Elias Khalifeh

E Piollet

Antoine Millecamps

A Batailly

Non-linear modeling of centrifugal stiffening effects for accurate bladed component reduced-order models

Keywords: rotor/stator interaction, rubbing, nonlinear dynamics, unilateral contact dynamics, friction, abradable coating

published or not. The documents may come

Non-linear modeling of centrifugal stiffening effects for accurate bladed component reduced-order models

Modélisation non-linéaire des effets centrifuges pour le développement de modèles réduits de composants aubagés E. Khalifeh1 , E. Piollet 1 , A. Millecamps2 , A. Batailly 1

Résumé

La modélisation des effets centrifuges, et notamment du raidissement centrifuge, pour les composants aubagés est particulièrement importante dans l'optique de représenter précisément leur comportement dynamique dépendamment de la vitesse de rotation. Les effets centrifuges se manifestent de deux façons : (1) par une modification de la raideur de la structure et, (2) par une modification de sa géométrie. Dans le cadre de l'hypothèse des petites perturbations, lorsque des modèles éléments finis linéaires sont utilisés, une hypothèse fréquente dans la littérature scientifique est l'utilisation d'une représentation polynômiale de degré quatre de la matrice de raideur en fonction de la vitesse de rotation. Ce développement polynômial peut toutefois s'avérer insuffisant pour représenter avec précision la modification de géométrie du composant. En effet, l'erreur effectuée avec un tel développement sur le déplacement des noeuds du sommet d'aube est typiquement du même ordre de grandeur que le jeu aube/carter lui-même, impliquant une erreur de 100 % sur le jeu aube/carter prédit. Cet article porte sur la présentation d'une méthodologie permettant de créer des modèles réduits précis de modèles éléments finis 3D de composants aubagés, en prenant en compte les effets centrifuges tout en assurant une description très précise des jeux aubes/carter. La qualité des résultats obtenus est démontrée dans la première section de l'article. Puis, dans une deuxième section, il est mis en évidence que cette méthodologie permet de prédire de nouvelles interactions que les modélisations linéaires usuelles ne permettent pas de prédire. Une attention particulière est portée à la sensibilité numérique de la méthodologie proposée. En contrepartie d'une précision accrue, cette méthodologie doit être mise en place avec prudence afin d'éviter de possibles instabilités numériques.

Mots-clés interaction rotor/stator; frottement aube/carter; dynamique non-linéaire; dynamique du contact; frottement; revêtement abradable

INTRODUCTION

The ability to accurately predict complex non-linear interactions within modern aircraft engines-such as those due to unilateral contact constraints between rotating components and their static counterpart [START_REF] Zilli | Nonlinear dynamics of a simplified model of an overhung rotor subjected to intermittent annular rubs[END_REF][START_REF] Williams | Simulation of blade casing interaction phenomena in gas turbines resulting from heavy tip rubs using an implicit time marching method[END_REF][START_REF] Padova | Casing treatment and blade-tip configuration effects on controlled gas turbine blade tip/shroud rubs at engine conditions[END_REF][START_REF] Ma | Vibration response analysis of blade-disk dovetail structure under blade tip rubbing condition[END_REF][START_REF] Batailly | Unilateral contact induced blade/casing vibratory interactions in impellers: analysis for rigid casings[END_REF]-is closely related to the efficient modeling of the structural components under investigation. In particular, accounting for inertial-or centrifugal-effects on bladed components remains challenging since it comes at a significant computational cost. The impact of inertial effects on bladed components is twofold: (1) the stiffness of the component changes with the angular speed Ω which yields a possibly large shift of its first eigenfrequencies and, (2) the blade geometry is modified which yields a modification of the blade-tip/casing clearance configuration. For most aircraft engine stages (fan, compressors and turbines, see Fig. 1), existing numerical strategies relying on the combination of reduced order models and a linear approximation of centrifugal effects [START_REF] Sternchüss | On the reduction of quasi-cyclic disks with variable rotation speeds[END_REF][START_REF] Batailly | Unilateral contact induced blade/casing vibratory interactions in impellers: analysis for rigid casings[END_REF] provide an accurate representation of the component's stiffness over a wide angular speed range. However, because it relies on linear finite element models, this strategy yields an approximated calculation of the blade-tip/casing clearance [START_REF] Batailly | Minimizing clearance consumption: a key factor for the design of blade robust to rotor/stator interactions ?[END_REF]. Indeed, the importance of the error on the blade-tip displacement associated to the use of a linear finite element model quickly reaches the same order of magnitude as the very tight blade-tip/casing clearance itself thus possibly yielding a 100 % error on the blade-tip/casing clearance configuration. Recent numerical investigations have underlined the key role of blade-tip/casing clearances both from a structural [START_REF] Batailly | Conjectural bifurcation analysis of the contact-induced vibratory response of an aircraft engine blade[END_REF] and an aerodynamic [START_REF] Erler | Desensitization of axial compressor performance and stability to tip clearance size[END_REF] standpoint. Hence, it is particularly important that numerical models be very accurate with respect to this quantity. In this article, the numerical strategy initially presented in [START_REF] Sternchüss | On the reduction of quasi-cyclic disks with variable rotation speeds[END_REF] and used in [START_REF] Batailly | Unilateral contact induced blade/casing vibratory interactions in impellers: analysis for rigid casings[END_REF] in combination with modal reduction and cyclic symmetry, is generalized and improved in order to obtain a much more accurate representation of blade-tip/casing clearance configuration. The limiting assumption involving a fourth-order polynomial development of the stiffness matrix in terms of the angular speed-which implies a linear computation of centrifugal effects [START_REF] Sternchüss | On the reduction of quasi-cyclic disks with variable rotation speeds[END_REF]-is here replaced by a more versatile formulation consistent with a non-linear calculation of centrifugal effects. The obtained reduced-order model thus advantageously combines an enhanced modeling of its stiffness variation as a function of the angular speed Ω and a very accurate representation of the blade-tip/casing clearance configuration.

In the first section of the article, details are given with respect to the blade model used for validation of the proposed method. In the second section, the employed numerical procedure is thoroughly presented, the exact values of the coefficients for a polynomial interpolation of degree six of the blade's stiffness are given explicitly. In the next section, the focus is made on the convergence of the blade's eigenfrequencies depending on: (1) the considered reduction parameter and (2) the accuracy of the predicted blade-tip/casing clearance configuration over a wide angular speed range. Then, in the final section, the computed reduced-order model is used for the simulation of rotor/stator interactions. The obtained results are compared with those obtained with the linear formulation. Lowpressure compressor blade.

Note: For the sake of confidentiality, all presented results have been normalized.

BLADE OF INTEREST

The proposed methodology is applicable to any bladed component of an aircraft engine or a large gas turbine. In this paper, it is applied on the finite element model of a low-pressure compressor blade, see Fig. 2. All computations carried out in this article (eigenfrequencies, blade/casing clearances, contact simulations, etc.) are based on this blade model. It should be underlined that the extension of the proposed methodology to full bladed disk structures is immediate since it is fully compatible with the developments introduced in [START_REF] Batailly | Unilateral contact induced blade/casing vibratory interactions in impellers: analysis for rigid casings[END_REF].

FORMULATION

The polynomial interpolation of the stiffness matrix K(Ω) presented in [START_REF] Sternchüss | On the reduction of quasi-cyclic disks with variable rotation speeds[END_REF] is the basis of the employed formulation:

K(Ω) = Np p=0 Ω 2p [K p ] (1)
This interpolation only relies on the following assumptions: (1) the angular speed Ω is constant and (2) the damping and the stiffness matrices do not depend on the excitation frequency. It is recalled in [START_REF] Sternchüss | On the reduction of quasi-cyclic disks with variable rotation speeds[END_REF] that for most industrial applications, a linear computation of inertial effects is acceptable. In that case, N p = 2 which yields the usual formulation [START_REF] Sternchüss | On the reduction of quasi-cyclic disks with variable rotation speeds[END_REF] used in [START_REF] Batailly | Unilateral contact induced blade/casing vibratory interactions in impellers: analysis for rigid casings[END_REF]:

K(Ω) = K 0 + Ω 2 K 1 + Ω 4 K 2 (2)
In the particular case where the angular speed range of interest is Ω ∈ [Ω a ; Ω c ] with Ω a = 0, the matrices K 0 , K 1 and K 2 may be expressed as functions of the stiffness matrices obtained for Ω = Ω a = 0, Ω = Ω c /2 and Ω = Ω c :

K 0 = K(Ω a = 0) K 1 = 1 3Ω 2 c 16K( 1 2 Ω c ) -K(Ω c ) -15K(Ω a = 0) K 2 = 4 3Ω 4 c K(Ω c ) -4K( 1 2 Ω c ) + 3K(Ω a = 0) (3)
When a non-linear computation of inertial effects is required however, higher values of N p must be considered so that the interpolation provides accurate results. Using a higher value of N p implies a new polynomial interpolation and thus new sets of coefficients. For the sake of brevity, only the coefficients obtained for N p = 3 are given explicitly in this article. For N p = 3, based on [START_REF] Sternchüss | On the reduction of quasi-cyclic disks with variable rotation speeds[END_REF], the polynomial development of the stiffness matrix is:

K(Ω) = K0 + Ω 2 K1 + Ω 4 K2 + Ω 6 K3 (4)
with Ω ∈ [Ω a ; Ω c ] and:

K0 = Ω 1 2 Ω 2 2 Ω c 2 (Ω 1 2 -Ω a 2 )(Ω 2 2 -Ω a 2 )(Ω c 2 -Ω a 2 ) [ K(Ω a )] - Ω a 2 Ω 2 2 Ω c 2 (Ω 1 2 -Ω a 2 )(Ω 2 2 -Ω 1 2 )(Ω c 2 -Ω 1 2 ) [ K(Ω 1 )] + Ω a 2 Ω 1 2 Ω c 2 (Ω 2 2 -Ω a 2 )(Ω 2 2 -Ω 1 2 )(Ω c 2 -Ω 2 2 ) [ K(Ω 2 )] - Ω a 2 Ω 1 2 Ω 2 2 (Ω c 2 -Ω a 2 )(Ω c 2 -Ω 1 2 )(Ω c 2 -Ω 2 2 ) [ K(Ω c )] (5)
K1 = - Ω 2 2 Ω c 2 + Ω 1 2 Ω c 2 + Ω 1 2 Ω 2 2 (Ω 1 2 -Ω a 2 )(Ω 2 2 -Ω a 2 )(Ω c 2 -Ω a 2 ) [ K(Ω a )] + Ω 2 2 Ω c 2 + Ω a 2 Ω c 2 + Ω a 2 Ω 2 2 (Ω 1 2 -Ω a 2 )(Ω 2 2 -Ω 1 2 )(Ω c 2 -Ω 1 2 ) [ K(Ω 1 )] - Ω 1 2 Ω c 2 + Ω a 2 Ω c 2 + Ω a 2 Ω 1 2 (Ω 2 2 -Ω a 2 )(Ω 2 2 -Ω 1 2 )(Ω c 2 -Ω 2 2 ) [ K(Ω 2 )] + Ω 1 2 Ω 2 2 + Ω a 2 Ω 2 2 + Ω a 2 Ω 1 2 (Ω c 2 -Ω a 2 )(Ω c 2 -Ω 1 2 )(Ω c 2 -Ω 2 2 ) [ K(Ω c )] (6)
K2 = Ω c 2 + Ω 2 2 + Ω 1 2 (Ω 1 2 -Ω a 2 )(Ω 2 2 -Ω a 2 )(Ω c 2 -Ω a 2 ) [ K(Ω a )] - Ω c 2 + Ω 2 2 + Ω a 2 (Ω 1 2 -Ω a 2 )(Ω 2 2 -Ω 1 2 )(Ω c 2 -Ω 1 2 ) [ K(Ω 1 )] + Ω c 2 + Ω 1 2 + Ω a 2 (Ω 2 2 -Ω a 2 )(Ω 2 2 -Ω 1 2 )(Ω 3 2 -Ω 2 2 ) [ K(Ω 2 )] - Ω 2 2 + Ω 1 2 + Ω a 2 (Ω c 2 -Ω a 2 )(Ω c 2 -Ω 1 2 )(Ω c 2 -Ω 2 2 ) [ K(Ω c )] (7)
K3 = - 1 (Ω 1 2 -Ω a 2 )(Ω 2 2 -Ω a 2 )(Ω c 2 -Ω a 2 ) [ K(Ω a )] + 1 (Ω 1 2 -Ω a 2 )(Ω 2 2 -Ω 1 2 )(Ω c 2 -Ω 1 2 ) [ K(Ω 1 )] - 1 (Ω 2 2 -Ω a 2 )(Ω 2 2 -Ω 1 2 )(Ω c 2 -Ω 2 2 ) [ K(Ω 2 )] + 1 (Ω c 2 -Ω a 2 )(Ω c 2 -Ω 1 2 )(Ω c 2 -Ω 2 2 ) [ K(Ω c )] (8)
where:

Ω a < Ω 1 < Ω 2 < Ω c (9)
It should be underlined that the fundamental difference between Eqs. ( 2) and ( 4) consists in the nature of the stiffness matrices K(Ω i ). While K(Ω i ) stems from linear finite element computations; in Eq. (4), K(Ω i ) is obtained from a static non-linear computation of the application of centrifugal effects.

In order to build numerically efficient models, the polynomial interpolations defined in Eqs. ( 2) and ( 4) may be projected into a suitable subspace using the Craig-Bampton method [START_REF] Craig | Coupling of substructures for dynamic analyses[END_REF]. This reduction procedure, already presented in [START_REF] Batailly | Unilateral contact induced blade/casing vibratory interactions in impellers: analysis for rigid casings[END_REF] is used here but not recalled for the sake of brevity. The reduction parameter η, which controls the number of constraint modes computed at each interpolation point, is set to η = 10. As a summary, key aspects of the proposed formulation-hereafter referred to as NLC-ROM (Non-Linear Centrifugal computation of inertial effects combined with a Reduced-Order Model)-are recalled in Tab. 1.

CONVERGENCE

Eigenfrequencies

Prior to any numerical investigation, it must be checked that the proposed formulation converges as the value of N p is increased. The eigenfrequencies of a reduced-order model ROM [ 1. Differences between the reduced order models (ROM) computed in [START_REF] Batailly | Unilateral contact induced blade/casing vibratory interactions in impellers: analysis for rigid casings[END_REF] and in this article.

computed with different values of N p and a non-linear representation of centrifugal effectsa NLC-ROM-are first compared with the eigenfrequencies of the usual reduced-order model-referred to as ROM-computed with a linear representation of centrifugal effects. This comparison is first carried out over a wide angular speed range Ω ∈ [Ω a ; Ω c ] with ∆Ω = Ω c -Ω a = 3 000 RPM, see Fig. 3. The graphs plotted in Fig. 3 focus on the first two eigenfrequencies of the blade model (which correspond to the first bending and torsion modes, namely 1B and 1T modes). They emphasize that the proposed formulation converges as N p increases. In particular, numerical oscillations of the eigenfrequencies that appear for a too low degree of the polynomial interpolation vanish when N p > 4. Also, it appears clearly in Fig. 3 that there is a smallyet significant-gap between the eigenfrequencies obtained with a converged NLC-ROM (

) and the eigenfrequencies obtained with a ROM ( ). The same gap is observed between the finite element reference solution of each model which underlines that it originates from the different modeling of inertial effects. For the investigated blade model, a linear computation of centrifugal effects yields an overestimation (about 2 to 5 %) of the first bending mode eigenfrequency and an underestimation (about 2 to 3 %) of the first torsion mode.

Similar depicted in Fig. 4. Because the angular speed range is narrower, it is found that numerical oscillations of the eigenfrequencies are significantly reduced, even for low values of N p . As a matter of fact, N p = 3 provides a very accurate approximation of the first eigenfrequencies.

In any case, this first convergence analysis suggests that N p should be sufficiently large in order to obtain an accurate approximation of the blade eigenfrequencies.

Blade-tip/casing clearance configuration

As mentioned in the introduction, one of the motivations for the development of the NLC-ROM is to obtain a much more accurate description of the blade-tip/casing clearance configuration, meaning a more accurate description of blade/casing clearances. The clearance variation at the leading edge due to centrifugal effects is depicted in Fig. 5 for a ROM and NLC-ROM featuring different values of N p . The same quantity observed at the trailing edge of the investigated blade is pictured in Fig. 6.

Both figures underline that over the angular speed range Ω ∈ [Ω a ; Ω c ], ROM and NLC-ROM lead to very distinct clearance variations. The good approximation obtained with a ROM only stands for low angular speeds Ω < Ω a , for Ω > Ω a , the error on the computation of clearance variation with a usual ROM becomes even larger than typical blade/casing nominal clearances. This observation is inherent to the way centrifugal effects are accounted for since the ROM and the NLC-ROM respectively match the linear and non-linear finite element reference solutions.

Contrary to the eigenfrequencies, it is found that the value of N p has low to no influence over the quality of the results: the difference between clearance variations for N p = 3 and N p = 6 is about 300 times lower than the error between a ROM and a NLC-ROM. Thus, this second convergence analysis reveals that for a NLC-ROM, N p has low to no impact on the accuracy of the predicted blade/casing clearance configuration.

Numerical stability

One of the key aspects of a reduced-order model dedicated to complex non-linear structural simulations is the conditioning of the reduced structural matrices. In particular, obtaining accurate and relevant results precludes ill-conditioned matrices.

In order to ensure that reduced structural matrices are not ill-conditioned, the reduction strategy proposed in [START_REF] Batailly | Unilateral contact induced blade/casing vibratory interactions in impellers: analysis for rigid casings[END_REF] relies on a fixed-interface component mode synthesis method. As an advantage towards free-interface methods, fixed-interface methods are known to be less prone to numerical sensitivity [START_REF] Bladh | Component-mode-based reduced order modeling techniques for mistuned bladed disks -part i: theoretical models[END_REF]. Nevertheless, the obtention of a NLC-ROM requires the computation of several sets of component modes: both constraint modes2 and static modes3 must be computed for each interpolation point as underlined in Tab. 1.

For the sake of understanding numerical issues that may arise depending on the reduction basis, let us consider a limit case where N p is very large. For such configuration, as mentioned in Tab. 1, there are (N p + 1)η component modes. The reduction matrix Ψ would thus contain n lines (n is the dimension of the finite element model) and (N p + 1)η columns. Because all the component modes are related to low-frequency constraint modes of the blade model, it is very likely that the span of these component modes will be lower than their numbers thus yielding a rank deficient reduction matrix. While an orthornormalization procedure of the component modes may correct the rank deficiency, it has been observed that the reduced structural matrices may still be ill-conditioned.

In the end, the reduction strategy employed in order to get accurate NLC-ROM imposes that the value of N p should not be too high so that: (1) the NLC-ROM remains computationally efficient and, (2) reduced structural matrices are not ill-conditioned.

Partial conclusion

In conclusion of the convergence analysis carried out in the previous sections, a compromise must be found: on one hand large values of N p will favor an accurate description of the blade eigenfrequencies. On the other hand, because the polynomial interpolation is projected into a lower-dimension subspace, N p must not be too high in order to avoid numerical issues.

The adopted solution relies on the computation of a NLC-ROM over a limited angular speed range ∆Ω. In the following, all NLC-ROMs are computed with ∆Ω = 1000 RPM and N p = 3. As underlined in Fig. 4, such parameters allow for a very accurate description of the blade eigenfrequencies. While specific results are not provided for the sake of brevity, these parameters also provide an excellent description of the blade/casing clearance configuration.

As an alternative to the proposed compromise between N p and the width of the considered angular speed range, one may mention the possibility to carry out a specific mathematical treatment of the reduction basis. Dedicated techniques are available in the literature [START_REF] Rixen | Dual craig-bampton with enrichment to avoid spurious modes[END_REF]. Such investigations go beyond the scope of the article but constitute promising avenues in case a much wider angular speed range would have to be considered.

MODAL ANALYSIS

In this section, the free-vibration modes obtained with a NLC-ROM are compared with those obtained with a ROM by means of the Modal Assurance Criterion (MAC) [START_REF] Allemang | A correlation coefficient for modal vector analysis[END_REF]. The i th free-vibration mode computed for the ROM and the NLC-ROM are respectively denoted Φ i and Φi , both vectors are expressed in the same initial finite element space for the sake of comparison. MAC values are computed by comparing the first ten free-vibration modes of each reduced model to one another:

MAC ij = (Φ i Φj ) 2 (Φ i Φ i )( Φ j Φj ) , i, j = 1, . . . 10 (10)
As mentioned above, both the ROM and NLC-ROM are computed with a Craig-Bampton reduction parameter η = 10. Since the associated reduction basis does not explicitly contain free-vibration modes of the full system, it is expected that the number of free-vibration modes accurately captured by the ROM or NLC-ROM is inferior to η (typically 7 or 8 if η = 10). Consequently, in order to avoid any misleading comparison, the scope of our analysis will be here limited to the first five MAC values (i ≤ 5 and j ≤ 5). Figures 7 and 8 underline the very good agreement between modal shapes obtained by the ROM and the NLC-ROM for Ω = 0 (which is obvious since inertial effects are not applied) and Ω = Ω a . For higher angular speeds such as Ω c and 1.7 × Ω c , non-negligible differences arise between the ROM and the NLC-ROM. This is patent for modes 2 and 3 (respectively the first and second torsional modes of the blade). The angular speed range over which ROM and NLC-ROM feature similar mode shapes is highly dependent on the considered blade model. As a reminder, the investigated blade belongs to the low-pressure compressor of an aircraft engine. It is expected that this domain would significantly shrink if fan blades were to be considered. For such blade, it is expected that a NLC-ROM would be particularly efficient. To the contrary, this domain would probably expand in the case of stiffer high-pressure compressor blades for which a classical ROM may be sufficient.

CASE STUDY

Going beyond the convergence analysis and the comparison between linear and non-linear formulations, the NLC-ROM is here employed for the simulation of blade/casing structural interactions accounting for abradable coating removal. Simulations are carried out with a dedicated computational tool developed for Safran Aircraft Engines [START_REF] Millecamps | Snecma's viewpoint on the numerical and experimental simulation of blade-tip/casing unilateral contacts[END_REF].

Interaction scenario

A rubbing interaction between the low-pressure compressor blade pictured in Fig. 2 and the surrounding abradable coating is simulated over the angular speed range Ω ∈ [Ω a -1000 RPM ; Ω a ]. The abradable layer deposited on the casing is 5 mm thick, the friction coefficient is µ = 0.15, blade-tip/casing clearances at rest are identical from the blade leading edge to the trailing edge and set to 1 mm. Numerical simulations are carried out assuming the angular speed Ω is constant. In total, 100 simulations are performed to accurately cover the considered angular speed range. Each simulation features 50 blade revolutions in a slightly deformed casing (two symmetric bumps are created) in order to initiate contact.

Convergence

Results obtained with a NLC-ROM are compared for different time integration steps: h = 10 -7 s, h = 5 • 10 -8 s, and h = 2.5 • 10 -8 s. A zoom over displacements and contact forces obtained for each of these time steps is ictured in Fig. 9 obtained for different time steps are perfectly superimposed thus underlining the convergence of the numerical time integration procedure when a NLC-ROM is employed. Similar results are obtained when increasing the reduction parameter η or refining the spatial discretization of the abradable coating. These results are not detailed in this article for the sake of brevity.

Wear maps

A wear map depicts the juxtaposition of the final wear profile on the casing predicted at the end of a simulation for each angular speed. The accuracy of these maps is thus directly related to the number of simulations (i.e. of angular speeds) carried out over the investigated angular speed range. These maps are plotted for two casing locations: in front of the blade leading edge (LE) and trailing edge (TE). For both a ROM and a NLC-ROM, two series of simulations are performed. (1) A first batch of results (see Figs. 10 and11) comes from simulations with a ROM and a NLC-ROM as presented above: inertial effects impact both the blade stiffness and the blade/casing clearance configuration. [START_REF] Williams | Simulation of blade casing interaction phenomena in gas turbines resulting from heavy tip rubs using an implicit time marching method[END_REF] The second batch of results (see Figs. 12 and13) has been obtained from simulations for which blade/casing clearances are independent on inertial effects. Accordingly, for these simulations, inertial effects only impact the blade stiffness. In the following. all wear maps feature the same colour code: from white (no wear) to black (maximum wear level: the 5 mm thick abradable layer is locally fully removed). The wear maps plotted in Figs. 10 and11 underline that over the considered angular speed range, there is no major wear predicted in front of the blade's leading edge. Indeed, both maps in Fig. 10 are very similar and do not feature any significant wear level. On the contrary, looking at the predicted wear levels in front of the blade's trailing edge in Fig. 11, specific wear patterns are detected. In particular, for Ω Ω a -∆Ω, six deep lobes are predicted both with a ROM and a NLC-ROM. A clear distinction can be made between the results obtained with a ROM and a NLC-ROM based on Fig. 11. As a matter of fact, it can be seen that for high angular speeds, there is no wear predicted in front of the trailing edge with a ROM. However, identical simulations with a NLC-ROM yield significant wear and even specific wear patterns (see, Ω Ω * in Fig. 11). These first results confirm the critical role that plays the accurate prediction of blade/casing clearance configurations for the prediction of rotor/stator interactions: it is evidenced that the approximation on blade/casing clearance configurations induced by a ROM may filter certain interactions.

In order to assess more specifically the differences between a ROM and a NLC-ROM with respect to the blade's stiffness, wear maps pictured in Figs. 12 and 13 stem from simulations for which blade/casing clearance configurations are fixed and independent from inertial effects. This second batch of results thus allows to assess the role of the blade stiffness matrix K(Ω) in the detection of critical interactions. Results pictured in Figs. 12 and 13 underline that for both the ROM and the NLC-ROM, two interactions are predicted over the angular speed range Ω ∈ a -∆Ω ; Ω a ]: a six-lobe interaction is predicted for lower speeds while a five-lobe interaction is found for higher speeds. In both cases, wear lobes are essentially visible in front of the blade trailing edge, see Fig. 13. Only low wear levels are predicted in front of the leading edge, see Fig. 12. Though more complex wear patterns are observed in front of the leading edge with a NLC-ROM for high angular speeds, more in-depth analysis of the blade time response would be required to confirm a distinct vibration behaviour between the ROM and the NLC-ROM. Indeed, these lobes may very well be related to the blade's transient response which amplitude may vary from the ROM to the NLC-ROM. Overall, results obtained with a ROM and a NLC-ROM are very close when inertial effects do not affect the blade/casing clearance configuration.

In the end, results presented in this section highlight that the NLC-ROM provides significant improvements to the usual ROM allowing for a much more precise prediction of blade/casing clearance configurations. The enriched polynomial interpolation of the stiffness yields consistent results with those obtained with a ROM. More differences are to be expected in the case of blades more sensitive to inertial effects such as fan blades.

CONCLUSION

This article focuses on the presentation of an enriched reduced-order model of bladed components within turbomachines. It aims at accurately capturing the impact of inertial effects on both the component's stiffness anf the blade/casing clearance configurations. It is thoroughly validated by investigating the convergence of the proposed formulation as well as its applicability for the numerical prediction of rotor/stator interactions stemming from structural contacts along the blade-tip. It is demonstrated that the proposed reduced-order model yields a much more accurate prediction of blade-tip/casing clearances, a quantity for which usual linear formulations may fail to provide accurate predictions. The quality of the obtained results (in terms of eigenfrequencies and clearance configurations) is also underlined with respect to full finite element computations. However, the proposed method must be employed carefully. Indeed, attention must be paid to the numerical sensitivity of the obtained reduced-order model, particularly with respect to the conditioning of reduced matrices. It is underlined that considering a too high degree of interpolation may yield ill-conditioned reduced matrices and inaccurate results. In order to avoid any numerical issue, a compromise must be found between the degree of interpolation and the width of the considered angular speed range. Nonetheless, the proposed methodology is applied successfully to the numerical prediction of rotor/stator interactions within a low-pressure compressor. Presented results confirm that accounting for a non-linear computation of inertial effects leads to more physically relevant reduced-order models: specific critical regimes that are predicted with the proposed method are filtered when using classical reduced-order models relying on a linear formulation of inertial effects. It should be underlined that the extension of the proposed approach to full bladed-disks is immediate.

Figure 1 .

 1 Figure 1. Schematic cut-view of an aircraft engine.

Figure 2 .

 2 Figure 2.

Figure 3 .

 3 Figure 3. Eigenfrequencies of the blade model over Ω ∈ [Ωa ; Ωc] for the linear model ( ) and the non-linear models with Np = 2 ( ), Np = 3 ( ), Np = 4 ( ) and Np = 6 ( ). Non-linear ( ) and linear ( ) finite element reference solutions are also plotted for comparison.

Figure 4 .

 4 Figure 4. Eigenfrequencies of the blade model over Ω ∈ [Ω b ; Ωc] for the linear model ( ) and the non-linear models with Np = 2 ( ), Np = 3 ( ) and Np = 4 ( ). Non-linear ( ) and linear ( ) finite element reference solutions are also plotted for comparison.

Figure 5 .

 5 fig.5b

Figure 6 .

 6 Figure 6. Clearance variation at the blade trailing edge over Ω ∈ [Ωa ; Ωc] for the linear model ( ) and the non-linear models with Np = 2 ( ), Np = 3 ( ), Np = 4 ( ) and Np = 6 ( ). Non-linear ( ) and linear ( ) finite element reference solutions are also plotted for comparison.

  ROM and NLC-ROM are compared over a very large angular speed range : Ω ∈ [0 ; 1.7 × Ω c ], MAC values are computed for Ω = 0, Ω a , Ω c and 1.7 × Ω c . Accordingly, four NLC-ROM are computed to carry out this comparison, one for each of the following angular speed ranges: [0 ; 1000] RPM, [Ω a ; Ω a + 1000 RPM], [Ω c -1000 RPM ; Ω c ], and [1.7 × Ω c -1000 RPM ; 1.7 × Ω c ]. MAC values plotted in Figs. 7 and 8 are respective to each of these NLC-ROMs.

Figure 7 .

 7 Figure 7. MAC of the first ten free vibration modes of a NLC-ROM with respect to those of a ROM (Ω = 0 and Ω = Ωa).

Figure 8 .

 8 Figure 8. MAC of the first ten free vibration modes of a NLC-ROM with respect to those of a ROM (Ω = Ωc and Ω = 1.7 × Ωc).

Figure 9 .

 9 Figure 9. Superimposition of results obtained with a NLC-ROM (Np = 3) for different values of h: h = 10 -7 s ( ), h = 5 • 10 -8 s ( ) and h = 2.5 • 10 -8 s ( ).

Figure 10 .Figure 11 .

 1011 Figure 10. Wear maps in front of the leading edge

Figure 12 .Figure 13 .

 1213 Figure 12. Wear maps in front of the leading edge, inertial effects only act on the blade's stiffness

  Table

		5]	NLC-ROM
	centrifugal effects reduction technique value of N p ROM dimension depending on η 1	linear computation Craig-Bampton non-linear computation N p = 2 N p ≥ 2 3η (N p + 1)η

-Department of Mechanical Engineering, École Polytechnique de Montréal, P.O. Box 6079, Succ. Centre-Ville, Montréal, Québec, Canada H3C 3A7

-Safran Aircraft Engines, site de Villaroche, Moissy-Cramayel, 77550, France

-Département de génie mécanique, École Polytechnique de Montréal, P.O. Box 6079, Succ. Centre-Ville, Montréal, Québec, Canada H3C 3A7

-Safran Aircraft Engines, site de Villaroche, Moissy-Cramayel, 77550, France

The actual dimension of both the ROM and the NLC-ROM may be slightly lower than 3η or (Np + 1)η due to an orthonormalization of the component modes that may lead to the elimination of linearly dependent vectors.

Constraint modes are free-vibration modes of the investigated system featuring clamped boundaries.

Static modes are specific static deformed shapes of the investigated system.

ACKNOWLEDGMENT

This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). The authors are also grateful to the industrial partner for supporting this project, Safran Aircraft Engines.


A Batailly

A Millecamps

Minimising clearance consumption: a key factor for the design of blades robust to rotor/stator interactions ?

Keywords: rotor/stator interaction, rubbing, nonlinear dynamics, unilateral contact dynamics, friction, abradable coating, clearance consumption 1B first bending mode

The recent development of a numerical strategy dedicated to the simulation of rotor/stator interactions stemming from structural contacts in modern aircraft engines led to the first optimization of a high-pressure compressor blade profile accounting for criteria related to nonlinear contact simulations. This optimization procedure revealed very significant improvements in terms of amplitudes of vibration but failed to identify key design parameters. Satisfying numerical results were obtained by a minor modification of a combination of many design parameters. Based on this observation, this contribution intends to shed a new light on this previous redesign operation focusing on one key quantity: the clearance consumption. This quantity is presented in the first section. In the second section, results of the redesign operation are recalled before the presentation of original results, featuring detailed interaction maps in the frequency domain, on which focuses the third section of the article. Finally, the blade profiles are extensively compared based on their specific clearance consumption and presented results suggest that this quantity may be key in discriminating acceptable from unacceptable blade profiles with respect to their vibratory behaviour when structural contacts occur.

Minimisation de la consommation de jeu : un facteur clé pour la conception d'aubes robustes aux interactions rotor/stator ? A. Batailly1 , A. Millecamps2

Résumé

Le développement récent d'un outil numérique dédié à la simulation d'interactions rotor/stator engendrées par des contacts dans les moteurs d'avion a permis de mener à bien la première optimisation d'une aube de compresseur haute-pression en prenant en compte des critères issus de la dynamique non-linéaire. Cette procédure d'optimisation a notamment mis en lumière des gains très significatifs en termes d'amplitudes de la réponse vibratoire de l'aube optimisée par rapport à l'aube initiale. En revanche, elle n'a pas permis d'identifier les paramètres de conception clés pour obtenir de tels résultats. L'aube optimisée a été obtenue en modifiant légèrement une combinaison de plusieurs paramètres de conception. Prenant acte de cette observation, cet article propose d'observer sous un autre angle les résultats précédemment obtenus et se focalise sur une quantité particulière : la consommation de jeu. La première section de l'article présente en détails cette quantité. Dans la section suivante, les résultats de la procédure d'optimisation sont brièvement rappelés et complétés par de nouveaux résultats, incluant des cartes d'interaction détaillées dans le domaine fréquentiel. Dans la dernière section, les profils d'aubes obtenus au cours de la procédure d'optimisation sont comparés relativement à leur consommation de jeu et les résultats présentés suggèrent que cette quantité pourrait permettre de discriminer les profils d'aubes acceptables de ceux qui ne le sont pas relativement à leur amplitude de vibration lorsque des contacts aubes/carter se produisent.

Mots-clés interaction rotor/stator; frottement aube/carter; dynamique non-linéaire; dynamique du contact; frottement; revêtement abradable; consommation de jeu

INTRODUCTION

This contribution intends to bring new insight regarding the redesign operation of a highpressure compressor blade within an aircraft engine previously published by the same authors [START_REF] Batailly | Redesign of a high-pressure compressor blade accounting for nonlinear structural interactions[END_REF]. This redesign was carried out on an industrial blade profile which was found sensitive to rotor/stator interactions stemming from blade-tip/casing structural contacts. Such interactions are of particular interest for aircraft engine manufacturers. In fact, an engine efficiency is intimately related to the reduction of parasitic leakages from a stage of the engine to another. Such reduction may only be achieved through a closing of operating blade-tip/casing clearances which leads to more frequent structural contacts that may initiate the aforementioned interaction phenomena. In this context, the development of blades and bladed components robust to structural contacts is a growing field of investigation and motivated many recent numerical investigations [START_REF] Jones | Forced vibrations of a turbine blade undergoing regularized unilateral contact conditions through the wavelet balance method[END_REF][START_REF] Zhou | A variable-coefficient harmonic balance method for the prediction of quasi-periodic response in nonlinear systems[END_REF][START_REF] Petrov | Analysis of bifurcations in multiharmonic analysis of nonlinear forced vibrations of gas-turbine engine structures with friction and gaps[END_REF]. The redesign operation that was previously carried out introduced a few optimization criteria [START_REF] Batailly | Redesign of a high-pressure compressor blade accounting for nonlinear structural interactions[END_REF] related to non-linear structural simulations between the blade of interest and a perfectly rigid casing on which was deposited an abradable coating. It led to an optimized blade profile that was shown, from a numerical standpoint, to be numerically insensitive to the investigated contact scenarios. The improvement in terms of the blade vibration amplitudes following structural contacts were made without impacting aerodynamic performances and the additional mass was limited to +4.15 %. Also, the impact on the blade first natural frequencies was limited to a few percents while the maximum Von Mises stress due to centrifugal effects was reduced by 31.90 %. In fact, it seems that typical design parameters underwent relatively small variations while amplitudes of vibration significantly dropped. Thus, there is a need for further analyses of this redesign operation in order to identify possibly key design factors for bladed components. In order to do so, this paper focuses on one particular quantity: the clearance consumption along the blade tip.

Studies focusing on blade-tip/casing clearances usually focus solely on aerodynamic performances [START_REF] Zhong | Effect of tip geometry and tip clearance on aerodynamic performance of a linear compressor cascade[END_REF][START_REF] Wei | Tip-leakage flow loss reduction in a two-stage turbine using axisymmetric-casing contouring[END_REF][START_REF] De Maesschalck | Aerothermodynamics of tight rotor tip clearance flows in high-speed unshrouded turbines[END_REF]. As structural interactions become of prime importance for aircraft engine manufacturers, there is a need for the convergence of the analyses carried out from aerodynamic and structural standpoints. As an example, recent optimization studies conducted from both an aerodynamic standpoint [START_REF] Erler | Desensitization of axial compressor performance and stability to tip clearance size[END_REF] and a structural standpoint [START_REF] Batailly | Redesign of a high-pressure compressor blade accounting for nonlinear structural interactions[END_REF] led to opposite conclusions: the former suggests that clearance closure as the blade vibrates increases aerodynamic performances while the latter underline that clearance closure should be avoided because of its negative consequences when structural contacts occur. In this context, this paper aims at introducing clearance consumption as a possibly relevant quantity for the optimization of blade profiles. Details regarding the computation of this quantity are given in the first section of the paper. In the second section, a brief summary of the redesign operation carried out in [START_REF] Batailly | Redesign of a high-pressure compressor blade accounting for nonlinear structural interactions[END_REF] is proposed. Then, in the third section, new results obtained from numerical simulations with the optimized blade profiles are provided. These results underline the distinct vibration behaviour of the iterated profiles. Finally, an in-depth comparison of clearance consumptions for each iterated profile is provided in the fourth section before the presentation of a case study for non-divergent interactions.

CLEARANCE CONSUMPTION: DEFINITION

Clearance consumption, noted c c , is a quantity that refers to the evolution of the bladetip/abradable coating clearance as the blade vibrates. The abradable coating, deposited along the casing circumference, may be worn out after being impacted by the blade which yield a local increase of clearances. This wear phenomenon is accounted for in the employed numerical strategy. The clearance consumption for the investigated high-pressure compressor blade will be scrutinized with an emphasis on its first bending (1B) and torsion (1T) modes. It will be computed in three locations: the leading edge (LE), the middle of chord (MC) and the trailing edge (TE). Exemplarily, Fig. 1 pictures the clearance consumption of a blade blade at rest bended blade ). For the computation of c c possible contacts between the blade and the abradable coating are not considered: only the blade is accounted for. The following convention is adopted: when c c < 0 the clearance between the blade and the coating is increased (this corresponds to the configuration pictured in Fig. 1), when c c > 0 the clearance between the blade and the coating is decreased. The value of c c is looked at over a certain amplitude of deformation δ ∈ [δ min ; δ max ] for both modes 1B and 1T, using the following criterion:

δ 1B = 0 δ 1B = -1 LE LE c 1B c (0) c 1B c (δ 1B ) c 1B c (+1) LE LE casing
u M LE,r ≤ 0.25 mm (1)
where u M LE,r is the radial displacement of the blade leading edge when the blade vibrates along its free-vibration mode M . The following notations are used in the paper:

δ 1B min = -1 ⇔ u 1B LE,r = -0.25 mm δ 1B max = +1 ⇔ u 1B LE,r = 0.25 mm δ 1T min = -1 ⇔ u 1T LE,r = -0.25 mm δ 1T max = +1 ⇔ u 1T LE,r = 0.25 mm (2)
One may note that the clearance consumption is usually distinct from the blade radial displacement (these quantities may only be equal if the contact surface of the abradable coating is perfectly clylindrical).

RECAPITULATION OF PREVIOUS WORK General overview

This section aims at providing the reader with key elements of the article [START_REF] Batailly | Redesign of a high-pressure compressor blade accounting for nonlinear structural interactions[END_REF] on which is based the current work. While the text has been reformulated for the sake of brevity, all the details-including the figures-contained in this section have already been published [START_REF] Batailly | Redesign of a high-pressure compressor blade accounting for nonlinear structural interactions[END_REF]. The blade of a high-pressure compressor underwent structural divergence after blade/abradable coating contact occurrences on a rig test. Thus, there was a need for a re-design operation of this blade in order to avoid potentially high amplitudes of vibration that may significantly shorten its lifespan. The same terminology is used in this article and in [START_REF] Batailly | Redesign of a high-pressure compressor blade accounting for nonlinear structural interactions[END_REF]: the defective blade is called the initial profile, it is pictured in Fig. 2. Through out the re-design operation, several blade profiles were generated, the focus is made on two of them: the first iterated profile and the final profile, respectively a first iteration providing satisfying results and the final result of the re-design operation. A few key design parameters related to the blade stacking law, see Fig. 2, were identified and used for the re-design operation. The criteria used for optimizing the blade were related to the following quantities: 1. the clearance consumption; 2. the stress field in the blade during an interaction; 3. the behaviour of the vibration amplitudes for an interaction speed. Very significant differences were obtained between the optimized profiles and the initial profile. As an example, the maximum Von-Mises stress within the blade due to centrifugal stiffening was reduced by almost 32 %, see Tab.

Linear and non-linear considerations

The numerical simulation of blade/casing or blade/abradable coating interactions are highly non-linear since contact areas are a priori unknown and both the contact pressure and the resulting displacements are part of the solution. This non-linearity is accounted for in the employed numerical strategy using a Lagrange-multiplier based algorithm. From a modeling standpoint however, linear finite element models are employed since the considered framework is the one of small perturbations: the focus is made on the arisal of interaction phenomena, where amplitudes of vibration remain small with respect to the charasteristics dimensions of the blade. Thus, the comparison of clearance consumptions between linear and non-linear models presented in section 4.1 intends to give additional insights on the numerical results by assessing up to which amplitude of deformation linear models may accurately capture the clearance consumption of non-linear models.

VIBRATION BEHAVIOR OF ITERATED PROFILES

The numerical strategy employed for the simulation of rotor/stator interactions has been detailed in previous publications [START_REF] Legrand | Full three-dimensional investigation of structural contact interactions in turbomachines[END_REF] and is not recalled here for the sake of brevity. It relies on an explicit time integration procedure [START_REF] Carpenter | Lagrange constraints for transcient finite element surface contact[END_REF] combined with a Lagrange multiplier based contact management technique in order to avoid residual penetrations between structures.

Convergence with respect to the parameters of the study (time step, reduction parameter of the blade model, density of abradable coating elements along the casing circumference. . . ) has been checked prior to the proposed analysis. At t = 0 s, the blade is at rest and the casing is perfectly circular. For t > 0 s, the blade angular speed is constant: Ω. Centrifugal stiffening as well as the casing deformation are progressively applied over the first 20 % of the time simulation. The imposed deformation of the casing is the key difference between the distinct contact scenarios considered in the following. Two contact scenarios were considered in [START_REF] Batailly | Redesign of a high-pressure compressor blade accounting for nonlinear structural interactions[END_REF]: [START_REF] Batailly | Redesign of a high-pressure compressor blade accounting for nonlinear structural interactions[END_REF] in the first one, the casing deformation is an ovalization featuring two symmetric lobes and (2) in the second scenario, the casing deformation features four symmetric contact areas. Results presented in the frequency domain in this paper complement the ones given in the time domain in [START_REF] Batailly | Redesign of a high-pressure compressor blade accounting for nonlinear structural interactions[END_REF]. For both contact scenarios, numerical simulations were carried out over a large angular speed range Ω ∈ [0.159 ; 0.3978]. For each angular speed, a Fourier transform of the leading edge radial displacement over the last 100 revolutions has been performed. The choice of the radial displacement is related to the fact that this quantity may be more representative of the wear level than tangential or axial displacements. Yet, considering a displacement type or another does not impact qualitatively the observations made below. Put side by side, these spectra create an interaction map where amplitudes of vibration are represented using the following colour code: lowest amplitudes of vibration in blue, highest amplitudes of vibration in red. The same colour code is used for all the interaction maps depicted in the remainder.

2-lobe contact scenario

The comparison between the interaction maps depicted in Figs. 3(a), 3(b) and 3(c) underlines the significant gain in terms of amplitudes of vibration between the initial profile and the iterated profiles. Indeed, the only modification between the carried out simulations is the blade profile. In the context of this article, it is noteworthy that highest amplitudes of vibration, namely the interaction areas, may be located around the modes 1B and 1T independently one from another, see Fig. 3(a), areas A and B. Thus, interactions may involve mode 1B only or mode 1T only and may be of different nature. While both 1B and 1T related interactions vanish with the final profile, 1B interactions are still predicted with the first iterated profile as it may be seen in Fig. 3(b),area C.

Repeated structural contacts between the casing and the blade logically lead to a synchronous vibration behaviour: that is why peaks of vibration are essentially located along engine order lines for which the frequency f is a multiple of the angular speed Ω : f = kΩ, k ∈ N. The aforementioned interaction areas are found in the vicinity of an intersection between an engine order line and an eigenfrequency of the blade. However, the highly non-linear nature of the simulated interaction does not allow for an obvious identification of interaction areas: an intersection between an engine order line and an eigenfrequency is not necessarily related to an interaction, see Fig. 3(c).

With this 2-lobe contact scenario, it is underlined that numerical simulations predict that three distinct blade profiles featuring identical aerodynamic performances may lead to extremely different vibration behaviour following structural contacts. It should be recalled here that the contact configuration (incursion depth, thickness of the abradable coating, clearance at rest. . . ) is identical from a profile to another. In order to assess the versatility of the iterated profiles, meaning that the presented results are not limited to a single contact scenario, a 4-lobe contact scenario is considered in the following section.

4-lobe contact scenario

Interaction maps obtained with a 4-lobe contact scenario confirm previous observations. As a matter of fact, bending or torsion induced interactions are predicted for the initial profile, see Fig. 4(a), areas D and E. Both types of interaction vanish with the final profile in Fig. 4(c) while bending induced interactions are still predicted for the first iterated profile as depicted in Fig. 4 In any case, the interaction maps drawn for both contact scenarios underline the very significant improvement in terms of amplitudes of vibration between the initial profile and the final (optimized) profile. In the remainder, the focus is made on the clearance

CLEARANCE CONSUMPTIONS OF ITERATED PROFILES

Clearance consumptions are presented for the initial profile (Fig. 5), the first iterated profile (Fig. 6) and the final blade profile (Fig. 7) for modes 1B and 1T. Linear / non-linear models Contact simulations carried out in [START_REF] Batailly | Redesign of a high-pressure compressor blade accounting for nonlinear structural interactions[END_REF] are highly non-linear due to the management of blade-tip/abradable coating and blade-tip/casing contacts. However, since the focus is made on the arisal of interaction phenomena, considering small perturbations provides an acceptable framework for conducting such analysis and advantageously allows for using linear finite element models which yields significant gain in computational times. Thus, the impact of using linear models instead of non-linear finite element models for the computation of clearance consumptions must be assessed: attention must be paid to potential differences between clearance consumptions based on the finite element models used in contact simulations and the actual blade profile. In order to ensure that the obtained results are relevant all clearance consumptions are pictured for both the linear model used in contact simulations (dashed lines ) and a non-linear finite element model (plain lines ). Clearance consumptions are computed in a similar fashion for both types of models. It is noticeable that for all the considered profiles, linear models yield a fairly good approximation of clearance consumptions obtained with the non-linear models, see Figs. 5, 6 and 7. Obviously, largest differences are observed for the largest deformations: in the vicinity of δ = -1 or δ = +1. Interestingly, the linear models of all blade profiles provide an excellent approximation of clearance consumptions for the middle of chord and the trailing edge. Indeed, largest differences are essentially found for the leading edge ( / ). Finally, it seems noteworthy to underline that the predicted clearance consumptions of linear models are generally higher than the ones of non-linear models. This is an expected result: indeed, for amplitudes of vibration going beyond the framework of small perturbations, the deformation of linear models will become invalid with a potentially infinite extension of the blade length-yielding an infinite clearance consumption-while the non-linear models will capture the blade contorsion. This observation suggests that for the blade of interest, the employed numerical strategy may exacerbate its sensitivity to contact interactions as it tends to increase blade/casing penetrations. Overall, the good agreement between linear and non-linear finite element models justifies a posteriori that linear finite element models be considered for the contact simulations carried out in [START_REF] Batailly | Redesign of a high-pressure compressor blade accounting for nonlinear structural interactions[END_REF].

Profile comparison

A close comparison of clearance consumptions obtained for each profiles highlights a few keypoints:

1. first of all, it is patent that clearance consumptions of the initial profile along the first bending mode (1B) are significantly higher than the clearance consumptions predicted for the other profiles. From the leading edge to the trailing edge, clearance consumptions of the initial profile are 5 to 10 times larger than the ones of the final profile, see Figs. 5(a) and 7(a), 2. the order of magnitude of clearance consumptions predicted for the first torsion mode are comparable for all blade profiles and are significantly higher than the ones predicted for the first bending mode. This observation hints that: (a) in agreement with previous numerical investigations [START_REF] Batailly | Numericalexperimental comparison in the simulation of rotor/stator interaction through bladetip/abradable coating contact[END_REF], the first torsion mode seems to play a less critical role in the blade dynamics when rubbing occurs compared to the first bending mode, (b) amplitudes of vibration related to the first torsion mode are generally much lower than the ones of the first bending mode, 3. specifically for mode 1T, the evolution of clearance consumptions is reversed between the initial profile (Fig. 5(b)) and the other profiles (Figs. 6(b) and 7(b)). In the end, the computation of clearance consumptions allows for a clear distinction between the intial profile and the final profile. Indeed, as it vibrates along its mode 1B, the amplitude of the clearance consumption for the final profile is negligible. This indicates that the final profile shape is in perfect agreement with its surrounding casing: its vibration along mode 1B does not generate any unwanted penetration within the casing. These results suggest that blade dynamics along its first bending mode should be looked at in two directions (δ < 0 and δ > 0). Looking at the blade deformation when contacts occur typically involve only one direction and might lead to inaccurate results for the design of blades robust with respect to contact conditions.

ANALYSIS FOR NON-DIVERGENT INTERACTIONS

The interaction areas highlighted in Figs. 3(a) and 4(a) lead to the divergence of the numerical simulations. In particular, the amplitudes of vibration may grow to a point that the small perturbation framework is not valid anymore. From a numerical standpoint, it has been observed that the arisal of such interaction is intimately related to the first bending mode of the blade [START_REF] Batailly | Numericalexperimental comparison in the simulation of rotor/stator interaction through bladetip/abradable coating contact[END_REF]. More recently, the focus has been made on other types of interactions [START_REF] Batailly | Conjectural bifurcation analysis of the contact-induced vibratory response of an aircraft engine blade[END_REF] which may involve higher frequency free vibration modes (such as the first torsion mode 1T) and do not yield the divergence of the numerical simulation. Instead, the amplitudes of vibration increase over a few revolutions before contact is lost. Though amplitudes of vibration may be relatively low, such interactions may involve higher frequency free vibration modes and yield crack initiation. The nature of the interaction may be scrutinized through an in-depth post-processing procedure of the acquired time responses (be it from numerical simulations or experimental data) and may be confirmed by the observation of a wear pattern along the abradable coating deposited on the casing. Such a non-divergent interaction is numerically predicted for the initial profile around the intersection of the first torsional free-vibration mode (1T) and certain high engine orders. The location of this interaction is outside of the angular speed range considered in Fig. 3(a), it appears for lower values of Ω.

Exemplarily, both the initial profile and the final profile are used for the simulation of this non-divergent interaction. A critical speed is found at the intersection of the 22 nd engine order with the first torsion mode. Because natural frequencies of both profiles are not perfectly identical, this interaction arises at slightly different angular speeds for each profile. The description of the pre-processing strategy employed for the detection of the interaction goes beyond the scope of this article. Results presented in the remainder are given at the critical speed for each profile. The radial displacements of the leading edge of each profile are pictured in Fig. 8. The improvement in terms of amplitude of vibration between the initial profile and the final profile is obvious. Indeed, amplitudes of vibration predicted for the final profile are about seven times lower than the ones predicted for the initial profile. Predicted wear levels on the surrounding abradable coating deposited along the casing contact surface are depicted in Fig. 9. These wear levels are consistent with the aforementioned radial displacements: a much lower wear level is predicted for the final profile. Twenty-two wear lobes are clearly distinguishible for the initial profile ( ) in agreement with the fact that the interaction of interest involves the 22 nd engine order. For the final profile however, these twenty-two lobes are not obviously visible ( ): the lower amplitudes of vibration did not yield significant wear levels on the casing.

CONCLUSION

This article focuses on a redesign operation that was carried out on a high-pressure compressor blade in a previous article [START_REF] Batailly | Redesign of a high-pressure compressor blade accounting for nonlinear structural interactions[END_REF]. The notion of clearance consumption, which may be seen as a quantification of how the blade tends to get closer to the casing as its vibrates along one of its free-vibration mode, allows for a discrimination of blade profiles depending on their sensitivity to rotor/stator interactions initiated by structural contacts. Indeed, blade profiles for which clearance consumption is minimal feature much lower amplitudes of vibration than other blade profiles. Thus, this study suggests that clearance consumption may be a relevant criterion for the design of robust blade profiles with respect to structural interactions. ). They are compared to the initial abradable coating profile ( )

Figure 1 .

 1 Figure 1. VISUAL REPRESENTATION OF THE CLEARANCE CONSUMPTION

Figure 2 .

 2 Figure 2. STACKING LAW AND MESH (CLAMPED AREA IN RED, CONTACT NODES IN GREEN) OF THE INITIAL PROFILE

Figure 3 .

 3 Figure 3. INTERACTION MAPS WITH 2 LOBES

Figure 4 .

 4 Figure 4. INTERACTION MAPS WITH 4 LOBES

Figure 5 .

 5 Figure 5. CLEARANCE CONSUMPTION FOR THE INTIAL PROFILE: LE (linear /nonlinear ), MC (linear /non-linear ), and TE (linear /non-linear )

Figure 6 .

 6 Figure 6. CLEARANCE CONSUMPTION FOR THE FIRST ITERATED PROFILE: LE (linear /non-linear ), MC (linear /non-linear ), and TE (linear /non-linear )

Figure 7 .

 7 Figure 7. CLEARANCE CONSUMPTION FOR THE FINAL PROFILE: LE (linear /non-linear ), MC (linear /non-linear ), and TE (linear /non-linear )

Figure 8 .Figure 9 .

 89 Figure 8. Superimposition of the amplitude increases observed for the initial profile () and the final profile ( )

Table 1 .

 1 1. EVOLUTION OF σc WITHIN THE BLADE PROFILE

	profile	initial first iterated	final
	σc evolution (%)	0	-28.32 -31.90

-Department of Mechanical Engineering, École Polytechnique de Montréal, P.O. Box 6079, Succ. Centre-Ville, Montréal, Québec, Canada H3C 3A7

-Safran Aircraft Engines, site de Villaroche, Moissy-Cramayel, 77550, France

-Département de génie mécanique, École Polytechnique de Montréal, P.O. Box 6079, Succ. Centre-Ville, Montréal, Québec, Canada H3C 3A7

-Safran Aircraft Engines, site de Villaroche, Moissy-Cramayel, 77550, France

Acknowledgement

Thanks go to Snecma for its technical and financial support.