Clément Jarroux
email: clement.jarroux@gmail.com

Jarir Mahfoud

Régis Dufour

Franck Legrand

Benjamin Defoy

Thomas Alban

Investigations on the dynamic behaviour of an on-board rotor-AMB system with touchdown bearing contacts: modelling and experimentation

Keywords: Rotordynamics, Active magnetic bearing, On-board rotor, Touchdown bearing, Rotor-stator interaction

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Global energy consumption constantly grows and turbomachinery have a crucial role in energy conversion. Depending on the desired applications, the external environment, such as swell, cobblestone road, or earthquake may be critical to their dynamic behaviour, aecting their operability and maintainability. In order to guaranty such performances, the research investigations of the base motion eects on rotordynamics 5 have begun since mid 1980s.

Earthquake may be approximated by non-stationary random process such as the Monte Carlo method and the eect of this kind of excitations on rotordynamics was studied in references [START_REF] El-Saeidy | Dynamics of a rigid rotor linear/nonlinear bearings system subject to rotating unbalance and base excitations[END_REF]. Even if the shaftbearing system natural frequencies are generally higher than the representative frequency of earthquake, the rotordynamics may become unstable when supported by oil-lm bearings. Lee et al. [START_REF] Lee | A nite element transient response analysis method of a rotor-bearing system to base shock excitations using the state-space newmark scheme and comparisons with experiments[END_REF] studied, numerically 10 and experimentally, the eects of vertical shock on a rotor-bearing system and showed that amplications of the rotor response are to be expected when the shock impulse is close to a system natural frequency.

Support rotations yielding parametric excitations may generate complex rotordynamics. At some specic support angular frequency, combined with the rotor-bearing system natural frequencies, instability zones are exhibited and their surfaces depend on the angle amplitude, as shown in [610]. The use of magnetic forces, through actuators, may overcome these instabilities by making the system stier using a PD feedforward loop, as demonstrated by Das et al. [START_REF] Das | Active vibration control of unbalanced exible rotor-shaft systems parametrically excited due to base motion[END_REF], or using various control strategies as shown by Soni et al. [START_REF] Soni | Active vibration control of ship mounted exible rotor-shaft-bearing system during seakeeping[END_REF].

The combined eects of support and shaft rotations were analysed previously by [START_REF] Driot | Theoretical and experimental analysis of a base-excited rotor[END_REF][START_REF] Briend | Eect of multi-frequency parametric excitations on the dynamics of on-board rotor-bearing systems[END_REF]1315]. The rotor orbits may be xed and perfectly periodic or quasi-periodic with more complex shapes depending on the ratio of the support angular frequency over the shaft speed of rotation. In case of high loads, the analysis of base motions combined with nonlinear bearing restoring forces exhibits the specic features of nonlinear dynamics, such as periodic, quasi-periodic or even chaotic regimes.

On the other hand, the use of active magnetic bearings (AMBs) is rising in industrial applications due to reduced costs but also to the improvement of electronic hardware performance and reliability. In addition to being an almost frictionless solution without lubrication device, it is convenient for high rotor speeds and a reduced system installation footprint. Being inherently unstable, the main diculty is to develop the proper controller managing AMBs, see [START_REF] Schweitzer | Magnetic Bearings, Theory, Design, and Application to Rotating Machinery[END_REF]. A wide variety of control strategies exist and most often, the conventional proportional-integral-derivative (PID) controller is employed in industrial applications. The use of touchdown bearings (TDBs) is systematic in rotor-AMB systems. These components are crucial to ensure safety in non-nominal operating conditions. They provide a support during transportation and a mechanical protection for AMBs against too large rotor deection amplitudes due to aerodynamic overload or base motions. In case of magnetic levitation loss, the rotor is forced by gravity to drop onto its touchdown bearings, ensuring a back-up support. This transient phenomenon has been intensively analysed in the literature, see references [1726] for more details. TDBs are generally composed of cageless ball bearings, comprising one or two rows, limiting the emergence of dangerous rotor behaviours induced by sliding friction. In most industrial designs, TDBs are tted into the stator through compliant mountings. Ribbon damper technology has been widely adopted in modern designs. It is made of a corrugated steel foil and permits limiting the transmitted loads and brings stability, with a properly designed stiness and damping respectively, when interactions with the rotor occur.

The eects of base motion on rotor-AMB systems are studied since the end of 90s when Murai et al. [START_REF] Murai | Seismic test on turbo-molecular pumps levitated by active magnetic bearing[END_REF] analysed the response to earthquake excitation. Since then, some works were dedicated to the implementation of control strategies to overcome these base motions and avoid TDBs interactions.

An eective solution is to implement a feedforward compensator added to a conventional feedback loop, as shown by Matsushita et al. [START_REF] Matsushita | Aseismic vibration control of exible rotors using active magnetic bearing[END_REF] where the vibration amplitudes of a rotor-AMB system subjected to Kobe earthquake were reduced by 60 %. Using less control current, Suzuki [START_REF] Suzuki | Acceleration feedforward control for active magnetic bearing systems excited by ground motion[END_REF] implemented the innite impulse response lter as feedforward control loop and showed that the rotor response could be reduced about the half of the response only using PID. Nonlinear dynamic systems generate sub and super harmonic responses when subjected to harmonic forcing. Based on this observation, Marx and Nataraj [START_REF] Marx | Suppression of base excitation of rotors on magnetic bearings[END_REF] developed a feedforward loop, where the correction current was set to be a sum of multiples and fractions of the base motion frequency. The acceleration feedforward compensation appears to be also eective in Kang and Yoon [START_REF] Kang | Acceleration feedforward control in active magnetic bearing system subject to base motion by ltered-x lms algorithm[END_REF] to limit harmonic base motions for applications such as electro-optical sight mounted on a moving vehicle. The compensator was designed experimentally thanks to the ltered-x least mean square algorithm, which was also used by Kim et al. [START_REF] Kim | Stabilization of active magnetic bearing system subject to base motion[END_REF].

Other works emphasized mostly on the AMB feedback control loop adapted to limit base motions. Kang et al. [START_REF] Kang | Sliding mode control for an active magnetic bearing system subject to base motion[END_REF] used the sliding mode control, which was able to achieve high performance robust control against parameter uncertainties and external disturbances. The eect of an horizontal impact was analysed by Cole et al. [START_REF] Cole | Vibration control of a exible rotor/magnetic bearing system subject to direct forcing and base motion disturbances[END_REF]. Three control strategies were developed to reduce the direct forcing as well as the motions of the support. If the controller was not suitable, the base motion generated rotor-TDB contacts. Some control strategies, however, can limit or even prevent the rotor-stator interaction such as the H ∞ controller according to Keogh et al. [START_REF] Keogh | Wavelet based adaptation of h-innity control in exible rotor/magnetic bearing systems[END_REF]. The combined wavelet -H ∞ controller gives the best performance in terms of transmitted forces and vibration limitations. The case of base rotations control was investigated by Maruyama et al. [START_REF] Maruyama | Study on control system for active magnetic bearing considering motions of stator[END_REF] where the stator motions, integrated as rotation disturbances, acted on the rotor-stator relatives angles. The dynamic behaviour of on-board rotor-AMB system was numerically investigated by Jarroux et al. [START_REF] Jarroux | On the dynamics of rotating machinery supported by amb during base motion[END_REF], considering rotation and translation motions of the base. The model was validated using previous experimental results where angular shocks were applied to a exible rotor with high gyroscopic eects supported by conventional ball bearings.

Most of AMB controllers employed in industrial applications are of PID type, without any base disturbance consideration. Therefore, some works analysed the eects of base disturbances when the control strategy was not specically tuned for that purpose. Hawkins et al. [START_REF] Hawkins | Shock and vibration testing of an amb supported energy storage ywheel[END_REF] tested a vertical ywheel-AMB system subjected to translational shocks for transit bus applications. Even if the controller was not specically tuned for base disturbance rejections, it was able to maintain the rotor and no TDB contact occurred.

The use of a shock isolator permitted reducing the transmitted axial shock by 65 %, corresponding with the work of Leilei et al. [START_REF] Leilei | Theoretical investigation on dynamic behavior of amb-rotor system subject to base motion disturbances[END_REF]: using exible supporting conditions, base disturbance rejection of rotor-AMB system can be improved. Hawkins et al. [START_REF] Hawkins | Test results and analytical predictions for mil-std-167 vibration testing of a direct drive compressor supported on magnetic bearings[END_REF] presented an experimental and numerical comparison of a magnetic bearing supported shipboard compressor subjected to sinusoidal translation up to 1.5 G. The vibration requirements were in accordance with MIL-STD-167 standards. AMBs remained in their linear range during these tests. This was not the case in the work of Kasarda et al. [START_REF] Kasarda | Eect of sinusoidal base motion on a magnetic bearing[END_REF] where a non-rotating mass mounted on magnetic bearings controlled by PID feedback and subjected to a sinusoidal base motion was analysed numerically and experimentally. In case of extremely low damping, nonlinear dynamics responses were observed.

In most of these previous studies, the rotor responses due to base motions remained limited and were not sucient to generate contact with the TDBs, either because a specic control strategy was applied to reject external disturbances or because the acceleration levels tested remained quite small. However, in case of large foundation displacements, rotor-TDBs contacts could occur while AMBs still operate. Indeed, even if the conventional procedure is to stop the machine in few seconds when the rotor displacements exceed a designed threshold, the dynamic behaviour of the shaft during this phase remains quite unknown and potential instabilities may be triggered. For example, the sliding friction forces produced at the rotor-TDB interface could lead to dangerous rotordynamics such as backward whirl followed by dry whip. More details about rotor-stator interactions and the wide variety of induced dynamic regimes, are provided in references [4252]. On the other hand, AMBs could destabilise the rotor by producing negative damping forces due to the emergence of undamped frequencies.

The literature shows few studies regarding the combined eect of base motions, TDB contact nonlinearities and magnetic forces on rotordynamics. Hawkins [START_REF] Hawkins | Shock analysis for a homopolar, permanent magnet bias magnetic bearing system[END_REF] studied numerically the transient dynamic behaviour of a rotor supported by homopolar permanent magnet bias magnetic bearings subject to vertical shocks and TDB contact interactions. More recently, Hawkins et al. [START_REF] Hawkins | Floating Shock Platform Testing of a Magnetic Bearing Supported Chiller Compressor: Measurements and Simulation Results[END_REF] analysed the eect of shock and TDB contacts on a magnetic bearing supported chiller compressor. After short period of rebounds, the rotor recovered its centered position and no particular damages were notied after the tests. Jarroux et al.

[55] also investigated those aspects and it was shown that the dynamic behaviour of the system remains mostly stable after rotor-TDB interactions. Su et al. [START_REF] Su | Nonlinear dynamic simulation and parametric analysis of a rotor-amb-tdb system experiencing strong base shock excitations[END_REF] numerically analysed the eects of TDB contact parameters onto the dynamic behaviour of a rigid rotor supported on AMBs subjected to strong base motion (up to 20 G). Periodic, nT-periodic, quasi-periodic responses as well as jump phenomena are revealed.

The presented work is a contribution to the understanding of complex phenomena associated with the dynamic behaviour of turbomachinery supported by AMBs facing critical external events. In particular, it investigates the exible rotordynamics exhibited when the eects of mass unbalance forces, rigid base motions and TDBs contact are combined while AMBs still operate. The numerical and experimental analyses are conducted both with harmonic and impulse foundation motions in vertical translation.

First is developed the modelling approach, using the nite-element method, required to perform the transient and nonlinear simulations. Then, the experimental set-up, comprising the lab-scale rotor-AMB system test rig mounted on a 6-DOF shaker, is described. Before carrying on base motion tests, two steps of validation are performed. At each step, the observed phenomena are explained, the numerical model is assessed and the prediction-measurement correlation is discussed.

On-board rotor-AMB system modelling

The dierent parts of the numerical model required to perform nonlinear and transient simulations are presented in the following subsections. The on-board exible rotor, the AMB and the TDB models are described separately. A typical rotor-AMB system is sketched in Fig. 1. The eects of exible support on rotordynamics was analysed by Cavalca et al. [START_REF] Cavalca | An investigation on the inuence of the supporting structure on the dynamics of the rotor system[END_REF]. However generally speaking, the stator of industrial turbomachinery support is very sti in comparison with the rotor. In this investigation, the support is assumed to be rigid but movable.

Considering the six motions of the rigid support, see Fig. The proposed model requires the use of three distinct frames of reference to set-up correctly the equations of motion. The rotor neutral line deections, represented by the non-inertial local frame R l (x l , y l , z l ), are described with respect to the rigid support, referred to as the non-inertial frame R (x, y, z), as for conventional rotordynamics. The motions of this rigid support are described with respect to the ground, Galilean or inertial frame, called R g (x g , y g , z g ). The rotor deections are therefore always described relatively to the support motions which makes it easy to integrate any type of bearings. Then, the method to derive the equations of motion is identical to the one proposed by Lalanne and Ferraris [START_REF] Lalanne | Rotordynamics Prediction in Engineering[END_REF], for conventional rotordynamics. Once the frames of reference established, the formulation of vectors describing the relative motions between each others is determined: the instantaneous angular velocity vector -→ ω Rg R and the position vector --→ O g O are needed. These vectors are expressed in Eq. ( 1), with respect to R g in R.

--→ O g O =    x 0 y 0 z 0    R ; -→ ω Rg R =    ω x ω y ω z    R (1)
where x 0 , y 0 , z 0 and ω x , ω y , ω z are the positions and the instantaneous angular velocities of the base, described in Fig. 2. The energies or the virtual work of each component is then calculated: the exible shaft modelled Timoshenko beam element, comprising 4-DOFs per node, contributes with its kinetic and strain energies, the one-node rigid disc element as well as the discrete mass unbalance with their kinetic energies, and the bearing with the virtual work of its restoring forces. The dierent contributions of the base are expressed in R in motion with respect to R g . The base motions modify only the kinetic energies.

Once the elementary energies are gathered, they are derived by using the Lagrange's equations:

d dt ∂T ∂ δi - ∂T ∂δ i + ∂U ∂δ i = F δi (2)
with δ i a DOF of the system, T and U , the kinetic and strain energies, respectively, and F δi the external force applied to the DOF δ i . The equations of motion are obtained as follows:

M δ + ΩC G + ω y C bm ωy δ + (K + ΩK G + ωy K bm ωy + Ω ω y K bm Ωωy + ω x 2 K bm ω x 2 + ω y 2 K bm ω y 2 + ω z 2 K bm ω z 2 + ω x ω z K bm ωxωz δ = F mu + F bm + F g + F amb + F c (3)
where M , K and C G are respectively the mass, the structural stiness and the gyroscopic matrices. The eects of the variable speed of rotation Ω are considered in the stiness matrix K G , see Bouziani and Ouelaa [START_REF] Bouziani | Simulation of the dynamic behavior of a rotor subject to base motion under variable rotational speed[END_REF]. The vector δ gathers all the DOFs of the mesh. The superscript bm stands for the base motion eects.

The support rotations bring C bm ω , K bm ω , the additional gyroscopic and stiness matrices which are responsible for parametric excitations, leading to potential lateral instabilities. The external force vector F bm includes the support translation eects combined with rotations, F mu is the mass unbalance force vector taking into account the normal centrifugal forces and the tangential eects associated with a variable speed of rotation.

The eect of gravity is considered in F g . The external force vectors F amb and F c represent the AMBs and the TDBs contact forces respectively. They are developed in the next section.

The modal superposition method is employed to perform low computational time simulations with a reduced system of equations while maintaining high reliability. A modal truncation is realised and the rst ten mode shapes are chosen in order to reduced the modal equations to be solved. The damping is introduced through the modal damping ratio α ψ evaluated from experimental modal analysis and the modal damping matrix C ψ is written as:

C ψ = 2α ψ K ψ M ψ (4)
To perform transient simulations using RK5 scheme, the Eq. ( 3), projected into the modal basis, is rewritten in its state space form Eq. ( 5). The subscript ψ stands for matrices and vectors established in the modal base. For practical reasons, the time dependent and parametric matrices are considered as external forces and placed in the second member.

Ẋ = AX + BU Y = CX + DU (5)
where X is the state vector, q is the modal variable vector and A the matrix of the system dynamics, such as:

X = q q (6) A = 0 I -M -1 ψ K ψ -M -1 ψ C ψ (7)
B is the input matrix and U is the input vector comprising the forces applied to the system, such as:

B = 0 M -1 ψ (8) U = -ΩC G,ψ + ω y C bm ωy,ψ q -( ΩK G,ψ + ωy K bm ωy,ψ + Ω ω y K bm Ωωy,ψ + ω x 2 K bm ω x 2 ,ψ + ω y 2 K bm ω y 2 ,ψ + ω z 2 K bm ω z 2 ,ψ + ω x ω z K bm ωxωz,ψ q +ψ t (F mu + F bm + F g + F amb + F c ) (9)
Finally, C is the output matrix and Y is the output vector comprising the DOFs of interest. The matrix D 155 is the feedforward matrix and is zero in this study.

Active magnetic bearings model

G co = G ampl × G so × G f o × G pid (10)
where G ampl , G so , G f o and G pid are respectively the amplier, the second order lead-lag lter, the rst order lead-lag lter and the PID transfer functions. The simplest way to formulate the PID transfer function is:

G pid (s) = K p + K d s + K i s (11)
The gain K p is equivalent to the controller stiness and is chosen to counteract the inherent negative stiness of AMBs. K i is the integral gain, it permits centering the rotor in the bearing. Finally, K d is the derivative gain, providing damping and stability into the system on the operating speed range. The rst order lag lter is used to limit the increase of damping generated by the derivative gain. Indeed, a too large increase of phase on the operating speed range may imply a low stiness and diculty to limit the gain in higher frequency. Its transfer function G f o is given by:

G f o (s) = s + 2πf n s + 2πf d (12)
where f n and f d are the numerator and denominator frequencies which characterized the lter.

The high frequency modes, outside the operating speed ranges but still comprised in the controller bandwidth are also damped to avoid spillover eect (high frequency instability). Second order lead-lag lters are used to this end, and the transfer function is given in Eq. ( 13). For robustness purpose, the gyroscopic eects on the dierent rotor modes are taken into account for the application of phase.

G so (s) = s 2 + 2ξ n (2πf n ) s + (2πf n ) 2 s 2 + 2ξ d (2πf d ) s + (2πf d ) 2 (13)
with f n , f d and ξ n , ξ d , the numerator, denominator lter frequencies and the damping ratios respectively. The amplier controls the electromagnet by applying the voltage u p on the coil, as shown in Fig. 3. Its transfer function G ampl is described with a rst order low pass lter. Then, the electromagnet, used in dierential driving mode for more stability, generates an attractive force, called f amb , along the action line which adopts the following formula:

f amb = 4 cos αµ 0 SN 2    (I 0 -i c ) 2 L µr + 2g 0 -2x cos α 2 - (I 0 + i c ) 2 L µr + 2g 0 + 2x cos α 2    (14)
with µ 0 and µ r the vacuum and iron core magnetic permeabilities respectively, S the pole area, N the number of turns in one coil, g 0 the nominal air gap, x the rotor position along the considered action line, L the average length of the magnetic ux lines and α the angle between the action line and the line joining the pole and the rotor centres.

To compensate the base motions, the augmented PID controller may have to generate large control currents i c to counteract large rotor displacements x. For this reason, the AMB forces are not linearised as done usually and the amplier saturation eects are taken into account.

Touchdown bearings model

The modelling of TDBs have been extensively developed in the literature, especially in the context of magnetic levitation loss. As explained by Kärkkäinen et al. [START_REF] Kärkkäinen | Dynamic simulation of a exible rotor during drop on retainer bearings[END_REF], the overall rotordynamics prediction of a turbomachinery using detailed models, where each rolling element has several DOF, is not necessary. The modelling approach is inspired by Schmied and Pradetto [START_REF] Schmied | Behaviour of a one ton rotor being dropped into auxiliary bearings[END_REF]. Though the ribbon damper has demonstrated dry friction damping under harmonic forcing, it was shown by Jarroux et al. [START_REF] Jarroux | Touchdown bearing models for rotor-amb systems[END_REF] that a specic contact law using viscous damping could correctly describe the overall transient dynamics and transmitted loads of an industrial-scale rotor drop. The same modelling approach is employed in this work.

Rotor-TDB interactions generate a normal contact force f n and an induced friction force µf n in tangential direction, as shown in Fig. 4. The shaft rst impacts the coupled ball bearing ribbon damper system in a series mode. Once the ribbon is completely crushed, it only has a static contribution and the rotor faces the ball bearing restoring force. The TDB model is characterized by softness, large damping and linear ball bearing when the ribbon is not crushed, then by high stiness, low damping and nonlinear bearing when the ribbon is crushed. The rotor-TDB misalignments are considered since they can seriously aect the rotordynamics during contact, as shown by Halminen et al. [START_REF] Halminen | Active magnetic bearing-supported rotor with misaligned cageless backup bearings: A dropdown event simulation model[END_REF] when analysing rotor drop response. The contact problem is solved using the penalty method and the related normal component f n is described in Eq. [START_REF] Dakel | Nonlinear dynamics of a support-excited exible rotor with hydrodynamic journal bearings[END_REF].

f n (u rs ) =      f brg (u rs -u rd ) + c brg urs + k eq u rd ; u rs ≥ u rd k eq u rs + c eq urs ; u rd > u rs ≥ 0 0 ; u rs < 0 (15)
with u rs , urs the rotor-TDB normal relative clearance and its time derivative, in Eq. ( 16) and (17), where x r , z r , x s , z s and u s are the rotor displacements, the TDB misalignments and the initial clearance respectively.

u rs = r rs -u s = (x r -x s ) 2 + (z r -z s ) 2 -u s (16) urs = du rs dt = ẋr (x r -x s ) + żr (z r -z s ) (x r -x s ) 2 + (z r -z s ) 2 (17)
with u rd the ribbon crushing capacity, k eq and c eq the coupled ball bearing ribbon damper dynamic parame-205 ters considered in series mode and computed using the linearised ball bearing k brg , c brg . The ribbon damper has k rd , c rd as dynamic parameters. The ball bearing restoring force f brg comes from the force-deection relationship, obtained from analytical results based on [START_REF] Harris | Rolling Bearing Analysis[END_REF][START_REF] Guay | Roulements calculs, Techniques de l'ingénieur Guidage mécanique[END_REF], as done in Jarroux et al. [START_REF] Jarroux | Touchdown bearing models for rotor-amb systems[END_REF]. As it is dicult to predict the depth of penetration, these forces are conserved nonlinear.

The tangential component f t , Eq. ( 18), applied on the rotor, considers the sliding friction force µ (v s ) f n generated at the rotor-TDB interface due to dierent speeds of rotation, v s being the relative rotor-TDB sliding velocity. The tangential damping c t provided by the ribbon damper counteracting the relative rotor whirl velocity w rs is also taken into account.

f t (u rs ) =      µ (v s ) f n ; u rs ≥ u rd µ (v s ) f n -c t w rs ; u rd > u rs ≥ 0 0 ; u rs < 0 (18)
where w rs is the relative whirl speed dened by Eq. ( 19) and where γ rs is the relative whirling angle given by Eq. [START_REF] Cole | The dynamic behavior of a rolling element auxiliary bearing following rotor impact[END_REF].

w rs = r rs γrs = ẋr (z r -z s ) -żr (x r -x s ) (x r -x s ) 2 + (z r -z s ) 2 (19)
γ rs = arctan z r -z s x r -x s (20)
The sliding friction coecient µ (v s ) is a polynomial formulation of the regularized Stribeck model, Eq. ( 21), used by Nguyen et al. [START_REF] Nguyen | Nonlinear rotordynamics of a drillstring in curved wells: Models and numerical techniques[END_REF].

µ (v s ) = v s 2v ref    1 -1 - µ dyn µ dry 1 + |vs| 2v ref µ dyn 1 -1 - µ dyn µ dry + 1 + 1 - µ dyn µ dry 1 + |vs| 2v ref µ dyn 1 -1 - µ dyn µ dry 2    (21)
where µ dry , µ dyn being respectively the dry and the dynamic friction coecient, depending on the value of the relative sliding velocity v s . v ref is a numerical parameter that species the slope of the transition between two slip states in opposite direction.

In order to avoid discontinuities inherent to linear viscous damping in the contact force, arctangent laws are employed to regularize the normal f n and the tangential f n components when the relative clearance exceeds zero u rd > u rs ≥ 0 and when it exceeds the ribbon crushing capacity u rs ≥ u rd .

The TDBs are driven in rotation by the previously described sliding friction forces µ (v s ). The modelling approach assumes a perfect rolling-without-slipping kinematic condition. This assumption seems not too strong according to Helfert et al. [START_REF] Helfert | High-speed video analysis of rotor-retainer-bearing-contacts due to failure of active magnetic bearings[END_REF]. They analysed with a rapid camera the rotation of the balls of a cageless TDB during a rotor drop and observed that the balls as well as the inner race reached the shaft speed almost at the same time: the sliding between the rolling elements has few inuence on the time needed for the TDB to reach the rotor speed of rotation.

The rotational equation of motion is provided by:

J eq Ωs + ΦΩ s = µ (v s ) f n R s -M f (22)
with Ω s the TDB speed of rotation, M f the resistive torque based on the Palmgren theory [START_REF] Palmgren | Ball and roller bearing engineering[END_REF], R s the inner race radius and Φ the rotational damping used to break the bearing when no more external loads are present. Assuming a perfect rolling-without-slipping kinematic condition between balls and races, the equivalent rotational inertia J eq , comprising balls and inner race, can be computed.

The complete numerical model is nally schemed in Fig. 5. Each modelling part developed in this section is represented by dedicated AMB, support motion and TDB blocks. All the contributions, not directly related to the rotor structure, are added as second members in the equations of motion. The approach is modular and each model can be employed or not in the simulations.

Test rig

The base motion experimentations are carried out using a lab-scale rotor-AMB system mounted on a 6-axis shaker.

Lab-scale rotor-AMB system

The lab-scale rotor-AMB system, see Fig. The AMBs are controlled with an augmented PID based on Eq. ( 10) and its complete transfer function is plotted in Fig. 8. Damping is brought over the operating speed range by applying a positive phase thanks to the derivative gain combined with a rst order lag lter. Concerning high frequency range, the green rectangles indicate the bending mode positions and their thickness are related to the gyroscopic eects.

To avoid spillover instabilities, positive phase is provided in these rectangles using second order lag lters.

When a vibration node is placed between the sensor and the actuator, the phase is turned by 180 degrees, as for the third bending mode, at the NDE side, see Fig. 7.

The rotor-AMB system is equipped with two TDBs which are cageless single row deep groove ball bearings composed of nitrogen steel races and ceramics balls. They are tted into the housing through a ribbon damper, characterized by its stiness, damping and dynamic crushing capacity, see Tab. 3.

6-DOF Shaker

To perform base motion tests, the lab-scale rotor-AMB system is xed on a 6-axis hydraulic shaker of 63 kN, as shown in Fig. 9. It is able to apply various combinations of solicitations along and around the Base motion tests were carried out considering harmonic and impulse test in vertical translation. The shaker stop was also recorded to analyse the capacity of the AMBs to control the rotor after contact situations. Base accelerations were recorded in the three directions (Xcube, Ycube, Zcube) using four triaxial accelerometers xed on the shaker. The displacement and current sensors for each action line (V13, W13, V24, W24), see 9, and the speed of rotation were also recorded. The sampling frequency was set to 24 756 Hz.

System identication

The base excitations is nil. Mass unbalance responses were rst realised to identify the rotor-AMB system dynamics and assess the reliability of the numerical model predictions for simple linear test cases.

Rotor drop response at rest were also performed since it is an ecient method to validate the TDB contact model.

Mass unbalance response

The speed of rotation was slowly increased up to 9 500 rpm with a rate of 50 rpm per seconds. AMBs are considered linear for these simulations since rotor deections are expected to be small. The numerical and experimental rotor synchronous responses, considering a 17 g.mm mass unbalance located at 0

• of the key phasor top dead centre, are depicted in DE side, are due to the electrical motor. Between 0 and 2 500 rpm, the motor was not able to produce a purely constant rotational speed. Investigations of base motion eects were then realised above 3 000 rpm.

Drop response

The rotor is at rest and initially levitated by the AMBs. At 1 s, the AMBs are shut-down. The results of the predicted and measured drop responses are provided in Fig. 11.

The transient dynamics seems to be well described and the rotor displacements are comparable. The measurements demonstrate more damping during this phase and the predicted rebounds are larger at the As the measured rotor drop dynamics is quite well described by the model, one can be condent on the predicted TDB contact loads which should be close to the real ones, and might overestimate them. The drop loads as well as the force-deection loops are displayed in Fig. 12. The maximal loads are found at DE side with almost 400 N which is far lower from the estimated static load capacity of the TDBs. The peaks appearing during the rst rebounds are related to the complete crushing of the ribbon damper, also revealed by the appearance of the tiny peaks in the force-deection loops. The latter demonstrate the large damping provided by the ribbon during the rst rebounds characterized by large areas. It is worth mentioning that when the rotor stabilizes at the bottom of its TDBs, the contact loads at both DE and NDE tend to 33 and 31 N respectively, which the sum is the rotor weight.

These preliminary tests show the eectiveness of the numerical model to reproduce quite accurately the measures for simple test cases.

Support motion investigations -Harmonic tests

The harmonic translation support motion was set to 20 Hz ranging from 0.1 to 1.1 G, see Fig. it is a representative frequency of earthquake whose energy is mainly comprised in the range [START_REF] Cole | The dynamic behavior of a rolling element auxiliary bearing following rotor impact[END_REF][START_REF] Sun | Detailed ball bearing model for magnetic suspension auxiliary service[END_REF][START_REF] Kärkkäinen | Dynamic simulation of a exible rotor during drop on retainer bearings[END_REF][START_REF] Jarroux | Touchdown bearing models for rotor-amb systems[END_REF][START_REF] Yu | Dynamic analysis of magnetic bearing rotor dropping on radial and axial integrated auxiliary bearing[END_REF][START_REF] Neisi | Performance evaluation of touchdown bearing using model-based approach[END_REF][START_REF] Kang | Auxiliary bearing squeeze lm dampers for magnetic bearing supported rotors[END_REF][START_REF] Murai | Seismic test on turbo-molecular pumps levitated by active magnetic bearing[END_REF][START_REF] Matsushita | Aseismic vibration control of exible rotors using active magnetic bearing[END_REF][START_REF] Suzuki | Acceleration feedforward control for active magnetic bearing systems excited by ground motion[END_REF][START_REF] Marx | Suppression of base excitation of rotors on magnetic bearings[END_REF] Hz, as explained by Matsushita et al. [START_REF] Matsushita | Aseismic vibration control of exible rotors using active magnetic bearing[END_REF]. The targeted acceleration level was applied progressively, during almost 30 s, to limit overshoots and was then maintained for approximately 40 s to analyse potential onset of nonlinear dynamic regimes. The rotor speeds of rotation were set to 3 000, 6 000 and 9 500 rpm. For all the running tested, the same level of mass unbalance was applied: 32 g.mm at 0 degree that corresponds to four times the recommended mass unbalance by the API standards. These base motion tests have been carried out three times and it appeared that the observed phenomena considering the rotordynamics were repeatable each time. For this reason, only one test is further investigated hereafter.

The accelerations recorded on the shaker, Fig. 13, were implemented into the numerical model.

Overall dynamics

This subsection is entirely treated using the case where the rotor is running at 6 000 rpm, the related observations being identical with respect to the 3 000 and 9 500 rpm cases. do not destabilize the rotor in both numerical and experimental results. The eects of contact generate high frequency current content, especially during the bottom rotor-TDB interactions. These eects are more noticeable in the measurements where the generated currents are amplied with respect to predictions, as if the controller gain was higher on these frequencies. This could be due to some dierences between the numerical and experimental controller transfer functions, especially at the NDE side.

The Fig. 19 depicts the predicted TDB contact loads. It shows that the rotor oscillates due to a resonance eect of the rotor-TDB system. This could explain the high frequency content observed in the currents provided by the controller. It was dicult to obtain a value of that frequency since the TDB model has a nonlinear behaviour. According to the force-deection relationship, the ribbon damper is not fully crushed, and the generated load remained limited.

Finally, it is shown that when the rotor exceeds the TDB clearance, loads are shared between AMBs and TDBs and the base motions are mainly counteracted by magnetic forces.

The spectral analysis gives further informations on the rotordynamics. In presence of contacts, nonlinear features such as sub-super harmonic responses or chaotic regimes may be highlighted. The time evolution of the NDE horizontal displacement frequency content is obtained through Short-Time Fourier Transform (STFT) and results are provided in Fig. 20 and 21, respectively.

The plots are composed of STFT and the average spectra, computed along the time axis. The predicted and measured results are qualitatively similar. The eects of contact are clearly highlighted, close to 30 s where a wide frequency bandwidth is excited, including the rst three bending modes located close to 200, 580 and 1 140 Hz. Gyroscopic eects split these frequencies into forward and backward whirl modes:

this is visible especially for the second and third bending modes. The level of response of these modes is, however, small. The full spectrum, usually computed to determine whether the rotor orbit motion frequency components are forward or backward, was not employed here since no particular backward whirl components 375 were observed. The rotor orbits are attened at the bottom and the top by the TDB interactions and specic periodicities can be observed. 385 Driot et al. [START_REF] Driot | Theoretical and experimental analysis of a base-excited rotor[END_REF] showed that when the ratio n between the base rotation frequency ω and the speed of rotation Ω was a rational number, periodic orbits were found and while when it was a ordinary one, they were quasi-periodic. This conclusion may be extended for the translation case since typical combinations of foundation motions and mass unbalance forces are exhibited. In this case, the TDB contacts do not change these periodicities. 390

Orbital responses

This can be demonstrated by the Eq. [START_REF] Jarroux | Touchdown bearing models for rotor-amb systems[END_REF].

ω = β χ Ω = nΩ (23)
where β and χ are integers. When Ω equals 3 000 rpm and 6 000 rpm, orbits are periodic and the ratio n Figure 22: Predicted (a) vs measured (b) rotor orbits (solid lines) and touchdown bearing clearances (dotted lines) -3 000 rpm and 1.1 G at 20 Hz is a rational whereas when Ω equals 9 500 rpm, orbits are quasi-periodic and n becomes an ordinary. Table 4 summarizes these results. AMBs, but also of weak rotor-TDB relative sliding velocities. Indeed, as the latter are driven in rotation by the friction forces produced by intermittent contacts, they quickly reach the rotor speed of rotation.

Support motion investigations -Impulse tests

The impulse tests were also carried out in vertical direction. The shock lasted 0.02 s (50 Hz) and increased from 0.5 to the set value of 3.1 G. Once the acceleration set level reached, ten shocks were repeated each second as shown in Fig. 25. The applied mass unbalance was the same according to harmonic tests, 32

g.mm at 0 degree. The rotor speed of rotation was set to 9 500 rpm. As for the harmonic case, these base motion tests have been carried out three times and it appeared that the observed phenomena considering the rotordynamics were repeatable each time. For this reason, only one test is further investigated hereafter.

The accelerations recorded on the shaker, see Fig. 25, were implemented into the numerical model. the point 1 and 2 on the Fig. 28. Then, the measured rotor rebounds show that for one base shock, three successive contacts are generated while two are predicted.

Sliding friction eects

The TDB contact loads are rather large with respect to the drop or the harmonic response. Moreover, the sliding friction eects are more expected to occur since TDB never reach the rotor full speed. This A zoom on the top contact is provided in Fig. 32. It seems that during a short period, the measured magnetic forces at DE side push the rotor in backward direction between the arrows 3 and 4, but not in the predictions. This could be due to negative damping provided by the experimental AMB transfer function or to the controller delay to apply the instruction. Obviously, tangential forces induced by friction contributes also to this backward whirl motion. However, the predicted contact loads, especially the tangential ones, are supposed to be small during the top contact.

Just after this event, the rotor is quickly recentred by the AMBs and no established instabilities are triggered.

Outcomes and discussions

The test congurations were chosen to be representative of high speed industrial turbomachinery where the rotor has no bending modes in the operating speed range and the PID controller was tuned for conventional rotordynamics. The numerical and experimental investigations for harmonic and impulse tests in vertical direction were conducted:

• Figure 32: Predicted (a) vs measured (b) rotor orbits and AMB forces during the top TDB contact -9 500 rpm and 3.1 G at 50 Hz -between the instants 1 and 2 in Fig. 28 the duration of each contact. Even if the PID controller was tuned without considering these particular frequencies, the rotor-AMB controller remains globally stable in both numerical and experimental tests. The rubbing contacts did not triggered dry whip instabilities in both experimental and numerical results. It seems that the TDBs are driven in rotation reaching the rotor rotational speed during 470 harmonic tests: tangential friction eects are limited. Moreover, the acceleration level did not generate large contact loads.

• Considering impulse tests, the sliding friction eects are more visible. On the one hand, because the acceleration levels tested were higher and on the other hand, because the TDBs are not driven in rotation up to the rotor speed. The sliding friction was then higher than for harmonic cases. At the bottom contact, short periods of backward whirl were exhibited both in predictions and measures with similar amplitude. The rotor was then quickly recentred. At the top contact, some eeting instabilities were recorded experimentally. The latter are most likely to be triggered by AMBs since during a short period, AMBs pushed the rotor in the sens of its trajectory. These instabilities failed to settle and the rotor was always quickly recentred just after.

• For all these analyses, no AMB nonlinearity was exhibited, in both predictions and measures, whether amplier saturation or strong magnetic force nonlinearity. This is mainly due to the fact that TDB contacts prevent actuator nonlinearities emergences. It should be mentioned that the TDB over AMB clearances ratio is close to 0.25 which is smaller than industrial values, closer to 0.5, making the nonlinearities of the actuator restoring forces more relevant.

Phenomena associated with rotor-TDB interactions are sensitive to contact parameters such as clearance, misalignments, damping and stiness, sliding friction, boundary conditions, etc. On the other hand, rotor-AMB systems are also sensitive to the electromagnetic forces driven by the controller. The combination of both these phenomena is complex and several discrepancies were noticed when comparing predicted and measured results:

• The eeting events observed during impulse tests were not predicted by the model. A slight dierence between the numerical and the experimental AMB transfer function, not plotted in this article, was noticed. The identication procedure carried-out revealed that the amplier was the most probable source of discrepancies.

• The measured rotor-TDB clearance was not exactly the provided one. Several assumptions can explain these dierences: manufacturing issues, the uncertainties of TDB internal clearance and ribbon parameters, the ribbon elastoplastic behaviour (dry-friction during the crush).

• Measured orbits at both DE and NDE sides are not perfectly vertical and are symmetrically tilted with respect to the vertical axis. This is probably due to the assembly, introducing a twist between the DE and NDE AMBs. Therefore, the base motion in vertical direction appears to be slightly inclined in the DE and NDE AMB sensors.

Conclusions

The aim of this research work was to provide further insights into the understanding of the dynamic behaviour of embedded rotor-AMB systems and to carried out an experimental validation of the proposed model. The eects of base motions, involving TDB interactions, on the system dynamics and on AMBs were of particular interest.

The developed numerical model permits performing nonlinear and transient simulations in order to understand phenomena and to have a reliable tool enabling the optimization of the system performance. It comprises an on-board rotor, active magnetic bearings (augmented PID controller and nonlinear actuator forces) and touchdown bearings (contact law and rotational dynamics). A lab-scale test rig, designed to be representative of industrial turbomachinery, was employed to perform harmonic and impulse testings. The control strategy was not specically tuned to withstand bases excitations.

Harmonic base excitation demonstrate periodic or quasi-periodic orbits depending on the ratio between the base motion and the speed of rotation. Rotor orbit shapes are attened by TDB interactions. Forward whirls were exhibited and no dry whip instabilities were noticed. Considering impulse excitations, short periods of backward whirl induced by TDB rubbing contacts were observed. The restoring magnetic forces generated by actuators remained in an acceptable linear range and no amplier saturations were exhibited.

Even if the PID controller was tuned without considering this particular excitation, it remained mostly stable in both numerical and experimental tests.

The model was able to reproduce the overall dynamics and the main observed phenomena which provides condence considering the prediction of the dynamic behaviour of on-board industrial turbomachinery. In particular, this model will be use to conduct further investigations regarding the inuence of TDBs, AMBs and mass unbalance parameters on the rotordynamics. The aim will be to extend the understanding of phenomena, to analyse the system sensitivity to parameter changes and to provide ecient design solutions to avoid dangerous dynamics, such as dry-whip.

Acknowledgement

Figure 1 :

 1 Figure 1: AMB and TDB assembly

Figure 2 :

 2 Figure 2: On-board rotor -set-up of the dierent frames of references

Figure 3 :

 3 Figure 3: Sketch of AMB supporting the rotor

Figure 4 :

 4 Figure 4: Sketch of the touchdown bearing model

6 ,Figure 5 :

 65 Figure 5: Diagram of the numerical model

Figure 6 :Figure 7 :

 67 Figure 6: Lab-scale rotor test rig

Figure 8 :Figure 9 :

 89 Figure 8: Controller transfer function -augmented PID Table 3: Touchdown bearing properties Parameter Value NDE touchdown bearing measured clearance -u snde 0.1 mm DE touchdown bearing measured clearance -u sde 0.11 mm Pitch diameter -d m 45.5 mm Ball diameter -D 4.7 mm Number of balls -Z 23 Computed ball bearing static capacity -C S 8200 N Computed ball bearing equivalent inertia -J eq 1.4e -5 kg.m 2 Computed ball bearing stiness -k brg 1.5e 8 N.m -1 Estimated ball bearing normal damping -c brg 4e 2 Ns.m -1 Computed ribbon stiness -k rd 3e 6 N.m -1 Estimated ribbon normal damping -c rd 1e 3 Ns.m -1 Estimated ribbon tangential damping -c t 5e 2 Ns.m -1 Estimated ribbon crushing capacity -u rd 0.02 mm

Fig. 10 .

 10 Due to the symmetry of the rotor-AMB system the numerical are identical along V and W axes. Trends are identical in terms of vibration amplitudes and frequency position of the suspension modes. The deviations revealed at low speed, particularly visible at Rotating speed (rpm)

Figure 10 :

 10 Figure 10: Predicted (dotted lines) vs measured mass unbalance response

Figure 11 :Figure 12 :

 1112 Figure 11: Predicted (a) vs measured (b) vertical rotor displacements (solid lines) and touchdown bearing clearances (dotted lines) -drop at rest

Figure 13 :

 13 Figure 13: Measured base vertical acceleration (Zcube) -0.1 to 1.1 G at 20 Hz

Fig. 14 Figure 14 :Figure 15 :Figure 16 :

 14141516 Figure 14: Predicted (a) vs measured (b) vertical rotor displacements (solid lines) and touchdown bearing clearances (dotted lines) -6 000 rpm and 1.1 G at 20 Hz

Figure 17 :Figure 18 :

 1718 Figure 17: Predicted (a) vs measured (b) horizontal rotor displacements (solid lines) -6 000 rpm and 1.1 G at 20 Hz -zoom in the range [60.7-60.85]s

TimeFigure 19 :Figure 20 :

 1920 Figure 19: Predicted normal contact loads -6 000 rpm and 1.1 G at 20 Hz -zoom in the range [60.7-60.85]s

Figure 21 :

 21 Figure 21: Measured NDE STFT and average spectrum in horizontal direction -6 000 rpm and 1.1 G at 20 Hz

Figure 23 :Figure 24 :

 2324 Figure 23: Predicted (a) vs measured (b) rotor orbits (solid lines) and touchdown bearing clearances (dotted lines) -6 000 rpm and 1.1 G at 20 Hz

Figure 25 :

 25 Figure 25: Measured repeated shocks of the base (Zcube) -3.1 G at 50 Hz

Figure 28 displays

 28 Figure 28 displays results during one impulse. The acceleration prole is also plotted in green: the related values have no physical meanings, the maximum value being 3.1 G. Numerical and experimental results are close during the rst phase of the shock, related to the maximum acceleration level, between

Figure 26 :Figure 27 :Figure 28 :Figure 29 :

 26272829 Figure 26: Predicted (a) vs measured (b) vertical rotor displacements (solid lines) and touchdown bearing clearances (dotted lines) -9 500 rpm and 3.1 G at 50 Hz

Fig. 29 since

 29 Fig. 29 since orbit has backward whirl during the contact. 6.3. Fleeting instabilities 435

Figure 30 :Fig. 31 .

 3031 Figure 30: Predicted (a) vs measured (b) rotor orbits -9 500 rpm and 3.1 G at 50 Hz -between the instants 1 and 3 in Fig. 28

Figure 31 : 28 x

 3128 Figure 31: Predicted (a) vs measured (b) rotor orbits and AMB forces during the bottom TDB contact -9 500 rpm and 3.1 G at 50 Hz -between the instants 1 and 2 in Fig. 28

  This work was supported by the ANRT (French National Research and Technology Agency), CIFRE [grant number 2013/1376]. The experimentations were carried out with the 6-axis shaker of Equipex PHARE 10-EQPX-0043 of the ANR (French National Research Agency) funded by the PIA (Programme d'Investissements d'Avenir).

Table 2 :

 2 Active magnetic bearings properties

	Parameter

Table 4 :

 4 Rotor orbit periodicities

	3 000 rpm 6 000 rpm 9 500 rpm

axes (translations and rotations) to a maximum payload of

kg in a range [0-250] Hz. A maximum acceleration of 10 G, ± 50 mm in translation and ± 4 degrees in rotation can be reached.


Frédérique Clément
email: frederique.clement@inria.fr

Danielle Monniaux

Mathematical modeling of ovarian follicle development : A population dynamics viewpoint

Keywords: population dynamics, regression models, structured populations, multiscale models ovarian follicles, cell fate, cell kinetics, morphogenesis

Amongst endocrine organs, ovaries are unique because their function is supported by an ever-changing number of histological entities, the ovarian follicles, which are themselves subject to asynchronous development from their formation up to ovulation. Follicle development combines finely controlled processes of cell dynamics with original morphogenesis steps. Understanding the dynamics of follicle populations, on the whole ovary level, and the coupled dynamics of the oocyte and somatic cells, on the follicle level, is crucial to understand ovarian physiology and ovarian aging. Mathematical modeling studies combined with proper experimental approaches can be particularly helpful for gaining insight into follicle dynamics on different scales.

Introduction

The ovary is a unique instance of dynamic, permanently remodeling endocrine organ in adulthood. The ovarian function is supported by spheroid, multi-layered and multi-phasic structures, the ovarian follicles, which shelter the oocyte and secrete a variety of peptide and steroid hormones. Understanding the dynamics of ovarian follicle populations is critical to characterize the reproductive physiological status of females from birth (or even prenatal life) to reproductive senescence. Follicle development involves coordinated processes of cell kinetics and morphogenesis, which are deployed in the framework of the hypothalamo-pituitary-gonadal (HPG) axis. Getting mechanistic insight into follicle development requires investigating the spatio-temporal dynamics of somatic follicle cells, coupled with oocyte maturation, and subject to hormonal and mechanical controls. In this review, we will focus on population dynamics issues raised by folliculogenesis in mammals, on the level of both the whole follicle population and the cell populations within developing follicles. We will highlight how historical and recent modeling approaches have tackled these issues, what insights have been gained or can be expected in the future from the models, and which experimental data are or will be critical to ensure a mutual benefit from modeling and experimental studies. We refer the interested reader to [START_REF] Yvinec | Advances in computational modeling approaches of pituitary gonadotropin signaling[END_REF][START_REF] Clément | Mathematical modeling approaches of cellular endocrinology within the hypothalamo-pituitarygonadal axis[END_REF] for a broader overview on mathematical models dedicated to the HPG axis.

Follicle population dynamics

At any given time, the status of the ovaries can be assessed by the distribution of follicles into different maturity stages and by the total number of follicles (Figure 1). These observations contain condensed information on long-term interacting processes. The total number is directly related to the remaining number of primordial follicles at a given age. In turn, the size of the primordial pool results from the combination of the initial (maximal) pool size and (possibly time -varying) rate of pool depletion by atresia and activation. The distribution of follicles is shaped by the asynchrony in the activation times and the relative rates of growth or atresia at different maturities, resulting from direct (paracrine) or indirect (endocrine, HPG-mediated) interactions between follicles (of similar or different maturities).

Mathematical models based on statistical regression have dealt mainly with the depletion of the "non growing follicles" (including primordial, intermediate and primary follicles) with aging (see e.g. [START_REF] Hansen | A new model of reproductive aging: the decline in ovarian non-growing follicle number from birth to menopause[END_REF][START_REF] Wallace | Human ovarian reserve from conception to the menopause[END_REF] and [START_REF] Coxworth | Ovarian follicle loss in humans and mice: lessons from statistical model comparison[END_REF] for a comparative review). A focus can also be put on antral follicle count -or rather on its hormonal counterpart, AMH (Anti-Müllerian Hormone) [START_REF] Kelsey | A validated model of serum anti-müllerian hormone from conception to menopause[END_REF][START_REF] De Kat | Back to the basics of ovarian aging: a population-based study on longitudinal anti-müllerian hormone decline[END_REF], to restrict to the maturity stages targeted by ovarian simulation treatments. The main advantage of these models is that they only need the adjustment of a few parameters, and can be used to predict the age at menopause in a "demographic" manner, for groups of patients in physiological or pathological conditions. They can also detect changes in the depletion rate, and have especially emphasized an acceleration in the premenopausal period. Using a biphasic loglinear model has led to the notion of break slope at a threshold age, that should be interpreted with caution, given that polynomial models can fit the data as well [START_REF] Hansen | A new model of reproductive aging: the decline in ovarian non-growing follicle number from birth to menopause[END_REF][START_REF] Coxworth | Ovarian follicle loss in humans and mice: lessons from statistical model comparison[END_REF], and that the selected threshold age is subject to significant statistical uncertainty. Finer data fitting can be expected from adding covariables, or at best, using mixed-effect models [START_REF] De Kat | Back to the basics of ovarian aging: a population-based study on longitudinal anti-müllerian hormone decline[END_REF], yet interindividual variability in the size of the initial pool and the (wasteful or thrifty) way of "consuming" the pool raises insurmountable obstacles for patient-specific prediction by regression models. Compartment-based models can encompass both the depletion of the total number of follicles and their distribution into growing stages, generally sorted by the follicle diameter and considered up to the antral stages [START_REF] Faddy | An analytical model for ovarian follicle dynamics[END_REF][START_REF] Faddy | Ovarian follicle dynamics in mice: a comparative study of three inbred strains and an F1 hybrid[END_REF]. Their main strength is that they fully exploit the available data on follicle numbers and introduce explicitly the dependence on time. They have also been used to obtain quantitative information on oogenesis before follicle formation [START_REF] Bristol-Gould | Fate of the initial follicle pool: empirical and mathematical evidence supporting its sufficiency for adult fertility[END_REF], and assess the number of founder primordial germ cells [START_REF] Zheng | On the number of founding germ cells in humans[END_REF]. Up to now, only linear models have been fitted to experimental data. The follicle trajectories are uncoupled, as such formulations do not account for interactions between follicles. As a consequence, they cannot be useful for understanding the link between the "static" reserve (primordial follicles) pool and the "dynamic" reserve (small antral follicles) [START_REF] Monniaux | The ovarian reserve of primordial follicles and the dynamic reserve of antral growing follicles: what is the link?[END_REF], possibly involving buffering mechanisms [START_REF] Bristol-Gould | Postnatal regulation of germ cells by activin: The establishment of the initial follicle pool[END_REF], or the difference between the first wave of follicle initiation and the next ones [START_REF] Zheng | Two classes of ovarian primordial follicles exhibit distinct developmental dynamics and physiological functions[END_REF]. They can neither help investigate mechanistically the main conditions perturbing these interactions (hypophysectomy [START_REF] Dufour | Short-and long-term effects of hypophysectomy and unilateral ovariectomy on ovarian follicular populations in sheep[END_REF], unilateral ovariectomy [START_REF] Baker | The number of oocytes and the rate of atresia in unilaterally ovariectomized mice up to 8 months after surgery[END_REF], AMH Knock-Out [START_REF] Durlinger | Control of Primordial Follicle Recruitment by Anti-Mullerian Hormone in the Mouse Ovary[END_REF] or overexpression [START_REF] Kano | AMH/MIS as a contraceptive that protects the ovarian reserve during chemotherapy[END_REF]). Another caveat is that the parameters critical for each compartment (death rate and exit rate to the next stage) are hardly identifiable from a parameter estimation viewpoint. This may seem a technical detail, yet it explains why attempts to use such models to decipher whether the primordial pool is or not replenished cannot lead to a decisive result, since conclusions are drawn from divergent assessment of the atresia rate [START_REF] Bristol-Gould | Fate of the initial follicle pool: empirical and mathematical evidence supporting its sufficiency for adult fertility[END_REF]. An extension of such models to a nonlinear framework is thus necessary. A comprehensive mathematical framework has been recently designed with this objective [START_REF] Bonnet | Multiscale population dynamics in reproductive biology: singular perturbation reduction in deterministic and stochastic models[END_REF]. Before being fully exploited for modeling purposes, ongoing mathematical works need to be completed, since the nonlinear formulation raises open mathematical questions. Finally, nonlinear models formulated in a continuous framework have tackled the specific issue of follicle selection. The follicle population corresponds to the cohort entering terminal development. The selection results from the dependency of follicles on FSH supply, whose levels are set in turn by the endocrine feedback of competing follicles. Nonlinearity is essential to widen the inequalities between follicles and separate the trajectories of ovulatory and atretic follicles. The first, phenomenological approaches have taken advantage of the mathematical properties of a specific class of functions (Abel functions) [START_REF] Lacker | How do the ovaries count?[END_REF]. They can reproduce different combinations of ovulation rates and even cyst-like situations [START_REF] Chavez-Ross | Modelling the control of ovulation and polycystic ovary syndrome[END_REF]. A more recent approach has considered selection as an emergent behavior emanating on the follicle level from the follicle cell level ( [START_REF] Aymard | Cellkinetics based calibration of a multiscale model of structured cell populations in ovarian follicles[END_REF][START_REF] Clément | Multiscale modelling of ovarian follicular selection[END_REF], see next section). Whatever the granularity of the models, their common main output is the change in the number of follicles with time (hence age), and, as a corollary, the cumulative number of follicles lost by atresia. Follicle counts are thus critical data to fit the model parameters and control the quantitative behavior [START_REF] Jones | The relationships between age, numbers of oocytes and fertility in virgin and multiparous mice[END_REF]. Unfortunately, there are many limits in follicle counts. First, except in the latest stages of development, counting follicles is an invasive measurement, so that there do no exist any true kinetic experimental data. Datasets retrieved from a given individual (animal or patient) provides only a snapshot of the follicle population at a given age, and time is somehow a statistical reconstruction obtained from different individuals observed at different ages. In addition, most studies are performed in 2D on sampled ovary slices, and the assessment of the whole population is based on stereological concepts, leading to a more or less reliable 2D/3D reconstruction [START_REF] Skodras | Computer-generated ovaries to assist follicle counting experiments[END_REF]. Up-to-date protocols, such as those based on transparization (clearing the tissue to gain ob-servation depth) [START_REF] Feng | CLARITY reveals dynamics of ovarian follicular architecture and vasculature in threedimensions[END_REF][START_REF] Mckey | Combined iDISCO and CUBIC tissue clearing and lightsheet microscopy for in toto analysis of the adult mouse ovary[END_REF] can get rid of the 2D constraints, yet they still acquire static data and are limited to small species compatible with whole ovary imaging. Great efforts should be put on improving the time resolution and reliability of follicle counts, through novel or improved in vivo imaging techniques and associated image analysis algorithms, possibly involving deep learning [START_REF] Sonigo | High-throughput ovarian follicle counting by an innovative deep learning approach[END_REF].

Cell populations dynamics

Somatic cells, and especially granulosa cells, undergo a sequence of cell events marking out the different steps of folliculogenesis (Figure 2). A couple of cell-centered models have investigated these critical cell biology issues [START_REF] Monniaux | Multiscale modelling of ovarian follicular development: From follicular morphogenesis to selection for ovulation[END_REF]:

• Activation [START_REF] Clément | Stochastic nonlinear model for somatic cell population dynamics during ovarian follicle activation[END_REF] Is cell proliferation concomitant or posterior to the flattened-to-cuboidal transformation? Is the transformation rate amplified locally or globally, or do transformation events occur independently?

• Preantral growth [START_REF] Clément | Coupled somatic cell kinetics and germ cell growth: multiscale model-based insight on ovarian follicular development[END_REF][START_REF] Clément | Analysis and calibration of a linear model for structured cell populations with unidirectional motion : Application to the morphogenesis of ovarian follicles[END_REF] How do granulosa cells organize into concentric layers, while sensing overcrowding? How does the molecular dialog between the oocyte and surrounding cells control the balance between oocyte enlargement and cell proliferation?

• Terminal development [START_REF] Clément | Multiscale modelling of ovarian follicular selection[END_REF][START_REF] Aymard | Cellkinetics based calibration of a multiscale model of structured cell populations in ovarian follicles[END_REF] Which is the good timing to switch from proliferation to differentiation, and get a preovulatory follicle endowed with a proper number of mature cells? How will the cell distribution and vulnerability to apoptosis determine whether a follicle is selected for ovulation, in a fluctuating endocrine environment?

From the methodological viewpoint, these models belong to the so-called structured population models. In addition to evolving with time, cells are distributed according to one or several variables, such as for instance the cytologic age (progression along the cell cycle), or the location with respect to the oocyte. A stochastic formalism, in which cell events are probabilistic, is required in the first stages, due to the small number of cells. A deterministic formalism can be handled for later stages, when the cell number can be considered as a continuous variable. Both formalism types are amenable to rigorous mathematical analysis certifying the predictions on the model behavior. They tackle the cell biology issues using appropriate biomathematical concepts such as (i) extinction time for the duration of activation (transformation of all flattened cells), (ii) doubling times and leading layer for cell proliferation in the preantral phase, and (iii) switching time and competition pressure for selection. The concept of extinction time has also been deployed in a mean field model, where each primordial follicle is assimilated to a single average cell subject to irreversible stochastic switch in phenotype [START_REF] Waldherr | Bridging time scales in cellular decision making with a stochastic bistable switch[END_REF]. These models have been designed separately, yet, when put one after the other, they cover the whole folliculogenesis, from activation to ovulation; the final output of one model can be used as initial condition for the next, and it would be worth building a comprehensive simulation environment.

Reducing the structuring variable to a discrete type amounts to dealing with subpopulations in a compartmental setup, as that introduced for follicle population dynamics. For instance, a compartment model has been used recently to sort cell subpopulations according to predefined classes of telomere length [START_REF] Portillo | Mathematical model to study the aging of the human follicle according to the telomerase activity[END_REF].

The main difficulty faced to calibrate the model parameters is the absence of kinetic information; cell numbers are provided as a function of morphological variables such as oocyte or follicle diameter [START_REF] Mcnatty | Hormonal correlates of follicular development in the human ovary[END_REF][START_REF] Lundy | Populations of granulosa cells in small follicles of the sheep ovary[END_REF]. Even worse, while there has been a "golden age" for cell kinetics in the 70s in many biological fields, including ovarian physiology (see e.g. [START_REF] Pedersen | Determination of follicle growth rate in the ovary of the immature mouse[END_REF]), studies dedicated to follicle cell populations have become scarcer and scarcer, compared to the ever increasing reports on molecular biology. In addition, despite of technical improvements (e.g. multiple fluorescent markers), there has been poor gain of information, especially on the quantitative ground. It would be all the more worth renewing the experimental approach, than the available models can be very helpful for interpreting cell kinetics experiments, or even analyzing the data. For instance, most studies assessing the cell cycle (and cell cycle phase) duration from mitotic and labeling indexes assumes implicitly that the population is fully proliferating, with no cell loss or cell cycle exit. Such an assumption is not valid for follicle development, while thorough in silico experiments have been performed in realistic situations [START_REF] Clément | Analysis and calibration of a linear model for structured cell populations with unidirectional motion : Application to the morphogenesis of ovarian follicles[END_REF][START_REF] Postel | A multiscale mathematical model of cell dynamics during neurogenesis in the mouse cerebral cortex[END_REF].

In the cell-centered models, the physical space may be explicitly accounted for, yet biophysical and mechanical processes are only considered implicitly in the rate for cell events such as division, death or migration. Fascinating open questions remain on how follicle cells are distributed within the follicle geometry, which is in turn shaped by cell dynamics. A key point would be to embed the orientation of cell division planes [START_REF] Silva-Buttkus | Effect of cell shape and packing density on granulosa cell proliferation and formation of multiple layers during early follicle development in the ovary[END_REF][START_REF] Rodgers | Dynamics of the membrana granulosa during expansion of the ovarian follicular antrum[END_REF] in relation to cell polarity, intercellular junctions [START_REF] Mora | Characterization and significance of adhesion and junction-related proteins in mouse ovarian follicles[END_REF], and cell cortical stress. Another would be to formulate follicle growth as a moving boundary prob-lem, in which the follicle outer boundary is driven by an internal time-varying pressure due to oocyte growth (preantral stage) or antrum inflation. Appropriate coordinate setups (dealing with the loss of spheric symmetry and oocyte eccentricity) have been proposed to model antrum growth [START_REF] Clark | Follicle structure influences the availability of oxygen to the oocyte in antral follicles[END_REF][START_REF] Bächler | Species-specific differences in follicular antral sizes result from diffusion-based limitations on the thickness of the granulosa cell layer[END_REF] or diffusion of soluble factors up to the cumulusenclosed oocyte [START_REF] Thompson | The ovarian antral follicle: Living on the edge of hypoxia or not?[END_REF], yet in static or quasi-steady state setups eluding the moving boundary difficulty. An interesting parallel could be drawn with the blastocyst to investigate the mechanisms underlying both antrum formation (concept of hydraulic fracturing [START_REF] Dumortier | Hydraulic fracturing and active coarsening position the lumen of the mouse blastocyst[END_REF]), or dynamic adaptation of the mural wall to antrum pressure (oscillations in the wall deepness around a steady-state [START_REF] Ruiz-Herrero | Organ size control via hydraulically gated oscillations[END_REF]). Also, since antral growth is associated with a dramatic increase in vascularization, and expansion of the outer wall into the ovarian stroma, a notable advance would result from embedding the dynamics of theca cells [START_REF] Richards | Ovarian follicular theca cell recruitment, differentiation, and impact on fertility: 2017 update[END_REF]. Similar problems coupling cell dynamics with morphogenesis and mechanical are also encountered during early oogenesis, with the combined migration and mitosis of primordial germ cells, and the formation and breakdown of germ cell cysts. An interesting phenomenological, cell-based modeling of oogenesis [START_REF] Wear | Cellbased computational model of early ovarian development in mice[END_REF] has been developed within an IBM (individual-based model) simulation environment (CompuCell3D), that would deserve further analysis and quantitative validation.

Conclusion

A promising perspective is to couple cell population dynamics with follicle population dynamics in a multiscale approach. The marker of follicle maturity is then directly related to the status of all follicle cells (the simplest being the total cell number), possibly in a multi-valued (e.g. considering different cell status) or even infinite dimension framework (continuous distribution of cells). Such a methodologically difficult approach has already been set up to study the emergence of follicle selection from cellular bases [START_REF] Clément | Multiscale modelling of ovarian follicular selection[END_REF][START_REF] Aymard | Cellkinetics based calibration of a multiscale model of structured cell populations in ovarian follicles[END_REF]. It would be worth extending it to earlier follicle stages, and ultimately designing a comprehensive, cellbiology based modeling framework of the whole folliculogenesis.

A parallel effort is expected on the experimental side to manage monitoring quantitatively and dynamically the ovarian follicle populations and follicle cell populations. A first step in this direction could come from exploiting ex vivo/in vitro devices such as cultures of ovarian cortex [START_REF] Telfer | A twostep serum-free culture system supports development of human oocytes from primordial follicles in the presence of activin[END_REF] or pooled follicles [START_REF] Hornick | Multiple follicle culture supports primary follicle growth through paracrine-acting signals[END_REF], organoid-like [START_REF] Jung | In vitro differentiation of human embryonic stem cells into ovarian follicle-like cells[END_REF], and microfluid systems [START_REF] Xiao | A microfluidic culture model of the human reproductive tract and 28day menstrual cycle[END_REF] to monitor follicle growth dy-namically and quantitatively, even if culture systems depart from physiological situations. Dealing with specific modeling issues devoted to the oocyte is beyond the scope of this review.

We would yet like to mention interesting modeling works dedicated to different steps of meiosisacentriolar spindle assembly during meiotic divisions [START_REF] Letort | A computational model of the early stages of acentriolar meiotic spindle assembly[END_REF], associated errors leading to aneuploidy [START_REF] Tyc | Mathematical modeling of human oocyte aneuploidy[END_REF], or resumption of meiosis II induced by calcium oscillations at fertilization [START_REF] Dupont | Oscillatory Ca2+ dynamics and cell cycle resumption at fertilization in mammals: a modelling approach[END_REF]-as well as to the control of nucleus positioning [START_REF] Ahmed | Active mechanics reveal molecular-scale force kinetics in living oocytes[END_REF] associated with oocyte fate.

Declarations of interest: none

References of outstanding interest

• [START_REF] Kano | AMH/MIS as a contraceptive that protects the ovarian reserve during chemotherapy[END_REF] This is a thorough study on the effect of transient AMH overexpression on the activation of primordial follicles. It includes particularly valuable kinetics on follicle counts (time to follicle "disappearance" after overexpression onset, time of "reappearance" of follicles in different stages after treatment).

• [START_REF] Zheng | Two classes of ovarian primordial follicles exhibit distinct developmental dynamics and physiological functions[END_REF] This study has settled the question of the involvement of the first formed primordial follicles, originating from the ovary medulla in the early reproductive life in rodents, which was raised in the early 70s by J.A. Hirshfield. It also provides very interesting comparative studies on follicle growth rates between the first activated follicles and the next ones, which can be assimilated respectively to an AMH-free environment and AMH-regulated one.

• [START_REF] Dumortier | Hydraulic fracturing and active coarsening position the lumen of the mouse blastocyst[END_REF] A nice, very integrative study mixing live imaging, physical modeling, cell and developmental biology. The newly discovered mechanism of hydraulic fracturing underlying blastocyst formation is very inspiring for the investigation on follicle antrum formation.

• [START_REF] Xiao | A microfluidic culture model of the human reproductive tract and 28day menstrual cycle[END_REF] An innovative and ingenious microfluidic device, integrating several components of the female reproductive system (of mixed human and mouse origin), and taking advantage of previous works that had led to optimized culture environments. The physiological relevance of this device is reinforced by dynamic perfusion of pituitary hormones and re-circulation of secreted factors emanating from cultured tissues. Combining in the future such a device with proper imaging sensors would offer a great potential for real-time monitoring of follicle dynamics.

References of special interest

• [START_REF] Sonigo | High-throughput ovarian follicle counting by an innovative deep learning approach[END_REF] This study applies cutting edge methodologies in deep learning to tackle the crucial question of ensuring the reliability of follicle counts.

• [START_REF] Mora | Characterization and significance of adhesion and junction-related proteins in mouse ovarian follicles[END_REF] This is one of the very few studies providing information on the types of intercellular junctions encountered along follicle development. Such cell biology-oriented studies, as others dedicated to the follicle cytoarchitecture, are essential to fill the gap between the molecular level and the tissular/organic level.

• [START_REF] Lundy | Populations of granulosa cells in small follicles of the sheep ovary[END_REF] A wealth of quantitative information on cell numbers in relation to follicle and oocyte growth, covering both follicle activation and early growth. As other works of KP McNatty lab (see e.g. [START_REF] Mcnatty | Hormonal correlates of follicular development in the human ovary[END_REF] for antral growth), such a study gives insight into the balance between cell proliferation and oocyte (or antrum) growth.

• [START_REF] Pedersen | Determination of follicle growth rate in the ovary of the immature mouse[END_REF] A seminal study on granulosa cell kinetics during follicle development, based on a former, still in use, classification of developmental stages established by the same author. The strong point in the cell cycle study is the achievement of labelled mitosis curves, which is a much more reliable approach than point MI (mitotic index) or LI (labeling index) measures. This transition is followed by the proliferation of cuboidal cells and initiation of the slow growth phase of the oocyte. On the cell population level, progression of activation can be monitored by the increase (resp. decrease) in the proportion of cuboidal (resp. flattened) cells in the whole cell population. On the follicle level, the more or less concomitant growth in oocyte diameter, with respect to the increase in the total cell number, preserves the unilaminar structuring of the follicle in these early stages. Subtle unbalance between proliferation and oocyte growth can yet be observed as soon as these very early stages in pathological conditions such as PCOS (PolyCystic Ovary Syndrome) [START_REF] Stubbs | Abnormal Preantral Folliculogenesis in Polycystic Ovaries Is Associated with Increased Granulosa Cell Division[END_REF]. Panel B : Follicle growth as a multi-layered structure. While the oocyte goes on enlarging, the increase in cell number associated with active cell proliferation exceeds the capacity of a single layer, and cells progressively occupy an increasing number of concentric layers. The balance between oocyte growth and granulosa cell proliferation is tuned by a tight molecular dialog. Somatic cells promote oocyte growth through the KIT signaling pathway, while growth factors of the BMP (Bone Morphogenetic Protein) family diffuse from the oocyte to the granulosa cells and stimulate proliferation. On the follicle level, this dynamic balance can be monitored by plotting the cell number against the oocyte diameter [START_REF] Clément | Coupled somatic cell kinetics and germ cell growth: multiscale model-based insight on ovarian follicular development[END_REF].

In sheep there exist natural mutations (FecII and FecBB affecting respectively BMP15 and BMPR1B) disturbing this balance and leading to anormally large oocytes. In contrast, the KO of Inha (inhibin subunit α) in mouse leads to proliferation in excess with respect to oocyte growth.

Figure 1 :

 1 Figure 1: Main stages of follicle formation and follicle development, and changes in the follicle pool along lifespan

Figure 2 :

 2 Figure 2: Cell fate and cell dynamics during early follicle development

Mitosis

N. Granulosa cells % of cells

Flattened Cuboidal