Sébastien Roques

Mathias Legrand

Patrice Cartraud

Carlo Stoisser

Christophe Pierre

Modeling of a rotor speed transient response with radial rubbing

Keywords: Speed transient, rotor, stator, contact, rubbing, finite element method, explicit time- marching technique

Introduction

In nuclear power plant turbosets similar to the one pictured in figure 1, the reference design-basis accident consists of a blade-off in the last stage of the low pressure turbine. During an accidental shutdown, FIGURE 1: A real power plant turboset. Courtesy of Électricité de France a severe rotor-stator interaction may occur at critical speeds due to large shaft line displacements originated by the lost blade high imbalance excitation. The potentially induced rubbing between the shaft and the stator, as illustrated in figure 2, leads to an important angular deceleration rate and highly modifies the dynamics of the turbogenerator. Risks of failure of the contacting components may occur due to the heavy friction torque. It is then of primary importance to ensure that the adopted turbine design is capable of going through critical speeds without catastrophic consequences for the shaft line.

Rubbing, known to be a serious malfunction in turbomachinery, has been the subject of a large amount of research and a detailed overview is provided in [START_REF] Muszynska | Influence of rubbing on rotor dynamics -part i & ii[END_REF][START_REF] Muszynska | Rotordynamics[END_REF]. First mathematical models dedicated to rubbing issues were as simple as Jeffcott rotors [START_REF] Pavlovskaia | Nonlinear dynamic interactions of a jeffcott rotor with preloaded snubber ring[END_REF]. They were then extended to flexible rotors through finite element approaches and/or modal synthesis techniques [START_REF] Lalanne | Rotordynamics Prediction in Engineering[END_REF] allowing for more realistic descriptions. However, these studies were limited to constant angular velocity steady states, which implies an increase of the driving torque in case of rubbing, for instance. In [START_REF] Lalanne | Rotordynamics Prediction in Engineering[END_REF], equations of motion were rewritten considering a non-constant but known angular velocity law. Only Dai [START_REF] Dai | Dynamic behavior of the full rotor/stop rubbing : numerical simulation and experimental verification[END_REF] has proposed a rigid rotor model where an additional equation computes the instantaneous rotational velocity depending on the contact forces.

The present work is a preliminary study of a design-basis accident in a nuclear power plant. A large-scale turbine operating at nominal conditions is suddenly disconnected from the electrical network after a blade-off. The main objective of this work is to suggest a predictive numerical tool in order to assess the ability of the designed turbine to go through critical speeds without catastrophic consequences. The present paper focuses on an accidental shutdown originated by a blade-off yielding radial rubbing between the shaft and the diaphragm as displayed in figure 2. Quite similar situations involving blade-tips to casing interaction have already been investigated in [START_REF] Legrand | Modèles de prédiction de l'interaction rotor/stator dans un moteur d'avion[END_REF][START_REF] Lesaffre | Model and stability analysis of a flexible bladed rotor[END_REF]. Nevertheless, it is here assumed that the latter is negligible compared to the shaft-diaphragm interaction. Previous works [START_REF] Muszynska | Synchronous and self-excited rotor vibrations caused by a full annular rub[END_REF][START_REF] Muszynska | Chaotic responses of unbalanced rotor/bearing/stator systems with looseness or rubs[END_REF][START_REF] Chu | Periodic quasi-periodic and chaotic vibration of a rub impact rotor system supported on oil film bearing[END_REF][START_REF] Chu | Bifurcation and chaos in a rub-impact jeffcott rotor system[END_REF][START_REF] Piccoli | Experimental observation of chaotic motion in a rotor with rubbing[END_REF][START_REF] Bartha | Dry friction backward whirl of rotors[END_REF][START_REF] Dai | The partial and full rubbing of a flywheel rotor-bearing-stop system[END_REF][START_REF] Muszynska | Rotor-to-stationary part full annular contact modelling[END_REF][START_REF] Dai | Dynamic behavior of the full rotor/stop rubbing : numerical simulation and experimental verification[END_REF] yield the following classification of rubbing phenomena:

• The duration of the contact involves either partial, i.e. intermittent, or full, i.e. continuous, annular rubbing. Stable vibrations with synchronous, sub-synchronous or super-synchronous responses and chaotic behaviors of the shaft-line with destructive damages of its components can be observed.

• The rotor features forward and reverse whirl motions, as well as oil and dry whip depending on running conditions.

• The contact load can cause heavy or light rubbing.

A reliable description of rubbing is only possible with a detailed finite element model together with a numerical approach capable of tackling transient dynamics and contact constraints. In that view, this paper contributes to a better description of shaft dynamics by incorporating rubbing with unknown angular velocity mainly affected by aerodynamical and contact forces. The related model includes gyroscopic effects and torsional displacements. The stator model is restricted in this study to the diaphragm part with an increasing level of complexity, from rigid to fully flexible. In the governing equations, the dynamics of the rotor and diaphragm are coupled through contact forces. Available approaches in contact mechanics [START_REF] Belytschko | Nonlinear Finite Elements for Continua and Structures[END_REF] are usually the penalty method, the Lagrange multiplier method or the augmented Lagrangian method [START_REF] Simo | An augmented lagrangian treatment of contact problems involving friction[END_REF][START_REF] Laursen | Computational contact and impact mechanics : fundamentals of modeling interfacial phenomena in nonlinear finite element analysis[END_REF][START_REF] Wriggers | Computational Contact Mechanics[END_REF]]. An accurate contact treatment strategy is strongly connected to the integration method. As our interest lies in the characterization of the transient response of the shaft during rubbing, approaches such as harmonic balance or shooting methods, more intended to forced and steady-state responses are discarded, and time marching procedures stand as a natural choice. Two main families are usually found in the literature: implicit versus explicit formulations [START_REF] Noels | Contributions aux algorithmes d'intégration temporelle conservant l'énergie en dynamique non-linéaire des structures (Contributions to energy-conserving time integration algorithms for nonlinear dynamics[END_REF][START_REF] Noels | Combined implicit/explicit algorithms for crashworthiness analysis[END_REF]. It is observed in [START_REF] Giraudon-Guilloteau | Modélisation du contact en dynamique: construction d'un élément simplifié de contact et application à l'interaction rotor/stator[END_REF] that the results highly depend on Newmark parameters for problems involving strong nonlinear terms such as direct contact constraints. Even though implicit methods with automatic control of the time step size [START_REF] Chen | Finite element simulation of nonlinear transient response due to rotor-stator contact[END_REF] may be used, conditionally stable explicit methods appear to be more relevant to non-differentiable terms such as contact. Accordingly, the well-known central finite difference scheme is adopted here. A previous study [START_REF] Legrand | Modèles de prédiction de l'interaction rotor/stator dans un moteur d'avion[END_REF] has shown that the penalty method is not always adapted to contact detection in crash analysis since residual penetrations between components is allowed: cumulative error and dependency of the results to the penalty parameter may be expected. Lagrange multipliers and augmented Lagrange multipliers are prone to Uzawa type algorithms that do not fit in explicit time marching techniques because of the required CPU. Nevertheless, the predictioncorrection algorithm forward increment Lagrange method developed in [START_REF] Taylor | Lagrange constraints for transient finite element surface contact[END_REF] embeds the Lagrange multiplier approach in an explicit technique keeping the advantages of both. It has been proved reliable for contact-impact problems [START_REF] Baillet | Finite element simulation of dynamic instabilities in frictional sliding contact[END_REF] by properly satisfying contact detection and ensuring displacement compatibility and is therefore preferred in this study.

Rotor speed transient modeling

The turbine model introduced in this work consists in Timoshenko beams for the rotating flexible shaft complemented by circular rigid disks for the bladed disks. Imbalances are originated by concentrated masses and the bearing behavior is linearized. It is assumed that the rotor is simply supported at its ends. This model is pictured in figure 3 and each of its components is thoroughly detailed in the sequel.

The geometrical and material parameters of the shaft line are chosen so that the first eigenfrequencies of the rotor match the corresponding ones of the real structures as explained later.

Shaft line components

An accidental blade-off in a nuclear power plant turboset leads to a controlled shutdown of the shaft line and subsequent high deceleration rates due to fluids friction and shaft-to-diaphragm resisting torques. Accordingly, attention must be paid to the modeling and formulation of such phenomena since the angular position of a cross-section becomes an unknown of the problem. In-plane as well as out-of-plane bending vibrations, torsional vibrations together with axial vibrations are considered. Gyroscopic terms are included and the following assumptions are also accounted for:

1. The shaft has a uniform circular cross-section along its length.

2. The shaft is initially balanced.

3. External torques with constant direction along the reference centroidal axis are applied at each end of the shaft.

Rigid disk

The kinetic energy T r d of a rigid circular disk rotating at angular velocity φr takes the form:

2T r d = M r d ur 2 + vr 2 + ẇr 2 + I r d x θ2 u r cos 2 θ v r + θ2 v r + I r d z θ2 u r sin 2 θ v r + φr + βr 2 + 2 φr + βr θu r sin θ v r (1)
where M r d , I r d x and I r d z respectively stand for the mass, the second moment of inertia and the polar moment of inertia of the considered rigid disk. Referring to figure 4, the notations used for the rotor kinematics are (u r , θ v r ) for the bending displacements in the (e X , e Z )-plane, (v r , θ u r ) for the bending displacements in the (e Y , e Z )-plane, w r for the axial displacement and β r for the torsional twist, both along e Z . These quantities are considered in equation ( 1) at the disk center. In the small perturbation framework, the kinetic energy, including gyroscopic effects, is rewritten as follows:

2T r d = M r d ur 2 + vr 2 + ẇr 2 + I r d x θ2 u r + θ2 v r + I r d z φr + βr 2 + 2 φr + βr θu r θ v r (2)

Shaft

The kinetic energy T s of the shaft corresponds to the integration of the disk energy along its longitudinal direction thus yielding: where ρ s , S s , I s x and I s p respectively refer to the mass density, the cross-section area, the moment of inertia and the polar moment of inertia of the shaft.

2T s =
The potential energy U s of a spinning Timoshenko beam is equal to:

2U s = l s 0 E s S s w 2 r,z + E s I s x θ 2 u r ,z + θ 2 v r ,z + G s I s p β 2 r ,z + k s G s S s u r ,z -θ v r 2 + v r,z + θ u r 2 dz (4)
where E s , G s and k s are respectively the Young modulus, the shear modulus and the transverse shear form factor of the shaft.

Linearized bearing

In real turbines, oil film bearings support the shaft and involve nonlinear terms described by Reynolds' equations. The virtual work δW of these external nonlinear forces acting on the shaft, F u r and F v r along u r and v r respectively, can be expressed in a general manner as:

δW = F u r F v r δu r δv r (5)
For small displacements with respect to the equilibrium position of the shaft in the bearings, stiffness and damping coefficients of the oil film, respectively k o i j and d o i j , can be calculated by linearizing Reynolds' equations [START_REF] Lee | Vibration Analysis of Rotors[END_REF] leading to:

F u r F v r = - k o x

Imbalance

In order to reflect a blade-off, a heavy imbalance is introduced through a concentrated mass m i located at a distance r i from the geometric center of the shaft. Its kinetic energy is:

2T i = m i ur 2 + vr 2 + ẇr 2 + r 2 i φr 2 + 2r i φr vr cos ϕ rur sin ϕ r (7) As mentioned previously, rotor-to-stator contact corresponds to a shaft-diaphragm interaction in this study. Therefore, the stator modeling reduces to the diaphragm representation. The real diaphragm or flow straightener comprises inner and outer rings and twisted blades as depicted in figure 5 and three different types of diaphragm have been developed:

Diaphragm

1. Diaphragm D1 is shown in figure 6(a): it is fully rigid and can be seen as a mathematical boundary.

Its main purpose is the validation of the contact algorithm.

2. Diaphragm D2 is a rigid ring with flexibility and damping as depicted in figure 6(b). This model is an extension of the first model with a formulation based on a previous EDF study [START_REF] Clément | Projet macadam -influence de la prise en compte du contact rotor-stator sur l'estimation des efforts aux paliers des groupes turbo-alternateurs de 900mw et 1300[END_REF]: it is augmented with stiffnesses, dampers and the mass of the inner ring through a mass matrix m d .

3. Diaphragm D3 is a flexible structure illustrated in figure 6(c). The inner ring is now flexible with the use of curved beams and the blades are considered as straight beams.

Inner ring: curved beams

The inner ring, initially in the (e X , e Y )-plane, is depicted in figure 7. Dimensions of the cross-section are small compared to the mean line length, so that Euler-Bernoulli theory holds. Strains are then written as follows [START_REF] Dubigeon | Mécanique des milieux continus (Continuum mechanics)[END_REF]:

ǫ s s = 1 r i r + r v i r + (r i r + r ) u i r,s -r r i r v i r,s s + z θ i r s -r i r w i r,s s γ r s = z r i r + r r i r θ i r s ,s + w i r,s γ z s = -r r i r + r r i r θ i r s ,s + w i r,s (8)
where u i r , v i r and w i r (resp. θ i r s , θ i r r and θ i r z ) respectively stand for radial, tangential and axial displacements (resp. rotations) of the centroidal line as displayed in figure 7. The path variable is s and r i r is the average radius of the inner ring. Assuming r ≪ r i r and a first-order expansion in r , ǫ s s in equation ( 8) becomes:

r i r r i
ǫ s s = v i r r i r + u i r,s -r v i r r i r 2 + v i r,s s + z θ i r s r i r -w i r,s s + O(r 2 ) (9)
Assuming a linear elastic behavior for the diaphragm, i.e. σ s s = E d ǫ s s , τ r s = G d γ r s and τ z s = G d γ z s (Hooke's law) where E d and G d are the Young's and shear moduli together with a uniform rectangular cross-section of the curved beam, the strain energy U i r of a finite element becomes:

2U i r = E d s 2 s 1 S i r v i r r i r + u i r,s 2 + I i r z v i r r i r 2 + v i r,s s 2 + I i r r θ i r s r i r -w i r,s s 2 + G d J i r E d θ i r s ,s + w i r,s r i r 2 ds ( 10
)
where I i r r , I i r z , J i r and S i r respectively stand for the two moments of inertia, the polar moment of inertia and the cross-section area of the inner ring. Similarly, the kinetic energy T i r is defined as:

2T i r = ρ d s 2 s 1 S i r u 2 i r + v 2 i r + ẇ 2 i r + I i r r ẇ 2 i r,s + I i r z ui r r i r -vi r,s 2 + I i r p θ 2 i r s ds (11)
with I i r p denoting the moment of inertia.

Blades: straight beams

The Euler Bernoulli theory is also used for the blades as slender structures. Their energies collapse to equations ( 3) and ( 4) where the angular velocity is equal to zero so that the shear energy vanishes. Consequently, the notations introduced in figure 8 together with the two usual kinematic relationships

θ w b = v b,x and θ v b = -w b,x yield: 2T b = ρ d l b 0 S b u 2 b + v 2 b + ẇ 2 b + I b z θ 2 w b + I b y θ 2 v b + I b x θ 2 u b dx 2U b = l b 0 ES b u 2 b,x + EI b z θ 2 w b ,x + EI b y θ 2 v b ,x + G d J b θ 2 u b ,x dx (12)
where I b z , I b y , I b x , J b and S b respectively stand for the two second moments of inertia, the polar moment of inertia, the cross-sectional polar moment of inertia and the cross-section area of the rectangular straight beam.

Curved beam-straight beam compatibility

The assembling procedure between the straight beam finite elements of the blade and curved beam finite elements of the inner ring at the connecting nodes (red and green finite elements in figure 6(c)) in order to compute the global matrices of the system of interest requires a special attention. This procedure stems from the compatibility conditions along the generalized displacements and forces to be satisfied at each node. As shown in figures 7 and 8, in the present study these conditions reduce to:

u b = v i r v b = u i r w b = w i r
and

θ u b = θ i r r θ v b = θ i r s θ w b = θ i r z ( 13
)
From equation ( 13), the rotation of a cross-section of a Euler-Bernoulli curved beam yields:

θ w b = u i r r i r -v i r,s and θ u b = w i r,s (14)
condition that has to be accounted for in the finite element discretization.

Finite element discretization

The different energies detailed above are discretized according to the finite element technique formulated in displacement. The elements used are detailed hereafter.

Shaft line

Traction and torsion are discretized using the usual linear shape functions. Denoting ξ = z l and q = (w r , β r ) yields:

q(z ) = [1 -ξ ξ] q 1 q 2 ( 15
)
The modified Hermite functions N i j used in [START_REF] Chen | Dynamic stability of a cantilever shaft-disk system[END_REF] for bending are selected. Degrees of freedom u r (z ), θ v r (z ), v (z ) and θ u r (z ) are thus written as follows:

u r (z ) θ v r (z ) = N 11 (z ) N 12 (z ) N 13 (z ) N 14 (z ) N 21 (z ) N 22 (z ) N 23 (z ) N 24 (z )      u r 1 θ v r 1 u r 2 θ v r 2      (16)
and:

v r (z ) θ u r (z ) = N 11 (z ) -N 12 (z ) N 13 (z ) -N 14 (z ) N 21 (z ) -N 22 (z ) N 23 (z ) -N 24 (z )      v r 1 θ u r 1 v r 2 θ u r 2      (17)
Since the angular position ϕ r is unknown, additional boundary conditions are necessary and detailed in section 4.3.

Diaphragm D3

Equations of motion are derived in a very general three-dimensional fashion. However, a numerical investigation conducted in a preliminary phase with a planar diaphragm model showed that conditions w i r = θ i r s = 0 for the inner ring and , where l i r e is the curvilinear length of the finite element, the shape functions are:

w b = θ u b = θ v b =
N 1 (s ) = 1 -3ζ 2 + 2ζ 3 N 2 (s ) = l i r e ζ 1 -2ζ + ζ 2 N 3 (s ) = ζ 3ζ -2ζ 2 N 4 (s ) = l i r e ζ -ζ + ζ 2 (18)
The discretized displacement field becomes, for an element whose nodes are 1 and 2:

u i r (s ) = N 1 (s )u i r 1 + N 2 (s )u i r 1,s + N 3 (s )u i r 2 + N 4 (s )u i r 2,s v i r (s ) = N 1 (s )v i r 1 + N 2 (s )v i r 1,s + N 3 (s )v i r 2 + N 4 (s )v i r 2,s (19)

Contact forces

The forces of particular interest in this study are the unilateral contact and friction forces acting between the shaft and the diaphragm. In order to simplify the contact detection, it is assumed that only one point of the beam cross-section area of the rotor comes into contact with one curved element of the diaphragm, as depicted in figure 10.

General framework

To derive the contact dynamic equations, the master-slave approach [4] is used. Figure 9 introduces the notations : the configurations of the master and slave bodies are denoted Ω (m ) and Ω (s ) with corresponding boundaries Γ (m ) and Γ (s ) and contact interface Γ c . The gap function g (x) between any point x of the master component and its closest counterpart x of the slave one can be computed such as:

g (x) = g 0 (x) + u (m ) (x) -u (s ) (x) • n ( 20
)
where g 0 denotes the initial gap, n the unit outward vector normal to the master surface, u (m ) (x) and u (s ) (x) displacements respective to the master and slaved structures. Every material point x ∈ Γ c must satisfy the following Kuhn-Tucker optimality conditions: where λ N stands for the positive contact force acting on the slave surface in the normal direction. These unilateral contact conditions are augmented with the Coulomb friction law assuming only sliding occurs:

λ N 0, g (x) 0 and λ N • g (x) = 0 (21) Γ c Γ (m ) Γ (s ) Ω (m ) Ω (m ) Ω (s ) Ω (s )
λ T = µ|λ N | ⇒ ∃α ∈ = 0 such as v T = α λ T λ T ( 22
)
for which µ is the coefficient of friction, v T the tangential slip velocity and λ T the contact force acting along the tangential direction. Since the finite element method is applied, the equations of motion of a mechanical model respective to contact dynamics can be written in a very general manner as follows:

M ü + D u + Ku + C T λ = 0 (23)
where M, D, K and u respectively stand for the mass, damping, stiffness matrices and displacement vector of the global system. Lagrange multipliers are stored in vector λ and C is the contact constraint matrix in the normal and tangential directions. Embedded in an explicit time marching approach, a solution method for solving equation ( 23) together with conditions ( 21) and ( 22) is proposed in [START_REF] Taylor | Lagrange constraints for transient finite element surface contact[END_REF][START_REF] Legrand | Two-dimensional modeling of an aircraft engine structural bladed disk-casing modal interaction[END_REF].

Application to rotor-diaphragm system

By choice in this study, the master component is the rotor whereas the diaphragm is the slave structure. Within the small perturbation framework with planar diaphragm models, it is legitimately assumed that a cross-section of the shaft always remains in the (e X , e Y ) plane. Accordingly, the determination of x of equation ( 20) can then be obtained through the following considerations:

1. diaphragms D1 or D2: the contact occurs at hot spot C in figure 10, i.e. highest rotor eccentricity.

Consequently the gap function in the radial direction is equal to the distance CD:

CD = Ō ′ D -Ō ′ C (24)
2. diaphragm D3: in figure 10, contact on the diaphragm occurs at point D with same angular position as point C. This approximation allows for CPU time savings together with accurate predictions of rotational velocities and bearing loads [START_REF] Roques | Development of beam-to-beam contact detection algorithms for rotor-stator rubbing applications[END_REF].

Boundary conditions and initial conditions

The initial conditions are given as follows: operating at normal conditions, the turbine is suddenly disconnected after the blade-off. The latter gives rise to a heavy mass imbalance while no driving torque holds anymore. Only aerodynamical and Newtonian fluid frictions act on the shaft thus initiating a slowdown with possible contact interaction: • newtonian fluid friction torque: C newt = -A newt φr ;

O O O ′ Ω O R C D g e X e X e Y λ T λ T λ N λ N C fric initi al c o n fi g u
• aerodynamical friction torque: C aero = -A aero φr 2 .

where A newt and A aero coefficients were identified in [START_REF] Fortin | Modélisation du ralentissement d'un groupe turbo-alternateur (modelling of a turbogenerator slowdown[END_REF]. Fluids forces acting on the turbine are uniformly distributed along the shaft as a first approach. The additional boundary conditions relative to the angular position are:

• Constant driving torque C max 0 at one end of the shaft.

• φ-dependent alternator resisting torque -C max φr Ω at the other end of the shaft.

It it noteworthy saying that C max > 0 refers to the run-up of the shaft line whereas C max = 0 refers to the accidental shutdown.

Governing equations

Rotor

Energy-based Hamilton's principle is used for the derivation of the system's equations of motion. Gyroscopic terms ( φr + βr ) θu r θ v r in equation ( 2) give rise to a strong nonlinear coupling between the flexural, torsional displacements and the angular position. Similarly, the term ( φr + βr ) 2 highly couples the angular position to the torsion angle whereas the contact constraints add even more complexity to the system dynamics. Neglecting torsional vibration, the governing equations are as follows:

M 1 Ẍr + (D o + φr D 2 ) Ẋr + K o + K 1 + φr K 2 X r + C T r λ = + N i j =1
m i j r i j φr sin ϕ r + φr 2 cos ϕ r F |u r +φr cos ϕ r + φr 2 sin ϕ r F |v r [START_REF] Muszynska | Influence of rubbing on rotor dynamics -part i & ii[END_REF] where X r is the vector of the rotor generalized displacements (bending, traction and torsion when considered), and M 1 , D 2 , K 1 , K 2 respectively the mass, gyroscopic and two stiffness matrices. Matrix C r stands for the rotor contribution to the contact matrix C. The linearized forces acting in the oil-film bearings appear through matrices K o and D o . The right-hand side of equation ( 25) stand for the imbalance forces originated by N i eccentric masses. Finally, the scalar equation governing the angular position ϕ r takes the form:

   ρ s I s p l s + N i j =1 m i j r 2 i j + N r d j =1 I r d Zj    φr + ẌT r MX r + ẊT r M Ẋr + N i j =1 m i j r i j v i r cos ϕ r -ü i r sin ϕ r |j = F |ϕ r = C(t ) -C f (26)
where N i is the number of rotor imbalances and u i r and v i r stand for their nodal vertical and horizontal displacements. N r d is the number of rigid disks, C(t ) corresponds to the driving torque and C f to the sum of the fluids friction and contact friction torques whose expression is C fric = µr s λ N , where r s is the outer radius of the shaft. When torsion is considered, terms that cannot be written in a matrix form and not mentioned here for the sake of simplicity, are added to the governing equations.

Diaphragm

The equation of motion respective to the diaphragm dynamics (diaphragm D2 or D3) is:

M d Ẍd + D d Ẋd + K d X d -C T d λ = 0 ( 27
)
where X d represents the vector of the generalized displacements of the diaphragm with consistent matrices M d , D d and K d for mass, damping and stiffness of the system. Matrix C d is the diaphragm contribution to the contact matrix C.

General algorithm

Coupled nonlinear equations of motion ( 25), ( 26) and ( 27) are now solved using the central finite difference approach. Using notation X = {X r , X d } storing the rotor and diaphragm generalized displacements (X reduces to X r for diaphragm D1), and a ≡ X or ϕ r , the discretization in time yields:

ȧ = a n +1 -a n -1 2∆t and ä = a n +1 -2a n + a n -1 ∆t 2 ( 28
)
where a n refers to quantity a at time t n and ∆t to the time step. The proposed time marching procedure simultaneously includes the calculation of the displacements and the Lagrange multipliers within three main steps [START_REF] Legrand | Two-dimensional modeling of an aircraft engine structural bladed disk-casing modal interaction[END_REF]:

1. Prediction: the governing equations are solved assuming that there is no contact between the two structures.

2. Contact detection: the gap function g p (x)2 is computed with the predicted displacements. A correction of the displacements is required if a penetration is detected between the contacting structures.

3. Correction: if a penetration is detected (the gap value is negative), the corresponding Lagrange multiplier in the normal direction is calculated in order to satisfy the non-penetrability condition.

The tangential contact force is then deduced from the Coulomb friction law eventually allowing for the calculation of the contact resisting torque within quasi-Newton loop.

At each time step, the matrices depending on the spinning speed and acceleration are updated. If a contact is detected, the rotational velocity and acceleration change according to the contact friction torque. Details of the time stepping procedure are provided in algorithm 1. The time step must guarantee the

Algorithm 1 Proposed solution method

Initialization of X and ϕ r at t 0 and t 1 : X 0 , X 1 and ϕ 0 r , ϕ 1 r for n = 2, . . . , n end do prediction of (X n+1 , ϕ n+1 r ) assuming a linear angular acceleration and no contact search for (X n+1 , ϕ n+1 r ) satisfying equations ( 25), ( 26) and ( 27)

while residual > ǫ do if no penetration then λ (k ) N = 0 else Lagrange multipliers calculation λ (k ) N ← λ p N so that (21) is satisfied 2 correction of (X n+1 , ϕ n+1 r ) through λ N , λ T and C fric end if new solution guess (X n+1 , ϕ n+1 r ) end while if ϕ n+1 r < ϕ n r then break 3 end if (X n-1 , ϕ n-1 r ) ← (X n , ϕ n r ) (X n , ϕ n r ) ← (X n+1 , ϕ n+1 r ) t ← t + ∆t end for
numerical stability of the integration scheme and satisfy the Courant-Friedrichs-Lewy (CFL) criterium for an explicit technique: a linear analysis proves that the Lagrange multiplier do not modify the linear time step size. However, numerical experience suggests that contact conditions do slightly reduce the time step [START_REF] Belytschko | Contact-impact by the pinball algorithm with penalty and lagrangian methods[END_REF], consideration which is accounted for in the present study.

Time integration results

The developed algorithm is validated through the computation of a linear speed transient where ϕ is known. Results were successfully compared with a commercial code. Consequently, calculations of transient run-down responses during accidental shutdowns are exclusively considered in what follows.

It is recalled from section 4.3 that the initial conditions in displacement and velocity of the shutdown correspond to the steady state at nominal operating conditions preceding the shutdown. From this initial state, the shutdown is initiated by setting C max = 0 and by introducing the Newtonian fluid and aerodynamical friction resisting torques together with a heavy blade-off mass imbalance. To authors' knowledge, there is neither analytical nor numerical reference solutions. Accordingly, subsequent simulations are validated through a time step size convergence analysis. The mechanical parameters of the study are listed in table 1: they are adopted so that the first eigenfrequency of the real investigated turbine is retrieved.

Diaphragm D1

Without torsion

In a preliminary study, the behavior of diaphragm D1 is explored without torsional vibrations. The imbalance mass is set to 45 kg. Figure 11(a) depicts the computed rotational velocity. It illustrates the convergence with respect to the time step. Initially spinning at its nominal speed, the turbine slows down

k oxx 2 • 10 5 N/m k oy y 5 • 10 5 N/m d oxx η × 2 • 10 5 N • s/m d oy y η × 5 • 10 5 N • s/m η 2 • 10 -4 (f ) k d x x 4 • 10 9 N/m k d y y 4 • 10 9 N/m d d x x 3.8 • 10 5 N • s/m d d y y 3.8 • 10 5 N • s/m m d x x 10 4 N • s 2 /m m d y y 10 4 N • s 2 /m
due to fluids friction with no contact detected between rotating and stationary parts. When approaching the first critical speed (19.6 Hz) at t = 3 s, the initial 8 mm gap is completely consumed as shown in figure 12(a) and re-opens at t = 6 s. During interaction, the angular deceleration of the shaft is clearly affected by the contact forces.

The clearance between the rotor and the diaphragm, depicted in figure 12(a), shows that the contact is well treated: even for a large time step still ensuring convergence, the residual penetration3 is negligible (lower than 0.2 µm, value to be compared with the initial gap of 8 mm). Further results which are given in Section 7.4 show that the residual penetration decreases with the time-step: this confirms the stability and consistence properties of the proposed algorithm.

In order to analyze the frequency content of the rotor vibrations, a fast Fourier transform (FFT) is performed. As expected, the first bending modes of the rotor are excited by the imbalance: the FFT of the rotor response before contact, from t = 0 to t = 3 s, depicted in figure 15(a), shows synchronous components together with natural frequencies of the shaft line in agreement with Campbell diagram 14(a).

The sensitivity of the response to the friction coefficient is depicted in figure 16. For a small value, only the forward whirl motion of the rotor is excited during interaction. When the friction coefficient reaches a threshold (µ ≃ 0.2), a backward whirl phenomenon is observed and the shaft line is violently stopped. Due to this specific reason, the friction coefficient is now set to µ = 0.1. Furthermore, only results for ∆t = 10 -5 s are now presented since convergence of the quantities of interest with respect to the time step is achieved (see gap function in figure 13 The angular speed, the clearance separating the shaft from the diaphragm as well as the vertical displacement are given in figure 11(b), figure 12(b) and figure 13(b) respectively. Comparison with the casestudy involving no torsion shows that these results are almost identical. Similar conclusions hold for other quantities of interest such as the load on bearing, see Section 7.4. Since this paper aims at providing a numerical tool capable to predict the overall response of the turbine, in the sequel, torsion is not accounted for. This allows for simplifying the model and saving subsequent computation time while preserving a good approximation of quantities of interest. The natural frequency of the diaphragm, equal to 100.7 Hz, is contained in the response. Concerning the other frequencies of the diaphragm response, these are common with the frequency content of the rotor during interaction (see figure 15(d) in comparison to figure 15(a)) and result from the coupling between rotor and diaphragm dynamics during contact.

Diaphragm D3

Once again, the rotational velocity evolution during shut-down is very similar to those obtained for diaphragm models D1 and D2, see figure 11(c). By contrast, the vibration levels of the shaft appear to be significantly higher than in the previous models, as shown in figure 13(d). From the gap function evolution displayed in figure 12(d), it is noticeable that the flexible diaphragm vibrates more during a longer contact interaction, up to 8 s. As illustrated in figures 15(d) and 15(e), the frequency content of the rotor is similar to the one interacting with diaphragm D2. Also, the frequency content of the new diaphragm D3 contains higher harmonics because of the chosen discretization but the magnitude of the main harmonics is very similar. Moreover, a beating vibration phenomenon is observed in responses and after interaction. This phenomenon more in figure 13(d) and is peculiar to situations where comparable frequency contents structures are combined, through contact efforts in the present study.

Influence of the diaphragm model

In a general manner, the rotor response during accidental situations depends on the geometrical and material properties of the diaphragm. Since the load acting in the bearing is the main quantity of interest when designing a turbine, its time evolution with respect to the diaphragm model is now investigated. A linear calculation, thus discarding contact constraints in the formulation, of the rotor speed transient provides the maximal possible load in bearings equal to 2 • 10 6 N. However, a better estimation of this value is obtained when including the rotor-to-diaphragm contact constraints and obviously depends on the diaphragm models detailed in table 2.

Diaphragm D1 corresponds to the fastest calculation in term of CPU time but underestimates the bearing load. Also, the influence of torsion appears to be negligible and only increases the computational time. Since an enhanced version of this model, diaphragm D2 represents a good compromise between CPU time and approximation of the bearing load but leads to a bouncing behavior of the rotor because of a non physical contact treatment inherent to rigid body modelings. Finally, flexible diaphragm D3 provides the most accurate estimation of the loads exerting in the bearings. Due to the flexibility of this model, the shaft and diaphragm displacements reach larger amplitudes directly affecting the bearing loads. The numerical cost is inevitably a significant increase of the CPU time.

Conclusions and prospects

The emphasis of the present study was twofold.

The purpose was first to establish a new theoretical formulation for a speed transient analysis of a rotor supported by linearized journal bearings together with gyroscopic effects, torsional and flexural couplings and contact/rubbing dynamics. The rotational velocity is not known and has to be simultaneously solved with the equations of motion. Discretizations in time and space are performed with the explicit and well-known central finite difference scheme and the usual finite element approach, respectively. Contact constraints are treated through the Lagrange multiplier technique in order to avoid unwanted residual penetrations between contacting structural components. Several diaphragm models were explored, starting from a fixed rigid one, up to a fully flexible one which incorporates the main features of the true structure. The related node-to-line contact algorithm has shown to be reliable since convergence with respect to the time step is achieved. Accordingly, a tool for analyzing rotor-diaphragm interactions is now available for the study of the turbine behavior, especially as it passes through critical speeds after a blade-off. Second, the developed numerical tool is capable of computing global quantities, such as bearing loads which are of great interest in the design process of a turbine, as well as usual structural displacements. Main results are: (1) torsional twist only slightly influences the maximum load acting in the bearings but allows for a reliable estimation of the torsional stresses along the shaft. As a matter of fact, bearing loads appear to be more dependent on the diaphragm modeling. (2) Rigid diaphragms overestimate contact loads with the shaft and consequently underestimate loads in the bearings. (3) Displacements of flexible diaphragms due to rubbing are larger than in the case of rigid models. This implies an augmentation of the shaft vibrations and of the bearing loads. The use of flexible diaphragm is therefore mandatory for an accurate prediction of the bearing loads.

FIGURE 3 :

 3 FIGURE 3: FE model schematic

FIGURE 5 :

 5 FIGURE 5: Schematic representation of a real diaphragm

FIGURE 6 :

 6 FIGURE 6: Investigated diaphragm models: (a) diaphragm D1, (b) diaphragm D2, (c) diaphragm D3

FIGURE 7 :

 7 FIGURE 7: Notations for a curved beam

FIGURE 8 :

 8 FIGURE 8: Notations for a straight beam in its local reference frame

FIGURE 9 :

 9 FIGURE 9: Notations for a contact problem formulation

TABLE 1 :

 1 Numerical characteristics of the shaft line. The following equality for diaphragm D3 holds: radius of the inner ring = outer radius of the shaft + gap. Coupling terms of stiffness and damping matrices of the bearing are such as k oxy = k oy x = d oxy = d oy x = 0. Diaphragm D2 model has been constructed based on a previous EDF study and simplifying assumptions namely k d x y = k d y x = d d x y = d d y x = 0; conditions m d x y = m d y x = 0 are also considered. (a) rigid disk, (b) shaft, (c) diaphragm D3, (d) imbalance, (e) oil-film bearing, (f ) diaphragm D2

  (a) and displacements in figure11(b) for instance).

FIGURE 11 :

 11 FIGURE 11: Rotational velocity for φr versus time in different configurations: (a) D1 without torsion: ∆t = 10 -5 s ( ), ∆t = 5 • 10 -6 s ( ), ∆t = 10 -6 s ( ), ∆t = 5 • 10 -7 s ( ), (b) D1 with torsion ( ) and without torsion ( ), (c) D1 ( ), D2 ( ), D3 ( ) without torsion

FIGURE 12 :

 12 FIGURE 12: Clearance at node 4 between the shaft and surrounding diaphragm in different configurations: (a) D1 without torsion: ∆t = 10 -5 s ( ) and ∆t = 5 • 10 -6 s ( ), (b) D1 with torsion, (c) D2 without torsion, (d) D3 without torsion

Figure 15 (

 15 Figure 15(d) depicts the spectrum of the vertical displacement of the diaphragm during interaction.The natural frequency of the diaphragm, equal to 100.7 Hz, is contained in the response. Concerning the other frequencies of the diaphragm response, these are common with the frequency content of the rotor during interaction (see figure15(d) in comparison to figure15(a)) and result from the coupling between rotor and diaphragm dynamics during contact.

FIGURE 13 :FIGURE 14 :FIGURE 15 :

 131415 FIGURE 13: Vertical displacement of rotor at node 2 for (a) D1 without torsion: ∆t = 10 -5 ( ∆t 5 • 10 -6 s ( ), (b) D1 with torsion, (c) D2 without torsion, (d) D3 without torsion

FIGURE 16 :FIGURE 17 :

 1617 FIGURE 16: Influence of friction coefficient on the computed rotational velocity: µ = 0.1 ( ), µ = 0.2 ( ), µ = 0.3 ( ), µ = 0.5 ( ), µ = 1 ( )

TABLE 2 :

 2 Sensitivity of the results to the diaphragm models -column 1: ∆t (s), column 2: CPU time (s), 3: on bearing (10 5 and 4: residual penetration (µm). (a) diaphragm D1 without torsion, (b) diaphragm D1 with torsion, D2 without torsion, (d) diaphragm D3 without torsion

		(a)		(b)
	10 -5	2,007	3.5424 0.76720	10 -5	2,738	3.4427 0.69885
	5 • 10 -6	4,389	3.5490 0.22468	5 • 10 -6	5,456	3.4632 0.24114
	10 -6 24,292 3.5309 0.01492	10 -6 26,684 3.4244 0.00807
	5 • 10 -7 47,980 3.5296 0.01636	5 • 10 -7 53,111 3.5539 0.00205
		(c)			(d)
	10 -5	2,012	3.7041 0.74075	10 -5	7,742	5.4562 58.8643
	5 • 10 -6	4,431	3.7437 0.14734	5 • 10 -6	15,918	5.4513 15.3205
	10 -6 22,458 3.7343 0.00836	10 -6	80,629	5.4522 0.62922
	5 • 10 -7 44,186 3.7295 0.00451	5 • 10 -7 161,265 5.4547 0.15778

Corresponding author: sebastien.roques@edf.fr

superscript p for prediction

The Lagrange multipliers are calculated assuming a constant angular position in the correction step. This approach is valid since the solution guess takes place in the neighborhood of the converged solution.

This condition means that the shaft is now rotating in a counter-clockwise direction: this is physically possible but not of interest in the present study.

The residual penetration depends on the numerical precision of the computer as well as the numerical error in inverting matrices and the precision ǫ in the nonlinear solver.

The model of diaphragm D3 is introduced in the next section


Christine Ferraris

Christian Martel

Laurence Vignollet

Modelling the case study with LDL and implementing it with LDI

Keywords: e-Learning, Learning Design Language (LDL), Modelling Learning Activity, Pedagogical Scenarios, IMS LD. Interactive Demonstration Observable Scenario Structure

LDL (Learning Design Language) is an education modelling language which was designed to model collaborative activities. This paper details the solution proposed to address, with LDL and the associated infrastructure LDI, the two steps presented in the workshop "Comparing Educational Modelling Languages on a case study". The modelling of the case study is described in detail. Its operationalization and execution are illustrated with videos. The second step is addressed in conclusion.

Introduction

LDL (Learning Design Language) is an EML (Educational Modelling Language) for modelling educational activities in order to produce executable activities in existing digital environments. It relies on a powerful meta-model which allows the representation of various situations, particularly collaborative ones, with few concepts. It is briefly described in the second section of this paper, and detailed in [START_REF] Martel | LDL: a Language to Model Collaborative Learning Activities[END_REF].

The main objectives of the Learning Design field is to give teachers and instructional designers the conceptual and the technological means to create and manage learning activities. Three steps can be distinguished in the life-cycle of a scenario: the design, the operationalization and the execution as shown in Figure 1. Considering LDL, a complete framework, from design to execution of the scenarios, is now available, associated with a methodology. To help the teachers and the instructional designers in charge of the design of collaborative learning activities using LDL, the authors of LDL propose a graphical notation of each concept defined in the meta-model. They also make recommendations for the design stage in an associated methodology. Finally, they propose a design tool called model editor [START_REF] Martel | Learning Design Language to model the activity supported by services[END_REF]) which produces the LDL-XML binding of the model.

The Learning Design Infrastructure (LDI) associated to LDL will assume the operationalization of the scenario: first the participants are chosen, the roles are distributed, the services and the contents selected ; then, a corresponding activity is created in the targeted digital environment (such as Plone or Moodle) for the chosen participants. When operationalized, the activity can be performed: LDI manages its execution. It also provides the means to observe the progression of the activity.

The LDL team was one of the challengers during the workshop at ICAL'T 2006. This paper describes how they cover all the steps proposed for the specified case study. The first step is covered in section 2 for the modelling of the case study and in section 3 for the operationalization and execution of the model built. Finally, the way the second step is covered is described in section 4.

LDL to design the case study

A scenario is a codified and formal description of a future activity. It is simply an arbitrary notation with which to describe a future activity planned by the author, independently of the context of its execution. Its production is based on an abstract model of a learning activity which is in turn constructed using a meta-model that defines all the concepts and their relationships that are useful for the description and modelling of the activity.

LDL meta-model

A scenario can be considered as a specification of the activity. Designing a scenario with LDL consists in describing the activity using the concepts of the associated meta-model (see Figure 2). The concepts are succinctly defined as follows:

• Interaction: The main concept of the model. The interactions specify the exchanges the participants will have during the learning activity.

• Structure: A structure specifies the order in which the interactions will be played out, i.e. the learning flow. Several types of structure exist, to specify whether the order is sequential or parallel. • Role: Held by the participants, a role is defined by the list of the interactions in which a participant having this role is involved.

• Arena: Where (and with whom) interactions are performed.

• Rules: Give the start-up and stopping conditions of the interactions or structures. Used to regulate the activity.

• Positions: Stated by participants/agents to make choices or to evaluate an activity.

• Observables: Every concept of LDL is observable. An observable requires an observer. An observed position is associated to each observable. Sensors are used to determine the value of an observable. Observed positions are also useful in producing structured traces of activities.

The LDL meta-model is "Interaction centred". The modelling is guided by the spatial metaphor: participants' activity is situated, in the context of a service or a content. So are their interactions in the corresponding arenas. In its simple form, an interaction is an action from one participant to another in a specific place. The initiator is called "the addresser", the other "the addressee".

LDL includes an original concept, that of position, which allows one to specify these reactions or actions and thus to adapt the activity according to them. Two types of position exist, depending on how the value of the position is obtained. A «declared» position means that its value is given explicitly by the participant involved. The value of an « observed » position is assigned by the system which deduces it from observation data on the activity's progression. The instructional designer will have to define which aspects are to be observed. These aspects are called the observables.

The value of a position is tested in the conditional part of a rule.

For a more complete description, see (Martel et al., 2006) and see also the description of the experimentations done with LDL in the Kaleidoscope project in [START_REF] Michelet | Adaptive learning scenarios for detection of misconceptions about electricity and remediation[END_REF] and in [START_REF] Lejeune | To set up pedagogical experiments in a virtual lab : methodology and first results[END_REF].

LDL's associated methodology

A methodology has been defined to facilitate the modelling of learning activities. It is described in detail in [START_REF] Ferraris | Handbook of Visual Languages in Instructional Design: Theories and Practices[END_REF].

To summarize, this methodology helps to formalize an informal version of a scenario, usually based on teachers' description of the activity they want to carry out with learners.

The LDL authors consider that the slightest educational activity is complex, and that it conceals several interwoven activities. They first propose to distinguish four types of activity: learning, observation, assessment, organization. Then, they recommend following the steps of the formalization process set out below:

• The first step is thus to identify the activities which constitute the overall one, determining the type of each of them.

• The second step is to define the relationships between the activities, that is to say the learning flow and what the activities share with each other.

• The third step is modelling each activity, following the steps mentioned below:

o Identify the interactions by naming them and define the learning flow;

o Specify each interaction using the LDL concepts roles and arenas;

o Specify the start-up and stopping conditions of each interaction;

2.3 Modelling the case study

Identification of the activities

This corresponds to the first step of the methodology. When analysing the informal scenario given in the case study description, we can easily identify the four kinds of activity (organization, learning, observation, assessment), which are usually described implicitly.

When analysing the informal scenario, it may be observed that the four kinds of activity we have mentioned in the methodology section actually exist in the planet game. For the learning activity, it is obvious. The informal scenario explicitly includes a lot of information about this activity. The learning objectives are listed, together with the way to reach these objectives (analysis of the clues, discussion in the chat room, negotiation with the other team, etc.). The aim is clearly for learners to acquire knowledge about the organisation of the solar system.

The informal scenario also mentions the interventions of the teacher in relation to the learning activity. Indeed, the teacher has to observe the teams whilst the learning activity proceeds. S/he may observe the team members' exchanges, their productions, the way they use the clues, etc. This is an activity per se during which the teacher may identify the difficulties encountered by the learners, may put new clues at their disposal, etc. This is the observation activity.

The assessment activity is also clearly mentioned. It consists of a summative assessment which occurs at the end of the learning period. It aims at checking the level and the solidity of their knowledge. These activities are represented graphically in Figure 3 , together with the learning flow between them (the schedule and the synchronisation points). Note that the learning activity appears twice, once for each team: the two activities correspond to the same scenario.

The activities' relationships

The description of the relationships between activities has to be supplemented by the definition of the shared objects: arenas and position. The ALearn1 and ALearn2 learning activities share the forum which supports the negotiation of the exchange of clues. This arena is also shared with the AObs observation activity, as the teacher involved in the observation activity has to be able to at least observe what happens in it (see

Figure 4).

Furthermore, the two learning activities, the observation one and the assessment one share the "end of the learning activities" position. This position is a decision by the teacher to be made explicitly in the AObs activity. Indeed, s/he observes the two ongoing learning activities and chooses the time at which to stop them, when s/he considers that the learners have worked sufficiently and have acquired enough clues and knowledge. Thus the AObs activity has to share this position with the two learning ones. As a consequence, these two activities will be aware of it and will be able to end. Note that it is also shared with the AAssess activity, as the end of Alearn1 and Alearn2 coincides with the beginning of AAssess.

Modelling of each activity

This section gives the specification of the activities, following the flow given in Figure 3.

The organization activity

When one analyses the informal scenario, the organization activity would seem to be dedicated to the choice of the group by the learners. Moreover, the organization activity allows one to prepare the learning activity. This means being able to operationalize and start the observation activity and the two learning activities. This leads to the definition of the interactions provided in the following section.

The interactions, associated to the roles and the arenas

The main flow is sequential and combines several interactions. The places where the activity takes place have also to be identified: the interactions are situated. The rules and the associated positions The organisation activity starts automatically with the reading of the statement. It has been stated that the choice of the group can only be made when all the learners have finished reading the statement. This is translated in the model by a start-up rule of the "Choose a group" interaction which tests whether each learner has taken the position "I have finished reading the statement". This leads to the introduction of the "Play the interlude" interaction in the scenario, so that the learner spends the time waiting for the other learners to finish reading the statement. The start-up of this interaction is also related to the end of the "Read the statement" one, when the learners are considered individually.

The interactions are summarized in

The operationalization of the learning activity can start when the learners have chosen their group. All the other interactions start automatically, as soon as the previous one is over.

The stopping of all the interactions of the organization activity depends on the user's action: when the user declares that s/he has finished, the interaction stops and the following one can start.

This is in the organisation activity where the teacher takes the position "I declare that the learning activity can start"; one position per group.

The learning activity

The learning objectives are listed in the informal scenario, together with the way to reach them. The learning scenario describes what may happen in the learning activities. There are two learning activities in the planet game, one for each team involved. The main participants in these activities are the learners. Thus most of the interactions specified in this scenario are intended for them.

The interactions, associated to the roles and the arenas

To model the learning activity, we have first considered the requirements described in the informal scenario. This had led to the design of a first minimal scenario including 3 interactions (see table 2): the "Negotiate clues with the other team", "Discuss with your team members", and "Read the interview with the expert" ones. This skeleton was then extended in order to propose funnier and richer interactions to the participants. It was also an opportunity to show LDL's expressiveness. For example, the "Complete the planets' properties table" interaction allows the learners to gather and share the clues discovered and the planets' properties. The arena supporting this interaction is a shared table in which clues and properties can be registered. Another example is the "Take a test to win a clue" interaction. It has been added as another, fun way for the learners to obtain clues. They can choose a quiz to answer in the "test bag" and, if the answer to the quiz is correct, they win a clue (the "Get clue" interaction in table 2). Finally, the "View the group members" interaction has been added to provide the members of a team with awareness. This interaction allows them to know the identity of their team mates involved in the same shared quest.

The teacher is in charge of deciding when the learning activities ends (the "Stop the learning activity" interaction). To help with taking such a decision, s/he observes what is going on in the activity. The observation is carried out in the "observation activity", described by the observation scenario (see following section). Teachers can also add new quizzes in the test bag, in order to enlarge the set of clues which can be discovered by the learners. To help them produce these quizzes, they may access the ESA (European Space Agency) website via the "Go and consult the ESA website" interaction.

Clue

The table

The list of the team mates Figures 5 and 6 are snapshots of the player put respectively at the learners' and teacher's disposal during the learning activity, once the scenario has been operationalized and the execution has begun.

The interactions to which they had access at the moment when the snapshot was taken are listed in the upper part of the interface. In Figure 5, the learner has chosen to perform the "Complete the planets' properties table" interaction. That is why the arena associated to this interaction appears on the interface. At that moment, the "Get clue" interaction is not visible, as the learner either has not yet answered a quiz or has done so, but has given a wrong answer. In figure 6, the teacher is currently performing the "Go to the QTI folder" interaction. It gives her/him access to the corresponding arena, allowing the addition of new quizzes. The arenas appearing in the two previous figures have been chosen during the operationalization phase. They reify the arenas described in a generic way in the scenario.

The rules and the related positions

Interactions are proposed simultaneously to their addressers, who can choose which one to perform. This means that they are organised in a main parallel structure and that they start automatically as soon as the learning activities are started ("automatic start-up" rule). The "Get clue" interaction is an exception. It has to be put at the learners' disposal only when -first -they have chosen the "Take a test to win a clue" interaction and -second -they have answered the test correctly. This has been modelled in the learning scenario as follows:

1. The interactions "Take a test to win a clue" and "Get clue" are embodied into a sequential structure which is itself embodied in the main parallel structure. An "automatic start-up" rule is attributed to this new structure.

2. A start-up rule is defined for the "Get clue" interaction. It tests the position "I have answered the quiz correctly " of the learner, the position named P_answer in Table 3. This position has thus to be defined.

3. The P_answer position is defined as being an observed one. It is associated to an observable defined in the scenario as "The answer to the question".

This is an example of an observed position. Observables, positions and rules are the means proposed by LDL to take into account what occurs in an ongoing activity and to influence the performance of the activity with regard to these events. It should be noted that several kinds of basic observables have been defined (see following section). They can be combined to compose more complex ones.

Table 3 shows the complete descriptions of some examples of observed and declared positions defined and used in the learning scenario. Some of these positions are shared by several activities. This is the case for example for P_start_learning and P_stop_learning. P_start_learning is a position taken by the teacher in the organisation activity. They are also checked in the condition part of the start-up rule of the learning activities. The observation activity is specified to allow the teacher to apprehend and verify the progress of another activity and to detect difficulties encountered by the participants. To manage this, it has to be performed in parallel with the activities to be observed.

The importance of allowing such observations has led the authors of LDL to propose that what is to be observed be specified in the model of the activity. To do so, they propose that each concept of the model be observable (Martel et al. 2006). For a description of the observable concept, see section 4.1.

Considering the Planet Game example, the teacher may be interested in the observation of the learning and of the assessment activities. So, the interesting concepts to be observed are: the productions (the identity cards of the celestials), exchanges between teams, exchanges within each team, the progression of the activity itself, etc.

Here are some examples of concepts defined as observable in the learning activity, associated with their semantics:

• For a given arena, who is inside? -This provides the localisation of the learners.

• The state of the interactions (started/active/stopped) -This gives information on the progress of the activity.

• The answer to a test, and more particularly who has answered the tests and is the answer correct? -This helps to know whether the learners manage to obtain more clues, and to decide whether more tests are to be proposed.

• The content of the table -This allows one to know whether the learners have acquired enough knowledge and to decide whether the learning activity can be stopped.

A dashboard is completed with the value of the observed positions associated to these observables.

The interactions, associated to the roles and the arenas

The observation activity gives the teacher the means to access this dashboard through a dedicated interaction. S/he can also access the content of the forum.

The rules and the associated positions

This interaction starts in conjunction with the learning activity. To manage this, both activities share the same position. This position is the one taken by the teacher in the organisation activity "I declare that the learning activity can start". It stops when the teacher so decides (another position is thus used).

2.3.3.4

The assessment activity

After the teacher has decided that the exchanges have ended, the evaluation of the knowledge acquired by the learners can be carried out. This is done through an individual questionnaire about the planet classification in which each learner has to associate the names of the planets to their position relative to the sun. The winner is the first learner to give the right associations: <Planet -Order>. The activity finishes when a winner is nominated.

The interactions, associated to the roles and the arenas Two interactions are executed sequentially: "To Do Test" performed by each learner in a questionnaire (the test), followed by "To Proclaim the Winner", allowing the teacher and the learners to have access to the prize list (from the fastest to the least rapid one who has answered the test correctly). The interaction "To Proclaim the Winner" is thus situated in the prize list.

The rules and the associated positions

As shown in Figure 7, the interaction "To Do Test" starts when the assessment activity is started. It stops when the learner validates the test.

The next interaction, "To Proclaim the Winner", starts when all the learners have validated the test. This means, as in the organization activity, that it starts when each learner has taken the corresponding position. 3 Operationalizing and running the activity An infrastructure, called LDI, and a player have been developed which are in charge of: (1) parsing the scenario, (2) requesting, through the player, the participants and their roles and the resources, (3) requesting, through the player, the targeted environment in which the teacher wants the activity to run, (4) deploying users and resources in the chosen environment, (5) orchestrating the scenario execution using a rule engine which controls the activity progress with regard to the structures and the rules, (6) storing the traces, ( 7) giving administration and supervision interfaces. Operationalization is concerned by steps (1) to (4).

The operationalization of the case study starts with the operationalization of the organisation activity. The operationalization of the other activities is realized through interactions. Part of this process is shown in the following video:

The execution of the operationalized scenario will provide the learners, teachers, tutors, etc. with the means (resources, services, tools, etc.) to take part in the activity. This is illustrated in the following videos. The first one corresponds to the learning activity performed by a learner. The second one to the activity carried out by the teacher, mainly in the observation activity.

The third one illustrates the assessment activity.

4

Step 2: observing, adapting and re-use

Observing the activity

In LDL, every concept is observable; thus, the designer of the scenario can specify what can be observed. The observable is introduced as a means of collecting relevant information on the progress of the activity, information which makes sense to the observer. The collected information is thus situated in the activity. This concept presupposes every time the presence of an observer who also takes part in the activity. This observer is a participant, considering also the system as a participant.

When an observable is declared, a corresponding value will be stored in the trace database when the activity is performed. The value of these observables could thus be used to state the value of an observed position.

Observed positions could therefore be used in the conditional part of a rule and/or to build a dashboard of the activity.

The type of what is observed depends on the type of the concept observed. For instance, for arenas, the intrusion/extrusion, the presence/absence, the answer (to a question) or the text produced in a forum can be observed. For the structures and on the interactions, its state (started/active/stopped) can be observed, but also, who has performed in the structure/interaction, when it was started/active/stopped, and its duration. Finally, it is possible to observe, for the rules, its state (triggered/not triggered) and when it is tested and by whom.

Adapting an activity and re-using a scenario

Two kinds of adaptation of an activity modelled with LDL can be considered. They are described in the following.

Adapting the context of use

The first kind corresponds to the adaptation to another context of use. The scenario is not changed: it is re-used as it has been modelled. What may change is the data that are set during the operationalization phase, that can be considered as the actual parameters of the activity. Such data include the participants to be involved, the contents to be involved -LO (Learning Objects) and services to associate to the arenas in order to reify them -the distribution of roles between participants but also the number of groups to be involved in the activity. Changing these data can be done very easily within the operationalization phase. For instance, changing the number of groups involved in the planet game is just a matter of creating as many learning activities as the desired number of groups.

When the contents (LO and services) are changed, there is a change in the learning context. For instance, the planet game scenario could be re-used in the context of learning traffic rules, with an individual examination at the end. This kind of adaptation definitely corresponds to the re-use of the scenario.

Adapting the activity as it runs

The second kind concerns the adaptation of the activity whilst it is going on, that is to say when its progress should be modified according to the events that happen. This is called the "adaptation on the fly". Once more, it is possible to distinguish between two ways of carrying out adaptation on the fly, considering the cases where the adaptations are predefined in the scenario or not.

First, events, as well as the way to react to them, may have been predicted by the designer and modelled in the scenario. The LDL concepts of rule, position and observable have been defined for that purpose. They allow one to specify what to observe (the observables), how to interpret these observations (the positions of the participants) and how to react (rules indicating which interactions to start for which participants according to their collective or individual positions). For example, this is the case of the "Get clue" interaction in the learning scenario. It becomes visible to participants when and only when a quiz has been completed successfully. This is also the case for the "Add new quiz" interaction. It is provided freely to the teachers who monitor the ongoing learning activities. As they are able to observe what is going on (this is done in the observation activity), they can evaluate the current situation and decide when it is relevant to add a quiz. This example is different from the "Get clue" one, in which what to observe, how to interpret the observations and how to react are completely modelled and predefined in the learning scenario. In the "Add new quiz" one, the teacher is in charge of observing, interpreting and deciding on the reaction (provide a new quiz). In both, the interactions are predefined and modelled in the scenarios. Here the scenario designer has anticipated a potential tension that may occur in the activity: the clues provided are not sufficient, and the learners are not able to achieve their goal.

But a designer cannot imagine every problem, tension or opportunity that could occur. So it is necessary to provide participants in an activity -at least the one who monitors it (e.g. the teacher in the example of the planet game) -with the means of modifying its progress dynamically, even if the situation faced has not been predefined. For example, the teacher could adapt the difficulty of a test by modifying the time-limit; s/he could put some help content at the learners' disposal, etc. This would lead to changing a rule dynamically, as the activity is running, or to creating new interactions, etc. In a general way, this requires the provision of means of modifying, adding or deleting any concept of an LDL scenario that describes an activity, whilst the activity is going on.

Dynamically adding, modifying or deleting instances of LDL concepts is technically supported by LDI. It can be done in an ad hoc way, via the administration functionalities provided by the infrastructure. But this remains a hazardous process and adaptation on the fly within LDL/LDI still has to be studied.

Conclusion

The challenge proposed during the workshop "Comparing Educational Modelling Languages on a case study" at ICALT in 2006 was the opportunity for the authors of LDL to show the expressiveness of their language and the efficiency of the associated infrastructure, LDI.

With respect to the modelling of the case study, details have been given in this paper, highlighting the more interesting ways of modelling.

First of all, one novelty is to propose to distinguish between four types of activity while analysing the informal scenario. For this case study, this leads to the breakdown of the activity into five activities, starting with an organization one, followed, when the teacher so decides, by two learning ones (collaborative, one per group) and an observation activity, and ending, when the teacher so decides, with an assessment activity (individual).

Then the detailed description of each activity underlines the original concepts of the languages and their use:

• The modelling of the organization activity emphasizes the way to automatically organize the initialisation of the other activities. It also shows that a position, "Read the statement" in this case, could be used either to start the following interaction, "Play the interlude", or to synchronise the start of the "Choose a group" interaction for all the learners.

• The modelling of the learning activity stresses how the activity could be adapted dynamically.

It also illustrates the use of observed positions.

• The modelling of the observation activity details the concept of observed positions that could be used to observe what is going on. It gives several examples of such positions and their use.

• The modelling of the assessment illustrates the use of positions to synchronise the activities.

6 References

Figure 1 :

 1 Figure 1: The life cycle of the model of the activity

Figure 2 :

 2 Figure 2: The life cycle of the model of the activity

Figure 3 :

 3 Figure 3: The four types of activities identified for the case studyFinally, there are some elements which are related to organisation. They are mentioned in a less explicit way. The most obvious one is to set up the two teams. To be carried out, the planet game requires this step. It also requires the teacher to prepare some instructions intended for the learners, to

Figure 4 :

 4 Figure 4: The forum is shared by the two learning activities and the observation one

Figure 5 :

 5 Figure 5: Learners' player during the learning activity.

Figure 6 :

 6 Figure 6: Teacher's player during the learning activity.

Figure 7 :

 7 Figure 7: The XML binding of the interaction "To Do Test"

Table 1 ,

 1 associated to the roles which are involved in the interaction.

	AObs
	ALearn1	Alearn2
	forum

Table 1 :

 1 The interactions of the organization activity

Table 2 :

 2 The interactions of the learning activity

Table 3 :

 3 Examples of positions used in the learning activity

	Position Id	Title	Value	Type	taken by… on…	Shared
	P_start_learning I start the	True or	Declared teacher	The learning	Yes
			learning	false			activity
			activity
	P_stop_learning I stop the	True or	Declared teacher	The learning	Yes
			learning	false			activity
			activity
	P_answer	The exactitude	True or	Observed learner	The question	No
			of my answer	false
	2.3.3.3	The observation activity