the determinant does not vanish. For the folded singularity, it is observed that det(α) = F 2 π 3 g/2 = 0, which means that, when f 2 and g are eliminated, the condition f 1 = 0 is automatically satisfied by Eq. ( 28). Thus the expression of f 2 = 0 can be studied in the following form:

α 2 21 + α 2 22 • cos(ρ -δ ) = β 2 , δ = arctan( α 21 α 22 ) (30)
Then the phase ρ can be deduced as:

ρ = arctan( α 21 α 22 ) + arccos( β 2 α 2 21 + α 2 22 ) (31)
According to Eq. ( 31), the critical condition of the excitation amplitude for the folded singularity to exist is obtained:

β 2 α 2 21 + α 2 22 = 1 (32)
Thus the threshold of the SMR is calculated as:

F c = 2(4 A 1 Γ 2 σ + A 1 π 2 σ + Γ π λ -π 2 σ + 4 A 1 ) π √ 4 Γ 2 + π 2 (33)
Substituting Eq. ( 22) into Eq. (33), the threshold can be expressed in a convenient form:

F c = 2(4 Γ 3 λ + Γ π 2 λ + 4 π) (4 Γ 2 + π 2 ) 3/2 (34)
Here, it is noted that the variable F c is a dimensionless variable of excitation, which is dependent only on the intrinsic properties of the LO and NES (i.e., the damping of the LO and the restitution coefficient).

Criteria for efficient targeted energy transfer

Optimal design criterion for SMR under periodic excitation

Based on the above analysis, the optimal design of VI NES is to make the target displacement amplitude of the LO locate at T 1 of the corresponding SIM under periodic excitation. For this, the cavity length b is chosen as the design parameter, and the objective is to obtain its optimal value so as to produce efficient TET for different types of excitation. According to Eq. ( 5), the relation between the cavity length and the amplitude of excitation can be written as:

b = G εF (35)
By introducing Eq. (34) into Eq. ( 35), the critical value of the cavity length is obtained as: and VI NES, and the projection of motion of the system into the SIM are presented in the first and second columns, respectively. As the cavity length is set to b > b c , a strongly modulated response and its flow jump on the SIM are observed. In this case, the response acts through successive synchronization between the LO and the VI NES. When the VI NES is not synchronized, the amplitude of the LO grows. Under a certain circumstance, the VI NES enters into 1:1 resonance capture with the LO, making the amplitude of the LO decrease fast by successive impacts until it decays at the transition point T 1 . As the cavity length is set as b < b c , a steady state response with two symmetric impacts per cycle is observed. As can be seen, after a short transient motion, the flow is rapidly attracted to the red fixed point.

b c = G 4 Γ 2 + π 2 3/2 2ε(4 Γ 3 λ + Γ π 2 λ + 4 π) (36)
The variation law of TET efficiency with different lengths of cavity b under a fixed periodic excitation is presented in Fig. 4, where A e and A m represent the mean and the maximum amplitude of the LO, respectively. The two examples used above are illustrated by b 1 and b 2 . As can be seen, when the clearance equals b c , the amplitude of the LO is minimum. Thus the calculation of b c can be used to provide the highest TET efficiency for the NES system.

Optimal design criterion for transient excitation

For the transient excitation, there does not exist SMR. The response regime transits continuously from one type to another with the decrease of the master energy. If the initial master energy is still set around the transition point T 1 , the NES system will rapidly pass the 1:1 resonance capture and enter the chaos with no duration of two impacts per cycle. In this case, the VI NES is only activated for a very short time, resulting in a low energy dissipation ratio being produced. In [START_REF] Li | Optimization mechanism of targeted energy transfer with vibro-impact energy sink under periodic and transient excitation[END_REF], it is demonstrated that the efficiency of TET should not only be high at the beginning but also last as long as possible. So in this paper, the transition point T 2 between the response with two symmetric and that with two asymmetric impacts per cycle is proposed for the position of the initial master energy. The analytical value of T 2 can be obtained from Fig. 2, with A 2 = 0.6. Based on this idea, the tuned parameter of the clearance b can be calculated in the following form:

b t = x m A 2 , x m = (x 2 0 + ẋ2 0 ω 2 0 ) (37)
where x m represents the initial master energy, x 0 and ẋ0 represents the initial position and velocity of the LO, respectively.

To demonstrate the criterion, three cases with different clearances are calculated under the conditions: G = 0, ẋ0 = 0 and x 0 = 30 mm. The tuned parameters are given in Table 1 and the optimal value b t is obtained by Eq. (37), where x 1 and x 2 represent the amplitude of the LO at the transition points T 1 and T 2 , t x=15 and t x=5 represent the time duration when the amplitude of the LO decreases to 15 mm and 5 mm, respectively. Transient responses of the first and second case with b < b t and b = b t are respectively presented in Fig. 5 (a) and(b). For the first case, four response regimes are observed during the whole process. A detailed view of these regimes can be found in Fig. 6. Firstly, the regime with three impacts per cycle is excited as shown in area a. Then the response of two asymmetric impacts per cycle appears consecutively as demonstrated in area b. When the amplitude of the system decreases to the transition point T 2 , the flow starts to follow the fixed points on the stable branch, which means that the response of two symmetric impacts per cycle with the 1:1 resonance capture is activated in area c. When the amplitude of the system decays to reach the transition point T 1 , the system escapes from resonance capture, and the VI NES performs chaotic motion, as shown in area d. For the second case, as the cavity length is set as b = b t , the master energy is directly decreased from T 2 . Under a fast phase of nonlinear beating, the system enters the 1:1 resonance capture and the amplitude of the LO undergoes a higher decay ratio than in the first case. Meanwhile, with the increase of the clearance b, the SIM structure shifts to the upper right side, and the band of the stable branch largely increases. For the third case, the cavity length increases to b > b t , making the initial master energy escape T 2 and approach T 1 more closely. As a result, its TET efficiency is large at the beginning period, as can be seen from the comparison of t x=15 in Table 1. However, as the the amplitude of the LO decreases to 5 mm, the TET efficiency of this case becomes low and the second case performs better in terms of energy dissipation for the whole process. Thus it can be concluded that tuning the cavity length of a VI NES to b t is a feasible way of achieving a high TET efficiency under transient excitation. However, optimal efficiency of TET is achieved only in the stable branch of the SIM (between T 1 and T 2 ), which means that the VI NES is effective only in a certain range of the external forcing amplitude. For engineering applications, the cavity length of VI NES is usually fixed. To improve the robustness of VI NES for different types of excitation, tuning the clearances of multiple-degree-of-freedom VI NESs in parallel would be an alternative approach. For this, the optimal design of multiple VI-NESs is discussed in the next section.

Appropriate design of multiple VI-NESs

In this section, the academic model is updated by replacing single VI NES attachments with a parallel configuration of n VI NESs, and the dynamic system with n = 2 is illustrated schematically in Fig. 7. Dynamic motion is consequently related to a set of n + 1 ordinary differential equations given by:

M ẍ + c 1 ẋ + k 1 x = k 1 x e + c 1 ẋe m ÿ1 = 0, ∀ |x -y i | < b 1 m ÿi = 0, ∀ |x -y i | < b i , i = 2..n (38)
System (38) then can be rewritten in dimensionless form:

Ẍ + ελ Ẋ + X = εF sin Ωτ + ε 2 λ FΩ cos Ωτ ε Ÿ1 = 0, ∀ |X -Y | < 1 ε Ÿi = 0, ∀ |X -Y | < ∆ i , i = 2..n (39)
where the variables of the displacements are dimensioned by x = Xb 1 and y i = Y i b 1 . The new physical parameters are expressed as follows:

ε = m 1 + m 2 + ..m n M , α i = m i εM , ω 0 2 = k 1 M , τ = ω 0 t, λ = c 1 εMω 0 , Ω = ω ω 0 , F = G εb 1 , ∆ i = b i b 1 (40)
In the same way, when |X -Y i | = ∆ i , an impact occurs. This yields:

X + = X -, Y + i = Y - i Ẋ+ + ε n ∑ i=1 α i Ẏi + = Ẋ-+ ε n ∑ i=1 α i Ẏi - Ẋ+ -Ẏi + = -r Ẋ--Ẏi -, Ẏj + = Ẏj -, ( j = i) (41)
Motions of the centre of mass and internal displacement are introduced in the following way:

V = X + ε n ∑ i=1 α i Y i , W i = X -Y i (42)
Then, substituting Eq. (42) in Eqs. ( 41) and (40) gives

V + ελ V + ε ∑ n i=1 Ẇi 1 + ε + V + ε ∑ n i=1 W i 1 + ε = εF sin Ωτ Ẅi + ελ V + ε ∑ n i=1 Ẇi 1 + ε + V + ε ∑ n i=1 W i 1 + ε = εF sin Ωτ ∀|W i | < ∆ i (43)
For this system, the impact condition (41) can be rewritten as:

V + = V -, W + i = W - i , V + = V - if |W 1 | = 1 : Ẇ1 + = -r Ẇ1 -, Ẇ + i = Ẇ - i - εα 1 (1 + r) 1 + εα 1 Ẇ1 -, (i = 1) if |W i | = ∆ i : Ẇi + = -r Ẇi -, Ẇ + j = Ẇ - j - εα i (1 + r) 1 + εα i Ẇi -, ( j = i) (44)
Then multiple scales Eq. ( 12) and Eq. ( 13) are introduced in Eq. (44), system approximated at order ε 0 and order ε 1 is obtained:

Order ε 0 :

D 2 0 V 0 +V 0 = 0 D 2 0 W i0 +V 0 = 0, ∀|W i0 | < ∆ i (45) V + 0 = V - 0 , W + 0 = W - 0 , for|W i0 | = ∆ i D 0 V + 0 = D 0 V - 0 , D 0 W + i0 = -rD 0 W - i0 (46)
Order ε 1 :

D 2 0 V 1 +V 1 = -2D 0 D 1 V 0 -λ D 0 V 0 - n ∑ i=1 α i W i0 +V 0 + F sin (τ 0 + σ τ 1 ) D 2 0 W i +V 1 = -2D 0 D 1 V 0 -λ D 0 V 0 - n ∑ i=1 α i W i0 +V 0 + F sin (τ 0 + σ τ 1 ) ∀|W i0 | < 1, i = 2..n (47)
With the same method as used in Eq. ( 21), the SIM expression for multi VI NESs is finally deduced as:

A 2 = (1 -C 1 ) 2 + 4C 2 1 Γ 2 π 2 A 2 = (∆ i -C i ) 2 + 4C 2 i Γ 2 π 2 , i = 2..n (48)
Here, it can be observed that each VI NES is dependent on the initial energy stored in the primary system and their performance is decoupled. With the principle of additivity and activities of VI NESs, the NESs efficiency is driven solely by the intrinsic properties of the LO and their own individual characteristics. Referring to the issue of optimal design for a single VI NES studied in section 3, it is fairly straightforward to extend the optimal criteria in the case of n-parallel VI NES by adjusting the two dimensionless variables ∆ i and α i , so as to ensure the VI NESs are activated in a large band of excitation.

The transition point T i1 , which corresponds to the optimal activation energy of each VI NES that allows 1:1 TET, is given by:

C i1 = π 2 ∆ i π 2 + 4Γ 2 , A 2 i1 = 4Γ 2 ∆ 2 i π 2 + 4Γ 2 , i = 2..n (49)
Thus the multiple VI NESs can be designed using the following rule: A 2 11 < A 2 21 < ... < A 2 n1 . In this case, the activation energy of each VI NES is monotonously increased, making VI-NESs work robustly in different types of energy. Here, it is important to emphasize that SIM expressions in Eq. ( 48) are studied in the vicinity of the 1:1 resonance. If the anticipated motions of multiple VI NESs are not located in the vicinity of the 1:1 resonance, the principle of additivity may not be valid and the chaotic response may produce. In this case, the VI NES is not a NES in the sense that the transfer of energy is no longer irreversible, and the energy may spread back to the primary system, which will jeopardize the vibration mitigation performance. To restrict this situation, the energy threshold of each VI NES (i.e. the curve of each SIM stable branch projected on y axis) should not be set far away from each other, so that each VI NES can execute oscillation with identical frequency of LO. To analyze the chaotic characteristic of multiple VI NESs, Lyapunov exponent is an alternative way [START_REF] Li | Chaotic characteristic of a linear oscillator coupled with vibro-impact nonlinear energy sink[END_REF] and further study will be carried out on this aspect.

Experimental validation

The objective of this section is to experimentally verify the optimal design criteria proposed in the above sections. For periodic excitation, the focus is initially to verify the periodic case at the resonance frequency and then to validate the results under a range of frequencies. As for transient excitation, the tuned method for optimal efficiency is verified, and two VI NESs in parallel are tested with the intention of improving robustness. The detailed procedures are as follows. The experimental setup of periodic excitation is presented in Fig. 8, where two clearances in parallel are used to decide whether one VI NES or two VI NESs are attached to the LO. The whole system was fixed to a 10 kN electrodynamic shaker. The raw signals were recorded using a digital oscilloscope and a bandpass filter was applied to correct biases and suppress high frequency noise. The displacement of the LO as well as the imposed displacement of the shaker were measured by two contactless laser displacement sensors. The acceleration was measured by an accelerometer and the impacts between LO and VI NES could be judged from sudden changes of the acceleration of the LO. The parameters identified on the experimental setup and used for the calculation are given in Table . 2. With the variation of the clearance, three different responses of the LO under periodic excitation G = 0.25 mm were obtained, as shown in Fig. 9(a-c). Where the excitation frequency is fixed at the resonance frequency. As can be seen, the amplitude of the LO with b = b c was smaller than in the other two cases (i.e., steady state response with b < b c and chaotic SMR with b > b c ), since its fixed point was targeted at the transition point T 1 . The optimal length of the cavity b c under different types of excitation was calculated by Eq. (36) and is presented in Fig. 9(d). It can be observed that the experimental points are close to the theoretical values and located almost in a line. The differences between the theoretical and experimental values were mainly caused by the weakly nonlinear damping of the LO. Thus it can be demonstrated that the analytical calculation of b c can be used to predict the efficient TET at resonance frequency for different types of excitation.

For the system in a range of frequencies, if the optimal clearance is still chosen by the value for resonance frequency, the responses of the other frequency points will not be optimal and may result in SMR or irregular response without any duration of two impacts per cycle. Therefore, there does not exist an optimal value of b for all frequencies. In [START_REF] Li | Optimization mechanism of targeted energy transfer with vibro-impact energy sink under periodic and transient excitation[END_REF], it is suggested that the objective of controlling the amplitude of the LO for a range of frequencies can be simplified to that of 10(a). The detailed time-displacement responses are presented in Fig. 10(b-c). When b < b c , the amplitude of LO is decreased with the addition of VI NES, and a small resonance peak still exists in this case. When b > b c , the resonance peak has vanished and a large band of SMR can be found. However, the average amplitude of LO is still large in a range around the resonance frequency. When b = b c , there exists a narrow range of frequency where SMR occurs. Although the VI NES cannot work at its optimal state for other frequencies except resonance frequency, the average amplitude of LO is smaller than the above two cases. Thus, the optimal design for a range of frequencies can directly use the analytical calculation of b c , so as to make VI NES achieve a high TET efficiency for vibration mitigation.

For the above experimental configuration, the single VI NES case with the clearances b = 34 and 23 mm, was observed to be optimal for the excitation with G = 0.25 mm and 0.15 mm, respectively. Consequently, the clearances of two VI NESs in parallel were selected by these two values. The frequency response curves under G = 0.25 mm were recorded for different combinations of b 1 and b 2 and are showed in Fig. 11(a). The detailed time-displacement responses are presented in Fig. 11(b-c). When b 1 = b 2 = 34 mm, the amplitude of the LO around the resonance frequency is decreased. However, its maximum amplitude shifts to the left side and still has the same value (7.9 mm) as that of a single VI NES with b = 34 mm. When b 1 = b 2 = 23 mm, the amplitudes of the LO at the other frequencies are decreased. Yet a small resonance peak exists at the resonance frequency, since the clearance is not optimal for this excitation. When b 1 = 23 mm and b 2 = 34 mm, the small resonance peak has vanished and the frequency range for response with two impacts per cycle is increased. Although there is still a narrow range of frequencies where SMR occurs, its maximum amplitude of the LO has decreased to 4.9 mm. This value is far lower than in the two cases above, which means that adding two VI NESs with different clearance can perform a better TET efficiency and improve the robustness in a range of resonance frequencies.

Thus, the design criteria for optimally tuned VI NES can be summarized. If a single VI NES is applied, it is recommended that the clearance of VI NES should be optimized at the point of resonance frequency. By using the calculation of Eq. (36), the maximum amplitude of the LO can be controlled well at resonance condition (i.e., a single resonance frequency or a range of frequencies). In this case, a semi-active control method can also be used: by adjusting the clearance, a VI NES can be tuned to work robustly with its best performance. If multiple VI NESs are adopted to improve the robustness, the objective should first be to calculate the optimized clearance for the maximum excitation and then to choose a smaller length of clearance for a lower level of energy. By choosing VI NESs that are activated at different types of energy, a high TET efficiency can be obtained in a large band of excitation.

Transient excitation

The experimental setup for transient excitation is presented in Fig. 12, and the corresponding parameters can be found in Table 2. Here, the initial displacement of LO was obtained by stretching the string to a fixed position (x 0 = 20 mm). Once the string was cut, both the LO and the VI NES started to vibrate, and the movement was recorded. The displacement and acceleration of the LO were measured by a laser and an accelerometer, respectively. By varying the length of cavity, the transition of response regimes was observed and the comparison of efficiency for different cavity lengths could be further studied. The experimental results of single VI NES are presented in Fig. 13. As can be seen in Fig. 13(a), without VI NES, the vibration extinction of LO follows a natural exponential decrease; while with VI NES, it follows two phases of quasi-linear decrease, much faster than the exponential one, during which the displacement of LO decreases fast until it reaches a transition point (T 1 ) and the decay rate after this point is obviously lower than the previous one. An enlarged view of acceleration around this transition point is shown in Fig. 13(b). The sudden pulse of acceleration denotes an impact moment, and the transition between response regime of two symmetric impacts per cycle and chaos with an infinite number of impacts per cycle can be clearly identified.

Pre-stretched springs

T 2a T 2b (c)
no VINES b1=10mm, b2=10mm b1=33mm, b2=10mm b1=33mm, b2=33mm 14(a)). If the amplitude of LO is higher than the transition point T 1 a, the first VI NES with a large cavity length is activated with two impacts per cycle. If the amplitude of LO is located in the range between T 1a and T 1b , the first VI NES escapes from the activation and the second VI NES with a small clearance is activated with 1:1 resonance capture. The principle of separate activation can also be found from the time history of the acceleration (see Fig. 14(b)). As can be seen, the impact strength of VI NES is related to the cavity length. The horizontal arrows show the two activations of two VI NESs with different cavity lengths, and the vertical line illustrates the sudden change between them.

Fig. 14(c) shows the comparison of the displacement envelope of LO attached to two VI NESs with different cavity lengths. With the addition of another ball, the three groups of two VI NESs perform better than the case of single VI NES. When b 1 = b 2 = 10 mm, the decay rate of first decrease phase is slightly improved, while the displacement of the transition point is almost the same as that of a single VI NES with b = 10 mm. When b 1 = b 2 = 33 mm, the decay rate of first decrease phase is optimal. However, the efficiency at low energy level is not improved, and the decay rate is far lower than the other two cases. When b 1 = 33 mm and b 2 = 10 mm, the decay rate of first decrease phase is close to that of VI NESs with b 1 = b 2 = 33 mm. At low energy level (the amplitude of LO is lower than T 1a ), the decay rate becomes better than in the other two cases. Thus this group of VI NESs can work robustly and efficiently for different kinds of transient excitation.

In summary, the tuned method for transient excitation is verified. The analytical calculation of Eq. ( 37) is demonstrated to predict the optimal clearance well. Moreover, the principles of additivity and separate activities of multiple VI NESs are observed, and the robustness of vibration control can be improved for a large band of transient excitation.

Conclusion

The ultimate goal of this paper is to propose the design criteria for optimally tuned Vibor-Impact (VI) NES to control the vibration under periodic and transient excitation. To this end, a generalized dimensionless model of a 2-degrees-offreedom system comprising a harmonically excited LO strongly coupled to a VI NES is studied. An analytically obtained Slow Invariant Manifold (SIM) is used to explain the different response regimes. The transition point (i.e., T 1 ) between the regime with two symmetrical impacts per cycles and that of Strongly Modulated Response (SMR) is demonstrated to be optimal for periodic excitation. Then, a bifurcation analysis is carried out. An activation energy threshold for targeted energy transfer is obtained and used to design an optimal criterion for a single NES.

For transient excitation, the critical point (i.e., T 2 ) between the regime with two symmetrical impacts per cycles and two asymmetrical impacts per cycles is adopted to calculate the optimal clearance, and it proves that tuning the initial master energy at this point can achieve a high TET efficiency during the whole vibration process. Thirdly, the procedure is extended to the case of multiple VI NESs in parallel. Two principles of additivity and separate activities of VI NESs are verified theoretically and experimentally.

Finally, experiments involving the whole system for periodic and transient excitation are performed. The results show that the analytical calculation of the clearance can not only predict the efficient TET at resonance frequency, but can also achieve an optimal performance to protect the primary system in a range of frequencies. Furthermore, the design criteria for a single VI NES can be straightforward for the application of multiple VI NESs, so as to make VI NES work robustly under different types of excitation.

Figure 1 :

 1 Figure 1: Schematic of the dynamic system: a harmonically excited LO coupled with a VI NES

2 Figure 2 :

 22 Figure 2: SIM of VI NES: on the stable branch in blue line and two unstable branches in green line with two transition point T 1 and T 2 . ε = 0.68%, λ = 1.91

Figure 3 :

 3 Figure 3: Time-displacement response of LO and VI NES, and the motion of the system projected into the SIM: (a) SMR with b > b c ; (b) steady state response with b < b c .

Figure 4 :

 4 Figure 4: TET efficiency explained by the amplitude of LO with the variation of clearance b: A e represents the mean amplitude and A m the maximum amplitude.

Figure 5 :

 5 Figure 5: Time-displacement response of LO and VI NES, and the motion of the system projected into the SIM: (a) four response regimes with b < b t ; (b) two response regimes with b = b t .

Figure 6 :

 6 Figure 6: Four response regimes of VI NES: (a) three impacts per cycle; (b) two asymmetric impacts per cycle; (c) two symmetric impacts per cycle; (d) chaos with no duration of two impacts per cycle.

Figure 7 :

 7 Figure 7: Schematic of the dynamic system: a harmonically excited LO coupled with two VI NESs in parallel

Figure 8 :

 8 Figure 8: Experimental setup for periodic excitation:(a) global view of the configuration; (b) detailed view of the VI NES

Figure 9 :

 9 Figure 9: Experimental results of single VI NES: response of the LO under periodic excitation G = 0.25 mm with (a) b = 25 mm (< b c ); (b) b = 34 mm (= b c ); (c) b = 45 mm (> b c ) and (d) optimal length of the cavity b c under different types of excitation.

Figure 10 :

 10 Figure 10: Experimental results of single VI NES: (a) frequency response curve of the LO with G = 0.25 mm and detailed view of response of the LO under sweep frequency test with (b) b = 25 mm (< b c ); (c) b = 34 mm (= b c ); (d) b = 45 mm (> b c ).

Figure 11 :

 11 Figure 11: Experimental results of two VI NESs: (a) frequency response curve of the LO with G = 0.25 mm and detailed view of response of the LO under sweep frequency test with (b) b 1 = 23 mm, b 2 = 23 mm; (c) b 1 = 23 mm, b 2 = 34 mm; (d) b 1 = 34 mm, b 2 = 34 mm.

Figure 12 :

 12 Figure 12: Experimental setup for transient excitation

Fig. 13 (Figure 13 :

 1313 Figure 13: Experimental results of single VI NES: (a) time history of the displacement; (b) time history of the acceleration; (c) comparison of the displacement envelope with different b.

Figure 14 :

 14 Figure 14: Experimental results of two VI NESs: (a) time history of the displacement; (b) time history of the acceleration; (c) comparison of the displacement envelope with different b.

Table 1 :

 1 Three cases with different clearance under transient excitation

			Tuned parameters
	case b (mm) x 1 (mm) x 2 (mm) t x=15 (s) t x=5 (s)
	< b t	20	3.1	12	1.8	3.7
	= b t	50	7.8	30	1.3	2.6
	> b t	80	11.8	48.2	1	3.3

Table 2 :

 2 Experimental parameters

		Physical Parameters
	M	4.7 kg	c 1 3.02 Ns/m
	k 1 1.147 × 10 4 N/m m	32 g
		Reduced Parameters
	ε	0.68 %	λ	1.91
	f 0	7.86 Hz	r	0.6


Donghai Qiu
email: donghai.qiu@insa-toulouse.fr

Manuel Paredes

Sébastien Seguy

S Ébastien Seguy

Variable pitch spring

Keywords: Variable pitch spring, Cubic stiffness, Nonlinear energy sink, Negative stiffness mechanism, Passive vibration control Nomenclature

published or not. The documents may come

x, y displacement of linear oscillator and NES

x e imposed harmonic displacement

Introduction

Over recent decades, nonlinear vibration absorbers have advanced rapidly and show considerable potential in passive vibration control applications [START_REF] Lee | Passive non-linear targeted energy transfer and its applications to vibration absorption: a review[END_REF]. Particularly in the areas of aero-space and civil engineering, absorbers are required to be light and to work over a broad spectrum of frequencies, leading to increased interests in exploiting the absorber with different nonlinearities (e.g. vibro-impact, monostable duffing, non-polynomial and bistable oscillators) [START_REF] Karayannis | Vibro-impact attachments as shock absorbers[END_REF][START_REF] Manevitch | Oscillatory models of vibroimpact type for essentially non-linear systems[END_REF][START_REF] Benarous | Nonlinear energy sink with combined nonlinearities: Enhanced mitigation of vibrations and amplitude locking phenomenon[END_REF][START_REF] Lamarque | Targeted energy transfer in mechanical systems by means of nonsmooth nonlinear energy sink[END_REF][START_REF] Su | Stabilisation of the highenergy orbit for a non-linear energy harvester with variable damping[END_REF][START_REF] Su | On electrical optimisation using a duffing-type vibrational energy harvester[END_REF][START_REF] Romeo | Dynamics of a linear oscillator coupled to a bistable light attachment: Numerical study[END_REF]. Among such devices, the nonlinear absorber with cubic stiffness, also called the Nonlinear Energy Sink (NES), has proved to be effective for moderate-energy vibration mitigation [START_REF] Lee | Passive non-linear targeted energy transfer and its applications to vibration absorption: a review[END_REF]. This type of absorber is characterized by a secondary mass strongly coupled via a cubic stiffness to the primary system that needs to be protected [START_REF] Vakakis | Energy pumping in nonlinear mechanical oscillators: part ii: resonance capture[END_REF]. Because of the nonlinearity, irreversible Targeted Energy Transfer (TET) takes place from the main structure to the secondary mass, enabling the NES to be effective in a broad band of frequencies.

Targeted Energy Transfer mechanisms and the feasibility of NES in different applications have been analyzed in detail by [START_REF] Gendelman | Targeted energy transfer in systems with external and self-excitation[END_REF][START_REF] Ahmadabadi | Self-excited oscillations attenuation of drill-string system using nonlinear energy sink[END_REF][START_REF] Gourc | Quenching chatter instability in turning process with a vibro-impact nonlinear energy sink[END_REF][START_REF] Luo | Large-scale experimental evaluation and numerical simulation of a system of nonlinear energy sinks for seismic mitigation[END_REF][START_REF] Luo | Realization of a strongly nonlinear vibration-mitigation device using elastomeric bumpers[END_REF][START_REF] Wang | Track nonlinear energy sink for rapid response reduction in building structures[END_REF]. In these approaches, the essential cubic stiffness was mostly achieved by adopting a construction of two linear springs with no pretension (e.g. a beam [START_REF] Kani | Design and performance analysis of a nonlinear energy sink attached to a beam with different support conditions[END_REF], a piano wire [START_REF] Mcfarland | Experimental study of non-linear energy pumping occurring at a single fast frequency[END_REF], a membrane [START_REF] Mariani | Toward an adjustable nonlinear low frequency acoustic absorber[END_REF] and helical springs [START_REF] Gourc | Experimental investigation and design optimization of targeted energy transfer under periodic forcing[END_REF]). Because of their self-geometric nonlinearity, the springs stretch in tension, thus creating the cubic force. However, this classical type of device cannot effectively take advantage of spring compression and tension, 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. Thus, implementing cubic stiffness elements in practice is still an important issue if the application of NES is to be broadened.

Nowadays, physical determination of the nonlinear forcedisplacement characteristics can be performed by two common methods. One is direct using nonlinear springs, such as conical springs, variable pitch springs, piecewise stiffness springs, and special sized beam segments [START_REF] Wahl | Mechanical springs[END_REF][START_REF] Paredes | Optimal design of conical springs[END_REF][START_REF] Jutte | Design of single, multiple, and scaled nonlinear springs for prescribed nonlinear responses[END_REF][START_REF] Paredes | Analytical and experimental study of conical telescoping springs with nonconstant pitch[END_REF][START_REF] Wang | A new computational model of large three-row roller slewing bearings using nonlinear springs[END_REF]. Various applications of these springs can be found in robotic joints, bandpass filters, vehicle suspensions and crashworthiness structures. The other method is to design a mechanism with variable stiffness. For example, Wu and Lan [START_REF] Wu | Linear variable-stiffness mechanisms based on preloaded curved beams[END_REF] proposed a linear variable stiffness mechanism with preloaded curved beams. Sönmez and Tutum [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 with 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. Stanton et al. [START_REF] Stanton | Nonlinear dynamics for broadband energy harvesting: Investigation of a bistable piezoelectric inertial generator[END_REF] developed a piezoelectric cantilever with a permanent magnet for the energy harvesting system. For these methods, 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] Ma | Nonlinear energy sinks with nontraditional kinds of nonlinear restoring forces[END_REF]. Whether it is possible to combine the advantage of nonlinear springs and variable stiffness mechanisms, so as to obtain strongly cubic stiffness without any linear part for the NES system. With this in mind, a generalized methodology for designing a novel NES with the proposed components is presented in this paper.

The structure of the paper is as follows. The first section is devoted to develop the generic model of the NES system, including the dynamical model for targeted energy transfer and the physical configuration. The next section provides a design theory of variable pitch spring under a given forcedisplacement function. In the third section, the designed springs and NES system are identified, and experiments for the whole system embedded on an electrodynamic shaker are performed. Finally, some concluding remarks are proposed.

Generic model of NES system

Dynamical model of NES system

The dynamic model of the NES system presented here is based on [START_REF] Gourc | Experimental investigation and design optimization of targeted energy transfer under periodic forcing[END_REF] and [START_REF] Gendelman | Attractors of harmonically forced linear oscillator with attached nonlinear energy sink i: Description of response regimes[END_REF]. The system is composed of a harmonically excited linear oscillator (LO) strongly coupled with a cubic NES (see Fig. 1) and is described by the following equations of motion: Substituting k 1 = m 1 ω 2 0 , t = τ /ω 0 and ω = Ωω 0 to equations (1):

     m 1 ẍ + k 1 x + 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)
     ẍ + x + c1 m1ω0 ẋ + c2 m1ω0 ( ẋ -ẏ) + k2 m1ω 2 0 (x -y) 3 = Gcos(Ωτ ) -c1Ω m1ω0 Gsin(Ωτ ) m2 m1 ÿ + c2 m1ω0 ( ẏ -ẋ) + k2 m1ω 2 0 (y -x) 3 = 0 (2)
Then introducing

c 1 = λ 1 m 2 ω 0 , c 2 = λ 2 m 2 ω 0 and k 2 = m 2 ω 2 0 K to Eqs. (2):      ẍ + x + λ1m2 m1 ẋ + λ2m2 m1 ( ẋ -ẏ) + Km2 m1 (x -y) 3 = Gcos(Ωτ ) -λ1m2Ω m1 Gsin(Ωτ ) m2 m1 ÿ + λ2m2 m1 ( ẏ -ẋ) + Km2 m1 (y -x) 3 = 0 (3)
Substituting m 2 /m 1 = and G = F to Eqs. (3):

     ẍ + x + λ 1 ẋ + λ 2 ( ẋ -ẏ) + K (x -y) 3 = F cos Ωτ -2 F λ 1 Ωsin(Ωτ ) ÿ + λ 2 ( ẏ -ẋ) + K (y -x) 3 = 0 (4)
where the term containing 2 is so small that it can be eliminated. The system of equations ( 1) can be finally reduced to the dimensionless form:

ẍ + x + λ 1 ẋ + λ 2 ( ẋ -ẏ) + K (x -y) 3 = F cos Ωτ ÿ + λ 2 ( ẏ -ẋ) + K (y -x) 3 = 0 (5)
The corresponding physical parameters of rescailing process are expressed as follows: v = x + y, w = x -y

= 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 , Ω = ω ω 0 , τ = ω 0 t (
Substituting Eqs. ( 7) into Eqs. ( 5):

         v + λ 1 v + ẇ 1 + + v + w 1 + = F cos Ωτ ẅ + λ 1 v + ẇ 1 + + v + w 1 + + λ 2 (1 + ) ẇ + K (1 + ) w 3 = F cos Ωt ( 8
)
The system is studied in the vicinity of the 1:1 resonance, where both the LO and the NES execute time-periodic 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 (9)
Substituting Eqs. (9) into Eqs. ( 8) and keeping only the secular term containing e iΩτ yields the following slowly modulated system:

             φ1 + iΩ 2 φ 1 + λ 1 (φ 1 + φ 2 ) 2(1 + ) - i(φ 1 + φ 2 ) 2Ω(1 + ) - F 2 = 0 φ2 + iΩ 2 φ 2 + λ 1 (φ 1 + φ 2 ) 2(1 + ) - i (φ 1 + εφ 2 ) 2Ω(1 + ) - F 2 + λ 2 (1 + )φ 2 2 - 3iK (1 + ) φ 2 2 φ2 8Ω 3 = 0 (10)
In the context of energy pumping, the mass ratio is taken to be small (≈ 1%). In this case, Eq. ( 10) can be analysed by a perturbation method with respect to this small parameter. For this purpose, the method of multiple scales [START_REF] Vakakis | Targeted Energy Transfer in Mechanical and Structural Systems[END_REF][START_REF] Nayfeh | Introduction to perturbation techniques[END_REF] is introduced in the following form:

φ i = φ i (τ 0 , τ 1 , + . . .), d dτ = ∂ ∂τ 0 + ∂ ∂τ 1 + . . . τ k = k τ, k = 0, 1, . . . (11)
Substituting Eqs. [START_REF] Ahmadabadi | Self-excited oscillations attenuation of drill-string system using nonlinear energy sink[END_REF] into Eqs. [START_REF] Gendelman | Targeted energy transfer in systems with external and self-excitation[END_REF] and equating coefficients of 0 gives:

∂ ∂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 (12)
Then we introduce the new variables as follows:

φ 1 (τ 1 ) = N 1 e iθ1 , φ 2 (τ 1 ) = N 2 e iθ2 (13)
With this change of variables in Eqs. [START_REF] Gourc | Quenching chatter instability in turning process with a vibro-impact nonlinear energy sink[END_REF], a topological structure of slow invariant manifold (SIM) is obtained 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 (14)
where N 10 , N 20 correspond to the amplitude of LO and NES in slow time scale, respectively. According to [START_REF] Starosvetsky | Strongly modulated response in forced 2dof oscillatory system with essential mass and potential asymmetry[END_REF], the SIM structure admits two extrema (N 21 and N 22 ) and can be divided into two stable branches and one unstable branch, where the unstable branch of the SIM is mainly responsible for the possible occurrence of energy pumping and may give rise to the strongly modulated response (SMR).

To illustrate this mechanism, a strongly modulated response and its corresponding wavelet transform are presented in Fig. 2. A quasi-periodic response with slow variation of the amplitudes of both oscillators is observed. In this regime, the procedure of energy pumping 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, with the frequency component of 1:1 resonance (see Fig. 2(b)), in this case large targeted energy is extracted and dissipated by NES, leading to a fast decrease of the LO amplitude; (3) escape from resonance capture, in which the NES crosses the bifurcation and is quickly attracted to the low branch of SIM (see Fig. 3), 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, which suppresses energy more efficiently than a steady state response.

Physical model of NES system

The physical NES model developed in this paper is presented in Fig. 4(a). It consists of a nonlinear characteristic part and a negative stiffness mechanism. The negative stiffness mechanism is created by two linear springs hinged together at one end with the mass of the NES, while the other ends, subjected to two equivalent preloads P , are allowed to move freely in frictionless horizontal channels.

As the two linear springs are subjected only to horizontal forces, their force on the axial spring is zero. From the equilibrium position, an external force (i.e. F ) is applied to generate an upward displacement (i.e. u) of the output link. Within an operating region of the linear spring, the stiffness curve depends on the preload p. The three F -u curves in Fig. 4(b) denote the NES system with different lateral preloads. The corresponding force-displacement curves of the two horizontal springs and the vertical spring are presented in Fig. 5 (b) and(d), respectively. From the curve with no preload, a positive preload decreases the linear stiffness, whereas a negative preload increases it. As can be seen from Fig. 4(b), the force characteristic of case p < 0 is almost dominated by its component of linear stiffness (i.e. the thick black line). Thus, to obtain strong (nonlinearizable) nonlinearity, the negative stiffness mechanism with p > 0 is adopted.

A detailed realization of the NES system is presented in Fig. 6, where the negative stiffness mechanism is implemented by two cylindrical compression springs having free length l 0 . After pre-compressing to the length l (see Fig. 6(a)), the force-displacement relation with the Taylor expansion is expressed as:

f p = 2k l p l • u -k (l + l p ) l 3 • u 3 (15)
For the axial direction, the nonlinear characteristic part is realized by two variable pitch springs. Unlike ordinary compression springs, variable pitch springs are designed to have a nonlinear spring rate. These springs can have multiple rates or a progressively increasing spring rate as the spring compresses. Since the pitch is varied, some of the coils close up faster than the rest and become inactive, meaning that they no longer absorb the compressive energy resulting from applied forces. When the amount of active energy absorbing coils is reduced, the spring becomes stiffer and the rate increases.

To benefit from the nonlinear performance of the variable pitch spring, a symmetrical connecting type of spring is proposed, as shown in Fig. 6(b). However, this configuration has a linear stiffness part that is hard to eliminate. To obtain the cubic stiffness of the NES system, the objective function of the variable pitch spring is defined as:

F = k 0 • u (u ≤ s t ) a 3 (u -s t ) 3 + a 1 (u -s t ) + P t (u > s t ) (16)
where k 0 is the stiffness of the linear phase, a 3 is the expected cubic value, a 1 is the spring rate after the group in linear phase is fully compressed to the ground block, P t and s t represent the force and displacement of the transition point, respectively.

To skip the linear phase, a method of pre-compressing spring at the transition point is adopted. By changing the initial origin point, the behaviors of two variable pitch springs can belong one to the linear and one to the nonlinear regime simultaneously.

By combining the two spring curves, a composed stiffness curve is obtained and the force relation is:

f K = (a 1 + k 0 ) • u + a 3 • u 3 (17)
Obviously, the new curve is smooth and no longer piecewise (see Fig. 7). By adding the force of two variable pitch springs (i.e. f K ) to the force of the negative stiffness mechanism (i.e. f p ), the composed force of the NES system is obtained:

F = (a 1 + k 0 -2k l p l ) • u + (a 3 + k (l + l p ) l 3 ) • u 3 (18)
As can be seen from Eq. ( 18), if a 1 + k 0 = 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.

Design theory of variable pitch spring

Since the stiffness of a variable pitch spring is piecewise, producing a smooth and nonlinear curve like that of a conical spring is hard. Here, a method using special coordinate points to fit the required stiffness curve is adopted. Depending on the method of curve fitting, two types of shape providing the polynomial components with only linear and cubic terms are proposed: (1) each coil having a different pitch; (2) each group of coils having a different pitch (see Fig. 8). The detailed force characteristics of the two types are presented in Fig. 9. For the first type, each coil is set with a different pitch. As one coil is fully compressed and becomes inactive, a transition point occurs and the spring rate increases considerably. By fitting the transition points on the objective function curve, the pitch of each coil can be calculated. However, the positions of these transition points cannot be selected, which leads to the last pitch being exceptionally large. As can be seen from Fig. 9(a), the last part is hard to fit on the objective function.

To overcome this limitation, the second type is adopted. Here, we allow some coils (usually the number is not an integer) to have the same pitch, which leads to several groups of ordinal linear springs being generated. By defining the transition points of the piecewise curve averagely on the nonlinear part, the stiffness in each piece will correspond well to the required value (see Fig. 9(b)). In this method, the number of transition points determines the number of groups possessing the same pitch. Then the number of active coils can be expressed by using the common formula:

n a = Ḡd 4 8D 3 • k 0 ( 19
)
where D, d and Ḡ represent the mean diameter of the coils, the wire diameter and shear modulus of elasticity, respectively. k 0 is the initial stiffness when all coils are active, as proposed in Eq. ( 16). Defining the total deflection length of the spring as l f (l f > 2s t ), and the number of spring groups in nonlinear regime as n, the interval of each part in the nonlinear phase is obtained: Within the interval of each part of the curve, the displacement and the force of each transition point are given by:

∆ = l f -s t n (20)
u i = s t + i • ∆, i = 1..n F i = a 3 i 3 • ∆ 3 + a 1 i • ∆ + P t ( 21
)
Then the detailed expression of each stiffness k 1 ...k n yields: As the stiffness is held at k i , the remaining active coils n i and the number of coils in each spring group ni can be obtained:

k i = F i -F i-1 u i -u i-1 = a 3 (3i 2 -3i + 1) • ∆ 2 + a 1 ( 22
) t P t s 0 k u 3 3 1 ( ) ( ) t t t a u s a u s P     u t P 0 k u t s u (a) (b) F F 3 3 1 ( ) ( ) t t t a u s a u s P    
n i = Ḡd 4 8D 3 • k i , ni = n i-1 -n i ( 23
)
Once the number of coils in each group has been obtained, the corresponding pitch can be calculated. When the force reaches the transition point between the linear and nonlinear parts, the first group of coils is fully compressed and its corresponding pitch is:

t 0 = s t n + d ( 24
)
When the force reaches the i th transition point of the nonlinear regime, the i th group of coils is fully compressed and its corresponding pitch is given by:

t i = u i -u i-1 ni + t i-1 = ∆ ni + t i-1 , i = 1..n (25)
By adding the length of each spring group to that of the closed ends, the free length of the variable pitch spring is obtained:

L f = na i=1 n i t i + 1.5 • d (26)
Here, it is important to highlight that the free length should satisfy the condition: λ = L f /D ≤ 2.6, so that the variable pitch spring will not buckle as the NES mass moves in a large displacement. To illustrate the detailed realization of a NES system, an example of the design of a variable pitch spring is presented in Table 1. The objective parameters are achieved under a given primary system specification, and detailed process can be referred from [START_REF] Qiu | Tuned nonlinear energy sink with conical spring: Design theory and sensitivity analysis[END_REF]. The force-displacement equation of the variable pitch spring corresponds to Eq. ( 16), in which the force at the transition point is given by P t = k 0 s t . We can note that the displacement of the transition point (i.e. s t ) determines the maximum amplitude of the NES mass. The number of spring groups in the nonlinear regime (i.e. n) controls the accuracy of the curve fitting. The variable pitch characteristics are detailed in a dedicated table as follows. The columns from left to right show the number of coils, the pitch, the helix angle, the stiffness and the displacement of the corresponding transition point. As can be seen, the pitch and helix angle of each spring group increase monotonously, meaning that the pitch distribution has an ascending order. To facilitate the manufacturing process, the final pitch distribution is split to symmetrical shape, as shown in Fig. 10. With this distribution, the number of spring groups is increased but the stiffness curve keeps the same shape.

Experimental validation

Static test

Based on the design parameters of Table 1, the variable pitch spring was manufactured, and its force characteristics test were executed, as shown in Fig. 11(a). With this spring, a NES system providing strongly nonlinear stiffness was designed, the components of which were spherical plain bearings, a linear guide, two variable pitch springs, two linear springs and a NES mass. It is important to highlight that the distance between each spring and the NES mass was adjustable so that a suitable force shape could be reached. Details of the NES experimental setup and the measuring equipment are presented in Fig. 11(b). The NES mass was held by a ring so that it could be connected to the internal load cell of the force gauge. 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. With this experimental setup, the force could be measured by turning the handle to control the deflection of spring.

Fig. 11 (c) and (d) show the force characteristics of the designed spring and the NES system. Based on curve fitting, polynomial parameters of the force-displacement curves are identified. Detailed comparison of parameters between the objective and static test curve is presented in Table 2. As can be seen, the manufactured spring curves and designed piecewise curve correspond to the objective curve well. The experimental curve of the NES system is close to the objective cubic curve. Thus it can be concluded that the design theory of variable pitch spring is efficient to produce the anticipated nonlinearity for the NES system. Combining variable pitch springs and a negative stiffness mechanism is a feasible way to generate pure cubic stiffness. To enable the anticipated nonlinearity to be obtained, enough precompressed length should be reserved in both the variable pitch and the linear springs. Moreover, buckling of the negative stiffness mechanism should not be neglected during the design procedure. , , m k c

F = k 0 • u (u ≤ s t ) a 3 (u -s t ) 3 + a 1 (u -s t ) + P t (u > s t ) k 0 (N/m) a 1 (N/m) a 3 (N/m

Dynamic test

To verify the efficiency of the designed NES, an experimental study to obtain the nonlinear frequency response function of the system around the 1:1 resonance was performed. The experimental setup is presented in Fig. 12. It consisted of a linear oscillator (LO), with an embedded NES. The whole system was fixed on a 10 kN electrodynamic shaker. The displacement of the LO and NES, and the acceleration of the shaker were measured by two contactless laser displacement sensors and an accelerometer, respectively. The raw signals were recorded using a digital oscilloscope and a bandpass filter was applied to correct biases and suppress high frequency noise. The parameters of the experimental setup are given in Table 3.

The frequency response functions (FRF) under excitation with G = 0.25 mm and G = 0.40 mm are illustrated in Fig. 13 (a) and (b), respectively, where the thick blue and the thin red lines represent the response of the LO with and without the NES, x is the amplitude of LO, and A is As the excitation increases further, the maximum amplitude of the LO remains at the same value (5.3 mm). This result is contrary to that for the traditional linear absorber, the tuned mass damper (TMD), where the maximum amplitude of the LO is proportional to the amplitude of excitation. The time-displacement response of a frequency sweep test under G = 0.25 mm is presented in Fig. 13(c). It shows that the unsmooth blue curve in Fig. 13(a) corresponds to the unstable fixed points, where a strongly modulated response (SMR) occurs. A detailed view of the SMR is presented in Fig. 13(d), 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 type of NES can produce energy pumping and is efficient to protect the primary system in a large band of frequencies. Moreover, it performs well in terms of controlling the maximum amplitude of the LO for different types of excitation.

Conclusion

The ultimate goal of this paper is to propose a generalized methodology for designing a novel NES with variable pitch spring, so as to control the vibration of any primary system. To this end, a generic NES system including the dynamical and physical model is firstly investigated. Targeted energy transfer (TET) of a 2 DOF system comprising a harmonically excited linear oscillator (LO) coupled with a cubic NES is studied theoretically. Strongly modulated response is shown to be beneficial for efficient TET. The physical configuration of the NES system for obtaining strongly cubic stiffness is introduced. Key features of the structure include: (i) specifically sizing two variable pitch springs to provide the force polynomial components with only linear and cubic terms; (ii) pre-compressing the two springs at the transition point to produce smooth nonlinear force characteristics; (iii) adding a negative stiffness mechanism to counterbalance the linear term. Secondly, to generate the variable pitch spring, design parametrization is implemented. Two types of shape are proposed to fit the objective force-displacement function. Among them, the type where each group of coils has a different pitch proves to produce the anticipated nonlinearity well for the NES system. Finally, a special sized NES system is developed, in which the distance between each spring and the NES mass is adjustable so that a suitable force shape can be reached. To verify the concept, identification of the forcedisplacement relation is performed, and a good agreement is observed between the theoretical and the experimental result. Then experiments for the whole system embedded on an electrodynamic shaker are studied. The results show that this NES is efficient to protect the primary system in a large band of frequencies. Moreover, it performs well in terms of controlling the maximum amplitude of the LO for different types of excitation, thus making it practical for the application of passive vibration control.

  ni number of coils of the i th spring group mass ratio between NES and linear oscillator λ 1 , λ 2 reduced damping of linear oscillator and NES a 1 , a 3 linear and cubic stiffness at the nonlinear regime c 1 , c 2 damping of linear oscillator and NES D mean diameter of variable pitch spring d wire diameter of variable pitch spring G amplitude of the harmonic excitation K reduced nonlinear stiffness of NES k stiffness of the linear spring k 0 stiffness at the linear regime k 1 , k 2 stiffness of linear oscillator and NES l remaining length of pre-compressed spring l 0 free length of linear spring L f free length of variable pitch spring l p pre-compressed length of linear spring m 1 , m 2 mass of linear oscillator and NES n number of spring groups in nonlinear regime n a number of active coils of variable pitch spring P t , s t force and displacement of the transition point t i pitch of the i th spring group

Figure 1 .

 1 Figure 1. Schematic of a harmonically excited LO coupled with a cubic NES

6 )Figure 2 .

 62 Figure 2. Cubic NES under periodic forcing with parameters K = 2400, λ1 = 0.8, λ2 = 0.2, = 0.015, G = 0.3 mm and initial conditions x0 = 0, ẋ0 = 0, y0 = 0 and ẏ0 = 0. (a) time-displacement of LO and NES; (b) wavelet transform of LO and NES. The states of 1, 2 and 3 represent the regime of nonlinear beating, transient resonance capture and escape from resonance capture, respectively.

22 Figure 3 .

 223 Figure 3. SIM structure (blue line) and the transient projection motion of SMR (red line). The states 1, 2 and 3 represent the regime of nonlinear beating, transient resonance capture and escape from resonance capture, respectively.

FFFigure 4 .FFFigure 5 .

 45 Figure 4. (a) Schematic of NES system with negative stiffness mechanism and nonlinear spring; (b) restoring force with respect to compressing length. Where the thick black line corresponds to the component of linear stiffness for the force characteristic of case p < 0.

Figure 6 .FFigure 7 .

 67 Figure 6. Detailed realization of NES system: (a) negative stiffness mechanism; (b) conical spring; (c) the composed system.

Figure 8 .

 8 Figure 8. Two types of shape for variable pitch spring: (a) each coil with a different pitch; (b) each group of coils with a different pitch.

Figure 9 .

 9 Figure 9. Force characteristics of variable pitch spring: (a) each coil with a different pitch; (b) each group of coils with a different pitch.

Figure 10 .

 10 Figure 10. Symmetrical type of pitch distribution for the spring

FFigure 11 .

 11 Figure 11. (a) Manufactured variable pitch spring and measuring equipment; (b) details of the NES experimental setup and measuring system; (c) force-displacement relation of the designed spring (d) force-displacement relation of the NES system.

Figure 12 .Figure 13 .

 1213 Figure 12. Experimental setup: (a) global scheme of the system and (b) detailed view of LO and NES.

Table 1 .

 1 Parameters of the variable pitch spring

		Objective parameters
	k 0	160 N/m	a 1 165 N/m
	a 3 2.3 × 10 5 N/m 3 s t	35 mm
		Designed parameters
	D		40 mm	d	2 mm
	G	7 × 10 4 Mpa	n	7
	L f	105.6 mm	n a	13.7
			Pitch parameters
	ni	t i (mm) α i ( o ) k i (N/m) u i (mm)
	1	4.56	2.08	160	35
	2.65	5.01	2.28	173	40.75
	2.96	5.59	2.55	218	46.5
	2.17	6.40	2.92	309	52.25
	1.42	7.57	3.45	446	58
	0.93	9.23	4.20	629	63.75
	0.62	11.48	5.22	857	69.5
	1.93	14.45	6.56	1131	75.25

Table 2 .

 2 Comparison of parameters between design and test

	Variable pitch spring

Table 3 .

 3 Parameters of the experiment

		Physical Parameters
	m 1	5.5 kg	m 2	0.1 kg
	k 1 1.15 × 10 4 N/m k 2 4 × 10 5 N/m 3
	c 1	3 Ns/m	c 2	0.4 Ns/m
		Reduced Parameters
	ε	1.8%	λ 2	0.088
	λ 1	0.66	K	1913

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