Claude Gibert
email: claude.gibert@ec-lyon.fr

Laurent Blanc

Patricio Almeida

Xavier Leblanc

Jean-Philippe Ousty

Fabrice Thouverez

Jean-Pierre Laîné

J.-P Ousty

J.-P Laîn

Modal Tests and Analysis of a Radial Impeller at Rest: Influence of Surrounding Air on Damping

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

INTRODUCTION

Turbomachinery blade vibration damping results from several energy dissipation phenomena. Three energy transfer paths can be distinguished for the most general cases of mechanical systems immersed in a fluid [START_REF] Blevins | Flow induced vibration[END_REF]: "fluid damping due to fluid drag, viscous dissipation and radiation to the surrounding fluid"; "internal material damping due to yielding, heating . . . and internal energy dissipation of materials"; "structural damping due to friction, impact, scraping and the motion of fluid trapped (squeeze film damping, etc.) in a joint".

Designing monobloc centrifugal impellers requires assessment for both the aerodynamic damping provided by the flow during operation and for material damping. Indeed, no structural damping occurs in such structures due to monobloc manufacturing and to the absence of any specific devices such as underplatform dampers, friction rings [START_REF] Laxalde | Experimental and numerical investigations of friction rings damping of blisks[END_REF], viscous material coatings, etc. Under such conditions, material damping assessment becomes all the more important as material damping -even low -is the only source of dissipation when aerodynamic damping becomes null or even negative, with the risk of instabilities. In addition to flutter margin prediction, material damping data is also required to calculate finely forced vibration levels when assessing HCF risk. In this study, we focus on material damping assessment.

Material damping refers to energy dissipation when a volume of the material is subjected to cyclic stresses. Many complex mechanisms are involved "associated with internal reconstruction such as molecular dislocations and stress changes at grain boundaries" [START_REF] Beards | Structural vibration: analysis and damping[END_REF]. Analysis is difficult due to non-linearity and variability inside the material. Nevertheless, indicative values of losses are obtained by experimental measurements performed on samples of specific materials. Influence parameters are a priori the composition and temperature of the material and stress distribution, which depends on the geometry and on the kind of loading mechanism, especially in terms of frequency and stress level [START_REF] Harris | Harris' Shock and Vibration Handbook[END_REF]. The loss factor, η, is used as the measurement of material damping for a given part. It is defined as the ratio of total energy dissipated D over the maximum strain energy U during one cycle of vibration:

η = D 2π U .
η is subjected to estimates varying from 0.00104% to 1.7% for a twisted titanium alloy plate as a function of the mode tested, the stress level and the presence or absence of air; in vacuum, the range of values ranged from 0.00104% to 0.051% [START_REF] Srinivasan | Turbojet engine blade damping[END_REF]. Such disparities have been observed in the literature and in studies of axial compressor blisks in unpublished works. Compared to axial configurations, radial compressor blisks present very different geometries and vibrate according to specific mode shapes, making it difficult to draw accurate analogies with axial machine material damping levels. Thus it is necessary to carry out a specific study.

A fair number of papers have dealt with this issue in the open literature. In [START_REF] Kielb | Experimental study of aerodynamic and structural damping in a full-scale rotating turbine[END_REF], non-aerodynamic damping was measured under vacuum conditions and substracted from overall damping obtained in rotating facilities to estimate aerodynamic damping. The experimental approach was based on [START_REF] Jeffers | A novel technique for the measurement of blade damping using piezoelectric actuators[END_REF] relating to piezoelectric excitation. In [START_REF] Kammerer | Experimental study on impeller blade vibration during resonance -Part 1: blade vibration due to inlet flow distortion[END_REF][START_REF] Kammerer | Experimental study on impeller blade vibration during resonance -Part 2: blade damping[END_REF], vacuum tests were performed on an aluminum impeller mounted on a shaft subjected to piezo-excitation and instrumented with strain gages and pressure transducers on the surface of the blades. The results focused on the first two eigenmodes of a main blade. Good agreement was obtained with the material damping extrapolated from measurements performed under aerodynamic excitation for decreasing inlet pressure values. In [START_REF] Zemp | Experimental investigation of forced response impeller blade vibration in a centrifugal compressor with variable inlet guide vanes -Part 1: blade damping[END_REF], with a specific engine representative of operating conditions, analogous tests revealed that aerodynamic damping could be about 3 times higher than material damping, then the latter is far from negligible.

Here, the purpose is to present and discuss experimental results obtained under vacuum conditions in view to assessing material damping under the "purest" possible experimental conditions. Contrary to other works constrained by the need to perform rotating or in-flow measurements in addition to structural measurements, here the setup is dedicated to removing the influence of boundary conditions and embarked instrumentation. To achieve this: 1/ the impeller is supported punctually instead of being clamped, to prevent energy from being lost in the mounting; 2/ the instrusivity of the measurement device is minimized as a laser Doppler vibrometer (LDV) is used.

In the first part of this paper, the principle of vibratory measurements in vacuum is recalled to provide a theoretical model of the influence of still air pressure on modal parameters. Then the experimental setup and procedure are described. Lastly, the results are presented and discussed in terms of structural damping quantification and also in terms of modal parameter identification, since lower damping in vacuum leads to the clear separation of FRF peaks and is useful for modal analysis. It is shown that levels at resonance of 1st flexural modes while testing at 10mbars are 30 dB higher, with damping ratios 30 times lower, compared to results obtained at ambient pressure.

PRINCIPLE OF VIBRATORY MEASUREMENTS IN VACUUM 1.Hypotheses on fluid -structure interaction

The aim is to develop a phenomenological model as the basis for formulating a measurement protocol. Let us consider the fluid-structure system composed of the impeller and the surrounding still air. The non-conservative effects associated with the fluid can be linked with the radiation of acoustic waves at "infinity", with viscosity effects and with turbulence if the amplitude of the blades is high enough. Here, air viscosity and turbulent losses are neglected compared to radiation losses. Thus air is modeled as a perfect gas that obeys linear acoustics laws. The impeller is assumed to be weakly damped in vacuum and able to freely oscillate. Under such hypotheses, it is conceivable to provide a modal description of the dynamical behaviour of the system. It is also assumed that intermodal coupling is negligible. In the vicinity of a natural frequency determined in vacuum, the system is therefore analogous with a classical SDOF oscillator.

Basic phenomena

A qualitative model is used in view to determining how vacuum affects modal quantities. It is taken from [START_REF] Lesueur | Rayonnement acoustique des structures (Acoustic Radiation of Structures[END_REF][START_REF] David | Caractérisation acoustique des structures vibrantes (Acoustic Characterization of Vibrating Structures[END_REF][START_REF] David | Using vacuum to measure the acoustical efficiency[END_REF]. The system is a rigid baffled flat piston of mass M, surface S, hanging from a spring of stiffness K and fitted with a viscous damper of coefficient C (see Fig. 1). It is placed in an infinite plane rigid baffle separating two half-spaces. Let w be the piston's displacement along the axis perpendicular to the baffle. When the mass is excited by a sinusoidal force F 0 e jωt , the equilibrium equation is:

(M + M a ) ẅ + (C + C a ) ẇ + Kw = F 0 e jωt . (1)
Terms M a and C a contain the contribution of air pressure force F p . Let p + and p -= -p + designate the fluid pressure on the piston faces. F p and piston speed ẇ are linked by radiation impedance Z r by:

- F p S = 2 S S p + dS = Z r ẇ = (R r + jX r ) ẇ. (2)
Then, M a = SX r /ω, with X r the reactive part of Z r , has an added mass effect and modifies the natural frequency. C a = SR r , along with R r , which is the resistive part of Z r , adds damping to structural damping. M a and C a are functions of ω. Both terms can modify the structural response and result in measurements that FIGURE 1: Baffled piston SDOF system, adapted from [START_REF] David | Caractérisation acoustique des structures vibrantes (Acoustic Characterization of Vibrating Structures[END_REF] would give erroneous natural frequencies and material damping if modal data identified in ambient air were assimilated with those expected in vacuum.

Here we presume -later confirmed by the experiments -that the influence of coupling with air is weak. Thus the complex natural frequency of the system in air is derived from the natural frequency in vacuum through a first order perturbation analysis. Moreover, it is shown that for linear acoustics, the radiated pressure generated by each point of the structure is proportional to the characteristic impedance of fluid ρ 0 c with ρ 0 being the den- sity and c the speed of the acoustic waves. For a perfect gas, ρ 0 is proportional to P and c depends only on the temperature. Then Z r is proportional to P and both natural frequency ω ′ 0 and critical damping ratio ζ in air are affine functions of static air pressure P. Their values are deduced from ω 0 -natural frequency without air or damping -and from ζ 0 -critical damping ratio of the system in vacuum by:

ζ = ζ 0 + α(ω 0 )P and ω ′ 0 = ω 0 + β (ω 0 )P, (3)
with α and β coefficients depending on S, M and Z r (ω 0 ). For the baffled piston (and also for the unbaffled piston). It can be shown analytically that an increase of air pressure is expected to lead to increased damping and a decrease in natural frequency compared to vacuum values.

Experimental quantification of material damping

Consequently, the general principle of the measurement is as follows [START_REF] David | Caractérisation acoustique des structures vibrantes (Acoustic Characterization of Vibrating Structures[END_REF]. Firstly, the vibratory signal is measured for decreasing values of pressure P in a vacuum chamber. Secondly, for each eigenmode considered, natural frequency ω ′ 0 (P) and critical damping ratio ζ (P) are estimated. The slopes of the straight lines fitting the best -in terms of least mean squares -ω ′ 0 (P) and ζ (P) are estimated and their values for a null pressure are extrapolated. ζ 0 = ζ (P = 0) stands for structural damping, i.e. material damping under the conditions of the experiment described below.

EXPERIMENTAL SETUP AND PROCEDURE 2.1 Test specimen

The impeller tested was a centrifugal compressor wheel analogous to that shown in Fig. 2. Its shaft was removed to concentrate on the blading so that it consisted of a single part with ten main blades and ten intermediate blades (or splitter blades). It was made of titanium alloy.

Free boundary conditions

In order to avoid any disturbances from clamping mounts, the impeller was supported close to its center of mass, as shown in Fig. 3. The impeller is simply supported at the end of a rigid cylindrical steel beam. This latter is clamped at its other end. A piece of rubber represented by a black rectangle in the sketch in Fig. 3 is placed between the beam and the impeller to provide a soft dynamic isolation between these two parts. The rubber dimensions are about 10 × 10 × 4mm. It is in contact with the inside of the cylindrical bore of the test piece. The main contact force applied to the impeller is the reaction of the rubber to its weight. Dynamic reaction forces are limited thanks to the very low stiffness of the rubber part. Also, the theoretical modal displacements of the free-free impeller at the bore is minimum and thus minimizing any effect of the support. This is illustrated in Fig. 4, which shows mode shapes obtained through pre-test FE modal calculations for one of the modes of interest. A closer insight at the bore surface would show that the modal displacements are less than 0.02 (compared to 10 on the backside and 300 on the balde).

The modes of the clamped-free beam supporting the impeller on the rubber part are outside the frequency range containing the first ten frequencies of the impeller.

This mounting method also had the advantage of being very

Excitation setup and measurement

Excitation was provided by a piezoelectric (PZT) translator stack P-802.00 [14] of dimensions 6 × 6 × 9mm, associated with a seismic mass (Fig. 5). The shaker was bonded to the upper side of a piezoelectric force sensor which in turn was attached to the structure. The elongation of the translator, essentially proportional to the tension provided by a piezoelectric driver amplifier, generated inertial reaction forces proportional to the square of the frequencies transmitted to the impeller. The translator was driven with a direct component of +50 Vdc superimposed with an alternating component (range +/-50 Vac) in order to operate within the prescribed 0-100 Volt range of the translator [START_REF] Szwedowicz | Numerical and experimental damping assessment of a thin-walled friction damper in the rotating set-up with high pressure turbine blades[END_REF].

The input point chosen was in a rather stiff and flat region on the structure but produced a sufficient response level at the measurement points when excited by the shaker (see Fig. 6 at midrange between the impeller center and outer radius). Also, the mass-loading induced by the force sensor (total mass 22.7 gm) was slight at this point. It was evaluated using a first order approximation of the sensitivity of natural frequencies w.r.t. a local punctual mass, and was found to be less than ∆ω the fre- quency resolution of the DSP unit. All the tests presented herein were performed with the same excitation system. The measurements have been repeated several times after checking the testing conditions and the results presented in the paper were repeatable.

Vacuum chamber

The impeller with its mounting was installed in a vacuum chamber to perform modal testing. The vacuum chamber was evacuated by a double stage volumetric vacuum pump. The pressure inside the chamber was measured using a Pirani pressure gauge. The vacuum pump was stopped once the desired level of vacuum was reached. The electrical wires necessary for exciting the structure and for measuring input force were passed through the chamber rear wall via a special seal. The front wall was made of transparent PMMA material and designed so that it could withstand external atmospheric pressure and allow the use of an optical measurement system outside the chamber. This feature was also useful for checking that no unwanted contact occurred, for example, between the wires and the blisk, during the measurement. The response of the impeller was measured by a commercial scanning laser vibrometry system (SLDV) [16]. Twenty measurement points were defined on the impeller rear surface close to the trailing edge of the blades (Fig. 6). The response at these locations reached its maximum when the structure vibrated on its modes. Small reflective targets (4 × 4mm) were fixed to improve the light returned back to the vibrometer head and therefore increase the signal to noise ratio. The axis of the scanning head was aligned with the axis of the impeller so that the incidence of the laser beam at all the targets was the same. Also, the angle between the laser beam and the normal vector at the targets centre was minimized by operating on the rear side of the impeller. The mass of the reflectors was less than 0.004 grams, which was the same for all the different tests. The evaluated mass-loading effect of these reflectors on natural frequencies was less than ∆ω/2. Also, it should be noted that this rig made it easier for the laser beam to reach the response close to the driving point.

FRF measurements

Acquisition and signal generation were carried out by a dualchannel dynamic signal analyzer associated with the SLDV system to measure FRFs relating velocity at each measurement point to the input force at the driving point. A pseudo-random excitation signal was used to avoid the use of time windowing. The FRFs were obtained classically using the H1 estimator with averaging and overlapping measurement blocks of data at frequency points within a band containing the first flexural mode family of the impeller. For this specific type of structure, it was necessary to use a high frequency resolution (∆ω), because of the high modal density, and to obtain 5 to 10 frequency points within the half power frequency band at resonances so that the damping factor could be identified accurately. This inevitably resulted in relatively long testing times during which it was possible to check that test conditions remained constant (essentially in terms of pressure and temperature).

Data reduction

The FRF data -velocity output at a given point over input force -were analysed by a modal parameters iterative extraction method based on [START_REF] Dennis | Numerical Methods for Unconstrained Optimization and Nonlinear Equations[END_REF]. First, a theoretical FRF form h th was choosen as the ratio of two polynomials:

h th (ω) = a 0 + a 1 λ + ... + a n λ n 1 + b 1 λ + ... + b n λ m ( 4
)
where λ = iω, and a j and b j are real unknown coefficients. The orders of the numerator and denominator (n,m) were chosen in accordance with the number of modes responding in the frequency band analysed. The form h th (ω) was fitted to frequency response data by minimizing a weighted error function solving for unknowns a j and b j

(a 0 , ..., a n , b 1 , ..., b m ) = argmin N ω ∑ k=1 w k |h th (ω k ) -h meas (ω k )| 2
(5) where N ω is the number of measured excitation frequencies in the range of interest and h meas the FRF data. The algorithm used the damped Gauss-Newton method for iterative search [START_REF] Dennis | Numerical Methods for Unconstrained Optimization and Nonlinear Equations[END_REF]. This solved the direct problem of minimizing the weighted sum of the squared error between the theoretical and the measured frequency response points. The weighting coefficients are user defined : w k = |h meas (ω k )| 2 . Once the polynomial coefficients had been found, the residues and the poles of the partial fraction expansion of the ratio of the two polynomials were obtained numerically. Real and imaginary parts of identified poles were derived afterward to extract natural frequencies and damping ratios as partial fraction expansion could be identified with the classical expression of h th in term of modal parameters:

h th (ω) = 10 ∑ k=1 A k λ -λ k + Āk λ -λk + U + V ω 2 (6)
where

λ k = -ζ ω k + iω k (1 -ζ 2 )
is the k th pole and λk its com- plex conjugate and A k is the corresponding complex modal residual and Āk its conjugate. Coefficients U and V aim to take into account the influence of out of range modes. The modal residues were used to identify mode shapes and their nodal diameter numbers (ND). As shown later, the use of vacuum conditions allows very good direct identification of the number of peaks. In fact, the expected total number of ten peaks for the impeller, which had ten sectors, was clearly seen on the FRFs. Expected doubles modes (pairs) with 1,2,3 and 4 NDs where found at lightly split frequencies. Both modes within a pair were excited by the single point excitation location chosen. It should also be mentioned that numerical analysis showed that the first mode family was also well separated from the others. The measured data were therefore reduced to ten damping ratios, natural frequencies and complex residues. An example of measured FRF data (displacement over force) superimposed with the FRF synthesized as the sum of the modal participations is shown in Fig. 7.

RESULTS AND DISCUSSION

Frequencies and damping ratios

The identified natural frequencies and damping ratios are gathered in table 1. These reduced data are sorted by ascending frequency and normalized. For each frequency, the ND of the corresponding mode shape is given. The relative distance in frequency within a pair of the same ND ranges from 0.06% to 0.1% depending on the ND. It is worth mentionning that damping ratios at ambient pressure are more than 30 times higher than at 10mbars. Uncertainties on modal parameters values have been evaluated by analyzing each one of the 20 measured FRFs separately and then by looking at the scattering of the results. Theoretically, these values should be the same. Although, possible errors in measurements or in identification process had lead to slightly different values. The mean over the 20 individual results obtained is used as an estimate of the true value for a modal parameter and these are reported in table 1. The amplitude of the scattering around mean values is represented on diagrams in Fig. 9 by vertical bars. The height of a vertical bar is equal to the difference between the highest and the smallest of the different identified values for a particular modal parameter. Several remarks can be made looking at these diagrams. First, the natural frequencies are very accurately estimated as the amplitude of the scattering is limited to ±0.001%, which is on the order the frequency resolution of the DSP unit ∆ω (bars height have been magnified 1000 times on the diagram for readability). Despite the uncertainty on damping ratios, their extremely relative low values are confirmed. The relative error on damping ratios at 100mbar are on the order of ±20%. Note that certain modes -0ND, 1NDb, 2NDb, 3NDb, 4NDa, 5ND -are estimated with more accuracy. As revealed by the modal extraction, these latter have relatively more important modal participations. This can be explained by the circumferantial of the shaker w.r.t. nodal diameters which makes some mode globally predominant in the set of FRFs (see a sample in Fig. 7). This could be optimized a posteriori concerning modes with 1, 2, 3, 4NDs. Note that modes with 0ND and 5ND are accurately estimated. Regarding the spread of damping ratios within a pair of modes, note that it is in the range of uncertainty. Interestingly, it can be mentioned from Fig. 9 that damping ratios for mode with ND numbers 1,3,5 are lower than the others (with ND = 0, 2, 4).

In the case where the pressure was 1 bar, the FRFs could not be fitted satisfactorily through all the frequency range. The damping ratios estimated at 1 atm scattered between 0.8 and 1.2. The results in terms of modal parameters are given for all the pressures tested for the first 2 modes only. Due to the difficulty of extracting modal parameters from the tests at 1 bar, a direct comparison of the FRFs is proposed (Fig. 8).

The influence of air on frequencies is quite mild as frequency shifts between 10 mbar and 1000 mbar does not exceed 0.3 %, validating the first order perturbation approach on which the principle of vacuum vibratory measurements is based. Such low variations justify a posteriori the care taken over the experimental setup. Nevertheless, due to the low number of pressure points available, it seems risky to assess the linearity with respect to pressure without extra measurements. Additional tests would be especially beneficial when starting at a high pressure limit of less than 1 bar and chosen so that identification is easy or at least feasible. In case an attempt of linear fitting were made, one could notice the tendency of points corresponding to damping measured at 10 mbar to be below the line joining the points of measurements at 100 or 1000 mbar (Fig. 10). Such deviation from a linear behaviour, if confirmed, could be compared to an analogous behaviour noticeable on similar measurements performed on a fan blade and reported in [START_REF] Srinivasan | Turbojet engine blade damping[END_REF]. This would weaken the hypothesis of linear acoustic behaviour. Interpolated normalized critical material damping ratios ζ 0norm in vacuum, based on the measurements performed at the indicated pressure levels

Comparison with other available data

In spite of confidentiality reasons that prevent us from publishing raw data, it seems important to place the present results in relation to other available data [START_REF] Kammerer | Experimental study on impeller blade vibration during resonance -Part 2: blade damping[END_REF][START_REF] Kammerer | Experimental Research into Resonant Vibration of Centrifugal Compressor Blades[END_REF]. The purpose is to provide tendencies of the conditions under which air is likely to have a significant impact on damping measurements, and thus indicate when experiment designers should be concerned about it.

A simple model taken from the literature [START_REF] Harris | Shock and Vibration Handbook[END_REF] was used to give an analytical framework. Let us consider the case of a plate oscillating harmonically. Its behavior is assumed to be similar to that of an impeller main blade. When replacing the plate surface by a distribution of harmonically oscillating sources, the air loading effect has different simplified expressions according to , where a is a characteristic length of the structure, ω the excitation frequency and c the acoustic wave speed in surrounding air. The distinction between low frequencies and high frequencies can be made when k < 1 and when k > 3, respectively. The intermediate frequency range represents a transition between both types of loadings.

An analogy is performed between the dynamical behaviours of a main blade -vibrating on its first flexural mode -and of a rectangular simply supported plate -also vibrating on its first flexural mode of natural frequency ω 1 . In spite of the fact that geometry and boundary conditions are different, the tendencies can be expected to be similar regarding the material parameters. This helps when comparing the aerodynamic dampings obtained between similar impellers made of different materials.

Let a, b and h be the plate length, width and thickness, ρ m the density of the plate material. a and b are chosen as equal to the average span and chord of the impeller main blade studied, and ω 1 is equal to the average frequency of the first flexural mode family. The case studied corresponds to a low frequency coupling because k < 1. The critical damping ratio associated with aerodynamical damping is the ratio between terms C a (of Eq.1) and C cr = 2ω 1 M [START_REF] Harris | Shock and Vibration Handbook[END_REF]:

C a C cr = 0.156 b a + a b ρ ρ m c m c ( 7
)
This equation is obtained for Poisson coefficient ν = 0.3.

c m = E ρ m
is the speed of sound in the material with Young modulus E and material density ρ m . If the temperature is constant, then c does not depend on P and ρ is proportional to P. Thus, for a given air pressure, air damping depends on the plate geometry and material, and on modal parameters (natural frequency and mode shape, since the pressure field depends on the latter).

Eq.7 provides clues to predict the impact of air on total damping. For example, air damping should not depend on the absolute dimensions of the plate but only on its aspect ratio a b . Another tendency is that increasing ρ m should result in decreas- ing c m and then lowering air damping, as expected. It is also noticeable that blade thickness does not seem to play a role in air damping. But this would be different for higher values of coefficient k.

At this stage, a comparison between the experimental results from the previous paragraph and those provided by [START_REF] Kammerer | Experimental Research into Resonant Vibration of Centrifugal Compressor Blades[END_REF] can be proposed. The characteristic dimensions are different, Kammerer's impeller is larger (400mm diameter compared to 260mm) but with a similar blade aspect ratio. It is made of aluminum instead of titanium. It has 7 main blades instead of 10 and they seem to be thicker. In terms of modal properties, the frequencies of the first mode family modes of the present impeller are located between the frequencies of the first and second family modes of Kammerer's (1350Hz, 2875Hz). The first family modes have similar mode shapes on both structures. The comparison is performed using data extrapolated from Kammerer's mode 1 (assumed identified on an isolated blade). Assuming that aerodynamical damping on impellers also depends on aspect ratios, Eq.7 allows predicting that the ratio between the air damping provided to Kammerer's impeller and air damping on the present one should be equal to 1.7. Averaging the air damping ratios identified herein on all the modes of the family provides a result in line with the tendency expected, with higher values on the aluminum impeller than on the titanium one. The discrepancy is about 60% at P = 1 bar and about 20% at P = 100 mbar. Therefore, in spite of the great simplicity of the analytical model used to correlate data obtained from quite different impellers, the real air damping ratio values measured here are in agreement with those measured and reported by other authors.

Concerning material damping, it can be concluded from the experiments described herein that the present material damping values are significantly lower than those given in [START_REF] Kammerer | Experimental Research into Resonant Vibration of Centrifugal Compressor Blades[END_REF] but of the same order of magnitude as the lowest values available in the literature [START_REF] Srinivasan | Turbojet engine blade damping[END_REF]. This can be explained by the fact that our impeller was tested with practically free boundary conditions and with contactless instrumentation rather than with clamping and more intrusive instrumentation, possibly leading to additional damp-ing. But the most important conclusion is that for such a structure the presence of air is highly likely to pollute material damping measurement as its relative contribution to total damping is much greater than previously reported. This makes using vacuum measurements essential.

CONCLUSIONS AND PROSPECTS

In this paper, a dedicated experimental setup was proposed to identify the dynamical characteristics of a radial impeller in terms of natural frequencies and nodal diameter mode shapes, and to assess material damping by trying to minimize different spurious effects. Although blisk structures are known to be very lightly damped, it has been noted that the direct location of resonances on measured FRFs was not obvious when modal tests on the impeller were conducted under ambiant pressure. The resonances seemed heavily damped and the high modal density was probably an additional difficulty for the analysis. Nonetheless, the blisk was tested carefully under practically free-free boundary conditions and its response measured by SLDV.

The decision to operate inside an evacuated chamber to improve testing conditions highlighted the remarkable property of vacuum. Then, all the expected resonance peaks in the frequency range could be clearly located on the FRF data, revealing modes with 0 to 5 nodal diameters with split pairs (1 to 4 ND). The analysis of data confirmed a strong dependency of damping ratios on pressure on the first flexural mode, as the ratio of damping in air and at vacuum ζ /ζ 0 was on the order of 30. Although, some published experimental works report a non linear evolution of damping w.r.t. pressure when pressure is close to zero, the principle of vibratory measurements in vacuum based on the linearity of modal parameter evolution as a function of pressure has been used to assess material modal damping, showing the limits of the method. Additional pressure points would be necessary to fit a nonlinear relationship -which remains to be determined -and to obtain more accurate material damping measurments.

A simple analytical model was used in order to show the influence of the different physical parameters involved in the damping ratio of a vibrating flat plate in surrounding air: aspect ratio, material density, frequency and mode shape, air density. Assuming that the analytical model was applicable, a quantitative comparison has been performed with results from a similar structure found in the literature. Considering the first flexural mode shapes, the ratio of damping in air and at vacuum was lesser (ζ /ζ 0 = 8) in these latter results but the analytical model gave a way to correlate satisfactorily both evaluations of air damping. Given the physical parameters of the two impellers, the simple model predicted that damping in air of the aluminium impeller was 1.7 times higher. Although, the fact that our damping ratios at vacuum are very much lower makes the ratios ζ /ζ 0 of the two studies consistant. Some differences from our case should be taken into account: especially, the blisk was clamped on a shaft made of different material, it instrumentation was probably more intrusive and aluminium may have more material damping than titanium.

Further measurements would focus on other mode families to perform analogous analysis with higher modes. The first two mode families for a similar structure have been tested in reference [START_REF] Kammerer | Experimental study on impeller blade vibration during resonance -Part 2: blade damping[END_REF]. It was found that the influence of surrounding air on the second family was two to three times lower w.r.t. the first one. Although our impeller geometry and material are different it seems that we could expect the same behavior.

The role of rotation also deserves re-examination. Indeed, in [START_REF] Srinivasan | Turbojet engine blade damping[END_REF], the authors concluded from rotating tests performed on a single plate-like blade that rotation effects were not significant, but suspected more complex material damping processes for wide chord and three-dimensional slender blisks. Specific experiments such as in [START_REF] Jean | Test-model correlation of dry-friction damping phenomena in aero-engines[END_REF] could be performed.

FIGURE 2 :

 2 FIGURE 2: Typical centrifugal compressor wheel, courtesy of Turbomeca

FIGURE 3 :

 3 FIGURE 3: Principle of impeller's free mount

FIGURE 4 :

 4 FIGURE 4: Mass normalised FE mode shapes; axial displacement of the 1st family mode with 4ND. (units : tons -1 2 )

FIGURE 5 :FIGURE 6 :

 56 FIGURE 5: Schematic of the piezoelectric shaker

FIGURE 7 :

 7 FIGURE 7: Typical FRF: measured FRF (dashed bold curve), synthesized FRF (solid bold curve), modal participation of each mode (thin curves)

FIGURE 9 :

 9 FIGURE 9: Damping ratios and natural frequencies over NDs at 10mbars

FIGURE 10 :

 10 FIGURE 10: Normalized critical material damping ratios vs pressure level for mode 2a (triangles) and mode 2b (circles)

TABLE 1 :

 1 Evolution of normalized modal frequencies f norm and normalized critical damping ratios ζ norm according to ambient pressure P. When double modes exist, index a or b is used to distinguish them. "ni" stands for "not identified"

	Another remark is

TABLE 2 :

 2

ACKNOWLEDGMENT

This work was partially funded by the French National Research Agency (ANR) in the framework of its Technological Research COSINUS program (IRINA project, ref. ANR 09 COSI 008 01 IRINA).


Christoph Dürr
email: christoph.durr@lip6.fr.

Shahin Kamali
email: shahin.kamali@umanitoba.ca.

Randomized Two-Valued Bounded Delay Online Buffer Management

Keywords: competitive ratio, oblivious adversary, buffer management, maximum throughput scheduling

In the bounded delay buffer management problem unit size packets arrive online to be sent over a network link. The objective is to maximize the total weight of packets sent before their deadline. In this paper we are interested in the two-valued variant of the problem, where every packet has either low (1) or high priority weight (α > 1). We show that the optimal randomized competitive ratio against an oblivious adversary is 1 + (α -1)/(α 2 + α).

Introduction

Online Buffer Management is a scheduling problem which arises in network routers. Packets arrive online to be sent on a specific link, the goal is to maximize the total weight of sent packets under various constraints. Different models have been studied in the past, the FIFO model, where the router stores pending packets in a limited capacity buffer, and the bounded delay model where packets can stay only limited time in the buffer.

We are particularly interested in the latter model, which can be formalized as the following scheduling problem. Time is partitioned into time slots. At each time slot, a (potentially empty) set of jobs of unit length arrive online on a single machine. Job j arrives at a release time r j ∈ N, has a deadline d j ∈ N and a weight w j ∈ R + . At every time slot t ∈ N, there is a set of pending jobs, which is updated by jobs released at time t and by jobs expiring at time t. The algorithm can choose to schedule one of the pending jobs, at every time slot The goal is to maximize the total weight of the scheduled jobs. The usual restriction on the online algorithm is that these decisions need to be made without knowledge of future arriving jobs. Its performance is measured by the competitive ratio, which compares the objective value reached by the algorithm with the objective value of the optimal schedule, maximizing the ratio over all possible instances. The deterministic competitive ratio of the problem is defined as the best competitive ratio among all deterministic online algorithms. As randomization usually helps in these online settings, we consider the randomized competitive ratio against an oblivious adversary. The term adversary comes from the game theoretical setting of an online problem, played between an online algorithm which needs to make decisions at every time slot and a malicious adversary which generates the instance over time. The adversary is "oblivious" in the sense that it is not aware of the random choices made by the randomized online algorithm. Other adversarial models exist but are not considered in this paper.

Related work

This problem has been introduced in 2001, together with a lower bound of ϕ ≈ 1.618 on the deterministic competitive ratio [START_REF] Hajek | On the competitiveness of on-line scheduling of unit-length packets with hard deadlines in slotted time[END_REF] and [START_REF] Kesselman | Buffer Overflow Management in QoS Switches[END_REF] and an upper bound of 2, achieved by the Greedy algorithm which schedules always the heaviest pending job (ties are broken by executing the most urgent job). This gap between lower and upper bound generated a series of studies of variants of this problem, mostly with restrictions on the deadlines of the job, such as agreeable deadlines (jobs can be ordered by release times and deadlines at the same time) or s-uniform or s-bounded instances (d jr j is s or at most s) for some parameter s> 1. Eventually in SODA 2019, the gap was closed, with a sophisticated algorithm and analysis [START_REF] Veselý | A Φ-Competitive Algorithm for Scheduling Packets with Deadlines[END_REF]. The randomized competitive ratio in the oblivious adversary model of this problem is still open, and the optimal competitive ratio is known to be between 1.25 [START_REF] Chin | Online Scheduling with Partial Job Values: Does Timesharing or Randomization Help?[END_REF] and 1.582 [START_REF] Bartal | Online Competitive Algorithms for Maximizing Weighted Throughput of Unit Jobs[END_REF][START_REF] Chin | Online competitive algorithms for maximizing weighted throughput of unit jobs[END_REF].

The paper [START_REF] Chin | Improved competitive algorithms for online scheduling with partial job values[END_REF] considered a variant of the problem, where every job has not unit processing time, but a processing time p ∈ N, all jobs are tight in the sense d j = r j + p, and the algorithm can preempt job executions. In this setting, the authors give a barely random algorithm with competitive ratio 2, and showed that this is optimal for barely random algorithms. By scaling, this is equivalent to the problem with unit processing time jobs by fractional release times and deadlines. In our problem these times are restricted to integers, and in this context preemption is of no help to the algorithm. We refer to [START_REF] Goldwasser | A survey of buffer management policies for packet switches[END_REF] for a survey on packet scheduling, as well as to the more recent introduction of [START_REF] Veselý | A Φ-Competitive Algorithm for Scheduling Packets with Deadlines[END_REF] and references therein.

In this paper we consider a particular variant of the bounded delay buffer management problem, where every job can have either low or high weight, that is w j ∈ {1, α} for some parameter α > 1. The 2-valued variant has been studied in [START_REF] Englert | Considering Suppressed Packets Improves Buffer Management in Quality of Service Switches[END_REF] who showed the deterministic competitive ratio is min{1 + 1/α, 1 + (α -1)/(α + 1)}. It is motivated by the fact that some applications, as for example online games, could send packets with two kind of informations: Major updates in the game configuration (such as the arrival or departure of participants), and minor updates (such as not critical moves).

Contributions

The lower bound of the deterministic competitive ratio for the 2-valued variant can be easily transformed into a randomized lower bound. In addition this transformation suggests a randomized online algorithm, which we show to have optimal competitive ratio 1 + (α -1)/(α 2 + α).

Preliminaries

Formally an instance of the bounded delay buffer management problem consists of a finite sequence of jobs. Each job j has a unit processing time, a release time r j ∈ N, a deadline d j ∈ N and a priority weight w j ∈ R + . The time line consists of time slots t ∈ N. During each time slot, at most one job can be executed. A job is pending at time t if r j ≤ t < d j and if it has not already been scheduled. An online algorithm A can schedule at each time slot t ∈ N at most one pending job, and has to make this decision without knowledge of future released jobs. The objective value of a schedule is the total priority weight of all scheduled jobs. This value is compared with the objective value of an optimal schedule, and the resulting value is called the competitive ratio. The optimal schedule can be computed offline by solving a maximum weight bipartite matching problem, matching jobs to time slots. The competitive ratio of algorithm A is the supremum of its competitive ratio over all instances. The competitive ratio of the problem is the infimum of the competitive ratio over all online algorithms. Usually the online computation paradigm is seen as a game played between the algorithm (making decisions which job to schedule) and the adversary (generating jobs).

If A is a randomized algorithm, then the performance of A is the expected objective value of the produced schedule. Different adversarial models have been studied in the literature. In this paper we focus on the oblivious adversary, where the input is generated without knowledge of the random choices made by the algorithm. R5cm Online algorithms for this problem heavily rely on the notion of a provisional schedule, which was introduced in [START_REF] Englert | Considering Suppressed Packets Improves Buffer Management in Quality of Service Switches[END_REF]. Recall, that for an algorithm A and a point in time t, we define the set of pending jobs as the set of jobs released prior or at time t, which have their deadline strictly after t, and which have not yet been scheduled by A. The provisional schedule S is a maximum weight subset of pending jobs, which can be scheduled from time t on. This set S can be computed greedily, by starting with S = ∅, and processing first all pending heavy jobs, then all pending light jobs, in arbitrary order within each category. Each considered job j is then added to S if and only if the number of jobs i ∈ S with d i ≤ d j is strictly less than d jt. This condition ensures that there is a provisional feasible schedule for S.

There are two natural deterministic algorithms that are defined in [START_REF] Englert | Considering Suppressed Packets Improves Buffer Management in Quality of Service Switches[END_REF]. The first algorithm, which we call Greedy, schedules at every time step, a job of the provisional schedule of maximum weight, breaking ties by the deadlines of job (earliest deadline has the highest priority). The second algorithm, we which we call Provisional Earliest Deadline First (PEDF), schedules at every time step, a job of the provisional schedule of minimum deadline. Note that it differs from the usual EDF algorithm (earliest deadline first), which is not restricted to select jobs from the provisional schedule.

The deterministic lower bound consists of an adversary which releases at time 0 two jobs, see Figure 1. A job of weight 1 and deadline 1 and a job of weight α and deadline 2. Any deterministic deterministic algorithm either schedules the urgent job, or the heavy job, or no job at all. Clearly the last option is suboptimal, hence we can focus on the first two options. If the algorithm schedules the heavy job, then no more jobs are released, and the competitive ratio is 1+α α . If the algorithm schedules the urgent light job, then at time 1 the adversary releases another job of weight α and deadline 2, leading to the ratio 2α 1+α . This shows a lower bound of min{ 1+α α , 2α 1+α } on the deterministic competitive ratio. It also suggests a simple deterministic algorithm. Let α * = 1 + √ 2 be the solution to the equation 1+α α = 2α 1+α . If α ≤ α * run PEDF, else run Greedy. In [START_REF] Englert | Considering Suppressed Packets Improves Buffer Management in Quality of Service Switches[END_REF] this strategy was shown to be optimal.

Randomized competitive ratio

The previous lower bound can be extended into a lower bound against an oblivious randomized adversary.

Proposition 1 No randomized online algorithm can achieve a competitive ratio better than R = 1 + α-1 α 2 +α .

Proof Let σ 1 , σ 2 be the two instances from the previous lower bound construction. Let y be a distribution on σ 1 , σ 2 . We denote E y [OPT(σ j )] the expected optimal profit, and by E y [A(σ j )] the expected profit of a given randomized algorithm A. By Yao's principle, see [START_REF] Borodin | Online computation and competitive analysis[END_REF]Theorem 8.3], the randomized competitive ratio against an oblivious adversary is lower bounded by max

y min A E y [OPT(σ j )]/E y [A(σ j )].
We choose y = ( α-1 α , 1/α), which leads to the expected optimal profit

E y [OPT(σ j )] = α -1 α (1 + α) + 1 α 2α = 2 + α -1/α.
We recall that at time 0, the instances σ 1 and σ 2 are indistinguishable. Any deterministic algorithm has 3 options at time 0, to execute the urgent light job, to execute the heavy job or to remain idle. We ignore the last option, as it is clearly suboptimal. Any deterministic algorithm, which starts by executing the urgent light job, has an expected profit of at most

y 1 (1 + α) + y 2 (1 + α) = 1 + α.
In addition any deterministic algorithm, which starts by executing the heavy job, has an expected profit of at most

y 1 α + y 2 2α = α -1 α α + 1 α 2α = 1 + α.
This shows the claimed lower bound.

Note that the above lower bound is maximized at α = 1+ √ 2 and gives the value 4-2 √ 2 ≈ 1.172.

Upper bound

The lower bound construction suggests a simple barely random algorithm. At time 0, the algorithm flips a biased coin and based on its outcome decides between the two already mentioned deterministic algorithms. In other words, Barely-Random will run Greedy with probability p and PEDF with probability 1p, for

p = α 2 -1 α 2 + 2α -1 .
This surprisingly simple algorithm achieves the optimal competitive ratio. Theorem 1 Algorithm Barely-Random has a competitive ratio of α 2 +2α-1

α 2 +α
against an oblivious adversary.

Proof Fix an instance, and denote by Opt an optimal schedule. We start with showing some structural properties of the three schedules Opt, Greedy and PEDF. Consider a time slot, where Greedy executes a heavy job j, while PEDF executes a light job. By the definition of PEDF we known that PEDF will eventually execute j as well.

In addition we observe that Greedy executes a maximum number of heavy jobs while PEDF executes a maximum number of jobs regardless of their weight. This follows by the earliest deadline policy applied by both algorithms, respectively on heavy pending jobs or on all pending jobs. Indeed this policy is known to maximize the number of executed jobs for the unweighted unit length job scheduling problem. Therefore OPT does not schedule more jobs than PEDF and does not schedule more heavy jobs than Greedy. Also since PEDF maximizes the number of pending jobs in every time slot, we know that whenever PEDF is idle, then Greedy is idle as well.

As a result, for the analysis of the competitive ratio it suffices to count the number of light and heavy jobs executed by respectively Greedy and PEDF. To formalize this intuition, we use the following notation. Note that the jobs executed by PEDF are H ∪ L ′ and the jobs executed by Greedy are H ∪ H * ∪ L. We use the corresponding lower case notation for the cardinality of those sets. The key to the proof is the following claim.

Claim 1 d + h * ≤ h.
The proof uses an injective mapping from L ′ to H ∪ L. This implies ℓ ′ ≤ h + ℓ, and by the following equality, the claim holds:

d + h * + h + ℓ = h + ℓ ′ (1)
The left and right sides of the above equality respectively indicate the number of time slots throughout the execution of Greedy and PEDF.

To define the mapping, let j ∈ L ′ be an arbitrary light job executed by PEDF at some moment t. If Greedy executes j as well and not later than t, then we map j to itself. Otherwise, we map j to the job k executed by Greedy at time t. We observe that if k is a heavy job, then it is common to both schedules. This follows by the definition of PEDF, for which k was pending at this point, hence k ∈ H. To show the injective nature of the mapping, for the sake of contradiction, we assume that two jobs j, k ∈ L ′ are mapped to the same job k ∈ L, executed by Greedy at some time t. Then by the definition of the mapping, k must be scheduled after j by PEDF, so both jobs j, k were pending at time t for PEDF but also for Greedy. This contradicts the fact that both schedules apply the same earliest deadline policy among the light jobs, and concludes the proof of the claim.

To summarize, we can express the objective values of the schedules as

profit(OPT) ≤ h * α + hα + ℓ + d profit(Greedy) = h * α + hα + ℓ profit(PEDF) = hα + ℓ + h * + d,
where the last expression uses (1). Claim 1 implies the following inequality, which through a sequence of equivalent transformations transforms into the required bound on the competitive ratio:

hα + ℓ ≥ h * α + dα hα(α -1) + ℓ(α -1) ≥ h * α(α -1) + dα(α -1) -(hα + h * α + ℓ) + α(hα + h * + ℓ + d) ≥ h * α(α -1) + dα 2 -(hα + h * α + ℓ) + 2α(hα + h * + ℓ + d) ≥ α(hα + h * + ℓ + d) + h * α(α -1) + dα 2 α 2 (hα + h * α + ℓ) -(hα + h * α + ℓ) + 2α(hα + h * + ℓ + d) ≥ α 2 (hα + h * α + ℓ) + α(hα + h * α + ℓ + d) + dα 2 (α 2 -1)(hα + h * α + ℓ) + 2α(hα + h * + ℓ + d) ≥ (α 2 + α)(hα + h * α + ℓ + d) α 2 -1 α 2 + α (hα + h * α + ℓ) + 2α α 2 + α (hα + h * + ℓ + d) ≥ hα + h * α + ℓ + d R • p • profit(Greedy) + R • (1 -p) • profit(PEDF) ≥ profit(Opt) R • profit(Barely-Random) ≥ profit(Opt).
(2)

Conclusion

Now that the randomized competitive ratio of the two valued bounded delay online buffer management problem is tackled -against the oblivious adversary -it would be interesting to see how it can be generalized to more values for the job weights, for example weights belonging to the set {1, α, α 2 }, or even to the general problem. Currently the only known randomized algorithms for the general problem are Rmix and Remix.

The algorithm Rmix was defined in [START_REF] Bartal | Online Competitive Algorithms for Maximizing Weighted Throughput of Unit Jobs[END_REF][START_REF] Chin | Online competitive algorithms for maximizing weighted throughput of unit jobs[END_REF] as follows. At every time slot, consider j the heaviest pending job. Denote w j its weight. Let x ∈ [-1, 0] be a uniformly chosen real number. Let f be a job with minimal deadline and weight w f ≥ e x w j . Execute f in this time slot. We can observe that, if executed on instance 1 of the lower bound instance from Proposition 1, at time 0, it will choose the heavy job with probability min{1, ln α}. Hence it has a larger tendency to choose the heavy job than Barely-Random, and is fooled by instance 1.

The algorithm ReMix was defined in [START_REF] Jeż | A Universal Randomized Packet Scheduling Algorithm[END_REF], and behaves even differently on instance 1. It will execute the heavy job with probability 1 -1/α and the light job with probability 1/α, but is also not optimal. These comparisons with Barely-Random are unfair in a sense, because they have been designed for the general weighted case, and mostly for an adaptive adversary. But the comparison indicates room for improvement. The optimal randomized algorithm for the general weighted problem, might combine ideas of all these mentioned algorithms, and in particular borrow from the optimal deterministic algorithm [START_REF] Veselý | A Φ-Competitive Algorithm for Scheduling Packets with Deadlines[END_REF].

Figure 1 :

 1 Figure 1: The deterministic lower bound from [9].

Figure 2 :

 2 Figure 2: Competitive ratios as function of α. The deterministic competitive ratio is the minimum of the competitive ratios of the algorithms PEDF and Greedy.

Figure 3 :

 3 Figure 3: The injective mapping from L ′ to H ∪ L. Each job in L ′ is mapped to either a heavy job in H (e.g., a, b) or it is mapped to itself (e.g., x, y, and z).

Acknowledgment

This work was partially supported by the French research agency (ANR-19-CE48-0016), the CNRS PEPS project ADVICE as well as by the EPSRC grant EP/S033483/1, and by the NSERC Discovery Grant.

We would like to thank Spyros Angelopoulos for helpful discussions as well as anonymous referees for spotting an error in a previous version of this paper.