Thanh Tung Pham

Stéphane Pernot
email: stephane.pernot@entpe.fr

Claude Henri Lamarque

Competitive energy transfer between a two degree-of-freedom dynamic system and an absorber with essential nonlinearity

Keywords: Passive vibration attenuation, energy transfer, nonlinear absorber, nonlinear normal modes, asymptotic analysis

The background of this work is related to passive vibration control of a two degree-of-freedom master system attached to an essentially nonlinear slave absorber aimed to attenuate vibrations by irreversibly transferring energy to a localized nonlinear normal mode 1 . Such nonlinear absorber which has no preferential frequency is theoretically able to capture several nonlinear resonances. The main purpose is here to bring an insight in what is actually going on when two linear modes are in competition for energy transfer. An original asymptotic analysis using two small parameters enables to build a scenario that improves the understanding of resonance mechanisms and to forecast which mode will be first attenuated by means of energy transfer.

Numerical benchmark simulations corroborate the reliability of obtained scenario.

Introduction

The concept of targeted energy transfer has been extensively studied and it proved to be very efficient to mitigate vibration in dynamic systems. Energy pumping mechanism consists in irreversibly transferring vibratory energy from a master system to an essentially nonlinear coupled auxiliary system -namely the nonlinear energy sink NES -by triggering resonances between related nonlinear normal modes [? ? ] followed by a bifurcation jump as explained in [? 1]. This resonance phenomenon is irreversible because of the existence of dissipation due for instance to internal viscous damping [START_REF] Gendelman | Energy Pumping in Nonlinear Mechanical Oscillators: Part I-Dynamics of the Underlying Hamiltonian Systems[END_REF][START_REF] Vakakis | Energy Pumping in Non-linear Mechanical Oscillators: Part II-Resonance Capture[END_REF][START_REF] Gendelman | Transition of Energy to a Nonlinear Localized Mode in a highly Asysmmetric System of Two Oscillators[END_REF] which means that vibration then remain trapped or localized into the NES slave system. Yet, energy transfer is permitted only when an intrinsic energy threshold is went through by master system. Substantial theoretical and experimental work has also been achieved to bring a better understanding of energy transfer mechanisms and to validate the feasibility of targeted energy transfer. In [START_REF] Vakakis | Use of nonlinear localization for isolating structures from earthquakeinduced motions[END_REF], Vakakis and al investigated the feasibility and efficiency of nonlinear localization for isolating structures from earthquakeinduced motions. In [START_REF] Mcfarland | Characterization of an essentially nonlinear 2-DOF vibration test apparatus[END_REF][START_REF] Mcfarland | Experimental verification of the performance of a non linear energy sink[END_REF][START_REF] Mcfarland | Energy pumping into passive nonlinear energy sinks attached to forced linear substructures: analytical and experimental results[END_REF][START_REF] Mcfarland | Experimantal study of non-linear energy pumping occurring at a single fast frequency[END_REF], McFarland and al developed an experimental setup to assess the reliability of a nonlinear energy sink device to mitigate the vibrations of a single degree-of-freedom system. Still in the context of mechanical engineering, Pernot and al and Gourdon and al led similar experiments [START_REF] Pernot | Experimental dynamics of a four storey building coupled with a nonlinear energy sink[END_REF][START_REF] Pernot | Optimal design and multi-modal control with passive nonlinear absorber(s)[END_REF][START_REF] Gourdon | Experimental Verification of Nonlinear Energy pumping under transient forcing with strongly nonlinear coupling[END_REF][START_REF] Gourdon | Contribution to efficiency of irreversible passive energy pumping with a strong nonlinear attachment[END_REF][START_REF] Gourdon | Energy pumping with multiple passive nonlinear absorbers[END_REF] with a four storey building model on top of which several NES absorbers were mounted to trigger irreversible energy transfer from the master modes towards the NESs. Configurations with NESs assembled in series were also considered to speed up modal localization. In [START_REF] Gendelman | A Degenerate Bifurcation Structure in the Dynamics of Coupled Oscillators with Essential Stiffness Nonlinearities[END_REF][START_REF] Vakakis | Dynamics og linear discrete systems connected to local, essentially non-linear attachements[END_REF], Gendelman and al and Vakakis and al investigated from a theoretical point of view a system of multiple linear coupled oscillators involving an essentially nonlinear attachment. These studies highlight many advantages of the concept of modal localization to trigger targeted resonance captures with one of the system's mode. The concept was soon extended to other fields of research. Manevitch and al [START_REF] Manevitch | Dynamics Interaction of a Semi-infinite Linear Chain of Coupled Oscillators with a Strongly Nonlinear Attachment[END_REF] used a strong nonlinear attachment to control the vibrations of semi-infinite chains of particules. In [START_REF] Vakakis | Shock Isolation Trough the Use of Nonlinear Energy Sinks[END_REF], Vakakis and al used NESs for shock isolation purposes. In [START_REF] Vakakis | Transient dynamics of a dispersive elastic wave guide weakly coupled to an essentially nonlinear end attachement[END_REF], energy transfer was seek to attenuate transient vibrations of a dispersive elastic wave guide.

Further academic works [START_REF] Vakakis | Isolated resonance captures and resonance capture cascades leading to single or multi-mode passive energy pumping in damped coupled oscillators[END_REF][START_REF] Gendelman | Bifurcations of Nonlinear Normal Modes of Linear Oscillator with Strongly Nonlinear damped Attachement[END_REF][START_REF] Gendelman | Dynamics of coupled linear and essentially nonlinear oscillators with substantially different masses[END_REF] soon focused on the ability of essential nonlinear absorbers to attenuate multiple resonance captures simultaneously or in cascade. In practice, a master system model with several degrees of freedom coupled with an auxiliary nonlinear system has been considered in [START_REF] Vakakis | Isolated resonance captures and resonance capture cascades leading to single or multi-mode passive energy pumping in damped coupled oscillators[END_REF] by proposing scenarios of isolated resonances and energy transfer within these scenarios. The analytical treatment has been made with the assumption of considering a single frequency of NES in each time interval and the numerical verification indicates the threshold energy necessary for targeted energy transfer. Latest developments aimed to study typical energy transfer mechanisms encountered for instance in the case of nonlinear systems involving multiple states of equilibrium [START_REF] Gendelman | Dynamics of linear oscillator coupled to strongly nonlinear attachement with multiple states of equilibrium[END_REF] or in strongly nonlinear systems featuring nonhomogeneous nonlinearity [START_REF] Manevitch | New analytical approach to energy pumping problem in strongly nonhomogeneous 2dof systems[END_REF]. Harmonically forced quasi-periodic regimes provoked by energy transfer were also studied in [START_REF] Gendelman | Quasiperiodic Energy Pumping in Coupled Oscillators under Periodic Forcing[END_REF][START_REF] Gendelman | Quasiperiodic reponse regimes of strongly nonlinear vibration absorber[END_REF]. A simplified theoretical understanding of nonlinear localization is finally proposed in [START_REF] Gendelman | Targeted energy transfer in systems with non-polynomial nonlinearity[END_REF] where the class of considered attachment is no longer restricted to polynomial like nonlinearity but also to non polynomial one.

Contrary to standard linear or weakly nonlinear tuned mass absorbers that are designed to attenuate a single frequency stationary regime of master system, NES systems have no preferential frequency and are also fitted to work with transient vibration regimes. They are therefore theoretically able to simultaneously attenuate several modes of a master system as long as feeding energy is sufficient to maintain an active energy transfer. In practical experiments, however, simultaneous energy transfer has to the best of authors knowledge never been observed. As an attempt of explanation, the answer is maybe to be sought by investigating the bassins of attraction of underlying nonlinear localized modes. Hence, let us consider the following ersatz stating that if multiple modal localization are likely to occur then a precedence mechanism must prevail.

Present work also aims to accurately understand energy transfer mechanisms likely to occur in the mitigation of a general linear two degree-of-freedom system attached to a NES device. The main purpose is first to understand what is going on when two nonlinear modal localizations compete against each other and second to build scenarios eventually permitting to forecast which resonance will be triggered first during transient regime -the idealistic case corresponding to large amplitude vibration localized in the NES absorber and small amplitude oscillation in the master system. Present paper is organized as follows: section 2 introduces the methodology upon the academic instance of a two degree-of-freedom master system attached to a purely cubic nonlinear oscillator. It is noticeable that case study is different from the one investigated in [START_REF] Vakakis | Isolated resonance captures and resonance capture cascades leading to single or multi-mode passive energy pumping in damped coupled oscillators[END_REF] for nonlinear attachment is not connected to the ground. After re-normalizing equations of motion in the linear modal base, an original asymptotic expansion with two independent scales is performed on previous equations. A superposition principle of nonlinear normal modes is introduced up to the first scale order to split motion into two new independent coordinates, the latter of which characterizing the amount of energy that is transferred between each master degree-of-freedom and the NES device. Manevitch's complexification method [START_REF] Manevitch | The description of localized normal modes in a chain of nonlinear coupled oscillators using complex variables[END_REF] is then used to investigate the dynamics response of related nonlinear normal modes close to a double one to one resonance. A multiple scale technique is finally applied to exhibit reduced nonlinear normal modes. Section 3 is dedicated to the study of targeted energy transfers (TET) that enable to forecast the transient dynamic response of the two dofs + NES system. The peculiar case of well separated master modes is investigated and allow to furnish an analytical trend of energy transfer versus time that is similar with results obtained by O. Gendelman in [START_REF] Gendelman | Bifurcations of Nonlinear Normal Modes of Linear Oscillator with Strongly Nonlinear damped Attachement[END_REF]. Going back to the general case, multiple scenarios are introduced to forecast energy transfers in the academic system -the approach being to slave one scale to the other in reduced equations of motion -Numerical evidences introduced in section 4 allow to corroborate the reliability of previous energy pumping scenarios. Conclusions are finally drawn and direction of prospective research works is given.

Two-scale asymptotic expansion of an academic model

The energy pumping phenomenon has been widely investigated in [START_REF] Vakakis | Use of nonlinear localization for isolating structures from earthquakeinduced motions[END_REF][START_REF] Vakakis | Shock Isolation Trough the Use of Nonlinear Energy Sinks[END_REF] for single degree-of-freedom master systems attached to slave mass by means of an essential nonlinear stiffness. Main features are a oneway and irreversible transfer of master system vibrations towards the NES device where energy is localized 3 NODY9745_source before vanishing thanks to the occurrence of damping [START_REF] Gendelman | Transition of Energy to a Nonlinear Localized Mode in a highly Asysmmetric System of Two Oscillators[END_REF][START_REF] Gendelman | Energy Pumping in Nonlinear Mechanical Oscillators: Part I-Dynamics of the Underlying Hamiltonian Systems[END_REF]. Hence nonlinear attachment is at the roots of a one to one resonance capture responsible for energy pumping phenomenon. Present study now aims to investigate mechanisms of energy transfer when master system includes two degree-of-freedom as depicted in Fig. 1.

Two-dofs master system coupled with a NES

Let us consider the following academic spring-mass-damper system obtained for instance by applying reduction theory and connected to a nonlinear spring/mass system namely the NES device. Applying Newton's second law of motion, it yields the following equations of motion written in the physical basis:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ M 1 ẍ + c 1 ẋ + c 2 ẋ -ẏ + k 1 x + k 2 x -y = 0 M 2 ÿ + c 2 ẏ -ẋ + k 2 y -x + c 3 ẏ -ż + k 3 y -z 3 = 0 mz + c 3 ż -ẏ + k 3 z -y 3 = 0 (1)
where x, y and z respectively stand for the displacement of the first, the second mass and the NES absorber. k 3 refers to a purely cubic nonlinear stiffness which means that related restoring force is nonlinear proportional with the internal displacement (z -y) and ẋ is the derivative of variable x with respect to time t.

Previous set of equations ( 1) may be rewritten in matrix form

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ M Ẍ + C Ẋ + K X + ⎡ ⎣ 0 c 3 ẏ -ż + k 3 y -z 3 ⎤ ⎦ = ⎡ ⎣ 0 0 ⎤ ⎦ mz + c 3 ż -ẏ + k 3 z -y 3 = 0 (2) involving mass matrix M = ⎡ ⎣ M 1 0 0 M 2 ⎤ ⎦ , stiffness matrix K = ⎡ ⎣ k 1 + k 2 -k 2 -k 2 k 1 ⎤ ⎦ , damping matrix C = ⎡ ⎣ c 1 + c 2 -c 2 -c 2 c 1 ⎤ ⎦ and displacement vector X = ⎡ ⎣ x y ⎤ ⎦ .

Model written in the modal basis

Assuming that nonlinear coupling does not change eigenmodes of master system so far, equations of motion (2) may be spanned in the modal basis by introducing the following modal change of coordinates:

X = Φ • q (3) involving mode shape vectors Φ = φ 1 φ 2 = ⎡ ⎣ φ 11 , φ 12 φ 12 , φ 22 ⎤ ⎦ and modal coordinates q = ⎡ ⎣ q 1 q 2 ⎤ ⎦
All simplifications done, one finally obtains:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Φ t M Φ q + Φ t C Φ q + Φ t K Φ q+ Φ t ⎡ ⎣ 0 c 3 φ 12 q1 + φ 22 q2 -ż + k 3 φ 12 q 1 + φ 22 q 2 -z 3 ⎤ ⎦ = ⎡ ⎣ 0 0 ⎤ ⎦ mz + c 3 ż -φ 12 q1 -φ 22 q2 + k 3 z -φ 12 q 1 -φ 22 q 2 3 = 0 (4)
Orthogonal properties of eigenmodes with respect to mass M and stiffness K matrices then enable to partly uncouple equations (4) by equating

Φ t M Φ = ⎡ ⎣ M 11 0 0 M 22 ⎤ ⎦ (5)
and

Φ t K Φ = ⎡ ⎣ k 11 0 0 k 22 ⎤ ⎦ (6)
and damping matrix

Φ t C Φ = ⎡ ⎣ c 11 c 21 c 12 c 22 ⎤ ⎦ (7)
Equations of motion consequently become

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ M 11 q1 + c 11 q1 + c 21 q2 + k 11 q 1 + φ 12 c 3 φ 12 q1 + φ 22 q2 -ż + φ 12 k 3 φ 12 q 1 + φ 22 q 2 -z 3 = 0 M 22 q2 + c 12 q1 + c 22 q2 + k 22 q 2 + φ 22 c 3 φ 12 q1 + φ 22 q2 -ż + φ 22 k 3 φ 12 q 1 + φ 22 q 2 -z 3 = 0 mz + c 3 ż -φ 12 q1 -φ 22 q2 + k 3 z -φ 12 q 1 -φ 22 q 2 3 = 0 (8)

Non-dimensional equations of motion

Introducing renormalization parameters

y 1 = φ 12 q 1 , y 2 = φ 22 q 2 , ω 2 1 = k 11 M 11 , ω 2 2 = k 22 M 22 , λ = c 3 m , K = k 3 m , 1 = φ 2 12 m M 11 , 2 = φ 2 22 m M 22 , 1 λ 1 = c 11 M 11 , 2 λ 2 = c 22 M
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ÿ1 + 1 λ 1 ẏ1 + 1 λ 21 ẏ2 + ω 2 1 y 1 + 1 λ ẏ1 + ẏ2 -ż + 1 K y 1 + y 2 -z 3 = 0 ÿ2 + 2 λ 12 ẏ1 + 2 λ 2 ẏ2 + ω 2 2 y 2 + 2 λ ẏ1 + ẏ2 -ż + 2 K y 1 + y 2 -z 3 = 0 z + λ ż -ẏ1 -ẏ2 + K z -y 1 -y 2 3 = 0 (9)
involving two small scale parameters 1

1 and 2 1 homogeneous to mass ratios.

Original problem (1) is hence transformed into a set of 3 weakly coupled differential equations. Modal coordinates y 1 (resp. y 2 ) are weakly coupled with y 2 and z (resp. with y 1 and z) in the first two equations while at the same time NES coordinate z remains strongly connected to y 1 and y 2 in the last equation.

In what follows, primary structure and NES device are assumed to be initially at rest. An impulse is then instantaneously exciting master coordinates y 1 and y 2 which is similar to consider the free oscillation response of master+NES system under some initial conditions given by:

y 1 (t = 0) = y 2 (t = 0) = z(t = 0) = 0, dy 1 dt = C I , dy 2 dt = C II , dz dt = 0 (10)

Splitting of NES response

Truncating two-scale series expansion of equations ( 9) at order 0 1 and 0 2 leads to a set of equations

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ÿ1 + ω 2 1 y 1 = 0 ÿ2 + ω 2 2 y 2 = 0 z + λ ż -ẏ1 -ẏ2 + K z -y 1 -y 2 3 = 0 (11)
whose solutions y 1 and y 2 are completely uncoupled from NES response z. As a result, y 1 and y 2 are pure oscillations vibrating at frequencies ω 1 and ω 2 respectively. Because the motion of z is dominated by components featuring y 1 and y 2 , it is reasonable to assume that z is mainly divided into two components z 1 and z 2 themselves vibrating at frequencies ω 1 and ω 2 . A splitting z = z 1 + z 2 is also introduced into equations (9) to account for this notice:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ÿ1 + 1 λ 1 ẏ1 + 1 λ 21 ẏ2 + ω 2 1 y 1 + 1 λ ẏ1 -ż1 + ẏ2 -ż2 + 1 K y 1 -z 1 + y 2 -z 2 3 = 0 ÿ2 + 2 λ 12 ẏ1 + 2 λ 2 ẏ2 + ω 2 2 y 2 + 2 λ ẏ1 -ż1 + ẏ2 -ż2 + 2 K y 1 -z 1 + y 2 -z 2 3 = 0 z1 + z2 + λ ż1 -ẏ1 + ż2 -ẏ2 + K z 1 -y 1 + z 2 -y 2 3 = 0 (12)
A new change of variables:

⎧ ⎨ ⎩ u 1 = y 1 + 1 z 1 , v 1 = y 1 -z 1 u 2 = y 2 + 2 z 2 , v 2 = y 2 -z 2 (13)
that exhibits the displacement of the center of inertia and the internal displacement of both modes 1 and 2 with respect to the NES is introduced and leads to equations:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ü1 + 1 v1 1 + 1 + 1 λ 1 u1 + 1 v1 1 + 1 + 1 λ 21 u2 + 2 v2 1 + 2 + ω 2 1 u 1 + 1 v 1 1 + 1 + 1 λ v1 + v2 + 1 K v 1 + v 2 3 = 0 ü2 + 2 v2 1 + 2 + 2 λ 2 u2 + 2 v2 1 + 2 + 2 λ 12 u1 + 1 v1 1 + 1 + ω 2 2 u 2 + 2 v 2 1 + 2 + 2 λ v1 + v2 + 2 K v 1 + v 2 3 = 0 ü1 -v1 1 + 1 + ü2 -v2 1 + 2 -λ v1 + v2 -K v 1 + v 2 3 = 0 (14) 6

Complexified problem

Manevitch's complexification technique [? ] is used to highlight low dynamics of studied system from fast oscillating response. Fast components also appear to match modal frequencies ω 1 and ω 2 modulated by slow changing amplitudes ϕ i i=1 • 4 .

By inserting complex change of variables written as:

⎧ ⎨ ⎩ ϕ 1 e iω1t = u1 ω1 + iu 1 , ϕ 2 e iω1t = v1 ω1 + iv 1 ϕ 3 e iω2t = u2 ω2 + iu 2 , ϕ 4 e iω2t = v2 ω2 + iv 2 (15)
one similarly obtains:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 1 + 1 ω 1 φ1 + 1 φ2 e iω1t + i 2 ω 2 1 ϕ 1 + 1 ϕ 2 e iω1t -ϕ 1 + 1 ϕ 2 e -iω1t + 1 λ 1 1 1 + 1 1 2 ω 1 ϕ 1 + 1 ϕ 2 e iω1t + ϕ 1 + 1 ϕ 2 e -iω1t + 1 λ 21 1 1 + 2 1 2 ω 2 ϕ 3 + 2 ϕ 4 e iω2t + ϕ 3 + 2 ϕ 4 e -iω2t +ω 2 1 1 1 + 1 1 2i ϕ 1 + 1 ϕ 2 e iω1t -ϕ 1 + 1 ϕ 2 e -iω1t + 1 λ 1 2 ω 1 ϕ 2 e iω1t + ϕ 2 e -iω1t + 1 2 ω 2 ϕ 4 e iω2t + ϕ 4 e -iω2t + 1 K 1 2i ϕ 2 e iω1t -ϕ 2 e -iω1t + 1 2i ϕ 4 e iω2t -ϕ 4 e -iω2t 3 = 0 1 1 + 2 ω 2 φ3 + 2 φ4 e iω2t + i 2 ω 2 2 ϕ 3 + 2 ϕ 4 e iω2t -ϕ 3 + 2 ϕ 4 e -iω2t + 2 λ 2 1 1 + 2 1 2 ω 2 ϕ 3 + 2 ϕ 4 e iω2t + ϕ 3 + 2 ϕ 4 e -iω2t + 2 λ 12 1 1 + 1 1 2 ω 1 ϕ 1 + 1 ϕ 2 e iω1t + ϕ 1 + 1 ϕ 2 e -iω1t +ω 2 2 1 1 + 2 1 2i ϕ 3 + 2 ϕ 4 e iω2t -ϕ 3 + 2 ϕ 4 e -iω2t + 2 λ 1 2 ω 2 ϕ 4 e iω2t + ϕ 4 e -iω2t + 1 2 ω 1 ϕ 2 e iω2t + ϕ 2 e -iω1t + 2 K 1 2i ϕ 4 e iω2t -ϕ 4 e -iω2t + 1 2i ϕ 2 e iω1t -ϕ 2 e -iω1t 3 = 0 1 1 + 1 ω 1 φ1 -φ2 e iω1t + i 2 ω 2 1 ϕ 1 -ϕ 2 e iω1t -ϕ 1 -ϕ 2 e -iω1t + 1 1 + 2 ω 2 φ3 -φ4 e iω2t + i 2 ω 2 2 ϕ 3 -ϕ 4 e iω2t -ϕ 3 -ϕ 4 e -iω2t -λ 1 2 ω 1 ϕ 2 e iω1t + ϕ 2 e -iω1t + 1 2 ω 2 ϕ 4 e iω2t + ϕ 4 e -iω2t -K 1 2i ϕ 2 e iω1t -ϕ 2 e -iω1t + 1 2i ϕ 4 e iω2t -ϕ 4 e -iω2t 3 = 0 (16)
The main goal is now to cancel secular components in e iωj t j=1, 2 within related multiple scale approximations for only stationary and bounded solutions are seek.

An averaging method with resonant frequency ω 1 is therefore applied on both first and third equations of system ( 16) which yields:

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ω 1 φ1 + 1 φ2 - 3 4 i 1 Kϕ 2 ϕ 4 2 - 3 8 i 1 Kϕ 2 ϕ 2 2 + 1 2 1 λω 1 ϕ 2 + 1 2 1 λ 1 ω 1 ϕ 1 = 0 1 + 2 ω 1 φ1 -φ2 + 3 4 i 1 + 1 + 2 Kϕ 2 ϕ 4 2 + 1 2 i 1 + 2 ω 2 1 ϕ 1 -ϕ 2 - 1 2 1 + 1 + 2 λω 1 ϕ 2 + 3 8 i 1 + 1 + 2 Kϕ 2 ϕ 2 2 = 0 (17)
Repeating similar averaging technique with resonant frequency ω 2 on second and third equations of system ( 16), it yields:

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ω 2 φ3 + 2 φ4 - 3 4 i 2 Kϕ 4 ϕ 2 2 - 3 8 i 2 Kϕ 4 ϕ 4 2 + 1 2 2 λω 2 ϕ 4 + 1 2 2 λ 2 ω 2 ϕ 3 = 0 1 + 1 ω 2 φ3 -φ4 + 3 4 i 1 + 1 + 2 Kϕ 4 ϕ 2 2 + 1 2 i 1 + 1 ω 2 2 ϕ 3 -ϕ 4 - 1 2 1 + 1 + 2 λω 2 ϕ 4 + 3 8 i 1 + 1 + 2 Kϕ 4 ϕ 4 2 = 0 (18)
By isolating φ1 and φ2 components and by considering a taylor series expansion up to order O( i ) i=1, 2 , system (17) may be written as:

⎧ ⎪ ⎨ ⎪ ⎩ 1 + 1 + 2 φ1 + 1 2 i 1 ω 1 ϕ 1 + 1 2 1 λ 1 ϕ 1 - 1 2 i 1 ω 1 ϕ 2 + O( 2 1 , 2 2 , 1 2 ) = 0 ω 1 φ2 - 3 8 iKϕ 2 ϕ 2 2 - 1 2 iω 2 1 ϕ 1 + 1 2 iω 2 1 ϕ 2 - 3 4 iKϕ 2 ϕ 4 2 + 1 2 λω 1 ϕ 2 + O( 1 , 2 ) = 0 (19)
Doing similarly with φ3 and φ4 in equations ( 18), one obtains:

⎧ ⎪ ⎨ ⎪ ⎩ 1 + 1 + 2 φ3 + 1 2 i 2 ω 2 ϕ 3 + 1 2 2 λ 2 ϕ 3 - 1 2 i 2 ω 2 ϕ 4 + O( 2 1 , 2 2 , 1 2 ) = 0 ω 2 φ4 - 3 8 iKϕ 4 ϕ 4 2 - 1 2 iω 2 2 ϕ 3 + 1 2 iω 2 2 ϕ 4 - 3 4 iKϕ 4 ϕ 2 2 + 1 2 λω 2 ϕ 4 + O( 1 , 2 ) = 0 (20)
It should be pointed out that such results only hold when following conditions are simultaneously valid:

ω 1 = ω 2 ( 21
)
3ω 1 = ω 2 (22)

Two-scale multiple scale analysis of complexified model

The method of multiple scales is applied on problem (20) by considering 1 and 2 like independent scale parameters. Introducing related time scales T i = i 1 t and τ i = i 2 t, new time derivatives may alternatively be derived:

d dt -→ ∞ i=0 i 1 ∂ ∂T i (23) or d dt -→ ∞ i=0 i 2 ∂ ∂τ i ( 24
)
Expanding complex amplitudes ϕ i i=1, •••, 4 by considering following changes of coordinates:

⎧ ⎨ ⎩ ϕ 1 (t) = ϕ 1 T 0 , T 1 , ϕ 2 (t) = ϕ 2 T 0 , T 1 ϕ 3 (t) = ϕ 3 τ 0 , τ 1 , ϕ 4 (t) = ϕ 4 τ 0 , τ 1 (25) 8
Because the energy drop provoked by energy pumping phenomenon is much faster than the one due to system's natural damping, it is reasonable to assume that following re-scaling rule holds:

λ 1 = 1 λ 10 , λ 2 = 2 λ 20 (26)
to minimize the influence of master system damping versus energy transfer.

Equations ( 19) and ( 20) may hence be expanded in telescopic series featuring scales j i i, j . Isolating latter expressions up to the first scale orders, one finally obtains:

• Equations with respect to scale orders i 1 order 0 1 :

∂ϕ 1 ∂T 0 = 0 (27) ω 1 ∂ϕ 2 ∂T 0 - 3 8 iKϕ 2 ϕ 2 2 - 1 2 iω 2 1 ϕ 1 + 1 2 iω 2 1 ϕ 2 - 3 4 iKϕ 2 ϕ 4 2 + 1 2 λω 1 ϕ 2 = 0 (28)
order 1 1 :

∂ϕ 1 ∂T 1 + 1 + 2 1 ∂ϕ 1 ∂T 0 + 1 2 iω 1 ϕ 1 - 1 2 iω 1 ϕ 2 = 0 (29)
• Equations with respect to scale orders i 2 order 0 2 : ∂ϕ 3 ∂τ 0 = 0 (30)

ω 1 ∂ϕ 4 ∂τ 0 - 3 8 iKϕ 4 ϕ 4 2 - 1 2 iω 2 2 ϕ 3 + 1 2 iω 2 2 ϕ 4 - 3 4 iKϕ 4 ϕ 2 2 + 1 2 λω 2 ϕ 4 = 0 (31)
order 1 2 :

∂ϕ 3 ∂τ 1 + 1 + 1 2 ∂ϕ 3 ∂τ 0 + 1 2 iω 2 ϕ 3 - 1 2 iω 2 ϕ 4 = 0 (32)
From equations ( 27) and ( 30), amplitudes ϕ 1 and ϕ 3 happen to be independent of T 0 and τ 0 respectively which gives:

ϕ 1 = ϕ 1 (T 1 ) (33)
ϕ 3 = ϕ 3 (τ 1 ) (34)
In addition, one may assume that

lim T0→∞ ∂ϕ 2 ∂T 0 = 0 (35) lim τ0→∞ ∂ϕ 4 ∂τ 0 = 0 (36)
for master system oscillations are asymptotically damped.

Let us also introduce following definitions in what follows:

ψ 2 (T 1 ) = lim T0→∞ ϕ 2 (T 0 , T 1 ) (37)
ψ 4 (τ 1 ) = lim τ0→∞ ϕ 4 (τ 0 , τ 1 ) (38)

Nonlinear modes of reduced model

Nonlinear modes are by the end solutions of a set of reduced equations of motion written with respect to two independent time scales T 1 and τ 1 .

From equations ( 28) and ( 29), one obtains:

⎧ ⎪ ⎨ ⎪ ⎩ 1 2 λω 1 ψ 2 -3 8 iKψ 2 ψ 2 2 -1 2 iω 2 1 ϕ 1 + 1 2 iω 2 1 ψ 2 -3 4 iKψ 2 ψ 4 2 = 0 ∂ϕ 1 ∂T 1 + 1 2 iω 1 ϕ 1 - 1 2 iω 1 ψ 2 = 0 (39)
From equations ( 31) and (32), one obtains:

⎧ ⎪ ⎨ ⎪ ⎩ 1 2 λω 2 ψ 4 -3 8 iKψ 4 ψ 4 2 -1 2 iω 2 2 ϕ 3 + 1 2 iω 2 2 ψ 4 -3 4 iKψ 4 ψ 2 2 = 0 ∂ϕ 3 ∂τ 1 + 1 2 iω 2 ϕ 3 - 1 2 iω 2 ψ 4 = 0 (40)

Targeted energy transfers in the two dofs + NES system

Multiple scenarios are introduced hereafter to forecast transient energy transfers likely to develop in the studied dynamical system including two degrees of freedom connected to a NES by means of an essentially nonlinear stiffness. The peculiar case of well separated master modes is investigated and allow to furnish an analytical trend of energy transfer versus time. A general approach is then extended by slaving one scale to another in previous reduced equations of motion.

TET mechanism for well separated master modes

Let's here consider the peculiar case 1 2 which may be translated for instance into the following ersatz 1 = χ 3 and 2 = χ2 with χ 1. Previous multi-scale expansions are not fundamentally modified 10 by these underlying scale changes and one can exploit related sets of equations. It is also reasonable to additionally assume that when time scale τ 1 begins to increase, time scale T 1 remains constant for a given duration.

Energy ψ 2 may hence be regarded as a constant mapping in the first equation of system (40) by writing:

ψ 2 2 = C 1 (41)
where C 1 stands for the 'initial' energy 2 embedded in the first differential mode.

After replacing statement (41) into equation ( 40) finally yields:

ϕ 3 = 1 - 3 2 C 1 Ω 2 ψ 4 -iγ 2 ψ 4 - 3 4 iΩ 2 ψ 4 ψ 4 2 (42)
featuring news parameters

Ω 2 = K ω 2 2 and γ 2 = λ ω 2 .
In order to better understand energy transfer towards second differential mode, variable ψ 4 is transformed into following polar coordinates:

ψ 4 = N 2 e iδ2 (43)
and equation ( 40) finally leads to the simplified expression:

dZ 2 dτ 1 = - 16ω 2 γ 2 Z 2 27Ω 2 2 Z 2 2 -48Ω 2 -72Ω 2 2 C 1 Z 2 + 16γ 2 2 + 16 + 36Ω 2 2 C 2 1 -48Ω 2 C 1 (44)
with

Z 2 = N 2 2 .
It is noticeable that equation ( 44) is similar with a commonly encountered differential equation that describes the transient evolution of targeted energy transfer within a single degree-of-freedom system attached to a NES. Multiple bifurcated solutions are obtained by seeking the roots of related polynomial mapping:

P (Z 2 ) = 27Ω 2 2 Z 2 2 -48Ω 2 -72Ω 2 2 C 1 Z 2 + 16γ 2 2 + 16 + 36Ω 2 2 C 2 1 -48Ω 2 C 1 (45)
Targeted energy transfer TET 2 in ψ 4 may be witnessed when energy drops down from one solution to another more stable one which means that P Z 2 must have at least two positive roots. This leads to the following conditions:

⎧ ⎨ ⎩ C 1 < 2 3Ω2 1 -3 2 Ω 2 C 1 + √ 3γ 2 1 -3 2 Ω 2 C 1 - √ 3γ 2 > 0 (46)
or similarly

C 1 < S 1 with S 1 = 2 3 1 - √ 3 λ ω 2 K ω 2 2 ( 47
)
involving an activation energy threshold S 1 .

As a consequence, targeted energy transfer TET 2 is triggered towards the second differential mode ψ 4

when condition (47) is satisfied. By considering the limit τ 1 → ∞, one trivially obtains that N 2 → 0 and equations (39) finally reduces to:

⎧ ⎪ ⎨ ⎪ ⎩ 1 2 λω 1 ψ 2 - 3 8 iKψ 2 ψ 2 2 - 1 2 iω 2 1 ϕ 1 + 1 2 iω 2 1 ψ 2 = 0 ∂ϕ 1 ∂T 1 + 1 2 iω 1 ϕ 1 - 1 2 iω 1 ψ 2 = 0 (48)
with

Ω 1 = K ω 2 1 and γ 1 = λ ω 1 .
Problem (39) is hence transformed into a classical targeted energy transfer problem between a single master degree-of-freedom and a NES. A simple explanation may here be given to improve the physical understanding of what is going on. Targeted energy transfer mechanism in ψ 4 is winning against its competitor in ψ 2 when characteristic times of both mechanisms are well separated i.e. nonlinear coupling in ψ 1 equations is much greater that its counterpart in ψ 3 equations. As a result, a relatively large coupling in ψ 1 equations prevents to feed energy to its related localized mode.

Generally speaking, ratio happens to be a little bit restrictive. Next sections also aim to investigate competitive energy transfers when small parameters 1 and 2 have the same order of magnitude and to provide TET scenarios likely to occur.

Basic mechanism for competitive TETs

Coming back to the general case with no assumption made upon natural frequencies ω 1 and ω 2 , one is now interested in how targeted energy transfer TET 1 in ψ 2 is competing against targeted energy transfer TET 2 in ψ 4 with respect to time. The basic idea is to consider that TET 1 and TET 2 events are triggered within almost instantaneous laps of time and that it is reasonable to claim that for instance ψ 4 remains piecewise constant time-dependent while TET 1 is active until a given activation energy threshold is went through.

Whatever natural frequencies ω 1 and ω 2 , one may always assume that time scale

ω 1 T 1 = 1 ω 1 t for TET 1 in ψ 2 is different from time scale ω 2 τ 1 = 2 ω 2 t according to TET 2 in ψ 4 .
Hence, it is possible to follow the simultaneous response of two quantities when time scales ω 1 T 1 and ω 2 τ 1 become neighbors.

By assuming that

C 2 = ψ 4 2 (49)
is a piecewise constant with respect to time, one may rewrite system of equations (39) like:

⎧ ⎪ ⎨ ⎪ ⎩ 1 2 λω 1 ψ 2 - 3 8 iKψ 2 ψ 2 2 - 1 2 iω 2 1 ϕ 1 + 1 2 iω 2 1 ψ 2 - 3 4 iKC 2 ψ 2 = 0 ∂ϕ 1 ∂T 1 + 1 2 iω 1 ϕ 1 - 1 2 iω 1 ψ 2 = 0 (50)
and Z 1 = N 2 1 appears to be a solution of equation:

dZ 1 dT 1 = - 16ω 1 γ 1 Z 1 27Ω 2 1 Z 2 1 -48Ω 1 -72Ω 2 1 C 2 Z 1 + 16γ 2 1 + 16 + 36Ω 2 1 C 2 1 -48Ω 1 C 2 (51)
Similarly to equation (47), one may exhibit another activation energy threshold S 2 defined by

C 2 < S 2 with S 2 = 2 3 1 - √ 3 λ ω 1 K ω 2 1 ( 52
)
Activation energy thresholds S 1 and S 2 that constitute upper bounds for targeted energy transfers and are invariant upon different choices of essential nonlinear stiffness or damping of NES, are prerequisite for triggering targeted energy transfers TET 2 and TET 1 respectively. By contrast, lower bound for energy thresholds indeed exist and they are time varying quantities likely to diminish mainly due to the occurrence of dissipation in related NNMs.

Several scenarios for targeted energy transfers detailed hereafter may also be derived qualitatively by exploring whether conditions (47) and (52) are satisfied or not.

3.3. Scenario C 1 ≥ S 1 ⇒ TET 2 disabled ; C 2 ≤ S 2 ⇒ TET 1 enabled
Here initial energy is assumed to be mainly concentrated upon the first internal mode with checked condition C 1 ≥ S 1 which prevents TET 2 mechanism to settle down. Condition C 2 ≤ S 2 is moreover 13 assumed which means that TET 1 is free to be triggered. In present configuration that is the most commonly encountered case, energy is not high enough to trigger a bifurcation jump in second internal mode and this latter is attenuated by its natural dissipation versus time. At the same time, the existence of multiple solutions for the first internal mode is ensured by condition C 2 ≤ S 2 and this latter undergoes a bifurcation jump from the higher energy solution towards the more stable solution, thus triggering TET 1 mechanism.

The time response of energy embedded in internal mode 1 merely depends on initial energy sprayed to internal mode 2 and on the choice of both nonlinear stiffness and damping of the NES attachment. As a result, dynamic equations may be uncoupled for C 2 plays a minor role in systems of equations ( 39) and (40). This scenario also brings back to the well known study of a single degree-of-freedom master system attached to a NES.

In that case, the trend of energy decay in internal mode 1 is given by:

dZ 1 dT 1 ≈ - 16ω 1 γ 1 Z 1 27Ω 2 1 Z 2 1 -48Ω 1 -72Ω 2 1 C 2 Z 1 + 16γ 2 1 + 16 + 36Ω 2 1 C 2 1 ( 53
)
or similarly by implicit formulae

27 2 Ω 2 1 Z 2 1 -Z 2 10 -48Ω 1 Z 1 -Z 10 + 16(1 + γ 2 1 ) + 36Ω 2 1 C 2 1 ln Z 1 Z 10 ≈ -16ω 1 γ 1 T 1 (54)
whereas energy decay in internal mode 2 is depicted by:

dZ 2 dτ 1 ≈ - λ 1 + λ 2 ω 2 2 Z 2 (55)
or similarly by:

Z 2 ≈ Z 20 exp λτ 1 1 + λ 2 ω 2 2 (56)
From a qualitative point of view, transient trend of energy decay Z 1 and Z 2 for internal modes 1 and 2 respectively is displayed in Fig. 2.

Scenario

C 1 ≥ S 1 ; C 2 ≥ S 2 ⇒ TET 1

and TET 2 in wait for being triggered

Here, initial energy C 1 and C 2 of internal modes 1 and 2 are assumed to be greater than their corresponding activation threshold S 1 and S 2 . Present scenario sketches what is very likely to occur in the triggering of TET 1 and TET 2 mechanisms: step 1 : At initial time, no TET mechanism is developing in the studied system. Due to the occurrence of dissipation in both internal modes 1 and 2, energies Z 1 (resp. Z 2 ) decay until one is going through its related activation threshold S 2 (resp. S 1 ), step 2 : One energy threshold S 1 or S 2 is reached. By considering that initial energy is equally spread onto internal modes 1 and 2 and assuming that modal frequencies satisfy relationship ω 1 < ω 2 which results in S 2 < S 1 , energy Z 1 is likely to go down its activation energy boundary S 1 first while at the same time Z 2 remains above S 2 . Hence, there is a possibility for TET 2 mechanism to settle down.

From this point, curve shape of Z 2 is virtually modified for future evolution and a new condition upon energy is required to trigger TET 2 , step 3 : Energy Z 1 goes on decaying due to damping and curve shape of Z 2 is modified accordingly through nonlinear coupling components. A virtual bifurcation jump for future evolution becomes steeper and the probability of TET 2 to occur is raised, step 4 : Energy Z 2 is high enough to trigger targeted energy transfer TET 2 involving internal mode 2 and Z 2 instantaneously drops down to a more stable energy level with no link with C 1 . Energy Z 2 fastly decays until condition Z 2 ≤ S 2 is encountered which again gives a chance to TET 1 to eventually occur, Main mechanisms involved in this scenario are step by step illustrated in Fig. 3. Because the decaying trend of energy Z 2 is closely related to the one of Z 1 , equations ( 39) and (40) may not be easily uncoupled.

Next section aims also introduces a degenerate case that enables to partly uncouple equations of trends.

Degenerate equations of motion using a single time scale

Previous analytical developments introduced independent time scales (T i ) i and (τ i ) i involving small non co-mensurable parameters 1 and 2 to derive equations of motion (39) and (40) in ϕ 1 , ϕ 3 , ψ 2 and ψ 4 . Yet, it is reasonable to claim that time scales T 1 and τ 1 are very close to each other and that they could be considered as dependant time variables constrained by for instance the following relationship:

α = τ 1 T 1 = 2 1 ( 57
)
The damping related to the primary mass λ 1 and λ 2 can be taken into account here by ignoring condition [START_REF] Gendelman | Quasiperiodic Energy Pumping in Coupled Oscillators under Periodic Forcing[END_REF], equations ( 39) and (40) hence become: featuring functions ϕ 3 and ψ 4 that now solely depend on time T 1 .

⎧ ⎪ ⎨ ⎪ ⎩ 1 2 λω 1 ψ 2 -3 8 iKψ 2 ψ 2 2 -1 2 iω 2 1 ϕ 1 + 1 2 iω 2 1 ψ 2 -3 4 iKψ 2 ψ 4 2 = 0 ∂ϕ 1 ∂T 1 + 1 2 λ 1 ϕ 1 + 1 2 iω 1 ϕ 1 - 1 2 iω 1 ψ 2 = 0 (58) and ⎧ ⎪ ⎨ ⎪ ⎩ 1 2 λω 2 ψ 4 -3 8 iKψ 4 ψ 4 2 -1 2 iω 2 2 ϕ 3 + 1 2 iω 2 2 ψ 4 -3 4 iKψ 4 ψ 2 2 = 0 1 α ∂ϕ 3 ∂T 1 + 1 2 λ 2 ϕ 3 + 1 2 iω 2 ϕ 3 - 1 2 iω 2 ψ 4 = 0 (59)
By expanding equations ( 39) and (59) using the following polar coordinates:

ψ 2 = N 1 e iδ1 ( 60
)
ψ 4 = N 2 e iδ2 ( 61
)
new equations of motion in energy amplitudes N 1 and N 2 may be obtained as:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ dN 1 dT 1 = P 1 N 1 , N 2 P N 1 , N 2 dN 2 dT 1 = P 2 N 1 , N 2 P N 1 , N 2 (62)
with

P 1 N 1 , N 2 = -324Ω 2 1 Ω 2 2 λ 1 N 9 1 + (432Ω 2 1 Ω 2 λ 1 + 864λ 1 Ω 2 2 Ω 1 )N 7 1 + (1296Ω 2 1 Ω 2 2 λ 2 -1944Ω 2 1 Ω 2 2 λ 1 )N 2 2 N 7 1 - (144Ω 2 1 γ 2 2 λ 1 +1152λ 1 Ω 1 Ω 2 +576λ 1 γ 2 1 Ω 2 2 +144Ω 2 1 λ 1 +576Ω 2 2 ω 1 γ 1 +576λ 1 Ω 2 2 )N 5 1 +Ω 2 1 Ω 2 2 (3888λ 2 -4131λ 1 )N 4 2 N 5 1 + (3456λ 1 Ω 2 2 Ω 1 + 2160Ω 2 1 Ω 2 λ 1 -1728Ω 2 1 λ 2 Ω 2 -1728Ω 1 Ω 2 2 λ 2 )N 2 2 N 5 1 + (768Ω 2 ω 1 γ 1 + 768λ 1 γ 2 1 Ω 2 + 384λ 1 Ω 1 + 384γ 2 2 λ 1 Ω 1 +768λ 1 Ω 2 )N 3 1 +(3456Ω 2 1 Ω 2 λ 1 +4104λ 1 Ω 2 2 Ω 1 -1728Ω 1 Ω 2 2 λ 2 -4320Ω 2 1 λ 2 Ω 2 )N 4 2 N 3 1 +(2916Ω 2 1 Ω 2 2 λ 2 - 3564Ω 2 1 Ω 2 2 λ 1 )N 6 2 N 3 1 + (-576Ω 2 1 γ 2 2 λ 1 -1152λ 1 Ω 2 2 -3456λ 1 Ω 1 Ω 2 -1152Ω 2 2 ω 1 γ 1 -576Ω 2 1 λ 1 + 576λ 2 γ 2 2 Ω 2 1 + 2304Ω 1 λ 2 Ω 2 + 576Ω 2 1 αω 2 γ 2 + 576Ω 2 1 λ 2 -1152λ 1 γ 2 1 Ω 2 2 )N 2 2 N 3 1 + (-256λ 1 -256ω 1 γ 1 -256λ 1 γ 2 1 -256λ 1 γ 2 1 γ 2 2 - 256λ 1 γ 2 2 -256γ 2 2 ω 1 γ 1 )N 1 + (-432λ 1 γ 2 1 Ω 2 2 + 1152Ω 2 1 αω 2 γ 2 -432Ω 2 2 ω 1 γ 1 + 1152Ω 1 λ 2 Ω 2 -2304λ 1 Ω 1 Ω 2 - 576Ω 2 1 λ 1 + 1152λ 2 γ 2 2 Ω 2 1 -576Ω 2 1 γ 2 2 λ 1 + 1152Ω 2 1 λ 2 -432λ 1 Ω 2 2 )N 4 2 N 1 + (-972Ω 2 1 Ω 2 2 λ 1 + 648Ω 2 1 Ω 2 2 λ 2 )N 8 2 N 1 + (-432Ω 1 Ω 2 2 λ 2 +1296λ 1 Ω 2 2 Ω 1 +1728Ω 2 1 Ω 2 λ 1 -1728Ω 2 1 λ 2 Ω 2 )N 6 2 N 1 +(768γ 2 2 λ 1 Ω 1 -768Ω 1 αω 2 γ 2 -768λ 2 γ 2 2 Ω 1 + 768λ 1 γ 2 1 Ω 2 + 768Ω 2 ω 1 γ 1 + 768λ 1 Ω 1 -768Ω 1 λ 2 + 768λ 1 Ω 2 )N 2 2 N 1 , P 2 N 1 , N 2 = (-972Ω 2 1 Ω 2 2 λ 2 + 648Ω 2 1 Ω 2 2 λ 1 )N 2 N 8 1 + (2916Ω 2 1 Ω 2 2 λ 1 -3564Ω 2 1 Ω 2 2 λ 2 )N 6 1 N 3 2 + (1728Ω 1 Ω 2 2 λ 2 - 1728λ 1 Ω 2 2 Ω 1 + 1296Ω 2 1 λ 2 Ω 2 -432Ω 2 1 Ω 2 λ 1 )N 6 1 N 2 + (3888Ω 2 1 Ω 2 2 λ 1 -4131Ω 2 1 Ω 2 2 λ 2 )N 4 1 N 5 2 + (-1728Ω 2 1 Ω 2 λ 1 - 4320λ 1 Ω 2 2 Ω 1 + 4104Ω 2 1 λ 2 Ω 2 + 3456Ω 1 Ω 2 2 λ 2 )N 4 1 N 3 2 + (1152Ω 2 2 ω 1 γ 1 -2304Ω 1 λ 2 Ω 2 -576λ 2 Ω 2 2 -432λ 2 γ 2 2 Ω 2 1 - 432Ω 2 1 λ 2 + 1152λ 1 Ω 2 2 -432Ω 2 1 αω 2 γ 2 + 1152λ 1 Ω 1 Ω 2 -576λ 2 Ω 2 2 γ 2 1 + 1152λ 1 γ 2 1 Ω 2 2 )N 4 1 N 2 + (-1944Ω 2 1 Ω 2 2 λ 2 + 1296Ω 2 1 Ω 2 2 λ 1 )N 2 1 N 7 2 + (2160Ω 1 Ω 2 2 λ 2 + 3456Ω 2 1 λ 2 Ω 2 -1728Ω 2 1 Ω 2 λ 1 -1728λ 1 Ω 2 2 Ω 1 )N 2 1 N 5 2 + (-3456Ω 1 λ 2 Ω 2 - 1152λ 2 γ 2 2 Ω 2 1 + 576λ 1 Ω 2 2 -576λ 2 Ω 2 2 -576λ 2 Ω 2 2 γ 2 1 -1152Ω 2 1 αω 2 γ 2 + 576Ω 2 2 ω 1 γ 1 -1152Ω 2 1 λ 2 + 576λ 1 γ 2 1 Ω 2 2 + 2304λ 1 Ω 1 Ω 2 )N 2 1 N 3 2 +(768Ω 1 λ 2 -768λ 1 Ω 2 +768λ 2 γ 2 2 Ω 1 -768λ 1 γ 2 1 Ω 2 -768Ω 2 ω 1 γ 1 +768λ 2 Ω 2 +768Ω 1 αω 2 γ 2 + 768λ 2 Ω 2 γ 2 1 )N 2 1 N 2 -324Ω 2 1 Ω 2 2 λ 2 N 9 2 + (432Ω 1 Ω 2 2 λ 2 + 864Ω 2 1 λ 2 Ω 2 )N 7 2 + (-576Ω 2 1 λ 2 -144λ 2 Ω 2 2 -576Ω 2 1 αω 2 γ 2 - 144λ 2 Ω 2 2 γ 2 1 -576λ 2 γ 2 2 Ω 2 1 -1152Ω 1 λ 2 Ω 2 )N 5 2 +(768Ω 1 λ 2 +384λ 2 Ω 2 γ 2 1 +384λ 2 Ω 2 +768λ 2 γ 2 2 Ω 1 +768Ω 1 αω 2 γ 2 )N 3 2 + (-256λ 2 -256λ 2 γ 2 2 -256αω 2 γ 2 -256λ 2 γ 2 2 γ 2 1 -256λ 2 γ 2 1 -256γ 2 1 αω 2 γ 2 )N 2 , P N 1 , N 2 = 512 + 512γ 2 2 γ 2 1 + 512γ 2 1 + 512γ 2 2 + 1458Ω 2 1 N 4 2 Ω 2 2 N 4 1 + 1944Ω 2 2 N 8 1 Ω 2 1 + (4608Ω 2 Ω 1 + 864Ω 2 2 γ 2 1 + 864Ω 2 2 + 1152Ω 2 1 + 1152γ 2 2 Ω 2 1 )N 4 2 + (-2592Ω 2 Ω 2 1 -3456Ω 2 2 Ω 1 )N 6 1 + 1944Ω 2 2 N 8 2 Ω 2 1 + (-1536Ω 2 -1536Ω 1 - 1536γ 2 2 Ω 1 -1536Ω 2 γ 2 1 )N 2 1 +(-3456Ω 2 Ω 2 1 -2592Ω 2 2 Ω 1 )N 6 2 +(-1536Ω 2 -1536Ω 1 -1536γ 2 2 Ω 1 -1536Ω 2 γ 2 1 )N 2 2 + (864Ω 2 1 + 4608Ω 2 Ω 1 + 864γ 2 2 Ω 2 1 + 1152Ω 2 2 + 1152Ω 2 2 γ 2 1 )N 4 1 + (2304γ 2 2 Ω 2 1 + 4608Ω 2 Ω 1 + 2304Ω 2 2 γ 2 1 + 2304Ω 2 2 + 2304Ω 2 1 )N 2 1 N 2 2 + 3888Ω 2 1 N 6 2 Ω 2 2 N 2 1 + 3888Ω 2 1 N 2 2 Ω 2 2 N 6 1 + (-3456Ω 2 Ω 2 1 -6048Ω 2 2 Ω 1 )N 4 2 N 2 1 + (-3456Ω 2 2 Ω 1 - 17 NODY9745_source 6048Ω 2 Ω 2 1 )N 2 2 N 4 1
Equations of motion (39) and (40) finally reduce to a set of partly uncoupled equations (62) that may easily be integrated to exhibit the dynamic behavior of N 1 and N 2 with respect to time scale T 1 . Eventual bifurcation jumps ie triggering of TET 1 and TET 2 mechanisms are here synchronous and related to the roots of polynomial P (N 1 , N 2 ) involved in equations (62). Next section introduces an academic numerical instance to highlight the relevance of TET s scenarios discussed here above.

Numerical experiments

A benchmark application featuring a two bay frame civil engineering structure is used to assess the reliability of proposed targeted energy transfers mechanisms. Here the two degrees-of-freedom building model is connected to essentially nonlinear absorber that can resonate with either the first or the second eigenmode of the master structure. Two targeted energy transfers are also in competition for mitigating vibration induced in the primary structure. Numerical simulations achieved in the case of a free oscillation regime or with a pulse-like excitation enable to assess the validity of proposed scenarios for energy pumping by comparing energy trends obtained by integrating both degenerate analytical equations of motion and original model.

A two storey building model

A reduced order model of an actual two storey building, presented in [START_REF] Gourdon | Contrôle passif de vibrations par pompage énergétique[END_REF], is here considered as a benchmark application. The building is made of two concrete slabs weighting 60 tons each as depicted in Fig. 4. Slabs are connected to a set of vertical steel beams themselves clamped on a concrete fondation. A purely nonlinear absorber (NES) is additionally mounted on first floor slab.

Modal properties

A discrete skewer model shown in Fig. 5 is obtained by considering concentrated masses at the center of slabs and equivalent bending stiffness given by:

k 1 = k 2 = 24EI h 3 = 18.6 × 10 3 kN/m (63)
A standard modal analysis exhibits eigen-modes φ 1 and φ 2 of master system associated to natural frequencies:

• f 1 = 1.73 Hz,
• f 2 = 4.53 Hz.

whose unit-normalized modes shapes are illustrated in Fig. 4. Mode shapes may firstly be written under matrix form:

Φ = φ 1 φ 2 = ⎡ ⎣ 1 -0.618 0.618 1 ⎤ ⎦ (64)
and secondly mass-normalized according to the following rule:

• M 11 = M 1 φ 2 11 + M 2 φ 2 12 = 82.91 tons, • M 22 = M 1 φ 2 21 + M 2 φ 2 22 = 82.91 tons.
Viscous damping matrix is assumed to be diagonal in the modal basis by using for instance Basile's rule with diagonal components chosen as in the realistic case:

• c 11 = 3.61 × 10 3 N.m -1 .s,

• c 22 = 23.6 × 10 3 N.m -1 .s.
Equations of motion are expanded onto the modal basis by using following change of coordinates

[x, y] T = Φ • [q 1 , q 2 ] T (65)
thus exhibiting modal coordinates q 1 and q 2 respectively for the first and the second mode; y 1 = φ 12 q 1 ,

y 2 = φ 22 q 2 .
All modal quantities such as mass, viscous damping and stiffness match variables introduced as in Fig. 1.

NES design

The nonlinear energy sink is designed by means of a small mass m attached to mass M 1 with a pure cubic geometric nonlinearity in parallel with a linear viscous damper. A quick glance at Fig. 4 A typical value k 3 = 4.05×10 6 kN/m 3 that is not precisely matching an optimal design value is chosen for cubic nonlinear stiffness or similarly K = 0.15 × 10 3 kN/m 3 /kg. Additionally a value c 3 = 4.2 × 10 3 N.m -1 .s is set arbitrarily for the NES or similarly λ = 1.55 s -1 .

TET energy thresholds

Activation energy thresholds S 1 and S 2 may then be estimated from the knowledge of previous modal and NES parameters:

S 1 = 2 3 1- √ 3 λ ω 2 K ω 2 2 = 2.53 × 10 -4 m 2 (67) S 2 = 2 3 1- √ 3 λ ω 1 K ω 2 1 = 1.15 × 10 -5 m 2 (68)

Validation of TET scenarios

Two cases are considered hereafter to assess the relevance of competitive scenarios for targeted energy transfer as introduced in sections 3.3 and 3.4. Energies (Z 1 , Z 2 ) are numerically integrated using a Runge Kutta scheme RK45 by using degenerated equations of motion (62).

6: scenario C 1 ≤ S 1 ; C 2 ≤ S 2 .
• Initial energy for first internal mode:

C 1 = 2.25 × 10 -4 m 2 ,
• Initial energy for second internal mode:

C 2 = 4 × 10 -6 m 2 .
In present configuration, C 2 is small enough so that TET 2 mechanism is inactive ; on the first hand, second internal mode is attenuated by its natural damping and on the other hand, TET 1 mechanism is enabled and a bifurcation jumped may be witnessed as depicted in Fig. 6. • Initial energy for first internal mode:

|ϕ 1 (0)| ≈ ẏ1(0) ω1 2 + y 1 (0) 2 ≈ 0.01564, (69)
• Initial energy for second internal mode:

|ϕ 3 (0)| ≈ ẏ2(0) ω2 2 + y 2 (0) 2 ≈ 0.008432. (70)
In order to validate previously mentioned analytical approach with numerical results, let's assume that the master system is without damping (λ 1 = 0 and λ 2 = 0). First of all |ψ 2 (0)| and |ψ 4 (0)| should be evaluated by solving the first equations of system (39) and (40) with initial energies taken as follows: In that case, it is noticeable in Fig. 7 that energy Z 2 decays fastly thanks to an active TET 2 mechanism that is triggered instantaneously 3 . In the meantime, energy Z 1 is increasing 4 . As soon as condition Z 2 ≤ S 2 is satisfied, TET 1 mechanism becomes active with a large energy jump.

|ψ 2 (0)| = 0.0067122 = N 1 (0) = C 1 , (71)
The energy variation of the master system related to two modes are evaluated analytically and numerically and presented in Fig. 8. Results show an obvious agreement between numerical and analytical forecasts.

Energy trends Z 1 and Z 2 computed numerically are therefore in good agreement with forecasts provided by scenarios explained in sections 3.3 and 3.4.

Results of energy transfer of the master system in the presence of the damping are introduced in Fig. 9.

Dynamic behavior in free vibration regime

Equations of motion (9) spanned onto the modal basis that involve quantities (y 1 , y 2 , z) are numerically integrated in a free vibration regime to investigate the reliability of proposed scenarios and to check the efficiency of targeted energy transfers.

Initial velocities given by

• y 1 (0) = 0.17 m/s, y 1 (0) = 0.0 m, 3 It is important to stress the difficulty in determining initial energies for internal modes for initial conditions are not set at time scale T 1 = 0 but after some laps of time T 0 such that condition T 0 → ∞ becomes valid. 4 Internal modes energies may increase at the beginning while energies related to the motion of the center of mass are decreasing. are chosen as initial conditions for the studied system.

A quick glance at Fig. 9.a) and 9.c) allows to claim that -apart a few peak oscillations remaining significantly not reduced at early times -the NES absorber efficiently mitigates time response of both structural modes 1 and 2 (blue curves) by comparison with their free response without any absorber link.

(red curves).

Scenario C 1 ≤ S 1 ; C 2 ≥ S 2 (Fig. 3) is clearly recognized when noticing the decaying behavior of modal energies E 1 and E 2 defined by

E 2 i = ẏ2 i ω 2 i + y 2 i with i = 1, 2
as shown in Fig. 9.b and 9.d. TET 2 mechanism is triggered at early time with a fastly decaying energy E 2 of mode 2 as witnessed in Fig. 9.d while at the same time, TET 1 is waiting for a delayed trigger and mode 1 dissipation is mainly due to its natural damping (see Fig. until times t ≈ 2 s as seen in Fig. 10.d. In Fig. 10.c, modal amplitude y 1 related to frequency f 1 = 1.73 Hz is continuously fed with energy until TET 2 mobilizing mode 2 vanishes then TET 1 mechanism is triggered and y 1 decay speed is increased after t ≈ 2 s. It is also noticeable in Fig. 10.b that frequencies mobilized by the NES response are both modal frequencies f 1 and f 2 . Amplitude z is first resonating at the vicinity of modal frequency f 2 at early times when TET 2 is active and then it is solely driven by frequency f 1 for later times as forecast by competitive targeted energy transfer scenario C 1 ≤ S 1 ; C 2 ≥ S 2 .

Dynamic behavior with a pulse-like excitation

A more complex transient regime is investigated in which both modes 1 and 2 initially at rest are excited with a gaussian pulse-like forcing vibrating respectively at the vicinity of frequencies f 1 = 1.73 Hz and f 2 = 4.53 Hz. Equations of motions are hence given by: 24

Conclusion

The transient dynamic of a two degree-of-freedom system linked to a nonlinear energy sink by means of a essential nonlinear stiffness has been investigated analytically by using an original asymptotic expansion based on Manevitch's complexification method and featuring two small independent scales. The analytical study naturally exhibited two activation energy thresholds for targeted energy transfer likely to develop.

Figure 1 :

 1 Figure 1: Two dofs + NES system.

  It is noticeable that equations (39) involve contributions depending mainly on time scale T 1 -nonlinear component -3 4 iKψ 2 ψ 4 2 excepted which also depends on time scale τ 1 -Similar result hold for equations (40) with respect to time scale τ 1 . Hence, competitive energy transfers between the first or the second mode and the NES device are ruled by the occurrence of nonlinear coupling featuring time scales T 1 , τ 1 . Next section aims to bring an insight in what is potentially going on by investigating several scenarios for energy transfer.

1 2

 1 is strongly depending on generalized modal masses and mode shape amplification factors at the attachment. Although very helpful to understand TET mechanisms, case study 1 2

Figure 2 :

 2 Figure 2: Scenario C 1 ≥ S 1 ; C 2 ≤ S 2 : TET 1 mechanism is solely active

step 5 : 6 :

 56 If energy Z 1 is sufficient, TET 1 mechanism is also triggered onto internal mode 1, and solution Z 1 jumps down instantaneously to a more stable energy level, step Neither conditions Z 1 ≥ S 1 nor Z 2 ≥ S 2 are satisfied. Both TET 1 and TET 2 mechanisms are disabled. Energies Z 1 and Z 2 continue to decay due to natural damping.

Figure 3 :

 3 Figure 3: Scenario C 1 ≥ S 1 ; C 2 ≥ S 2 : TET 1 and TET 2 mechanisms eventually active in sequence ; steps 1 to 6.

Figure 4 :

 4 Figure 4: left: A two storey building model coupled with a NES ; center: modal deflection φ 1 ; right: modal deflection φ 2 .

Figure 5 :

 5 Figure 5: Discrete skewer model.

scenario C 1 ≤ S 1 ; C 2 ≤ S 2 :

 1122 initial energies C 1 and C 2 are chosen such as:

Figure

  Figure 6: scenario C 1 ≤ S 1 ; C 2 ≤ S 2 .

scenario C 1 ≤ S 1 ; C 2 ≥ S 2 :

 1122 initial energies |ϕ 1 (0)| and |ϕ 3 (0)| are chosen such as:

and |ψ 4 2 Figure 7 :

 427 Figure 7: Evolution of the energy of relative modes.

Figure 8 :

 8 Figure 8: Comparison of energy transfer in the master system by using analytical and numerical methods

  9.b). As soon as modal energy E 2 pass downward its activation threshold S 2 at time t ≈ 2 s, TET 1 mechanism is to its turn triggered until t ≈ 17 s with a quasi-linear decaying trend. Beyond times t ≥ 17 s, both competitive energy pumping mechanisms TET 1 and TET 2 give up being active. A wavelet time-frequency analysis of modal y 1 , y 2 and NES z responses gathered in Fig 10.a is additionally achieved (see Figs. 10.b though 10.d) to track changes into the instantaneous frequency content of previous responses. Amplitude y 2 of modal frequency f 2 = 4.53 Hz soon decreases with an active TET 2 mechanism energy E2 versus time

Figure 9 := ẏ2 i ω 2 i + y 2 i

 922 Figure 9: Mitigation of free oscillation response of modal quantities(y i , E i ) i=1, 2 with E 2 i = ẏ2 i ω 2 i

  displacements y1, y2, z versus time b) scalogram of z c) scalogram of y1 d) scalogram of y2

Figure 10 :t c ≈ 5 s 2 • A 2

 10522 Figure 10: Free oscillation response of modal displacements y 1 , y 2 , z and their related wavelet time-frequency scalogram.

Figure 11 := ẏ2 i ω 2 i + y 2 i

 1122 Figure 11: Attenuation of modal quantities(y i , E i ) i=1, 2 with E 2 i = ẏ2 i ω 2 i

  displacements y1, y2, z versus time b) scalogram of z c) scalogram of y1 d) scalogram of y2

Figure 12 :

 12 Figure 12: Response of modal displacements y 1 , y 2 , z to a pulse-like excitation and their related wavelet time-frequency scalogram.

  Scenarios were also introduced to a priori forecast which and how competitive TET mechanisms are triggered knowing initial energies. An academic instance involving a two-storey building model was numerically simulated for comparison with analytical trends. Numerical results are in good agreement with expected analytical behaviors and thus validate TET scenarios. Futur prospect are now focusing on design methodologies to optimize energy pumping efficiency in the case of multiple TET mechanisms in competition.[START_REF] Gendelman | Quasiperiodic reponse regimes of strongly nonlinear vibration absorber[END_REF]

  allows to notice that the NES is physically attached to a body of mode 2 and far away from the node of mode 1 which practically means that it is able to resonate with both modes. NES mass is chosen such that

			m = 1	M 11 12 φ 2	= 2.7 tons	(66)
	corresponding to a mass ratio 1 =	1 80	and 2 =	φ 2 22 φ 2 12	M 11 M 22	1 =	1 31	.

NODY9745_source

In fact, C 1 may indeed be understood as a time delayed initial energy for ψ 2 = lim T 0 →∞ ϕ 2 which practically means after a certain amount of time T 0 has lasted.

Acknowledgment

This work has been supported by French National Research Agency under contract ANR-07-BLAN-0193.


Rachel ; Roberts
email: rachel.roberts@kcl.ac.uk

King

R E Roberts

C Glicksman

J Alaghband-Zadeh

R A Sherwood

N Akuji

C W Le Roux

Rachel E Roberts

Keywords: Bile acids, Peptide YY (PYY), Glucagon-like peptide Bile acids, peptide YY (PYY), glucagon-like peptide-1 (GLP-1), ghrelin

Background: Gut hormones peptide YY (PYY) and glucagon-like peptide-1 (GLP-1) play an integral role in appetite control and energy homeostasis. Entero-endocrine L-cells can be stimulated by nutrients and or bile acids to co-secrete PYY and GLP-1. The aim of this study was to determine the response of bile acids, PYY,GLP-1 and ghrelin after a test meal.

Design: Twelve subjects with a BMI of 22.8 (0.52) kg/m 2 (mean (SEM)) received a 400 kcal test meal after which blood samples were taken every 30 minutes from 0 to 180 minutes. PYY, GLP-1 and ghrelin were measured by radioimmunoassays. Fractionated bile acids were measured by HPLC-MSMS.

Results: PYY positively correlated with glycochenodeoxycholic acid (GCDCA) (rs = 0.23, p = 0.03) and taurochenodeoxycholic acid (TCDCA) (rs = 0.26, p = 0.02). GLP-1 positively correlated with GCDCA (rs = 0.22, p = 0.047) and glycodeoxycholic acid (GDCA) (rs = 0.3, p = 0.005). Ghrelin negatively correlated with GDCA (rs = -0.45, p = < 0.0001), TCDCA (rs = -0.23, p = 0.034) and taurodeoxycholic acid (TDCA) (rs = -0.44, p = < 0.0001).

Conclusion: PYY and GLP-1 responses correlated with chenodeoxycholic acid (CDCA) counterparts, whereas ghrelin negatively correlated with deoxycholic acid (DCA) counterparts. Specific bile acids may thus differentially affect entero-endocrine cells.

Introduction

Bile acids have several endocrine effects and are involved in three main signalling pathways; mitogenactivated protein kinase (MAPK) pathways [START_REF] Gupta | Down-regulation of cholesterol 7alpha-hydroxylase (CYP7A1) gene expression by bile acids in primary rat hepatocytes is mediated by the c-Jun N-terminal kinase pathway[END_REF][START_REF] Qiao | Bile acid regulation of C/EBPbeta, CREB, and c-Jun function, via the extracellular signal-regulated kinase and c-Jun NH2-terminal kinase pathways, modulates the apoptotic response of hepatocytes[END_REF]; via the G-protein coupled receptor TGR5 [START_REF] Maruyama | Identification of membrane-type receptor for bile acids (M-BAR)[END_REF][START_REF] Kawamata | A G protein-coupled receptor responsive to bile acids[END_REF], and via nuclear hormone receptors such as farnesoid X receptor α (FXRα) [START_REF] Parks | Bile acids: natural ligands for an orphan nuclear receptor[END_REF][START_REF] Makishima | Identification of a nuclear receptor for bile acids[END_REF][START_REF] Mangelsdorf | The nuclear receptor superfamily: the second decade[END_REF]. Administration of bile acids to mice increases energy expenditure in brown adipose tissue via induction of the cyclic-AMP-dependent thyroid hormone activating enzyme type 2 iodothyronine deiodinase (D2) [START_REF] Watanabe | Bile acids induce energy expenditure by promoting intracellular thyroid hormone activation[END_REF]. In vitro treatment of brown adipocytes and human skeletal myocytes with bile acids increases D2 activity and oxygen consumption [START_REF] Watanabe | Bile acids induce energy expenditure by promoting intracellular thyroid hormone activation[END_REF]. In addition bile acid stimulation of FXRα activates a signalling pathway culminating in increased energy expenditure and decreased adiposity [START_REF] Holt | Definition of a novel growth factor-dependent signal cascade for the suppression of bile acid biosynthesis[END_REF][START_REF] Inagaki | Fibroblast growth factor 15 functions as an enterohepatic signal to regulate bile acid homeostasis[END_REF][START_REF] Tomlinson | Transgenic mice expressing human fibroblast growth factor-19 display increased metabolic rate and decreased adiposity[END_REF]. In humans circulating bile acid concentrations correlate with measures of insulin sensitivity (12) and administration of bile acids has been found to prevent high fat diet-induced obesity and development of insulin resistance in mice [START_REF] Ikemoto | Cholate inhibits high-fat diet-induced hyperglycemia and obesity with acyl-CoA synthetase mRNA decrease[END_REF]. Modulation of bile acid homeostasis using bile acid sequestrants such as cholestyramine and colesevelam have been shown to improve glycaemic control in patients with type 2 diabetes mellitus [START_REF] Garg | Cholestyramine therapy for dyslipidemia in non-insulin-dependent diabetes mellitus. A short-term, double-blind, crossover trial[END_REF][START_REF] Goldberg | Efficacy and safety of colesevelam in patients with type 2 diabetes mellitus and inadequate glycemic control receiving insulin-based therapy[END_REF][START_REF] Fonseca | Colesevelam HCl improves glycemic control and reduces LDL cholesterol in patients with inadequately controlled type 2 diabetes on sulfonylurea-based therapy[END_REF]. The mechanisms by which bile acids improve glycaemic control are largely thought to be FXRα-independent and are instead mediated by binding to TGR5, leading to cAMP generation and activation of the intracellular type 2 thyroid hormone deiodinase [START_REF] Yamagata | Bile acids regulate gluconeogenic gene expression via small heterodimer partner-mediated repression of hepatocyte nuclear factor 4 and Foxo1[END_REF]. Bile acids also act via the phosphoinositide 3-kinase (PI3 kinase)/AKT pathway, directly promoting insulin signalling and glycogen synthase activation, thus aiding insulin-dependent control of glucose metabolism in the liver [START_REF] Han | Bile acids enhance the activity of the insulin receptor and glycogen synthase in primary rodent hepatocytes[END_REF]. In addition bile acid activation of TGR5 has been found to stimulate GLP-1 production in vitro, which promotes insulin secretion and thus improves glucose metabolism.

Bile acids constitute a large family of molecules composed of a steroid structure with four rings and a five or eight carbon side-chain terminating in a carboxylic acid. Each bile acid has a specific number and orientation of hydroxyl groups (-OH). The principal primary bile acids in humans are cholic acid (CA) and chenodeoxycholic acid (CDCA). Ursodeoxycholic acid (UDCA) is the 7β epimer of CDCA. Bile acids can be conjugated to either glycine or taurine, forming bile salts. Glycochenodeoxycholic acid (GCDCA) and taurochenodeoxycholic acid (TCDCA) are the glycine and taurine conjugates of CDCA, respectively and glycocholic acid (GCA) and taurocholic acid (TCA) are the glycine and taurine conjugates of CA, respectively.

Bile acids can also be conjugated in phase II metabolic reactions such as N-acetylglucosaminidation at the C7 position with a beta-hydroxyl group (19). Bacteria in the intestinal tract can convert primary bile acids into secondary bile acids by removing a hydroxyl group, for example CDCA becomes lithocholic acid (LA) and Gut hormones peptide YY (PYY) and glucagon-like peptide-1 (GLP-1) play an integral role in appetite control and energy homeostasis. PYY and GLP-1 are secreted by L-cells of the distal small intestine in response to nutrients and bile acids [START_REF] Anini | Comparison of the postprandial release of peptide YY and proglucagon-derived peptides in the rat[END_REF][START_REF] Ballantyne | Deoxycholate is an important releaser of peptide YY and enteroglucagon from the human colon[END_REF][START_REF] Katsuma | Bile acids promote glucagon-like peptide-1 secretion through TGR5 in a murine enteroendocrine cell line STC-1[END_REF]. Exaggerated responses of PYY and GLP-1 may contribute to the weight loss and improvement in glycaemic control following gastric bypass surgery for the treatment of obesity. The mechanisms for increased concentrations of PYY and GLP-1 after gastric bypass are not entirely understood.

The 'hindgut hypothesis', described by Mason (23), proposes that altered anatomy following bariatric surgery accelerates the delivery of nutrient-dense chyme to the distal intestine which stimulates increased gut hormones secretion. In addition plasma bile acid concentrations are doubled in individuals after gastric bypass surgery compared to obese individuals who have not had bariatric surgery [START_REF] Houten | Serum bile acids are higher in humans with prior gastric bypass: potential contribution to improved glucose and lipid metabolism[END_REF] and it is possible that co-delivery of these bile acids within the chyme enhances the stimulation of gut hormone release postgastric bypass.

A role of bile acids in appetite suppression was proposed as early as 1968 [START_REF] Bray | Suppression of appetite by bile acids[END_REF]. Several studies indicate that bile acids deoxycholic acid and taurocholic acid combined with fatty acids are more significant than fatty acids alone in stimulating PYY release in the ileum [START_REF] Ballantyne | Deoxycholate is an important releaser of peptide YY and enteroglucagon from the human colon[END_REF][START_REF] Soper | The 'ileal brake' after ileal pouch-anal anastomosis[END_REF][START_REF] Ballantyne | Deoxycholate-stimulated release of peptide YY from the isolated perfused rabbit left colon[END_REF][START_REF] Phillips | PYY and GLP-1 contribute to feedback inhibition from the canine ileum and colon[END_REF][START_REF] Aponte | Regional distribution and release of peptide YY with fatty acids of different chain length[END_REF]. Bile acid infusions of taurochenodeoxycholic acid and deoxycholic acid have been shown to increase plasma PYY in rabbits [START_REF] Ballantyne | Deoxycholate-stimulated release of peptide YY from the isolated perfused rabbit left colon[END_REF][START_REF] Krenz | Bile salt inhibition of motility in the isolated perfused rabbit terminal ileum[END_REF] and deoxycholic acid infusion in the human colon stimulates PYY release in a dose-dependent manner [START_REF] Ballantyne | Deoxycholate is an important releaser of peptide YY and enteroglucagon from the human colon[END_REF]. Importantly the concentrations of bile acids used in these studies are consistent with the physiological concentrations of bile acids found in the intestinal lumen post-prandially [START_REF] Dietschy | Mechanisms for the intestinal absorption of bile acids[END_REF], supporting a physiological role of bile acids in the stimulation of PYY release. However, not all bile acids have been found to stimulate PYY release with hyodeoxycholic acid and ursodeoxycholic acid having no effect on PYY release [START_REF] Ballantyne | Deoxycholate-stimulated release of peptide YY from the isolated perfused rabbit left colon[END_REF][START_REF] Plaisancié | Luminal peptide YY-releasing factors in the isolated vascularly perfused rat colon[END_REF]. The reason why PYY would be stimulated by specific bile acids is not clear. The post-prandial peak concentrations of GLP-1 and total bile acid concentrations inversely correlate with fasting and post-prandial glucose after gastric bypass [START_REF] Houten | Serum bile acids are higher in humans with prior gastric bypass: potential contribution to improved glucose and lipid metabolism[END_REF].

Ghrelin is secreted from X/A-like cells of the fundus of the stomach [START_REF] Cummings | Plasma ghrelin levels after diet-induced weight loss or gastric bypass surgery[END_REF][START_REF] Date | Ghrelin, a novel growth-hormone-releasing acylated peptide, is synthesized in a distinct endocrine cell type in the gastrointestinal tracts of rats and humans[END_REF] and may play a role in regulating pre-meal hunger, meal initiation and long-term energy balance [START_REF] Cummings | A preprandial rise in plasma ghrelin levels suggests a role in meal initiation in humans[END_REF][START_REF] Cummings | Roles for ghrelin in the regulation of appetite and body weight[END_REF]. Circulating ghrelin concentrations increase during fasting and decrease after eating (37) while anticipation of meals may contribute to its secretion [START_REF] Drazen | Effects of a fixed meal pattern on ghrelin secretion: evidence for a learned response independent of nutrient status[END_REF]. The mechanisms by which ghrelin promotes hunger and increased energy intake are not entirely clear. There is some evidence to suggest that ghrelin may exert its effects by acting on the central nervous system [START_REF] Theander-Carrillo | Ghrelin action in the brain controls adipocyte metabolism[END_REF]. Obese subjects have lower fasting plasma ghrelin concentrations than lean controls [START_REF] Tschöp | Circulating ghrelin levels are decreased in human obesity[END_REF], although appetite is commonly increased, suggesting that higher body mass is associated with increased gastric responsiveness to ghrelin [START_REF] Cremonini | Obesity does not increase effects of synthetic ghrelin on human gastric motor functions[END_REF]. A relationship between ghrelin and bile acids has not been described.

The aim of this study was to investigate the associations between the responses of bile acids, PYY, GLP-1 and ghrelin after a standard 400 kcal test meal.

Methods

All studies were performed according to the principles of the Declaration of Helsinki. The Local Research and Ethics Committee at King's College Hospital approved the study (05/Q0703/4). Written informed consent was obtained.

Subjects and Sample Collection

We studied 12 healthy male subjects (age 21.2 (0.46) years, BMI 22.8 (0.52) kg/m 2 (mean (SEM)).

Subjects fasted overnight from 23:00 before ingesting a 400 kcal test meal (46.6g carbohydrate, 28.5 g fat and 10.4 g protein) at 09:00. EDTA blood samples were taken every 30 minutes from 0 to 180 minutes.

Bile acid and hormone measurement

Fractionated bile acids were measured using HPLC-MSMS as previously described by Tagliacozzi et al [START_REF] Tagliacozzi | Quantitative analysis of bile acids in human plasma by liquid chromatography-electrospray tandem mass spectrometry: a simple and rapid one-step method[END_REF]. Bile acids analysed were chenodeoxycholic acid (CDCA), deoxycholic acid (DCA), cholic acid (CA), ursodeoxycholic acid (UDCA), lithocholic acid (LA) and their respective glycine (G-) and taurine (T-)

conjugates. The method was linear between 0.1-10 µmol/L for all bile acids and their conjugates with intraassay coefficient of variations (CV) ranging from 3.6 -8.0 % and inter-assay CVs ranging from 1.5 -6.8 %.

The lower limit of quantitation was 0.1 µmol/L. LCA and UDCA and their conjugates and CDCA were below the detection limit. Glucose was measured using the glucose oxidase method on the Advia 2400 analyser (Siemens Healthcare Diagnostics, Frimley, UK). Insulin was measured using a two site sandwich chemiluminescent immunoassay on the Advia Centaur analyser (Siemens Healthcare). The intra-assay CV was 3.3% and inter-assay CV was 4.8%. PYY was measured by an established in-house radioimmunassay as previously described by Adrian et al [START_REF] Ferri | Human distribution and release of a putative new gut hormone, peptide YY[END_REF]. The assay measured the biologically active components, both the full length (PYY 1-36 ) and the fragment (PYY 3-36 ). The detection limit was 5 pmol/L and the intra-and interassay CVs were 5.8% and 9.8%, respectively. GLP-1 was measured by an established in-house radioimmunoassay as previously described by Kreymann et al [START_REF] Kreymann | Glucagon-like peptide-1 7-36: a physiological incretin in man[END_REF]. The detection limit was 7.5 pmol/L and the intra-assay CV was 6.1%. Ghrelin was measured by the commercially available Phoenix Pharmaceutical [START_REF] Patterson | Characterization of ghrelin-like immunoreactivity in human plasma[END_REF]. The assay measured total human ghrelin (acylated and des-acylated) and the intra-assay CV was 8.8%

Statistical Analysis

Results are reported as mean (SEM) and correlation co-efficient (rs). Statistical analysis was performed using Analyse-It for Excel (Analyse-It, Leeds, UK). Data were normally distributed and a Spearman correlation was used. Significance is shown by a P < 0.05.

Results

None of the unconjugated bile acids changed significantly throughout the study period but all glycine and taurine conjugated bile acids and total bile acids increased significantly (Table . 1.). The predominant bile acid measured was GCDCA, followed by GCA and GDCA. All other bile acids had peak concentrations of < 1.00 µmol/L. Table.

Discussion

The CDCA group of bile acids GCDCA and TCDCA correlated with PYY. GCDCA also correlated with GLP- The DCA group of bile acids GDCA, TDCA and TCDCA correlated negatively with ghrelin. DCA counterparts are hydroxylated at both C7 and C12. Bile acids may contribute through an endocrine mechanism to the inhibition of ghrelin secretion from P/D1 cells of the fundus of the stomach post-prandially. Ghrelin and cholecystokinin (CCK) are negatively correlated [START_REF] Date | Peripheral interaction of ghrelin with cholecystokinin on feeding regulation[END_REF]. CCK is a hormone released from the I-cells of the small intestine in response to lipid ingestion which stimulates the release of bile from gallbladder, and also acts as a satiety signal. It is possible that increased CCK levels post-prandially stimulate increased bile acid secretion which may in turn inhibit ghrelin synthesis through neural or endocrine mechanisms.

Limitations of our study included the homogenous nature of our volunteers and therefore the data can not be extrapolated to females or older people. The detection limit of the assay was insufficient for determination of a correlation with UDCA, its conjugates and CDCA. The number of subjects were too small to draw firm conclusions, but the data suggested that the total bile acid and GCDCA response may have been biphasic in

  CA becomes deoxycholic acid (DCA) (Figure.1.).

  previously described by Patterson et al

  2. and Figure. 2. show the changes in plasma hormones and glucose before and after the standardised 400 kcal meal. Correlations were sought between individual types of bile acids, PYY, GLP-1 and ghrelin. Bile acids which significantly correlated with PYY were GCDCA (rs = 0.23, p = 0.03), total glycine conjugated bile acids (rs = 0.23, p = 0.03), TCDCA (rs = 0.26, p = 0.02) (Figure. 3.), total taurine conjugated bile acids (rs = 0.24, p = 0.03) and total bile acids (rs = 0.26, p = 0.02). Bile acids which significantly correlated with GLP-1 were GCDCA (rs = 0.22, p = 0.047), GDCA (rs = 0.3, p = 0.005), total glycine conjugated bile acids (rs = 0.26, p = 0.02) and total bile acids (rs = 0.26, p = 0.02) (Figure. 3.). Bile acids which significantly correlated with ghrelin were GDCA (rs = -0.45, p = < 0.0001) (Figure. 3.), TCDCA (rs = -0.23, p = 0.03), TDCA (rs = -0.44, p = < 0.0001), total taurine conjugated bile acids (rs = -0.23, p = 0.038) and total bile acids (rs = -0.22, p = 0.044).

1 .

 1 GCDCA and TCDCA share a common chemical structure with -OH hydroxylation at C7 and a single hydrogen ion at C12 (Figure.1.). The positive correlation found between GCDCA and PYY and GLP-1 and TCDCA and PYY suggests that the specific chemical structure of bile acids may play a role in stimulating Lcells. UDCA and its conjugates and CDCA also share this chemical structure but concentrations were below of our assay and therefore we are unable to comment whether a similar correlation is possible with PYY and GLP-1.

9

  of 12 subjects (Figure.4.). It is possible that with 30 minute sampling the peaks and troughs may have been missed in all subjects. The biphasic response of bile acids in some of our subjects may also have accounted for the low rs values of our correlations.ConclusionThis study suggests that post-prandial conjugated bile acids correlate with appetite regulating gut peptides PYY, GLP-1 and ghrelin. The post-prandial responses of the CDCA counterparts primarily correlated with PYY and GLP-1 responses, whereas the post-prandial DCA counterparts correlated with changes in ghrelin. Specific bile acids may play a part in entero-endocrine cell stimulation or inhibition. Targeted experiments, specifically with bile acids hydroxylated at C7 may now be considered to determine the therapeutic value of these bile acids in reducing weight through stimulation of PYY and GLP-1 and inhibition of ghrelin.

Figure. 2 . 13 Figure. 3 . 15 Figure. 4 .

 2133154 Figure. 2. Changes in plasma levels of gut hormones following a 400 kcal test meal in 12 subjects. Expressed as mean (SEM). a) PYY, b) GLP-1 and c) ghrelin a)

Table . 1

 . . Baseline and peak values (mean (SEM)) of glycine and taurine conjugated bile acids in plasma following a 400 kcal test meal.

	Table. 2. Baseline and peak values (mean (SEM)) of plasma hormones and glucose following a 400 kcal test
	meal.
	Bile Acid	Baseline (µmol/L) Baseline	Peak (µmol/L) Peak	P value P value
	GCDCA PYY (pmol/L)	0.38 (0.05) 21.3 (1.98)	3.82 (0.39) 34.9 (2.54)	< 0.0001 < 0.001
	GDCA GLP-1 (pmol/L)	0.11 (0.02) 52.1 (4.63)	1.13 (0.22) 72.5 (4.54)	0.0006 < 0.001
	GCA Ghrelin (pmol/L)	0.15 (0.02) 1026 (133.5)	1.45 (0.19) 550 (75.3)	< 0.0001 < 0.001
	Total glycine Glucose (mmol/L)	0.77 (0.08) 4.6 (0.08)		6.97 (0.77) 5.6 (0.10)	< 0.0001 < 0.001
	conjugated bile acids Insulin (mU/L)	5.7 (1.57)		48.8 (6.42)	< 0.001
	TCDCA	0.03 (0.01)		0.34 (0.14)	0.048
	TDCA TCA Total taurine R1	0.02 (0.01) 0.01 (0.01) 0.16 (0.09) F o r F o r F o r	0.16 (0.04) 0.14 (0.05) 0.66 (0.21) R2	0.007 0.022 0.012
	conjugated bile acids Total bile acids Bile acid CA CDCA DCA	1.41 (0.14) R1 (C3) OH OH OH	8.33 (0.85) e P e r P e e r R2 (C7) OH P OH (α) e H e r	< 0.0001 R3 (C12) OH H OH
	LCA UDCA	OH OH		R R H OH (β) R	H H
				e e e
	v i e v i e X unconjugated -OH, X tauro-conjugated -NHCH 2 CH 2 CH 2 SO 3 H, X glyco-conjugated -NHCH 2 COOH, CA -Legend: v i e
	w w cholic acid, CDCA -chenodeoxycholic acid, DCA -deoxycholic acid, LCA -lithocholic acid, UDCA -ursodeoxycholic acid w
					9 11

10

Figure. 1. The structure of bile acids.

Clinical Endocrinology

The authors have nothing to declare.


Eric Blond

Gasser Alain

Matthieu Landreau

Daniel Jean-Luc

Nguyen Thi My Hanh

Modelling of joint effect in refractory structures

Keywords: masonry modelling, homogenization, refractory, joints

Some refractory linings of metallurgical vessels consist of masonry. To model these large-sized structures, it is convenient to use an equivalent material instead of a model that involves all bricks and joints. The properties of the equivalent material depend on geometrical and mechanical properties of the bricks, the joint states and the temperature. In this paper, several joint states which are the combinations of open/closed states of bed and head joints are considered. The equivalent material properties are determined thanks to periodic homogenization techniques. Results obtained with this approach are used to model different masonries walls, with or without mortar, subjected to bi-axial compression and bending. Finally, an application to a coke oven heatingwall is presented.

Introduction

Masonry structures are researched extensively owing to their wide application in many fields such as civil engineering or cultural heritage restoration. To take into account joint effects in such large structures, two different approaches are possible [1]: the micro-modelling which describes each brick and joint with their own behaviour, the macro-modelling which pleads for replacing them with a homogeneous equivalent material. In micro-modelling approach, there are two subcategories: the simplified approach where only units and joints are represented and the detailed approach where units, joints and the unit/mortar interface are included.

The micro-modelling strategy is the most accurate but its computational cost is very onerous because of the necessity to model separately units and mortar. Then, to reduce the size of the numerical model, the mortar is classically reduced to an interface element (i.e. thickness of mortar close to zero and no real shearing in the sense of classical Cauchy media), as initiated by Page [2]. Despite of these assumptions this method will be a long-term task if the bricks which compose the structures don't have all the same shape, size, and architectural organization. Moreover, for mortarless walls, other factors should be additionally accounted for such as sliding between bricks, which cannot help the convergence for the computation by the classical Finite Element Method (FEM). Over the recent years, authors like Rafiee [3] manage to model large structures (for example: an arena and an aqueduct) using a Discrete Element Method (DEM) with stone blocks considered as rigid elements. It is really an interesting and promising numerical method. The actual recent developments focus on the coupling between DEM and the FEM to account for deformable parts, today limited to elastic behaviour. However, this method considers only bricks with simple shape and not a tongue-and-groove ones. Moreover, interaction between masonry and material with complex behaviour like a ramming-mix [START_REF] Brulin | UNited International TEchnical Congress on Refractory[END_REF] is not actually possible in such computational codes.

In the macro-modelling alternative approach, all the masonry is represented by an equivalent continuum media. Thus, techniques of homogenization are becoming more and more popular among the masonry community. To account for many effects such as temperature dependence of the constitutive materials behaviour or complex 3D brick shape and architectural arrangement, numerical approaches are better adapted than analytical ones. However, analytical expressions proposed by Cecchi [START_REF] Cecchi | [END_REF]6] give a good estimation of the effective parameters for simple 2D shape and arrangement. From a numerical point of view, an alternative to the classical FEM approach, based on Cauchy's assumption, is to use enriched kinematics of the Cosserat continuum [7]. Unfortunately, the complexity of the Cosserat's kinematics is a great handicap for the development of new numerical codes. Then, it does not seem to be as promising as the DEM. Finally, the large majority of the literature focuses on the periodic homogenization of masonry by FEM as initially proposed by Anthoine [8]. The main difference between the numerous recent works concerns the definition of the joints behaviour. The dominant idea is that the non-linear behaviour of mortar (or contact for mortarless masonry) is the key point of the structure behaviour. However, in the work presented hereafter, the joint is purely elastic and can only be closed (perfect joint) or open (broken joint) without any other refinement. Then, the constitutive equation of equivalent material is linear elastic with different joint states and, as it was already demonstrated in the pioneering work of Luciano [9], it seems to be enough to reproduce the whole behaviour of the structure.

The presentation proposed here is a summary of the work initiated by Gasser and Boisse [10] on the modelling of refractory masonry by homogenisation. The objective is to derive a homogeneous material whose mechanical characteristics depend on the joint state, on the basis of the joint opening/closure mechanism. For each joint state, the equivalent mechanical properties are determined by homogenization using an energy approach and the finite element method. The transition criterion between joint states allows to make the difference between masonry with or without mortar. The unilateral contact conditions written in terms of macroscopic strain give the closure criterion for mortarless masonry while a local tensile cut-off combined with a local Mohr-Coulomb strength criterion defines the breakage / degradation (i.e., opening) of the mortar for "classical" masonry.

In the first part of the paper, the philosophy which drives the build of the different reference joint states is presented. In a second part, the necessary theoretical developments of periodical homogenization are quickly reminded. Then, the proposed method is used to simulate a bi-axial compression test of mortarless masonry and a bending test of masonry with mortar. Finally, an example of the simulation of an industrial refractory structure is presented: the case of a heating wall in coke oven battery.

Joint state definition

Whatever is the mechanical behaviour of bricks, joints are often the weak link of the masonry, mainly because of the low strength of the brick / mortar interface. Obviously, in service, a diffused damage in mortar, which results to its bulk breakage, is seldom observed. But, at the scale of the structure, it is not so different than an interface fracture because the mortar is thin regard to the brick size. Then, it is usually assumed (and often observed) that fractures develop only in the brick/mortar interface [9]. With this assumption, it becomes possible to simplify the representation of the joint damage by considering only two possibilities: the joint is closed until an ultimate strength is not reached and then the joint is open.

Masonry is a composite medium characterized by a periodic microstructure. Consequently, a repetitive cell, as presented in figure 1a, is employed to proceed to the evaluation of the effective parameters of the equivalent material (i.e. periodic homogenization). According to opening or closure of bed and head joints, different states for the elementary cell can be defined. Selecting a unit cell symmetrical with respect of three orthogonal planes passing through the origin, the problem can be reduced to an eighth of unit cell. Then, the number of possible states is reduced to symmetric joints opening/closure combination, as proposed on figure 1b. Of course, for each state, different effective parameters must be evaluated. Once different patterns were defined, a transition criterion is established to go from one state to another. Here, it is possible to adapt the criterion for the two different cases of masonry with or without mortar. For mortarless masonry, the bricks are laid in contact one another via soft interfaces (due to their asperities and shape defaults), then initially the joints are opened and the closure criterion should be linked to the local relative displacement of bricks. Conversely, in the case of masonry with mortar, joints are initially closed and the opening criterion should be linked to the local stress field. Thus, the difference between the two types of masonry is resumed to a change in the joint closure/opening criterion.

Homogenization of masonry

Because the distribution of joints is different in the three directions, the equivalent material is assumed orthotropic. The behaviour of joints and bricks is assumed elastic with temperature dependent Young modulus, and so the equivalent material too. Since masonry arrangement is periodical, a periodical homogenization combined with an energy approach is well adapted. In order to evaluate the effective parameters, the strain energy bulk density is computed for the heterogeneous cell through finite element software (ABAQUS) and compared to the strain energy bulk density of the equivalent material submitted to the same load. Boundary conditions which much be applied on the cell are defined as regard to the expression of bulk density of strain energy. For example, in the case of a two dimensional problem, the shearing modulus G 12 of the equivalent material is given by:

2 12 12 2 E W G = (1)
Where W is the bulk density of strain energy in the heterogeneous cell computed by F.E.M. and E 12 the homogeneous macroscopic shearing strain field. Then the strain tensor can be written:

( ) 12
1 2 2 1 E e e e e E ⊗ + ⊗ = (2)
Where e i are the base vectors and E 12 the shearing component of the strain tensor. The local displacement U is the sum of an average field, equal to the macroscopic one, and a local perturbation which must respect the periodicity of the problem. Then, the local displacement can be written as:

( ) y U y E y U per + = . ) ( ( 3
)
Where y is the position vector and U per the local periodic fluctuation vector. Let us note per i u the component of the vector of the periodic fluctuation, then:

⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ + + = per per per u u y E u y E U 3 2 1 12 1 2 12 (4)
Symmetry conditions with respect to the plane lead to: (), gives the boundary conditions corresponding to a perfect shearing: The reader interested in details on the definition of boundary conditions for different loads should report to [8] and [11]. To identify automatically the effective parameters for many different equivalent materials, corresponding to many different cells (i.e. different architectures and/or different joint states) at different temperatures, the F.E.M. Code ABAQUS has been coupled with a Fortran routine.

⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎨ ⎧ = = = = = 0 ) , , ( 0 ) , , ( 0 ) , , ( 0 ) , , ( 0 ) , , ( 3 1 3
⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎨ ⎧ = = = = = 0 ) , 0 , ( 0 ) , 0 , ( 0 ) 0 , , ( 0 ) , , 0 ( 0 )

Bi-axial compression of a mortarless masonry

Figure 2 presents a typical result of compression of two bricks of same material laid in direct contact with each others. The load displacement curve is non-linear but could be reasonably approximated by two straight lines. The first, in blue, corresponds to the crushing of the shape default and roughness. The second, in red, corresponds to the brick behaviour. The joint will be called "opened" until the global rigidity be lower than the brick's one. Then, the intersection of the two lines allows to define the initial size of the open joint. In mortarless masonry, there are two main reasons responsible of joint closures, first the deformation of the brick and second their sliding. Then, the first criterion for the joint closure is based on the initial joint thickness g n between the surfaces which are potentially in contact:

( ) (1)
( 2) .

n u u n g - = (7)
where u (1) and u (2) are the displacements of the nearest points of the two surfaces and n is the normal vector to the two surfaces. This local criterion, to use in the homogeneous equivalent material, must be written in function of global strain. Then, accounting that the effective behaviour of the equivalent continuous media is piece-wise linear, it is possible to write [11]:

[ ] n z k k n k k jp njp k g T E m = Δ - Δ ∑ = - - 1 2 1 2 1 θ (8)
Where z is the number of joint states changes in the cell and The second criterion, accounting for the brick sliding possibilities, is based on the Coulomb friction law. In the same way than for the displacement, the local inequality between the ratio of tangential to normal loads and friction coefficient could be expressed in term of global strain:

( ) ( ) ∫ ∑ ∫ ∑ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ Δ + Δ ≤ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ Δ + Δ = = S z k k ii k pq k iipq k s S z k k ij k pq k ijpq k dS T P E C f dS T P E C 1 1 (9)
Where k C and k P are the tensors which link the macroscopic strain increment and temperature increment to the local stress increment and S is the surface contact. Finally, to take into account the possibility of the re-opening of joints during unloading, a criterion based on the impossibility to transfer tensile stress (i.e., cap in local tension) is added:

( )0 1 ≥ Δ + Δ ∑ = z k k ii k pq k iipq k T P E C ( 10
)
This re-opening criterion does not need any more parameters than those previously defined. To validate the proposed model, results of the simulation of a biaxial compression of the equivalent material was compared with those issued from a biaxial compression test on a real mortarless masonry performed by the center of research of RHI (Leoben). The wall laid on the ground and is bi-axially loaded by two pistons via ceramic layers. Displacements at four points, corresponding to the positions of the sensors, are recorded (Fig. 3). The wall has a square form with each side of 1100 mm and thickness of 185 mm. All details of the experimental set up are presented in [12]. The test was realized at constant room temperature. The results of the first comparisons show the excellent ability of the model to reproduce the results obtained during the loading stage if the initial joint thickness is well known [11,[START_REF] Nguyen | UNited International TEchnical Congress on Refractory[END_REF]. In a second time, assuming an average initial joint thickness of 0.1 mm and using the same effective parameters than in [11], the cyclic loading and unloading have been simulated. Figure 4 presents the comparison between the computation results and the experimental data from the cyclic biaxial compression test performed at RHI. Figure 4 shows that the model well reproduces the high anisotropy of the behaviour of the masonry. Moreover, the shapes of the first load are close for the model and the test. For the first unloading, the model captures the global shape but follows not exactly the experimental curve. Despite this discrepancy, the model seems able to reproduce the hysteresis of the cyclic loading curve and the evolution of the maximum displacement along axis 2 with each consecutive load. In conclusion, the proposed model reproduces well the whole behaviour of mortarless masonry but could be improved to better reproduce the reopening of joints during unloading stage.

Bending of masonry with mortar

In this study and as presented previously, it is assumed that fractures might develop only in the brick/mortar interface due to the small thickness of the mortar [1,9]. The damage or breaking of mortar interface is due to local shearing or tensile strength, so the criterion for joint opening is the combination of a cut-off criterion for tension and a modified Mohr-Coulomb for shear:

⎪ ⎩ ⎪ ⎨ ⎧ > + > shear ki ji tension ii f or f 2 2 σ σ σ (11)
Where f shear and f tension are the tensile and shear strength of brick-mortar interface. A modification in Mohr-Coulomb criterion allows taking into account hardening of shear limit strength only in compression:

ii ii shear sign c f σ σ )) ( 1 ( tan 2 1 - Φ - = (12)
With c the unit-interface cohesion, and Φ the internal friction angle. Using the same way and the same notation than for mortarless masonry, the criterion is written in term of macroscopic strain:

⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎨ ⎧ > ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Δ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Δ > Δ ∑ ∑ ∑ = = = shear lm k n k hilm k lm k n k jilm k t lm k n k iilm k f E C E C or f E C 2 1 2 1 1 (13)
And the modified Mohr-Coulomb limits can be written:

lm k n k iilm k lm k n k iilm k cis E C E C c f Δ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Δ - - = ∑ ∑ = = 1 1 sgn 1 tan 2 1 φ (14)
In [START_REF] Landreau | UNited International TEchnical Congress on Refractory[END_REF] this model was used to compute a shearing wall test. The comparisons between test and simulation is in good agreement. Hereafter, the "deep beam test" test of [2] is considered. The beam is 757 × 457 mm² wide. As represented on figure 5a, it is supported at each side over a support length equal to 188 mm. The top load P is applied through a stiff steel beam. Although the brick size is not the same than for the shearing test, the effective parameters used herein are the same than in [START_REF] Landreau | UNited International TEchnical Congress on Refractory[END_REF] because the mortar / brick ratio is similar. Figure 5b presents the stress deduced form strain gauges pasted on the bricks (see fig. 5a) and stress calculated from the model. Results are in fairly good agreement. [2] These results show that replacing the masonry by a homogeneous effective media with effective elastic parameters depending on the local cell joint states allows to predict reasonably the behaviour of masonry structures. Then, it becomes possible to use such approach to study the design of refractory structures.

Coke oven heating wall modelling

Several studies summed up in [START_REF] Dürselen | [END_REF]16], were already performed on heating walls. Analytical works were very simplified since the heating wall was modelled as a beam. An improvement was brought by numerical methods. Nevertheless, all existing models present some simplifications: masonry joints are not taken into account; the heating wall is represented by an orthotropic shell or in 2D. Thus complete models, which take into account all phenomena, do not exist until now. To build this model, all parameters have been taken into account:

• Complexity of the structure (figure 6a): heating walls include flues and are made of masonries with various types of bricks (with various shapes and materials). • Brick and joint material behaviours: refractory thermo-mechanical behaviour is temperature dependent • Knowledge of thermo-mechanical loading: temperature field, weight of walls, roof and larry car, pre-stresses (anchoring system), lateral pressure due to coal pushing In order to keep the 3D effective properties of the equivalent material of heating wall, two elementary cells were chosen: the first one for the chamber wall and the second one for the binder wall as shown on figure 6b. To accurately identify the effective thermomechanical properties, mechanical and thermal tests were carried out on both silica brick and mortar samples [START_REF] Landreau | UNited International TEchnical Congress on Refractory[END_REF]17].

As presented before, unit cell mechanical behaviour is function of the joint states and, here, is also function of the temperature. Thus, equivalent behaviours were determined for each joint state at three temperatures: 800, 1080 and 1350°C. Some numerical values of effective mechanical parameters are presented in [START_REF] Landreau | UNited International TEchnical Congress on Refractory[END_REF] and all are summarized in [17]. Equivalent thermal parameters were determined for different states too. The effective thermal conductivity was bounded by the Hashin-Shtrikman (HS) bounds [18]. The expansion coefficients were identified by the same method than mechanical parameters and the density and specific heat were estimated by an average per volume unit.

The calculation of this large structure was realised in two steps. The first one is the modelling of thermal loads in order to obtain the thermal field in the heating wall. All thermal exchanges have been taken into account: exchanges between the chamber wall and the coal and those between the protection plate and insulating bricks in the coke oven heads and the heating of flues. Results of the thermal computation are presented in [START_REF] Landreau | UNited International TEchnical Congress on Refractory[END_REF]. The temperature field was validated thanks to coke oven heating wall instrumentation. In fact different thermocouples were put at different positions in the structure and the temperature in flues were measured by the CPM with Pyrofil©.

In the second step, the previously obtained temperature field is used in the thermomechanical model as an imposed temperature field. In this mechanical step, all loads previously quoted (i.e., weights of roof and larry car, anchoring system, lateral pressure) are applied on the structure. Due to the discrete damage evolution (i.e., closed joint toward open one) an explicit integration scheme is used. Despite of the large reduction of the size of the model induced by the use of equivalent material, the computation is time consuming and requires important computers resource: around two weeks on 24 CPUs with 20 Gb of RAM. First results are presented on figure 7. Localization of vertical joints openings presented on figure 7 is in fairly good agreement with observations on site. Indeed, vertical fractures, localized near the binder, are often observed on the chamber wall of heating wall. Moreover, as assumed by previous models [START_REF] Dürselen | [END_REF]16], the centre of the heating wall is the most damaged zone. These results confirm that the replacement of the masonry by an equivalent homogeneous material allows to reproduce a realistic degradation of a complex industrial structures. This ongoing work focuses now on the estimation of the limit for coal pushing.

Conclusion

In this paper the philosophy of the masonry model developed by Gasser et al. has been presented and some theoretical aspects were briefly summarized. Then, to demonstrate the capability of this model to capture the behaviour of different type of masonries, two different tests were simulated: bi-axial compression of mortarless masonry and a bending of masonry with mortar. The simulation results are in good agreement with experimental data.

The proposed model is based on the simple hypothesis on the constituant material: elasticity for mortar and brick and discrete damage for mortar, joints are open or closed. This simplification reduces the number of parameters to be identified but is capable to capture the complex behaviour of industrial structures. Moreover, the results have shown that the main part of the non linearity of the masonry behaviour is well reproduced. Nevertheless the model could be refined to better reproduce some particular points like the opening of joints during unloading for example.

  of a periodic cell b) Joint states definition at the cell scale Figure 1: Periodic cell and associated joint state

  h and L are the depth, the height and the length of the eighth of the cell. Substituting the periodic fluctuation in equation

Figure 2 :

 2 Figure 2: Typical result of compression test of two bricks laid in direct contact

θ

  the change of global strain tensor and the change of temperature during state number k. are the components of the tensor which link the global strain and temperature change to the local one in state k. The superscript (1-2) denotes the solids which are concerned. Moreover, the Einstein convention is used for summation when twice subscripts are repeated .

Figure 3 :

 3 Figure 3: Installed test lining segment

Figure 4 :

 4 Figure 4 : Cyclic biaxial compression, comparison between simulation and test

  . a) Test architecture b) Stress along courses, simulation vs experimental data Figure 5: Deep beam test, simulation vs. experiment

Figure 6 :

 6 Figure 6: Heating wall architecture and elementary cells.

  a) Coke oven face b) flue faces (inner of the heating wall)

Figure 7 :

 7 Figure 7: vertical joint opening on the liner (a) view from oven and (b) view from flue